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We expanded these evaluations after performance on our existing mathematical evaluations saturated. Some outputs build upon earlier results produced by the models.\n\nThis collection includes results at different stages of verification. Not all have accompanying Lean formalizations. We will continue to update this repository with Lean formalizations as we obtain them. \n\nSome of the unformalized results could have issues. We will endeavor to fix any such issues quickly.\nWe are also exploring community-hosted repositories for these materials. \n\n## Navigating the collection\n\nThe current catalogue contains 719 manuscripts organized into 372 families. A family groups related papers, which may include a principal result, companion arguments, consequences, or alternative proofs. Each family is classified by mathematical discipline.\n\n- Start with the [overview](overview.pdf) for descriptions of the families.\n- Use the [manuscript map](CONTENTS.md) to find individual papers and their supporting materials.\n- The [`preprints/`](preprints/) directory contains PDFs, source files, and manuscript-specific citation and build instructions.\n- The [Lean library](lean/README.md) and [formalization catalogue](lean/formalization.yaml) describe the available formal proofs, their associated papers, and verification configurations. See the [Comparator instructions](lean/ComparatorChallenges/README.md) for additional checking instructions. The repository has ~42% top-line results formalized.\n- Updates to the repo are described in the [history](history.md).\n\n### Reasoning summaries\n\nWe are also releasing abridged summaries of the model's reasoning, covering the following results:\n\n| Family | Subject |\n|---|---|\n| 007 | [Ordinary two-point correlations of multiplicative functions](reasoning_traces/ordinary-two-point-correlations.pdf) |\n| 017 | [The irrationality exponent of π](reasoning_traces/irrationality-exponent-of-pi.pdf) |\n| 087 | [Symmetric and general Mahler conjectures](reasoning_traces/symmetric-and-general-mahler-conjectures.pdf) |\n| 102 | [Ordinary NP-hardness at the basic semidefinite threshold](reasoning_traces/basic-semidefinite-threshold-np-hardness.pdf) |\n| 159 | [Quasipolynomial bounds for arithmetic progressions](reasoning_traces/quasipolynomial-arithmetic-progressions.pdf) |\n| 197 | [Kaplansky's direct-finiteness conjecture in characteristic two](reasoning_traces/kaplansky-direct-finiteness-characteristic-two.pdf) |\n| 221 | [The Mézard–Parisi formula for diluted spin glasses](reasoning_traces/mezard-parisi-formula.pdf) |\n| 271 | [Spontaneous magnetization in the quantum Heisenberg ferromagnet](reasoning_traces/spontaneous-magnetization-quantum-heisenberg-ferromagnet.pdf) |\n| 287 | [Isomorphism of free group factors](reasoning_traces/free-group-factor-isomorphism.pdf) |\n| 362 | [The three-dimensional relativistic Vlasov–Maxwell system](reasoning_traces/relativistic-vlasov-maxwell.pdf) |\n\n\n## How the results were produced\n\nThe vast majority of results were obtained with the same procedure using an unreleased internal OpenAI model. On average, each result used three hours of ChatGPT Pro thinking compute with that model. Over the course of the evaluation, the model was posed approximately 4,000 problems. Aggregating the output into result families and manuscripts and requiring an appropriate level of significance led to the catalog outlined above.\n\nExceptions to this fixed procedure include work on a zero-free region for the Riemann zeta function and proof of the Hodge Conjecture for CM abelian varieties. Additionally, the writeup for the Re(s) > 11/12 zero-free region for the Riemann zeta function was human edited for readability. \n\n\n## Versions and citations\n\nWe will preserve the public release history of this collection. Corrections and revisions will be recorded as new versions, with previously released versions remaining accessible. \n\nTo cite the individual manuscript, use the BibTeX block in its directory. \n\n\nDOCUMENT lean/README.md \n # Lean formalizations\n\nThis directory contains Lean formalizations of some results in this repository.\nThey are organized in a single large library, so we recommend compiling only small portions at a time.\n\nSee the [Comparator README](ComparatorChallenges/README.md) for more information about how to verify the results.\n\n## Technical note: mmap\n\nCompiling the entire library may fail if Linux's `vm.max_map_count` is too low.\nOne workaround is to build Lean with the CMake option `-DMMAP=OFF`; it may also be necessary to set the environment variable `GLIBC_TUNABLES` to `glibc.malloc.mmap_max=0:glibc.malloc.arena_max=1` when running Lean or Lake.\n\n\nDOCUMENT lean/ComparatorChallenges/README.md \n # [Comparator](https://github.com/leanprover/comparator) challenges\n\nInstall `comparator`, `landrun`, and `lean4export`, and make them available on `PATH`.\nThen, from `lean/`:\n\n```sh\nlake update\nlake exe cache get\nlake env comparator ComparatorChallenges/QuasiRiemannHypothesis.json\n```\n\n## Supporting-result comparisons\n\nThese setups verify supporting results, not the corresponding papers’ main\ntheorems:\n\n- `CharacterVarietiesAllSeamsSupport.json`: compatibility of the produced marked\n  solution with every seam equation.\n- `CartierChartCompactnessSupport.json`: compactness of normalized chart-field\n  retractions.\n- `SurfaceConeCandidateSupport.json`: ring-theoretic properties of the completed\n  surface cone.\n- `TraceIdealTransportSupport.json`: transport of finite and zero ideals, with\n  the construction-provider statements required by the original comparison.\n- `HoneycombBridgeMassSupport.json`: finiteness and quarter-power bounds for the\n  strict honeycomb bridge mass.\n\nSee the [scope notes](../docs/) for per-paper coverage and limitations.\n\n\nCONTENTS preview\n # Mathematics manuscript collection\n\n**719 manuscripts covering 372 result families.**\n\n[**Read the overview PDF**](overview.pdf).\n\n## Manuscript map\n\nEach result description is followed by its constituent manuscripts and their abstracts. Paper titles link directly to PDFs.\n\n<table>\n<thead><tr><th>Result</th></tr></thead>\n<tbody><tr>\n<td>\n\n**001. Milne’s rationality conjecture and algebraic specialization.** Proves Milne's rationality conjecture for abelian varieties over $`\\overline{\\mathbb Q}`$ with good reduction: specialized Hodge classes pair rationally with complementary divisor products, independently of cohomology theory. Together with result 032, every specialized Hodge class is represented by a single rational algebraic cycle simultaneously in all prime-to-<i>p</i> and crystalline realizations, for every residue characteristic <i>p</i>.\n\n</td>\n</tr></tbody>\n<tbody><tr>\n<td>\n\n&emsp;[Milne's rationality conjecture for abelian varieties](preprints/Milnes-rationality-conjecture-for-abelian-varieties-October-7-2026/paper.pdf)\n\nWe prove Milne's rationality conjecture for abelian varieties, including residue characteristic 2. After good reduction, the pairing of a rational Hodge class with any complementary product of divisor classes on the reduction is the same rational number in every prime-to-<i>p</i> realization and in crystalline cohomology. Using the Hodge theorem for CM abelian varieties, we also show that every such specialized Hodge class is represented by a single rational algebraic cycle in all these realizations.\n\n</td>\n</tr></tbody>\n<tbody><tr>\n<td>\n\n**002. The full BSD formula from low Selmer corank.** Proves the full Birch–Swinnerton-Dyer leading-term formula for every elliptic curve over ℚ whose full <i>q</i>-power Selmer group has corank zero or one for some prime <i>q</i>, including finiteness of the Tate–Shafarevich group. With result 006, this gives full BSD for a density-one set of quadratic twists of every elliptic curve over ℚ.\n\n</td>\n</tr></tbody>\n<tbody><tr>\n<td>\n\n&emsp;[Exact Birch–Swinnerton-Dyer Formula from Low Selmer Corank](preprints/Exact-Birch-Swinnerton-Dyer-Formula-from-Low-Selmer-Corank-October-7-2026/exact-bsd-low-selmer-corank.pdf)\n\nWe prove the full Birch–Swinnerton-Dyer leading-term formula for every elliptic curve over ℚ whose full <i>q</i>-power Selmer group has corank zero or one at some prime <i>q</i>. The analytic and Mordell–Weil ranks equal that corank, and the Tate–Shafarevich group is finite. The formula includes all prime factors and requires no additional hypotheses on reduction, rational torsion, isogenies, complex multiplication, or residual Galois representations.\n\n</td>\n</tr></tbody>\n<tbody><tr>\n<td>\n\n&emsp;[The Selmer converse for elliptic curves at every prime](preprints/The-Selmer-converse-for-elliptic-curves-at-every-prime-October-7-2026/main.pdf)\n\nWe prove the Selmer converse in coranks zero and one for every elliptic curve over ℚ and every prime <i>p</i>: if the full <i>p</i>-power Selmer group has ℤ<sub><i>p</i></sub>-corank $`r\\in\\{0,1\\}`$, then the analytic and Mordell–Weil ranks both equal <i>r</i>, and the entire Tate–Shafarevich group is finite. As an application at the additive prime 3, we prove that for every prime $`\\ell\\equiv4,7,8\\pmod9`$, the cubic $`X^3+Y^3=\\ell Z^3`$ has analytic and Mordell–Weil rank one and finite Tate–Shafarevich group. In particular, every such <i>ℓ</i> is a sum of two rational cubes.\n\n</td>\n</tr></tbody>\n<tbody><tr>\n<td>\n\n&emsp;[The two-primary Birch–Swinnerton-Dyer formula in Selmer corank at most one](preprints/The-two-primary-Birch-Swinnerton-Dyer-formula-in-Selmer-corank-at-most-one-October-6-2026/paper.pdf)\n\nWe prove the two-primary Birch and Swinnerton-Dyer leading-term formula for every elliptic curve over the rationals whose two-power Selmer group has corank at most one. In this range, the algebraic rank, analytic rank, and Selmer corank are equal, and the Tate–Shafarevich group is finite. Combined with the quadratic-twist Selmer distribution, this gives the exact two-primary formula for a density-one set of signed squarefree twists of each fixed curve, ordered by absolute value; the common rank is zero or one, with each value having density one half.\n\n</td>\n</tr></tbody>\n<tbody><tr>\n<td>\n\n**003. The quasi-Riemann hypothesis.** Proves that every Dirichlet <i>L</i>-function, including $`\\zeta(s)`$, is zero-free in $`\\Re s\\gt 7/8`$, resolving the quasi-Riemann hypothesis. The same half-plane is zero-free for every finite-order Hecke <i>L</i>-function over $`\\mathbb Q(\\sqrt{-3})`$. A companion gives a different proof of the zero-free half-plane $`\\Re s\\gt 11/12`$. ([Lean](lean/docs/003.md))\n\n</td>\n</tr></tbody>\n<tbody><tr>\n<td>\n\n&emsp;[The Quasi-Riemann Hypothesis: A Zero-Free Half-Plane $`\\Re s\\gt 7/8`$ ](preprints/The-Quasi-Riemann-Hypothesis-September-30-2026/paper.pdf)\n\nWe prove that all finite-order Hecke <i>L</i>-functions over $`\\mathbb Q(\\sqrt{-3})`$ and all Dirichlet <i>L</i>-functions are zero-free in the half-plane $`\\Re s\\gt 7/8`$, with the principal pole at <i>s</i> = 1 allowed. In particular, the Riemann zeta function is zero-free in this half-plane, proving the quasi-Riemann hypothesis.\n\n</td>\n</tr></tbody>\n<tbody><tr>\n<td>\n\n&emsp;[The Quasi-Riemann Hypothesis (alternate 11/12 proof)](preprints/The-Quasi-Riemann-Hypothesis-October-5-2026/paper2.pdf)\n\nWe establish the quasi-Riemann hypothesis by proving that every Dirichlet <i>L</i>-function, including Riemann's zeta function, has no zeros in the half-plane $`\\mathop{\\mathrm{Re}}\\nolimits s\\gt 11/12`$. More generally, we prove the same zero-free half-plane for every finite-order Hecke <i>L</i>-function over $`K=\\mathbb Q(\\sqrt{-3})`$. In particular, this rules out the existence of Landau–Siegel zeros.\n\n</td>\n</tr></tbody>\n<tbody><tr>\n<td>\n\n&emsp;[Uniform exclusion of Landau–Siegel zeros](preprints/Uniform-exclusion-of-Landau-Siegel-zeros-October-1-2026/paper.pdf)\n\nWe prove the uniform exclusion of Landau–Siegel zeros. There is an absolute constant <i>c</i> &gt; 0 such that every real zero $`\\beta\\in(0,1)`$ of every primitive nonprincipal real Dirichlet <i>L</i>-function of conductor <i>q</i> ≥ 3 satisfies $`(1-\\beta)\\log q\\ge c`$.\n\n</td>\n</tr></tbody>\n<tbody><tr>\n<td>\n\n**004. Hilbert’s tenth problem over ℚ.** Proves that no algorithm decides whether an integer-coefficient polynomial in an arbitrary number of variables has a rational zero, resolving Hilbert's tenth problem over ℚ negatively.\n\n</td>\n</tr></tbody>\n<tbody><tr>\n<td>\n\n&emsp;[Hilbert’s tenth problem over the rational numbers](preprints/Hilberts-tenth-problem-over-the-rational-numbers-October-6-2026/main.pdf)\n\nWe give a negative answer to Hilbert's tenth problem over the rational numbers: no algorithm decides whether a polynomial with integer coefficients has a rational zero. The number of variables is part of the input.\n\n</td>\n</tr></tbody>\n<tbody><tr>\n<td>\n\n&emsp;[A pointwise 2-converse for elliptic curves with rational two-torsion](preprints/A-pointwise-2-converse-for-elliptic-curves-with-rational-two-torsion-October-7-2026/paper.pdf)\n\nWe prove a pointwise 2-converse for elliptic curves over $`\\mathbf Q`$ with nonzero rational two-torsion: if the $`2^\\infty`$-Selmer corank is zero or one, then the analytic rank and Mordell–Weil rank equal that corank, and the Shafarevich–Tate group is finite. The result allows arbitrary reduction at 2.\n\n</td>\n</tr></tbody>\n<tbody><tr>\n<td>\n\n**005. Irrationality of Catalan’s constant.** Proves that Catalan's constant $`G=\\sum_{j\\ge0}(-1)^j/(2j+1)^2`$ is irrational. ([Lean](lean/docs/005.md))\n\n</td>\n</tr></tbody>\n<tbody><tr>\n<td>\n\n&emsp;[Catalan's constant is irrational](preprints/Catalans-constant-is-irrational-September-24-2026/paper.pdf)\n\nWe prove that Catalan's constant $`G=\\sum_{j\\geq0}(-1)^j/(2j+1)^2`$ is irrational.\n\n</td>\n</tr></tbody>\n<tbody><tr>\n<td>\n\n**006. Goldfeld’s conjecture: densities and mean analytic rank.** Proves Goldfeld's conjecture for quadratic twists of every elliptic curve over ℚ: analytic ranks zero and one each have density 1/2, and the mean analytic rank tends to 1/2. Both statements order signed squarefree twist parameters by absolute value.\n\n</td>\n</tr></tbody>\n<tbody><tr>\n<td>\n\n&emsp;[Goldfeld's analytic density conjecture and the 2-converse for elliptic curves](preprints/Goldfelds-analytic-density-conjecture-and-the-2-converse-for-elliptic-curves-October-7-2026/paper.pdf)\n\nWe prove Goldfeld's analytic density conjecture: for every elliptic curve <i>E</i> over ℚ, the quadratic twists of <i>E</i> with analytic rank zero and one each have density 1/2 among signed squarefree twist parameters ordered by absolute value. We also prove the low-corank 2-converse: if the $`2^\\infty`$-Selmer corank of <i>E</i> is zero or one, then it equals the analytic and Mordell–Weil ranks, and the Tate–Shafarevich group is finite.\n\n</td>\n</tr></tbody>\n<tbody><tr>\n<td>\n\n&emsp;[The mean analytic rank of quadratic twists of elliptic curves](preprints/The-mean-analytic-rank-of-quadratic-twists-of-elliptic-curves-October-6-2026/paper.pdf)\n\nFor every elliptic curve over ℚ, we prove that the average analytic rank of its quadratic twists tends to 1/2 when signed squarefree twist parameters are ordered by absolute value. This resolves Goldfeld's mean analytic-rank conjecture in this counting convention.\n\n</td>\n</tr></tbody>\n<tbody><tr>\n<td>\n\n**007. Ordinary two-point correlations and the corrected Elliott conjecture.** Proves the ordinary two-point Chowla conjecture, with a bound $`O(X/(\\log X)^c)`$ for Liouville correlation sums along fixed nonproportional affine forms, where <i>c</i> &gt; 0 is absolute. More generally, proves the binary corrected Elliott conjecture for complex multiplicative functions bounded by one when one factor is uniformly nonpretentious against each fixed Dirichlet character times $`n^{it}`$ for $`|t|\\le X`$. ([Lean](lean/docs/007.md))\n\n</td>\n</tr></tbody>\n<tbody><tr>\n<td>\n\n&emsp;[Ordinary two-point correlations of multiplicative functions](preprints/Ordinary-two-point-correlations-of-multiplicative-functions-September-24-2026/final.pdf)\n\nWe prove the ordinary two-point Chowla conjecture. For every fixed pair of nonproportional affine forms, the Liouville correlation has a power-of-logarithm saving at every cutoff, with an absolute exponent. We also prove the binary corrected Elliott conjecture for ordinary averages of complex multiplicative functions of modulus at most one, under uniform nonpretentiousness of at least one original factor. This qualitative conclusion holds in fixed residue classes and for fixed nonproportional affine forms.\n\n</td>\n</tr></tbody>\n<tbody><tr>\n<td>\n\n**008. The Deligne–Drinfeld conjecture.** Proves that the rational Grothendieck–Teichmüller Lie algebra, with the Ihara bracket, is freely generated by one element in each odd weight 3, 5, 7, …, resolving the Deligne–Drinfeld conjecture. ([Lean](lean/docs/008.md))\n\n</td>\n</tr></tbody>\n<tbody><tr>\n<td>\n\n&emsp;[The Deligne-Drinfeld conjecture](preprints/The-Deligne-Drinfeld-conjecture-September-23-2026/paper.pdf)\n\nWe prove the Deligne–Drinfeld conjecture: the rational Grothendieck–Teichmüller Lie algebra is freely generated by one element in every odd weight at least three, with the corresponding isomorphism after weight completion.\n\n</td>\n</tr></tbody>\n<tbody><tr>\n<td>\n\n**009. Function-field reconstruction from Milnor K-theory and Galois data.** Reconstructs function fields of transcendence degree at least two over algebraically closed constants from $`K^{\\mathrm M}_1/\\ell`$, $`K^{\\mathrm M}_2/\\ell`$, and their product. These data recover the perfect closure and constants when <i>ℓ</i> differs from the characteristic, and the original field and its named base in equal characteristic. Also proves Bogomolov–Pop reconstruction from abelian-by-central pro-<i>ℓ</i> Galois data away from the characteristic. ([Lean](lean/docs/009.md))\n\n</td>\n</tr></tbody>\n<tbody><tr>\n<td>\n\n&emsp;[Reconstruction of Function Fields from Mod-ℓ Milnor K-Theory](preprints/Reconstruction-of-Function-Fields-from-Mod-ell-Milnor-K-Theory-October-5-2026/mod-ell-bogomolov-pop.pdf)\n\nWe prove a mod-<i>ℓ</i> Bogomolov–Pop reconstruction theorem for function fields of transcendence degree at least two over arbitrary algebraically closed fields of characteristic different from <i>ℓ</i>. The groups $`K^{\\mathrm M}_1/\\ell`$ and $`K^{\\mathrm M}_2/\\ell`$, together with their full bilinear product, determine the perfect closure and its constant field. Every compatible isomorphism of these data is induced by a field isomorphism up to a single scalar in $`\\mathbb F_\\ell^\\times`$, with only Frobenius ambiguity in the field isomorphism in positive characteristic.\n\n</td>\n</tr></tbody>\n<tbody><tr>\n<td>\n\n&emsp;[Reconstruction from Milnor K-theory modulo the characteristic](preprints/Reconstruction-from-Milnor-K-theory-modulo-the-characteristic-October-5-2026/paper.pdf)\n\nLet <i>K</i> and <i>L</i> be finitely generated extensions of transcendence degree at least two over algebraically closed fields of characteristic <i>p</i>. We prove that every isomorphism of their first Milnor <i>K</i>-groups modulo <i>p</i> preserving the degree-two Steinberg relations is a nonzero scalar multiple of the map induced by a unique field isomorphism. The proof recovers projective lines over the subfields of <i>p</i>th powers by an elementary calculation with derivations.\n\n</td>\n</tr></tbody>\n<tbody><tr>\n<td>\n\n&emsp;[The Bogomolov-Pop reconstruction theorem](preprints/The-Bogomolov-Pop-reconstruction-theorem-September-23-2026/paper.pdf)\n\nWe prove the Bogomolov–Pop reconstruction conjecture for function fields of transcendence degree at least two over arbitrary algebraically closed fields of characteristic different from <i>ℓ</i>. The pro-<i>ℓ</i> abelian-by-central datum determines the perfect closure and its constant field, with precisely the Frobenius and <i>ℓ</i>-adic unit ambiguities.\n\n</td>\n</tr></tbody>\n<tbody><tr>\n<td>\n\n**010. Unrestricte\n",
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We expanded these evaluations after performance on our existing mathematical evaluations saturated. Some outputs build upon earlier results produced by the models.\n\nThis collection includes results at different stages of verification. Not all have accompanying Lean formalizations. We will continue to update this repository with Lean formalizations as we obtain them. \n\nSome of the unformalized results could have issues. We will endeavor to fix any such issues quickly.\nWe are also exploring community-hosted repositories for these materials. \n\n## Navigating the collection\n\nThe current catalogue contains 719 manuscripts organized into 372 families. A family groups related papers, which may include a principal result, companion arguments, consequences, or alternative proofs. Each family is classified by mathematical discipline.\n\n- Start with the [overview](overview.pdf) for descriptions of the families.\n- Use the [manuscript map](CONTENTS.md) to find individual papers and their supporting materials.\n- The [`preprints/`](preprints/) directory contains PDFs, source files, and manuscript-specific citation and build instructions.\n- The [Lean library](lean/README.md) and [formalization catalogue](lean/formalization.yaml) describe the available formal proofs, their associated papers, and verification configurations. See the [Comparator instructions](lean/ComparatorChallenges/README.md) for additional checking instructions. The repository has ~42% top-line results formalized.\n- Updates to the repo are described in the [history](history.md).\n\n### Reasoning summaries\n\nWe are also releasing abridged summaries of the model's reasoning, covering the following results:\n\n| Family | Subject |\n|---|---|\n| 007 | [Ordinary two-point correlations of multiplicative functions](reasoning_traces/ordinary-two-point-correlations.pdf) |\n| 017 | [The irrationality exponent of π](reasoning_traces/irrationality-exponent-of-pi.pdf) |\n| 087 | [Symmetric and general Mahler conjectures](reasoning_traces/symmetric-and-general-mahler-conjectures.pdf) |\n| 102 | [Ordinary NP-hardness at the basic semidefinite threshold](reasoning_traces/basic-semidefinite-threshold-np-hardness.pdf) |\n| 159 | [Quasipolynomial bounds for arithmetic progressions](reasoning_traces/quasipolynomial-arithmetic-progressions.pdf) |\n| 197 | [Kaplansky's direct-finiteness conjecture in characteristic two](reasoning_traces/kaplansky-direct-finiteness-characteristic-two.pdf) |\n| 221 | [The Mézard–Parisi formula for diluted spin glasses](reasoning_traces/mezard-parisi-formula.pdf) |\n| 271 | [Spontaneous magnetization in the quantum Heisenberg ferromagnet](reasoning_traces/spontaneous-magnetization-quantum-heisenberg-ferromagnet.pdf) |\n| 287 | [Isomorphism of free group factors](reasoning_traces/free-group-factor-isomorphism.pdf) |\n| 362 | [The three-dimensional relativistic Vlasov–Maxwell system](reasoning_traces/relativistic-vlasov-maxwell.pdf) |\n\n\n## How the results were produced\n\nThe vast majority of results were obtained with the same procedure using an unreleased internal OpenAI model. On average, each result used three hours of ChatGPT Pro thinking compute with that model. Over the course of the evaluation, the model was posed approximately 4,000 problems. Aggregating the output into result families and manuscripts and requiring an appropriate level of significance led to the catalog outlined above.\n\nExceptions to this fixed procedure include work on a zero-free region for the Riemann zeta function and proof of the Hodge Conjecture for CM abelian varieties. Additionally, the writeup for the Re(s) > 11/12 zero-free region for the Riemann zeta function was human edited for readability. \n\n\n## Versions and citations\n\nWe will preserve the public release history of this collection. Corrections and revisions will be recorded as new versions, with previously released versions remaining accessible. \n\nTo cite the individual manuscript, use the BibTeX block in its directory. \n\n\nDOCUMENT lean/README.md \n # Lean formalizations\n\nThis directory contains Lean formalizations of some results in this repository.\nThey are organized in a single large library, so we recommend compiling only small portions at a time.\n\nSee the [Comparator README](ComparatorChallenges/README.md) for more information about how to verify the results.\n\n## Technical note: mmap\n\nCompiling the entire library may fail if Linux's `vm.max_map_count` is too low.\nOne workaround is to build Lean with the CMake option `-DMMAP=OFF`; it may also be necessary to set the environment variable `GLIBC_TUNABLES` to `glibc.malloc.mmap_max=0:glibc.malloc.arena_max=1` when running Lean or Lake.\n\n\nDOCUMENT lean/ComparatorChallenges/README.md \n # [Comparator](https://github.com/leanprover/comparator) challenges\n\nInstall `comparator`, `landrun`, and `lean4export`, and make them available on `PATH`.\nThen, from `lean/`:\n\n```sh\nlake update\nlake exe cache get\nlake env comparator ComparatorChallenges/QuasiRiemannHypothesis.json\n```\n\n## Supporting-result comparisons\n\nThese setups verify supporting results, not the corresponding papers’ main\ntheorems:\n\n- `CharacterVarietiesAllSeamsSupport.json`: compatibility of the produced marked\n  solution with every seam equation.\n- `CartierChartCompactnessSupport.json`: compactness of normalized chart-field\n  retractions.\n- `SurfaceConeCandidateSupport.json`: ring-theoretic properties of the completed\n  surface cone.\n- `TraceIdealTransportSupport.json`: transport of finite and zero ideals, with\n  the construction-provider statements required by the original comparison.\n- `HoneycombBridgeMassSupport.json`: finiteness and quarter-power bounds for the\n  strict honeycomb bridge mass.\n\nSee the [scope notes](../docs/) for per-paper coverage and limitations.\n\n\nCONTENTS preview\n # Mathematics manuscript collection\n\n**719 manuscripts covering 372 result families.**\n\n[**Read the overview PDF**](overview.pdf).\n\n## Manuscript map\n\nEach result description is followed by its constituent manuscripts and their abstracts. Paper titles link directly to PDFs.\n\n<table>\n<thead><tr><th>Result</th></tr></thead>\n<tbody><tr>\n<td>\n\n**001. Milne’s rationality conjecture and algebraic specialization.** Proves Milne's rationality conjecture for abelian varieties over $`\\overline{\\mathbb Q}`$ with good reduction: specialized Hodge classes pair rationally with complementary divisor products, independently of cohomology theory. Together with result 032, every specialized Hodge class is represented by a single rational algebraic cycle simultaneously in all prime-to-<i>p</i> and crystalline realizations, for every residue characteristic <i>p</i>.\n\n</td>\n</tr></tbody>\n<tbody><tr>\n<td>\n\n&emsp;[Milne's rationality conjecture for abelian varieties](preprints/Milnes-rationality-conjecture-for-abelian-varieties-October-7-2026/paper.pdf)\n\nWe prove Milne's rationality conjecture for abelian varieties, including residue characteristic 2. After good reduction, the pairing of a rational Hodge class with any complementary product of divisor classes on the reduction is the same rational number in every prime-to-<i>p</i> realization and in crystalline cohomology. Using the Hodge theorem for CM abelian varieties, we also show that every such specialized Hodge class is represented by a single rational algebraic cycle in all these realizations.\n\n</td>\n</tr></tbody>\n<tbody><tr>\n<td>\n\n**002. The full BSD formula from low Selmer corank.** Proves the full Birch–Swinnerton-Dyer leading-term formula for every elliptic curve over ℚ whose full <i>q</i>-power Selmer group has corank zero or one for some prime <i>q</i>, including finiteness of the Tate–Shafarevich group. With result 006, this gives full BSD for a density-one set of quadratic twists of every elliptic curve over ℚ.\n\n</td>\n</tr></tbody>\n<tbody><tr>\n<td>\n\n&emsp;[Exact Birch–Swinnerton-Dyer Formula from Low Selmer Corank](preprints/Exact-Birch-Swinnerton-Dyer-Formula-from-Low-Selmer-Corank-October-7-2026/exact-bsd-low-selmer-corank.pdf)\n\nWe prove the full Birch–Swinnerton-Dyer leading-term formula for every elliptic curve over ℚ whose full <i>q</i>-power Selmer group has corank zero or one at some prime <i>q</i>. The analytic and Mordell–Weil ranks equal that corank, and the Tate–Shafarevich group is finite. The formula includes all prime factors and requires no additional hypotheses on reduction, rational torsion, isogenies, complex multiplication, or residual Galois representations.\n\n</td>\n</tr></tbody>\n<tbody><tr>\n<td>\n\n&emsp;[The Selmer converse for elliptic curves at every prime](preprints/The-Selmer-converse-for-elliptic-curves-at-every-prime-October-7-2026/main.pdf)\n\nWe prove the Selmer converse in coranks zero and one for every elliptic curve over ℚ and every prime <i>p</i>: if the full <i>p</i>-power Selmer group has ℤ<sub><i>p</i></sub>-corank $`r\\in\\{0,1\\}`$, then the analytic and Mordell–Weil ranks both equal <i>r</i>, and the entire Tate–Shafarevich group is finite. As an application at the additive prime 3, we prove that for every prime $`\\ell\\equiv4,7,8\\pmod9`$, the cubic $`X^3+Y^3=\\ell Z^3`$ has analytic and Mordell–Weil rank one and finite Tate–Shafarevich group. In particular, every such <i>ℓ</i> is a sum of two rational cubes.\n\n</td>\n</tr></tbody>\n<tbody><tr>\n<td>\n\n&emsp;[The two-primary Birch–Swinnerton-Dyer formula in Selmer corank at most one](preprints/The-two-primary-Birch-Swinnerton-Dyer-formula-in-Selmer-corank-at-most-one-October-6-2026/paper.pdf)\n\nWe prove the two-primary Birch and Swinnerton-Dyer leading-term formula for every elliptic curve over the rationals whose two-power Selmer group has corank at most one. In this range, the algebraic rank, analytic rank, and Selmer corank are equal, and the Tate–Shafarevich group is finite. Combined with the quadratic-twist Selmer distribution, this gives the exact two-primary formula for a density-one set of signed squarefree twists of each fixed curve, ordered by absolute value; the common rank is zero or one, with each value having density one half.\n\n</td>\n</tr></tbody>\n<tbody><tr>\n<td>\n\n**003. The quasi-Riemann hypothesis.** Proves that every Dirichlet <i>L</i>-function, including $`\\zeta(s)`$, is zero-free in $`\\Re s\\gt 7/8`$, resolving the quasi-Riemann hypothesis. The same half-plane is zero-free for every finite-order Hecke <i>L</i>-function over $`\\mathbb Q(\\sqrt{-3})`$. A companion gives a different proof of the zero-free half-plane $`\\Re s\\gt 11/12`$. ([Lean](lean/docs/003.md))\n\n</td>\n</tr></tbody>\n<tbody><tr>\n<td>\n\n&emsp;[The Quasi-Riemann Hypothesis: A Zero-Free Half-Plane $`\\Re s\\gt 7/8`$ ](preprints/The-Quasi-Riemann-Hypothesis-September-30-2026/paper.pdf)\n\nWe prove that all finite-order Hecke <i>L</i>-functions over $`\\mathbb Q(\\sqrt{-3})`$ and all Dirichlet <i>L</i>-functions are zero-free in the half-plane $`\\Re s\\gt 7/8`$, with the principal pole at <i>s</i> = 1 allowed. In particular, the Riemann zeta function is zero-free in this half-plane, proving the quasi-Riemann hypothesis.\n\n</td>\n</tr></tbody>\n<tbody><tr>\n<td>\n\n&emsp;[The Quasi-Riemann Hypothesis (alternate 11/12 proof)](preprints/The-Quasi-Riemann-Hypothesis-October-5-2026/paper2.pdf)\n\nWe establish the quasi-Riemann hypothesis by proving that every Dirichlet <i>L</i>-function, including Riemann's zeta function, has no zeros in the half-plane $`\\mathop{\\mathrm{Re}}\\nolimits s\\gt 11/12`$. More generally, we prove the same zero-free half-plane for every finite-order Hecke <i>L</i>-function over $`K=\\mathbb Q(\\sqrt{-3})`$. In particular, this rules out the existence of Landau–Siegel zeros.\n\n</td>\n</tr></tbody>\n<tbody><tr>\n<td>\n\n&emsp;[Uniform exclusion of Landau–Siegel zeros](preprints/Uniform-exclusion-of-Landau-Siegel-zeros-October-1-2026/paper.pdf)\n\nWe prove the uniform exclusion of Landau–Siegel zeros. There is an absolute constant <i>c</i> &gt; 0 such that every real zero $`\\beta\\in(0,1)`$ of every primitive nonprincipal real Dirichlet <i>L</i>-function of conductor <i>q</i> ≥ 3 satisfies $`(1-\\beta)\\log q\\ge c`$.\n\n</td>\n</tr></tbody>\n<tbody><tr>\n<td>\n\n**004. Hilbert’s tenth problem over ℚ.** Proves that no algorithm decides whether an integer-coefficient polynomial in an arbitrary number of variables has a rational zero, resolving Hilbert's tenth problem over ℚ negatively.\n\n</td>\n</tr></tbody>\n<tbody><tr>\n<td>\n\n&emsp;[Hilbert’s tenth problem over the rational numbers](preprints/Hilberts-tenth-problem-over-the-rational-numbers-October-6-2026/main.pdf)\n\nWe give a negative answer to Hilbert's tenth problem over the rational numbers: no algorithm decides whether a polynomial with integer coefficients has a rational zero. The number of variables is part of the input.\n\n</td>\n</tr></tbody>\n<tbody><tr>\n<td>\n\n&emsp;[A pointwise 2-converse for elliptic curves with rational two-torsion](preprints/A-pointwise-2-converse-for-elliptic-curves-with-rational-two-torsion-October-7-2026/paper.pdf)\n\nWe prove a pointwise 2-converse for elliptic curves over $`\\mathbf Q`$ with nonzero rational two-torsion: if the $`2^\\infty`$-Selmer corank is zero or one, then the analytic rank and Mordell–Weil rank equal that corank, and the Shafarevich–Tate group is finite. The result allows arbitrary reduction at 2.\n\n</td>\n</tr></tbody>\n<tbody><tr>\n<td>\n\n**005. Irrationality of Catalan’s constant.** Proves that Catalan's constant $`G=\\sum_{j\\ge0}(-1)^j/(2j+1)^2`$ is irrational. ([Lean](lean/docs/005.md))\n\n</td>\n</tr></tbody>\n<tbody><tr>\n<td>\n\n&emsp;[Catalan's constant is irrational](preprints/Catalans-constant-is-irrational-September-24-2026/paper.pdf)\n\nWe prove that Catalan's constant $`G=\\sum_{j\\geq0}(-1)^j/(2j+1)^2`$ is irrational.\n\n</td>\n</tr></tbody>\n<tbody><tr>\n<td>\n\n**006. Goldfeld’s conjecture: densities and mean analytic rank.** Proves Goldfeld's conjecture for quadratic twists of every elliptic curve over ℚ: analytic ranks zero and one each have density 1/2, and the mean analytic rank tends to 1/2. Both statements order signed squarefree twist parameters by absolute value.\n\n</td>\n</tr></tbody>\n<tbody><tr>\n<td>\n\n&emsp;[Goldfeld's analytic density conjecture and the 2-converse for elliptic curves](preprints/Goldfelds-analytic-density-conjecture-and-the-2-converse-for-elliptic-curves-October-7-2026/paper.pdf)\n\nWe prove Goldfeld's analytic density conjecture: for every elliptic curve <i>E</i> over ℚ, the quadratic twists of <i>E</i> with analytic rank zero and one each have density 1/2 among signed squarefree twist parameters ordered by absolute value. We also prove the low-corank 2-converse: if the $`2^\\infty`$-Selmer corank of <i>E</i> is zero or one, then it equals the analytic and Mordell–Weil ranks, and the Tate–Shafarevich group is finite.\n\n</td>\n</tr></tbody>\n<tbody><tr>\n<td>\n\n&emsp;[The mean analytic rank of quadratic twists of elliptic curves](preprints/The-mean-analytic-rank-of-quadratic-twists-of-elliptic-curves-October-6-2026/paper.pdf)\n\nFor every elliptic curve over ℚ, we prove that the average analytic rank of its quadratic twists tends to 1/2 when signed squarefree twist parameters are ordered by absolute value. This resolves Goldfeld's mean analytic-rank conjecture in this counting convention.\n\n</td>\n</tr></tbody>\n<tbody><tr>\n<td>\n\n**007. Ordinary two-point correlations and the corrected Elliott conjecture.** Proves the ordinary two-point Chowla conjecture, with a bound $`O(X/(\\log X)^c)`$ for Liouville correlation sums along fixed nonproportional affine forms, where <i>c</i> &gt; 0 is absolute. More generally, proves the binary corrected Elliott conjecture for complex multiplicative functions bounded by one when one factor is uniformly nonpretentious against each fixed Dirichlet character times $`n^{it}`$ for $`|t|\\le X`$. ([Lean](lean/docs/007.md))\n\n</td>\n</tr></tbody>\n<tbody><tr>\n<td>\n\n&emsp;[Ordinary two-point correlations of multiplicative functions](preprints/Ordinary-two-point-correlations-of-multiplicative-functions-September-24-2026/final.pdf)\n\nWe prove the ordinary two-point Chowla conjecture. For every fixed pair of nonproportional affine forms, the Liouville correlation has a power-of-logarithm saving at every cutoff, with an absolute exponent. We also prove the binary corrected Elliott conjecture for ordinary averages of complex multiplicative functions of modulus at most one, under uniform nonpretentiousness of at least one original factor. This qualitative conclusion holds in fixed residue classes and for fixed nonproportional affine forms.\n\n</td>\n</tr></tbody>\n<tbody><tr>\n<td>\n\n**008. The Deligne–Drinfeld conjecture.** Proves that the rational Grothendieck–Teichmüller Lie algebra, with the Ihara bracket, is freely generated by one element in each odd weight 3, 5, 7, …, resolving the Deligne–Drinfeld conjecture. ([Lean](lean/docs/008.md))\n\n</td>\n</tr></tbody>\n<tbody><tr>\n<td>\n\n&emsp;[The Deligne-Drinfeld conjecture](preprints/The-Deligne-Drinfeld-conjecture-September-23-2026/paper.pdf)\n\nWe prove the Deligne–Drinfeld conjecture: the rational Grothendieck–Teichmüller Lie algebra is freely generated by one element in every odd weight at least three, with the corresponding isomorphism after weight completion.\n\n</td>\n</tr></tbody>\n<tbody><tr>\n<td>\n\n**009. Function-field reconstruction from Milnor K-theory and Galois data.** Reconstructs function fields of transcendence degree at least two over algebraically closed constants from $`K^{\\mathrm M}_1/\\ell`$, $`K^{\\mathrm M}_2/\\ell`$, and their product. These data recover the perfect closure and constants when <i>ℓ</i> differs from the characteristic, and the original field and its named base in equal characteristic. Also proves Bogomolov–Pop reconstruction from abelian-by-central pro-<i>ℓ</i> Galois data away from the characteristic. ([Lean](lean/docs/009.md))\n\n</td>\n</tr></tbody>\n<tbody><tr>\n<td>\n\n&emsp;[Reconstruction of Function Fields from Mod-ℓ Milnor K-Theory](preprints/Reconstruction-of-Function-Fields-from-Mod-ell-Milnor-K-Theory-October-5-2026/mod-ell-bogomolov-pop.pdf)\n\nWe prove a mod-<i>ℓ</i> Bogomolov–Pop reconstruction theorem for function fields of transcendence degree at least two over arbitrary algebraically closed fields of characteristic different from <i>ℓ</i>. The groups $`K^{\\mathrm M}_1/\\ell`$ and $`K^{\\mathrm M}_2/\\ell`$, together with their full bilinear product, determine the perfect closure and its constant field. Every compatible isomorphism of these data is induced by a field isomorphism up to a single scalar in $`\\mathbb F_\\ell^\\times`$, with only Frobenius ambiguity in the field isomorphism in positive characteristic.\n\n</td>\n</tr></tbody>\n<tbody><tr>\n<td>\n\n&emsp;[Reconstruction from Milnor K-theory modulo the characteristic](preprints/Reconstruction-from-Milnor-K-theory-modulo-the-characteristic-October-5-2026/paper.pdf)\n\nLet <i>K</i> and <i>L</i> be finitely generated extensions of transcendence degree at least two over algebraically closed fields of characteristic <i>p</i>. We prove that every isomorphism of their first Milnor <i>K</i>-groups modulo <i>p</i> preserving the degree-two Steinberg relations is a nonzero scalar multiple of the map induced by a unique field isomorphism. The proof recovers projective lines over the subfields of <i>p</i>th powers by an elementary calculation with derivations.\n\n</td>\n</tr></tbody>\n<tbody><tr>\n<td>\n\n&emsp;[The Bogomolov-Pop reconstruction theorem](preprints/The-Bogomolov-Pop-reconstruction-theorem-September-23-2026/paper.pdf)\n\nWe prove the Bogomolov–Pop reconstruction conjecture for function fields of transcendence degree at least two over arbitrary algebraically closed fields of characteristic different from <i>ℓ</i>. The pro-<i>ℓ</i> abelian-by-central datum determines the perfect closure and its constant field, with precisely the Frobenius and <i>ℓ</i>-adic unit ambiguities.\n\n</td>\n</tr></tbody>\n<tbody><tr>\n<td>\n\n**010. Unrestricte\n"
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We expanded these evaluations after performance on our existing mathematical evaluations saturated. Some outputs build upon earlier results produced by the models.\n\nThis collection includes results at different stages of verification. Not all have accompanying Lean formalizations. We will continue to update this repository with Lean formalizations as we obtain them. \n\nSome of the unformalized results could have issues. We will endeavor to fix any such issues quickly.\nWe are also exploring community-hosted repositories for these materials. \n\n## Navigating the collection\n\nThe current catalogue contains 719 manuscripts organized into 372 families. A family groups related papers, which may include a principal result, companion arguments, consequences, or alternative proofs. Each family is classified by mathematical discipline.\n\n- Start with the [overview](overview.pdf) for descriptions of the families.\n- Use the [manuscript map](CONTENTS.md) to find individual papers and their supporting materials.\n- The [`preprints/`](preprints/) directory contains PDFs, source files, and manuscript-specific citation and build instructions.\n- The [Lean library](lean/README.md) and [formalization catalogue](lean/formalization.yaml) describe the available formal proofs, their associated papers, and verification configurations. See the [Comparator instructions](lean/ComparatorChallenges/README.md) for additional checking instructions. The repository has ~42% top-line results formalized.\n- Updates to the repo are described in the [history](history.md).\n\n### Reasoning summaries\n\nWe are also releasing abridged summaries of the model's reasoning, covering the following results:\n\n| Family | Subject |\n|---|---|\n| 007 | [Ordinary two-point correlations of multiplicative functions](reasoning_traces/ordinary-two-point-correlations.pdf) |\n| 017 | [The irrationality exponent of π](reasoning_traces/irrationality-exponent-of-pi.pdf) |\n| 087 | [Symmetric and general Mahler conjectures](reasoning_traces/symmetric-and-general-mahler-conjectures.pdf) |\n| 102 | [Ordinary NP-hardness at the basic semidefinite threshold](reasoning_traces/basic-semidefinite-threshold-np-hardness.pdf) |\n| 159 | [Quasipolynomial bounds for arithmetic progressions](reasoning_traces/quasipolynomial-arithmetic-progressions.pdf) |\n| 197 | [Kaplansky's direct-finiteness conjecture in characteristic two](reasoning_traces/kaplansky-direct-finiteness-characteristic-two.pdf) |\n| 221 | [The Mézard–Parisi formula for diluted spin glasses](reasoning_traces/mezard-parisi-formula.pdf) |\n| 271 | [Spontaneous magnetization in the quantum Heisenberg ferromagnet](reasoning_traces/spontaneous-magnetization-quantum-heisenberg-ferromagnet.pdf) |\n| 287 | [Isomorphism of free group factors](reasoning_traces/free-group-factor-isomorphism.pdf) |\n| 362 | [The three-dimensional relativistic Vlasov–Maxwell system](reasoning_traces/relativistic-vlasov-maxwell.pdf) |\n\n\n## How the results were produced\n\nThe vast majority of results were obtained with the same procedure using an unreleased internal OpenAI model. On average, each result used three hours of ChatGPT Pro thinking compute with that model. Over the course of the evaluation, the model was posed approximately 4,000 problems. Aggregating the output into result families and manuscripts and requiring an appropriate level of significance led to the catalog outlined above.\n\nExceptions to this fixed procedure include work on a zero-free region for the Riemann zeta function and proof of the Hodge Conjecture for CM abelian varieties. Additionally, the writeup for the Re(s) > 11/12 zero-free region for the Riemann zeta function was human edited for readability. \n\n\n## Versions and citations\n\nWe will preserve the public release history of this collection. Corrections and revisions will be recorded as new versions, with previously released versions remaining accessible. \n\nTo cite the individual manuscript, use the BibTeX block in its directory. \n\n\nDOCUMENT lean/README.md \n # Lean formalizations\n\nThis directory contains Lean formalizations of some results in this repository.\nThey are organized in a single large library, so we recommend compiling only small portions at a time.\n\nSee the [Comparator README](ComparatorChallenges/README.md) for more information about how to verify the results.\n\n## Technical note: mmap\n\nCompiling the entire library may fail if Linux's `vm.max_map_count` is too low.\nOne workaround is to build Lean with the CMake option `-DMMAP=OFF`; it may also be necessary to set the environment variable `GLIBC_TUNABLES` to `glibc.malloc.mmap_max=0:glibc.malloc.arena_max=1` when running Lean or Lake.\n\n\nDOCUMENT lean/ComparatorChallenges/README.md \n # [Comparator](https://github.com/leanprover/comparator) challenges\n\nInstall `comparator`, `landrun`, and `lean4export`, and make them available on `PATH`.\nThen, from `lean/`:\n\n```sh\nlake update\nlake exe cache get\nlake env comparator ComparatorChallenges/QuasiRiemannHypothesis.json\n```\n\n## Supporting-result comparisons\n\nThese setups verify supporting results, not the corresponding papers’ main\ntheorems:\n\n- `CharacterVarietiesAllSeamsSupport.json`: compatibility of the produced marked\n  solution with every seam equation.\n- `CartierChartCompactnessSupport.json`: compactness of normalized chart-field\n  retractions.\n- `SurfaceConeCandidateSupport.json`: ring-theoretic properties of the completed\n  surface cone.\n- `TraceIdealTransportSupport.json`: transport of finite and zero ideals, with\n  the construction-provider statements required by the original comparison.\n- `HoneycombBridgeMassSupport.json`: finiteness and quarter-power bounds for the\n  strict honeycomb bridge mass.\n\nSee the [scope notes](../docs/) for per-paper coverage and limitations.\n\n\nCONTENTS preview\n # Mathematics manuscript collection\n\n**719 manuscripts covering 372 result families.**\n\n[**Read the overview PDF**](overview.pdf).\n\n## Manuscript map\n\nEach result description is followed by its constituent manuscripts and their abstracts. Paper titles link directly to PDFs.\n\n<table>\n<thead><tr><th>Result</th></tr></thead>\n<tbody><tr>\n<td>\n\n**001. Milne’s rationality conjecture and algebraic specialization.** Proves Milne's rationality conjecture for abelian varieties over $`\\overline{\\mathbb Q}`$ with good reduction: specialized Hodge classes pair rationally with complementary divisor products, independently of cohomology theory. Together with result 032, every specialized Hodge class is represented by a single rational algebraic cycle simultaneously in all prime-to-<i>p</i> and crystalline realizations, for every residue characteristic <i>p</i>.\n\n</td>\n</tr></tbody>\n<tbody><tr>\n<td>\n\n&emsp;[Milne's rationality conjecture for abelian varieties](preprints/Milnes-rationality-conjecture-for-abelian-varieties-October-7-2026/paper.pdf)\n\nWe prove Milne's rationality conjecture for abelian varieties, including residue characteristic 2. After good reduction, the pairing of a rational Hodge class with any complementary product of divisor classes on the reduction is the same rational number in every prime-to-<i>p</i> realization and in crystalline cohomology. Using the Hodge theorem for CM abelian varieties, we also show that every such specialized Hodge class is represented by a single rational algebraic cycle in all these realizations.\n\n</td>\n</tr></tbody>\n<tbody><tr>\n<td>\n\n**002. The full BSD formula from low Selmer corank.** Proves the full Birch–Swinnerton-Dyer leading-term formula for every elliptic curve over ℚ whose full <i>q</i>-power Selmer group has corank zero or one for some prime <i>q</i>, including finiteness of the Tate–Shafarevich group. With result 006, this gives full BSD for a density-one set of quadratic twists of every elliptic curve over ℚ.\n\n</td>\n</tr></tbody>\n<tbody><tr>\n<td>\n\n&emsp;[Exact Birch–Swinnerton-Dyer Formula from Low Selmer Corank](preprints/Exact-Birch-Swinnerton-Dyer-Formula-from-Low-Selmer-Corank-October-7-2026/exact-bsd-low-selmer-corank.pdf)\n\nWe prove the full Birch–Swinnerton-Dyer leading-term formula for every elliptic curve over ℚ whose full <i>q</i>-power Selmer group has corank zero or one at some prime <i>q</i>. The analytic and Mordell–Weil ranks equal that corank, and the Tate–Shafarevich group is finite. The formula includes all prime factors and requires no additional hypotheses on reduction, rational torsion, isogenies, complex multiplication, or residual Galois representations.\n\n</td>\n</tr></tbody>\n<tbody><tr>\n<td>\n\n&emsp;[The Selmer converse for elliptic curves at every prime](preprints/The-Selmer-converse-for-elliptic-curves-at-every-prime-October-7-2026/main.pdf)\n\nWe prove the Selmer converse in coranks zero and one for every elliptic curve over ℚ and every prime <i>p</i>: if the full <i>p</i>-power Selmer group has ℤ<sub><i>p</i></sub>-corank $`r\\in\\{0,1\\}`$, then the analytic and Mordell–Weil ranks both equal <i>r</i>, and the entire Tate–Shafarevich group is finite. As an application at the additive prime 3, we prove that for every prime $`\\ell\\equiv4,7,8\\pmod9`$, the cubic $`X^3+Y^3=\\ell Z^3`$ has analytic and Mordell–Weil rank one and finite Tate–Shafarevich group. In particular, every such <i>ℓ</i> is a sum of two rational cubes.\n\n</td>\n</tr></tbody>\n<tbody><tr>\n<td>\n\n&emsp;[The two-primary Birch–Swinnerton-Dyer formula in Selmer corank at most one](preprints/The-two-primary-Birch-Swinnerton-Dyer-formula-in-Selmer-corank-at-most-one-October-6-2026/paper.pdf)\n\nWe prove the two-primary Birch and Swinnerton-Dyer leading-term formula for every elliptic curve over the rationals whose two-power Selmer group has corank at most one. In this range, the algebraic rank, analytic rank, and Selmer corank are equal, and the Tate–Shafarevich group is finite. Combined with the quadratic-twist Selmer distribution, this gives the exact two-primary formula for a density-one set of signed squarefree twists of each fixed curve, ordered by absolute value; the common rank is zero or one, with each value having density one half.\n\n</td>\n</tr></tbody>\n<tbody><tr>\n<td>\n\n**003. The quasi-Riemann hypothesis.** Proves that every Dirichlet <i>L</i>-function, including $`\\zeta(s)`$, is zero-free in $`\\Re s\\gt 7/8`$, resolving the quasi-Riemann hypothesis. The same half-plane is zero-free for every finite-order Hecke <i>L</i>-function over $`\\mathbb Q(\\sqrt{-3})`$. A companion gives a different proof of the zero-free half-plane $`\\Re s\\gt 11/12`$. ([Lean](lean/docs/003.md))\n\n</td>\n</tr></tbody>\n<tbody><tr>\n<td>\n\n&emsp;[The Quasi-Riemann Hypothesis: A Zero-Free Half-Plane $`\\Re s\\gt 7/8`$ ](preprints/The-Quasi-Riemann-Hypothesis-September-30-2026/paper.pdf)\n\nWe prove that all finite-order Hecke <i>L</i>-functions over $`\\mathbb Q(\\sqrt{-3})`$ and all Dirichlet <i>L</i>-functions are zero-free in the half-plane $`\\Re s\\gt 7/8`$, with the principal pole at <i>s</i> = 1 allowed. In particular, the Riemann zeta function is zero-free in this half-plane, proving the quasi-Riemann hypothesis.\n\n</td>\n</tr></tbody>\n<tbody><tr>\n<td>\n\n&emsp;[The Quasi-Riemann Hypothesis (alternate 11/12 proof)](preprints/The-Quasi-Riemann-Hypothesis-October-5-2026/paper2.pdf)\n\nWe establish the quasi-Riemann hypothesis by proving that every Dirichlet <i>L</i>-function, including Riemann's zeta function, has no zeros in the half-plane $`\\mathop{\\mathrm{Re}}\\nolimits s\\gt 11/12`$. More generally, we prove the same zero-free half-plane for every finite-order Hecke <i>L</i>-function over $`K=\\mathbb Q(\\sqrt{-3})`$. In particular, this rules out the existence of Landau–Siegel zeros.\n\n</td>\n</tr></tbody>\n<tbody><tr>\n<td>\n\n&emsp;[Uniform exclusion of Landau–Siegel zeros](preprints/Uniform-exclusion-of-Landau-Siegel-zeros-October-1-2026/paper.pdf)\n\nWe prove the uniform exclusion of Landau–Siegel zeros. There is an absolute constant <i>c</i> &gt; 0 such that every real zero $`\\beta\\in(0,1)`$ of every primitive nonprincipal real Dirichlet <i>L</i>-function of conductor <i>q</i> ≥ 3 satisfies $`(1-\\beta)\\log q\\ge c`$.\n\n</td>\n</tr></tbody>\n<tbody><tr>\n<td>\n\n**004. Hilbert’s tenth problem over ℚ.** Proves that no algorithm decides whether an integer-coefficient polynomial in an arbitrary number of variables has a rational zero, resolving Hilbert's tenth problem over ℚ negatively.\n\n</td>\n</tr></tbody>\n<tbody><tr>\n<td>\n\n&emsp;[Hilbert’s tenth problem over the rational numbers](preprints/Hilberts-tenth-problem-over-the-rational-numbers-October-6-2026/main.pdf)\n\nWe give a negative answer to Hilbert's tenth problem over the rational numbers: no algorithm decides whether a polynomial with integer coefficients has a rational zero. The number of variables is part of the input.\n\n</td>\n</tr></tbody>\n<tbody><tr>\n<td>\n\n&emsp;[A pointwise 2-converse for elliptic curves with rational two-torsion](preprints/A-pointwise-2-converse-for-elliptic-curves-with-rational-two-torsion-October-7-2026/paper.pdf)\n\nWe prove a pointwise 2-converse for elliptic curves over $`\\mathbf Q`$ with nonzero rational two-torsion: if the $`2^\\infty`$-Selmer corank is zero or one, then the analytic rank and Mordell–Weil rank equal that corank, and the Shafarevich–Tate group is finite. The result allows arbitrary reduction at 2.\n\n</td>\n</tr></tbody>\n<tbody><tr>\n<td>\n\n**005. Irrationality of Catalan’s constant.** Proves that Catalan's constant $`G=\\sum_{j\\ge0}(-1)^j/(2j+1)^2`$ is irrational. ([Lean](lean/docs/005.md))\n\n</td>\n</tr></tbody>\n<tbody><tr>\n<td>\n\n&emsp;[Catalan's constant is irrational](preprints/Catalans-constant-is-irrational-September-24-2026/paper.pdf)\n\nWe prove that Catalan's constant $`G=\\sum_{j\\geq0}(-1)^j/(2j+1)^2`$ is irrational.\n\n</td>\n</tr></tbody>\n<tbody><tr>\n<td>\n\n**006. Goldfeld’s conjecture: densities and mean analytic rank.** Proves Goldfeld's conjecture for quadratic twists of every elliptic curve over ℚ: analytic ranks zero and one each have density 1/2, and the mean analytic rank tends to 1/2. Both statements order signed squarefree twist parameters by absolute value.\n\n</td>\n</tr></tbody>\n<tbody><tr>\n<td>\n\n&emsp;[Goldfeld's analytic density conjecture and the 2-converse for elliptic curves](preprints/Goldfelds-analytic-density-conjecture-and-the-2-converse-for-elliptic-curves-October-7-2026/paper.pdf)\n\nWe prove Goldfeld's analytic density conjecture: for every elliptic curve <i>E</i> over ℚ, the quadratic twists of <i>E</i> with analytic rank zero and one each have density 1/2 among signed squarefree twist parameters ordered by absolute value. We also prove the low-corank 2-converse: if the $`2^\\infty`$-Selmer corank of <i>E</i> is zero or one, then it equals the analytic and Mordell–Weil ranks, and the Tate–Shafarevich group is finite.\n\n</td>\n</tr></tbody>\n<tbody><tr>\n<td>\n\n&emsp;[The mean analytic rank of quadratic twists of elliptic curves](preprints/The-mean-analytic-rank-of-quadratic-twists-of-elliptic-curves-October-6-2026/paper.pdf)\n\nFor every elliptic curve over ℚ, we prove that the average analytic rank of its quadratic twists tends to 1/2 when signed squarefree twist parameters are ordered by absolute value. This resolves Goldfeld's mean analytic-rank conjecture in this counting convention.\n\n</td>\n</tr></tbody>\n<tbody><tr>\n<td>\n\n**007. Ordinary two-point correlations and the corrected Elliott conjecture.** Proves the ordinary two-point Chowla conjecture, with a bound $`O(X/(\\log X)^c)`$ for Liouville correlation sums along fixed nonproportional affine forms, where <i>c</i> &gt; 0 is absolute. More generally, proves the binary corrected Elliott conjecture for complex multiplicative functions bounded by one when one factor is uniformly nonpretentious against each fixed Dirichlet character times $`n^{it}`$ for $`|t|\\le X`$. ([Lean](lean/docs/007.md))\n\n</td>\n</tr></tbody>\n<tbody><tr>\n<td>\n\n&emsp;[Ordinary two-point correlations of multiplicative functions](preprints/Ordinary-two-point-correlations-of-multiplicative-functions-September-24-2026/final.pdf)\n\nWe prove the ordinary two-point Chowla conjecture. For every fixed pair of nonproportional affine forms, the Liouville correlation has a power-of-logarithm saving at every cutoff, with an absolute exponent. We also prove the binary corrected Elliott conjecture for ordinary averages of complex multiplicative functions of modulus at most one, under uniform nonpretentiousness of at least one original factor. This qualitative conclusion holds in fixed residue classes and for fixed nonproportional affine forms.\n\n</td>\n</tr></tbody>\n<tbody><tr>\n<td>\n\n**008. The Deligne–Drinfeld conjecture.** Proves that the rational Grothendieck–Teichmüller Lie algebra, with the Ihara bracket, is freely generated by one element in each odd weight 3, 5, 7, …, resolving the Deligne–Drinfeld conjecture. ([Lean](lean/docs/008.md))\n\n</td>\n</tr></tbody>\n<tbody><tr>\n<td>\n\n&emsp;[The Deligne-Drinfeld conjecture](preprints/The-Deligne-Drinfeld-conjecture-September-23-2026/paper.pdf)\n\nWe prove the Deligne–Drinfeld conjecture: the rational Grothendieck–Teichmüller Lie algebra is freely generated by one element in every odd weight at least three, with the corresponding isomorphism after weight completion.\n\n</td>\n</tr></tbody>\n<tbody><tr>\n<td>\n\n**009. Function-field reconstruction from Milnor K-theory and Galois data.** Reconstructs function fields of transcendence degree at least two over algebraically closed constants from $`K^{\\mathrm M}_1/\\ell`$, $`K^{\\mathrm M}_2/\\ell`$, and their product. These data recover the perfect closure and constants when <i>ℓ</i> differs from the characteristic, and the original field and its named base in equal characteristic. Also proves Bogomolov–Pop reconstruction from abelian-by-central pro-<i>ℓ</i> Galois data away from the characteristic. ([Lean](lean/docs/009.md))\n\n</td>\n</tr></tbody>\n<tbody><tr>\n<td>\n\n&emsp;[Reconstruction of Function Fields from Mod-ℓ Milnor K-Theory](preprints/Reconstruction-of-Function-Fields-from-Mod-ell-Milnor-K-Theory-October-5-2026/mod-ell-bogomolov-pop.pdf)\n\nWe prove a mod-<i>ℓ</i> Bogomolov–Pop reconstruction theorem for function fields of transcendence degree at least two over arbitrary algebraically closed fields of characteristic different from <i>ℓ</i>. The groups $`K^{\\mathrm M}_1/\\ell`$ and $`K^{\\mathrm M}_2/\\ell`$, together with their full bilinear product, determine the perfect closure and its constant field. Every compatible isomorphism of these data is induced by a field isomorphism up to a single scalar in $`\\mathbb F_\\ell^\\times`$, with only Frobenius ambiguity in the field isomorphism in positive characteristic.\n\n</td>\n</tr></tbody>\n<tbody><tr>\n<td>\n\n&emsp;[Reconstruction from Milnor K-theory modulo the characteristic](preprints/Reconstruction-from-Milnor-K-theory-modulo-the-characteristic-October-5-2026/paper.pdf)\n\nLet <i>K</i> and <i>L</i> be finitely generated extensions of transcendence degree at least two over algebraically closed fields of characteristic <i>p</i>. We prove that every isomorphism of their first Milnor <i>K</i>-groups modulo <i>p</i> preserving the degree-two Steinberg relations is a nonzero scalar multiple of the map induced by a unique field isomorphism. The proof recovers projective lines over the subfields of <i>p</i>th powers by an elementary calculation with derivations.\n\n</td>\n</tr></tbody>\n<tbody><tr>\n<td>\n\n&emsp;[The Bogomolov-Pop reconstruction theorem](preprints/The-Bogomolov-Pop-reconstruction-theorem-September-23-2026/paper.pdf)\n\nWe prove the Bogomolov–Pop reconstruction conjecture for function fields of transcendence degree at least two over arbitrary algebraically closed fields of characteristic different from <i>ℓ</i>. The pro-<i>ℓ</i> abelian-by-central datum determines the perfect closure and its constant field, with precisely the Frobenius and <i>ℓ</i>-adic unit ambiguities.\n\n</td>\n</tr></tbody>\n<tbody><tr>\n<td>\n\n**010. Unrestricte\n"
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        "stdout": "lean/README.md # Lean formalizations\n\nThis directory contains Lean formalizations of some results in this repository.\nThey are organized in a single large library, so we recommend compiling only small portions at a time.\n\nSee the [Comparator README](ComparatorChallenges/README.md) for more information about how to verify the results.\n\n## Technical note: mmap\n\nCompiling the entire library may fail if Linux's `vm.max_map_count` is too low.\nOne workaround is to build Lean with the CMake option `-DMMAP=OFF`; it may also be necessary to set the environment variable `GLIBC_TUNABLES` to `glibc.malloc.mmap_max=0:glibc.malloc.arena_max=1` when running Lean or Lake.\n\nlean/ComparatorChallenges/README.md # [Comparator](https://github.com/leanprover/comparator) challenges\n\nInstall `comparator`, `landrun`, and `lean4export`, and make them available on `PATH`.\nThen, from `lean/`:\n\n```sh\nlake update\nlake exe cache get\nlake env comparator ComparatorChallenges/QuasiRiemannHypothesis.json\n```\n\n## Supporting-result comparisons\n\nThese setups verify supporting results, not the corresponding papers’ main\ntheorems:\n\n- `CharacterVarietiesAllSeamsSupport.json`: compatibility of the produced marked\n  solution with every seam equation.\n- `CartierChartCompactnessSupport.json`: compactness of normalized chart-field\n  retractions.\n- `SurfaceConeCandidateSupport.json`: ring-theoretic properties of the completed\n  surface cone.\n- `TraceIdealTransportSupport.json`: transport of finite and zero ideals, with\n  the construction-provider statements required by the original comparison.\n- `HoneycombBridgeMassSupport.json`: finiteness and quarter-power bounds for the\n  strict honeycomb bridge mass.\n\nSee the [scope notes](../docs/) for per-paper coverage and limitations.\n\nlean/lean-toolchain leanprover/lean4:v4.34.1\n\nCONTENTS start # Mathematics manuscript collection\n\n**719 manuscripts covering 372 result families.**\n\n[**Read the overview PDF**](overview.pdf).\n\n## Manuscript map\n\nEach result description is followed by its constituent manuscripts and their abstracts. Paper titles link directly to PDFs.\n\n<table>\n<thead><tr><th>Result</th></tr></thead>\n<tbody><tr>\n<td>\n\n**001. Milne’s rationality conjecture and algebraic specialization.** Proves Milne's rationality conjecture for abelian varieties over $`\\overline{\\mathbb Q}`$ with good reduction: specialized Hodge classes pair rationally with complementary divisor products, independently of cohomology theory. Together with result 032, every specialized Hodge class is represented by a single rational algebraic cycle simultaneously in all prime-to-<i>p</i> and crystalline realizations, for every residue characteristic <i>p</i>.\n\n</td>\n</tr></tbody>\n<tbody><tr>\n<td>\n\n&emsp;[Milne's rationality conjecture for abelian varieties](preprints/Milnes-rationality-conjecture-for-abelian-varieties-October-7-2026/paper.pdf)\n\nWe prove Milne's rationality conjecture for abelian varieties, including residue characteristic 2. After good reduction, the pairing of a rational Hodge class with any complementary product of divisor classes on the reduction is the same rational number in every prime-to-<i>p</i> realization and in crystalline cohomology. Using the Hodge theorem for CM abelian varieties, we also show that every such specialized Hodge class is represented by a single rational algebraic cycle in all these realizations.\n\n</td>\n</tr></tbody>\n<tbody><tr>\n<td>\n\n**002. The full BSD formula from low Selmer corank.** Proves the full Birch–Swinnerton-Dyer leading-term formula for every elliptic curve over ℚ whose full <i>q</i>-power Selmer group has corank zero or one for some prime <i>q</i>, including finiteness of the Tate–Shafarevich group. With result 006, this gives full BSD for a density-one set of quadratic twists of every elliptic curve over ℚ.\n\n</td>\n</tr></tbody>\n<tbody><tr>\n<td>\n\n&emsp;[Exact Birch–Swinnerton-Dyer Formula from Low Selmer Corank](preprints/Exact-Birch-Swinnerton-Dyer-Formula-from-Low-Selmer-Corank-October-7-2026/exact-bsd-low-selmer-corank.pdf)\n\nWe prove the full Birch–Swinnerton-Dyer leading-term formula for every elliptic curve over ℚ whose full <i>q</i>-power Selmer group has corank zero or one at some prime <i>q</i>. The analytic and Mordell–Weil ranks equal that corank, and the Tate–Shafarevich group is finite. The formula includes all prime factors and requires no additional hypotheses on reduction, rational torsion, isogenies, complex multiplication, or residual Galois representations.\n\n</td>\n</tr></tbody>\n<tbody><tr>\n<td>\n\n&emsp;[The Selmer converse for elliptic curves at every prime](preprints/The-Selmer-converse-for-elliptic-curves-at-every-prime-October-7-2026/main.pdf)\n\nWe prove the Selmer converse in coranks zero and one for every elliptic curve over ℚ and every prime <i>p</i>: if the full <i>p</i>-power Selmer group has ℤ<sub><i>p</i></sub>-corank $`r\\in\\{0,1\\}`$, then the analytic and Mordell–Weil ranks both equal <i>r</i>, and the entire Tate–Shafarevich group is finite. As an application at the additive prime 3, we prove that for every prime $`\\ell\\equiv4,7,8\\pmod9`$, the cubic $`X^3+Y^3=\\ell Z^3`$ has analytic and Mordell–Weil rank one and finite Tate–Shafarevich group. In particular, every such <i>ℓ</i> is a sum of two rational cubes.\n\n</td>\n</tr></tbody>\n<tbody><tr>\n<td>\n\n&emsp;[The two-primary Birch–Swinnerton-Dyer formula in Selmer corank at most one](preprints/The-two-primary-Birch-Swinnerton-Dyer-formula-in-Selmer-corank-at-most-one-October-6-2026/paper.pdf)\n\nWe prove the two-primary Birch and Swinnerton-Dyer leading-term formula for every elliptic curve over the rationals whose two-power Selmer group has corank at most one. In this range, the algebraic rank, analytic rank, and Selmer corank are equal, and the Tate–Shafarevich group is finite. Combined with the quadratic-twist Selmer distribution, this gives the exact two-primary formula for a density-one set of signed squarefree twists of each fixed curve, ordered by absolute value; the common rank is zero or one, with each value having density one half.\n\n</td>\n</tr></tbody>\n<tbody><tr>\n<td>\n\n**003. The quasi-Riemann hypothesis.** Proves that every Dirichlet <i>L</i>-function, including $`\\zeta(s)`$, is zero-free in $`\\Re s\\gt 7/8`$, resolving the quasi-Riemann hypothesis. The same half-plane is zero-free for every finite-order Hecke <i>L</i>-function over $`\\mathbb Q(\\sqrt{-3})`$. A companion gives a different proof of the zero-free half-plane $`\\Re s\\gt 11/12`$. ([Lean](lean/docs/003.md))\n\n</td>\n</tr></tbody>\n<tbody><tr>\n<td>\n\n&emsp;[The Quasi-Riemann Hypothesis: A Zero-Free Half-Plane $`\\Re s\\gt 7/8`$ ](preprints/The-Quasi-Riemann-Hypothesis-September-30-2026/paper.pdf)\n\nWe prove that all finite-order Hecke <i>L</i>-functions over $`\\mathbb Q(\\sqrt{-3})`$ and all Dirichlet <i>L</i>-functions are zero-free in the half-plane $`\\Re s\\gt 7/8`$, with the principal pole at <i>s</i> = 1 allowed. In particular, the Riemann zeta function is zero-free in this half-plane, proving the quasi-Riemann hypothesis.\n\n</td>\n</tr></tbody>\n<tbody><tr>\n<td>\n\n&emsp;[The Quasi-Riemann Hypothesis (alternate 11/12 proof)](preprints/The-Quasi-Riemann-Hypothesis-October-5-2026/paper2.pdf)\n\nWe establish the quasi-Riemann hypothesis by proving that every Dirichlet <i>L</i>-function, including Riemann's zeta function, has no zeros in the half-plane $`\\mathop{\\mathrm{Re}}\\nolimits s\\gt 11/12`$. More generally, we prove the same zero-free half-plane for every finite-order Hecke <i>L</i>-function over $`K=\\mathbb Q(\\sqrt{-3})`$. In particular, this rules out the existence of Landau–Siegel zeros.\n\n</td>\n</tr></tbody>\n<tbody><tr>\n<td>\n\n&emsp;[Uniform exclusion of Landau–Siegel zeros](preprints/Uniform-exclusion-of-Landau-Siegel-zeros-October-1-2026/paper.pdf)\n\nWe prove the uniform exclusion of Landau–Siegel zeros. There is an absolute constant <i>c</i> &gt; 0 such that every real zero $`\\beta\\in(0,1)`$ of every primitive nonprincipal real Dirichlet <i>L</i>-function of conductor <i>q</i> ≥ 3 satisfies $`(1-\\beta)\\log q\\ge c`$.\n\n</td>\n</tr></tbody>\n<tbody><tr>\n<td>\n\n**004. Hilbert’s tenth problem over ℚ.** Proves that no algorithm decides whether an integer-coefficient polynomial in an arbitrary number of variables has a rational zero, resolving Hilbert's tenth problem over ℚ negatively.\n\n</td>\n</tr></tbody>\n\nYAML start # yaml-language-server: $schema=https://raw.githubusercontent.com/mathlib-initiative/formalization.yaml/main/schema/formalization.schema.json\n# Catalog of papers with a formalized main result. Paths are relative to lean/.\nversion: \"v0.4\"\n\nproject:\n  name: \"OpenAI math repository\"\n  description: \"Lean formalizations accompanying a mathematics manuscript collection.\"\n  authors: [\"OpenAI\"]\n  license: \"Apache-2.0\"\n\nsources:\n  - title: \"$L_1$ Embeddings of Graphs of Bounded Treewidth\"\n    authors: [\"OpenAI\"]\n    id: ../preprints/L1-Embeddings-of-Graphs-of-Bounded-Treewidth-September-23-2026/paper.pdf\n    type: article\n\n  - title: \"A CH obstruction to a prescribed categoricity threshold\"\n    authors: [\"OpenAI\"]\n    id: ../preprints/A-CH-Obstruction-to-a-Prescribed-Categoricity-Threshold-September-24-2026/paper.pdf\n    type: article\n\n  - title: \"A classification of finite Euclidean Ramsey configurations\"\n    authors: [\"OpenAI\"]\n    id: ../preprints/A-classification-of-finite-Euclidean-Ramsey-configurations-September-23-2026/paper.pdf\n    type: article\n\n  - title: \"A Complete Local Domain without a Small Cohen–Macaulay Module\"\n    authors: [\"OpenAI\"]\n    id: ../preprints/A-Complete-Local-Domain-Without-a-Small-Cohen-Macaulay-Module-September-23-2026/paper.pdf\n    type: article\n\n  - title: \"A continuum temperature singularity for a radial pair potential\"\n    authors: [\"OpenAI\"]\n    id: ../preprints/A-continuum-temperature-singularity-for-a-radial-pair-potential-September-24-2026/paper.pdf\n    type: article\n\n  - title: \"A counterexample to Hadwiger's conjecture\"\n    authors: [\"OpenAI\"]\n    id: ../preprints/A-counterexample-to-Hadwigers-conjecture-September-23-2026/paper.pdf\n    type: article\n\n  - title: \"A counterexample to integer-degree harmonic dimension comparison\"\n    authors: [\"OpenAI\"]\n    id: ../preprints/A-counterexample-to-integer-degree-harmonic-dimension-comparison-September-25-2026/paper.pdf\n    type: article\n\n  - title: \"A Counterexample to Kaplansky's Direct-Finiteness Conjecture in Characteristic Two\"\n    authors: [\"OpenAI\"]\n    id: ../preprints/A-Counterexample-to-Kaplanskys-Direct-Finiteness-Conjecture-in-Characteristic-Two-September-23-2026/paper.pdf\n    type: article\n\n  - title: \"A Counterexample to Kaplansky's Direct-Finiteness Conjecture in Odd Characteristic\"\n    authors: [\"OpenAI\"]\n    id: ../preprints/A-Counterexample-to-Kaplanskys-Direct-Finiteness-Conjecture-in-Odd-Characteristic-September-26-2026/paper.pdf\n    type: article\n\n  - title: \"A counterexample to Kaplansky's quasitrace conjecture and failure of tensor-product stable finiteness\"\n    authors: [\"OpenAI\"]\n    id: ../preprints/A-counterexample-to-Kaplanskys-quasitrace-conjecture-September-23-2026/paper.pdf\n    type: article\n\n  - title: \"A counterexample to Ryser's covering conjecture\"\n    authors: [\"OpenAI\"]\n    id: ../preprints/A-Counterexample-to-Rysers-Covering-Conjecture-September-23-2026/paper.pdf\n    type: article\n\n  - title: \"A counterexample to Tachikawa's second conjecture\"\n    authors: [\"OpenAI\"]\n    id: ../preprints/A-counterexample-to-Tachikawas-second-conjecture-September-23-2026/paper.pdf\n    type: article\n\n  - title: \"A Counterexample to the Group-Ring Determinant Conjecture\"\n    authors: [\"OpenAI\"]\n    id: ../preprints/A-Counterexample-to-the-Group-Ring-Determinant-Conjecture-September-23-2026/paper.pdf\n    type: article\n\n  - title: \"A Counterexample to the Infinite Matroid Packing/Covering Conjecture\"\n    authors: [\"OpenAI\"]\n    id: ../preprints/A\ntrees [('preprints_tree', 13223, False), ('comparator_tree', 833, False)]\n",
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          "lean/README.md # Lean formalizations\n\nThis directory contains Lean formalizations of some results in this repository.\nThey are organized in a single large library, so we recommend compiling only small portions at a time.\n\nSee the [Comparator README](ComparatorChallenges/README.md) for more information about how to verify the results.\n\n## Technical note: mmap\n\nCompiling the entire library may fail if Linux's `vm.max_map_count` is too low.\nOne workaround is to build Lean with the CMake option `-DMMAP=OFF`; it may also be necessary to set the environment variable `GLIBC_TUNABLES` to `glibc.malloc.mmap_max=0:glibc.malloc.arena_max=1` when running Lean or Lake.\n\nlean/ComparatorChallenges/README.md # [Comparator](https://github.com/leanprover/comparator) challenges\n\nInstall `comparator`, `landrun`, and `lean4export`, and make them available on `PATH`.\nThen, from `lean/`:\n\n```sh\nlake update\nlake exe cache get\nlake env comparator ComparatorChallenges/QuasiRiemannHypothesis.json\n```\n\n## Supporting-result comparisons\n\nThese setups verify supporting results, not the corresponding papers’ main\ntheorems:\n\n- `CharacterVarietiesAllSeamsSupport.json`: compatibility of the produced marked\n  solution with every seam equation.\n- `CartierChartCompactnessSupport.json`: compactness of normalized chart-field\n  retractions.\n- `SurfaceConeCandidateSupport.json`: ring-theoretic properties of the completed\n  surface cone.\n- `TraceIdealTransportSupport.json`: transport of finite and zero ideals, with\n  the construction-provider statements required by the original comparison.\n- `HoneycombBridgeMassSupport.json`: finiteness and quarter-power bounds for the\n  strict honeycomb bridge mass.\n\nSee the [scope notes](../docs/) for per-paper coverage and limitations.\n\nlean/lean-toolchain leanprover/lean4:v4.34.1\n\nCONTENTS start # Mathematics manuscript collection\n\n**719 manuscripts covering 372 result families.**\n\n[**Read the overview PDF**](overview.pdf).\n\n## Manuscript map\n\nEach result description is followed by its constituent manuscripts and their abstracts. Paper titles link directly to PDFs.\n\n<table>\n<thead><tr><th>Result</th></tr></thead>\n<tbody><tr>\n<td>\n\n**001. Milne’s rationality conjecture and algebraic specialization.** Proves Milne's rationality conjecture for abelian varieties over $`\\overline{\\mathbb Q}`$ with good reduction: specialized Hodge classes pair rationally with complementary divisor products, independently of cohomology theory. Together with result 032, every specialized Hodge class is represented by a single rational algebraic cycle simultaneously in all prime-to-<i>p</i> and crystalline realizations, for every residue characteristic <i>p</i>.\n\n</td>\n</tr></tbody>\n<tbody><tr>\n<td>\n\n&emsp;[Milne's rationality conjecture for abelian varieties](preprints/Milnes-rationality-conjecture-for-abelian-varieties-October-7-2026/paper.pdf)\n\nWe prove Milne's rationality conjecture for abelian varieties, including residue characteristic 2. After good reduction, the pairing of a rational Hodge class with any complementary product of divisor classes on the reduction is the same rational number in every prime-to-<i>p</i> realization and in crystalline cohomology. Using the Hodge theorem for CM abelian varieties, we also show that every such specialized Hodge class is represented by a single rational algebraic cycle in all these realizations.\n\n</td>\n</tr></tbody>\n<tbody><tr>\n<td>\n\n**002. The full BSD formula from low Selmer corank.** Proves the full Birch–Swinnerton-Dyer leading-term formula for every elliptic curve over ℚ whose full <i>q</i>-power Selmer group has corank zero or one for some prime <i>q</i>, including finiteness of the Tate–Shafarevich group. With result 006, this gives full BSD for a density-one set of quadratic twists of every elliptic curve over ℚ.\n\n</td>\n</tr></tbody>\n<tbody><tr>\n<td>\n\n&emsp;[Exact Birch–Swinnerton-Dyer Formula from Low Selmer Corank](preprints/Exact-Birch-Swinnerton-Dyer-Formula-from-Low-Selmer-Corank-October-7-2026/exact-bsd-low-selmer-corank.pdf)\n\nWe prove the full Birch–Swinnerton-Dyer leading-term formula for every elliptic curve over ℚ whose full <i>q</i>-power Selmer group has corank zero or one at some prime <i>q</i>. The analytic and Mordell–Weil ranks equal that corank, and the Tate–Shafarevich group is finite. The formula includes all prime factors and requires no additional hypotheses on reduction, rational torsion, isogenies, complex multiplication, or residual Galois representations.\n\n</td>\n</tr></tbody>\n<tbody><tr>\n<td>\n\n&emsp;[The Selmer converse for elliptic curves at every prime](preprints/The-Selmer-converse-for-elliptic-curves-at-every-prime-October-7-2026/main.pdf)\n\nWe prove the Selmer converse in coranks zero and one for every elliptic curve over ℚ and every prime <i>p</i>: if the full <i>p</i>-power Selmer group has ℤ<sub><i>p</i></sub>-corank $`r\\in\\{0,1\\}`$, then the analytic and Mordell–Weil ranks both equal <i>r</i>, and the entire Tate–Shafarevich group is finite. As an application at the additive prime 3, we prove that for every prime $`\\ell\\equiv4,7,8\\pmod9`$, the cubic $`X^3+Y^3=\\ell Z^3`$ has analytic and Mordell–Weil rank one and finite Tate–Shafarevich group. In particular, every such <i>ℓ</i> is a sum of two rational cubes.\n\n</td>\n</tr></tbody>\n<tbody><tr>\n<td>\n\n&emsp;[The two-primary Birch–Swinnerton-Dyer formula in Selmer corank at most one](preprints/The-two-primary-Birch-Swinnerton-Dyer-formula-in-Selmer-corank-at-most-one-October-6-2026/paper.pdf)\n\nWe prove the two-primary Birch and Swinnerton-Dyer leading-term formula for every elliptic curve over the rationals whose two-power Selmer group has corank at most one. In this range, the algebraic rank, analytic rank, and Selmer corank are equal, and the Tate–Shafarevich group is finite. Combined with the quadratic-twist Selmer distribution, this gives the exact two-primary formula for a density-one set of signed squarefree twists of each fixed curve, ordered by absolute value; the common rank is zero or one, with each value having density one half.\n\n</td>\n</tr></tbody>\n<tbody><tr>\n<td>\n\n**003. The quasi-Riemann hypothesis.** Proves that every Dirichlet <i>L</i>-function, including $`\\zeta(s)`$, is zero-free in $`\\Re s\\gt 7/8`$, resolving the quasi-Riemann hypothesis. The same half-plane is zero-free for every finite-order Hecke <i>L</i>-function over $`\\mathbb Q(\\sqrt{-3})`$. A companion gives a different proof of the zero-free half-plane $`\\Re s\\gt 11/12`$. ([Lean](lean/docs/003.md))\n\n</td>\n</tr></tbody>\n<tbody><tr>\n<td>\n\n&emsp;[The Quasi-Riemann Hypothesis: A Zero-Free Half-Plane $`\\Re s\\gt 7/8`$ ](preprints/The-Quasi-Riemann-Hypothesis-September-30-2026/paper.pdf)\n\nWe prove that all finite-order Hecke <i>L</i>-functions over $`\\mathbb Q(\\sqrt{-3})`$ and all Dirichlet <i>L</i>-functions are zero-free in the half-plane $`\\Re s\\gt 7/8`$, with the principal pole at <i>s</i> = 1 allowed. In particular, the Riemann zeta function is zero-free in this half-plane, proving the quasi-Riemann hypothesis.\n\n</td>\n</tr></tbody>\n<tbody><tr>\n<td>\n\n&emsp;[The Quasi-Riemann Hypothesis (alternate 11/12 proof)](preprints/The-Quasi-Riemann-Hypothesis-October-5-2026/paper2.pdf)\n\nWe establish the quasi-Riemann hypothesis by proving that every Dirichlet <i>L</i>-function, including Riemann's zeta function, has no zeros in the half-plane $`\\mathop{\\mathrm{Re}}\\nolimits s\\gt 11/12`$. More generally, we prove the same zero-free half-plane for every finite-order Hecke <i>L</i>-function over $`K=\\mathbb Q(\\sqrt{-3})`$. In particular, this rules out the existence of Landau–Siegel zeros.\n\n</td>\n</tr></tbody>\n<tbody><tr>\n<td>\n\n&emsp;[Uniform exclusion of Landau–Siegel zeros](preprints/Uniform-exclusion-of-Landau-Siegel-zeros-October-1-2026/paper.pdf)\n\nWe prove the uniform exclusion of Landau–Siegel zeros. There is an absolute constant <i>c</i> &gt; 0 such that every real zero $`\\beta\\in(0,1)`$ of every primitive nonprincipal real Dirichlet <i>L</i>-function of conductor <i>q</i> ≥ 3 satisfies $`(1-\\beta)\\log q\\ge c`$.\n\n</td>\n</tr></tbody>\n<tbody><tr>\n<td>\n\n**004. Hilbert’s tenth problem over ℚ.** Proves that no algorithm decides whether an integer-coefficient polynomial in an arbitrary number of variables has a rational zero, resolving Hilbert's tenth problem over ℚ negatively.\n\n</td>\n</tr></tbody>\n\nYAML start # yaml-language-server: $schema=https://raw.githubusercontent.com/mathlib-initiative/formalization.yaml/main/schema/formalization.schema.json\n# Catalog of papers with a formalized main result. Paths are relative to lean/.\nversion: \"v0.4\"\n\nproject:\n  name: \"OpenAI math repository\"\n  description: \"Lean formalizations accompanying a mathematics manuscript collection.\"\n  authors: [\"OpenAI\"]\n  license: \"Apache-2.0\"\n\nsources:\n  - title: \"$L_1$ Embeddings of Graphs of Bounded Treewidth\"\n    authors: [\"OpenAI\"]\n    id: ../preprints/L1-Embeddings-of-Graphs-of-Bounded-Treewidth-September-23-2026/paper.pdf\n    type: article\n\n  - title: \"A CH obstruction to a prescribed categoricity threshold\"\n    authors: [\"OpenAI\"]\n    id: ../preprints/A-CH-Obstruction-to-a-Prescribed-Categoricity-Threshold-September-24-2026/paper.pdf\n    type: article\n\n  - title: \"A classification of finite Euclidean Ramsey configurations\"\n    authors: [\"OpenAI\"]\n    id: ../preprints/A-classification-of-finite-Euclidean-Ramsey-configurations-September-23-2026/paper.pdf\n    type: article\n\n  - title: \"A Complete Local Domain without a Small Cohen–Macaulay Module\"\n    authors: [\"OpenAI\"]\n    id: ../preprints/A-Complete-Local-Domain-Without-a-Small-Cohen-Macaulay-Module-September-23-2026/paper.pdf\n    type: article\n\n  - title: \"A continuum temperature singularity for a radial pair potential\"\n    authors: [\"OpenAI\"]\n    id: ../preprints/A-continuum-temperature-singularity-for-a-radial-pair-potential-September-24-2026/paper.pdf\n    type: article\n\n  - title: \"A counterexample to Hadwiger's conjecture\"\n    authors: [\"OpenAI\"]\n    id: ../preprints/A-counterexample-to-Hadwigers-conjecture-September-23-2026/paper.pdf\n    type: article\n\n  - title: \"A counterexample to integer-degree harmonic dimension comparison\"\n    authors: [\"OpenAI\"]\n    id: ../preprints/A-counterexample-to-integer-degree-harmonic-dimension-comparison-September-25-2026/paper.pdf\n    type: article\n\n  - title: \"A Counterexample to Kaplansky's Direct-Finiteness Conjecture in Characteristic Two\"\n    authors: [\"OpenAI\"]\n    id: ../preprints/A-Counterexample-to-Kaplanskys-Direct-Finiteness-Conjecture-in-Characteristic-Two-September-23-2026/paper.pdf\n    type: article\n\n  - title: \"A Counterexample to Kaplansky's Direct-Finiteness Conjecture in Odd Characteristic\"\n    authors: [\"OpenAI\"]\n    id: ../preprints/A-Counterexample-to-Kaplanskys-Direct-Finiteness-Conjecture-in-Odd-Characteristic-September-26-2026/paper.pdf\n    type: article\n\n  - title: \"A counterexample to Kaplansky's quasitrace conjecture and failure of tensor-product stable finiteness\"\n    authors: [\"OpenAI\"]\n    id: ../preprints/A-counterexample-to-Kaplanskys-quasitrace-conjecture-September-23-2026/paper.pdf\n    type: article\n\n  - title: \"A counterexample to Ryser's covering conjecture\"\n    authors: [\"OpenAI\"]\n    id: ../preprints/A-Counterexample-to-Rysers-Covering-Conjecture-September-23-2026/paper.pdf\n    type: article\n\n  - title: \"A counterexample to Tachikawa's second conjecture\"\n    authors: [\"OpenAI\"]\n    id: ../preprints/A-counterexample-to-Tachikawas-second-conjecture-September-23-2026/paper.pdf\n    type: article\n\n  - title: \"A Counterexample to the Group-Ring Determinant Conjecture\"\n    authors: [\"OpenAI\"]\n    id: ../preprints/A-Counterexample-to-the-Group-Ring-Determinant-Conjecture-September-23-2026/paper.pdf\n    type: article\n\n  - title: \"A Counterexample to the Infinite Matroid Packing/Covering Conjecture\"\n    authors: [\"OpenAI\"]\n    id: ../preprints/A\ntrees [('preprints_tree', 13223, False), ('comparator_tree', 833, False)]\n"
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          "text": "lean/README.md # Lean formalizations\n\nThis directory contains Lean formalizations of some results in this repository.\nThey are organized in a single large library, so we recommend compiling only small portions at a time.\n\nSee the [Comparator README](ComparatorChallenges/README.md) for more information about how to verify the results.\n\n## Technical note: mmap\n\nCompiling the entire library may fail if Linux's `vm.max_map_count` is too low.\nOne workaround is to build Lean with the CMake option `-DMMAP=OFF`; it may also be necessary to set the environment variable `GLIBC_TUNABLES` to `glibc.malloc.mmap_max=0:glibc.malloc.arena_max=1` when running Lean or Lake.\n\nlean/ComparatorChallenges/README.md # [Comparator](https://github.com/leanprover/comparator) challenges\n\nInstall `comparator`, `landrun`, and `lean4export`, and make them available on `PATH`.\nThen, from `lean/`:\n\n```sh\nlake update\nlake exe cache get\nlake env comparator ComparatorChallenges/QuasiRiemannHypothesis.json\n```\n\n## Supporting-result comparisons\n\nThese setups verify supporting results, not the corresponding papers’ main\ntheorems:\n\n- `CharacterVarietiesAllSeamsSupport.json`: compatibility of the produced marked\n  solution with every seam equation.\n- `CartierChartCompactnessSupport.json`: compactness of normalized chart-field\n  retractions.\n- `SurfaceConeCandidateSupport.json`: ring-theoretic properties of the completed\n  surface cone.\n- `TraceIdealTransportSupport.json`: transport of finite and zero ideals, with\n  the construction-provider statements required by the original comparison.\n- `HoneycombBridgeMassSupport.json`: finiteness and quarter-power bounds for the\n  strict honeycomb bridge mass.\n\nSee the [scope notes](../docs/) for per-paper coverage and limitations.\n\nlean/lean-toolchain leanprover/lean4:v4.34.1\n\nCONTENTS start # Mathematics manuscript collection\n\n**719 manuscripts covering 372 result families.**\n\n[**Read the overview PDF**](overview.pdf).\n\n## Manuscript map\n\nEach result description is followed by its constituent manuscripts and their abstracts. Paper titles link directly to PDFs.\n\n<table>\n<thead><tr><th>Result</th></tr></thead>\n<tbody><tr>\n<td>\n\n**001. Milne’s rationality conjecture and algebraic specialization.** Proves Milne's rationality conjecture for abelian varieties over $`\\overline{\\mathbb Q}`$ with good reduction: specialized Hodge classes pair rationally with complementary divisor products, independently of cohomology theory. Together with result 032, every specialized Hodge class is represented by a single rational algebraic cycle simultaneously in all prime-to-<i>p</i> and crystalline realizations, for every residue characteristic <i>p</i>.\n\n</td>\n</tr></tbody>\n<tbody><tr>\n<td>\n\n&emsp;[Milne's rationality conjecture for abelian varieties](preprints/Milnes-rationality-conjecture-for-abelian-varieties-October-7-2026/paper.pdf)\n\nWe prove Milne's rationality conjecture for abelian varieties, including residue characteristic 2. After good reduction, the pairing of a rational Hodge class with any complementary product of divisor classes on the reduction is the same rational number in every prime-to-<i>p</i> realization and in crystalline cohomology. Using the Hodge theorem for CM abelian varieties, we also show that every such specialized Hodge class is represented by a single rational algebraic cycle in all these realizations.\n\n</td>\n</tr></tbody>\n<tbody><tr>\n<td>\n\n**002. The full BSD formula from low Selmer corank.** Proves the full Birch–Swinnerton-Dyer leading-term formula for every elliptic curve over ℚ whose full <i>q</i>-power Selmer group has corank zero or one for some prime <i>q</i>, including finiteness of the Tate–Shafarevich group. With result 006, this gives full BSD for a density-one set of quadratic twists of every elliptic curve over ℚ.\n\n</td>\n</tr></tbody>\n<tbody><tr>\n<td>\n\n&emsp;[Exact Birch–Swinnerton-Dyer Formula from Low Selmer Corank](preprints/Exact-Birch-Swinnerton-Dyer-Formula-from-Low-Selmer-Corank-October-7-2026/exact-bsd-low-selmer-corank.pdf)\n\nWe prove the full Birch–Swinnerton-Dyer leading-term formula for every elliptic curve over ℚ whose full <i>q</i>-power Selmer group has corank zero or one at some prime <i>q</i>. The analytic and Mordell–Weil ranks equal that corank, and the Tate–Shafarevich group is finite. The formula includes all prime factors and requires no additional hypotheses on reduction, rational torsion, isogenies, complex multiplication, or residual Galois representations.\n\n</td>\n</tr></tbody>\n<tbody><tr>\n<td>\n\n&emsp;[The Selmer converse for elliptic curves at every prime](preprints/The-Selmer-converse-for-elliptic-curves-at-every-prime-October-7-2026/main.pdf)\n\nWe prove the Selmer converse in coranks zero and one for every elliptic curve over ℚ and every prime <i>p</i>: if the full <i>p</i>-power Selmer group has ℤ<sub><i>p</i></sub>-corank $`r\\in\\{0,1\\}`$, then the analytic and Mordell–Weil ranks both equal <i>r</i>, and the entire Tate–Shafarevich group is finite. As an application at the additive prime 3, we prove that for every prime $`\\ell\\equiv4,7,8\\pmod9`$, the cubic $`X^3+Y^3=\\ell Z^3`$ has analytic and Mordell–Weil rank one and finite Tate–Shafarevich group. In particular, every such <i>ℓ</i> is a sum of two rational cubes.\n\n</td>\n</tr></tbody>\n<tbody><tr>\n<td>\n\n&emsp;[The two-primary Birch–Swinnerton-Dyer formula in Selmer corank at most one](preprints/The-two-primary-Birch-Swinnerton-Dyer-formula-in-Selmer-corank-at-most-one-October-6-2026/paper.pdf)\n\nWe prove the two-primary Birch and Swinnerton-Dyer leading-term formula for every elliptic curve over the rationals whose two-power Selmer group has corank at most one. In this range, the algebraic rank, analytic rank, and Selmer corank are equal, and the Tate–Shafarevich group is finite. Combined with the quadratic-twist Selmer distribution, this gives the exact two-primary formula for a density-one set of signed squarefree twists of each fixed curve, ordered by absolute value; the common rank is zero or one, with each value having density one half.\n\n</td>\n</tr></tbody>\n<tbody><tr>\n<td>\n\n**003. The quasi-Riemann hypothesis.** Proves that every Dirichlet <i>L</i>-function, including $`\\zeta(s)`$, is zero-free in $`\\Re s\\gt 7/8`$, resolving the quasi-Riemann hypothesis. The same half-plane is zero-free for every finite-order Hecke <i>L</i>-function over $`\\mathbb Q(\\sqrt{-3})`$. A companion gives a different proof of the zero-free half-plane $`\\Re s\\gt 11/12`$. ([Lean](lean/docs/003.md))\n\n</td>\n</tr></tbody>\n<tbody><tr>\n<td>\n\n&emsp;[The Quasi-Riemann Hypothesis: A Zero-Free Half-Plane $`\\Re s\\gt 7/8`$ ](preprints/The-Quasi-Riemann-Hypothesis-September-30-2026/paper.pdf)\n\nWe prove that all finite-order Hecke <i>L</i>-functions over $`\\mathbb Q(\\sqrt{-3})`$ and all Dirichlet <i>L</i>-functions are zero-free in the half-plane $`\\Re s\\gt 7/8`$, with the principal pole at <i>s</i> = 1 allowed. In particular, the Riemann zeta function is zero-free in this half-plane, proving the quasi-Riemann hypothesis.\n\n</td>\n</tr></tbody>\n<tbody><tr>\n<td>\n\n&emsp;[The Quasi-Riemann Hypothesis (alternate 11/12 proof)](preprints/The-Quasi-Riemann-Hypothesis-October-5-2026/paper2.pdf)\n\nWe establish the quasi-Riemann hypothesis by proving that every Dirichlet <i>L</i>-function, including Riemann's zeta function, has no zeros in the half-plane $`\\mathop{\\mathrm{Re}}\\nolimits s\\gt 11/12`$. More generally, we prove the same zero-free half-plane for every finite-order Hecke <i>L</i>-function over $`K=\\mathbb Q(\\sqrt{-3})`$. In particular, this rules out the existence of Landau–Siegel zeros.\n\n</td>\n</tr></tbody>\n<tbody><tr>\n<td>\n\n&emsp;[Uniform exclusion of Landau–Siegel zeros](preprints/Uniform-exclusion-of-Landau-Siegel-zeros-October-1-2026/paper.pdf)\n\nWe prove the uniform exclusion of Landau–Siegel zeros. There is an absolute constant <i>c</i> &gt; 0 such that every real zero $`\\beta\\in(0,1)`$ of every primitive nonprincipal real Dirichlet <i>L</i>-function of conductor <i>q</i> ≥ 3 satisfies $`(1-\\beta)\\log q\\ge c`$.\n\n</td>\n</tr></tbody>\n<tbody><tr>\n<td>\n\n**004. Hilbert’s tenth problem over ℚ.** Proves that no algorithm decides whether an integer-coefficient polynomial in an arbitrary number of variables has a rational zero, resolving Hilbert's tenth problem over ℚ negatively.\n\n</td>\n</tr></tbody>\n\nYAML start # yaml-language-server: $schema=https://raw.githubusercontent.com/mathlib-initiative/formalization.yaml/main/schema/formalization.schema.json\n# Catalog of papers with a formalized main result. Paths are relative to lean/.\nversion: \"v0.4\"\n\nproject:\n  name: \"OpenAI math repository\"\n  description: \"Lean formalizations accompanying a mathematics manuscript collection.\"\n  authors: [\"OpenAI\"]\n  license: \"Apache-2.0\"\n\nsources:\n  - title: \"$L_1$ Embeddings of Graphs of Bounded Treewidth\"\n    authors: [\"OpenAI\"]\n    id: ../preprints/L1-Embeddings-of-Graphs-of-Bounded-Treewidth-September-23-2026/paper.pdf\n    type: article\n\n  - title: \"A CH obstruction to a prescribed categoricity threshold\"\n    authors: [\"OpenAI\"]\n    id: ../preprints/A-CH-Obstruction-to-a-Prescribed-Categoricity-Threshold-September-24-2026/paper.pdf\n    type: article\n\n  - title: \"A classification of finite Euclidean Ramsey configurations\"\n    authors: [\"OpenAI\"]\n    id: ../preprints/A-classification-of-finite-Euclidean-Ramsey-configurations-September-23-2026/paper.pdf\n    type: article\n\n  - title: \"A Complete Local Domain without a Small Cohen–Macaulay Module\"\n    authors: [\"OpenAI\"]\n    id: ../preprints/A-Complete-Local-Domain-Without-a-Small-Cohen-Macaulay-Module-September-23-2026/paper.pdf\n    type: article\n\n  - title: \"A continuum temperature singularity for a radial pair potential\"\n    authors: [\"OpenAI\"]\n    id: ../preprints/A-continuum-temperature-singularity-for-a-radial-pair-potential-September-24-2026/paper.pdf\n    type: article\n\n  - title: \"A counterexample to Hadwiger's conjecture\"\n    authors: [\"OpenAI\"]\n    id: ../preprints/A-counterexample-to-Hadwigers-conjecture-September-23-2026/paper.pdf\n    type: article\n\n  - title: \"A counterexample to integer-degree harmonic dimension comparison\"\n    authors: [\"OpenAI\"]\n    id: ../preprints/A-counterexample-to-integer-degree-harmonic-dimension-comparison-September-25-2026/paper.pdf\n    type: article\n\n  - title: \"A Counterexample to Kaplansky's Direct-Finiteness Conjecture in Characteristic Two\"\n    authors: [\"OpenAI\"]\n    id: ../preprints/A-Counterexample-to-Kaplanskys-Direct-Finiteness-Conjecture-in-Characteristic-Two-September-23-2026/paper.pdf\n    type: article\n\n  - title: \"A Counterexample to Kaplansky's Direct-Finiteness Conjecture in Odd Characteristic\"\n    authors: [\"OpenAI\"]\n    id: ../preprints/A-Counterexample-to-Kaplanskys-Direct-Finiteness-Conjecture-in-Odd-Characteristic-September-26-2026/paper.pdf\n    type: article\n\n  - title: \"A counterexample to Kaplansky's quasitrace conjecture and failure of tensor-product stable finiteness\"\n    authors: [\"OpenAI\"]\n    id: ../preprints/A-counterexample-to-Kaplanskys-quasitrace-conjecture-September-23-2026/paper.pdf\n    type: article\n\n  - title: \"A counterexample to Ryser's covering conjecture\"\n    authors: [\"OpenAI\"]\n    id: ../preprints/A-Counterexample-to-Rysers-Covering-Conjecture-September-23-2026/paper.pdf\n    type: article\n\n  - title: \"A counterexample to Tachikawa's second conjecture\"\n    authors: [\"OpenAI\"]\n    id: ../preprints/A-counterexample-to-Tachikawas-second-conjecture-September-23-2026/paper.pdf\n    type: article\n\n  - title: \"A Counterexample to the Group-Ring Determinant Conjecture\"\n    authors: [\"OpenAI\"]\n    id: ../preprints/A-Counterexample-to-the-Group-Ring-Determinant-Conjecture-September-23-2026/paper.pdf\n    type: article\n\n  - title: \"A Counterexample to the Infinite Matroid Packing/Covering Conjecture\"\n    authors: [\"OpenAI\"]\n    id: ../preprints/A\ntrees [('preprints_tree', 13223, False), ('comparator_tree', 833, False)]\n"
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Paths are relative to lean/.\nversion: \"v0.4\"\n\nproject:\n  name: \"OpenAI math repository\"\n  description: \"Lean formalizations accompanying a mathematics manuscript collection.\"\n  authors: [\"OpenAI\"]\n  license: \"Apache-2.0\"\n\nsources:\n  - title: \"$L_1$ Embeddings of Graphs of Bounded Treewidth\"\n    authors: [\"OpenAI\"]\n    id: ../preprints/L1-Embeddings-of-Graphs-of-Bounded-Treewidth-September-23-2026/paper.pdf\n    type: article\n\n  - title: \"A CH obstruction to a prescribed categoricity threshold\"\n    authors: [\"OpenAI\"]\n    id: ../preprints/A-CH-Obstruction-to-a-Prescribed-Categoricity-Threshold-September-24-2026/paper.pdf\n    type: article\n\n  - title: \"A classification of finite Euclidean Ramsey configurations\"\n    authors: [\"OpenAI\"]\n    id: ../preprints/A-classification-of-finite-Euclidean-Ramsey-configurations-September-23-2026/paper.pdf\n    type: article\n\n  - title: \"A Complete Local Domain without a Small Cohen–Macaulay Module\"\n    authors: [\"OpenAI\"]\n    id: ../preprints/A-Complete-Local-Domain-Without-a-Small-Cohen-Macaulay-Module-September-23-2026/paper.pdf\n    type: article\n\n  - title: \"A continuum temperature singularity for a radial pair potential\"\n    authors: [\"OpenAI\"]\n    id: ../preprints/A-continuum-temperature-singularity-for-a-radial-pair-potential-September-24-2026/paper.pdf\n    type: article\n\n  - title: \"A counterexample to Hadwiger's conjecture\"\n    authors: [\"OpenAI\"]\n    id: ../preprints/A-counterexample-to-Hadwigers-conjecture-September-23-2026/paper.pdf\n    type: article\n\n  - title: \"A counterexample to integer-degree harmonic dimension comparison\"\n    authors: [\"OpenAI\"]\n    id: ../preprints/A-counterexample-to-integer-degree-harmonic-dimension-comparison-September-25-2026/paper.pdf\n    type: article\n\n  - title: \"A Counterexample to Kaplansky's Direct-Finiteness Conjecture in Characteristic Two\"\n    authors: [\"OpenAI\"]\n    id: ../preprints/A-Counterexample-to-Kaplanskys-Direct-Finiteness-Conjecture-in-Characteristic-Two-September-23-2026/paper.pdf\n    type: article\n\n  - title: \"A Counterexample to Kaplansky's Direct-Finiteness Conjecture in Odd Characteristic\"\n    authors: [\"OpenAI\"]\n    id: ../preprints/A-Counterexample-to-Kaplanskys-Direct-Finiteness-Conjecture-in-Odd-Characteristic-September-26-2026/paper.pdf\n    type: article\n\n  - title: \"A counterexample to Kaplansky's quasitrace conjecture and failure of tensor-product stable finiteness\"\n    authors: [\"OpenAI\"]\n    id: ../preprints/A-counterexample-to-Kaplanskys-quasitrace-conjecture-September-23-2026/paper.pdf\n    type: article\n\n  - title: \"A counterexample to Ryser's covering conjecture\"\n    authors: [\"OpenAI\"]\n    id: ../preprints/A-Counterexample-to-Rysers-Covering-Conjecture-September-23-2026/paper.pdf\n    type: article\n\n  - title: \"A counterexample to Tachikawa's second conjecture\"\n    authors: [\"OpenAI\"]\n    id: ../preprints/A-counterexample-to-Tachikawas-second-conjecture-September-23-2026/paper.pdf\n    type: article\n\n  - title: \"A Counterexample to the Group-Ring Determinant Conjecture\"\n    authors: [\"OpenAI\"]\n    id: ../preprints/A-Counterexample-to-the-Group-Ring-Determinant-Conjecture-September-23-2026/paper.pdf\n    type: article\n\n  - title: \"A Counterexample to the Infinite Matroid Packing/Covering Conjecture\"\n    authors: [\"OpenAI\"]\n    id: ../preprints/A-Counterexample-to-the-Infinite-Matroid-Packing-Covering-Conjecture-September-24-2026/paper.pdf\n    type: article\n\n  - title: \"A Cyclic Polytabloid Proof of Saxl's Conjecture\"\n    authors: [\"OpenAI\"]\n    id: ../preprints/A-Cyclic-Polytabloid-Proof-of-Saxls-Conjecture-September-24-2026/paper.pdf\n    type: article\n\n  - title: \"A Direct Proof of Optimal Max-Cut Hardness\"\n    authors: [\"OpenAI\"]\n    id: ../preprints/A-Direct-Proof-of-Optimal-Max-Cut-Hardness-September-23-2026/paper.pdf\n    type: article\n\n  - title: \"A direct proof of the complete Crouzeix inequality\"\n    authors: [\"OpenAI\"]\n    id: ../preprints/A-direct-proof-of-the-complete-Crouzeix-inequality-September-26-2026/paper.pdf\n    type: article\n\n  - title: \"A doubling Hilbert subset with no finite-dimensional bi-Lipschitz embedding\"\n    authors: [\"OpenAI\"]\n    id: ../preprints/A-doubling-Hilbert-subset-with-no-finite-dimensional-bi-Lipschitz-embedding-September-25-2026/main.pdf\n    type: article\n\n  - title: \"A Fock-space inequality and the Laughlin spectral gap\"\n    authors: [\"OpenAI\"]\n    id: ../preprints/A-Fock-space-inequality-and-the-Laughlin-spectral-gap-September-24-2026/A-Fock-space-inequality-and-the-Laughlin-spectral-gap-September-24-2026.pdf\n    type: article\n\n  - title: \"A hyperbolic group with no geometric CAT(0) action\"\n    authors: [\"OpenAI\"]\n    id: ../preprints/A-hyperbolic-group-with-no-geometric-CAT0-action-September-25-2026/paper.pdf\n    type: article\n\n  - title: \"A linear cycle-and-edge decompositi\ntrees [('preprints_tree', 13223, False), ('comparator_tree', 833, False)]\n",
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          "# yaml-language-server: $schema=https://raw.githubusercontent.com/mathlib-initiative/formalization.yaml/main/schema/formalization.schema.json\n# Catalog of papers with a formalized main result. Paths are relative to lean/.\nversion: \"v0.4\"\n\nproject:\n  name: \"OpenAI math repository\"\n  description: \"Lean formalizations accompanying a mathematics manuscript collection.\"\n  authors: [\"OpenAI\"]\n  license: \"Apache-2.0\"\n\nsources:\n  - title: \"$L_1$ Embeddings of Graphs of Bounded Treewidth\"\n    authors: [\"OpenAI\"]\n    id: ../preprints/L1-Embeddings-of-Graphs-of-Bounded-Treewidth-September-23-2026/paper.pdf\n    type: article\n\n  - title: \"A CH obstruction to a prescribed categoricity threshold\"\n    authors: [\"OpenAI\"]\n    id: ../preprints/A-CH-Obstruction-to-a-Prescribed-Categoricity-Threshold-September-24-2026/paper.pdf\n    type: article\n\n  - title: \"A classification of finite Euclidean Ramsey configurations\"\n    authors: [\"OpenAI\"]\n    id: ../preprints/A-classification-of-finite-Euclidean-Ramsey-configurations-September-23-2026/paper.pdf\n    type: article\n\n  - title: \"A Complete Local Domain without a Small Cohen–Macaulay Module\"\n    authors: [\"OpenAI\"]\n    id: ../preprints/A-Complete-Local-Domain-Without-a-Small-Cohen-Macaulay-Module-September-23-2026/paper.pdf\n    type: article\n\n  - title: \"A continuum temperature singularity for a radial pair potential\"\n    authors: [\"OpenAI\"]\n    id: ../preprints/A-continuum-temperature-singularity-for-a-radial-pair-potential-September-24-2026/paper.pdf\n    type: article\n\n  - title: \"A counterexample to Hadwiger's conjecture\"\n    authors: [\"OpenAI\"]\n    id: ../preprints/A-counterexample-to-Hadwigers-conjecture-September-23-2026/paper.pdf\n    type: article\n\n  - title: \"A counterexample to integer-degree harmonic dimension comparison\"\n    authors: [\"OpenAI\"]\n    id: ../preprints/A-counterexample-to-integer-degree-harmonic-dimension-comparison-September-25-2026/paper.pdf\n    type: article\n\n  - title: \"A Counterexample to Kaplansky's Direct-Finiteness Conjecture in Characteristic Two\"\n    authors: [\"OpenAI\"]\n    id: ../preprints/A-Counterexample-to-Kaplanskys-Direct-Finiteness-Conjecture-in-Characteristic-Two-September-23-2026/paper.pdf\n    type: article\n\n  - title: \"A Counterexample to Kaplansky's Direct-Finiteness Conjecture in Odd Characteristic\"\n    authors: [\"OpenAI\"]\n    id: ../preprints/A-Counterexample-to-Kaplanskys-Direct-Finiteness-Conjecture-in-Odd-Characteristic-September-26-2026/paper.pdf\n    type: article\n\n  - title: \"A counterexample to Kaplansky's quasitrace conjecture and failure of tensor-product stable finiteness\"\n    authors: [\"OpenAI\"]\n    id: ../preprints/A-counterexample-to-Kaplanskys-quasitrace-conjecture-September-23-2026/paper.pdf\n    type: article\n\n  - title: \"A counterexample to Ryser's covering conjecture\"\n    authors: [\"OpenAI\"]\n    id: ../preprints/A-Counterexample-to-Rysers-Covering-Conjecture-September-23-2026/paper.pdf\n    type: article\n\n  - title: \"A counterexample to Tachikawa's second conjecture\"\n    authors: [\"OpenAI\"]\n    id: ../preprints/A-counterexample-to-Tachikawas-second-conjecture-September-23-2026/paper.pdf\n    type: article\n\n  - title: \"A Counterexample to the Group-Ring Determinant Conjecture\"\n    authors: [\"OpenAI\"]\n    id: ../preprints/A-Counterexample-to-the-Group-Ring-Determinant-Conjecture-September-23-2026/paper.pdf\n    type: article\n\n  - title: \"A Counterexample to the Infinite Matroid Packing/Covering Conjecture\"\n    authors: [\"OpenAI\"]\n    id: ../preprints/A-Counterexample-to-the-Infinite-Matroid-Packing-Covering-Conjecture-September-24-2026/paper.pdf\n    type: article\n\n  - title: \"A Cyclic Polytabloid Proof of Saxl's Conjecture\"\n    authors: [\"OpenAI\"]\n    id: ../preprints/A-Cyclic-Polytabloid-Proof-of-Saxls-Conjecture-September-24-2026/paper.pdf\n    type: article\n\n  - title: \"A Direct Proof of Optimal Max-Cut Hardness\"\n    authors: [\"OpenAI\"]\n    id: ../preprints/A-Direct-Proof-of-Optimal-Max-Cut-Hardness-September-23-2026/paper.pdf\n    type: article\n\n  - title: \"A direct proof of the complete Crouzeix inequality\"\n    authors: [\"OpenAI\"]\n    id: ../preprints/A-direct-proof-of-the-complete-Crouzeix-inequality-September-26-2026/paper.pdf\n    type: article\n\n  - title: \"A doubling Hilbert subset with no finite-dimensional bi-Lipschitz embedding\"\n    authors: [\"OpenAI\"]\n    id: ../preprints/A-doubling-Hilbert-subset-with-no-finite-dimensional-bi-Lipschitz-embedding-September-25-2026/main.pdf\n    type: article\n\n  - title: \"A Fock-space inequality and the Laughlin spectral gap\"\n    authors: [\"OpenAI\"]\n    id: ../preprints/A-Fock-space-inequality-and-the-Laughlin-spectral-gap-September-24-2026/A-Fock-space-inequality-and-the-Laughlin-spectral-gap-September-24-2026.pdf\n    type: article\n\n  - title: \"A hyperbolic group with no geometric CAT(0) action\"\n    authors: [\"OpenAI\"]\n    id: ../preprints/A-hyperbolic-group-with-no-geometric-CAT0-action-September-25-2026/paper.pdf\n    type: article\n\n  - title: \"A linear cycle-and-edge decompositi\ntrees [('preprints_tree', 13223, False), ('comparator_tree', 833, False)]\n"
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Paths are relative to lean/.\nversion: \"v0.4\"\n\nproject:\n  name: \"OpenAI math repository\"\n  description: \"Lean formalizations accompanying a mathematics manuscript collection.\"\n  authors: [\"OpenAI\"]\n  license: \"Apache-2.0\"\n\nsources:\n  - title: \"$L_1$ Embeddings of Graphs of Bounded Treewidth\"\n    authors: [\"OpenAI\"]\n    id: ../preprints/L1-Embeddings-of-Graphs-of-Bounded-Treewidth-September-23-2026/paper.pdf\n    type: article\n\n  - title: \"A CH obstruction to a prescribed categoricity threshold\"\n    authors: [\"OpenAI\"]\n    id: ../preprints/A-CH-Obstruction-to-a-Prescribed-Categoricity-Threshold-September-24-2026/paper.pdf\n    type: article\n\n  - title: \"A classification of finite Euclidean Ramsey configurations\"\n    authors: [\"OpenAI\"]\n    id: ../preprints/A-classification-of-finite-Euclidean-Ramsey-configurations-September-23-2026/paper.pdf\n    type: article\n\n  - title: \"A Complete Local Domain without a Small Cohen–Macaulay Module\"\n    authors: [\"OpenAI\"]\n    id: ../preprints/A-Complete-Local-Domain-Without-a-Small-Cohen-Macaulay-Module-September-23-2026/paper.pdf\n    type: article\n\n  - title: \"A continuum temperature singularity for a radial pair potential\"\n    authors: [\"OpenAI\"]\n    id: ../preprints/A-continuum-temperature-singularity-for-a-radial-pair-potential-September-24-2026/paper.pdf\n    type: article\n\n  - title: \"A counterexample to Hadwiger's conjecture\"\n    authors: [\"OpenAI\"]\n    id: ../preprints/A-counterexample-to-Hadwigers-conjecture-September-23-2026/paper.pdf\n    type: article\n\n  - title: \"A counterexample to integer-degree harmonic dimension comparison\"\n    authors: [\"OpenAI\"]\n    id: ../preprints/A-counterexample-to-integer-degree-harmonic-dimension-comparison-September-25-2026/paper.pdf\n    type: article\n\n  - title: \"A Counterexample to Kaplansky's Direct-Finiteness Conjecture in Characteristic Two\"\n    authors: [\"OpenAI\"]\n    id: ../preprints/A-Counterexample-to-Kaplanskys-Direct-Finiteness-Conjecture-in-Characteristic-Two-September-23-2026/paper.pdf\n    type: article\n\n  - title: \"A Counterexample to Kaplansky's Direct-Finiteness Conjecture in Odd Characteristic\"\n    authors: [\"OpenAI\"]\n    id: ../preprints/A-Counterexample-to-Kaplanskys-Direct-Finiteness-Conjecture-in-Odd-Characteristic-September-26-2026/paper.pdf\n    type: article\n\n  - title: \"A counterexample to Kaplansky's quasitrace conjecture and failure of tensor-product stable finiteness\"\n    authors: [\"OpenAI\"]\n    id: ../preprints/A-counterexample-to-Kaplanskys-quasitrace-conjecture-September-23-2026/paper.pdf\n    type: article\n\n  - title: \"A counterexample to Ryser's covering conjecture\"\n    authors: [\"OpenAI\"]\n    id: ../preprints/A-Counterexample-to-Rysers-Covering-Conjecture-September-23-2026/paper.pdf\n    type: article\n\n  - title: \"A counterexample to Tachikawa's second conjecture\"\n    authors: [\"OpenAI\"]\n    id: ../preprints/A-counterexample-to-Tachikawas-second-conjecture-September-23-2026/paper.pdf\n    type: article\n\n  - title: \"A Counterexample to the Group-Ring Determinant Conjecture\"\n    authors: [\"OpenAI\"]\n    id: ../preprints/A-Counterexample-to-the-Group-Ring-Determinant-Conjecture-September-23-2026/paper.pdf\n    type: article\n\n  - title: \"A Counterexample to the Infinite Matroid Packing/Covering Conjecture\"\n    authors: [\"OpenAI\"]\n    id: ../preprints/A-Counterexample-to-the-Infinite-Matroid-Packing-Covering-Conjecture-September-24-2026/paper.pdf\n    type: article\n\n  - title: \"A Cyclic Polytabloid Proof of Saxl's Conjecture\"\n    authors: [\"OpenAI\"]\n    id: ../preprints/A-Cyclic-Polytabloid-Proof-of-Saxls-Conjecture-September-24-2026/paper.pdf\n    type: article\n\n  - title: \"A Direct Proof of Optimal Max-Cut Hardness\"\n    authors: [\"OpenAI\"]\n    id: ../preprints/A-Direct-Proof-of-Optimal-Max-Cut-Hardness-September-23-2026/paper.pdf\n    type: article\n\n  - title: \"A direct proof of the complete Crouzeix inequality\"\n    authors: [\"OpenAI\"]\n    id: ../preprints/A-direct-proof-of-the-complete-Crouzeix-inequality-September-26-2026/paper.pdf\n    type: article\n\n  - title: \"A doubling Hilbert subset with no finite-dimensional bi-Lipschitz embedding\"\n    authors: [\"OpenAI\"]\n    id: ../preprints/A-doubling-Hilbert-subset-with-no-finite-dimensional-bi-Lipschitz-embedding-September-25-2026/main.pdf\n    type: article\n\n  - title: \"A Fock-space inequality and the Laughlin spectral gap\"\n    authors: [\"OpenAI\"]\n    id: ../preprints/A-Fock-space-inequality-and-the-Laughlin-spectral-gap-September-24-2026/A-Fock-space-inequality-and-the-Laughlin-spectral-gap-September-24-2026.pdf\n    type: article\n\n  - title: \"A hyperbolic group with no geometric CAT(0) action\"\n    authors: [\"OpenAI\"]\n    id: ../preprints/A-hyperbolic-group-with-no-geometric-CAT0-action-September-25-2026/paper.pdf\n    type: article\n\n  - title: \"A linear cycle-and-edge decompositi\ntrees [('preprints_tree', 13223, False), ('comparator_tree', 833, False)]\n"
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      "script": "txt=rec['CONTENTS.md']['content']\nheads=list(re.finditer(r'^\\*\\*(\\d{3})\\. (.*?)\\*\\*',txt,re.M))\nfamilies=[]\nfor i,m in enumerate(heads):\n block=txt[m.start():heads[i+1].start() if i+1<len(heads) else len(txt)]\n papers=re.findall(r'&emsp;\\[(.*?)\\]\\((preprints/[^)]+\\.pdf)\\)',block)\n families.append({'id':m.group(1),'title':m.group(2),'block':block,'papers':papers})\nprint('families',len(families),'papers',sum(len(f['papers']) for f in families),'unique ids',len({f['id'] for f in families}))\nprint('yaml structure',[(i+1,l) for i,l in enumerate(rec['lean/formalization.yaml']['content'].splitlines()) if re.match(r'^\\w',l)])\ny=rec['lean/formalization.yaml']['content']; idx=y.find('\\ndeclarations:'); print(y[idx:idx+6000] if idx>=0 else y[-6500:])",
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      "text": {
        "stdout": "families 372 papers 718 unique ids 372\nyaml structure [(3, 'version: \"v0.4\"'), (5, 'project:'), (11, 'sources:'), (877, 'related_formalizations:'), (969, 'status:'), (1773, 'automation:'), (1777, 'review:'), (1780, 'acknowledgements: >-')]\ne_main\n      file: OAI/Analysis/LpDimension/Main.lean\n\n    - comparator_config: ComparatorChallenges/SubsphereCurrent.json\n      declaration: OAI.SubsphereCurrent.full_current_main_scope\n      file: OAI/Probability/Subsphere/Main.lean\n\n    - comparator_config: ComparatorChallenges/Superstring.json\n      declaration: OAI.Superstring.main\n      file: OAI/Computability/Superstring/Main.lean\n\n    - comparator_config: ComparatorChallenges/SurfaceConeCandidateSupport.json\n      declaration: OAI.SmallCM.candidate_ring_properties\n      file: OAI/AlgebraicGeometry/SurfaceCones/CandidateRing.lean\n\n    - comparator_config: ComparatorChallenges/SymmetricDomains.json\n      declaration: OAI.Release061.main\n      file: OAI/Analysis/SymmetricDomains/Main.lean\n\n    - comparator_config: ComparatorChallenges/SymmetricMahlerEquality.json\n      declaration: OAI.SymmetricMahler.symmetric_mahler_equality\n      file: OAI/Analysis/Mahler/Classification.lean\n\n    - comparator_config: ComparatorChallenges/SymmetricPolar.json\n      declaration: OAI.SymmetricPolar.symmetric_polar_main\n      file: OAI/Geometry/PolarProducts/Main.lean\n\n    - comparator_config: ComparatorChallenges/Tachikawa.json\n      declaration: OAI.Tachikawa.main_theorem\n      file: OAI/RingTheory/Tachikawa/Counterexample.lean\n\n    - comparator_config: ComparatorChallenges/TalagrandDiscreteConvexity.json\n      declaration: OAI.TalagrandDiscreteConvexity.talagrand_discrete_convexity\n      file: OAI/Combinatorics/DiscreteConvexity/Main.lean\n\n    - comparator_config: ComparatorChallenges/TalagrandExpectationThreshold.json\n      declaration: OAI.TalagrandThreshold.talagrand_expectation_threshold_equivalence\n      file: OAI/Combinatorics/ExpectationThreshold/Main.lean\n\n    - comparator_config: ComparatorChallenges/TamingCompatibility.json\n      declaration: OAI.TamingCompatibility.taming_implies_compatibility\n      file: OAI/Geometry/TamingCompatibility/Main.lean\n\n    - comparator_config: ComparatorChallenges/ThompsonNonamenability.json\n      declaration: OAI.ThompsonNonamenability.thompson_F_nonamenable_composition\n      file: OAI/GroupTheory/Thompson/Main.lean\n\n    - comparator_config: ComparatorChallenges/ThorpRouting.json\n      declaration: OAI.ThorpNine.main\n      file: OAI/Probability/ThorpRouting/Main.lean\n\n    - comparator_config: ComparatorChallenges/ThorpWeightedCompatibility.json\n      declaration: OAI.ThorpCompatibility.weightedCompatibility_main\n      file: OAI/Probability/ThorpCompatibility/Main.lean\n\n    - comparator_config: ComparatorChallenges/TingleySphereIsometry.json\n      declaration: OAI.Tingley.tingley_sphere_isometry_full\n      file: OAI/Analysis/SphereIsometry/Extension.lean\n\n    - comparator_config: ComparatorChallenges/TorsionFreeHyperbolic.json\n      declaration: OAI.Release075.main\n      file: OAI/GroupTheory/Hyperbolic/Main.lean\n\n    - comparator_config: ComparatorChallenges/TorsionFreeZeroDivisors.json\n      declaration: OAI.TorsionFreeZeroDivisors.main\n      file: OAI/Algebra/GroupRing/Main.lean\n\n    - comparator_config: ComparatorChallenges/TraceIdealTransportSupport.json\n      declaration: OAI.TraceConeClassification.TraceConeEquiv.idealOrderIso_finIdeal\n      file: OAI/Analysis/TraceCone/IdealTransport.lean\n\n    - comparator_config: ComparatorChallenges/TraceIdealTransportSupport.json\n      declaration: OAI.TraceConeClassification.TraceConeEquiv.idealOrderIso_zeroIdeal\n      file: OAI/Analysis/TraceCone/IdealTransport.lean\n\n    - comparator_config: ComparatorChallenges/TraceIdealTransportSupport.json\n      declaration: OAI.TraceConeClassification.closed_ideal_weight_exists\n      file: OAI/Analysis/TraceCone/FiniteIdeal.lean\n\n    - comparator_config: ComparatorChallenges/TriangularHilbert.json\n      declaration: OAI.TriangularHilbert.main_estimate\n      file: OAI/Analysis/TriangularHilbert/Main.lean\n\n    - comparator_config: ComparatorChallenges/TwoWayComplementation.json\n      declaration: OAI.TwoWayComplementation.complementation_lower_bound_finite\n      file: OAI/Combinatorics/TwoWayAutomata/Main.lean\n\n    - comparator_config: ComparatorChallenges/TwoWayDeterminization.json\n      declaration: OAI.TwoWayComplementation.explicit_family_main\n      file: OAI/Combinatorics/TwoWayAutomata/ExplicitFamily.lean\n\n    - comparator_config: ComparatorChallenges/UniformFourier.json\n      declaration: OAI.PowerSaving.transform_main\n      file: OAI/Computability/FourierTransform/Main.lean\n\n    - comparator_config: ComparatorChallenges/UniformFourier.json\n      declaration: OAI.PowerSaving.convolution_main\n      file: OAI/Computability/FourierTransform/Main.lean\n\n    - comparator_config: ComparatorChallenges/UniformGamma.json\n      declaration: OAI.ComparatorModel.CurrentMain.main\n      file: OAI/Analysis/TracialSplitting/Challenge.lean\n\n    - comparator_config: ComparatorChallenges/UniformSparsestCut.json\n      declaration: OAI.UniformSparsestCut.mainGap\n      file: OAI/Combinatorics/SparsestCut/Main.lean\n\n    - comparator_config: ComparatorChallenges/UniqueGamesTheorem.json\n      declaration: OAI.UniqueGamesTheorem.theorem11\n      file: OAI/Computability/UniqueGames/Theorem.lean\n\n    - comparator_config: ComparatorChallenges/UniversalFInfinity.json\n      declaration: OAI.UniversalFInfinity.universal_group_of_type_FInfinity\n      file: OAI/GroupTheory/UniversalGroup/Main.lean\n\n    - comparator_config: ComparatorChallenges/VertexCover.json\n      declaration: OAI.VertexCover.every_fixed_factor_below_two\n      file: OAI/Computability/VertexCover/Main.lean\n\n    - comparator_config: ComparatorChallenges/VlasovMaxwell.json\n      declaration: OAI.RVM.global_classical_solution\n      file: OAI/Analysis/VlasovMaxwell/Main.lean\n\n    - comparator_config: ComparatorChallenges/WeakHessian.json\n      declaration: OAI.WeakHessian.every_geodesic\n      file: OAI/Geometry/WeakHessian/Main.lean\n\n    - comparator_config: ComparatorChallenges/WeakMTWGlobalSupport.json\n      declaration: OAI.WeakMTWGlobalSupport.current_main_convexity\n      file: OAI/Geometry/WeakMTW/Convexity.lean\n\n    - comparator_config: ComparatorChallenges/YauCounterexample.json\n      declaration: OAI.Yau.Target.yau_nodal_set_upper_bound_counterexample\n      file: OAI/Geometry/NodalSets/Main.lean\n\nautomation:\n  methods:\n    - method: agent\n\nreview:\n  status: unchecked\n\nacknowledgements: >-\n  Thank you very much to the authors of Lean 4, mathlib, and the other imported\n  libraries listed above, as well as to the authors of Lake, Comparator,\n  lean4export, and related tools.\n\n",
        "stderr": "",
        "traceback": "",
        "plain": [
          "families 372 papers 718 unique ids 372\nyaml structure [(3, 'version: \"v0.4\"'), (5, 'project:'), (11, 'sources:'), (877, 'related_formalizations:'), (969, 'status:'), (1773, 'automation:'), (1777, 'review:'), (1780, 'acknowledgements: >-')]\ne_main\n      file: OAI/Analysis/LpDimension/Main.lean\n\n    - comparator_config: ComparatorChallenges/SubsphereCurrent.json\n      declaration: OAI.SubsphereCurrent.full_current_main_scope\n      file: OAI/Probability/Subsphere/Main.lean\n\n    - comparator_config: ComparatorChallenges/Superstring.json\n      declaration: OAI.Superstring.main\n      file: OAI/Computability/Superstring/Main.lean\n\n    - comparator_config: ComparatorChallenges/SurfaceConeCandidateSupport.json\n      declaration: OAI.SmallCM.candidate_ring_properties\n      file: OAI/AlgebraicGeometry/SurfaceCones/CandidateRing.lean\n\n    - comparator_config: ComparatorChallenges/SymmetricDomains.json\n      declaration: OAI.Release061.main\n      file: OAI/Analysis/SymmetricDomains/Main.lean\n\n    - comparator_config: ComparatorChallenges/SymmetricMahlerEquality.json\n      declaration: OAI.SymmetricMahler.symmetric_mahler_equality\n      file: OAI/Analysis/Mahler/Classification.lean\n\n    - comparator_config: ComparatorChallenges/SymmetricPolar.json\n      declaration: OAI.SymmetricPolar.symmetric_polar_main\n      file: OAI/Geometry/PolarProducts/Main.lean\n\n    - comparator_config: ComparatorChallenges/Tachikawa.json\n      declaration: OAI.Tachikawa.main_theorem\n      file: OAI/RingTheory/Tachikawa/Counterexample.lean\n\n    - comparator_config: ComparatorChallenges/TalagrandDiscreteConvexity.json\n      declaration: OAI.TalagrandDiscreteConvexity.talagrand_discrete_convexity\n      file: OAI/Combinatorics/DiscreteConvexity/Main.lean\n\n    - comparator_config: ComparatorChallenges/TalagrandExpectationThreshold.json\n      declaration: OAI.TalagrandThreshold.talagrand_expectation_threshold_equivalence\n      file: OAI/Combinatorics/ExpectationThreshold/Main.lean\n\n    - comparator_config: ComparatorChallenges/TamingCompatibility.json\n      declaration: OAI.TamingCompatibility.taming_implies_compatibility\n      file: OAI/Geometry/TamingCompatibility/Main.lean\n\n    - comparator_config: ComparatorChallenges/ThompsonNonamenability.json\n      declaration: OAI.ThompsonNonamenability.thompson_F_nonamenable_composition\n      file: OAI/GroupTheory/Thompson/Main.lean\n\n    - comparator_config: ComparatorChallenges/ThorpRouting.json\n      declaration: OAI.ThorpNine.main\n      file: OAI/Probability/ThorpRouting/Main.lean\n\n    - comparator_config: ComparatorChallenges/ThorpWeightedCompatibility.json\n      declaration: OAI.ThorpCompatibility.weightedCompatibility_main\n      file: OAI/Probability/ThorpCompatibility/Main.lean\n\n    - comparator_config: ComparatorChallenges/TingleySphereIsometry.json\n      declaration: OAI.Tingley.tingley_sphere_isometry_full\n      file: OAI/Analysis/SphereIsometry/Extension.lean\n\n    - comparator_config: ComparatorChallenges/TorsionFreeHyperbolic.json\n      declaration: OAI.Release075.main\n      file: OAI/GroupTheory/Hyperbolic/Main.lean\n\n    - comparator_config: ComparatorChallenges/TorsionFreeZeroDivisors.json\n      declaration: OAI.TorsionFreeZeroDivisors.main\n      file: OAI/Algebra/GroupRing/Main.lean\n\n    - comparator_config: ComparatorChallenges/TraceIdealTransportSupport.json\n      declaration: OAI.TraceConeClassification.TraceConeEquiv.idealOrderIso_finIdeal\n      file: OAI/Analysis/TraceCone/IdealTransport.lean\n\n    - comparator_config: ComparatorChallenges/TraceIdealTransportSupport.json\n      declaration: OAI.TraceConeClassification.TraceConeEquiv.idealOrderIso_zeroIdeal\n      file: OAI/Analysis/TraceCone/IdealTransport.lean\n\n    - comparator_config: ComparatorChallenges/TraceIdealTransportSupport.json\n      declaration: OAI.TraceConeClassification.closed_ideal_weight_exists\n      file: OAI/Analysis/TraceCone/FiniteIdeal.lean\n\n    - comparator_config: ComparatorChallenges/TriangularHilbert.json\n      declaration: OAI.TriangularHilbert.main_estimate\n      file: OAI/Analysis/TriangularHilbert/Main.lean\n\n    - comparator_config: ComparatorChallenges/TwoWayComplementation.json\n      declaration: OAI.TwoWayComplementation.complementation_lower_bound_finite\n      file: OAI/Combinatorics/TwoWayAutomata/Main.lean\n\n    - comparator_config: ComparatorChallenges/TwoWayDeterminization.json\n      declaration: OAI.TwoWayComplementation.explicit_family_main\n      file: OAI/Combinatorics/TwoWayAutomata/ExplicitFamily.lean\n\n    - comparator_config: ComparatorChallenges/UniformFourier.json\n      declaration: OAI.PowerSaving.transform_main\n      file: OAI/Computability/FourierTransform/Main.lean\n\n    - comparator_config: ComparatorChallenges/UniformFourier.json\n      declaration: OAI.PowerSaving.convolution_main\n      file: OAI/Computability/FourierTransform/Main.lean\n\n    - comparator_config: ComparatorChallenges/UniformGamma.json\n      declaration: OAI.ComparatorModel.CurrentMain.main\n      file: OAI/Analysis/TracialSplitting/Challenge.lean\n\n    - comparator_config: ComparatorChallenges/UniformSparsestCut.json\n      declaration: OAI.UniformSparsestCut.mainGap\n      file: OAI/Combinatorics/SparsestCut/Main.lean\n\n    - comparator_config: ComparatorChallenges/UniqueGamesTheorem.json\n      declaration: OAI.UniqueGamesTheorem.theorem11\n      file: OAI/Computability/UniqueGames/Theorem.lean\n\n    - comparator_config: ComparatorChallenges/UniversalFInfinity.json\n      declaration: OAI.UniversalFInfinity.universal_group_of_type_FInfinity\n      file: OAI/GroupTheory/UniversalGroup/Main.lean\n\n    - comparator_config: ComparatorChallenges/VertexCover.json\n      declaration: OAI.VertexCover.every_fixed_factor_below_two\n      file: OAI/Computability/VertexCover/Main.lean\n\n    - comparator_config: ComparatorChallenges/VlasovMaxwell.json\n      declaration: OAI.RVM.global_classical_solution\n      file: OAI/Analysis/VlasovMaxwell/Main.lean\n\n    - comparator_config: ComparatorChallenges/WeakHessian.json\n      declaration: OAI.WeakHessian.every_geodesic\n      file: OAI/Geometry/WeakHessian/Main.lean\n\n    - comparator_config: ComparatorChallenges/WeakMTWGlobalSupport.json\n      declaration: OAI.WeakMTWGlobalSupport.current_main_convexity\n      file: OAI/Geometry/WeakMTW/Convexity.lean\n\n    - comparator_config: ComparatorChallenges/YauCounterexample.json\n      declaration: OAI.Yau.Target.yau_nodal_set_upper_bound_counterexample\n      file: OAI/Geometry/NodalSets/Main.lean\n\nautomation:\n  methods:\n    - method: agent\n\nreview:\n  status: unchecked\n\nacknowledgements: >-\n  Thank you very much to the authors of Lean 4, mathlib, and the other imported\n  libraries listed above, as well as to the authors of Lake, Comparator,\n  lean4export, and related tools.\n\n"
        ]
      },
      "outputs": [
        {
          "type": "stream",
          "name": "stdout",
          "text": "families 372 papers 718 unique ids 372\nyaml structure [(3, 'version: \"v0.4\"'), (5, 'project:'), (11, 'sources:'), (877, 'related_formalizations:'), (969, 'status:'), (1773, 'automation:'), (1777, 'review:'), (1780, 'acknowledgements: >-')]\ne_main\n      file: OAI/Analysis/LpDimension/Main.lean\n\n    - comparator_config: ComparatorChallenges/SubsphereCurrent.json\n      declaration: OAI.SubsphereCurrent.full_current_main_scope\n      file: OAI/Probability/Subsphere/Main.lean\n\n    - comparator_config: ComparatorChallenges/Superstring.json\n      declaration: OAI.Superstring.main\n      file: OAI/Computability/Superstring/Main.lean\n\n    - comparator_config: ComparatorChallenges/SurfaceConeCandidateSupport.json\n      declaration: OAI.SmallCM.candidate_ring_properties\n      file: OAI/AlgebraicGeometry/SurfaceCones/CandidateRing.lean\n\n    - comparator_config: ComparatorChallenges/SymmetricDomains.json\n      declaration: OAI.Release061.main\n      file: OAI/Analysis/SymmetricDomains/Main.lean\n\n    - comparator_config: ComparatorChallenges/SymmetricMahlerEquality.json\n      declaration: OAI.SymmetricMahler.symmetric_mahler_equality\n      file: OAI/Analysis/Mahler/Classification.lean\n\n    - comparator_config: ComparatorChallenges/SymmetricPolar.json\n      declaration: OAI.SymmetricPolar.symmetric_polar_main\n      file: OAI/Geometry/PolarProducts/Main.lean\n\n    - comparator_config: ComparatorChallenges/Tachikawa.json\n      declaration: OAI.Tachikawa.main_theorem\n      file: OAI/RingTheory/Tachikawa/Counterexample.lean\n\n    - comparator_config: ComparatorChallenges/TalagrandDiscreteConvexity.json\n      declaration: OAI.TalagrandDiscreteConvexity.talagrand_discrete_convexity\n      file: OAI/Combinatorics/DiscreteConvexity/Main.lean\n\n    - comparator_config: ComparatorChallenges/TalagrandExpectationThreshold.json\n      declaration: OAI.TalagrandThreshold.talagrand_expectation_threshold_equivalence\n      file: OAI/Combinatorics/ExpectationThreshold/Main.lean\n\n    - comparator_config: ComparatorChallenges/TamingCompatibility.json\n      declaration: OAI.TamingCompatibility.taming_implies_compatibility\n      file: OAI/Geometry/TamingCompatibility/Main.lean\n\n    - comparator_config: ComparatorChallenges/ThompsonNonamenability.json\n      declaration: OAI.ThompsonNonamenability.thompson_F_nonamenable_composition\n      file: OAI/GroupTheory/Thompson/Main.lean\n\n    - comparator_config: ComparatorChallenges/ThorpRouting.json\n      declaration: OAI.ThorpNine.main\n      file: OAI/Probability/ThorpRouting/Main.lean\n\n    - comparator_config: ComparatorChallenges/ThorpWeightedCompatibility.json\n      declaration: OAI.ThorpCompatibility.weightedCompatibility_main\n      file: OAI/Probability/ThorpCompatibility/Main.lean\n\n    - comparator_config: ComparatorChallenges/TingleySphereIsometry.json\n      declaration: OAI.Tingley.tingley_sphere_isometry_full\n      file: OAI/Analysis/SphereIsometry/Extension.lean\n\n    - comparator_config: ComparatorChallenges/TorsionFreeHyperbolic.json\n      declaration: OAI.Release075.main\n      file: OAI/GroupTheory/Hyperbolic/Main.lean\n\n    - comparator_config: ComparatorChallenges/TorsionFreeZeroDivisors.json\n      declaration: OAI.TorsionFreeZeroDivisors.main\n      file: OAI/Algebra/GroupRing/Main.lean\n\n    - comparator_config: ComparatorChallenges/TraceIdealTransportSupport.json\n      declaration: OAI.TraceConeClassification.TraceConeEquiv.idealOrderIso_finIdeal\n      file: OAI/Analysis/TraceCone/IdealTransport.lean\n\n    - comparator_config: ComparatorChallenges/TraceIdealTransportSupport.json\n      declaration: OAI.TraceConeClassification.TraceConeEquiv.idealOrderIso_zeroIdeal\n      file: OAI/Analysis/TraceCone/IdealTransport.lean\n\n    - comparator_config: ComparatorChallenges/TraceIdealTransportSupport.json\n      declaration: OAI.TraceConeClassification.closed_ideal_weight_exists\n      file: OAI/Analysis/TraceCone/FiniteIdeal.lean\n\n    - comparator_config: ComparatorChallenges/TriangularHilbert.json\n      declaration: OAI.TriangularHilbert.main_estimate\n      file: OAI/Analysis/TriangularHilbert/Main.lean\n\n    - comparator_config: ComparatorChallenges/TwoWayComplementation.json\n      declaration: OAI.TwoWayComplementation.complementation_lower_bound_finite\n      file: OAI/Combinatorics/TwoWayAutomata/Main.lean\n\n    - 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method: agent\n\nreview:\n  status: unchecked\n\nacknowledgements: >-\n  Thank you very much to the authors of Lean 4, mathlib, and the other imported\n  libraries listed above, as well as to the authors of Lake, Comparator,\n  lean4export, and related tools.\n\n"
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      "script": "import random\nprint('paper link discrepancies',[(f['id'],len(re.findall(r'&emsp;',f['block'])),len(f['papers'])) for f in families if len(re.findall(r'&emsp;',f['block']))!=len(f['papers'])])\nprint('missing family papers',[f['id'] for f in families if not f['papers']])\nchosen=random.Random(20261008).sample(families,20)\nprint('draw order',[f['id'] for f in chosen])\nfor f in sorted(chosen,key=lambda f:f['id']):\n print('\\n',f['id'],f['title'],'\\nFIRST',f['papers'][0], '\\nFAMILY SCOPE',f['block'].split('</td>')[0][:1400])\nprint('\\nYAML related/status\\n','\\n'.join(y.splitlines()[875:1005]))",
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        "stdout": "paper link discrepancies [('337', 2, 1)]\nmissing family papers []\ndraw order ['154', '024', '230', '127', '082', '148', '046', '372', '113', '164', '351', '369', '115', '215', '065', '296', '245', '015', '210', '192']\n\n 015 Torus-packet equidistribution in prime, quartic, and sextic degrees. \nFIRST ('Equidistribution of Prime-Degree Torus Packets with Arbitrary Local Type', 'preprints/Equidistribution-of-Prime-Degree-Torus-Packets-with-Arbitrary-Local-Type-September-24-2026/paper.pdf') \nFAMILY SCOPE **015. Torus-packet equidistribution in prime, quartic, and sextic degrees.** Proves Haar equidistribution without escape of mass for complete volume-weighted torus packets from totally real fields: arbitrary lattices and prescribed local types in fixed prime degree at least five, arbitrary-order Picard packets in primitive quartic fields, and maximal-order ideal-class packets in primitive sextic fields. Here primitive means having no proper intermediate field; the relevant order or field discriminant tends to infinity. ([Lean](lean/docs/015.md))\n\n\n\n 024 An asymptotic formula for the number of totients. \nFIRST ('An asymptotic formula for the number of totients', 'preprints/An-asymptotic-formula-for-the-number-of-totients-September-25-2026/An-asymptotic-formula-for-the-number-of-totients-September-25-2026.pdf') \nFAMILY SCOPE **024. An asymptotic formula for the number of totients.** Gives an asymptotic equivalent for the number $`V(x)`$ of distinct totient values up to <i>x</i>, with a positive bounded phase-dependent factor determined by convergent arithmetic approximations. In particular, $`V(cx)/V(x)\\to c`$ for every fixed <i>c</i> &gt; 0, answering Erdős and Hall’s scaling question. ([Lean](lean/docs/024.md))\n\n\n\n 046 Shafarevich counterexamples in dimension two and with large fundamental group. \nFIRST ('A projective fourfold with large fundamental group and non-Stein universal cover', 'preprints/A-projective-fourfold-with-large-fundamental-group-and-non-Stein-universal-cover-October-5-2026/large-fundamental-group-non-stein.pdf') \nFAMILY SCOPE **046. Shafarevich counterexamples in dimension two and with large fundamental group.** Constructs a smooth projective complex fourfold with large fundamental group whose universal cover contains no positive-dimensional compact analytic subvariety but is neither Stein nor holomorphically convex. A separate smooth projective complex surface already disproves unrestricted Shafarevich holomorphic convexity in dimension two.\n\n\n\n 065 Virasoro constraints for complete intersections and projective-bundle towers. \nFIRST ('Virasoro Constraints under Projectivization', 'preprints/Virasoro-Constraints-under-Projectivization-October-5-2026/virasoro-constraints-under-projectivization.pdf') \nFAMILY SCOPE **065. Virasoro constraints for complete intersections and projective-bundle towers.** Proves the full ordinary unreduced descendant Virasoro conjecture for smooth complete intersections in complex projective space, in every genus and curve class with arbitrary cohomology insertions. The constraints also pass from any smooth projective complex base satisfying them to the projectivization of every algebraic vector bundle of rank at least two, and hence to projective-bundle towers.\n\n\n\n 082 Annular variation and dyadic absolute bounds for the triangular Hilbert transform. \nFIRST ('Annular variation of the triangular Hilbert transform at the symmetric point', 'preprints/Annular-variation-of-the-triangular-Hilbert-transform-at-the-symmetric-point-October-5-2026/annular-variation.pdf') \nFAMILY SCOPE **082. Annular variation and dyadic absolute bounds for the triangular Hilbert transform.** Proves maximal and annular <i>r</i>-variation bounds, for every <i>r</i> &gt; 2, from complex $`L^3(\\mathbb R^2)\\times L^3(\\mathbb R^2)`$ to $`L^{3/2}(\\mathbb R^2)`$. The maximal estimate controls both hard truncation endpoints and gives almost-everywhere and norm convergence. Pairing with a third input settles the triangular Hilbert transform estimate at the symmetric $`L^3\\times L^3\\times L^3`$ point. ([Lean](lean/docs/082.md))\n\n\n\n 113 Approximate counting and entropy of perfect matchings. \nFIRST ('A Fully Polynomial Randomized Approximation Scheme for Perfect Matchings in General Graphs', 'preprints/A-Fully-Polynomial-Randomized-Approximation-Scheme-for-Perfect-Matchings-in-General-Graphs-September-23-2026/main.pdf') \nFAMILY SCOPE **113. Approximate counting and entropy of perfect matchings.** Gives a fully polynomial randomized approximation scheme for counting perfect matchings in arbitrary finite simple graphs, with exact detection of zero counts. Also proves the perfect-matching entropy conjecture of Anari, Oveis Gharan, and Vinzant, bounding the maximum entropy of a matching law at every feasible edge-marginal vector in a loopless labelled multigraph, including boundary points. ([Lean](lean/docs/113.md))\n\n\n\n 115 Sampling and counting contingency tables with arbitrary margins. \nFIRST ('Exact Uniform Sampling of Contingency Tables with Arbitrary Margins', 'preprints/Exact-Uniform-Sampling-of-Contingency-Tables-with-Arbitrary-Margins-September-24-2026/main.pdf') \nFAMILY SCOPE **115. Sampling and counting contingency tables with arbitrary margins.** For nonnegative integer matrices with prescribed row and column sums, gives exact uniform sampling in expected polynomial bit time and almost-uniform sampling in worst-case polynomial bit time. The dimensions and binary-encoded margins are unrestricted. Also gives a fully polynomial randomized approximation scheme for counting such tables with arbitrary individual cell bounds, including structural zeros, with polynomial cost on every execution. ([Lean](lean/docs/115.md))\n\n\n\n 127 Average sensitivity of polynomial threshold functions. \nFIRST ('Average sensitivity of polynomial threshold functions', 'preprints/Average-Sensitivity-of-Polynomial-Threshold-Functions-September-25-2026/main.pdf') \nFAMILY SCOPE **127. Average sensitivity of polynomial threshold functions.** Proves that a degree-at-most-<i>d</i> polynomial threshold function on the uniform <i>n</i>-dimensional Boolean cube has average sensitivity at most $`8d\\sqrt n`$, uniformly for $`1\\le d\\le n`$. Average sensitivity counts expected output changes under single-bit flips. This establishes the asymptotic Gotsman–Linial conjecture, allowing polynomial zeros with $`\\mathop{\\mathrm{sign}}\\nolimits (0)=1`$. ([Lean](lean/docs/127.md))\n\n\n\n 148 The entropy-rate dimension formula for self-similar measures. \nFIRST ('The entropy-rate dimension formula for self-similar measures on the line', 'preprints/The-entropy-rate-dimension-formula-for-self-similar-measures-on-the-line-September-24-2026/main.pdf') \nFAMILY SCOPE **148. The entropy-rate dimension formula for self-similar measures.** For every self-similar measure on the line generated by finitely many contracting similarities, proves $`\\dim_{\\mathrm H}\\mu=\\min\\{1,h_{\\mathrm{RW}}/\\chi\\}`$, where $`h_{\\mathrm{RW}}`$ is the entropy rate of random composed maps and <i>χ</i> the average logarithmic contraction. This resolves the entropy-rate dimension conjecture without a separation assumption, allowing exact overlaps and unequal contraction ratios. ([Lean](lean/docs/148.md))\n\n\n\n 154 Pointwise multiple ergodic averages for mixing transformations. \nFIRST ('Pointwise Multiple Ergodic Averages for Mixing Transformations', 'preprints/Pointwise-Multiple-Ergodic-Averages-for-Mixing-Transformations-October-4-2026/multiple-ergodic-averages.pdf') \nFAMILY SCOPE **154. Pointwise multiple ergodic averages for mixing transformations.** Proves almost-everywhere convergence of consecutive multiple ergodic averages of every finite length for invertible mixing probability-preserving transformations. For each fixed tuple of bounded functions, the limit is the product of their integrals, along all positive averaging lengths. No mixing rate or standardness assumption on the probability space is required.\n\n\n\n 164 Hindman’s finite sums and products conjecture. \nFIRST ('Monochromatic finite sums and products in the positive integers', 'preprints/Monochromatic-finite-sums-and-products-in-the-positive-integers-September-23-2026/paper.pdf') \nFAMILY SCOPE **164. Hindman’s finite sums and products conjecture.** Proves Hindman's finite sums and products conjecture: every finite coloring of the positive integers contains sets of any prescribed finite size whose nonempty subset sums and nonempty subset products all have one common color.\n\n\n\n 192 Boolean functions violate the square-root degree bound by arbitrary factors. \nFIRST ('Unbounded Violations of the Square-Root Degree Bound', 'preprints/Unbounded-Violations-of-the-Square-Root-Degree-Bound-September-26-2026/Unbounded-Violations-of-the-Square-Root-Degree-Bound-September-26-2026.pdf') \nFAMILY SCOPE **192. Boolean functions violate the square-root degree bound by arbitrary factors.** Disproves the proposed square-root bound relating a Boolean function's linear Fourier coefficients to its polynomial degree. For every <i>C</i> &gt; 0, there is a sign-valued Boolean function <i>f</i> with $`\\sum_i\\widehat f(\\{i\\})\\gt C\\sqrt{\\deg(f)}`$. Thus its total signed correlation with individual input bits can exceed the proposed bound by an arbitrary factor. ([Lean](lean/docs/192.md))\n\n\n\n 210 Foulkes' conjecture for sixth powers and quadratic stabilization. \nFIRST (\"Foulkes' conjecture for the sixth symmetric power\", 'preprints/Foulkes-Conjecture-for-the-Sixth-Symmetric-Power-September-25-2026/main.pdf') \nFAMILY SCOPE **210. Foulkes' conjecture for sixth powers and quadratic stabilization.** Proves the sixth case of Foulkes’ conjecture: $`\\mathop{\\mathrm{Sym}}\\nolimits ^6(\\mathop{\\mathrm{Sym}}\\nolimits ^bV)`$ embeds equivariantly in $`\\mathop{\\mathrm{Sym}}\\nolimits ^b(\\mathop{\\mathrm{Sym}}\\nolimits ^6V)`$ for every <i>b</i> ≥ 6 and finite-dimensional complex <i>V</i>. More generally, the canonical multiplication map $`\\mathop{\\mathrm{Sym}}\\nolimits ^b(\\mathop{\\mathrm{Sym}}\\nolimits ^aV)\\to\\mathop{\\mathrm{Sym}}\\nolimits ^a(\\mathop{\\mathrm{Sym}}\\nolimits ^bV)`$ is surjective for <i>a</i> ≥ 2 and $`b\\ge a(a-1)`$, giving dimension-independent quadratic stabilization. ([Lean](lean/docs/210.md))\n\n\n\n 215 Canonical $`O(3)`$ continuum limit and exact $`O(4)`$ mass asymptotics. \nFIRST ('The canonical massive continuum limit of the two-dimensional O(3) model', 'preprints/The-canonical-massive-continuum-limit-of-the-two-dimensional-O3-model-October-4-2026/massive-continuum-o3.pdf') \nFAMILY SCOPE **215. Canonical $`O(3)`$ continuum limit and exact $`O(4)`$ mass asymptotics.** Constructs the canonical continuum limit of the two-dimensional nearest-neighbor $`O(3)`$ model: a non-Gaussian local relativistic theory with a unique vacuum and a positive mass gap. For the square-lattice $`O(4)`$ model, determines the exact leading asymptotic of the full transfer gap, $`m_{\\mathrm{lat}}(\\beta)\\sim32e^{\\pi/4-1/2}\\sqrt\\beta\\,e^{-\\pi\\beta}`$. The family also proves exponential spin-correlation decay for two-dimensional nearest-neighbor $`O(n)`$ models with <i>n</i> ≥ 3 at every positive temperature. ([Lean](lean/docs/215.md))\n\n\n\n 230 Exact Hausdorff gauges for SLE. \nFIRST ('An exact Hausdorff gauge for SLE', 'preprints/An-exact-Hausdorff-gauge-for-SLE-September-25-2026/An-exact-Hausdorff-gauge-for-SLE-September-25-2026.pdf') \nFAMILY SCOPE **230. Exact Hausdorff gauges for SLE.** Resolves Schramm’s Hausdorff-measure question for chordal SLE<sub><i>κ</i></sub>, $`0\\lt \\kappa\\lt 8`$. The explicit gauge $`r^d(\\log\\log(1/r))^{(2-d)/2}`$, $`d=1+\\kappa/8`$, gives almost surely positive finite measure to every trace segment $`\\gamma([s,t])`$ with $`0\\lt s\\lt t\\lt \\infty`$, and finite expected measure to the trace in every bounded disk. ([Lean](lean/docs/230.md))\n\n\n\n 245 Weak normalization implies strong normalization in pure type systems. \nFIRST ('Weak and strong normalization in pure type systems', 'preprints/Weak-and-strong-normalization-in-pure-type-systems-September-25-2026/paper.pdf') \nFAMILY SCOPE **245. Weak normalization implies strong normalization in pure type systems.** Proves that weak normalization implies strong normalization for every pure type system: if every legal expression in every valid context has a <i>β</i>-normal form, every <i>β</i>-reduction sequence terminates. This resolves the <i>β</i>-Barendregt–Geuvers–Klop conjecture, including nonfunctional rules and open contexts. ([Lean](lean/docs/245.md))\n\n\n\n 296 The generator problem for finite factors. \nFIRST ('Relative generation and the generator problem for finite factors', 'preprints/Relative-generation-and-the-generator-problem-for-finite-factors-September-23-2026/paper.pdf') \nFAMILY SCOPE **296. The generator problem for finite factors.** Proves that every type II<sub>1</sub> factor with separable predual is generated by a single operator, equivalently by two self-adjoint operators, resolving the generator problem. More strongly, for every irreducible inclusion $`P\\subset M`$ of such factors, the unitaries <i>u</i> with $`M=W^*(P,u)`$ form a dense <i>G</i><sub><i>δ</i></sub> subset in the trace 2-norm topology. ([Lean](lean/docs/296.md))\n\n\n\n 351 Scalar curvature and finite-time Ricci-flow singularities. \nFIRST ('A closed Ricci flow with bounded scalar curvature and finite-time curvature blowup', 'preprints/A-closed-Ricci-flow-with-bounded-scalar-curvature-and-finite-time-curvature-blowup-September-24-2026/paper.pdf') \nFAMILY SCOPE **351. Scalar curvature and finite-time Ricci-flow singularities.** Proves that a smooth Ricci flow on a closed four-manifold extends past any finite time at which scalar curvature remains uniformly bounded. A higher-dimensional counterexample has bounded scalar curvature but unbounded full curvature at its finite maximal time, disproving the unrestricted scalar-curvature extension conjecture.\n\n\n\n 369 The hot spots conjecture for simply connected planar domains. \nFIRST ('Strict hot spots and absence of interior critical points on smooth simply connected planar domains', 'preprints/Strict-hot-spots-and-absence-of-interior-critical-points-on-smooth-simply-connected-planar-domains-September-24-2026/main.pdf') \nFAMILY SCOPE **369. The hot spots conjecture for simply connected planar domains.** Proves a strict form of Burdzy's simply connected hot spots conjecture. On every smooth bounded simply connected planar domain, each nonzero eigenfunction for the first positive Neumann eigenvalue has no interior critical point, so all global extrema lie on the boundary. Eigenvalue multiplicity is allowed. ([Lean](lean/docs/369.md))\n\n\n\n 372 Global uniqueness in smooth isotropic elasticity. \nFIRST ('Global Uniqueness for the Smooth Isotropic Elasticity Inverse Problem', 'preprints/Global-Uniqueness-for-the-Smooth-Isotropic-Elasticity-Inverse-Problem-September-24-2026/article.pdf') \nFAMILY SCOPE **372. Global uniqueness in smooth isotropic elasticity.** Proves that full static boundary displacement-to-traction data determine both smooth real Lamé moduli on every bounded connected smooth domain in ℝ<sup>3</sup>, provided <i>μ</i> &gt; 0 and $`3\\lambda+2\\mu\\gt 0`$ on the closure. Neither analyticity, proximity to constant coefficients nor prior knowledge near the boundary is required. ([Lean](lean/docs/372.md))\n\n\n\nYAML related/status\n \nrelated_formalizations:\n  # Imported libraries, ordered by direct import count in project Lean sources.\n  - id: https://github.com/leanprover-community/mathlib4\n    relationship: builds-on\n\n  - id: https://github.com/AlexKontorovich/PrimeNumberTheoremAnd\n    relationship: builds-on\n\n  - id: https://github.com/n-yamaguchi-0729/ClassFieldTheory\n    relationship: builds-on\n\n  - id: https://github.com/CBirkbeck/AINTLIB\n    relationship: builds-on\n\n  - id: https://github.com/math-inc/strongpnt\n    relationship: builds-on\n\n  - id: https://github.com/abenenson/rellich-kondrachov\n    relationship: builds-on\n\n  - id: https://github.com/harfe/fixed-point-theorems-lean4\n    relationship: builds-on\n\n  - id: https://github.com/TauCetiProject/TauCeti\n    relationship: builds-on\n\n  - id: https://github.com/fpvandoorn/carleson\n    relationship: builds-on\n\n  - id: https://github.com/lana-agents/heights\n    relationship: builds-on\n\n  - id: https://github.com/Aaron1011/gromov\n    relationship: builds-on\n\n  - id: https://github.com/lana-agents/iut\n    relationship: builds-on\n\n  - id: https://github.com/alonamaloh/schoenflies-lean\n    relationship: builds-on\n\n  - id: https://github.com/leanprover-community/sphere-eversion\n    relationship: builds-on\n\n  - id: https://github.com/ahhwuhu/zeta_3_irrational\n    relationship: builds-on\n\n  - id: https://github.com/BennyAvelin/AbsorptionCutoff\n    relationship: builds-on\n\n  - id: https://github.com/lana-agents/belyi\n    relationship: builds-on\n\n  - id: https://github.com/PatrickMassot/checkdecls\n    relationship: builds-on\n\n  - id: https://github.com/leanprover/doc-gen4\n    relationship: builds-on\n\n  - id: https://github.com/lana-agents/elliptic-curves\n    relationship: builds-on\n\n  - id: https://github.com/lana-agents/formal-schemes\n    relationship: builds-on\n\n  - id: https://github.com/lana-agents/genl\n    relationship: builds-on\n\n  - id: https://github.com/hanwenzhu/LeanArchitect\n    relationship: builds-on\n\n  - id: https://github.com/alerad/leancert\n    relationship: builds-on\n\n  - id: https://github.com/lana-agents/oka\n    relationship: builds-on\n\n  - id: https://github.com/lana-agents/orbicurve-cores\n    relationship: builds-on\n\n  - id: https://github.com/lana-agents/pi1\n    relationship: builds-on\n\n  - id: https://github.com/b-mehta/PrimeCert\n    relationship: builds-on\n\n  - id: https://github.com/lana-agents/tate-curves-theta\n    relationship: builds-on\n\n  - id: https://github.com/lana-agents/tempered-fundamental-groups\n    relationship: builds-on\n\nstatus:\n  scope: \"Partial progress.\"\n  # Entries use source-qualified declaration names and the files containing them.\n  main_results:\n    - comparator_config: ComparatorChallenges/AbhyankarSathaye.json\n      declaration: OAI.AbhyankarSathaye.exists_noncoordinate_polynomial\n      file: OAI/AlgebraicGeometry/AbhyankarSathaye/Counterexample.lean\n\n    - comparator_config: ComparatorChallenges/AmplitudeDamping.json\n      declaration: OAI.GAD.main\n      file: OAI/InformationTheory/AmplitudeDamping/Main.lean\n\n    - comparator_config: ComparatorChallenges/ArnoldCounterexample.json\n      declaration: OAI.ArnoldCounterexample.main\n      file: OAI/Geometry/Arnold/Main.lean\n\n    - comparator_config: ComparatorChallenges/ArtinCAT0.json\n      declaration: OAI.ArtinCAT0.main\n      file: OAI/GroupTheory/ArtinCAT0/Main.lean\n\n    - comparator_config: ComparatorChallenges/AsymptoticallyMinimalLittlewood.json\n      declaration: OAI.AsymptoticallyMinimalLittlewood.main\n      file: OAI/Analysis/Littlewood/Main.lean\n\n    - comparator_config: ComparatorChallenges/AuslanderReiten.json\n      declaration: OAI.ArExplicit.Statement.main\n      file: OAI/Algebra/AuslanderReiten/All.lean\n\n    - comparator_config: ComparatorChallenges/BackwardIntertwiners.json\n      declaration: OAI.BackwardIntertwiners.direct_algebra_corollary\n      file: OAI/Analysis/BackwardIntertwiners/AlgebraCorollary.lean\n\n    - comparator_config: ComparatorChallenges/BassTrace.json\n      declaration: OAI.BassTrace.RightProjective.bassTraceModules_vanishing_and_support\n      file: OAI/RingTheory/BassTrace/Main.lean\n\n    - comparator_config: ComparatorChallenges/BiholderTransport.json\n",
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          "paper link discrepancies [('337', 2, 1)]\nmissing family papers []\ndraw order ['154', '024', '230', '127', '082', '148', '046', '372', '113', '164', '351', '369', '115', '215', '065', '296', '245', '015', '210', '192']\n\n 015 Torus-packet equidistribution in prime, quartic, and sextic degrees. \nFIRST ('Equidistribution of Prime-Degree Torus Packets with Arbitrary Local Type', 'preprints/Equidistribution-of-Prime-Degree-Torus-Packets-with-Arbitrary-Local-Type-September-24-2026/paper.pdf') \nFAMILY SCOPE **015. Torus-packet equidistribution in prime, quartic, and sextic degrees.** Proves Haar equidistribution without escape of mass for complete volume-weighted torus packets from totally real fields: arbitrary lattices and prescribed local types in fixed prime degree at least five, arbitrary-order Picard packets in primitive quartic fields, and maximal-order ideal-class packets in primitive sextic fields. Here primitive means having no proper intermediate field; the relevant order or field discriminant tends to infinity. ([Lean](lean/docs/015.md))\n\n\n\n 024 An asymptotic formula for the number of totients. \nFIRST ('An asymptotic formula for the number of totients', 'preprints/An-asymptotic-formula-for-the-number-of-totients-September-25-2026/An-asymptotic-formula-for-the-number-of-totients-September-25-2026.pdf') \nFAMILY SCOPE **024. An asymptotic formula for the number of totients.** Gives an asymptotic equivalent for the number $`V(x)`$ of distinct totient values up to <i>x</i>, with a positive bounded phase-dependent factor determined by convergent arithmetic approximations. In particular, $`V(cx)/V(x)\\to c`$ for every fixed <i>c</i> &gt; 0, answering Erdős and Hall’s scaling question. ([Lean](lean/docs/024.md))\n\n\n\n 046 Shafarevich counterexamples in dimension two and with large fundamental group. \nFIRST ('A projective fourfold with large fundamental group and non-Stein universal cover', 'preprints/A-projective-fourfold-with-large-fundamental-group-and-non-Stein-universal-cover-October-5-2026/large-fundamental-group-non-stein.pdf') \nFAMILY SCOPE **046. Shafarevich counterexamples in dimension two and with large fundamental group.** Constructs a smooth projective complex fourfold with large fundamental group whose universal cover contains no positive-dimensional compact analytic subvariety but is neither Stein nor holomorphically convex. A separate smooth projective complex surface already disproves unrestricted Shafarevich holomorphic convexity in dimension two.\n\n\n\n 065 Virasoro constraints for complete intersections and projective-bundle towers. \nFIRST ('Virasoro Constraints under Projectivization', 'preprints/Virasoro-Constraints-under-Projectivization-October-5-2026/virasoro-constraints-under-projectivization.pdf') \nFAMILY SCOPE **065. Virasoro constraints for complete intersections and projective-bundle towers.** Proves the full ordinary unreduced descendant Virasoro conjecture for smooth complete intersections in complex projective space, in every genus and curve class with arbitrary cohomology insertions. The constraints also pass from any smooth projective complex base satisfying them to the projectivization of every algebraic vector bundle of rank at least two, and hence to projective-bundle towers.\n\n\n\n 082 Annular variation and dyadic absolute bounds for the triangular Hilbert transform. \nFIRST ('Annular variation of the triangular Hilbert transform at the symmetric point', 'preprints/Annular-variation-of-the-triangular-Hilbert-transform-at-the-symmetric-point-October-5-2026/annular-variation.pdf') \nFAMILY SCOPE **082. Annular variation and dyadic absolute bounds for the triangular Hilbert transform.** Proves maximal and annular <i>r</i>-variation bounds, for every <i>r</i> &gt; 2, from complex $`L^3(\\mathbb R^2)\\times L^3(\\mathbb R^2)`$ to $`L^{3/2}(\\mathbb R^2)`$. The maximal estimate controls both hard truncation endpoints and gives almost-everywhere and norm convergence. Pairing with a third input settles the triangular Hilbert transform estimate at the symmetric $`L^3\\times L^3\\times L^3`$ point. ([Lean](lean/docs/082.md))\n\n\n\n 113 Approximate counting and entropy of perfect matchings. \nFIRST ('A Fully Polynomial Randomized Approximation Scheme for Perfect Matchings in General Graphs', 'preprints/A-Fully-Polynomial-Randomized-Approximation-Scheme-for-Perfect-Matchings-in-General-Graphs-September-23-2026/main.pdf') \nFAMILY SCOPE **113. Approximate counting and entropy of perfect matchings.** Gives a fully polynomial randomized approximation scheme for counting perfect matchings in arbitrary finite simple graphs, with exact detection of zero counts. Also proves the perfect-matching entropy conjecture of Anari, Oveis Gharan, and Vinzant, bounding the maximum entropy of a matching law at every feasible edge-marginal vector in a loopless labelled multigraph, including boundary points. ([Lean](lean/docs/113.md))\n\n\n\n 115 Sampling and counting contingency tables with arbitrary margins. \nFIRST ('Exact Uniform Sampling of Contingency Tables with Arbitrary Margins', 'preprints/Exact-Uniform-Sampling-of-Contingency-Tables-with-Arbitrary-Margins-September-24-2026/main.pdf') \nFAMILY SCOPE **115. Sampling and counting contingency tables with arbitrary margins.** For nonnegative integer matrices with prescribed row and column sums, gives exact uniform sampling in expected polynomial bit time and almost-uniform sampling in worst-case polynomial bit time. The dimensions and binary-encoded margins are unrestricted. Also gives a fully polynomial randomized approximation scheme for counting such tables with arbitrary individual cell bounds, including structural zeros, with polynomial cost on every execution. ([Lean](lean/docs/115.md))\n\n\n\n 127 Average sensitivity of polynomial threshold functions. \nFIRST ('Average sensitivity of polynomial threshold functions', 'preprints/Average-Sensitivity-of-Polynomial-Threshold-Functions-September-25-2026/main.pdf') \nFAMILY SCOPE **127. Average sensitivity of polynomial threshold functions.** Proves that a degree-at-most-<i>d</i> polynomial threshold function on the uniform <i>n</i>-dimensional Boolean cube has average sensitivity at most $`8d\\sqrt n`$, uniformly for $`1\\le d\\le n`$. Average sensitivity counts expected output changes under single-bit flips. This establishes the asymptotic Gotsman–Linial conjecture, allowing polynomial zeros with $`\\mathop{\\mathrm{sign}}\\nolimits (0)=1`$. ([Lean](lean/docs/127.md))\n\n\n\n 148 The entropy-rate dimension formula for self-similar measures. \nFIRST ('The entropy-rate dimension formula for self-similar measures on the line', 'preprints/The-entropy-rate-dimension-formula-for-self-similar-measures-on-the-line-September-24-2026/main.pdf') \nFAMILY SCOPE **148. The entropy-rate dimension formula for self-similar measures.** For every self-similar measure on the line generated by finitely many contracting similarities, proves $`\\dim_{\\mathrm H}\\mu=\\min\\{1,h_{\\mathrm{RW}}/\\chi\\}`$, where $`h_{\\mathrm{RW}}`$ is the entropy rate of random composed maps and <i>χ</i> the average logarithmic contraction. This resolves the entropy-rate dimension conjecture without a separation assumption, allowing exact overlaps and unequal contraction ratios. ([Lean](lean/docs/148.md))\n\n\n\n 154 Pointwise multiple ergodic averages for mixing transformations. \nFIRST ('Pointwise Multiple Ergodic Averages for Mixing Transformations', 'preprints/Pointwise-Multiple-Ergodic-Averages-for-Mixing-Transformations-October-4-2026/multiple-ergodic-averages.pdf') \nFAMILY SCOPE **154. Pointwise multiple ergodic averages for mixing transformations.** Proves almost-everywhere convergence of consecutive multiple ergodic averages of every finite length for invertible mixing probability-preserving transformations. For each fixed tuple of bounded functions, the limit is the product of their integrals, along all positive averaging lengths. No mixing rate or standardness assumption on the probability space is required.\n\n\n\n 164 Hindman’s finite sums and products conjecture. \nFIRST ('Monochromatic finite sums and products in the positive integers', 'preprints/Monochromatic-finite-sums-and-products-in-the-positive-integers-September-23-2026/paper.pdf') \nFAMILY SCOPE **164. Hindman’s finite sums and products conjecture.** Proves Hindman's finite sums and products conjecture: every finite coloring of the positive integers contains sets of any prescribed finite size whose nonempty subset sums and nonempty subset products all have one common color.\n\n\n\n 192 Boolean functions violate the square-root degree bound by arbitrary factors. \nFIRST ('Unbounded Violations of the Square-Root Degree Bound', 'preprints/Unbounded-Violations-of-the-Square-Root-Degree-Bound-September-26-2026/Unbounded-Violations-of-the-Square-Root-Degree-Bound-September-26-2026.pdf') \nFAMILY SCOPE **192. Boolean functions violate the square-root degree bound by arbitrary factors.** Disproves the proposed square-root bound relating a Boolean function's linear Fourier coefficients to its polynomial degree. For every <i>C</i> &gt; 0, there is a sign-valued Boolean function <i>f</i> with $`\\sum_i\\widehat f(\\{i\\})\\gt C\\sqrt{\\deg(f)}`$. Thus its total signed correlation with individual input bits can exceed the proposed bound by an arbitrary factor. ([Lean](lean/docs/192.md))\n\n\n\n 210 Foulkes' conjecture for sixth powers and quadratic stabilization. \nFIRST (\"Foulkes' conjecture for the sixth symmetric power\", 'preprints/Foulkes-Conjecture-for-the-Sixth-Symmetric-Power-September-25-2026/main.pdf') \nFAMILY SCOPE **210. Foulkes' conjecture for sixth powers and quadratic stabilization.** Proves the sixth case of Foulkes’ conjecture: $`\\mathop{\\mathrm{Sym}}\\nolimits ^6(\\mathop{\\mathrm{Sym}}\\nolimits ^bV)`$ embeds equivariantly in $`\\mathop{\\mathrm{Sym}}\\nolimits ^b(\\mathop{\\mathrm{Sym}}\\nolimits ^6V)`$ for every <i>b</i> ≥ 6 and finite-dimensional complex <i>V</i>. More generally, the canonical multiplication map $`\\mathop{\\mathrm{Sym}}\\nolimits ^b(\\mathop{\\mathrm{Sym}}\\nolimits ^aV)\\to\\mathop{\\mathrm{Sym}}\\nolimits ^a(\\mathop{\\mathrm{Sym}}\\nolimits ^bV)`$ is surjective for <i>a</i> ≥ 2 and $`b\\ge a(a-1)`$, giving dimension-independent quadratic stabilization. ([Lean](lean/docs/210.md))\n\n\n\n 215 Canonical $`O(3)`$ continuum limit and exact $`O(4)`$ mass asymptotics. \nFIRST ('The canonical massive continuum limit of the two-dimensional O(3) model', 'preprints/The-canonical-massive-continuum-limit-of-the-two-dimensional-O3-model-October-4-2026/massive-continuum-o3.pdf') \nFAMILY SCOPE **215. Canonical $`O(3)`$ continuum limit and exact $`O(4)`$ mass asymptotics.** Constructs the canonical continuum limit of the two-dimensional nearest-neighbor $`O(3)`$ model: a non-Gaussian local relativistic theory with a unique vacuum and a positive mass gap. For the square-lattice $`O(4)`$ model, determines the exact leading asymptotic of the full transfer gap, $`m_{\\mathrm{lat}}(\\beta)\\sim32e^{\\pi/4-1/2}\\sqrt\\beta\\,e^{-\\pi\\beta}`$. The family also proves exponential spin-correlation decay for two-dimensional nearest-neighbor $`O(n)`$ models with <i>n</i> ≥ 3 at every positive temperature. ([Lean](lean/docs/215.md))\n\n\n\n 230 Exact Hausdorff gauges for SLE. \nFIRST ('An exact Hausdorff gauge for SLE', 'preprints/An-exact-Hausdorff-gauge-for-SLE-September-25-2026/An-exact-Hausdorff-gauge-for-SLE-September-25-2026.pdf') \nFAMILY SCOPE **230. Exact Hausdorff gauges for SLE.** Resolves Schramm’s Hausdorff-measure question for chordal SLE<sub><i>κ</i></sub>, $`0\\lt \\kappa\\lt 8`$. The explicit gauge $`r^d(\\log\\log(1/r))^{(2-d)/2}`$, $`d=1+\\kappa/8`$, gives almost surely positive finite measure to every trace segment $`\\gamma([s,t])`$ with $`0\\lt s\\lt t\\lt \\infty`$, and finite expected measure to the trace in every bounded disk. ([Lean](lean/docs/230.md))\n\n\n\n 245 Weak normalization implies strong normalization in pure type systems. \nFIRST ('Weak and strong normalization in pure type systems', 'preprints/Weak-and-strong-normalization-in-pure-type-systems-September-25-2026/paper.pdf') \nFAMILY SCOPE **245. Weak normalization implies strong normalization in pure type systems.** Proves that weak normalization implies strong normalization for every pure type system: if every legal expression in every valid context has a <i>β</i>-normal form, every <i>β</i>-reduction sequence terminates. This resolves the <i>β</i>-Barendregt–Geuvers–Klop conjecture, including nonfunctional rules and open contexts. ([Lean](lean/docs/245.md))\n\n\n\n 296 The generator problem for finite factors. \nFIRST ('Relative generation and the generator problem for finite factors', 'preprints/Relative-generation-and-the-generator-problem-for-finite-factors-September-23-2026/paper.pdf') \nFAMILY SCOPE **296. The generator problem for finite factors.** Proves that every type II<sub>1</sub> factor with separable predual is generated by a single operator, equivalently by two self-adjoint operators, resolving the generator problem. More strongly, for every irreducible inclusion $`P\\subset M`$ of such factors, the unitaries <i>u</i> with $`M=W^*(P,u)`$ form a dense <i>G</i><sub><i>δ</i></sub> subset in the trace 2-norm topology. ([Lean](lean/docs/296.md))\n\n\n\n 351 Scalar curvature and finite-time Ricci-flow singularities. \nFIRST ('A closed Ricci flow with bounded scalar curvature and finite-time curvature blowup', 'preprints/A-closed-Ricci-flow-with-bounded-scalar-curvature-and-finite-time-curvature-blowup-September-24-2026/paper.pdf') \nFAMILY SCOPE **351. Scalar curvature and finite-time Ricci-flow singularities.** Proves that a smooth Ricci flow on a closed four-manifold extends past any finite time at which scalar curvature remains uniformly bounded. A higher-dimensional counterexample has bounded scalar curvature but unbounded full curvature at its finite maximal time, disproving the unrestricted scalar-curvature extension conjecture.\n\n\n\n 369 The hot spots conjecture for simply connected planar domains. \nFIRST ('Strict hot spots and absence of interior critical points on smooth simply connected planar domains', 'preprints/Strict-hot-spots-and-absence-of-interior-critical-points-on-smooth-simply-connected-planar-domains-September-24-2026/main.pdf') \nFAMILY SCOPE **369. The hot spots conjecture for simply connected planar domains.** Proves a strict form of Burdzy's simply connected hot spots conjecture. On every smooth bounded simply connected planar domain, each nonzero eigenfunction for the first positive Neumann eigenvalue has no interior critical point, so all global extrema lie on the boundary. Eigenvalue multiplicity is allowed. ([Lean](lean/docs/369.md))\n\n\n\n 372 Global uniqueness in smooth isotropic elasticity. \nFIRST ('Global Uniqueness for the Smooth Isotropic Elasticity Inverse Problem', 'preprints/Global-Uniqueness-for-the-Smooth-Isotropic-Elasticity-Inverse-Problem-September-24-2026/article.pdf') \nFAMILY SCOPE **372. Global uniqueness in smooth isotropic elasticity.** Proves that full static boundary displacement-to-traction data determine both smooth real Lamé moduli on every bounded connected smooth domain in ℝ<sup>3</sup>, provided <i>μ</i> &gt; 0 and $`3\\lambda+2\\mu\\gt 0`$ on the closure. Neither analyticity, proximity to constant coefficients nor prior knowledge near the boundary is required. ([Lean](lean/docs/372.md))\n\n\n\nYAML related/status\n \nrelated_formalizations:\n  # Imported libraries, ordered by direct import count in project Lean sources.\n  - id: https://github.com/leanprover-community/mathlib4\n    relationship: builds-on\n\n  - id: https://github.com/AlexKontorovich/PrimeNumberTheoremAnd\n    relationship: builds-on\n\n  - id: https://github.com/n-yamaguchi-0729/ClassFieldTheory\n    relationship: builds-on\n\n  - id: https://github.com/CBirkbeck/AINTLIB\n    relationship: builds-on\n\n  - id: https://github.com/math-inc/strongpnt\n    relationship: builds-on\n\n  - id: https://github.com/abenenson/rellich-kondrachov\n    relationship: builds-on\n\n  - id: https://github.com/harfe/fixed-point-theorems-lean4\n    relationship: builds-on\n\n  - id: https://github.com/TauCetiProject/TauCeti\n    relationship: builds-on\n\n  - id: https://github.com/fpvandoorn/carleson\n    relationship: builds-on\n\n  - id: https://github.com/lana-agents/heights\n    relationship: builds-on\n\n  - id: https://github.com/Aaron1011/gromov\n    relationship: builds-on\n\n  - id: https://github.com/lana-agents/iut\n    relationship: builds-on\n\n  - id: https://github.com/alonamaloh/schoenflies-lean\n    relationship: builds-on\n\n  - id: https://github.com/leanprover-community/sphere-eversion\n    relationship: builds-on\n\n  - id: https://github.com/ahhwuhu/zeta_3_irrational\n    relationship: builds-on\n\n  - id: https://github.com/BennyAvelin/AbsorptionCutoff\n    relationship: builds-on\n\n  - id: https://github.com/lana-agents/belyi\n    relationship: builds-on\n\n  - id: https://github.com/PatrickMassot/checkdecls\n    relationship: builds-on\n\n  - id: https://github.com/leanprover/doc-gen4\n    relationship: builds-on\n\n  - id: https://github.com/lana-agents/elliptic-curves\n    relationship: builds-on\n\n  - id: https://github.com/lana-agents/formal-schemes\n    relationship: builds-on\n\n  - id: https://github.com/lana-agents/genl\n    relationship: builds-on\n\n  - id: https://github.com/hanwenzhu/LeanArchitect\n    relationship: builds-on\n\n  - id: https://github.com/alerad/leancert\n    relationship: builds-on\n\n  - id: https://github.com/lana-agents/oka\n    relationship: builds-on\n\n  - id: https://github.com/lana-agents/orbicurve-cores\n    relationship: builds-on\n\n  - id: https://github.com/lana-agents/pi1\n    relationship: builds-on\n\n  - id: https://github.com/b-mehta/PrimeCert\n    relationship: builds-on\n\n  - id: https://github.com/lana-agents/tate-curves-theta\n    relationship: builds-on\n\n  - id: https://github.com/lana-agents/tempered-fundamental-groups\n    relationship: builds-on\n\nstatus:\n  scope: \"Partial progress.\"\n  # Entries use source-qualified declaration names and the files containing them.\n  main_results:\n    - comparator_config: ComparatorChallenges/AbhyankarSathaye.json\n      declaration: OAI.AbhyankarSathaye.exists_noncoordinate_polynomial\n      file: OAI/AlgebraicGeometry/AbhyankarSathaye/Counterexample.lean\n\n    - comparator_config: ComparatorChallenges/AmplitudeDamping.json\n      declaration: OAI.GAD.main\n      file: OAI/InformationTheory/AmplitudeDamping/Main.lean\n\n    - comparator_config: ComparatorChallenges/ArnoldCounterexample.json\n      declaration: OAI.ArnoldCounterexample.main\n      file: OAI/Geometry/Arnold/Main.lean\n\n    - comparator_config: ComparatorChallenges/ArtinCAT0.json\n      declaration: OAI.ArtinCAT0.main\n      file: OAI/GroupTheory/ArtinCAT0/Main.lean\n\n    - comparator_config: ComparatorChallenges/AsymptoticallyMinimalLittlewood.json\n      declaration: OAI.AsymptoticallyMinimalLittlewood.main\n      file: OAI/Analysis/Littlewood/Main.lean\n\n    - comparator_config: ComparatorChallenges/AuslanderReiten.json\n      declaration: OAI.ArExplicit.Statement.main\n      file: OAI/Algebra/AuslanderReiten/All.lean\n\n    - comparator_config: ComparatorChallenges/BackwardIntertwiners.json\n      declaration: OAI.BackwardIntertwiners.direct_algebra_corollary\n      file: OAI/Analysis/BackwardIntertwiners/AlgebraCorollary.lean\n\n    - comparator_config: ComparatorChallenges/BassTrace.json\n      declaration: OAI.BassTrace.RightProjective.bassTraceModules_vanishing_and_support\n      file: OAI/RingTheory/BassTrace/Main.lean\n\n    - comparator_config: ComparatorChallenges/BiholderTransport.json\n"
        ]
      },
      "outputs": [
        {
          "type": "stream",
          "name": "stdout",
          "text": "paper link discrepancies [('337', 2, 1)]\nmissing family papers []\ndraw order ['154', '024', '230', '127', '082', '148', '046', '372', '113', '164', '351', '369', '115', '215', '065', '296', '245', '015', '210', '192']\n\n 015 Torus-packet equidistribution in prime, quartic, and sextic degrees. \nFIRST ('Equidistribution of Prime-Degree Torus Packets with Arbitrary Local Type', 'preprints/Equidistribution-of-Prime-Degree-Torus-Packets-with-Arbitrary-Local-Type-September-24-2026/paper.pdf') \nFAMILY SCOPE **015. Torus-packet equidistribution in prime, quartic, and sextic degrees.** Proves Haar equidistribution without escape of mass for complete volume-weighted torus packets from totally real fields: arbitrary lattices and prescribed local types in fixed prime degree at least five, arbitrary-order Picard packets in primitive quartic fields, and maximal-order ideal-class packets in primitive sextic fields. Here primitive means having no proper intermediate field; the relevant order or field discriminant tends to infinity. ([Lean](lean/docs/015.md))\n\n\n\n 024 An asymptotic formula for the number of totients. \nFIRST ('An asymptotic formula for the number of totients', 'preprints/An-asymptotic-formula-for-the-number-of-totients-September-25-2026/An-asymptotic-formula-for-the-number-of-totients-September-25-2026.pdf') \nFAMILY SCOPE **024. An asymptotic formula for the number of totients.** Gives an asymptotic equivalent for the number $`V(x)`$ of distinct totient values up to <i>x</i>, with a positive bounded phase-dependent factor determined by convergent arithmetic approximations. In particular, $`V(cx)/V(x)\\to c`$ for every fixed <i>c</i> &gt; 0, answering Erdős and Hall’s scaling question. ([Lean](lean/docs/024.md))\n\n\n\n 046 Shafarevich counterexamples in dimension two and with large fundamental group. \nFIRST ('A projective fourfold with large fundamental group and non-Stein universal cover', 'preprints/A-projective-fourfold-with-large-fundamental-group-and-non-Stein-universal-cover-October-5-2026/large-fundamental-group-non-stein.pdf') \nFAMILY SCOPE **046. Shafarevich counterexamples in dimension two and with large fundamental group.** Constructs a smooth projective complex fourfold with large fundamental group whose universal cover contains no positive-dimensional compact analytic subvariety but is neither Stein nor holomorphically convex. A separate smooth projective complex surface already disproves unrestricted Shafarevich holomorphic convexity in dimension two.\n\n\n\n 065 Virasoro constraints for complete intersections and projective-bundle towers. \nFIRST ('Virasoro Constraints under Projectivization', 'preprints/Virasoro-Constraints-under-Projectivization-October-5-2026/virasoro-constraints-under-projectivization.pdf') \nFAMILY SCOPE **065. Virasoro constraints for complete intersections and projective-bundle towers.** Proves the full ordinary unreduced descendant Virasoro conjecture for smooth complete intersections in complex projective space, in every genus and curve class with arbitrary cohomology insertions. The constraints also pass from any smooth projective complex base satisfying them to the projectivization of every algebraic vector bundle of rank at least two, and hence to projective-bundle towers.\n\n\n\n 082 Annular variation and dyadic absolute bounds for the triangular Hilbert transform. \nFIRST ('Annular variation of the triangular Hilbert transform at the symmetric point', 'preprints/Annular-variation-of-the-triangular-Hilbert-transform-at-the-symmetric-point-October-5-2026/annular-variation.pdf') \nFAMILY SCOPE **082. Annular variation and dyadic absolute bounds for the triangular Hilbert transform.** Proves maximal and annular <i>r</i>-variation bounds, for every <i>r</i> &gt; 2, from complex $`L^3(\\mathbb R^2)\\times L^3(\\mathbb R^2)`$ to $`L^{3/2}(\\mathbb R^2)`$. The maximal estimate controls both hard truncation endpoints and gives almost-everywhere and norm convergence. Pairing with a third input settles the triangular Hilbert transform estimate at the symmetric $`L^3\\times L^3\\times L^3`$ point. ([Lean](lean/docs/082.md))\n\n\n\n 113 Approximate counting and entropy of perfect matchings. \nFIRST ('A Fully Polynomial Randomized Approximation Scheme for Perfect Matchings in General Graphs', 'preprints/A-Fully-Polynomial-Randomized-Approximation-Scheme-for-Perfect-Matchings-in-General-Graphs-September-23-2026/main.pdf') \nFAMILY SCOPE **113. Approximate counting and entropy of perfect matchings.** Gives a fully polynomial randomized approximation scheme for counting perfect matchings in arbitrary finite simple graphs, with exact detection of zero counts. Also proves the perfect-matching entropy conjecture of Anari, Oveis Gharan, and Vinzant, bounding the maximum entropy of a matching law at every feasible edge-marginal vector in a loopless labelled multigraph, including boundary points. ([Lean](lean/docs/113.md))\n\n\n\n 115 Sampling and counting contingency tables with arbitrary margins. \nFIRST ('Exact Uniform Sampling of Contingency Tables with Arbitrary Margins', 'preprints/Exact-Uniform-Sampling-of-Contingency-Tables-with-Arbitrary-Margins-September-24-2026/main.pdf') \nFAMILY SCOPE **115. Sampling and counting contingency tables with arbitrary margins.** For nonnegative integer matrices with prescribed row and column sums, gives exact uniform sampling in expected polynomial bit time and almost-uniform sampling in worst-case polynomial bit time. The dimensions and binary-encoded margins are unrestricted. Also gives a fully polynomial randomized approximation scheme for counting such tables with arbitrary individual cell bounds, including structural zeros, with polynomial cost on every execution. ([Lean](lean/docs/115.md))\n\n\n\n 127 Average sensitivity of polynomial threshold functions. \nFIRST ('Average sensitivity of polynomial threshold functions', 'preprints/Average-Sensitivity-of-Polynomial-Threshold-Functions-September-25-2026/main.pdf') \nFAMILY SCOPE **127. Average sensitivity of polynomial threshold functions.** Proves that a degree-at-most-<i>d</i> polynomial threshold function on the uniform <i>n</i>-dimensional Boolean cube has average sensitivity at most $`8d\\sqrt n`$, uniformly for $`1\\le d\\le n`$. Average sensitivity counts expected output changes under single-bit flips. This establishes the asymptotic Gotsman–Linial conjecture, allowing polynomial zeros with $`\\mathop{\\mathrm{sign}}\\nolimits (0)=1`$. ([Lean](lean/docs/127.md))\n\n\n\n 148 The entropy-rate dimension formula for self-similar measures. \nFIRST ('The entropy-rate dimension formula for self-similar measures on the line', 'preprints/The-entropy-rate-dimension-formula-for-self-similar-measures-on-the-line-September-24-2026/main.pdf') \nFAMILY SCOPE **148. The entropy-rate dimension formula for self-similar measures.** For every self-similar measure on the line generated by finitely many contracting similarities, proves $`\\dim_{\\mathrm H}\\mu=\\min\\{1,h_{\\mathrm{RW}}/\\chi\\}`$, where $`h_{\\mathrm{RW}}`$ is the entropy rate of random composed maps and <i>χ</i> the average logarithmic contraction. This resolves the entropy-rate dimension conjecture without a separation assumption, allowing exact overlaps and unequal contraction ratios. ([Lean](lean/docs/148.md))\n\n\n\n 154 Pointwise multiple ergodic averages for mixing transformations. \nFIRST ('Pointwise Multiple Ergodic Averages for Mixing Transformations', 'preprints/Pointwise-Multiple-Ergodic-Averages-for-Mixing-Transformations-October-4-2026/multiple-ergodic-averages.pdf') \nFAMILY SCOPE **154. Pointwise multiple ergodic averages for mixing transformations.** Proves almost-everywhere convergence of consecutive multiple ergodic averages of every finite length for invertible mixing probability-preserving transformations. For each fixed tuple of bounded functions, the limit is the product of their integrals, along all positive averaging lengths. No mixing rate or standardness assumption on the probability space is required.\n\n\n\n 164 Hindman’s finite sums and products conjecture. \nFIRST ('Monochromatic finite sums and products in the positive integers', 'preprints/Monochromatic-finite-sums-and-products-in-the-positive-integers-September-23-2026/paper.pdf') \nFAMILY SCOPE **164. Hindman’s finite sums and products conjecture.** Proves Hindman's finite sums and products conjecture: every finite coloring of the positive integers contains sets of any prescribed finite size whose nonempty subset sums and nonempty subset products all have one common color.\n\n\n\n 192 Boolean functions violate the square-root degree bound by arbitrary factors. \nFIRST ('Unbounded Violations of the Square-Root Degree Bound', 'preprints/Unbounded-Violations-of-the-Square-Root-Degree-Bound-September-26-2026/Unbounded-Violations-of-the-Square-Root-Degree-Bound-September-26-2026.pdf') \nFAMILY SCOPE **192. Boolean functions violate the square-root degree bound by arbitrary factors.** Disproves the proposed square-root bound relating a Boolean function's linear Fourier coefficients to its polynomial degree. For every <i>C</i> &gt; 0, there is a sign-valued Boolean function <i>f</i> with $`\\sum_i\\widehat f(\\{i\\})\\gt C\\sqrt{\\deg(f)}`$. Thus its total signed correlation with individual input bits can exceed the proposed bound by an arbitrary factor. ([Lean](lean/docs/192.md))\n\n\n\n 210 Foulkes' conjecture for sixth powers and quadratic stabilization. \nFIRST (\"Foulkes' conjecture for the sixth symmetric power\", 'preprints/Foulkes-Conjecture-for-the-Sixth-Symmetric-Power-September-25-2026/main.pdf') \nFAMILY SCOPE **210. Foulkes' conjecture for sixth powers and quadratic stabilization.** Proves the sixth case of Foulkes’ conjecture: $`\\mathop{\\mathrm{Sym}}\\nolimits ^6(\\mathop{\\mathrm{Sym}}\\nolimits ^bV)`$ embeds equivariantly in $`\\mathop{\\mathrm{Sym}}\\nolimits ^b(\\mathop{\\mathrm{Sym}}\\nolimits ^6V)`$ for every <i>b</i> ≥ 6 and finite-dimensional complex <i>V</i>. More generally, the canonical multiplication map $`\\mathop{\\mathrm{Sym}}\\nolimits ^b(\\mathop{\\mathrm{Sym}}\\nolimits ^aV)\\to\\mathop{\\mathrm{Sym}}\\nolimits ^a(\\mathop{\\mathrm{Sym}}\\nolimits ^bV)`$ is surjective for <i>a</i> ≥ 2 and $`b\\ge a(a-1)`$, giving dimension-independent quadratic stabilization. ([Lean](lean/docs/210.md))\n\n\n\n 215 Canonical $`O(3)`$ continuum limit and exact $`O(4)`$ mass asymptotics. \nFIRST ('The canonical massive continuum limit of the two-dimensional O(3) model', 'preprints/The-canonical-massive-continuum-limit-of-the-two-dimensional-O3-model-October-4-2026/massive-continuum-o3.pdf') \nFAMILY SCOPE **215. Canonical $`O(3)`$ continuum limit and exact $`O(4)`$ mass asymptotics.** Constructs the canonical continuum limit of the two-dimensional nearest-neighbor $`O(3)`$ model: a non-Gaussian local relativistic theory with a unique vacuum and a positive mass gap. For the square-lattice $`O(4)`$ model, determines the exact leading asymptotic of the full transfer gap, $`m_{\\mathrm{lat}}(\\beta)\\sim32e^{\\pi/4-1/2}\\sqrt\\beta\\,e^{-\\pi\\beta}`$. The family also proves exponential spin-correlation decay for two-dimensional nearest-neighbor $`O(n)`$ models with <i>n</i> ≥ 3 at every positive temperature. ([Lean](lean/docs/215.md))\n\n\n\n 230 Exact Hausdorff gauges for SLE. \nFIRST ('An exact Hausdorff gauge for SLE', 'preprints/An-exact-Hausdorff-gauge-for-SLE-September-25-2026/An-exact-Hausdorff-gauge-for-SLE-September-25-2026.pdf') \nFAMILY SCOPE **230. Exact Hausdorff gauges for SLE.** Resolves Schramm’s Hausdorff-measure question for chordal SLE<sub><i>κ</i></sub>, $`0\\lt \\kappa\\lt 8`$. The explicit gauge $`r^d(\\log\\log(1/r))^{(2-d)/2}`$, $`d=1+\\kappa/8`$, gives almost surely positive finite measure to every trace segment $`\\gamma([s,t])`$ with $`0\\lt s\\lt t\\lt \\infty`$, and finite expected measure to the trace in every bounded disk. ([Lean](lean/docs/230.md))\n\n\n\n 245 Weak normalization implies strong normalization in pure type systems. \nFIRST ('Weak and strong normalization in pure type systems', 'preprints/Weak-and-strong-normalization-in-pure-type-systems-September-25-2026/paper.pdf') \nFAMILY SCOPE **245. Weak normalization implies strong normalization in pure type systems.** Proves that weak normalization implies strong normalization for every pure type system: if every legal expression in every valid context has a <i>β</i>-normal form, every <i>β</i>-reduction sequence terminates. This resolves the <i>β</i>-Barendregt–Geuvers–Klop conjecture, including nonfunctional rules and open contexts. ([Lean](lean/docs/245.md))\n\n\n\n 296 The generator problem for finite factors. \nFIRST ('Relative generation and the generator problem for finite factors', 'preprints/Relative-generation-and-the-generator-problem-for-finite-factors-September-23-2026/paper.pdf') \nFAMILY SCOPE **296. The generator problem for finite factors.** Proves that every type II<sub>1</sub> factor with separable predual is generated by a single operator, equivalently by two self-adjoint operators, resolving the generator problem. More strongly, for every irreducible inclusion $`P\\subset M`$ of such factors, the unitaries <i>u</i> with $`M=W^*(P,u)`$ form a dense <i>G</i><sub><i>δ</i></sub> subset in the trace 2-norm topology. ([Lean](lean/docs/296.md))\n\n\n\n 351 Scalar curvature and finite-time Ricci-flow singularities. \nFIRST ('A closed Ricci flow with bounded scalar curvature and finite-time curvature blowup', 'preprints/A-closed-Ricci-flow-with-bounded-scalar-curvature-and-finite-time-curvature-blowup-September-24-2026/paper.pdf') \nFAMILY SCOPE **351. Scalar curvature and finite-time Ricci-flow singularities.** Proves that a smooth Ricci flow on a closed four-manifold extends past any finite time at which scalar curvature remains uniformly bounded. A higher-dimensional counterexample has bounded scalar curvature but unbounded full curvature at its finite maximal time, disproving the unrestricted scalar-curvature extension conjecture.\n\n\n\n 369 The hot spots conjecture for simply connected planar domains. \nFIRST ('Strict hot spots and absence of interior critical points on smooth simply connected planar domains', 'preprints/Strict-hot-spots-and-absence-of-interior-critical-points-on-smooth-simply-connected-planar-domains-September-24-2026/main.pdf') \nFAMILY SCOPE **369. The hot spots conjecture for simply connected planar domains.** Proves a strict form of Burdzy's simply connected hot spots conjecture. On every smooth bounded simply connected planar domain, each nonzero eigenfunction for the first positive Neumann eigenvalue has no interior critical point, so all global extrema lie on the boundary. Eigenvalue multiplicity is allowed. ([Lean](lean/docs/369.md))\n\n\n\n 372 Global uniqueness in smooth isotropic elasticity. \nFIRST ('Global Uniqueness for the Smooth Isotropic Elasticity Inverse Problem', 'preprints/Global-Uniqueness-for-the-Smooth-Isotropic-Elasticity-Inverse-Problem-September-24-2026/article.pdf') \nFAMILY SCOPE **372. Global uniqueness in smooth isotropic elasticity.** Proves that full static boundary displacement-to-traction data determine both smooth real Lamé moduli on every bounded connected smooth domain in ℝ<sup>3</sup>, provided <i>μ</i> &gt; 0 and $`3\\lambda+2\\mu\\gt 0`$ on the closure. Neither analyticity, proximity to constant coefficients nor prior knowledge near the boundary is required. ([Lean](lean/docs/372.md))\n\n\n\nYAML related/status\n \nrelated_formalizations:\n  # Imported libraries, ordered by direct import count in project Lean sources.\n  - id: https://github.com/leanprover-community/mathlib4\n    relationship: builds-on\n\n  - id: https://github.com/AlexKontorovich/PrimeNumberTheoremAnd\n    relationship: builds-on\n\n  - id: https://github.com/n-yamaguchi-0729/ClassFieldTheory\n    relationship: builds-on\n\n  - id: https://github.com/CBirkbeck/AINTLIB\n    relationship: builds-on\n\n  - id: https://github.com/math-inc/strongpnt\n    relationship: builds-on\n\n  - id: https://github.com/abenenson/rellich-kondrachov\n    relationship: builds-on\n\n  - id: https://github.com/harfe/fixed-point-theorems-lean4\n    relationship: builds-on\n\n  - id: https://github.com/TauCetiProject/TauCeti\n    relationship: builds-on\n\n  - id: https://github.com/fpvandoorn/carleson\n    relationship: builds-on\n\n  - id: https://github.com/lana-agents/heights\n    relationship: builds-on\n\n  - id: https://github.com/Aaron1011/gromov\n    relationship: builds-on\n\n  - id: https://github.com/lana-agents/iut\n    relationship: builds-on\n\n  - id: https://github.com/alonamaloh/schoenflies-lean\n    relationship: builds-on\n\n  - id: https://github.com/leanprover-community/sphere-eversion\n    relationship: builds-on\n\n  - id: https://github.com/ahhwuhu/zeta_3_irrational\n    relationship: builds-on\n\n  - id: https://github.com/BennyAvelin/AbsorptionCutoff\n    relationship: builds-on\n\n  - id: https://github.com/lana-agents/belyi\n    relationship: builds-on\n\n  - id: https://github.com/PatrickMassot/checkdecls\n    relationship: builds-on\n\n  - id: https://github.com/leanprover/doc-gen4\n    relationship: builds-on\n\n  - id: https://github.com/lana-agents/elliptic-curves\n    relationship: builds-on\n\n  - id: https://github.com/lana-agents/formal-schemes\n    relationship: builds-on\n\n  - id: https://github.com/lana-agents/genl\n    relationship: builds-on\n\n  - id: https://github.com/hanwenzhu/LeanArchitect\n    relationship: builds-on\n\n  - id: https://github.com/alerad/leancert\n    relationship: builds-on\n\n  - id: https://github.com/lana-agents/oka\n    relationship: builds-on\n\n  - id: https://github.com/lana-agents/orbicurve-cores\n    relationship: builds-on\n\n  - id: https://github.com/lana-agents/pi1\n    relationship: builds-on\n\n  - id: https://github.com/b-mehta/PrimeCert\n    relationship: builds-on\n\n  - id: https://github.com/lana-agents/tate-curves-theta\n    relationship: builds-on\n\n  - id: https://github.com/lana-agents/tempered-fundamental-groups\n    relationship: builds-on\n\nstatus:\n  scope: \"Partial progress.\"\n  # Entries use source-qualified declaration names and the files containing them.\n  main_results:\n    - comparator_config: ComparatorChallenges/AbhyankarSathaye.json\n      declaration: OAI.AbhyankarSathaye.exists_noncoordinate_polynomial\n      file: OAI/AlgebraicGeometry/AbhyankarSathaye/Counterexample.lean\n\n    - comparator_config: ComparatorChallenges/AmplitudeDamping.json\n      declaration: OAI.GAD.main\n      file: OAI/InformationTheory/AmplitudeDamping/Main.lean\n\n    - comparator_config: ComparatorChallenges/ArnoldCounterexample.json\n      declaration: OAI.ArnoldCounterexample.main\n      file: OAI/Geometry/Arnold/Main.lean\n\n    - comparator_config: ComparatorChallenges/ArtinCAT0.json\n      declaration: OAI.ArtinCAT0.main\n      file: OAI/GroupTheory/ArtinCAT0/Main.lean\n\n    - comparator_config: ComparatorChallenges/AsymptoticallyMinimalLittlewood.json\n      declaration: OAI.AsymptoticallyMinimalLittlewood.main\n      file: OAI/Analysis/Littlewood/Main.lean\n\n    - comparator_config: ComparatorChallenges/AuslanderReiten.json\n      declaration: OAI.ArExplicit.Statement.main\n      file: OAI/Algebra/AuslanderReiten/All.lean\n\n    - comparator_config: ComparatorChallenges/BackwardIntertwiners.json\n      declaration: OAI.BackwardIntertwiners.direct_algebra_corollary\n      file: OAI/Analysis/BackwardIntertwiners/AlgebraCorollary.lean\n\n    - comparator_config: ComparatorChallenges/BassTrace.json\n      declaration: OAI.BassTrace.RightProjective.bassTraceModules_vanishing_and_support\n      file: OAI/RingTheory/BassTrace/Main.lean\n\n    - comparator_config: ComparatorChallenges/BiholderTransport.json\n"
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        "stdout": "\nrelated_formalizations:\n  # Imported libraries, ordered by direct import count in project Lean sources.\n  - id: https://github.com/leanprover-community/mathlib4\n    relationship: builds-on\n\n  - id: https://github.com/AlexKontorovich/PrimeNumberTheoremAnd\n    relationship: builds-on\n\n  - id: https://github.com/n-yamaguchi-0729/ClassFieldTheory\n    relationship: builds-on\n\n  - id: https://github.com/CBirkbeck/AINTLIB\n    relationship: builds-on\n\n  - id: https://github.com/math-inc/strongpnt\n    relationship: builds-on\n\n  - id: https://github.com/abenenson/rellich-kondrachov\n    relationship: builds-on\n\n  - id: https://github.com/harfe/fixed-point-theorems-lean4\n    relationship: builds-on\n\n  - id: https://github.com/TauCetiProject/TauCeti\n    relationship: builds-on\n\n  - id: https://github.com/fpvandoorn/carleson\n    relationship: builds-on\n\n  - id: https://github.com/lana-agents/heights\n    relationship: builds-on\n\n  - id: https://github.com/Aaron1011/gromov\n    relationship: builds-on\n\n  - id: https://github.com/lana-agents/iut\n    relationship: builds-on\n\n  - id: https://github.com/alonamaloh/schoenflies-lean\n    relationship: builds-on\n\n  - id: https://github.com/leanprover-community/sphere-eversion\n    relationship: builds-on\n\n  - id: https://github.com/ahhwuhu/zeta_3_irrational\n    relationship: builds-on\n\n  - id: https://github.com/BennyAvelin/AbsorptionCutoff\n    relationship: builds-on\n\n  - id: https://github.com/lana-agents/belyi\n    relationship: builds-on\n\n  - id: https://github.com/PatrickMassot/checkdecls\n    relationship: builds-on\n\n  - id: https://github.com/leanprover/doc-gen4\n    relationship: builds-on\n\n  - id: https://github.com/lana-agents/elliptic-curves\n    relationship: builds-on\n\n  - id: https://github.com/lana-agents/formal-schemes\n    relationship: builds-on\n\n  - id: https://github.com/lana-agents/genl\n    relationship: builds-on\n\n  - id: https://github.com/hanwenzhu/LeanArchitect\n    relationship: builds-on\n\n  - id: https://github.com/alerad/leancert\n    relationship: builds-on\n\n  - id: https://github.com/lana-agents/oka\n    relationship: builds-on\n\n  - id: https://github.com/lana-agents/orbicurve-cores\n    relationship: builds-on\n\n  - id: https://github.com/lana-agents/pi1\n    relationship: builds-on\n\n  - id: https://github.com/b-mehta/PrimeCert\n    relationship: builds-on\n\n  - id: https://github.com/lana-agents/tate-curves-theta\n    relationship: builds-on\n\n  - id: https://github.com/lana-agents/tempered-fundamental-groups\n    relationship: builds-on\n\nstatus:\n  scope: \"Partial progress.\"\n  # Entries use source-qualified declaration names and the files containing them.\n  main_results:\n    - comparator_config: ComparatorChallenges/AbhyankarSathaye.json\n      declaration: OAI.AbhyankarSathaye.exists_noncoordinate_polynomial\n      file: OAI/AlgebraicGeometry/AbhyankarSathaye/Counterexample.lean\n\n    - comparator_config: ComparatorChallenges/AmplitudeDamping.json\n      declaration: OAI.GAD.main\n      file: OAI/InformationTheory/AmplitudeDamping/Main.lean\n\n    - comparator_config: ComparatorChallenges/ArnoldCounterexample.json\n      declaration: OAI.ArnoldCounterexample.main\n      file: OAI/Geometry/Arnold/Main.lean\n\n    - comparator_config: ComparatorChallenges/ArtinCAT0.json\n      declaration: OAI.ArtinCAT0.main\n      file: OAI/GroupTheory/ArtinCAT0/Main.lean\n\n    - comparator_config: ComparatorChallenges/AsymptoticallyMinimalLittlewood.json\n      declaration: OAI.AsymptoticallyMinimalLittlewood.main\nunparsed &emsp;[Taming implies compatibility on four-manifolds](preprints/Taming-implies-compatibility-on-four-manifolds-October-6-2026/paper.pdf)\nselected remainder [('164', ('Monochromatic finite sums and products in the positive integers', 'preprints/Monochromatic-finite-sums-and-products-in-the-positive-integers-September-23-2026/paper.pdf'), False), ('192', ('Unbounded Violations of the Square-Root Degree Bound', 'preprints/Unbounded-Violations-of-the-Square-Root-Degree-Bound-September-26-2026/Unbounded-Violations-of-the-Square-Root-Degree-Bound-September-26-2026.pdf'), True), ('210', (\"Foulkes' conjecture for the sixth symmetric power\", 'preprints/Foulkes-Conjecture-for-the-Sixth-Symmetric-Power-September-25-2026/main.pdf'), True), ('215', ('The canonical massive continuum limit of the two-dimensional O(3) model', 'preprints/The-canonical-massive-continuum-limit-of-the-two-dimensional-O3-model-October-4-2026/massive-continuum-o3.pdf'), True), ('230', ('An exact Hausdorff gauge for SLE', 'preprints/An-exact-Hausdorff-gauge-for-SLE-September-25-2026/An-exact-Hausdorff-gauge-for-SLE-September-25-2026.pdf'), True), ('245', ('Weak and strong normalization in pure type systems', 'preprints/Weak-and-strong-normalization-in-pure-type-systems-September-25-2026/paper.pdf'), True), ('296', ('Relative generation and the generator problem for finite factors', 'preprints/Relative-generation-and-the-generator-problem-for-finite-factors-September-23-2026/paper.pdf'), True), ('351', ('A closed Ricci flow with bounded scalar curvature and finite-time curvature blowup', 'preprints/A-closed-Ricci-flow-with-bounded-scalar-curvature-and-finite-time-curvature-blowup-September-24-2026/paper.pdf'), False), ('369', ('Strict hot spots and absence of interior critical points on smooth simply connected planar domains', 'preprints/Strict-hot-spots-and-absence-of-interior-critical-points-on-smooth-simply-connected-planar-domains-September-24-2026/main.pdf'), True), ('372', ('Global Uniqueness for the Smooth Isotropic Elasticity Inverse Problem', 'preprints/Global-Uniqueness-for-the-Smooth-Isotropic-Elasticity-Inverse-Problem-September-24-2026/article.pdf'), True)]\n",
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          "\nrelated_formalizations:\n  # Imported libraries, ordered by direct import count in project Lean sources.\n  - id: https://github.com/leanprover-community/mathlib4\n    relationship: builds-on\n\n  - id: https://github.com/AlexKontorovich/PrimeNumberTheoremAnd\n    relationship: builds-on\n\n  - id: https://github.com/n-yamaguchi-0729/ClassFieldTheory\n    relationship: builds-on\n\n  - id: https://github.com/CBirkbeck/AINTLIB\n    relationship: builds-on\n\n  - id: https://github.com/math-inc/strongpnt\n    relationship: builds-on\n\n  - id: https://github.com/abenenson/rellich-kondrachov\n    relationship: builds-on\n\n  - id: https://github.com/harfe/fixed-point-theorems-lean4\n    relationship: builds-on\n\n  - id: https://github.com/TauCetiProject/TauCeti\n    relationship: builds-on\n\n  - id: https://github.com/fpvandoorn/carleson\n    relationship: builds-on\n\n  - id: https://github.com/lana-agents/heights\n    relationship: builds-on\n\n  - id: https://github.com/Aaron1011/gromov\n    relationship: builds-on\n\n  - id: https://github.com/lana-agents/iut\n    relationship: builds-on\n\n  - id: https://github.com/alonamaloh/schoenflies-lean\n    relationship: builds-on\n\n  - id: https://github.com/leanprover-community/sphere-eversion\n    relationship: builds-on\n\n  - id: https://github.com/ahhwuhu/zeta_3_irrational\n    relationship: builds-on\n\n  - id: https://github.com/BennyAvelin/AbsorptionCutoff\n    relationship: builds-on\n\n  - id: https://github.com/lana-agents/belyi\n    relationship: builds-on\n\n  - id: https://github.com/PatrickMassot/checkdecls\n    relationship: builds-on\n\n  - id: https://github.com/leanprover/doc-gen4\n    relationship: builds-on\n\n  - id: https://github.com/lana-agents/elliptic-curves\n    relationship: builds-on\n\n  - id: https://github.com/lana-agents/formal-schemes\n    relationship: builds-on\n\n  - id: https://github.com/lana-agents/genl\n    relationship: builds-on\n\n  - id: https://github.com/hanwenzhu/LeanArchitect\n    relationship: builds-on\n\n  - id: https://github.com/alerad/leancert\n    relationship: builds-on\n\n  - id: https://github.com/lana-agents/oka\n    relationship: builds-on\n\n  - id: https://github.com/lana-agents/orbicurve-cores\n    relationship: builds-on\n\n  - id: https://github.com/lana-agents/pi1\n    relationship: builds-on\n\n  - id: https://github.com/b-mehta/PrimeCert\n    relationship: builds-on\n\n  - id: https://github.com/lana-agents/tate-curves-theta\n    relationship: builds-on\n\n  - id: https://github.com/lana-agents/tempered-fundamental-groups\n    relationship: builds-on\n\nstatus:\n  scope: \"Partial progress.\"\n  # Entries use source-qualified declaration names and the files containing them.\n  main_results:\n    - comparator_config: ComparatorChallenges/AbhyankarSathaye.json\n      declaration: OAI.AbhyankarSathaye.exists_noncoordinate_polynomial\n      file: OAI/AlgebraicGeometry/AbhyankarSathaye/Counterexample.lean\n\n    - comparator_config: ComparatorChallenges/AmplitudeDamping.json\n      declaration: OAI.GAD.main\n      file: OAI/InformationTheory/AmplitudeDamping/Main.lean\n\n    - comparator_config: ComparatorChallenges/ArnoldCounterexample.json\n      declaration: OAI.ArnoldCounterexample.main\n      file: OAI/Geometry/Arnold/Main.lean\n\n    - comparator_config: ComparatorChallenges/ArtinCAT0.json\n      declaration: OAI.ArtinCAT0.main\n      file: OAI/GroupTheory/ArtinCAT0/Main.lean\n\n    - comparator_config: ComparatorChallenges/AsymptoticallyMinimalLittlewood.json\n      declaration: OAI.AsymptoticallyMinimalLittlewood.main\nunparsed &emsp;[Taming implies compatibility on four-manifolds](preprints/Taming-implies-compatibility-on-four-manifolds-October-6-2026/paper.pdf)\nselected remainder [('164', ('Monochromatic finite sums and products in the positive integers', 'preprints/Monochromatic-finite-sums-and-products-in-the-positive-integers-September-23-2026/paper.pdf'), False), ('192', ('Unbounded Violations of the Square-Root Degree Bound', 'preprints/Unbounded-Violations-of-the-Square-Root-Degree-Bound-September-26-2026/Unbounded-Violations-of-the-Square-Root-Degree-Bound-September-26-2026.pdf'), True), ('210', (\"Foulkes' conjecture for the sixth symmetric power\", 'preprints/Foulkes-Conjecture-for-the-Sixth-Symmetric-Power-September-25-2026/main.pdf'), True), ('215', ('The canonical massive continuum limit of the two-dimensional O(3) model', 'preprints/The-canonical-massive-continuum-limit-of-the-two-dimensional-O3-model-October-4-2026/massive-continuum-o3.pdf'), True), ('230', ('An exact Hausdorff gauge for SLE', 'preprints/An-exact-Hausdorff-gauge-for-SLE-September-25-2026/An-exact-Hausdorff-gauge-for-SLE-September-25-2026.pdf'), True), ('245', ('Weak and strong normalization in pure type systems', 'preprints/Weak-and-strong-normalization-in-pure-type-systems-September-25-2026/paper.pdf'), True), ('296', ('Relative generation and the generator problem for finite factors', 'preprints/Relative-generation-and-the-generator-problem-for-finite-factors-September-23-2026/paper.pdf'), True), ('351', ('A closed Ricci flow with bounded scalar curvature and finite-time curvature blowup', 'preprints/A-closed-Ricci-flow-with-bounded-scalar-curvature-and-finite-time-curvature-blowup-September-24-2026/paper.pdf'), False), ('369', ('Strict hot spots and absence of interior critical points on smooth simply connected planar domains', 'preprints/Strict-hot-spots-and-absence-of-interior-critical-points-on-smooth-simply-connected-planar-domains-September-24-2026/main.pdf'), True), ('372', ('Global Uniqueness for the Smooth Isotropic Elasticity Inverse Problem', 'preprints/Global-Uniqueness-for-the-Smooth-Isotropic-Elasticity-Inverse-Problem-September-24-2026/article.pdf'), True)]\n"
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          "text": "\nrelated_formalizations:\n  # Imported libraries, ordered by direct import count in project Lean sources.\n  - id: https://github.com/leanprover-community/mathlib4\n    relationship: builds-on\n\n  - id: https://github.com/AlexKontorovich/PrimeNumberTheoremAnd\n    relationship: builds-on\n\n  - id: https://github.com/n-yamaguchi-0729/ClassFieldTheory\n    relationship: builds-on\n\n  - id: https://github.com/CBirkbeck/AINTLIB\n    relationship: builds-on\n\n  - id: https://github.com/math-inc/strongpnt\n    relationship: builds-on\n\n  - id: https://github.com/abenenson/rellich-kondrachov\n    relationship: builds-on\n\n  - id: https://github.com/harfe/fixed-point-theorems-lean4\n    relationship: builds-on\n\n  - id: https://github.com/TauCetiProject/TauCeti\n    relationship: builds-on\n\n  - id: https://github.com/fpvandoorn/carleson\n    relationship: builds-on\n\n  - id: https://github.com/lana-agents/heights\n    relationship: builds-on\n\n  - id: https://github.com/Aaron1011/gromov\n    relationship: builds-on\n\n  - 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comparator_config: ComparatorChallenges/ArnoldCounterexample.json\n      declaration: OAI.ArnoldCounterexample.main\n      file: OAI/Geometry/Arnold/Main.lean\n\n    - comparator_config: ComparatorChallenges/ArtinCAT0.json\n      declaration: OAI.ArtinCAT0.main\n      file: OAI/GroupTheory/ArtinCAT0/Main.lean\n\n    - comparator_config: ComparatorChallenges/AsymptoticallyMinimalLittlewood.json\n      declaration: OAI.AsymptoticallyMinimalLittlewood.main\nunparsed &emsp;[Taming implies compatibility on four-manifolds](preprints/Taming-implies-compatibility-on-four-manifolds-October-6-2026/paper.pdf)\nselected remainder [('164', ('Monochromatic finite sums and products in the positive integers', 'preprints/Monochromatic-finite-sums-and-products-in-the-positive-integers-September-23-2026/paper.pdf'), False), ('192', ('Unbounded Violations of the Square-Root Degree Bound', 'preprints/Unbounded-Violations-of-the-Square-Root-Degree-Bound-September-26-2026/Unbounded-Violations-of-the-Square-Root-Degree-Bound-September-26-2026.pdf'), True), ('210', (\"Foulkes' conjecture for the sixth symmetric power\", 'preprints/Foulkes-Conjecture-for-the-Sixth-Symmetric-Power-September-25-2026/main.pdf'), True), ('215', ('The canonical massive continuum limit of the two-dimensional O(3) model', 'preprints/The-canonical-massive-continuum-limit-of-the-two-dimensional-O3-model-October-4-2026/massive-continuum-o3.pdf'), True), ('230', ('An exact Hausdorff gauge for SLE', 'preprints/An-exact-Hausdorff-gauge-for-SLE-September-25-2026/An-exact-Hausdorff-gauge-for-SLE-September-25-2026.pdf'), True), ('245', ('Weak and strong normalization in pure type systems', 'preprints/Weak-and-strong-normalization-in-pure-type-systems-September-25-2026/paper.pdf'), True), ('296', ('Relative generation and the generator problem for finite factors', 'preprints/Relative-generation-and-the-generator-problem-for-finite-factors-September-23-2026/paper.pdf'), True), ('351', ('A closed Ricci flow with bounded scalar curvature and finite-time curvature blowup', 'preprints/A-closed-Ricci-flow-with-bounded-scalar-curvature-and-finite-time-curvature-blowup-September-24-2026/paper.pdf'), False), ('369', ('Strict hot spots and absence of interior critical points on smooth simply connected planar domains', 'preprints/Strict-hot-spots-and-absence-of-interior-critical-points-on-smooth-simply-connected-planar-domains-September-24-2026/main.pdf'), True), ('372', ('Global Uniqueness for the Smooth Isotropic Elasticity Inverse Problem', 'preprints/Global-Uniqueness-for-the-Smooth-Isotropic-Elasticity-Inverse-Problem-September-24-2026/article.pdf'), True)]\n"
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        "stdout": "**342. Donaldson's tamed-to-compatible conjecture.** Proves Donaldson's tamed-to-compatible conjecture: every smooth almost complex structure on a closed four-manifold that is tamed by a symplectic form admits a compatible symplectic form. The almost complex structure stays fixed; the form's cohomology class may change. ([Lean](lean/docs/342.md))\n\n</td>\n</tr></tbody>\n<tbody><tr>\n<td>\n\n&emsp;[Taming implies compatibility on four-manifolds](preprints/Taming-implies-compatibility-on-four-manifolds-October-6-2026/paper.pdf)\n\nWe prove that every smooth almost complex structure on a closed four-manifold which is tamed by a symplectic form is compatible with a symplectic form. This gives a positive solution to Donaldson's tamed-to-compatible conjecture. We also characterize the compatible cone by strict intersection positivity on the closure of the invariant part of the taming cone.\n\n</td>\n</tr></tbody>\n<tbody><tr>\n<td>\n\n\n015 catalogued main: False family papers: 3 files: ['README.md', 'build/adelic.tex', 'build/analytic.tex', 'build/entropy.tex', 'build/geometry.tex', 'build/local.tex', 'build/packets.tex', 'build/paper.tex', 'build/references.bib', 'paper.pdf']\n024 catalogued main: False family papers: 1 files: ['An-asymptotic-formula-for-the-number-of-totients-September-25-2026.pdf', 'README.md', 'build/source/collisions.tex', 'build/source/companion.tex', 'build/source/extraction.tex', 'build/source/introduction.tex', 'build/source/layers.tex', 'build/source/limits.tex', 'build/source/main.tex', 'build/source/references.bib', 'build/source/statements.tex']\n046 catalogued main: False family papers: 2 files: ['README.md', 'build/figures/arrangement.tex', 'build/figures/blowdown.tex', 'build/figures/path-ladder.tex', 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'build/sections/consequences.tex', 'build/sections/frequency.tex', 'build/sections/heat.tex', 'build/sections/introduction.tex', 'build/sections/matrix.tex', 'build/sections/preliminaries.tex', 'build/sections/rough.tex', 'build/sections/smooth.tex']\n113 catalogued main: False family papers: 2 files: ['README.md', 'build/main.tex', 'main.pdf']\n115 catalogued main: False family papers: 2 files: ['README.md', 'build/figures/repair.tex', 'build/macros.tex', 'build/main.tex', 'build/refs/sources.bib', 'build/sections/algorithms.tex', 'build/sections/dense.tex', 'build/sections/introduction.tex', 'build/sections/model-and-graph.tex', 'build/sections/signatures.tex', 'build/sections/transport.tex', 'main.pdf']\n127 catalogued main: True family papers: 1 files: ['.gitignore', 'README.md', 'build/figures/grade-coupling.tex', 'build/main.tex', 'build/references.tex', 'build/sections/01-introduction.tex', 'build/sections/02-coupling.tex', 'build/sections/03-weighted-grading.tex', 'build/sections/04-edge-recovery.tex', 'build/sections/05-parity-coupling.tex', 'build/sections/06-consequences.tex', 'main.pdf']\n148 catalogued main: True family papers: 1 files: ['README.md', 'build/main.tex', 'build/references.bib', 'build/sections/corollaries.tex', 'build/sections/entropy.tex', 'build/sections/introduction.tex', 'build/sections/pairs.tex', 'build/sections/types.tex', 'build/sections/windows.tex', 'main.pdf']\n154 catalogued main: False family papers: 4 files: ['README.md', 'build/figures/square.tex', 'build/main.tex', 'build/references.bib', 'build/sections/channels.tex', 'build/sections/correlations.tex', 'build/sections/inputs.tex', 'build/sections/introduction.tex', 'build/sections/packing.tex', 'build/sections/selected.tex', 'build/sections/square.tex', 'multiple-ergodic-averages.pdf']\n164 catalogued main: False family papers: 1 files: ['README.md', 'build/main.tex', 'build/references.tex', 'build/sections/01_introduction.tex', 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'build/sections/shooting.tex', 'build/sections/sources.tex', 'build/sections/trace.tex', 'build/sections/trajectories.tex', 'build/sections/uniqueness.tex', 'build/sections/volume.tex', 'massive-continuum-o3.pdf']\n230 catalogued main: False family papers: 2 files: ['An-exact-Hausdorff-gauge-for-SLE-September-25-2026.pdf', 'README.md', 'build/figures/batch-timeline.tex', 'build/figures/individual-stops.tex', 'build/macros.tex', 'build/main.tex', 'build/refs/sources.bib', 'build/sections/coefficients.tex', 'build/sections/gauge.tex', 'build/sections/hausdorff.tex', 'build/sections/introduction.tex', 'build/sections/mass.tex', 'build/sections/multipoint.tex', 'build/sections/onepoint.tex', 'build/sections/upper-cover.tex']\n245 catalogued main: False family papers: 1 files: ['.gitignore', 'README.md', 'build/paper.tex', 'build/references.bib', 'build/sections/channels.tex', 'build/sections/erasure.tex', 'build/sections/introduction.tex', 'build/sections/metatheory.tex', 'build/sections/observation-bounds.inc.tex', 'build/sections/observation-free-spaces.inc.tex', 'build/sections/observations.tex', 'build/sections/profiles.tex', 'build/sections/relational.tex', 'build/sections/wrappers.tex', 'paper.pdf']\n296 catalogued main: True family papers: 1 files: ['README.md', 'build/main.tex', 'build/preamble.tex', 'build/references.bib', 'build/sections/consequences.tex', 'build/sections/generation.tex', 'build/sections/independence.tex', 'build/sections/introduction.tex', 'build/sections/perturbation.tex', 'paper.pdf']\n351 catalogued main: False family papers: 3 files: ['README.md', 'build/figures/drift-cylinder.tex', 'build/figures/scales.tex', 'build/paper.tex', 'build/references/bibliography.tex', 'build/sections/01-introduction.tex', 'build/sections/02-construction.tex', 'build/sections/03-geometry.tex', 'build/sections/04-cap.tex', 'build/sections/05-reaction.tex', 'build/sections/06-barriers.tex', 'paper.pdf']\n369 catalogued main: False family 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          "**342. Donaldson's tamed-to-compatible conjecture.** Proves Donaldson's tamed-to-compatible conjecture: every smooth almost complex structure on a closed four-manifold that is tamed by a symplectic form admits a compatible symplectic form. The almost complex structure stays fixed; the form's cohomology class may change. ([Lean](lean/docs/342.md))\n\n</td>\n</tr></tbody>\n<tbody><tr>\n<td>\n\n&emsp;[Taming implies compatibility on four-manifolds](preprints/Taming-implies-compatibility-on-four-manifolds-October-6-2026/paper.pdf)\n\nWe prove that every smooth almost complex structure on a closed four-manifold which is tamed by a symplectic form is compatible with a symplectic form. This gives a positive solution to Donaldson's tamed-to-compatible conjecture. We also characterize the compatible cone by strict intersection positivity on the closure of the invariant part of the taming cone.\n\n</td>\n</tr></tbody>\n<tbody><tr>\n<td>\n\n\n015 catalogued main: False family papers: 3 files: ['README.md', 'build/adelic.tex', 'build/analytic.tex', 'build/entropy.tex', 'build/geometry.tex', 'build/local.tex', 'build/packets.tex', 'build/paper.tex', 'build/references.bib', 'paper.pdf']\n024 catalogued main: False family papers: 1 files: ['An-asymptotic-formula-for-the-number-of-totients-September-25-2026.pdf', 'README.md', 'build/source/collisions.tex', 'build/source/companion.tex', 'build/source/extraction.tex', 'build/source/introduction.tex', 'build/source/layers.tex', 'build/source/limits.tex', 'build/source/main.tex', 'build/source/references.bib', 'build/source/statements.tex']\n046 catalogued main: False family papers: 2 files: ['README.md', 'build/figures/arrangement.tex', 'build/figures/blowdown.tex', 'build/figures/path-ladder.tex', 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          "text": "**342. Donaldson's tamed-to-compatible conjecture.** Proves Donaldson's tamed-to-compatible conjecture: every smooth almost complex structure on a closed four-manifold that is tamed by a symplectic form admits a compatible symplectic form. The almost complex structure stays fixed; the form's cohomology class may change. ([Lean](lean/docs/342.md))\n\n</td>\n</tr></tbody>\n<tbody><tr>\n<td>\n\n&emsp;[Taming implies compatibility on four-manifolds](preprints/Taming-implies-compatibility-on-four-manifolds-October-6-2026/paper.pdf)\n\nWe prove that every smooth almost complex structure on a closed four-manifold which is tamed by a symplectic form is compatible with a symplectic form. This gives a positive solution to Donaldson's tamed-to-compatible conjecture. We also characterize the compatible cone by strict intersection positivity on the closure of the invariant part of the taming cone.\n\n</td>\n</tr></tbody>\n<tbody><tr>\n<td>\n\n\n015 catalogued main: False family papers: 3 files: ['README.md', 'build/adelic.tex', 'build/analytic.tex', 'build/entropy.tex', 'build/geometry.tex', 'build/local.tex', 'build/packets.tex', 'build/paper.tex', 'build/references.bib', 'paper.pdf']\n024 catalogued main: False family papers: 1 files: ['An-asymptotic-formula-for-the-number-of-totients-September-25-2026.pdf', 'README.md', 'build/source/collisions.tex', 'build/source/companion.tex', 'build/source/extraction.tex', 'build/source/introduction.tex', 'build/source/layers.tex', 'build/source/limits.tex', 'build/source/main.tex', 'build/source/references.bib', 'build/source/statements.tex']\n046 catalogued main: False family papers: 2 files: ['README.md', 'build/figures/arrangement.tex', 'build/figures/blowdown.tex', 'build/figures/path-ladder.tex', 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        "stdout": "\"Beyond the Square-Root Exponent for Depth-Three Boolean Circuits\"\n    authors: [\"OpenAI\"]\n    id: ../preprints/Beyond-the-Square-Root-Exponent-for-Depth-Three-Boolean-Circuits-September-23-2026/main.pdf\n    type: article\n\n\"Near-square-root logarithmic integrality gaps for uniform sparsest cut\"\n    authors: [\"OpenAI\"]\n    id: ../preprints/Near-square-root-logarithmic-integrality-gaps-for-uniform-sparsest-cut-September-24-2026/Near-square-root-logarithmic-integrality-gaps-for-uniform-sparsest-cut-September-24-2026.pdf\n    type: article\n\n\"Quadratic stabilization of the canonical Foulkes–Howe map\"\n    authors: [\"OpenAI\"]\n    id: ../preprints/Quadratic-Stabilization-of-the-Canonical-Foulkes-Howe-Map-September-25-2026/paper.pdf\n    type: article\n\n\"The maximal triangular Hilbert transform at the symmetric point\"\n    authors: [\"OpenAI\"]\n    id: ../preprints/The-maximal-triangular-Hilbert-transform-at-the-symmetric-point-September-24-2026/paper.pdf\n    type: article\n\nfamily337 actual **337. Sharp Cartan–Hadamard isoperimetry and rigidity.** Proves generalized Cartan–Hadamard isoperimetry in every dimension: in a complete simply connected manifold with sectional curvature at most <i>κ</i> ≤ 0, every finite-volume finite-perimeter set satisfies the sharp comparison with the equal-volume model ball. Bounded positive-volume equality regions for <i>κ</i> = 0 are Euclidean balls. Also proves sharp Euclidean filling bounds for compactly supported integral <i>n</i>-cycles, <i>n</i> ≥ 2, in arbitrary proper CAT$`(0)`$ spaces. ([Lean](lean/docs/337.md))\n\n</td>\n</tr></tbody>\n<tbody><tr>\n<td>\n\n&emsp;[Generalized Cartan–Hadamard isoperimetry and Euclidean equality rigidity](preprints/Generalized-Cartan-Hadamard-isoperimetry-and-Euclidean-equality-rigidity-September-23-2026/paper.pdf)\n\nWe resolve the generalized Cartan–Hadamard isoperimetric conjecture in every dimension. In a complete simply connected smooth manifold with sectional curvature at most <i>κ</i> ≤ 0, every finite-volume set of finite ambient perimeter has perimeter at least that of the equal-volume ball in curvature <i>κ</i>. For bounded positive-volume sets, equality in the Euclidean comparison holds precisely when the set agrees up to null sets with an open region isometric, with its induced metric, to a round Euclidean ball.\n\n</td>\n</tr></tbody>\n<tbody><tr>\n<td>\n\n&emsp;[Sharp integral fillings in CAT(0) spaces](preprints/Sharp-integral-fillings-in-CAT(0)-spaces-September-23-2026/paper.pdf)\n\nEvery compactly supported integral <i>n</i>-cycle, <i>n</i> ≥ 2, in a proper CAT(0) space bounds a compactly supported integral current with the sharp Euclidean mass bound. The theorem allows arbitrary integer multiplicities and unrestricted ambient dimension. In particular, we prove the Euclidean Cartan–Hadamard isoperimetric conjecture in dimensions at least three.\n\n</td>\n</tr></tbody>\n<tbody><tr>\n<td>\n\n\nselected last statuses [('296', True), ('351', False), ('369', False), ('372', False)]\n",
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          "\"Beyond the Square-Root Exponent for Depth-Three Boolean Circuits\"\n    authors: [\"OpenAI\"]\n    id: ../preprints/Beyond-the-Square-Root-Exponent-for-Depth-Three-Boolean-Circuits-September-23-2026/main.pdf\n    type: article\n\n\"Near-square-root logarithmic integrality gaps for uniform sparsest cut\"\n    authors: [\"OpenAI\"]\n    id: ../preprints/Near-square-root-logarithmic-integrality-gaps-for-uniform-sparsest-cut-September-24-2026/Near-square-root-logarithmic-integrality-gaps-for-uniform-sparsest-cut-September-24-2026.pdf\n    type: article\n\n\"Quadratic stabilization of the canonical Foulkes–Howe map\"\n    authors: [\"OpenAI\"]\n    id: ../preprints/Quadratic-Stabilization-of-the-Canonical-Foulkes-Howe-Map-September-25-2026/paper.pdf\n    type: article\n\n\"The maximal triangular Hilbert transform at the symmetric point\"\n    authors: [\"OpenAI\"]\n    id: ../preprints/The-maximal-triangular-Hilbert-transform-at-the-symmetric-point-September-24-2026/paper.pdf\n    type: article\n\nfamily337 actual **337. Sharp Cartan–Hadamard isoperimetry and rigidity.** Proves generalized Cartan–Hadamard isoperimetry in every dimension: in a complete simply connected manifold with sectional curvature at most <i>κ</i> ≤ 0, every finite-volume finite-perimeter set satisfies the sharp comparison with the equal-volume model ball. Bounded positive-volume equality regions for <i>κ</i> = 0 are Euclidean balls. Also proves sharp Euclidean filling bounds for compactly supported integral <i>n</i>-cycles, <i>n</i> ≥ 2, in arbitrary proper CAT$`(0)`$ spaces. ([Lean](lean/docs/337.md))\n\n</td>\n</tr></tbody>\n<tbody><tr>\n<td>\n\n&emsp;[Generalized Cartan–Hadamard isoperimetry and Euclidean equality rigidity](preprints/Generalized-Cartan-Hadamard-isoperimetry-and-Euclidean-equality-rigidity-September-23-2026/paper.pdf)\n\nWe resolve the generalized Cartan–Hadamard isoperimetric conjecture in every dimension. In a complete simply connected smooth manifold with sectional curvature at most <i>κ</i> ≤ 0, every finite-volume set of finite ambient perimeter has perimeter at least that of the equal-volume ball in curvature <i>κ</i>. For bounded positive-volume sets, equality in the Euclidean comparison holds precisely when the set agrees up to null sets with an open region isometric, with its induced metric, to a round Euclidean ball.\n\n</td>\n</tr></tbody>\n<tbody><tr>\n<td>\n\n&emsp;[Sharp integral fillings in CAT(0) spaces](preprints/Sharp-integral-fillings-in-CAT(0)-spaces-September-23-2026/paper.pdf)\n\nEvery compactly supported integral <i>n</i>-cycle, <i>n</i> ≥ 2, in a proper CAT(0) space bounds a compactly supported integral current with the sharp Euclidean mass bound. The theorem allows arbitrary integer multiplicities and unrestricted ambient dimension. In particular, we prove the Euclidean Cartan–Hadamard isoperimetric conjecture in dimensions at least three.\n\n</td>\n</tr></tbody>\n<tbody><tr>\n<td>\n\n\nselected last statuses [('296', True), ('351', False), ('369', False), ('372', False)]\n"
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          "text": "\"Beyond the Square-Root Exponent for Depth-Three Boolean Circuits\"\n    authors: [\"OpenAI\"]\n    id: ../preprints/Beyond-the-Square-Root-Exponent-for-Depth-Three-Boolean-Circuits-September-23-2026/main.pdf\n    type: article\n\n\"Near-square-root logarithmic integrality gaps for uniform sparsest cut\"\n    authors: [\"OpenAI\"]\n    id: ../preprints/Near-square-root-logarithmic-integrality-gaps-for-uniform-sparsest-cut-September-24-2026/Near-square-root-logarithmic-integrality-gaps-for-uniform-sparsest-cut-September-24-2026.pdf\n    type: article\n\n\"Quadratic stabilization of the canonical Foulkes–Howe map\"\n    authors: [\"OpenAI\"]\n    id: ../preprints/Quadratic-Stabilization-of-the-Canonical-Foulkes-Howe-Map-September-25-2026/paper.pdf\n    type: article\n\n\"The maximal triangular Hilbert transform at the symmetric point\"\n    authors: [\"OpenAI\"]\n    id: ../preprints/The-maximal-triangular-Hilbert-transform-at-the-symmetric-point-September-24-2026/paper.pdf\n    type: article\n\nfamily337 actual **337. Sharp Cartan–Hadamard isoperimetry and rigidity.** Proves generalized Cartan–Hadamard isoperimetry in every dimension: in a complete simply connected manifold with sectional curvature at most <i>κ</i> ≤ 0, every finite-volume finite-perimeter set satisfies the sharp comparison with the equal-volume model ball. Bounded positive-volume equality regions for <i>κ</i> = 0 are Euclidean balls. Also proves sharp Euclidean filling bounds for compactly supported integral <i>n</i>-cycles, <i>n</i> ≥ 2, in arbitrary proper CAT$`(0)`$ spaces. ([Lean](lean/docs/337.md))\n\n</td>\n</tr></tbody>\n<tbody><tr>\n<td>\n\n&emsp;[Generalized Cartan–Hadamard isoperimetry and Euclidean equality rigidity](preprints/Generalized-Cartan-Hadamard-isoperimetry-and-Euclidean-equality-rigidity-September-23-2026/paper.pdf)\n\nWe resolve the generalized Cartan–Hadamard isoperimetric conjecture in every dimension. In a complete simply connected smooth manifold with sectional curvature at most <i>κ</i> ≤ 0, every finite-volume set of finite ambient perimeter has perimeter at least that of the equal-volume ball in curvature <i>κ</i>. For bounded positive-volume sets, equality in the Euclidean comparison holds precisely when the set agrees up to null sets with an open region isometric, with its induced metric, to a round Euclidean ball.\n\n</td>\n</tr></tbody>\n<tbody><tr>\n<td>\n\n&emsp;[Sharp integral fillings in CAT(0) spaces](preprints/Sharp-integral-fillings-in-CAT(0)-spaces-September-23-2026/paper.pdf)\n\nEvery compactly supported integral <i>n</i>-cycle, <i>n</i> ≥ 2, in a proper CAT(0) space bounds a compactly supported integral current with the sharp Euclidean mass bound. The theorem allows arbitrary integer multiplicities and unrestricted ambient dimension. In particular, we prove the Euclidean Cartan–Hadamard isoperimetric conjecture in dimensions at least three.\n\n</td>\n</tr></tbody>\n<tbody><tr>\n<td>\n\n\nselected last statuses [('296', True), ('351', False), ('369', False), ('372', False)]\n"
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        "stdout": "corrected papers 719\nsource count 173 main entries 200\ncomparator names AbhyankarSathaye AffineBernstein AlgorithmicThinTrees AmplitudeDamping ArnoldCounterexample ArtinCAT0 ArtinParabolicIntersections AsymptoticallyMinimalLittlewood AtomicGaussian AuslanderReiten BackwardIntertwiners BalancedBoxRouting BalancedRyser BalancedThreeStack BallPacking BallPackingNecessity BarnetteHamiltonian BassTorsionFree BassTrace BenjaminiSchramm BernoulliNonflatInSeven BiholderTransport BinPackingGap BinaryEditLower BinaryMatching BinarySweep BipartiteCrossing BogomolovPopInjectivity BoltzmannNonuniqueness BooneHigman BorsukNine BoundedRecovery BoundedTreePotentials BoundedTreewidthL1 Brenier Brennan BrennanSharp BufferedIsing C0Absorption C1EntropyCounterexample CHObstruction CKSBondiPenrose CannonGeometricAction CartierChartCompactnessSupport Catalan CayleyPercolation CharacterCriterion CharacterVarietiesAllSeamsSupport ChoicelessPolynomialTime CirculantHadamard ClassicalON 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          "corrected papers 719\nsource count 173 main entries 200\ncomparator names AbhyankarSathaye AffineBernstein AlgorithmicThinTrees AmplitudeDamping ArnoldCounterexample ArtinCAT0 ArtinParabolicIntersections AsymptoticallyMinimalLittlewood AtomicGaussian AuslanderReiten BackwardIntertwiners BalancedBoxRouting BalancedRyser BalancedThreeStack BallPacking BallPackingNecessity BarnetteHamiltonian BassTorsionFree BassTrace BenjaminiSchramm BernoulliNonflatInSeven BiholderTransport BinPackingGap BinaryEditLower BinaryMatching BinarySweep BipartiteCrossing BogomolovPopInjectivity BoltzmannNonuniqueness BooneHigman BorsukNine BoundedRecovery BoundedTreePotentials BoundedTreewidthL1 Brenier Brennan BrennanSharp BufferedIsing C0Absorption C1EntropyCounterexample CHObstruction CKSBondiPenrose CannonGeometricAction CartierChartCompactnessSupport Catalan CayleyPercolation CharacterCriterion CharacterVarietiesAllSeamsSupport ChoicelessPolynomialTime CirculantHadamard ClassicalON CliqueFreeLog CoarseAssembly CoboundaryExpanders CommonBasesFPRAS CommutingDerivations CompactBanach CompleteCrossing CompleteCrouzeix ComplexCancellation Conductivity ConjugatePoints ContingencyTables ContinuousCircleWeight ContinuumCoulombHardness ContinuumTransition CoordinateSweeps CoordinateTrace Cotype CoulombCounterexample CoulombIonization CoulombRadii CourtadeKumar CoveringDensity CriticalPercolation CriticalSK CriticalSKMixing CriticalStripMass CriticalZ3 CrouzeixHilbert CubicTorus CycleCliqueRamsey CycleDecomposition CylinderCovering DaugavetModuli DefocusingNLS DegreeRigidity DeligneDrinfeld DepthFive DepthThree DiamondDistortion DilutedSpin DimensionTenChannel DimensionTenPair DirectCrouzeix DirectedFeedback DirectionalBallisticity DirectionalWalk DirichletSevenEighths DiscreteLipschitzFree DiskMaximal Dixmier DixmierAllDiscrete DoublingHilbert DukePrimeDegree DuttaDomain DyadicAvoidance EditApproximation EditDistance EgyptianFractions EilenbergGanea EinsteinFour ElasticityUniqueness ElementaryPositivity EntangledGames EntropyPhotonNumber ErdosReciprocal EuclideanFiveColor EuclideanRamsey EuclideanRamseyCircle EuclideanRamseyNine EuclideanRamseyQuadratic EuclideanRamseySpherical EuclideanRamseyTransitive EvenBarker ExactFourier ExactInfiniteness ExactQuantumFactoring FKCRT FactorGeneration FalconerAllDimensions FillingCoefficient FiniteCircle FiniteCongruenceGraph FiniteEntropySeparation FiniteFactor FinitisticAsymmetry FixedClauseThreshold ForcedNavierStokesComputation ForestSpace FormulaHitting FoulkesHowe FourRowPermanent FourierLLogL FreeIsing FreeUniformSpanningForest GammaPassage GaussianFiniteEntropy GaussianInformation GaussianMoat GaussianPropeller GaussianReplacement GeneralMahler GeneralizedStarHeight GotsmanLinial GrahamSpherical GrothendieckElementaryExpansion GroupRingDeterminant HadamardCubeFiber HadwigerCounterexample HalvingLines HardSphere HarmonicArtin HarmonicGrowth HeckeSevenEighths HeilbronnTriangle Heisenberg HenonEmden HilbertCrouzeix HoneycombBridgeFiniteness HoneycombBridgeMassSupport HoneycombFreeEnergy HotSpots HyperbolicCones HyperbolicObstruction HyperinvariantSubspaces IndependentProducts IndependentSets IndividualStrongInfiniteness InfiniteMatroid InfiniteMatroidCorollaries InterpolatedFactors InvariantIsing IrrationalRotation IrrationalTriangleBilliard IsingFiniteness IsometricImmersion Jacobsthal JacobsthalImproved JointDickman KLInvariance KMedianRecovery KMedianRefinedRecovery KMedianThreshold KServer KadisonRingrose KadisonSimilarity KahlerSplitting KaplanskyDirectFiniteness KaplanskyFinitelyPresented KaplanskyQuasitrace KaplanskyStableFiniteness Kervaire KoebeCircleDomains Laughlin LaughlinFock LaughlinGap LaughlinPlanar LiebThirring LipschitzEquivalence LipschitzHilbert ListHadwiger LittleFinitistic LittlewoodFiniteFlatness LogBrunnMinkowski LogConcaveQuery LogKodairaFiberNegative LogspaceEquality MUBSix MahlerConjecture MarkovType MassAction MatchingAffineLift MatchingEntropy MatchingEntropyBounds MatchingFPRAS MatchingPSD MatrixFields MatrixMultiplication MatrixRemoval MatroidProphet MatroidSecretary MaximalSeshadriConstants MemoryPrecision MetricEntropyDuality MetricKMedian MidpointLenses MinUncut Nagata Naimark NavierStokesAlternating NavierStokesVelocity NodalLength NoiselessRegression NuclearUltrapower OccupiedOverlap OddKaplansky OneTapeSpace OneWayLiveness OptimalMaxCut OrdinaryElliott OrdinaryTwoPointCorrelations OstmannComplete OstmannPrimes ParityLifts PartialPermutation PartitionConsistency PartitionPrinciple PattersonFirstMoment PerceptronFreeEnergy PerfectCompleteness PeriodicGroup PeriodicTilingThree PermanentCubic PettyProjectionVolume PiExponent PinchedKahler PinnedDistances PlanarAndersonSpectrum PlanarFalconer PlanarFirstPassage PlanarL1 PlanarPacking PlanarUnitDistances PlaneColoring PolycyclicRecognition PosteriorReplicas PowerFreeValues PrimeGaps ProductBrown ProjectionCounterexample ProjectionMoments ProjectionVolume QACParity QuadricBundles QuantitativeVanDerWaerden QuasiRiemannHypothesis QuinticLienard RadialDensityInterval RadialTransition RamseyFive RandomizedMeanPayoff RationalHitting RealL1Renorming RecursivePotentials ReflexiveFixedPoints RegularParity RelativeGeneration RieszQuantitative Rokhlin RuledCovering RyserCovering RyserOddExtensions SATComputability SATSharpness SATVariance SKBarriers SKFullSupport SKHighTemperature SKRatio SKValue SLELowerPositivity Saxl SecondKahnKalai SelfSimilar SelfSimilarCorollaries SensitivitySeparation SeparableQuotientNegative SeymourSecondNeighborhood SharpCAT0Filling SharpLogRamsey SharpThreshold ShortEgyptianFractions SidorenkoCounterexample SiegelZeros SignedFiniteBand SignedSweepMoment SimpleAmenable SimpleOvergroups SingleFold SingleLatticeCovering SingletonLoopMatching SmoothInitialForm SmoothObstruction SmoothYau SnakyCertificate SnakyConditional SnakyTwentyOne SoftChannel204 SolenoidalSheetPrograms SphericalField SpinAngle SplitTangentIntegrability SquareDifference SquareRootDegree StableCoordinateFour StandardMapComponents StandardMapEntropy StandardMapLyapunov SteinitzBergstrom StrictMeans StrongKadisonKastler StrongThinTree StronglyRayleighDPP StructuralCrouzeix SubpolynomialLp SubsphereCurrent Superstring SurfaceConeCandidateSupport SurfaceImmersion SwitchChain SymmetricDomains SymmetricMahlerEquality SymmetricPolar Tachikawa TalagrandDiscreteConvexity TalagrandExpectationThreshold TamingCompatibility ThomasonModelStructures ThompsonNonamenability ThorpFirstReciprocal ThorpRemaining ThorpRouting ThorpWeightedCompatibility ThreeMachine ThreeStateSupercritical ThreeStateTreeClauses ThreeTorus TingleySphereIsometry TorsionFreeHyperbolic TorsionFreeZeroDivisors TotientAsymptotic TotientCompanionZero TraceIdealTransportSupport TraceReconstruction TreeEdit TriangleFace TriangleRemoval TriangularCovering TriangularEnergy TriangularGaussian TriangularHilbert TruffetCounterexample TwoWayComplementation TwoWayDeterminization TypeSystemNormalization UniformCommutator UniformFourier UniformGamma UniformKServer UniformSparsestCut UniqueGamesTheorem UniversalFInfinity UniversalTensorSquares VariableWL VelocityHemisphere VertexAlgebraNet VertexCover VlasovMaxwell WLIdentification WeakHessian WeakMTWGlobalSupport WeightedSweepMoments WeisfeilerLeman WitnessedChoice YauCounterexample\nsource relevant titles ['\"A superquadratic separation between sensitivity and block sensitivity\"', '\"Average sensitivity of polynomial threshold functions\"', '\"Entropy and Face Dimension of the Perfect-Matching Polytope\"', '\"Relative generation and the generator problem for finite factors\"', '\"The entropy-rate dimension formula for self-similar measures on the line\"']\n"
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          "text": "corrected papers 719\nsource count 173 main entries 200\ncomparator names AbhyankarSathaye AffineBernstein AlgorithmicThinTrees AmplitudeDamping ArnoldCounterexample ArtinCAT0 ArtinParabolicIntersections AsymptoticallyMinimalLittlewood AtomicGaussian AuslanderReiten BackwardIntertwiners BalancedBoxRouting BalancedRyser BalancedThreeStack BallPacking BallPackingNecessity BarnetteHamiltonian BassTorsionFree BassTrace BenjaminiSchramm BernoulliNonflatInSeven BiholderTransport BinPackingGap BinaryEditLower BinaryMatching BinarySweep BipartiteCrossing BogomolovPopInjectivity BoltzmannNonuniqueness BooneHigman BorsukNine BoundedRecovery BoundedTreePotentials BoundedTreewidthL1 Brenier Brennan BrennanSharp BufferedIsing C0Absorption C1EntropyCounterexample CHObstruction CKSBondiPenrose CannonGeometricAction CartierChartCompactnessSupport Catalan CayleyPercolation CharacterCriterion CharacterVarietiesAllSeamsSupport ChoicelessPolynomialTime CirculantHadamard ClassicalON CliqueFreeLog CoarseAssembly CoboundaryExpanders CommonBasesFPRAS CommutingDerivations CompactBanach CompleteCrossing CompleteCrouzeix ComplexCancellation Conductivity ConjugatePoints ContingencyTables ContinuousCircleWeight ContinuumCoulombHardness ContinuumTransition CoordinateSweeps CoordinateTrace Cotype CoulombCounterexample CoulombIonization CoulombRadii CourtadeKumar CoveringDensity CriticalPercolation CriticalSK CriticalSKMixing CriticalStripMass CriticalZ3 CrouzeixHilbert CubicTorus CycleCliqueRamsey CycleDecomposition CylinderCovering DaugavetModuli DefocusingNLS DegreeRigidity DeligneDrinfeld DepthFive DepthThree DiamondDistortion DilutedSpin DimensionTenChannel DimensionTenPair DirectCrouzeix DirectedFeedback DirectionalBallisticity DirectionalWalk DirichletSevenEighths DiscreteLipschitzFree DiskMaximal Dixmier DixmierAllDiscrete DoublingHilbert DukePrimeDegree DuttaDomain DyadicAvoidance EditApproximation EditDistance EgyptianFractions EilenbergGanea EinsteinFour ElasticityUniqueness ElementaryPositivity EntangledGames EntropyPhotonNumber ErdosReciprocal EuclideanFiveColor EuclideanRamsey EuclideanRamseyCircle EuclideanRamseyNine EuclideanRamseyQuadratic EuclideanRamseySpherical EuclideanRamseyTransitive EvenBarker ExactFourier ExactInfiniteness ExactQuantumFactoring FKCRT FactorGeneration FalconerAllDimensions FillingCoefficient FiniteCircle FiniteCongruenceGraph FiniteEntropySeparation FiniteFactor FinitisticAsymmetry FixedClauseThreshold ForcedNavierStokesComputation ForestSpace FormulaHitting FoulkesHowe FourRowPermanent FourierLLogL FreeIsing FreeUniformSpanningForest GammaPassage GaussianFiniteEntropy GaussianInformation GaussianMoat GaussianPropeller GaussianReplacement GeneralMahler GeneralizedStarHeight GotsmanLinial GrahamSpherical GrothendieckElementaryExpansion GroupRingDeterminant HadamardCubeFiber HadwigerCounterexample HalvingLines HardSphere HarmonicArtin HarmonicGrowth HeckeSevenEighths HeilbronnTriangle Heisenberg HenonEmden HilbertCrouzeix HoneycombBridgeFiniteness HoneycombBridgeMassSupport HoneycombFreeEnergy HotSpots HyperbolicCones HyperbolicObstruction HyperinvariantSubspaces IndependentProducts IndependentSets IndividualStrongInfiniteness InfiniteMatroid InfiniteMatroidCorollaries InterpolatedFactors InvariantIsing IrrationalRotation IrrationalTriangleBilliard IsingFiniteness IsometricImmersion Jacobsthal JacobsthalImproved JointDickman KLInvariance KMedianRecovery KMedianRefinedRecovery KMedianThreshold KServer KadisonRingrose KadisonSimilarity KahlerSplitting KaplanskyDirectFiniteness KaplanskyFinitelyPresented KaplanskyQuasitrace KaplanskyStableFiniteness Kervaire KoebeCircleDomains Laughlin LaughlinFock LaughlinGap LaughlinPlanar LiebThirring LipschitzEquivalence LipschitzHilbert ListHadwiger LittleFinitistic LittlewoodFiniteFlatness LogBrunnMinkowski LogConcaveQuery LogKodairaFiberNegative LogspaceEquality MUBSix MahlerConjecture MarkovType MassAction MatchingAffineLift MatchingEntropy MatchingEntropyBounds MatchingFPRAS MatchingPSD MatrixFields MatrixMultiplication MatrixRemoval MatroidProphet MatroidSecretary MaximalSeshadriConstants MemoryPrecision MetricEntropyDuality MetricKMedian MidpointLenses MinUncut Nagata Naimark NavierStokesAlternating NavierStokesVelocity NodalLength NoiselessRegression NuclearUltrapower OccupiedOverlap OddKaplansky OneTapeSpace OneWayLiveness OptimalMaxCut OrdinaryElliott OrdinaryTwoPointCorrelations OstmannComplete OstmannPrimes ParityLifts PartialPermutation PartitionConsistency PartitionPrinciple PattersonFirstMoment PerceptronFreeEnergy PerfectCompleteness PeriodicGroup PeriodicTilingThree PermanentCubic PettyProjectionVolume PiExponent PinchedKahler PinnedDistances PlanarAndersonSpectrum PlanarFalconer PlanarFirstPassage PlanarL1 PlanarPacking PlanarUnitDistances PlaneColoring PolycyclicRecognition PosteriorReplicas PowerFreeValues PrimeGaps ProductBrown ProjectionCounterexample ProjectionMoments ProjectionVolume QACParity QuadricBundles QuantitativeVanDerWaerden QuasiRiemannHypothesis QuinticLienard RadialDensityInterval RadialTransition RamseyFive RandomizedMeanPayoff RationalHitting RealL1Renorming RecursivePotentials ReflexiveFixedPoints RegularParity RelativeGeneration RieszQuantitative Rokhlin RuledCovering RyserCovering RyserOddExtensions SATComputability SATSharpness SATVariance SKBarriers SKFullSupport SKHighTemperature SKRatio SKValue SLELowerPositivity Saxl SecondKahnKalai SelfSimilar SelfSimilarCorollaries SensitivitySeparation SeparableQuotientNegative SeymourSecondNeighborhood SharpCAT0Filling SharpLogRamsey SharpThreshold ShortEgyptianFractions SidorenkoCounterexample SiegelZeros SignedFiniteBand SignedSweepMoment SimpleAmenable SimpleOvergroups SingleFold SingleLatticeCovering SingletonLoopMatching SmoothInitialForm SmoothObstruction SmoothYau SnakyCertificate SnakyConditional SnakyTwentyOne SoftChannel204 SolenoidalSheetPrograms SphericalField SpinAngle SplitTangentIntegrability SquareDifference SquareRootDegree StableCoordinateFour StandardMapComponents StandardMapEntropy StandardMapLyapunov SteinitzBergstrom StrictMeans StrongKadisonKastler StrongThinTree StronglyRayleighDPP StructuralCrouzeix SubpolynomialLp SubsphereCurrent Superstring SurfaceConeCandidateSupport SurfaceImmersion SwitchChain SymmetricDomains SymmetricMahlerEquality SymmetricPolar Tachikawa TalagrandDiscreteConvexity TalagrandExpectationThreshold TamingCompatibility ThomasonModelStructures ThompsonNonamenability ThorpFirstReciprocal ThorpRemaining ThorpRouting ThorpWeightedCompatibility ThreeMachine ThreeStateSupercritical ThreeStateTreeClauses ThreeTorus TingleySphereIsometry TorsionFreeHyperbolic TorsionFreeZeroDivisors TotientAsymptotic TotientCompanionZero TraceIdealTransportSupport TraceReconstruction TreeEdit TriangleFace TriangleRemoval TriangularCovering TriangularEnergy TriangularGaussian TriangularHilbert TruffetCounterexample TwoWayComplementation TwoWayDeterminization TypeSystemNormalization UniformCommutator UniformFourier UniformGamma UniformKServer UniformSparsestCut UniqueGamesTheorem UniversalFInfinity UniversalTensorSquares VariableWL VelocityHemisphere VertexAlgebraNet VertexCover VlasovMaxwell WLIdentification WeakHessian WeakMTWGlobalSupport WeightedSweepMoments WeisfeilerLeman WitnessedChoice YauCounterexample\nsource relevant titles ['\"A superquadratic separation between sensitivity and block sensitivity\"', '\"Average sensitivity of polynomial threshold functions\"', '\"Entropy and Face Dimension of the Perfect-Matching Polytope\"', '\"Relative generation and the generator problem for finite factors\"', '\"The entropy-rate dimension formula for self-similar measures on the line\"']\n"
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For any updated papers, previously published editions remain accessible through the version notes in their README.\n\n## October 7, 2026\n\n**Withdrawals**\n\nIn “Algebraicity of Weil classes on split abelian eightfolds” a sign error invalidates a stabilization-trace cancellation argument and the construction used by two dependent papers. As a result, we have withdrawn the following three manuscripts:\n\n- Algebraicity of Weil classes on split abelian eightfolds\n- Algebraicity of Kuga–Satake Correspondences for K3 Surfaces\n- The rational Hodge conjecture for products of K3 surfaces\n\nThe withdrawn papers now carry notices explaining the gap and linking to the archived manuscripts.\n\n**Fixes**\n\nWe have revised 14 other manuscripts with proof repairs, corrected statements, clearer hypotheses and dependencies, and one correction to an obsolete citation. The changes include:\n\n- Lipschitz heights and Ashkin–Teller currents (4 manuscripts): repaired crossing, boundary-attachment, conditioning and convergence arguments, including additional work on the real-Lipschitz interface proof.\n- Kähler minimal model programs and abundance (6 manuscripts): expanded positivity and contraction arguments and clarified which results are used as inputs, with their required hypotheses.\n- Taming and hypersymplectic deformation (2 manuscripts): corrected the cone-equality claim in Taming implies compatibility, added a strict-inclusion example, and removed an unnecessary cone-comparison dependency from the hypersymplectic paper.\n- Incompressible Box Transport and Finite Computation: revised the torus-projection and common-clock estimates.\n- Exact Birch–Swinnerton-Dyer Formula from Low Selmer Corank: removed an obsolete introductory citation to a removed supporting manuscript.\n\nAlso, as a consequence of these fixes we updated 13 additional manuscripts to cite the revised editions of companion papers. These changes update references and version dates.\n\n**Additional Formalizations**\n\nWe have added an additional 6 formalizations and 5 other additions covering supporting results. This brings the total percentage of top-line results formalized to 300 / 719 = ~42%.\n\n",
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For any updated papers, previously published editions remain accessible through the version notes in their README.\n\n## October 7, 2026\n\n**Withdrawals**\n\nIn “Algebraicity of Weil classes on split abelian eightfolds” a sign error invalidates a stabilization-trace cancellation argument and the construction used by two dependent papers. As a result, we have withdrawn the following three manuscripts:\n\n- Algebraicity of Weil classes on split abelian eightfolds\n- Algebraicity of Kuga–Satake Correspondences for K3 Surfaces\n- The rational Hodge conjecture for products of K3 surfaces\n\nThe withdrawn papers now carry notices explaining the gap and linking to the archived manuscripts.\n\n**Fixes**\n\nWe have revised 14 other manuscripts with proof repairs, corrected statements, clearer hypotheses and dependencies, and one correction to an obsolete citation. The changes include:\n\n- Lipschitz heights and Ashkin–Teller currents (4 manuscripts): repaired crossing, boundary-attachment, conditioning and convergence arguments, including additional work on the real-Lipschitz interface proof.\n- Kähler minimal model programs and abundance (6 manuscripts): expanded positivity and contraction arguments and clarified which results are used as inputs, with their required hypotheses.\n- Taming and hypersymplectic deformation (2 manuscripts): corrected the cone-equality claim in Taming implies compatibility, added a strict-inclusion example, and removed an unnecessary cone-comparison dependency from the hypersymplectic paper.\n- Incompressible Box Transport and Finite Computation: revised the torus-projection and common-clock estimates.\n- Exact Birch–Swinnerton-Dyer Formula from Low Selmer Corank: removed an obsolete introductory citation to a removed supporting manuscript.\n\nAlso, as a consequence of these fixes we updated 13 additional manuscripts to cite the revised editions of companion papers. These changes update references and version dates.\n\n**Additional Formalizations**\n\nWe have added an additional 6 formalizations and 5 other additions covering supporting results. This brings the total percentage of top-line results formalized to 300 / 719 = ~42%.\n\n"
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Boolean functions violate the square-root degree bound by arbitrary factors.** Disproves the proposed square-root bound relating a Boolean function's linear Fourier coefficients to its polynomial degree. For every <i>C</i> &gt; 0, there is a sign-valued Boolean function <i>f</i> with $`\\sum_i\\widehat f(\\{i\\})\\gt C\\sqrt{\\deg(f)}`$. Thus its total signed correlation with individual input bits can exceed the proposed bound by an arbitrary factor. ([Lean](lean/docs/192.md))\n\n\n215 **215. Canonical $`O(3)`$ continuum limit and exact $`O(4)`$ mass asymptotics.** Constructs the canonical continuum limit of the two-dimensional nearest-neighbor $`O(3)`$ model: a non-Gaussian local relativistic theory with a unique vacuum and a positive mass gap. For the square-lattice $`O(4)`$ model, determines the exact leading asymptotic of the full transfer gap, $`m_{\\mathrm{lat}}(\\beta)\\sim32e^{\\pi/4-1/2}\\sqrt\\beta\\,e^{-\\pi\\beta}`$. 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This resolves the <i>β</i>-Barendregt–Geuvers–Klop conjecture, including nonfunctional rules and open contexts. ([Lean](lean/docs/245.md))\n\n\n296 **296. The generator problem for finite factors.** Proves that every type II<sub>1</sub> factor with separable predual is generated by a single operator, equivalently by two self-adjoint operators, resolving the generator problem. More strongly, for every irreducible inclusion $`P\\subset M`$ of such factors, the unitaries <i>u</i> with $`M=W^*(P,u)`$ form a dense <i>G</i><sub><i>δ</i></sub> subset in the trace 2-norm topology. ([Lean](lean/docs/296.md))\n\n\n369 **369. The hot spots conjecture for simply connected planar domains.** Proves a strict form of Burdzy's simply connected hot spots conjecture. On every smooth bounded simply connected planar domain, each nonzero eigenfunction for the first positive Neumann eigenvalue has no interior critical point, so all global extrema lie on the boundary. Eigenvalue multiplicity is allowed. ([Lean](lean/docs/369.md))\n\n\n372 **372. Global uniqueness in smooth isotropic elasticity.** Proves that full static boundary displacement-to-traction data determine both smooth real Lamé moduli on every bounded connected smooth domain in ℝ<sup>3</sup>, provided <i>μ</i> &gt; 0 and $`3\\lambda+2\\mu\\gt 0`$ on the closure. Neither analyticity, proximity to constant coefficients nor prior knowledge near the boundary is required. 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Boolean functions violate the square-root degree bound by arbitrary factors.** Disproves the proposed square-root bound relating a Boolean function's linear Fourier coefficients to its polynomial degree. For every <i>C</i> &gt; 0, there is a sign-valued Boolean function <i>f</i> with $`\\sum_i\\widehat f(\\{i\\})\\gt C\\sqrt{\\deg(f)}`$. Thus its total signed correlation with individual input bits can exceed the proposed bound by an arbitrary factor. ([Lean](lean/docs/192.md))\n\n\n215 **215. Canonical $`O(3)`$ continuum limit and exact $`O(4)`$ mass asymptotics.** Constructs the canonical continuum limit of the two-dimensional nearest-neighbor $`O(3)`$ model: a non-Gaussian local relativistic theory with a unique vacuum and a positive mass gap. For the square-lattice $`O(4)`$ model, determines the exact leading asymptotic of the full transfer gap, $`m_{\\mathrm{lat}}(\\beta)\\sim32e^{\\pi/4-1/2}\\sqrt\\beta\\,e^{-\\pi\\beta}`$. The family also proves exponential spin-correlation decay for two-dimensional nearest-neighbor $`O(n)`$ models with <i>n</i> ≥ 3 at every positive temperature. ([Lean](lean/docs/215.md))\n\n\n230 **230. Exact Hausdorff gauges for SLE.** Resolves Schramm’s Hausdorff-measure question for chordal SLE<sub><i>κ</i></sub>, $`0\\lt \\kappa\\lt 8`$. The explicit gauge $`r^d(\\log\\log(1/r))^{(2-d)/2}`$, $`d=1+\\kappa/8`$, gives almost surely positive finite measure to every trace segment $`\\gamma([s,t])`$ with $`0\\lt s\\lt t\\lt \\infty`$, and finite expected measure to the trace in every bounded disk. ([Lean](lean/docs/230.md))\n\n\n245 **245. Weak normalization implies strong normalization in pure type systems.** Proves that weak normalization implies strong normalization for every pure type system: if every legal expression in every valid context has a <i>β</i>-normal form, every <i>β</i>-reduction sequence terminates. This resolves the <i>β</i>-Barendregt–Geuvers–Klop conjecture, including nonfunctional rules and open contexts. ([Lean](lean/docs/245.md))\n\n\n296 **296. The generator problem for finite factors.** Proves that every type II<sub>1</sub> factor with separable predual is generated by a single operator, equivalently by two self-adjoint operators, resolving the generator problem. More strongly, for every irreducible inclusion $`P\\subset M`$ of such factors, the unitaries <i>u</i> with $`M=W^*(P,u)`$ form a dense <i>G</i><sub><i>δ</i></sub> subset in the trace 2-norm topology. ([Lean](lean/docs/296.md))\n\n\n369 **369. The hot spots conjecture for simply connected planar domains.** Proves a strict form of Burdzy's simply connected hot spots conjecture. On every smooth bounded simply connected planar domain, each nonzero eigenfunction for the first positive Neumann eigenvalue has no interior critical point, so all global extrema lie on the boundary. Eigenvalue multiplicity is allowed. ([Lean](lean/docs/369.md))\n\n\n372 **372. Global uniqueness in smooth isotropic elasticity.** Proves that full static boundary displacement-to-traction data determine both smooth real Lamé moduli on every bounded connected smooth domain in ℝ<sup>3</sup>, provided <i>μ</i> &gt; 0 and $`3\\lambda+2\\mu\\gt 0`$ on the closure. Neither analyticity, proximity to constant coefficients nor prior knowledge near the boundary is required. 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Boolean functions violate the square-root degree bound by arbitrary factors.** Disproves the proposed square-root bound relating a Boolean function's linear Fourier coefficients to its polynomial degree. For every <i>C</i> &gt; 0, there is a sign-valued Boolean function <i>f</i> with $`\\sum_i\\widehat f(\\{i\\})\\gt C\\sqrt{\\deg(f)}`$. Thus its total signed correlation with individual input bits can exceed the proposed bound by an arbitrary factor. ([Lean](lean/docs/192.md))\n\n\n215 **215. Canonical $`O(3)`$ continuum limit and exact $`O(4)`$ mass asymptotics.** Constructs the canonical continuum limit of the two-dimensional nearest-neighbor $`O(3)`$ model: a non-Gaussian local relativistic theory with a unique vacuum and a positive mass gap. For the square-lattice $`O(4)`$ model, determines the exact leading asymptotic of the full transfer gap, $`m_{\\mathrm{lat}}(\\beta)\\sim32e^{\\pi/4-1/2}\\sqrt\\beta\\,e^{-\\pi\\beta}`$. 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([Lean](lean/docs/369.md))\n\n\n372 **372. Global uniqueness in smooth isotropic elasticity.** Proves that full static boundary displacement-to-traction data determine both smooth real Lamé moduli on every bounded connected smooth domain in ℝ<sup>3</sup>, provided <i>μ</i> &gt; 0 and $`3\\lambda+2\\mu\\gt 0`$ on the closure. Neither analyticity, proximity to constant coefficients nor prior knowledge near the boundary is required. 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      "script": "import csv, sys\nfrom urllib.parse import quote\ncommit=catalogue['commit']; base=f'https://github.com/openai/math/blob/{commit}/'\ncandidates={'015':['DukePrimeDegree'],'024':['TotientAsymptotic','TotientCompanionZero'],'082':['TriangularHilbert'],'113':['MatchingFPRAS','MatchingEntropy'],'115':['ContingencyTables'],'127':['GotsmanLinial'],'148':['SelfSimilar','SelfSimilarCorollaries'],'192':['SquareRootDegree'],'210':['FoulkesHowe'],'215':['ClassicalON'],'230':['SLELowerPositivity'],'245':['TypeSystemNormalization'],'296':['FactorGeneration','RelativeGeneration'],'369':['HotSpots'],'372':['ElasticityUniqueness']}\nrows=[]\nfor rank,f in enumerate(chosen,1):\n title,pdf=f['papers'][0];folder=pdf.rsplit('/',1)[0];prefix=folder.removeprefix('preprints/')+'/'\n files=[{'path':'preprints/'+q,**{k:v for k,v in r.items() if k!='path'}} for q,r in pre_tree.items() if q.startswith(prefix)]\n tex=[r['path'] for r in files if r['path'].endswith('.tex')]\n companion=[{'title':t,'pdf':p,'source_line':next(i+1 for i,l in enumerate(y.splitlines()) if l.strip()=='id: ../'+p)} for t,p in f['papers'][1:] if p in source_ids]\n main=pdf in source_ids;lean_link=f'lean/docs/{f[\"id\"]}.md' if '[Lean]' in f['block'] else None\n status='representative_listed' if main else ('companion_listed_representative_unresolved' if companion else ('family_lean_link_scope_unresolved' if lean_link else 'no_family_lean_link_found'))\n configs=[]\n for name in candidates.get(f['id'],[]):\n  assert name+'.json' in comp_tree and name+'.lean' in comp_tree\n  configs.append({'config_path':'lean/ComparatorChallenges/'+name+'.json','challenge_path':'lean/ComparatorChallenges/'+name+'.lean','config_blob_sha':comp_tree[name+'.json']['sha'],'config_bytes':comp_tree[name+'.json']['size'],'challenge_blob_sha':comp_tree[name+'.lean']['sha'],'mapping_basis':'Name-based candidate, not source-verified paper-to-target mapping','catalogue_entries':[e for e in main_entries if e['comparator_config']=='ComparatorChallenges/'+name+'.json']})\n row={'draw_rank':rank,'family_id':f['id'],'family_title':f['title'],'family_manuscript_count':len(f['papers']),'paper_title':title,'pdf_path':pdf,'pdf_url':base+quote(pdf,safe='/'),'pdf_bytes':pre_tree[pdf.removeprefix('preprints/')]['size'],'readme_path':folder+'/README.md','tex_paths':tex,'contents_line':txt[:txt.index(f['block'])].count('\\n')+1,'lean_scope_link':lean_link,'status':status,'representative_source_line':next((i+1 for i,l in enumerate(y.splitlines()) if l.strip()=='id: ../'+pdf),None),'catalogued_companions':companion,'candidate_comparators':configs,'file_inventory':files,'checker_execution':'not_run','proof_content_inspected':False}\n assert any(r['path']==row['readme_path'] for r in files) and tex and row['pdf_bytes']>0\n rows.append(row)\nassert len(rows)==len({r['family_id'] for r in rows})==20\nprint('status counts',dict(collections.Counter(r['status'] for r in rows)))\nprint('written material',len(rows),'family lean links',sum(bool(r['lean_scope_link']) for r in rows),'main catalogue',sum(bool(r['representative_source_line']) for r in rows),'companion catalogue',sum(bool(r['catalogued_companions']) for r in rows))\nprint('locations',[(r['family_id'],r['contents_line'],r['representative_source_line'],len(r['tex_paths'])) for r in sorted(rows,key=lambda r:r['family_id'])])\nprint('python',sys.version.split()[0])",
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        "stdout": "status counts {'no_family_lean_link_found': 5, 'family_lean_link_scope_unresolved': 8, 'representative_listed': 3, 'companion_listed_representative_unresolved': 4}\nwritten material 20 family lean links 15 main catalogue 3 companion catalogue 4\nlocations [('015', 411, None, 7), ('024', 618, None, 8), ('046', 1233, None, 12), ('065', 1629, None, 11), ('082', 2020, None, 10), ('113', 2768, None, 1), ('115', 2822, None, 9), ('127', 3070, 304, 9), ('148', 3541, 724, 7), ('154', 3655, None, 9), ('164', 3861, None, 10), ('192', 4407, None, 9), ('210', 4789, None, 8), ('215', 4950, None, 21), ('230', 5548, None, 12), ('245', 6078, None, 11), ('296', 7366, 579, 7), ('351', 8533, None, 10), ('369', 8929, None, 8), ('372', 8977, None, 7)]\npython 3.12.14\n",
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          "status counts {'no_family_lean_link_found': 5, 'family_lean_link_scope_unresolved': 8, 'representative_listed': 3, 'companion_listed_representative_unresolved': 4}\nwritten material 20 family lean links 15 main catalogue 3 companion catalogue 4\nlocations [('015', 411, None, 7), ('024', 618, None, 8), ('046', 1233, None, 12), ('065', 1629, None, 11), ('082', 2020, None, 10), ('113', 2768, None, 1), ('115', 2822, None, 9), ('127', 3070, 304, 9), ('148', 3541, 724, 7), ('154', 3655, None, 9), ('164', 3861, None, 10), ('192', 4407, None, 9), ('210', 4789, None, 8), ('215', 4950, None, 21), ('230', 5548, None, 12), ('245', 6078, None, 11), ('296', 7366, 579, 7), ('351', 8533, None, 10), ('369', 8929, None, 8), ('372', 8977, None, 7)]\npython 3.12.14\n"
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          "text": "status counts {'no_family_lean_link_found': 5, 'family_lean_link_scope_unresolved': 8, 'representative_listed': 3, 'companion_listed_representative_unresolved': 4}\nwritten material 20 family lean links 15 main catalogue 3 companion catalogue 4\nlocations [('015', 411, None, 7), ('024', 618, None, 8), ('046', 1233, None, 12), ('065', 1629, None, 11), ('082', 2020, None, 10), ('113', 2768, None, 1), ('115', 2822, None, 9), ('127', 3070, 304, 9), ('148', 3541, 724, 7), ('154', 3655, None, 9), ('164', 3861, None, 10), ('192', 4407, None, 9), ('210', 4789, None, 8), ('215', 4950, None, 21), ('230', 5548, None, 12), ('245', 6078, None, 11), ('296', 7366, 579, 7), ('351', 8533, None, 10), ('369', 8929, None, 8), ('372', 8977, None, 7)]\npython 3.12.14\n"
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    {
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      "script": "audit={'repository':catalogue['repository'],'commit':commit,'retrieved_at':catalogue['retrieved_at'],'provenance':catalogue['provenance'],'catalogue_sha256':hashlib.sha256(p.read_bytes()).hexdigest() if isinstance(p,Path) else hashlib.sha256(Path('inputs/math-repository-catalogue-142dce48093d.json').read_bytes()).hexdigest(),'embedded_record_checks':{'count':len(records),'all_sha256_and_size_match':all(hashlib.sha256(r['content'].encode()).hexdigest()==r['sha256'] and len(r['content'].encode())==r['bytes'] for r in records)},'sampling':{'seed':20261008,'algorithm':'random.Random(20261008).sample(families_in_CONTENTS_order,20)','python':sys.version.split()[0],'family_count':len(families),'manuscript_count':sum(len(f['papers']) for f in families),'representative_rule':'First-listed manuscript within each sampled family','draw_order':[r['family_id'] for r in rows],'excluded_absent_family_ids':['045','061','070','123','163']},'counts':{'manuscript_packages_present':20,'family_lean_links':15,'selected_manuscript_explicitly_source_listed':3,'companion_source_listed_selected_unresolved':4,'other_family_lean_scope_unresolved':8,'no_family_lean_link':5,'checker_runs':0},'limitations':['User-supplied snapshot, not independently retrieved from GitHub in this audit.','Trees marked non-truncated but root commit-to-tree relationship not independently authenticated.','PDF, TeX, per-paper README, scope-note, Comparator configuration and proof-source contents unavailable in attachment.','Name-based Comparator candidates are not verified paper-to-theorem mappings.','Formalization catalogue is partial; absence does not establish absence of a main-result formalization.','No mathematical proof validation, Lean build, Comparator run or computation verification performed.'],'sampling_frame':[{'id':f['id'],'title':f['title'],'papers':[{'title':t,'path':q} for t,q in f['papers']]} for f in families],'selected':rows}\nPath('math-proof-audit.json').write_text(json.dumps(audit,ensure_ascii=False,indent=2))\nfields=['draw_rank','family_id','paper_title','pdf_url','pdf_bytes','tex_file_count','readme_url','contents_url','lean_scope_url','status','representative_catalogue_url','catalogued_companion_titles','candidate_comparators','checker_execution','proof_content_inspected']\nwith Path('math-proof-audit.csv').open('w',newline='') as fh:\n writer=csv.DictWriter(fh,fieldnames=fields);writer.writeheader()\n for r in sorted(rows,key=lambda r:r['family_id']):\n  writer.writerow({'draw_rank':r['draw_rank'],'family_id':r['family_id'],'paper_title':r['paper_title'],'pdf_url':r['pdf_url'],'pdf_bytes':r['pdf_bytes'],'tex_file_count':len(r['tex_paths']),'readme_url':base+quote(r['readme_path'],safe='/'),'contents_url':base+'CONTENTS.md#L'+str(r['contents_line']),'lean_scope_url':base+r['lean_scope_link'] if r['lean_scope_link'] else '', 'status':r['status'],'representative_catalogue_url':base+'lean/formalization.yaml#L'+str(r['representative_source_line']) if r['representative_source_line'] else '', 'catalogued_companion_titles':' | '.join(c['title'] for c in r['catalogued_companions']),'candidate_comparators':' | '.join(c['config_path'] for c in r['candidate_comparators']),'checker_execution':'not_run','proof_content_inspected':False})\nprint('saved',[(fn,Path(fn).stat().st_size) for fn in ['math-proof-audit.json','math-proof-audit.csv']])",
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        "stdout": "saved [('math-proof-audit.json', 378379), ('math-proof-audit.csv', 15104)]\n",
        "stderr": "",
        "traceback": "",
        "plain": [
          "saved [('math-proof-audit.json', 378379), ('math-proof-audit.csv', 15104)]\n"
        ]
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        {
          "type": "stream",
          "name": "stdout",
          "text": "saved [('math-proof-audit.json', 378379), ('math-proof-audit.csv', 15104)]\n"
        }
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        "stdout": "hash failures [] commit matches True\nlean files [('lean/ComparatorChallenges/BinaryMatching.lean', 2246), ('lean/ComparatorChallenges/ClassicalON.lean', 1812), ('lean/ComparatorChallenges/ContingencyTables.lean', 8921), ('lean/ComparatorChallenges/DukePrimeDegree.lean', 9385), ('lean/ComparatorChallenges/ElasticityUniqueness.lean', 3365), ('lean/ComparatorChallenges/FactorGeneration.lean', 1622), ('lean/ComparatorChallenges/FoulkesHowe.lean', 1686), ('lean/ComparatorChallenges/GotsmanLinial.lean', 1576), ('lean/ComparatorChallenges/HotSpots.lean', 1869), ('lean/ComparatorChallenges/MatchingEntropy.lean', 2010), ('lean/ComparatorChallenges/MatchingEntropyBounds.lean', 2354), ('lean/ComparatorChallenges/MatchingFPRAS.lean', 4224), ('lean/ComparatorChallenges/RelativeGeneration.lean', 5724), ('lean/ComparatorChallenges/SLELowerPositivity.lean', 2992), ('lean/ComparatorChallenges/SelfSimilar.lean', 2104), ('lean/ComparatorChallenges/SelfSimilarCorollaries.lean', 2481), ('lean/ComparatorChallenges/SingletonLoopMatching.lean', 3926), ('lean/ComparatorChallenges/SquareRootDegree.lean', 1803), ('lean/ComparatorChallenges/TotientAsymptotic.lean', 4930), ('lean/ComparatorChallenges/TotientCompanionZero.lean', 3232), ('lean/ComparatorChallenges/TriangleFace.lean', 6109), ('lean/ComparatorChallenges/TriangularHilbert.lean', 1356), ('lean/ComparatorChallenges/TypeSystemNormalization.lean', 4288), ('lean/OAI/Analysis/FactorGeneration/Main.lean', 2017), ('lean/OAI/Analysis/TriangularHilbert/Main.lean', 2003), ('lean/OAI/Combinatorics/GotsmanLinial/Main.lean', 6267), ('lean/OAI/Combinatorics/PerfectMatching/Main.lean', 9489), ('lean/OAI/Computability/LoopMatching/Main.lean', 1083), ('lean/OAI/Computability/MatchingCount/BinarySolve.lean', 6431), ('lean/OAI/MeasureTheory/SelfSimilar/Main.lean', 2935), ('lean/OAI/Probability/ClassicalON/Main.lean', 1980), ('lean/OAI/RepresentationTheory/FoulkesHowe/Stabilization.lean', 2544)]\n\nSCOPE 015 \n# Torus-packet equidistribution in prime, quartic, and sextic degrees\n\nThe following describes the scope of the Lean formalization related to the following accompanying paper(s):\n\n- [Equidistribution of Prime-Degree Torus Packets with Arbitrary Local Type](../../preprints/Equidistribution-of-Prime-Degree-Torus-Packets-with-Arbitrary-Local-Type-September-24-2026/paper.pdf)\n\n## Scope\n\nThe formalization proves equidistribution of volume-weighted torus packets for totally real number fields of every fixed prime degree at least five. For any sequence of full lattices whose multiplier-order discriminants tend to infinity, the packet measures converge weakly to Haar probability measure and form a tight family, so no mass escapes. Arbitrary local homothety types are allowed, and the fields may vary along the sequence.\n\n## Comparator links\n\n| Result | Comparator statement |\n| --- | --- |\n| Equidistribution of prime-degree torus packets | [DukePrimeDegree.lean](../ComparatorChallenges/DukePrimeDegree.lean) |\n\n\nSCOPE 024 \n# An asymptotic formula for the number of totients\n\nThe following describes the scope of the Lean formalization related to the following accompanying paper(s):\n\n- [An asymptotic formula for the number of totients](../../preprints/An-asymptotic-formula-for-the-number-of-totients-September-25-2026/An-asymptotic-formula-for-the-number-of-totients-September-25-2026.pdf)\n\n## Scope\n\nLet $V(x)$ count the distinct values of Euler's totient function up to $x$. The formalization constructs the paper's explicit positive main term from finite arithmetic approximants and proves that their ratio tends to one. In particular, $V(cx)/V(x)\\to c$ for every fixed $c>0$, answering the Erdős–Hall regular-variation question.\n\nThe formalization also gives asymptotics for totients $v\\le x$ whose least preimage $\\ell(v)=\\min\\{n\\ge1:\\varphi(n)=v\\}$ lies between $kx$ and $(k+1)x$. The associated coefficient is positive under the stated existence condition; when no totient $d$ satisfies $kd<\\ell(d)$, the count and coefficient are identically zero. The cases $k=1,2$ have positive coefficients.\n\n## Comparator links\n\n| Result | Comparator statement |\n| --- | --- |\n| Totient-count asymptotics and regular variation | [TotientAsymptotic.lean](../ComparatorChallenges/TotientAsymptotic.lean) |\n| Exact zero case for the companion totient counts | [TotientCompanionZero.lean](../ComparatorChallenges/TotientCompanionZero.lean) |\n\n\nSCOPE 082 \n# Annular variation and dyadic absolute bounds for the triangular Hilbert transform\n\nThe following describes the scope of the Lean formalization related to the following accompanying paper(s):\n\n- [The maximal triangular Hilbert transform at the symmetric point](../../preprints/The-maximal-triangular-Hilbert-transform-at-the-symmetric-point-September-24-2026/paper.pdf)\n\n## Scope\n\nThe formalized result proves the symmetric maximal estimate for the triangular Hilbert transform: for arbitrary complex $F,G\\in L^3(\\mathbb R^2)$, the $L^{3/2}$ norm of the supremum over all finite hard-truncation intervals is at most $C\\|F\\|_3\\|G\\|_3$ for one absolute constant $C$. It also establishes a common full-measure set on which the truncated integrals are defined and almost-everywhere measurability of the maximal output.\n\n## Comparator links\n\n| Result | Comparator statement |\n| --- | --- |\n| Maximal triangular Hilbert transform bound | [TriangularHilbert.lean](../ComparatorChallenges/TriangularHilbert.lean) |\n\n\nSCOPE 113 \n# Approximate counting and entropy of perfect matchings\n\nThe following describes the scope of the Lean formalization related to the following accompanying paper(s):\n\n- [A Fully Polynomial Randomized Approximation Scheme for Perfect Matchings in General Graphs](../../preprints/A-Fully-Polynomial-Randomized-Approximation-Scheme-for-Perfect-Matchings-in-General-Graphs-September-23-2026/main.pdf)\n- [Entropy and Face Dimension of the Perfect-Matching Polytope](../../preprints/Entropy-and-Face-Dimension-of-the-Perfect-Matching-Polytope-September-23-2026/main.pdf)\n\n## Scope\n\nThe formalization gives a fully polynomial randomized approximation scheme for counting perfect matchings in every finite simple undirected graph. For rational $0<\\varepsilon<1$ and $0<\\delta<1/2$, it returns a nonnegative rational estimate with relative error at most $\\varepsilon$ with probability at least $1-\\delta$. If the graph has no perfect matching, every execution returns zero.\n\nThe algorithm is a fixed finite-alphabet randomized machine. Its worst-case running time is polynomial in the binary input length, $\\varepsilon^{-1}$, and $\\log(\\delta^{-1})$, including on unsuccessful random tapes.\n\nFor feasible edge marginals $x$ of perfect matchings in a loopless multigraph on $2m$ vertices, let $F(x)=-\\sum_e x_e\\log x_e$, $B(x)=-\\sum_e(1-x_e)\\log(1-x_e)$, and let $H(x)$ be the maximum entropy of a matching law with those marginals. The formalization proves $F(x)-(2-2/m)B(x)\\le H(x)\\le F(x)$ for $m\\ge1$, including boundary points of the polytope. It retains the earlier coefficient-eight entropy bound and the deterministic counting approximations with factors $512^n$ for loopless graphs and $2^{18n}$ with singleton loops.\n\nA further statement identifies the exact face obtained by expanding a degree-three vertex into a triangle and shows that minimal faces are carried to minimal faces by the expansion. The paper's sharp global face-dimension bound is not part of that selected statement.\n\n## Comparator links\n\n| Result | Comparator statement |\n| --- | --- |\n| Randomized approximation of the perfect-matching count | [MatchingFPRAS.lean](../ComparatorChallenges/MatchingFPRAS.lean) |\n| Perfect-matching entropy bound | [MatchingEntropy.lean](../ComparatorChallenges/MatchingEntropy.lean) |\n| Deterministic approximate matching count | [BinaryMatching.lean](../ComparatorChallenges/BinaryMatching.lean) |\n| Approximate counting with singleton loops | [SingletonLoopMatching.lean](../ComparatorChallenges/SingletonLoopMatching.lean) |\n| Refined pointwise perfect-matching entropy bounds | [MatchingEntropyBounds.lean](../ComparatorChallenges/MatchingEntropyBounds.lean) |\n| Face correspondence under triangle expansion | [TriangleFace.lean](../ComparatorChallenges/TriangleFace.lean) |\n\n\nSCOPE 115 \n# Sampling and counting contingency tables with arbitrary margins\n\nThe following describes the scope of the Lean formalization related to the following accompanying paper(s):\n\n- [Exact Uniform Sampling of Contingency Tables with Arbitrary Margins](../../preprints/Exact-Uniform-Sampling-of-Contingency-Tables-with-Arbitrary-Margins-September-24-2026/main.pdf)\n- [An FPRAS for Cell-Bounded Contingency Tables](../../preprints/An-FPRAS-for-Cell-Bounded-Contingency-Tables-September-24-2026/main.pdf)\n\n## Scope\n\nThe formalization gives an exact uniform sampler for nonnegative integer contingency tables with arbitrary prescribed row and column sums having equal totals. It terminates almost surely, every halted output is feasible, and each table has exactly the uniform limiting probability. Expected bit complexity is polynomial in the dimensions and the binary length of the margins.\n\nIt also gives a sampler with a fixed polynomial time bound on every execution whose total-variation error is at most $2^{-k}$ for requested precision $k\\ge1$. The same Comparator file includes the companion approximation scheme for counting tables with individual cell bounds.\n\nThe formalization gives a fully polynomial randomized approximation scheme for counting nonnegative integer matrices with prescribed row sums, column sums, and individual entry bounds. Both dimensions may vary, the margins and bounds are binary encoded, and zero entry bounds are allowed. For rational $0<\\varepsilon,\\delta<1$, the algorithm returns a nonnegative estimate with relative error at most $\\varepsilon$ with probability at least $1-\\delta$, and returns zero on every execution when no table exists.\n\nThe running time is polynomial in the encoded input size, $\\varepsilon^{-1}$, and $\\log(\\delta^{-1})$ on every random tape. The same Comparator file also contains the companion sampling results for tables without cell bounds.\n\n## Comparator links\n\n| Result | Comparator statement |\n| --- | --- |\n| Exact and bounded-time sampling of contingency tables | [ContingencyTables.lean](../ComparatorChallenges/ContingencyTables.lean) |\n| Randomized approximate counting of cell-bounded contingency tables | [ContingencyTables.lean](../ComparatorChallenges/ContingencyTables.lean) |\n\n",
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          "hash failures [] commit matches True\nlean files [('lean/ComparatorChallenges/BinaryMatching.lean', 2246), ('lean/ComparatorChallenges/ClassicalON.lean', 1812), ('lean/ComparatorChallenges/ContingencyTables.lean', 8921), ('lean/ComparatorChallenges/DukePrimeDegree.lean', 9385), ('lean/ComparatorChallenges/ElasticityUniqueness.lean', 3365), ('lean/ComparatorChallenges/FactorGeneration.lean', 1622), ('lean/ComparatorChallenges/FoulkesHowe.lean', 1686), ('lean/ComparatorChallenges/GotsmanLinial.lean', 1576), ('lean/ComparatorChallenges/HotSpots.lean', 1869), ('lean/ComparatorChallenges/MatchingEntropy.lean', 2010), ('lean/ComparatorChallenges/MatchingEntropyBounds.lean', 2354), ('lean/ComparatorChallenges/MatchingFPRAS.lean', 4224), ('lean/ComparatorChallenges/RelativeGeneration.lean', 5724), ('lean/ComparatorChallenges/SLELowerPositivity.lean', 2992), ('lean/ComparatorChallenges/SelfSimilar.lean', 2104), ('lean/ComparatorChallenges/SelfSimilarCorollaries.lean', 2481), ('lean/ComparatorChallenges/SingletonLoopMatching.lean', 3926), ('lean/ComparatorChallenges/SquareRootDegree.lean', 1803), ('lean/ComparatorChallenges/TotientAsymptotic.lean', 4930), ('lean/ComparatorChallenges/TotientCompanionZero.lean', 3232), ('lean/ComparatorChallenges/TriangleFace.lean', 6109), ('lean/ComparatorChallenges/TriangularHilbert.lean', 1356), ('lean/ComparatorChallenges/TypeSystemNormalization.lean', 4288), ('lean/OAI/Analysis/FactorGeneration/Main.lean', 2017), ('lean/OAI/Analysis/TriangularHilbert/Main.lean', 2003), ('lean/OAI/Combinatorics/GotsmanLinial/Main.lean', 6267), ('lean/OAI/Combinatorics/PerfectMatching/Main.lean', 9489), ('lean/OAI/Computability/LoopMatching/Main.lean', 1083), ('lean/OAI/Computability/MatchingCount/BinarySolve.lean', 6431), ('lean/OAI/MeasureTheory/SelfSimilar/Main.lean', 2935), ('lean/OAI/Probability/ClassicalON/Main.lean', 1980), ('lean/OAI/RepresentationTheory/FoulkesHowe/Stabilization.lean', 2544)]\n\nSCOPE 015 \n# Torus-packet equidistribution in prime, quartic, and sextic degrees\n\nThe following describes the scope of the Lean formalization related to the following accompanying paper(s):\n\n- [Equidistribution of Prime-Degree Torus Packets with Arbitrary Local Type](../../preprints/Equidistribution-of-Prime-Degree-Torus-Packets-with-Arbitrary-Local-Type-September-24-2026/paper.pdf)\n\n## Scope\n\nThe formalization proves equidistribution of volume-weighted torus packets for totally real number fields of every fixed prime degree at least five. For any sequence of full lattices whose multiplier-order discriminants tend to infinity, the packet measures converge weakly to Haar probability measure and form a tight family, so no mass escapes. Arbitrary local homothety types are allowed, and the fields may vary along the sequence.\n\n## Comparator links\n\n| Result | Comparator statement |\n| --- | --- |\n| Equidistribution of prime-degree torus packets | [DukePrimeDegree.lean](../ComparatorChallenges/DukePrimeDegree.lean) |\n\n\nSCOPE 024 \n# An asymptotic formula for the number of totients\n\nThe following describes the scope of the Lean formalization related to the following accompanying paper(s):\n\n- [An asymptotic formula for the number of totients](../../preprints/An-asymptotic-formula-for-the-number-of-totients-September-25-2026/An-asymptotic-formula-for-the-number-of-totients-September-25-2026.pdf)\n\n## Scope\n\nLet $V(x)$ count the distinct values of Euler's totient function up to $x$. The formalization constructs the paper's explicit positive main term from finite arithmetic approximants and proves that their ratio tends to one. In particular, $V(cx)/V(x)\\to c$ for every fixed $c>0$, answering the Erdős–Hall regular-variation question.\n\nThe formalization also gives asymptotics for totients $v\\le x$ whose least preimage $\\ell(v)=\\min\\{n\\ge1:\\varphi(n)=v\\}$ lies between $kx$ and $(k+1)x$. The associated coefficient is positive under the stated existence condition; when no totient $d$ satisfies $kd<\\ell(d)$, the count and coefficient are identically zero. The cases $k=1,2$ have positive coefficients.\n\n## Comparator links\n\n| Result | Comparator statement |\n| --- | --- |\n| Totient-count asymptotics and regular variation | [TotientAsymptotic.lean](../ComparatorChallenges/TotientAsymptotic.lean) |\n| Exact zero case for the companion totient counts | [TotientCompanionZero.lean](../ComparatorChallenges/TotientCompanionZero.lean) |\n\n\nSCOPE 082 \n# Annular variation and dyadic absolute bounds for the triangular Hilbert transform\n\nThe following describes the scope of the Lean formalization related to the following accompanying paper(s):\n\n- [The maximal triangular Hilbert transform at the symmetric point](../../preprints/The-maximal-triangular-Hilbert-transform-at-the-symmetric-point-September-24-2026/paper.pdf)\n\n## Scope\n\nThe formalized result proves the symmetric maximal estimate for the triangular Hilbert transform: for arbitrary complex $F,G\\in L^3(\\mathbb R^2)$, the $L^{3/2}$ norm of the supremum over all finite hard-truncation intervals is at most $C\\|F\\|_3\\|G\\|_3$ for one absolute constant $C$. It also establishes a common full-measure set on which the truncated integrals are defined and almost-everywhere measurability of the maximal output.\n\n## Comparator links\n\n| Result | Comparator statement |\n| --- | --- |\n| Maximal triangular Hilbert transform bound | [TriangularHilbert.lean](../ComparatorChallenges/TriangularHilbert.lean) |\n\n\nSCOPE 113 \n# Approximate counting and entropy of perfect matchings\n\nThe following describes the scope of the Lean formalization related to the following accompanying paper(s):\n\n- [A Fully Polynomial Randomized Approximation Scheme for Perfect Matchings in General Graphs](../../preprints/A-Fully-Polynomial-Randomized-Approximation-Scheme-for-Perfect-Matchings-in-General-Graphs-September-23-2026/main.pdf)\n- [Entropy and Face Dimension of the Perfect-Matching Polytope](../../preprints/Entropy-and-Face-Dimension-of-the-Perfect-Matching-Polytope-September-23-2026/main.pdf)\n\n## Scope\n\nThe formalization gives a fully polynomial randomized approximation scheme for counting perfect matchings in every finite simple undirected graph. For rational $0<\\varepsilon<1$ and $0<\\delta<1/2$, it returns a nonnegative rational estimate with relative error at most $\\varepsilon$ with probability at least $1-\\delta$. If the graph has no perfect matching, every execution returns zero.\n\nThe algorithm is a fixed finite-alphabet randomized machine. Its worst-case running time is polynomial in the binary input length, $\\varepsilon^{-1}$, and $\\log(\\delta^{-1})$, including on unsuccessful random tapes.\n\nFor feasible edge marginals $x$ of perfect matchings in a loopless multigraph on $2m$ vertices, let $F(x)=-\\sum_e x_e\\log x_e$, $B(x)=-\\sum_e(1-x_e)\\log(1-x_e)$, and let $H(x)$ be the maximum entropy of a matching law with those marginals. The formalization proves $F(x)-(2-2/m)B(x)\\le H(x)\\le F(x)$ for $m\\ge1$, including boundary points of the polytope. It retains the earlier coefficient-eight entropy bound and the deterministic counting approximations with factors $512^n$ for loopless graphs and $2^{18n}$ with singleton loops.\n\nA further statement identifies the exact face obtained by expanding a degree-three vertex into a triangle and shows that minimal faces are carried to minimal faces by the expansion. The paper's sharp global face-dimension bound is not part of that selected statement.\n\n## Comparator links\n\n| Result | Comparator statement |\n| --- | --- |\n| Randomized approximation of the perfect-matching count | [MatchingFPRAS.lean](../ComparatorChallenges/MatchingFPRAS.lean) |\n| Perfect-matching entropy bound | [MatchingEntropy.lean](../ComparatorChallenges/MatchingEntropy.lean) |\n| Deterministic approximate matching count | [BinaryMatching.lean](../ComparatorChallenges/BinaryMatching.lean) |\n| Approximate counting with singleton loops | [SingletonLoopMatching.lean](../ComparatorChallenges/SingletonLoopMatching.lean) |\n| Refined pointwise perfect-matching entropy bounds | [MatchingEntropyBounds.lean](../ComparatorChallenges/MatchingEntropyBounds.lean) |\n| Face correspondence under triangle expansion | [TriangleFace.lean](../ComparatorChallenges/TriangleFace.lean) |\n\n\nSCOPE 115 \n# Sampling and counting contingency tables with arbitrary margins\n\nThe following describes the scope of the Lean formalization related to the following accompanying paper(s):\n\n- [Exact Uniform Sampling of Contingency Tables with Arbitrary Margins](../../preprints/Exact-Uniform-Sampling-of-Contingency-Tables-with-Arbitrary-Margins-September-24-2026/main.pdf)\n- [An FPRAS for Cell-Bounded Contingency Tables](../../preprints/An-FPRAS-for-Cell-Bounded-Contingency-Tables-September-24-2026/main.pdf)\n\n## Scope\n\nThe formalization gives an exact uniform sampler for nonnegative integer contingency tables with arbitrary prescribed row and column sums having equal totals. It terminates almost surely, every halted output is feasible, and each table has exactly the uniform limiting probability. Expected bit complexity is polynomial in the dimensions and the binary length of the margins.\n\nIt also gives a sampler with a fixed polynomial time bound on every execution whose total-variation error is at most $2^{-k}$ for requested precision $k\\ge1$. The same Comparator file includes the companion approximation scheme for counting tables with individual cell bounds.\n\nThe formalization gives a fully polynomial randomized approximation scheme for counting nonnegative integer matrices with prescribed row sums, column sums, and individual entry bounds. Both dimensions may vary, the margins and bounds are binary encoded, and zero entry bounds are allowed. For rational $0<\\varepsilon,\\delta<1$, the algorithm returns a nonnegative estimate with relative error at most $\\varepsilon$ with probability at least $1-\\delta$, and returns zero on every execution when no table exists.\n\nThe running time is polynomial in the encoded input size, $\\varepsilon^{-1}$, and $\\log(\\delta^{-1})$ on every random tape. The same Comparator file also contains the companion sampling results for tables without cell bounds.\n\n## Comparator links\n\n| Result | Comparator statement |\n| --- | --- |\n| Exact and bounded-time sampling of contingency tables | [ContingencyTables.lean](../ComparatorChallenges/ContingencyTables.lean) |\n| Randomized approximate counting of cell-bounded contingency tables | [ContingencyTables.lean](../ComparatorChallenges/ContingencyTables.lean) |\n\n"
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          "text": "hash failures [] commit matches True\nlean files [('lean/ComparatorChallenges/BinaryMatching.lean', 2246), ('lean/ComparatorChallenges/ClassicalON.lean', 1812), ('lean/ComparatorChallenges/ContingencyTables.lean', 8921), ('lean/ComparatorChallenges/DukePrimeDegree.lean', 9385), ('lean/ComparatorChallenges/ElasticityUniqueness.lean', 3365), ('lean/ComparatorChallenges/FactorGeneration.lean', 1622), ('lean/ComparatorChallenges/FoulkesHowe.lean', 1686), ('lean/ComparatorChallenges/GotsmanLinial.lean', 1576), ('lean/ComparatorChallenges/HotSpots.lean', 1869), ('lean/ComparatorChallenges/MatchingEntropy.lean', 2010), ('lean/ComparatorChallenges/MatchingEntropyBounds.lean', 2354), ('lean/ComparatorChallenges/MatchingFPRAS.lean', 4224), ('lean/ComparatorChallenges/RelativeGeneration.lean', 5724), ('lean/ComparatorChallenges/SLELowerPositivity.lean', 2992), ('lean/ComparatorChallenges/SelfSimilar.lean', 2104), ('lean/ComparatorChallenges/SelfSimilarCorollaries.lean', 2481), ('lean/ComparatorChallenges/SingletonLoopMatching.lean', 3926), ('lean/ComparatorChallenges/SquareRootDegree.lean', 1803), ('lean/ComparatorChallenges/TotientAsymptotic.lean', 4930), ('lean/ComparatorChallenges/TotientCompanionZero.lean', 3232), ('lean/ComparatorChallenges/TriangleFace.lean', 6109), ('lean/ComparatorChallenges/TriangularHilbert.lean', 1356), ('lean/ComparatorChallenges/TypeSystemNormalization.lean', 4288), ('lean/OAI/Analysis/FactorGeneration/Main.lean', 2017), ('lean/OAI/Analysis/TriangularHilbert/Main.lean', 2003), ('lean/OAI/Combinatorics/GotsmanLinial/Main.lean', 6267), ('lean/OAI/Combinatorics/PerfectMatching/Main.lean', 9489), ('lean/OAI/Computability/LoopMatching/Main.lean', 1083), ('lean/OAI/Computability/MatchingCount/BinarySolve.lean', 6431), ('lean/OAI/MeasureTheory/SelfSimilar/Main.lean', 2935), ('lean/OAI/Probability/ClassicalON/Main.lean', 1980), ('lean/OAI/RepresentationTheory/FoulkesHowe/Stabilization.lean', 2544)]\n\nSCOPE 015 \n# Torus-packet equidistribution in prime, quartic, and sextic degrees\n\nThe following describes the scope of the Lean formalization related to the following accompanying paper(s):\n\n- [Equidistribution of Prime-Degree Torus Packets with Arbitrary Local Type](../../preprints/Equidistribution-of-Prime-Degree-Torus-Packets-with-Arbitrary-Local-Type-September-24-2026/paper.pdf)\n\n## Scope\n\nThe formalization proves equidistribution of volume-weighted torus packets for totally real number fields of every fixed prime degree at least five. For any sequence of full lattices whose multiplier-order discriminants tend to infinity, the packet measures converge weakly to Haar probability measure and form a tight family, so no mass escapes. Arbitrary local homothety types are allowed, and the fields may vary along the sequence.\n\n## Comparator links\n\n| Result | Comparator statement |\n| --- | --- |\n| Equidistribution of prime-degree torus packets | [DukePrimeDegree.lean](../ComparatorChallenges/DukePrimeDegree.lean) |\n\n\nSCOPE 024 \n# An asymptotic formula for the number of totients\n\nThe following describes the scope of the Lean formalization related to the following accompanying paper(s):\n\n- [An asymptotic formula for the number of totients](../../preprints/An-asymptotic-formula-for-the-number-of-totients-September-25-2026/An-asymptotic-formula-for-the-number-of-totients-September-25-2026.pdf)\n\n## Scope\n\nLet $V(x)$ count the distinct values of Euler's totient function up to $x$. The formalization constructs the paper's explicit positive main term from finite arithmetic approximants and proves that their ratio tends to one. In particular, $V(cx)/V(x)\\to c$ for every fixed $c>0$, answering the Erdős–Hall regular-variation question.\n\nThe formalization also gives asymptotics for totients $v\\le x$ whose least preimage $\\ell(v)=\\min\\{n\\ge1:\\varphi(n)=v\\}$ lies between $kx$ and $(k+1)x$. The associated coefficient is positive under the stated existence condition; when no totient $d$ satisfies $kd<\\ell(d)$, the count and coefficient are identically zero. The cases $k=1,2$ have positive coefficients.\n\n## Comparator links\n\n| Result | Comparator statement |\n| --- | --- |\n| Totient-count asymptotics and regular variation | [TotientAsymptotic.lean](../ComparatorChallenges/TotientAsymptotic.lean) |\n| Exact zero case for the companion totient counts | [TotientCompanionZero.lean](../ComparatorChallenges/TotientCompanionZero.lean) |\n\n\nSCOPE 082 \n# Annular variation and dyadic absolute bounds for the triangular Hilbert transform\n\nThe following describes the scope of the Lean formalization related to the following accompanying paper(s):\n\n- [The maximal triangular Hilbert transform at the symmetric point](../../preprints/The-maximal-triangular-Hilbert-transform-at-the-symmetric-point-September-24-2026/paper.pdf)\n\n## Scope\n\nThe formalized result proves the symmetric maximal estimate for the triangular Hilbert transform: for arbitrary complex $F,G\\in L^3(\\mathbb R^2)$, the $L^{3/2}$ norm of the supremum over all finite hard-truncation intervals is at most $C\\|F\\|_3\\|G\\|_3$ for one absolute constant $C$. It also establishes a common full-measure set on which the truncated integrals are defined and almost-everywhere measurability of the maximal output.\n\n## Comparator links\n\n| Result | Comparator statement |\n| --- | --- |\n| Maximal triangular Hilbert transform bound | [TriangularHilbert.lean](../ComparatorChallenges/TriangularHilbert.lean) |\n\n\nSCOPE 113 \n# Approximate counting and entropy of perfect matchings\n\nThe following describes the scope of the Lean formalization related to the following accompanying paper(s):\n\n- [A Fully Polynomial Randomized Approximation Scheme for Perfect Matchings in General Graphs](../../preprints/A-Fully-Polynomial-Randomized-Approximation-Scheme-for-Perfect-Matchings-in-General-Graphs-September-23-2026/main.pdf)\n- [Entropy and Face Dimension of the Perfect-Matching Polytope](../../preprints/Entropy-and-Face-Dimension-of-the-Perfect-Matching-Polytope-September-23-2026/main.pdf)\n\n## Scope\n\nThe formalization gives a fully polynomial randomized approximation scheme for counting perfect matchings in every finite simple undirected graph. For rational $0<\\varepsilon<1$ and $0<\\delta<1/2$, it returns a nonnegative rational estimate with relative error at most $\\varepsilon$ with probability at least $1-\\delta$. If the graph has no perfect matching, every execution returns zero.\n\nThe algorithm is a fixed finite-alphabet randomized machine. Its worst-case running time is polynomial in the binary input length, $\\varepsilon^{-1}$, and $\\log(\\delta^{-1})$, including on unsuccessful random tapes.\n\nFor feasible edge marginals $x$ of perfect matchings in a loopless multigraph on $2m$ vertices, let $F(x)=-\\sum_e x_e\\log x_e$, $B(x)=-\\sum_e(1-x_e)\\log(1-x_e)$, and let $H(x)$ be the maximum entropy of a matching law with those marginals. The formalization proves $F(x)-(2-2/m)B(x)\\le H(x)\\le F(x)$ for $m\\ge1$, including boundary points of the polytope. It retains the earlier coefficient-eight entropy bound and the deterministic counting approximations with factors $512^n$ for loopless graphs and $2^{18n}$ with singleton loops.\n\nA further statement identifies the exact face obtained by expanding a degree-three vertex into a triangle and shows that minimal faces are carried to minimal faces by the expansion. The paper's sharp global face-dimension bound is not part of that selected statement.\n\n## Comparator links\n\n| Result | Comparator statement |\n| --- | --- |\n| Randomized approximation of the perfect-matching count | [MatchingFPRAS.lean](../ComparatorChallenges/MatchingFPRAS.lean) |\n| Perfect-matching entropy bound | [MatchingEntropy.lean](../ComparatorChallenges/MatchingEntropy.lean) |\n| Deterministic approximate matching count | [BinaryMatching.lean](../ComparatorChallenges/BinaryMatching.lean) |\n| Approximate counting with singleton loops | [SingletonLoopMatching.lean](../ComparatorChallenges/SingletonLoopMatching.lean) |\n| Refined pointwise perfect-matching entropy bounds | [MatchingEntropyBounds.lean](../ComparatorChallenges/MatchingEntropyBounds.lean) |\n| Face correspondence under triangle expansion | [TriangleFace.lean](../ComparatorChallenges/TriangleFace.lean) |\n\n\nSCOPE 115 \n# Sampling and counting contingency tables with arbitrary margins\n\nThe following describes the scope of the Lean formalization related to the following accompanying paper(s):\n\n- [Exact Uniform Sampling of Contingency Tables with Arbitrary Margins](../../preprints/Exact-Uniform-Sampling-of-Contingency-Tables-with-Arbitrary-Margins-September-24-2026/main.pdf)\n- [An FPRAS for Cell-Bounded Contingency Tables](../../preprints/An-FPRAS-for-Cell-Bounded-Contingency-Tables-September-24-2026/main.pdf)\n\n## Scope\n\nThe formalization gives an exact uniform sampler for nonnegative integer contingency tables with arbitrary prescribed row and column sums having equal totals. It terminates almost surely, every halted output is feasible, and each table has exactly the uniform limiting probability. Expected bit complexity is polynomial in the dimensions and the binary length of the margins.\n\nIt also gives a sampler with a fixed polynomial time bound on every execution whose total-variation error is at most $2^{-k}$ for requested precision $k\\ge1$. The same Comparator file includes the companion approximation scheme for counting tables with individual cell bounds.\n\nThe formalization gives a fully polynomial randomized approximation scheme for counting nonnegative integer matrices with prescribed row sums, column sums, and individual entry bounds. Both dimensions may vary, the margins and bounds are binary encoded, and zero entry bounds are allowed. For rational $0<\\varepsilon,\\delta<1$, the algorithm returns a nonnegative estimate with relative error at most $\\varepsilon$ with probability at least $1-\\delta$, and returns zero on every execution when no table exists.\n\nThe running time is polynomial in the encoded input size, $\\varepsilon^{-1}$, and $\\log(\\delta^{-1})$ on every random tape. The same Comparator file also contains the companion sampling results for tables without cell bounds.\n\n## Comparator links\n\n| Result | Comparator statement |\n| --- | --- |\n| Exact and bounded-time sampling of contingency tables | [ContingencyTables.lean](../ComparatorChallenges/ContingencyTables.lean) |\n| Randomized approximate counting of cell-bounded contingency tables | [ContingencyTables.lean](../ComparatorChallenges/ContingencyTables.lean) |\n\n"
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        "stdout": "\nSCOPE 115 \n# Sampling and counting contingency tables with arbitrary margins\n\nThe following describes the scope of the Lean formalization related to the following accompanying paper(s):\n\n- [Exact Uniform Sampling of Contingency Tables with Arbitrary Margins](../../preprints/Exact-Uniform-Sampling-of-Contingency-Tables-with-Arbitrary-Margins-September-24-2026/main.pdf)\n- [An FPRAS for Cell-Bounded Contingency Tables](../../preprints/An-FPRAS-for-Cell-Bounded-Contingency-Tables-September-24-2026/main.pdf)\n\n## Scope\n\nThe formalization gives an exact uniform sampler for nonnegative integer contingency tables with arbitrary prescribed row and column sums having equal totals. It terminates almost surely, every halted output is feasible, and each table has exactly the uniform limiting probability. Expected bit complexity is polynomial in the dimensions and the binary length of the margins.\n\nIt also gives a sampler with a fixed polynomial time bound on every execution whose total-variation error is at most $2^{-k}$ for requested precision $k\\ge1$. The same Comparator file includes the companion approximation scheme for counting tables with individual cell bounds.\n\nThe formalization gives a fully polynomial randomized approximation scheme for counting nonnegative integer matrices with prescribed row sums, column sums, and individual entry bounds. Both dimensions may vary, the margins and bounds are binary encoded, and zero entry bounds are allowed. For rational $0<\\varepsilon,\\delta<1$, the algorithm returns a nonnegative estimate with relative error at most $\\varepsilon$ with probability at least $1-\\delta$, and returns zero on every execution when no table exists.\n\nThe running time is polynomial in the encoded input size, $\\varepsilon^{-1}$, and $\\log(\\delta^{-1})$ on every random tape. The same Comparator file also contains the companion sampling results for tables without cell bounds.\n\n## Comparator links\n\n| Result | Comparator statement |\n| --- | --- |\n| Exact and bounded-time sampling of contingency tables | [ContingencyTables.lean](../ComparatorChallenges/ContingencyTables.lean) |\n| Randomized approximate counting of cell-bounded contingency tables | [ContingencyTables.lean](../ComparatorChallenges/ContingencyTables.lean) |\n\n\nSCOPE 127 \n# Average sensitivity of polynomial threshold functions\n\nThe following describes the scope of the Lean formalization related to the following accompanying paper(s):\n\n- [Average sensitivity of polynomial threshold functions](../../preprints/Average-Sensitivity-of-Polynomial-Threshold-Functions-September-25-2026/main.pdf)\n\n## Scope\n\nThe formalized result proves the average-sensitivity bound $8d\\sqrt n$ for every Boolean threshold function defined by a real multilinear polynomial of degree at most $d$ on the $n$-dimensional cube. The sign convention assigns value $1$ at zero. This establishes the stated asymptotic bound, rather than the separate conjecture identifying exact extremizers.\n\n## Comparator links\n\n| Result | Comparator statement |\n| --- | --- |\n| Average sensitivity of polynomial threshold functions | [GotsmanLinial.lean](../ComparatorChallenges/GotsmanLinial.lean) |\n\n\nSCOPE 148 \n# The entropy-rate dimension formula for self-similar measures\n\nThe following describes the scope of the Lean formalization related to the following accompanying paper(s):\n\n- [The entropy-rate dimension formula for self-similar measures on the line](../../preprints/The-entropy-rate-dimension-formula-for-self-similar-measures-on-the-line-September-24-2026/main.pdf)\n\n## Scope\n\nThe formalization proves $\\dim_H\\mu=\\min\\{1,h_{\\mathrm{RW}}/\\chi\\}$ for the lower Hausdorff dimension of every finite real self-similar probability measure with positive weights and nonzero contraction ratios of absolute value below one. Here $h_{\\mathrm{RW}}$ is the entropy rate of random affine compositions and $\\chi$ is the corresponding Lyapunov exponent in the same logarithmic base. Ratios may be signed and unequal, and exact overlaps are allowed.\n\nIt also gives two consequences without exact overlaps: the usual entropy-over-Lyapunov formula for a common positive contraction ratio, and the attractor formula $\\dim_H K=\\min\\{1,s\\}$, where $s\\ge0$ is the unique solution of $\\sum_i|r_i|^s=1$. Absolute continuity is outside these statements.\n\n## Comparator links\n\n| Result | Comparator statement |\n| --- | --- |\n| Entropy-rate dimension formula | [SelfSimilar.lean](../ComparatorChallenges/SelfSimilar.lean) |\n| Homogeneous-measure and self-similar-attractor dimension formulas | [SelfSimilarCorollaries.lean](../ComparatorChallenges/SelfSimilarCorollaries.lean) |\n\n\nSCOPE 192 \n# Boolean functions violate the square-root degree bound by arbitrary factors\n\nThe following describes the scope of the Lean formalization related to the following accompanying paper(s):\n\n- [Unbounded Violations of the Square-Root Degree Bound](../../preprints/Unbounded-Violations-of-the-Square-Root-Degree-Bound-September-26-2026/Unbounded-Violations-of-the-Square-Root-Degree-Bound-September-26-2026.pdf)\n\n## Scope\n\nThe formalization disproves the Gopalan–Servedio square-root degree conjecture by an unbounded factor. For every $C>0$, it gives a nonconstant Boolean function on a finite sign cube for which the sum of its linear Fourier coefficients exceeds $C\\sqrt{\\deg f}$, where $\\deg f$ is its real multilinear degree.\n\nIt also proves that the ratio of the sum of the absolute values of those coefficients to $\\sqrt{\\deg f}$ is unbounded among positive-degree Boolean functions.\n\n## Comparator links\n\n| Result | Comparator statement |\n| --- | --- |\n| Unbounded violations of the square-root Fourier-degree bound | [SquareRootDegree.lean](../ComparatorChallenges/SquareRootDegree.lean) |\n\n",
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          "\nSCOPE 115 \n# Sampling and counting contingency tables with arbitrary margins\n\nThe following describes the scope of the Lean formalization related to the following accompanying paper(s):\n\n- [Exact Uniform Sampling of Contingency Tables with Arbitrary Margins](../../preprints/Exact-Uniform-Sampling-of-Contingency-Tables-with-Arbitrary-Margins-September-24-2026/main.pdf)\n- [An FPRAS for Cell-Bounded Contingency Tables](../../preprints/An-FPRAS-for-Cell-Bounded-Contingency-Tables-September-24-2026/main.pdf)\n\n## Scope\n\nThe formalization gives an exact uniform sampler for nonnegative integer contingency tables with arbitrary prescribed row and column sums having equal totals. It terminates almost surely, every halted output is feasible, and each table has exactly the uniform limiting probability. Expected bit complexity is polynomial in the dimensions and the binary length of the margins.\n\nIt also gives a sampler with a fixed polynomial time bound on every execution whose total-variation error is at most $2^{-k}$ for requested precision $k\\ge1$. The same Comparator file includes the companion approximation scheme for counting tables with individual cell bounds.\n\nThe formalization gives a fully polynomial randomized approximation scheme for counting nonnegative integer matrices with prescribed row sums, column sums, and individual entry bounds. Both dimensions may vary, the margins and bounds are binary encoded, and zero entry bounds are allowed. For rational $0<\\varepsilon,\\delta<1$, the algorithm returns a nonnegative estimate with relative error at most $\\varepsilon$ with probability at least $1-\\delta$, and returns zero on every execution when no table exists.\n\nThe running time is polynomial in the encoded input size, $\\varepsilon^{-1}$, and $\\log(\\delta^{-1})$ on every random tape. The same Comparator file also contains the companion sampling results for tables without cell bounds.\n\n## Comparator links\n\n| Result | Comparator statement |\n| --- | --- |\n| Exact and bounded-time sampling of contingency tables | [ContingencyTables.lean](../ComparatorChallenges/ContingencyTables.lean) |\n| Randomized approximate counting of cell-bounded contingency tables | [ContingencyTables.lean](../ComparatorChallenges/ContingencyTables.lean) |\n\n\nSCOPE 127 \n# Average sensitivity of polynomial threshold functions\n\nThe following describes the scope of the Lean formalization related to the following accompanying paper(s):\n\n- [Average sensitivity of polynomial threshold functions](../../preprints/Average-Sensitivity-of-Polynomial-Threshold-Functions-September-25-2026/main.pdf)\n\n## Scope\n\nThe formalized result proves the average-sensitivity bound $8d\\sqrt n$ for every Boolean threshold function defined by a real multilinear polynomial of degree at most $d$ on the $n$-dimensional cube. The sign convention assigns value $1$ at zero. This establishes the stated asymptotic bound, rather than the separate conjecture identifying exact extremizers.\n\n## Comparator links\n\n| Result | Comparator statement |\n| --- | --- |\n| Average sensitivity of polynomial threshold functions | [GotsmanLinial.lean](../ComparatorChallenges/GotsmanLinial.lean) |\n\n\nSCOPE 148 \n# The entropy-rate dimension formula for self-similar measures\n\nThe following describes the scope of the Lean formalization related to the following accompanying paper(s):\n\n- [The entropy-rate dimension formula for self-similar measures on the line](../../preprints/The-entropy-rate-dimension-formula-for-self-similar-measures-on-the-line-September-24-2026/main.pdf)\n\n## Scope\n\nThe formalization proves $\\dim_H\\mu=\\min\\{1,h_{\\mathrm{RW}}/\\chi\\}$ for the lower Hausdorff dimension of every finite real self-similar probability measure with positive weights and nonzero contraction ratios of absolute value below one. Here $h_{\\mathrm{RW}}$ is the entropy rate of random affine compositions and $\\chi$ is the corresponding Lyapunov exponent in the same logarithmic base. Ratios may be signed and unequal, and exact overlaps are allowed.\n\nIt also gives two consequences without exact overlaps: the usual entropy-over-Lyapunov formula for a common positive contraction ratio, and the attractor formula $\\dim_H K=\\min\\{1,s\\}$, where $s\\ge0$ is the unique solution of $\\sum_i|r_i|^s=1$. Absolute continuity is outside these statements.\n\n## Comparator links\n\n| Result | Comparator statement |\n| --- | --- |\n| Entropy-rate dimension formula | [SelfSimilar.lean](../ComparatorChallenges/SelfSimilar.lean) |\n| Homogeneous-measure and self-similar-attractor dimension formulas | [SelfSimilarCorollaries.lean](../ComparatorChallenges/SelfSimilarCorollaries.lean) |\n\n\nSCOPE 192 \n# Boolean functions violate the square-root degree bound by arbitrary factors\n\nThe following describes the scope of the Lean formalization related to the following accompanying paper(s):\n\n- [Unbounded Violations of the Square-Root Degree Bound](../../preprints/Unbounded-Violations-of-the-Square-Root-Degree-Bound-September-26-2026/Unbounded-Violations-of-the-Square-Root-Degree-Bound-September-26-2026.pdf)\n\n## Scope\n\nThe formalization disproves the Gopalan–Servedio square-root degree conjecture by an unbounded factor. For every $C>0$, it gives a nonconstant Boolean function on a finite sign cube for which the sum of its linear Fourier coefficients exceeds $C\\sqrt{\\deg f}$, where $\\deg f$ is its real multilinear degree.\n\nIt also proves that the ratio of the sum of the absolute values of those coefficients to $\\sqrt{\\deg f}$ is unbounded among positive-degree Boolean functions.\n\n## Comparator links\n\n| Result | Comparator statement |\n| --- | --- |\n| Unbounded violations of the square-root Fourier-degree bound | [SquareRootDegree.lean](../ComparatorChallenges/SquareRootDegree.lean) |\n\n"
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          "text": "\nSCOPE 115 \n# Sampling and counting contingency tables with arbitrary margins\n\nThe following describes the scope of the Lean formalization related to the following accompanying paper(s):\n\n- [Exact Uniform Sampling of Contingency Tables with Arbitrary Margins](../../preprints/Exact-Uniform-Sampling-of-Contingency-Tables-with-Arbitrary-Margins-September-24-2026/main.pdf)\n- [An FPRAS for Cell-Bounded Contingency Tables](../../preprints/An-FPRAS-for-Cell-Bounded-Contingency-Tables-September-24-2026/main.pdf)\n\n## Scope\n\nThe formalization gives an exact uniform sampler for nonnegative integer contingency tables with arbitrary prescribed row and column sums having equal totals. It terminates almost surely, every halted output is feasible, and each table has exactly the uniform limiting probability. Expected bit complexity is polynomial in the dimensions and the binary length of the margins.\n\nIt also gives a sampler with a fixed polynomial time bound on every execution whose total-variation error is at most $2^{-k}$ for requested precision $k\\ge1$. The same Comparator file includes the companion approximation scheme for counting tables with individual cell bounds.\n\nThe formalization gives a fully polynomial randomized approximation scheme for counting nonnegative integer matrices with prescribed row sums, column sums, and individual entry bounds. Both dimensions may vary, the margins and bounds are binary encoded, and zero entry bounds are allowed. For rational $0<\\varepsilon,\\delta<1$, the algorithm returns a nonnegative estimate with relative error at most $\\varepsilon$ with probability at least $1-\\delta$, and returns zero on every execution when no table exists.\n\nThe running time is polynomial in the encoded input size, $\\varepsilon^{-1}$, and $\\log(\\delta^{-1})$ on every random tape. The same Comparator file also contains the companion sampling results for tables without cell bounds.\n\n## Comparator links\n\n| Result | Comparator statement |\n| --- | --- |\n| Exact and bounded-time sampling of contingency tables | [ContingencyTables.lean](../ComparatorChallenges/ContingencyTables.lean) |\n| Randomized approximate counting of cell-bounded contingency tables | [ContingencyTables.lean](../ComparatorChallenges/ContingencyTables.lean) |\n\n\nSCOPE 127 \n# Average sensitivity of polynomial threshold functions\n\nThe following describes the scope of the Lean formalization related to the following accompanying paper(s):\n\n- [Average sensitivity of polynomial threshold functions](../../preprints/Average-Sensitivity-of-Polynomial-Threshold-Functions-September-25-2026/main.pdf)\n\n## Scope\n\nThe formalized result proves the average-sensitivity bound $8d\\sqrt n$ for every Boolean threshold function defined by a real multilinear polynomial of degree at most $d$ on the $n$-dimensional cube. The sign convention assigns value $1$ at zero. This establishes the stated asymptotic bound, rather than the separate conjecture identifying exact extremizers.\n\n## Comparator links\n\n| Result | Comparator statement |\n| --- | --- |\n| Average sensitivity of polynomial threshold functions | [GotsmanLinial.lean](../ComparatorChallenges/GotsmanLinial.lean) |\n\n\nSCOPE 148 \n# The entropy-rate dimension formula for self-similar measures\n\nThe following describes the scope of the Lean formalization related to the following accompanying paper(s):\n\n- [The entropy-rate dimension formula for self-similar measures on the line](../../preprints/The-entropy-rate-dimension-formula-for-self-similar-measures-on-the-line-September-24-2026/main.pdf)\n\n## Scope\n\nThe formalization proves $\\dim_H\\mu=\\min\\{1,h_{\\mathrm{RW}}/\\chi\\}$ for the lower Hausdorff dimension of every finite real self-similar probability measure with positive weights and nonzero contraction ratios of absolute value below one. Here $h_{\\mathrm{RW}}$ is the entropy rate of random affine compositions and $\\chi$ is the corresponding Lyapunov exponent in the same logarithmic base. Ratios may be signed and unequal, and exact overlaps are allowed.\n\nIt also gives two consequences without exact overlaps: the usual entropy-over-Lyapunov formula for a common positive contraction ratio, and the attractor formula $\\dim_H K=\\min\\{1,s\\}$, where $s\\ge0$ is the unique solution of $\\sum_i|r_i|^s=1$. Absolute continuity is outside these statements.\n\n## Comparator links\n\n| Result | Comparator statement |\n| --- | --- |\n| Entropy-rate dimension formula | [SelfSimilar.lean](../ComparatorChallenges/SelfSimilar.lean) |\n| Homogeneous-measure and self-similar-attractor dimension formulas | [SelfSimilarCorollaries.lean](../ComparatorChallenges/SelfSimilarCorollaries.lean) |\n\n\nSCOPE 192 \n# Boolean functions violate the square-root degree bound by arbitrary factors\n\nThe following describes the scope of the Lean formalization related to the following accompanying paper(s):\n\n- [Unbounded Violations of the Square-Root Degree Bound](../../preprints/Unbounded-Violations-of-the-Square-Root-Degree-Bound-September-26-2026/Unbounded-Violations-of-the-Square-Root-Degree-Bound-September-26-2026.pdf)\n\n## Scope\n\nThe formalization disproves the Gopalan–Servedio square-root degree conjecture by an unbounded factor. For every $C>0$, it gives a nonconstant Boolean function on a finite sign cube for which the sum of its linear Fourier coefficients exceeds $C\\sqrt{\\deg f}$, where $\\deg f$ is its real multilinear degree.\n\nIt also proves that the ratio of the sum of the absolute values of those coefficients to $\\sqrt{\\deg f}$ is unbounded among positive-degree Boolean functions.\n\n## Comparator links\n\n| Result | Comparator statement |\n| --- | --- |\n| Unbounded violations of the square-root Fourier-degree bound | [SquareRootDegree.lean](../ComparatorChallenges/SquareRootDegree.lean) |\n\n"
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        "stdout": "\nSCOPE 210 \n# Foulkes' conjecture for sixth powers and quadratic stabilization\n\nThe following describes the scope of the Lean formalization related to the following accompanying paper(s):\n\n- [Quadratic stabilization of the canonical Foulkes–Howe map](../../preprints/Quadratic-Stabilization-of-the-Canonical-Foulkes-Howe-Map-September-25-2026/paper.pdf)\n\n## Scope\n\nThe formalized result proves surjectivity of the canonical averaged Foulkes–Howe map $\\mathrm{Sym}^b(\\mathrm{Sym}^a\\,V)\\to\\mathrm{Sym}^a(\\mathrm{Sym}^b\\,V)$ for every finite-dimensional complex vector space, $a\\ge2$, and $b\\ge a(a-1)$. It also covers the bijective $a=1$ case, the vanishing-on-products consequence, and the corresponding equivariant embedding in the reverse direction. The sixth-power specialization is covered for $b\\ge30$; the companion's full range $b\\ge6$ is not included.\n\n## Comparator links\n\n| Result | Comparator statement |\n| --- | --- |\n| Quadratic stabilization of the canonical Foulkes–Howe map | [FoulkesHowe.lean](../ComparatorChallenges/FoulkesHowe.lean) |\n\n\nSCOPE 215 \n# Canonical $`O(3)`$ continuum limit and exact $`O(4)`$ mass asymptotics\n\nThe following describes the scope of the Lean formalization related to the following accompanying paper(s):\n\n- [Exponential decay in two-dimensional classical $O(n)$ models](../../preprints/Exponential-decay-in-two-dimensional-classical-On-models-September-23-2026/paper.pdf)\n\n## Scope\n\nThe formalized result proves exponential decay of spin correlations for the two-dimensional classical $O(n)$ model at every temperature when $n\\ge3$. For each interaction bound $\\beta>0$, constants $A$ and $m>0$ work for every finite square-lattice subgraph with free boundary and nonnegative edge strengths at most $\\beta$: the correlation between sites $x,y$ is at most $Ae^{-m|x-y|}$. The infinite-volume limit and the separate $O(4)$ spectral-gap claim are not included.\n\n## Comparator links\n\n| Result | Comparator statement |\n| --- | --- |\n| Exponential correlation decay for classical $O(n)$ models | [ClassicalON.lean](../ComparatorChallenges/ClassicalON.lean) |\n\n\nSCOPE 230 \n# Exact Hausdorff gauges for SLE\n\nThe following describes the scope of the Lean formalization related to the following accompanying paper(s):\n\n- [An exact Hausdorff gauge for SLE: A moment-integral and finite-batch construction](../../preprints/An-exact-Hausdorff-gauge-for-SLE-September-25-2026/An-exact-Hausdorff-gauge-for-SLE-September-25-2026.pdf)\n- [An explicit exact Hausdorff gauge for SLE: A regular formula from quantitative tails and dense visits](../../preprints/An-explicit-exact-Hausdorff-gauge-for-SLE-September-26-2026/An-explicit-exact-Hausdorff-gauge-for-SLE-September-26-2026.pdf)\n\n## Scope\n\nFor $0<\\kappa<8$, put $d=1+\\kappa/8$. The formalization proves that every continuous nondecreasing Hausdorff gauge agreeing with $h(r)=r^d(\\log\\log(1/r))^{(2-d)/2}$ at sufficiently small positive radii almost surely assigns positive measure to every nontrivial compact positive-time segment of ordinary chordal $\\mathrm{SLE}_\\kappa$.\n\n## Comparator links\n\n| Result | Comparator statement |\n| --- | --- |\n| Positivity of the explicit gauge on every positive-time segment | [SLELowerPositivity.lean](../ComparatorChallenges/SLELowerPositivity.lean) |\n\n\nSCOPE 245 \n# Weak normalization implies strong normalization in pure type systems\n\nThe following describes the scope of the Lean formalization related to the following accompanying paper(s):\n\n- [Weak and strong normalization in pure type systems](../../preprints/Weak-and-strong-normalization-in-pure-type-systems-September-25-2026/paper.pdf)\n\n## Scope\n\nThe formalization proves the $\\beta$-Barendregt–Geuvers–Klop conjecture for pure type systems: if every legal expression in every valid context has some terminating $\\beta$-reduction sequence, then every $\\beta$-reduction sequence from every such expression terminates. Reduction is allowed inside type annotations, and the specification may have arbitrary sorts and nonfunctional axioms or rules.\n\n## Comparator links\n\n| Result | Comparator statement |\n| --- | --- |\n| Weak normalization implies strong normalization in every pure type system | [TypeSystemNormalization.lean](../ComparatorChallenges/TypeSystemNormalization.lean) |\n\n\nSCOPE 296 \n# The generator problem for finite factors\n\nThe following describes the scope of the Lean formalization related to the following accompanying paper(s):\n\n- [Relative generation and the generator problem for finite factors](../../preprints/Relative-generation-and-the-generator-problem-for-finite-factors-September-23-2026/paper.pdf)\n\n## Scope\n\nThe formalization answers the generator problem affirmatively for type $\\mathrm{II}_1$ factors with separable predual: one bounded operator generates the factor under adjoints and weak-operator closure. It imposes no separability assumption on the representing Hilbert space or on the operator-norm topology.\n\nIt also proves the paper's relative-generation result. For an irreducible inclusion of type $\\mathrm{II}_1$ factors with separable predual for the larger factor, the unitaries that generate the larger factor together with the smaller one form a dense $G_\\delta$ set in the normalized trace two-norm topology. The normalized faithful normal trace is included in the assertion.\n\n## Comparator links\n\n| Result | Comparator statement |\n| --- | --- |\n| Single generation of separable-predual $\\mathrm{II}_1$ factors | [FactorGeneration.lean](../ComparatorChallenges/FactorGeneration.lean) |\n| Dense relative generators for irreducible inclusions of finite factors | [RelativeGeneration.lean](../ComparatorChallenges/RelativeGeneration.lean) |\n\n\nSCOPE 369 \n# The hot spots conjecture for simply connected planar domains\n\nThe following describes the scope of the Lean formalization related to the following accompanying paper(s):\n\n- [Strict hot spots and absence of interior critical points on smooth simply connected planar domains](../../preprints/Strict-hot-spots-and-absence-of-interior-critical-points-on-smooth-simply-connected-planar-domains-September-24-2026/main.pdf)\n\n## Scope\n\nThe strict hot-spots conjecture asks whether extrema of a first nonconstant Neumann eigenfunction occur only on the boundary. The formalization proves the stronger interior statement on every nonempty smooth bounded simply connected planar domain: every nonzero eigenfunction in the first positive Neumann eigenspace has nonvanishing gradient throughout the interior. Consequently all global maxima and minima lie on the boundary. The conclusion applies to every eigenfunction even when the eigenvalue is multiple.\n\n## Comparator links\n\n| Result | Comparator statement |\n| --- | --- |\n| Strict hot spots and absence of interior critical points | [HotSpots.lean](../ComparatorChallenges/HotSpots.lean) |\n\n\nSCOPE 372 \n# Global uniqueness in smooth isotropic elasticity\n\nThe following describes the scope of the Lean formalization related to the following accompanying paper(s):\n\n- [Global Uniqueness for the Smooth Isotropic Elasticity Inverse Problem](../../preprints/Global-Uniqueness-for-the-Smooth-Isotropic-Elasticity-Inverse-Problem-September-24-2026/article.pdf)\n\n## Scope\n\nThe formalization proves global uniqueness in the three-dimensional static isotropic elasticity inverse problem. On a bounded connected smooth domain, let two pairs of smooth real Lamé moduli satisfy $\\mu>0$ and $3\\lambda+2\\mu>0$ on the closure. If their full displacement-to-traction maps agree, then both Lamé moduli agree throughout the domain. The selected statement is uniqueness; it does not supply a reconstruction algorithm.\n\n## Comparator links\n\n| Result | Comparator statement |\n| --- | --- |\n| Global uniqueness of smooth isotropic Lamé moduli | [ElasticityUniqueness.lean](../ComparatorChallenges/ElasticityUniqueness.lean) |\n\n",
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          "\nSCOPE 210 \n# Foulkes' conjecture for sixth powers and quadratic stabilization\n\nThe following describes the scope of the Lean formalization related to the following accompanying paper(s):\n\n- [Quadratic stabilization of the canonical Foulkes–Howe map](../../preprints/Quadratic-Stabilization-of-the-Canonical-Foulkes-Howe-Map-September-25-2026/paper.pdf)\n\n## Scope\n\nThe formalized result proves surjectivity of the canonical averaged Foulkes–Howe map $\\mathrm{Sym}^b(\\mathrm{Sym}^a\\,V)\\to\\mathrm{Sym}^a(\\mathrm{Sym}^b\\,V)$ for every finite-dimensional complex vector space, $a\\ge2$, and $b\\ge a(a-1)$. It also covers the bijective $a=1$ case, the vanishing-on-products consequence, and the corresponding equivariant embedding in the reverse direction. The sixth-power specialization is covered for $b\\ge30$; the companion's full range $b\\ge6$ is not included.\n\n## Comparator links\n\n| Result | Comparator statement |\n| --- | --- |\n| Quadratic stabilization of the canonical Foulkes–Howe map | [FoulkesHowe.lean](../ComparatorChallenges/FoulkesHowe.lean) |\n\n\nSCOPE 215 \n# Canonical $`O(3)`$ continuum limit and exact $`O(4)`$ mass asymptotics\n\nThe following describes the scope of the Lean formalization related to the following accompanying paper(s):\n\n- [Exponential decay in two-dimensional classical $O(n)$ models](../../preprints/Exponential-decay-in-two-dimensional-classical-On-models-September-23-2026/paper.pdf)\n\n## Scope\n\nThe formalized result proves exponential decay of spin correlations for the two-dimensional classical $O(n)$ model at every temperature when $n\\ge3$. For each interaction bound $\\beta>0$, constants $A$ and $m>0$ work for every finite square-lattice subgraph with free boundary and nonnegative edge strengths at most $\\beta$: the correlation between sites $x,y$ is at most $Ae^{-m|x-y|}$. The infinite-volume limit and the separate $O(4)$ spectral-gap claim are not included.\n\n## Comparator links\n\n| Result | Comparator statement |\n| --- | --- |\n| Exponential correlation decay for classical $O(n)$ models | [ClassicalON.lean](../ComparatorChallenges/ClassicalON.lean) |\n\n\nSCOPE 230 \n# Exact Hausdorff gauges for SLE\n\nThe following describes the scope of the Lean formalization related to the following accompanying paper(s):\n\n- [An exact Hausdorff gauge for SLE: A moment-integral and finite-batch construction](../../preprints/An-exact-Hausdorff-gauge-for-SLE-September-25-2026/An-exact-Hausdorff-gauge-for-SLE-September-25-2026.pdf)\n- [An explicit exact Hausdorff gauge for SLE: A regular formula from quantitative tails and dense visits](../../preprints/An-explicit-exact-Hausdorff-gauge-for-SLE-September-26-2026/An-explicit-exact-Hausdorff-gauge-for-SLE-September-26-2026.pdf)\n\n## Scope\n\nFor $0<\\kappa<8$, put $d=1+\\kappa/8$. The formalization proves that every continuous nondecreasing Hausdorff gauge agreeing with $h(r)=r^d(\\log\\log(1/r))^{(2-d)/2}$ at sufficiently small positive radii almost surely assigns positive measure to every nontrivial compact positive-time segment of ordinary chordal $\\mathrm{SLE}_\\kappa$.\n\n## Comparator links\n\n| Result | Comparator statement |\n| --- | --- |\n| Positivity of the explicit gauge on every positive-time segment | [SLELowerPositivity.lean](../ComparatorChallenges/SLELowerPositivity.lean) |\n\n\nSCOPE 245 \n# Weak normalization implies strong normalization in pure type systems\n\nThe following describes the scope of the Lean formalization related to the following accompanying paper(s):\n\n- [Weak and strong normalization in pure type systems](../../preprints/Weak-and-strong-normalization-in-pure-type-systems-September-25-2026/paper.pdf)\n\n## Scope\n\nThe formalization proves the $\\beta$-Barendregt–Geuvers–Klop conjecture for pure type systems: if every legal expression in every valid context has some terminating $\\beta$-reduction sequence, then every $\\beta$-reduction sequence from every such expression terminates. Reduction is allowed inside type annotations, and the specification may have arbitrary sorts and nonfunctional axioms or rules.\n\n## Comparator links\n\n| Result | Comparator statement |\n| --- | --- |\n| Weak normalization implies strong normalization in every pure type system | [TypeSystemNormalization.lean](../ComparatorChallenges/TypeSystemNormalization.lean) |\n\n\nSCOPE 296 \n# The generator problem for finite factors\n\nThe following describes the scope of the Lean formalization related to the following accompanying paper(s):\n\n- [Relative generation and the generator problem for finite factors](../../preprints/Relative-generation-and-the-generator-problem-for-finite-factors-September-23-2026/paper.pdf)\n\n## Scope\n\nThe formalization answers the generator problem affirmatively for type $\\mathrm{II}_1$ factors with separable predual: one bounded operator generates the factor under adjoints and weak-operator closure. It imposes no separability assumption on the representing Hilbert space or on the operator-norm topology.\n\nIt also proves the paper's relative-generation result. For an irreducible inclusion of type $\\mathrm{II}_1$ factors with separable predual for the larger factor, the unitaries that generate the larger factor together with the smaller one form a dense $G_\\delta$ set in the normalized trace two-norm topology. The normalized faithful normal trace is included in the assertion.\n\n## Comparator links\n\n| Result | Comparator statement |\n| --- | --- |\n| Single generation of separable-predual $\\mathrm{II}_1$ factors | [FactorGeneration.lean](../ComparatorChallenges/FactorGeneration.lean) |\n| Dense relative generators for irreducible inclusions of finite factors | [RelativeGeneration.lean](../ComparatorChallenges/RelativeGeneration.lean) |\n\n\nSCOPE 369 \n# The hot spots conjecture for simply connected planar domains\n\nThe following describes the scope of the Lean formalization related to the following accompanying paper(s):\n\n- [Strict hot spots and absence of interior critical points on smooth simply connected planar domains](../../preprints/Strict-hot-spots-and-absence-of-interior-critical-points-on-smooth-simply-connected-planar-domains-September-24-2026/main.pdf)\n\n## Scope\n\nThe strict hot-spots conjecture asks whether extrema of a first nonconstant Neumann eigenfunction occur only on the boundary. The formalization proves the stronger interior statement on every nonempty smooth bounded simply connected planar domain: every nonzero eigenfunction in the first positive Neumann eigenspace has nonvanishing gradient throughout the interior. Consequently all global maxima and minima lie on the boundary. The conclusion applies to every eigenfunction even when the eigenvalue is multiple.\n\n## Comparator links\n\n| Result | Comparator statement |\n| --- | --- |\n| Strict hot spots and absence of interior critical points | [HotSpots.lean](../ComparatorChallenges/HotSpots.lean) |\n\n\nSCOPE 372 \n# Global uniqueness in smooth isotropic elasticity\n\nThe following describes the scope of the Lean formalization related to the following accompanying paper(s):\n\n- [Global Uniqueness for the Smooth Isotropic Elasticity Inverse Problem](../../preprints/Global-Uniqueness-for-the-Smooth-Isotropic-Elasticity-Inverse-Problem-September-24-2026/article.pdf)\n\n## Scope\n\nThe formalization proves global uniqueness in the three-dimensional static isotropic elasticity inverse problem. On a bounded connected smooth domain, let two pairs of smooth real Lamé moduli satisfy $\\mu>0$ and $3\\lambda+2\\mu>0$ on the closure. If their full displacement-to-traction maps agree, then both Lamé moduli agree throughout the domain. The selected statement is uniqueness; it does not supply a reconstruction algorithm.\n\n## Comparator links\n\n| Result | Comparator statement |\n| --- | --- |\n| Global uniqueness of smooth isotropic Lamé moduli | [ElasticityUniqueness.lean](../ComparatorChallenges/ElasticityUniqueness.lean) |\n\n"
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          "text": "\nSCOPE 210 \n# Foulkes' conjecture for sixth powers and quadratic stabilization\n\nThe following describes the scope of the Lean formalization related to the following accompanying paper(s):\n\n- [Quadratic stabilization of the canonical Foulkes–Howe map](../../preprints/Quadratic-Stabilization-of-the-Canonical-Foulkes-Howe-Map-September-25-2026/paper.pdf)\n\n## Scope\n\nThe formalized result proves surjectivity of the canonical averaged Foulkes–Howe map $\\mathrm{Sym}^b(\\mathrm{Sym}^a\\,V)\\to\\mathrm{Sym}^a(\\mathrm{Sym}^b\\,V)$ for every finite-dimensional complex vector space, $a\\ge2$, and $b\\ge a(a-1)$. It also covers the bijective $a=1$ case, the vanishing-on-products consequence, and the corresponding equivariant embedding in the reverse direction. The sixth-power specialization is covered for $b\\ge30$; the companion's full range $b\\ge6$ is not included.\n\n## Comparator links\n\n| Result | Comparator statement |\n| --- | --- |\n| Quadratic stabilization of the canonical Foulkes–Howe map | [FoulkesHowe.lean](../ComparatorChallenges/FoulkesHowe.lean) |\n\n\nSCOPE 215 \n# Canonical $`O(3)`$ continuum limit and exact $`O(4)`$ mass asymptotics\n\nThe following describes the scope of the Lean formalization related to the following accompanying paper(s):\n\n- [Exponential decay in two-dimensional classical $O(n)$ models](../../preprints/Exponential-decay-in-two-dimensional-classical-On-models-September-23-2026/paper.pdf)\n\n## Scope\n\nThe formalized result proves exponential decay of spin correlations for the two-dimensional classical $O(n)$ model at every temperature when $n\\ge3$. For each interaction bound $\\beta>0$, constants $A$ and $m>0$ work for every finite square-lattice subgraph with free boundary and nonnegative edge strengths at most $\\beta$: the correlation between sites $x,y$ is at most $Ae^{-m|x-y|}$. The infinite-volume limit and the separate $O(4)$ spectral-gap claim are not included.\n\n## Comparator links\n\n| Result | Comparator statement |\n| --- | --- |\n| Exponential correlation decay for classical $O(n)$ models | [ClassicalON.lean](../ComparatorChallenges/ClassicalON.lean) |\n\n\nSCOPE 230 \n# Exact Hausdorff gauges for SLE\n\nThe following describes the scope of the Lean formalization related to the following accompanying paper(s):\n\n- [An exact Hausdorff gauge for SLE: A moment-integral and finite-batch construction](../../preprints/An-exact-Hausdorff-gauge-for-SLE-September-25-2026/An-exact-Hausdorff-gauge-for-SLE-September-25-2026.pdf)\n- [An explicit exact Hausdorff gauge for SLE: A regular formula from quantitative tails and dense visits](../../preprints/An-explicit-exact-Hausdorff-gauge-for-SLE-September-26-2026/An-explicit-exact-Hausdorff-gauge-for-SLE-September-26-2026.pdf)\n\n## Scope\n\nFor $0<\\kappa<8$, put $d=1+\\kappa/8$. The formalization proves that every continuous nondecreasing Hausdorff gauge agreeing with $h(r)=r^d(\\log\\log(1/r))^{(2-d)/2}$ at sufficiently small positive radii almost surely assigns positive measure to every nontrivial compact positive-time segment of ordinary chordal $\\mathrm{SLE}_\\kappa$.\n\n## Comparator links\n\n| Result | Comparator statement |\n| --- | --- |\n| Positivity of the explicit gauge on every positive-time segment | [SLELowerPositivity.lean](../ComparatorChallenges/SLELowerPositivity.lean) |\n\n\nSCOPE 245 \n# Weak normalization implies strong normalization in pure type systems\n\nThe following describes the scope of the Lean formalization related to the following accompanying paper(s):\n\n- [Weak and strong normalization in pure type systems](../../preprints/Weak-and-strong-normalization-in-pure-type-systems-September-25-2026/paper.pdf)\n\n## Scope\n\nThe formalization proves the $\\beta$-Barendregt–Geuvers–Klop conjecture for pure type systems: if every legal expression in every valid context has some terminating $\\beta$-reduction sequence, then every $\\beta$-reduction sequence from every such expression terminates. Reduction is allowed inside type annotations, and the specification may have arbitrary sorts and nonfunctional axioms or rules.\n\n## Comparator links\n\n| Result | Comparator statement |\n| --- | --- |\n| Weak normalization implies strong normalization in every pure type system | [TypeSystemNormalization.lean](../ComparatorChallenges/TypeSystemNormalization.lean) |\n\n\nSCOPE 296 \n# The generator problem for finite factors\n\nThe following describes the scope of the Lean formalization related to the following accompanying paper(s):\n\n- [Relative generation and the generator problem for finite factors](../../preprints/Relative-generation-and-the-generator-problem-for-finite-factors-September-23-2026/paper.pdf)\n\n## Scope\n\nThe formalization answers the generator problem affirmatively for type $\\mathrm{II}_1$ factors with separable predual: one bounded operator generates the factor under adjoints and weak-operator closure. It imposes no separability assumption on the representing Hilbert space or on the operator-norm topology.\n\nIt also proves the paper's relative-generation result. For an irreducible inclusion of type $\\mathrm{II}_1$ factors with separable predual for the larger factor, the unitaries that generate the larger factor together with the smaller one form a dense $G_\\delta$ set in the normalized trace two-norm topology. The normalized faithful normal trace is included in the assertion.\n\n## Comparator links\n\n| Result | Comparator statement |\n| --- | --- |\n| Single generation of separable-predual $\\mathrm{II}_1$ factors | [FactorGeneration.lean](../ComparatorChallenges/FactorGeneration.lean) |\n| Dense relative generators for irreducible inclusions of finite factors | [RelativeGeneration.lean](../ComparatorChallenges/RelativeGeneration.lean) |\n\n\nSCOPE 369 \n# The hot spots conjecture for simply connected planar domains\n\nThe following describes the scope of the Lean formalization related to the following accompanying paper(s):\n\n- [Strict hot spots and absence of interior critical points on smooth simply connected planar domains](../../preprints/Strict-hot-spots-and-absence-of-interior-critical-points-on-smooth-simply-connected-planar-domains-September-24-2026/main.pdf)\n\n## Scope\n\nThe strict hot-spots conjecture asks whether extrema of a first nonconstant Neumann eigenfunction occur only on the boundary. The formalization proves the stronger interior statement on every nonempty smooth bounded simply connected planar domain: every nonzero eigenfunction in the first positive Neumann eigenspace has nonvanishing gradient throughout the interior. Consequently all global maxima and minima lie on the boundary. The conclusion applies to every eigenfunction even when the eigenvalue is multiple.\n\n## Comparator links\n\n| Result | Comparator statement |\n| --- | --- |\n| Strict hot spots and absence of interior critical points | [HotSpots.lean](../ComparatorChallenges/HotSpots.lean) |\n\n\nSCOPE 372 \n# Global uniqueness in smooth isotropic elasticity\n\nThe following describes the scope of the Lean formalization related to the following accompanying paper(s):\n\n- [Global Uniqueness for the Smooth Isotropic Elasticity Inverse Problem](../../preprints/Global-Uniqueness-for-the-Smooth-Isotropic-Elasticity-Inverse-Problem-September-24-2026/article.pdf)\n\n## Scope\n\nThe formalization proves global uniqueness in the three-dimensional static isotropic elasticity inverse problem. On a bounded connected smooth domain, let two pairs of smooth real Lamé moduli satisfy $\\mu>0$ and $3\\lambda+2\\mu>0$ on the closure. If their full displacement-to-traction maps agree, then both Lamé moduli agree throughout the domain. The selected statement is uniqueness; it does not supply a reconstruction algorithm.\n\n## Comparator links\n\n| Result | Comparator statement |\n| --- | --- |\n| Global uniqueness of smooth isotropic Lamé moduli | [ElasticityUniqueness.lean](../ComparatorChallenges/ElasticityUniqueness.lean) |\n\n"
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        "stdout": "BinaryMatching.json => ['OAI.BinaryMatching.deterministic_approximate_counting'] solution supplied True axioms ['propext', 'Quot.sound', 'Classical.choice'] nanoda False\nClassicalON.json => ['OAI.ClassicalON.main'] solution supplied True axioms ['propext', 'Quot.sound', 'Classical.choice'] nanoda False\nContingencyTables.json => ['OAI.ContingencyTables.boundedSampling', 'OAI.ContingencyTables.exactSampling', 'OAI.ContingencyTables.counting'] solution supplied False axioms ['propext', 'Quot.sound', 'Classical.choice'] nanoda False\nDukePrimeDegree.json => ['OAI.DukePrimeDegree.prime_degree_packet_measure_equidistribution_unconditional', 'OAI.DukePrimeDegree.prime_degree_packet_measures_tendsto_haar_unconditional'] solution supplied False axioms ['propext', 'Quot.sound', 'Classical.choice'] nanoda False\nElasticityUniqueness.json => ['OAI.Elasticity.global_uniqueness'] solution supplied False axioms ['propext', 'Quot.sound', 'Classical.choice'] nanoda False\nFactorGeneration.json => ['OAI.Generator.single_generation_of_II1_separable_predual'] solution supplied True axioms ['propext', 'Quot.sound', 'Classical.choice'] nanoda False\nFoulkesHowe.json => ['OAI.Problem346.canonical_foulkes_howe_surjective'] solution supplied True axioms ['propext', 'Quot.sound', 'Classical.choice'] nanoda False\nGotsmanLinial.json => ['OAI.LeanBlast.GotsmanLinial.gotsmanLinialStatement'] solution supplied True axioms ['propext', 'Quot.sound', 'Classical.choice'] nanoda False\nHotSpots.json => ['OAI.StrictHotSpots.main_theorem'] solution supplied False axioms ['propext', 'Quot.sound', 'Classical.choice'] nanoda False\nMatchingEntropy.json => ['OAI.MatchingEntropy.entropy_main'] solution supplied True axioms ['propext', 'Quot.sound', 'Classical.choice'] nanoda False\nMatchingEntropyBounds.json => ['OAI.MatchingEntropyBounds.Refined.pointwise_entropy'] solution supplied False axioms ['propext', 'Quot.sound', 'Classical.choice'] nanoda False\nMatchingFPRAS.json => ['OAI.MatchingFPRAS.thm_main'] solution supplied False axioms ['propext', 'Quot.sound', 'Classical.choice'] nanoda False\nRelativeGeneration.json => ['OAI.RelativeGeneration.main_theorem'] solution supplied False axioms ['propext', 'Quot.sound', 'Classical.choice'] nanoda False\nSLELowerPositivity.json => ['OAI.SLEExactGauge.sourceLowerMain_proved'] solution supplied False axioms ['propext', 'Quot.sound', 'Classical.choice'] nanoda False\nSelfSimilar.json => ['OAI.EntropyRateDimension.entropy_rate_dimension'] solution supplied True axioms ['propext', 'Quot.sound', 'Classical.choice'] nanoda False\nSelfSimilarCorollaries.json => ['OAI.EntropyRateDimension.Extensions.homogeneous_dimension_direct', 'OAI.CorSetReference.cor_set'] solution supplied False axioms ['propext', 'Classical.choice', 'Quot.sound'] nanoda False\nSingletonLoopMatching.json => ['OAI.LoopMatching.fullEndpoint'] solution supplied False axioms ['propext', 'Quot.sound', 'Classical.choice'] nanoda False\nSquareRootDegree.json => ['OAI.SquareRootDegree.main'] solution supplied False axioms ['propext', 'Quot.sound', 'Classical.choice'] nanoda False\nTotientAsymptotic.json => ['OAI.TotientAsymptotic.totient_asymptotic_formula', 'OAI.TotientAsymptotic.weighted_totient_asymptotic', 'OAI.TotientAsymptotic.weighted_totient_one_two', 'OAI.TotientAsymptotic.coefficient_nonnegative'] solution supplied False axioms ['propext', 'Quot.sound', 'Classical.choice'] nanoda False\nTotientCompanionZero.json => ['OAI.TotientAsymptotic.companion_zero_case'] solution supplied False axioms ['propext', 'Quot.sound', 'Classical.choice'] nanoda False\nTriangleFace.json => ['OAI.TriangleFace198.triangle_expansion_face'] solution supplied False axioms ['propext', 'Quot.sound', 'Classical.choice'] nanoda False\nTriangularHilbert.json => ['OAI.TriangularHilbert.main_estimate'] solution supplied True axioms ['propext', 'Quot.sound', 'Classical.choice'] nanoda False\nTypeSystemNormalization.json => ['OAI.PureTypeSystem.weak_implies_strong'] solution supplied False axioms ['propext', 'Quot.sound', 'Classical.choice'] nanoda False\n\nPAPER 015 [('preprints/Equidistribution-of-Prime-Degree-Torus-Packets-with-Arbitrary-Local-Type-September-24-2026/build/paper.tex', 41, 'enAI}\\n\\\\date{September 24, 2026}\\n\\\\begin{document}\\n\\\\maketitle\\n\\\\begin{abstract}\\nWe prove the packet form of the higher-dimensional Duke equidistribution\\nproblem for totally real fields of any fixed prime degree at least five,\\nallowing arbitrary local homothety types of full lattices. As the\\nmultiplier-order discriminant tends to infinity, the volume-weighted packet\\nmeasures converge to Haar probability measure, with no escape of mass.\\n\\\\end{abstract}\\n\\n\\\\section{Introduction}\\\\label{sec:intro}\\n\\nCompact diagonal orbits in the space of unimodular lattices encode\\nideal classes and units of totally real number fields. The higher-dimensional\\nDuke equidistribution problem asks whether arithmetically complete families\\nof these orbits become uniformly distributed as their discriminants grow.\\nThe grouping matters: one local homothety type specifies a complete packet,\\nwhereas a multiplier order alone can combine several packets. We prove the\\nprime-degree packet statement for every such local type.\\n\\nFix a prime $n\\\\ge5$, and write\\n\\\\[\\n X_n=\\\\SL_n(\\\\Z)\\\\backslash\\\\SL_n(\\\\R),\\\\qquad\\n A_n=\\\\{\\\\diag(e^{u_1},\\\\ldots,e^{u_n}):\\\\textstyle\\\\sum_j u_j=0\\\\}.\\n\\\\]\\nWe regard $X_n$ as the space of covolume-one row lattices, with the\\nright action of $A_n$, and let $m_n$ be its $\\\\SL_n(\\\\R)$-invariant\\nprobability measure. Haar measure on $A_n$ is\\n$du_1\\\\cdots du_{n-1}$ in the displayed logarithmic coordinates.\\n\\nLet $K$ be a totally real field of degree $n$, let\\n$\\\\sigma=(\\\\sigma_1,\\\\ldots,\\\\sigma_n)$ order ')]\n\nPAPER 024 [('preprints/An-asymptotic-formula-for-the-number-of-totients-September-25-2026/build/source/main.tex', 32, \"enAI}\\n\\\\date{September 25, 2026}\\n\\\\begin{document}\\n\\\\maketitle\\n\\\\begin{abstract}\\nLet $V(x)$ count the distinct values of Euler's totient function up to~$x$. We give an explicit asymptotic equivalent for $V(x)$. Its coefficient is a uniform limit of functions defined from finite arithmetic data. We also prove that $V(cx)/V(x)\\\\to c$ as $x\\\\to\\\\infty$ for every fixed $c>0$, answering a question of Erd\\\\H{o}s and Hall.\\n\\\\end{abstract}\\n\\\\begingroup\\n\\\\small\\n\\\\setlength{\\\\baselineskip}{10.5pt}\\n\\\\tableofcontents\\n\\\\endgroup\\n\\\\clearpage\\n\\\\input{introduction}\\n\\\\input{statements}\\n\\\\input{extraction}\\n\\\\input{layers}\\n\\\\input{collisions}\\n\\\\input{limits}\\n\\\\input{companion}\\n\\\\clearpage\\n\\\\bibliographystyle{plain}\\n\\\\bibliography{references}\\n\\\\end{document}\\n\")]\n\nPAPER 046 [('preprints/A-projective-fourfold-with-large-fundamental-group-and-non-Stein-universal-cover-October-5-2026/build/main.tex', 12, \"ex hyperbolic arrangement, Stein manifold}\\n\\\\begin{document}\\n\\\\begin{abstract}\\nWe construct a smooth projective complex fourfold with large fundamental group\\nwhose universal cover is not Stein. The universal cover contains no\\npositive-dimensional compact complex-analytic subvariety, so this disproves\\nShafarevich's holomorphic-convexity conjecture even under the\\nlarge-fundamental-group hypothesis.\\n\\\\end{abstract}\\n\\\\maketitle\\n\\\\tableofcontents\\n\\\\input{sections/01-introduction}\\n\\\\input{sections/02-reduction}\\n\\\\input{sections/03-paths}\\n\\\\input{sections/04-geometry}\\n\\\\input{sections/05-models}\\n\\\\input{sections/06-image}\\n\\\\input{sections/07-projective}\\n\\\\bibliographystyle{amsplain}\\n\\\\bibliography{references}\\n\\\\end{document}\\n\")]\n\nPAPER 065 [('preprints/Virasoro-Constraints-under-Projectivization-October-5-2026/build/paper.tex', 8, '{OpenAI}\\n\\\\date{October 5, 2026}\\n\\\\begin{document}\\n\\\\maketitle\\n\\\\begin{abstract}\\nWe prove that full ordinary descendant Virasoro constraints pass from a\\nsmooth projective complex base to the projectivization of any algebraic\\nvector bundle of rank at least two. The bundle need not split and satisfies\\nno positivity requirement. Assuming the full constraints on the base, the\\nconclusion includes every genus, each individual integral curve class, and\\nall cohomology insertions, including primitive and odd classes. The result\\nalso applies successively to towers of projective bundles.\\n\\\\end{abstract}\\n\\\\tableofcontents\\n\\\\input{sections/01-introduction}\\n\\\\input{sections/02-conventions}\\n\\\\input{sections/03-master-space}\\n\\\\input{sections/04-localization}\\n\\\\input{sections/05-shift}\\n\\\\input{sections/06-spectral}\\n\\\\input{sections/07-fixed-block}\\n\\\\input{sections/08-continuation}\\n\\\\input{references}\\n\\\\end{document}\\n')]\n\nPAPER 082 [('preprints/Annular-variation-of-the-triangular-Hilbert-transform-at-the-symmetric-point-October-5-2026/build/main.tex', 44, '{OpenAI}\\n\\\\date{October 5, 2026}\\n\\\\begin{document}\\n\\\\maketitle\\n\\\\begin{abstract}\\nWe prove the annular $r$-variation estimate for the triangular Hilbert\\ntransform from complex $L^3\\\\times L^3$ to $L^{3/2}$ for every $r>2$.\\nThe partitions may depend on the output point and range over all positive\\nscales. The estimate yields the two-endpoint maximal bound and joint\\nalmost-everywhere and norm principal values, and resolves the symmetric\\nscalar triangular Hilbert transform problem.\\n\\\\end{abstract}\\n\\\\tableofcontents\\n\\\\input{sections/introduction}\\n\\\\input{sections/preliminaries}\\n\\\\input{sections/matrix}\\n\\\\input{sections/heat}\\n\\\\input{sections/smooth}\\n\\\\input{sections/frequency}\\n\\\\input{sections/rough}\\n\\\\input{sections/completion}\\n\\\\input{sections/consequences}\\n\\\\bibliographystyle{plain}\\n\\\\bibliography{references}\\n\\\\end{document}\\n')]\n",
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          "BinaryMatching.json => ['OAI.BinaryMatching.deterministic_approximate_counting'] solution supplied True axioms ['propext', 'Quot.sound', 'Classical.choice'] nanoda False\nClassicalON.json => ['OAI.ClassicalON.main'] solution supplied True axioms ['propext', 'Quot.sound', 'Classical.choice'] nanoda False\nContingencyTables.json => ['OAI.ContingencyTables.boundedSampling', 'OAI.ContingencyTables.exactSampling', 'OAI.ContingencyTables.counting'] solution supplied False axioms ['propext', 'Quot.sound', 'Classical.choice'] nanoda False\nDukePrimeDegree.json => ['OAI.DukePrimeDegree.prime_degree_packet_measure_equidistribution_unconditional', 'OAI.DukePrimeDegree.prime_degree_packet_measures_tendsto_haar_unconditional'] solution supplied False axioms ['propext', 'Quot.sound', 'Classical.choice'] nanoda False\nElasticityUniqueness.json => ['OAI.Elasticity.global_uniqueness'] solution supplied False axioms ['propext', 'Quot.sound', 'Classical.choice'] nanoda False\nFactorGeneration.json => ['OAI.Generator.single_generation_of_II1_separable_predual'] solution supplied True axioms ['propext', 'Quot.sound', 'Classical.choice'] nanoda False\nFoulkesHowe.json => ['OAI.Problem346.canonical_foulkes_howe_surjective'] solution supplied True axioms ['propext', 'Quot.sound', 'Classical.choice'] nanoda False\nGotsmanLinial.json => ['OAI.LeanBlast.GotsmanLinial.gotsmanLinialStatement'] solution supplied True axioms ['propext', 'Quot.sound', 'Classical.choice'] nanoda False\nHotSpots.json => ['OAI.StrictHotSpots.main_theorem'] solution supplied False axioms ['propext', 'Quot.sound', 'Classical.choice'] nanoda False\nMatchingEntropy.json => ['OAI.MatchingEntropy.entropy_main'] solution supplied True axioms ['propext', 'Quot.sound', 'Classical.choice'] nanoda False\nMatchingEntropyBounds.json => ['OAI.MatchingEntropyBounds.Refined.pointwise_entropy'] solution supplied False axioms ['propext', 'Quot.sound', 'Classical.choice'] nanoda False\nMatchingFPRAS.json => ['OAI.MatchingFPRAS.thm_main'] solution supplied False axioms ['propext', 'Quot.sound', 'Classical.choice'] nanoda False\nRelativeGeneration.json => ['OAI.RelativeGeneration.main_theorem'] solution supplied False axioms ['propext', 'Quot.sound', 'Classical.choice'] nanoda False\nSLELowerPositivity.json => ['OAI.SLEExactGauge.sourceLowerMain_proved'] solution supplied False axioms ['propext', 'Quot.sound', 'Classical.choice'] nanoda False\nSelfSimilar.json => ['OAI.EntropyRateDimension.entropy_rate_dimension'] solution supplied True axioms ['propext', 'Quot.sound', 'Classical.choice'] nanoda False\nSelfSimilarCorollaries.json => ['OAI.EntropyRateDimension.Extensions.homogeneous_dimension_direct', 'OAI.CorSetReference.cor_set'] solution supplied False axioms ['propext', 'Classical.choice', 'Quot.sound'] nanoda False\nSingletonLoopMatching.json => ['OAI.LoopMatching.fullEndpoint'] solution supplied False axioms ['propext', 'Quot.sound', 'Classical.choice'] nanoda False\nSquareRootDegree.json => ['OAI.SquareRootDegree.main'] solution supplied False axioms ['propext', 'Quot.sound', 'Classical.choice'] nanoda False\nTotientAsymptotic.json => ['OAI.TotientAsymptotic.totient_asymptotic_formula', 'OAI.TotientAsymptotic.weighted_totient_asymptotic', 'OAI.TotientAsymptotic.weighted_totient_one_two', 'OAI.TotientAsymptotic.coefficient_nonnegative'] solution supplied False axioms ['propext', 'Quot.sound', 'Classical.choice'] nanoda False\nTotientCompanionZero.json => ['OAI.TotientAsymptotic.companion_zero_case'] solution supplied False axioms ['propext', 'Quot.sound', 'Classical.choice'] nanoda False\nTriangleFace.json => ['OAI.TriangleFace198.triangle_expansion_face'] solution supplied False axioms ['propext', 'Quot.sound', 'Classical.choice'] nanoda False\nTriangularHilbert.json => ['OAI.TriangularHilbert.main_estimate'] solution supplied True axioms ['propext', 'Quot.sound', 'Classical.choice'] nanoda False\nTypeSystemNormalization.json => ['OAI.PureTypeSystem.weak_implies_strong'] solution supplied False axioms ['propext', 'Quot.sound', 'Classical.choice'] nanoda False\n\nPAPER 015 [('preprints/Equidistribution-of-Prime-Degree-Torus-Packets-with-Arbitrary-Local-Type-September-24-2026/build/paper.tex', 41, 'enAI}\\n\\\\date{September 24, 2026}\\n\\\\begin{document}\\n\\\\maketitle\\n\\\\begin{abstract}\\nWe prove the packet form of the higher-dimensional Duke equidistribution\\nproblem for totally real fields of any fixed prime degree at least five,\\nallowing arbitrary local homothety types of full lattices. As the\\nmultiplier-order discriminant tends to infinity, the volume-weighted packet\\nmeasures converge to Haar probability measure, with no escape of mass.\\n\\\\end{abstract}\\n\\n\\\\section{Introduction}\\\\label{sec:intro}\\n\\nCompact diagonal orbits in the space of unimodular lattices encode\\nideal classes and units of totally real number fields. The higher-dimensional\\nDuke equidistribution problem asks whether arithmetically complete families\\nof these orbits become uniformly distributed as their discriminants grow.\\nThe grouping matters: one local homothety type specifies a complete packet,\\nwhereas a multiplier order alone can combine several packets. We prove the\\nprime-degree packet statement for every such local type.\\n\\nFix a prime $n\\\\ge5$, and write\\n\\\\[\\n X_n=\\\\SL_n(\\\\Z)\\\\backslash\\\\SL_n(\\\\R),\\\\qquad\\n A_n=\\\\{\\\\diag(e^{u_1},\\\\ldots,e^{u_n}):\\\\textstyle\\\\sum_j u_j=0\\\\}.\\n\\\\]\\nWe regard $X_n$ as the space of covolume-one row lattices, with the\\nright action of $A_n$, and let $m_n$ be its $\\\\SL_n(\\\\R)$-invariant\\nprobability measure. Haar measure on $A_n$ is\\n$du_1\\\\cdots du_{n-1}$ in the displayed logarithmic coordinates.\\n\\nLet $K$ be a totally real field of degree $n$, let\\n$\\\\sigma=(\\\\sigma_1,\\\\ldots,\\\\sigma_n)$ order ')]\n\nPAPER 024 [('preprints/An-asymptotic-formula-for-the-number-of-totients-September-25-2026/build/source/main.tex', 32, \"enAI}\\n\\\\date{September 25, 2026}\\n\\\\begin{document}\\n\\\\maketitle\\n\\\\begin{abstract}\\nLet $V(x)$ count the distinct values of Euler's totient function up to~$x$. We give an explicit asymptotic equivalent for $V(x)$. Its coefficient is a uniform limit of functions defined from finite arithmetic data. We also prove that $V(cx)/V(x)\\\\to c$ as $x\\\\to\\\\infty$ for every fixed $c>0$, answering a question of Erd\\\\H{o}s and Hall.\\n\\\\end{abstract}\\n\\\\begingroup\\n\\\\small\\n\\\\setlength{\\\\baselineskip}{10.5pt}\\n\\\\tableofcontents\\n\\\\endgroup\\n\\\\clearpage\\n\\\\input{introduction}\\n\\\\input{statements}\\n\\\\input{extraction}\\n\\\\input{layers}\\n\\\\input{collisions}\\n\\\\input{limits}\\n\\\\input{companion}\\n\\\\clearpage\\n\\\\bibliographystyle{plain}\\n\\\\bibliography{references}\\n\\\\end{document}\\n\")]\n\nPAPER 046 [('preprints/A-projective-fourfold-with-large-fundamental-group-and-non-Stein-universal-cover-October-5-2026/build/main.tex', 12, \"ex hyperbolic arrangement, Stein manifold}\\n\\\\begin{document}\\n\\\\begin{abstract}\\nWe construct a smooth projective complex fourfold with large fundamental group\\nwhose universal cover is not Stein. The universal cover contains no\\npositive-dimensional compact complex-analytic subvariety, so this disproves\\nShafarevich's holomorphic-convexity conjecture even under the\\nlarge-fundamental-group hypothesis.\\n\\\\end{abstract}\\n\\\\maketitle\\n\\\\tableofcontents\\n\\\\input{sections/01-introduction}\\n\\\\input{sections/02-reduction}\\n\\\\input{sections/03-paths}\\n\\\\input{sections/04-geometry}\\n\\\\input{sections/05-models}\\n\\\\input{sections/06-image}\\n\\\\input{sections/07-projective}\\n\\\\bibliographystyle{amsplain}\\n\\\\bibliography{references}\\n\\\\end{document}\\n\")]\n\nPAPER 065 [('preprints/Virasoro-Constraints-under-Projectivization-October-5-2026/build/paper.tex', 8, '{OpenAI}\\n\\\\date{October 5, 2026}\\n\\\\begin{document}\\n\\\\maketitle\\n\\\\begin{abstract}\\nWe prove that full ordinary descendant Virasoro constraints pass from a\\nsmooth projective complex base to the projectivization of any algebraic\\nvector bundle of rank at least two. The bundle need not split and satisfies\\nno positivity requirement. Assuming the full constraints on the base, the\\nconclusion includes every genus, each individual integral curve class, and\\nall cohomology insertions, including primitive and odd classes. The result\\nalso applies successively to towers of projective bundles.\\n\\\\end{abstract}\\n\\\\tableofcontents\\n\\\\input{sections/01-introduction}\\n\\\\input{sections/02-conventions}\\n\\\\input{sections/03-master-space}\\n\\\\input{sections/04-localization}\\n\\\\input{sections/05-shift}\\n\\\\input{sections/06-spectral}\\n\\\\input{sections/07-fixed-block}\\n\\\\input{sections/08-continuation}\\n\\\\input{references}\\n\\\\end{document}\\n')]\n\nPAPER 082 [('preprints/Annular-variation-of-the-triangular-Hilbert-transform-at-the-symmetric-point-October-5-2026/build/main.tex', 44, '{OpenAI}\\n\\\\date{October 5, 2026}\\n\\\\begin{document}\\n\\\\maketitle\\n\\\\begin{abstract}\\nWe prove the annular $r$-variation estimate for the triangular Hilbert\\ntransform from complex $L^3\\\\times L^3$ to $L^{3/2}$ for every $r>2$.\\nThe partitions may depend on the output point and range over all positive\\nscales. The estimate yields the two-endpoint maximal bound and joint\\nalmost-everywhere and norm principal values, and resolves the symmetric\\nscalar triangular Hilbert transform problem.\\n\\\\end{abstract}\\n\\\\tableofcontents\\n\\\\input{sections/introduction}\\n\\\\input{sections/preliminaries}\\n\\\\input{sections/matrix}\\n\\\\input{sections/heat}\\n\\\\input{sections/smooth}\\n\\\\input{sections/frequency}\\n\\\\input{sections/rough}\\n\\\\input{sections/completion}\\n\\\\input{sections/consequences}\\n\\\\bibliographystyle{plain}\\n\\\\bibliography{references}\\n\\\\end{document}\\n')]\n"
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          "text": "BinaryMatching.json => ['OAI.BinaryMatching.deterministic_approximate_counting'] solution supplied True axioms ['propext', 'Quot.sound', 'Classical.choice'] nanoda False\nClassicalON.json => ['OAI.ClassicalON.main'] solution supplied True axioms ['propext', 'Quot.sound', 'Classical.choice'] nanoda False\nContingencyTables.json => ['OAI.ContingencyTables.boundedSampling', 'OAI.ContingencyTables.exactSampling', 'OAI.ContingencyTables.counting'] solution supplied False axioms ['propext', 'Quot.sound', 'Classical.choice'] nanoda False\nDukePrimeDegree.json => ['OAI.DukePrimeDegree.prime_degree_packet_measure_equidistribution_unconditional', 'OAI.DukePrimeDegree.prime_degree_packet_measures_tendsto_haar_unconditional'] solution supplied False axioms ['propext', 'Quot.sound', 'Classical.choice'] nanoda False\nElasticityUniqueness.json => ['OAI.Elasticity.global_uniqueness'] solution supplied False axioms ['propext', 'Quot.sound', 'Classical.choice'] nanoda False\nFactorGeneration.json => ['OAI.Generator.single_generation_of_II1_separable_predual'] solution supplied True axioms ['propext', 'Quot.sound', 'Classical.choice'] nanoda False\nFoulkesHowe.json => ['OAI.Problem346.canonical_foulkes_howe_surjective'] solution supplied True axioms ['propext', 'Quot.sound', 'Classical.choice'] nanoda False\nGotsmanLinial.json => ['OAI.LeanBlast.GotsmanLinial.gotsmanLinialStatement'] solution supplied True axioms ['propext', 'Quot.sound', 'Classical.choice'] nanoda False\nHotSpots.json => ['OAI.StrictHotSpots.main_theorem'] solution supplied False axioms ['propext', 'Quot.sound', 'Classical.choice'] nanoda False\nMatchingEntropy.json => ['OAI.MatchingEntropy.entropy_main'] solution supplied True axioms ['propext', 'Quot.sound', 'Classical.choice'] nanoda False\nMatchingEntropyBounds.json => ['OAI.MatchingEntropyBounds.Refined.pointwise_entropy'] solution supplied False axioms ['propext', 'Quot.sound', 'Classical.choice'] nanoda False\nMatchingFPRAS.json => ['OAI.MatchingFPRAS.thm_main'] solution supplied False axioms ['propext', 'Quot.sound', 'Classical.choice'] nanoda False\nRelativeGeneration.json => ['OAI.RelativeGeneration.main_theorem'] solution supplied False axioms ['propext', 'Quot.sound', 'Classical.choice'] nanoda False\nSLELowerPositivity.json => ['OAI.SLEExactGauge.sourceLowerMain_proved'] solution supplied False axioms ['propext', 'Quot.sound', 'Classical.choice'] nanoda False\nSelfSimilar.json => ['OAI.EntropyRateDimension.entropy_rate_dimension'] solution supplied True axioms ['propext', 'Quot.sound', 'Classical.choice'] nanoda False\nSelfSimilarCorollaries.json => ['OAI.EntropyRateDimension.Extensions.homogeneous_dimension_direct', 'OAI.CorSetReference.cor_set'] solution supplied False axioms ['propext', 'Classical.choice', 'Quot.sound'] nanoda False\nSingletonLoopMatching.json => ['OAI.LoopMatching.fullEndpoint'] solution supplied False axioms ['propext', 'Quot.sound', 'Classical.choice'] nanoda False\nSquareRootDegree.json => ['OAI.SquareRootDegree.main'] solution supplied False axioms ['propext', 'Quot.sound', 'Classical.choice'] nanoda False\nTotientAsymptotic.json => ['OAI.TotientAsymptotic.totient_asymptotic_formula', 'OAI.TotientAsymptotic.weighted_totient_asymptotic', 'OAI.TotientAsymptotic.weighted_totient_one_two', 'OAI.TotientAsymptotic.coefficient_nonnegative'] solution supplied False axioms ['propext', 'Quot.sound', 'Classical.choice'] nanoda False\nTotientCompanionZero.json => ['OAI.TotientAsymptotic.companion_zero_case'] solution supplied False axioms ['propext', 'Quot.sound', 'Classical.choice'] nanoda False\nTriangleFace.json => ['OAI.TriangleFace198.triangle_expansion_face'] solution supplied False axioms ['propext', 'Quot.sound', 'Classical.choice'] nanoda False\nTriangularHilbert.json => ['OAI.TriangularHilbert.main_estimate'] solution supplied True axioms ['propext', 'Quot.sound', 'Classical.choice'] nanoda False\nTypeSystemNormalization.json => ['OAI.PureTypeSystem.weak_implies_strong'] solution supplied False axioms ['propext', 'Quot.sound', 'Classical.choice'] nanoda False\n\nPAPER 015 [('preprints/Equidistribution-of-Prime-Degree-Torus-Packets-with-Arbitrary-Local-Type-September-24-2026/build/paper.tex', 41, 'enAI}\\n\\\\date{September 24, 2026}\\n\\\\begin{document}\\n\\\\maketitle\\n\\\\begin{abstract}\\nWe prove the packet form of the higher-dimensional Duke equidistribution\\nproblem for totally real fields of any fixed prime degree at least five,\\nallowing arbitrary local homothety types of full lattices. As the\\nmultiplier-order discriminant tends to infinity, the volume-weighted packet\\nmeasures converge to Haar probability measure, with no escape of mass.\\n\\\\end{abstract}\\n\\n\\\\section{Introduction}\\\\label{sec:intro}\\n\\nCompact diagonal orbits in the space of unimodular lattices encode\\nideal classes and units of totally real number fields. The higher-dimensional\\nDuke equidistribution problem asks whether arithmetically complete families\\nof these orbits become uniformly distributed as their discriminants grow.\\nThe grouping matters: one local homothety type specifies a complete packet,\\nwhereas a multiplier order alone can combine several packets. We prove the\\nprime-degree packet statement for every such local type.\\n\\nFix a prime $n\\\\ge5$, and write\\n\\\\[\\n X_n=\\\\SL_n(\\\\Z)\\\\backslash\\\\SL_n(\\\\R),\\\\qquad\\n A_n=\\\\{\\\\diag(e^{u_1},\\\\ldots,e^{u_n}):\\\\textstyle\\\\sum_j u_j=0\\\\}.\\n\\\\]\\nWe regard $X_n$ as the space of covolume-one row lattices, with the\\nright action of $A_n$, and let $m_n$ be its $\\\\SL_n(\\\\R)$-invariant\\nprobability measure. Haar measure on $A_n$ is\\n$du_1\\\\cdots du_{n-1}$ in the displayed logarithmic coordinates.\\n\\nLet $K$ be a totally real field of degree $n$, let\\n$\\\\sigma=(\\\\sigma_1,\\\\ldots,\\\\sigma_n)$ order ')]\n\nPAPER 024 [('preprints/An-asymptotic-formula-for-the-number-of-totients-September-25-2026/build/source/main.tex', 32, \"enAI}\\n\\\\date{September 25, 2026}\\n\\\\begin{document}\\n\\\\maketitle\\n\\\\begin{abstract}\\nLet $V(x)$ count the distinct values of Euler's totient function up to~$x$. We give an explicit asymptotic equivalent for $V(x)$. Its coefficient is a uniform limit of functions defined from finite arithmetic data. We also prove that $V(cx)/V(x)\\\\to c$ as $x\\\\to\\\\infty$ for every fixed $c>0$, answering a question of Erd\\\\H{o}s and Hall.\\n\\\\end{abstract}\\n\\\\begingroup\\n\\\\small\\n\\\\setlength{\\\\baselineskip}{10.5pt}\\n\\\\tableofcontents\\n\\\\endgroup\\n\\\\clearpage\\n\\\\input{introduction}\\n\\\\input{statements}\\n\\\\input{extraction}\\n\\\\input{layers}\\n\\\\input{collisions}\\n\\\\input{limits}\\n\\\\input{companion}\\n\\\\clearpage\\n\\\\bibliographystyle{plain}\\n\\\\bibliography{references}\\n\\\\end{document}\\n\")]\n\nPAPER 046 [('preprints/A-projective-fourfold-with-large-fundamental-group-and-non-Stein-universal-cover-October-5-2026/build/main.tex', 12, \"ex hyperbolic arrangement, Stein manifold}\\n\\\\begin{document}\\n\\\\begin{abstract}\\nWe construct a smooth projective complex fourfold with large fundamental group\\nwhose universal cover is not Stein. The universal cover contains no\\npositive-dimensional compact complex-analytic subvariety, so this disproves\\nShafarevich's holomorphic-convexity conjecture even under the\\nlarge-fundamental-group hypothesis.\\n\\\\end{abstract}\\n\\\\maketitle\\n\\\\tableofcontents\\n\\\\input{sections/01-introduction}\\n\\\\input{sections/02-reduction}\\n\\\\input{sections/03-paths}\\n\\\\input{sections/04-geometry}\\n\\\\input{sections/05-models}\\n\\\\input{sections/06-image}\\n\\\\input{sections/07-projective}\\n\\\\bibliographystyle{amsplain}\\n\\\\bibliography{references}\\n\\\\end{document}\\n\")]\n\nPAPER 065 [('preprints/Virasoro-Constraints-under-Projectivization-October-5-2026/build/paper.tex', 8, '{OpenAI}\\n\\\\date{October 5, 2026}\\n\\\\begin{document}\\n\\\\maketitle\\n\\\\begin{abstract}\\nWe prove that full ordinary descendant Virasoro constraints pass from a\\nsmooth projective complex base to the projectivization of any algebraic\\nvector bundle of rank at least two. The bundle need not split and satisfies\\nno positivity requirement. Assuming the full constraints on the base, the\\nconclusion includes every genus, each individual integral curve class, and\\nall cohomology insertions, including primitive and odd classes. The result\\nalso applies successively to towers of projective bundles.\\n\\\\end{abstract}\\n\\\\tableofcontents\\n\\\\input{sections/01-introduction}\\n\\\\input{sections/02-conventions}\\n\\\\input{sections/03-master-space}\\n\\\\input{sections/04-localization}\\n\\\\input{sections/05-shift}\\n\\\\input{sections/06-spectral}\\n\\\\input{sections/07-fixed-block}\\n\\\\input{sections/08-continuation}\\n\\\\input{references}\\n\\\\end{document}\\n')]\n\nPAPER 082 [('preprints/Annular-variation-of-the-triangular-Hilbert-transform-at-the-symmetric-point-October-5-2026/build/main.tex', 44, '{OpenAI}\\n\\\\date{October 5, 2026}\\n\\\\begin{document}\\n\\\\maketitle\\n\\\\begin{abstract}\\nWe prove the annular $r$-variation estimate for the triangular Hilbert\\ntransform from complex $L^3\\\\times L^3$ to $L^{3/2}$ for every $r>2$.\\nThe partitions may depend on the output point and range over all positive\\nscales. The estimate yields the two-endpoint maximal bound and joint\\nalmost-everywhere and norm principal values, and resolves the symmetric\\nscalar triangular Hilbert transform problem.\\n\\\\end{abstract}\\n\\\\tableofcontents\\n\\\\input{sections/introduction}\\n\\\\input{sections/preliminaries}\\n\\\\input{sections/matrix}\\n\\\\input{sections/heat}\\n\\\\input{sections/smooth}\\n\\\\input{sections/frequency}\\n\\\\input{sections/rough}\\n\\\\input{sections/completion}\\n\\\\input{sections/consequences}\\n\\\\bibliographystyle{plain}\\n\\\\bibliography{references}\\n\\\\end{document}\\n')]\n"
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        "stdout": "\nPAPER 065\npreprints/Virasoro-Constraints-under-Projectivization-October-5-2026/build/paper.tex 8 \nWe prove that full ordinary descendant Virasoro constraints pass from a\nsmooth projective complex base to the projectivization of any algebraic\nvector bundle of rank at least two. The bundle need not split and satisfies\nno positivity requirement. Assuming the full constraints on the base, the\nconclusion includes every genus, each individual integral curve class, and\nall cohomology insertions, including primitive and odd classes. The result\nalso applies successively to towers of projective bundles.\n\n\nPAPER 082\npreprints/Annular-variation-of-the-triangular-Hilbert-transform-at-the-symmetric-point-October-5-2026/build/main.tex 44 \nWe prove the annular $r$-variation estimate for the triangular Hilbert\ntransform from complex $L^3\\times L^3$ to $L^{3/2}$ for every $r>2$.\nThe partitions may depend on the output point and range over all positive\nscales. The estimate yields the two-endpoint maximal bound and joint\nalmost-everywhere and norm principal values, and resolves the symmetric\nscalar triangular Hilbert transform problem.\n\n\nPAPER 113\npreprints/A-Fully-Polynomial-Randomized-Approximation-Scheme-for-Perfect-Matchings-in-General-Graphs-September-23-2026/build/main.tex 108 \nWe give a fully polynomial randomized approximation scheme (FPRAS) for\ncounting perfect matchings in arbitrary finite simple undirected graphs,\nresolving the general-graph perfect-matching approximation problem.\nThe algorithm returns zero with certainty when no perfect matching exists.\nOtherwise, it achieves relative error $\\eps$ with failure probability at\nmost $\\delta$ in worst-case bit time polynomial in the input length,\n$\\eps^{-1}$, and $\\log\\delta^{-1}$.\n\n\nPAPER 115\npreprints/Exact-Uniform-Sampling-of-Contingency-Tables-with-Arbitrary-Margins-September-24-2026/build/main.tex 22 \nWe give an exact uniform sampler for nonnegative integer contingency tables\nwith arbitrary prescribed margins. It terminates almost surely and has expected\nbit complexity polynomial in both dimensions and the binary length of the\nmargins. No positivity, balance, sparsity, or fixed-dimension assumption is\nrequired.\n\n\nPAPER 127\npreprints/Average-Sensitivity-of-Polynomial-Threshold-Functions-September-25-2026/build/main.tex 61 \nFor every $n\\ge1$ and $1\\le d\\le n$, we prove that a polynomial\nthreshold function of degree at most $d$ on the uniform Boolean cube has\naverage sensitivity at most $8d\\sqrt n$. This proves the asymptotic form\nof the Gotsman--Linial conjecture. The bound is uniform in both parameters\nand uses the convention $\\sgn(0)=1$.\n\n",
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          "text": "\nPAPER 065\npreprints/Virasoro-Constraints-under-Projectivization-October-5-2026/build/paper.tex 8 \nWe prove that full ordinary descendant Virasoro constraints pass from a\nsmooth projective complex base to the projectivization of any algebraic\nvector bundle of rank at least two. The bundle need not split and satisfies\nno positivity requirement. Assuming the full constraints on the base, the\nconclusion includes every genus, each individual integral curve class, and\nall cohomology insertions, including primitive and odd classes. The result\nalso applies successively to towers of projective bundles.\n\n\nPAPER 082\npreprints/Annular-variation-of-the-triangular-Hilbert-transform-at-the-symmetric-point-October-5-2026/build/main.tex 44 \nWe prove the annular $r$-variation estimate for the triangular Hilbert\ntransform from complex $L^3\\times L^3$ to $L^{3/2}$ for every $r>2$.\nThe partitions may depend on the output point and range over all positive\nscales. The estimate yields the two-endpoint maximal bound and joint\nalmost-everywhere and norm principal values, and resolves the symmetric\nscalar triangular Hilbert transform problem.\n\n\nPAPER 113\npreprints/A-Fully-Polynomial-Randomized-Approximation-Scheme-for-Perfect-Matchings-in-General-Graphs-September-23-2026/build/main.tex 108 \nWe give a fully polynomial randomized approximation scheme (FPRAS) for\ncounting perfect matchings in arbitrary finite simple undirected graphs,\nresolving the general-graph perfect-matching approximation problem.\nThe algorithm returns zero with certainty when no perfect matching exists.\nOtherwise, it achieves relative error $\\eps$ with failure probability at\nmost $\\delta$ in worst-case bit time polynomial in the input length,\n$\\eps^{-1}$, and $\\log\\delta^{-1}$.\n\n\nPAPER 115\npreprints/Exact-Uniform-Sampling-of-Contingency-Tables-with-Arbitrary-Margins-September-24-2026/build/main.tex 22 \nWe give an exact uniform sampler for nonnegative integer contingency tables\nwith arbitrary prescribed margins. It terminates almost surely and has expected\nbit complexity polynomial in both dimensions and the binary length of the\nmargins. No positivity, balance, sparsity, or fixed-dimension assumption is\nrequired.\n\n\nPAPER 127\npreprints/Average-Sensitivity-of-Polynomial-Threshold-Functions-September-25-2026/build/main.tex 61 \nFor every $n\\ge1$ and $1\\le d\\le n$, we prove that a polynomial\nthreshold function of degree at most $d$ on the uniform Boolean cube has\naverage sensitivity at most $8d\\sqrt n$. This proves the asymptotic form\nof the Gotsman--Linial conjecture. The bound is uniform in both parameters\nand uses the convention $\\sgn(0)=1$.\n\n"
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        "stdout": "\nPAPER 148\npreprints/The-entropy-rate-dimension-formula-for-self-similar-measures-on-the-line-September-24-2026/build/main.tex 45 \nWe prove that the Hausdorff dimension of every finite real self-similar\nmeasure equals the minimum of one and its random-walk entropy rate divided\nby its Lyapunov exponent. Exact overlaps are allowed, and the contraction\nratios may be unequal and negative. This resolves the entropy-rate dimension\nconjecture.\n\n\nPAPER 154\npreprints/Pointwise-Multiple-Ergodic-Averages-for-Mixing-Transformations-October-4-2026/build/main.tex 53 \nLet $T$ be an invertible mixing probability-preserving transformation.\nFor every integer $n\\ge2$ and every fixed tuple of bounded measurable\nfunctions, we prove that the consecutive multiple ergodic averages of\nlength $n$ converge almost everywhere to the product of the integrals,\nas the averaging length tends to infinity through all positive integers.\nThe probability space need not be standard, and no rate of mixing is\nrequired.\n\n\nPAPER 164\npreprints/Monochromatic-finite-sums-and-products-in-the-positive-integers-September-23-2026/build/main.tex 66 \nWe prove Hindman's finite sums and products conjecture: for every finite\ncoloring of the positive integers and every positive integer $k$, there is a\n$k$-element set whose nonempty subset sums and nonempty subset products all\nhave the same color.\n\n\nPAPER 192\npreprints/Unbounded-Violations-of-the-Square-Root-Degree-Bound-September-26-2026/build/main.tex 26 \nWe disprove the Gopalan--Servedio square-root conjecture, even up to an\narbitrary constant factor. For every real \\(C>0\\), there is a nonconstant\nBoolean function \\(f:\\{-1,1\\}^n\\to\\{-1,1\\}\\) on a finite sign cube such that\n\\[\n\\sum_{i=1}^n \\widehat f(\\{i\\})>C\\sqrt{\\deg(f)}.\n\\]\nHere \\(\\widehat f(\\{i\\})\\) is the linear Fourier coefficient associated with\nthe \\(i\\)th input, and \\(\\deg(f)\\) is the degree of the real multilinear\npolynomial representing \\(f\\).\n\n\nPAPER 210\npreprints/Foulkes-Conjecture-for-the-Sixth-Symmetric-Power-September-25-2026/build/main.tex 90 \nWe prove the sixth-symmetric-power case of Foulkes' conjecture. For every\ninteger \\(b\\ge6\\) and every finite-dimensional complex vector space \\(V\\),\nthere is a \\(\\GL(V)\\)-equivariant injection\n\\(\\Sym^6(\\Sym^b V)\\hookrightarrow\\Sym^b(\\Sym^6 V)\\).\n\n",
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          "\nPAPER 148\npreprints/The-entropy-rate-dimension-formula-for-self-similar-measures-on-the-line-September-24-2026/build/main.tex 45 \nWe prove that the Hausdorff dimension of every finite real self-similar\nmeasure equals the minimum of one and its random-walk entropy rate divided\nby its Lyapunov exponent. Exact overlaps are allowed, and the contraction\nratios may be unequal and negative. This resolves the entropy-rate dimension\nconjecture.\n\n\nPAPER 154\npreprints/Pointwise-Multiple-Ergodic-Averages-for-Mixing-Transformations-October-4-2026/build/main.tex 53 \nLet $T$ be an invertible mixing probability-preserving transformation.\nFor every integer $n\\ge2$ and every fixed tuple of bounded measurable\nfunctions, we prove that the consecutive multiple ergodic averages of\nlength $n$ converge almost everywhere to the product of the integrals,\nas the averaging length tends to infinity through all positive integers.\nThe probability space need not be standard, and no rate of mixing is\nrequired.\n\n\nPAPER 164\npreprints/Monochromatic-finite-sums-and-products-in-the-positive-integers-September-23-2026/build/main.tex 66 \nWe prove Hindman's finite sums and products conjecture: for every finite\ncoloring of the positive integers and every positive integer $k$, there is a\n$k$-element set whose nonempty subset sums and nonempty subset products all\nhave the same color.\n\n\nPAPER 192\npreprints/Unbounded-Violations-of-the-Square-Root-Degree-Bound-September-26-2026/build/main.tex 26 \nWe disprove the Gopalan--Servedio square-root conjecture, even up to an\narbitrary constant factor. For every real \\(C>0\\), there is a nonconstant\nBoolean function \\(f:\\{-1,1\\}^n\\to\\{-1,1\\}\\) on a finite sign cube such that\n\\[\n\\sum_{i=1}^n \\widehat f(\\{i\\})>C\\sqrt{\\deg(f)}.\n\\]\nHere \\(\\widehat f(\\{i\\})\\) is the linear Fourier coefficient associated with\nthe \\(i\\)th input, and \\(\\deg(f)\\) is the degree of the real multilinear\npolynomial representing \\(f\\).\n\n\nPAPER 210\npreprints/Foulkes-Conjecture-for-the-Sixth-Symmetric-Power-September-25-2026/build/main.tex 90 \nWe prove the sixth-symmetric-power case of Foulkes' conjecture. For every\ninteger \\(b\\ge6\\) and every finite-dimensional complex vector space \\(V\\),\nthere is a \\(\\GL(V)\\)-equivariant injection\n\\(\\Sym^6(\\Sym^b V)\\hookrightarrow\\Sym^b(\\Sym^6 V)\\).\n\n"
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          "text": "\nPAPER 148\npreprints/The-entropy-rate-dimension-formula-for-self-similar-measures-on-the-line-September-24-2026/build/main.tex 45 \nWe prove that the Hausdorff dimension of every finite real self-similar\nmeasure equals the minimum of one and its random-walk entropy rate divided\nby its Lyapunov exponent. Exact overlaps are allowed, and the contraction\nratios may be unequal and negative. This resolves the entropy-rate dimension\nconjecture.\n\n\nPAPER 154\npreprints/Pointwise-Multiple-Ergodic-Averages-for-Mixing-Transformations-October-4-2026/build/main.tex 53 \nLet $T$ be an invertible mixing probability-preserving transformation.\nFor every integer $n\\ge2$ and every fixed tuple of bounded measurable\nfunctions, we prove that the consecutive multiple ergodic averages of\nlength $n$ converge almost everywhere to the product of the integrals,\nas the averaging length tends to infinity through all positive integers.\nThe probability space need not be standard, and no rate of mixing is\nrequired.\n\n\nPAPER 164\npreprints/Monochromatic-finite-sums-and-products-in-the-positive-integers-September-23-2026/build/main.tex 66 \nWe prove Hindman's finite sums and products conjecture: for every finite\ncoloring of the positive integers and every positive integer $k$, there is a\n$k$-element set whose nonempty subset sums and nonempty subset products all\nhave the same color.\n\n\nPAPER 192\npreprints/Unbounded-Violations-of-the-Square-Root-Degree-Bound-September-26-2026/build/main.tex 26 \nWe disprove the Gopalan--Servedio square-root conjecture, even up to an\narbitrary constant factor. For every real \\(C>0\\), there is a nonconstant\nBoolean function \\(f:\\{-1,1\\}^n\\to\\{-1,1\\}\\) on a finite sign cube such that\n\\[\n\\sum_{i=1}^n \\widehat f(\\{i\\})>C\\sqrt{\\deg(f)}.\n\\]\nHere \\(\\widehat f(\\{i\\})\\) is the linear Fourier coefficient associated with\nthe \\(i\\)th input, and \\(\\deg(f)\\) is the degree of the real multilinear\npolynomial representing \\(f\\).\n\n\nPAPER 210\npreprints/Foulkes-Conjecture-for-the-Sixth-Symmetric-Power-September-25-2026/build/main.tex 90 \nWe prove the sixth-symmetric-power case of Foulkes' conjecture. For every\ninteger \\(b\\ge6\\) and every finite-dimensional complex vector space \\(V\\),\nthere is a \\(\\GL(V)\\)-equivariant injection\n\\(\\Sym^6(\\Sym^b V)\\hookrightarrow\\Sym^b(\\Sym^6 V)\\).\n\n"
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        "stdout": "\nPAPER 215\npreprints/The-canonical-massive-continuum-limit-of-the-two-dimensional-O3-model-October-4-2026/build/main.tex 48 \nWe construct a canonical interacting massive continuum limit of the\ntwo-dimensional nearest-neighbor $O(3)$ model with unit-length spins,\nno external field, and no topological term. Normalized by susceptibility\nand second-moment correlation length, the limit exists as the bare coupling\ntends to infinity through all positive real values, without selecting\nsubsequences. The limiting fields satisfy the Osterwalder--Schrader axioms\nand have a nonzero connected four-point correlation on separated time\nsupports. Their reconstructed theory has a unique vacuum, a nonzero vacuum\ncomplement, and a positive Hamiltonian gap on that entire complement.\n\n\nPAPER 230\npreprints/An-exact-Hausdorff-gauge-for-SLE-September-25-2026/build/main.tex 29 \nFor each \\(0<\\kappa<8\\), we construct a deterministic Hausdorff gauge\nthat almost surely assigns positive finite measure to every nontrivial\ncompact positive-time segment of chordal Schramm--Loewner evolution.\nThis answers Schramm's Hausdorff-measure existence problem in this\nparameter range. The entire trace has finite expected gauge measure\nin each bounded box.\n\n\nPAPER 245\npreprints/Weak-and-strong-normalization-in-pure-type-systems-September-25-2026/build/paper.tex 30 \nWe prove that every weakly $\\beta$-normalizing pure type system is strongly\n$\\beta$-normalizing. Both properties quantify over all legal expressions\nin all valid contexts, and reduction acts inside type annotations.\nNo functionality hypothesis is required. This resolves the\n$\\beta$-Barendregt--Geuvers--Klop conjecture.\n\n\nPAPER 296\npreprints/Relative-generation-and-the-generator-problem-for-finite-factors-September-23-2026/build/main.tex 8 \nWe prove that every type $\\mathrm{II}_1$ factor with separable predual is\ngenerated by one operator, equivalently by two self-adjoint operators.\nCombined with the established direct-integral reduction to this case,\nthis gives an affirmative solution of the generator problem for von\nNeumann algebras with separable predual.\n\n\nPAPER 351\npreprints/A-closed-Ricci-flow-with-bounded-scalar-curvature-and-finite-time-curvature-blowup-September-24-2026/build/sections/01-introduction.tex 1 \nWe disprove the scalar-curvature extension conjecture in its unrestricted\nall-dimensions form. In sufficiently high dimension, we construct a Ricci\nflow on a closed manifold whose scalar curvature remains uniformly bounded\nwhile full curvature diverges at a finite maximal time. In one fixed\nsufficiently high dimension, the examples have two-sided power-law curvature\nblowup with arbitrarily large exponents.\n\n\nPAPER 369\npreprints/Strict-hot-spots-and-absence-of-interior-critical-points-on-smooth-simply-connected-planar-domains-September-24-2026/build/main.tex 59 \nWe prove the strict hot spots conjecture for smooth bounded simply connected planar domains. More precisely, every nonzero eigenfunction for the first positive Neumann eigenvalue has nonvanishing gradient in the interior, so all its global maxima and minima lie on the boundary. This holds even when the eigenvalue is multiple.\n\n\nPAPER 372\npreprints/Global-Uniqueness-for-the-Smooth-Isotropic-Elasticity-Inverse-Problem-September-24-2026/build/main.tex 45 \nWe prove that the full static displacement-to-traction map uniquely determines both real smooth Lam\\'e moduli on every bounded connected smooth domain in $\\R^3$, provided $\\mu>0$ and $3\\lambda+2\\mu>0$ on the closure. This resolves the smooth three-dimensional isotropic elastic Calder\\'on uniqueness problem under these positivity assumptions.\n\n",
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          "\nPAPER 215\npreprints/The-canonical-massive-continuum-limit-of-the-two-dimensional-O3-model-October-4-2026/build/main.tex 48 \nWe construct a canonical interacting massive continuum limit of the\ntwo-dimensional nearest-neighbor $O(3)$ model with unit-length spins,\nno external field, and no topological term. Normalized by susceptibility\nand second-moment correlation length, the limit exists as the bare coupling\ntends to infinity through all positive real values, without selecting\nsubsequences. The limiting fields satisfy the Osterwalder--Schrader axioms\nand have a nonzero connected four-point correlation on separated time\nsupports. Their reconstructed theory has a unique vacuum, a nonzero vacuum\ncomplement, and a positive Hamiltonian gap on that entire complement.\n\n\nPAPER 230\npreprints/An-exact-Hausdorff-gauge-for-SLE-September-25-2026/build/main.tex 29 \nFor each \\(0<\\kappa<8\\), we construct a deterministic Hausdorff gauge\nthat almost surely assigns positive finite measure to every nontrivial\ncompact positive-time segment of chordal Schramm--Loewner evolution.\nThis answers Schramm's Hausdorff-measure existence problem in this\nparameter range. The entire trace has finite expected gauge measure\nin each bounded box.\n\n\nPAPER 245\npreprints/Weak-and-strong-normalization-in-pure-type-systems-September-25-2026/build/paper.tex 30 \nWe prove that every weakly $\\beta$-normalizing pure type system is strongly\n$\\beta$-normalizing. Both properties quantify over all legal expressions\nin all valid contexts, and reduction acts inside type annotations.\nNo functionality hypothesis is required. This resolves the\n$\\beta$-Barendregt--Geuvers--Klop conjecture.\n\n\nPAPER 296\npreprints/Relative-generation-and-the-generator-problem-for-finite-factors-September-23-2026/build/main.tex 8 \nWe prove that every type $\\mathrm{II}_1$ factor with separable predual is\ngenerated by one operator, equivalently by two self-adjoint operators.\nCombined with the established direct-integral reduction to this case,\nthis gives an affirmative solution of the generator problem for von\nNeumann algebras with separable predual.\n\n\nPAPER 351\npreprints/A-closed-Ricci-flow-with-bounded-scalar-curvature-and-finite-time-curvature-blowup-September-24-2026/build/sections/01-introduction.tex 1 \nWe disprove the scalar-curvature extension conjecture in its unrestricted\nall-dimensions form. In sufficiently high dimension, we construct a Ricci\nflow on a closed manifold whose scalar curvature remains uniformly bounded\nwhile full curvature diverges at a finite maximal time. In one fixed\nsufficiently high dimension, the examples have two-sided power-law curvature\nblowup with arbitrarily large exponents.\n\n\nPAPER 369\npreprints/Strict-hot-spots-and-absence-of-interior-critical-points-on-smooth-simply-connected-planar-domains-September-24-2026/build/main.tex 59 \nWe prove the strict hot spots conjecture for smooth bounded simply connected planar domains. More precisely, every nonzero eigenfunction for the first positive Neumann eigenvalue has nonvanishing gradient in the interior, so all its global maxima and minima lie on the boundary. This holds even when the eigenvalue is multiple.\n\n\nPAPER 372\npreprints/Global-Uniqueness-for-the-Smooth-Isotropic-Elasticity-Inverse-Problem-September-24-2026/build/main.tex 45 \nWe prove that the full static displacement-to-traction map uniquely determines both real smooth Lam\\'e moduli on every bounded connected smooth domain in $\\R^3$, provided $\\mu>0$ and $3\\lambda+2\\mu>0$ on the closure. This resolves the smooth three-dimensional isotropic elastic Calder\\'on uniqueness problem under these positivity assumptions.\n\n"
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          "text": "\nPAPER 215\npreprints/The-canonical-massive-continuum-limit-of-the-two-dimensional-O3-model-October-4-2026/build/main.tex 48 \nWe construct a canonical interacting massive continuum limit of the\ntwo-dimensional nearest-neighbor $O(3)$ model with unit-length spins,\nno external field, and no topological term. Normalized by susceptibility\nand second-moment correlation length, the limit exists as the bare coupling\ntends to infinity through all positive real values, without selecting\nsubsequences. The limiting fields satisfy the Osterwalder--Schrader axioms\nand have a nonzero connected four-point correlation on separated time\nsupports. Their reconstructed theory has a unique vacuum, a nonzero vacuum\ncomplement, and a positive Hamiltonian gap on that entire complement.\n\n\nPAPER 230\npreprints/An-exact-Hausdorff-gauge-for-SLE-September-25-2026/build/main.tex 29 \nFor each \\(0<\\kappa<8\\), we construct a deterministic Hausdorff gauge\nthat almost surely assigns positive finite measure to every nontrivial\ncompact positive-time segment of chordal Schramm--Loewner evolution.\nThis answers Schramm's Hausdorff-measure existence problem in this\nparameter range. The entire trace has finite expected gauge measure\nin each bounded box.\n\n\nPAPER 245\npreprints/Weak-and-strong-normalization-in-pure-type-systems-September-25-2026/build/paper.tex 30 \nWe prove that every weakly $\\beta$-normalizing pure type system is strongly\n$\\beta$-normalizing. Both properties quantify over all legal expressions\nin all valid contexts, and reduction acts inside type annotations.\nNo functionality hypothesis is required. This resolves the\n$\\beta$-Barendregt--Geuvers--Klop conjecture.\n\n\nPAPER 296\npreprints/Relative-generation-and-the-generator-problem-for-finite-factors-September-23-2026/build/main.tex 8 \nWe prove that every type $\\mathrm{II}_1$ factor with separable predual is\ngenerated by one operator, equivalently by two self-adjoint operators.\nCombined with the established direct-integral reduction to this case,\nthis gives an affirmative solution of the generator problem for von\nNeumann algebras with separable predual.\n\n\nPAPER 351\npreprints/A-closed-Ricci-flow-with-bounded-scalar-curvature-and-finite-time-curvature-blowup-September-24-2026/build/sections/01-introduction.tex 1 \nWe disprove the scalar-curvature extension conjecture in its unrestricted\nall-dimensions form. In sufficiently high dimension, we construct a Ricci\nflow on a closed manifold whose scalar curvature remains uniformly bounded\nwhile full curvature diverges at a finite maximal time. In one fixed\nsufficiently high dimension, the examples have two-sided power-law curvature\nblowup with arbitrarily large exponents.\n\n\nPAPER 369\npreprints/Strict-hot-spots-and-absence-of-interior-critical-points-on-smooth-simply-connected-planar-domains-September-24-2026/build/main.tex 59 \nWe prove the strict hot spots conjecture for smooth bounded simply connected planar domains. More precisely, every nonzero eigenfunction for the first positive Neumann eigenvalue has nonvanishing gradient in the interior, so all its global maxima and minima lie on the boundary. This holds even when the eigenvalue is multiple.\n\n\nPAPER 372\npreprints/Global-Uniqueness-for-the-Smooth-Isotropic-Elasticity-Inverse-Problem-September-24-2026/build/main.tex 45 \nWe prove that the full static displacement-to-traction map uniquely determines both real smooth Lam\\'e moduli on every bounded connected smooth domain in $\\R^3$, provided $\\mu>0$ and $3\\lambda+2\\mu>0$ on the closure. This resolves the smooth three-dimensional isotropic elastic Calder\\'on uniqueness problem under these positivity assumptions.\n\n"
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        "stdout": "\nFILE lean/ComparatorChallenges/GotsmanLinial.lean \nimport Mathlib.Algebra.MvPolynomial.Degrees\nimport Mathlib.Algebra.BigOperators.Field\nimport Mathlib.Data.Fintype.BigOperators\nimport Mathlib.Analysis.Real.Sqrt\nimport Mathlib.Tactic.NormNum\n\nnamespace OAI\n\nopen scoped BigOperators\n\nnamespace LeanBlast.GotsmanLinial\n\nabbrev Cube (n : ℕ) := Fin n → Bool\n\ndef cubeCoord {n : ℕ} (x : Cube n) (i : Fin n) : ℝ :=\n  if x i then 1 else -1\n\ndef flip {n : ℕ} (i : Fin n) (x : Cube n) : Cube n :=\n  Function.update x i (!(x i))\n\nnoncomputable def thresholdSign (t : ℝ) : ℝ :=\n  if 0 ≤ t then 1 else -1\n\nnoncomputable def sensitiveVertices {n : ℕ} (f : Cube n → ℝ) (i : Fin n) :\n    Finset (Cube n) := by\n  classical\n  exact Finset.univ.filter fun x => f x ≠ f (flip i x)\n\nnoncomputable def averageSensitivity {n : ℕ} (f : Cube n → ℝ) : ℝ :=\n  ∑ i : Fin n, ((sensitiveVertices f i).card : ℝ) / (2 : ℝ) ^ n\n\ndef IsMultilinear {n : ℕ} (p : MvPolynomial (Fin n) ℝ) : Prop :=\n  ∀ m ∈ p.support, ∀ i : Fin n, m i ≤ 1\n\nnoncomputable def polynomialValue {n : ℕ} (p : MvPolynomial (Fin n) ℝ)\n    (x : Cube n) : ℝ :=\n  MvPolynomial.eval (cubeCoord x) p\n\nnoncomputable def polynomialThreshold {n : ℕ} (p : MvPolynomial (Fin n) ℝ) :\n    Cube n → ℝ :=\n  fun x => thresholdSign (polynomialValue p x)\n\ndef GotsmanLinialStatement : Prop :=\n  ∀ (n d : ℕ), 1 ≤ n → 1 ≤ d → d ≤ n →\n    ∀ p : MvPolynomial (Fin n) ℝ,\n      IsMultilinear p → p.totalDegree ≤ d →\n        averageSensitivity (polynomialThreshold p) ≤ 8 * (d : ℝ) * Real.sqrt (n : ℝ)\n\ntheorem gotsmanLinialStatement : GotsmanLinialStatement := by\n  sorry\n\nend LeanBlast.GotsmanLinial\n\nend OAI\n\n\nFILE lean/ComparatorChallenges/SelfSimilar.lean \nimport Mathlib\n\nnamespace OAI\n\nuniverse uι\n\nopen scoped BigOperators ENNReal\nopen MeasureTheory\n\nnamespace EntropyRateDimension\n\nstructure System (ι : Type uι) [Fintype ι] where\n  ratio : ι → ℝ\n  offset : ι → ℝ\n  weight : ι → ℝ\n  ratio_pos : ∀ i, 0 < |ratio i|\n  ratio_lt_one : ∀ i, |ratio i| < 1\n  weight_pos : ∀ i, 0 < weight i\n  weight_sum : ∑ i, weight i = 1\n\nnamespace System\n\nvariable {ι : Type uι} [Fintype ι]\n\ndef affine (S : System ι) (i : ι) (x : ℝ) : ℝ := S.ratio i * x + S.offset i\n\ndef wordAffine (S : System ι) : List ι → ℝ × ℝ\n  | [] => (1, 0)\n  | i :: w => (S.ratio i * (S.wordAffine w).1,\n      S.ratio i * (S.wordAffine w).2 + S.offset i)\n\ndef completeMap (S : System ι) {n : ℕ} (w : Fin n → ι) : ℝ × ℝ :=\n  S.wordAffine (List.ofFn w)\n\nnoncomputable def wordWeight (S : System ι) {n : ℕ} (w : Fin n → ι) : ℝ :=\n  ∏ j, S.weight (w j)\n\nnoncomputable def mapSupport (S : System ι) (n : ℕ) : Finset (ℝ × ℝ) := by\n  classical\n  exact Finset.univ.image (S.completeMap (n := n))\n\nnoncomputable def mapMass (S : System ι) (n : ℕ) (g : ℝ × ℝ) : ℝ := by\n  classical\n  exact ∑ w ∈ (Finset.univ : Finset (Fin n → ι)).filter (S.completeMap · = g),\n    S.wordWeight w\n\nnoncomputable def walkEntropy (S : System ι) (n : ℕ) : ℝ :=\n  (∑ g ∈ S.mapSupport n, Real.negMulLog (S.mapMass n g)) / Real.log 2\n\nnoncomputable def entropyRate (S : System ι) : ℝ :=\n  sInf (Set.range fun n : ℕ => S.walkEntropy (n + 1) / (n + 1 : ℕ))\n\nnoncomputable def lyapunov (S : System ι) : ℝ :=\n  -(∑ i, S.weight i * (Real.log |S.ratio i| / Real.log 2))\n\ndef SelfSimilar (S : System ι) (μ : Measure ℝ) : Prop :=\n  μ = ∑ i, ENNReal.ofReal (S.weight i) • μ.map (S.affine i)\n\nend System\n\nnoncomputable def lowerHausdorffDimension (μ : Measure ℝ) : ℝ≥0∞ :=\n  ⨅ (E : Set ℝ) (_ : MeasurableSet E) (_ : 0 < μ E), dimH E\nopen MeasureTheory\n\ntheorem entropy_rate_dimension {ι : Type uι} [Fintype ι] [Nonempty ι]\n    (S : System ι) (μ : Measure ℝ) [IsProbabilityMeasure μ]\n    (hμ : S.SelfSimilar μ) :\n    lowerHausdorffDimension μ = ENNReal.ofReal (min 1 (S.entropyRate / S.lyapunov)) := by\n  sorry\n\nend EntropyRateDimension\n\nend OAI\n\n\nFILE lean/ComparatorChallenges/FactorGeneration.lean \nimport Mathlib.Analysis.CStarAlgebra.ContinuousLinearMap\nimport Mathlib.Analysis.InnerProductSpace.WeakOperatorTopology\nimport Mathlib.Analysis.VonNeumannAlgebra.Basic\n\nnamespace OAI\n\nnoncomputable section\n\nnamespace Generator\n\nuniverse u\n\nvariable {H : Type u} [NormedAddCommGroup H] [InnerProductSpace ℂ H] [CompleteSpace H]\n\nattribute [-instance] instCStarAlgebraContinuousLinearMapComplexIdOfCompleteSpace\n\ndef WOTClosed (S : StarSubalgebra ℂ (H →L[ℂ] H)) : Prop :=\n  IsClosed {T : H →WOT[ℂ] H | T.toCLM ∈ S}\n\ndef wstar (s : Set (H →L[ℂ] H)) : StarSubalgebra ℂ (H →L[ℂ] H) :=\n  sInf {S | WOTClosed S ∧ s ⊆ S}\n\nstructure IsII1Factor (S : StarSubalgebra ℂ (H →L[ℂ] H)) : Prop where\n  nonzero : (0 : H →L[ℂ] H) ≠ 1\n  weaklyClosed : WOTClosed S\n  factor : ∀ x : S, (∀ y : S, y * x = x * y) → ∃ c : ℂ, x = algebraMap ℂ S c\n  finite : ∀ v : S, star v * v = 1 → v * star v = 1\n  diffuse : ∀ p : S, IsStarProjection p → p ≠ 0 →\n    ∃ q : S, IsStarProjection q ∧ q ≠ 0 ∧ q ≠ p ∧ p * q = q ∧ q * p = q\n\ndef HasSeparablePredual (S : StarSubalgebra ℂ (H →L[ℂ] H)) : Prop :=\n  ∃ (X : Type u) (_ : NormedAddCommGroup X) (_ : NormedSpace ℂ X)\n    (_ : CompleteSpace X) (_ : TopologicalSpace.SeparableSpace X),\n    Nonempty (StrongDual ℂ X ≃ₗᵢ⋆[ℂ] S)\n\ndef SinglyGenerated (S : StarSubalgebra ℂ (H →L[ℂ] H)) : Prop :=\n  ∃ x ∈ S, wstar {x} = S\n\nattribute [instance] instCStarAlgebraContinuousLinearMapComplexIdOfCompleteSpace\n\ntheorem single_generation_of_II1_separable_predual\n    (S : StarSubalgebra ℂ (H →L[ℂ] H)) (hS : IsII1Factor S)\n    (hsep : HasSeparablePredual S) : SinglyGenerated S := by\n  sorry\n\nend Generator\n\nend\n\nend OAI\n\n\nFILE lean/ComparatorChallenges/FoulkesHowe.lean \nimport Mathlib\n\nnamespace OAI\n\nnoncomputable section\n\nopen scoped BigOperators\nuniverse u\n\nnamespace Problem346\n\ndef symMonomialRaw (n : ℕ) (V : Type u) [AddCommGroup V] [Module ℂ V]\n    (v : Fin n → V) : SymmetricAlgebra ℂ V :=\n  ∏ i : Fin n, SymmetricAlgebra.ι ℂ V (v i)\n\ndef symPowSubmodule (n : ℕ) (V : Type u) [AddCommGroup V] [Module ℂ V] :\n    Submodule ℂ (SymmetricAlgebra ℂ V) :=\n  Submodule.span ℂ (Set.range (symMonomialRaw n V))\n\nabbrev SymPow (n : ℕ) (V : Type u) [AddCommGroup V] [Module ℂ V] :=\n  ↥(symPowSubmodule n V)\n\ndef symMonomial (n : ℕ) (V : Type u) [AddCommGroup V] [Module ℂ V]\n    (v : Fin n → V) : SymPow n V :=\n  ⟨symMonomialRaw n V v, Submodule.subset_span (Set.mem_range_self v)⟩\n\ndef foulkesFormula (a b : ℕ) (V : Type u) [AddCommGroup V] [Module ℂ V]\n    (v : Fin b → Fin a → V) : SymPow a (SymPow b V) := by\n  classical\n  exact\n    ((a.factorial : ℂ) ^ b)⁻¹ •\n      ∑ σ : Fin b → Equiv.Perm (Fin a),\n        symMonomial a (SymPow b V)\n          (fun i => symMonomial b V (fun j => v j ((σ j) i)))\n\ndef IsFoulkesMap (a b : ℕ) (V : Type u) [AddCommGroup V] [Module ℂ V]\n    (μ : SymPow b (SymPow a V) →ₗ[ℂ] SymPow a (SymPow b V)) : Prop :=\n  ∀ v : Fin b → Fin a → V,\n    μ (symMonomial b (SymPow a V) (fun j => symMonomial a V (v j))) =\n      foulkesFormula a b V v\n\ntheorem canonical_foulkes_howe_surjective :\n    ∀ (V : Type u) [AddCommGroup V] [Module ℂ V] [FiniteDimensional ℂ V] (a b : ℕ), 2 ≤ a → a * (a - 1) ≤ b → ∃ μ : SymPow b (SymPow a V) →ₗ[ℂ] SymPow a (SymPow b V), IsFoulkesMap a b V μ ∧ Function.Surjective μ ∧ ∀ ν : SymPow b (SymPow a V) →ₗ[ℂ] SymPow a (SymPow b V), IsFoulkesMap a b V ν → ν = μ := by\n  sorry\n\nend Problem346\n\nend\n\nend OAI\n\n",
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          "\nFILE lean/ComparatorChallenges/GotsmanLinial.lean \nimport Mathlib.Algebra.MvPolynomial.Degrees\nimport Mathlib.Algebra.BigOperators.Field\nimport Mathlib.Data.Fintype.BigOperators\nimport Mathlib.Analysis.Real.Sqrt\nimport Mathlib.Tactic.NormNum\n\nnamespace OAI\n\nopen scoped BigOperators\n\nnamespace LeanBlast.GotsmanLinial\n\nabbrev Cube (n : ℕ) := Fin n → Bool\n\ndef cubeCoord {n : ℕ} (x : Cube n) (i : Fin n) : ℝ :=\n  if x i then 1 else -1\n\ndef flip {n : ℕ} (i : Fin n) (x : Cube n) : Cube n :=\n  Function.update x i (!(x i))\n\nnoncomputable def thresholdSign (t : ℝ) : ℝ :=\n  if 0 ≤ t then 1 else -1\n\nnoncomputable def sensitiveVertices {n : ℕ} (f : Cube n → ℝ) (i : Fin n) :\n    Finset (Cube n) := by\n  classical\n  exact Finset.univ.filter fun x => f x ≠ f (flip i x)\n\nnoncomputable def averageSensitivity {n : ℕ} (f : Cube n → ℝ) : ℝ :=\n  ∑ i : Fin n, ((sensitiveVertices f i).card : ℝ) / (2 : ℝ) ^ n\n\ndef IsMultilinear {n : ℕ} (p : MvPolynomial (Fin n) ℝ) : Prop :=\n  ∀ m ∈ p.support, ∀ i : Fin n, m i ≤ 1\n\nnoncomputable def polynomialValue {n : ℕ} (p : MvPolynomial (Fin n) ℝ)\n    (x : Cube n) : ℝ :=\n  MvPolynomial.eval (cubeCoord x) p\n\nnoncomputable def polynomialThreshold {n : ℕ} (p : MvPolynomial (Fin n) ℝ) :\n    Cube n → ℝ :=\n  fun x => thresholdSign (polynomialValue p x)\n\ndef GotsmanLinialStatement : Prop :=\n  ∀ (n d : ℕ), 1 ≤ n → 1 ≤ d → d ≤ n →\n    ∀ p : MvPolynomial (Fin n) ℝ,\n      IsMultilinear p → p.totalDegree ≤ d →\n        averageSensitivity (polynomialThreshold p) ≤ 8 * (d : ℝ) * Real.sqrt (n : ℝ)\n\ntheorem gotsmanLinialStatement : GotsmanLinialStatement := by\n  sorry\n\nend LeanBlast.GotsmanLinial\n\nend OAI\n\n\nFILE lean/ComparatorChallenges/SelfSimilar.lean \nimport Mathlib\n\nnamespace OAI\n\nuniverse uι\n\nopen scoped BigOperators ENNReal\nopen MeasureTheory\n\nnamespace EntropyRateDimension\n\nstructure System (ι : Type uι) [Fintype ι] where\n  ratio : ι → ℝ\n  offset : ι → ℝ\n  weight : ι → ℝ\n  ratio_pos : ∀ i, 0 < |ratio i|\n  ratio_lt_one : ∀ i, |ratio i| < 1\n  weight_pos : ∀ i, 0 < weight i\n  weight_sum : ∑ i, weight i = 1\n\nnamespace System\n\nvariable {ι : Type uι} [Fintype ι]\n\ndef affine (S : System ι) (i : ι) (x : ℝ) : ℝ := S.ratio i * x + S.offset i\n\ndef wordAffine (S : System ι) : List ι → ℝ × ℝ\n  | [] => (1, 0)\n  | i :: w => (S.ratio i * (S.wordAffine w).1,\n      S.ratio i * (S.wordAffine w).2 + S.offset i)\n\ndef completeMap (S : System ι) {n : ℕ} (w : Fin n → ι) : ℝ × ℝ :=\n  S.wordAffine (List.ofFn w)\n\nnoncomputable def wordWeight (S : System ι) {n : ℕ} (w : Fin n → ι) : ℝ :=\n  ∏ j, S.weight (w j)\n\nnoncomputable def mapSupport (S : System ι) (n : ℕ) : Finset (ℝ × ℝ) := by\n  classical\n  exact Finset.univ.image (S.completeMap (n := n))\n\nnoncomputable def mapMass (S : System ι) (n : ℕ) (g : ℝ × ℝ) : ℝ := by\n  classical\n  exact ∑ w ∈ (Finset.univ : Finset (Fin n → ι)).filter (S.completeMap · = g),\n    S.wordWeight w\n\nnoncomputable def walkEntropy (S : System ι) (n : ℕ) : ℝ :=\n  (∑ g ∈ S.mapSupport n, Real.negMulLog (S.mapMass n g)) / Real.log 2\n\nnoncomputable def entropyRate (S : System ι) : ℝ :=\n  sInf (Set.range fun n : ℕ => S.walkEntropy (n + 1) / (n + 1 : ℕ))\n\nnoncomputable def lyapunov (S : System ι) : ℝ :=\n  -(∑ i, S.weight i * (Real.log |S.ratio i| / Real.log 2))\n\ndef SelfSimilar (S : System ι) (μ : Measure ℝ) : Prop :=\n  μ = ∑ i, ENNReal.ofReal (S.weight i) • μ.map (S.affine i)\n\nend System\n\nnoncomputable def lowerHausdorffDimension (μ : Measure ℝ) : ℝ≥0∞ :=\n  ⨅ (E : Set ℝ) (_ : MeasurableSet E) (_ : 0 < μ E), dimH E\nopen MeasureTheory\n\ntheorem entropy_rate_dimension {ι : Type uι} [Fintype ι] [Nonempty ι]\n    (S : System ι) (μ : Measure ℝ) [IsProbabilityMeasure μ]\n    (hμ : S.SelfSimilar μ) :\n    lowerHausdorffDimension μ = ENNReal.ofReal (min 1 (S.entropyRate / S.lyapunov)) := by\n  sorry\n\nend EntropyRateDimension\n\nend OAI\n\n\nFILE lean/ComparatorChallenges/FactorGeneration.lean \nimport Mathlib.Analysis.CStarAlgebra.ContinuousLinearMap\nimport Mathlib.Analysis.InnerProductSpace.WeakOperatorTopology\nimport Mathlib.Analysis.VonNeumannAlgebra.Basic\n\nnamespace OAI\n\nnoncomputable section\n\nnamespace Generator\n\nuniverse u\n\nvariable {H : Type u} [NormedAddCommGroup H] [InnerProductSpace ℂ H] [CompleteSpace H]\n\nattribute [-instance] instCStarAlgebraContinuousLinearMapComplexIdOfCompleteSpace\n\ndef WOTClosed (S : StarSubalgebra ℂ (H →L[ℂ] H)) : Prop :=\n  IsClosed {T : H →WOT[ℂ] H | T.toCLM ∈ S}\n\ndef wstar (s : Set (H →L[ℂ] H)) : StarSubalgebra ℂ (H →L[ℂ] H) :=\n  sInf {S | WOTClosed S ∧ s ⊆ S}\n\nstructure IsII1Factor (S : StarSubalgebra ℂ (H →L[ℂ] H)) : Prop where\n  nonzero : (0 : H →L[ℂ] H) ≠ 1\n  weaklyClosed : WOTClosed S\n  factor : ∀ x : S, (∀ y : S, y * x = x * y) → ∃ c : ℂ, x = algebraMap ℂ S c\n  finite : ∀ v : S, star v * v = 1 → v * star v = 1\n  diffuse : ∀ p : S, IsStarProjection p → p ≠ 0 →\n    ∃ q : S, IsStarProjection q ∧ q ≠ 0 ∧ q ≠ p ∧ p * q = q ∧ q * p = q\n\ndef HasSeparablePredual (S : StarSubalgebra ℂ (H →L[ℂ] H)) : Prop :=\n  ∃ (X : Type u) (_ : NormedAddCommGroup X) (_ : NormedSpace ℂ X)\n    (_ : CompleteSpace X) (_ : TopologicalSpace.SeparableSpace X),\n    Nonempty (StrongDual ℂ X ≃ₗᵢ⋆[ℂ] S)\n\ndef SinglyGenerated (S : StarSubalgebra ℂ (H →L[ℂ] H)) : Prop :=\n  ∃ x ∈ S, wstar {x} = S\n\nattribute [instance] instCStarAlgebraContinuousLinearMapComplexIdOfCompleteSpace\n\ntheorem single_generation_of_II1_separable_predual\n    (S : StarSubalgebra ℂ (H →L[ℂ] H)) (hS : IsII1Factor S)\n    (hsep : HasSeparablePredual S) : SinglyGenerated S := by\n  sorry\n\nend Generator\n\nend\n\nend OAI\n\n\nFILE lean/ComparatorChallenges/FoulkesHowe.lean \nimport Mathlib\n\nnamespace OAI\n\nnoncomputable section\n\nopen scoped BigOperators\nuniverse u\n\nnamespace Problem346\n\ndef symMonomialRaw (n : ℕ) (V : Type u) [AddCommGroup V] [Module ℂ V]\n    (v : Fin n → V) : SymmetricAlgebra ℂ V :=\n  ∏ i : Fin n, SymmetricAlgebra.ι ℂ V (v i)\n\ndef symPowSubmodule (n : ℕ) (V : Type u) [AddCommGroup V] [Module ℂ V] :\n    Submodule ℂ (SymmetricAlgebra ℂ V) :=\n  Submodule.span ℂ (Set.range (symMonomialRaw n V))\n\nabbrev SymPow (n : ℕ) (V : Type u) [AddCommGroup V] [Module ℂ V] :=\n  ↥(symPowSubmodule n V)\n\ndef symMonomial (n : ℕ) (V : Type u) [AddCommGroup V] [Module ℂ V]\n    (v : Fin n → V) : SymPow n V :=\n  ⟨symMonomialRaw n V v, Submodule.subset_span (Set.mem_range_self v)⟩\n\ndef foulkesFormula (a b : ℕ) (V : Type u) [AddCommGroup V] [Module ℂ V]\n    (v : Fin b → Fin a → V) : SymPow a (SymPow b V) := by\n  classical\n  exact\n    ((a.factorial : ℂ) ^ b)⁻¹ •\n      ∑ σ : Fin b → Equiv.Perm (Fin a),\n        symMonomial a (SymPow b V)\n          (fun i => symMonomial b V (fun j => v j ((σ j) i)))\n\ndef IsFoulkesMap (a b : ℕ) (V : Type u) [AddCommGroup V] [Module ℂ V]\n    (μ : SymPow b (SymPow a V) →ₗ[ℂ] SymPow a (SymPow b V)) : Prop :=\n  ∀ v : Fin b → Fin a → V,\n    μ (symMonomial b (SymPow a V) (fun j => symMonomial a V (v j))) =\n      foulkesFormula a b V v\n\ntheorem canonical_foulkes_howe_surjective :\n    ∀ (V : Type u) [AddCommGroup V] [Module ℂ V] [FiniteDimensional ℂ V] (a b : ℕ), 2 ≤ a → a * (a - 1) ≤ b → ∃ μ : SymPow b (SymPow a V) →ₗ[ℂ] SymPow a (SymPow b V), IsFoulkesMap a b V μ ∧ Function.Surjective μ ∧ ∀ ν : SymPow b (SymPow a V) →ₗ[ℂ] SymPow a (SymPow b V), IsFoulkesMap a b V ν → ν = μ := by\n  sorry\n\nend Problem346\n\nend\n\nend OAI\n\n"
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          "text": "\nFILE lean/ComparatorChallenges/GotsmanLinial.lean \nimport Mathlib.Algebra.MvPolynomial.Degrees\nimport Mathlib.Algebra.BigOperators.Field\nimport Mathlib.Data.Fintype.BigOperators\nimport Mathlib.Analysis.Real.Sqrt\nimport Mathlib.Tactic.NormNum\n\nnamespace OAI\n\nopen scoped BigOperators\n\nnamespace LeanBlast.GotsmanLinial\n\nabbrev Cube (n : ℕ) := Fin n → Bool\n\ndef cubeCoord {n : ℕ} (x : Cube n) (i : Fin n) : ℝ :=\n  if x i then 1 else -1\n\ndef flip {n : ℕ} (i : Fin n) (x : Cube n) : Cube n :=\n  Function.update x i (!(x i))\n\nnoncomputable def thresholdSign (t : ℝ) : ℝ :=\n  if 0 ≤ t then 1 else -1\n\nnoncomputable def sensitiveVertices {n : ℕ} (f : Cube n → ℝ) (i : Fin n) :\n    Finset (Cube n) := by\n  classical\n  exact Finset.univ.filter fun x => f x ≠ f (flip i x)\n\nnoncomputable def averageSensitivity {n : ℕ} (f : Cube n → ℝ) : ℝ :=\n  ∑ i : Fin n, ((sensitiveVertices f i).card : ℝ) / (2 : ℝ) ^ n\n\ndef IsMultilinear {n : ℕ} (p : MvPolynomial (Fin n) ℝ) : Prop :=\n  ∀ m ∈ p.support, ∀ i : Fin n, m i ≤ 1\n\nnoncomputable def polynomialValue {n : ℕ} (p : MvPolynomial (Fin n) ℝ)\n    (x : Cube n) : ℝ :=\n  MvPolynomial.eval (cubeCoord x) p\n\nnoncomputable def polynomialThreshold {n : ℕ} (p : MvPolynomial (Fin n) ℝ) :\n    Cube n → ℝ :=\n  fun x => thresholdSign (polynomialValue p x)\n\ndef GotsmanLinialStatement : Prop :=\n  ∀ (n d : ℕ), 1 ≤ n → 1 ≤ d → d ≤ n →\n    ∀ p : MvPolynomial (Fin n) ℝ,\n      IsMultilinear p → p.totalDegree ≤ d →\n        averageSensitivity (polynomialThreshold p) ≤ 8 * (d : ℝ) * Real.sqrt (n : ℝ)\n\ntheorem gotsmanLinialStatement : GotsmanLinialStatement := by\n  sorry\n\nend LeanBlast.GotsmanLinial\n\nend OAI\n\n\nFILE lean/ComparatorChallenges/SelfSimilar.lean \nimport Mathlib\n\nnamespace OAI\n\nuniverse uι\n\nopen scoped BigOperators ENNReal\nopen MeasureTheory\n\nnamespace EntropyRateDimension\n\nstructure System (ι : Type uι) [Fintype ι] where\n  ratio : ι → ℝ\n  offset : ι → ℝ\n  weight : ι → ℝ\n  ratio_pos : ∀ i, 0 < |ratio i|\n  ratio_lt_one : ∀ i, |ratio i| < 1\n  weight_pos : ∀ i, 0 < weight i\n  weight_sum : ∑ i, weight i = 1\n\nnamespace System\n\nvariable {ι : Type uι} [Fintype ι]\n\ndef affine (S : System ι) (i : ι) (x : ℝ) : ℝ := S.ratio i * x + S.offset i\n\ndef wordAffine (S : System ι) : List ι → ℝ × ℝ\n  | [] => (1, 0)\n  | i :: w => (S.ratio i * (S.wordAffine w).1,\n      S.ratio i * (S.wordAffine w).2 + S.offset i)\n\ndef completeMap (S : System ι) {n : ℕ} (w : Fin n → ι) : ℝ × ℝ :=\n  S.wordAffine (List.ofFn w)\n\nnoncomputable def wordWeight (S : System ι) {n : ℕ} (w : Fin n → ι) : ℝ :=\n  ∏ j, S.weight (w j)\n\nnoncomputable def mapSupport (S : System ι) (n : ℕ) : Finset (ℝ × ℝ) := by\n  classical\n  exact Finset.univ.image (S.completeMap (n := n))\n\nnoncomputable def mapMass (S : System ι) (n : ℕ) (g : ℝ × ℝ) : ℝ := by\n  classical\n  exact ∑ w ∈ (Finset.univ : Finset (Fin n → ι)).filter (S.completeMap · = g),\n    S.wordWeight w\n\nnoncomputable def walkEntropy (S : System ι) (n : ℕ) : ℝ :=\n  (∑ g ∈ S.mapSupport n, Real.negMulLog (S.mapMass n g)) / Real.log 2\n\nnoncomputable def entropyRate (S : System ι) : ℝ :=\n  sInf (Set.range fun n : ℕ => S.walkEntropy (n + 1) / (n + 1 : ℕ))\n\nnoncomputable def lyapunov (S : System ι) : ℝ :=\n  -(∑ i, S.weight i * (Real.log |S.ratio i| / Real.log 2))\n\ndef SelfSimilar (S : System ι) (μ : Measure ℝ) : Prop :=\n  μ = ∑ i, ENNReal.ofReal (S.weight i) • μ.map (S.affine i)\n\nend System\n\nnoncomputable def lowerHausdorffDimension (μ : Measure ℝ) : ℝ≥0∞ :=\n  ⨅ (E : Set ℝ) (_ : MeasurableSet E) (_ : 0 < μ E), dimH E\nopen MeasureTheory\n\ntheorem entropy_rate_dimension {ι : Type uι} [Fintype ι] [Nonempty ι]\n    (S : System ι) (μ : Measure ℝ) [IsProbabilityMeasure μ]\n    (hμ : S.SelfSimilar μ) :\n    lowerHausdorffDimension μ = ENNReal.ofReal (min 1 (S.entropyRate / S.lyapunov)) := by\n  sorry\n\nend EntropyRateDimension\n\nend OAI\n\n\nFILE lean/ComparatorChallenges/FactorGeneration.lean \nimport Mathlib.Analysis.CStarAlgebra.ContinuousLinearMap\nimport Mathlib.Analysis.InnerProductSpace.WeakOperatorTopology\nimport Mathlib.Analysis.VonNeumannAlgebra.Basic\n\nnamespace OAI\n\nnoncomputable section\n\nnamespace Generator\n\nuniverse u\n\nvariable {H : Type u} [NormedAddCommGroup H] [InnerProductSpace ℂ H] [CompleteSpace H]\n\nattribute [-instance] instCStarAlgebraContinuousLinearMapComplexIdOfCompleteSpace\n\ndef WOTClosed (S : StarSubalgebra ℂ (H →L[ℂ] H)) : Prop :=\n  IsClosed {T : H →WOT[ℂ] H | T.toCLM ∈ S}\n\ndef wstar (s : Set (H →L[ℂ] H)) : StarSubalgebra ℂ (H →L[ℂ] H) :=\n  sInf {S | WOTClosed S ∧ s ⊆ S}\n\nstructure IsII1Factor (S : StarSubalgebra ℂ (H →L[ℂ] H)) : Prop where\n  nonzero : (0 : H →L[ℂ] H) ≠ 1\n  weaklyClosed : WOTClosed S\n  factor : ∀ x : S, (∀ y : S, y * x = x * y) → ∃ c : ℂ, x = algebraMap ℂ S c\n  finite : ∀ v : S, star v * v = 1 → v * star v = 1\n  diffuse : ∀ p : S, IsStarProjection p → p ≠ 0 →\n    ∃ q : S, IsStarProjection q ∧ q ≠ 0 ∧ q ≠ p ∧ p * q = q ∧ q * p = q\n\ndef HasSeparablePredual (S : StarSubalgebra ℂ (H →L[ℂ] H)) : Prop :=\n  ∃ (X : Type u) (_ : NormedAddCommGroup X) (_ : NormedSpace ℂ X)\n    (_ : CompleteSpace X) (_ : TopologicalSpace.SeparableSpace X),\n    Nonempty (StrongDual ℂ X ≃ₗᵢ⋆[ℂ] S)\n\ndef SinglyGenerated (S : StarSubalgebra ℂ (H →L[ℂ] H)) : Prop :=\n  ∃ x ∈ S, wstar {x} = S\n\nattribute [instance] instCStarAlgebraContinuousLinearMapComplexIdOfCompleteSpace\n\ntheorem single_generation_of_II1_separable_predual\n    (S : StarSubalgebra ℂ (H →L[ℂ] H)) (hS : IsII1Factor S)\n    (hsep : HasSeparablePredual S) : SinglyGenerated S := by\n  sorry\n\nend Generator\n\nend\n\nend OAI\n\n\nFILE lean/ComparatorChallenges/FoulkesHowe.lean \nimport Mathlib\n\nnamespace OAI\n\nnoncomputable section\n\nopen scoped BigOperators\nuniverse u\n\nnamespace Problem346\n\ndef symMonomialRaw (n : ℕ) (V : Type u) [AddCommGroup V] [Module ℂ V]\n    (v : Fin n → V) : SymmetricAlgebra ℂ V :=\n  ∏ i : Fin n, SymmetricAlgebra.ι ℂ V (v i)\n\ndef symPowSubmodule (n : ℕ) (V : Type u) [AddCommGroup V] [Module ℂ V] :\n    Submodule ℂ (SymmetricAlgebra ℂ V) :=\n  Submodule.span ℂ (Set.range (symMonomialRaw n V))\n\nabbrev SymPow (n : ℕ) (V : Type u) [AddCommGroup V] [Module ℂ V] :=\n  ↥(symPowSubmodule n V)\n\ndef symMonomial (n : ℕ) (V : Type u) [AddCommGroup V] [Module ℂ V]\n    (v : Fin n → V) : SymPow n V :=\n  ⟨symMonomialRaw n V v, Submodule.subset_span (Set.mem_range_self v)⟩\n\ndef foulkesFormula (a b : ℕ) (V : Type u) [AddCommGroup V] [Module ℂ V]\n    (v : Fin b → Fin a → V) : SymPow a (SymPow b V) := by\n  classical\n  exact\n    ((a.factorial : ℂ) ^ b)⁻¹ •\n      ∑ σ : Fin b → Equiv.Perm (Fin a),\n        symMonomial a (SymPow b V)\n          (fun i => symMonomial b V (fun j => v j ((σ j) i)))\n\ndef IsFoulkesMap (a b : ℕ) (V : Type u) [AddCommGroup V] [Module ℂ V]\n    (μ : SymPow b (SymPow a V) →ₗ[ℂ] SymPow a (SymPow b V)) : Prop :=\n  ∀ v : Fin b → Fin a → V,\n    μ (symMonomial b (SymPow a V) (fun j => symMonomial a V (v j))) =\n      foulkesFormula a b V v\n\ntheorem canonical_foulkes_howe_surjective :\n    ∀ (V : Type u) [AddCommGroup V] [Module ℂ V] [FiniteDimensional ℂ V] (a b : ℕ), 2 ≤ a → a * (a - 1) ≤ b → ∃ μ : SymPow b (SymPow a V) →ₗ[ℂ] SymPow a (SymPow b V), IsFoulkesMap a b V μ ∧ Function.Surjective μ ∧ ∀ ν : SymPow b (SymPow a V) →ₗ[ℂ] SymPow a (SymPow b V), IsFoulkesMap a b V ν → ν = μ := by\n  sorry\n\nend Problem346\n\nend\n\nend OAI\n\n"
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        "stdout": "\nFILE lean/ComparatorChallenges/TriangularHilbert.lean \nimport Mathlib\n\nnamespace OAI\n\nnoncomputable section\n\nopen MeasureTheory\nopen scoped ENNReal NNReal\n\nnamespace TriangularHilbert\n\nabbrev Plane := ℝ × ℝ\n\nstructure Endpoints where\n  lower : ℝ\n  upper : ℝ\n  lower_pos : 0 < lower\n  lower_lt_upper : lower < upper\n\ndef annulus (ε R : ℝ) : Set ℝ := {t | ε < |t| ∧ |t| < R}\n\ndef integrand (F G : Plane → ℂ) (z : Plane) (t : ℝ) : ℂ :=\n  F (z.1 + t, z.2) * G (z.1, z.2 + t) / (t : ℂ)\n\ndef truncation (F G : Plane → ℂ) (E : Endpoints) (z : Plane) : ℂ :=\n  ∫ t in annulus E.lower E.upper, integrand F G z t\n\ndef GoodPoint (F G : Plane → ℂ) (z : Plane) : Prop :=\n  ∀ n : ℕ, IntegrableOn (integrand F G z)\n    (annulus (1 / ((n : ℝ) + 2)) ((n : ℝ) + 2))\n\ndef maximal (F G : Plane → ℂ) (z : Plane) : ℝ≥0∞ := by\n  classical\n  exact if GoodPoint F G z then\n    ⨆ E : Endpoints, ENNReal.ofReal ‖truncation F G E z‖ else 0\n\ndef maximalNorm (F G : Plane → ℂ) : ℝ≥0∞ :=\n  (∫⁻ z : Plane, maximal F G z ^ ((3 : ℝ) / 2)) ^ ((2 : ℝ) / 3)\n\ndef MainEstimate : Prop :=\n  ∃ C : ℝ≥0, ∀ F G : Plane → ℂ,\n    MemLp F 3 volume → MemLp G 3 volume →\n    (∀ᵐ z ∂volume, GoodPoint F G z) ∧\n    AEMeasurable (maximal F G) volume ∧\n    maximalNorm F G ≤ (C : ℝ≥0∞) * eLpNorm F 3 volume * eLpNorm G 3 volume\n\nopen MeasureTheory Filter\nopen scoped ENNReal\n\ntheorem main_estimate : MainEstimate := by\n  sorry\n\nend TriangularHilbert\n\nend\n\nend OAI\n\n\nFILE lean/ComparatorChallenges/ClassicalON.lean \nimport Mathlib\n\nnamespace OAI\n\nnoncomputable section\nopen MeasureTheory\nopen scoped BigOperators InnerProductSpace\n\nnamespace ClassicalON\n\nabbrev Site := ℤ × ℤ\n\ndef PositiveNeighbor (x y : Site) : Prop :=\n  (y.1 = x.1 + 1 ∧ y.2 = x.2) ∨ (y.1 = x.1 ∧ y.2 = x.2 + 1)\n\nstructure LatticeGraph where\n  vertices : Finset Site\n  edges : Finset (vertices × vertices)\n  nearest : ∀ e ∈ edges, PositiveNeighbor e.1.val e.2.val\n\nabbrev Spin (n : ℕ) := ↥(Metric.sphere (0 : EuclideanSpace ℝ (Fin n)) 1)\nabbrev Configuration (n : ℕ) (G : LatticeGraph) := G.vertices → Spin n\n\ndef sphereProbability (n : ℕ) : Measure (Spin n) :=\n  let surface := (volume : Measure (EuclideanSpace ℝ (Fin n))).toSphere\n  (surface Set.univ)⁻¹ • surface\n\ndef referenceLaw (n : ℕ) (G : LatticeGraph) : Measure (Configuration n G) :=\n  Measure.pi (fun _ => sphereProbability n)\n\ndef interaction (n : ℕ) (G : LatticeGraph) (b : G.edges → ℝ)\n    (σ : Configuration n G) : ℝ :=\n  ∑ e : G.edges, b e * ⟪(σ e.val.1).val, (σ e.val.2).val⟫_ℝ\n\ndef partition (n : ℕ) (G : LatticeGraph) (b : G.edges → ℝ) : ℝ :=\n  ∫ σ, Real.exp (interaction n G b σ) ∂referenceLaw n G\n\ndef correlation (n : ℕ) (G : LatticeGraph) (b : G.edges → ℝ)\n    (x y : G.vertices) : ℝ :=\n  (∫ σ, ⟪(σ x).val, (σ y).val⟫_ℝ * Real.exp (interaction n G b σ)\n    ∂referenceLaw n G) / partition n G b\n\ndef siteDistance (x y : Site) : ℝ :=\n  Real.sqrt (((x.1 : ℝ) - (y.1 : ℝ)) ^ 2 + ((x.2 : ℝ) - (y.2 : ℝ)) ^ 2)\n\ndef ExponentialDecay : Prop :=\n  ∀ (n : ℕ), 3 ≤ n → ∀ (β : ℝ), 0 < β →\n    ∃ A m : ℝ, 0 < m ∧ ∀ (G : LatticeGraph) (b : G.edges → ℝ),\n      (∀ e, 0 ≤ b e ∧ b e ≤ β) → ∀ x y : G.vertices,\n        0 ≤ correlation n G b x y ∧\n          correlation n G b x y ≤ A * Real.exp (-m * siteDistance x.val y.val)\n\ntheorem main : ExponentialDecay := by\n  sorry\n\nend ClassicalON\n\nend\n\nend OAI\n\n\nFILE lean/ComparatorChallenges/SLELowerPositivity.lean \nimport Mathlib\n\nnamespace OAI\n\n/-! Positivity of the explicit Hausdorff gauge on all positive-time SLE segments. -/\n\nnoncomputable section\nopen Set Filter MeasureTheory\nopen scoped Topology ENNReal NNReal\nnamespace SLEExactGauge\nuniverse v_lower\n\nnoncomputable def dimension (κ : ℝ) : ℝ := 1 + κ / 8\n\nnoncomputable def codimension (κ : ℝ) : ℝ := ((2 : ℕ) : ℝ) - dimension κ\n\ndef upperHalfPlane : Set ℂ := {z | 0 < z.im}\n\ndef survivingDomain (γ : ℝ≥0 → ℂ) (t : ℝ≥0) : Set ℂ :=\n  {z | z ∈ upperHalfPlane \\ (γ '' Icc 0 t) ∧\n    ¬ Bornology.IsBounded (connectedComponentIn (upperHalfPlane \\ (γ '' Icc 0 t)) z)}\n\ndef IsCapacityTwoWitness (U : ℝ≥0 → ℝ) (γ : ℝ≥0 → ℂ)\n    (G F : ℝ≥0 → ℂ → ℂ) : Prop :=\n  Continuous γ ∧ γ 0 = 0 ∧ (∀ t, 0 ≤ (γ t).im) ∧\n    (∀ t, DifferentiableOn ℂ (G t) (survivingDomain γ t)) ∧\n    (∀ t, DifferentiableOn ℂ (F t) upperHalfPlane) ∧\n    (∀ t, MapsTo (G t) (survivingDomain γ t) upperHalfPlane) ∧\n    (∀ t, MapsTo (F t) upperHalfPlane (survivingDomain γ t)) ∧\n    (∀ t z, z ∈ survivingDomain γ t → F t (G t z) = z) ∧\n    (∀ t z, z ∈ upperHalfPlane → G t (F t z) = z) ∧\n    (∀ z ∈ upperHalfPlane, G 0 z = z) ∧\n    (∀ t z, z ∈ survivingDomain γ t →\n      HasDerivWithinAt (fun u : ℝ => G (Real.toNNReal u) z)\n        (2 / (G t z - (U t : ℂ))) (Ici 0) (t : ℝ)) ∧\n    (∀ t, Tendsto (fun z : ℂ => z * (G t z - z))\n      (cocompact ℂ ⊓ 𝓟 (survivingDomain γ t)) (𝓝 (2 * (t : ℂ)))) ∧\n    (∀ t, Tendsto (fun y : ℝ => F t ((U t : ℂ) + (y : ℂ) * Complex.I))\n      (𝓝[>] 0) (𝓝 (γ t)))\n\ndef IsCapacityTwoTrace (U : ℝ≥0 → ℝ) (γ : ℝ≥0 → ℂ) : Prop :=\n  ∃ G F : ℝ≥0 → ℂ → ℂ, IsCapacityTwoWitness U γ G F\n\ndef IsOrdinaryChordalSLE {Ω : Type*} [MeasurableSpace Ω]\n    (κ : ℝ) (γ : Ω → ℝ≥0 → ℂ) (P : Measure Ω) : Prop :=\n  (∀ t, Measurable (fun ω => γ ω t)) ∧\n  ∃ B : ℝ≥0 → Ω → ℝ, ProbabilityTheory.IsBrownianReal B P ∧\n    ∀ᵐ ω ∂P, IsCapacityTwoTrace (fun t => Real.sqrt κ * B t ω) (γ ω)\n\nstructure IsGauge (h : ℝ → ℝ) : Prop where\n  continuousOn : ContinuousOn h (Ici 0)\n  monotoneOn : MonotoneOn h (Ici 0)\n  zero : h 0 = 0\n  positive : ∀ r, 0 < r → 0 < h r\n\nnoncomputable def hFormula (κ r : ℝ) : ℝ :=\n  r ^ dimension κ * (Real.log (Real.log (1 / r))) ^ (codimension κ / ((2 : ℕ) : ℝ))\n\ndef HasSmallRadiusFormula (h f : ℝ → ℝ) : Prop :=\n  h =ᶠ[𝓝[>] 0] f\n\nnoncomputable def extendedGauge (h : ℝ → ℝ) (r : ℝ≥0∞) : ℝ≥0∞ :=\n  ENNReal.ofReal (h r.toReal)\n\nnoncomputable def hausdorffGauge (h : ℝ → ℝ) : Measure ℂ :=\n  Measure.mkMetric (extendedGauge h)\n\ndef segment (γ : ℝ≥0 → ℂ) (s t : ℝ≥0) : Set ℂ := γ '' Icc s t\n\ndef SourceLowerMainTarget : Prop :=\n  ∀ (Ω : Type v_lower) (_ : MeasurableSpace Ω) (P : Measure Ω), IsProbabilityMeasure P →\n  ∀ κ : ℝ, 0 < κ → κ < ((8 : ℕ) : ℝ) →\n  ∀ (γ : Ω → ℝ≥0 → ℂ), IsOrdinaryChordalSLE κ γ P →\n  ∀ h : ℝ → ℝ, IsGauge h → HasSmallRadiusFormula h (hFormula κ) →\n    ∀ᵐ ω ∂P, ∀ s t : ℝ≥0, 0 < s → s < t →\n      0 < hausdorffGauge h (segment (γ ω) s t)\n\ntheorem sourceLowerMain_proved : SourceLowerMainTarget := by\n  sorry\n\nend SLEExactGauge\n\nend\n\nend OAI\n\n",
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          "\nFILE lean/ComparatorChallenges/TriangularHilbert.lean \nimport Mathlib\n\nnamespace OAI\n\nnoncomputable section\n\nopen MeasureTheory\nopen scoped ENNReal NNReal\n\nnamespace TriangularHilbert\n\nabbrev Plane := ℝ × ℝ\n\nstructure Endpoints where\n  lower : ℝ\n  upper : ℝ\n  lower_pos : 0 < lower\n  lower_lt_upper : lower < upper\n\ndef annulus (ε R : ℝ) : Set ℝ := {t | ε < |t| ∧ |t| < R}\n\ndef integrand (F G : Plane → ℂ) (z : Plane) (t : ℝ) : ℂ :=\n  F (z.1 + t, z.2) * G (z.1, z.2 + t) / (t : ℂ)\n\ndef truncation (F G : Plane → ℂ) (E : Endpoints) (z : Plane) : ℂ :=\n  ∫ t in annulus E.lower E.upper, integrand F G z t\n\ndef GoodPoint (F G : Plane → ℂ) (z : Plane) : Prop :=\n  ∀ n : ℕ, IntegrableOn (integrand F G z)\n    (annulus (1 / ((n : ℝ) + 2)) ((n : ℝ) + 2))\n\ndef maximal (F G : Plane → ℂ) (z : Plane) : ℝ≥0∞ := by\n  classical\n  exact if GoodPoint F G z then\n    ⨆ E : Endpoints, ENNReal.ofReal ‖truncation F G E z‖ else 0\n\ndef maximalNorm (F G : Plane → ℂ) : ℝ≥0∞ :=\n  (∫⁻ z : Plane, maximal F G z ^ ((3 : ℝ) / 2)) ^ ((2 : ℝ) / 3)\n\ndef MainEstimate : Prop :=\n  ∃ C : ℝ≥0, ∀ F G : Plane → ℂ,\n    MemLp F 3 volume → MemLp G 3 volume →\n    (∀ᵐ z ∂volume, GoodPoint F G z) ∧\n    AEMeasurable (maximal F G) volume ∧\n    maximalNorm F G ≤ (C : ℝ≥0∞) * eLpNorm F 3 volume * eLpNorm G 3 volume\n\nopen MeasureTheory Filter\nopen scoped ENNReal\n\ntheorem main_estimate : MainEstimate := by\n  sorry\n\nend TriangularHilbert\n\nend\n\nend OAI\n\n\nFILE lean/ComparatorChallenges/ClassicalON.lean \nimport Mathlib\n\nnamespace OAI\n\nnoncomputable section\nopen MeasureTheory\nopen scoped BigOperators InnerProductSpace\n\nnamespace ClassicalON\n\nabbrev Site := ℤ × ℤ\n\ndef PositiveNeighbor (x y : Site) : Prop :=\n  (y.1 = x.1 + 1 ∧ y.2 = x.2) ∨ (y.1 = x.1 ∧ y.2 = x.2 + 1)\n\nstructure LatticeGraph where\n  vertices : Finset Site\n  edges : Finset (vertices × vertices)\n  nearest : ∀ e ∈ edges, PositiveNeighbor e.1.val e.2.val\n\nabbrev Spin (n : ℕ) := ↥(Metric.sphere (0 : EuclideanSpace ℝ (Fin n)) 1)\nabbrev Configuration (n : ℕ) (G : LatticeGraph) := G.vertices → Spin n\n\ndef sphereProbability (n : ℕ) : Measure (Spin n) :=\n  let surface := (volume : Measure (EuclideanSpace ℝ (Fin n))).toSphere\n  (surface Set.univ)⁻¹ • surface\n\ndef referenceLaw (n : ℕ) (G : LatticeGraph) : Measure (Configuration n G) :=\n  Measure.pi (fun _ => sphereProbability n)\n\ndef interaction (n : ℕ) (G : LatticeGraph) (b : G.edges → ℝ)\n    (σ : Configuration n G) : ℝ :=\n  ∑ e : G.edges, b e * ⟪(σ e.val.1).val, (σ e.val.2).val⟫_ℝ\n\ndef partition (n : ℕ) (G : LatticeGraph) (b : G.edges → ℝ) : ℝ :=\n  ∫ σ, Real.exp (interaction n G b σ) ∂referenceLaw n G\n\ndef correlation (n : ℕ) (G : LatticeGraph) (b : G.edges → ℝ)\n    (x y : G.vertices) : ℝ :=\n  (∫ σ, ⟪(σ x).val, (σ y).val⟫_ℝ * Real.exp (interaction n G b σ)\n    ∂referenceLaw n G) / partition n G b\n\ndef siteDistance (x y : Site) : ℝ :=\n  Real.sqrt (((x.1 : ℝ) - (y.1 : ℝ)) ^ 2 + ((x.2 : ℝ) - (y.2 : ℝ)) ^ 2)\n\ndef ExponentialDecay : Prop :=\n  ∀ (n : ℕ), 3 ≤ n → ∀ (β : ℝ), 0 < β →\n    ∃ A m : ℝ, 0 < m ∧ ∀ (G : LatticeGraph) (b : G.edges → ℝ),\n      (∀ e, 0 ≤ b e ∧ b e ≤ β) → ∀ x y : G.vertices,\n        0 ≤ correlation n G b x y ∧\n          correlation n G b x y ≤ A * Real.exp (-m * siteDistance x.val y.val)\n\ntheorem main : ExponentialDecay := by\n  sorry\n\nend ClassicalON\n\nend\n\nend OAI\n\n\nFILE lean/ComparatorChallenges/SLELowerPositivity.lean \nimport Mathlib\n\nnamespace OAI\n\n/-! Positivity of the explicit Hausdorff gauge on all positive-time SLE segments. -/\n\nnoncomputable section\nopen Set Filter MeasureTheory\nopen scoped Topology ENNReal NNReal\nnamespace SLEExactGauge\nuniverse v_lower\n\nnoncomputable def dimension (κ : ℝ) : ℝ := 1 + κ / 8\n\nnoncomputable def codimension (κ : ℝ) : ℝ := ((2 : ℕ) : ℝ) - dimension κ\n\ndef upperHalfPlane : Set ℂ := {z | 0 < z.im}\n\ndef survivingDomain (γ : ℝ≥0 → ℂ) (t : ℝ≥0) : Set ℂ :=\n  {z | z ∈ upperHalfPlane \\ (γ '' Icc 0 t) ∧\n    ¬ Bornology.IsBounded (connectedComponentIn (upperHalfPlane \\ (γ '' Icc 0 t)) z)}\n\ndef IsCapacityTwoWitness (U : ℝ≥0 → ℝ) (γ : ℝ≥0 → ℂ)\n    (G F : ℝ≥0 → ℂ → ℂ) : Prop :=\n  Continuous γ ∧ γ 0 = 0 ∧ (∀ t, 0 ≤ (γ t).im) ∧\n    (∀ t, DifferentiableOn ℂ (G t) (survivingDomain γ t)) ∧\n    (∀ t, DifferentiableOn ℂ (F t) upperHalfPlane) ∧\n    (∀ t, MapsTo (G t) (survivingDomain γ t) upperHalfPlane) ∧\n    (∀ t, MapsTo (F t) upperHalfPlane (survivingDomain γ t)) ∧\n    (∀ t z, z ∈ survivingDomain γ t → F t (G t z) = z) ∧\n    (∀ t z, z ∈ upperHalfPlane → G t (F t z) = z) ∧\n    (∀ z ∈ upperHalfPlane, G 0 z = z) ∧\n    (∀ t z, z ∈ survivingDomain γ t →\n      HasDerivWithinAt (fun u : ℝ => G (Real.toNNReal u) z)\n        (2 / (G t z - (U t : ℂ))) (Ici 0) (t : ℝ)) ∧\n    (∀ t, Tendsto (fun z : ℂ => z * (G t z - z))\n      (cocompact ℂ ⊓ 𝓟 (survivingDomain γ t)) (𝓝 (2 * (t : ℂ)))) ∧\n    (∀ t, Tendsto (fun y : ℝ => F t ((U t : ℂ) + (y : ℂ) * Complex.I))\n      (𝓝[>] 0) (𝓝 (γ t)))\n\ndef IsCapacityTwoTrace (U : ℝ≥0 → ℝ) (γ : ℝ≥0 → ℂ) : Prop :=\n  ∃ G F : ℝ≥0 → ℂ → ℂ, IsCapacityTwoWitness U γ G F\n\ndef IsOrdinaryChordalSLE {Ω : Type*} [MeasurableSpace Ω]\n    (κ : ℝ) (γ : Ω → ℝ≥0 → ℂ) (P : Measure Ω) : Prop :=\n  (∀ t, Measurable (fun ω => γ ω t)) ∧\n  ∃ B : ℝ≥0 → Ω → ℝ, ProbabilityTheory.IsBrownianReal B P ∧\n    ∀ᵐ ω ∂P, IsCapacityTwoTrace (fun t => Real.sqrt κ * B t ω) (γ ω)\n\nstructure IsGauge (h : ℝ → ℝ) : Prop where\n  continuousOn : ContinuousOn h (Ici 0)\n  monotoneOn : MonotoneOn h (Ici 0)\n  zero : h 0 = 0\n  positive : ∀ r, 0 < r → 0 < h r\n\nnoncomputable def hFormula (κ r : ℝ) : ℝ :=\n  r ^ dimension κ * (Real.log (Real.log (1 / r))) ^ (codimension κ / ((2 : ℕ) : ℝ))\n\ndef HasSmallRadiusFormula (h f : ℝ → ℝ) : Prop :=\n  h =ᶠ[𝓝[>] 0] f\n\nnoncomputable def extendedGauge (h : ℝ → ℝ) (r : ℝ≥0∞) : ℝ≥0∞ :=\n  ENNReal.ofReal (h r.toReal)\n\nnoncomputable def hausdorffGauge (h : ℝ → ℝ) : Measure ℂ :=\n  Measure.mkMetric (extendedGauge h)\n\ndef segment (γ : ℝ≥0 → ℂ) (s t : ℝ≥0) : Set ℂ := γ '' Icc s t\n\ndef SourceLowerMainTarget : Prop :=\n  ∀ (Ω : Type v_lower) (_ : MeasurableSpace Ω) (P : Measure Ω), IsProbabilityMeasure P →\n  ∀ κ : ℝ, 0 < κ → κ < ((8 : ℕ) : ℝ) →\n  ∀ (γ : Ω → ℝ≥0 → ℂ), IsOrdinaryChordalSLE κ γ P →\n  ∀ h : ℝ → ℝ, IsGauge h → HasSmallRadiusFormula h (hFormula κ) →\n    ∀ᵐ ω ∂P, ∀ s t : ℝ≥0, 0 < s → s < t →\n      0 < hausdorffGauge h (segment (γ ω) s t)\n\ntheorem sourceLowerMain_proved : SourceLowerMainTarget := by\n  sorry\n\nend SLEExactGauge\n\nend\n\nend OAI\n\n"
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          "text": "\nFILE lean/ComparatorChallenges/TriangularHilbert.lean \nimport Mathlib\n\nnamespace OAI\n\nnoncomputable section\n\nopen MeasureTheory\nopen scoped ENNReal NNReal\n\nnamespace TriangularHilbert\n\nabbrev Plane := ℝ × ℝ\n\nstructure Endpoints where\n  lower : ℝ\n  upper : ℝ\n  lower_pos : 0 < lower\n  lower_lt_upper : lower < upper\n\ndef annulus (ε R : ℝ) : Set ℝ := {t | ε < |t| ∧ |t| < R}\n\ndef integrand (F G : Plane → ℂ) (z : Plane) (t : ℝ) : ℂ :=\n  F (z.1 + t, z.2) * G (z.1, z.2 + t) / (t : ℂ)\n\ndef truncation (F G : Plane → ℂ) (E : Endpoints) (z : Plane) : ℂ :=\n  ∫ t in annulus E.lower E.upper, integrand F G z t\n\ndef GoodPoint (F G : Plane → ℂ) (z : Plane) : Prop :=\n  ∀ n : ℕ, IntegrableOn (integrand F G z)\n    (annulus (1 / ((n : ℝ) + 2)) ((n : ℝ) + 2))\n\ndef maximal (F G : Plane → ℂ) (z : Plane) : ℝ≥0∞ := by\n  classical\n  exact if GoodPoint F G z then\n    ⨆ E : Endpoints, ENNReal.ofReal ‖truncation F G E z‖ else 0\n\ndef maximalNorm (F G : Plane → ℂ) : ℝ≥0∞ :=\n  (∫⁻ z : Plane, maximal F G z ^ ((3 : ℝ) / 2)) ^ ((2 : ℝ) / 3)\n\ndef MainEstimate : Prop :=\n  ∃ C : ℝ≥0, ∀ F G : Plane → ℂ,\n    MemLp F 3 volume → MemLp G 3 volume →\n    (∀ᵐ z ∂volume, GoodPoint F G z) ∧\n    AEMeasurable (maximal F G) volume ∧\n    maximalNorm F G ≤ (C : ℝ≥0∞) * eLpNorm F 3 volume * eLpNorm G 3 volume\n\nopen MeasureTheory Filter\nopen scoped ENNReal\n\ntheorem main_estimate : MainEstimate := by\n  sorry\n\nend TriangularHilbert\n\nend\n\nend OAI\n\n\nFILE lean/ComparatorChallenges/ClassicalON.lean \nimport Mathlib\n\nnamespace OAI\n\nnoncomputable section\nopen MeasureTheory\nopen scoped BigOperators InnerProductSpace\n\nnamespace ClassicalON\n\nabbrev Site := ℤ × ℤ\n\ndef PositiveNeighbor (x y : Site) : Prop :=\n  (y.1 = x.1 + 1 ∧ y.2 = x.2) ∨ (y.1 = x.1 ∧ y.2 = x.2 + 1)\n\nstructure LatticeGraph where\n  vertices : Finset Site\n  edges : Finset (vertices × vertices)\n  nearest : ∀ e ∈ edges, PositiveNeighbor e.1.val e.2.val\n\nabbrev Spin (n : ℕ) := ↥(Metric.sphere (0 : EuclideanSpace ℝ (Fin n)) 1)\nabbrev Configuration (n : ℕ) (G : LatticeGraph) := G.vertices → Spin n\n\ndef sphereProbability (n : ℕ) : Measure (Spin n) :=\n  let surface := (volume : Measure (EuclideanSpace ℝ (Fin n))).toSphere\n  (surface Set.univ)⁻¹ • surface\n\ndef referenceLaw (n : ℕ) (G : LatticeGraph) : Measure (Configuration n G) :=\n  Measure.pi (fun _ => sphereProbability n)\n\ndef interaction (n : ℕ) (G : LatticeGraph) (b : G.edges → ℝ)\n    (σ : Configuration n G) : ℝ :=\n  ∑ e : G.edges, b e * ⟪(σ e.val.1).val, (σ e.val.2).val⟫_ℝ\n\ndef partition (n : ℕ) (G : LatticeGraph) (b : G.edges → ℝ) : ℝ :=\n  ∫ σ, Real.exp (interaction n G b σ) ∂referenceLaw n G\n\ndef correlation (n : ℕ) (G : LatticeGraph) (b : G.edges → ℝ)\n    (x y : G.vertices) : ℝ :=\n  (∫ σ, ⟪(σ x).val, (σ y).val⟫_ℝ * Real.exp (interaction n G b σ)\n    ∂referenceLaw n G) / partition n G b\n\ndef siteDistance (x y : Site) : ℝ :=\n  Real.sqrt (((x.1 : ℝ) - (y.1 : ℝ)) ^ 2 + ((x.2 : ℝ) - (y.2 : ℝ)) ^ 2)\n\ndef ExponentialDecay : Prop :=\n  ∀ (n : ℕ), 3 ≤ n → ∀ (β : ℝ), 0 < β →\n    ∃ A m : ℝ, 0 < m ∧ ∀ (G : LatticeGraph) (b : G.edges → ℝ),\n      (∀ e, 0 ≤ b e ∧ b e ≤ β) → ∀ x y : G.vertices,\n        0 ≤ correlation n G b x y ∧\n          correlation n G b x y ≤ A * Real.exp (-m * siteDistance x.val y.val)\n\ntheorem main : ExponentialDecay := by\n  sorry\n\nend ClassicalON\n\nend\n\nend OAI\n\n\nFILE lean/ComparatorChallenges/SLELowerPositivity.lean \nimport Mathlib\n\nnamespace OAI\n\n/-! Positivity of the explicit Hausdorff gauge on all positive-time SLE segments. -/\n\nnoncomputable section\nopen Set Filter MeasureTheory\nopen scoped Topology ENNReal NNReal\nnamespace SLEExactGauge\nuniverse v_lower\n\nnoncomputable def dimension (κ : ℝ) : ℝ := 1 + κ / 8\n\nnoncomputable def codimension (κ : ℝ) : ℝ := ((2 : ℕ) : ℝ) - dimension κ\n\ndef upperHalfPlane : Set ℂ := {z | 0 < z.im}\n\ndef survivingDomain (γ : ℝ≥0 → ℂ) (t : ℝ≥0) : Set ℂ :=\n  {z | z ∈ upperHalfPlane \\ (γ '' Icc 0 t) ∧\n    ¬ Bornology.IsBounded (connectedComponentIn (upperHalfPlane \\ (γ '' Icc 0 t)) z)}\n\ndef IsCapacityTwoWitness (U : ℝ≥0 → ℝ) (γ : ℝ≥0 → ℂ)\n    (G F : ℝ≥0 → ℂ → ℂ) : Prop :=\n  Continuous γ ∧ γ 0 = 0 ∧ (∀ t, 0 ≤ (γ t).im) ∧\n    (∀ t, DifferentiableOn ℂ (G t) (survivingDomain γ t)) ∧\n    (∀ t, DifferentiableOn ℂ (F t) upperHalfPlane) ∧\n    (∀ t, MapsTo (G t) (survivingDomain γ t) upperHalfPlane) ∧\n    (∀ t, MapsTo (F t) upperHalfPlane (survivingDomain γ t)) ∧\n    (∀ t z, z ∈ survivingDomain γ t → F t (G t z) = z) ∧\n    (∀ t z, z ∈ upperHalfPlane → G t (F t z) = z) ∧\n    (∀ z ∈ upperHalfPlane, G 0 z = z) ∧\n    (∀ t z, z ∈ survivingDomain γ t →\n      HasDerivWithinAt (fun u : ℝ => G (Real.toNNReal u) z)\n        (2 / (G t z - (U t : ℂ))) (Ici 0) (t : ℝ)) ∧\n    (∀ t, Tendsto (fun z : ℂ => z * (G t z - z))\n      (cocompact ℂ ⊓ 𝓟 (survivingDomain γ t)) (𝓝 (2 * (t : ℂ)))) ∧\n    (∀ t, Tendsto (fun y : ℝ => F t ((U t : ℂ) + (y : ℂ) * Complex.I))\n      (𝓝[>] 0) (𝓝 (γ t)))\n\ndef IsCapacityTwoTrace (U : ℝ≥0 → ℝ) (γ : ℝ≥0 → ℂ) : Prop :=\n  ∃ G F : ℝ≥0 → ℂ → ℂ, IsCapacityTwoWitness U γ G F\n\ndef IsOrdinaryChordalSLE {Ω : Type*} [MeasurableSpace Ω]\n    (κ : ℝ) (γ : Ω → ℝ≥0 → ℂ) (P : Measure Ω) : Prop :=\n  (∀ t, Measurable (fun ω => γ ω t)) ∧\n  ∃ B : ℝ≥0 → Ω → ℝ, ProbabilityTheory.IsBrownianReal B P ∧\n    ∀ᵐ ω ∂P, IsCapacityTwoTrace (fun t => Real.sqrt κ * B t ω) (γ ω)\n\nstructure IsGauge (h : ℝ → ℝ) : Prop where\n  continuousOn : ContinuousOn h (Ici 0)\n  monotoneOn : MonotoneOn h (Ici 0)\n  zero : h 0 = 0\n  positive : ∀ r, 0 < r → 0 < h r\n\nnoncomputable def hFormula (κ r : ℝ) : ℝ :=\n  r ^ dimension κ * (Real.log (Real.log (1 / r))) ^ (codimension κ / ((2 : ℕ) : ℝ))\n\ndef HasSmallRadiusFormula (h f : ℝ → ℝ) : Prop :=\n  h =ᶠ[𝓝[>] 0] f\n\nnoncomputable def extendedGauge (h : ℝ → ℝ) (r : ℝ≥0∞) : ℝ≥0∞ :=\n  ENNReal.ofReal (h r.toReal)\n\nnoncomputable def hausdorffGauge (h : ℝ → ℝ) : Measure ℂ :=\n  Measure.mkMetric (extendedGauge h)\n\ndef segment (γ : ℝ≥0 → ℂ) (s t : ℝ≥0) : Set ℂ := γ '' Icc s t\n\ndef SourceLowerMainTarget : Prop :=\n  ∀ (Ω : Type v_lower) (_ : MeasurableSpace Ω) (P : Measure Ω), IsProbabilityMeasure P →\n  ∀ κ : ℝ, 0 < κ → κ < ((8 : ℕ) : ℝ) →\n  ∀ (γ : Ω → ℝ≥0 → ℂ), IsOrdinaryChordalSLE κ γ P →\n  ∀ h : ℝ → ℝ, IsGauge h → HasSmallRadiusFormula h (hFormula κ) →\n    ∀ᵐ ω ∂P, ∀ s t : ℝ≥0, 0 < s → s < t →\n      0 < hausdorffGauge h (segment (γ ω) s t)\n\ntheorem sourceLowerMain_proved : SourceLowerMainTarget := by\n  sorry\n\nend SLEExactGauge\n\nend\n\nend OAI\n\n"
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        "stdout": "\nFILE lean/ComparatorChallenges/SquareRootDegree.lean \nimport Mathlib\n\nnamespace OAI\n\n/-!\nBool encodes a sign: false is +1 and true is -1. The average is uniform,\nand degree is ordinary real Fourier degree, not F₂ degree or threshold degree.\n-/\n\nopen scoped BigOperators\n\nnoncomputable section\n\nnamespace SquareRootDegree\n\nabbrev Cube (ι : Type*) := ι → Bool\n\ndef sign (b : Bool) : ℝ := if b then -1 else 1\n\ndef average {α : Type*} [Fintype α] (f : α → ℝ) : ℝ :=\n  (Fintype.card α : ℝ)⁻¹ * ∑ x, f x\n\ndef character {ι : Type*} (s : Finset ι) (x : Cube ι) : ℝ :=\n  ∏ i ∈ s, sign (x i)\n\ndef fourierCoeff {ι : Type*} [Fintype ι] (f : Cube ι → ℝ) (s : Finset ι) : ℝ := by\n  classical\n  exact average (fun x => f x * character s x)\n\ndef fourierDegree {ι : Type*} [Fintype ι] (f : Cube ι → ℝ) : ℕ := by\n  classical\n  exact (Finset.univ.filter (fun s : Finset ι => fourierCoeff f s ≠ 0)).sup Finset.card\n\ndef IsBoolean {α : Type*} (f : α → ℝ) : Prop :=\n  ∀ x, f x = -1 ∨ f x = 1\n\ndef Nonconstant {α : Type*} (f : α → ℝ) : Prop :=\n  ∃ x y, f x ≠ f y\n\ndef singletonSum {ι : Type*} [Fintype ι] (f : Cube ι → ℝ) : ℝ :=\n  ∑ i, fourierCoeff f {i}\n\ndef absoluteSingletonSum {ι : Type*} [Fintype ι] (f : Cube ι → ℝ) : ℝ :=\n  ∑ i, |fourierCoeff f {i}|\n\ndef SignedViolations : Prop :=\n  ∀ C : ℝ, 0 < C → ∃ n : ℕ, 0 < n ∧ ∃ f : Cube (Fin n) → ℝ,\n    IsBoolean f ∧ Nonconstant f ∧ C * Real.sqrt (fourierDegree f) < singletonSum f\n\ndef AbsoluteRatios : Set ℝ :=\n  {r | ∃ n : ℕ, 0 < n ∧ ∃ f : Cube (Fin n) → ℝ,\n    IsBoolean f ∧ 0 < fourierDegree f ∧\n      r = absoluteSingletonSum f / Real.sqrt (fourierDegree f)}\n\n/-- The two clauses of the literal main theorem. Unboundedness above is the\nreal-valued formulation of the displayed supremum being positive infinity. -/\ntheorem main : SignedViolations ∧ ¬ BddAbove AbsoluteRatios := by\n  sorry\n\nend SquareRootDegree\n\nend\n\nend OAI\n\n\nFILE lean/ComparatorChallenges/HotSpots.lean \nimport Mathlib\n\nnamespace OAI\n\nnoncomputable section\nopen Set MeasureTheory\nopen scoped ContDiff\n\nnamespace StrictHotSpots\n\nabbrev Plane := EuclideanSpace ℝ (Fin 2)\n\ndef SmoothBoundary (Ω : Set Plane) : Prop :=\n  ∀ p ∈ frontier Ω, ∃ (U : Set Plane) (ρ : Plane → ℝ),\n    IsOpen U ∧ p ∈ U ∧ ContDiffOn ℝ ∞ ρ U ∧\n    fderiv ℝ ρ p ≠ 0 ∧ ρ p = 0 ∧\n    Ω ∩ U = {x | ρ x < 0} ∩ U\n\ndef AdmissibleDomain (Ω : Set Plane) : Prop :=\n  Ω.Nonempty ∧ IsOpen Ω ∧ Bornology.IsBounded Ω ∧\n    IsSimplyConnected Ω ∧ SmoothBoundary Ω\n\ndef HasH1Gradient (Ω : Set Plane) (v : Plane → ℝ) (g : Plane → Plane) : Prop :=\n  MemLp v 2 (volume.restrict Ω) ∧ MemLp g 2 (volume.restrict Ω) ∧\n  ∀ φ : Plane → ℝ, ContDiff ℝ ∞ φ → HasCompactSupport φ → tsupport φ ⊆ Ω →\n    ∀ e : Plane,\n      (∫ x in Ω, v x * (fderiv ℝ φ x) e) =\n        -(∫ x in Ω, (inner ℝ (g x) e) * φ x)\n\ndef rayleighValues (Ω : Set Plane) : Set ℝ :=\n  {r | ∃ (v : Plane → ℝ) (g : Plane → Plane),\n    HasH1Gradient Ω v g ∧ (∫ x in Ω, v x) = 0 ∧\n    0 < (∫ x in Ω, (v x) ^ 2) ∧\n    r = (∫ x in Ω, ‖g x‖ ^ 2) / (∫ x in Ω, (v x) ^ 2)}\n\ndef firstPositiveNeumannValue (Ω : Set Plane) : ℝ := sInf (rayleighValues Ω)\n\ndef InFirstNeumannEigenspace (Ω : Set Plane) (u : Plane → ℝ) : Prop :=\n  ContDiffOn ℝ ∞ u (closure Ω) ∧ HasH1Gradient Ω u (gradient u) ∧\n    (∫ x in Ω, u x) = 0 ∧\n    ∀ (v : Plane → ℝ) (g : Plane → Plane), HasH1Gradient Ω v g →\n      (∫ x in Ω, inner ℝ (gradient u x) (g x)) =\n        firstPositiveNeumannValue Ω * (∫ x in Ω, u x * v x)\n\ndef MainConclusion (Ω : Set Plane) (u : Plane → ℝ) : Prop :=\n  (∀ x ∈ Ω, gradient u x ≠ 0) ∧\n    ∀ x ∈ Ω, sInf (u '' frontier Ω) < u x ∧ u x < sSup (u '' frontier Ω)\n\ntheorem main_theorem (Ω : Set Plane) (hΩ : AdmissibleDomain Ω)\n    (u : Plane → ℝ) (hu : InFirstNeumannEigenspace Ω u)\n    (hne : ∃ x ∈ Ω, u x ≠ 0) : MainConclusion Ω u := by\n  sorry\n\nend StrictHotSpots\nend\n\nend OAI\n\n\nFILE lean/ComparatorChallenges/ElasticityUniqueness.lean \nimport Mathlib\n\nnamespace OAI\n\n/-!\n# Global uniqueness for smooth isotropic elasticity\n\nThe physical Dirichlet-to-Neumann pairing acts on the variational trace\nquotient. Its quotient norm is not identified here with boundary-chart\nfractional Sobolev norms.\n-/\n\nnoncomputable section\nopen MeasureTheory Set\nopen scoped BigOperators\n\nnamespace Elasticity\n\nabbrev X := EuclideanSpace ℝ (Fin 3)\nabbrev V := X\n\ndef coordVector (i : Fin 3) : X := EuclideanSpace.single i 1\n\ndef SmoothBoundary (Ω : Set X) : Prop :=\n  ∀ x ∈ frontier Ω, ∃ e : OpenPartialHomeomorph X X,\n    x ∈ e.source ∧\n    ContDiffOn ℝ (⊤ : ℕ∞) e e.source ∧\n    ContDiffOn ℝ (⊤ : ℕ∞) e.symm e.target ∧\n    e x 0 = 0 ∧\n    ∀ y ∈ e.source, y ∈ Ω ↔ 0 < e y 0\n\ndef Domain (Ω : Set X) : Prop :=\n  IsOpen Ω ∧ IsConnected Ω ∧ Bornology.IsBounded Ω ∧ SmoothBoundary Ω\n\ndef SmoothUpTo (Ω : Set X) (f : X → ℝ) : Prop :=\n  ∃ U : Set X, IsOpen U ∧ closure Ω ⊆ U ∧ ContDiffOn ℝ (⊤ : ℕ∞) f U\n\ndef Admissible (Ω : Set X) (lam mu : X → ℝ) : Prop :=\n  SmoothUpTo Ω lam ∧ SmoothUpTo Ω mu ∧\n    ∀ x ∈ closure Ω, 0 < mu x ∧ 0 < 3 * lam x + 2 * mu x\n\nabbrev Ambient (Ω : Set X) :=\n  Lp V 2 (volume.restrict Ω) × (Fin 3 → Lp V 2 (volume.restrict Ω))\n\ndef coordDeriv (f : X → V) (i : Fin 3) (x : X) : V :=\n  fderiv ℝ f x (coordVector i)\n\ndef SmoothJets (Ω : Set X) : Set (Ambient Ω) :=\n  {j | ∃ (f : X → V) (hf : MemLp f 2 (volume.restrict Ω))\n      (hd : ∀ i, MemLp (coordDeriv f i) 2 (volume.restrict Ω)),\n    ContDiff ℝ (⊤ : ℕ∞) f ∧\n    j.1 = hf.toLp f ∧ j.2 = fun i => (hd i).toLp (coordDeriv f i)}\n\ndef TestJets (Ω : Set X) : Set (Ambient Ω) :=\n  {j | ∃ (f : X → V) (hf : MemLp f 2 (volume.restrict Ω))\n      (hd : ∀ i, MemLp (coordDeriv f i) 2 (volume.restrict Ω)),\n    ContDiff ℝ (⊤ : ℕ∞) f ∧ HasCompactSupport f ∧ tsupport f ⊆ Ω ∧\n    j.1 = hf.toLp f ∧ j.2 = fun i => (hd i).toLp (coordDeriv f i)}\n\nabbrev H1 (Ω : Set X) := {j : Ambient Ω // j ∈ closure (SmoothJets Ω)}\n\ndef HasZeroTrace (Ω : Set X) (u : H1 Ω) : Prop :=\n  u.val ∈ closure (TestJets Ω)\n\ndef SameTrace (Ω : Set X) (u v : H1 Ω) : Prop :=\n  u.val - v.val ∈ closure (TestJets Ω)\n\nabbrev BoundaryData (Ω : Set X) := Quot (SameTrace Ω)\n\ndef trace (Ω : Set X) (u : H1 Ω) : BoundaryData Ω := Quot.mk _ u\n\ndef div (Ω : Set X) (u : H1 Ω) (x : X) : ℝ :=\n  ∑ i, (u.val.2 i x) i\n\ndef strain (Ω : Set X) (u : H1 Ω) (x : X) (i j : Fin 3) : ℝ :=\n  ((u.val.2 j x) i + (u.val.2 i x) j) / 2\n\ndef energy (Ω : Set X) (lam mu : X → ℝ) (u v : H1 Ω) : ℝ :=\n  ∫ x, (lam x * div Ω u x * div Ω v x +\n    2 * mu x * ∑ i, ∑ j, strain Ω u x i j * strain Ω v x i j)\n    ∂(volume.restrict Ω)\n\ndef WeakSolution (Ω : Set X) (lam mu : X → ℝ) (u : H1 Ω) : Prop :=\n  ∀ v : H1 Ω, HasZeroTrace Ω v → energy Ω lam mu u v = 0\n\ndef dirichletSolution (Ω : Set X) (lam mu : X → ℝ)\n    (f : BoundaryData Ω) : H1 Ω := by\n  classical\n  exact if h : ∃ u : H1 Ω, trace Ω u = f ∧ WeakSolution Ω lam mu u\n  then Classical.choose h\n  else f.out\n\ndef DN (Ω : Set X) (lam mu : X → ℝ) : BoundaryData Ω → BoundaryData Ω → ℝ :=\n  fun f g => energy Ω lam mu (dirichletSolution Ω lam mu f) g.out\n\ndef MainClaim : Prop :=\n  ∀ (Ω : Set X), Domain Ω →\n  ∀ (lam₁ mu₁ lam₂ mu₂ : X → ℝ),\n    Admissible Ω lam₁ mu₁ → Admissible Ω lam₂ mu₂ →\n    DN Ω lam₁ mu₁ = DN Ω lam₂ mu₂ →\n    (∀ x ∈ Ω, lam₁ x = lam₂ x) ∧ (∀ x ∈ Ω, mu₁ x = mu₂ x)\n\ntheorem global_uniqueness : MainClaim := by\n  sorry\n\nend Elasticity\n\nend\n\nend OAI\n\n",
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          "\nFILE lean/ComparatorChallenges/SquareRootDegree.lean \nimport Mathlib\n\nnamespace OAI\n\n/-!\nBool encodes a sign: false is +1 and true is -1. The average is uniform,\nand degree is ordinary real Fourier degree, not F₂ degree or threshold degree.\n-/\n\nopen scoped BigOperators\n\nnoncomputable section\n\nnamespace SquareRootDegree\n\nabbrev Cube (ι : Type*) := ι → Bool\n\ndef sign (b : Bool) : ℝ := if b then -1 else 1\n\ndef average {α : Type*} [Fintype α] (f : α → ℝ) : ℝ :=\n  (Fintype.card α : ℝ)⁻¹ * ∑ x, f x\n\ndef character {ι : Type*} (s : Finset ι) (x : Cube ι) : ℝ :=\n  ∏ i ∈ s, sign (x i)\n\ndef fourierCoeff {ι : Type*} [Fintype ι] (f : Cube ι → ℝ) (s : Finset ι) : ℝ := by\n  classical\n  exact average (fun x => f x * character s x)\n\ndef fourierDegree {ι : Type*} [Fintype ι] (f : Cube ι → ℝ) : ℕ := by\n  classical\n  exact (Finset.univ.filter (fun s : Finset ι => fourierCoeff f s ≠ 0)).sup Finset.card\n\ndef IsBoolean {α : Type*} (f : α → ℝ) : Prop :=\n  ∀ x, f x = -1 ∨ f x = 1\n\ndef Nonconstant {α : Type*} (f : α → ℝ) : Prop :=\n  ∃ x y, f x ≠ f y\n\ndef singletonSum {ι : Type*} [Fintype ι] (f : Cube ι → ℝ) : ℝ :=\n  ∑ i, fourierCoeff f {i}\n\ndef absoluteSingletonSum {ι : Type*} [Fintype ι] (f : Cube ι → ℝ) : ℝ :=\n  ∑ i, |fourierCoeff f {i}|\n\ndef SignedViolations : Prop :=\n  ∀ C : ℝ, 0 < C → ∃ n : ℕ, 0 < n ∧ ∃ f : Cube (Fin n) → ℝ,\n    IsBoolean f ∧ Nonconstant f ∧ C * Real.sqrt (fourierDegree f) < singletonSum f\n\ndef AbsoluteRatios : Set ℝ :=\n  {r | ∃ n : ℕ, 0 < n ∧ ∃ f : Cube (Fin n) → ℝ,\n    IsBoolean f ∧ 0 < fourierDegree f ∧\n      r = absoluteSingletonSum f / Real.sqrt (fourierDegree f)}\n\n/-- The two clauses of the literal main theorem. Unboundedness above is the\nreal-valued formulation of the displayed supremum being positive infinity. -/\ntheorem main : SignedViolations ∧ ¬ BddAbove AbsoluteRatios := by\n  sorry\n\nend SquareRootDegree\n\nend\n\nend OAI\n\n\nFILE lean/ComparatorChallenges/HotSpots.lean \nimport Mathlib\n\nnamespace OAI\n\nnoncomputable section\nopen Set MeasureTheory\nopen scoped ContDiff\n\nnamespace StrictHotSpots\n\nabbrev Plane := EuclideanSpace ℝ (Fin 2)\n\ndef SmoothBoundary (Ω : Set Plane) : Prop :=\n  ∀ p ∈ frontier Ω, ∃ (U : Set Plane) (ρ : Plane → ℝ),\n    IsOpen U ∧ p ∈ U ∧ ContDiffOn ℝ ∞ ρ U ∧\n    fderiv ℝ ρ p ≠ 0 ∧ ρ p = 0 ∧\n    Ω ∩ U = {x | ρ x < 0} ∩ U\n\ndef AdmissibleDomain (Ω : Set Plane) : Prop :=\n  Ω.Nonempty ∧ IsOpen Ω ∧ Bornology.IsBounded Ω ∧\n    IsSimplyConnected Ω ∧ SmoothBoundary Ω\n\ndef HasH1Gradient (Ω : Set Plane) (v : Plane → ℝ) (g : Plane → Plane) : Prop :=\n  MemLp v 2 (volume.restrict Ω) ∧ MemLp g 2 (volume.restrict Ω) ∧\n  ∀ φ : Plane → ℝ, ContDiff ℝ ∞ φ → HasCompactSupport φ → tsupport φ ⊆ Ω →\n    ∀ e : Plane,\n      (∫ x in Ω, v x * (fderiv ℝ φ x) e) =\n        -(∫ x in Ω, (inner ℝ (g x) e) * φ x)\n\ndef rayleighValues (Ω : Set Plane) : Set ℝ :=\n  {r | ∃ (v : Plane → ℝ) (g : Plane → Plane),\n    HasH1Gradient Ω v g ∧ (∫ x in Ω, v x) = 0 ∧\n    0 < (∫ x in Ω, (v x) ^ 2) ∧\n    r = (∫ x in Ω, ‖g x‖ ^ 2) / (∫ x in Ω, (v x) ^ 2)}\n\ndef firstPositiveNeumannValue (Ω : Set Plane) : ℝ := sInf (rayleighValues Ω)\n\ndef InFirstNeumannEigenspace (Ω : Set Plane) (u : Plane → ℝ) : Prop :=\n  ContDiffOn ℝ ∞ u (closure Ω) ∧ HasH1Gradient Ω u (gradient u) ∧\n    (∫ x in Ω, u x) = 0 ∧\n    ∀ (v : Plane → ℝ) (g : Plane → Plane), HasH1Gradient Ω v g →\n      (∫ x in Ω, inner ℝ (gradient u x) (g x)) =\n        firstPositiveNeumannValue Ω * (∫ x in Ω, u x * v x)\n\ndef MainConclusion (Ω : Set Plane) (u : Plane → ℝ) : Prop :=\n  (∀ x ∈ Ω, gradient u x ≠ 0) ∧\n    ∀ x ∈ Ω, sInf (u '' frontier Ω) < u x ∧ u x < sSup (u '' frontier Ω)\n\ntheorem main_theorem (Ω : Set Plane) (hΩ : AdmissibleDomain Ω)\n    (u : Plane → ℝ) (hu : InFirstNeumannEigenspace Ω u)\n    (hne : ∃ x ∈ Ω, u x ≠ 0) : MainConclusion Ω u := by\n  sorry\n\nend StrictHotSpots\nend\n\nend OAI\n\n\nFILE lean/ComparatorChallenges/ElasticityUniqueness.lean \nimport Mathlib\n\nnamespace OAI\n\n/-!\n# Global uniqueness for smooth isotropic elasticity\n\nThe physical Dirichlet-to-Neumann pairing acts on the variational trace\nquotient. Its quotient norm is not identified here with boundary-chart\nfractional Sobolev norms.\n-/\n\nnoncomputable section\nopen MeasureTheory Set\nopen scoped BigOperators\n\nnamespace Elasticity\n\nabbrev X := EuclideanSpace ℝ (Fin 3)\nabbrev V := X\n\ndef coordVector (i : Fin 3) : X := EuclideanSpace.single i 1\n\ndef SmoothBoundary (Ω : Set X) : Prop :=\n  ∀ x ∈ frontier Ω, ∃ e : OpenPartialHomeomorph X X,\n    x ∈ e.source ∧\n    ContDiffOn ℝ (⊤ : ℕ∞) e e.source ∧\n    ContDiffOn ℝ (⊤ : ℕ∞) e.symm e.target ∧\n    e x 0 = 0 ∧\n    ∀ y ∈ e.source, y ∈ Ω ↔ 0 < e y 0\n\ndef Domain (Ω : Set X) : Prop :=\n  IsOpen Ω ∧ IsConnected Ω ∧ Bornology.IsBounded Ω ∧ SmoothBoundary Ω\n\ndef SmoothUpTo (Ω : Set X) (f : X → ℝ) : Prop :=\n  ∃ U : Set X, IsOpen U ∧ closure Ω ⊆ U ∧ ContDiffOn ℝ (⊤ : ℕ∞) f U\n\ndef Admissible (Ω : Set X) (lam mu : X → ℝ) : Prop :=\n  SmoothUpTo Ω lam ∧ SmoothUpTo Ω mu ∧\n    ∀ x ∈ closure Ω, 0 < mu x ∧ 0 < 3 * lam x + 2 * mu x\n\nabbrev Ambient (Ω : Set X) :=\n  Lp V 2 (volume.restrict Ω) × (Fin 3 → Lp V 2 (volume.restrict Ω))\n\ndef coordDeriv (f : X → V) (i : Fin 3) (x : X) : V :=\n  fderiv ℝ f x (coordVector i)\n\ndef SmoothJets (Ω : Set X) : Set (Ambient Ω) :=\n  {j | ∃ (f : X → V) (hf : MemLp f 2 (volume.restrict Ω))\n      (hd : ∀ i, MemLp (coordDeriv f i) 2 (volume.restrict Ω)),\n    ContDiff ℝ (⊤ : ℕ∞) f ∧\n    j.1 = hf.toLp f ∧ j.2 = fun i => (hd i).toLp (coordDeriv f i)}\n\ndef TestJets (Ω : Set X) : Set (Ambient Ω) :=\n  {j | ∃ (f : X → V) (hf : MemLp f 2 (volume.restrict Ω))\n      (hd : ∀ i, MemLp (coordDeriv f i) 2 (volume.restrict Ω)),\n    ContDiff ℝ (⊤ : ℕ∞) f ∧ HasCompactSupport f ∧ tsupport f ⊆ Ω ∧\n    j.1 = hf.toLp f ∧ j.2 = fun i => (hd i).toLp (coordDeriv f i)}\n\nabbrev H1 (Ω : Set X) := {j : Ambient Ω // j ∈ closure (SmoothJets Ω)}\n\ndef HasZeroTrace (Ω : Set X) (u : H1 Ω) : Prop :=\n  u.val ∈ closure (TestJets Ω)\n\ndef SameTrace (Ω : Set X) (u v : H1 Ω) : Prop :=\n  u.val - v.val ∈ closure (TestJets Ω)\n\nabbrev BoundaryData (Ω : Set X) := Quot (SameTrace Ω)\n\ndef trace (Ω : Set X) (u : H1 Ω) : BoundaryData Ω := Quot.mk _ u\n\ndef div (Ω : Set X) (u : H1 Ω) (x : X) : ℝ :=\n  ∑ i, (u.val.2 i x) i\n\ndef strain (Ω : Set X) (u : H1 Ω) (x : X) (i j : Fin 3) : ℝ :=\n  ((u.val.2 j x) i + (u.val.2 i x) j) / 2\n\ndef energy (Ω : Set X) (lam mu : X → ℝ) (u v : H1 Ω) : ℝ :=\n  ∫ x, (lam x * div Ω u x * div Ω v x +\n    2 * mu x * ∑ i, ∑ j, strain Ω u x i j * strain Ω v x i j)\n    ∂(volume.restrict Ω)\n\ndef WeakSolution (Ω : Set X) (lam mu : X → ℝ) (u : H1 Ω) : Prop :=\n  ∀ v : H1 Ω, HasZeroTrace Ω v → energy Ω lam mu u v = 0\n\ndef dirichletSolution (Ω : Set X) (lam mu : X → ℝ)\n    (f : BoundaryData Ω) : H1 Ω := by\n  classical\n  exact if h : ∃ u : H1 Ω, trace Ω u = f ∧ WeakSolution Ω lam mu u\n  then Classical.choose h\n  else f.out\n\ndef DN (Ω : Set X) (lam mu : X → ℝ) : BoundaryData Ω → BoundaryData Ω → ℝ :=\n  fun f g => energy Ω lam mu (dirichletSolution Ω lam mu f) g.out\n\ndef MainClaim : Prop :=\n  ∀ (Ω : Set X), Domain Ω →\n  ∀ (lam₁ mu₁ lam₂ mu₂ : X → ℝ),\n    Admissible Ω lam₁ mu₁ → Admissible Ω lam₂ mu₂ →\n    DN Ω lam₁ mu₁ = DN Ω lam₂ mu₂ →\n    (∀ x ∈ Ω, lam₁ x = lam₂ x) ∧ (∀ x ∈ Ω, mu₁ x = mu₂ x)\n\ntheorem global_uniqueness : MainClaim := by\n  sorry\n\nend Elasticity\n\nend\n\nend OAI\n\n"
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          "text": "\nFILE lean/ComparatorChallenges/SquareRootDegree.lean \nimport Mathlib\n\nnamespace OAI\n\n/-!\nBool encodes a sign: false is +1 and true is -1. The average is uniform,\nand degree is ordinary real Fourier degree, not F₂ degree or threshold degree.\n-/\n\nopen scoped BigOperators\n\nnoncomputable section\n\nnamespace SquareRootDegree\n\nabbrev Cube (ι : Type*) := ι → Bool\n\ndef sign (b : Bool) : ℝ := if b then -1 else 1\n\ndef average {α : Type*} [Fintype α] (f : α → ℝ) : ℝ :=\n  (Fintype.card α : ℝ)⁻¹ * ∑ x, f x\n\ndef character {ι : Type*} (s : Finset ι) (x : Cube ι) : ℝ :=\n  ∏ i ∈ s, sign (x i)\n\ndef fourierCoeff {ι : Type*} [Fintype ι] (f : Cube ι → ℝ) (s : Finset ι) : ℝ := by\n  classical\n  exact average (fun x => f x * character s x)\n\ndef fourierDegree {ι : Type*} [Fintype ι] (f : Cube ι → ℝ) : ℕ := by\n  classical\n  exact (Finset.univ.filter (fun s : Finset ι => fourierCoeff f s ≠ 0)).sup Finset.card\n\ndef IsBoolean {α : Type*} (f : α → ℝ) : Prop :=\n  ∀ x, f x = -1 ∨ f x = 1\n\ndef Nonconstant {α : Type*} (f : α → ℝ) : Prop :=\n  ∃ x y, f x ≠ f y\n\ndef singletonSum {ι : Type*} [Fintype ι] (f : Cube ι → ℝ) : ℝ :=\n  ∑ i, fourierCoeff f {i}\n\ndef absoluteSingletonSum {ι : Type*} [Fintype ι] (f : Cube ι → ℝ) : ℝ :=\n  ∑ i, |fourierCoeff f {i}|\n\ndef SignedViolations : Prop :=\n  ∀ C : ℝ, 0 < C → ∃ n : ℕ, 0 < n ∧ ∃ f : Cube (Fin n) → ℝ,\n    IsBoolean f ∧ Nonconstant f ∧ C * Real.sqrt (fourierDegree f) < singletonSum f\n\ndef AbsoluteRatios : Set ℝ :=\n  {r | ∃ n : ℕ, 0 < n ∧ ∃ f : Cube (Fin n) → ℝ,\n    IsBoolean f ∧ 0 < fourierDegree f ∧\n      r = absoluteSingletonSum f / Real.sqrt (fourierDegree f)}\n\n/-- The two clauses of the literal main theorem. Unboundedness above is the\nreal-valued formulation of the displayed supremum being positive infinity. -/\ntheorem main : SignedViolations ∧ ¬ BddAbove AbsoluteRatios := by\n  sorry\n\nend SquareRootDegree\n\nend\n\nend OAI\n\n\nFILE lean/ComparatorChallenges/HotSpots.lean \nimport Mathlib\n\nnamespace OAI\n\nnoncomputable section\nopen Set MeasureTheory\nopen scoped ContDiff\n\nnamespace StrictHotSpots\n\nabbrev Plane := EuclideanSpace ℝ (Fin 2)\n\ndef SmoothBoundary (Ω : Set Plane) : Prop :=\n  ∀ p ∈ frontier Ω, ∃ (U : Set Plane) (ρ : Plane → ℝ),\n    IsOpen U ∧ p ∈ U ∧ ContDiffOn ℝ ∞ ρ U ∧\n    fderiv ℝ ρ p ≠ 0 ∧ ρ p = 0 ∧\n    Ω ∩ U = {x | ρ x < 0} ∩ U\n\ndef AdmissibleDomain (Ω : Set Plane) : Prop :=\n  Ω.Nonempty ∧ IsOpen Ω ∧ Bornology.IsBounded Ω ∧\n    IsSimplyConnected Ω ∧ SmoothBoundary Ω\n\ndef HasH1Gradient (Ω : Set Plane) (v : Plane → ℝ) (g : Plane → Plane) : Prop :=\n  MemLp v 2 (volume.restrict Ω) ∧ MemLp g 2 (volume.restrict Ω) ∧\n  ∀ φ : Plane → ℝ, ContDiff ℝ ∞ φ → HasCompactSupport φ → tsupport φ ⊆ Ω →\n    ∀ e : Plane,\n      (∫ x in Ω, v x * (fderiv ℝ φ x) e) =\n        -(∫ x in Ω, (inner ℝ (g x) e) * φ x)\n\ndef rayleighValues (Ω : Set Plane) : Set ℝ :=\n  {r | ∃ (v : Plane → ℝ) (g : Plane → Plane),\n    HasH1Gradient Ω v g ∧ (∫ x in Ω, v x) = 0 ∧\n    0 < (∫ x in Ω, (v x) ^ 2) ∧\n    r = (∫ x in Ω, ‖g x‖ ^ 2) / (∫ x in Ω, (v x) ^ 2)}\n\ndef firstPositiveNeumannValue (Ω : Set Plane) : ℝ := sInf (rayleighValues Ω)\n\ndef InFirstNeumannEigenspace (Ω : Set Plane) (u : Plane → ℝ) : Prop :=\n  ContDiffOn ℝ ∞ u (closure Ω) ∧ HasH1Gradient Ω u (gradient u) ∧\n    (∫ x in Ω, u x) = 0 ∧\n    ∀ (v : Plane → ℝ) (g : Plane → Plane), HasH1Gradient Ω v g →\n      (∫ x in Ω, inner ℝ (gradient u x) (g x)) =\n        firstPositiveNeumannValue Ω * (∫ x in Ω, u x * v x)\n\ndef MainConclusion (Ω : Set Plane) (u : Plane → ℝ) : Prop :=\n  (∀ x ∈ Ω, gradient u x ≠ 0) ∧\n    ∀ x ∈ Ω, sInf (u '' frontier Ω) < u x ∧ u x < sSup (u '' frontier Ω)\n\ntheorem main_theorem (Ω : Set Plane) (hΩ : AdmissibleDomain Ω)\n    (u : Plane → ℝ) (hu : InFirstNeumannEigenspace Ω u)\n    (hne : ∃ x ∈ Ω, u x ≠ 0) : MainConclusion Ω u := by\n  sorry\n\nend StrictHotSpots\nend\n\nend OAI\n\n\nFILE lean/ComparatorChallenges/ElasticityUniqueness.lean \nimport Mathlib\n\nnamespace OAI\n\n/-!\n# Global uniqueness for smooth isotropic elasticity\n\nThe physical Dirichlet-to-Neumann pairing acts on the variational trace\nquotient. Its quotient norm is not identified here with boundary-chart\nfractional Sobolev norms.\n-/\n\nnoncomputable section\nopen MeasureTheory Set\nopen scoped BigOperators\n\nnamespace Elasticity\n\nabbrev X := EuclideanSpace ℝ (Fin 3)\nabbrev V := X\n\ndef coordVector (i : Fin 3) : X := EuclideanSpace.single i 1\n\ndef SmoothBoundary (Ω : Set X) : Prop :=\n  ∀ x ∈ frontier Ω, ∃ e : OpenPartialHomeomorph X X,\n    x ∈ e.source ∧\n    ContDiffOn ℝ (⊤ : ℕ∞) e e.source ∧\n    ContDiffOn ℝ (⊤ : ℕ∞) e.symm e.target ∧\n    e x 0 = 0 ∧\n    ∀ y ∈ e.source, y ∈ Ω ↔ 0 < e y 0\n\ndef Domain (Ω : Set X) : Prop :=\n  IsOpen Ω ∧ IsConnected Ω ∧ Bornology.IsBounded Ω ∧ SmoothBoundary Ω\n\ndef SmoothUpTo (Ω : Set X) (f : X → ℝ) : Prop :=\n  ∃ U : Set X, IsOpen U ∧ closure Ω ⊆ U ∧ ContDiffOn ℝ (⊤ : ℕ∞) f U\n\ndef Admissible (Ω : Set X) (lam mu : X → ℝ) : Prop :=\n  SmoothUpTo Ω lam ∧ SmoothUpTo Ω mu ∧\n    ∀ x ∈ closure Ω, 0 < mu x ∧ 0 < 3 * lam x + 2 * mu x\n\nabbrev Ambient (Ω : Set X) :=\n  Lp V 2 (volume.restrict Ω) × (Fin 3 → Lp V 2 (volume.restrict Ω))\n\ndef coordDeriv (f : X → V) (i : Fin 3) (x : X) : V :=\n  fderiv ℝ f x (coordVector i)\n\ndef SmoothJets (Ω : Set X) : Set (Ambient Ω) :=\n  {j | ∃ (f : X → V) (hf : MemLp f 2 (volume.restrict Ω))\n      (hd : ∀ i, MemLp (coordDeriv f i) 2 (volume.restrict Ω)),\n    ContDiff ℝ (⊤ : ℕ∞) f ∧\n    j.1 = hf.toLp f ∧ j.2 = fun i => (hd i).toLp (coordDeriv f i)}\n\ndef TestJets (Ω : Set X) : Set (Ambient Ω) :=\n  {j | ∃ (f : X → V) (hf : MemLp f 2 (volume.restrict Ω))\n      (hd : ∀ i, MemLp (coordDeriv f i) 2 (volume.restrict Ω)),\n    ContDiff ℝ (⊤ : ℕ∞) f ∧ HasCompactSupport f ∧ tsupport f ⊆ Ω ∧\n    j.1 = hf.toLp f ∧ j.2 = fun i => (hd i).toLp (coordDeriv f i)}\n\nabbrev H1 (Ω : Set X) := {j : Ambient Ω // j ∈ closure (SmoothJets Ω)}\n\ndef HasZeroTrace (Ω : Set X) (u : H1 Ω) : Prop :=\n  u.val ∈ closure (TestJets Ω)\n\ndef SameTrace (Ω : Set X) (u v : H1 Ω) : Prop :=\n  u.val - v.val ∈ closure (TestJets Ω)\n\nabbrev BoundaryData (Ω : Set X) := Quot (SameTrace Ω)\n\ndef trace (Ω : Set X) (u : H1 Ω) : BoundaryData Ω := Quot.mk _ u\n\ndef div (Ω : Set X) (u : H1 Ω) (x : X) : ℝ :=\n  ∑ i, (u.val.2 i x) i\n\ndef strain (Ω : Set X) (u : H1 Ω) (x : X) (i j : Fin 3) : ℝ :=\n  ((u.val.2 j x) i + (u.val.2 i x) j) / 2\n\ndef energy (Ω : Set X) (lam mu : X → ℝ) (u v : H1 Ω) : ℝ :=\n  ∫ x, (lam x * div Ω u x * div Ω v x +\n    2 * mu x * ∑ i, ∑ j, strain Ω u x i j * strain Ω v x i j)\n    ∂(volume.restrict Ω)\n\ndef WeakSolution (Ω : Set X) (lam mu : X → ℝ) (u : H1 Ω) : Prop :=\n  ∀ v : H1 Ω, HasZeroTrace Ω v → energy Ω lam mu u v = 0\n\ndef dirichletSolution (Ω : Set X) (lam mu : X → ℝ)\n    (f : BoundaryData Ω) : H1 Ω := by\n  classical\n  exact if h : ∃ u : H1 Ω, trace Ω u = f ∧ WeakSolution Ω lam mu u\n  then Classical.choose h\n  else f.out\n\ndef DN (Ω : Set X) (lam mu : X → ℝ) : BoundaryData Ω → BoundaryData Ω → ℝ :=\n  fun f g => energy Ω lam mu (dirichletSolution Ω lam mu f) g.out\n\ndef MainClaim : Prop :=\n  ∀ (Ω : Set X), Domain Ω →\n  ∀ (lam₁ mu₁ lam₂ mu₂ : X → ℝ),\n    Admissible Ω lam₁ mu₁ → Admissible Ω lam₂ mu₂ →\n    DN Ω lam₁ mu₁ = DN Ω lam₂ mu₂ →\n    (∀ x ∈ Ω, lam₁ x = lam₂ x) ∧ (∀ x ∈ Ω, mu₁ x = mu₂ x)\n\ntheorem global_uniqueness : MainClaim := by\n  sorry\n\nend Elasticity\n\nend\n\nend OAI\n\n"
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        "stdout": "\nFILE lean/ComparatorChallenges/DukePrimeDegree.lean TAIL\ntice {n : ℕ} {K : Type*} [Field K] [NumberField K]\n    (p : ℕ) [Fact p.Prime] (M : FullLattice n K) :\n    Submodule (PadicInt p) (LocalAlgebra p K) :=\n  Submodule.span (PadicInt p)\n    ((Algebra.TensorProduct.includeRight : K →ₐ[ℚ] LocalAlgebra p K) '' (M.carrier : Set K))\n\ndef LocallyHomothetic {n : ℕ} {K : Type*} [Field K] [NumberField K]\n    (M M' : FullLattice n K) : Prop :=\n  ∀ (p : ℕ) (hp : p.Prime), letI : Fact p.Prime := ⟨hp⟩\n    ∃ c : (LocalAlgebra p K)ˣ,\n      (completedLattice p M' : Set (LocalAlgebra p K)) =\n        (fun z => (c : LocalAlgebra p K) * z) '' (completedLattice p M : Set (LocalAlgebra p K))\n\ndef packetOrbits {d : ℕ} {K : Type*} [Field K] [NumberField K]\n    (M : FullLattice (d + 1) K) (σ : OrderedEmbeddings (d + 1) K) :\n    Set (Set (X (d + 1))) :=\n  {O | ∃ M' : FullLattice (d + 1) K, LocallyHomothetic M M' ∧\n    ∃ w ∈ signMatrices (d + 1), O = diagonalOrbit (rightAction (latticePoint M' σ) w)}\n\ndef orbitBasepoint {d : ℕ} {K : Type*} [Field K] [NumberField K]\n    {M : FullLattice (d + 1) K} {σ : OrderedEmbeddings (d + 1) K}\n    (O : packetOrbits M σ) : X (d + 1) :=\n  Classical.epsilon (fun x => diagonalOrbit x = O.1)\n\ndef packetMeasure {d : ℕ} {K : Type*} [Field K] [NumberField K]\n    (M : FullLattice (d + 1) K) (σ : OrderedEmbeddings (d + 1) K) : Measure (X (d + 1)) :=\n  (∑' O : packetOrbits M σ, orbitVolume (orbitBasepoint O))⁻¹ •\n    Measure.sum (fun O : packetOrbits M σ =>\n      orbitVolume (orbitBasepoint O) • orbitProbability (orbitBasepoint O))\n\ndef IsHaarProbability {n : ℕ} (μ : Measure (X n)) : Prop :=\n  IsProbabilityMeasure μ ∧ ∀ g : G n, Measure.map (fun x => rightAction x g) μ = μ\n\ndef WeakProbabilityConvergence {n : ℕ} (μ : ℕ → Measure (X n)) (m : Measure (X n)) : Prop :=\n  (∀ i, IsProbabilityMeasure (μ i)) ∧ IsProbabilityMeasure m ∧\n    ∀ f : BoundedContinuousFunction (X n) ℝ,\n    Tendsto (fun i => ∫ x, f x ∂μ i) atTop (𝓝 (∫ x, f x ∂m))\n\nend PrimeDegreePackets\nend\nend\n\nopen MeasureTheory Filter\nopen scoped Topology\nnamespace DukePrimeDegree\nopen PrimeDegreePackets\n\nuniverse u\n\n/-- Theorem 1.1 as weak convergence of the actual complete packet measures,\nincluding tightness of the entire family. -/\ntheorem prime_degree_packet_measure_equidistribution_unconditional\n    (d : ℕ) (hprime : Nat.Prime (d+1)) (hfive : 5 ≤ d+1)\n    (K : ℕ → Type u) [∀ i, Field (K i)] [∀ i, NumberField (K i)]\n    [∀ i, NumberField.IsTotallyReal (K i)]\n    (M : ∀ i, PrimeDegreePackets.FullLattice (d+1) (K i))\n    (σ : ∀ i, OrderedEmbeddings (d+1) (K i))\n    (hn : ∀ i, Module.finrank ℚ (K i)=d+1)\n    (hD : Tendsto (fun i => PrimeDegreePackets.multiplierDiscriminant (M i)) atTop atTop) :\n    ∃ ν : Measure (X (d+1)), IsHaarProbability ν ∧\n      WeakProbabilityConvergence (fun i => packetMeasure (M i) (σ i)) ν ∧\n      IsTightMeasureSet (Set.range (fun i => packetMeasure (M i) (σ i))) := by\n  sorry\n\n/-- The complete packet measures converge weakly to any specified Haar\nprobability, in the formulation by bounded continuous test functions. -/\ntheorem prime_degree_packet_measures_tendsto_haar_unconditional\n    (d : ℕ) (hprime : Nat.Prime (d+1)) (hfive : 5 ≤ d+1)\n    (K : ℕ → Type u) [∀ i, Field (K i)] [∀ i, NumberField (K i)]\n    [∀ i, NumberField.IsTotallyReal (K i)]\n    (M : ∀ i, PrimeDegreePackets.FullLattice (d+1) (K i))\n    (σ : ∀ i, OrderedEmbeddings (d+1) (K i))\n    (hn : ∀ i, Module.finrank ℚ (K i)=d+1)\n    (hD : Tendsto (fun i => PrimeDegreePackets.multiplierDiscriminant (M i)) atTop atTop)\n    (m : ProbabilityMeasure (X (d+1)))\n    (hm : IsHaarProbability (m : Measure (X (d+1)))) :\n    WeakProbabilityConvergence (fun i => packetMeasure (M i) (σ i))\n      (m : Measure (X (d+1))) := by\n  sorry\n\nend DukePrimeDegree\n\nend OAI\n\n\nFILE lean/ComparatorChallenges/TotientAsymptotic.lean TAIL\nrimeFactor (n : ℕ) : ℕ := max 1 (n.primeFactors.sup id)\n\nstructure TailDatum (H : ℕ) where\n  Q : Fin H → ℕ\n  cofactor : ℕ\n\ndef tailPrime {H : ℕ} (η : TailDatum H) (h : ℕ) : ℕ :=\n  if hH : h < H then η.Q ⟨h, hH⟩ else 1\n\ndef tailLog {H : ℕ} (η : TailDatum H) (h : ℕ) : ℝ :=\n  Real.log (Real.log (tailPrime η h : ℝ))\n\ndef IsWitness (H : ℕ) (s : ℝ) (η : TailDatum H) : Prop :=\n  (∀ h : Fin H, h.val < P H → η.Q h = 1) ∧\n  0 < η.cofactor ∧\n  (∀ h ∈ Finset.Ico (P H) H, (tailPrime η h).Prime ∧\n    (9 / 10 : ℝ) * alpha s * h * (rho^h)⁻¹ ≤ tailLog η h ∧\n    tailLog η h ≤ (11 / 10 : ℝ) * alpha s * h * (rho^h)⁻¹ ∧\n    (∑ l ∈ Finset.Ico (P H) h, a (h-l) * tailLog η l) ≤\n      (1 + (1 / 10000 : ℝ) * Real.exp (-(h : ℝ) / 40)) * tailLog η h) ∧\n  largestPrimeFactor η.cofactor ≤ tailPrime η (P H) ∧\n  Real.log (η.cofactor : ℝ) ≤\n    Real.exp (2 * alpha s * (P H : ℝ) * (rho^(P H))⁻¹)\n\ndef w {H : ℕ} (η : TailDatum H) : ℕ :=\n  η.cofactor * ∏ h ∈ Finset.Ico (P H) H, tailPrime η h\n\ndef witnesses (H : ℕ) (s : ℝ) (d : ℕ) : Set (TailDatum H) :=\n  {η | IsWitness H s η ∧ (w η).totient = d}\n\ndef D {H : ℕ} (h : ℕ) (η : TailDatum H) : ℝ :=\n  ∑ l ∈ Finset.Ico (P H) H, a (h-l) * tailLog η l\n\ndef maxD {H : ℕ} (h : ℕ) (T : Finset (TailDatum H)) : ℝ := by\n  classical\n  exact if ht : T.Nonempty then T.sup' ht (D h) else 0\n\ndef AH (H : ℕ) (f : ℝ → ℝ) (s : ℝ) : ℝ := by\n  classical\n  exact rho^(H*(H-1)/2) * (gamma / alpha s)^H *\n    ∑ᶠ d : ℕ,\n      if IsTotient d then\n        f ((ell d : ℝ) / d) / d *\n          ∑ᶠ T : Finset (TailDatum H),\n            if T.Nonempty ∧ (↑T : Set (TailDatum H)) ⊆ witnesses H s d then\n              (-1 : ℝ)^(T.card-1) *\n               Real.exp (-(gamma / alpha s) *\n                 ∑' n : ℕ, rho^(H+n) * maxD (H+n) T)\n            else 0\n      else 0\n\ndef A (f : ℝ → ℝ) (s : ℝ) : ℝ := limUnder atTop (fun H : ℕ => AH H f s)\n\ndef mainTerm (x : ℝ) : ℝ := x / Real.log x * G x (m x) * A (fun _ => 1) (theta x)\n\ndef N (k : ℕ) (x : ℝ) : ℝ := by\n  classical\n  exact (((Finset.Icc 1 ⌊x⌋₊).filter fun v =>\n    IsTotient v ∧ (k : ℝ) * x < ell v ∧ (ell v : ℝ) ≤ (k+1 : ℝ) * x).card : ℝ)\n\ndef fk (k : ℕ) (r : ℝ) : ℝ := min 1 ((k+1 : ℝ) / r) - min 1 ((k : ℝ) / r)\n\ndef normalizedCount (F : ℝ → ℝ) (x : ℝ) : ℝ :=\n  F x / (x / Real.log x * G x (m x))\n\ndef tupleNormalization (x : ℝ) : ℝ := x / Real.log x * G x (m x)\n\ntheorem totient_asymptotic_formula\n    :\n    TendstoUniformlyOn (fun H => AH H (fun _ => 1)) (A (fun _ => 1))\n      atTop (Set.Ico (0 : ℝ) 1) ∧\n    (∃ cMinus cPlus : ℝ, 0 < cMinus ∧ ∀ s ∈ Set.Ico (0 : ℝ) 1,\n      cMinus ≤ A (fun _ => 1) s ∧ A (fun _ => 1) s ≤ cPlus) ∧\n    Tendsto (fun x => V x/mainTerm x) atTop (nhds 1) ∧\n    ∀ c : ℝ, 0 < c → Tendsto (fun x => V (c*x)/V x) atTop (nhds c) := by\n  sorry\n\ntheorem weighted_totient_asymptotic\n    (k : ℕ) (hk : 1≤k) :\n    TendstoUniformlyOn (fun H => AH H (fk k)) (A (fk k)) atTop (Set.Ico (0 : ℝ) 1) ∧\n    Tendsto (fun x => normalizedCount (N k) x-A (fk k) (theta x)) atTop (nhds 0) ∧\n    ((∃ d : ℕ, IsTotient d ∧ k*d<ell d) →\n      ∃ cMinus cPlus : ℝ, 0<cMinus ∧ 0<cPlus ∧\n        (∀ s∈Set.Ico (0 : ℝ) 1, cMinus≤A (fk k) s ∧ A (fk k) s≤cPlus) ∧\n        Tendsto (fun x => N k x/(tupleNormalization x*A (fk k) (theta x))) atTop (nhds 1) ∧\n        ∀ᶠ x : ℝ in atTop, (cMinus/cPlus)*V x≤N k x ∧ N k x≤V x) ∧\n    ((¬∃ d : ℕ, IsTotient d ∧ k*d<ell d) →\n      (∀ x : ℝ, 0<x → N k x=0) ∧ (∀ s∈Set.Ico (0 : ℝ) 1, A (fk k) s=0)) := by\n  sorry\n\ntheorem weighted_totient_one_two\n    :\n    ∀ k∈({1,2} : Finset ℕ), ∃ c : ℝ, 0<c ∧\n      ∀ s∈Set.Ico (0 : ℝ) 1, c≤A (fk k) s := by\n  sorry\n\ntheorem coefficient_nonnegative (H : ℕ) {s : ℝ}\n    (hs : s ∈ Set.Ico (0 : ℝ) 1) : 0 ≤ AH H (fun _ => 1) s := by\n  sorry\n\nend TotientAsymptotic\n\nend\n\nend OAI\n\n\nFILE lean/ComparatorChallenges/ContingencyTables.lean TAIL\nact ((Finset.univ.filter fun bits : Fin t → Bool =>\n      (MatchingFPRAS.run A input bits).state = none).card : ℝ) / (2 : ℝ) ^ t\n\ndef samplingSize (n : ℕ) {m : ℕ} (r : Fin m → ℕ) (k : ℕ) : ℕ :=\n  m + n + Nat.clog 2 ((∑ i, r i) + 1) + k + 1\n\n/-- Paper 200-01, Theorem main(i): bounded runtime on every random tape\nand total variation at most `2⁻ᵏ`, with freely varying dimensions. -/\ndef BoundedSamplingStatement : Prop :=\n  ∃ (A : Machine) (C d : ℕ), 0 < C ∧\n    ∀ (m n : ℕ) (r : Fin m → ℕ) (c : Fin n → ℕ),\n      (∑ i, r i) = ∑ j, c j → ∀ k : ℕ, 1 ≤ k →\n        let input := encodeSamplingInput r c k\n        let t := C * (samplingSize n r k) ^ d\n        (∀ bits : Fin t → Bool, ∃ X : Table r c, OutputsTable A input bits X.val) ∧\n        (∑ X : Table r c,\n          |tableMass A input t X.val - (Fintype.card (Table r c) : ℝ)⁻¹|) / 2 ≤\n            ((2 : ℝ) ^ k)⁻¹\n\n/-- Paper 200-01, Theorem main(ii): every halted output is feasible, each\ntable has limiting mass `1/|Ω|`, termination has probability one, and the\nexpected number of bit operations is bounded by a uniform polynomial. -/\ndef ExactSamplingStatement : Prop :=\n  ∃ (A : Machine) (C d : ℕ), 0 < C ∧\n    ∀ (m n : ℕ) (r : Fin m → ℕ) (c : Fin n → ℕ),\n      (∑ i, r i) = ∑ j, c j →\n        let input := encodeMargins r c\n        let tail : ℕ → ℝ := fun t => 1 - haltMass A input t\n        (∀ t (bits : Fin t → Bool), (MatchingFPRAS.run A input bits).state = none →\n          ∃ X : Table r c, OutputsTable A input bits X.val) ∧\n        (∀ X : Table r c, Tendsto (fun t => tableMass A input t X.val) atTop\n          (nhds ((Fintype.card (Table r c) : ℝ)⁻¹))) ∧\n        Tendsto (haltMass A input) atTop (nhds 1) ∧\n        Summable tail ∧\n        tsum tail ≤ C * (samplingSize n r 0) ^ d\n\nnoncomputable def count {m n : ℕ} (r : Fin m → ℕ) (c : Fin n → ℕ)\n    (b : Fin m → Fin n → ℕ) : ℕ := Fintype.card (BoundedTable r c b)\n\ndef countingTime (C d : ℕ) {m n : ℕ} (r : Fin m → ℕ) (c : Fin n → ℕ)\n    (b : Fin m → Fin n → ℕ) (ε δ : ℚ) : ℕ :=\n  C * ((encodeCountingInput r c b ε δ).length + ⌈ε⁻¹⌉₊ +\n    Nat.clog 2 ⌈δ⁻¹⌉₊ + 1) ^ d\n\nnoncomputable def goodCountingTapes (A : Machine) {m n : ℕ}\n    (r : Fin m → ℕ) (c : Fin n → ℕ) (b : Fin m → Fin n → ℕ)\n    (ε δ : ℚ) (t : ℕ) : Finset (Fin t → Bool) := by\n  classical\n  exact Finset.univ.filter fun bits => ∃ q : ℚ,\n    MatchingFPRAS.Outputs A (encodeCountingInput r c b ε δ) bits q ∧\n      (1 - ε) * count r c b ≤ q ∧ q ≤ (1 + ε) * count r c b\n\n/-- Paper 200-02, Theorem main. The input length includes the rational\naccuracy parameters, and infeasible inputs always produce exactly zero. -/\ndef CountingStatement : Prop :=\n  ∃ (A : Machine) (C d : ℕ), 0 < C ∧\n    ∀ (m n : ℕ), 0 < m → 0 < n →\n    ∀ (r : Fin m → ℕ) (c : Fin n → ℕ) (b : Fin m → Fin n → ℕ),\n      (∑ i, r i) = ∑ j, c j →\n      ∀ ε δ : ℚ, 0 < ε → ε < 1 → 0 < δ → δ < 1 →\n        let input := encodeCountingInput r c b ε δ\n        let t := countingTime C d r c b ε δ\n        (∀ bits : Fin t → Bool, ∃ q : ℚ, 0 ≤ q ∧ MatchingFPRAS.Outputs A input bits q) ∧\n        (count r c b = 0 → ∀ bits : Fin t → Bool, MatchingFPRAS.Outputs A input bits 0) ∧\n        ((goodCountingTapes A r c b ε δ t).card : ℚ) / (2 : ℚ) ^ t ≥ 1 - δ\n\nend ContingencyTables.Algorithms\n\nnamespace ContingencyTables\n\n/-- An almost-uniform sampler with polynomial cost on every execution. -/\ntheorem boundedSampling : Algorithms.BoundedSamplingStatement := by sorry\n\n/-- Exact uniform sampling in polynomial expected bit time. -/\ntheorem exactSampling : Algorithms.ExactSamplingStatement := by sorry\n\n/-- A cell-bounded FPRAS, with zero output on every infeasible execution. -/\ntheorem counting : Algorithms.CountingStatement := by sorry\n\nend ContingencyTables\n\nend OAI\n\n",
        "stderr": "",
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        "plain": [
          "\nFILE lean/ComparatorChallenges/DukePrimeDegree.lean TAIL\ntice {n : ℕ} {K : Type*} [Field K] [NumberField K]\n    (p : ℕ) [Fact p.Prime] (M : FullLattice n K) :\n    Submodule (PadicInt p) (LocalAlgebra p K) :=\n  Submodule.span (PadicInt p)\n    ((Algebra.TensorProduct.includeRight : K →ₐ[ℚ] LocalAlgebra p K) '' (M.carrier : Set K))\n\ndef LocallyHomothetic {n : ℕ} {K : Type*} [Field K] [NumberField K]\n    (M M' : FullLattice n K) : Prop :=\n  ∀ (p : ℕ) (hp : p.Prime), letI : Fact p.Prime := ⟨hp⟩\n    ∃ c : (LocalAlgebra p K)ˣ,\n      (completedLattice p M' : Set (LocalAlgebra p K)) =\n        (fun z => (c : LocalAlgebra p K) * z) '' (completedLattice p M : Set (LocalAlgebra p K))\n\ndef packetOrbits {d : ℕ} {K : Type*} [Field K] [NumberField K]\n    (M : FullLattice (d + 1) K) (σ : OrderedEmbeddings (d + 1) K) :\n    Set (Set (X (d + 1))) :=\n  {O | ∃ M' : FullLattice (d + 1) K, LocallyHomothetic M M' ∧\n    ∃ w ∈ signMatrices (d + 1), O = diagonalOrbit (rightAction (latticePoint M' σ) w)}\n\ndef orbitBasepoint {d : ℕ} {K : Type*} [Field K] [NumberField K]\n    {M : FullLattice (d + 1) K} {σ : OrderedEmbeddings (d + 1) K}\n    (O : packetOrbits M σ) : X (d + 1) :=\n  Classical.epsilon (fun x => diagonalOrbit x = O.1)\n\ndef packetMeasure {d : ℕ} {K : Type*} [Field K] [NumberField K]\n    (M : FullLattice (d + 1) K) (σ : OrderedEmbeddings (d + 1) K) : Measure (X (d + 1)) :=\n  (∑' O : packetOrbits M σ, orbitVolume (orbitBasepoint O))⁻¹ •\n    Measure.sum (fun O : packetOrbits M σ =>\n      orbitVolume (orbitBasepoint O) • orbitProbability (orbitBasepoint O))\n\ndef IsHaarProbability {n : ℕ} (μ : Measure (X n)) : Prop :=\n  IsProbabilityMeasure μ ∧ ∀ g : G n, Measure.map (fun x => rightAction x g) μ = μ\n\ndef WeakProbabilityConvergence {n : ℕ} (μ : ℕ → Measure (X n)) (m : Measure (X n)) : Prop :=\n  (∀ i, IsProbabilityMeasure (μ i)) ∧ IsProbabilityMeasure m ∧\n    ∀ f : BoundedContinuousFunction (X n) ℝ,\n    Tendsto (fun i => ∫ x, f x ∂μ i) atTop (𝓝 (∫ x, f x ∂m))\n\nend PrimeDegreePackets\nend\nend\n\nopen MeasureTheory Filter\nopen scoped Topology\nnamespace DukePrimeDegree\nopen PrimeDegreePackets\n\nuniverse u\n\n/-- Theorem 1.1 as weak convergence of the actual complete packet measures,\nincluding tightness of the entire family. -/\ntheorem prime_degree_packet_measure_equidistribution_unconditional\n    (d : ℕ) (hprime : Nat.Prime (d+1)) (hfive : 5 ≤ d+1)\n    (K : ℕ → Type u) [∀ i, Field (K i)] [∀ i, NumberField (K i)]\n    [∀ i, NumberField.IsTotallyReal (K i)]\n    (M : ∀ i, PrimeDegreePackets.FullLattice (d+1) (K i))\n    (σ : ∀ i, OrderedEmbeddings (d+1) (K i))\n    (hn : ∀ i, Module.finrank ℚ (K i)=d+1)\n    (hD : Tendsto (fun i => PrimeDegreePackets.multiplierDiscriminant (M i)) atTop atTop) :\n    ∃ ν : Measure (X (d+1)), IsHaarProbability ν ∧\n      WeakProbabilityConvergence (fun i => packetMeasure (M i) (σ i)) ν ∧\n      IsTightMeasureSet (Set.range (fun i => packetMeasure (M i) (σ i))) := by\n  sorry\n\n/-- The complete packet measures converge weakly to any specified Haar\nprobability, in the formulation by bounded continuous test functions. -/\ntheorem prime_degree_packet_measures_tendsto_haar_unconditional\n    (d : ℕ) (hprime : Nat.Prime (d+1)) (hfive : 5 ≤ d+1)\n    (K : ℕ → Type u) [∀ i, Field (K i)] [∀ i, NumberField (K i)]\n    [∀ i, NumberField.IsTotallyReal (K i)]\n    (M : ∀ i, PrimeDegreePackets.FullLattice (d+1) (K i))\n    (σ : ∀ i, OrderedEmbeddings (d+1) (K i))\n    (hn : ∀ i, Module.finrank ℚ (K i)=d+1)\n    (hD : Tendsto (fun i => PrimeDegreePackets.multiplierDiscriminant (M i)) atTop atTop)\n    (m : ProbabilityMeasure (X (d+1)))\n    (hm : IsHaarProbability (m : Measure (X (d+1)))) :\n    WeakProbabilityConvergence (fun i => packetMeasure (M i) (σ i))\n      (m : Measure (X (d+1))) := by\n  sorry\n\nend DukePrimeDegree\n\nend OAI\n\n\nFILE lean/ComparatorChallenges/TotientAsymptotic.lean TAIL\nrimeFactor (n : ℕ) : ℕ := max 1 (n.primeFactors.sup id)\n\nstructure TailDatum (H : ℕ) where\n  Q : Fin H → ℕ\n  cofactor : ℕ\n\ndef tailPrime {H : ℕ} (η : TailDatum H) (h : ℕ) : ℕ :=\n  if hH : h < H then η.Q ⟨h, hH⟩ else 1\n\ndef tailLog {H : ℕ} (η : TailDatum H) (h : ℕ) : ℝ :=\n  Real.log (Real.log (tailPrime η h : ℝ))\n\ndef IsWitness (H : ℕ) (s : ℝ) (η : TailDatum H) : Prop :=\n  (∀ h : Fin H, h.val < P H → η.Q h = 1) ∧\n  0 < η.cofactor ∧\n  (∀ h ∈ Finset.Ico (P H) H, (tailPrime η h).Prime ∧\n    (9 / 10 : ℝ) * alpha s * h * (rho^h)⁻¹ ≤ tailLog η h ∧\n    tailLog η h ≤ (11 / 10 : ℝ) * alpha s * h * (rho^h)⁻¹ ∧\n    (∑ l ∈ Finset.Ico (P H) h, a (h-l) * tailLog η l) ≤\n      (1 + (1 / 10000 : ℝ) * Real.exp (-(h : ℝ) / 40)) * tailLog η h) ∧\n  largestPrimeFactor η.cofactor ≤ tailPrime η (P H) ∧\n  Real.log (η.cofactor : ℝ) ≤\n    Real.exp (2 * alpha s * (P H : ℝ) * (rho^(P H))⁻¹)\n\ndef w {H : ℕ} (η : TailDatum H) : ℕ :=\n  η.cofactor * ∏ h ∈ Finset.Ico (P H) H, tailPrime η h\n\ndef witnesses (H : ℕ) (s : ℝ) (d : ℕ) : Set (TailDatum H) :=\n  {η | IsWitness H s η ∧ (w η).totient = d}\n\ndef D {H : ℕ} (h : ℕ) (η : TailDatum H) : ℝ :=\n  ∑ l ∈ Finset.Ico (P H) H, a (h-l) * tailLog η l\n\ndef maxD {H : ℕ} (h : ℕ) (T : Finset (TailDatum H)) : ℝ := by\n  classical\n  exact if ht : T.Nonempty then T.sup' ht (D h) else 0\n\ndef AH (H : ℕ) (f : ℝ → ℝ) (s : ℝ) : ℝ := by\n  classical\n  exact rho^(H*(H-1)/2) * (gamma / alpha s)^H *\n    ∑ᶠ d : ℕ,\n      if IsTotient d then\n        f ((ell d : ℝ) / d) / d *\n          ∑ᶠ T : Finset (TailDatum H),\n            if T.Nonempty ∧ (↑T : Set (TailDatum H)) ⊆ witnesses H s d then\n              (-1 : ℝ)^(T.card-1) *\n               Real.exp (-(gamma / alpha s) *\n                 ∑' n : ℕ, rho^(H+n) * maxD (H+n) T)\n            else 0\n      else 0\n\ndef A (f : ℝ → ℝ) (s : ℝ) : ℝ := limUnder atTop (fun H : ℕ => AH H f s)\n\ndef mainTerm (x : ℝ) : ℝ := x / Real.log x * G x (m x) * A (fun _ => 1) (theta x)\n\ndef N (k : ℕ) (x : ℝ) : ℝ := by\n  classical\n  exact (((Finset.Icc 1 ⌊x⌋₊).filter fun v =>\n    IsTotient v ∧ (k : ℝ) * x < ell v ∧ (ell v : ℝ) ≤ (k+1 : ℝ) * x).card : ℝ)\n\ndef fk (k : ℕ) (r : ℝ) : ℝ := min 1 ((k+1 : ℝ) / r) - min 1 ((k : ℝ) / r)\n\ndef normalizedCount (F : ℝ → ℝ) (x : ℝ) : ℝ :=\n  F x / (x / Real.log x * G x (m x))\n\ndef tupleNormalization (x : ℝ) : ℝ := x / Real.log x * G x (m x)\n\ntheorem totient_asymptotic_formula\n    :\n    TendstoUniformlyOn (fun H => AH H (fun _ => 1)) (A (fun _ => 1))\n      atTop (Set.Ico (0 : ℝ) 1) ∧\n    (∃ cMinus cPlus : ℝ, 0 < cMinus ∧ ∀ s ∈ Set.Ico (0 : ℝ) 1,\n      cMinus ≤ A (fun _ => 1) s ∧ A (fun _ => 1) s ≤ cPlus) ∧\n    Tendsto (fun x => V x/mainTerm x) atTop (nhds 1) ∧\n    ∀ c : ℝ, 0 < c → Tendsto (fun x => V (c*x)/V x) atTop (nhds c) := by\n  sorry\n\ntheorem weighted_totient_asymptotic\n    (k : ℕ) (hk : 1≤k) :\n    TendstoUniformlyOn (fun H => AH H (fk k)) (A (fk k)) atTop (Set.Ico (0 : ℝ) 1) ∧\n    Tendsto (fun x => normalizedCount (N k) x-A (fk k) (theta x)) atTop (nhds 0) ∧\n    ((∃ d : ℕ, IsTotient d ∧ k*d<ell d) →\n      ∃ cMinus cPlus : ℝ, 0<cMinus ∧ 0<cPlus ∧\n        (∀ s∈Set.Ico (0 : ℝ) 1, cMinus≤A (fk k) s ∧ A (fk k) s≤cPlus) ∧\n        Tendsto (fun x => N k x/(tupleNormalization x*A (fk k) (theta x))) atTop (nhds 1) ∧\n        ∀ᶠ x : ℝ in atTop, (cMinus/cPlus)*V x≤N k x ∧ N k x≤V x) ∧\n    ((¬∃ d : ℕ, IsTotient d ∧ k*d<ell d) →\n      (∀ x : ℝ, 0<x → N k x=0) ∧ (∀ s∈Set.Ico (0 : ℝ) 1, A (fk k) s=0)) := by\n  sorry\n\ntheorem weighted_totient_one_two\n    :\n    ∀ k∈({1,2} : Finset ℕ), ∃ c : ℝ, 0<c ∧\n      ∀ s∈Set.Ico (0 : ℝ) 1, c≤A (fk k) s := by\n  sorry\n\ntheorem coefficient_nonnegative (H : ℕ) {s : ℝ}\n    (hs : s ∈ Set.Ico (0 : ℝ) 1) : 0 ≤ AH H (fun _ => 1) s := by\n  sorry\n\nend TotientAsymptotic\n\nend\n\nend OAI\n\n\nFILE lean/ComparatorChallenges/ContingencyTables.lean TAIL\nact ((Finset.univ.filter fun bits : Fin t → Bool =>\n      (MatchingFPRAS.run A input bits).state = none).card : ℝ) / (2 : ℝ) ^ t\n\ndef samplingSize (n : ℕ) {m : ℕ} (r : Fin m → ℕ) (k : ℕ) : ℕ :=\n  m + n + Nat.clog 2 ((∑ i, r i) + 1) + k + 1\n\n/-- Paper 200-01, Theorem main(i): bounded runtime on every random tape\nand total variation at most `2⁻ᵏ`, with freely varying dimensions. -/\ndef BoundedSamplingStatement : Prop :=\n  ∃ (A : Machine) (C d : ℕ), 0 < C ∧\n    ∀ (m n : ℕ) (r : Fin m → ℕ) (c : Fin n → ℕ),\n      (∑ i, r i) = ∑ j, c j → ∀ k : ℕ, 1 ≤ k →\n        let input := encodeSamplingInput r c k\n        let t := C * (samplingSize n r k) ^ d\n        (∀ bits : Fin t → Bool, ∃ X : Table r c, OutputsTable A input bits X.val) ∧\n        (∑ X : Table r c,\n          |tableMass A input t X.val - (Fintype.card (Table r c) : ℝ)⁻¹|) / 2 ≤\n            ((2 : ℝ) ^ k)⁻¹\n\n/-- Paper 200-01, Theorem main(ii): every halted output is feasible, each\ntable has limiting mass `1/|Ω|`, termination has probability one, and the\nexpected number of bit operations is bounded by a uniform polynomial. -/\ndef ExactSamplingStatement : Prop :=\n  ∃ (A : Machine) (C d : ℕ), 0 < C ∧\n    ∀ (m n : ℕ) (r : Fin m → ℕ) (c : Fin n → ℕ),\n      (∑ i, r i) = ∑ j, c j →\n        let input := encodeMargins r c\n        let tail : ℕ → ℝ := fun t => 1 - haltMass A input t\n        (∀ t (bits : Fin t → Bool), (MatchingFPRAS.run A input bits).state = none →\n          ∃ X : Table r c, OutputsTable A input bits X.val) ∧\n        (∀ X : Table r c, Tendsto (fun t => tableMass A input t X.val) atTop\n          (nhds ((Fintype.card (Table r c) : ℝ)⁻¹))) ∧\n        Tendsto (haltMass A input) atTop (nhds 1) ∧\n        Summable tail ∧\n        tsum tail ≤ C * (samplingSize n r 0) ^ d\n\nnoncomputable def count {m n : ℕ} (r : Fin m → ℕ) (c : Fin n → ℕ)\n    (b : Fin m → Fin n → ℕ) : ℕ := Fintype.card (BoundedTable r c b)\n\ndef countingTime (C d : ℕ) {m n : ℕ} (r : Fin m → ℕ) (c : Fin n → ℕ)\n    (b : Fin m → Fin n → ℕ) (ε δ : ℚ) : ℕ :=\n  C * ((encodeCountingInput r c b ε δ).length + ⌈ε⁻¹⌉₊ +\n    Nat.clog 2 ⌈δ⁻¹⌉₊ + 1) ^ d\n\nnoncomputable def goodCountingTapes (A : Machine) {m n : ℕ}\n    (r : Fin m → ℕ) (c : Fin n → ℕ) (b : Fin m → Fin n → ℕ)\n    (ε δ : ℚ) (t : ℕ) : Finset (Fin t → Bool) := by\n  classical\n  exact Finset.univ.filter fun bits => ∃ q : ℚ,\n    MatchingFPRAS.Outputs A (encodeCountingInput r c b ε δ) bits q ∧\n      (1 - ε) * count r c b ≤ q ∧ q ≤ (1 + ε) * count r c b\n\n/-- Paper 200-02, Theorem main. The input length includes the rational\naccuracy parameters, and infeasible inputs always produce exactly zero. -/\ndef CountingStatement : Prop :=\n  ∃ (A : Machine) (C d : ℕ), 0 < C ∧\n    ∀ (m n : ℕ), 0 < m → 0 < n →\n    ∀ (r : Fin m → ℕ) (c : Fin n → ℕ) (b : Fin m → Fin n → ℕ),\n      (∑ i, r i) = ∑ j, c j →\n      ∀ ε δ : ℚ, 0 < ε → ε < 1 → 0 < δ → δ < 1 →\n        let input := encodeCountingInput r c b ε δ\n        let t := countingTime C d r c b ε δ\n        (∀ bits : Fin t → Bool, ∃ q : ℚ, 0 ≤ q ∧ MatchingFPRAS.Outputs A input bits q) ∧\n        (count r c b = 0 → ∀ bits : Fin t → Bool, MatchingFPRAS.Outputs A input bits 0) ∧\n        ((goodCountingTapes A r c b ε δ t).card : ℚ) / (2 : ℚ) ^ t ≥ 1 - δ\n\nend ContingencyTables.Algorithms\n\nnamespace ContingencyTables\n\n/-- An almost-uniform sampler with polynomial cost on every execution. -/\ntheorem boundedSampling : Algorithms.BoundedSamplingStatement := by sorry\n\n/-- Exact uniform sampling in polynomial expected bit time. -/\ntheorem exactSampling : Algorithms.ExactSamplingStatement := by sorry\n\n/-- A cell-bounded FPRAS, with zero output on every infeasible execution. -/\ntheorem counting : Algorithms.CountingStatement := by sorry\n\nend ContingencyTables\n\nend OAI\n\n"
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          "text": "\nFILE lean/ComparatorChallenges/DukePrimeDegree.lean TAIL\ntice {n : ℕ} {K : Type*} [Field K] [NumberField K]\n    (p : ℕ) [Fact p.Prime] (M : FullLattice n K) :\n    Submodule (PadicInt p) (LocalAlgebra p K) :=\n  Submodule.span (PadicInt p)\n    ((Algebra.TensorProduct.includeRight : K →ₐ[ℚ] LocalAlgebra p K) '' (M.carrier : Set K))\n\ndef LocallyHomothetic {n : ℕ} {K : Type*} [Field K] [NumberField K]\n    (M M' : FullLattice n K) : Prop :=\n  ∀ (p : ℕ) (hp : p.Prime), letI : Fact p.Prime := ⟨hp⟩\n    ∃ c : (LocalAlgebra p K)ˣ,\n      (completedLattice p M' : Set (LocalAlgebra p K)) =\n        (fun z => (c : LocalAlgebra p K) * z) '' (completedLattice p M : Set (LocalAlgebra p K))\n\ndef packetOrbits {d : ℕ} {K : Type*} [Field K] [NumberField K]\n    (M : FullLattice (d + 1) K) (σ : OrderedEmbeddings (d + 1) K) :\n    Set (Set (X (d + 1))) :=\n  {O | ∃ M' : FullLattice (d + 1) K, LocallyHomothetic M M' ∧\n    ∃ w ∈ signMatrices (d + 1), O = diagonalOrbit (rightAction (latticePoint M' σ) w)}\n\ndef orbitBasepoint {d : ℕ} {K : Type*} [Field K] [NumberField K]\n    {M : FullLattice (d + 1) K} {σ : OrderedEmbeddings (d + 1) K}\n    (O : packetOrbits M σ) : X (d + 1) :=\n  Classical.epsilon (fun x => diagonalOrbit x = O.1)\n\ndef packetMeasure {d : ℕ} {K : Type*} [Field K] [NumberField K]\n    (M : FullLattice (d + 1) K) (σ : OrderedEmbeddings (d + 1) K) : Measure (X (d + 1)) :=\n  (∑' O : packetOrbits M σ, orbitVolume (orbitBasepoint O))⁻¹ •\n    Measure.sum (fun O : packetOrbits M σ =>\n      orbitVolume (orbitBasepoint O) • orbitProbability (orbitBasepoint O))\n\ndef IsHaarProbability {n : ℕ} (μ : Measure (X n)) : Prop :=\n  IsProbabilityMeasure μ ∧ ∀ g : G n, Measure.map (fun x => rightAction x g) μ = μ\n\ndef WeakProbabilityConvergence {n : ℕ} (μ : ℕ → Measure (X n)) (m : Measure (X n)) : Prop :=\n  (∀ i, IsProbabilityMeasure (μ i)) ∧ IsProbabilityMeasure m ∧\n    ∀ f : BoundedContinuousFunction (X n) ℝ,\n    Tendsto (fun i => ∫ x, f x ∂μ i) atTop (𝓝 (∫ x, f x ∂m))\n\nend PrimeDegreePackets\nend\nend\n\nopen MeasureTheory Filter\nopen scoped Topology\nnamespace DukePrimeDegree\nopen PrimeDegreePackets\n\nuniverse u\n\n/-- Theorem 1.1 as weak convergence of the actual complete packet measures,\nincluding tightness of the entire family. -/\ntheorem prime_degree_packet_measure_equidistribution_unconditional\n    (d : ℕ) (hprime : Nat.Prime (d+1)) (hfive : 5 ≤ d+1)\n    (K : ℕ → Type u) [∀ i, Field (K i)] [∀ i, NumberField (K i)]\n    [∀ i, NumberField.IsTotallyReal (K i)]\n    (M : ∀ i, PrimeDegreePackets.FullLattice (d+1) (K i))\n    (σ : ∀ i, OrderedEmbeddings (d+1) (K i))\n    (hn : ∀ i, Module.finrank ℚ (K i)=d+1)\n    (hD : Tendsto (fun i => PrimeDegreePackets.multiplierDiscriminant (M i)) atTop atTop) :\n    ∃ ν : Measure (X (d+1)), IsHaarProbability ν ∧\n      WeakProbabilityConvergence (fun i => packetMeasure (M i) (σ i)) ν ∧\n      IsTightMeasureSet (Set.range (fun i => packetMeasure (M i) (σ i))) := by\n  sorry\n\n/-- The complete packet measures converge weakly to any specified Haar\nprobability, in the formulation by bounded continuous test functions. -/\ntheorem prime_degree_packet_measures_tendsto_haar_unconditional\n    (d : ℕ) (hprime : Nat.Prime (d+1)) (hfive : 5 ≤ d+1)\n    (K : ℕ → Type u) [∀ i, Field (K i)] [∀ i, NumberField (K i)]\n    [∀ i, NumberField.IsTotallyReal (K i)]\n    (M : ∀ i, PrimeDegreePackets.FullLattice (d+1) (K i))\n    (σ : ∀ i, OrderedEmbeddings (d+1) (K i))\n    (hn : ∀ i, Module.finrank ℚ (K i)=d+1)\n    (hD : Tendsto (fun i => PrimeDegreePackets.multiplierDiscriminant (M i)) atTop atTop)\n    (m : ProbabilityMeasure (X (d+1)))\n    (hm : IsHaarProbability (m : Measure (X (d+1)))) :\n    WeakProbabilityConvergence (fun i => packetMeasure (M i) (σ i))\n      (m : Measure (X (d+1))) := by\n  sorry\n\nend DukePrimeDegree\n\nend OAI\n\n\nFILE lean/ComparatorChallenges/TotientAsymptotic.lean TAIL\nrimeFactor (n : ℕ) : ℕ := max 1 (n.primeFactors.sup id)\n\nstructure TailDatum (H : ℕ) where\n  Q : Fin H → ℕ\n  cofactor : ℕ\n\ndef tailPrime {H : ℕ} (η : TailDatum H) (h : ℕ) : ℕ :=\n  if hH : h < H then η.Q ⟨h, hH⟩ else 1\n\ndef tailLog {H : ℕ} (η : TailDatum H) (h : ℕ) : ℝ :=\n  Real.log (Real.log (tailPrime η h : ℝ))\n\ndef IsWitness (H : ℕ) (s : ℝ) (η : TailDatum H) : Prop :=\n  (∀ h : Fin H, h.val < P H → η.Q h = 1) ∧\n  0 < η.cofactor ∧\n  (∀ h ∈ Finset.Ico (P H) H, (tailPrime η h).Prime ∧\n    (9 / 10 : ℝ) * alpha s * h * (rho^h)⁻¹ ≤ tailLog η h ∧\n    tailLog η h ≤ (11 / 10 : ℝ) * alpha s * h * (rho^h)⁻¹ ∧\n    (∑ l ∈ Finset.Ico (P H) h, a (h-l) * tailLog η l) ≤\n      (1 + (1 / 10000 : ℝ) * Real.exp (-(h : ℝ) / 40)) * tailLog η h) ∧\n  largestPrimeFactor η.cofactor ≤ tailPrime η (P H) ∧\n  Real.log (η.cofactor : ℝ) ≤\n    Real.exp (2 * alpha s * (P H : ℝ) * (rho^(P H))⁻¹)\n\ndef w {H : ℕ} (η : TailDatum H) : ℕ :=\n  η.cofactor * ∏ h ∈ Finset.Ico (P H) H, tailPrime η h\n\ndef witnesses (H : ℕ) (s : ℝ) (d : ℕ) : Set (TailDatum H) :=\n  {η | IsWitness H s η ∧ (w η).totient = d}\n\ndef D {H : ℕ} (h : ℕ) (η : TailDatum H) : ℝ :=\n  ∑ l ∈ Finset.Ico (P H) H, a (h-l) * tailLog η l\n\ndef maxD {H : ℕ} (h : ℕ) (T : Finset (TailDatum H)) : ℝ := by\n  classical\n  exact if ht : T.Nonempty then T.sup' ht (D h) else 0\n\ndef AH (H : ℕ) (f : ℝ → ℝ) (s : ℝ) : ℝ := by\n  classical\n  exact rho^(H*(H-1)/2) * (gamma / alpha s)^H *\n    ∑ᶠ d : ℕ,\n      if IsTotient d then\n        f ((ell d : ℝ) / d) / d *\n          ∑ᶠ T : Finset (TailDatum H),\n            if T.Nonempty ∧ (↑T : Set (TailDatum H)) ⊆ witnesses H s d then\n              (-1 : ℝ)^(T.card-1) *\n               Real.exp (-(gamma / alpha s) *\n                 ∑' n : ℕ, rho^(H+n) * maxD (H+n) T)\n            else 0\n      else 0\n\ndef A (f : ℝ → ℝ) (s : ℝ) : ℝ := limUnder atTop (fun H : ℕ => AH H f s)\n\ndef mainTerm (x : ℝ) : ℝ := x / Real.log x * G x (m x) * A (fun _ => 1) (theta x)\n\ndef N (k : ℕ) (x : ℝ) : ℝ := by\n  classical\n  exact (((Finset.Icc 1 ⌊x⌋₊).filter fun v =>\n    IsTotient v ∧ (k : ℝ) * x < ell v ∧ (ell v : ℝ) ≤ (k+1 : ℝ) * x).card : ℝ)\n\ndef fk (k : ℕ) (r : ℝ) : ℝ := min 1 ((k+1 : ℝ) / r) - min 1 ((k : ℝ) / r)\n\ndef normalizedCount (F : ℝ → ℝ) (x : ℝ) : ℝ :=\n  F x / (x / Real.log x * G x (m x))\n\ndef tupleNormalization (x : ℝ) : ℝ := x / Real.log x * G x (m x)\n\ntheorem totient_asymptotic_formula\n    :\n    TendstoUniformlyOn (fun H => AH H (fun _ => 1)) (A (fun _ => 1))\n      atTop (Set.Ico (0 : ℝ) 1) ∧\n    (∃ cMinus cPlus : ℝ, 0 < cMinus ∧ ∀ s ∈ Set.Ico (0 : ℝ) 1,\n      cMinus ≤ A (fun _ => 1) s ∧ A (fun _ => 1) s ≤ cPlus) ∧\n    Tendsto (fun x => V x/mainTerm x) atTop (nhds 1) ∧\n    ∀ c : ℝ, 0 < c → Tendsto (fun x => V (c*x)/V x) atTop (nhds c) := by\n  sorry\n\ntheorem weighted_totient_asymptotic\n    (k : ℕ) (hk : 1≤k) :\n    TendstoUniformlyOn (fun H => AH H (fk k)) (A (fk k)) atTop (Set.Ico (0 : ℝ) 1) ∧\n    Tendsto (fun x => normalizedCount (N k) x-A (fk k) (theta x)) atTop (nhds 0) ∧\n    ((∃ d : ℕ, IsTotient d ∧ k*d<ell d) →\n      ∃ cMinus cPlus : ℝ, 0<cMinus ∧ 0<cPlus ∧\n        (∀ s∈Set.Ico (0 : ℝ) 1, cMinus≤A (fk k) s ∧ A (fk k) s≤cPlus) ∧\n        Tendsto (fun x => N k x/(tupleNormalization x*A (fk k) (theta x))) atTop (nhds 1) ∧\n        ∀ᶠ x : ℝ in atTop, (cMinus/cPlus)*V x≤N k x ∧ N k x≤V x) ∧\n    ((¬∃ d : ℕ, IsTotient d ∧ k*d<ell d) →\n      (∀ x : ℝ, 0<x → N k x=0) ∧ (∀ s∈Set.Ico (0 : ℝ) 1, A (fk k) s=0)) := by\n  sorry\n\ntheorem weighted_totient_one_two\n    :\n    ∀ k∈({1,2} : Finset ℕ), ∃ c : ℝ, 0<c ∧\n      ∀ s∈Set.Ico (0 : ℝ) 1, c≤A (fk k) s := by\n  sorry\n\ntheorem coefficient_nonnegative (H : ℕ) {s : ℝ}\n    (hs : s ∈ Set.Ico (0 : ℝ) 1) : 0 ≤ AH H (fun _ => 1) s := by\n  sorry\n\nend TotientAsymptotic\n\nend\n\nend OAI\n\n\nFILE lean/ComparatorChallenges/ContingencyTables.lean TAIL\nact ((Finset.univ.filter fun bits : Fin t → Bool =>\n      (MatchingFPRAS.run A input bits).state = none).card : ℝ) / (2 : ℝ) ^ t\n\ndef samplingSize (n : ℕ) {m : ℕ} (r : Fin m → ℕ) (k : ℕ) : ℕ :=\n  m + n + Nat.clog 2 ((∑ i, r i) + 1) + k + 1\n\n/-- Paper 200-01, Theorem main(i): bounded runtime on every random tape\nand total variation at most `2⁻ᵏ`, with freely varying dimensions. -/\ndef BoundedSamplingStatement : Prop :=\n  ∃ (A : Machine) (C d : ℕ), 0 < C ∧\n    ∀ (m n : ℕ) (r : Fin m → ℕ) (c : Fin n → ℕ),\n      (∑ i, r i) = ∑ j, c j → ∀ k : ℕ, 1 ≤ k →\n        let input := encodeSamplingInput r c k\n        let t := C * (samplingSize n r k) ^ d\n        (∀ bits : Fin t → Bool, ∃ X : Table r c, OutputsTable A input bits X.val) ∧\n        (∑ X : Table r c,\n          |tableMass A input t X.val - (Fintype.card (Table r c) : ℝ)⁻¹|) / 2 ≤\n            ((2 : ℝ) ^ k)⁻¹\n\n/-- Paper 200-01, Theorem main(ii): every halted output is feasible, each\ntable has limiting mass `1/|Ω|`, termination has probability one, and the\nexpected number of bit operations is bounded by a uniform polynomial. -/\ndef ExactSamplingStatement : Prop :=\n  ∃ (A : Machine) (C d : ℕ), 0 < C ∧\n    ∀ (m n : ℕ) (r : Fin m → ℕ) (c : Fin n → ℕ),\n      (∑ i, r i) = ∑ j, c j →\n        let input := encodeMargins r c\n        let tail : ℕ → ℝ := fun t => 1 - haltMass A input t\n        (∀ t (bits : Fin t → Bool), (MatchingFPRAS.run A input bits).state = none →\n          ∃ X : Table r c, OutputsTable A input bits X.val) ∧\n        (∀ X : Table r c, Tendsto (fun t => tableMass A input t X.val) atTop\n          (nhds ((Fintype.card (Table r c) : ℝ)⁻¹))) ∧\n        Tendsto (haltMass A input) atTop (nhds 1) ∧\n        Summable tail ∧\n        tsum tail ≤ C * (samplingSize n r 0) ^ d\n\nnoncomputable def count {m n : ℕ} (r : Fin m → ℕ) (c : Fin n → ℕ)\n    (b : Fin m → Fin n → ℕ) : ℕ := Fintype.card (BoundedTable r c b)\n\ndef countingTime (C d : ℕ) {m n : ℕ} (r : Fin m → ℕ) (c : Fin n → ℕ)\n    (b : Fin m → Fin n → ℕ) (ε δ : ℚ) : ℕ :=\n  C * ((encodeCountingInput r c b ε δ).length + ⌈ε⁻¹⌉₊ +\n    Nat.clog 2 ⌈δ⁻¹⌉₊ + 1) ^ d\n\nnoncomputable def goodCountingTapes (A : Machine) {m n : ℕ}\n    (r : Fin m → ℕ) (c : Fin n → ℕ) (b : Fin m → Fin n → ℕ)\n    (ε δ : ℚ) (t : ℕ) : Finset (Fin t → Bool) := by\n  classical\n  exact Finset.univ.filter fun bits => ∃ q : ℚ,\n    MatchingFPRAS.Outputs A (encodeCountingInput r c b ε δ) bits q ∧\n      (1 - ε) * count r c b ≤ q ∧ q ≤ (1 + ε) * count r c b\n\n/-- Paper 200-02, Theorem main. The input length includes the rational\naccuracy parameters, and infeasible inputs always produce exactly zero. -/\ndef CountingStatement : Prop :=\n  ∃ (A : Machine) (C d : ℕ), 0 < C ∧\n    ∀ (m n : ℕ), 0 < m → 0 < n →\n    ∀ (r : Fin m → ℕ) (c : Fin n → ℕ) (b : Fin m → Fin n → ℕ),\n      (∑ i, r i) = ∑ j, c j →\n      ∀ ε δ : ℚ, 0 < ε → ε < 1 → 0 < δ → δ < 1 →\n        let input := encodeCountingInput r c b ε δ\n        let t := countingTime C d r c b ε δ\n        (∀ bits : Fin t → Bool, ∃ q : ℚ, 0 ≤ q ∧ MatchingFPRAS.Outputs A input bits q) ∧\n        (count r c b = 0 → ∀ bits : Fin t → Bool, MatchingFPRAS.Outputs A input bits 0) ∧\n        ((goodCountingTapes A r c b ε δ t).card : ℚ) / (2 : ℚ) ^ t ≥ 1 - δ\n\nend ContingencyTables.Algorithms\n\nnamespace ContingencyTables\n\n/-- An almost-uniform sampler with polynomial cost on every execution. -/\ntheorem boundedSampling : Algorithms.BoundedSamplingStatement := by sorry\n\n/-- Exact uniform sampling in polynomial expected bit time. -/\ntheorem exactSampling : Algorithms.ExactSamplingStatement := by sorry\n\n/-- A cell-bounded FPRAS, with zero output on every infeasible execution. -/\ntheorem counting : Algorithms.CountingStatement := by sorry\n\nend ContingencyTables\n\nend OAI\n\n"
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        "stdout": "\nFILE lean/ComparatorChallenges/ContingencyTables.lean TAIL\nact ((Finset.univ.filter fun bits : Fin t → Bool =>\n      (MatchingFPRAS.run A input bits).state = none).card : ℝ) / (2 : ℝ) ^ t\n\ndef samplingSize (n : ℕ) {m : ℕ} (r : Fin m → ℕ) (k : ℕ) : ℕ :=\n  m + n + Nat.clog 2 ((∑ i, r i) + 1) + k + 1\n\n/-- Paper 200-01, Theorem main(i): bounded runtime on every random tape\nand total variation at most `2⁻ᵏ`, with freely varying dimensions. -/\ndef BoundedSamplingStatement : Prop :=\n  ∃ (A : Machine) (C d : ℕ), 0 < C ∧\n    ∀ (m n : ℕ) (r : Fin m → ℕ) (c : Fin n → ℕ),\n      (∑ i, r i) = ∑ j, c j → ∀ k : ℕ, 1 ≤ k →\n        let input := encodeSamplingInput r c k\n        let t := C * (samplingSize n r k) ^ d\n        (∀ bits : Fin t → Bool, ∃ X : Table r c, OutputsTable A input bits X.val) ∧\n        (∑ X : Table r c,\n          |tableMass A input t X.val - (Fintype.card (Table r c) : ℝ)⁻¹|) / 2 ≤\n            ((2 : ℝ) ^ k)⁻¹\n\n/-- Paper 200-01, Theorem main(ii): every halted output is feasible, each\ntable has limiting mass `1/|Ω|`, termination has probability one, and the\nexpected number of bit operations is bounded by a uniform polynomial. -/\ndef ExactSamplingStatement : Prop :=\n  ∃ (A : Machine) (C d : ℕ), 0 < C ∧\n    ∀ (m n : ℕ) (r : Fin m → ℕ) (c : Fin n → ℕ),\n      (∑ i, r i) = ∑ j, c j →\n        let input := encodeMargins r c\n        let tail : ℕ → ℝ := fun t => 1 - haltMass A input t\n        (∀ t (bits : Fin t → Bool), (MatchingFPRAS.run A input bits).state = none →\n          ∃ X : Table r c, OutputsTable A input bits X.val) ∧\n        (∀ X : Table r c, Tendsto (fun t => tableMass A input t X.val) atTop\n          (nhds ((Fintype.card (Table r c) : ℝ)⁻¹))) ∧\n        Tendsto (haltMass A input) atTop (nhds 1) ∧\n        Summable tail ∧\n        tsum tail ≤ C * (samplingSize n r 0) ^ d\n\nnoncomputable def count {m n : ℕ} (r : Fin m → ℕ) (c : Fin n → ℕ)\n    (b : Fin m → Fin n → ℕ) : ℕ := Fintype.card (BoundedTable r c b)\n\ndef countingTime (C d : ℕ) {m n : ℕ} (r : Fin m → ℕ) (c : Fin n → ℕ)\n    (b : Fin m → Fin n → ℕ) (ε δ : ℚ) : ℕ :=\n  C * ((encodeCountingInput r c b ε δ).length + ⌈ε⁻¹⌉₊ +\n    Nat.clog 2 ⌈δ⁻¹⌉₊ + 1) ^ d\n\nnoncomputable def goodCountingTapes (A : Machine) {m n : ℕ}\n    (r : Fin m → ℕ) (c : Fin n → ℕ) (b : Fin m → Fin n → ℕ)\n    (ε δ : ℚ) (t : ℕ) : Finset (Fin t → Bool) := by\n  classical\n  exact Finset.univ.filter fun bits => ∃ q : ℚ,\n    MatchingFPRAS.Outputs A (encodeCountingInput r c b ε δ) bits q ∧\n      (1 - ε) * count r c b ≤ q ∧ q ≤ (1 + ε) * count r c b\n\n/-- Paper 200-02, Theorem main. The input length includes the rational\naccuracy parameters, and infeasible inputs always produce exactly zero. -/\ndef CountingStatement : Prop :=\n  ∃ (A : Machine) (C d : ℕ), 0 < C ∧\n    ∀ (m n : ℕ), 0 < m → 0 < n →\n    ∀ (r : Fin m → ℕ) (c : Fin n → ℕ) (b : Fin m → Fin n → ℕ),\n      (∑ i, r i) = ∑ j, c j →\n      ∀ ε δ : ℚ, 0 < ε → ε < 1 → 0 < δ → δ < 1 →\n        let input := encodeCountingInput r c b ε δ\n        let t := countingTime C d r c b ε δ\n        (∀ bits : Fin t → Bool, ∃ q : ℚ, 0 ≤ q ∧ MatchingFPRAS.Outputs A input bits q) ∧\n        (count r c b = 0 → ∀ bits : Fin t → Bool, MatchingFPRAS.Outputs A input bits 0) ∧\n        ((goodCountingTapes A r c b ε δ t).card : ℚ) / (2 : ℚ) ^ t ≥ 1 - δ\n\nend ContingencyTables.Algorithms\n\nnamespace ContingencyTables\n\n/-- An almost-uniform sampler with polynomial cost on every execution. -/\ntheorem boundedSampling : Algorithms.BoundedSamplingStatement := by sorry\n\n/-- Exact uniform sampling in polynomial expected bit time. -/\ntheorem exactSampling : Algorithms.ExactSamplingStatement := by sorry\n\n/-- A cell-bounded FPRAS, with zero output on every infeasible execution. -/\ntheorem counting : Algorithms.CountingStatement := by sorry\n\nend ContingencyTables\n\nend OAI\n\n\nFILE lean/ComparatorChallenges/MatchingFPRAS.lean TAIL\n : Configuration A) (bit : Bool) : Configuration A :=\n  match c.state with\n  | none => c\n  | some s => match A.transition s c.tape.head bit with\n    | none => ⟨none, c.tape⟩\n    | some (s', action) =>\n      ⟨some s', match action with\n        | .move d => c.tape.move d\n        | .write a => c.tape.write a⟩\n\ndef initial (A : RandomMachine) (input : List Symbol) : Configuration A :=\n  ⟨some 0, Turing.Tape.mk₁ input⟩\n\n/-- Bounded execution on a finite prefix of the random tape. -/\ndef run (A : RandomMachine) (input : List Symbol) {t : ℕ}\n    (bits : Fin t → Bool) : Configuration A :=\n  (List.ofFn bits).foldl (tick A) (initial A input)\n\ndef Outputs (A : RandomMachine) (input : List Symbol) {t : ℕ}\n    (bits : Fin t → Bool) (q : ℚ) : Prop :=\n  (run A input bits).state = none ∧\n    (run A input bits).tape.right₀ = Turing.ListBlank.mk (encodeRat q)\n\n/-- A fixed-degree polynomial in input bits, ε⁻¹, and log δ⁻¹.\nNatural ceilings and binary ceiling-log change this bound only by constants. -/\ndef timeBound (C d : ℕ) (G : GraphInput) (ε δ : ℚ) : ℕ :=\n  C * ((encodeInput G ε δ).length + ⌈ε⁻¹⌉₊ + Nat.clog 2 ⌈δ⁻¹⌉₊ + 1) ^ d\n\nnoncomputable def goodTapes (A : RandomMachine) (G : GraphInput) (ε δ : ℚ)\n    (t : ℕ) : Finset (Fin t → Bool) := by\n  classical\n  exact Finset.univ.filter fun bits => ∃ q : ℚ,\n    Outputs A (encodeInput G ε δ) bits q ∧\n    (1 - ε) * (Z G : ℚ) ≤ q ∧ q ≤ (1 + ε) * (Z G : ℚ)\n\n/-- A uniform approximation scheme using independent fair bits. All tapes halt\nwithin the fixed polynomial bound and return a nonnegative rational.\nA zero count always produces encoded zero. -/\ndef MainStatement : Prop :=\n  ∃ (A : RandomMachine) (C d : ℕ), 0 < C ∧\n    ∀ (G : GraphInput) (ε δ : ℚ), 0 < ε → ε < 1 → 0 < δ → δ < 1 / 2 →\n      let t := timeBound C d G ε δ\n      (∀ bits : Fin t → Bool, ∃ q : ℚ, 0 ≤ q ∧ Outputs A (encodeInput G ε δ) bits q) ∧\n      (Z G = 0 → ∀ bits : Fin t → Bool, Outputs A (encodeInput G ε δ) bits 0) ∧\n      ((goodTapes A G ε δ t).card : ℚ) / (2 : ℚ) ^ t ≥ 1 - δ\n\n\n/-- A fully polynomial randomized approximation scheme for perfect matchings\nin every finite simple undirected graph. -/\ntheorem thm_main : MainStatement := by\n  sorry\n\nend MatchingFPRAS\n\nend OAI\n\n\nFILE lean/ComparatorChallenges/TypeSystemNormalization.lean TAIL\n  List (Expr S) → Expr S → Expr S → Prop\n  | ax {s t} : P.axioms s t → HasType P [] (.sort s) (.sort t)\n  | var {Γ A s} : HasType P Γ A (.sort s) →\n      HasType P (A :: Γ) (.var 0) (A.rename Nat.succ)\n  | weaken {Γ M B A s} : HasType P Γ M B → HasType P Γ A (.sort s) →\n      HasType P (A :: Γ) (M.rename Nat.succ) (B.rename Nat.succ)\n  | product {Γ A B s₁ s₂ s₃} : HasType P Γ A (.sort s₁) →\n      HasType P (A :: Γ) B (.sort s₂) → P.rule s₁ s₂ s₃ →\n      HasType P Γ (.pi A B) (.sort s₃)\n  | abstraction {Γ A b B s} : HasType P (A :: Γ) b B →\n      HasType P Γ (.pi A B) (.sort s) →\n      HasType P Γ (.lam A b) (.pi A B)\n  | application {Γ f a A B} : HasType P Γ f (.pi A B) →\n      HasType P Γ a A → HasType P Γ (.app f a) (B.instantiate a)\n  | conversion {Γ M A B s} : HasType P Γ M A → HasType P Γ B (.sort s) →\n      Converts A B → HasType P Γ M B\n\ninductive ValidContext {S : Type u} (P : Specification S) : List (Expr S) → Prop\n  | nil : ValidContext P []\n  | cons {Γ A s} : ValidContext P Γ → HasType P Γ A (.sort s) →\n      ValidContext P (A :: Γ)\n\ndef Legal {S : Type u} (P : Specification S) (Γ : List (Expr S)) (M : Expr S) : Prop :=\n  ∃ A, HasType P Γ M A ∨ HasType P Γ A M\n\ndef SystemWeaklyNormalizing {S : Type u} (P : Specification S) : Prop :=\n  ∀ Γ, ValidContext P Γ → ∀ M, Legal P Γ M → WeaklyNormalizing M\n\ndef SystemStronglyNormalizing {S : Type u} (P : Specification S) : Prop :=\n  ∀ Γ, ValidContext P Γ → ∀ M, Legal P Γ M → StronglyNormalizing M\n\n/-- System-wide weak normalization implies strong normalization for every PTS,\nincluding arbitrary sort types, nonfunctional rules and annotated terms. -/\ntheorem weak_implies_strong {S : Type u} (P : Specification S)\n    (h : SystemWeaklyNormalizing P) : SystemStronglyNormalizing P := by\n  sorry\n\nend PureTypeSystem\n\nend OAI\n\n",
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          "\nFILE lean/ComparatorChallenges/ContingencyTables.lean TAIL\nact ((Finset.univ.filter fun bits : Fin t → Bool =>\n      (MatchingFPRAS.run A input bits).state = none).card : ℝ) / (2 : ℝ) ^ t\n\ndef samplingSize (n : ℕ) {m : ℕ} (r : Fin m → ℕ) (k : ℕ) : ℕ :=\n  m + n + Nat.clog 2 ((∑ i, r i) + 1) + k + 1\n\n/-- Paper 200-01, Theorem main(i): bounded runtime on every random tape\nand total variation at most `2⁻ᵏ`, with freely varying dimensions. -/\ndef BoundedSamplingStatement : Prop :=\n  ∃ (A : Machine) (C d : ℕ), 0 < C ∧\n    ∀ (m n : ℕ) (r : Fin m → ℕ) (c : Fin n → ℕ),\n      (∑ i, r i) = ∑ j, c j → ∀ k : ℕ, 1 ≤ k →\n        let input := encodeSamplingInput r c k\n        let t := C * (samplingSize n r k) ^ d\n        (∀ bits : Fin t → Bool, ∃ X : Table r c, OutputsTable A input bits X.val) ∧\n        (∑ X : Table r c,\n          |tableMass A input t X.val - (Fintype.card (Table r c) : ℝ)⁻¹|) / 2 ≤\n            ((2 : ℝ) ^ k)⁻¹\n\n/-- Paper 200-01, Theorem main(ii): every halted output is feasible, each\ntable has limiting mass `1/|Ω|`, termination has probability one, and the\nexpected number of bit operations is bounded by a uniform polynomial. -/\ndef ExactSamplingStatement : Prop :=\n  ∃ (A : Machine) (C d : ℕ), 0 < C ∧\n    ∀ (m n : ℕ) (r : Fin m → ℕ) (c : Fin n → ℕ),\n      (∑ i, r i) = ∑ j, c j →\n        let input := encodeMargins r c\n        let tail : ℕ → ℝ := fun t => 1 - haltMass A input t\n        (∀ t (bits : Fin t → Bool), (MatchingFPRAS.run A input bits).state = none →\n          ∃ X : Table r c, OutputsTable A input bits X.val) ∧\n        (∀ X : Table r c, Tendsto (fun t => tableMass A input t X.val) atTop\n          (nhds ((Fintype.card (Table r c) : ℝ)⁻¹))) ∧\n        Tendsto (haltMass A input) atTop (nhds 1) ∧\n        Summable tail ∧\n        tsum tail ≤ C * (samplingSize n r 0) ^ d\n\nnoncomputable def count {m n : ℕ} (r : Fin m → ℕ) (c : Fin n → ℕ)\n    (b : Fin m → Fin n → ℕ) : ℕ := Fintype.card (BoundedTable r c b)\n\ndef countingTime (C d : ℕ) {m n : ℕ} (r : Fin m → ℕ) (c : Fin n → ℕ)\n    (b : Fin m → Fin n → ℕ) (ε δ : ℚ) : ℕ :=\n  C * ((encodeCountingInput r c b ε δ).length + ⌈ε⁻¹⌉₊ +\n    Nat.clog 2 ⌈δ⁻¹⌉₊ + 1) ^ d\n\nnoncomputable def goodCountingTapes (A : Machine) {m n : ℕ}\n    (r : Fin m → ℕ) (c : Fin n → ℕ) (b : Fin m → Fin n → ℕ)\n    (ε δ : ℚ) (t : ℕ) : Finset (Fin t → Bool) := by\n  classical\n  exact Finset.univ.filter fun bits => ∃ q : ℚ,\n    MatchingFPRAS.Outputs A (encodeCountingInput r c b ε δ) bits q ∧\n      (1 - ε) * count r c b ≤ q ∧ q ≤ (1 + ε) * count r c b\n\n/-- Paper 200-02, Theorem main. The input length includes the rational\naccuracy parameters, and infeasible inputs always produce exactly zero. -/\ndef CountingStatement : Prop :=\n  ∃ (A : Machine) (C d : ℕ), 0 < C ∧\n    ∀ (m n : ℕ), 0 < m → 0 < n →\n    ∀ (r : Fin m → ℕ) (c : Fin n → ℕ) (b : Fin m → Fin n → ℕ),\n      (∑ i, r i) = ∑ j, c j →\n      ∀ ε δ : ℚ, 0 < ε → ε < 1 → 0 < δ → δ < 1 →\n        let input := encodeCountingInput r c b ε δ\n        let t := countingTime C d r c b ε δ\n        (∀ bits : Fin t → Bool, ∃ q : ℚ, 0 ≤ q ∧ MatchingFPRAS.Outputs A input bits q) ∧\n        (count r c b = 0 → ∀ bits : Fin t → Bool, MatchingFPRAS.Outputs A input bits 0) ∧\n        ((goodCountingTapes A r c b ε δ t).card : ℚ) / (2 : ℚ) ^ t ≥ 1 - δ\n\nend ContingencyTables.Algorithms\n\nnamespace ContingencyTables\n\n/-- An almost-uniform sampler with polynomial cost on every execution. -/\ntheorem boundedSampling : Algorithms.BoundedSamplingStatement := by sorry\n\n/-- Exact uniform sampling in polynomial expected bit time. -/\ntheorem exactSampling : Algorithms.ExactSamplingStatement := by sorry\n\n/-- A cell-bounded FPRAS, with zero output on every infeasible execution. -/\ntheorem counting : Algorithms.CountingStatement := by sorry\n\nend ContingencyTables\n\nend OAI\n\n\nFILE lean/ComparatorChallenges/MatchingFPRAS.lean TAIL\n : Configuration A) (bit : Bool) : Configuration A :=\n  match c.state with\n  | none => c\n  | some s => match A.transition s c.tape.head bit with\n    | none => ⟨none, c.tape⟩\n    | some (s', action) =>\n      ⟨some s', match action with\n        | .move d => c.tape.move d\n        | .write a => c.tape.write a⟩\n\ndef initial (A : RandomMachine) (input : List Symbol) : Configuration A :=\n  ⟨some 0, Turing.Tape.mk₁ input⟩\n\n/-- Bounded execution on a finite prefix of the random tape. -/\ndef run (A : RandomMachine) (input : List Symbol) {t : ℕ}\n    (bits : Fin t → Bool) : Configuration A :=\n  (List.ofFn bits).foldl (tick A) (initial A input)\n\ndef Outputs (A : RandomMachine) (input : List Symbol) {t : ℕ}\n    (bits : Fin t → Bool) (q : ℚ) : Prop :=\n  (run A input bits).state = none ∧\n    (run A input bits).tape.right₀ = Turing.ListBlank.mk (encodeRat q)\n\n/-- A fixed-degree polynomial in input bits, ε⁻¹, and log δ⁻¹.\nNatural ceilings and binary ceiling-log change this bound only by constants. -/\ndef timeBound (C d : ℕ) (G : GraphInput) (ε δ : ℚ) : ℕ :=\n  C * ((encodeInput G ε δ).length + ⌈ε⁻¹⌉₊ + Nat.clog 2 ⌈δ⁻¹⌉₊ + 1) ^ d\n\nnoncomputable def goodTapes (A : RandomMachine) (G : GraphInput) (ε δ : ℚ)\n    (t : ℕ) : Finset (Fin t → Bool) := by\n  classical\n  exact Finset.univ.filter fun bits => ∃ q : ℚ,\n    Outputs A (encodeInput G ε δ) bits q ∧\n    (1 - ε) * (Z G : ℚ) ≤ q ∧ q ≤ (1 + ε) * (Z G : ℚ)\n\n/-- A uniform approximation scheme using independent fair bits. All tapes halt\nwithin the fixed polynomial bound and return a nonnegative rational.\nA zero count always produces encoded zero. -/\ndef MainStatement : Prop :=\n  ∃ (A : RandomMachine) (C d : ℕ), 0 < C ∧\n    ∀ (G : GraphInput) (ε δ : ℚ), 0 < ε → ε < 1 → 0 < δ → δ < 1 / 2 →\n      let t := timeBound C d G ε δ\n      (∀ bits : Fin t → Bool, ∃ q : ℚ, 0 ≤ q ∧ Outputs A (encodeInput G ε δ) bits q) ∧\n      (Z G = 0 → ∀ bits : Fin t → Bool, Outputs A (encodeInput G ε δ) bits 0) ∧\n      ((goodTapes A G ε δ t).card : ℚ) / (2 : ℚ) ^ t ≥ 1 - δ\n\n\n/-- A fully polynomial randomized approximation scheme for perfect matchings\nin every finite simple undirected graph. -/\ntheorem thm_main : MainStatement := by\n  sorry\n\nend MatchingFPRAS\n\nend OAI\n\n\nFILE lean/ComparatorChallenges/TypeSystemNormalization.lean TAIL\n  List (Expr S) → Expr S → Expr S → Prop\n  | ax {s t} : P.axioms s t → HasType P [] (.sort s) (.sort t)\n  | var {Γ A s} : HasType P Γ A (.sort s) →\n      HasType P (A :: Γ) (.var 0) (A.rename Nat.succ)\n  | weaken {Γ M B A s} : HasType P Γ M B → HasType P Γ A (.sort s) →\n      HasType P (A :: Γ) (M.rename Nat.succ) (B.rename Nat.succ)\n  | product {Γ A B s₁ s₂ s₃} : HasType P Γ A (.sort s₁) →\n      HasType P (A :: Γ) B (.sort s₂) → P.rule s₁ s₂ s₃ →\n      HasType P Γ (.pi A B) (.sort s₃)\n  | abstraction {Γ A b B s} : HasType P (A :: Γ) b B →\n      HasType P Γ (.pi A B) (.sort s) →\n      HasType P Γ (.lam A b) (.pi A B)\n  | application {Γ f a A B} : HasType P Γ f (.pi A B) →\n      HasType P Γ a A → HasType P Γ (.app f a) (B.instantiate a)\n  | conversion {Γ M A B s} : HasType P Γ M A → HasType P Γ B (.sort s) →\n      Converts A B → HasType P Γ M B\n\ninductive ValidContext {S : Type u} (P : Specification S) : List (Expr S) → Prop\n  | nil : ValidContext P []\n  | cons {Γ A s} : ValidContext P Γ → HasType P Γ A (.sort s) →\n      ValidContext P (A :: Γ)\n\ndef Legal {S : Type u} (P : Specification S) (Γ : List (Expr S)) (M : Expr S) : Prop :=\n  ∃ A, HasType P Γ M A ∨ HasType P Γ A M\n\ndef SystemWeaklyNormalizing {S : Type u} (P : Specification S) : Prop :=\n  ∀ Γ, ValidContext P Γ → ∀ M, Legal P Γ M → WeaklyNormalizing M\n\ndef SystemStronglyNormalizing {S : Type u} (P : Specification S) : Prop :=\n  ∀ Γ, ValidContext P Γ → ∀ M, Legal P Γ M → StronglyNormalizing M\n\n/-- System-wide weak normalization implies strong normalization for every PTS,\nincluding arbitrary sort types, nonfunctional rules and annotated terms. -/\ntheorem weak_implies_strong {S : Type u} (P : Specification S)\n    (h : SystemWeaklyNormalizing P) : SystemStronglyNormalizing P := by\n  sorry\n\nend PureTypeSystem\n\nend OAI\n\n"
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          "text": "\nFILE lean/ComparatorChallenges/ContingencyTables.lean TAIL\nact ((Finset.univ.filter fun bits : Fin t → Bool =>\n      (MatchingFPRAS.run A input bits).state = none).card : ℝ) / (2 : ℝ) ^ t\n\ndef samplingSize (n : ℕ) {m : ℕ} (r : Fin m → ℕ) (k : ℕ) : ℕ :=\n  m + n + Nat.clog 2 ((∑ i, r i) + 1) + k + 1\n\n/-- Paper 200-01, Theorem main(i): bounded runtime on every random tape\nand total variation at most `2⁻ᵏ`, with freely varying dimensions. -/\ndef BoundedSamplingStatement : Prop :=\n  ∃ (A : Machine) (C d : ℕ), 0 < C ∧\n    ∀ (m n : ℕ) (r : Fin m → ℕ) (c : Fin n → ℕ),\n      (∑ i, r i) = ∑ j, c j → ∀ k : ℕ, 1 ≤ k →\n        let input := encodeSamplingInput r c k\n        let t := C * (samplingSize n r k) ^ d\n        (∀ bits : Fin t → Bool, ∃ X : Table r c, OutputsTable A input bits X.val) ∧\n        (∑ X : Table r c,\n          |tableMass A input t X.val - (Fintype.card (Table r c) : ℝ)⁻¹|) / 2 ≤\n            ((2 : ℝ) ^ k)⁻¹\n\n/-- Paper 200-01, Theorem main(ii): every halted output is feasible, each\ntable has limiting mass `1/|Ω|`, termination has probability one, and the\nexpected number of bit operations is bounded by a uniform polynomial. -/\ndef ExactSamplingStatement : Prop :=\n  ∃ (A : Machine) (C d : ℕ), 0 < C ∧\n    ∀ (m n : ℕ) (r : Fin m → ℕ) (c : Fin n → ℕ),\n      (∑ i, r i) = ∑ j, c j →\n        let input := encodeMargins r c\n        let tail : ℕ → ℝ := fun t => 1 - haltMass A input t\n        (∀ t (bits : Fin t → Bool), (MatchingFPRAS.run A input bits).state = none →\n          ∃ X : Table r c, OutputsTable A input bits X.val) ∧\n        (∀ X : Table r c, Tendsto (fun t => tableMass A input t X.val) atTop\n          (nhds ((Fintype.card (Table r c) : ℝ)⁻¹))) ∧\n        Tendsto (haltMass A input) atTop (nhds 1) ∧\n        Summable tail ∧\n        tsum tail ≤ C * (samplingSize n r 0) ^ d\n\nnoncomputable def count {m n : ℕ} (r : Fin m → ℕ) (c : Fin n → ℕ)\n    (b : Fin m → Fin n → ℕ) : ℕ := Fintype.card (BoundedTable r c b)\n\ndef countingTime (C d : ℕ) {m n : ℕ} (r : Fin m → ℕ) (c : Fin n → ℕ)\n    (b : Fin m → Fin n → ℕ) (ε δ : ℚ) : ℕ :=\n  C * ((encodeCountingInput r c b ε δ).length + ⌈ε⁻¹⌉₊ +\n    Nat.clog 2 ⌈δ⁻¹⌉₊ + 1) ^ d\n\nnoncomputable def goodCountingTapes (A : Machine) {m n : ℕ}\n    (r : Fin m → ℕ) (c : Fin n → ℕ) (b : Fin m → Fin n → ℕ)\n    (ε δ : ℚ) (t : ℕ) : Finset (Fin t → Bool) := by\n  classical\n  exact Finset.univ.filter fun bits => ∃ q : ℚ,\n    MatchingFPRAS.Outputs A (encodeCountingInput r c b ε δ) bits q ∧\n      (1 - ε) * count r c b ≤ q ∧ q ≤ (1 + ε) * count r c b\n\n/-- Paper 200-02, Theorem main. The input length includes the rational\naccuracy parameters, and infeasible inputs always produce exactly zero. -/\ndef CountingStatement : Prop :=\n  ∃ (A : Machine) (C d : ℕ), 0 < C ∧\n    ∀ (m n : ℕ), 0 < m → 0 < n →\n    ∀ (r : Fin m → ℕ) (c : Fin n → ℕ) (b : Fin m → Fin n → ℕ),\n      (∑ i, r i) = ∑ j, c j →\n      ∀ ε δ : ℚ, 0 < ε → ε < 1 → 0 < δ → δ < 1 →\n        let input := encodeCountingInput r c b ε δ\n        let t := countingTime C d r c b ε δ\n        (∀ bits : Fin t → Bool, ∃ q : ℚ, 0 ≤ q ∧ MatchingFPRAS.Outputs A input bits q) ∧\n        (count r c b = 0 → ∀ bits : Fin t → Bool, MatchingFPRAS.Outputs A input bits 0) ∧\n        ((goodCountingTapes A r c b ε δ t).card : ℚ) / (2 : ℚ) ^ t ≥ 1 - δ\n\nend ContingencyTables.Algorithms\n\nnamespace ContingencyTables\n\n/-- An almost-uniform sampler with polynomial cost on every execution. -/\ntheorem boundedSampling : Algorithms.BoundedSamplingStatement := by sorry\n\n/-- Exact uniform sampling in polynomial expected bit time. -/\ntheorem exactSampling : Algorithms.ExactSamplingStatement := by sorry\n\n/-- A cell-bounded FPRAS, with zero output on every infeasible execution. -/\ntheorem counting : Algorithms.CountingStatement := by sorry\n\nend ContingencyTables\n\nend OAI\n\n\nFILE lean/ComparatorChallenges/MatchingFPRAS.lean TAIL\n : Configuration A) (bit : Bool) : Configuration A :=\n  match c.state with\n  | none => c\n  | some s => match A.transition s c.tape.head bit with\n    | none => ⟨none, c.tape⟩\n    | some (s', action) =>\n      ⟨some s', match action with\n        | .move d => c.tape.move d\n        | .write a => c.tape.write a⟩\n\ndef initial (A : RandomMachine) (input : List Symbol) : Configuration A :=\n  ⟨some 0, Turing.Tape.mk₁ input⟩\n\n/-- Bounded execution on a finite prefix of the random tape. -/\ndef run (A : RandomMachine) (input : List Symbol) {t : ℕ}\n    (bits : Fin t → Bool) : Configuration A :=\n  (List.ofFn bits).foldl (tick A) (initial A input)\n\ndef Outputs (A : RandomMachine) (input : List Symbol) {t : ℕ}\n    (bits : Fin t → Bool) (q : ℚ) : Prop :=\n  (run A input bits).state = none ∧\n    (run A input bits).tape.right₀ = Turing.ListBlank.mk (encodeRat q)\n\n/-- A fixed-degree polynomial in input bits, ε⁻¹, and log δ⁻¹.\nNatural ceilings and binary ceiling-log change this bound only by constants. -/\ndef timeBound (C d : ℕ) (G : GraphInput) (ε δ : ℚ) : ℕ :=\n  C * ((encodeInput G ε δ).length + ⌈ε⁻¹⌉₊ + Nat.clog 2 ⌈δ⁻¹⌉₊ + 1) ^ d\n\nnoncomputable def goodTapes (A : RandomMachine) (G : GraphInput) (ε δ : ℚ)\n    (t : ℕ) : Finset (Fin t → Bool) := by\n  classical\n  exact Finset.univ.filter fun bits => ∃ q : ℚ,\n    Outputs A (encodeInput G ε δ) bits q ∧\n    (1 - ε) * (Z G : ℚ) ≤ q ∧ q ≤ (1 + ε) * (Z G : ℚ)\n\n/-- A uniform approximation scheme using independent fair bits. All tapes halt\nwithin the fixed polynomial bound and return a nonnegative rational.\nA zero count always produces encoded zero. -/\ndef MainStatement : Prop :=\n  ∃ (A : RandomMachine) (C d : ℕ), 0 < C ∧\n    ∀ (G : GraphInput) (ε δ : ℚ), 0 < ε → ε < 1 → 0 < δ → δ < 1 / 2 →\n      let t := timeBound C d G ε δ\n      (∀ bits : Fin t → Bool, ∃ q : ℚ, 0 ≤ q ∧ Outputs A (encodeInput G ε δ) bits q) ∧\n      (Z G = 0 → ∀ bits : Fin t → Bool, Outputs A (encodeInput G ε δ) bits 0) ∧\n      ((goodTapes A G ε δ t).card : ℚ) / (2 : ℚ) ^ t ≥ 1 - δ\n\n\n/-- A fully polynomial randomized approximation scheme for perfect matchings\nin every finite simple undirected graph. -/\ntheorem thm_main : MainStatement := by\n  sorry\n\nend MatchingFPRAS\n\nend OAI\n\n\nFILE lean/ComparatorChallenges/TypeSystemNormalization.lean TAIL\n  List (Expr S) → Expr S → Expr S → Prop\n  | ax {s t} : P.axioms s t → HasType P [] (.sort s) (.sort t)\n  | var {Γ A s} : HasType P Γ A (.sort s) →\n      HasType P (A :: Γ) (.var 0) (A.rename Nat.succ)\n  | weaken {Γ M B A s} : HasType P Γ M B → HasType P Γ A (.sort s) →\n      HasType P (A :: Γ) (M.rename Nat.succ) (B.rename Nat.succ)\n  | product {Γ A B s₁ s₂ s₃} : HasType P Γ A (.sort s₁) →\n      HasType P (A :: Γ) B (.sort s₂) → P.rule s₁ s₂ s₃ →\n      HasType P Γ (.pi A B) (.sort s₃)\n  | abstraction {Γ A b B s} : HasType P (A :: Γ) b B →\n      HasType P Γ (.pi A B) (.sort s) →\n      HasType P Γ (.lam A b) (.pi A B)\n  | application {Γ f a A B} : HasType P Γ f (.pi A B) →\n      HasType P Γ a A → HasType P Γ (.app f a) (B.instantiate a)\n  | conversion {Γ M A B s} : HasType P Γ M A → HasType P Γ B (.sort s) →\n      Converts A B → HasType P Γ M B\n\ninductive ValidContext {S : Type u} (P : Specification S) : List (Expr S) → Prop\n  | nil : ValidContext P []\n  | cons {Γ A s} : ValidContext P Γ → HasType P Γ A (.sort s) →\n      ValidContext P (A :: Γ)\n\ndef Legal {S : Type u} (P : Specification S) (Γ : List (Expr S)) (M : Expr S) : Prop :=\n  ∃ A, HasType P Γ M A ∨ HasType P Γ A M\n\ndef SystemWeaklyNormalizing {S : Type u} (P : Specification S) : Prop :=\n  ∀ Γ, ValidContext P Γ → ∀ M, Legal P Γ M → WeaklyNormalizing M\n\ndef SystemStronglyNormalizing {S : Type u} (P : Specification S) : Prop :=\n  ∀ Γ, ValidContext P Γ → ∀ M, Legal P Γ M → StronglyNormalizing M\n\n/-- System-wide weak normalization implies strong normalization for every PTS,\nincluding arbitrary sort types, nonfunctional rules and annotated terms. -/\ntheorem weak_implies_strong {S : Type u} (P : Specification S)\n    (h : SystemWeaklyNormalizing P) : SystemStronglyNormalizing P := by\n  sorry\n\nend PureTypeSystem\n\nend OAI\n\n"
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        "stdout": "ℂ (left index) (operator (right index))) inferInstance\n\n/-- The concrete algebra carries the induced ultraweak topology. -/\nabbrev algebraUltraweakTopology (algebra : StarSubalgebra ℂ (Hilbert →L[ℂ] Hilbert)) :\n    TopologicalSpace algebra :=\n  TopologicalSpace.induced (fun operator : algebra => (operator : Hilbert →L[ℂ] Hilbert))\n    (ultraweakTopology Hilbert)\n\n/-- A faithful normal normalized trace, not an additional hypothesis on the inclusion. -/\nstructure IsNormalizedTrace (algebra : StarSubalgebra ℂ (Hilbert →L[ℂ] Hilbert))\n    (trace : algebra →L[ℂ] ℂ) : Prop where\n  normalized : trace 1 = 1\n  positive : ∀ element : algebra, 0 ≤ (trace (star element * element)).re\n  star_preserving : ∀ element : algebra, trace (star element) = star (trace element)\n  tracial : ∀ left right : algebra, trace (left * right) = trace (right * left)\n  faithful : ∀ element : algebra, trace (star element * element) = 0 → element = 0\n  normal : @Continuous algebra ℂ (algebraUltraweakTopology algebra) inferInstance trace\n\n/-- The normalized trace two-distance between genuine unitaries of the original algebra. -/\ndef traceDistance (algebra : StarSubalgebra ℂ (Hilbert →L[ℂ] Hilbert))\n    (trace : algebra →L[ℂ] ℂ) (left right : unitary algebra) : ℝ :=\n  Real.sqrt (trace (star ((left : algebra) - (right : algebra)) *\n    ((left : algebra) - (right : algebra)))).re\n\n/-- The topology generated by trace two-distance balls, not the inherited operator norm topology. -/\nabbrev traceTopology (algebra : StarSubalgebra ℂ (Hilbert →L[ℂ] Hilbert))\n    (trace : algebra →L[ℂ] ℂ) : TopologicalSpace (unitary algebra) :=\n  TopologicalSpace.generateFrom {ball | ∃ center : unitary algebra, ∃ radius : ℝ,\n    0 < radius ∧ ball = {point | traceDistance algebra trace center point < radius}}\n\n/-- The entire relative-generator locus in the original unitary group. -/\ndef relativeGeneratorLocus (small large : StarSubalgebra ℂ (Hilbert →L[ℂ] Hilbert)) :\n    Set (unitary large) :=\n  {element | wstar ((small : Set (Hilbert →L[ℂ] Hilbert)) ∪\n    {((element : large) : Hilbert →L[ℂ] Hilbert)}) = large}\n\n/-- Every irreducible inclusion of concrete type II₁ factors with separable predual\nof the larger factor has a dense Gδ full relative-generator locus in the normalized\ntrace two-topology. The normalized faithful normal trace is asserted to exist;\nno representation, transport, cyclicity, or perturbation premise is imposed. -/\ndef MainTarget : Prop :=\n  ∀ (Hilbert : Type level) [NormedAddCommGroup Hilbert] [InnerProductSpace ℂ Hilbert]\n    [CompleteSpace Hilbert] (small large : StarSubalgebra ℂ (Hilbert →L[ℂ] Hilbert)),\n    IsII1Factor small → IsII1Factor large → ∀ inclusion : small ≤ large,\n      (∀ element : large,\n        (∀ member : small, StarSubalgebra.inclusion inclusion member * element =\n          element * StarSubalgebra.inclusion inclusion member) →\n        ∃ scalar : ℂ, element = algebraMap ℂ large scalar) →\n      HasSeparablePredual large →\n      ∃ trace : large →L[ℂ] ℂ, IsNormalizedTrace large trace ∧\n        @IsGδ (unitary large) (traceTopology large trace) (relativeGeneratorLocus small large) ∧\n        @Dense (unitary large) (traceTopology large trace) (relativeGeneratorLocus small large)\n\ntheorem main_theorem : MainTarget.{level} := by\n  sorry\n\nend RelativeGeneration\n\nend\n\nend OAI\n\n\nIMPLEMENTATION lean/OAI/Analysis/FactorGeneration/Main.lean \nimport OAI.Analysis.FactorGeneration.Generation\nimport OAI.Analysis.FactorGeneration.PredualContinuity\nimport OAI.Analysis.FactorGeneration.TraceTransport\n\nnamespace OAI\n\nnamespace Generator\n\n/-! Single generation of a finite diffuse factor with a separable Banach predual,\nusing its faithful normal trace and given-predual continuity. -/\nnoncomputable section\nuniverse u\nvariable {H : Type u} [NormedAddCommGroup H] [InnerProductSpace ℂ H] [CompleteSpace H]\n\ntheorem single_generation_of_II1_separable_predual\n    (S : StarSubalgebra ℂ (H →L[ℂ] H)) (hS : IsII1Factor S)\n    (hsep : HasSeparablePredual S) : SinglyGenerated S := by\n  let M := vonNeumannOfWOTClosed S hS.weaklyClosed\n  have hf : ∀ c : M.toStarSubalgebra,\n      (∀ a : M.toStarSubalgebra, a*c=c*a) → ∃ z : ℂ, c=z • (1 : M.toStarSubalgebra) := by\n    intro c hc\n    obtain ⟨z, hz⟩ := hS.factor c hc\n    exact ⟨z, hz.trans (Algebra.algebraMap_eq_smul_one z)⟩\n  have hunit : (1 : M.toStarSubalgebra) ≠ 0 := by\n    intro h\n    exact hS.nonzero (congrArg Subtype.val h).symm\n  let E := Tracial.TraceFreePreHilbert hf hS.diffuse hS.finite hunit\n  let : NormedAddCommGroup E :=\n    Tracial.traceFreePreHilbertNormedAddCommGroup hf hS.diffuse hS.finite hunit\n  let : InnerProductSpace ℂ E :=\n    Tracial.traceFreePreHilbertInnerProductSpace hf hS.diffuse hS.finite hunit\n  let N := Tracial.traceFreeHilbertClosure hf hS.diffuse hS.finite hunit\n  let τ := Tracial.traceFreeHilbertClosureTrace hf hS.diffuse hS.finite hunit\n  have hN := Tracial.traceFreeHilbertClosure_isII1Factor hf hS.diffuse hS.finite hunit\n  have hpre := Tracial.traceFreeHilbertClosure_hasSeparablePredual hf hS.diffuse hS.finite hunit hsep\n  have htrace := Tracial.traceOrbit_isSeparable_of_separablePredual τ hpre\n  exact Tracial.traceFreeHilbert_singlyGenerated_pullback hf hS.diffuse hS.finite hunit\n    (Tracial.singlyGenerated_of_tracialRepresentation hN τ htrace)\n\ntheorem mainTarget : @MainTarget H _ _ _ :=\n  single_generation_of_II1_separable_predual\n\nend\n\nend Generator\n\nend OAI\n\n\nIMPLEMENTATION lean/OAI/MeasureTheory/SelfSimilar/Main.lean \nimport OAI.MeasureTheory.SelfSimilar.EntropyGap\n\nnamespace OAI\n\nuniverse uι\n\nnamespace EntropyRateDimension\nopen MeasureTheory MeasureEntropy\nopen scoped BigOperators ENNReal\n\ntheorem exists_slightly_larger_depth {d χ h : ℝ} (hd : 0 ≤ d) (hgap : d*χ < h) :\n    ∃ a : ℝ, χ < a ∧ d*a < h := by\n  let t := (h-d*χ)/(2*(d+1))\n  have hp : 0 < 2*(d+1) := by linarith\n  have ht : 0 < t := div_pos (sub_pos.mpr hgap) hp\n  have he : t*(2*(d+1))=h-d*χ := div_mul_cancel₀ _ hp.ne'\n  refine ⟨χ+t,by linarith,?_⟩\n  nlinarith\n\nnamespace System\nvariable {ι : Type uι} [Fintype ι] [Nonempty ι]\nlocal instance : MeasurableSpace ι := ⊤\nlocal instance : MeasurableSingletonClass ι := ⟨fun _ => trivial⟩\n\ntheorem exponent_lower_bound (S : System ι) (μ : Measure ℝ) [IsProbabilityMeasure μ]\n    (hμ : S.SelfSimilar μ) : min 1 (S.entropyRate/S.lyapunov) ≤ Coding.exponent S := by\n  classical\n  by_contra! hstrict\n  obtain ⟨d,hed,hdm⟩ := exists_between hstrict\n  have hd : 0 ≤ d := (Coding.exponent_nonneg S).trans hed.le\n  have hd1 : d < 1 := hdm.trans_le (min_le_left _ _)\n  have hdr : d < S.entropyRate/S.lyapunov := hdm.trans_le (min_le_right _ _)\n  have hdh : d*S.lyapunov < S.entropyRate := (lt_div_iff₀ S.lyapunov_pos).mp hdr\n  obtain ⟨a,hχa,hda⟩ := exists_slightly_larger_depth hd hdh\n  have hap : 0 < a := S.lyapunov_pos.trans hχa\n  have hl : 0 < Real.log 2 := Real.log_pos (by norm_num)\n  have hε : 0 < (S.entropyRate-d*a)*Real.log 2 := mul_pos (sub_pos.mpr hda) hl\n  obtain ⟨b,hb,hbtype⟩ := exists_nat_log_small hε (Fintype.card ι)\n  have hbR : (0 : ℝ)<b := by exact_mod_cast hb\n  have htype : Real.log (Fintype.card (ι → Fin (b+1)))=\n      (Fintype.card ι : ℝ)*Real.log ((b : ℝ)+1) := by\n    simp only [Fintype.card_fun,Fintype.card_fin,Nat.cast_pow,Real.log_pow,Nat.cast_add,Nat.cast_one]\n  have hblock : (b : ℝ)*S.entropyRate ≤ (S.blocks hb).entropyRate := S.blocks_entropyRate_lower hb\n  have hgap : d*((b : ℝ)*a)*Real.log 2 < (S.blocks hb).entropyRate*Real.log 2-\n      Real.log (Fintype.card (ι → Fin (b+1))) := by\n    rw [htype]\n    have hh := mul_le_mul_of_nonneg_right hblock hl.le\n    nlinarith\n  obtain ⟨C,hC⟩ := S.selfSimilar_entropy_bound μ hμ hed\n  apply Typed.no_strict_entropy_gap (S.blocks hb) (wordType (ι := ι)) (S.ratio_wordType hb)\n    μ (S.selfSimilar_blocks μ hμ hb) hd hd1 (mul_pos hbR hap)\n    (A' := (b : ℝ)*((S.lyapunov+a)/2)) _ _ hC hgap\n  · rw [S.blocks_lyapunov]\n    exact mul_lt_mul_of_pos_left (by linarith) hbR\n  · exact mul_lt_mul_of_pos_left (by linarith) hbR\n\nend System\n\ntheorem entropy_rate_dimension {ι : Type uι} [Fintype ι] [Nonempty ι]\n    (S : System ι) (μ : Measure ℝ) [IsProbabilityMeasure μ]\n    (hμ : S.SelfSimilar μ) :\n    lowerHausdorffDimension μ = ENNReal.ofReal (min 1 (S.entropyRate / S.lyapunov)) := by\n  rw [S.selfSimilar_dimension μ hμ]\n  congr 1\n  exact le_antisymm (le_min (S.exponent_le_one μ hμ) (S.exponent_le_rate μ hμ)) (S.exponent_lower_bound μ hμ)\n\nend EntropyRateDimension\n\nend OAI\n\n",
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          "ℂ (left index) (operator (right index))) inferInstance\n\n/-- The concrete algebra carries the induced ultraweak topology. -/\nabbrev algebraUltraweakTopology (algebra : StarSubalgebra ℂ (Hilbert →L[ℂ] Hilbert)) :\n    TopologicalSpace algebra :=\n  TopologicalSpace.induced (fun operator : algebra => (operator : Hilbert →L[ℂ] Hilbert))\n    (ultraweakTopology Hilbert)\n\n/-- A faithful normal normalized trace, not an additional hypothesis on the inclusion. -/\nstructure IsNormalizedTrace (algebra : StarSubalgebra ℂ (Hilbert →L[ℂ] Hilbert))\n    (trace : algebra →L[ℂ] ℂ) : Prop where\n  normalized : trace 1 = 1\n  positive : ∀ element : algebra, 0 ≤ (trace (star element * element)).re\n  star_preserving : ∀ element : algebra, trace (star element) = star (trace element)\n  tracial : ∀ left right : algebra, trace (left * right) = trace (right * left)\n  faithful : ∀ element : algebra, trace (star element * element) = 0 → element = 0\n  normal : @Continuous algebra ℂ (algebraUltraweakTopology algebra) inferInstance trace\n\n/-- The normalized trace two-distance between genuine unitaries of the original algebra. -/\ndef traceDistance (algebra : StarSubalgebra ℂ (Hilbert →L[ℂ] Hilbert))\n    (trace : algebra →L[ℂ] ℂ) (left right : unitary algebra) : ℝ :=\n  Real.sqrt (trace (star ((left : algebra) - (right : algebra)) *\n    ((left : algebra) - (right : algebra)))).re\n\n/-- The topology generated by trace two-distance balls, not the inherited operator norm topology. -/\nabbrev traceTopology (algebra : StarSubalgebra ℂ (Hilbert →L[ℂ] Hilbert))\n    (trace : algebra →L[ℂ] ℂ) : TopologicalSpace (unitary algebra) :=\n  TopologicalSpace.generateFrom {ball | ∃ center : unitary algebra, ∃ radius : ℝ,\n    0 < radius ∧ ball = {point | traceDistance algebra trace center point < radius}}\n\n/-- The entire relative-generator locus in the original unitary group. -/\ndef relativeGeneratorLocus (small large : StarSubalgebra ℂ (Hilbert →L[ℂ] Hilbert)) :\n    Set (unitary large) :=\n  {element | wstar ((small : Set (Hilbert →L[ℂ] Hilbert)) ∪\n    {((element : large) : Hilbert →L[ℂ] Hilbert)}) = large}\n\n/-- Every irreducible inclusion of concrete type II₁ factors with separable predual\nof the larger factor has a dense Gδ full relative-generator locus in the normalized\ntrace two-topology. The normalized faithful normal trace is asserted to exist;\nno representation, transport, cyclicity, or perturbation premise is imposed. -/\ndef MainTarget : Prop :=\n  ∀ (Hilbert : Type level) [NormedAddCommGroup Hilbert] [InnerProductSpace ℂ Hilbert]\n    [CompleteSpace Hilbert] (small large : StarSubalgebra ℂ (Hilbert →L[ℂ] Hilbert)),\n    IsII1Factor small → IsII1Factor large → ∀ inclusion : small ≤ large,\n      (∀ element : large,\n        (∀ member : small, StarSubalgebra.inclusion inclusion member * element =\n          element * StarSubalgebra.inclusion inclusion member) →\n        ∃ scalar : ℂ, element = algebraMap ℂ large scalar) →\n      HasSeparablePredual large →\n      ∃ trace : large →L[ℂ] ℂ, IsNormalizedTrace large trace ∧\n        @IsGδ (unitary large) (traceTopology large trace) (relativeGeneratorLocus small large) ∧\n        @Dense (unitary large) (traceTopology large trace) (relativeGeneratorLocus small large)\n\ntheorem main_theorem : MainTarget.{level} := by\n  sorry\n\nend RelativeGeneration\n\nend\n\nend OAI\n\n\nIMPLEMENTATION lean/OAI/Analysis/FactorGeneration/Main.lean \nimport OAI.Analysis.FactorGeneration.Generation\nimport OAI.Analysis.FactorGeneration.PredualContinuity\nimport OAI.Analysis.FactorGeneration.TraceTransport\n\nnamespace OAI\n\nnamespace Generator\n\n/-! Single generation of a finite diffuse factor with a separable Banach predual,\nusing its faithful normal trace and given-predual continuity. -/\nnoncomputable section\nuniverse u\nvariable {H : Type u} [NormedAddCommGroup H] [InnerProductSpace ℂ H] [CompleteSpace H]\n\ntheorem single_generation_of_II1_separable_predual\n    (S : StarSubalgebra ℂ (H →L[ℂ] H)) (hS : IsII1Factor S)\n    (hsep : HasSeparablePredual S) : SinglyGenerated S := by\n  let M := vonNeumannOfWOTClosed S hS.weaklyClosed\n  have hf : ∀ c : M.toStarSubalgebra,\n      (∀ a : M.toStarSubalgebra, a*c=c*a) → ∃ z : ℂ, c=z • (1 : M.toStarSubalgebra) := by\n    intro c hc\n    obtain ⟨z, hz⟩ := hS.factor c hc\n    exact ⟨z, hz.trans (Algebra.algebraMap_eq_smul_one z)⟩\n  have hunit : (1 : M.toStarSubalgebra) ≠ 0 := by\n    intro h\n    exact hS.nonzero (congrArg Subtype.val h).symm\n  let E := Tracial.TraceFreePreHilbert hf hS.diffuse hS.finite hunit\n  let : NormedAddCommGroup E :=\n    Tracial.traceFreePreHilbertNormedAddCommGroup hf hS.diffuse hS.finite hunit\n  let : InnerProductSpace ℂ E :=\n    Tracial.traceFreePreHilbertInnerProductSpace hf hS.diffuse hS.finite hunit\n  let N := Tracial.traceFreeHilbertClosure hf hS.diffuse hS.finite hunit\n  let τ := Tracial.traceFreeHilbertClosureTrace hf hS.diffuse hS.finite hunit\n  have hN := Tracial.traceFreeHilbertClosure_isII1Factor hf hS.diffuse hS.finite hunit\n  have hpre := Tracial.traceFreeHilbertClosure_hasSeparablePredual hf hS.diffuse hS.finite hunit hsep\n  have htrace := Tracial.traceOrbit_isSeparable_of_separablePredual τ hpre\n  exact Tracial.traceFreeHilbert_singlyGenerated_pullback hf hS.diffuse hS.finite hunit\n    (Tracial.singlyGenerated_of_tracialRepresentation hN τ htrace)\n\ntheorem mainTarget : @MainTarget H _ _ _ :=\n  single_generation_of_II1_separable_predual\n\nend\n\nend Generator\n\nend OAI\n\n\nIMPLEMENTATION lean/OAI/MeasureTheory/SelfSimilar/Main.lean \nimport OAI.MeasureTheory.SelfSimilar.EntropyGap\n\nnamespace OAI\n\nuniverse uι\n\nnamespace EntropyRateDimension\nopen MeasureTheory MeasureEntropy\nopen scoped BigOperators ENNReal\n\ntheorem exists_slightly_larger_depth {d χ h : ℝ} (hd : 0 ≤ d) (hgap : d*χ < h) :\n    ∃ a : ℝ, χ < a ∧ d*a < h := by\n  let t := (h-d*χ)/(2*(d+1))\n  have hp : 0 < 2*(d+1) := by linarith\n  have ht : 0 < t := div_pos (sub_pos.mpr hgap) hp\n  have he : t*(2*(d+1))=h-d*χ := div_mul_cancel₀ _ hp.ne'\n  refine ⟨χ+t,by linarith,?_⟩\n  nlinarith\n\nnamespace System\nvariable {ι : Type uι} [Fintype ι] [Nonempty ι]\nlocal instance : MeasurableSpace ι := ⊤\nlocal instance : MeasurableSingletonClass ι := ⟨fun _ => trivial⟩\n\ntheorem exponent_lower_bound (S : System ι) (μ : Measure ℝ) [IsProbabilityMeasure μ]\n    (hμ : S.SelfSimilar μ) : min 1 (S.entropyRate/S.lyapunov) ≤ Coding.exponent S := by\n  classical\n  by_contra! hstrict\n  obtain ⟨d,hed,hdm⟩ := exists_between hstrict\n  have hd : 0 ≤ d := (Coding.exponent_nonneg S).trans hed.le\n  have hd1 : d < 1 := hdm.trans_le (min_le_left _ _)\n  have hdr : d < S.entropyRate/S.lyapunov := hdm.trans_le (min_le_right _ _)\n  have hdh : d*S.lyapunov < S.entropyRate := (lt_div_iff₀ S.lyapunov_pos).mp hdr\n  obtain ⟨a,hχa,hda⟩ := exists_slightly_larger_depth hd hdh\n  have hap : 0 < a := S.lyapunov_pos.trans hχa\n  have hl : 0 < Real.log 2 := Real.log_pos (by norm_num)\n  have hε : 0 < (S.entropyRate-d*a)*Real.log 2 := mul_pos (sub_pos.mpr hda) hl\n  obtain ⟨b,hb,hbtype⟩ := exists_nat_log_small hε (Fintype.card ι)\n  have hbR : (0 : ℝ)<b := by exact_mod_cast hb\n  have htype : Real.log (Fintype.card (ι → Fin (b+1)))=\n      (Fintype.card ι : ℝ)*Real.log ((b : ℝ)+1) := by\n    simp only [Fintype.card_fun,Fintype.card_fin,Nat.cast_pow,Real.log_pow,Nat.cast_add,Nat.cast_one]\n  have hblock : (b : ℝ)*S.entropyRate ≤ (S.blocks hb).entropyRate := S.blocks_entropyRate_lower hb\n  have hgap : d*((b : ℝ)*a)*Real.log 2 < (S.blocks hb).entropyRate*Real.log 2-\n      Real.log (Fintype.card (ι → Fin (b+1))) := by\n    rw [htype]\n    have hh := mul_le_mul_of_nonneg_right hblock hl.le\n    nlinarith\n  obtain ⟨C,hC⟩ := S.selfSimilar_entropy_bound μ hμ hed\n  apply Typed.no_strict_entropy_gap (S.blocks hb) (wordType (ι := ι)) (S.ratio_wordType hb)\n    μ (S.selfSimilar_blocks μ hμ hb) hd hd1 (mul_pos hbR hap)\n    (A' := (b : ℝ)*((S.lyapunov+a)/2)) _ _ hC hgap\n  · rw [S.blocks_lyapunov]\n    exact mul_lt_mul_of_pos_left (by linarith) hbR\n  · exact mul_lt_mul_of_pos_left (by linarith) hbR\n\nend System\n\ntheorem entropy_rate_dimension {ι : Type uι} [Fintype ι] [Nonempty ι]\n    (S : System ι) (μ : Measure ℝ) [IsProbabilityMeasure μ]\n    (hμ : S.SelfSimilar μ) :\n    lowerHausdorffDimension μ = ENNReal.ofReal (min 1 (S.entropyRate / S.lyapunov)) := by\n  rw [S.selfSimilar_dimension μ hμ]\n  congr 1\n  exact le_antisymm (le_min (S.exponent_le_one μ hμ) (S.exponent_le_rate μ hμ)) (S.exponent_lower_bound μ hμ)\n\nend EntropyRateDimension\n\nend OAI\n\n"
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          "text": "ℂ (left index) (operator (right index))) inferInstance\n\n/-- The concrete algebra carries the induced ultraweak topology. -/\nabbrev algebraUltraweakTopology (algebra : StarSubalgebra ℂ (Hilbert →L[ℂ] Hilbert)) :\n    TopologicalSpace algebra :=\n  TopologicalSpace.induced (fun operator : algebra => (operator : Hilbert →L[ℂ] Hilbert))\n    (ultraweakTopology Hilbert)\n\n/-- A faithful normal normalized trace, not an additional hypothesis on the inclusion. -/\nstructure IsNormalizedTrace (algebra : StarSubalgebra ℂ (Hilbert →L[ℂ] Hilbert))\n    (trace : algebra →L[ℂ] ℂ) : Prop where\n  normalized : trace 1 = 1\n  positive : ∀ element : algebra, 0 ≤ (trace (star element * element)).re\n  star_preserving : ∀ element : algebra, trace (star element) = star (trace element)\n  tracial : ∀ left right : algebra, trace (left * right) = trace (right * left)\n  faithful : ∀ element : algebra, trace (star element * element) = 0 → element = 0\n  normal : @Continuous algebra ℂ (algebraUltraweakTopology algebra) inferInstance trace\n\n/-- The normalized trace two-distance between genuine unitaries of the original algebra. -/\ndef traceDistance (algebra : StarSubalgebra ℂ (Hilbert →L[ℂ] Hilbert))\n    (trace : algebra →L[ℂ] ℂ) (left right : unitary algebra) : ℝ :=\n  Real.sqrt (trace (star ((left : algebra) - (right : algebra)) *\n    ((left : algebra) - (right : algebra)))).re\n\n/-- The topology generated by trace two-distance balls, not the inherited operator norm topology. -/\nabbrev traceTopology (algebra : StarSubalgebra ℂ (Hilbert →L[ℂ] Hilbert))\n    (trace : algebra →L[ℂ] ℂ) : TopologicalSpace (unitary algebra) :=\n  TopologicalSpace.generateFrom {ball | ∃ center : unitary algebra, ∃ radius : ℝ,\n    0 < radius ∧ ball = {point | traceDistance algebra trace center point < radius}}\n\n/-- The entire relative-generator locus in the original unitary group. -/\ndef relativeGeneratorLocus (small large : StarSubalgebra ℂ (Hilbert →L[ℂ] Hilbert)) :\n    Set (unitary large) :=\n  {element | wstar ((small : Set (Hilbert →L[ℂ] Hilbert)) ∪\n    {((element : large) : Hilbert →L[ℂ] Hilbert)}) = large}\n\n/-- Every irreducible inclusion of concrete type II₁ factors with separable predual\nof the larger factor has a dense Gδ full relative-generator locus in the normalized\ntrace two-topology. The normalized faithful normal trace is asserted to exist;\nno representation, transport, cyclicity, or perturbation premise is imposed. -/\ndef MainTarget : Prop :=\n  ∀ (Hilbert : Type level) [NormedAddCommGroup Hilbert] [InnerProductSpace ℂ Hilbert]\n    [CompleteSpace Hilbert] (small large : StarSubalgebra ℂ (Hilbert →L[ℂ] Hilbert)),\n    IsII1Factor small → IsII1Factor large → ∀ inclusion : small ≤ large,\n      (∀ element : large,\n        (∀ member : small, StarSubalgebra.inclusion inclusion member * element =\n          element * StarSubalgebra.inclusion inclusion member) →\n        ∃ scalar : ℂ, element = algebraMap ℂ large scalar) →\n      HasSeparablePredual large →\n      ∃ trace : large →L[ℂ] ℂ, IsNormalizedTrace large trace ∧\n        @IsGδ (unitary large) (traceTopology large trace) (relativeGeneratorLocus small large) ∧\n        @Dense (unitary large) (traceTopology large trace) (relativeGeneratorLocus small large)\n\ntheorem main_theorem : MainTarget.{level} := by\n  sorry\n\nend RelativeGeneration\n\nend\n\nend OAI\n\n\nIMPLEMENTATION lean/OAI/Analysis/FactorGeneration/Main.lean \nimport OAI.Analysis.FactorGeneration.Generation\nimport OAI.Analysis.FactorGeneration.PredualContinuity\nimport OAI.Analysis.FactorGeneration.TraceTransport\n\nnamespace OAI\n\nnamespace Generator\n\n/-! Single generation of a finite diffuse factor with a separable Banach predual,\nusing its faithful normal trace and given-predual continuity. -/\nnoncomputable section\nuniverse u\nvariable {H : Type u} [NormedAddCommGroup H] [InnerProductSpace ℂ H] [CompleteSpace H]\n\ntheorem single_generation_of_II1_separable_predual\n    (S : StarSubalgebra ℂ (H →L[ℂ] H)) (hS : IsII1Factor S)\n    (hsep : HasSeparablePredual S) : SinglyGenerated S := by\n  let M := vonNeumannOfWOTClosed S hS.weaklyClosed\n  have hf : ∀ c : M.toStarSubalgebra,\n      (∀ a : M.toStarSubalgebra, a*c=c*a) → ∃ z : ℂ, c=z • (1 : M.toStarSubalgebra) := by\n    intro c hc\n    obtain ⟨z, hz⟩ := hS.factor c hc\n    exact ⟨z, hz.trans (Algebra.algebraMap_eq_smul_one z)⟩\n  have hunit : (1 : M.toStarSubalgebra) ≠ 0 := by\n    intro h\n    exact hS.nonzero (congrArg Subtype.val h).symm\n  let E := Tracial.TraceFreePreHilbert hf hS.diffuse hS.finite hunit\n  let : NormedAddCommGroup E :=\n    Tracial.traceFreePreHilbertNormedAddCommGroup hf hS.diffuse hS.finite hunit\n  let : InnerProductSpace ℂ E :=\n    Tracial.traceFreePreHilbertInnerProductSpace hf hS.diffuse hS.finite hunit\n  let N := Tracial.traceFreeHilbertClosure hf hS.diffuse hS.finite hunit\n  let τ := Tracial.traceFreeHilbertClosureTrace hf hS.diffuse hS.finite hunit\n  have hN := Tracial.traceFreeHilbertClosure_isII1Factor hf hS.diffuse hS.finite hunit\n  have hpre := Tracial.traceFreeHilbertClosure_hasSeparablePredual hf hS.diffuse hS.finite hunit hsep\n  have htrace := Tracial.traceOrbit_isSeparable_of_separablePredual τ hpre\n  exact Tracial.traceFreeHilbert_singlyGenerated_pullback hf hS.diffuse hS.finite hunit\n    (Tracial.singlyGenerated_of_tracialRepresentation hN τ htrace)\n\ntheorem mainTarget : @MainTarget H _ _ _ :=\n  single_generation_of_II1_separable_predual\n\nend\n\nend Generator\n\nend OAI\n\n\nIMPLEMENTATION lean/OAI/MeasureTheory/SelfSimilar/Main.lean \nimport OAI.MeasureTheory.SelfSimilar.EntropyGap\n\nnamespace OAI\n\nuniverse uι\n\nnamespace EntropyRateDimension\nopen MeasureTheory MeasureEntropy\nopen scoped BigOperators ENNReal\n\ntheorem exists_slightly_larger_depth {d χ h : ℝ} (hd : 0 ≤ d) (hgap : d*χ < h) :\n    ∃ a : ℝ, χ < a ∧ d*a < h := by\n  let t := (h-d*χ)/(2*(d+1))\n  have hp : 0 < 2*(d+1) := by linarith\n  have ht : 0 < t := div_pos (sub_pos.mpr hgap) hp\n  have he : t*(2*(d+1))=h-d*χ := div_mul_cancel₀ _ hp.ne'\n  refine ⟨χ+t,by linarith,?_⟩\n  nlinarith\n\nnamespace System\nvariable {ι : Type uι} [Fintype ι] [Nonempty ι]\nlocal instance : MeasurableSpace ι := ⊤\nlocal instance : MeasurableSingletonClass ι := ⟨fun _ => trivial⟩\n\ntheorem exponent_lower_bound (S : System ι) (μ : Measure ℝ) [IsProbabilityMeasure μ]\n    (hμ : S.SelfSimilar μ) : min 1 (S.entropyRate/S.lyapunov) ≤ Coding.exponent S := by\n  classical\n  by_contra! hstrict\n  obtain ⟨d,hed,hdm⟩ := exists_between hstrict\n  have hd : 0 ≤ d := (Coding.exponent_nonneg S).trans hed.le\n  have hd1 : d < 1 := hdm.trans_le (min_le_left _ _)\n  have hdr : d < S.entropyRate/S.lyapunov := hdm.trans_le (min_le_right _ _)\n  have hdh : d*S.lyapunov < S.entropyRate := (lt_div_iff₀ S.lyapunov_pos).mp hdr\n  obtain ⟨a,hχa,hda⟩ := exists_slightly_larger_depth hd hdh\n  have hap : 0 < a := S.lyapunov_pos.trans hχa\n  have hl : 0 < Real.log 2 := Real.log_pos (by norm_num)\n  have hε : 0 < (S.entropyRate-d*a)*Real.log 2 := mul_pos (sub_pos.mpr hda) hl\n  obtain ⟨b,hb,hbtype⟩ := exists_nat_log_small hε (Fintype.card ι)\n  have hbR : (0 : ℝ)<b := by exact_mod_cast hb\n  have htype : Real.log (Fintype.card (ι → Fin (b+1)))=\n      (Fintype.card ι : ℝ)*Real.log ((b : ℝ)+1) := by\n    simp only [Fintype.card_fun,Fintype.card_fin,Nat.cast_pow,Real.log_pow,Nat.cast_add,Nat.cast_one]\n  have hblock : (b : ℝ)*S.entropyRate ≤ (S.blocks hb).entropyRate := S.blocks_entropyRate_lower hb\n  have hgap : d*((b : ℝ)*a)*Real.log 2 < (S.blocks hb).entropyRate*Real.log 2-\n      Real.log (Fintype.card (ι → Fin (b+1))) := by\n    rw [htype]\n    have hh := mul_le_mul_of_nonneg_right hblock hl.le\n    nlinarith\n  obtain ⟨C,hC⟩ := S.selfSimilar_entropy_bound μ hμ hed\n  apply Typed.no_strict_entropy_gap (S.blocks hb) (wordType (ι := ι)) (S.ratio_wordType hb)\n    μ (S.selfSimilar_blocks μ hμ hb) hd hd1 (mul_pos hbR hap)\n    (A' := (b : ℝ)*((S.lyapunov+a)/2)) _ _ hC hgap\n  · rw [S.blocks_lyapunov]\n    exact mul_lt_mul_of_pos_left (by linarith) hbR\n  · exact mul_lt_mul_of_pos_left (by linarith) hbR\n\nend System\n\ntheorem entropy_rate_dimension {ι : Type uι} [Fintype ι] [Nonempty ι]\n    (S : System ι) (μ : Measure ℝ) [IsProbabilityMeasure μ]\n    (hμ : S.SelfSimilar μ) :\n    lowerHausdorffDimension μ = ENNReal.ofReal (min 1 (S.entropyRate / S.lyapunov)) := by\n  rw [S.selfSimilar_dimension μ hμ]\n  congr 1\n  exact le_antisymm (le_min (S.exponent_le_one μ hμ) (S.exponent_le_rate μ hμ)) (S.exponent_lower_bound μ hμ)\n\nend EntropyRateDimension\n\nend OAI\n\n"
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        "stdout": "\nIMPLEMENTATION lean/OAI/Combinatorics/GotsmanLinial/Main.lean \n import OAI.Combinatorics.GotsmanLinial.Statement\nimport OAI.Combinatorics.GotsmanLinial.BlockCoupling\nimport OAI.Combinatorics.GotsmanLinial.Commutator\nimport OAI.Combinatorics.GotsmanLinial.CubeOperators\nimport OAI.Combinatorics.GotsmanLinial.EdgeRe \nFINAL\n : ℝ) * Real.sqrt (n : ℝ) := by\n  let w : Cube n → ℝ := fun x => |polynomialValue p x|\n  have hw (x : Cube n) : 0 < w x := abs_pos.mpr (hnz x)\n  apply sensitivity_bound_of_cube_grading (polynomialThreshold p)\n    (fun x => thresholdSign_cases (polynomialValue p x))\n    (weightedProjectionMatrix w hw)\n    (weightedProjectionMatrix_family w hw)\n    (weightedProjectionMatrix_trace_re w hw)\n  · intro i\n    exact weightedProjectionMatrix_tridiagonal w hw i\n  · intro r s hrs\n    exact weightedProjectionMatrix_block_eq_zero_of_polynomial_degree p hp hd hnz r s hrs\n\n/-- Every real multilinear polynomial of degree at most `d` has average\nsensitivity at most `8*d*sqrt(n)`, including cube zeros under `sign(0)=1`.\nThe argument in fact covers all natural `n` and `d`. -/\ntheorem polynomialThreshold_averageSensitivity_le {n d : ℕ}\n    (p : MvPolynomial (Fin n) ℝ) (hp : IsMultilinear p) (hd : p.totalDegree ≤ d) :\n    averageSensitivity (polynomialThreshold p) ≤ 8 * (d : ℝ) * Real.sqrt (n : ℝ) := by\n  obtain ⟨q, hq, hqd, hqnz, hqf⟩ := exists_nonzero_polynomial_same_threshold hp hd\n  rw [← hqf]\n  exact polynomialThreshold_averageSensitivity_le_of_nonzero q hq hqd hqnz\n\n/-- The average-sensitivity bound for `1 ≤ d ≤ n`, without auxiliary\nconstruction hypotheses. -/\ntheorem gotsmanLinialStatement : GotsmanLinialStatement := by\n  intro n d _ _ _ p hp hd\n  exact polynomialThreshold_averageSensitivity_le p hp hd\n\nend LeanBlast.GotsmanLinial\n\nend OAI\n\n\nIMPLEMENTATION lean/OAI/MeasureTheory/SelfSimilar/Main.lean \n import OAI.MeasureTheory.SelfSimilar.EntropyGap\n\nnamespace OAI\n\nuniverse uι\n\nnamespace EntropyRateDimension\nopen MeasureTheory MeasureEntropy\nopen scoped BigOperators ENNReal\n\ntheorem exists_slightly_larger_depth {d χ h : ℝ} (hd : 0 ≤ d) (hgap : d*χ  \nFINAL\n nat_log_small hε (Fintype.card ι)\n  have hbR : (0 : ℝ)<b := by exact_mod_cast hb\n  have htype : Real.log (Fintype.card (ι → Fin (b+1)))=\n      (Fintype.card ι : ℝ)*Real.log ((b : ℝ)+1) := by\n    simp only [Fintype.card_fun,Fintype.card_fin,Nat.cast_pow,Real.log_pow,Nat.cast_add,Nat.cast_one]\n  have hblock : (b : ℝ)*S.entropyRate ≤ (S.blocks hb).entropyRate := S.blocks_entropyRate_lower hb\n  have hgap : d*((b : ℝ)*a)*Real.log 2 < (S.blocks hb).entropyRate*Real.log 2-\n      Real.log (Fintype.card (ι → Fin (b+1))) := by\n    rw [htype]\n    have hh := mul_le_mul_of_nonneg_right hblock hl.le\n    nlinarith\n  obtain ⟨C,hC⟩ := S.selfSimilar_entropy_bound μ hμ hed\n  apply Typed.no_strict_entropy_gap (S.blocks hb) (wordType (ι := ι)) (S.ratio_wordType hb)\n    μ (S.selfSimilar_blocks μ hμ hb) hd hd1 (mul_pos hbR hap)\n    (A' := (b : ℝ)*((S.lyapunov+a)/2)) _ _ hC hgap\n  · rw [S.blocks_lyapunov]\n    exact mul_lt_mul_of_pos_left (by linarith) hbR\n  · exact mul_lt_mul_of_pos_left (by linarith) hbR\n\nend System\n\ntheorem entropy_rate_dimension {ι : Type uι} [Fintype ι] [Nonempty ι]\n    (S : System ι) (μ : Measure ℝ) [IsProbabilityMeasure μ]\n    (hμ : S.SelfSimilar μ) :\n    lowerHausdorffDimension μ = ENNReal.ofReal (min 1 (S.entropyRate / S.lyapunov)) := by\n  rw [S.selfSimilar_dimension μ hμ]\n  congr 1\n  exact le_antisymm (le_min (S.exponent_le_one μ hμ) (S.exponent_le_rate μ hμ)) (S.exponent_lower_bound μ hμ)\n\nend EntropyRateDimension\n\nend OAI\n\n\nIMPLEMENTATION lean/OAI/Analysis/TriangularHilbert/Main.lean \n import OAI.Analysis.TriangularHilbert.TruncationCoordinates\n\nnamespace OAI\n\nsection\nnamespace TriangularHilbert\nopen MeasureTheory Heat Filter\nopen scoped ENNReal\nnoncomputable section\n\nlemma maximal_eq_cyclicMax_ae (F G : Plane → ℂ) (hF : MemLp F 3  \nFINAL\n q_left hz.goodPoint]\n  simp_rw [truncation_eq_cyclicHard]\n  exact (cyclicMax_eq_iSup hz).symm\n\nlemma aemeasurable_maximal (F G : Plane → ℂ) (hF : MemLp F 3 volume) (hG : MemLp G 3 volume) :\n    AEMeasurable (maximal F G) volume := by\n  have h := (aemeasurable_cyclicMax (cyclicB F) (cyclicC G) (memLp_cyclicB hF) (memLp_cyclicC hG)).comp_quasiMeasurePreserving\n    measurePreserving_reflect.quasiMeasurePreserving\n  exact h.congr (maximal_eq_cyclicMax_ae F G hF hG).symm\n\nlemma maximalNorm_eq_cyclicNorm (F G : Plane → ℂ) (hF : MemLp F 3 volume) (hG : MemLp G 3 volume) :\n    maximalNorm F G = cyclicNorm (cyclicB F) (cyclicC G) := by\n  unfold maximalNorm cyclicNorm\n  congr 1\n  calc\n    (∫⁻ z : Plane, maximal F G z ^ ((3:ℝ)/2)) =\n        ∫⁻ z : Plane, cyclicMax (cyclicB F) (cyclicC G) (-z)^((3:ℝ)/2) := by\n      apply lintegral_congr_ae\n      filter_upwards [maximal_eq_cyclicMax_ae F G hF hG] with z hz\n      rw [hz]\n    _ = _ := measurePreserving_reflect.lintegral_comp_emb (Homeomorph.neg Plane).measurableEmbedding\n      (fun z => cyclicMax (cyclicB F) (cyclicC G) z^((3:ℝ)/2))\n\ntheorem main_estimate : MainEstimate := by\n  refine ⟨mainConstant,fun F G hF hG => ⟨ae_goodPoint F G hF hG,aemeasurable_maximal F G hF hG,?_⟩⟩\n  rw [maximalNorm_eq_cyclicNorm F G hF hG]\n  simpa only [eLpNorm_cyclicB hF,eLpNorm_cyclicC hG] using\n    cyclicNorm_le (cyclicB F) (cyclicC G) (memLp_cyclicB hF) (memLp_cyclicC hG)\nend\nend TriangularHilbert\nend\n\nend OAI\n\n\nIMPLEMENTATION lean/OAI/Probability/ClassicalON/Main.lean \n import OAI.Probability.ClassicalON.BlockReduction\n\nnamespace OAI\n\nnoncomputable section\nopen MeasureTheory\nopen scoped InnerProductSpace BigOperators Classical\nnamespace ClassicalON\n\ntheorem correlation_eq_freeSpinMean (n : ℕ) (G : LatticeGraph) (b : \nFINAL\n ference\n  congr 1\n  apply integral_congr_ae\n  filter_upwards with s\n  exact mul_comm _ _\n\ntheorem main : ExponentialDecay := by\n  intro n hn β hβ\n  obtain ⟨k,rfl⟩ := Nat.exists_eq_add_of_le hn\n  let : NeZero (3+k) := ⟨by omega⟩\n  obtain ⟨m,hm,hexp⟩ := LatticeGraph.connection_exponential β hβ.le\n  refine ⟨(3+k:ℕ)*4,m,hm,?_⟩\n  intro G b hb x y\n  have hthree (c : G.edges → ℝ) (hc : ∀ e,0 ≤ c e ∧ c e ≤ b e) :\n      0 ≤ freeSpinMean 3 (fun e : G.edges => e.val.1) (fun e => e.val.2) c (firstSpinProduct x y) ∧\n      freeSpinMean 3 (fun e : G.edges => e.val.1) (fun e => e.val.2) c (firstSpinProduct x y) ≤\n        4*Real.exp (-m*siteDistance x.val y.val) := by\n    have hfirst := firstSpinMean_bounds (fun e : G.edges => e.val.1) (fun e => e.val.2) c\n      (fun e => (hc e).1) x y\n    exact ⟨hfirst.1,hfirst.2.trans (hexp G c (fun e => ⟨(hc e).1,(hc e).2.trans (hb e).2⟩) x y)⟩\n  have hbound := coordinateSpinMean_block_bounds k (fun e : G.edges => e.val.1) (fun e => e.val.2) b\n    (fun e => (hb e).1) x y (4*Real.exp (-m*siteDistance x.val y.val)) hthree\n  rw [correlation_eq_freeSpinMean,innerSpinMean_eq_coordinate (3+k)\n    (fun e : G.edges => e.val.1) (fun e => e.val.2) b ⟨0,by omega⟩ x y]\n  constructor\n  · exact mul_nonneg (Nat.cast_nonneg _) hbound.1\n  · calc\n      _ ≤ (3+k:ℕ)*(4*Real.exp (-m*siteDistance x.val y.val)) :=\n        mul_le_mul_of_nonneg_left hbound.2 (Nat.cast_nonneg _)\n      _ = _ := by ring\n\nend ClassicalON\n\nend\n\nend OAI\n\n\nIMPLEMENTATION lean/OAI/RepresentationTheory/FoulkesHowe/Stabilization.lean \n import OAI.RepresentationTheory.FoulkesHowe.Model\nimport OAI.RepresentationTheory.FoulkesHowe.LinearCase\nimport OAI.RepresentationTheory.FoulkesHowe.PairingConstruction\nimport OAI.RepresentationTheory.FoulkesHowe.Surjectivity\nimport OAI.Representatio \nFINAL\n ymPow b (SymPow 1 V) →ₗ[ℂ] SymPow 1 (SymPow b V), IsFoulkesMap 1 b V μ ∧ Function.Bijective μ ∧ ∀ ν : SymPow b (SymPow 1 V) →ₗ[ℂ] SymPow 1 (SymPow b V), IsFoulkesMap 1 b V ν → ν = μ := by\n  intro V _ _ _ b\n  exact canonical_foulkes_howe_a_one_aux V b\n\ntheorem vanishing_on_products :\n    ∀ (V : Type u) [AddCommGroup V] [Module ℂ V] [FiniteDimensional ℂ V] (a b : ℕ), 1 ≤ a → 1 ≤ b → a * (a - 1) ≤ b → ∀ T : SymmetricMultilinearForm a b V, IsSymmetricMultilinearForm a b V T → (∀ u : Fin b → V, T (fun _ => symMonomial b V u) = 0) → T = 0 := by\n  intro V _ _ _ a b ha hb hab T hT hdiag\n  exact vanishing_on_products_aux V a b ha hb hab T hT hdiag\n\ntheorem foulkes_comparison_embedding :\n    ∀ (V : Type u) [AddCommGroup V] [Module ℂ V] [FiniteDimensional ℂ V] (a b : ℕ), 2 ≤ a → a * (a - 1) ≤ b → ∃ ι : SymPow a (SymPow b V) →ₗ[ℂ] SymPow b (SymPow a V), IsGLEquivariantEmbedding a b V ι := by\n  intro V _ _ _ a b ha hab\n  obtain ⟨μ, hμ, hsurj, _⟩ :=\n    canonical_foulkes_howe_surjective (Module.Dual ℂ V) a b ha hab\n  exact PairingConstruction.comparison_of_surjection V a b ⟨μ, hμ, hsurj⟩\n\ntheorem sixth_symmetric_power_comparison :\n    ∀ (V : Type u) [AddCommGroup V] [Module ℂ V] [FiniteDimensional ℂ V] (b : ℕ), 30 ≤ b → ∃ ι : SymPow 6 (SymPow b V) →ₗ[ℂ] SymPow b (SymPow 6 V), IsGLEquivariantEmbedding 6 b V ι := by\n  intro V _ _ _ b hb\n  exact foulkes_comparison_embedding V 6 b (by norm_num) (by simpa using hb)\n\nend Problem346\n\nend\n\nend OAI\n\nimplementation scan [('lean/OAI/Analysis/FactorGeneration/Main.lean', 0, ['OAI.Analysis.FactorGeneration.Generation', 'OAI.Analysis.FactorGeneration.PredualContinuity', 'OAI.Analysis.FactorGeneration.TraceTransport']), ('lean/OAI/Analysis/TriangularHilbert/Main.lean', 0, ['OAI.Analysis.TriangularHilbert.TruncationCoordinates']), ('lean/OAI/Combinatorics/GotsmanLinial/Main.lean', 0, ['OAI.Combinatorics.GotsmanLinial.Statement', 'OAI.Combinatorics.GotsmanLinial.BlockCoupling', 'OAI.Combinatorics.GotsmanLinial.Commutator', 'OAI.Combinatorics.GotsmanLinial.CubeOperators', 'OAI.Combinatorics.GotsmanLinial.EdgeRecovery', 'OAI.Combinatorics.GotsmanLinial.GradingShift', 'OAI.Combinatorics.GotsmanLinial.BlockVanishing', 'Mathlib.Tactic.Ring']), ('lean/OAI/Combinatorics/PerfectMatching/Main.lean', 0, ['OAI.Combinatorics.PerfectMatching.MatchingCovariance']), ('lean/OAI/Computability/LoopMatching/Main.lean', 0, ['OAI.Computability.LoopMatching.MachineMain', 'OAI.Computability.LoopMatching.ExpandCount', 'OAI.Computability.LoopMatching.SparseBoundary']), ('lean/OAI/Computability/MatchingCount/BinarySolve.lean', 0, ['OAI.Computability.MatchingCount.RawInput']), ('lean/OAI/MeasureTheory/SelfSimilar/Main.lean', 0, ['OAI.MeasureTheory.SelfSimilar.EntropyGap']), ('lean/OAI/Probability/ClassicalON/Main.lean', 0, ['OAI.Probability.ClassicalON.BlockReduction']), ('lean/OAI/RepresentationTheory/FoulkesHowe/Stabilization.lean', 0, ['OAI.RepresentationTheory.FoulkesHowe.Model', 'OAI.RepresentationTheory.FoulkesHowe.LinearCase', 'OAI.RepresentationTheory.FoulkesHowe.PairingConstruction', 'OAI.RepresentationTheory.FoulkesHowe.Surjectivity', 'OAI.RepresentationTheory.FoulkesHowe.ProductVanishing'])]\n",
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          "\nIMPLEMENTATION lean/OAI/Combinatorics/GotsmanLinial/Main.lean \n import OAI.Combinatorics.GotsmanLinial.Statement\nimport OAI.Combinatorics.GotsmanLinial.BlockCoupling\nimport OAI.Combinatorics.GotsmanLinial.Commutator\nimport OAI.Combinatorics.GotsmanLinial.CubeOperators\nimport OAI.Combinatorics.GotsmanLinial.EdgeRe \nFINAL\n : ℝ) * Real.sqrt (n : ℝ) := by\n  let w : Cube n → ℝ := fun x => |polynomialValue p x|\n  have hw (x : Cube n) : 0 < w x := abs_pos.mpr (hnz x)\n  apply sensitivity_bound_of_cube_grading (polynomialThreshold p)\n    (fun x => thresholdSign_cases (polynomialValue p x))\n    (weightedProjectionMatrix w hw)\n    (weightedProjectionMatrix_family w hw)\n    (weightedProjectionMatrix_trace_re w hw)\n  · intro i\n    exact weightedProjectionMatrix_tridiagonal w hw i\n  · intro r s hrs\n    exact weightedProjectionMatrix_block_eq_zero_of_polynomial_degree p hp hd hnz r s hrs\n\n/-- Every real multilinear polynomial of degree at most `d` has average\nsensitivity at most `8*d*sqrt(n)`, including cube zeros under `sign(0)=1`.\nThe argument in fact covers all natural `n` and `d`. -/\ntheorem polynomialThreshold_averageSensitivity_le {n d : ℕ}\n    (p : MvPolynomial (Fin n) ℝ) (hp : IsMultilinear p) (hd : p.totalDegree ≤ d) :\n    averageSensitivity (polynomialThreshold p) ≤ 8 * (d : ℝ) * Real.sqrt (n : ℝ) := by\n  obtain ⟨q, hq, hqd, hqnz, hqf⟩ := exists_nonzero_polynomial_same_threshold hp hd\n  rw [← hqf]\n  exact polynomialThreshold_averageSensitivity_le_of_nonzero q hq hqd hqnz\n\n/-- The average-sensitivity bound for `1 ≤ d ≤ n`, without auxiliary\nconstruction hypotheses. -/\ntheorem gotsmanLinialStatement : GotsmanLinialStatement := by\n  intro n d _ _ _ p hp hd\n  exact polynomialThreshold_averageSensitivity_le p hp hd\n\nend LeanBlast.GotsmanLinial\n\nend OAI\n\n\nIMPLEMENTATION lean/OAI/MeasureTheory/SelfSimilar/Main.lean \n import OAI.MeasureTheory.SelfSimilar.EntropyGap\n\nnamespace OAI\n\nuniverse uι\n\nnamespace EntropyRateDimension\nopen MeasureTheory MeasureEntropy\nopen scoped BigOperators ENNReal\n\ntheorem exists_slightly_larger_depth {d χ h : ℝ} (hd : 0 ≤ d) (hgap : d*χ  \nFINAL\n nat_log_small hε (Fintype.card ι)\n  have hbR : (0 : ℝ)<b := by exact_mod_cast hb\n  have htype : Real.log (Fintype.card (ι → Fin (b+1)))=\n      (Fintype.card ι : ℝ)*Real.log ((b : ℝ)+1) := by\n    simp only [Fintype.card_fun,Fintype.card_fin,Nat.cast_pow,Real.log_pow,Nat.cast_add,Nat.cast_one]\n  have hblock : (b : ℝ)*S.entropyRate ≤ (S.blocks hb).entropyRate := S.blocks_entropyRate_lower hb\n  have hgap : d*((b : ℝ)*a)*Real.log 2 < (S.blocks hb).entropyRate*Real.log 2-\n      Real.log (Fintype.card (ι → Fin (b+1))) := by\n    rw [htype]\n    have hh := mul_le_mul_of_nonneg_right hblock hl.le\n    nlinarith\n  obtain ⟨C,hC⟩ := S.selfSimilar_entropy_bound μ hμ hed\n  apply Typed.no_strict_entropy_gap (S.blocks hb) (wordType (ι := ι)) (S.ratio_wordType hb)\n    μ (S.selfSimilar_blocks μ hμ hb) hd hd1 (mul_pos hbR hap)\n    (A' := (b : ℝ)*((S.lyapunov+a)/2)) _ _ hC hgap\n  · rw [S.blocks_lyapunov]\n    exact mul_lt_mul_of_pos_left (by linarith) hbR\n  · exact mul_lt_mul_of_pos_left (by linarith) hbR\n\nend System\n\ntheorem entropy_rate_dimension {ι : Type uι} [Fintype ι] [Nonempty ι]\n    (S : System ι) (μ : Measure ℝ) [IsProbabilityMeasure μ]\n    (hμ : S.SelfSimilar μ) :\n    lowerHausdorffDimension μ = ENNReal.ofReal (min 1 (S.entropyRate / S.lyapunov)) := by\n  rw [S.selfSimilar_dimension μ hμ]\n  congr 1\n  exact le_antisymm (le_min (S.exponent_le_one μ hμ) (S.exponent_le_rate μ hμ)) (S.exponent_lower_bound μ hμ)\n\nend EntropyRateDimension\n\nend OAI\n\n\nIMPLEMENTATION lean/OAI/Analysis/TriangularHilbert/Main.lean \n import OAI.Analysis.TriangularHilbert.TruncationCoordinates\n\nnamespace OAI\n\nsection\nnamespace TriangularHilbert\nopen MeasureTheory Heat Filter\nopen scoped ENNReal\nnoncomputable section\n\nlemma maximal_eq_cyclicMax_ae (F G : Plane → ℂ) (hF : MemLp F 3  \nFINAL\n q_left hz.goodPoint]\n  simp_rw [truncation_eq_cyclicHard]\n  exact (cyclicMax_eq_iSup hz).symm\n\nlemma aemeasurable_maximal (F G : Plane → ℂ) (hF : MemLp F 3 volume) (hG : MemLp G 3 volume) :\n    AEMeasurable (maximal F G) volume := by\n  have h := (aemeasurable_cyclicMax (cyclicB F) (cyclicC G) (memLp_cyclicB hF) (memLp_cyclicC hG)).comp_quasiMeasurePreserving\n    measurePreserving_reflect.quasiMeasurePreserving\n  exact h.congr (maximal_eq_cyclicMax_ae F G hF hG).symm\n\nlemma maximalNorm_eq_cyclicNorm (F G : Plane → ℂ) (hF : MemLp F 3 volume) (hG : MemLp G 3 volume) :\n    maximalNorm F G = cyclicNorm (cyclicB F) (cyclicC G) := by\n  unfold maximalNorm cyclicNorm\n  congr 1\n  calc\n    (∫⁻ z : Plane, maximal F G z ^ ((3:ℝ)/2)) =\n        ∫⁻ z : Plane, cyclicMax (cyclicB F) (cyclicC G) (-z)^((3:ℝ)/2) := by\n      apply lintegral_congr_ae\n      filter_upwards [maximal_eq_cyclicMax_ae F G hF hG] with z hz\n      rw [hz]\n    _ = _ := measurePreserving_reflect.lintegral_comp_emb (Homeomorph.neg Plane).measurableEmbedding\n      (fun z => cyclicMax (cyclicB F) (cyclicC G) z^((3:ℝ)/2))\n\ntheorem main_estimate : MainEstimate := by\n  refine ⟨mainConstant,fun F G hF hG => ⟨ae_goodPoint F G hF hG,aemeasurable_maximal F G hF hG,?_⟩⟩\n  rw [maximalNorm_eq_cyclicNorm F G hF hG]\n  simpa only [eLpNorm_cyclicB hF,eLpNorm_cyclicC hG] using\n    cyclicNorm_le (cyclicB F) (cyclicC G) (memLp_cyclicB hF) (memLp_cyclicC hG)\nend\nend TriangularHilbert\nend\n\nend OAI\n\n\nIMPLEMENTATION lean/OAI/Probability/ClassicalON/Main.lean \n import OAI.Probability.ClassicalON.BlockReduction\n\nnamespace OAI\n\nnoncomputable section\nopen MeasureTheory\nopen scoped InnerProductSpace BigOperators Classical\nnamespace ClassicalON\n\ntheorem correlation_eq_freeSpinMean (n : ℕ) (G : LatticeGraph) (b : \nFINAL\n ference\n  congr 1\n  apply integral_congr_ae\n  filter_upwards with s\n  exact mul_comm _ _\n\ntheorem main : ExponentialDecay := by\n  intro n hn β hβ\n  obtain ⟨k,rfl⟩ := Nat.exists_eq_add_of_le hn\n  let : NeZero (3+k) := ⟨by omega⟩\n  obtain ⟨m,hm,hexp⟩ := LatticeGraph.connection_exponential β hβ.le\n  refine ⟨(3+k:ℕ)*4,m,hm,?_⟩\n  intro G b hb x y\n  have hthree (c : G.edges → ℝ) (hc : ∀ e,0 ≤ c e ∧ c e ≤ b e) :\n      0 ≤ freeSpinMean 3 (fun e : G.edges => e.val.1) (fun e => e.val.2) c (firstSpinProduct x y) ∧\n      freeSpinMean 3 (fun e : G.edges => e.val.1) (fun e => e.val.2) c (firstSpinProduct x y) ≤\n        4*Real.exp (-m*siteDistance x.val y.val) := by\n    have hfirst := firstSpinMean_bounds (fun e : G.edges => e.val.1) (fun e => e.val.2) c\n      (fun e => (hc e).1) x y\n    exact ⟨hfirst.1,hfirst.2.trans (hexp G c (fun e => ⟨(hc e).1,(hc e).2.trans (hb e).2⟩) x y)⟩\n  have hbound := coordinateSpinMean_block_bounds k (fun e : G.edges => e.val.1) (fun e => e.val.2) b\n    (fun e => (hb e).1) x y (4*Real.exp (-m*siteDistance x.val y.val)) hthree\n  rw [correlation_eq_freeSpinMean,innerSpinMean_eq_coordinate (3+k)\n    (fun e : G.edges => e.val.1) (fun e => e.val.2) b ⟨0,by omega⟩ x y]\n  constructor\n  · exact mul_nonneg (Nat.cast_nonneg _) hbound.1\n  · calc\n      _ ≤ (3+k:ℕ)*(4*Real.exp (-m*siteDistance x.val y.val)) :=\n        mul_le_mul_of_nonneg_left hbound.2 (Nat.cast_nonneg _)\n      _ = _ := by ring\n\nend ClassicalON\n\nend\n\nend OAI\n\n\nIMPLEMENTATION lean/OAI/RepresentationTheory/FoulkesHowe/Stabilization.lean \n import OAI.RepresentationTheory.FoulkesHowe.Model\nimport OAI.RepresentationTheory.FoulkesHowe.LinearCase\nimport OAI.RepresentationTheory.FoulkesHowe.PairingConstruction\nimport OAI.RepresentationTheory.FoulkesHowe.Surjectivity\nimport OAI.Representatio \nFINAL\n ymPow b (SymPow 1 V) →ₗ[ℂ] SymPow 1 (SymPow b V), IsFoulkesMap 1 b V μ ∧ Function.Bijective μ ∧ ∀ ν : SymPow b (SymPow 1 V) →ₗ[ℂ] SymPow 1 (SymPow b V), IsFoulkesMap 1 b V ν → ν = μ := by\n  intro V _ _ _ b\n  exact canonical_foulkes_howe_a_one_aux V b\n\ntheorem vanishing_on_products :\n    ∀ (V : Type u) [AddCommGroup V] [Module ℂ V] [FiniteDimensional ℂ V] (a b : ℕ), 1 ≤ a → 1 ≤ b → a * (a - 1) ≤ b → ∀ T : SymmetricMultilinearForm a b V, IsSymmetricMultilinearForm a b V T → (∀ u : Fin b → V, T (fun _ => symMonomial b V u) = 0) → T = 0 := by\n  intro V _ _ _ a b ha hb hab T hT hdiag\n  exact vanishing_on_products_aux V a b ha hb hab T hT hdiag\n\ntheorem foulkes_comparison_embedding :\n    ∀ (V : Type u) [AddCommGroup V] [Module ℂ V] [FiniteDimensional ℂ V] (a b : ℕ), 2 ≤ a → a * (a - 1) ≤ b → ∃ ι : SymPow a (SymPow b V) →ₗ[ℂ] SymPow b (SymPow a V), IsGLEquivariantEmbedding a b V ι := by\n  intro V _ _ _ a b ha hab\n  obtain ⟨μ, hμ, hsurj, _⟩ :=\n    canonical_foulkes_howe_surjective (Module.Dual ℂ V) a b ha hab\n  exact PairingConstruction.comparison_of_surjection V a b ⟨μ, hμ, hsurj⟩\n\ntheorem sixth_symmetric_power_comparison :\n    ∀ (V : Type u) [AddCommGroup V] [Module ℂ V] [FiniteDimensional ℂ V] (b : ℕ), 30 ≤ b → ∃ ι : SymPow 6 (SymPow b V) →ₗ[ℂ] SymPow b (SymPow 6 V), IsGLEquivariantEmbedding 6 b V ι := by\n  intro V _ _ _ b hb\n  exact foulkes_comparison_embedding V 6 b (by norm_num) (by simpa using hb)\n\nend Problem346\n\nend\n\nend OAI\n\nimplementation scan [('lean/OAI/Analysis/FactorGeneration/Main.lean', 0, ['OAI.Analysis.FactorGeneration.Generation', 'OAI.Analysis.FactorGeneration.PredualContinuity', 'OAI.Analysis.FactorGeneration.TraceTransport']), ('lean/OAI/Analysis/TriangularHilbert/Main.lean', 0, ['OAI.Analysis.TriangularHilbert.TruncationCoordinates']), ('lean/OAI/Combinatorics/GotsmanLinial/Main.lean', 0, ['OAI.Combinatorics.GotsmanLinial.Statement', 'OAI.Combinatorics.GotsmanLinial.BlockCoupling', 'OAI.Combinatorics.GotsmanLinial.Commutator', 'OAI.Combinatorics.GotsmanLinial.CubeOperators', 'OAI.Combinatorics.GotsmanLinial.EdgeRecovery', 'OAI.Combinatorics.GotsmanLinial.GradingShift', 'OAI.Combinatorics.GotsmanLinial.BlockVanishing', 'Mathlib.Tactic.Ring']), ('lean/OAI/Combinatorics/PerfectMatching/Main.lean', 0, ['OAI.Combinatorics.PerfectMatching.MatchingCovariance']), ('lean/OAI/Computability/LoopMatching/Main.lean', 0, ['OAI.Computability.LoopMatching.MachineMain', 'OAI.Computability.LoopMatching.ExpandCount', 'OAI.Computability.LoopMatching.SparseBoundary']), ('lean/OAI/Computability/MatchingCount/BinarySolve.lean', 0, ['OAI.Computability.MatchingCount.RawInput']), ('lean/OAI/MeasureTheory/SelfSimilar/Main.lean', 0, ['OAI.MeasureTheory.SelfSimilar.EntropyGap']), ('lean/OAI/Probability/ClassicalON/Main.lean', 0, ['OAI.Probability.ClassicalON.BlockReduction']), ('lean/OAI/RepresentationTheory/FoulkesHowe/Stabilization.lean', 0, ['OAI.RepresentationTheory.FoulkesHowe.Model', 'OAI.RepresentationTheory.FoulkesHowe.LinearCase', 'OAI.RepresentationTheory.FoulkesHowe.PairingConstruction', 'OAI.RepresentationTheory.FoulkesHowe.Surjectivity', 'OAI.RepresentationTheory.FoulkesHowe.ProductVanishing'])]\n"
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          "text": "\nIMPLEMENTATION lean/OAI/Combinatorics/GotsmanLinial/Main.lean \n import OAI.Combinatorics.GotsmanLinial.Statement\nimport OAI.Combinatorics.GotsmanLinial.BlockCoupling\nimport OAI.Combinatorics.GotsmanLinial.Commutator\nimport OAI.Combinatorics.GotsmanLinial.CubeOperators\nimport OAI.Combinatorics.GotsmanLinial.EdgeRe \nFINAL\n : ℝ) * Real.sqrt (n : ℝ) := by\n  let w : Cube n → ℝ := fun x => |polynomialValue p x|\n  have hw (x : Cube n) : 0 < w x := abs_pos.mpr (hnz x)\n  apply sensitivity_bound_of_cube_grading (polynomialThreshold p)\n    (fun x => thresholdSign_cases (polynomialValue p x))\n    (weightedProjectionMatrix w hw)\n    (weightedProjectionMatrix_family w hw)\n    (weightedProjectionMatrix_trace_re w hw)\n  · intro i\n    exact weightedProjectionMatrix_tridiagonal w hw i\n  · intro r s hrs\n    exact weightedProjectionMatrix_block_eq_zero_of_polynomial_degree p hp hd hnz r s hrs\n\n/-- Every real multilinear polynomial of degree at most `d` has average\nsensitivity at most `8*d*sqrt(n)`, including cube zeros under `sign(0)=1`.\nThe argument in fact covers all natural `n` and `d`. -/\ntheorem polynomialThreshold_averageSensitivity_le {n d : ℕ}\n    (p : MvPolynomial (Fin n) ℝ) (hp : IsMultilinear p) (hd : p.totalDegree ≤ d) :\n    averageSensitivity (polynomialThreshold p) ≤ 8 * (d : ℝ) * Real.sqrt (n : ℝ) := by\n  obtain ⟨q, hq, hqd, hqnz, hqf⟩ := exists_nonzero_polynomial_same_threshold hp hd\n  rw [← hqf]\n  exact polynomialThreshold_averageSensitivity_le_of_nonzero q hq hqd hqnz\n\n/-- The average-sensitivity bound for `1 ≤ d ≤ n`, without auxiliary\nconstruction hypotheses. -/\ntheorem gotsmanLinialStatement : GotsmanLinialStatement := by\n  intro n d _ _ _ p hp hd\n  exact polynomialThreshold_averageSensitivity_le p hp hd\n\nend LeanBlast.GotsmanLinial\n\nend OAI\n\n\nIMPLEMENTATION lean/OAI/MeasureTheory/SelfSimilar/Main.lean \n import OAI.MeasureTheory.SelfSimilar.EntropyGap\n\nnamespace OAI\n\nuniverse uι\n\nnamespace EntropyRateDimension\nopen MeasureTheory MeasureEntropy\nopen scoped BigOperators ENNReal\n\ntheorem exists_slightly_larger_depth {d χ h : ℝ} (hd : 0 ≤ d) (hgap : d*χ  \nFINAL\n nat_log_small hε (Fintype.card ι)\n  have hbR : (0 : ℝ)<b := by exact_mod_cast hb\n  have htype : Real.log (Fintype.card (ι → Fin (b+1)))=\n      (Fintype.card ι : ℝ)*Real.log ((b : ℝ)+1) := by\n    simp only [Fintype.card_fun,Fintype.card_fin,Nat.cast_pow,Real.log_pow,Nat.cast_add,Nat.cast_one]\n  have hblock : (b : ℝ)*S.entropyRate ≤ (S.blocks hb).entropyRate := S.blocks_entropyRate_lower hb\n  have hgap : d*((b : ℝ)*a)*Real.log 2 < (S.blocks hb).entropyRate*Real.log 2-\n      Real.log (Fintype.card (ι → Fin (b+1))) := by\n    rw [htype]\n    have hh := mul_le_mul_of_nonneg_right hblock hl.le\n    nlinarith\n  obtain ⟨C,hC⟩ := S.selfSimilar_entropy_bound μ hμ hed\n  apply Typed.no_strict_entropy_gap (S.blocks hb) (wordType (ι := ι)) (S.ratio_wordType hb)\n    μ (S.selfSimilar_blocks μ hμ hb) hd hd1 (mul_pos hbR hap)\n    (A' := (b : ℝ)*((S.lyapunov+a)/2)) _ _ hC hgap\n  · rw [S.blocks_lyapunov]\n    exact mul_lt_mul_of_pos_left (by linarith) hbR\n  · exact mul_lt_mul_of_pos_left (by linarith) hbR\n\nend System\n\ntheorem entropy_rate_dimension {ι : Type uι} [Fintype ι] [Nonempty ι]\n    (S : System ι) (μ : Measure ℝ) [IsProbabilityMeasure μ]\n    (hμ : S.SelfSimilar μ) :\n    lowerHausdorffDimension μ = ENNReal.ofReal (min 1 (S.entropyRate / S.lyapunov)) := by\n  rw [S.selfSimilar_dimension μ hμ]\n  congr 1\n  exact le_antisymm (le_min (S.exponent_le_one μ hμ) (S.exponent_le_rate μ hμ)) (S.exponent_lower_bound μ hμ)\n\nend EntropyRateDimension\n\nend OAI\n\n\nIMPLEMENTATION lean/OAI/Analysis/TriangularHilbert/Main.lean \n import OAI.Analysis.TriangularHilbert.TruncationCoordinates\n\nnamespace OAI\n\nsection\nnamespace TriangularHilbert\nopen MeasureTheory Heat Filter\nopen scoped ENNReal\nnoncomputable section\n\nlemma maximal_eq_cyclicMax_ae (F G : Plane → ℂ) (hF : MemLp F 3  \nFINAL\n q_left hz.goodPoint]\n  simp_rw [truncation_eq_cyclicHard]\n  exact (cyclicMax_eq_iSup hz).symm\n\nlemma aemeasurable_maximal (F G : Plane → ℂ) (hF : MemLp F 3 volume) (hG : MemLp G 3 volume) :\n    AEMeasurable (maximal F G) volume := by\n  have h := (aemeasurable_cyclicMax (cyclicB F) (cyclicC G) (memLp_cyclicB hF) (memLp_cyclicC hG)).comp_quasiMeasurePreserving\n    measurePreserving_reflect.quasiMeasurePreserving\n  exact h.congr (maximal_eq_cyclicMax_ae F G hF hG).symm\n\nlemma maximalNorm_eq_cyclicNorm (F G : Plane → ℂ) (hF : MemLp F 3 volume) (hG : MemLp G 3 volume) :\n    maximalNorm F G = cyclicNorm (cyclicB F) (cyclicC G) := by\n  unfold maximalNorm cyclicNorm\n  congr 1\n  calc\n    (∫⁻ z : Plane, maximal F G z ^ ((3:ℝ)/2)) =\n        ∫⁻ z : Plane, cyclicMax (cyclicB F) (cyclicC G) (-z)^((3:ℝ)/2) := by\n      apply lintegral_congr_ae\n      filter_upwards [maximal_eq_cyclicMax_ae F G hF hG] with z hz\n      rw [hz]\n    _ = _ := measurePreserving_reflect.lintegral_comp_emb (Homeomorph.neg Plane).measurableEmbedding\n      (fun z => cyclicMax (cyclicB F) (cyclicC G) z^((3:ℝ)/2))\n\ntheorem main_estimate : MainEstimate := by\n  refine ⟨mainConstant,fun F G hF hG => ⟨ae_goodPoint F G hF hG,aemeasurable_maximal F G hF hG,?_⟩⟩\n  rw [maximalNorm_eq_cyclicNorm F G hF hG]\n  simpa only [eLpNorm_cyclicB hF,eLpNorm_cyclicC hG] using\n    cyclicNorm_le (cyclicB F) (cyclicC G) (memLp_cyclicB hF) (memLp_cyclicC hG)\nend\nend TriangularHilbert\nend\n\nend OAI\n\n\nIMPLEMENTATION lean/OAI/Probability/ClassicalON/Main.lean \n import OAI.Probability.ClassicalON.BlockReduction\n\nnamespace OAI\n\nnoncomputable section\nopen MeasureTheory\nopen scoped InnerProductSpace BigOperators Classical\nnamespace ClassicalON\n\ntheorem correlation_eq_freeSpinMean (n : ℕ) (G : LatticeGraph) (b : \nFINAL\n ference\n  congr 1\n  apply integral_congr_ae\n  filter_upwards with s\n  exact mul_comm _ _\n\ntheorem main : ExponentialDecay := by\n  intro n hn β hβ\n  obtain ⟨k,rfl⟩ := Nat.exists_eq_add_of_le hn\n  let : NeZero (3+k) := ⟨by omega⟩\n  obtain ⟨m,hm,hexp⟩ := LatticeGraph.connection_exponential β hβ.le\n  refine ⟨(3+k:ℕ)*4,m,hm,?_⟩\n  intro G b hb x y\n  have hthree (c : G.edges → ℝ) (hc : ∀ e,0 ≤ c e ∧ c e ≤ b e) :\n      0 ≤ freeSpinMean 3 (fun e : G.edges => e.val.1) (fun e => e.val.2) c (firstSpinProduct x y) ∧\n      freeSpinMean 3 (fun e : G.edges => e.val.1) (fun e => e.val.2) c (firstSpinProduct x y) ≤\n        4*Real.exp (-m*siteDistance x.val y.val) := by\n    have hfirst := firstSpinMean_bounds (fun e : G.edges => e.val.1) (fun e => e.val.2) c\n      (fun e => (hc e).1) x y\n    exact ⟨hfirst.1,hfirst.2.trans (hexp G c (fun e => ⟨(hc e).1,(hc e).2.trans (hb e).2⟩) x y)⟩\n  have hbound := coordinateSpinMean_block_bounds k (fun e : G.edges => e.val.1) (fun e => e.val.2) b\n    (fun e => (hb e).1) x y (4*Real.exp (-m*siteDistance x.val y.val)) hthree\n  rw [correlation_eq_freeSpinMean,innerSpinMean_eq_coordinate (3+k)\n    (fun e : G.edges => e.val.1) (fun e => e.val.2) b ⟨0,by omega⟩ x y]\n  constructor\n  · exact mul_nonneg (Nat.cast_nonneg _) hbound.1\n  · calc\n      _ ≤ (3+k:ℕ)*(4*Real.exp (-m*siteDistance x.val y.val)) :=\n        mul_le_mul_of_nonneg_left hbound.2 (Nat.cast_nonneg _)\n      _ = _ := by ring\n\nend ClassicalON\n\nend\n\nend OAI\n\n\nIMPLEMENTATION lean/OAI/RepresentationTheory/FoulkesHowe/Stabilization.lean \n import OAI.RepresentationTheory.FoulkesHowe.Model\nimport OAI.RepresentationTheory.FoulkesHowe.LinearCase\nimport OAI.RepresentationTheory.FoulkesHowe.PairingConstruction\nimport OAI.RepresentationTheory.FoulkesHowe.Surjectivity\nimport OAI.Representatio \nFINAL\n ymPow b (SymPow 1 V) →ₗ[ℂ] SymPow 1 (SymPow b V), IsFoulkesMap 1 b V μ ∧ Function.Bijective μ ∧ ∀ ν : SymPow b (SymPow 1 V) →ₗ[ℂ] SymPow 1 (SymPow b V), IsFoulkesMap 1 b V ν → ν = μ := by\n  intro V _ _ _ b\n  exact canonical_foulkes_howe_a_one_aux V b\n\ntheorem vanishing_on_products :\n    ∀ (V : Type u) [AddCommGroup V] [Module ℂ V] [FiniteDimensional ℂ V] (a b : ℕ), 1 ≤ a → 1 ≤ b → a * (a - 1) ≤ b → ∀ T : SymmetricMultilinearForm a b V, IsSymmetricMultilinearForm a b V T → (∀ u : Fin b → V, T (fun _ => symMonomial b V u) = 0) → T = 0 := by\n  intro V _ _ _ a b ha hb hab T hT hdiag\n  exact vanishing_on_products_aux V a b ha hb hab T hT hdiag\n\ntheorem foulkes_comparison_embedding :\n    ∀ (V : Type u) [AddCommGroup V] [Module ℂ V] [FiniteDimensional ℂ V] (a b : ℕ), 2 ≤ a → a * (a - 1) ≤ b → ∃ ι : SymPow a (SymPow b V) →ₗ[ℂ] SymPow b (SymPow a V), IsGLEquivariantEmbedding a b V ι := by\n  intro V _ _ _ a b ha hab\n  obtain ⟨μ, hμ, hsurj, _⟩ :=\n    canonical_foulkes_howe_surjective (Module.Dual ℂ V) a b ha hab\n  exact PairingConstruction.comparison_of_surjection V a b ⟨μ, hμ, hsurj⟩\n\ntheorem sixth_symmetric_power_comparison :\n    ∀ (V : Type u) [AddCommGroup V] [Module ℂ V] [FiniteDimensional ℂ V] (b : ℕ), 30 ≤ b → ∃ ι : SymPow 6 (SymPow b V) →ₗ[ℂ] SymPow b (SymPow 6 V), IsGLEquivariantEmbedding 6 b V ι := by\n  intro V _ _ _ b hb\n  exact foulkes_comparison_embedding V 6 b (by norm_num) (by simpa using hb)\n\nend Problem346\n\nend\n\nend OAI\n\nimplementation scan [('lean/OAI/Analysis/FactorGeneration/Main.lean', 0, ['OAI.Analysis.FactorGeneration.Generation', 'OAI.Analysis.FactorGeneration.PredualContinuity', 'OAI.Analysis.FactorGeneration.TraceTransport']), ('lean/OAI/Analysis/TriangularHilbert/Main.lean', 0, ['OAI.Analysis.TriangularHilbert.TruncationCoordinates']), ('lean/OAI/Combinatorics/GotsmanLinial/Main.lean', 0, ['OAI.Combinatorics.GotsmanLinial.Statement', 'OAI.Combinatorics.GotsmanLinial.BlockCoupling', 'OAI.Combinatorics.GotsmanLinial.Commutator', 'OAI.Combinatorics.GotsmanLinial.CubeOperators', 'OAI.Combinatorics.GotsmanLinial.EdgeRecovery', 'OAI.Combinatorics.GotsmanLinial.GradingShift', 'OAI.Combinatorics.GotsmanLinial.BlockVanishing', 'Mathlib.Tactic.Ring']), ('lean/OAI/Combinatorics/PerfectMatching/Main.lean', 0, 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        "stdout": "implementation scan [('lean/OAI/Analysis/FactorGeneration/Main.lean', 0), ('lean/OAI/Analysis/TriangularHilbert/Main.lean', 0), ('lean/OAI/Combinatorics/GotsmanLinial/Main.lean', 0), ('lean/OAI/Combinatorics/PerfectMatching/Main.lean', 0), ('lean/OAI/Computability/LoopMatching/Main.lean', 0), ('lean/OAI/Computability/MatchingCount/BinarySolve.lean', 0), ('lean/OAI/MeasureTheory/SelfSimilar/Main.lean', 0), ('lean/OAI/Probability/ClassicalON/Main.lean', 0), ('lean/OAI/RepresentationTheory/FoulkesHowe/Stabilization.lean', 0)]\nmissing OAI imports 23\nreadme build excerpts\n015 []\n024 []\n046 []\n065 []\n082 []\n113 []\n115 []\n127 []\n148 []\n154 []\n164 []\n192 []\n210 ['python3 -B verification/verify_computations.py --check\\n', 'python3 -B verification/verify_computations.py --run --serial\\n']\n215 []\n230 []\n245 []\n296 []\n351 []\n369 []\n372 []\n",
        "stderr": "",
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        "stdout": "\nREADME 015 \n # [Equidistribution of Prime-Degree Torus Packets with Arbitrary Local Type](paper.pdf)\n\n**Author:** OpenAI\n\n**Date:** September 24, 2026\n\n## Citation\n\n```bibtex\n@misc{OAI:Equidistribution-of-Prime-Degree-Torus-Packets-with-Arbitrary-Local-Type-September-24-2026,\n  author = {{OpenAI}},\n  title = {{Equidistribution of Prime-Degree Torus Packets with Arbitrary Local Type}},\n  howpublished = {OpenAI Math Release preprint\n                  \\href{https://github.com/openai/math/blob/main/preprints/Equidistribution-of-Prime-Degree-Torus-Packets-with-Arbitrary-Local-Type-September-24-2026/paper.pdf}{OAI:Equidistribution-of-Prime-Degree-Torus-Packets-with-Arbitrary-Local-Type-September-24-2026}},\n  year = {2026}\n}\n```\n\n\n\nREADME 024 \n # [An asymptotic formula for the number of totients](An-asymptotic-formula-for-the-number-of-totients-September-25-2026.pdf)\n\n**Author:** OpenAI\n\n**Date:** September 25, 2026\n\n## Citation\n\n```bibtex\n@misc{OAI:An-asymptotic-formula-for-the-number-of-totients-September-25-2026,\n  author = {{OpenAI}},\n  title = {{An asymptotic formula for the number of totients}},\n  howpublished = {OpenAI Math Release preprint\n                  \\href{https://github.com/openai/math/blob/main/preprints/An-asymptotic-formula-for-the-number-of-totients-September-25-2026/An-asymptotic-formula-for-the-number-of-totients-September-25-2026.pdf}{OAI:An-asymptotic-formula-for-the-number-of-totients-September-25-2026}},\n  year = {2026}\n}\n```\n\n\n\nREADME 046 \n # [A projective fourfold with large fundamental group and non-Stein universal cover](large-fundamental-group-non-stein.pdf)\n\n**Author:** OpenAI\n\n**Date:** October 5, 2026\n\n## Citation\n\n```bibtex\n@misc{OAI:A-projective-fourfold-with-large-fundamental-group-and-non-Stein-universal-cover-October-5-2026,\n  author = {{OpenAI}},\n  title = {{A projective fourfold with large fundamental group and non-Stein universal cover}},\n  howpublished = {OpenAI Math Release preprint\n                  \\href{https://github.com/openai/math/blob/main/preprints/A-projective-fourfold-with-large-fundamental-group-and-non-Stein-universal-cover-October-5-2026/large-fundamental-group-non-stein.pdf}{OAI:A-projective-fourfold-with-large-fundamental-group-and-non-Stein-universal-cover-October-5-2026}},\n  year = {2026}\n}\n```\n\n\n\nREADME 065 \n # [Virasoro Constraints under Projectivization](virasoro-constraints-under-projectivization.pdf)\n\n**Author:** OpenAI\n\n**Date:** October 5, 2026\n\n## Citation\n\n```bibtex\n@misc{OAI:Virasoro-Constraints-under-Projectivization-October-5-2026,\n  author = {{OpenAI}},\n  title = {{Virasoro Constraints under Projectivization}},\n  howpublished = {OpenAI Math Release preprint\n                  \\href{https://github.com/openai/math/blob/main/preprints/Virasoro-Constraints-under-Projectivization-October-5-2026/virasoro-constraints-under-projectivization.pdf}{OAI:Virasoro-Constraints-under-Projectivization-October-5-2026}},\n  year = {2026}\n}\n```\n\n\n\nREADME 082 \n # [Annular variation of the triangular Hilbert transform at the symmetric point](annular-variation.pdf)\n\n**Author:** OpenAI\n\n**Date:** October 5, 2026\n\n## Citation\n\n```bibtex\n@misc{OAI:Annular-variation-of-the-triangular-Hilbert-transform-at-the-symmetric-point-October-5-2026,\n  author = {{OpenAI}},\n  title = {{Annular variation of the triangular Hilbert transform at the symmetric point}},\n  howpublished = {OpenAI Math Release preprint\n                  \\href{https://github.com/openai/math/blob/main/preprints/Annular-variation-of-the-triangular-Hilbert-transform-at-the-symmetric-point-October-5-2026/annular-variation.pdf}{OAI:Annular-variation-of-the-triangular-Hilbert-transform-at-the-symmetric-point-October-5-2026}},\n  year = {2026}\n}\n```\n\n\n\nREADME 210\n # [Foulkes' conjecture for the sixth symmetric power](main.pdf)\n\n**Author:** OpenAI\n\n**Date:** September 25, 2026\n\n## Citation\n\n```bibtex\n@misc{OAI:Foulkes-Conjecture-for-the-Sixth-Symmetric-Power-September-25-2026,\n  author = {{OpenAI}},\n  title = {{Foulkes' conjecture for the sixth symmetric power}},\n  howpublished = {OpenAI Math Release preprint\n                  \\href{https://github.com/openai/math/blob/main/preprints/Foulkes-Conjecture-for-the-Sixth-Symmetric-Power-September-25-2026/main.pdf}{OAI:Foulkes-Conjecture-for-the-Sixth-Symmetric-Power-September-25-2026}},\n  year = {2026}\n}\n```\n\n## Verification\n\nRequires Python 3.9 or later. Run the supplied read-only checks from this paper\ndirectory:\n\n```sh\npython3 -B verification/verify_computations.py --check\n```\n\nSuccess prints `PASS` and confirms the bundled source and reference-stream\nidentities, all 62 expected numeric rows, agreement with the manuscript tables,\nand the stated numerical bounds. This command does not execute the C++ programs.\n\nTo compile and run both complete exact computations, GNU C++ with signed\n128-bit integers and Boost headers is also required:\n\n```sh\npython3 -B verification/verify_computations.py --run --serial\n```\n\nAlternatively, replace `--serial` with `--threads 2` to use OpenMP for Appendix A.\nAssertions remain enabled. A successful full run prints `PASS` after both\nprograms exit successfully and their complete output streams match the reference\nbytes. It records compiler and execution details in a new temporary directory\noutside the paper; `--output PATH` selects a new external directory instead.\n\nall README lengths [('015', 720), ('024', 722), ('046', 804), ('065', 639), ('082', 752), ('113', 808), ('115', 719), ('127', 622), ('148', 718), ('154', 698), ('164', 675), ('192', 749), ('210', 1605), ('215', 727), ('230', 687), ('245', 610), ('296', 680), ('351', 749), ('369', 848), ('372', 709)]\n",
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          "\nREADME 015 \n # [Equidistribution of Prime-Degree Torus Packets with Arbitrary Local Type](paper.pdf)\n\n**Author:** OpenAI\n\n**Date:** September 24, 2026\n\n## Citation\n\n```bibtex\n@misc{OAI:Equidistribution-of-Prime-Degree-Torus-Packets-with-Arbitrary-Local-Type-September-24-2026,\n  author = {{OpenAI}},\n  title = {{Equidistribution of Prime-Degree Torus Packets with Arbitrary Local Type}},\n  howpublished = {OpenAI Math Release preprint\n                  \\href{https://github.com/openai/math/blob/main/preprints/Equidistribution-of-Prime-Degree-Torus-Packets-with-Arbitrary-Local-Type-September-24-2026/paper.pdf}{OAI:Equidistribution-of-Prime-Degree-Torus-Packets-with-Arbitrary-Local-Type-September-24-2026}},\n  year = {2026}\n}\n```\n\n\n\nREADME 024 \n # [An asymptotic formula for the number of totients](An-asymptotic-formula-for-the-number-of-totients-September-25-2026.pdf)\n\n**Author:** OpenAI\n\n**Date:** September 25, 2026\n\n## Citation\n\n```bibtex\n@misc{OAI:An-asymptotic-formula-for-the-number-of-totients-September-25-2026,\n  author = {{OpenAI}},\n  title = {{An asymptotic formula for the number of totients}},\n  howpublished = {OpenAI Math Release preprint\n                  \\href{https://github.com/openai/math/blob/main/preprints/An-asymptotic-formula-for-the-number-of-totients-September-25-2026/An-asymptotic-formula-for-the-number-of-totients-September-25-2026.pdf}{OAI:An-asymptotic-formula-for-the-number-of-totients-September-25-2026}},\n  year = {2026}\n}\n```\n\n\n\nREADME 046 \n # [A projective fourfold with large fundamental group and non-Stein universal cover](large-fundamental-group-non-stein.pdf)\n\n**Author:** OpenAI\n\n**Date:** October 5, 2026\n\n## Citation\n\n```bibtex\n@misc{OAI:A-projective-fourfold-with-large-fundamental-group-and-non-Stein-universal-cover-October-5-2026,\n  author = {{OpenAI}},\n  title = {{A projective fourfold with large fundamental group and non-Stein universal cover}},\n  howpublished = {OpenAI Math Release preprint\n                  \\href{https://github.com/openai/math/blob/main/preprints/A-projective-fourfold-with-large-fundamental-group-and-non-Stein-universal-cover-October-5-2026/large-fundamental-group-non-stein.pdf}{OAI:A-projective-fourfold-with-large-fundamental-group-and-non-Stein-universal-cover-October-5-2026}},\n  year = {2026}\n}\n```\n\n\n\nREADME 065 \n # [Virasoro Constraints under Projectivization](virasoro-constraints-under-projectivization.pdf)\n\n**Author:** OpenAI\n\n**Date:** October 5, 2026\n\n## Citation\n\n```bibtex\n@misc{OAI:Virasoro-Constraints-under-Projectivization-October-5-2026,\n  author = {{OpenAI}},\n  title = {{Virasoro Constraints under Projectivization}},\n  howpublished = {OpenAI Math Release preprint\n                  \\href{https://github.com/openai/math/blob/main/preprints/Virasoro-Constraints-under-Projectivization-October-5-2026/virasoro-constraints-under-projectivization.pdf}{OAI:Virasoro-Constraints-under-Projectivization-October-5-2026}},\n  year = {2026}\n}\n```\n\n\n\nREADME 082 \n # [Annular variation of the triangular Hilbert transform at the symmetric point](annular-variation.pdf)\n\n**Author:** OpenAI\n\n**Date:** October 5, 2026\n\n## Citation\n\n```bibtex\n@misc{OAI:Annular-variation-of-the-triangular-Hilbert-transform-at-the-symmetric-point-October-5-2026,\n  author = {{OpenAI}},\n  title = {{Annular variation of the triangular Hilbert transform at the symmetric point}},\n  howpublished = {OpenAI Math Release preprint\n                  \\href{https://github.com/openai/math/blob/main/preprints/Annular-variation-of-the-triangular-Hilbert-transform-at-the-symmetric-point-October-5-2026/annular-variation.pdf}{OAI:Annular-variation-of-the-triangular-Hilbert-transform-at-the-symmetric-point-October-5-2026}},\n  year = {2026}\n}\n```\n\n\n\nREADME 210\n # [Foulkes' conjecture for the sixth symmetric power](main.pdf)\n\n**Author:** OpenAI\n\n**Date:** September 25, 2026\n\n## Citation\n\n```bibtex\n@misc{OAI:Foulkes-Conjecture-for-the-Sixth-Symmetric-Power-September-25-2026,\n  author = {{OpenAI}},\n  title = {{Foulkes' conjecture for the sixth symmetric power}},\n  howpublished = {OpenAI Math Release preprint\n                  \\href{https://github.com/openai/math/blob/main/preprints/Foulkes-Conjecture-for-the-Sixth-Symmetric-Power-September-25-2026/main.pdf}{OAI:Foulkes-Conjecture-for-the-Sixth-Symmetric-Power-September-25-2026}},\n  year = {2026}\n}\n```\n\n## Verification\n\nRequires Python 3.9 or later. Run the supplied read-only checks from this paper\ndirectory:\n\n```sh\npython3 -B verification/verify_computations.py --check\n```\n\nSuccess prints `PASS` and confirms the bundled source and reference-stream\nidentities, all 62 expected numeric rows, agreement with the manuscript tables,\nand the stated numerical bounds. This command does not execute the C++ programs.\n\nTo compile and run both complete exact computations, GNU C++ with signed\n128-bit integers and Boost headers is also required:\n\n```sh\npython3 -B verification/verify_computations.py --run --serial\n```\n\nAlternatively, replace `--serial` with `--threads 2` to use OpenMP for Appendix A.\nAssertions remain enabled. A successful full run prints `PASS` after both\nprograms exit successfully and their complete output streams match the reference\nbytes. It records compiler and execution details in a new temporary directory\noutside the paper; `--output PATH` selects a new external directory instead.\n\nall README lengths [('015', 720), ('024', 722), ('046', 804), ('065', 639), ('082', 752), ('113', 808), ('115', 719), ('127', 622), ('148', 718), ('154', 698), ('164', 675), ('192', 749), ('210', 1605), ('215', 727), ('230', 687), ('245', 610), ('296', 680), ('351', 749), ('369', 848), ('372', 709)]\n"
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          "text": "\nREADME 015 \n # [Equidistribution of Prime-Degree Torus Packets with Arbitrary Local Type](paper.pdf)\n\n**Author:** OpenAI\n\n**Date:** September 24, 2026\n\n## Citation\n\n```bibtex\n@misc{OAI:Equidistribution-of-Prime-Degree-Torus-Packets-with-Arbitrary-Local-Type-September-24-2026,\n  author = {{OpenAI}},\n  title = {{Equidistribution of Prime-Degree Torus Packets with Arbitrary Local Type}},\n  howpublished = {OpenAI Math Release preprint\n                  \\href{https://github.com/openai/math/blob/main/preprints/Equidistribution-of-Prime-Degree-Torus-Packets-with-Arbitrary-Local-Type-September-24-2026/paper.pdf}{OAI:Equidistribution-of-Prime-Degree-Torus-Packets-with-Arbitrary-Local-Type-September-24-2026}},\n  year = {2026}\n}\n```\n\n\n\nREADME 024 \n # [An asymptotic formula for the number of totients](An-asymptotic-formula-for-the-number-of-totients-September-25-2026.pdf)\n\n**Author:** OpenAI\n\n**Date:** September 25, 2026\n\n## Citation\n\n```bibtex\n@misc{OAI:An-asymptotic-formula-for-the-number-of-totients-September-25-2026,\n  author = {{OpenAI}},\n  title = {{An asymptotic formula for the number of totients}},\n  howpublished = {OpenAI Math Release preprint\n                  \\href{https://github.com/openai/math/blob/main/preprints/An-asymptotic-formula-for-the-number-of-totients-September-25-2026/An-asymptotic-formula-for-the-number-of-totients-September-25-2026.pdf}{OAI:An-asymptotic-formula-for-the-number-of-totients-September-25-2026}},\n  year = {2026}\n}\n```\n\n\n\nREADME 046 \n # [A projective fourfold with large fundamental group and non-Stein universal cover](large-fundamental-group-non-stein.pdf)\n\n**Author:** OpenAI\n\n**Date:** October 5, 2026\n\n## Citation\n\n```bibtex\n@misc{OAI:A-projective-fourfold-with-large-fundamental-group-and-non-Stein-universal-cover-October-5-2026,\n  author = {{OpenAI}},\n  title = {{A projective fourfold with large fundamental group and non-Stein universal cover}},\n  howpublished = {OpenAI Math Release preprint\n                  \\href{https://github.com/openai/math/blob/main/preprints/A-projective-fourfold-with-large-fundamental-group-and-non-Stein-universal-cover-October-5-2026/large-fundamental-group-non-stein.pdf}{OAI:A-projective-fourfold-with-large-fundamental-group-and-non-Stein-universal-cover-October-5-2026}},\n  year = {2026}\n}\n```\n\n\n\nREADME 065 \n # [Virasoro Constraints under Projectivization](virasoro-constraints-under-projectivization.pdf)\n\n**Author:** OpenAI\n\n**Date:** October 5, 2026\n\n## Citation\n\n```bibtex\n@misc{OAI:Virasoro-Constraints-under-Projectivization-October-5-2026,\n  author = {{OpenAI}},\n  title = {{Virasoro Constraints under Projectivization}},\n  howpublished = {OpenAI Math Release preprint\n                  \\href{https://github.com/openai/math/blob/main/preprints/Virasoro-Constraints-under-Projectivization-October-5-2026/virasoro-constraints-under-projectivization.pdf}{OAI:Virasoro-Constraints-under-Projectivization-October-5-2026}},\n  year = {2026}\n}\n```\n\n\n\nREADME 082 \n # [Annular variation of the triangular Hilbert transform at the symmetric point](annular-variation.pdf)\n\n**Author:** OpenAI\n\n**Date:** October 5, 2026\n\n## Citation\n\n```bibtex\n@misc{OAI:Annular-variation-of-the-triangular-Hilbert-transform-at-the-symmetric-point-October-5-2026,\n  author = {{OpenAI}},\n  title = {{Annular variation of the triangular Hilbert transform at the symmetric point}},\n  howpublished = {OpenAI Math Release preprint\n                  \\href{https://github.com/openai/math/blob/main/preprints/Annular-variation-of-the-triangular-Hilbert-transform-at-the-symmetric-point-October-5-2026/annular-variation.pdf}{OAI:Annular-variation-of-the-triangular-Hilbert-transform-at-the-symmetric-point-October-5-2026}},\n  year = {2026}\n}\n```\n\n\n\nREADME 210\n # [Foulkes' conjecture for the sixth symmetric power](main.pdf)\n\n**Author:** OpenAI\n\n**Date:** September 25, 2026\n\n## Citation\n\n```bibtex\n@misc{OAI:Foulkes-Conjecture-for-the-Sixth-Symmetric-Power-September-25-2026,\n  author = {{OpenAI}},\n  title = {{Foulkes' conjecture for the sixth symmetric power}},\n  howpublished = {OpenAI Math Release preprint\n                  \\href{https://github.com/openai/math/blob/main/preprints/Foulkes-Conjecture-for-the-Sixth-Symmetric-Power-September-25-2026/main.pdf}{OAI:Foulkes-Conjecture-for-the-Sixth-Symmetric-Power-September-25-2026}},\n  year = {2026}\n}\n```\n\n## Verification\n\nRequires Python 3.9 or later. Run the supplied read-only checks from this paper\ndirectory:\n\n```sh\npython3 -B verification/verify_computations.py --check\n```\n\nSuccess prints `PASS` and confirms the bundled source and reference-stream\nidentities, all 62 expected numeric rows, agreement with the manuscript tables,\nand the stated numerical bounds. This command does not execute the C++ programs.\n\nTo compile and run both complete exact computations, GNU C++ with signed\n128-bit integers and Boost headers is also required:\n\n```sh\npython3 -B verification/verify_computations.py --run --serial\n```\n\nAlternatively, replace `--serial` with `--threads 2` to use OpenMP for Appendix A.\nAssertions remain enabled. A successful full run prints `PASS` after both\nprograms exit successfully and their complete output streams match the reference\nbytes. It records compiler and execution details in a new temporary directory\noutside the paper; `--output PATH` selects a new external directory instead.\n\nall README lengths [('015', 720), ('024', 722), ('046', 804), ('065', 639), ('082', 752), ('113', 808), ('115', 719), ('127', 622), ('148', 718), ('154', 698), ('164', 675), ('192', 749), ('210', 1605), ('215', 727), ('230', 687), ('245', 610), ('296', 680), ('351', 749), ('369', 848), ('372', 709)]\n"
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        "stdout": "\nTHEOREM 015 preprints/Equidistribution-of-Prime-Degree-Torus-Packets-with-Arbitrary-Local-Type-September-24-2026/build/paper.tex 103 \n\\begin{theorem}\\label{thm:main}\nFor every fixed prime $n\\ge5$ and every sequence\n$(K_i,M_i,\\sigma_i)$ as above with \\mbox{$D(M_i)\\to\\infty$},\n\\[\n \\mu_{K_i,M_i,\\sigma_i}\\longrightarrow m_n.\n\\]\nThe convergence is weak convergence of probability measures. In\nparticular, it holds against every continuous compactly supported\nfunction, and no mass escapes into the cusp.\n\\end{theorem}\n\nTHEOREM 024 preprints/An-asymptotic-formula-for-the-number-of-totients-September-25-2026/build/source/statements.tex 45 \n\\begin{theorem}\\label{thm:main}\nFor sufficiently large positive integers $H$, there are nonnegative\nfunctions $A_H(1;\\cdot):[0,1)\\to\\mathbb R$, given by the finite\narithmetic formula \\eqref{eq:AH} below,\nwhich converge uniformly as $H\\to\\infty$ to a function $A(1;\\cdot)$\nsatisfying\n\\[\n 0<\\inf_{0\\le s<1}A(1;s)\n   \\le\\sup_{0\\le s<1}A(1;s)<\\infty.\n\\]\nWith the parameters \\eqref{eq:renewal}--\\eqref{eq:Gj},\n\\[\n V(x)\\sim\\frac{x}{\\log x}\\,G_m A(1;\\theta).\n\\]\nMoreover, for every fixed real $c>0$,\n\\[\n \\frac{V(cx)}{V(x)}\\longrightarrow c.\n\\]\n\\end{theorem}\n\nTHEOREM 113 preprints/A-Fully-Polynomial-Randomized-Approximation-Scheme-for-Perfect-Matchings-in-General-Graphs-September-23-2026/build/main.tex 132 \n\\begin{theorem}\\label{thm:main}\nThere is a uniform classical randomized algorithm which, given $G$ and\nrational parameters $0<\\eps<1$ and $0<\\delta<1/2$, returns a nonnegative\nrational number $\\widehat Z$ such that\n\\[\n \\Prb\\bigl[(1-\\eps)Z(G)\\le \\widehat Z\\le(1+\\eps)Z(G)\\bigr]\n \\ge 1-\\delta.\n\\]\nIf $Z(G)=0$, the output is zero with certainty. The worst-case bit running\ntime is polynomial in the input encoding length, $\\eps^{-1}$, and\n$\\log\\delta^{-1}$.\n\\end{theorem}\n\nTHEOREM 115 preprints/Exact-Uniform-Sampling-of-Contingency-Tables-with-Arbitrary-Margins-September-24-2026/build/sections/introduction.tex 18 \n\\begin{theorem}\\label{thm:main}\nFor arbitrary nonnegative integer margins of equal total, the following\nalgorithms exist in the model of unbiased random bits and bit operations.\n\\begin{enumerate}[label=\\textup{(\\roman*)}]\n \\item Given an integer $k\\ge1$, a bounded-time algorithm outputs a table\n in $\\Omega(r,c)$ whose law is within $2^{-k}$ in total variation of the\n uniform law. Its running time is polynomial in\n $m,n,\\log(N+1),k$.\n \\item An algorithm outputs an exactly uniform table in $\\Omega(r,c)$,\n terminates almost surely, and has expected running time polynomial in\n $m,n,\\log(N+1)$.\n\\end{enumerate}\nThe polynomials are uniform over all margin vectors. No positivity,\nsparsity, balance, or fixed-dimension hypothesis is imposed.\n\\end{theorem}\n\nTHEOREM 127 preprints/Average-Sensitivity-of-Polynomial-Threshold-Functions-September-25-2026/build/sections/01-introduction.tex 20 \n\\begin{theorem}\\label{thm:main}\nLet $n\\ge1$ and $1\\le d\\le n$ be integers. Let $p$ be a real multilinear\npolynomial of degree at most $d$, and define\n$f:\\{-1,1\\}^n\\to\\{-1,1\\}$ by $f(x)=\\sgn(p(x))$, with $\\sgn(0)=1$.\nThen, for average sensitivity under the uniform law,\n\\[\n I(f)\\le 8d\\sqrt n.\n\\]\n\\end{theorem}\n\nTHEOREM 148 preprints/The-entropy-rate-dimension-formula-for-self-similar-measures-on-the-line-September-24-2026/build/sections/introduction.tex 78 \n\\begin{theorem}\\label{thm:main}\nFor every finite nonempty indexed family\n$\\varphi_i(x)=r_i x+t_i$ on $\\R$ with $0<|r_i|<1$, and every\nstrictly positive probability vector $p$, its self-similar measure satisfies\n\\[\n \\dim_H\\mu_{\\Phi,p}\n =\\min\\left\\{1,\\frac{h_{\\mathrm{RW}}(\\Phi,p)}{\\chi(\\Phi,p)}\\right\\}.\n\\]\nNo separation assumption is required; exact overlaps and repeated\ngenerators are allowed.\n\\end{theorem}\n",
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          "\nTHEOREM 015 preprints/Equidistribution-of-Prime-Degree-Torus-Packets-with-Arbitrary-Local-Type-September-24-2026/build/paper.tex 103 \n\\begin{theorem}\\label{thm:main}\nFor every fixed prime $n\\ge5$ and every sequence\n$(K_i,M_i,\\sigma_i)$ as above with \\mbox{$D(M_i)\\to\\infty$},\n\\[\n \\mu_{K_i,M_i,\\sigma_i}\\longrightarrow m_n.\n\\]\nThe convergence is weak convergence of probability measures. In\nparticular, it holds against every continuous compactly supported\nfunction, and no mass escapes into the cusp.\n\\end{theorem}\n\nTHEOREM 024 preprints/An-asymptotic-formula-for-the-number-of-totients-September-25-2026/build/source/statements.tex 45 \n\\begin{theorem}\\label{thm:main}\nFor sufficiently large positive integers $H$, there are nonnegative\nfunctions $A_H(1;\\cdot):[0,1)\\to\\mathbb R$, given by the finite\narithmetic formula \\eqref{eq:AH} below,\nwhich converge uniformly as $H\\to\\infty$ to a function $A(1;\\cdot)$\nsatisfying\n\\[\n 0<\\inf_{0\\le s<1}A(1;s)\n   \\le\\sup_{0\\le s<1}A(1;s)<\\infty.\n\\]\nWith the parameters \\eqref{eq:renewal}--\\eqref{eq:Gj},\n\\[\n V(x)\\sim\\frac{x}{\\log x}\\,G_m A(1;\\theta).\n\\]\nMoreover, for every fixed real $c>0$,\n\\[\n \\frac{V(cx)}{V(x)}\\longrightarrow c.\n\\]\n\\end{theorem}\n\nTHEOREM 113 preprints/A-Fully-Polynomial-Randomized-Approximation-Scheme-for-Perfect-Matchings-in-General-Graphs-September-23-2026/build/main.tex 132 \n\\begin{theorem}\\label{thm:main}\nThere is a uniform classical randomized algorithm which, given $G$ and\nrational parameters $0<\\eps<1$ and $0<\\delta<1/2$, returns a nonnegative\nrational number $\\widehat Z$ such that\n\\[\n \\Prb\\bigl[(1-\\eps)Z(G)\\le \\widehat Z\\le(1+\\eps)Z(G)\\bigr]\n \\ge 1-\\delta.\n\\]\nIf $Z(G)=0$, the output is zero with certainty. The worst-case bit running\ntime is polynomial in the input encoding length, $\\eps^{-1}$, and\n$\\log\\delta^{-1}$.\n\\end{theorem}\n\nTHEOREM 115 preprints/Exact-Uniform-Sampling-of-Contingency-Tables-with-Arbitrary-Margins-September-24-2026/build/sections/introduction.tex 18 \n\\begin{theorem}\\label{thm:main}\nFor arbitrary nonnegative integer margins of equal total, the following\nalgorithms exist in the model of unbiased random bits and bit operations.\n\\begin{enumerate}[label=\\textup{(\\roman*)}]\n \\item Given an integer $k\\ge1$, a bounded-time algorithm outputs a table\n in $\\Omega(r,c)$ whose law is within $2^{-k}$ in total variation of the\n uniform law. Its running time is polynomial in\n $m,n,\\log(N+1),k$.\n \\item An algorithm outputs an exactly uniform table in $\\Omega(r,c)$,\n terminates almost surely, and has expected running time polynomial in\n $m,n,\\log(N+1)$.\n\\end{enumerate}\nThe polynomials are uniform over all margin vectors. No positivity,\nsparsity, balance, or fixed-dimension hypothesis is imposed.\n\\end{theorem}\n\nTHEOREM 127 preprints/Average-Sensitivity-of-Polynomial-Threshold-Functions-September-25-2026/build/sections/01-introduction.tex 20 \n\\begin{theorem}\\label{thm:main}\nLet $n\\ge1$ and $1\\le d\\le n$ be integers. Let $p$ be a real multilinear\npolynomial of degree at most $d$, and define\n$f:\\{-1,1\\}^n\\to\\{-1,1\\}$ by $f(x)=\\sgn(p(x))$, with $\\sgn(0)=1$.\nThen, for average sensitivity under the uniform law,\n\\[\n I(f)\\le 8d\\sqrt n.\n\\]\n\\end{theorem}\n\nTHEOREM 148 preprints/The-entropy-rate-dimension-formula-for-self-similar-measures-on-the-line-September-24-2026/build/sections/introduction.tex 78 \n\\begin{theorem}\\label{thm:main}\nFor every finite nonempty indexed family\n$\\varphi_i(x)=r_i x+t_i$ on $\\R$ with $0<|r_i|<1$, and every\nstrictly positive probability vector $p$, its self-similar measure satisfies\n\\[\n \\dim_H\\mu_{\\Phi,p}\n =\\min\\left\\{1,\\frac{h_{\\mathrm{RW}}(\\Phi,p)}{\\chi(\\Phi,p)}\\right\\}.\n\\]\nNo separation assumption is required; exact overlaps and repeated\ngenerators are allowed.\n\\end{theorem}\n"
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          "text": "\nTHEOREM 015 preprints/Equidistribution-of-Prime-Degree-Torus-Packets-with-Arbitrary-Local-Type-September-24-2026/build/paper.tex 103 \n\\begin{theorem}\\label{thm:main}\nFor every fixed prime $n\\ge5$ and every sequence\n$(K_i,M_i,\\sigma_i)$ as above with \\mbox{$D(M_i)\\to\\infty$},\n\\[\n \\mu_{K_i,M_i,\\sigma_i}\\longrightarrow m_n.\n\\]\nThe convergence is weak convergence of probability measures. In\nparticular, it holds against every continuous compactly supported\nfunction, and no mass escapes into the cusp.\n\\end{theorem}\n\nTHEOREM 024 preprints/An-asymptotic-formula-for-the-number-of-totients-September-25-2026/build/source/statements.tex 45 \n\\begin{theorem}\\label{thm:main}\nFor sufficiently large positive integers $H$, there are nonnegative\nfunctions $A_H(1;\\cdot):[0,1)\\to\\mathbb R$, given by the finite\narithmetic formula \\eqref{eq:AH} below,\nwhich converge uniformly as $H\\to\\infty$ to a function $A(1;\\cdot)$\nsatisfying\n\\[\n 0<\\inf_{0\\le s<1}A(1;s)\n   \\le\\sup_{0\\le s<1}A(1;s)<\\infty.\n\\]\nWith the parameters \\eqref{eq:renewal}--\\eqref{eq:Gj},\n\\[\n V(x)\\sim\\frac{x}{\\log x}\\,G_m A(1;\\theta).\n\\]\nMoreover, for every fixed real $c>0$,\n\\[\n \\frac{V(cx)}{V(x)}\\longrightarrow c.\n\\]\n\\end{theorem}\n\nTHEOREM 113 preprints/A-Fully-Polynomial-Randomized-Approximation-Scheme-for-Perfect-Matchings-in-General-Graphs-September-23-2026/build/main.tex 132 \n\\begin{theorem}\\label{thm:main}\nThere is a uniform classical randomized algorithm which, given $G$ and\nrational parameters $0<\\eps<1$ and $0<\\delta<1/2$, returns a nonnegative\nrational number $\\widehat Z$ such that\n\\[\n \\Prb\\bigl[(1-\\eps)Z(G)\\le \\widehat Z\\le(1+\\eps)Z(G)\\bigr]\n \\ge 1-\\delta.\n\\]\nIf $Z(G)=0$, the output is zero with certainty. The worst-case bit running\ntime is polynomial in the input encoding length, $\\eps^{-1}$, and\n$\\log\\delta^{-1}$.\n\\end{theorem}\n\nTHEOREM 115 preprints/Exact-Uniform-Sampling-of-Contingency-Tables-with-Arbitrary-Margins-September-24-2026/build/sections/introduction.tex 18 \n\\begin{theorem}\\label{thm:main}\nFor arbitrary nonnegative integer margins of equal total, the following\nalgorithms exist in the model of unbiased random bits and bit operations.\n\\begin{enumerate}[label=\\textup{(\\roman*)}]\n \\item Given an integer $k\\ge1$, a bounded-time algorithm outputs a table\n in $\\Omega(r,c)$ whose law is within $2^{-k}$ in total variation of the\n uniform law. Its running time is polynomial in\n $m,n,\\log(N+1),k$.\n \\item An algorithm outputs an exactly uniform table in $\\Omega(r,c)$,\n terminates almost surely, and has expected running time polynomial in\n $m,n,\\log(N+1)$.\n\\end{enumerate}\nThe polynomials are uniform over all margin vectors. No positivity,\nsparsity, balance, or fixed-dimension hypothesis is imposed.\n\\end{theorem}\n\nTHEOREM 127 preprints/Average-Sensitivity-of-Polynomial-Threshold-Functions-September-25-2026/build/sections/01-introduction.tex 20 \n\\begin{theorem}\\label{thm:main}\nLet $n\\ge1$ and $1\\le d\\le n$ be integers. Let $p$ be a real multilinear\npolynomial of degree at most $d$, and define\n$f:\\{-1,1\\}^n\\to\\{-1,1\\}$ by $f(x)=\\sgn(p(x))$, with $\\sgn(0)=1$.\nThen, for average sensitivity under the uniform law,\n\\[\n I(f)\\le 8d\\sqrt n.\n\\]\n\\end{theorem}\n\nTHEOREM 148 preprints/The-entropy-rate-dimension-formula-for-self-similar-measures-on-the-line-September-24-2026/build/sections/introduction.tex 78 \n\\begin{theorem}\\label{thm:main}\nFor every finite nonempty indexed family\n$\\varphi_i(x)=r_i x+t_i$ on $\\R$ with $0<|r_i|<1$, and every\nstrictly positive probability vector $p$, its self-similar measure satisfies\n\\[\n \\dim_H\\mu_{\\Phi,p}\n =\\min\\left\\{1,\\frac{h_{\\mathrm{RW}}(\\Phi,p)}{\\chi(\\Phi,p)}\\right\\}.\n\\]\nNo separation assumption is required; exact overlaps and repeated\ngenerators are allowed.\n\\end{theorem}\n"
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        "stdout": "\nTHEOREM 082 preprints/Annular-variation-of-the-triangular-Hilbert-transform-at-the-symmetric-point-October-5-2026/build/sections/introduction.tex 27 \n\\begin{theorem}[Full annular variation]\\label{thm:variation}\nFor every real $r>2$ there is a finite constant $C_r$ such that\n\\[\n \\|V_r(F,G)\\|_{L^{3/2}(\\R^2)}\n \\le C_r\\|F\\|_{L^3(\\R^2)}\\|G\\|_{L^3(\\R^2)}\n\\]\nfor all complex $F,G\\in L^3(\\R^2)$.\n\\end{theorem}\n\nTHEOREM 210 preprints/Foulkes-Conjecture-for-the-Sixth-Symmetric-Power-September-25-2026/build/sections/introduction.tex 19 \n\\begin{theorem}\\label{thm:main}\nFor every integer \\(b\\ge6\\) and every finite-dimensional complex vector\nspace \\(V\\), there is a \\(\\GL(V)\\)-equivariant injection\n\\[\n                 \\Sym^6(\\Sym^b V)\\ \\hookrightarrow\\ \\Sym^b(\\Sym^6 V).\n\\]\nEquivalently, \\(h_b[h_6]-h_6[h_b]\\) is Schur-positive for every \\(b\\ge6\\).\n\\end{theorem}\n\nTHEOREM 215 preprints/The-canonical-massive-continuum-limit-of-the-two-dimensional-O3-model-October-4-2026/build/sections/introduction.tex 61 \n\\begin{theorem}[Canonical continuum limit]\\label{thm:canonical}\nFor the nearest-neighbor probability \\eqref{eq:lattice-model}, the\nperiodic thermodynamic limit $\\mu_\\beta$ exists for all sufficiently\nlarge $\\beta$.  Both sums in \\eqref{eq:canonical-lattice-scales} are\nfinite, $\\chi_\\beta,\\xi_\\beta>0$, and $\\xi_\\beta\\to\\infty$ as\n$\\beta\\to\\infty$.  The following limits hold through all real values\nof $\\beta$ tending to infinity.\n\\begin{enumerate}[label=\\textup{(\\roman*)}]\n\\item For every finite list of vector tests in\n$C_c^\\infty(\\R^2;\\R^3)$, the fields \\eqref{eq:canonical-field} converge\njointly in law and in all joint moments to one common limiting hierarchy.\n\\item For every order and every choice of spin components, the associated\nSchwinger distributions converge in the strong topology of\n$\\mathcal S'((\\R^2)^n)$, that is, uniformly on bounded sets of Schwartz\ntests.\n\\item The limiting hierarchy has susceptibility and second-moment length\nequal to one.  It is the hierarchy constructed in Theorem~\\ref{thm:main},\nafter the unique positive constant length and field rescalings imposing\nthese two normalizations.\n\\end{enumerate}\n\\end{theorem}\n\nTHEOREM 230 preprints/An-exact-Hausdorff-gauge-for-SLE-September-25-2026/build/sections/introduction.tex 38 \n\\begin{theorem}[An exact moment gauge]\\label{thm:main}\nFix $0<\\kappa<8$, and let $\\gamma$ be chordal $\\SLE_\\kappa$ from\n$0$ to $\\infty$ in the upper half-plane $\\HH$, parametrized by half-plane capacity $2t$.\nThere is a deterministic Hausdorff gauge $h_\\kappa$, depending only on\n$\\kappa$, such that almost surely, simultaneously for every real\n$0<s<t<\\infty$,\n\\[\n 0<\\Haus^{h_\\kappa}(\\gamma([s,t]))<\\infty.\n\\]\n\\end{theorem}\n\nTHEOREM 296 preprints/Relative-generation-and-the-generator-problem-for-finite-factors-September-23-2026/build/sections/introduction.tex 29 \n\\begin{theorem}\\label{thm:main}\nEvery type $\\mathrm{II}_1$ factor $M$ with separable predual is\ngenerated by two self-adjoint elements. Equivalently, there is an\n$x\\in M$ such that $M=W^*(x)$.\n\\end{theorem}\n",
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          "\nTHEOREM 082 preprints/Annular-variation-of-the-triangular-Hilbert-transform-at-the-symmetric-point-October-5-2026/build/sections/introduction.tex 27 \n\\begin{theorem}[Full annular variation]\\label{thm:variation}\nFor every real $r>2$ there is a finite constant $C_r$ such that\n\\[\n \\|V_r(F,G)\\|_{L^{3/2}(\\R^2)}\n \\le C_r\\|F\\|_{L^3(\\R^2)}\\|G\\|_{L^3(\\R^2)}\n\\]\nfor all complex $F,G\\in L^3(\\R^2)$.\n\\end{theorem}\n\nTHEOREM 210 preprints/Foulkes-Conjecture-for-the-Sixth-Symmetric-Power-September-25-2026/build/sections/introduction.tex 19 \n\\begin{theorem}\\label{thm:main}\nFor every integer \\(b\\ge6\\) and every finite-dimensional complex vector\nspace \\(V\\), there is a \\(\\GL(V)\\)-equivariant injection\n\\[\n                 \\Sym^6(\\Sym^b V)\\ \\hookrightarrow\\ \\Sym^b(\\Sym^6 V).\n\\]\nEquivalently, \\(h_b[h_6]-h_6[h_b]\\) is Schur-positive for every \\(b\\ge6\\).\n\\end{theorem}\n\nTHEOREM 215 preprints/The-canonical-massive-continuum-limit-of-the-two-dimensional-O3-model-October-4-2026/build/sections/introduction.tex 61 \n\\begin{theorem}[Canonical continuum limit]\\label{thm:canonical}\nFor the nearest-neighbor probability \\eqref{eq:lattice-model}, the\nperiodic thermodynamic limit $\\mu_\\beta$ exists for all sufficiently\nlarge $\\beta$.  Both sums in \\eqref{eq:canonical-lattice-scales} are\nfinite, $\\chi_\\beta,\\xi_\\beta>0$, and $\\xi_\\beta\\to\\infty$ as\n$\\beta\\to\\infty$.  The following limits hold through all real values\nof $\\beta$ tending to infinity.\n\\begin{enumerate}[label=\\textup{(\\roman*)}]\n\\item For every finite list of vector tests in\n$C_c^\\infty(\\R^2;\\R^3)$, the fields \\eqref{eq:canonical-field} converge\njointly in law and in all joint moments to one common limiting hierarchy.\n\\item For every order and every choice of spin components, the associated\nSchwinger distributions converge in the strong topology of\n$\\mathcal S'((\\R^2)^n)$, that is, uniformly on bounded sets of Schwartz\ntests.\n\\item The limiting hierarchy has susceptibility and second-moment length\nequal to one.  It is the hierarchy constructed in Theorem~\\ref{thm:main},\nafter the unique positive constant length and field rescalings imposing\nthese two normalizations.\n\\end{enumerate}\n\\end{theorem}\n\nTHEOREM 230 preprints/An-exact-Hausdorff-gauge-for-SLE-September-25-2026/build/sections/introduction.tex 38 \n\\begin{theorem}[An exact moment gauge]\\label{thm:main}\nFix $0<\\kappa<8$, and let $\\gamma$ be chordal $\\SLE_\\kappa$ from\n$0$ to $\\infty$ in the upper half-plane $\\HH$, parametrized by half-plane capacity $2t$.\nThere is a deterministic Hausdorff gauge $h_\\kappa$, depending only on\n$\\kappa$, such that almost surely, simultaneously for every real\n$0<s<t<\\infty$,\n\\[\n 0<\\Haus^{h_\\kappa}(\\gamma([s,t]))<\\infty.\n\\]\n\\end{theorem}\n\nTHEOREM 296 preprints/Relative-generation-and-the-generator-problem-for-finite-factors-September-23-2026/build/sections/introduction.tex 29 \n\\begin{theorem}\\label{thm:main}\nEvery type $\\mathrm{II}_1$ factor $M$ with separable predual is\ngenerated by two self-adjoint elements. Equivalently, there is an\n$x\\in M$ such that $M=W^*(x)$.\n\\end{theorem}\n"
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          "text": "\nTHEOREM 082 preprints/Annular-variation-of-the-triangular-Hilbert-transform-at-the-symmetric-point-October-5-2026/build/sections/introduction.tex 27 \n\\begin{theorem}[Full annular variation]\\label{thm:variation}\nFor every real $r>2$ there is a finite constant $C_r$ such that\n\\[\n \\|V_r(F,G)\\|_{L^{3/2}(\\R^2)}\n \\le C_r\\|F\\|_{L^3(\\R^2)}\\|G\\|_{L^3(\\R^2)}\n\\]\nfor all complex $F,G\\in L^3(\\R^2)$.\n\\end{theorem}\n\nTHEOREM 210 preprints/Foulkes-Conjecture-for-the-Sixth-Symmetric-Power-September-25-2026/build/sections/introduction.tex 19 \n\\begin{theorem}\\label{thm:main}\nFor every integer \\(b\\ge6\\) and every finite-dimensional complex vector\nspace \\(V\\), there is a \\(\\GL(V)\\)-equivariant injection\n\\[\n                 \\Sym^6(\\Sym^b V)\\ \\hookrightarrow\\ \\Sym^b(\\Sym^6 V).\n\\]\nEquivalently, \\(h_b[h_6]-h_6[h_b]\\) is Schur-positive for every \\(b\\ge6\\).\n\\end{theorem}\n\nTHEOREM 215 preprints/The-canonical-massive-continuum-limit-of-the-two-dimensional-O3-model-October-4-2026/build/sections/introduction.tex 61 \n\\begin{theorem}[Canonical continuum limit]\\label{thm:canonical}\nFor the nearest-neighbor probability \\eqref{eq:lattice-model}, the\nperiodic thermodynamic limit $\\mu_\\beta$ exists for all sufficiently\nlarge $\\beta$.  Both sums in \\eqref{eq:canonical-lattice-scales} are\nfinite, $\\chi_\\beta,\\xi_\\beta>0$, and $\\xi_\\beta\\to\\infty$ as\n$\\beta\\to\\infty$.  The following limits hold through all real values\nof $\\beta$ tending to infinity.\n\\begin{enumerate}[label=\\textup{(\\roman*)}]\n\\item For every finite list of vector tests in\n$C_c^\\infty(\\R^2;\\R^3)$, the fields \\eqref{eq:canonical-field} converge\njointly in law and in all joint moments to one common limiting hierarchy.\n\\item For every order and every choice of spin components, the associated\nSchwinger distributions converge in the strong topology of\n$\\mathcal S'((\\R^2)^n)$, that is, uniformly on bounded sets of Schwartz\ntests.\n\\item The limiting hierarchy has susceptibility and second-moment length\nequal to one.  It is the hierarchy constructed in Theorem~\\ref{thm:main},\nafter the unique positive constant length and field rescalings imposing\nthese two normalizations.\n\\end{enumerate}\n\\end{theorem}\n\nTHEOREM 230 preprints/An-exact-Hausdorff-gauge-for-SLE-September-25-2026/build/sections/introduction.tex 38 \n\\begin{theorem}[An exact moment gauge]\\label{thm:main}\nFix $0<\\kappa<8$, and let $\\gamma$ be chordal $\\SLE_\\kappa$ from\n$0$ to $\\infty$ in the upper half-plane $\\HH$, parametrized by half-plane capacity $2t$.\nThere is a deterministic Hausdorff gauge $h_\\kappa$, depending only on\n$\\kappa$, such that almost surely, simultaneously for every real\n$0<s<t<\\infty$,\n\\[\n 0<\\Haus^{h_\\kappa}(\\gamma([s,t]))<\\infty.\n\\]\n\\end{theorem}\n\nTHEOREM 296 preprints/Relative-generation-and-the-generator-problem-for-finite-factors-September-23-2026/build/sections/introduction.tex 29 \n\\begin{theorem}\\label{thm:main}\nEvery type $\\mathrm{II}_1$ factor $M$ with separable predual is\ngenerated by two self-adjoint elements. Equivalently, there is an\n$x\\in M$ such that $M=W^*(x)$.\n\\end{theorem}\n"
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        "stdout": "\nTHEOREM 046 17 \n\\begin{theorem}\\label{thm:main}\nThere exist a smooth connected projective complex fourfold $X$ and a\nsimple abelian surface $A\\subset X$ such that\n\\begin{enumerate}\n\\item $X$ has large fundamental group;\n\\item $\\im\\bigl(\\pi_1(A)\\to\\pi_1(X)\\bigr)\\simeq\\Z$.\n\\end{enumerate}\nThe universal cover $\\widetilde X$ contains no positive-dimensional\ncompact complex analytic subvariety and is not Stein.\n\\end{theorem}\n\nTHEOREM 065 36 \n\\begin{theorem}\n\\label{thm:main}\nLet $B$ be a smooth connected projective variety over $\\C$, and\nlet $E$ be an algebraic vector bundle of rank $r\\geq2$ on $B$.\nLet $X=\\mathbb P_B(E)$ parametrize one-dimensional subspaces of\nthe fibers of $E$. Then\n\\[\n                           \\V(B)\\ \\Longrightarrow\\ \\V(X).\n\\]\n\\end{theorem}\n\nTHEOREM 154 20 \n\\begin{theorem}\\label{thm:main}\nSuppose that $T$ satisfies \\eqref{eq:mixing}. For every integer\n$n\\ge2$ and every fixed $f_1,\\ldots,f_n\\in L^\\infty(\\mu)$,\n\\[\n A_N(f_1,\\ldots,f_n)(x)\n \\longrightarrow\\prod_{j=1}^n\\int_X f_j\\dd\\mu\n \\qquad\\text{for $\\mu$-almost every }x,\n\\]\nas $N$ tends to infinity through all positive integers. The probability\nspace need not be standard, and no rate of mixing is required.\n\\end{theorem}\n\nTHEOREM 164 13 \n\\begin{theorem}\\label{thm:main}\nLet $r,m\\ge1$ be integers, and fix real numbers $R\\ge2$ and $D\\ge1$.\nFor every coloring $\\chi:\\N\\to[r]$,\nthere are distinct positive integers $a_1<\\cdots<a_m$ and a color\n$c\\in[r]$ such that\n\\[\n \\chi\\left(\\sum_{j\\in J}a_j\\right)\n =\\chi\\left(\\prod_{j\\in J}a_j\\right)=c\n \\quad\\text{for every }\\varnothing\\ne J\\subseteq[m].\n\\]\nEquivalently, $\\FS(A)\\cup\\FP(A)$ is monochromatic for a set\n$A\\subset\\N$ of cardinality $m$.\nThe elements can additionally be chosen to satisfy\n\\begin{equation}\\label{eq:separated-elements}\n a_1>R,\\qquad\n a_d>R\\left(\\sum_{k<d}a_k+\\prod_{k<d}a_k\\right)^D\n \\quad(2\\le d\\le m).\n\\end{equation}\n\\end{theorem}\n\nTHEOREM 192 23 \n\\begin{theorem}\\label{thm:main}\nFor every real \\(C>0\\), there are a positive integer \\(n\\) and a\nnonconstant Boolean function \\(f:\\{-1,1\\}^n\\to\\{-1,1\\}\\) such that\n\\[\n \\sum_{i=1}^n\\widehat f(\\{i\\})>C\\sqrt{\\deg(f)}.\n\\]\nConsequently,\n\\[\n \\sup_{\\substack{n\\ge1,\\ f:\\{-1,1\\}^n\\to\\{-1,1\\}\\\\\\deg(f)>0}}\n \\frac{\\sum_i|\\widehat f(\\{i\\})|}{\\sqrt{\\deg(f)}}=\\infty.\n\\]\n\\end{theorem}\n\nTHEOREM 245 116 \n\\begin{theorem}[The $\\beta$-Barendregt--Geuvers--Klop conjecture]\n\\label{thm:main}\nLet $\\mathcal P=(\\mathcal S,\\mathcal A,\\mathcal R)$ be any pure type system.\nIf every legal expression of $\\mathcal P$, in every valid context, is weakly\n$\\beta$-normalizing, then every such expression is strongly\n$\\beta$-normalizing. No functionality assumption is imposed on\n$\\mathcal A$ or $\\mathcal R$.\n\\end{theorem}\n\nTHEOREM 351 68 \n\\begin{theorem}\\label{thm:main}\nThere are an integer \\(q\\ge10\\), a time \\(T>0\\), and a smooth Ricci flow\n\\(g(t)\\), \\(0\\le t<T\\), on the closed connected manifold\n\\(M=\\Sp^2\\times\\Sp^{q+1}\\), with smooth initial metric, such that\n\\[\n\\sup_{M\\times[0,T)}|R_{g(t)}|<\\infty,\n\\qquad\n\\lim_{t\\uparrow T}\\max_M|\\Rm_{g(t)}|=\\infty.\n\\]\nIn particular, \\(T\\) is the finite maximal existence time of this flow.\n\\end{theorem}\n\nTHEOREM 369 13 \n\\begin{theorem}\\label{thm:main}\nFor every $0\\ne u\\in\\V$,\n\\[\n\\grad u(x)\\ne0\\qquad(x\\in\\Omega).\n\\]\nIn particular,\n\\begin{equation}\\label{eq:hotspots}\n\\min_{y\\in\\partial\\Omega}u(y)<u(x)<\\max_{y\\in\\partial\\Omega}u(y)\n\\qquad(x\\in\\Omega).\n\\end{equation}\n\\end{theorem}\n\nTHEOREM 372 31 \n\\begin{theorem}\\label{thm:main}\nLet $\\Omega\\subset\\R^3$ be any bounded connected domain with\n$C^\\infty$ boundary. For $j=1,2$, let\n$\\lambda_j,\\mu_j\\in C^\\infty(\\overline\\Omega;\\R)$ satisfy\n$\\mu_j>0$ and $3\\lambda_j+2\\mu_j>0$ on $\\overline\\Omega$.\nIf\n\\[\n \\Lambda_{\\lambda_1,\\mu_1}\n =\\Lambda_{\\lambda_2,\\mu_2}\n \\colon H^{1/2}(\\partial\\Omega;\\R^3)\n       \\longrightarrow H^{-1/2}(\\partial\\Omega;\\R^3),\n\\]\nthen $\\lambda_1=\\lambda_2$ and $\\mu_1=\\mu_2$ throughout $\\Omega$.\n\\end{theorem}\ncitation-only readmes [('015', ['Citation']), ('024', ['Citation']), ('046', ['Citation']), ('065', ['Citation']), ('082', ['Citation']), ('113', ['Citation']), ('115', ['Citation']), ('127', ['Citation']), ('148', ['Citation']), ('154', ['Citation']), ('164', ['Citation']), ('192', ['Citation']), ('215', ['Citation']), ('230', ['Citation']), ('245', ['Citation']), ('296', ['Citation']), ('351', ['Citation']), ('369', ['Citation']), ('372', ['Citation'])]\nPDF headers Counter({'%PDF-1.7': 18, '%PDF-1.5': 2})\n",
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          "\nTHEOREM 046 17 \n\\begin{theorem}\\label{thm:main}\nThere exist a smooth connected projective complex fourfold $X$ and a\nsimple abelian surface $A\\subset X$ such that\n\\begin{enumerate}\n\\item $X$ has large fundamental group;\n\\item $\\im\\bigl(\\pi_1(A)\\to\\pi_1(X)\\bigr)\\simeq\\Z$.\n\\end{enumerate}\nThe universal cover $\\widetilde X$ contains no positive-dimensional\ncompact complex analytic subvariety and is not Stein.\n\\end{theorem}\n\nTHEOREM 065 36 \n\\begin{theorem}\n\\label{thm:main}\nLet $B$ be a smooth connected projective variety over $\\C$, and\nlet $E$ be an algebraic vector bundle of rank $r\\geq2$ on $B$.\nLet $X=\\mathbb P_B(E)$ parametrize one-dimensional subspaces of\nthe fibers of $E$. Then\n\\[\n                           \\V(B)\\ \\Longrightarrow\\ \\V(X).\n\\]\n\\end{theorem}\n\nTHEOREM 154 20 \n\\begin{theorem}\\label{thm:main}\nSuppose that $T$ satisfies \\eqref{eq:mixing}. For every integer\n$n\\ge2$ and every fixed $f_1,\\ldots,f_n\\in L^\\infty(\\mu)$,\n\\[\n A_N(f_1,\\ldots,f_n)(x)\n \\longrightarrow\\prod_{j=1}^n\\int_X f_j\\dd\\mu\n \\qquad\\text{for $\\mu$-almost every }x,\n\\]\nas $N$ tends to infinity through all positive integers. The probability\nspace need not be standard, and no rate of mixing is required.\n\\end{theorem}\n\nTHEOREM 164 13 \n\\begin{theorem}\\label{thm:main}\nLet $r,m\\ge1$ be integers, and fix real numbers $R\\ge2$ and $D\\ge1$.\nFor every coloring $\\chi:\\N\\to[r]$,\nthere are distinct positive integers $a_1<\\cdots<a_m$ and a color\n$c\\in[r]$ such that\n\\[\n \\chi\\left(\\sum_{j\\in J}a_j\\right)\n =\\chi\\left(\\prod_{j\\in J}a_j\\right)=c\n \\quad\\text{for every }\\varnothing\\ne J\\subseteq[m].\n\\]\nEquivalently, $\\FS(A)\\cup\\FP(A)$ is monochromatic for a set\n$A\\subset\\N$ of cardinality $m$.\nThe elements can additionally be chosen to satisfy\n\\begin{equation}\\label{eq:separated-elements}\n a_1>R,\\qquad\n a_d>R\\left(\\sum_{k<d}a_k+\\prod_{k<d}a_k\\right)^D\n \\quad(2\\le d\\le m).\n\\end{equation}\n\\end{theorem}\n\nTHEOREM 192 23 \n\\begin{theorem}\\label{thm:main}\nFor every real \\(C>0\\), there are a positive integer \\(n\\) and a\nnonconstant Boolean function \\(f:\\{-1,1\\}^n\\to\\{-1,1\\}\\) such that\n\\[\n \\sum_{i=1}^n\\widehat f(\\{i\\})>C\\sqrt{\\deg(f)}.\n\\]\nConsequently,\n\\[\n \\sup_{\\substack{n\\ge1,\\ f:\\{-1,1\\}^n\\to\\{-1,1\\}\\\\\\deg(f)>0}}\n \\frac{\\sum_i|\\widehat f(\\{i\\})|}{\\sqrt{\\deg(f)}}=\\infty.\n\\]\n\\end{theorem}\n\nTHEOREM 245 116 \n\\begin{theorem}[The $\\beta$-Barendregt--Geuvers--Klop conjecture]\n\\label{thm:main}\nLet $\\mathcal P=(\\mathcal S,\\mathcal A,\\mathcal R)$ be any pure type system.\nIf every legal expression of $\\mathcal P$, in every valid context, is weakly\n$\\beta$-normalizing, then every such expression is strongly\n$\\beta$-normalizing. No functionality assumption is imposed on\n$\\mathcal A$ or $\\mathcal R$.\n\\end{theorem}\n\nTHEOREM 351 68 \n\\begin{theorem}\\label{thm:main}\nThere are an integer \\(q\\ge10\\), a time \\(T>0\\), and a smooth Ricci flow\n\\(g(t)\\), \\(0\\le t<T\\), on the closed connected manifold\n\\(M=\\Sp^2\\times\\Sp^{q+1}\\), with smooth initial metric, such that\n\\[\n\\sup_{M\\times[0,T)}|R_{g(t)}|<\\infty,\n\\qquad\n\\lim_{t\\uparrow T}\\max_M|\\Rm_{g(t)}|=\\infty.\n\\]\nIn particular, \\(T\\) is the finite maximal existence time of this flow.\n\\end{theorem}\n\nTHEOREM 369 13 \n\\begin{theorem}\\label{thm:main}\nFor every $0\\ne u\\in\\V$,\n\\[\n\\grad u(x)\\ne0\\qquad(x\\in\\Omega).\n\\]\nIn particular,\n\\begin{equation}\\label{eq:hotspots}\n\\min_{y\\in\\partial\\Omega}u(y)<u(x)<\\max_{y\\in\\partial\\Omega}u(y)\n\\qquad(x\\in\\Omega).\n\\end{equation}\n\\end{theorem}\n\nTHEOREM 372 31 \n\\begin{theorem}\\label{thm:main}\nLet $\\Omega\\subset\\R^3$ be any bounded connected domain with\n$C^\\infty$ boundary. For $j=1,2$, let\n$\\lambda_j,\\mu_j\\in C^\\infty(\\overline\\Omega;\\R)$ satisfy\n$\\mu_j>0$ and $3\\lambda_j+2\\mu_j>0$ on $\\overline\\Omega$.\nIf\n\\[\n \\Lambda_{\\lambda_1,\\mu_1}\n =\\Lambda_{\\lambda_2,\\mu_2}\n \\colon H^{1/2}(\\partial\\Omega;\\R^3)\n       \\longrightarrow H^{-1/2}(\\partial\\Omega;\\R^3),\n\\]\nthen $\\lambda_1=\\lambda_2$ and $\\mu_1=\\mu_2$ throughout $\\Omega$.\n\\end{theorem}\ncitation-only readmes [('015', ['Citation']), ('024', ['Citation']), ('046', ['Citation']), ('065', ['Citation']), ('082', ['Citation']), ('113', ['Citation']), ('115', ['Citation']), ('127', ['Citation']), ('148', ['Citation']), ('154', ['Citation']), ('164', ['Citation']), ('192', ['Citation']), ('215', ['Citation']), ('230', ['Citation']), ('245', ['Citation']), ('296', ['Citation']), ('351', ['Citation']), ('369', ['Citation']), ('372', ['Citation'])]\nPDF headers Counter({'%PDF-1.7': 18, '%PDF-1.5': 2})\n"
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          "text": "\nTHEOREM 046 17 \n\\begin{theorem}\\label{thm:main}\nThere exist a smooth connected projective complex fourfold $X$ and a\nsimple abelian surface $A\\subset X$ such that\n\\begin{enumerate}\n\\item $X$ has large fundamental group;\n\\item $\\im\\bigl(\\pi_1(A)\\to\\pi_1(X)\\bigr)\\simeq\\Z$.\n\\end{enumerate}\nThe universal cover $\\widetilde X$ contains no positive-dimensional\ncompact complex analytic subvariety and is not Stein.\n\\end{theorem}\n\nTHEOREM 065 36 \n\\begin{theorem}\n\\label{thm:main}\nLet $B$ be a smooth connected projective variety over $\\C$, and\nlet $E$ be an algebraic vector bundle of rank $r\\geq2$ on $B$.\nLet $X=\\mathbb P_B(E)$ parametrize one-dimensional subspaces of\nthe fibers of $E$. Then\n\\[\n                           \\V(B)\\ \\Longrightarrow\\ \\V(X).\n\\]\n\\end{theorem}\n\nTHEOREM 154 20 \n\\begin{theorem}\\label{thm:main}\nSuppose that $T$ satisfies \\eqref{eq:mixing}. For every integer\n$n\\ge2$ and every fixed $f_1,\\ldots,f_n\\in L^\\infty(\\mu)$,\n\\[\n A_N(f_1,\\ldots,f_n)(x)\n \\longrightarrow\\prod_{j=1}^n\\int_X f_j\\dd\\mu\n \\qquad\\text{for $\\mu$-almost every }x,\n\\]\nas $N$ tends to infinity through all positive integers. The probability\nspace need not be standard, and no rate of mixing is required.\n\\end{theorem}\n\nTHEOREM 164 13 \n\\begin{theorem}\\label{thm:main}\nLet $r,m\\ge1$ be integers, and fix real numbers $R\\ge2$ and $D\\ge1$.\nFor every coloring $\\chi:\\N\\to[r]$,\nthere are distinct positive integers $a_1<\\cdots<a_m$ and a color\n$c\\in[r]$ such that\n\\[\n \\chi\\left(\\sum_{j\\in J}a_j\\right)\n =\\chi\\left(\\prod_{j\\in J}a_j\\right)=c\n \\quad\\text{for every }\\varnothing\\ne J\\subseteq[m].\n\\]\nEquivalently, $\\FS(A)\\cup\\FP(A)$ is monochromatic for a set\n$A\\subset\\N$ of cardinality $m$.\nThe elements can additionally be chosen to satisfy\n\\begin{equation}\\label{eq:separated-elements}\n a_1>R,\\qquad\n a_d>R\\left(\\sum_{k<d}a_k+\\prod_{k<d}a_k\\right)^D\n \\quad(2\\le d\\le m).\n\\end{equation}\n\\end{theorem}\n\nTHEOREM 192 23 \n\\begin{theorem}\\label{thm:main}\nFor every real \\(C>0\\), there are a positive integer \\(n\\) and a\nnonconstant Boolean function \\(f:\\{-1,1\\}^n\\to\\{-1,1\\}\\) such that\n\\[\n \\sum_{i=1}^n\\widehat f(\\{i\\})>C\\sqrt{\\deg(f)}.\n\\]\nConsequently,\n\\[\n \\sup_{\\substack{n\\ge1,\\ f:\\{-1,1\\}^n\\to\\{-1,1\\}\\\\\\deg(f)>0}}\n \\frac{\\sum_i|\\widehat f(\\{i\\})|}{\\sqrt{\\deg(f)}}=\\infty.\n\\]\n\\end{theorem}\n\nTHEOREM 245 116 \n\\begin{theorem}[The $\\beta$-Barendregt--Geuvers--Klop conjecture]\n\\label{thm:main}\nLet $\\mathcal P=(\\mathcal S,\\mathcal A,\\mathcal R)$ be any pure type system.\nIf every legal expression of $\\mathcal P$, in every valid context, is weakly\n$\\beta$-normalizing, then every such expression is strongly\n$\\beta$-normalizing. No functionality assumption is imposed on\n$\\mathcal A$ or $\\mathcal R$.\n\\end{theorem}\n\nTHEOREM 351 68 \n\\begin{theorem}\\label{thm:main}\nThere are an integer \\(q\\ge10\\), a time \\(T>0\\), and a smooth Ricci flow\n\\(g(t)\\), \\(0\\le t<T\\), on the closed connected manifold\n\\(M=\\Sp^2\\times\\Sp^{q+1}\\), with smooth initial metric, such that\n\\[\n\\sup_{M\\times[0,T)}|R_{g(t)}|<\\infty,\n\\qquad\n\\lim_{t\\uparrow T}\\max_M|\\Rm_{g(t)}|=\\infty.\n\\]\nIn particular, \\(T\\) is the finite maximal existence time of this flow.\n\\end{theorem}\n\nTHEOREM 369 13 \n\\begin{theorem}\\label{thm:main}\nFor every $0\\ne u\\in\\V$,\n\\[\n\\grad u(x)\\ne0\\qquad(x\\in\\Omega).\n\\]\nIn particular,\n\\begin{equation}\\label{eq:hotspots}\n\\min_{y\\in\\partial\\Omega}u(y)<u(x)<\\max_{y\\in\\partial\\Omega}u(y)\n\\qquad(x\\in\\Omega).\n\\end{equation}\n\\end{theorem}\n\nTHEOREM 372 31 \n\\begin{theorem}\\label{thm:main}\nLet $\\Omega\\subset\\R^3$ be any bounded connected domain with\n$C^\\infty$ boundary. For $j=1,2$, let\n$\\lambda_j,\\mu_j\\in C^\\infty(\\overline\\Omega;\\R)$ satisfy\n$\\mu_j>0$ and $3\\lambda_j+2\\mu_j>0$ on $\\overline\\Omega$.\nIf\n\\[\n \\Lambda_{\\lambda_1,\\mu_1}\n =\\Lambda_{\\lambda_2,\\mu_2}\n \\colon H^{1/2}(\\partial\\Omega;\\R^3)\n       \\longrightarrow H^{-1/2}(\\partial\\Omega;\\R^3),\n\\]\nthen $\\lambda_1=\\lambda_2$ and $\\mu_1=\\mu_2$ throughout $\\Omega$.\n\\end{theorem}\ncitation-only readmes [('015', ['Citation']), ('024', ['Citation']), ('046', ['Citation']), ('065', ['Citation']), ('082', ['Citation']), ('113', ['Citation']), ('115', ['Citation']), ('127', ['Citation']), ('148', ['Citation']), ('154', ['Citation']), ('164', ['Citation']), ('192', ['Citation']), ('215', ['Citation']), ('230', ['Citation']), ('245', ['Citation']), ('296', ['Citation']), ('351', ['Citation']), ('369', ['Citation']), ('372', ['Citation'])]\nPDF headers Counter({'%PDF-1.7': 18, '%PDF-1.5': 2})\n"
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      "script": "# Record source-level evidence and exact target mappings, preserving the fixed sample.\nformal_map={'015':['DukePrimeDegree'],'024':['TotientAsymptotic','TotientCompanionZero'],'082':['TriangularHilbert'],'113':['MatchingFPRAS','MatchingEntropy','MatchingEntropyBounds','BinaryMatching','SingletonLoopMatching','TriangleFace'],'115':['ContingencyTables'],'127':['GotsmanLinial'],'148':['SelfSimilar','SelfSimilarCorollaries'],'192':['SquareRootDegree'],'210':['FoulkesHowe'],'215':['ClassicalON'],'230':['SLELowerPositivity'],'245':['TypeSystemNormalization'],'296':['FactorGeneration','RelativeGeneration'],'369':['HotSpots'],'372':['ElasticityUniqueness']}\npartial_ids={'082','210','215','230'}\ncoverage_notes={\n'015':'Main target: prime degree at least 5; arbitrary full lattices/local types; discriminant tends to infinity; Haar weak convergence and tightness.',\n'024':'Main target: explicit positive asymptotic main term, ratio tends to 1, and V(cx)/V(x) tends to c; companion count statements also included.',\n'046':'Written construction and proof of a projective fourfold with large fundamental group and non-Stein universal cover; no linked Lean material found.',\n'065':'Written proof that Virasoro constraints pass to projectivization of arbitrary rank-at-least-2 bundles; no linked Lean material found.',\n'082':'Partial/companion only: maximal hard-truncation estimate. Selected paper main theorem is annular r-variation for every r>2; no such target is supplied.',\n'113':'Main target thm_main asserts FPRAS, polynomial worst-case bit time on every random tape, relative error/failure guarantees and exact zero output. Entropy targets are additional, not substitutes.',\n'115':'Main targets exactSampling and boundedSampling include exact uniform law, almost-sure termination, polynomial expected bit time and bounded-time approximation.',\n'127':'Main target matches 8*d*sqrt(n) sensitivity bound for multilinear degree <=d, including sign(0)=1.',\n'148':'Main target matches entropy-rate/lower-Hausdorff-dimension formula with positive weights, signed unequal ratios and exact overlaps. Absolute continuity is not claimed.',\n'154':'Written proof of pointwise multiple ergodic convergence for mixing transformations; no linked Lean material found.',\n'164':'Written proof of monochromatic finite sums and products for every finite set size; no linked Lean material found.',\n'192':'Main target asserts unbounded signed square-root-degree violations and unbounded absolute-coefficient ratios.',\n'210':'Partial range: a*(a-1)<=b gives b>=30 at a=6; selected paper claims every b>=6. Cases 6<=b<30 are not covered by this Lean target.',\n'215':'Supporting/companion only: finite-volume O(n) exponential correlation decay. No canonical interacting O(3) continuum-limit or reconstructed Hamiltonian-gap target.',\n'230':'Supporting component only: positivity for gauges with the explicit small-radius formula. No finite-measure upper bound or finite expectation, and no gauge-existence conclusion in this target.',\n'245':'Main target: system-wide weak normalization implies strong normalization for arbitrary PTS specifications, including nonfunctional rules and annotated terms.',\n'296':'Main target covers single generation of II1 factors with separable predual; separate RelativeGeneration target covers the dense G-delta relative-generator locus in trace two-topology.',\n'351':'Written proof of closed Ricci-flow example with bounded scalar curvature and finite-time full-curvature blowup; no linked Lean material found.',\n'369':'Main target includes nonzero interior gradient and strict boundary-extrema inequalities for every first-positive Neumann eigenfunction, without simplicity requirement.',\n'372':'Main target asserts Lamé-modulus uniqueness with the stated positivity and smoothness. Boundary pairing uses a variational trace quotient; equivalence to the paper Sobolev trace presentation was not independently proved.'}\nupdated=[]\nfor old in sorted(rows,key=lambda r:r['family_id']):\n fid=old['family_id'];targets=[]\n for name in formal_map.get(fid,[]):\n  cp='lean/ComparatorChallenges/'+name+'.json';ch='lean/ComparatorChallenges/'+name+'.lean';cfg=configs[cp];solution='lean/'+cfg['solution_module'].replace('.','/')+'.lean'\n  declarations=[{'name':m.group(1),'line':src[ch]['content'][:m.start()].count('\\n')+1} for m in re.finditer(r'^theorem\\s+(\\S+)',src[ch]['content'],re.M)]\n  targets.append({'name':name,'configuration_path':cp,'challenge_path':ch,'theorem_names':cfg['theorem_names'],'declarations':declarations,'solution_path':solution,'solution_file_supplied':solution in src,'permitted_axioms':cfg['permitted_axioms'],'enable_nanoda':cfg['enable_nanoda']})\n status='supporting_partial' if fid in partial_ids else ('main_target' if targets else 'no_linked_lean')\n rr={'family_id':fid,'draw_rank':old['draw_rank'],'paper_title':old['paper_title'],'pdf_path':old['pdf_path'],'readme_path':old['readme_path'],'coverage':status,'coverage_note':coverage_notes[fid],'paper_abstract':paper_evidence[fid][0],'main_statement':main_paper_statements[fid],'proof_environment_count':len(paper_statements[fid]['proofs']),'example_written_proof':paper_statements[fid]['proofs'][0],'scope_path':f'lean/docs/{fid}.md' if targets else None,'targets':targets,'checker_execution':'not_run','manuscript_verification_instructions':fid=='210','pdf_sha256':src[old['pdf_path']]['sha256']}\n updated.append(rr)\nprint('coverage',dict(collections.Counter(r['coverage'] for r in updated)))\nprint('configs',len(configs),'solution modules supplied',sum('lean/'+c['solution_module'].replace('.','/')+'.lean' in src for c in configs.values()))\nprint('proof source evidence examples')\nfor r in updated[:5]:print(r['family_id'],r['example_written_proof']['path'],r['example_written_proof']['line'],r['example_written_proof']['excerpt'][:160])\n# Check content against the older blob inventories, where available.\nblob_matches=[]\nfor k,r in src.items():\n ref=pre_tree.get(k.removeprefix('preprints/')) if k.startswith('preprints/') else comp_tree.get(k.removeprefix('lean/ComparatorChallenges/')) if k.startswith('lean/ComparatorChallenges/') else None\n if ref:\n  b=raw_record(r);blob_matches.append(hashlib.sha1(b'blob '+str(len(b)).encode()+b'\\0'+b).hexdigest()==ref['sha'])\nprint('cross-catalogue Git blob matches',sum(blob_matches),len(blob_matches))",
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        "stdout": "coverage {'main_target': 11, 'no_linked_lean': 5, 'supporting_partial': 4}\nconfigs 23 solution modules supplied 8\nproof source evidence examples\n015 preprints/Equidistribution-of-Prime-Degree-Torus-Packets-with-Arbitrary-Local-Type-September-24-2026/build/adelic.tex 60 \\begin{proof}\nWrite $M=M(g_f)$ and $\\widehat M=\\widehat M(g_f)$.\nFirst identify the ambient quotient with $X_n$. Given a finite matrix\nlift $g_f$, choose a basi\n024 preprints/An-asymptotic-formula-for-the-number-of-totients-September-25-2026/build/source/collisions.tex 41 \\begin{proof}\nChoose one basic witness for each tuple under consideration.  All choices\nsatisfy \\eqref{eq:good-conditions}, by the definition of $\\mathcal G(t)$\n046 preprints/A-projective-fourfold-with-large-fundamental-group-and-non-Stein-universal-cover-October-5-2026/build/sections/02-reduction.tex 32 \\begin{proof}\nSuppose that $Z\\subset X$ has positive dimension and that\n$\\pi_1(Z^\\nu)$ has finite image in $\\pi_1(X)$.  The covering of $Z^\\nu$\ncorresponding to\n065 preprints/Virasoro-Constraints-under-Projectivization-October-5-2026/build/sections/02-conventions.tex 174 \\begin{proof}\nThe commutator of two normally ordered quadratic operators is the\nquantization of the corresponding quadratic Hamiltonian bracket, together\nwith t\n082 preprints/Annular-variation-of-the-triangular-Hilbert-transform-at-the-symmetric-point-October-5-2026/build/sections/completion.tex 8 \\begin{proof}[Proof of Theorem~\\ref{thm:count}]\nFirst let $F,G\\in C_c^\\infty(\\R^2)$, use the pair maps\n\\eqref{eq:pair-maps}, and take $G_0\\in C_c^\\infty(\\R^2)$.\ncross-catalogue Git blob matches 287 287\n",
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          "coverage {'main_target': 11, 'no_linked_lean': 5, 'supporting_partial': 4}\nconfigs 23 solution modules supplied 8\nproof source evidence examples\n015 preprints/Equidistribution-of-Prime-Degree-Torus-Packets-with-Arbitrary-Local-Type-September-24-2026/build/adelic.tex 60 \\begin{proof}\nWrite $M=M(g_f)$ and $\\widehat M=\\widehat M(g_f)$.\nFirst identify the ambient quotient with $X_n$. Given a finite matrix\nlift $g_f$, choose a basi\n024 preprints/An-asymptotic-formula-for-the-number-of-totients-September-25-2026/build/source/collisions.tex 41 \\begin{proof}\nChoose one basic witness for each tuple under consideration.  All choices\nsatisfy \\eqref{eq:good-conditions}, by the definition of $\\mathcal G(t)$\n046 preprints/A-projective-fourfold-with-large-fundamental-group-and-non-Stein-universal-cover-October-5-2026/build/sections/02-reduction.tex 32 \\begin{proof}\nSuppose that $Z\\subset X$ has positive dimension and that\n$\\pi_1(Z^\\nu)$ has finite image in $\\pi_1(X)$.  The covering of $Z^\\nu$\ncorresponding to\n065 preprints/Virasoro-Constraints-under-Projectivization-October-5-2026/build/sections/02-conventions.tex 174 \\begin{proof}\nThe commutator of two normally ordered quadratic operators is the\nquantization of the corresponding quadratic Hamiltonian bracket, together\nwith t\n082 preprints/Annular-variation-of-the-triangular-Hilbert-transform-at-the-symmetric-point-October-5-2026/build/sections/completion.tex 8 \\begin{proof}[Proof of Theorem~\\ref{thm:count}]\nFirst let $F,G\\in C_c^\\infty(\\R^2)$, use the pair maps\n\\eqref{eq:pair-maps}, and take $G_0\\in C_c^\\infty(\\R^2)$.\ncross-catalogue Git blob matches 287 287\n"
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          "text": "coverage {'main_target': 11, 'no_linked_lean': 5, 'supporting_partial': 4}\nconfigs 23 solution modules supplied 8\nproof source evidence examples\n015 preprints/Equidistribution-of-Prime-Degree-Torus-Packets-with-Arbitrary-Local-Type-September-24-2026/build/adelic.tex 60 \\begin{proof}\nWrite $M=M(g_f)$ and $\\widehat M=\\widehat M(g_f)$.\nFirst identify the ambient quotient with $X_n$. Given a finite matrix\nlift $g_f$, choose a basi\n024 preprints/An-asymptotic-formula-for-the-number-of-totients-September-25-2026/build/source/collisions.tex 41 \\begin{proof}\nChoose one basic witness for each tuple under consideration.  All choices\nsatisfy \\eqref{eq:good-conditions}, by the definition of $\\mathcal G(t)$\n046 preprints/A-projective-fourfold-with-large-fundamental-group-and-non-Stein-universal-cover-October-5-2026/build/sections/02-reduction.tex 32 \\begin{proof}\nSuppose that $Z\\subset X$ has positive dimension and that\n$\\pi_1(Z^\\nu)$ has finite image in $\\pi_1(X)$.  The covering of $Z^\\nu$\ncorresponding to\n065 preprints/Virasoro-Constraints-under-Projectivization-October-5-2026/build/sections/02-conventions.tex 174 \\begin{proof}\nThe commutator of two normally ordered quadratic operators is the\nquantization of the corresponding quadratic Hamiltonian bracket, together\nwith t\n082 preprints/Annular-variation-of-the-triangular-Hilbert-transform-at-the-symmetric-point-October-5-2026/build/sections/completion.tex 8 \\begin{proof}[Proof of Theorem~\\ref{thm:count}]\nFirst let $F,G\\in C_c^\\infty(\\R^2)$, use the pair maps\n\\eqref{eq:pair-maps}, and take $G_0\\in C_c^\\infty(\\R^2)$.\ncross-catalogue Git blob matches 287 287\n"
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        ],
        "complete": true,
        "captureStatus": "complete",
        "warnings": [
          "inventory-cache-best-effort"
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      "environmentManifestChecksum": "263853ab0d78701fbb69ec408980c6f1ca3063d0dc69bdfd99531604231e036c"
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    {
      "runId": "notebook-run-1791455866206-93",
      "executionMode": "foreground",
      "submissionIdentity": "56bd595d-1976-4c28-bfda-a20ef407baec",
      "submissionFingerprint": "f452a3d8283a3e02ee9891e4d5b0f447b703a913f3af998ea154a10a39afe227",
      "admittedAt": 1791455866206,
      "kernelEpochId": "30b941a1-2823-4570-addf-fc45a3e9f366",
      "executionInvocationId": "56bd595d-1976-4c28-bfda-a20ef407baec",
      "cellId": "cell-27b0a73d-f1a1-43d3-b0ba-0e8cdf053531",
      "source": "agent",
      "inputKind": "cell",
      "kernelKind": "python",
      "script": "import matplotlib as mpl\nimport matplotlib.pyplot as plt\nimport numpy as np\napply_figure_style()\ncounts=collections.Counter(r['coverage'] for r in updated)\nfig,ax=plt.subplots(figsize=(9,4.8),dpi=180)\nfig.subplots_adjust(left=0.34,right=0.94,top=0.76,bottom=0.29)\nlabels=['Main-result target','Supporting / partial target only','No linked Lean material found']\nvalues=[counts['main_target'],counts['supporting_partial'],counts['no_linked_lean']]\ncolors=['#0072B2','#D58B00','#767E87']\nfor j,(v,c) in enumerate(zip(values,colors)):\n ax.hlines(j,0,v,color=c,linewidth=4)\n ax.scatter([v],[j],s=95,color=c,zorder=3)\n ax.text(v+0.45,j,f'{v}/20 ({v/20:.0%})',va='center',fontsize=11,color='#222222')\nax.set_yticks(range(3),labels,fontsize=11)\nax.set_ylim(2.55,-0.55);ax.set_xlim(0,14)\nax.set_xticks([0,2,4,6,8,10,12]);ax.tick_params(axis='x',labelsize=10)\nax.set_xlabel('Number of sampled papers',fontsize=11,labelpad=8)\nax.tick_params(axis='y',length=0,pad=12)\nfor spine in ['top','right','left']:ax.spines[spine].set_visible(False)\nax.grid(axis='x',color='#E8E8E8',linewidth=0.6);ax.set_axisbelow(True)\nfig.text(0.06,0.92,'11 of 20 papers have main-result Lean targets',fontsize=13,ha='left')\nfig.text(0.06,0.85,'One fixed representative per research family · seed 20261008',fontsize=11,ha='left',color='#444444')\nfig.text(0.06,0.14,'Written proof materials: 20/20 available     |     Proof checker: not run for any paper',fontsize=11)\nfig.text(0.06,0.065,'Statement coverage was inspected; successful proof verification was not established.\\nPinned commit: fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb',fontsize=10,color='#444444')\nfig.canvas.draw()\nrenderer=fig.canvas.get_renderer()\ntexts=[(t,t.get_window_extent(renderer)) for t in fig.findobj(mpl.text.Text) if t.get_text().strip() and t.get_visible()]\noverlap=[(a.get_text(),b.get_text()) for i,(a,ba) in enumerate(texts) for b,bb in texts[i+1:] if ba.overlaps(bb)]\noutside=[t.get_text() for t,b in texts if not fig.bbox.contains(b.x0,b.y0) or not fig.bbox.contains(b.x1,b.y1)]\nprint('QA text overlaps',overlap,'outside canvas',outside)\nfig.savefig('proof-coverage-summary.png',dpi=180)\nfig.savefig('proof-coverage-summary.svg')\nplt.show()",
      "status": "completed",
      "startedAt": 1791455866613,
      "cwdBefore": "$DATA/notebooks/cmuzdubm90002rxoq52hkpl0k/01a11b04-b537-7300-9a27-fde070942335/data",
      "executionCount": 43,
      "environment": "default-python",
      "rootFrameId": "root-frame-01a11b04-b537-7300-9a27-fde070942335",
      "agentFrameId": "root-frame-01a11b04-b537-7300-9a27-fde070942335",
      "messageBranchId": "message-branch-01a11b04-b537-7300-9a27-fde070942335",
      "runtimeSegmentId": "runtime-segment-1791455444904-13",
      "promptMessageId": "message-7be8bde6-4ace-4a18-bf05-f1047b76da11",
      "text": {
        "stdout": "QA text overlaps [] outside canvas []\n",
        "stderr": "",
        "traceback": "",
        "plain": [
          "QA text overlaps [] outside canvas []\n"
        ]
      },
      "outputs": [
        {
          "type": "stream",
          "name": "stdout",
          "text": "QA text overlaps [] outside canvas []\n"
        },
        {
          "type": "display",
          "data": {
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eHXbYIbuoX9LKi9DtLMJgeTPPPHPWArvWGegAAED9rv7nTS2vJNZZZbFYj2232Khf16OdVU+qaVblpfD55593eGzKKafs9jnVga5oURihq3pbBHb2up093ig33HBDxf38uXorfPbZZ3Vv61bLr0+EwN5999009dRTd/nzo0aN6vS5eW+88UY6/fTTy/d32223NPvss6dmWWCBBdL0009fXrcYK3nggQda0toUAABoD2NMUCxKZX/77bfl+xEU23fffdOgQYP6db0GsrjIEa3SAKC3ZpllluxGpUceeaTi/jbbbCMkBgAATfT4U8PTE08Nb5t1Wfip4WmRhebv71VpS9GGsSSqLzXz+uSrr75aU5Wx6opneRH6qXfSTzwngkX5ilrPPPNMapaoKHX33XdXPLbMMsukVnrppZfq3tatFiGrvEcffTStvvrqnf5sBLEiBNbVc0uOOuqockguqqrtvffeqdnivc23Gb3jjjsExQAAYAwyxrSe3GSTTdKEE05Yvh9ltI877rh+XScAgM6MHj264v4888xjQwEAQJN8/Mmn6Zp/3tRW2/ea62/O1mugiOpK0Tpv2LBh6eGHH85CPx988H/s3QXYFNX///8joNKgpKQgIQiihAIWBigq2CJgI3Z3YHegGFhgd35QLERFARUFSRGxlZSWEkW4/9fr/P6735nZmtl7c+7n47r2gt17Z3f2TJ95n/f7r4x/z+LFi83s2bOjz7t3726qVKlicpW9rEWLFjYbUzJ//PFHTIboTJRinDdvnsnmYKU1a9ZEn6sffeeddza59O677wZu61yvk126dHGVOX399dcTvvfVV191Pe/atWvMe8aPH28+/PDD6PMhQ4a47mFky2677RYzHwAAAADKjjKTUSwyGkeZxCIeeughc9FFFxXcyCQAAFC2OTvoJRcdxQAAAEBZ9cFHn5l162PL3uXTunXr7XwdfVhvU8g2btxoBg0aZL7++muzfn1sYFvTpk1tucTDDjssIxmLPv74Y1NSUhJ9fsABB5hsUfDUTz/9FLjMpbPcYLyAL78aNGjgCopbtGiRyRYFUnkHK1WokLtbByp9OHfu3IyUFM3mOlmnTh2z//77R4O7xowZY0aPHm369Onjet+sWbPM448/7loH9t5775hAthtvvDH6XH/v1auXyYW2bdu6nms918D66tWr5+T7AQAAAORXmQkUk6uuusqMGDEievNVKZ2V2vnhhx/O6XwsXbrUXkzqAlgdB8uWLbMj3+rWrWtHSh144IFml112saUdy6Lly5ebN99803z55Zdm/vz50YvwnXbayRxyyCGmQ4cOpth+j0YfRpa3lv+WW25pf1OzZs1s54JGlOm10vrxxx/NG2+8YVPcK7X51ltvberXr2/XqyOOOMK0atXKlEXr1q0zn376qR0dp041LYNNmzbZZaCOmh49eph99tmn1CNQtaz1HTNnzjQ//PCDWblypdmwYYMtFaAyt+3bt7ffpX+ztX1rPXvnnXfMt99+a8sTqINHwbD6fb1797bzUihUDviTTz6x+0N1BqpDqlq1arazqm/fvnZUcCoquzBq1Cib6l/70kqVKtllqu1K+4swLNNiWPe91Gk/adIk22H63Xff2Xb8559/oiOEg5bcCPtyikfLKdPLRB3/WiY///yzzVim86DatWvbEdo6DqlDOkwluXOxP9RxXSPItQ/ScVf7NR13VU710EMPjRmlDQAAgMKjALFpM/8vGKeQaL4O7tnDVK5cyRQqBeXoujNZ6UY91F+14447muuuu65UJQ2dGb50fafr/2yVYhw2bFjMAJ4TTzwx6XRr1661AUBO6V5neadzlqHMNF03OWlZ5XIdUh990LbO1zp56aWX2lKNkX4ODUSfMGGC2XfffW3/7uTJk83zzz9v+yWc9yWcmchE9ymU8SxSQvX66683uaI+4nLlytlr2EifgYLbMhHMCQAAAKDwlalAMd0M1YXbTTfdFH1t5MiR5rLLLrMBO9mmQIgrr7zSvP3229GLsHh04aiRTZrP448/3l60FSJ1euy3336u1/73v//ZAIRENDrp5JNPjj5XIJNGAkY+T79ZWd908zqea665xgbSPfjggzkrw6Xgq3POOcc1Aq1y5cpm+PDhSedBow41v7opn2x5a+SYbtJruZ955pm2TdKZx7POOivalvFo3VN69KFDh5q99trLlAVK9X/DDTeYF154IaaTzunuu++2y/S8884zl19+udl22219f8eKFSts58/LL79sgxKSLesIBT2qE+qYY44JFLSiTjOtTxEnnXSSGTx4sP2/AmS0/ijAMp7HHnvMBmFde+215pJLLsnJfiXZ/CpA7Nxzz7WBXl4aianSwNq/PPnkk3Z/6KVAl/PPP9+89957cb9bozYVrKHtVKWHg8j2Mr3vvvtsZ6REOgQjFNSy5557Jp1eQSn6bflc95Mt23Hjxtlj7YwZMxJ22Do9++yzrlG2Gg2ueS+kbS8XtAx03IjwjqRWmz/66KNxp33xxRdNkyZN4v5N7fL000/bDudI8HU8DzzwgN0vaHu5/fbbbYCxXxqFrWC9iEceecQG5QWhmzuRTnZ57rnnkp6b5Xt/qN+raTWfidY9zWPz5s3tvwMGDDBlyfRZc8yM2XNjboQAKIAgZGVg2WILtk+ggLBt5p/6exQUVIg0X1OmzzJ7dw/HAAQNbFRfp/oDLrjggsDTKwhL138RKo2ogbfZoOsa9bc5qe8t1fdpwJZXOn198TJL67MV0JON61nvNej2229vckX9JN7+IT9tna91cocddjC33Xab7UfRPlzLRP08kb4eL12rarCSkwYaaR1zvieXba6BluqPdmapU1sQKAYAAACUDWUqUEx0U0+BA5EbmrphrZunulGdTQp+uuKKK3x3PGlUk2566mbmW2+9ZYPcCo1ujH7++eeu15w3eRN16DiniXSUKPuLsv8kuqnrpBvD3bp1Mx999JHNvJZNX3zxhc1s5LwBrkA4BbIkChLTMr7wwgttpjpnKvxUGUk0zTPPPGMDCRs1auR7HhU0csIJJ7hGqSWiEW0a3aYOGGUYCzMFPajDJtU66ewYVnCS9gVaBkpBn4o6sXbddVff3xGhsgH9+vWzgZVPPfWU75Jy6rRzbj/KkCTqiBo4cGDK+VA2RbWJMt/od2b7Bn6i+VVHmDrgUgX2KJhMWcU0StMZtKLXFczy119/JZ1+8eLF5qijjrJBSJEAkkJYpgpy8+47nfsCb4kKr1Tbei7W/UTLVvsWHWeT7fu8f1NQm/Oz/Oz/cr3t5YJG7iZaL0SZ0vSIJ14pDdG6pPIX2vf7oW1SxxSdd2jkvEp2+xHJmBmRatuMR7/duTzj3VwplP2hAvoOOuggm5UtlV9++cXOjzLeOW8ChN36vzeY5StW5ns2AAAAQmHS5Glmz65dTLlyhTXYxUklCpVVV1l79X9dE8ybNy/uQFBdd+i6VYIGi+m82jkYKltlJ5X5XINCvAOPzjjjjJTTegdHSbrXnvGm0+9PN/AsmQULFrieK4goF9TWzsFjQdo6n+vk4YcfbrOla6BbokFZymp98cUX235bLw0oisyLMr4nu/5Wv7kCy/R+DfDTQLFMlAXVIERnoFiywWUAAAAAwqXMBYrpAk0BW7o5GKGMK8q2pHJn2aDvu+uuu9IOVNJIHgVG6KIxjNTBofJMfoLEIhS4pWlU0kzLNBt0s1kjypxBGQoOe//99+3IsUSBcEcffbQry0kQuvmsMlW66a7OjFRUcq9///6BRr5qpJsyMQUNsCgWCkLR73vooYfSml4dJHvvvbddhqkyrymQoTTtqExIKreoLFLpjgZVwORxxx0XaB1Q9qHWrVvbzEq5pgAdZejzG0SpjjAFe02ZMsWm7//mm29s8GaqIBInZRbq2LGj6dSpU8r3FsIyLYZ1Px4dS9UB6mc+S6uYl1OuKOBcWfkUqBSUbmxoO1Xwl7JeFotc7A+VrUw3o5TRLggF/m+zzTaBpgEAAABEAfgLFi4yjRs1KKgGUbYn9c0p878yCnuDl9T/pME66l/TgBTvoCcF5qgvtmfPnoHO+Z2yESimEozeAVBVq1a186t+iXSkO1AvVxl6ly1bFrN8cpHNK9Ntnet1UoNxNcBRZS7Vr60Biwo6U8lQDcLT+qlgsngBjwqQixgyZEjcoMB3333XPPHEE/Y61NlG+kwNXtKg49IsJ++06oMDAAAAUDaUuUAxUSYbZT2JjJjRBZwuyN58882Mf9crr7wSN0hMF4wKKNJNeY3QUmYLZcd47bXXbCCEkzKIKBuFgsUKtQxlaVx99dU2IE40GkoX9L169bKjo3RBr9FMKi+nC3jnRbFGuukGtkopZprWj0svvdSV8UhZzJRtJ1l2t9NOOy0mSExBCPvss4+9Ya/OBt0o1k14zf/EiRPNq6++6gp60Xp57LHH2k6DZJ0iahdlnfPeEFfqcGULU6eGggvV6aEMRgrQ0Wc62z2M7rjjjriBMh06dLBp3lWWQNufOtsUBKGgPG2nzmw4GqGnYAMF7gXpcFGGu913391+x3bbbWc7btQJpYACZQtSJjp17jhp3VaWQ+2Xglq6dKktZxZZB1SiUUFVCjZU5jsFLur7lKlOZV+dlCJfIxpzUXY3Qm186qmnRrdjjRBVCUB13mm7UHYedfoqu49zhLB+g4IsVLZW2aAi24uWo7YVlWrUflRZlRRQNmLECNeISG3HGhGq7S2oXC/TYl331Zl4//33u7KCaV1U22nZbLXVVtG/ZSNYppiWUy5on6DlGC9IrGvXrjYjX6tWrWwHvDqy1amtY+yqVatc71U5aGVu00jpQpeL/aGCE9Wu3iAxnZupk17nLyoVouO+MuUpsFzrni0nZYzN3BfG8zgAAABk3/yFiwsmUEzXW7fccos933Ze63np2lPXaHroel4DmzTg03sdq+zAfoKCdK6va5cI9RvquibTA24GDRrkytqsvsoHH3zQd1nAeAE/6Q508gYyqU2zkU1MA6m8dL2YTfHaWutBkLbO9zopWh4KTNPDD/V36Vo7QvcG1A/ufY8C6NRvEY+yo6k/R4Fk6gPTdW86qlSpknI9AAAAABBOZTJQTIE0CgxTtowIBdEowMBPyS2/dPNdmWy8lKVKI5Ti3SxX9jFlZTn99NNdF8oqvRYp6RUmuvDV75IuXbrYtO7K7OGli/cJEybYm7DOi9Ynn3zS3HrrrRkrH6aAEo3GUqeEk26qP//883bdSURp0nWR7h1ZphKUO+64Y9xpFDRzzz332PTizmknTZpkA2OSBTBoGu8FvEaxqQ2bN28e8379LnUg6Ga4UpbHS4Vf7BRw6M0Ko7ZXua9IWTIvBRYomFOdNLfffns0iEmBLwqmGzlyZNLvVDDeeeedZzO7qZMyEQU4qdPqgw8+sGUQnenc9d16LWhnn9Z/dZQq8EAp6xXc6O0QU/YtZVDU+qJRiM5OSj3XPOXK008/bedXnawKKtL+0Rs0oXbSeq9RlxrRGqH3K7hSQY9y4okn2tdUQsBJZfa0n1QpW2dgmErUKWBIQWmpZHuZKuuWgk1EgUrKbhWhfZyOA8koEKoQ1n0n7XciATpXXXWVnZdsl3XM57aXDTruOEs2KvBYAeQR1157bUzncYSCopwUkKTjiJO2Fe0z4pUdVjC6gqV07qFMlU6aD2WYU5BhIcvF/lDnGyp76qQbGC+99JINwPNSeyroUuuolqW2sUjQGAAAABDE/IX/Nxgq33T94b0GSUWDMnTdqMFeket6+e233+yAS10Lp/L111+7rpkynU1Mg2jU1+Dsi9D1hUoLKpgn3eCb0gSKeadL1i9ZGvHKMWbzmj5RW6uPIkhb53udTIf6Wn799Vf7f12zxsturcon3iAxVblQVQ0NfIpsBxoQdcopp5i33nrLtGjRIvC8eJdxvPUAAAAAQDiV2bQGujHszRyR6QxLCgDwZudQYJCy5STLqKIbtiqH5R25dPfdd7uy7ISBbpoqOEuZYMaNGxc3SCxCN6oVPOWk9tXN/0zQxbBGnnmDxJSJSJneknXGaISfbuJ7gzDGjh2bMEgsYtttt7WBIrqwd1IAWaLyWUqTriwlTiqrp46MeEFiEQqe0TxVrlzZhJE6UpxtphF1CpZIFCgToSARBQAMGzbM9bo6jBSclIgyYqlzR4E9yQJVnJR1RpndnEEXyoSjdSyoyG9VgIP2X4lGTep1BTIqGNPJGaCUC5H5VVCFAjUSZdZRBiwFWDqpIyySnVGBK8oK5A0Si1DHmTL1eTtnvYGc+Vqm2kaVBU0Pb0lhZd+K/C3RI15QU67Xfa/IdytASQFH2Q4Sy/e2lw0KYnQuZ+/obY2UT7ROOI9POpZFArAjtM/Xvj9ekFiEsm4pq5gCo72lnr3H3kKU7f2hgvcV5O/dXnXuEi9ILEIZ2fSeoCPiAQAAAKcFCxcXfYPoGj7e4AxnlrBkPv74Y9fzTAaKKWBJAyudA4tEA9H8ZoqK0PWZt79DAT3pcFYgSBSElgnxBrRkKxtyorZW1QgN7immdTIoDcxz9nepf8t7f0LruQK/IhQApz5g9X+rr0slLs866yxXn7QGRGWitGmifmgAAAAA4VNmA8UUhKURYU66iarMXZmgbE26EekNDlBZMJUkSkWZqLwXecpQpgvCsFGgggJH/HR2KAvPDjvsEDOisLQULKA2d2ZSUYfIvffeawMoUnWOKFOSyuZFaBSXRoh5L7iT0Q3o+vXru1KwKwtSPN6b9lqfFTjjJy28Asq8634YqIyess45AyMUGKT0834p5bwCEiMUmKnAiWSdf35T0TupE8g7YjBROvlUlKlGGfdS0TrsDYbViElvCbVs07yq7G4qeo/KFzopqFTBFt5gjURZtzQqNOi+ohCWaTGs+/Hsscce5rLLLjO5UIzLKVd0PFVwl5Myp/nJmKpjlgKtnMciUUd2MWTCyub+UIFkzuwFon2RnwAw7Y+KIdgOAAAAhevPpcujWaCLmQY1KTOS0+TJkwMHiinAJ1NVITQI9aSTTjK//PKL63VlsE422CYZ78C2RYsWpR1Y5B1smg2ZLJeZTlurD1wDbottnQxKQWmRrF0aOKhBlF4qJekc6KRr9LZt27r60ZVB+/DDD4++NmPGDNs3VKgZ6wAAAAAUnjIbKBYpAem80MpkVjGVsVRglzfjS5ALLr3fG/ij0oFhoxu73tFTyXhHl82cObNU3//DDz+Ybt26uS6o1UGiQIuLLrrI12e8+OKLMWUeg15ca1mrLZwSBS56s4kpA4wy7Pilzq5kWe2KkUq2OqmDKZ0MLhrN55Sp4FEvb7BUup1Q11xzje/39u7dOya4prTbT1CpyipGKKC2Z8+ecQOa/G5byiCVy9+aqWVarOu+lq2fQOh8y9dyyhXveULt2rXtPj/IDQ1vmWtll5s+fbopdNncH3qPu8oW6g1GTUZlQ3WuAQAAAKRDA6f0CANvdl/nwMtEvv/+ezNv3rzocw32DDI4M5E1a9bYwCV9vpMCeFQaMV3eQa7egC+/vAFmyaoIlEa8wbvKVJVJydr6jDPOMMW2TgalAX6qBOG8fvX2bynbmrOPQn3gifrM1ffs9P777weeJ+8yDmsFCgAAAACxynSgmDJKKMuG0xdffJGRYCylgXbSzUhlwwoaOOS9oe393DAIOjpP5bmcvAF5QWh5d+/e3WYSiVBZMo1S9JP1KHJR7c1UpBvQ6ejYsaPr+bRp02Leo86K3377LSboMQgFwh1zzDEmTLxBLdlcBpnQoEEDVwlBlVaMjCr0SyMegwQI6vu8oyRLs/0EpdKyqUqxOrVp0ybmtSBlCLzTK3NgNkdgZ2KZFuu6r3Kf6X5vruVrOeWK9zyhX79+CcswJqLOe2/QX6Gff2R7f+gdna0y4UGpvAoAAACQrv/+K/wsv36o7L2TMlp7Syx6qQpEpstO6jtPOeUU8+2338ZcD3kHz5Q2UEz9jkED/davXx8TYKYKBtmgAUbea0BvpupCbet8rZNB6PNUWjNCGd3jlTT1DtBSQGQijRs3dl3TptN/6V3GdevWDfwZAAAAAIpTBVPGHXXUUTZVuTKARQwZMsQcfPDBpcqM4r04U3BTtWrVAn+OSnmprKGzHKFKI2Ur1Xi+UnyX5uJ99erVaX2vyqrppq1z9JRG5mkEVqtWrXx/ztSpU2NSdZ9++un230hQip9/9fCWvdJIMq94F/5du3b1Pb/OabzlUYvV2rVrYzqb7rzzTlsuzdm+kf8n+9e7LOMtg2QdPxohqKw0c+bMsctTIybVuRcvQOm///5zPV+5cmWgTHRBt51Mbj/pCFoWwrufU6Yjb2drkOnVKat1Jci+ONfLtFjX/V133dVUqJC/U4pCX065okAn74hznUeks5/QcXDu3LlZD5rNlGzuD+O1azrZwdI5VgMAAAARFSqUPoNWIfAG4Kj/1TmYJ1XZSQ2EUaBNaWiwkLJqe69zNKjy2muvNaXlDejS9akqGgQZPKd58167ZitQTO2v6yNnJq3Fixdn5LOz3db5WieDUPnIX3/9Nbr+Xn/99XHf573uTNUHpvXhxx9/jDutH97BUiqHCQAAAKBsKPOBYnLLLbe4SpRp9M6rr75qs3Cky3uD3Vvi0q942TE02idMgWIatRaEtxynAgGCuvfee82ll17qCiDQTWaVlgo6eipe+nhnB1ZpxBu9p6xITurICdqGibI1FSt1XnlHZnqzvJUmEEdBKMky8vzxxx82O6GCD//666+0v2vVqlU221G6QQ652n7SFXR+vaUPgq7n8Uon6Pf6CRTL1zIttnU/253VYVlOuRIvuK805x/OQLEggYP5kM39ofe4m+4xNEzHXQAAAOS+KoIeYeDNkl+jRo2kA4903escIKXKAPGu9/3SICmVOvReOx966KHmtttuK9XA4WQDdlTVIEigmN7vpFKb2Rx8ohKHzkCxBQsWlPozc9HW+Vgng/YbDx8+PPp80KBBCctJegcvpeq/cv496EBQ9Yl7g8sSzRcAAACA8AlHD0MpKdXz3nvv7XrtuuuuM5s2pZ/SXdlJnLbZZpu0PifedN7PLmYanVXaC++gpeTUSaG05s7p+vTpY8aNG5dWiu1MpmL3UpBGquWvLEvpSHe6QpTNZZBoOUQoK5s6+jQ6sDSBKhK0/F1pOkYjslmKMdPzm6vfm89lWkzrvrcTNdeKaTnlSrzzg0ydfxT6uUc29w/xfns6x1CVfc7kqHQAAACUHfXq1CqYoJrSUGatzz//PFBwigZjOs/VS1N2Ute4Z599dsw86DOHDh2asWA8ZWhu0qSJ6zUNSvZLmcTefPNN12sdO3bM6sDhdu3auZ4rA1pp5Kqt87FOBnHrrbdG+xw0QO2cc85J+F7vQL2NGzf67rMJeq05b968mExqqogCAAAAoGwojCuyAqCLNu/FsLPkY2lvrvvJyBJPvIs8Z6lEZIZuAnvTuQfJupNL2Vy3ilWul0HEo48+akdGZirIJJdBWwjHMs3Xuu+V67KNxbacciVeYF+mjhFl+dwjm+0KAAAA+NGwQf1QNNQDDzwQcx3Xo0ePpNOMHTs2+n8Fy+2///5pfbf6/S644ALz6aeful5XGUvNV6YySEX06tXL9VwlAj/55BNf044aNcqV3Ssy0DmbvEFCzgzThd7WuV4n/ZowYYL54IMPos+HDBmStP/EO2DLWxrSy/n3oIOZvMtXGbfJKAYAAACUHYVzVZZne+65p+ndu7d5//33o6+pnNXxxx+f1k0978VZ0PTPEfGypKSbHQT/j5bn+eefb+6+++5ok0ycONF2NI0ZMybw6DzvslZ5vP/9738Zae54nQfezD2ZXLeKVbzOEJURzVTWtMqVK8e89vvvv9vMdPGy2uy7775ml112MY0bNzb16tWzy1HrncoEOGmfUyiBPijOZZqPdT/finE55Uq85a5jRP369Ut9jMj2uUchB+vFy5indg16vqDfGMb1DgAAANnXqMF2BdHMK1asSDur1csvv2yeeuop12u6Vjv44IMTTqPz56+++ir6fOedd06rGsDmzZvNpZdeaj788EPX67vttpt55JFHsjKg46STTjLPPvusa+DJtddeazp16pQ0K7dKbaoso/d67OijjzbZ1K1bNxuIF7k2U6CayiYqC1Yht3Wu10m/tNxvvPFGV5BcqmA/b2nSb775xnTu3Dnh5ztLsgYpayrTpk1zPVdZ0zBkLQQAAADgD4FinqxiGuUTuSD+448/bNYSjYAKyntDdcGCBYE/I9F0BIqV3l133WWDCm644Yboa1OmTLEjxjRSUQEGfikwzNtB0aVLl6xlDfEGAqjjRiP1go7GU2dPWHiXgWgZajlki/YN69evjylZe8UVV/gOrvFb1g+5UYzLNB/rfr4V43LKlXjnBzqPUNmT0p5/JDv38JYM0Y2BIDR6u5CXSbwAPB1Dg96MWLRoUeC2AQAAAKRRgWQUu+OOO8wvv/xiBgwYYPbbbz9fg5R07nzvvffGHVR5zDHHmObNmyecdvz48a5rhXTKTqqf95prrrGDqpw6dOhgRowYkbUM2Qqw6t+/v3nmmWdcQWB6beTIkXEDsH766SczePDgmIE7yqhdrVo1k+3+hZ122skVfDRp0iRz5JFHFnRb53qd9OuJJ54wv/76azQj9fXXX59yGgV7af5XrVpln6v86GmnnRYz+E002F1lM52BfkE4AzAzmUUNAAAAQHEgUMxh1113NUcddZR5/fXXo69pBJcuyBRUFIQ3VbOCkNLhnU4X1OlkBkEsXaArrbZGuUXMmjXL7LPPPuajjz4yjRo18tVs3s4D3QT++uuv7UixbGjRooXr+T///GO+++47O6oyiKlTp5qw2G677ey24UwVr86sbAbLvPPOO67nylLnHCmYyvLlyws6MKIsKsZlmo91P9+KcTnlim40qAPa+ft0HqFMa0Goc997jEhWgkLHUidnZ7Uf8+bNM4VMGeq87ar2adeuXZk97gIAACB3am27jWlYIBnFdK2gTER6KHhFfVEKblFfVa1ataLXBrom+O2332xGpC+//NJs2rQp5rOaNm0aN1u0k/rnnNIJFLvzzjvNq6++6npNQVfq750xY0bC6ZRFWPOt31m9evWYeffTb3juuefagB5nGUmV/FNVg759+9psUQrQUglBtZMGMGswqFPLli3NiSeeaHJB7esMFPv8888DBYql29bJpGrrXK+TfigQ7eGHH44+HzRokK+yjhoErEA1BdVFAgfvu+8+V991ZGCXAuQi1C90xBFH+J4/tYVzOWvwV9B+AwAAAADFjUAxj5tvvtmOJopcLOpCftiwYXY0VBDeUTz6HAUPKc12EG+99ZbruW7+B80chcR08a8gwLPPPjuaSU4dNgry+vjjj32NINONYqW9d3b6vPbaa1kLFGvfvr3NmuPMqPPuu+8GDhTzBlsUM93A32OPPVwdiFoG5513Xta+MzIqMOLUU08NNL1GxaKw5GOZljatfz7W/Xxj20tMmSw7duxogwUjNJL7sssuC9TG48aNiylr3L1794Tv95ZNUXnQMO0P1a4qaarzOOdxN+jNmjAdd1OpXKmivaEZb+Q7gPyx17i65tliC7ZPoICwbeaf+lfWrnNnLS4kXbvsasqV26Ig191IgE5QGgSrsozJsvQqYOrTTz+NPm/SpEla2ZIVFBQvUKY0180XXXSRDQJLRb/vwQcfNCeffLJrgJcGoWigsnOwcjzK7Dx8+PCsVS3wOvzww839998f7SPVteHGjRvNlltuWfBtnYt1MkjVkki/rQZ0nXPOOb6nPf30082oUaPM0qVL7XOV65w9e7Y57LDDbMCiBgs//fTTZuXKla4BdMnKmXqpz9sZkKj7GAxMBwAAAMoWIo48NOLohBNOsBdcEffcc48NJApCN+41GsdZYkgX9kECxXRx7b2w3XPPPQPNB1I788wzbbDYKaecEg0Q1Aizvffe2wZfaJ1IFejRs2dP88ILL0RfUwr5yy+/3HdWsiB001UX8Lqod6Yz1+gyvx03Wq+8KcaL3YEHHugKlpkwYYJtI43SzDR16K1bty5lCcBknCMLkX/5WqYVK1aMyRBYyOt+vrHtpaYgZWegmNaHmTNnBgom1vlKvONOIttvv70rA6q+P8h50+OPP26KoV2dgWIK5FfZGL+d6Sod8vLLL5uyYpf2bUyXjjtn5CYLgMxZsWJF9GYv2ydQONg282/d+r/NbUOHx2RyKgQaLNp5l/YmTA4++GBz0003JS1vL5MnT3aVYEwnm1ghUNawp556yl4jaXv3S32KulbaYYcdTC6zKWt+1fai9lf/tPpIw8zvOunHxIkTbWa4iCFDhgQqualzNAXrKQtbJNhMg6sSDbDSvOu9QTjnT4JkIwMAAAAQDuXyPQOF6IYbbrBZWpw39+66665An6Ebh71793a99txzz7lGwqW6Ga4AJicFngXNcAN/FByoG7jOQCul8VYZSt1gT+WCCy5wPdcowYEDB6YV9OGHgtqcfv75ZxvQ6IeC4YIGPhYDbRtKZ++kjpJFixZl/Lu0f/AG+DhTtqeioEJv+QTkV76WqXeddWYmLMR1P9/Y9vwdH7yZ6s4444y4ZTXiUaasN9980/WaRi4nC5xUFjMnjX7WuZMfCnSO3IQoZN7jro7v3mN/Mgoed97kAgAAQOGoUrmS2XXnnUwh0nxVruw/yCTblNUrnWBj9bcpoEXXzsqw5ScgJxNlJwuFKkSoBOWxxx7r6nOOR1UEVKpQGYlbt25tck39mcmqXRSaXK6Tfvrzb7zxRteAIw3uC2r33Xe389W2bduE71GWOWVaU1CZ7hn4pWv1zz77LPpcbad2AAAAAFC2kFEsjqZNm9o0zw899FD0tQceeCDQ6B/RDUTdcI1Q2u6jjjrKdnTsuuuuCafTzUd1HHgDlA455JCcjiIra44++mjbGaNltGHDhmjQRo8ePexIq2TZ4NThc+ihh7rKSmmk10EHHWSef/5507Bhw0DzoqwlutF+2223JZxXpV6PpCGXa6+91jRr1swcd9xxCT9XgQK62e3MNhMW6tjQNnfLLbdEX1NmOC0/BQEm2+bi+emnn8yIESPM1VdfHTd9uzrrZsyY4RohqA4grUPJvPTSS7bDD4UnH8tUHZpOysoUpKxDPtb9fGPbS65NmzY2y+WHH34YfU37fAVEq4xGsvLVX3zxhenfv3/M66kConTDRuuLs7SIOqwVIJ+svKpKffstIZJvO+20k90fKENbxKuvvmpatmzp2vbiufvuu+02BQAAgMJ10AH7mO++/8FmFysUVapUtvNVSFRCUI9ly5aZH3/80Q5cVKbdtWvX2usAPZSZrWrVqrZMnjJUtW/f3mY41mtBODPp16xZ02a6SofKyGtegtK1ufpydU3jvUbX7wpKg29uv/12c9lll9mBxOr31QAv9QOrz1l9h5pXDVr1DgbLJQ181oDphQsX2ucKcFOfo5ZBtto6mVRtnct1MhVdU2vwuB5ab66//vq0P0vzpyA9ZXRTYJeWhwYmqw9If1MAWt26dQN/rgaGKaDNGRiYq9KmAAAAAAoHgWIJXHPNNebJJ5+MpnjWv5H/+6UbtQo6euONN6KvKcW4yjepTKCC0ZxBAgpOeu+99+x3f//9967P0oXrvffeG+j7EZxGUCm4r2/fvtEydCtXrrQ3wfW6bhIn8sgjj9ggD3VGRKjjR6O/lPFHF97qMIl3k17ZyxSgMnbsWNsBM3fuXNtBlChQTBfwQ4cONSeeeKIrCGzAgAE2OOCSSy6xN7Uj1CGiDjbdyJ86dWpoVw1tOwrqc5ZA++GHH+xIvH79+tnl0LVr17hBn8uXL7cZpD755BP7Gcpwow5BdeDFo8BAZ1CRvlOffccdd5hevXq5lrM6YD7//HNz3333mdGjR2f8dyMz8rFMtU9w0jHi/PPPt53Hfjph87Hu5xvbXmrDhg0znTp1sp3IzoBGLWcFNSmI2TmK/ddff7XHME2nmyFOOnalKjOiYGkdc2bPnh19TaOfV69ebddl5/FI65aOQ/ouvUfP9Xetr97vLjQaqa2gcWdJoltvvdWWclY22u7du7sC4xT0ffPNN7uCyAEAAFCYqlapbA4/9EDzwqujTKE4/JBedr4KkYKe9EhWor401C82b9686HMNhCpfvnxan+XM8FQIZWH1WUceeaR9FCL1fagPITIgRoFs6tv2M0Au3bYuhnXSD62nemSKMoXtscce9pEJuv5+5ZVXos/VR3T88cdn5LMBAAAAFBcCxRLQyB+NRrrzzjtL1cCPP/64vVHo7NzQBbZuLOqhQLF69erZUU6///57wmA0pcBu0aJFqeYF/uy33342YEsj6CJlojT6TDfWlf1EASPxNGrUyI7K2n///V0353WjXDfE9VBmIr1PASAK7NLf/vzzT/tvUMoOo44aZwp4XfA/9dRT9qF1WMFmWt+0buk3OOlmt9bNfFJbBkmPHi84xlniVaUDlYlNwTEKvotQx54yu+mhDj5td1oG+m61vQJlNPIwCAXzKNOgs11nzZplM/+po0Xbq5a31iG1v3OdkA4dOtjMTZGARORfPpbpjjvuaB/O4OBHH33UjBw50jRv3tx2IHs7w19//XW7fTvlct3PN7Y9f1nFFFx+1llnuV7X+qwyklWqVLHrgv7VMch5juKkdfDhhx/2tVx0E+GII45wvaYgSj10LGrQoIENiJ8/f74NwHauu88880zGOr6zSdn5FHB90003uV5Xplg9VKpE7apgMf1O77albV2jwNM55gMAACD7dt5pRzNzpx3NrNnuwZv5mhc9yir1y3kH4yJ3NBBV/RKRwbDqZ9Rg1SDZz1F4NEBQ/VYRJ510kg2uAwAAAFD2ECiWxBVXXGFv2EeChdKhm/wqU6QgI2+WMPnjjz/sIxEFCOgm7cknn5z2PCA4jT4bN26cDWSK3OhVEJ8yjanUlP5NNJ3SjGtUoDK0eOkzlDUlU5QhRkEsmlcvdeY4s5s5KcW6OnyUqjy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          ],
          "source": "META_GREY = \"#888888\"\n\n\ndef apply_figure_style(*, frame=\"open\", font=None, sizes=(8, 7, 6), grid=False):\n    \"\"\"Set matplotlib rcParams for publication-grade output. Call once before plotting.\n\n    This sets mechanics (role-mapped font-size ladder, outward ticks, frameless\n    legends, 300-dpi save, Type-42 embedded fonts) — not a house aesthetic.\n    Frame, font and the size ladder are parameters.\n\n    frame : 'open' (bottom+left spines, default) | 'boxed' (all four) | 'none'\n    font  : sans-serif family name; None = system default sans-serif\n    sizes : (base, secondary, tick) — titles/axis-labels, legend/annotation, ticks\n    grid  : whether to draw axes.grid (default False)\n    \"\"\"\n    import matplotlib as mpl\n    if frame not in (\"open\", \"boxed\", \"none\"):\n        raise ValueError(f\"frame must be 'open'|'boxed'|'none', got {frame!r}\")\n\n    try:\n        import os, sys, glob, matplotlib.font_manager as fm\n        fdir = os.path.join(os.environ.get(\"CONDA_PREFIX\") or sys.prefix, \"fonts\")\n        if os.path.isdir(fdir):\n            known = {f.fname for f in fm.fontManager.ttflist}\n            for f in glob.glob(os.path.join(fdir, \"*.ttf\")):\n                if f not in known:\n                    fm.fontManager.addfont(f)\n    except Exception:\n        pass\n    base, secondary, tick = sizes\n    boxed = (frame == \"boxed\")\n    rc = {\n        \"font.family\": \"sans-serif\",\n        \"font.size\": base,\n        \"axes.labelsize\": base,\n        \"axes.titlesize\": base,\n        \"legend.fontsize\": secondary,\n        \"xtick.labelsize\": tick,\n        \"ytick.labelsize\": tick,\n        \"axes.linewidth\": 0.6,\n        \"xtick.direction\": \"out\", \"ytick.direction\": \"out\",\n        \"xtick.major.size\": 3, \"ytick.major.size\": 3,\n        \"xtick.major.width\": 0.6, \"ytick.major.width\": 0.6,\n        \"axes.spines.top\": boxed, \"axes.spines.right\": boxed,\n        \"axes.spines.left\": frame != \"none\", \"axes.spines.bottom\": frame != \"none\",\n        \"axes.grid\": bool(grid),\n        \"legend.frameon\": False,\n        \"figure.dpi\": 200,\n        \"savefig.dpi\": 300,\n        \"savefig.bbox\": \"tight\",\n        \"axes.titleweight\": \"normal\",\n        \"axes.titlelocation\": \"left\",\n        \"axes.labelweight\": \"normal\",\n        \"lines.linewidth\": 1.2,\n        \"patch.linewidth\": 0.6,\n        \"pdf.fonttype\": 42, \"ps.fonttype\": 42,\n    }\n    if font:\n        rc[\"font.sans-serif\"] = [font, \"DejaVu Sans\"]\n    mpl.rcParams.update(rc)\n\n\ndef set_frame(ax, style=\"open\"):\n    \"\"\"§3: set spine visibility on an existing axes. style ∈ {'open','boxed','none'}.\"\"\"\n    show = {\"open\": (False, False, True, True),\n            \"boxed\": (True, True, True, True),\n            \"none\": (False, False, False, False)}[style]\n    for side, vis in zip((\"top\", \"right\", \"bottom\", \"left\"), show):\n        ax.spines[side].set_visible(vis)\n        if vis:\n            ax.spines[side].set_linewidth(0.6)\n    ax.tick_params(direction=\"out\", length=0 if style == \"none\" else 3, width=0.6)\n\n\ndef panel_letter(ax, letter, dx=-0.18, dy=1.02, case=\"lower\", fontsize=None):\n    \"\"\"§5.7: bold panel letter outside top-left of axes. case ∈ {'lower','upper'}.\"\"\"\n    import matplotlib.pyplot as plt\n    if fontsize is None:\n        fontsize = plt.rcParams.get(\"font.size\", 8) + 1\n    s = letter.lower() if case == \"lower\" else letter.upper()\n    artist = ax.text(dx, dy, s, transform=ax.transAxes,\n                     fontweight=\"bold\", fontsize=fontsize, va=\"bottom\", ha=\"left\")\n    artist.set_gid(\"figure-style-panel-letter\")\n\n\ndef focal_palette(labels, focal, focal_color, other=\"muted\", base_colors=None):\n    \"\"\"§4.2: map labels → colors with the focal series visually dominant.\n\n    other='muted'   — desaturate base_colors (or a default cycle) toward gray\n    other='grey'    — uniform light gray for all non-focal\n    other='ordinal' — non-focal on a single light→dark gray ramp (input order)\n    \"\"\"\n    import matplotlib.colors as mcolors\n    import matplotlib.pyplot as plt\n    focal_set = {focal} if isinstance(focal, str) else set(focal)\n    n = len(labels)\n    if not focal_set & set(labels):\n        raise ValueError(f\"focal {focal!r} not found in labels\")\n    base_colors = [] if base_colors is None else list(base_colors)\n    if not base_colors:\n        base_colors = plt.rcParams[\"axes.prop_cycle\"].by_key().get(\"color\", [\"#444444\"])\n    if not base_colors:\n        base_colors = [\"#444444\"]\n    base_colors = [base_colors[i % len(base_colors)] for i in range(n)]\n    if other == \"grey\":\n        rest = [\"#BCBCBC\"] * n\n    elif other == \"ordinal\":\n        nf = max(1, n - len(focal_set))\n        ramp = [mcolors.to_hex((v, v, v)) for v in\n                ([0.55] if nf == 1 else [0.80 - 0.35 * i / (nf - 1) for i in range(nf)])]\n        rest, k = [], 0\n        for l in labels:\n            rest.append(ramp[min(k, nf - 1)]); k += (l not in focal_set)\n    elif other == \"muted\":\n        def mute(c):\n            r, g, b = mcolors.to_rgb(c)\n            m = (r + g + b) / 3\n            return mcolors.to_hex((0.3 * r + 0.7 * m, 0.3 * g + 0.7 * m, 0.3 * b + 0.7 * m))\n        rest = [mute(c) for c in base_colors]\n    else:\n        raise ValueError(\"other must be 'muted', 'grey', or 'ordinal'\")\n    return [focal_color if l in focal_set else rest[i] for i, l in enumerate(labels)]\n\n\ndef bar_with_points(ax, x, ymat, labels, colors, jitter=0.08, show_points=True,\n                    errorbar=None, point_alpha=0.5, point_size=8):\n    \"\"\"§6.1: bar = mean; optionally overlay raw points or draw an interval.\n\n    colors   : per-label color list (e.g. from focal_palette)\n    x, ymat and labels must have equal lengths; mismatches raise before drawing.\n    Each group must be a nonempty, finite, one-dimensional real numeric array.\n    Requested SD/CI intervals require at least two observations per group.\n    errorbar : None | 'sd' | 'ci95' — drawn only when show_points is False.\n               'ci95' is the t-distribution 95% CI of the mean\n               (half-width t_{0.975,n-1} · s/√n); correct at small n where the\n               z-approximation (1.96·s/√n) is markedly too narrow.\n    \"\"\"\n    import numpy as np\n    x = np.asarray(x)\n    if x.ndim != 1 or not np.issubdtype(x.dtype, np.number):\n        raise TypeError(\n            \"bar_with_points x must be one-dimensional numeric positions; \"\n            \"pass category names through labels\"\n        )\n    ymat = list(ymat)\n    labels = list(labels)\n    if len(x) != len(ymat) or len(x) != len(labels):\n        raise ValueError(\"bar_with_points x, ymat and labels must have equal lengths\")\n    if colors is not None and not isinstance(colors, str):\n        colors = list(colors)\n        if not colors:\n            colors = None\n    if x.dtype.kind not in \"iuf\" or not np.isfinite(x).all():\n        raise ValueError(\"bar_with_points x must contain finite real positions\")\n    for i, y in enumerate(ymat):\n        if np.ma.isMaskedArray(y) and np.ma.getmaskarray(y).any():\n            raise ValueError(f\"bar_with_points group {i} contains masked observations; resolve exclusions explicitly before plotting\")\n    ymat = [np.asarray(y) for y in ymat]\n    for i, y in enumerate(ymat):\n        if y.ndim != 1 or y.size == 0 or y.dtype.kind not in \"iuf\" or not np.isfinite(y).all():\n            raise ValueError(f\"bar_with_points group {i} must be nonempty, one-dimensional and finite real numeric data\")\n    if errorbar and not show_points:\n        if errorbar not in (\"sd\", \"ci95\"):\n            raise ValueError(\"errorbar must be None, 'sd', or 'ci95'\")\n        if any(y.size < 2 for y in ymat):\n            raise ValueError(\"SD/CI cannot be estimated from one observation; use raw points without an interval\")\n    means = np.array([np.mean(y) for y in ymat], float)\n    err = None\n    if errorbar and not show_points:\n        if errorbar == \"sd\":\n            err = np.array([np.std(y, ddof=1) for y in ymat])\n        elif errorbar == \"ci95\":\n            from scipy.stats import t\n            def _hw(y):\n                n = np.asarray(y).size\n                return t.ppf(0.975, n - 1) * np.std(y, ddof=1) / np.sqrt(n)\n            err = np.array([_hw(y) for y in ymat])\n    ax.bar(x, means, color=colors, width=0.7, edgecolor=\"none\",\n           yerr=err, error_kw={\"elinewidth\": 0.8, \"capsize\": 0})\n    if show_points:\n        for xi, ys in zip(x, ymat):\n            ys = np.asarray(ys)\n            if ys.ndim == 1 and ys.size > 0:\n                jit = (np.random.rand(ys.size) - 0.5) * 2 * jitter\n                ax.scatter(np.full(ys.size, xi) + jit, ys, s=point_size, color=\"black\",\n                           alpha=point_alpha, zorder=3, linewidths=0)\n    ax.set_xticks(x); ax.set_xticklabels(labels)\n    return ax\n\n\ndef strip_with_median(ax, groups, values, colors=None, jitter=0.12):\n    \"\"\"§6.1: jittered points + bold horizontal median tick per group.\n\n    groups and values must have equal lengths. Mismatches raise before drawing.\n    colors=None or an empty color sequence supplies default gray; a nonempty\n    color sequence cycles across groups.\n    Each group must be nonempty, one-dimensional and finite real numeric data;\n    resolve missing observations explicitly before plotting.\n    \"\"\"\n    import numpy as np\n    labs = list(groups)\n    values = list(values)\n    if len(labs) != len(values):\n        raise ValueError(\"strip_with_median groups and values must have equal lengths\")\n    for i, y in enumerate(values):\n        if np.ma.isMaskedArray(y) and np.ma.getmaskarray(y).any():\n            raise ValueError(f\"strip_with_median group {i} contains masked observations; resolve exclusions explicitly before plotting\")\n    values = [np.asarray(y) for y in values]\n    for i, y in enumerate(values):\n        if y.ndim != 1 or y.size == 0 or y.dtype.kind not in \"iuf\" or not np.isfinite(y).all():\n            raise ValueError(f\"strip_with_median group {i} must be nonempty, one-dimensional and finite real numeric data\")\n    colors = [] if colors is None else list(colors)\n    if not colors:\n        colors = [\"#444444\"]\n    for i, ys in enumerate(values):\n        c = colors[i % len(colors)]\n        ys = np.asarray(ys)\n        jit = (np.random.rand(ys.size) - 0.5) * 2 * jitter\n        ax.scatter(np.full(ys.size, i) + jit, ys, s=10, color=c, alpha=0.6, linewidths=0, zorder=2)\n        m = np.median(ys)\n        ax.plot([i - 0.22, i + 0.22], [m, m], color=\"black\", lw=1.6, zorder=3)\n    ax.set_xticks(range(len(labs))); ax.set_xticklabels(labs)\n    return ax\n\n\ndef goodness_arrow(ax, text=\"higher = better\", loc=\"upper left\", axis=\"y\", fontsize=None):\n    \"\"\"§3.6: small upright direction-of-goodness cue in the margin.\"\"\"\n    import matplotlib.pyplot as plt\n    if fontsize is None:\n        fontsize = plt.rcParams[\"legend.fontsize\"]\n    pos = {\"upper left\": (0.02, 0.98), \"upper right\": (0.98, 0.98),\n           \"lower left\": (0.02, 0.02), \"lower right\": (0.98, 0.02)}[loc]\n    ha = \"left\" if \"left\" in loc else \"right\"\n    va = \"top\" if \"upper\" in loc else \"bottom\"\n    arrow = \"↑ \" if axis == \"y\" else \"→ \"\n    ax.text(pos[0], pos[1], arrow + text, transform=ax.transAxes,\n            fontsize=fontsize, color=META_GREY, ha=ha, va=va)\n\n\ndef two_tier_label(name, meta):\n    \"\"\"§5: two-line label string (name / metadata). Meta line styled separately by caller.\"\"\"\n    return f\"{name}\\n{meta}\"\n\n\ndef end_of_line_labels(ax, xs, ys, labels, colors=None, dx=0.01, fontsize=None):\n    \"\"\"§6.3 / §7.3: label each line series at its right end instead of a legend box.\n\n    Each label requires a nonempty, matched x/y series. Validate all series\n    before drawing. Nonempty colors cycle; None or empty uses default text color.\n    Inner sequences must support len() and indexing; they are not copied.\n    Nonfinite endpoints are rejected rather than producing an invisible label.\n    \"\"\"\n    import matplotlib.pyplot as plt\n    import math\n    xs, ys, labels = list(xs), list(ys), list(labels)\n    if len(xs) != len(ys) or len(xs) != len(labels):\n        raise ValueError(\"end_of_line_labels xs, ys and labels must have equal lengths\")\n    if any(len(x) == 0 or len(x) != len(y) for x, y in zip(xs, ys)):\n        raise ValueError(\"end_of_line_labels each x/y series must be nonempty and have equal lengths\")\n    if any(not math.isfinite(x[-1]) or not math.isfinite(y[-1]) for x, y in zip(xs, ys)):\n        raise ValueError(\"end_of_line_labels endpoints must be finite; choose valid endpoints explicitly\")\n    if fontsize is None:\n        fontsize = plt.rcParams[\"font.size\"]\n    colors = [] if colors is None else list(colors)\n    if not colors:\n        colors = [None]\n    span = ax.get_xlim()[1] - ax.get_xlim()[0]\n    for i, (x, y, lab) in enumerate(zip(xs, ys, labels)):\n        c = colors[i % len(colors)]\n        ax.text(x[-1] + dx * span, y[-1], lab, color=c, va=\"center\", ha=\"left\", fontsize=fontsize)\n\n\ndef panel_crops(fig, dpi=None, pad_px=6, bbox_inches=None, pad_inches=None):\n    \"\"\"§9.2: pixel-space crop boxes for each lettered panel in the SAVED PNG.\n\n    Returns ``{letter: (x0, y0, x1, y1)}`` in image-space pixels (origin\n    top-left, matching PIL's ``Image.crop``). Panels are detected as bold\n    single-character ``Text``\n    objects placed by :func:`panel_letter`; each panel's crop is its axes'\n    tightbbox mapped into the saved file's pixel space, padded by ``pad_px``.\n    For §3.4 composites (abutting subplots sharing an axis, letter on the\n    leftmost only) the crop unions in letterless ``sharex``/``sharey`` siblings\n    on the same grid row/col so the whole composite is covered. When no axes\n    carries a panel letter (standalone plot, or a figure-composer sub-agent),\n    falls back to one crop per axes keyed by index.\n\n    ``bbox_inches`` mirrors ``Figure.savefig`` semantics: ``None`` means\n    *consult rcParams* (so under :func:`apply_figure_style` it resolves to\n    ``'tight'``); pass an explicit ``Bbox`` only if you saved with one. The\n    boxes are clamped to the saved image extent; panels with no intersection\n    are omitted.\n\n        >>> from PIL import Image\n        >>> fig.savefig(\"fig.png\")  # bbox_inches='tight' via rcParams\n        >>> with Image.open(\"fig.png\") as saved:\n        ...     for letter, box in panel_crops(fig).items():\n        ...         saved.crop(box).save(f\"panel_{letter}.png\")\n\n    Match savefig's DPI, bbox_inches and pad_inches when overriding defaults.\n    Returned boxes are pixels, never savefig bbox_inches (inches). Passing them\n    back to savefig as inches can allocate an enormous raster.\n    \"\"\"\n    import matplotlib as mpl\n    import matplotlib.text\n    from matplotlib.backends.backend_agg import FigureCanvasAgg\n    if isinstance(pad_px, bool) or not isinstance(pad_px, int) or pad_px < 0:\n        raise TypeError(\"panel_crops pad_px must be a non-negative integer\")\n    if dpi is None:\n        dpi = mpl.rcParams.get(\"savefig.dpi\", fig.dpi)\n        if dpi == \"figure\":\n            dpi = fig.dpi\n    dpi = float(dpi)\n    if bbox_inches is None:\n        bbox_inches = mpl.rcParams.get(\"savefig.bbox\")\n    # The Open-Science notebook captures and closes pyplot figures after every\n    # cell. Matplotlib 3.11 decouples a closed Figure from its backend by\n    # replacing FigureCanvasAgg with FigureCanvasBase, while the persistent\n    # notebook namespace can still retain the Figure itself. Reattach the\n    # headless canvas before asking for saved-image geometry so this helper is\n    # safe in a later QA cell as well as the producer cell.\n    if not hasattr(fig.canvas, \"get_renderer\"):\n        FigureCanvasAgg(fig)\n    fig.canvas.draw()\n    r = fig.canvas.get_renderer()\n\n    if bbox_inches == \"tight\":\n        if pad_inches is None:\n            pad_inches = mpl.rcParams.get(\"savefig.pad_inches\", 0.1)\n        tb = fig.get_tightbbox(r).padded(pad_inches)\n        ox_in, oy_in = tb.x0, tb.y0\n        W_in, H_in = tb.width, tb.height\n    elif isinstance(bbox_inches, mpl.transforms.BboxBase):\n        ox_in, oy_in = bbox_inches.x0, bbox_inches.y0\n        W_in, H_in = bbox_inches.width, bbox_inches.height\n    elif bbox_inches is None:\n        ox_in, oy_in = 0.0, 0.0\n        W_in, H_in = fig.get_size_inches()\n    else:\n        raise TypeError(\"panel_crops bbox_inches must be None, 'tight', or a matplotlib Bbox\")\n    W_px, H_px = int(round(W_in * dpi)), int(round(H_in * dpi))\n    lettered = {}\n    seen_letters = set()\n    for ax in fig.axes:\n        for t in ax.findobj(matplotlib.text.Text):\n            s = (t.get_text() or \"\").strip()\n            if t.get_gid() == \"figure-style-panel-letter\":\n                if s in seen_letters:\n                    raise ValueError(f\"Duplicate panel letter: {s!r}\")\n                seen_letters.add(s)\n                lettered[ax] = s\n                break\n\n\n\n    if not lettered:\n        lettered = {ax: str(i) for i, ax in enumerate(fig.axes)}\n    out = {}\n    for ax, letter in lettered.items():\n        bbs = [ax.get_tightbbox(r)]\n\n\n\n\n        ss = ax.get_subplotspec()\n        for sib in fig.axes:\n            if sib is ax or sib in lettered:\n                continue\n            ssib = sib.get_subplotspec()\n            same_row = ss is None or ssib is None or ss.rowspan == ssib.rowspan\n            same_col = ss is None or ssib is None or ss.colspan == ssib.colspan\n            if ((ax.get_shared_y_axes().joined(ax, sib) and same_row)\n                    or (ax.get_shared_x_axes().joined(ax, sib) and same_col)):\n                bbs.append(sib.get_tightbbox(r))\n        bb = mpl.transforms.Bbox.union(bbs)\n\n        bx0 = (bb.x0 / fig.dpi - ox_in) * dpi\n        bx1 = (bb.x1 / fig.dpi - ox_in) * dpi\n        by0 = H_px - (bb.y1 / fig.dpi - oy_in) * dpi\n        by1 = H_px - (bb.y0 / fig.dpi - oy_in) * dpi\n        box = (\n            min(max(int(bx0) - pad_px, 0), W_px),\n            min(max(int(by0) - pad_px, 0), H_px),\n            max(min(int(bx1) + pad_px, W_px), 0),\n            max(min(int(by1) + pad_px, H_px), 0),\n        )\n        if box[0] < box[2] and box[1] < box[3]:\n            out[letter] = box\n    return out\n",
          "sourceDigest": "8020d409bfb60f0c8dfa43d0dfe7bb97f7458e1a3908fefbcb6e4551e0feae4e"
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          "helperId": "figure-style",
          "skillIdentity": "figure-style",
          "packageOrigin": "builtin",
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          "registeredGeneration": "sha256:90258343aec7bb96c2e129c39e1beb23230ee947c0db563f96913458d1795f1d",
          "exports": [
            "apply_figure_style",
            "set_frame",
            "panel_letter",
            "focal_palette",
            "bar_with_points",
            "strip_with_median",
            "goodness_arrow",
            "two_tier_label",
            "end_of_line_labels",
            "panel_crops"
          ],
          "source": "META_GREY = \"#888888\"\n\n\ndef apply_figure_style(*, frame=\"open\", font=None, sizes=(8, 7, 6), grid=False):\n    \"\"\"Set matplotlib rcParams for publication-grade output. Call once before plotting.\n\n    This sets mechanics (role-mapped font-size ladder, outward ticks, frameless\n    legends, 300-dpi save, Type-42 embedded fonts) — not a house aesthetic.\n    Frame, font and the size ladder are parameters.\n\n    frame : 'open' (bottom+left spines, default) | 'boxed' (all four) | 'none'\n    font  : sans-serif family name; None = system default sans-serif\n    sizes : (base, secondary, tick) — titles/axis-labels, legend/annotation, ticks\n    grid  : whether to draw axes.grid (default False)\n    \"\"\"\n    import matplotlib as mpl\n    if frame not in (\"open\", \"boxed\", \"none\"):\n        raise ValueError(f\"frame must be 'open'|'boxed'|'none', got {frame!r}\")\n\n    try:\n        import os, sys, glob, matplotlib.font_manager as fm\n        fdir = os.path.join(os.environ.get(\"CONDA_PREFIX\") or sys.prefix, \"fonts\")\n        if os.path.isdir(fdir):\n            known = {f.fname for f in fm.fontManager.ttflist}\n            for f in glob.glob(os.path.join(fdir, \"*.ttf\")):\n                if f not in known:\n                    fm.fontManager.addfont(f)\n    except Exception:\n        pass\n    base, secondary, tick = sizes\n    boxed = (frame == \"boxed\")\n    rc = {\n        \"font.family\": \"sans-serif\",\n        \"font.size\": base,\n        \"axes.labelsize\": base,\n        \"axes.titlesize\": base,\n        \"legend.fontsize\": secondary,\n        \"xtick.labelsize\": tick,\n        \"ytick.labelsize\": tick,\n        \"axes.linewidth\": 0.6,\n        \"xtick.direction\": \"out\", \"ytick.direction\": \"out\",\n        \"xtick.major.size\": 3, \"ytick.major.size\": 3,\n        \"xtick.major.width\": 0.6, \"ytick.major.width\": 0.6,\n        \"axes.spines.top\": boxed, \"axes.spines.right\": boxed,\n        \"axes.spines.left\": frame != \"none\", \"axes.spines.bottom\": frame != \"none\",\n        \"axes.grid\": bool(grid),\n        \"legend.frameon\": False,\n        \"figure.dpi\": 200,\n        \"savefig.dpi\": 300,\n        \"savefig.bbox\": \"tight\",\n        \"axes.titleweight\": \"normal\",\n        \"axes.titlelocation\": \"left\",\n        \"axes.labelweight\": \"normal\",\n        \"lines.linewidth\": 1.2,\n        \"patch.linewidth\": 0.6,\n        \"pdf.fonttype\": 42, \"ps.fonttype\": 42,\n    }\n    if font:\n        rc[\"font.sans-serif\"] = [font, \"DejaVu Sans\"]\n    mpl.rcParams.update(rc)\n\n\ndef set_frame(ax, style=\"open\"):\n    \"\"\"§3: set spine visibility on an existing axes. style ∈ {'open','boxed','none'}.\"\"\"\n    show = {\"open\": (False, False, True, True),\n            \"boxed\": (True, True, True, True),\n            \"none\": (False, False, False, False)}[style]\n    for side, vis in zip((\"top\", \"right\", \"bottom\", \"left\"), show):\n        ax.spines[side].set_visible(vis)\n        if vis:\n            ax.spines[side].set_linewidth(0.6)\n    ax.tick_params(direction=\"out\", length=0 if style == \"none\" else 3, width=0.6)\n\n\ndef panel_letter(ax, letter, dx=-0.18, dy=1.02, case=\"lower\", fontsize=None):\n    \"\"\"§5.7: bold panel letter outside top-left of axes. case ∈ {'lower','upper'}.\"\"\"\n    import matplotlib.pyplot as plt\n    if fontsize is None:\n        fontsize = plt.rcParams.get(\"font.size\", 8) + 1\n    s = letter.lower() if case == \"lower\" else letter.upper()\n    artist = ax.text(dx, dy, s, transform=ax.transAxes,\n                     fontweight=\"bold\", fontsize=fontsize, va=\"bottom\", ha=\"left\")\n    artist.set_gid(\"figure-style-panel-letter\")\n\n\ndef focal_palette(labels, focal, focal_color, other=\"muted\", base_colors=None):\n    \"\"\"§4.2: map labels → colors with the focal series visually dominant.\n\n    other='muted'   — desaturate base_colors (or a default cycle) toward gray\n    other='grey'    — uniform light gray for all non-focal\n    other='ordinal' — non-focal on a single light→dark gray ramp (input order)\n    \"\"\"\n    import matplotlib.colors as mcolors\n    import matplotlib.pyplot as plt\n    focal_set = {focal} if isinstance(focal, str) else set(focal)\n    n = len(labels)\n    if not focal_set & set(labels):\n        raise ValueError(f\"focal {focal!r} not found in labels\")\n    base_colors = [] if base_colors is None else list(base_colors)\n    if not base_colors:\n        base_colors = plt.rcParams[\"axes.prop_cycle\"].by_key().get(\"color\", [\"#444444\"])\n    if not base_colors:\n        base_colors = [\"#444444\"]\n    base_colors = [base_colors[i % len(base_colors)] for i in range(n)]\n    if other == \"grey\":\n        rest = [\"#BCBCBC\"] * n\n    elif other == \"ordinal\":\n        nf = max(1, n - len(focal_set))\n        ramp = [mcolors.to_hex((v, v, v)) for v in\n                ([0.55] if nf == 1 else [0.80 - 0.35 * i / (nf - 1) for i in range(nf)])]\n        rest, k = [], 0\n        for l in labels:\n            rest.append(ramp[min(k, nf - 1)]); k += (l not in focal_set)\n    elif other == \"muted\":\n        def mute(c):\n            r, g, b = mcolors.to_rgb(c)\n            m = (r + g + b) / 3\n            return mcolors.to_hex((0.3 * r + 0.7 * m, 0.3 * g + 0.7 * m, 0.3 * b + 0.7 * m))\n        rest = [mute(c) for c in base_colors]\n    else:\n        raise ValueError(\"other must be 'muted', 'grey', or 'ordinal'\")\n    return [focal_color if l in focal_set else rest[i] for i, l in enumerate(labels)]\n\n\ndef bar_with_points(ax, x, ymat, labels, colors, jitter=0.08, show_points=True,\n                    errorbar=None, point_alpha=0.5, point_size=8):\n    \"\"\"§6.1: bar = mean; optionally overlay raw points or draw an interval.\n\n    colors   : per-label color list (e.g. from focal_palette)\n    x, ymat and labels must have equal lengths; mismatches raise before drawing.\n    Each group must be a nonempty, finite, one-dimensional real numeric array.\n    Requested SD/CI intervals require at least two observations per group.\n    errorbar : None | 'sd' | 'ci95' — drawn only when show_points is False.\n               'ci95' is the t-distribution 95% CI of the mean\n               (half-width t_{0.975,n-1} · s/√n); correct at small n where the\n               z-approximation (1.96·s/√n) is markedly too narrow.\n    \"\"\"\n    import numpy as np\n    x = np.asarray(x)\n    if x.ndim != 1 or not np.issubdtype(x.dtype, np.number):\n        raise TypeError(\n            \"bar_with_points x must be one-dimensional numeric positions; \"\n            \"pass category names through labels\"\n        )\n    ymat = list(ymat)\n    labels = list(labels)\n    if len(x) != len(ymat) or len(x) != len(labels):\n        raise ValueError(\"bar_with_points x, ymat and labels must have equal lengths\")\n    if colors is not None and not isinstance(colors, str):\n        colors = list(colors)\n        if not colors:\n            colors = None\n    if x.dtype.kind not in \"iuf\" or not np.isfinite(x).all():\n        raise ValueError(\"bar_with_points x must contain finite real positions\")\n    for i, y in enumerate(ymat):\n        if np.ma.isMaskedArray(y) and np.ma.getmaskarray(y).any():\n            raise ValueError(f\"bar_with_points group {i} contains masked observations; resolve exclusions explicitly before plotting\")\n    ymat = [np.asarray(y) for y in ymat]\n    for i, y in enumerate(ymat):\n        if y.ndim != 1 or y.size == 0 or y.dtype.kind not in \"iuf\" or not np.isfinite(y).all():\n            raise ValueError(f\"bar_with_points group {i} must be nonempty, one-dimensional and finite real numeric data\")\n    if errorbar and not show_points:\n        if errorbar not in (\"sd\", \"ci95\"):\n            raise ValueError(\"errorbar must be None, 'sd', or 'ci95'\")\n        if any(y.size < 2 for y in ymat):\n            raise ValueError(\"SD/CI cannot be estimated from one observation; use raw points without an interval\")\n    means = np.array([np.mean(y) for y in ymat], float)\n    err = None\n    if errorbar and not show_points:\n        if errorbar == \"sd\":\n            err = np.array([np.std(y, ddof=1) for y in ymat])\n        elif errorbar == \"ci95\":\n            from scipy.stats import t\n            def _hw(y):\n                n = np.asarray(y).size\n                return t.ppf(0.975, n - 1) * np.std(y, ddof=1) / np.sqrt(n)\n            err = np.array([_hw(y) for y in ymat])\n    ax.bar(x, means, color=colors, width=0.7, edgecolor=\"none\",\n           yerr=err, error_kw={\"elinewidth\": 0.8, \"capsize\": 0})\n    if show_points:\n        for xi, ys in zip(x, ymat):\n            ys = np.asarray(ys)\n            if ys.ndim == 1 and ys.size > 0:\n                jit = (np.random.rand(ys.size) - 0.5) * 2 * jitter\n                ax.scatter(np.full(ys.size, xi) + jit, ys, s=point_size, color=\"black\",\n                           alpha=point_alpha, zorder=3, linewidths=0)\n    ax.set_xticks(x); ax.set_xticklabels(labels)\n    return ax\n\n\ndef strip_with_median(ax, groups, values, colors=None, jitter=0.12):\n    \"\"\"§6.1: jittered points + bold horizontal median tick per group.\n\n    groups and values must have equal lengths. Mismatches raise before drawing.\n    colors=None or an empty color sequence supplies default gray; a nonempty\n    color sequence cycles across groups.\n    Each group must be nonempty, one-dimensional and finite real numeric data;\n    resolve missing observations explicitly before plotting.\n    \"\"\"\n    import numpy as np\n    labs = list(groups)\n    values = list(values)\n    if len(labs) != len(values):\n        raise ValueError(\"strip_with_median groups and values must have equal lengths\")\n    for i, y in enumerate(values):\n        if np.ma.isMaskedArray(y) and np.ma.getmaskarray(y).any():\n            raise ValueError(f\"strip_with_median group {i} contains masked observations; resolve exclusions explicitly before plotting\")\n    values = [np.asarray(y) for y in values]\n    for i, y in enumerate(values):\n        if y.ndim != 1 or y.size == 0 or y.dtype.kind not in \"iuf\" or not np.isfinite(y).all():\n            raise ValueError(f\"strip_with_median group {i} must be nonempty, one-dimensional and finite real numeric data\")\n    colors = [] if colors is None else list(colors)\n    if not colors:\n        colors = [\"#444444\"]\n    for i, ys in enumerate(values):\n        c = colors[i % len(colors)]\n        ys = np.asarray(ys)\n        jit = (np.random.rand(ys.size) - 0.5) * 2 * jitter\n        ax.scatter(np.full(ys.size, i) + jit, ys, s=10, color=c, alpha=0.6, linewidths=0, zorder=2)\n        m = np.median(ys)\n        ax.plot([i - 0.22, i + 0.22], [m, m], color=\"black\", lw=1.6, zorder=3)\n    ax.set_xticks(range(len(labs))); ax.set_xticklabels(labs)\n    return ax\n\n\ndef goodness_arrow(ax, text=\"higher = better\", loc=\"upper left\", axis=\"y\", fontsize=None):\n    \"\"\"§3.6: small upright direction-of-goodness cue in the margin.\"\"\"\n    import matplotlib.pyplot as plt\n    if fontsize is None:\n        fontsize = plt.rcParams[\"legend.fontsize\"]\n    pos = {\"upper left\": (0.02, 0.98), \"upper right\": (0.98, 0.98),\n           \"lower left\": (0.02, 0.02), \"lower right\": (0.98, 0.02)}[loc]\n    ha = \"left\" if \"left\" in loc else \"right\"\n    va = \"top\" if \"upper\" in loc else \"bottom\"\n    arrow = \"↑ \" if axis == \"y\" else \"→ \"\n    ax.text(pos[0], pos[1], arrow + text, transform=ax.transAxes,\n            fontsize=fontsize, color=META_GREY, ha=ha, va=va)\n\n\ndef two_tier_label(name, meta):\n    \"\"\"§5: two-line label string (name / metadata). Meta line styled separately by caller.\"\"\"\n    return f\"{name}\\n{meta}\"\n\n\ndef end_of_line_labels(ax, xs, ys, labels, colors=None, dx=0.01, fontsize=None):\n    \"\"\"§6.3 / §7.3: label each line series at its right end instead of a legend box.\n\n    Each label requires a nonempty, matched x/y series. Validate all series\n    before drawing. Nonempty colors cycle; None or empty uses default text color.\n    Inner sequences must support len() and indexing; they are not copied.\n    Nonfinite endpoints are rejected rather than producing an invisible label.\n    \"\"\"\n    import matplotlib.pyplot as plt\n    import math\n    xs, ys, labels = list(xs), list(ys), list(labels)\n    if len(xs) != len(ys) or len(xs) != len(labels):\n        raise ValueError(\"end_of_line_labels xs, ys and labels must have equal lengths\")\n    if any(len(x) == 0 or len(x) != len(y) for x, y in zip(xs, ys)):\n        raise ValueError(\"end_of_line_labels each x/y series must be nonempty and have equal lengths\")\n    if any(not math.isfinite(x[-1]) or not math.isfinite(y[-1]) for x, y in zip(xs, ys)):\n        raise ValueError(\"end_of_line_labels endpoints must be finite; choose valid endpoints explicitly\")\n    if fontsize is None:\n        fontsize = plt.rcParams[\"font.size\"]\n    colors = [] if colors is None else list(colors)\n    if not colors:\n        colors = [None]\n    span = ax.get_xlim()[1] - ax.get_xlim()[0]\n    for i, (x, y, lab) in enumerate(zip(xs, ys, labels)):\n        c = colors[i % len(colors)]\n        ax.text(x[-1] + dx * span, y[-1], lab, color=c, va=\"center\", ha=\"left\", fontsize=fontsize)\n\n\ndef panel_crops(fig, dpi=None, pad_px=6, bbox_inches=None, pad_inches=None):\n    \"\"\"§9.2: pixel-space crop boxes for each lettered panel in the SAVED PNG.\n\n    Returns ``{letter: (x0, y0, x1, y1)}`` in image-space pixels (origin\n    top-left, matching PIL's ``Image.crop``). Panels are detected as bold\n    single-character ``Text``\n    objects placed by :func:`panel_letter`; each panel's crop is its axes'\n    tightbbox mapped into the saved file's pixel space, padded by ``pad_px``.\n    For §3.4 composites (abutting subplots sharing an axis, letter on the\n    leftmost only) the crop unions in letterless ``sharex``/``sharey`` siblings\n    on the same grid row/col so the whole composite is covered. When no axes\n    carries a panel letter (standalone plot, or a figure-composer sub-agent),\n    falls back to one crop per axes keyed by index.\n\n    ``bbox_inches`` mirrors ``Figure.savefig`` semantics: ``None`` means\n    *consult rcParams* (so under :func:`apply_figure_style` it resolves to\n    ``'tight'``); pass an explicit ``Bbox`` only if you saved with one. The\n    boxes are clamped to the saved image extent; panels with no intersection\n    are omitted.\n\n        >>> from PIL import Image\n        >>> fig.savefig(\"fig.png\")  # bbox_inches='tight' via rcParams\n        >>> with Image.open(\"fig.png\") as saved:\n        ...     for letter, box in panel_crops(fig).items():\n        ...         saved.crop(box).save(f\"panel_{letter}.png\")\n\n    Match savefig's DPI, bbox_inches and pad_inches when overriding defaults.\n    Returned boxes are pixels, never savefig bbox_inches (inches). Passing them\n    back to savefig as inches can allocate an enormous raster.\n    \"\"\"\n    import matplotlib as mpl\n    import matplotlib.text\n    from matplotlib.backends.backend_agg import FigureCanvasAgg\n    if isinstance(pad_px, bool) or not isinstance(pad_px, int) or pad_px < 0:\n        raise TypeError(\"panel_crops pad_px must be a non-negative integer\")\n    if dpi is None:\n        dpi = mpl.rcParams.get(\"savefig.dpi\", fig.dpi)\n        if dpi == \"figure\":\n            dpi = fig.dpi\n    dpi = float(dpi)\n    if bbox_inches is None:\n        bbox_inches = mpl.rcParams.get(\"savefig.bbox\")\n    # The Open-Science notebook captures and closes pyplot figures after every\n    # cell. Matplotlib 3.11 decouples a closed Figure from its backend by\n    # replacing FigureCanvasAgg with FigureCanvasBase, while the persistent\n    # notebook namespace can still retain the Figure itself. Reattach the\n    # headless canvas before asking for saved-image geometry so this helper is\n    # safe in a later QA cell as well as the producer cell.\n    if not hasattr(fig.canvas, \"get_renderer\"):\n        FigureCanvasAgg(fig)\n    fig.canvas.draw()\n    r = fig.canvas.get_renderer()\n\n    if bbox_inches == \"tight\":\n        if pad_inches is None:\n            pad_inches = mpl.rcParams.get(\"savefig.pad_inches\", 0.1)\n        tb = fig.get_tightbbox(r).padded(pad_inches)\n        ox_in, oy_in = tb.x0, tb.y0\n        W_in, H_in = tb.width, tb.height\n    elif isinstance(bbox_inches, mpl.transforms.BboxBase):\n        ox_in, oy_in = bbox_inches.x0, bbox_inches.y0\n        W_in, H_in = bbox_inches.width, bbox_inches.height\n    elif bbox_inches is None:\n        ox_in, oy_in = 0.0, 0.0\n        W_in, H_in = fig.get_size_inches()\n    else:\n        raise TypeError(\"panel_crops bbox_inches must be None, 'tight', or a matplotlib Bbox\")\n    W_px, H_px = int(round(W_in * dpi)), int(round(H_in * dpi))\n    lettered = {}\n    seen_letters = set()\n    for ax in fig.axes:\n        for t in ax.findobj(matplotlib.text.Text):\n            s = (t.get_text() or \"\").strip()\n            if t.get_gid() == \"figure-style-panel-letter\":\n                if s in seen_letters:\n                    raise ValueError(f\"Duplicate panel letter: {s!r}\")\n                seen_letters.add(s)\n                lettered[ax] = s\n                break\n\n\n\n    if not lettered:\n        lettered = {ax: str(i) for i, ax in enumerate(fig.axes)}\n    out = {}\n    for ax, letter in lettered.items():\n        bbs = [ax.get_tightbbox(r)]\n\n\n\n\n        ss = ax.get_subplotspec()\n        for sib in fig.axes:\n            if sib is ax or sib in lettered:\n                continue\n            ssib = sib.get_subplotspec()\n            same_row = ss is None or ssib is None or ss.rowspan == ssib.rowspan\n            same_col = ss is None or ssib is None or ss.colspan == ssib.colspan\n            if ((ax.get_shared_y_axes().joined(ax, sib) and same_row)\n                    or (ax.get_shared_x_axes().joined(ax, sib) and same_col)):\n                bbs.append(sib.get_tightbbox(r))\n        bb = mpl.transforms.Bbox.union(bbs)\n\n        bx0 = (bb.x0 / fig.dpi - ox_in) * dpi\n        bx1 = (bb.x1 / fig.dpi - ox_in) * dpi\n        by0 = H_px - (bb.y1 / fig.dpi - oy_in) * dpi\n        by1 = H_px - (bb.y0 / fig.dpi - oy_in) * dpi\n        box = (\n            min(max(int(bx0) - pad_px, 0), W_px),\n            min(max(int(by0) - pad_px, 0), H_px),\n            max(min(int(bx1) + pad_px, W_px), 0),\n            max(min(int(by1) + pad_px, H_px), 0),\n        )\n        if box[0] < box[2] and box[1] < box[3]:\n            out[letter] = box\n    return out\n",
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          "source": "META_GREY = \"#888888\"\n\n\ndef apply_figure_style(*, frame=\"open\", font=None, sizes=(8, 7, 6), grid=False):\n    \"\"\"Set matplotlib rcParams for publication-grade output. Call once before plotting.\n\n    This sets mechanics (role-mapped font-size ladder, outward ticks, frameless\n    legends, 300-dpi save, Type-42 embedded fonts) — not a house aesthetic.\n    Frame, font and the size ladder are parameters.\n\n    frame : 'open' (bottom+left spines, default) | 'boxed' (all four) | 'none'\n    font  : sans-serif family name; None = system default sans-serif\n    sizes : (base, secondary, tick) — titles/axis-labels, legend/annotation, ticks\n    grid  : whether to draw axes.grid (default False)\n    \"\"\"\n    import matplotlib as mpl\n    if frame not in (\"open\", \"boxed\", \"none\"):\n        raise ValueError(f\"frame must be 'open'|'boxed'|'none', got {frame!r}\")\n\n    try:\n        import os, sys, glob, matplotlib.font_manager as fm\n        fdir = os.path.join(os.environ.get(\"CONDA_PREFIX\") or sys.prefix, \"fonts\")\n        if os.path.isdir(fdir):\n            known = {f.fname for f in fm.fontManager.ttflist}\n            for f in glob.glob(os.path.join(fdir, \"*.ttf\")):\n                if f not in known:\n                    fm.fontManager.addfont(f)\n    except Exception:\n        pass\n    base, secondary, tick = sizes\n    boxed = (frame == \"boxed\")\n    rc = {\n        \"font.family\": \"sans-serif\",\n        \"font.size\": base,\n        \"axes.labelsize\": base,\n        \"axes.titlesize\": base,\n        \"legend.fontsize\": secondary,\n        \"xtick.labelsize\": tick,\n        \"ytick.labelsize\": tick,\n        \"axes.linewidth\": 0.6,\n        \"xtick.direction\": \"out\", \"ytick.direction\": \"out\",\n        \"xtick.major.size\": 3, \"ytick.major.size\": 3,\n        \"xtick.major.width\": 0.6, \"ytick.major.width\": 0.6,\n        \"axes.spines.top\": boxed, \"axes.spines.right\": boxed,\n        \"axes.spines.left\": frame != \"none\", \"axes.spines.bottom\": frame != \"none\",\n        \"axes.grid\": bool(grid),\n        \"legend.frameon\": False,\n        \"figure.dpi\": 200,\n        \"savefig.dpi\": 300,\n        \"savefig.bbox\": \"tight\",\n        \"axes.titleweight\": \"normal\",\n        \"axes.titlelocation\": \"left\",\n        \"axes.labelweight\": \"normal\",\n        \"lines.linewidth\": 1.2,\n        \"patch.linewidth\": 0.6,\n        \"pdf.fonttype\": 42, \"ps.fonttype\": 42,\n    }\n    if font:\n        rc[\"font.sans-serif\"] = [font, \"DejaVu Sans\"]\n    mpl.rcParams.update(rc)\n\n\ndef set_frame(ax, style=\"open\"):\n    \"\"\"§3: set spine visibility on an existing axes. style ∈ {'open','boxed','none'}.\"\"\"\n    show = {\"open\": (False, False, True, True),\n            \"boxed\": (True, True, True, True),\n            \"none\": (False, False, False, False)}[style]\n    for side, vis in zip((\"top\", \"right\", \"bottom\", \"left\"), show):\n        ax.spines[side].set_visible(vis)\n        if vis:\n            ax.spines[side].set_linewidth(0.6)\n    ax.tick_params(direction=\"out\", length=0 if style == \"none\" else 3, width=0.6)\n\n\ndef panel_letter(ax, letter, dx=-0.18, dy=1.02, case=\"lower\", fontsize=None):\n    \"\"\"§5.7: bold panel letter outside top-left of axes. case ∈ {'lower','upper'}.\"\"\"\n    import matplotlib.pyplot as plt\n    if fontsize is None:\n        fontsize = plt.rcParams.get(\"font.size\", 8) + 1\n    s = letter.lower() if case == \"lower\" else letter.upper()\n    artist = ax.text(dx, dy, s, transform=ax.transAxes,\n                     fontweight=\"bold\", fontsize=fontsize, va=\"bottom\", ha=\"left\")\n    artist.set_gid(\"figure-style-panel-letter\")\n\n\ndef focal_palette(labels, focal, focal_color, other=\"muted\", base_colors=None):\n    \"\"\"§4.2: map labels → colors with the focal series visually dominant.\n\n    other='muted'   — desaturate base_colors (or a default cycle) toward gray\n    other='grey'    — uniform light gray for all non-focal\n    other='ordinal' — non-focal on a single light→dark gray ramp (input order)\n    \"\"\"\n    import matplotlib.colors as mcolors\n    import matplotlib.pyplot as plt\n    focal_set = {focal} if isinstance(focal, str) else set(focal)\n    n = len(labels)\n    if not focal_set & set(labels):\n        raise ValueError(f\"focal {focal!r} not found in labels\")\n    base_colors = [] if base_colors is None else list(base_colors)\n    if not base_colors:\n        base_colors = plt.rcParams[\"axes.prop_cycle\"].by_key().get(\"color\", [\"#444444\"])\n    if not base_colors:\n        base_colors = [\"#444444\"]\n    base_colors = [base_colors[i % len(base_colors)] for i in range(n)]\n    if other == \"grey\":\n        rest = [\"#BCBCBC\"] * n\n    elif other == \"ordinal\":\n        nf = max(1, n - len(focal_set))\n        ramp = [mcolors.to_hex((v, v, v)) for v in\n                ([0.55] if nf == 1 else [0.80 - 0.35 * i / (nf - 1) for i in range(nf)])]\n        rest, k = [], 0\n        for l in labels:\n            rest.append(ramp[min(k, nf - 1)]); k += (l not in focal_set)\n    elif other == \"muted\":\n        def mute(c):\n            r, g, b = mcolors.to_rgb(c)\n            m = (r + g + b) / 3\n            return mcolors.to_hex((0.3 * r + 0.7 * m, 0.3 * g + 0.7 * m, 0.3 * b + 0.7 * m))\n        rest = [mute(c) for c in base_colors]\n    else:\n        raise ValueError(\"other must be 'muted', 'grey', or 'ordinal'\")\n    return [focal_color if l in focal_set else rest[i] for i, l in enumerate(labels)]\n\n\ndef bar_with_points(ax, x, ymat, labels, colors, jitter=0.08, show_points=True,\n                    errorbar=None, point_alpha=0.5, point_size=8):\n    \"\"\"§6.1: bar = mean; optionally overlay raw points or draw an interval.\n\n    colors   : per-label color list (e.g. from focal_palette)\n    x, ymat and labels must have equal lengths; mismatches raise before drawing.\n    Each group must be a nonempty, finite, one-dimensional real numeric array.\n    Requested SD/CI intervals require at least two observations per group.\n    errorbar : None | 'sd' | 'ci95' — drawn only when show_points is False.\n               'ci95' is the t-distribution 95% CI of the mean\n               (half-width t_{0.975,n-1} · s/√n); correct at small n where the\n               z-approximation (1.96·s/√n) is markedly too narrow.\n    \"\"\"\n    import numpy as np\n    x = np.asarray(x)\n    if x.ndim != 1 or not np.issubdtype(x.dtype, np.number):\n        raise TypeError(\n            \"bar_with_points x must be one-dimensional numeric positions; \"\n            \"pass category names through labels\"\n        )\n    ymat = list(ymat)\n    labels = list(labels)\n    if len(x) != len(ymat) or len(x) != len(labels):\n        raise ValueError(\"bar_with_points x, ymat and labels must have equal lengths\")\n    if colors is not None and not isinstance(colors, str):\n        colors = list(colors)\n        if not colors:\n            colors = None\n    if x.dtype.kind not in \"iuf\" or not np.isfinite(x).all():\n        raise ValueError(\"bar_with_points x must contain finite real positions\")\n    for i, y in enumerate(ymat):\n        if np.ma.isMaskedArray(y) and np.ma.getmaskarray(y).any():\n            raise ValueError(f\"bar_with_points group {i} contains masked observations; resolve exclusions explicitly before plotting\")\n    ymat = [np.asarray(y) for y in ymat]\n    for i, y in enumerate(ymat):\n        if y.ndim != 1 or y.size == 0 or y.dtype.kind not in \"iuf\" or not np.isfinite(y).all():\n            raise ValueError(f\"bar_with_points group {i} must be nonempty, one-dimensional and finite real numeric data\")\n    if errorbar and not show_points:\n        if errorbar not in (\"sd\", \"ci95\"):\n            raise ValueError(\"errorbar must be None, 'sd', or 'ci95'\")\n        if any(y.size < 2 for y in ymat):\n            raise ValueError(\"SD/CI cannot be estimated from one observation; use raw points without an interval\")\n    means = np.array([np.mean(y) for y in ymat], float)\n    err = None\n    if errorbar and not show_points:\n        if errorbar == \"sd\":\n            err = np.array([np.std(y, ddof=1) for y in ymat])\n        elif errorbar == \"ci95\":\n            from scipy.stats import t\n            def _hw(y):\n                n = np.asarray(y).size\n                return t.ppf(0.975, n - 1) * np.std(y, ddof=1) / np.sqrt(n)\n            err = np.array([_hw(y) for y in ymat])\n    ax.bar(x, means, color=colors, width=0.7, edgecolor=\"none\",\n           yerr=err, error_kw={\"elinewidth\": 0.8, \"capsize\": 0})\n    if show_points:\n        for xi, ys in zip(x, ymat):\n            ys = np.asarray(ys)\n            if ys.ndim == 1 and ys.size > 0:\n                jit = (np.random.rand(ys.size) - 0.5) * 2 * jitter\n                ax.scatter(np.full(ys.size, xi) + jit, ys, s=point_size, color=\"black\",\n                           alpha=point_alpha, zorder=3, linewidths=0)\n    ax.set_xticks(x); ax.set_xticklabels(labels)\n    return ax\n\n\ndef strip_with_median(ax, groups, values, colors=None, jitter=0.12):\n    \"\"\"§6.1: jittered points + bold horizontal median tick per group.\n\n    groups and values must have equal lengths. Mismatches raise before drawing.\n    colors=None or an empty color sequence supplies default gray; a nonempty\n    color sequence cycles across groups.\n    Each group must be nonempty, one-dimensional and finite real numeric data;\n    resolve missing observations explicitly before plotting.\n    \"\"\"\n    import numpy as np\n    labs = list(groups)\n    values = list(values)\n    if len(labs) != len(values):\n        raise ValueError(\"strip_with_median groups and values must have equal lengths\")\n    for i, y in enumerate(values):\n        if np.ma.isMaskedArray(y) and np.ma.getmaskarray(y).any():\n            raise ValueError(f\"strip_with_median group {i} contains masked observations; resolve exclusions explicitly before plotting\")\n    values = [np.asarray(y) for y in values]\n    for i, y in enumerate(values):\n        if y.ndim != 1 or y.size == 0 or y.dtype.kind not in \"iuf\" or not np.isfinite(y).all():\n            raise ValueError(f\"strip_with_median group {i} must be nonempty, one-dimensional and finite real numeric data\")\n    colors = [] if colors is None else list(colors)\n    if not colors:\n        colors = [\"#444444\"]\n    for i, ys in enumerate(values):\n        c = colors[i % len(colors)]\n        ys = np.asarray(ys)\n        jit = (np.random.rand(ys.size) - 0.5) * 2 * jitter\n        ax.scatter(np.full(ys.size, i) + jit, ys, s=10, color=c, alpha=0.6, linewidths=0, zorder=2)\n        m = np.median(ys)\n        ax.plot([i - 0.22, i + 0.22], [m, m], color=\"black\", lw=1.6, zorder=3)\n    ax.set_xticks(range(len(labs))); ax.set_xticklabels(labs)\n    return ax\n\n\ndef goodness_arrow(ax, text=\"higher = better\", loc=\"upper left\", axis=\"y\", fontsize=None):\n    \"\"\"§3.6: small upright direction-of-goodness cue in the margin.\"\"\"\n    import matplotlib.pyplot as plt\n    if fontsize is None:\n        fontsize = plt.rcParams[\"legend.fontsize\"]\n    pos = {\"upper left\": (0.02, 0.98), \"upper right\": (0.98, 0.98),\n           \"lower left\": (0.02, 0.02), \"lower right\": (0.98, 0.02)}[loc]\n    ha = \"left\" if \"left\" in loc else \"right\"\n    va = \"top\" if \"upper\" in loc else \"bottom\"\n    arrow = \"↑ \" if axis == \"y\" else \"→ \"\n    ax.text(pos[0], pos[1], arrow + text, transform=ax.transAxes,\n            fontsize=fontsize, color=META_GREY, ha=ha, va=va)\n\n\ndef two_tier_label(name, meta):\n    \"\"\"§5: two-line label string (name / metadata). Meta line styled separately by caller.\"\"\"\n    return f\"{name}\\n{meta}\"\n\n\ndef end_of_line_labels(ax, xs, ys, labels, colors=None, dx=0.01, fontsize=None):\n    \"\"\"§6.3 / §7.3: label each line series at its right end instead of a legend box.\n\n    Each label requires a nonempty, matched x/y series. Validate all series\n    before drawing. Nonempty colors cycle; None or empty uses default text color.\n    Inner sequences must support len() and indexing; they are not copied.\n    Nonfinite endpoints are rejected rather than producing an invisible label.\n    \"\"\"\n    import matplotlib.pyplot as plt\n    import math\n    xs, ys, labels = list(xs), list(ys), list(labels)\n    if len(xs) != len(ys) or len(xs) != len(labels):\n        raise ValueError(\"end_of_line_labels xs, ys and labels must have equal lengths\")\n    if any(len(x) == 0 or len(x) != len(y) for x, y in zip(xs, ys)):\n        raise ValueError(\"end_of_line_labels each x/y series must be nonempty and have equal lengths\")\n    if any(not math.isfinite(x[-1]) or not math.isfinite(y[-1]) for x, y in zip(xs, ys)):\n        raise ValueError(\"end_of_line_labels endpoints must be finite; choose valid endpoints explicitly\")\n    if fontsize is None:\n        fontsize = plt.rcParams[\"font.size\"]\n    colors = [] if colors is None else list(colors)\n    if not colors:\n        colors = [None]\n    span = ax.get_xlim()[1] - ax.get_xlim()[0]\n    for i, (x, y, lab) in enumerate(zip(xs, ys, labels)):\n        c = colors[i % len(colors)]\n        ax.text(x[-1] + dx * span, y[-1], lab, color=c, va=\"center\", ha=\"left\", fontsize=fontsize)\n\n\ndef panel_crops(fig, dpi=None, pad_px=6, bbox_inches=None, pad_inches=None):\n    \"\"\"§9.2: pixel-space crop boxes for each lettered panel in the SAVED PNG.\n\n    Returns ``{letter: (x0, y0, x1, y1)}`` in image-space pixels (origin\n    top-left, matching PIL's ``Image.crop``). Panels are detected as bold\n    single-character ``Text``\n    objects placed by :func:`panel_letter`; each panel's crop is its axes'\n    tightbbox mapped into the saved file's pixel space, padded by ``pad_px``.\n    For §3.4 composites (abutting subplots sharing an axis, letter on the\n    leftmost only) the crop unions in letterless ``sharex``/``sharey`` siblings\n    on the same grid row/col so the whole composite is covered. When no axes\n    carries a panel letter (standalone plot, or a figure-composer sub-agent),\n    falls back to one crop per axes keyed by index.\n\n    ``bbox_inches`` mirrors ``Figure.savefig`` semantics: ``None`` means\n    *consult rcParams* (so under :func:`apply_figure_style` it resolves to\n    ``'tight'``); pass an explicit ``Bbox`` only if you saved with one. The\n    boxes are clamped to the saved image extent; panels with no intersection\n    are omitted.\n\n        >>> from PIL import Image\n        >>> fig.savefig(\"fig.png\")  # bbox_inches='tight' via rcParams\n        >>> with Image.open(\"fig.png\") as saved:\n        ...     for letter, box in panel_crops(fig).items():\n        ...         saved.crop(box).save(f\"panel_{letter}.png\")\n\n    Match savefig's DPI, bbox_inches and pad_inches when overriding defaults.\n    Returned boxes are pixels, never savefig bbox_inches (inches). Passing them\n    back to savefig as inches can allocate an enormous raster.\n    \"\"\"\n    import matplotlib as mpl\n    import matplotlib.text\n    from matplotlib.backends.backend_agg import FigureCanvasAgg\n    if isinstance(pad_px, bool) or not isinstance(pad_px, int) or pad_px < 0:\n        raise TypeError(\"panel_crops pad_px must be a non-negative integer\")\n    if dpi is None:\n        dpi = mpl.rcParams.get(\"savefig.dpi\", fig.dpi)\n        if dpi == \"figure\":\n            dpi = fig.dpi\n    dpi = float(dpi)\n    if bbox_inches is None:\n        bbox_inches = mpl.rcParams.get(\"savefig.bbox\")\n    # The Open-Science notebook captures and closes pyplot figures after every\n    # cell. Matplotlib 3.11 decouples a closed Figure from its backend by\n    # replacing FigureCanvasAgg with FigureCanvasBase, while the persistent\n    # notebook namespace can still retain the Figure itself. Reattach the\n    # headless canvas before asking for saved-image geometry so this helper is\n    # safe in a later QA cell as well as the producer cell.\n    if not hasattr(fig.canvas, \"get_renderer\"):\n        FigureCanvasAgg(fig)\n    fig.canvas.draw()\n    r = fig.canvas.get_renderer()\n\n    if bbox_inches == \"tight\":\n        if pad_inches is None:\n            pad_inches = mpl.rcParams.get(\"savefig.pad_inches\", 0.1)\n        tb = fig.get_tightbbox(r).padded(pad_inches)\n        ox_in, oy_in = tb.x0, tb.y0\n        W_in, H_in = tb.width, tb.height\n    elif isinstance(bbox_inches, mpl.transforms.BboxBase):\n        ox_in, oy_in = bbox_inches.x0, bbox_inches.y0\n        W_in, H_in = bbox_inches.width, bbox_inches.height\n    elif bbox_inches is None:\n        ox_in, oy_in = 0.0, 0.0\n        W_in, H_in = fig.get_size_inches()\n    else:\n        raise TypeError(\"panel_crops bbox_inches must be None, 'tight', or a matplotlib Bbox\")\n    W_px, H_px = int(round(W_in * dpi)), int(round(H_in * dpi))\n    lettered = {}\n    seen_letters = set()\n    for ax in fig.axes:\n        for t in ax.findobj(matplotlib.text.Text):\n            s = (t.get_text() or \"\").strip()\n            if t.get_gid() == \"figure-style-panel-letter\":\n                if s in seen_letters:\n                    raise ValueError(f\"Duplicate panel letter: {s!r}\")\n                seen_letters.add(s)\n                lettered[ax] = s\n                break\n\n\n\n    if not lettered:\n        lettered = {ax: str(i) for i, ax in enumerate(fig.axes)}\n    out = {}\n    for ax, letter in lettered.items():\n        bbs = [ax.get_tightbbox(r)]\n\n\n\n\n        ss = ax.get_subplotspec()\n        for sib in fig.axes:\n            if sib is ax or sib in lettered:\n                continue\n            ssib = sib.get_subplotspec()\n            same_row = ss is None or ssib is None or ss.rowspan == ssib.rowspan\n            same_col = ss is None or ssib is None or ss.colspan == ssib.colspan\n            if ((ax.get_shared_y_axes().joined(ax, sib) and same_row)\n                    or (ax.get_shared_x_axes().joined(ax, sib) and same_col)):\n                bbs.append(sib.get_tightbbox(r))\n        bb = mpl.transforms.Bbox.union(bbs)\n\n        bx0 = (bb.x0 / fig.dpi - ox_in) * dpi\n        bx1 = (bb.x1 / fig.dpi - ox_in) * dpi\n        by0 = H_px - (bb.y1 / fig.dpi - oy_in) * dpi\n        by1 = H_px - (bb.y0 / fig.dpi - oy_in) * dpi\n        box = (\n            min(max(int(bx0) - pad_px, 0), W_px),\n            min(max(int(by0) - pad_px, 0), H_px),\n            max(min(int(bx1) + pad_px, W_px), 0),\n            max(min(int(by1) + pad_px, H_px), 0),\n        )\n        if box[0] < box[2] and box[1] < box[3]:\n            out[letter] = box\n    return out\n",
          "sourceDigest": "8020d409bfb60f0c8dfa43d0dfe7bb97f7458e1a3908fefbcb6e4551e0feae4e"
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        "stdout": "[{\"id\": \"148\", \"coverage\": \"main_target\", \"note\": \"Main target matches entropy-rate/lower-Hausdorff-dimension formula with positive weights, signed unequal ratios and exact overlaps. Absolute continuity is not claimed.\", \"statement_path\": \"preprints/The-entropy-rate-dimension-formula-for-self-similar-measures-on-the-line-September-24-2026/build/sections/introduction.tex\", \"statement_line\": 78, \"targets\": [{\"name\": \"SelfSimilar\", \"declarations\": [{\"name\": \"entropy_rate_dimension\", \"line\": 65}], \"solution\": \"lean/OAI/MeasureTheory/SelfSimilar/Main.lean\", \"supplied\": true}, {\"name\": \"SelfSimilarCorollaries\", \"declarations\": [{\"name\": \"homogeneous_dimension_direct\", \"line\": 49}, {\"name\": \"cor_set\", \"line\": 78}], \"solution\": \"lean/OAI/MeasureTheory/SelfSimilar/DimensionCorollaries.lean\", \"supplied\": false}]}, {\"id\": \"154\", \"coverage\": \"no_linked_lean\", \"note\": \"Written proof of pointwise multiple ergodic convergence for mixing transformations; no linked Lean material found.\", \"statement_path\": \"preprints/Pointwise-Multiple-Ergodic-Averages-for-Mixing-Transformations-October-4-2026/build/sections/introduction.tex\", \"statement_line\": 20, \"targets\": []}, {\"id\": \"164\", \"coverage\": \"no_linked_lean\", \"note\": \"Written proof of monochromatic finite sums and products for every finite set size; no linked Lean material found.\", \"statement_path\": \"preprints/Monochromatic-finite-sums-and-products-in-the-positive-integers-September-23-2026/build/sections/01_introduction.tex\", \"statement_line\": 13, \"targets\": []}, {\"id\": \"192\", \"coverage\": \"main_target\", \"note\": \"Main target asserts unbounded signed square-root-degree violations and unbounded absolute-coefficient ratios.\", \"statement_path\": \"preprints/Unbounded-Violations-of-the-Square-Root-Degree-Bound-September-26-2026/build/sections/00-introduction.tex\", \"statement_line\": 23, \"targets\": [{\"name\": \"SquareRootDegree\", \"declarations\": [{\"name\": \"main\", \"line\": 57}], \"solution\": \"lean/OAI/Combinatorics/BooleanFunctions/Main.lean\", \"supplied\": false}]}]\n",
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          "text": "[{\"id\": \"148\", \"coverage\": \"main_target\", \"note\": \"Main target matches entropy-rate/lower-Hausdorff-dimension formula with positive weights, signed unequal ratios and exact overlaps. Absolute continuity is not claimed.\", \"statement_path\": \"preprints/The-entropy-rate-dimension-formula-for-self-similar-measures-on-the-line-September-24-2026/build/sections/introduction.tex\", \"statement_line\": 78, \"targets\": [{\"name\": \"SelfSimilar\", \"declarations\": [{\"name\": \"entropy_rate_dimension\", \"line\": 65}], \"solution\": \"lean/OAI/MeasureTheory/SelfSimilar/Main.lean\", \"supplied\": true}, {\"name\": \"SelfSimilarCorollaries\", \"declarations\": [{\"name\": \"homogeneous_dimension_direct\", \"line\": 49}, {\"name\": \"cor_set\", \"line\": 78}], \"solution\": \"lean/OAI/MeasureTheory/SelfSimilar/DimensionCorollaries.lean\", \"supplied\": false}]}, {\"id\": \"154\", \"coverage\": \"no_linked_lean\", \"note\": \"Written proof of pointwise multiple ergodic convergence for mixing transformations; no linked Lean material found.\", \"statement_path\": \"preprints/Pointwise-Multiple-Ergodic-Averages-for-Mixing-Transformations-October-4-2026/build/sections/introduction.tex\", \"statement_line\": 20, \"targets\": []}, {\"id\": \"164\", \"coverage\": \"no_linked_lean\", \"note\": \"Written proof of monochromatic finite sums and products for every finite set size; no linked Lean material found.\", \"statement_path\": \"preprints/Monochromatic-finite-sums-and-products-in-the-positive-integers-September-23-2026/build/sections/01_introduction.tex\", \"statement_line\": 13, \"targets\": []}, {\"id\": \"192\", \"coverage\": \"main_target\", \"note\": \"Main target asserts unbounded signed square-root-degree violations and unbounded absolute-coefficient ratios.\", \"statement_path\": \"preprints/Unbounded-Violations-of-the-Square-Root-Degree-Bound-September-26-2026/build/sections/00-introduction.tex\", \"statement_line\": 23, \"targets\": [{\"name\": \"SquareRootDegree\", \"declarations\": [{\"name\": \"main\", \"line\": 57}], \"solution\": \"lean/OAI/Combinatorics/BooleanFunctions/Main.lean\", \"supplied\": false}]}]\n"
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          "source": "META_GREY = \"#888888\"\n\n\ndef apply_figure_style(*, frame=\"open\", font=None, sizes=(8, 7, 6), grid=False):\n    \"\"\"Set matplotlib rcParams for publication-grade output. Call once before plotting.\n\n    This sets mechanics (role-mapped font-size ladder, outward ticks, frameless\n    legends, 300-dpi save, Type-42 embedded fonts) — not a house aesthetic.\n    Frame, font and the size ladder are parameters.\n\n    frame : 'open' (bottom+left spines, default) | 'boxed' (all four) | 'none'\n    font  : sans-serif family name; None = system default sans-serif\n    sizes : (base, secondary, tick) — titles/axis-labels, legend/annotation, ticks\n    grid  : whether to draw axes.grid (default False)\n    \"\"\"\n    import matplotlib as mpl\n    if frame not in (\"open\", \"boxed\", \"none\"):\n        raise ValueError(f\"frame must be 'open'|'boxed'|'none', got {frame!r}\")\n\n    try:\n        import os, sys, glob, matplotlib.font_manager as fm\n        fdir = os.path.join(os.environ.get(\"CONDA_PREFIX\") or sys.prefix, \"fonts\")\n        if os.path.isdir(fdir):\n            known = {f.fname for f in fm.fontManager.ttflist}\n            for f in glob.glob(os.path.join(fdir, \"*.ttf\")):\n                if f not in known:\n                    fm.fontManager.addfont(f)\n    except Exception:\n        pass\n    base, secondary, tick = sizes\n    boxed = (frame == \"boxed\")\n    rc = {\n        \"font.family\": \"sans-serif\",\n        \"font.size\": base,\n        \"axes.labelsize\": base,\n        \"axes.titlesize\": base,\n        \"legend.fontsize\": secondary,\n        \"xtick.labelsize\": tick,\n        \"ytick.labelsize\": tick,\n        \"axes.linewidth\": 0.6,\n        \"xtick.direction\": \"out\", \"ytick.direction\": \"out\",\n        \"xtick.major.size\": 3, \"ytick.major.size\": 3,\n        \"xtick.major.width\": 0.6, \"ytick.major.width\": 0.6,\n        \"axes.spines.top\": boxed, \"axes.spines.right\": boxed,\n        \"axes.spines.left\": frame != \"none\", \"axes.spines.bottom\": frame != \"none\",\n        \"axes.grid\": bool(grid),\n        \"legend.frameon\": False,\n        \"figure.dpi\": 200,\n        \"savefig.dpi\": 300,\n        \"savefig.bbox\": \"tight\",\n        \"axes.titleweight\": \"normal\",\n        \"axes.titlelocation\": \"left\",\n        \"axes.labelweight\": \"normal\",\n        \"lines.linewidth\": 1.2,\n        \"patch.linewidth\": 0.6,\n        \"pdf.fonttype\": 42, \"ps.fonttype\": 42,\n    }\n    if font:\n        rc[\"font.sans-serif\"] = [font, \"DejaVu Sans\"]\n    mpl.rcParams.update(rc)\n\n\ndef set_frame(ax, style=\"open\"):\n    \"\"\"§3: set spine visibility on an existing axes. style ∈ {'open','boxed','none'}.\"\"\"\n    show = {\"open\": (False, False, True, True),\n            \"boxed\": (True, True, True, True),\n            \"none\": (False, False, False, False)}[style]\n    for side, vis in zip((\"top\", \"right\", \"bottom\", \"left\"), show):\n        ax.spines[side].set_visible(vis)\n        if vis:\n            ax.spines[side].set_linewidth(0.6)\n    ax.tick_params(direction=\"out\", length=0 if style == \"none\" else 3, width=0.6)\n\n\ndef panel_letter(ax, letter, dx=-0.18, dy=1.02, case=\"lower\", fontsize=None):\n    \"\"\"§5.7: bold panel letter outside top-left of axes. case ∈ {'lower','upper'}.\"\"\"\n    import matplotlib.pyplot as plt\n    if fontsize is None:\n        fontsize = plt.rcParams.get(\"font.size\", 8) + 1\n    s = letter.lower() if case == \"lower\" else letter.upper()\n    artist = ax.text(dx, dy, s, transform=ax.transAxes,\n                     fontweight=\"bold\", fontsize=fontsize, va=\"bottom\", ha=\"left\")\n    artist.set_gid(\"figure-style-panel-letter\")\n\n\ndef focal_palette(labels, focal, focal_color, other=\"muted\", base_colors=None):\n    \"\"\"§4.2: map labels → colors with the focal series visually dominant.\n\n    other='muted'   — desaturate base_colors (or a default cycle) toward gray\n    other='grey'    — uniform light gray for all non-focal\n    other='ordinal' — non-focal on a single light→dark gray ramp (input order)\n    \"\"\"\n    import matplotlib.colors as mcolors\n    import matplotlib.pyplot as plt\n    focal_set = {focal} if isinstance(focal, str) else set(focal)\n    n = len(labels)\n    if not focal_set & set(labels):\n        raise ValueError(f\"focal {focal!r} not found in labels\")\n    base_colors = [] if base_colors is None else list(base_colors)\n    if not base_colors:\n        base_colors = plt.rcParams[\"axes.prop_cycle\"].by_key().get(\"color\", [\"#444444\"])\n    if not base_colors:\n        base_colors = [\"#444444\"]\n    base_colors = [base_colors[i % len(base_colors)] for i in range(n)]\n    if other == \"grey\":\n        rest = [\"#BCBCBC\"] * n\n    elif other == \"ordinal\":\n        nf = max(1, n - len(focal_set))\n        ramp = [mcolors.to_hex((v, v, v)) for v in\n                ([0.55] if nf == 1 else [0.80 - 0.35 * i / (nf - 1) for i in range(nf)])]\n        rest, k = [], 0\n        for l in labels:\n            rest.append(ramp[min(k, nf - 1)]); k += (l not in focal_set)\n    elif other == \"muted\":\n        def mute(c):\n            r, g, b = mcolors.to_rgb(c)\n            m = (r + g + b) / 3\n            return mcolors.to_hex((0.3 * r + 0.7 * m, 0.3 * g + 0.7 * m, 0.3 * b + 0.7 * m))\n        rest = [mute(c) for c in base_colors]\n    else:\n        raise ValueError(\"other must be 'muted', 'grey', or 'ordinal'\")\n    return [focal_color if l in focal_set else rest[i] for i, l in enumerate(labels)]\n\n\ndef bar_with_points(ax, x, ymat, labels, colors, jitter=0.08, show_points=True,\n                    errorbar=None, point_alpha=0.5, point_size=8):\n    \"\"\"§6.1: bar = mean; optionally overlay raw points or draw an interval.\n\n    colors   : per-label color list (e.g. from focal_palette)\n    x, ymat and labels must have equal lengths; mismatches raise before drawing.\n    Each group must be a nonempty, finite, one-dimensional real numeric array.\n    Requested SD/CI intervals require at least two observations per group.\n    errorbar : None | 'sd' | 'ci95' — drawn only when show_points is False.\n               'ci95' is the t-distribution 95% CI of the mean\n               (half-width t_{0.975,n-1} · s/√n); correct at small n where the\n               z-approximation (1.96·s/√n) is markedly too narrow.\n    \"\"\"\n    import numpy as np\n    x = np.asarray(x)\n    if x.ndim != 1 or not np.issubdtype(x.dtype, np.number):\n        raise TypeError(\n            \"bar_with_points x must be one-dimensional numeric positions; \"\n            \"pass category names through labels\"\n        )\n    ymat = list(ymat)\n    labels = list(labels)\n    if len(x) != len(ymat) or len(x) != len(labels):\n        raise ValueError(\"bar_with_points x, ymat and labels must have equal lengths\")\n    if colors is not None and not isinstance(colors, str):\n        colors = list(colors)\n        if not colors:\n            colors = None\n    if x.dtype.kind not in \"iuf\" or not np.isfinite(x).all():\n        raise ValueError(\"bar_with_points x must contain finite real positions\")\n    for i, y in enumerate(ymat):\n        if np.ma.isMaskedArray(y) and np.ma.getmaskarray(y).any():\n            raise ValueError(f\"bar_with_points group {i} contains masked observations; resolve exclusions explicitly before plotting\")\n    ymat = [np.asarray(y) for y in ymat]\n    for i, y in enumerate(ymat):\n        if y.ndim != 1 or y.size == 0 or y.dtype.kind not in \"iuf\" or not np.isfinite(y).all():\n            raise ValueError(f\"bar_with_points group {i} must be nonempty, one-dimensional and finite real numeric data\")\n    if errorbar and not show_points:\n        if errorbar not in (\"sd\", \"ci95\"):\n            raise ValueError(\"errorbar must be None, 'sd', or 'ci95'\")\n        if any(y.size < 2 for y in ymat):\n            raise ValueError(\"SD/CI cannot be estimated from one observation; use raw points without an interval\")\n    means = np.array([np.mean(y) for y in ymat], float)\n    err = None\n    if errorbar and not show_points:\n        if errorbar == \"sd\":\n            err = np.array([np.std(y, ddof=1) for y in ymat])\n        elif errorbar == \"ci95\":\n            from scipy.stats import t\n            def _hw(y):\n                n = np.asarray(y).size\n                return t.ppf(0.975, n - 1) * np.std(y, ddof=1) / np.sqrt(n)\n            err = np.array([_hw(y) for y in ymat])\n    ax.bar(x, means, color=colors, width=0.7, edgecolor=\"none\",\n           yerr=err, error_kw={\"elinewidth\": 0.8, \"capsize\": 0})\n    if show_points:\n        for xi, ys in zip(x, ymat):\n            ys = np.asarray(ys)\n            if ys.ndim == 1 and ys.size > 0:\n                jit = (np.random.rand(ys.size) - 0.5) * 2 * jitter\n                ax.scatter(np.full(ys.size, xi) + jit, ys, s=point_size, color=\"black\",\n                           alpha=point_alpha, zorder=3, linewidths=0)\n    ax.set_xticks(x); ax.set_xticklabels(labels)\n    return ax\n\n\ndef strip_with_median(ax, groups, values, colors=None, jitter=0.12):\n    \"\"\"§6.1: jittered points + bold horizontal median tick per group.\n\n    groups and values must have equal lengths. Mismatches raise before drawing.\n    colors=None or an empty color sequence supplies default gray; a nonempty\n    color sequence cycles across groups.\n    Each group must be nonempty, one-dimensional and finite real numeric data;\n    resolve missing observations explicitly before plotting.\n    \"\"\"\n    import numpy as np\n    labs = list(groups)\n    values = list(values)\n    if len(labs) != len(values):\n        raise ValueError(\"strip_with_median groups and values must have equal lengths\")\n    for i, y in enumerate(values):\n        if np.ma.isMaskedArray(y) and np.ma.getmaskarray(y).any():\n            raise ValueError(f\"strip_with_median group {i} contains masked observations; resolve exclusions explicitly before plotting\")\n    values = [np.asarray(y) for y in values]\n    for i, y in enumerate(values):\n        if y.ndim != 1 or y.size == 0 or y.dtype.kind not in \"iuf\" or not np.isfinite(y).all():\n            raise ValueError(f\"strip_with_median group {i} must be nonempty, one-dimensional and finite real numeric data\")\n    colors = [] if colors is None else list(colors)\n    if not colors:\n        colors = [\"#444444\"]\n    for i, ys in enumerate(values):\n        c = colors[i % len(colors)]\n        ys = np.asarray(ys)\n        jit = (np.random.rand(ys.size) - 0.5) * 2 * jitter\n        ax.scatter(np.full(ys.size, i) + jit, ys, s=10, color=c, alpha=0.6, linewidths=0, zorder=2)\n        m = np.median(ys)\n        ax.plot([i - 0.22, i + 0.22], [m, m], color=\"black\", lw=1.6, zorder=3)\n    ax.set_xticks(range(len(labs))); ax.set_xticklabels(labs)\n    return ax\n\n\ndef goodness_arrow(ax, text=\"higher = better\", loc=\"upper left\", axis=\"y\", fontsize=None):\n    \"\"\"§3.6: small upright direction-of-goodness cue in the margin.\"\"\"\n    import matplotlib.pyplot as plt\n    if fontsize is None:\n        fontsize = plt.rcParams[\"legend.fontsize\"]\n    pos = {\"upper left\": (0.02, 0.98), \"upper right\": (0.98, 0.98),\n           \"lower left\": (0.02, 0.02), \"lower right\": (0.98, 0.02)}[loc]\n    ha = \"left\" if \"left\" in loc else \"right\"\n    va = \"top\" if \"upper\" in loc else \"bottom\"\n    arrow = \"↑ \" if axis == \"y\" else \"→ \"\n    ax.text(pos[0], pos[1], arrow + text, transform=ax.transAxes,\n            fontsize=fontsize, color=META_GREY, ha=ha, va=va)\n\n\ndef two_tier_label(name, meta):\n    \"\"\"§5: two-line label string (name / metadata). Meta line styled separately by caller.\"\"\"\n    return f\"{name}\\n{meta}\"\n\n\ndef end_of_line_labels(ax, xs, ys, labels, colors=None, dx=0.01, fontsize=None):\n    \"\"\"§6.3 / §7.3: label each line series at its right end instead of a legend box.\n\n    Each label requires a nonempty, matched x/y series. Validate all series\n    before drawing. Nonempty colors cycle; None or empty uses default text color.\n    Inner sequences must support len() and indexing; they are not copied.\n    Nonfinite endpoints are rejected rather than producing an invisible label.\n    \"\"\"\n    import matplotlib.pyplot as plt\n    import math\n    xs, ys, labels = list(xs), list(ys), list(labels)\n    if len(xs) != len(ys) or len(xs) != len(labels):\n        raise ValueError(\"end_of_line_labels xs, ys and labels must have equal lengths\")\n    if any(len(x) == 0 or len(x) != len(y) for x, y in zip(xs, ys)):\n        raise ValueError(\"end_of_line_labels each x/y series must be nonempty and have equal lengths\")\n    if any(not math.isfinite(x[-1]) or not math.isfinite(y[-1]) for x, y in zip(xs, ys)):\n        raise ValueError(\"end_of_line_labels endpoints must be finite; choose valid endpoints explicitly\")\n    if fontsize is None:\n        fontsize = plt.rcParams[\"font.size\"]\n    colors = [] if colors is None else list(colors)\n    if not colors:\n        colors = [None]\n    span = ax.get_xlim()[1] - ax.get_xlim()[0]\n    for i, (x, y, lab) in enumerate(zip(xs, ys, labels)):\n        c = colors[i % len(colors)]\n        ax.text(x[-1] + dx * span, y[-1], lab, color=c, va=\"center\", ha=\"left\", fontsize=fontsize)\n\n\ndef panel_crops(fig, dpi=None, pad_px=6, bbox_inches=None, pad_inches=None):\n    \"\"\"§9.2: pixel-space crop boxes for each lettered panel in the SAVED PNG.\n\n    Returns ``{letter: (x0, y0, x1, y1)}`` in image-space pixels (origin\n    top-left, matching PIL's ``Image.crop``). Panels are detected as bold\n    single-character ``Text``\n    objects placed by :func:`panel_letter`; each panel's crop is its axes'\n    tightbbox mapped into the saved file's pixel space, padded by ``pad_px``.\n    For §3.4 composites (abutting subplots sharing an axis, letter on the\n    leftmost only) the crop unions in letterless ``sharex``/``sharey`` siblings\n    on the same grid row/col so the whole composite is covered. When no axes\n    carries a panel letter (standalone plot, or a figure-composer sub-agent),\n    falls back to one crop per axes keyed by index.\n\n    ``bbox_inches`` mirrors ``Figure.savefig`` semantics: ``None`` means\n    *consult rcParams* (so under :func:`apply_figure_style` it resolves to\n    ``'tight'``); pass an explicit ``Bbox`` only if you saved with one. The\n    boxes are clamped to the saved image extent; panels with no intersection\n    are omitted.\n\n        >>> from PIL import Image\n        >>> fig.savefig(\"fig.png\")  # bbox_inches='tight' via rcParams\n        >>> with Image.open(\"fig.png\") as saved:\n        ...     for letter, box in panel_crops(fig).items():\n        ...         saved.crop(box).save(f\"panel_{letter}.png\")\n\n    Match savefig's DPI, bbox_inches and pad_inches when overriding defaults.\n    Returned boxes are pixels, never savefig bbox_inches (inches). Passing them\n    back to savefig as inches can allocate an enormous raster.\n    \"\"\"\n    import matplotlib as mpl\n    import matplotlib.text\n    from matplotlib.backends.backend_agg import FigureCanvasAgg\n    if isinstance(pad_px, bool) or not isinstance(pad_px, int) or pad_px < 0:\n        raise TypeError(\"panel_crops pad_px must be a non-negative integer\")\n    if dpi is None:\n        dpi = mpl.rcParams.get(\"savefig.dpi\", fig.dpi)\n        if dpi == \"figure\":\n            dpi = fig.dpi\n    dpi = float(dpi)\n    if bbox_inches is None:\n        bbox_inches = mpl.rcParams.get(\"savefig.bbox\")\n    # The Open-Science notebook captures and closes pyplot figures after every\n    # cell. Matplotlib 3.11 decouples a closed Figure from its backend by\n    # replacing FigureCanvasAgg with FigureCanvasBase, while the persistent\n    # notebook namespace can still retain the Figure itself. Reattach the\n    # headless canvas before asking for saved-image geometry so this helper is\n    # safe in a later QA cell as well as the producer cell.\n    if not hasattr(fig.canvas, \"get_renderer\"):\n        FigureCanvasAgg(fig)\n    fig.canvas.draw()\n    r = fig.canvas.get_renderer()\n\n    if bbox_inches == \"tight\":\n        if pad_inches is None:\n            pad_inches = mpl.rcParams.get(\"savefig.pad_inches\", 0.1)\n        tb = fig.get_tightbbox(r).padded(pad_inches)\n        ox_in, oy_in = tb.x0, tb.y0\n        W_in, H_in = tb.width, tb.height\n    elif isinstance(bbox_inches, mpl.transforms.BboxBase):\n        ox_in, oy_in = bbox_inches.x0, bbox_inches.y0\n        W_in, H_in = bbox_inches.width, bbox_inches.height\n    elif bbox_inches is None:\n        ox_in, oy_in = 0.0, 0.0\n        W_in, H_in = fig.get_size_inches()\n    else:\n        raise TypeError(\"panel_crops bbox_inches must be None, 'tight', or a matplotlib Bbox\")\n    W_px, H_px = int(round(W_in * dpi)), int(round(H_in * dpi))\n    lettered = {}\n    seen_letters = set()\n    for ax in fig.axes:\n        for t in ax.findobj(matplotlib.text.Text):\n            s = (t.get_text() or \"\").strip()\n            if t.get_gid() == \"figure-style-panel-letter\":\n                if s in seen_letters:\n                    raise ValueError(f\"Duplicate panel letter: {s!r}\")\n                seen_letters.add(s)\n                lettered[ax] = s\n                break\n\n\n\n    if not lettered:\n        lettered = {ax: str(i) for i, ax in enumerate(fig.axes)}\n    out = {}\n    for ax, letter in lettered.items():\n        bbs = [ax.get_tightbbox(r)]\n\n\n\n\n        ss = ax.get_subplotspec()\n        for sib in fig.axes:\n            if sib is ax or sib in lettered:\n                continue\n            ssib = sib.get_subplotspec()\n            same_row = ss is None or ssib is None or ss.rowspan == ssib.rowspan\n            same_col = ss is None or ssib is None or ss.colspan == ssib.colspan\n            if ((ax.get_shared_y_axes().joined(ax, sib) and same_row)\n                    or (ax.get_shared_x_axes().joined(ax, sib) and same_col)):\n                bbs.append(sib.get_tightbbox(r))\n        bb = mpl.transforms.Bbox.union(bbs)\n\n        bx0 = (bb.x0 / fig.dpi - ox_in) * dpi\n        bx1 = (bb.x1 / fig.dpi - ox_in) * dpi\n        by0 = H_px - (bb.y1 / fig.dpi - oy_in) * dpi\n        by1 = H_px - (bb.y0 / fig.dpi - oy_in) * dpi\n        box = (\n            min(max(int(bx0) - pad_px, 0), W_px),\n            min(max(int(by0) - pad_px, 0), H_px),\n            max(min(int(bx1) + pad_px, W_px), 0),\n            max(min(int(by1) + pad_px, H_px), 0),\n        )\n        if box[0] < box[2] and box[1] < box[3]:\n            out[letter] = box\n    return out\n",
          "sourceDigest": "8020d409bfb60f0c8dfa43d0dfe7bb97f7458e1a3908fefbcb6e4551e0feae4e"
        }
      ],
      "helperEvidenceStatus": {
        "state": "complete"
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      "environmentManifest": {
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        "captureKind": "completed-run",
        "capturedAt": "2026-10-08T10:38:27.417Z",
        "installedInventory": {
          "capturedAt": "2026-10-08T07:31:03.005Z",
          "source": "cache-reused",
          "validation": "best-effort"
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          "text": "[{\"id\": \"210\", \"coverage\": \"supporting_partial\", \"note\": \"Partial range: a*(a-1)<=b gives b>=30 at a=6; selected paper claims every b>=6. Cases 6<=b<30 are not covered by this Lean target.\", \"statement_path\": \"preprints/Foulkes-Conjecture-for-the-Sixth-Symmetric-Power-September-25-2026/build/sections/introduction.tex\", \"statement_line\": 19, \"targets\": [{\"name\": \"FoulkesHowe\", \"declarations\": [{\"name\": \"canonical_foulkes_howe_surjective\", \"line\": 42}], \"solution\": \"lean/OAI/RepresentationTheory/FoulkesHowe/Stabilization.lean\", \"supplied\": true}]}, {\"id\": \"215\", \"coverage\": \"supporting_partial\", \"note\": \"Supporting/companion only: finite-volume O(n) exponential correlation decay. No canonical interacting O(3) continuum-limit or reconstructed Hamiltonian-gap target.\", \"statement_path\": \"preprints/The-canonical-massive-continuum-limit-of-the-two-dimensional-O3-model-October-4-2026/build/sections/introduction.tex\", \"statement_line\": 61, \"targets\": [{\"name\": \"ClassicalON\", \"declarations\": [{\"name\": \"main\", \"line\": 53}], \"solution\": \"lean/OAI/Probability/ClassicalON/Main.lean\", \"supplied\": true}]}, {\"id\": \"230\", \"coverage\": \"supporting_partial\", \"note\": \"Supporting component only: positivity for gauges with the explicit small-radius formula. No finite-measure upper bound or finite expectation, and no gauge-existence conclusion in this target.\", \"statement_path\": \"preprints/An-exact-Hausdorff-gauge-for-SLE-September-25-2026/build/sections/introduction.tex\", \"statement_line\": 38, \"targets\": [{\"name\": \"SLELowerPositivity\", \"declarations\": [{\"name\": \"sourceLowerMain_proved\", \"line\": 78}], \"solution\": \"lean/OAI/Probability/SLE/LowerPositivity.lean\", \"supplied\": false}]}, {\"id\": \"245\", \"coverage\": \"main_target\", \"note\": \"Main target: system-wide weak normalization implies strong normalization for arbitrary PTS specifications, including nonfunctional rules and annotated terms.\", \"statement_path\": \"preprints/Weak-and-strong-normalization-in-pure-type-systems-September-25-2026/build/sections/introduction.tex\", \"statement_line\": 116, \"targets\": [{\"name\": \"TypeSystemNormalization\", \"declarations\": [{\"name\": \"weak_implies_strong\", \"line\": 116}], \"solution\": \"lean/OAI/Computability/TypeSystem/Normalization.lean\", \"supplied\": false}]}]\n"
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          "exports": [
            "apply_figure_style",
            "set_frame",
            "panel_letter",
            "focal_palette",
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            "two_tier_label",
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          "source": "META_GREY = \"#888888\"\n\n\ndef apply_figure_style(*, frame=\"open\", font=None, sizes=(8, 7, 6), grid=False):\n    \"\"\"Set matplotlib rcParams for publication-grade output. Call once before plotting.\n\n    This sets mechanics (role-mapped font-size ladder, outward ticks, frameless\n    legends, 300-dpi save, Type-42 embedded fonts) — not a house aesthetic.\n    Frame, font and the size ladder are parameters.\n\n    frame : 'open' (bottom+left spines, default) | 'boxed' (all four) | 'none'\n    font  : sans-serif family name; None = system default sans-serif\n    sizes : (base, secondary, tick) — titles/axis-labels, legend/annotation, ticks\n    grid  : whether to draw axes.grid (default False)\n    \"\"\"\n    import matplotlib as mpl\n    if frame not in (\"open\", \"boxed\", \"none\"):\n        raise ValueError(f\"frame must be 'open'|'boxed'|'none', got {frame!r}\")\n\n    try:\n        import os, sys, glob, matplotlib.font_manager as fm\n        fdir = os.path.join(os.environ.get(\"CONDA_PREFIX\") or sys.prefix, \"fonts\")\n        if os.path.isdir(fdir):\n            known = {f.fname for f in fm.fontManager.ttflist}\n            for f in glob.glob(os.path.join(fdir, \"*.ttf\")):\n                if f not in known:\n                    fm.fontManager.addfont(f)\n    except Exception:\n        pass\n    base, secondary, tick = sizes\n    boxed = (frame == \"boxed\")\n    rc = {\n        \"font.family\": \"sans-serif\",\n        \"font.size\": base,\n        \"axes.labelsize\": base,\n        \"axes.titlesize\": base,\n        \"legend.fontsize\": secondary,\n        \"xtick.labelsize\": tick,\n        \"ytick.labelsize\": tick,\n        \"axes.linewidth\": 0.6,\n        \"xtick.direction\": \"out\", \"ytick.direction\": \"out\",\n        \"xtick.major.size\": 3, \"ytick.major.size\": 3,\n        \"xtick.major.width\": 0.6, \"ytick.major.width\": 0.6,\n        \"axes.spines.top\": boxed, \"axes.spines.right\": boxed,\n        \"axes.spines.left\": frame != \"none\", \"axes.spines.bottom\": frame != \"none\",\n        \"axes.grid\": bool(grid),\n        \"legend.frameon\": False,\n        \"figure.dpi\": 200,\n        \"savefig.dpi\": 300,\n        \"savefig.bbox\": \"tight\",\n        \"axes.titleweight\": \"normal\",\n        \"axes.titlelocation\": \"left\",\n        \"axes.labelweight\": \"normal\",\n        \"lines.linewidth\": 1.2,\n        \"patch.linewidth\": 0.6,\n        \"pdf.fonttype\": 42, \"ps.fonttype\": 42,\n    }\n    if font:\n        rc[\"font.sans-serif\"] = [font, \"DejaVu Sans\"]\n    mpl.rcParams.update(rc)\n\n\ndef set_frame(ax, style=\"open\"):\n    \"\"\"§3: set spine visibility on an existing axes. style ∈ {'open','boxed','none'}.\"\"\"\n    show = {\"open\": (False, False, True, True),\n            \"boxed\": (True, True, True, True),\n            \"none\": (False, False, False, False)}[style]\n    for side, vis in zip((\"top\", \"right\", \"bottom\", \"left\"), show):\n        ax.spines[side].set_visible(vis)\n        if vis:\n            ax.spines[side].set_linewidth(0.6)\n    ax.tick_params(direction=\"out\", length=0 if style == \"none\" else 3, width=0.6)\n\n\ndef panel_letter(ax, letter, dx=-0.18, dy=1.02, case=\"lower\", fontsize=None):\n    \"\"\"§5.7: bold panel letter outside top-left of axes. case ∈ {'lower','upper'}.\"\"\"\n    import matplotlib.pyplot as plt\n    if fontsize is None:\n        fontsize = plt.rcParams.get(\"font.size\", 8) + 1\n    s = letter.lower() if case == \"lower\" else letter.upper()\n    artist = ax.text(dx, dy, s, transform=ax.transAxes,\n                     fontweight=\"bold\", fontsize=fontsize, va=\"bottom\", ha=\"left\")\n    artist.set_gid(\"figure-style-panel-letter\")\n\n\ndef focal_palette(labels, focal, focal_color, other=\"muted\", base_colors=None):\n    \"\"\"§4.2: map labels → colors with the focal series visually dominant.\n\n    other='muted'   — desaturate base_colors (or a default cycle) toward gray\n    other='grey'    — uniform light gray for all non-focal\n    other='ordinal' — non-focal on a single light→dark gray ramp (input order)\n    \"\"\"\n    import matplotlib.colors as mcolors\n    import matplotlib.pyplot as plt\n    focal_set = {focal} if isinstance(focal, str) else set(focal)\n    n = len(labels)\n    if not focal_set & set(labels):\n        raise ValueError(f\"focal {focal!r} not found in labels\")\n    base_colors = [] if base_colors is None else list(base_colors)\n    if not base_colors:\n        base_colors = plt.rcParams[\"axes.prop_cycle\"].by_key().get(\"color\", [\"#444444\"])\n    if not base_colors:\n        base_colors = [\"#444444\"]\n    base_colors = [base_colors[i % len(base_colors)] for i in range(n)]\n    if other == \"grey\":\n        rest = [\"#BCBCBC\"] * n\n    elif other == \"ordinal\":\n        nf = max(1, n - len(focal_set))\n        ramp = [mcolors.to_hex((v, v, v)) for v in\n                ([0.55] if nf == 1 else [0.80 - 0.35 * i / (nf - 1) for i in range(nf)])]\n        rest, k = [], 0\n        for l in labels:\n            rest.append(ramp[min(k, nf - 1)]); k += (l not in focal_set)\n    elif other == \"muted\":\n        def mute(c):\n            r, g, b = mcolors.to_rgb(c)\n            m = (r + g + b) / 3\n            return mcolors.to_hex((0.3 * r + 0.7 * m, 0.3 * g + 0.7 * m, 0.3 * b + 0.7 * m))\n        rest = [mute(c) for c in base_colors]\n    else:\n        raise ValueError(\"other must be 'muted', 'grey', or 'ordinal'\")\n    return [focal_color if l in focal_set else rest[i] for i, l in enumerate(labels)]\n\n\ndef bar_with_points(ax, x, ymat, labels, colors, jitter=0.08, show_points=True,\n                    errorbar=None, point_alpha=0.5, point_size=8):\n    \"\"\"§6.1: bar = mean; optionally overlay raw points or draw an interval.\n\n    colors   : per-label color list (e.g. from focal_palette)\n    x, ymat and labels must have equal lengths; mismatches raise before drawing.\n    Each group must be a nonempty, finite, one-dimensional real numeric array.\n    Requested SD/CI intervals require at least two observations per group.\n    errorbar : None | 'sd' | 'ci95' — drawn only when show_points is False.\n               'ci95' is the t-distribution 95% CI of the mean\n               (half-width t_{0.975,n-1} · s/√n); correct at small n where the\n               z-approximation (1.96·s/√n) is markedly too narrow.\n    \"\"\"\n    import numpy as np\n    x = np.asarray(x)\n    if x.ndim != 1 or not np.issubdtype(x.dtype, np.number):\n        raise TypeError(\n            \"bar_with_points x must be one-dimensional numeric positions; \"\n            \"pass category names through labels\"\n        )\n    ymat = list(ymat)\n    labels = list(labels)\n    if len(x) != len(ymat) or len(x) != len(labels):\n        raise ValueError(\"bar_with_points x, ymat and labels must have equal lengths\")\n    if colors is not None and not isinstance(colors, str):\n        colors = list(colors)\n        if not colors:\n            colors = None\n    if x.dtype.kind not in \"iuf\" or not np.isfinite(x).all():\n        raise ValueError(\"bar_with_points x must contain finite real positions\")\n    for i, y in enumerate(ymat):\n        if np.ma.isMaskedArray(y) and np.ma.getmaskarray(y).any():\n            raise ValueError(f\"bar_with_points group {i} contains masked observations; resolve exclusions explicitly before plotting\")\n    ymat = [np.asarray(y) for y in ymat]\n    for i, y in enumerate(ymat):\n        if y.ndim != 1 or y.size == 0 or y.dtype.kind not in \"iuf\" or not np.isfinite(y).all():\n            raise ValueError(f\"bar_with_points group {i} must be nonempty, one-dimensional and finite real numeric data\")\n    if errorbar and not show_points:\n        if errorbar not in (\"sd\", \"ci95\"):\n            raise ValueError(\"errorbar must be None, 'sd', or 'ci95'\")\n        if any(y.size < 2 for y in ymat):\n            raise ValueError(\"SD/CI cannot be estimated from one observation; use raw points without an interval\")\n    means = np.array([np.mean(y) for y in ymat], float)\n    err = None\n    if errorbar and not show_points:\n        if errorbar == \"sd\":\n            err = np.array([np.std(y, ddof=1) for y in ymat])\n        elif errorbar == \"ci95\":\n            from scipy.stats import t\n            def _hw(y):\n                n = np.asarray(y).size\n                return t.ppf(0.975, n - 1) * np.std(y, ddof=1) / np.sqrt(n)\n            err = np.array([_hw(y) for y in ymat])\n    ax.bar(x, means, color=colors, width=0.7, edgecolor=\"none\",\n           yerr=err, error_kw={\"elinewidth\": 0.8, \"capsize\": 0})\n    if show_points:\n        for xi, ys in zip(x, ymat):\n            ys = np.asarray(ys)\n            if ys.ndim == 1 and ys.size > 0:\n                jit = (np.random.rand(ys.size) - 0.5) * 2 * jitter\n                ax.scatter(np.full(ys.size, xi) + jit, ys, s=point_size, color=\"black\",\n                           alpha=point_alpha, zorder=3, linewidths=0)\n    ax.set_xticks(x); ax.set_xticklabels(labels)\n    return ax\n\n\ndef strip_with_median(ax, groups, values, colors=None, jitter=0.12):\n    \"\"\"§6.1: jittered points + bold horizontal median tick per group.\n\n    groups and values must have equal lengths. Mismatches raise before drawing.\n    colors=None or an empty color sequence supplies default gray; a nonempty\n    color sequence cycles across groups.\n    Each group must be nonempty, one-dimensional and finite real numeric data;\n    resolve missing observations explicitly before plotting.\n    \"\"\"\n    import numpy as np\n    labs = list(groups)\n    values = list(values)\n    if len(labs) != len(values):\n        raise ValueError(\"strip_with_median groups and values must have equal lengths\")\n    for i, y in enumerate(values):\n        if np.ma.isMaskedArray(y) and np.ma.getmaskarray(y).any():\n            raise ValueError(f\"strip_with_median group {i} contains masked observations; resolve exclusions explicitly before plotting\")\n    values = [np.asarray(y) for y in values]\n    for i, y in enumerate(values):\n        if y.ndim != 1 or y.size == 0 or y.dtype.kind not in \"iuf\" or not np.isfinite(y).all():\n            raise ValueError(f\"strip_with_median group {i} must be nonempty, one-dimensional and finite real numeric data\")\n    colors = [] if colors is None else list(colors)\n    if not colors:\n        colors = [\"#444444\"]\n    for i, ys in enumerate(values):\n        c = colors[i % len(colors)]\n        ys = np.asarray(ys)\n        jit = (np.random.rand(ys.size) - 0.5) * 2 * jitter\n        ax.scatter(np.full(ys.size, i) + jit, ys, s=10, color=c, alpha=0.6, linewidths=0, zorder=2)\n        m = np.median(ys)\n        ax.plot([i - 0.22, i + 0.22], [m, m], color=\"black\", lw=1.6, zorder=3)\n    ax.set_xticks(range(len(labs))); ax.set_xticklabels(labs)\n    return ax\n\n\ndef goodness_arrow(ax, text=\"higher = better\", loc=\"upper left\", axis=\"y\", fontsize=None):\n    \"\"\"§3.6: small upright direction-of-goodness cue in the margin.\"\"\"\n    import matplotlib.pyplot as plt\n    if fontsize is None:\n        fontsize = plt.rcParams[\"legend.fontsize\"]\n    pos = {\"upper left\": (0.02, 0.98), \"upper right\": (0.98, 0.98),\n           \"lower left\": (0.02, 0.02), \"lower right\": (0.98, 0.02)}[loc]\n    ha = \"left\" if \"left\" in loc else \"right\"\n    va = \"top\" if \"upper\" in loc else \"bottom\"\n    arrow = \"↑ \" if axis == \"y\" else \"→ \"\n    ax.text(pos[0], pos[1], arrow + text, transform=ax.transAxes,\n            fontsize=fontsize, color=META_GREY, ha=ha, va=va)\n\n\ndef two_tier_label(name, meta):\n    \"\"\"§5: two-line label string (name / metadata). Meta line styled separately by caller.\"\"\"\n    return f\"{name}\\n{meta}\"\n\n\ndef end_of_line_labels(ax, xs, ys, labels, colors=None, dx=0.01, fontsize=None):\n    \"\"\"§6.3 / §7.3: label each line series at its right end instead of a legend box.\n\n    Each label requires a nonempty, matched x/y series. Validate all series\n    before drawing. Nonempty colors cycle; None or empty uses default text color.\n    Inner sequences must support len() and indexing; they are not copied.\n    Nonfinite endpoints are rejected rather than producing an invisible label.\n    \"\"\"\n    import matplotlib.pyplot as plt\n    import math\n    xs, ys, labels = list(xs), list(ys), list(labels)\n    if len(xs) != len(ys) or len(xs) != len(labels):\n        raise ValueError(\"end_of_line_labels xs, ys and labels must have equal lengths\")\n    if any(len(x) == 0 or len(x) != len(y) for x, y in zip(xs, ys)):\n        raise ValueError(\"end_of_line_labels each x/y series must be nonempty and have equal lengths\")\n    if any(not math.isfinite(x[-1]) or not math.isfinite(y[-1]) for x, y in zip(xs, ys)):\n        raise ValueError(\"end_of_line_labels endpoints must be finite; choose valid endpoints explicitly\")\n    if fontsize is None:\n        fontsize = plt.rcParams[\"font.size\"]\n    colors = [] if colors is None else list(colors)\n    if not colors:\n        colors = [None]\n    span = ax.get_xlim()[1] - ax.get_xlim()[0]\n    for i, (x, y, lab) in enumerate(zip(xs, ys, labels)):\n        c = colors[i % len(colors)]\n        ax.text(x[-1] + dx * span, y[-1], lab, color=c, va=\"center\", ha=\"left\", fontsize=fontsize)\n\n\ndef panel_crops(fig, dpi=None, pad_px=6, bbox_inches=None, pad_inches=None):\n    \"\"\"§9.2: pixel-space crop boxes for each lettered panel in the SAVED PNG.\n\n    Returns ``{letter: (x0, y0, x1, y1)}`` in image-space pixels (origin\n    top-left, matching PIL's ``Image.crop``). Panels are detected as bold\n    single-character ``Text``\n    objects placed by :func:`panel_letter`; each panel's crop is its axes'\n    tightbbox mapped into the saved file's pixel space, padded by ``pad_px``.\n    For §3.4 composites (abutting subplots sharing an axis, letter on the\n    leftmost only) the crop unions in letterless ``sharex``/``sharey`` siblings\n    on the same grid row/col so the whole composite is covered. When no axes\n    carries a panel letter (standalone plot, or a figure-composer sub-agent),\n    falls back to one crop per axes keyed by index.\n\n    ``bbox_inches`` mirrors ``Figure.savefig`` semantics: ``None`` means\n    *consult rcParams* (so under :func:`apply_figure_style` it resolves to\n    ``'tight'``); pass an explicit ``Bbox`` only if you saved with one. The\n    boxes are clamped to the saved image extent; panels with no intersection\n    are omitted.\n\n        >>> from PIL import Image\n        >>> fig.savefig(\"fig.png\")  # bbox_inches='tight' via rcParams\n        >>> with Image.open(\"fig.png\") as saved:\n        ...     for letter, box in panel_crops(fig).items():\n        ...         saved.crop(box).save(f\"panel_{letter}.png\")\n\n    Match savefig's DPI, bbox_inches and pad_inches when overriding defaults.\n    Returned boxes are pixels, never savefig bbox_inches (inches). Passing them\n    back to savefig as inches can allocate an enormous raster.\n    \"\"\"\n    import matplotlib as mpl\n    import matplotlib.text\n    from matplotlib.backends.backend_agg import FigureCanvasAgg\n    if isinstance(pad_px, bool) or not isinstance(pad_px, int) or pad_px < 0:\n        raise TypeError(\"panel_crops pad_px must be a non-negative integer\")\n    if dpi is None:\n        dpi = mpl.rcParams.get(\"savefig.dpi\", fig.dpi)\n        if dpi == \"figure\":\n            dpi = fig.dpi\n    dpi = float(dpi)\n    if bbox_inches is None:\n        bbox_inches = mpl.rcParams.get(\"savefig.bbox\")\n    # The Open-Science notebook captures and closes pyplot figures after every\n    # cell. Matplotlib 3.11 decouples a closed Figure from its backend by\n    # replacing FigureCanvasAgg with FigureCanvasBase, while the persistent\n    # notebook namespace can still retain the Figure itself. Reattach the\n    # headless canvas before asking for saved-image geometry so this helper is\n    # safe in a later QA cell as well as the producer cell.\n    if not hasattr(fig.canvas, \"get_renderer\"):\n        FigureCanvasAgg(fig)\n    fig.canvas.draw()\n    r = fig.canvas.get_renderer()\n\n    if bbox_inches == \"tight\":\n        if pad_inches is None:\n            pad_inches = mpl.rcParams.get(\"savefig.pad_inches\", 0.1)\n        tb = fig.get_tightbbox(r).padded(pad_inches)\n        ox_in, oy_in = tb.x0, tb.y0\n        W_in, H_in = tb.width, tb.height\n    elif isinstance(bbox_inches, mpl.transforms.BboxBase):\n        ox_in, oy_in = bbox_inches.x0, bbox_inches.y0\n        W_in, H_in = bbox_inches.width, bbox_inches.height\n    elif bbox_inches is None:\n        ox_in, oy_in = 0.0, 0.0\n        W_in, H_in = fig.get_size_inches()\n    else:\n        raise TypeError(\"panel_crops bbox_inches must be None, 'tight', or a matplotlib Bbox\")\n    W_px, H_px = int(round(W_in * dpi)), int(round(H_in * dpi))\n    lettered = {}\n    seen_letters = set()\n    for ax in fig.axes:\n        for t in ax.findobj(matplotlib.text.Text):\n            s = (t.get_text() or \"\").strip()\n            if t.get_gid() == \"figure-style-panel-letter\":\n                if s in seen_letters:\n                    raise ValueError(f\"Duplicate panel letter: {s!r}\")\n                seen_letters.add(s)\n                lettered[ax] = s\n                break\n\n\n\n    if not lettered:\n        lettered = {ax: str(i) for i, ax in enumerate(fig.axes)}\n    out = {}\n    for ax, letter in lettered.items():\n        bbs = [ax.get_tightbbox(r)]\n\n\n\n\n        ss = ax.get_subplotspec()\n        for sib in fig.axes:\n            if sib is ax or sib in lettered:\n                continue\n            ssib = sib.get_subplotspec()\n            same_row = ss is None or ssib is None or ss.rowspan == ssib.rowspan\n            same_col = ss is None or ssib is None or ss.colspan == ssib.colspan\n            if ((ax.get_shared_y_axes().joined(ax, sib) and same_row)\n                    or (ax.get_shared_x_axes().joined(ax, sib) and same_col)):\n                bbs.append(sib.get_tightbbox(r))\n        bb = mpl.transforms.Bbox.union(bbs)\n\n        bx0 = (bb.x0 / fig.dpi - ox_in) * dpi\n        bx1 = (bb.x1 / fig.dpi - ox_in) * dpi\n        by0 = H_px - (bb.y1 / fig.dpi - oy_in) * dpi\n        by1 = H_px - (bb.y0 / fig.dpi - oy_in) * dpi\n        box = (\n            min(max(int(bx0) - pad_px, 0), W_px),\n            min(max(int(by0) - pad_px, 0), H_px),\n            max(min(int(bx1) + pad_px, W_px), 0),\n            max(min(int(by1) + pad_px, H_px), 0),\n        )\n        if box[0] < box[2] and box[1] < box[3]:\n            out[letter] = box\n    return out\n",
          "sourceDigest": "8020d409bfb60f0c8dfa43d0dfe7bb97f7458e1a3908fefbcb6e4551e0feae4e"
        }
      ],
      "helperEvidenceStatus": {
        "state": "complete"
      },
      "environmentManifest": {
        "schemaVersion": 1,
        "captureKind": "completed-run",
        "capturedAt": "2026-10-08T10:38:28.884Z",
        "installedInventory": {
          "capturedAt": "2026-10-08T07:31:03.005Z",
          "source": "cache-reused",
          "validation": "best-effort"
        },
        "kernelKind": "python",
        "environmentName": "default-python",
        "runtimeSource": "managed",
        "runtimeVersion": "3.12.14",
        "executionContext": {
          "schemaVersion": 1,
          "before": {
            "locale": "C/C.UTF-8/C/C/C/C",
            "timezone": "CST/CST",
            "threadLimits": {},
            "randomLibraries": [
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            ],
            "pythonRandomState": {
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          "source": "META_GREY = \"#888888\"\n\n\ndef apply_figure_style(*, frame=\"open\", font=None, sizes=(8, 7, 6), grid=False):\n    \"\"\"Set matplotlib rcParams for publication-grade output. Call once before plotting.\n\n    This sets mechanics (role-mapped font-size ladder, outward ticks, frameless\n    legends, 300-dpi save, Type-42 embedded fonts) — not a house aesthetic.\n    Frame, font and the size ladder are parameters.\n\n    frame : 'open' (bottom+left spines, default) | 'boxed' (all four) | 'none'\n    font  : sans-serif family name; None = system default sans-serif\n    sizes : (base, secondary, tick) — titles/axis-labels, legend/annotation, ticks\n    grid  : whether to draw axes.grid (default False)\n    \"\"\"\n    import matplotlib as mpl\n    if frame not in (\"open\", \"boxed\", \"none\"):\n        raise ValueError(f\"frame must be 'open'|'boxed'|'none', got {frame!r}\")\n\n    try:\n        import os, sys, glob, matplotlib.font_manager as fm\n        fdir = os.path.join(os.environ.get(\"CONDA_PREFIX\") or sys.prefix, \"fonts\")\n        if os.path.isdir(fdir):\n            known = {f.fname for f in fm.fontManager.ttflist}\n            for f in glob.glob(os.path.join(fdir, \"*.ttf\")):\n                if f not in known:\n                    fm.fontManager.addfont(f)\n    except Exception:\n        pass\n    base, secondary, tick = sizes\n    boxed = (frame == \"boxed\")\n    rc = {\n        \"font.family\": \"sans-serif\",\n        \"font.size\": base,\n        \"axes.labelsize\": base,\n        \"axes.titlesize\": base,\n        \"legend.fontsize\": secondary,\n        \"xtick.labelsize\": tick,\n        \"ytick.labelsize\": tick,\n        \"axes.linewidth\": 0.6,\n        \"xtick.direction\": \"out\", \"ytick.direction\": \"out\",\n        \"xtick.major.size\": 3, \"ytick.major.size\": 3,\n        \"xtick.major.width\": 0.6, \"ytick.major.width\": 0.6,\n        \"axes.spines.top\": boxed, \"axes.spines.right\": boxed,\n        \"axes.spines.left\": frame != \"none\", \"axes.spines.bottom\": frame != \"none\",\n        \"axes.grid\": bool(grid),\n        \"legend.frameon\": False,\n        \"figure.dpi\": 200,\n        \"savefig.dpi\": 300,\n        \"savefig.bbox\": \"tight\",\n        \"axes.titleweight\": \"normal\",\n        \"axes.titlelocation\": \"left\",\n        \"axes.labelweight\": \"normal\",\n        \"lines.linewidth\": 1.2,\n        \"patch.linewidth\": 0.6,\n        \"pdf.fonttype\": 42, \"ps.fonttype\": 42,\n    }\n    if font:\n        rc[\"font.sans-serif\"] = [font, \"DejaVu Sans\"]\n    mpl.rcParams.update(rc)\n\n\ndef set_frame(ax, style=\"open\"):\n    \"\"\"§3: set spine visibility on an existing axes. style ∈ {'open','boxed','none'}.\"\"\"\n    show = {\"open\": (False, False, True, True),\n            \"boxed\": (True, True, True, True),\n            \"none\": (False, False, False, False)}[style]\n    for side, vis in zip((\"top\", \"right\", \"bottom\", \"left\"), show):\n        ax.spines[side].set_visible(vis)\n        if vis:\n            ax.spines[side].set_linewidth(0.6)\n    ax.tick_params(direction=\"out\", length=0 if style == \"none\" else 3, width=0.6)\n\n\ndef panel_letter(ax, letter, dx=-0.18, dy=1.02, case=\"lower\", fontsize=None):\n    \"\"\"§5.7: bold panel letter outside top-left of axes. case ∈ {'lower','upper'}.\"\"\"\n    import matplotlib.pyplot as plt\n    if fontsize is None:\n        fontsize = plt.rcParams.get(\"font.size\", 8) + 1\n    s = letter.lower() if case == \"lower\" else letter.upper()\n    artist = ax.text(dx, dy, s, transform=ax.transAxes,\n                     fontweight=\"bold\", fontsize=fontsize, va=\"bottom\", ha=\"left\")\n    artist.set_gid(\"figure-style-panel-letter\")\n\n\ndef focal_palette(labels, focal, focal_color, other=\"muted\", base_colors=None):\n    \"\"\"§4.2: map labels → colors with the focal series visually dominant.\n\n    other='muted'   — desaturate base_colors (or a default cycle) toward gray\n    other='grey'    — uniform light gray for all non-focal\n    other='ordinal' — non-focal on a single light→dark gray ramp (input order)\n    \"\"\"\n    import matplotlib.colors as mcolors\n    import matplotlib.pyplot as plt\n    focal_set = {focal} if isinstance(focal, str) else set(focal)\n    n = len(labels)\n    if not focal_set & set(labels):\n        raise ValueError(f\"focal {focal!r} not found in labels\")\n    base_colors = [] if base_colors is None else list(base_colors)\n    if not base_colors:\n        base_colors = plt.rcParams[\"axes.prop_cycle\"].by_key().get(\"color\", [\"#444444\"])\n    if not base_colors:\n        base_colors = [\"#444444\"]\n    base_colors = [base_colors[i % len(base_colors)] for i in range(n)]\n    if other == \"grey\":\n        rest = [\"#BCBCBC\"] * n\n    elif other == \"ordinal\":\n        nf = max(1, n - len(focal_set))\n        ramp = [mcolors.to_hex((v, v, v)) for v in\n                ([0.55] if nf == 1 else [0.80 - 0.35 * i / (nf - 1) for i in range(nf)])]\n        rest, k = [], 0\n        for l in labels:\n            rest.append(ramp[min(k, nf - 1)]); k += (l not in focal_set)\n    elif other == \"muted\":\n        def mute(c):\n            r, g, b = mcolors.to_rgb(c)\n            m = (r + g + b) / 3\n            return mcolors.to_hex((0.3 * r + 0.7 * m, 0.3 * g + 0.7 * m, 0.3 * b + 0.7 * m))\n        rest = [mute(c) for c in base_colors]\n    else:\n        raise ValueError(\"other must be 'muted', 'grey', or 'ordinal'\")\n    return [focal_color if l in focal_set else rest[i] for i, l in enumerate(labels)]\n\n\ndef bar_with_points(ax, x, ymat, labels, colors, jitter=0.08, show_points=True,\n                    errorbar=None, point_alpha=0.5, point_size=8):\n    \"\"\"§6.1: bar = mean; optionally overlay raw points or draw an interval.\n\n    colors   : per-label color list (e.g. from focal_palette)\n    x, ymat and labels must have equal lengths; mismatches raise before drawing.\n    Each group must be a nonempty, finite, one-dimensional real numeric array.\n    Requested SD/CI intervals require at least two observations per group.\n    errorbar : None | 'sd' | 'ci95' — drawn only when show_points is False.\n               'ci95' is the t-distribution 95% CI of the mean\n               (half-width t_{0.975,n-1} · s/√n); correct at small n where the\n               z-approximation (1.96·s/√n) is markedly too narrow.\n    \"\"\"\n    import numpy as np\n    x = np.asarray(x)\n    if x.ndim != 1 or not np.issubdtype(x.dtype, np.number):\n        raise TypeError(\n            \"bar_with_points x must be one-dimensional numeric positions; \"\n            \"pass category names through labels\"\n        )\n    ymat = list(ymat)\n    labels = list(labels)\n    if len(x) != len(ymat) or len(x) != len(labels):\n        raise ValueError(\"bar_with_points x, ymat and labels must have equal lengths\")\n    if colors is not None and not isinstance(colors, str):\n        colors = list(colors)\n        if not colors:\n            colors = None\n    if x.dtype.kind not in \"iuf\" or not np.isfinite(x).all():\n        raise ValueError(\"bar_with_points x must contain finite real positions\")\n    for i, y in enumerate(ymat):\n        if np.ma.isMaskedArray(y) and np.ma.getmaskarray(y).any():\n            raise ValueError(f\"bar_with_points group {i} contains masked observations; resolve exclusions explicitly before plotting\")\n    ymat = [np.asarray(y) for y in ymat]\n    for i, y in enumerate(ymat):\n        if y.ndim != 1 or y.size == 0 or y.dtype.kind not in \"iuf\" or not np.isfinite(y).all():\n            raise ValueError(f\"bar_with_points group {i} must be nonempty, one-dimensional and finite real numeric data\")\n    if errorbar and not show_points:\n        if errorbar not in (\"sd\", \"ci95\"):\n            raise ValueError(\"errorbar must be None, 'sd', or 'ci95'\")\n        if any(y.size < 2 for y in ymat):\n            raise ValueError(\"SD/CI cannot be estimated from one observation; use raw points without an interval\")\n    means = np.array([np.mean(y) for y in ymat], float)\n    err = None\n    if errorbar and not show_points:\n        if errorbar == \"sd\":\n            err = np.array([np.std(y, ddof=1) for y in ymat])\n        elif errorbar == \"ci95\":\n            from scipy.stats import t\n            def _hw(y):\n                n = np.asarray(y).size\n                return t.ppf(0.975, n - 1) * np.std(y, ddof=1) / np.sqrt(n)\n            err = np.array([_hw(y) for y in ymat])\n    ax.bar(x, means, color=colors, width=0.7, edgecolor=\"none\",\n           yerr=err, error_kw={\"elinewidth\": 0.8, \"capsize\": 0})\n    if show_points:\n        for xi, ys in zip(x, ymat):\n            ys = np.asarray(ys)\n            if ys.ndim == 1 and ys.size > 0:\n                jit = (np.random.rand(ys.size) - 0.5) * 2 * jitter\n                ax.scatter(np.full(ys.size, xi) + jit, ys, s=point_size, color=\"black\",\n                           alpha=point_alpha, zorder=3, linewidths=0)\n    ax.set_xticks(x); ax.set_xticklabels(labels)\n    return ax\n\n\ndef strip_with_median(ax, groups, values, colors=None, jitter=0.12):\n    \"\"\"§6.1: jittered points + bold horizontal median tick per group.\n\n    groups and values must have equal lengths. Mismatches raise before drawing.\n    colors=None or an empty color sequence supplies default gray; a nonempty\n    color sequence cycles across groups.\n    Each group must be nonempty, one-dimensional and finite real numeric data;\n    resolve missing observations explicitly before plotting.\n    \"\"\"\n    import numpy as np\n    labs = list(groups)\n    values = list(values)\n    if len(labs) != len(values):\n        raise ValueError(\"strip_with_median groups and values must have equal lengths\")\n    for i, y in enumerate(values):\n        if np.ma.isMaskedArray(y) and np.ma.getmaskarray(y).any():\n            raise ValueError(f\"strip_with_median group {i} contains masked observations; resolve exclusions explicitly before plotting\")\n    values = [np.asarray(y) for y in values]\n    for i, y in enumerate(values):\n        if y.ndim != 1 or y.size == 0 or y.dtype.kind not in \"iuf\" or not np.isfinite(y).all():\n            raise ValueError(f\"strip_with_median group {i} must be nonempty, one-dimensional and finite real numeric data\")\n    colors = [] if colors is None else list(colors)\n    if not colors:\n        colors = [\"#444444\"]\n    for i, ys in enumerate(values):\n        c = colors[i % len(colors)]\n        ys = np.asarray(ys)\n        jit = (np.random.rand(ys.size) - 0.5) * 2 * jitter\n        ax.scatter(np.full(ys.size, i) + jit, ys, s=10, color=c, alpha=0.6, linewidths=0, zorder=2)\n        m = np.median(ys)\n        ax.plot([i - 0.22, i + 0.22], [m, m], color=\"black\", lw=1.6, zorder=3)\n    ax.set_xticks(range(len(labs))); ax.set_xticklabels(labs)\n    return ax\n\n\ndef goodness_arrow(ax, text=\"higher = better\", loc=\"upper left\", axis=\"y\", fontsize=None):\n    \"\"\"§3.6: small upright direction-of-goodness cue in the margin.\"\"\"\n    import matplotlib.pyplot as plt\n    if fontsize is None:\n        fontsize = plt.rcParams[\"legend.fontsize\"]\n    pos = {\"upper left\": (0.02, 0.98), \"upper right\": (0.98, 0.98),\n           \"lower left\": (0.02, 0.02), \"lower right\": (0.98, 0.02)}[loc]\n    ha = \"left\" if \"left\" in loc else \"right\"\n    va = \"top\" if \"upper\" in loc else \"bottom\"\n    arrow = \"↑ \" if axis == \"y\" else \"→ \"\n    ax.text(pos[0], pos[1], arrow + text, transform=ax.transAxes,\n            fontsize=fontsize, color=META_GREY, ha=ha, va=va)\n\n\ndef two_tier_label(name, meta):\n    \"\"\"§5: two-line label string (name / metadata). Meta line styled separately by caller.\"\"\"\n    return f\"{name}\\n{meta}\"\n\n\ndef end_of_line_labels(ax, xs, ys, labels, colors=None, dx=0.01, fontsize=None):\n    \"\"\"§6.3 / §7.3: label each line series at its right end instead of a legend box.\n\n    Each label requires a nonempty, matched x/y series. Validate all series\n    before drawing. Nonempty colors cycle; None or empty uses default text color.\n    Inner sequences must support len() and indexing; they are not copied.\n    Nonfinite endpoints are rejected rather than producing an invisible label.\n    \"\"\"\n    import matplotlib.pyplot as plt\n    import math\n    xs, ys, labels = list(xs), list(ys), list(labels)\n    if len(xs) != len(ys) or len(xs) != len(labels):\n        raise ValueError(\"end_of_line_labels xs, ys and labels must have equal lengths\")\n    if any(len(x) == 0 or len(x) != len(y) for x, y in zip(xs, ys)):\n        raise ValueError(\"end_of_line_labels each x/y series must be nonempty and have equal lengths\")\n    if any(not math.isfinite(x[-1]) or not math.isfinite(y[-1]) for x, y in zip(xs, ys)):\n        raise ValueError(\"end_of_line_labels endpoints must be finite; choose valid endpoints explicitly\")\n    if fontsize is None:\n        fontsize = plt.rcParams[\"font.size\"]\n    colors = [] if colors is None else list(colors)\n    if not colors:\n        colors = [None]\n    span = ax.get_xlim()[1] - ax.get_xlim()[0]\n    for i, (x, y, lab) in enumerate(zip(xs, ys, labels)):\n        c = colors[i % len(colors)]\n        ax.text(x[-1] + dx * span, y[-1], lab, color=c, va=\"center\", ha=\"left\", fontsize=fontsize)\n\n\ndef panel_crops(fig, dpi=None, pad_px=6, bbox_inches=None, pad_inches=None):\n    \"\"\"§9.2: pixel-space crop boxes for each lettered panel in the SAVED PNG.\n\n    Returns ``{letter: (x0, y0, x1, y1)}`` in image-space pixels (origin\n    top-left, matching PIL's ``Image.crop``). Panels are detected as bold\n    single-character ``Text``\n    objects placed by :func:`panel_letter`; each panel's crop is its axes'\n    tightbbox mapped into the saved file's pixel space, padded by ``pad_px``.\n    For §3.4 composites (abutting subplots sharing an axis, letter on the\n    leftmost only) the crop unions in letterless ``sharex``/``sharey`` siblings\n    on the same grid row/col so the whole composite is covered. When no axes\n    carries a panel letter (standalone plot, or a figure-composer sub-agent),\n    falls back to one crop per axes keyed by index.\n\n    ``bbox_inches`` mirrors ``Figure.savefig`` semantics: ``None`` means\n    *consult rcParams* (so under :func:`apply_figure_style` it resolves to\n    ``'tight'``); pass an explicit ``Bbox`` only if you saved with one. The\n    boxes are clamped to the saved image extent; panels with no intersection\n    are omitted.\n\n        >>> from PIL import Image\n        >>> fig.savefig(\"fig.png\")  # bbox_inches='tight' via rcParams\n        >>> with Image.open(\"fig.png\") as saved:\n        ...     for letter, box in panel_crops(fig).items():\n        ...         saved.crop(box).save(f\"panel_{letter}.png\")\n\n    Match savefig's DPI, bbox_inches and pad_inches when overriding defaults.\n    Returned boxes are pixels, never savefig bbox_inches (inches). Passing them\n    back to savefig as inches can allocate an enormous raster.\n    \"\"\"\n    import matplotlib as mpl\n    import matplotlib.text\n    from matplotlib.backends.backend_agg import FigureCanvasAgg\n    if isinstance(pad_px, bool) or not isinstance(pad_px, int) or pad_px < 0:\n        raise TypeError(\"panel_crops pad_px must be a non-negative integer\")\n    if dpi is None:\n        dpi = mpl.rcParams.get(\"savefig.dpi\", fig.dpi)\n        if dpi == \"figure\":\n            dpi = fig.dpi\n    dpi = float(dpi)\n    if bbox_inches is None:\n        bbox_inches = mpl.rcParams.get(\"savefig.bbox\")\n    # The Open-Science notebook captures and closes pyplot figures after every\n    # cell. Matplotlib 3.11 decouples a closed Figure from its backend by\n    # replacing FigureCanvasAgg with FigureCanvasBase, while the persistent\n    # notebook namespace can still retain the Figure itself. Reattach the\n    # headless canvas before asking for saved-image geometry so this helper is\n    # safe in a later QA cell as well as the producer cell.\n    if not hasattr(fig.canvas, \"get_renderer\"):\n        FigureCanvasAgg(fig)\n    fig.canvas.draw()\n    r = fig.canvas.get_renderer()\n\n    if bbox_inches == \"tight\":\n        if pad_inches is None:\n            pad_inches = mpl.rcParams.get(\"savefig.pad_inches\", 0.1)\n        tb = fig.get_tightbbox(r).padded(pad_inches)\n        ox_in, oy_in = tb.x0, tb.y0\n        W_in, H_in = tb.width, tb.height\n    elif isinstance(bbox_inches, mpl.transforms.BboxBase):\n        ox_in, oy_in = bbox_inches.x0, bbox_inches.y0\n        W_in, H_in = bbox_inches.width, bbox_inches.height\n    elif bbox_inches is None:\n        ox_in, oy_in = 0.0, 0.0\n        W_in, H_in = fig.get_size_inches()\n    else:\n        raise TypeError(\"panel_crops bbox_inches must be None, 'tight', or a matplotlib Bbox\")\n    W_px, H_px = int(round(W_in * dpi)), int(round(H_in * dpi))\n    lettered = {}\n    seen_letters = set()\n    for ax in fig.axes:\n        for t in ax.findobj(matplotlib.text.Text):\n            s = (t.get_text() or \"\").strip()\n            if t.get_gid() == \"figure-style-panel-letter\":\n                if s in seen_letters:\n                    raise ValueError(f\"Duplicate panel letter: {s!r}\")\n                seen_letters.add(s)\n                lettered[ax] = s\n                break\n\n\n\n    if not lettered:\n        lettered = {ax: str(i) for i, ax in enumerate(fig.axes)}\n    out = {}\n    for ax, letter in lettered.items():\n        bbs = [ax.get_tightbbox(r)]\n\n\n\n\n        ss = ax.get_subplotspec()\n        for sib in fig.axes:\n            if sib is ax or sib in lettered:\n                continue\n            ssib = sib.get_subplotspec()\n            same_row = ss is None or ssib is None or ss.rowspan == ssib.rowspan\n            same_col = ss is None or ssib is None or ss.colspan == ssib.colspan\n            if ((ax.get_shared_y_axes().joined(ax, sib) and same_row)\n                    or (ax.get_shared_x_axes().joined(ax, sib) and same_col)):\n                bbs.append(sib.get_tightbbox(r))\n        bb = mpl.transforms.Bbox.union(bbs)\n\n        bx0 = (bb.x0 / fig.dpi - ox_in) * dpi\n        bx1 = (bb.x1 / fig.dpi - ox_in) * dpi\n        by0 = H_px - (bb.y1 / fig.dpi - oy_in) * dpi\n        by1 = H_px - (bb.y0 / fig.dpi - oy_in) * dpi\n        box = (\n            min(max(int(bx0) - pad_px, 0), W_px),\n            min(max(int(by0) - pad_px, 0), H_px),\n            max(min(int(bx1) + pad_px, W_px), 0),\n            max(min(int(by1) + pad_px, H_px), 0),\n        )\n        if box[0] < box[2] and box[1] < box[3]:\n            out[letter] = box\n    return out\n",
          "sourceDigest": "8020d409bfb60f0c8dfa43d0dfe7bb97f7458e1a3908fefbcb6e4551e0feae4e"
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          "inventory-cache-best-effort"
        ]
      },
      "environmentManifestChecksum": "36e43041c544f3a22c1f07913dfc5f7a86cffae02607fe4763590e6bee0d18a4"
    },
    {
      "runId": "notebook-run-1791455954277-101",
      "executionMode": "foreground",
      "submissionIdentity": "43ac78eb-3697-49db-a7b9-f5a697cd3601",
      "submissionFingerprint": "f950608c924608de2458ab68b9439b5ac4a8738492cb5728626961d560c2a671",
      "admittedAt": 1791455954277,
      "kernelEpochId": "30b941a1-2823-4570-addf-fc45a3e9f366",
      "executionInvocationId": "43ac78eb-3697-49db-a7b9-f5a697cd3601",
      "cellId": "cell-f4dc70b3-32f1-4475-9cd9-bb3ba0e27029",
      "source": "agent",
      "inputKind": "cell",
      "kernelKind": "python",
      "script": "import csv\nfrom urllib.parse import quote\nlink=lambda q:base+quote(q,safe='/')\nsource_manifest=[{'path':k,'sha256':r['sha256'],'bytes':r['bytes'],'encoding':'base64' if 'base64' in r else 'utf8'} for k,r in src.items()]\nupdated_audit={'repository':pack['repository'],'commit':commit,'sample_seed':20261008,'selection_rule':'Same 20 actual family records as previous audit; first-listed manuscript; no resampling','draw_order':[r['family_id'] for r in sorted(updated,key=lambda r:r['draw_rank'])],'input_sha256':hashlib.sha256(srcpath.read_bytes()).hexdigest(),'source_provenance':pack['provenance'],'retrieved_at':pack['retrieved_at'],'checks':{'records':len(src),'all_size_sha256_pass':not failures,'matching_git_blob_hashes':sum(blob_matches),'comparable_git_blobs':len(blob_matches)},'counts':dict(counts),'written_material_papers':20,'checker_runs':0,'method':'Static inspection of manuscript TeX main statements and proof excerpts, scope notes, exact Comparator configuration/target statements and supplied solution entry files. PDF bytes checked; PDFs not independently rendered. Not a proof-correctness review or a transitive dependency audit.','limitations':['Scope alignment concerns advertised main conclusions, not independent certification of every encoding or auxiliary corollary.','Nine OAI implementation files supplied; only 8 of 23 Comparator configurations have their exact solution-module file supplied.','23 distinct direct OAI imports from the supplied implementation files are absent; the bundle is not a complete build environment.','Comparator challenge files use sorry placeholders; actual solutions are separate modules.','No proof checker, paper verification program or LaTeX build was run.','Nineteen individual READMEs contain citation information only; Foulkes README contains computational verification instructions.'],'records':source_manifest,'papers':updated}\nPath('proof-content-audit.json').write_text(json.dumps(updated_audit,ensure_ascii=False,indent=2))\nfields=['family_id','draw_rank','paper_title','pdf_url','paper_statement_url','scope_url','formal_coverage','coverage_evidence','comparator_targets','supplied_solution_entry_files','missing_solution_entry_files','written_proof_material','individual_readme_instructions','checker_execution']\nwith Path('proof-content-evidence.csv').open('w',newline='') as fh:\n w=csv.DictWriter(fh,fieldnames=fields);w.writeheader()\n for r in updated:\n  w.writerow({'family_id':r['family_id'],'draw_rank':r['draw_rank'],'paper_title':r['paper_title'],'pdf_url':link(r['pdf_path']),'paper_statement_url':link(r['main_statement']['path'])+'#L'+str(r['main_statement']['line']),'scope_url':link(r['scope_path']) if r['scope_path'] else '', 'formal_coverage':r['coverage'],'coverage_evidence':r['coverage_note'],'comparator_targets':' | '.join(link(t['challenge_path']) for t in r['targets']),'supplied_solution_entry_files':' | '.join(t['solution_path'] for t in r['targets'] if t['solution_file_supplied']),'missing_solution_entry_files':' | '.join(t['solution_path'] for t in r['targets'] if not t['solution_file_supplied']),'written_proof_material':True,'individual_readme_instructions':'Foulkes computational check/run instructions' if r['family_id']=='210' else 'Citation information only','checker_execution':'not_run'})\nassert sum(counts.values())==20 and set(counts)=={'main_target','supporting_partial','no_linked_lean'}\nassert {r['family_id'] for r in updated}=={r['family_id'] for r in rows}\nassert all(src[r['pdf_path']]['bytes']>0 and r['proof_environment_count']>0 for r in updated)\nprint('validated 20 unchanged representatives; saved',[(x,Path(x).stat().st_size) for x in ['proof-content-audit.json','proof-content-evidence.csv']])",
      "status": "completed",
      "startedAt": 1791455954502,
      "cwdBefore": "$DATA/notebooks/cmuzdubm90002rxoq52hkpl0k/01a11b04-b537-7300-9a27-fde070942335/data",
      "executionCount": 49,
      "environment": "default-python",
      "rootFrameId": "root-frame-01a11b04-b537-7300-9a27-fde070942335",
      "agentFrameId": "root-frame-01a11b04-b537-7300-9a27-fde070942335",
      "messageBranchId": "message-branch-01a11b04-b537-7300-9a27-fde070942335",
      "runtimeSegmentId": "runtime-segment-1791455444904-13",
      "promptMessageId": "message-7be8bde6-4ace-4a18-bf05-f1047b76da11",
      "text": {
        "stdout": "validated 20 unchanged representatives; saved [('proof-content-audit.json', 160677), ('proof-content-evidence.csv', 19348)]\n",
        "stderr": "",
        "traceback": "",
        "plain": [
          "validated 20 unchanged representatives; saved [('proof-content-audit.json', 160677), ('proof-content-evidence.csv', 19348)]\n"
        ]
      },
      "outputs": [
        {
          "type": "stream",
          "name": "stdout",
          "text": "validated 20 unchanged representatives; saved [('proof-content-audit.json', 160677), ('proof-content-evidence.csv', 19348)]\n"
        }
      ],
      "workingFiles": [
        {
          "path": "$DATA/notebooks/cmuzdubm90002rxoq52hkpl0k/01a11b04-b537-7300-9a27-fde070942335/data/proof-content-audit.json",
          "relativePath": "data/proof-content-audit.json",
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          "size": 160677,
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          "change": "created",
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        {
          "path": "$DATA/notebooks/cmuzdubm90002rxoq52hkpl0k/01a11b04-b537-7300-9a27-fde070942335/data/proof-content-evidence.csv",
          "relativePath": "data/proof-content-evidence.csv",
          "kind": "other",
          "size": 19348,
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          "change": "created",
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      "inputFiles": [
        {
          "inputFileVersionId": "e90a943c-32ea-4e4d-8396-fd3712a0e6d6",
          "sourceKind": "upload-version",
          "sourceFileId": "6e5827ac-14b1-4cdc-b511-a97f89eb61d8",
          "sourceVersionNumber": 1,
          "sourceCreatedAt": "2026-10-08T10:30:44.697Z",
          "sourceProjectId": "cmuzdubm90002rxoq52hkpl0k",
          "sourceSessionId": "01a11b04-b537-7300-9a27-fde070942335",
          "filename": "sampled-proof-sources.json",
          "contentType": "application/json",
          "sizeBytes": 16614108,
          "checksum": "4aa791ebc7778974755c8c7701f51a4f8c6a88468962132cc680349a848912e2",
          "storageKey": "uploads/cmuzdubm90002rxoq52hkpl0k/01a11b04-b537-7300-9a27-fde070942335/6e5827ac-14b1-4cdc-b511-a97f89eb61d8/versions/e90a943c-32ea-4e4d-8396-fd3712a0e6d6/content",
          "association": "turn-attached"
        }
      ],
      "frozenPermissionScope": {
        "allowedHelperSkillIds": []
      },
      "frozenRuntimeTarget": {
        "language": "python",
        "environment": "default-python",
        "processKey": "python:default-python"
      },
      "endedAt": 1791455956850,
      "cwdAfter": "$DATA/notebooks/cmuzdubm90002rxoq52hkpl0k/01a11b04-b537-7300-9a27-fde070942335/data",
      "fileEvidence": {
        "schemaVersion": 1,
        "activityId": "notebook-run-1791455954277-101",
        "activityKind": "notebook-run",
        "evidenceId": "execution-file-evidence-notebook-run-1791455954277-101",
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        "checksum": "24b582ebb43dcea48343c517a5b669af1f869bf70af7884fee5fa0a347f1ed33",
        "storageKey": "execution-file-evidence/cmuzdubm90002rxoq52hkpl0k/01a11b04-b537-7300-9a27-fde070942335/activity-notebook-run-1791455954277-101/evidence.json",
        "relationCount": 8,
        "generationCount": 8,
        "scientificOutputCount": 2,
        "initialViewState": "complete",
        "managedRootsFinalState": "partial",
        "scientificOutputAnalysis": "partial",
        "fileReads": "partial",
        "externalPaths": "partial",
        "writerAttribution": "partial",
        "reasonCodes": [
          "delayed-writes-not-observed",
          "dynamic-path-unresolved",
          "external-paths-not-observed",
          "file-reads-not-observed",
          "remote-outputs-not-observed",
          "source-analysis-unsupported-call",
          "transient-files-not-captured",
          "writer-not-isolated"
        ]
      },
      "environmentCapture": {
        "state": "available",
        "manifestChecksum": "b67485b3902f6d9f18319cafd8f3dad744d91a8a1c86b4de6117e7ef3d17103f",
        "warnings": [
          "inventory-cache-best-effort"
        ]
      },
      "environmentLock": {
        "state": "partial",
        "format": "environment-lock-bundle",
        "lockChecksum": "4bc194a59542cea2d1d61315dce7c4a1e010daf97c18179a77a707abd8e38fc7",
        "partialReasons": [
          "non-conda-package-detected"
        ],
        "diagnostics": [
          {
            "reason": "package-lock-missing",
            "packageName": "fonttools",
            "observedVersion": "4.63.0"
          },
          {
            "reason": "package-lock-missing",
            "packageName": "kiwisolver",
            "observedVersion": "1.5.1"
          },
          {
            "reason": "package-lock-missing",
            "packageName": "matplotlib",
            "observedVersion": "3.11.1"
          },
          {
            "reason": "package-lock-missing",
            "packageName": "numpy",
            "observedVersion": "2.5.3"
          },
          {
            "reason": "package-lock-missing",
            "packageName": "packaging",
            "observedVersion": "26.3"
          },
          {
            "reason": "package-lock-missing",
            "packageName": "pip",
            "observedVersion": "26.2.1"
          },
          {
            "reason": "package-lock-missing",
            "packageName": "setuptools",
            "observedVersion": "83.0.0"
          }
        ]
      },
      "kernelDispatched": true,
      "helperModules": [
        {
          "helperId": "figure-style",
          "skillIdentity": "figure-style",
          "packageOrigin": "builtin",
          "interfaceRevision": "1",
          "registeredGeneration": "sha256:90258343aec7bb96c2e129c39e1beb23230ee947c0db563f96913458d1795f1d",
          "exports": [
            "apply_figure_style",
            "set_frame",
            "panel_letter",
            "focal_palette",
            "bar_with_points",
            "strip_with_median",
            "goodness_arrow",
            "two_tier_label",
            "end_of_line_labels",
            "panel_crops"
          ],
          "source": "META_GREY = \"#888888\"\n\n\ndef apply_figure_style(*, frame=\"open\", font=None, sizes=(8, 7, 6), grid=False):\n    \"\"\"Set matplotlib rcParams for publication-grade output. Call once before plotting.\n\n    This sets mechanics (role-mapped font-size ladder, outward ticks, frameless\n    legends, 300-dpi save, Type-42 embedded fonts) — not a house aesthetic.\n    Frame, font and the size ladder are parameters.\n\n    frame : 'open' (bottom+left spines, default) | 'boxed' (all four) | 'none'\n    font  : sans-serif family name; None = system default sans-serif\n    sizes : (base, secondary, tick) — titles/axis-labels, legend/annotation, ticks\n    grid  : whether to draw axes.grid (default False)\n    \"\"\"\n    import matplotlib as mpl\n    if frame not in (\"open\", \"boxed\", \"none\"):\n        raise ValueError(f\"frame must be 'open'|'boxed'|'none', got {frame!r}\")\n\n    try:\n        import os, sys, glob, matplotlib.font_manager as fm\n        fdir = os.path.join(os.environ.get(\"CONDA_PREFIX\") or sys.prefix, \"fonts\")\n        if os.path.isdir(fdir):\n            known = {f.fname for f in fm.fontManager.ttflist}\n            for f in glob.glob(os.path.join(fdir, \"*.ttf\")):\n                if f not in known:\n                    fm.fontManager.addfont(f)\n    except Exception:\n        pass\n    base, secondary, tick = sizes\n    boxed = (frame == \"boxed\")\n    rc = {\n        \"font.family\": \"sans-serif\",\n        \"font.size\": base,\n        \"axes.labelsize\": base,\n        \"axes.titlesize\": base,\n        \"legend.fontsize\": secondary,\n        \"xtick.labelsize\": tick,\n        \"ytick.labelsize\": tick,\n        \"axes.linewidth\": 0.6,\n        \"xtick.direction\": \"out\", \"ytick.direction\": \"out\",\n        \"xtick.major.size\": 3, \"ytick.major.size\": 3,\n        \"xtick.major.width\": 0.6, \"ytick.major.width\": 0.6,\n        \"axes.spines.top\": boxed, \"axes.spines.right\": boxed,\n        \"axes.spines.left\": frame != \"none\", \"axes.spines.bottom\": frame != \"none\",\n        \"axes.grid\": bool(grid),\n        \"legend.frameon\": False,\n        \"figure.dpi\": 200,\n        \"savefig.dpi\": 300,\n        \"savefig.bbox\": \"tight\",\n        \"axes.titleweight\": \"normal\",\n        \"axes.titlelocation\": \"left\",\n        \"axes.labelweight\": \"normal\",\n        \"lines.linewidth\": 1.2,\n        \"patch.linewidth\": 0.6,\n        \"pdf.fonttype\": 42, \"ps.fonttype\": 42,\n    }\n    if font:\n        rc[\"font.sans-serif\"] = [font, \"DejaVu Sans\"]\n    mpl.rcParams.update(rc)\n\n\ndef set_frame(ax, style=\"open\"):\n    \"\"\"§3: set spine visibility on an existing axes. style ∈ {'open','boxed','none'}.\"\"\"\n    show = {\"open\": (False, False, True, True),\n            \"boxed\": (True, True, True, True),\n            \"none\": (False, False, False, False)}[style]\n    for side, vis in zip((\"top\", \"right\", \"bottom\", \"left\"), show):\n        ax.spines[side].set_visible(vis)\n        if vis:\n            ax.spines[side].set_linewidth(0.6)\n    ax.tick_params(direction=\"out\", length=0 if style == \"none\" else 3, width=0.6)\n\n\ndef panel_letter(ax, letter, dx=-0.18, dy=1.02, case=\"lower\", fontsize=None):\n    \"\"\"§5.7: bold panel letter outside top-left of axes. case ∈ {'lower','upper'}.\"\"\"\n    import matplotlib.pyplot as plt\n    if fontsize is None:\n        fontsize = plt.rcParams.get(\"font.size\", 8) + 1\n    s = letter.lower() if case == \"lower\" else letter.upper()\n    artist = ax.text(dx, dy, s, transform=ax.transAxes,\n                     fontweight=\"bold\", fontsize=fontsize, va=\"bottom\", ha=\"left\")\n    artist.set_gid(\"figure-style-panel-letter\")\n\n\ndef focal_palette(labels, focal, focal_color, other=\"muted\", base_colors=None):\n    \"\"\"§4.2: map labels → colors with the focal series visually dominant.\n\n    other='muted'   — desaturate base_colors (or a default cycle) toward gray\n    other='grey'    — uniform light gray for all non-focal\n    other='ordinal' — non-focal on a single light→dark gray ramp (input order)\n    \"\"\"\n    import matplotlib.colors as mcolors\n    import matplotlib.pyplot as plt\n    focal_set = {focal} if isinstance(focal, str) else set(focal)\n    n = len(labels)\n    if not focal_set & set(labels):\n        raise ValueError(f\"focal {focal!r} not found in labels\")\n    base_colors = [] if base_colors is None else list(base_colors)\n    if not base_colors:\n        base_colors = plt.rcParams[\"axes.prop_cycle\"].by_key().get(\"color\", [\"#444444\"])\n    if not base_colors:\n        base_colors = [\"#444444\"]\n    base_colors = [base_colors[i % len(base_colors)] for i in range(n)]\n    if other == \"grey\":\n        rest = [\"#BCBCBC\"] * n\n    elif other == \"ordinal\":\n        nf = max(1, n - len(focal_set))\n        ramp = [mcolors.to_hex((v, v, v)) for v in\n                ([0.55] if nf == 1 else [0.80 - 0.35 * i / (nf - 1) for i in range(nf)])]\n        rest, k = [], 0\n        for l in labels:\n            rest.append(ramp[min(k, nf - 1)]); k += (l not in focal_set)\n    elif other == \"muted\":\n        def mute(c):\n            r, g, b = mcolors.to_rgb(c)\n            m = (r + g + b) / 3\n            return mcolors.to_hex((0.3 * r + 0.7 * m, 0.3 * g + 0.7 * m, 0.3 * b + 0.7 * m))\n        rest = [mute(c) for c in base_colors]\n    else:\n        raise ValueError(\"other must be 'muted', 'grey', or 'ordinal'\")\n    return [focal_color if l in focal_set else rest[i] for i, l in enumerate(labels)]\n\n\ndef bar_with_points(ax, x, ymat, labels, colors, jitter=0.08, show_points=True,\n                    errorbar=None, point_alpha=0.5, point_size=8):\n    \"\"\"§6.1: bar = mean; optionally overlay raw points or draw an interval.\n\n    colors   : per-label color list (e.g. from focal_palette)\n    x, ymat and labels must have equal lengths; mismatches raise before drawing.\n    Each group must be a nonempty, finite, one-dimensional real numeric array.\n    Requested SD/CI intervals require at least two observations per group.\n    errorbar : None | 'sd' | 'ci95' — drawn only when show_points is False.\n               'ci95' is the t-distribution 95% CI of the mean\n               (half-width t_{0.975,n-1} · s/√n); correct at small n where the\n               z-approximation (1.96·s/√n) is markedly too narrow.\n    \"\"\"\n    import numpy as np\n    x = np.asarray(x)\n    if x.ndim != 1 or not np.issubdtype(x.dtype, np.number):\n        raise TypeError(\n            \"bar_with_points x must be one-dimensional numeric positions; \"\n            \"pass category names through labels\"\n        )\n    ymat = list(ymat)\n    labels = list(labels)\n    if len(x) != len(ymat) or len(x) != len(labels):\n        raise ValueError(\"bar_with_points x, ymat and labels must have equal lengths\")\n    if colors is not None and not isinstance(colors, str):\n        colors = list(colors)\n        if not colors:\n            colors = None\n    if x.dtype.kind not in \"iuf\" or not np.isfinite(x).all():\n        raise ValueError(\"bar_with_points x must contain finite real positions\")\n    for i, y in enumerate(ymat):\n        if np.ma.isMaskedArray(y) and np.ma.getmaskarray(y).any():\n            raise ValueError(f\"bar_with_points group {i} contains masked observations; resolve exclusions explicitly before plotting\")\n    ymat = [np.asarray(y) for y in ymat]\n    for i, y in enumerate(ymat):\n        if y.ndim != 1 or y.size == 0 or y.dtype.kind not in \"iuf\" or not np.isfinite(y).all():\n            raise ValueError(f\"bar_with_points group {i} must be nonempty, one-dimensional and finite real numeric data\")\n    if errorbar and not show_points:\n        if errorbar not in (\"sd\", \"ci95\"):\n            raise ValueError(\"errorbar must be None, 'sd', or 'ci95'\")\n        if any(y.size < 2 for y in ymat):\n            raise ValueError(\"SD/CI cannot be estimated from one observation; use raw points without an interval\")\n    means = np.array([np.mean(y) for y in ymat], float)\n    err = None\n    if errorbar and not show_points:\n        if errorbar == \"sd\":\n            err = np.array([np.std(y, ddof=1) for y in ymat])\n        elif errorbar == \"ci95\":\n            from scipy.stats import t\n            def _hw(y):\n                n = np.asarray(y).size\n                return t.ppf(0.975, n - 1) * np.std(y, ddof=1) / np.sqrt(n)\n            err = np.array([_hw(y) for y in ymat])\n    ax.bar(x, means, color=colors, width=0.7, edgecolor=\"none\",\n           yerr=err, error_kw={\"elinewidth\": 0.8, \"capsize\": 0})\n    if show_points:\n        for xi, ys in zip(x, ymat):\n            ys = np.asarray(ys)\n            if ys.ndim == 1 and ys.size > 0:\n                jit = (np.random.rand(ys.size) - 0.5) * 2 * jitter\n                ax.scatter(np.full(ys.size, xi) + jit, ys, s=point_size, color=\"black\",\n                           alpha=point_alpha, zorder=3, linewidths=0)\n    ax.set_xticks(x); ax.set_xticklabels(labels)\n    return ax\n\n\ndef strip_with_median(ax, groups, values, colors=None, jitter=0.12):\n    \"\"\"§6.1: jittered points + bold horizontal median tick per group.\n\n    groups and values must have equal lengths. Mismatches raise before drawing.\n    colors=None or an empty color sequence supplies default gray; a nonempty\n    color sequence cycles across groups.\n    Each group must be nonempty, one-dimensional and finite real numeric data;\n    resolve missing observations explicitly before plotting.\n    \"\"\"\n    import numpy as np\n    labs = list(groups)\n    values = list(values)\n    if len(labs) != len(values):\n        raise ValueError(\"strip_with_median groups and values must have equal lengths\")\n    for i, y in enumerate(values):\n        if np.ma.isMaskedArray(y) and np.ma.getmaskarray(y).any():\n            raise ValueError(f\"strip_with_median group {i} contains masked observations; resolve exclusions explicitly before plotting\")\n    values = [np.asarray(y) for y in values]\n    for i, y in enumerate(values):\n        if y.ndim != 1 or y.size == 0 or y.dtype.kind not in \"iuf\" or not np.isfinite(y).all():\n            raise ValueError(f\"strip_with_median group {i} must be nonempty, one-dimensional and finite real numeric data\")\n    colors = [] if colors is None else list(colors)\n    if not colors:\n        colors = [\"#444444\"]\n    for i, ys in enumerate(values):\n        c = colors[i % len(colors)]\n        ys = np.asarray(ys)\n        jit = (np.random.rand(ys.size) - 0.5) * 2 * jitter\n        ax.scatter(np.full(ys.size, i) + jit, ys, s=10, color=c, alpha=0.6, linewidths=0, zorder=2)\n        m = np.median(ys)\n        ax.plot([i - 0.22, i + 0.22], [m, m], color=\"black\", lw=1.6, zorder=3)\n    ax.set_xticks(range(len(labs))); ax.set_xticklabels(labs)\n    return ax\n\n\ndef goodness_arrow(ax, text=\"higher = better\", loc=\"upper left\", axis=\"y\", fontsize=None):\n    \"\"\"§3.6: small upright direction-of-goodness cue in the margin.\"\"\"\n    import matplotlib.pyplot as plt\n    if fontsize is None:\n        fontsize = plt.rcParams[\"legend.fontsize\"]\n    pos = {\"upper left\": (0.02, 0.98), \"upper right\": (0.98, 0.98),\n           \"lower left\": (0.02, 0.02), \"lower right\": (0.98, 0.02)}[loc]\n    ha = \"left\" if \"left\" in loc else \"right\"\n    va = \"top\" if \"upper\" in loc else \"bottom\"\n    arrow = \"↑ \" if axis == \"y\" else \"→ \"\n    ax.text(pos[0], pos[1], arrow + text, transform=ax.transAxes,\n            fontsize=fontsize, color=META_GREY, ha=ha, va=va)\n\n\ndef two_tier_label(name, meta):\n    \"\"\"§5: two-line label string (name / metadata). Meta line styled separately by caller.\"\"\"\n    return f\"{name}\\n{meta}\"\n\n\ndef end_of_line_labels(ax, xs, ys, labels, colors=None, dx=0.01, fontsize=None):\n    \"\"\"§6.3 / §7.3: label each line series at its right end instead of a legend box.\n\n    Each label requires a nonempty, matched x/y series. Validate all series\n    before drawing. Nonempty colors cycle; None or empty uses default text color.\n    Inner sequences must support len() and indexing; they are not copied.\n    Nonfinite endpoints are rejected rather than producing an invisible label.\n    \"\"\"\n    import matplotlib.pyplot as plt\n    import math\n    xs, ys, labels = list(xs), list(ys), list(labels)\n    if len(xs) != len(ys) or len(xs) != len(labels):\n        raise ValueError(\"end_of_line_labels xs, ys and labels must have equal lengths\")\n    if any(len(x) == 0 or len(x) != len(y) for x, y in zip(xs, ys)):\n        raise ValueError(\"end_of_line_labels each x/y series must be nonempty and have equal lengths\")\n    if any(not math.isfinite(x[-1]) or not math.isfinite(y[-1]) for x, y in zip(xs, ys)):\n        raise ValueError(\"end_of_line_labels endpoints must be finite; choose valid endpoints explicitly\")\n    if fontsize is None:\n        fontsize = plt.rcParams[\"font.size\"]\n    colors = [] if colors is None else list(colors)\n    if not colors:\n        colors = [None]\n    span = ax.get_xlim()[1] - ax.get_xlim()[0]\n    for i, (x, y, lab) in enumerate(zip(xs, ys, labels)):\n        c = colors[i % len(colors)]\n        ax.text(x[-1] + dx * span, y[-1], lab, color=c, va=\"center\", ha=\"left\", fontsize=fontsize)\n\n\ndef panel_crops(fig, dpi=None, pad_px=6, bbox_inches=None, pad_inches=None):\n    \"\"\"§9.2: pixel-space crop boxes for each lettered panel in the SAVED PNG.\n\n    Returns ``{letter: (x0, y0, x1, y1)}`` in image-space pixels (origin\n    top-left, matching PIL's ``Image.crop``). Panels are detected as bold\n    single-character ``Text``\n    objects placed by :func:`panel_letter`; each panel's crop is its axes'\n    tightbbox mapped into the saved file's pixel space, padded by ``pad_px``.\n    For §3.4 composites (abutting subplots sharing an axis, letter on the\n    leftmost only) the crop unions in letterless ``sharex``/``sharey`` siblings\n    on the same grid row/col so the whole composite is covered. When no axes\n    carries a panel letter (standalone plot, or a figure-composer sub-agent),\n    falls back to one crop per axes keyed by index.\n\n    ``bbox_inches`` mirrors ``Figure.savefig`` semantics: ``None`` means\n    *consult rcParams* (so under :func:`apply_figure_style` it resolves to\n    ``'tight'``); pass an explicit ``Bbox`` only if you saved with one. The\n    boxes are clamped to the saved image extent; panels with no intersection\n    are omitted.\n\n        >>> from PIL import Image\n        >>> fig.savefig(\"fig.png\")  # bbox_inches='tight' via rcParams\n        >>> with Image.open(\"fig.png\") as saved:\n        ...     for letter, box in panel_crops(fig).items():\n        ...         saved.crop(box).save(f\"panel_{letter}.png\")\n\n    Match savefig's DPI, bbox_inches and pad_inches when overriding defaults.\n    Returned boxes are pixels, never savefig bbox_inches (inches). Passing them\n    back to savefig as inches can allocate an enormous raster.\n    \"\"\"\n    import matplotlib as mpl\n    import matplotlib.text\n    from matplotlib.backends.backend_agg import FigureCanvasAgg\n    if isinstance(pad_px, bool) or not isinstance(pad_px, int) or pad_px < 0:\n        raise TypeError(\"panel_crops pad_px must be a non-negative integer\")\n    if dpi is None:\n        dpi = mpl.rcParams.get(\"savefig.dpi\", fig.dpi)\n        if dpi == \"figure\":\n            dpi = fig.dpi\n    dpi = float(dpi)\n    if bbox_inches is None:\n        bbox_inches = mpl.rcParams.get(\"savefig.bbox\")\n    # The Open-Science notebook captures and closes pyplot figures after every\n    # cell. Matplotlib 3.11 decouples a closed Figure from its backend by\n    # replacing FigureCanvasAgg with FigureCanvasBase, while the persistent\n    # notebook namespace can still retain the Figure itself. Reattach the\n    # headless canvas before asking for saved-image geometry so this helper is\n    # safe in a later QA cell as well as the producer cell.\n    if not hasattr(fig.canvas, \"get_renderer\"):\n        FigureCanvasAgg(fig)\n    fig.canvas.draw()\n    r = fig.canvas.get_renderer()\n\n    if bbox_inches == \"tight\":\n        if pad_inches is None:\n            pad_inches = mpl.rcParams.get(\"savefig.pad_inches\", 0.1)\n        tb = fig.get_tightbbox(r).padded(pad_inches)\n        ox_in, oy_in = tb.x0, tb.y0\n        W_in, H_in = tb.width, tb.height\n    elif isinstance(bbox_inches, mpl.transforms.BboxBase):\n        ox_in, oy_in = bbox_inches.x0, bbox_inches.y0\n        W_in, H_in = bbox_inches.width, bbox_inches.height\n    elif bbox_inches is None:\n        ox_in, oy_in = 0.0, 0.0\n        W_in, H_in = fig.get_size_inches()\n    else:\n        raise TypeError(\"panel_crops bbox_inches must be None, 'tight', or a matplotlib Bbox\")\n    W_px, H_px = int(round(W_in * dpi)), int(round(H_in * dpi))\n    lettered = {}\n    seen_letters = set()\n    for ax in fig.axes:\n        for t in ax.findobj(matplotlib.text.Text):\n            s = (t.get_text() or \"\").strip()\n            if t.get_gid() == \"figure-style-panel-letter\":\n                if s in seen_letters:\n                    raise ValueError(f\"Duplicate panel letter: {s!r}\")\n                seen_letters.add(s)\n                lettered[ax] = s\n                break\n\n\n\n    if not lettered:\n        lettered = {ax: str(i) for i, ax in enumerate(fig.axes)}\n    out = {}\n    for ax, letter in lettered.items():\n        bbs = [ax.get_tightbbox(r)]\n\n\n\n\n        ss = ax.get_subplotspec()\n        for sib in fig.axes:\n            if sib is ax or sib in lettered:\n                continue\n            ssib = sib.get_subplotspec()\n            same_row = ss is None or ssib is None or ss.rowspan == ssib.rowspan\n            same_col = ss is None or ssib is None or ss.colspan == ssib.colspan\n            if ((ax.get_shared_y_axes().joined(ax, sib) and same_row)\n                    or (ax.get_shared_x_axes().joined(ax, sib) and same_col)):\n                bbs.append(sib.get_tightbbox(r))\n        bb = mpl.transforms.Bbox.union(bbs)\n\n        bx0 = (bb.x0 / fig.dpi - ox_in) * dpi\n        bx1 = (bb.x1 / fig.dpi - ox_in) * dpi\n        by0 = H_px - (bb.y1 / fig.dpi - oy_in) * dpi\n        by1 = H_px - (bb.y0 / fig.dpi - oy_in) * dpi\n        box = (\n            min(max(int(bx0) - pad_px, 0), W_px),\n            min(max(int(by0) - pad_px, 0), H_px),\n            max(min(int(bx1) + pad_px, W_px), 0),\n            max(min(int(by1) + pad_px, H_px), 0),\n        )\n        if box[0] < box[2] and box[1] < box[3]:\n            out[letter] = box\n    return out\n",
          "sourceDigest": "8020d409bfb60f0c8dfa43d0dfe7bb97f7458e1a3908fefbcb6e4551e0feae4e"
        }
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        "captureKind": "completed-run",
        "capturedAt": "2026-10-08T10:39:16.228Z",
        "installedInventory": {
          "capturedAt": "2026-10-08T07:31:03.005Z",
          "source": "cache-reused",
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        "stdout": "metadata {'repository': 'https://github.com/openai/math', 'commit': 'fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb', 'retrieved_at': '2026-10-08T10:34:03.179531+00:00', 'provenance': 'Public official files retrieved separately on the host and supplied as inputs. Not an Open-Science network fetch or proof verification.', 'errors': []}\ncount 15 hash e112b2a78c2acb58ae7b2f38e233842d33926dcf5c96533a2e8ffefb4065c871\nintegrity failures []\nexisting state 311 23 20\nlean/OAI/Analysis/HotSpots/Main.lean lines 89 tokens []\nlean/OAI/Analysis/RelativeGeneration/Main.lean lines 117 tokens []\nlean/OAI/Combinatorics/BooleanFunctions/Main.lean lines 27 tokens []\nlean/OAI/Combinatorics/ContingencyTables/UnconditionalMain.lean lines 35 tokens []\nlean/OAI/Combinatorics/MatchingCount/Main.lean lines 33 tokens []\nlean/OAI/Combinatorics/MatchingEntropy/Main.lean lines 16 tokens []\nlean/OAI/Combinatorics/TriangleFace/Main.lean lines 73 tokens []\nlean/OAI/Computability/LoopMatching/Endpoint.lean lines 25 tokens []\nlean/OAI/Computability/TypeSystem/Normalization.lean lines 87 tokens ['axiom']\nlean/OAI/MathematicalPhysics/Elasticity/Uniqueness.lean lines 98 tokens []\nlean/OAI/MeasureTheory/SelfSimilar/DimensionCorollaries.lean lines 2 tokens []\nlean/OAI/NumberTheory/DukePrimeDegree/MainUnconditional.lean lines 99 tokens []\nlean/OAI/NumberTheory/TotientAsymptotic/Main.lean lines 208 tokens []\nlean/OAI/NumberTheory/TotientAsymptotic/UnconditionalMain.lean lines 52 tokens []\nlean/OAI/Probability/SLE/LowerPositivity.lean lines 72 tokens []\nexact config modules present 23 / 23\n",
        "stderr": "",
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        "plain": [
          "metadata {'repository': 'https://github.com/openai/math', 'commit': 'fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb', 'retrieved_at': '2026-10-08T10:34:03.179531+00:00', 'provenance': 'Public official files retrieved separately on the host and supplied as inputs. Not an Open-Science network fetch or proof verification.', 'errors': []}\ncount 15 hash e112b2a78c2acb58ae7b2f38e233842d33926dcf5c96533a2e8ffefb4065c871\nintegrity failures []\nexisting state 311 23 20\nlean/OAI/Analysis/HotSpots/Main.lean lines 89 tokens []\nlean/OAI/Analysis/RelativeGeneration/Main.lean lines 117 tokens []\nlean/OAI/Combinatorics/BooleanFunctions/Main.lean lines 27 tokens []\nlean/OAI/Combinatorics/ContingencyTables/UnconditionalMain.lean lines 35 tokens []\nlean/OAI/Combinatorics/MatchingCount/Main.lean lines 33 tokens []\nlean/OAI/Combinatorics/MatchingEntropy/Main.lean lines 16 tokens []\nlean/OAI/Combinatorics/TriangleFace/Main.lean lines 73 tokens []\nlean/OAI/Computability/LoopMatching/Endpoint.lean lines 25 tokens []\nlean/OAI/Computability/TypeSystem/Normalization.lean lines 87 tokens ['axiom']\nlean/OAI/MathematicalPhysics/Elasticity/Uniqueness.lean lines 98 tokens []\nlean/OAI/MeasureTheory/SelfSimilar/DimensionCorollaries.lean lines 2 tokens []\nlean/OAI/NumberTheory/DukePrimeDegree/MainUnconditional.lean lines 99 tokens []\nlean/OAI/NumberTheory/TotientAsymptotic/Main.lean lines 208 tokens []\nlean/OAI/NumberTheory/TotientAsymptotic/UnconditionalMain.lean lines 52 tokens []\nlean/OAI/Probability/SLE/LowerPositivity.lean lines 72 tokens []\nexact config modules present 23 / 23\n"
        ]
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      "outputs": [
        {
          "type": "stream",
          "name": "stdout",
          "text": "metadata {'repository': 'https://github.com/openai/math', 'commit': 'fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb', 'retrieved_at': '2026-10-08T10:34:03.179531+00:00', 'provenance': 'Public official files retrieved separately on the host and supplied as inputs. Not an Open-Science network fetch or proof verification.', 'errors': []}\ncount 15 hash e112b2a78c2acb58ae7b2f38e233842d33926dcf5c96533a2e8ffefb4065c871\nintegrity failures []\nexisting state 311 23 20\nlean/OAI/Analysis/HotSpots/Main.lean lines 89 tokens []\nlean/OAI/Analysis/RelativeGeneration/Main.lean lines 117 tokens []\nlean/OAI/Combinatorics/BooleanFunctions/Main.lean lines 27 tokens []\nlean/OAI/Combinatorics/ContingencyTables/UnconditionalMain.lean lines 35 tokens []\nlean/OAI/Combinatorics/MatchingCount/Main.lean lines 33 tokens []\nlean/OAI/Combinatorics/MatchingEntropy/Main.lean lines 16 tokens []\nlean/OAI/Combinatorics/TriangleFace/Main.lean lines 73 tokens []\nlean/OAI/Computability/LoopMatching/Endpoint.lean lines 25 tokens []\nlean/OAI/Computability/TypeSystem/Normalization.lean lines 87 tokens ['axiom']\nlean/OAI/MathematicalPhysics/Elasticity/Uniqueness.lean lines 98 tokens []\nlean/OAI/MeasureTheory/SelfSimilar/DimensionCorollaries.lean lines 2 tokens []\nlean/OAI/NumberTheory/DukePrimeDegree/MainUnconditional.lean lines 99 tokens []\nlean/OAI/NumberTheory/TotientAsymptotic/Main.lean lines 208 tokens []\nlean/OAI/NumberTheory/TotientAsymptotic/UnconditionalMain.lean lines 52 tokens []\nlean/OAI/Probability/SLE/LowerPositivity.lean lines 72 tokens []\nexact config modules present 23 / 23\n"
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          "exports": [
            "apply_figure_style",
            "set_frame",
            "panel_letter",
            "focal_palette",
            "bar_with_points",
            "strip_with_median",
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          "source": "META_GREY = \"#888888\"\n\n\ndef apply_figure_style(*, frame=\"open\", font=None, sizes=(8, 7, 6), grid=False):\n    \"\"\"Set matplotlib rcParams for publication-grade output. Call once before plotting.\n\n    This sets mechanics (role-mapped font-size ladder, outward ticks, frameless\n    legends, 300-dpi save, Type-42 embedded fonts) — not a house aesthetic.\n    Frame, font and the size ladder are parameters.\n\n    frame : 'open' (bottom+left spines, default) | 'boxed' (all four) | 'none'\n    font  : sans-serif family name; None = system default sans-serif\n    sizes : (base, secondary, tick) — titles/axis-labels, legend/annotation, ticks\n    grid  : whether to draw axes.grid (default False)\n    \"\"\"\n    import matplotlib as mpl\n    if frame not in (\"open\", \"boxed\", \"none\"):\n        raise ValueError(f\"frame must be 'open'|'boxed'|'none', got {frame!r}\")\n\n    try:\n        import os, sys, glob, matplotlib.font_manager as fm\n        fdir = os.path.join(os.environ.get(\"CONDA_PREFIX\") or sys.prefix, \"fonts\")\n        if os.path.isdir(fdir):\n            known = {f.fname for f in fm.fontManager.ttflist}\n            for f in glob.glob(os.path.join(fdir, \"*.ttf\")):\n                if f not in known:\n                    fm.fontManager.addfont(f)\n    except Exception:\n        pass\n    base, secondary, tick = sizes\n    boxed = (frame == \"boxed\")\n    rc = {\n        \"font.family\": \"sans-serif\",\n        \"font.size\": base,\n        \"axes.labelsize\": base,\n        \"axes.titlesize\": base,\n        \"legend.fontsize\": secondary,\n        \"xtick.labelsize\": tick,\n        \"ytick.labelsize\": tick,\n        \"axes.linewidth\": 0.6,\n        \"xtick.direction\": \"out\", \"ytick.direction\": \"out\",\n        \"xtick.major.size\": 3, \"ytick.major.size\": 3,\n        \"xtick.major.width\": 0.6, \"ytick.major.width\": 0.6,\n        \"axes.spines.top\": boxed, \"axes.spines.right\": boxed,\n        \"axes.spines.left\": frame != \"none\", \"axes.spines.bottom\": frame != \"none\",\n        \"axes.grid\": bool(grid),\n        \"legend.frameon\": False,\n        \"figure.dpi\": 200,\n        \"savefig.dpi\": 300,\n        \"savefig.bbox\": \"tight\",\n        \"axes.titleweight\": \"normal\",\n        \"axes.titlelocation\": \"left\",\n        \"axes.labelweight\": \"normal\",\n        \"lines.linewidth\": 1.2,\n        \"patch.linewidth\": 0.6,\n        \"pdf.fonttype\": 42, \"ps.fonttype\": 42,\n    }\n    if font:\n        rc[\"font.sans-serif\"] = [font, \"DejaVu Sans\"]\n    mpl.rcParams.update(rc)\n\n\ndef set_frame(ax, style=\"open\"):\n    \"\"\"§3: set spine visibility on an existing axes. style ∈ {'open','boxed','none'}.\"\"\"\n    show = {\"open\": (False, False, True, True),\n            \"boxed\": (True, True, True, True),\n            \"none\": (False, False, False, False)}[style]\n    for side, vis in zip((\"top\", \"right\", \"bottom\", \"left\"), show):\n        ax.spines[side].set_visible(vis)\n        if vis:\n            ax.spines[side].set_linewidth(0.6)\n    ax.tick_params(direction=\"out\", length=0 if style == \"none\" else 3, width=0.6)\n\n\ndef panel_letter(ax, letter, dx=-0.18, dy=1.02, case=\"lower\", fontsize=None):\n    \"\"\"§5.7: bold panel letter outside top-left of axes. case ∈ {'lower','upper'}.\"\"\"\n    import matplotlib.pyplot as plt\n    if fontsize is None:\n        fontsize = plt.rcParams.get(\"font.size\", 8) + 1\n    s = letter.lower() if case == \"lower\" else letter.upper()\n    artist = ax.text(dx, dy, s, transform=ax.transAxes,\n                     fontweight=\"bold\", fontsize=fontsize, va=\"bottom\", ha=\"left\")\n    artist.set_gid(\"figure-style-panel-letter\")\n\n\ndef focal_palette(labels, focal, focal_color, other=\"muted\", base_colors=None):\n    \"\"\"§4.2: map labels → colors with the focal series visually dominant.\n\n    other='muted'   — desaturate base_colors (or a default cycle) toward gray\n    other='grey'    — uniform light gray for all non-focal\n    other='ordinal' — non-focal on a single light→dark gray ramp (input order)\n    \"\"\"\n    import matplotlib.colors as mcolors\n    import matplotlib.pyplot as plt\n    focal_set = {focal} if isinstance(focal, str) else set(focal)\n    n = len(labels)\n    if not focal_set & set(labels):\n        raise ValueError(f\"focal {focal!r} not found in labels\")\n    base_colors = [] if base_colors is None else list(base_colors)\n    if not base_colors:\n        base_colors = plt.rcParams[\"axes.prop_cycle\"].by_key().get(\"color\", [\"#444444\"])\n    if not base_colors:\n        base_colors = [\"#444444\"]\n    base_colors = [base_colors[i % len(base_colors)] for i in range(n)]\n    if other == \"grey\":\n        rest = [\"#BCBCBC\"] * n\n    elif other == \"ordinal\":\n        nf = max(1, n - len(focal_set))\n        ramp = [mcolors.to_hex((v, v, v)) for v in\n                ([0.55] if nf == 1 else [0.80 - 0.35 * i / (nf - 1) for i in range(nf)])]\n        rest, k = [], 0\n        for l in labels:\n            rest.append(ramp[min(k, nf - 1)]); k += (l not in focal_set)\n    elif other == \"muted\":\n        def mute(c):\n            r, g, b = mcolors.to_rgb(c)\n            m = (r + g + b) / 3\n            return mcolors.to_hex((0.3 * r + 0.7 * m, 0.3 * g + 0.7 * m, 0.3 * b + 0.7 * m))\n        rest = [mute(c) for c in base_colors]\n    else:\n        raise ValueError(\"other must be 'muted', 'grey', or 'ordinal'\")\n    return [focal_color if l in focal_set else rest[i] for i, l in enumerate(labels)]\n\n\ndef bar_with_points(ax, x, ymat, labels, colors, jitter=0.08, show_points=True,\n                    errorbar=None, point_alpha=0.5, point_size=8):\n    \"\"\"§6.1: bar = mean; optionally overlay raw points or draw an interval.\n\n    colors   : per-label color list (e.g. from focal_palette)\n    x, ymat and labels must have equal lengths; mismatches raise before drawing.\n    Each group must be a nonempty, finite, one-dimensional real numeric array.\n    Requested SD/CI intervals require at least two observations per group.\n    errorbar : None | 'sd' | 'ci95' — drawn only when show_points is False.\n               'ci95' is the t-distribution 95% CI of the mean\n               (half-width t_{0.975,n-1} · s/√n); correct at small n where the\n               z-approximation (1.96·s/√n) is markedly too narrow.\n    \"\"\"\n    import numpy as np\n    x = np.asarray(x)\n    if x.ndim != 1 or not np.issubdtype(x.dtype, np.number):\n        raise TypeError(\n            \"bar_with_points x must be one-dimensional numeric positions; \"\n            \"pass category names through labels\"\n        )\n    ymat = list(ymat)\n    labels = list(labels)\n    if len(x) != len(ymat) or len(x) != len(labels):\n        raise ValueError(\"bar_with_points x, ymat and labels must have equal lengths\")\n    if colors is not None and not isinstance(colors, str):\n        colors = list(colors)\n        if not colors:\n            colors = None\n    if x.dtype.kind not in \"iuf\" or not np.isfinite(x).all():\n        raise ValueError(\"bar_with_points x must contain finite real positions\")\n    for i, y in enumerate(ymat):\n        if np.ma.isMaskedArray(y) and np.ma.getmaskarray(y).any():\n            raise ValueError(f\"bar_with_points group {i} contains masked observations; resolve exclusions explicitly before plotting\")\n    ymat = [np.asarray(y) for y in ymat]\n    for i, y in enumerate(ymat):\n        if y.ndim != 1 or y.size == 0 or y.dtype.kind not in \"iuf\" or not np.isfinite(y).all():\n            raise ValueError(f\"bar_with_points group {i} must be nonempty, one-dimensional and finite real numeric data\")\n    if errorbar and not show_points:\n        if errorbar not in (\"sd\", \"ci95\"):\n            raise ValueError(\"errorbar must be None, 'sd', or 'ci95'\")\n        if any(y.size < 2 for y in ymat):\n            raise ValueError(\"SD/CI cannot be estimated from one observation; use raw points without an interval\")\n    means = np.array([np.mean(y) for y in ymat], float)\n    err = None\n    if errorbar and not show_points:\n        if errorbar == \"sd\":\n            err = np.array([np.std(y, ddof=1) for y in ymat])\n        elif errorbar == \"ci95\":\n            from scipy.stats import t\n            def _hw(y):\n                n = np.asarray(y).size\n                return t.ppf(0.975, n - 1) * np.std(y, ddof=1) / np.sqrt(n)\n            err = np.array([_hw(y) for y in ymat])\n    ax.bar(x, means, color=colors, width=0.7, edgecolor=\"none\",\n           yerr=err, error_kw={\"elinewidth\": 0.8, \"capsize\": 0})\n    if show_points:\n        for xi, ys in zip(x, ymat):\n            ys = np.asarray(ys)\n            if ys.ndim == 1 and ys.size > 0:\n                jit = (np.random.rand(ys.size) - 0.5) * 2 * jitter\n                ax.scatter(np.full(ys.size, xi) + jit, ys, s=point_size, color=\"black\",\n                           alpha=point_alpha, zorder=3, linewidths=0)\n    ax.set_xticks(x); ax.set_xticklabels(labels)\n    return ax\n\n\ndef strip_with_median(ax, groups, values, colors=None, jitter=0.12):\n    \"\"\"§6.1: jittered points + bold horizontal median tick per group.\n\n    groups and values must have equal lengths. Mismatches raise before drawing.\n    colors=None or an empty color sequence supplies default gray; a nonempty\n    color sequence cycles across groups.\n    Each group must be nonempty, one-dimensional and finite real numeric data;\n    resolve missing observations explicitly before plotting.\n    \"\"\"\n    import numpy as np\n    labs = list(groups)\n    values = list(values)\n    if len(labs) != len(values):\n        raise ValueError(\"strip_with_median groups and values must have equal lengths\")\n    for i, y in enumerate(values):\n        if np.ma.isMaskedArray(y) and np.ma.getmaskarray(y).any():\n            raise ValueError(f\"strip_with_median group {i} contains masked observations; resolve exclusions explicitly before plotting\")\n    values = [np.asarray(y) for y in values]\n    for i, y in enumerate(values):\n        if y.ndim != 1 or y.size == 0 or y.dtype.kind not in \"iuf\" or not np.isfinite(y).all():\n            raise ValueError(f\"strip_with_median group {i} must be nonempty, one-dimensional and finite real numeric data\")\n    colors = [] if colors is None else list(colors)\n    if not colors:\n        colors = [\"#444444\"]\n    for i, ys in enumerate(values):\n        c = colors[i % len(colors)]\n        ys = np.asarray(ys)\n        jit = (np.random.rand(ys.size) - 0.5) * 2 * jitter\n        ax.scatter(np.full(ys.size, i) + jit, ys, s=10, color=c, alpha=0.6, linewidths=0, zorder=2)\n        m = np.median(ys)\n        ax.plot([i - 0.22, i + 0.22], [m, m], color=\"black\", lw=1.6, zorder=3)\n    ax.set_xticks(range(len(labs))); ax.set_xticklabels(labs)\n    return ax\n\n\ndef goodness_arrow(ax, text=\"higher = better\", loc=\"upper left\", axis=\"y\", fontsize=None):\n    \"\"\"§3.6: small upright direction-of-goodness cue in the margin.\"\"\"\n    import matplotlib.pyplot as plt\n    if fontsize is None:\n        fontsize = plt.rcParams[\"legend.fontsize\"]\n    pos = {\"upper left\": (0.02, 0.98), \"upper right\": (0.98, 0.98),\n           \"lower left\": (0.02, 0.02), \"lower right\": (0.98, 0.02)}[loc]\n    ha = \"left\" if \"left\" in loc else \"right\"\n    va = \"top\" if \"upper\" in loc else \"bottom\"\n    arrow = \"↑ \" if axis == \"y\" else \"→ \"\n    ax.text(pos[0], pos[1], arrow + text, transform=ax.transAxes,\n            fontsize=fontsize, color=META_GREY, ha=ha, va=va)\n\n\ndef two_tier_label(name, meta):\n    \"\"\"§5: two-line label string (name / metadata). Meta line styled separately by caller.\"\"\"\n    return f\"{name}\\n{meta}\"\n\n\ndef end_of_line_labels(ax, xs, ys, labels, colors=None, dx=0.01, fontsize=None):\n    \"\"\"§6.3 / §7.3: label each line series at its right end instead of a legend box.\n\n    Each label requires a nonempty, matched x/y series. Validate all series\n    before drawing. Nonempty colors cycle; None or empty uses default text color.\n    Inner sequences must support len() and indexing; they are not copied.\n    Nonfinite endpoints are rejected rather than producing an invisible label.\n    \"\"\"\n    import matplotlib.pyplot as plt\n    import math\n    xs, ys, labels = list(xs), list(ys), list(labels)\n    if len(xs) != len(ys) or len(xs) != len(labels):\n        raise ValueError(\"end_of_line_labels xs, ys and labels must have equal lengths\")\n    if any(len(x) == 0 or len(x) != len(y) for x, y in zip(xs, ys)):\n        raise ValueError(\"end_of_line_labels each x/y series must be nonempty and have equal lengths\")\n    if any(not math.isfinite(x[-1]) or not math.isfinite(y[-1]) for x, y in zip(xs, ys)):\n        raise ValueError(\"end_of_line_labels endpoints must be finite; choose valid endpoints explicitly\")\n    if fontsize is None:\n        fontsize = plt.rcParams[\"font.size\"]\n    colors = [] if colors is None else list(colors)\n    if not colors:\n        colors = [None]\n    span = ax.get_xlim()[1] - ax.get_xlim()[0]\n    for i, (x, y, lab) in enumerate(zip(xs, ys, labels)):\n        c = colors[i % len(colors)]\n        ax.text(x[-1] + dx * span, y[-1], lab, color=c, va=\"center\", ha=\"left\", fontsize=fontsize)\n\n\ndef panel_crops(fig, dpi=None, pad_px=6, bbox_inches=None, pad_inches=None):\n    \"\"\"§9.2: pixel-space crop boxes for each lettered panel in the SAVED PNG.\n\n    Returns ``{letter: (x0, y0, x1, y1)}`` in image-space pixels (origin\n    top-left, matching PIL's ``Image.crop``). Panels are detected as bold\n    single-character ``Text``\n    objects placed by :func:`panel_letter`; each panel's crop is its axes'\n    tightbbox mapped into the saved file's pixel space, padded by ``pad_px``.\n    For §3.4 composites (abutting subplots sharing an axis, letter on the\n    leftmost only) the crop unions in letterless ``sharex``/``sharey`` siblings\n    on the same grid row/col so the whole composite is covered. When no axes\n    carries a panel letter (standalone plot, or a figure-composer sub-agent),\n    falls back to one crop per axes keyed by index.\n\n    ``bbox_inches`` mirrors ``Figure.savefig`` semantics: ``None`` means\n    *consult rcParams* (so under :func:`apply_figure_style` it resolves to\n    ``'tight'``); pass an explicit ``Bbox`` only if you saved with one. The\n    boxes are clamped to the saved image extent; panels with no intersection\n    are omitted.\n\n        >>> from PIL import Image\n        >>> fig.savefig(\"fig.png\")  # bbox_inches='tight' via rcParams\n        >>> with Image.open(\"fig.png\") as saved:\n        ...     for letter, box in panel_crops(fig).items():\n        ...         saved.crop(box).save(f\"panel_{letter}.png\")\n\n    Match savefig's DPI, bbox_inches and pad_inches when overriding defaults.\n    Returned boxes are pixels, never savefig bbox_inches (inches). Passing them\n    back to savefig as inches can allocate an enormous raster.\n    \"\"\"\n    import matplotlib as mpl\n    import matplotlib.text\n    from matplotlib.backends.backend_agg import FigureCanvasAgg\n    if isinstance(pad_px, bool) or not isinstance(pad_px, int) or pad_px < 0:\n        raise TypeError(\"panel_crops pad_px must be a non-negative integer\")\n    if dpi is None:\n        dpi = mpl.rcParams.get(\"savefig.dpi\", fig.dpi)\n        if dpi == \"figure\":\n            dpi = fig.dpi\n    dpi = float(dpi)\n    if bbox_inches is None:\n        bbox_inches = mpl.rcParams.get(\"savefig.bbox\")\n    # The Open-Science notebook captures and closes pyplot figures after every\n    # cell. Matplotlib 3.11 decouples a closed Figure from its backend by\n    # replacing FigureCanvasAgg with FigureCanvasBase, while the persistent\n    # notebook namespace can still retain the Figure itself. Reattach the\n    # headless canvas before asking for saved-image geometry so this helper is\n    # safe in a later QA cell as well as the producer cell.\n    if not hasattr(fig.canvas, \"get_renderer\"):\n        FigureCanvasAgg(fig)\n    fig.canvas.draw()\n    r = fig.canvas.get_renderer()\n\n    if bbox_inches == \"tight\":\n        if pad_inches is None:\n            pad_inches = mpl.rcParams.get(\"savefig.pad_inches\", 0.1)\n        tb = fig.get_tightbbox(r).padded(pad_inches)\n        ox_in, oy_in = tb.x0, tb.y0\n        W_in, H_in = tb.width, tb.height\n    elif isinstance(bbox_inches, mpl.transforms.BboxBase):\n        ox_in, oy_in = bbox_inches.x0, bbox_inches.y0\n        W_in, H_in = bbox_inches.width, bbox_inches.height\n    elif bbox_inches is None:\n        ox_in, oy_in = 0.0, 0.0\n        W_in, H_in = fig.get_size_inches()\n    else:\n        raise TypeError(\"panel_crops bbox_inches must be None, 'tight', or a matplotlib Bbox\")\n    W_px, H_px = int(round(W_in * dpi)), int(round(H_in * dpi))\n    lettered = {}\n    seen_letters = set()\n    for ax in fig.axes:\n        for t in ax.findobj(matplotlib.text.Text):\n            s = (t.get_text() or \"\").strip()\n            if t.get_gid() == \"figure-style-panel-letter\":\n                if s in seen_letters:\n                    raise ValueError(f\"Duplicate panel letter: {s!r}\")\n                seen_letters.add(s)\n                lettered[ax] = s\n                break\n\n\n\n    if not lettered:\n        lettered = {ax: str(i) for i, ax in enumerate(fig.axes)}\n    out = {}\n    for ax, letter in lettered.items():\n        bbs = [ax.get_tightbbox(r)]\n\n\n\n\n        ss = ax.get_subplotspec()\n        for sib in fig.axes:\n            if sib is ax or sib in lettered:\n                continue\n            ssib = sib.get_subplotspec()\n            same_row = ss is None or ssib is None or ss.rowspan == ssib.rowspan\n            same_col = ss is None or ssib is None or ss.colspan == ssib.colspan\n            if ((ax.get_shared_y_axes().joined(ax, sib) and same_row)\n                    or (ax.get_shared_x_axes().joined(ax, sib) and same_col)):\n                bbs.append(sib.get_tightbbox(r))\n        bb = mpl.transforms.Bbox.union(bbs)\n\n        bx0 = (bb.x0 / fig.dpi - ox_in) * dpi\n        bx1 = (bb.x1 / fig.dpi - ox_in) * dpi\n        by0 = H_px - (bb.y1 / fig.dpi - oy_in) * dpi\n        by1 = H_px - (bb.y0 / fig.dpi - oy_in) * dpi\n        box = (\n            min(max(int(bx0) - pad_px, 0), W_px),\n            min(max(int(by0) - pad_px, 0), H_px),\n            max(min(int(bx1) + pad_px, W_px), 0),\n            max(min(int(by1) + pad_px, H_px), 0),\n        )\n        if box[0] < box[2] and box[1] < box[3]:\n            out[letter] = box\n    return out\n",
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        "stdout": "\nFILE lean/OAI/Computability/TypeSystem/Normalization.lean \nimport OAI.Computability.TypeSystem.Restriction\nimport OAI.Computability.TypeSystem.Components\nimport OAI.Computability.TypeSystem.Signed\nimport OAI.Computability.TypeSystem.OneChild\nimport OAI.Computability.TypeSystem.ObservationCandidates\nimport OAI.Computability.TypeSystem.Diagonal\n\nnamespace OAI\n\nnamespace PureTypeSystem\n\nsection\nuniverse u\nopen Classical\n\n/-- Relational exclusion for finite profile components. This internal\nproposition is proved from weak normalization, not assumed in the main theorem. -/\ndef RelationalExclusion {S : Type u} (P : Specification S) : Prop :=\n  ∀ C, IsComponent (PrimaryEdge P) C →\n    (∃ I ∈ C, SignedPath (PrimaryEdge P) I I true) →\n    (∃ I J K, ProfileTriple P I J K ∧ I ∈ C ∧ J ∈ C ∧ K ∈ C) →\n    ¬ ForbiddenConfiguration P C\n\n/-- All three actual candidate models are selected using the exact source case\nsplit. The exclusion interface is isolated, not hidden in a candidate law. -/\ntheorem interfaces_of_exclusion {S : Type u} [Fintype S] {P : Specification S}\n    (Ω : ContextTower P) (hw : SystemWeaklyNormalizing P) (he : RelationalExclusion P)\n    (C : Set (Set S)) (hC : IsComponent (PrimaryEdge P) C) :\n    Nonempty (ComponentInterface.{u,u} Ω hw C) := by\n  by_cases ho : ∃ I ∈ C, SignedPath (PrimaryEdge P) I I true\n  · by_cases ht : ∃ I J K, ProfileTriple P I J K ∧ I ∈ C ∧ J ∈ C ∧ K ∈ C\n    · exact ⟨ObservationValues.componentInterface Ω C (he C hC ho ht) hC⟩\n    · exact ⟨OneChildInterpretation.componentInterface Ω C hw hC\n        (fun h hi hj hk => ht ⟨_,_,_,h,hi,hj,hk⟩)⟩\n  · exact ⟨SignedInterpretation.componentInterface Ω hC hw\n      (fun I hi hp => ho ⟨I,hi,hp⟩)⟩\n\ntheorem finite_normalization_of_exclusion {S : Type u} [Fintype S] {P : Specification S}\n    (hw : SystemWeaklyNormalizing P) (he : RelationalExclusion P) :\n    SystemStronglyNormalizing P := by\n  apply stronglyNormalizing_of_application\n  intro Γ f a D E hf ha hsf hsa\n  obtain ⟨s,t,hat⟩ := hf.has_axiom\n  let Ω := ContextTowerConstruction.tower P hat\n  exact application_sn_of_ground Ω hw\n    (all_ground_eq_top Ω hw (interfaces_of_exclusion Ω hw he)) hf ha hsf hsa\n\n/-- Finite restriction retains open contexts, annotations and every source rule.\nThis closes the main theorem once the typed relational exclusion is proved for\nfinite specifications. No finite-only conclusion is substituted for the main. -/\ntheorem normalization_of_finite_exclusion\n    (exclude : ∀ (S : Type u) [Fintype S] (P : Specification S),\n      SystemWeaklyNormalizing P → RelationalExclusion P)\n    {S : Type u} {P : Specification S} (hw : SystemWeaklyNormalizing P) :\n    SystemStronglyNormalizing P := by\n  by_contra hn\n  obtain ⟨Γ,f,a,D,E,hf,ha,hsf,hsa,hbad⟩ := exists_bad_application hn\n  have htype := HasType.application hf ha\n  obtain ⟨F,Γ',M',T',ht,hn',hw'⟩ := htype.finite_counterexample hbad hw\n  have hsn := finite_normalization_of_exclusion hw' (exclude _ _ hw')\n  exact hn' (hsn Γ' ht.valid M' ⟨T',.inl ht⟩)\n\nend\n\nuniverse u\nopen Labelled\n/-- Relational exclusion via the concrete illative encoding\nin the actual finite ambient context, not supplied as an axiom. -/\ntheorem relational_exclusion {S : Type u} [Fintype S] {P : Specification S}\n    (hw : SystemWeaklyNormalizing P) : RelationalExclusion P := by\n  intro C hC ho ht hf\n  obtain ⟨Γ,γ,hΓ,⟨L⟩⟩ := LogicSetup.exists_setup C hC ho ht\n  obtain ⟨K,⟨R⟩⟩ := ChannelSetup.exists_setup_return (L := L) hC ho hf\n  let KY := L.quantifiers K.logicalVertex K.directLogical\n  let KP := L.quantifiers K.predVertex K.directPred\n  obtain ⟨b,hb,mb⟩ := R.specialized_formed.relational_bottom KY KP\n  exact no_proof_at_target hw hΓ\n    ⟨.terminal (L.witnesses.allData L.base L.triple.codomain),L.witnesses.normal L.base L.triple.codomain⟩ hb mb\n\n/-- System-wide weak normalization implies strong normalization for every PTS,\nincluding arbitrary sort types, nonfunctional rules and annotated terms. -/\ntheorem weak_implies_strong {S : Type u} (P : Specification S)\n    (h : SystemWeaklyNormalizing P) : SystemStronglyNormalizing P :=\n  normalization_of_finite_exclusion (fun _ _ _ hw => relational_exclusion hw) h\nend PureTypeSystem\n\nend OAI\n\n\nFILE lean/OAI/MeasureTheory/SelfSimilar/DimensionCorollaries.lean \nimport OAI.MeasureTheory.SelfSimilar.HomogeneousDimension\nimport OAI.MeasureTheory.SelfSimilar.AttractorFormula\n\n\nFILE lean/OAI/Combinatorics/ContingencyTables/UnconditionalMain.lean \nimport OAI.Combinatorics.ContingencyTables.Sampling.AllSamplingMachine\nimport OAI.Combinatorics.ContingencyTables.Sampling.ExactSamplingMachine\nimport OAI.Combinatorics.ContingencyTables.Counting.CountingMainMachine\nimport OAI.Combinatorics.ContingencyTables.Dense.FiniteBoxPrekopa\nimport OAI.Combinatorics.ContingencyTables.Dense.FiniteCheegerProof\nimport OAI.Combinatorics.ContingencyTables.Transport.EdmondsKarpBound\n\nnamespace OAI\n\n/-! # Sampling and counting contingency tables with arbitrary margins\n\nThe finite-box Prékopa–Leindler inequality, finite Cheeger inequality\nand Edmonds–Karp bound yield sampling and counting algorithms with\nuniform polynomial bit-cost guarantees.\n-/\nnamespace ContingencyTables\n\n/-- Paper 200-01: an almost-uniform sampler with polynomial cost on every execution. -/\ntheorem boundedSampling : Algorithms.BoundedSamplingStatement :=\n  SmallGraphProfiles.boundedSampling_conditional\n    PublishedInputs.finiteBoxPrekopaLeindler (fun _ _ => CheegerProof.finiteCheeger)\n\n/-- Paper 200-01: exact uniform sampling in expected polynomial bit time. -/\ntheorem exactSampling : Algorithms.ExactSamplingStatement :=\n  SmallGraphProfiles.exactSampling_conditional\n    PublishedInputs.finiteBoxPrekopaLeindler (fun _ _ => CheegerProof.finiteCheeger)\n\n/-- Paper 200-02: the cell-bounded counting FPRAS with a public fair-bit machine. -/\ntheorem counting : Algorithms.CountingStatement :=\n  CountingExecutedMachine.counting_conditional EdmondsKarpBound.edmondsKarp1972Theorem1\n    PublishedInputs.finiteBoxPrekopaLeindler (fun _ _ => CheegerProof.finiteCheeger)\n\nend ContingencyTables\n\nend OAI\n\n\nFILE lean/OAI/Combinatorics/MatchingCount/Main.lean \nimport Mathlib\nimport OAI.Combinatorics.MatchingCount.Sampling.Sewn\nimport OAI.Combinatorics.MatchingCount.Sampling.FiniteSampler\nimport OAI.Combinatorics.MatchingCount.Sampling.FiniteSampler2\nimport OAI.Combinatorics.MatchingCount.Sampling.FiniteSampler3\nimport OAI.Combinatorics.MatchingCount.Sampling.FiniteSampler4\nimport OAI.Combinatorics.MatchingCount.Algorithm.GuardBudget\nimport OAI.Combinatorics.MatchingCount.Machines.ProgramComposition\nimport OAI.Combinatorics.MatchingCount.Graphs.Output\nimport OAI.Combinatorics.MatchingCount.Arithmetic.TreeDivision\nimport OAI.Combinatorics.MatchingCount.Machines.TreeLanguage\nimport OAI.Combinatorics.MatchingCount.Sampling.TreeUnitCorrect\nimport OAI.Combinatorics.MatchingCount.Tapes.RandomReservoir\nimport OAI.Combinatorics.MatchingCount.Complexity.PhysicalCost\nimport OAI.Combinatorics.MatchingCount.Probability.MainLaw\n\nnamespace OAI\n\nnamespace MatchingFPRAS\n\ntheorem thm_main : MainStatement := by\n  obtain ⟨C,d,hC,hbound⟩:=LiteralPhysical.mainTime_bound\n  refine ⟨LiteralPhysical.mainProgram.machine,C,d,hC,?_⟩\n  intro G ε δ he he1 hd _hd1\n  dsimp only\n  have hn:=hbound G ε δ he he1 hd.le\n  obtain ⟨hp,hz⟩:=LiteralPhysical.mainProgram_outputs G ε δ he he1 hd.le\n    (timeBound C d G ε δ) hn\n  exact ⟨hp,hz,LiteralPhysical.mainProgram_success G ε δ he he1 hd\n    (timeBound C d G ε δ) hn⟩\nend MatchingFPRAS\n\nend OAI\n\n\nFILE lean/OAI/Combinatorics/BooleanFunctions/Main.lean \nimport OAI.Combinatorics.BooleanFunctions.Amplification\nimport OAI.Combinatorics.BooleanFunctions.Readout\n\nnamespace OAI\n\nopen scoped BigOperators\n\nnoncomputable section\n\nnamespace SquareRootDegree\n\n/-- Every stage and the final sign readout use finitely many actual Boolean inputs. -/\ntheorem signed_violations : SignedViolations := by\n  intro C hC\n  obtain ⟨F,hF⟩ := Amplification.exists_large_ratio ((C/ReadoutCLT.constant)^2)\n  exact Observations.violation_of_observation F.map F.cost F.cost_pos F.cell_le\n    F.variance_pos C hC hF\n\n/-- Unbounded violations for ordinary real Fourier degree and uniform cube averages. -/\ntheorem main : SignedViolations ∧ ¬ BddAbove AbsoluteRatios := by\n  exact ⟨signed_violations,absolute_unbounded_of_signed signed_violations⟩\n\nend SquareRootDegree\n\nend\n\nend OAI\n\n\nFILE lean/OAI/Combinatorics/MatchingEntropy/Main.lean \nimport OAI.Combinatorics.MatchingEntropy.ExpansionSequences\n\n/-!\n# Sharp analytic bounds for perfect-matching entropy\n\nFor a finite loopless labelled multigraph on `2m` vertices, with `m ≥ 1` and a perfect\nmatching, every point of the matching polytope satisfies\n`F - (2 - 2 / m) * B ≤ H ≤ F`. Here `F` and `B` are the coordinate entropy and\ncomplement entropy with natural logarithms, and `H` is the attained supremum of entropy\nover all perfect-matching laws with the prescribed mean.\n\nThe development includes the odd-cut description of the matching polytope, sharp face\nrank bounds, minimal faces, fractional and connected-component reductions, attained\nentropy and weighted partition objectives, matching counts, degenerate cases, colored\ntriangle-expansion families, and concrete coefficient obstructions.\n-/\n\n",
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          "\nFILE lean/OAI/Computability/TypeSystem/Normalization.lean \nimport OAI.Computability.TypeSystem.Restriction\nimport OAI.Computability.TypeSystem.Components\nimport OAI.Computability.TypeSystem.Signed\nimport OAI.Computability.TypeSystem.OneChild\nimport OAI.Computability.TypeSystem.ObservationCandidates\nimport OAI.Computability.TypeSystem.Diagonal\n\nnamespace OAI\n\nnamespace PureTypeSystem\n\nsection\nuniverse u\nopen Classical\n\n/-- Relational exclusion for finite profile components. This internal\nproposition is proved from weak normalization, not assumed in the main theorem. -/\ndef RelationalExclusion {S : Type u} (P : Specification S) : Prop :=\n  ∀ C, IsComponent (PrimaryEdge P) C →\n    (∃ I ∈ C, SignedPath (PrimaryEdge P) I I true) →\n    (∃ I J K, ProfileTriple P I J K ∧ I ∈ C ∧ J ∈ C ∧ K ∈ C) →\n    ¬ ForbiddenConfiguration P C\n\n/-- All three actual candidate models are selected using the exact source case\nsplit. The exclusion interface is isolated, not hidden in a candidate law. -/\ntheorem interfaces_of_exclusion {S : Type u} [Fintype S] {P : Specification S}\n    (Ω : ContextTower P) (hw : SystemWeaklyNormalizing P) (he : RelationalExclusion P)\n    (C : Set (Set S)) (hC : IsComponent (PrimaryEdge P) C) :\n    Nonempty (ComponentInterface.{u,u} Ω hw C) := by\n  by_cases ho : ∃ I ∈ C, SignedPath (PrimaryEdge P) I I true\n  · by_cases ht : ∃ I J K, ProfileTriple P I J K ∧ I ∈ C ∧ J ∈ C ∧ K ∈ C\n    · exact ⟨ObservationValues.componentInterface Ω C (he C hC ho ht) hC⟩\n    · exact ⟨OneChildInterpretation.componentInterface Ω C hw hC\n        (fun h hi hj hk => ht ⟨_,_,_,h,hi,hj,hk⟩)⟩\n  · exact ⟨SignedInterpretation.componentInterface Ω hC hw\n      (fun I hi hp => ho ⟨I,hi,hp⟩)⟩\n\ntheorem finite_normalization_of_exclusion {S : Type u} [Fintype S] {P : Specification S}\n    (hw : SystemWeaklyNormalizing P) (he : RelationalExclusion P) :\n    SystemStronglyNormalizing P := by\n  apply stronglyNormalizing_of_application\n  intro Γ f a D E hf ha hsf hsa\n  obtain ⟨s,t,hat⟩ := hf.has_axiom\n  let Ω := ContextTowerConstruction.tower P hat\n  exact application_sn_of_ground Ω hw\n    (all_ground_eq_top Ω hw (interfaces_of_exclusion Ω hw he)) hf ha hsf hsa\n\n/-- Finite restriction retains open contexts, annotations and every source rule.\nThis closes the main theorem once the typed relational exclusion is proved for\nfinite specifications. No finite-only conclusion is substituted for the main. -/\ntheorem normalization_of_finite_exclusion\n    (exclude : ∀ (S : Type u) [Fintype S] (P : Specification S),\n      SystemWeaklyNormalizing P → RelationalExclusion P)\n    {S : Type u} {P : Specification S} (hw : SystemWeaklyNormalizing P) :\n    SystemStronglyNormalizing P := by\n  by_contra hn\n  obtain ⟨Γ,f,a,D,E,hf,ha,hsf,hsa,hbad⟩ := exists_bad_application hn\n  have htype := HasType.application hf ha\n  obtain ⟨F,Γ',M',T',ht,hn',hw'⟩ := htype.finite_counterexample hbad hw\n  have hsn := finite_normalization_of_exclusion hw' (exclude _ _ hw')\n  exact hn' (hsn Γ' ht.valid M' ⟨T',.inl ht⟩)\n\nend\n\nuniverse u\nopen Labelled\n/-- Relational exclusion via the concrete illative encoding\nin the actual finite ambient context, not supplied as an axiom. -/\ntheorem relational_exclusion {S : Type u} [Fintype S] {P : Specification S}\n    (hw : SystemWeaklyNormalizing P) : RelationalExclusion P := by\n  intro C hC ho ht hf\n  obtain ⟨Γ,γ,hΓ,⟨L⟩⟩ := LogicSetup.exists_setup C hC ho ht\n  obtain ⟨K,⟨R⟩⟩ := ChannelSetup.exists_setup_return (L := L) hC ho hf\n  let KY := L.quantifiers K.logicalVertex K.directLogical\n  let KP := L.quantifiers K.predVertex K.directPred\n  obtain ⟨b,hb,mb⟩ := R.specialized_formed.relational_bottom KY KP\n  exact no_proof_at_target hw hΓ\n    ⟨.terminal (L.witnesses.allData L.base L.triple.codomain),L.witnesses.normal L.base L.triple.codomain⟩ hb mb\n\n/-- System-wide weak normalization implies strong normalization for every PTS,\nincluding arbitrary sort types, nonfunctional rules and annotated terms. -/\ntheorem weak_implies_strong {S : Type u} (P : Specification S)\n    (h : SystemWeaklyNormalizing P) : SystemStronglyNormalizing P :=\n  normalization_of_finite_exclusion (fun _ _ _ hw => relational_exclusion hw) h\nend PureTypeSystem\n\nend OAI\n\n\nFILE lean/OAI/MeasureTheory/SelfSimilar/DimensionCorollaries.lean \nimport OAI.MeasureTheory.SelfSimilar.HomogeneousDimension\nimport OAI.MeasureTheory.SelfSimilar.AttractorFormula\n\n\nFILE lean/OAI/Combinatorics/ContingencyTables/UnconditionalMain.lean \nimport OAI.Combinatorics.ContingencyTables.Sampling.AllSamplingMachine\nimport OAI.Combinatorics.ContingencyTables.Sampling.ExactSamplingMachine\nimport OAI.Combinatorics.ContingencyTables.Counting.CountingMainMachine\nimport OAI.Combinatorics.ContingencyTables.Dense.FiniteBoxPrekopa\nimport OAI.Combinatorics.ContingencyTables.Dense.FiniteCheegerProof\nimport OAI.Combinatorics.ContingencyTables.Transport.EdmondsKarpBound\n\nnamespace OAI\n\n/-! # Sampling and counting contingency tables with arbitrary margins\n\nThe finite-box Prékopa–Leindler inequality, finite Cheeger inequality\nand Edmonds–Karp bound yield sampling and counting algorithms with\nuniform polynomial bit-cost guarantees.\n-/\nnamespace ContingencyTables\n\n/-- Paper 200-01: an almost-uniform sampler with polynomial cost on every execution. -/\ntheorem boundedSampling : Algorithms.BoundedSamplingStatement :=\n  SmallGraphProfiles.boundedSampling_conditional\n    PublishedInputs.finiteBoxPrekopaLeindler (fun _ _ => CheegerProof.finiteCheeger)\n\n/-- Paper 200-01: exact uniform sampling in expected polynomial bit time. -/\ntheorem exactSampling : Algorithms.ExactSamplingStatement :=\n  SmallGraphProfiles.exactSampling_conditional\n    PublishedInputs.finiteBoxPrekopaLeindler (fun _ _ => CheegerProof.finiteCheeger)\n\n/-- Paper 200-02: the cell-bounded counting FPRAS with a public fair-bit machine. -/\ntheorem counting : Algorithms.CountingStatement :=\n  CountingExecutedMachine.counting_conditional EdmondsKarpBound.edmondsKarp1972Theorem1\n    PublishedInputs.finiteBoxPrekopaLeindler (fun _ _ => CheegerProof.finiteCheeger)\n\nend ContingencyTables\n\nend OAI\n\n\nFILE lean/OAI/Combinatorics/MatchingCount/Main.lean \nimport Mathlib\nimport OAI.Combinatorics.MatchingCount.Sampling.Sewn\nimport OAI.Combinatorics.MatchingCount.Sampling.FiniteSampler\nimport OAI.Combinatorics.MatchingCount.Sampling.FiniteSampler2\nimport OAI.Combinatorics.MatchingCount.Sampling.FiniteSampler3\nimport OAI.Combinatorics.MatchingCount.Sampling.FiniteSampler4\nimport OAI.Combinatorics.MatchingCount.Algorithm.GuardBudget\nimport OAI.Combinatorics.MatchingCount.Machines.ProgramComposition\nimport OAI.Combinatorics.MatchingCount.Graphs.Output\nimport OAI.Combinatorics.MatchingCount.Arithmetic.TreeDivision\nimport OAI.Combinatorics.MatchingCount.Machines.TreeLanguage\nimport OAI.Combinatorics.MatchingCount.Sampling.TreeUnitCorrect\nimport OAI.Combinatorics.MatchingCount.Tapes.RandomReservoir\nimport OAI.Combinatorics.MatchingCount.Complexity.PhysicalCost\nimport OAI.Combinatorics.MatchingCount.Probability.MainLaw\n\nnamespace OAI\n\nnamespace MatchingFPRAS\n\ntheorem thm_main : MainStatement := by\n  obtain ⟨C,d,hC,hbound⟩:=LiteralPhysical.mainTime_bound\n  refine ⟨LiteralPhysical.mainProgram.machine,C,d,hC,?_⟩\n  intro G ε δ he he1 hd _hd1\n  dsimp only\n  have hn:=hbound G ε δ he he1 hd.le\n  obtain ⟨hp,hz⟩:=LiteralPhysical.mainProgram_outputs G ε δ he he1 hd.le\n    (timeBound C d G ε δ) hn\n  exact ⟨hp,hz,LiteralPhysical.mainProgram_success G ε δ he he1 hd\n    (timeBound C d G ε δ) hn⟩\nend MatchingFPRAS\n\nend OAI\n\n\nFILE lean/OAI/Combinatorics/BooleanFunctions/Main.lean \nimport OAI.Combinatorics.BooleanFunctions.Amplification\nimport OAI.Combinatorics.BooleanFunctions.Readout\n\nnamespace OAI\n\nopen scoped BigOperators\n\nnoncomputable section\n\nnamespace SquareRootDegree\n\n/-- Every stage and the final sign readout use finitely many actual Boolean inputs. -/\ntheorem signed_violations : SignedViolations := by\n  intro C hC\n  obtain ⟨F,hF⟩ := Amplification.exists_large_ratio ((C/ReadoutCLT.constant)^2)\n  exact Observations.violation_of_observation F.map F.cost F.cost_pos F.cell_le\n    F.variance_pos C hC hF\n\n/-- Unbounded violations for ordinary real Fourier degree and uniform cube averages. -/\ntheorem main : SignedViolations ∧ ¬ BddAbove AbsoluteRatios := by\n  exact ⟨signed_violations,absolute_unbounded_of_signed signed_violations⟩\n\nend SquareRootDegree\n\nend\n\nend OAI\n\n\nFILE lean/OAI/Combinatorics/MatchingEntropy/Main.lean \nimport OAI.Combinatorics.MatchingEntropy.ExpansionSequences\n\n/-!\n# Sharp analytic bounds for perfect-matching entropy\n\nFor a finite loopless labelled multigraph on `2m` vertices, with `m ≥ 1` and a perfect\nmatching, every point of the matching polytope satisfies\n`F - (2 - 2 / m) * B ≤ H ≤ F`. Here `F` and `B` are the coordinate entropy and\ncomplement entropy with natural logarithms, and `H` is the attained supremum of entropy\nover all perfect-matching laws with the prescribed mean.\n\nThe development includes the odd-cut description of the matching polytope, sharp face\nrank bounds, minimal faces, fractional and connected-component reductions, attained\nentropy and weighted partition objectives, matching counts, degenerate cases, colored\ntriangle-expansion families, and concrete coefficient obstructions.\n-/\n\n"
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          "text": "\nFILE lean/OAI/Computability/TypeSystem/Normalization.lean \nimport OAI.Computability.TypeSystem.Restriction\nimport OAI.Computability.TypeSystem.Components\nimport OAI.Computability.TypeSystem.Signed\nimport OAI.Computability.TypeSystem.OneChild\nimport OAI.Computability.TypeSystem.ObservationCandidates\nimport OAI.Computability.TypeSystem.Diagonal\n\nnamespace OAI\n\nnamespace PureTypeSystem\n\nsection\nuniverse u\nopen Classical\n\n/-- Relational exclusion for finite profile components. This internal\nproposition is proved from weak normalization, not assumed in the main theorem. -/\ndef RelationalExclusion {S : Type u} (P : Specification S) : Prop :=\n  ∀ C, IsComponent (PrimaryEdge P) C →\n    (∃ I ∈ C, SignedPath (PrimaryEdge P) I I true) →\n    (∃ I J K, ProfileTriple P I J K ∧ I ∈ C ∧ J ∈ C ∧ K ∈ C) →\n    ¬ ForbiddenConfiguration P C\n\n/-- All three actual candidate models are selected using the exact source case\nsplit. The exclusion interface is isolated, not hidden in a candidate law. -/\ntheorem interfaces_of_exclusion {S : Type u} [Fintype S] {P : Specification S}\n    (Ω : ContextTower P) (hw : SystemWeaklyNormalizing P) (he : RelationalExclusion P)\n    (C : Set (Set S)) (hC : IsComponent (PrimaryEdge P) C) :\n    Nonempty (ComponentInterface.{u,u} Ω hw C) := by\n  by_cases ho : ∃ I ∈ C, SignedPath (PrimaryEdge P) I I true\n  · by_cases ht : ∃ I J K, ProfileTriple P I J K ∧ I ∈ C ∧ J ∈ C ∧ K ∈ C\n    · exact ⟨ObservationValues.componentInterface Ω C (he C hC ho ht) hC⟩\n    · exact ⟨OneChildInterpretation.componentInterface Ω C hw hC\n        (fun h hi hj hk => ht ⟨_,_,_,h,hi,hj,hk⟩)⟩\n  · exact ⟨SignedInterpretation.componentInterface Ω hC hw\n      (fun I hi hp => ho ⟨I,hi,hp⟩)⟩\n\ntheorem finite_normalization_of_exclusion {S : Type u} [Fintype S] {P : Specification S}\n    (hw : SystemWeaklyNormalizing P) (he : RelationalExclusion P) :\n    SystemStronglyNormalizing P := by\n  apply stronglyNormalizing_of_application\n  intro Γ f a D E hf ha hsf hsa\n  obtain ⟨s,t,hat⟩ := hf.has_axiom\n  let Ω := ContextTowerConstruction.tower P hat\n  exact application_sn_of_ground Ω hw\n    (all_ground_eq_top Ω hw (interfaces_of_exclusion Ω hw he)) hf ha hsf hsa\n\n/-- Finite restriction retains open contexts, annotations and every source rule.\nThis closes the main theorem once the typed relational exclusion is proved for\nfinite specifications. No finite-only conclusion is substituted for the main. -/\ntheorem normalization_of_finite_exclusion\n    (exclude : ∀ (S : Type u) [Fintype S] (P : Specification S),\n      SystemWeaklyNormalizing P → RelationalExclusion P)\n    {S : Type u} {P : Specification S} (hw : SystemWeaklyNormalizing P) :\n    SystemStronglyNormalizing P := by\n  by_contra hn\n  obtain ⟨Γ,f,a,D,E,hf,ha,hsf,hsa,hbad⟩ := exists_bad_application hn\n  have htype := HasType.application hf ha\n  obtain ⟨F,Γ',M',T',ht,hn',hw'⟩ := htype.finite_counterexample hbad hw\n  have hsn := finite_normalization_of_exclusion hw' (exclude _ _ hw')\n  exact hn' (hsn Γ' ht.valid M' ⟨T',.inl ht⟩)\n\nend\n\nuniverse u\nopen Labelled\n/-- Relational exclusion via the concrete illative encoding\nin the actual finite ambient context, not supplied as an axiom. -/\ntheorem relational_exclusion {S : Type u} [Fintype S] {P : Specification S}\n    (hw : SystemWeaklyNormalizing P) : RelationalExclusion P := by\n  intro C hC ho ht hf\n  obtain ⟨Γ,γ,hΓ,⟨L⟩⟩ := LogicSetup.exists_setup C hC ho ht\n  obtain ⟨K,⟨R⟩⟩ := ChannelSetup.exists_setup_return (L := L) hC ho hf\n  let KY := L.quantifiers K.logicalVertex K.directLogical\n  let KP := L.quantifiers K.predVertex K.directPred\n  obtain ⟨b,hb,mb⟩ := R.specialized_formed.relational_bottom KY KP\n  exact no_proof_at_target hw hΓ\n    ⟨.terminal (L.witnesses.allData L.base L.triple.codomain),L.witnesses.normal L.base L.triple.codomain⟩ hb mb\n\n/-- System-wide weak normalization implies strong normalization for every PTS,\nincluding arbitrary sort types, nonfunctional rules and annotated terms. -/\ntheorem weak_implies_strong {S : Type u} (P : Specification S)\n    (h : SystemWeaklyNormalizing P) : SystemStronglyNormalizing P :=\n  normalization_of_finite_exclusion (fun _ _ _ hw => relational_exclusion hw) h\nend PureTypeSystem\n\nend OAI\n\n\nFILE lean/OAI/MeasureTheory/SelfSimilar/DimensionCorollaries.lean \nimport OAI.MeasureTheory.SelfSimilar.HomogeneousDimension\nimport OAI.MeasureTheory.SelfSimilar.AttractorFormula\n\n\nFILE lean/OAI/Combinatorics/ContingencyTables/UnconditionalMain.lean \nimport OAI.Combinatorics.ContingencyTables.Sampling.AllSamplingMachine\nimport OAI.Combinatorics.ContingencyTables.Sampling.ExactSamplingMachine\nimport OAI.Combinatorics.ContingencyTables.Counting.CountingMainMachine\nimport OAI.Combinatorics.ContingencyTables.Dense.FiniteBoxPrekopa\nimport OAI.Combinatorics.ContingencyTables.Dense.FiniteCheegerProof\nimport OAI.Combinatorics.ContingencyTables.Transport.EdmondsKarpBound\n\nnamespace OAI\n\n/-! # Sampling and counting contingency tables with arbitrary margins\n\nThe finite-box Prékopa–Leindler inequality, finite Cheeger inequality\nand Edmonds–Karp bound yield sampling and counting algorithms with\nuniform polynomial bit-cost guarantees.\n-/\nnamespace ContingencyTables\n\n/-- Paper 200-01: an almost-uniform sampler with polynomial cost on every execution. -/\ntheorem boundedSampling : Algorithms.BoundedSamplingStatement :=\n  SmallGraphProfiles.boundedSampling_conditional\n    PublishedInputs.finiteBoxPrekopaLeindler (fun _ _ => CheegerProof.finiteCheeger)\n\n/-- Paper 200-01: exact uniform sampling in expected polynomial bit time. -/\ntheorem exactSampling : Algorithms.ExactSamplingStatement :=\n  SmallGraphProfiles.exactSampling_conditional\n    PublishedInputs.finiteBoxPrekopaLeindler (fun _ _ => CheegerProof.finiteCheeger)\n\n/-- Paper 200-02: the cell-bounded counting FPRAS with a public fair-bit machine. -/\ntheorem counting : Algorithms.CountingStatement :=\n  CountingExecutedMachine.counting_conditional EdmondsKarpBound.edmondsKarp1972Theorem1\n    PublishedInputs.finiteBoxPrekopaLeindler (fun _ _ => CheegerProof.finiteCheeger)\n\nend ContingencyTables\n\nend OAI\n\n\nFILE lean/OAI/Combinatorics/MatchingCount/Main.lean \nimport Mathlib\nimport OAI.Combinatorics.MatchingCount.Sampling.Sewn\nimport OAI.Combinatorics.MatchingCount.Sampling.FiniteSampler\nimport OAI.Combinatorics.MatchingCount.Sampling.FiniteSampler2\nimport OAI.Combinatorics.MatchingCount.Sampling.FiniteSampler3\nimport OAI.Combinatorics.MatchingCount.Sampling.FiniteSampler4\nimport OAI.Combinatorics.MatchingCount.Algorithm.GuardBudget\nimport OAI.Combinatorics.MatchingCount.Machines.ProgramComposition\nimport OAI.Combinatorics.MatchingCount.Graphs.Output\nimport OAI.Combinatorics.MatchingCount.Arithmetic.TreeDivision\nimport OAI.Combinatorics.MatchingCount.Machines.TreeLanguage\nimport OAI.Combinatorics.MatchingCount.Sampling.TreeUnitCorrect\nimport OAI.Combinatorics.MatchingCount.Tapes.RandomReservoir\nimport OAI.Combinatorics.MatchingCount.Complexity.PhysicalCost\nimport OAI.Combinatorics.MatchingCount.Probability.MainLaw\n\nnamespace OAI\n\nnamespace MatchingFPRAS\n\ntheorem thm_main : MainStatement := by\n  obtain ⟨C,d,hC,hbound⟩:=LiteralPhysical.mainTime_bound\n  refine ⟨LiteralPhysical.mainProgram.machine,C,d,hC,?_⟩\n  intro G ε δ he he1 hd _hd1\n  dsimp only\n  have hn:=hbound G ε δ he he1 hd.le\n  obtain ⟨hp,hz⟩:=LiteralPhysical.mainProgram_outputs G ε δ he he1 hd.le\n    (timeBound C d G ε δ) hn\n  exact ⟨hp,hz,LiteralPhysical.mainProgram_success G ε δ he he1 hd\n    (timeBound C d G ε δ) hn⟩\nend MatchingFPRAS\n\nend OAI\n\n\nFILE lean/OAI/Combinatorics/BooleanFunctions/Main.lean \nimport OAI.Combinatorics.BooleanFunctions.Amplification\nimport OAI.Combinatorics.BooleanFunctions.Readout\n\nnamespace OAI\n\nopen scoped BigOperators\n\nnoncomputable section\n\nnamespace SquareRootDegree\n\n/-- Every stage and the final sign readout use finitely many actual Boolean inputs. -/\ntheorem signed_violations : SignedViolations := by\n  intro C hC\n  obtain ⟨F,hF⟩ := Amplification.exists_large_ratio ((C/ReadoutCLT.constant)^2)\n  exact Observations.violation_of_observation F.map F.cost F.cost_pos F.cell_le\n    F.variance_pos C hC hF\n\n/-- Unbounded violations for ordinary real Fourier degree and uniform cube averages. -/\ntheorem main : SignedViolations ∧ ¬ BddAbove AbsoluteRatios := by\n  exact ⟨signed_violations,absolute_unbounded_of_signed signed_violations⟩\n\nend SquareRootDegree\n\nend\n\nend OAI\n\n\nFILE lean/OAI/Combinatorics/MatchingEntropy/Main.lean \nimport OAI.Combinatorics.MatchingEntropy.ExpansionSequences\n\n/-!\n# Sharp analytic bounds for perfect-matching entropy\n\nFor a finite loopless labelled multigraph on `2m` vertices, with `m ≥ 1` and a perfect\nmatching, every point of the matching polytope satisfies\n`F - (2 - 2 / m) * B ≤ H ≤ F`. Here `F` and `B` are the coordinate entropy and\ncomplement entropy with natural logarithms, and `H` is the attained supremum of entropy\nover all perfect-matching laws with the prescribed mean.\n\nThe development includes the odd-cut description of the matching polytope, sharp face\nrank bounds, minimal faces, fractional and connected-component reductions, attained\nentropy and weighted partition objectives, matching counts, degenerate cases, colored\ntriangle-expansion families, and concrete coefficient obstructions.\n-/\n\n"
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          "source": "META_GREY = \"#888888\"\n\n\ndef apply_figure_style(*, frame=\"open\", font=None, sizes=(8, 7, 6), grid=False):\n    \"\"\"Set matplotlib rcParams for publication-grade output. Call once before plotting.\n\n    This sets mechanics (role-mapped font-size ladder, outward ticks, frameless\n    legends, 300-dpi save, Type-42 embedded fonts) — not a house aesthetic.\n    Frame, font and the size ladder are parameters.\n\n    frame : 'open' (bottom+left spines, default) | 'boxed' (all four) | 'none'\n    font  : sans-serif family name; None = system default sans-serif\n    sizes : (base, secondary, tick) — titles/axis-labels, legend/annotation, ticks\n    grid  : whether to draw axes.grid (default False)\n    \"\"\"\n    import matplotlib as mpl\n    if frame not in (\"open\", \"boxed\", \"none\"):\n        raise ValueError(f\"frame must be 'open'|'boxed'|'none', got {frame!r}\")\n\n    try:\n        import os, sys, glob, matplotlib.font_manager as fm\n        fdir = os.path.join(os.environ.get(\"CONDA_PREFIX\") or sys.prefix, \"fonts\")\n        if os.path.isdir(fdir):\n            known = {f.fname for f in fm.fontManager.ttflist}\n            for f in glob.glob(os.path.join(fdir, \"*.ttf\")):\n                if f not in known:\n                    fm.fontManager.addfont(f)\n    except Exception:\n        pass\n    base, secondary, tick = sizes\n    boxed = (frame == \"boxed\")\n    rc = {\n        \"font.family\": \"sans-serif\",\n        \"font.size\": base,\n        \"axes.labelsize\": base,\n        \"axes.titlesize\": base,\n        \"legend.fontsize\": secondary,\n        \"xtick.labelsize\": tick,\n        \"ytick.labelsize\": tick,\n        \"axes.linewidth\": 0.6,\n        \"xtick.direction\": \"out\", \"ytick.direction\": \"out\",\n        \"xtick.major.size\": 3, \"ytick.major.size\": 3,\n        \"xtick.major.width\": 0.6, \"ytick.major.width\": 0.6,\n        \"axes.spines.top\": boxed, \"axes.spines.right\": boxed,\n        \"axes.spines.left\": frame != \"none\", \"axes.spines.bottom\": frame != \"none\",\n        \"axes.grid\": bool(grid),\n        \"legend.frameon\": False,\n        \"figure.dpi\": 200,\n        \"savefig.dpi\": 300,\n        \"savefig.bbox\": \"tight\",\n        \"axes.titleweight\": \"normal\",\n        \"axes.titlelocation\": \"left\",\n        \"axes.labelweight\": \"normal\",\n        \"lines.linewidth\": 1.2,\n        \"patch.linewidth\": 0.6,\n        \"pdf.fonttype\": 42, \"ps.fonttype\": 42,\n    }\n    if font:\n        rc[\"font.sans-serif\"] = [font, \"DejaVu Sans\"]\n    mpl.rcParams.update(rc)\n\n\ndef set_frame(ax, style=\"open\"):\n    \"\"\"§3: set spine visibility on an existing axes. style ∈ {'open','boxed','none'}.\"\"\"\n    show = {\"open\": (False, False, True, True),\n            \"boxed\": (True, True, True, True),\n            \"none\": (False, False, False, False)}[style]\n    for side, vis in zip((\"top\", \"right\", \"bottom\", \"left\"), show):\n        ax.spines[side].set_visible(vis)\n        if vis:\n            ax.spines[side].set_linewidth(0.6)\n    ax.tick_params(direction=\"out\", length=0 if style == \"none\" else 3, width=0.6)\n\n\ndef panel_letter(ax, letter, dx=-0.18, dy=1.02, case=\"lower\", fontsize=None):\n    \"\"\"§5.7: bold panel letter outside top-left of axes. case ∈ {'lower','upper'}.\"\"\"\n    import matplotlib.pyplot as plt\n    if fontsize is None:\n        fontsize = plt.rcParams.get(\"font.size\", 8) + 1\n    s = letter.lower() if case == \"lower\" else letter.upper()\n    artist = ax.text(dx, dy, s, transform=ax.transAxes,\n                     fontweight=\"bold\", fontsize=fontsize, va=\"bottom\", ha=\"left\")\n    artist.set_gid(\"figure-style-panel-letter\")\n\n\ndef focal_palette(labels, focal, focal_color, other=\"muted\", base_colors=None):\n    \"\"\"§4.2: map labels → colors with the focal series visually dominant.\n\n    other='muted'   — desaturate base_colors (or a default cycle) toward gray\n    other='grey'    — uniform light gray for all non-focal\n    other='ordinal' — non-focal on a single light→dark gray ramp (input order)\n    \"\"\"\n    import matplotlib.colors as mcolors\n    import matplotlib.pyplot as plt\n    focal_set = {focal} if isinstance(focal, str) else set(focal)\n    n = len(labels)\n    if not focal_set & set(labels):\n        raise ValueError(f\"focal {focal!r} not found in labels\")\n    base_colors = [] if base_colors is None else list(base_colors)\n    if not base_colors:\n        base_colors = plt.rcParams[\"axes.prop_cycle\"].by_key().get(\"color\", [\"#444444\"])\n    if not base_colors:\n        base_colors = [\"#444444\"]\n    base_colors = [base_colors[i % len(base_colors)] for i in range(n)]\n    if other == \"grey\":\n        rest = [\"#BCBCBC\"] * n\n    elif other == \"ordinal\":\n        nf = max(1, n - len(focal_set))\n        ramp = [mcolors.to_hex((v, v, v)) for v in\n                ([0.55] if nf == 1 else [0.80 - 0.35 * i / (nf - 1) for i in range(nf)])]\n        rest, k = [], 0\n        for l in labels:\n            rest.append(ramp[min(k, nf - 1)]); k += (l not in focal_set)\n    elif other == \"muted\":\n        def mute(c):\n            r, g, b = mcolors.to_rgb(c)\n            m = (r + g + b) / 3\n            return mcolors.to_hex((0.3 * r + 0.7 * m, 0.3 * g + 0.7 * m, 0.3 * b + 0.7 * m))\n        rest = [mute(c) for c in base_colors]\n    else:\n        raise ValueError(\"other must be 'muted', 'grey', or 'ordinal'\")\n    return [focal_color if l in focal_set else rest[i] for i, l in enumerate(labels)]\n\n\ndef bar_with_points(ax, x, ymat, labels, colors, jitter=0.08, show_points=True,\n                    errorbar=None, point_alpha=0.5, point_size=8):\n    \"\"\"§6.1: bar = mean; optionally overlay raw points or draw an interval.\n\n    colors   : per-label color list (e.g. from focal_palette)\n    x, ymat and labels must have equal lengths; mismatches raise before drawing.\n    Each group must be a nonempty, finite, one-dimensional real numeric array.\n    Requested SD/CI intervals require at least two observations per group.\n    errorbar : None | 'sd' | 'ci95' — drawn only when show_points is False.\n               'ci95' is the t-distribution 95% CI of the mean\n               (half-width t_{0.975,n-1} · s/√n); correct at small n where the\n               z-approximation (1.96·s/√n) is markedly too narrow.\n    \"\"\"\n    import numpy as np\n    x = np.asarray(x)\n    if x.ndim != 1 or not np.issubdtype(x.dtype, np.number):\n        raise TypeError(\n            \"bar_with_points x must be one-dimensional numeric positions; \"\n            \"pass category names through labels\"\n        )\n    ymat = list(ymat)\n    labels = list(labels)\n    if len(x) != len(ymat) or len(x) != len(labels):\n        raise ValueError(\"bar_with_points x, ymat and labels must have equal lengths\")\n    if colors is not None and not isinstance(colors, str):\n        colors = list(colors)\n        if not colors:\n            colors = None\n    if x.dtype.kind not in \"iuf\" or not np.isfinite(x).all():\n        raise ValueError(\"bar_with_points x must contain finite real positions\")\n    for i, y in enumerate(ymat):\n        if np.ma.isMaskedArray(y) and np.ma.getmaskarray(y).any():\n            raise ValueError(f\"bar_with_points group {i} contains masked observations; resolve exclusions explicitly before plotting\")\n    ymat = [np.asarray(y) for y in ymat]\n    for i, y in enumerate(ymat):\n        if y.ndim != 1 or y.size == 0 or y.dtype.kind not in \"iuf\" or not np.isfinite(y).all():\n            raise ValueError(f\"bar_with_points group {i} must be nonempty, one-dimensional and finite real numeric data\")\n    if errorbar and not show_points:\n        if errorbar not in (\"sd\", \"ci95\"):\n            raise ValueError(\"errorbar must be None, 'sd', or 'ci95'\")\n        if any(y.size < 2 for y in ymat):\n            raise ValueError(\"SD/CI cannot be estimated from one observation; use raw points without an interval\")\n    means = np.array([np.mean(y) for y in ymat], float)\n    err = None\n    if errorbar and not show_points:\n        if errorbar == \"sd\":\n            err = np.array([np.std(y, ddof=1) for y in ymat])\n        elif errorbar == \"ci95\":\n            from scipy.stats import t\n            def _hw(y):\n                n = np.asarray(y).size\n                return t.ppf(0.975, n - 1) * np.std(y, ddof=1) / np.sqrt(n)\n            err = np.array([_hw(y) for y in ymat])\n    ax.bar(x, means, color=colors, width=0.7, edgecolor=\"none\",\n           yerr=err, error_kw={\"elinewidth\": 0.8, \"capsize\": 0})\n    if show_points:\n        for xi, ys in zip(x, ymat):\n            ys = np.asarray(ys)\n            if ys.ndim == 1 and ys.size > 0:\n                jit = (np.random.rand(ys.size) - 0.5) * 2 * jitter\n                ax.scatter(np.full(ys.size, xi) + jit, ys, s=point_size, color=\"black\",\n                           alpha=point_alpha, zorder=3, linewidths=0)\n    ax.set_xticks(x); ax.set_xticklabels(labels)\n    return ax\n\n\ndef strip_with_median(ax, groups, values, colors=None, jitter=0.12):\n    \"\"\"§6.1: jittered points + bold horizontal median tick per group.\n\n    groups and values must have equal lengths. Mismatches raise before drawing.\n    colors=None or an empty color sequence supplies default gray; a nonempty\n    color sequence cycles across groups.\n    Each group must be nonempty, one-dimensional and finite real numeric data;\n    resolve missing observations explicitly before plotting.\n    \"\"\"\n    import numpy as np\n    labs = list(groups)\n    values = list(values)\n    if len(labs) != len(values):\n        raise ValueError(\"strip_with_median groups and values must have equal lengths\")\n    for i, y in enumerate(values):\n        if np.ma.isMaskedArray(y) and np.ma.getmaskarray(y).any():\n            raise ValueError(f\"strip_with_median group {i} contains masked observations; resolve exclusions explicitly before plotting\")\n    values = [np.asarray(y) for y in values]\n    for i, y in enumerate(values):\n        if y.ndim != 1 or y.size == 0 or y.dtype.kind not in \"iuf\" or not np.isfinite(y).all():\n            raise ValueError(f\"strip_with_median group {i} must be nonempty, one-dimensional and finite real numeric data\")\n    colors = [] if colors is None else list(colors)\n    if not colors:\n        colors = [\"#444444\"]\n    for i, ys in enumerate(values):\n        c = colors[i % len(colors)]\n        ys = np.asarray(ys)\n        jit = (np.random.rand(ys.size) - 0.5) * 2 * jitter\n        ax.scatter(np.full(ys.size, i) + jit, ys, s=10, color=c, alpha=0.6, linewidths=0, zorder=2)\n        m = np.median(ys)\n        ax.plot([i - 0.22, i + 0.22], [m, m], color=\"black\", lw=1.6, zorder=3)\n    ax.set_xticks(range(len(labs))); ax.set_xticklabels(labs)\n    return ax\n\n\ndef goodness_arrow(ax, text=\"higher = better\", loc=\"upper left\", axis=\"y\", fontsize=None):\n    \"\"\"§3.6: small upright direction-of-goodness cue in the margin.\"\"\"\n    import matplotlib.pyplot as plt\n    if fontsize is None:\n        fontsize = plt.rcParams[\"legend.fontsize\"]\n    pos = {\"upper left\": (0.02, 0.98), \"upper right\": (0.98, 0.98),\n           \"lower left\": (0.02, 0.02), \"lower right\": (0.98, 0.02)}[loc]\n    ha = \"left\" if \"left\" in loc else \"right\"\n    va = \"top\" if \"upper\" in loc else \"bottom\"\n    arrow = \"↑ \" if axis == \"y\" else \"→ \"\n    ax.text(pos[0], pos[1], arrow + text, transform=ax.transAxes,\n            fontsize=fontsize, color=META_GREY, ha=ha, va=va)\n\n\ndef two_tier_label(name, meta):\n    \"\"\"§5: two-line label string (name / metadata). Meta line styled separately by caller.\"\"\"\n    return f\"{name}\\n{meta}\"\n\n\ndef end_of_line_labels(ax, xs, ys, labels, colors=None, dx=0.01, fontsize=None):\n    \"\"\"§6.3 / §7.3: label each line series at its right end instead of a legend box.\n\n    Each label requires a nonempty, matched x/y series. Validate all series\n    before drawing. Nonempty colors cycle; None or empty uses default text color.\n    Inner sequences must support len() and indexing; they are not copied.\n    Nonfinite endpoints are rejected rather than producing an invisible label.\n    \"\"\"\n    import matplotlib.pyplot as plt\n    import math\n    xs, ys, labels = list(xs), list(ys), list(labels)\n    if len(xs) != len(ys) or len(xs) != len(labels):\n        raise ValueError(\"end_of_line_labels xs, ys and labels must have equal lengths\")\n    if any(len(x) == 0 or len(x) != len(y) for x, y in zip(xs, ys)):\n        raise ValueError(\"end_of_line_labels each x/y series must be nonempty and have equal lengths\")\n    if any(not math.isfinite(x[-1]) or not math.isfinite(y[-1]) for x, y in zip(xs, ys)):\n        raise ValueError(\"end_of_line_labels endpoints must be finite; choose valid endpoints explicitly\")\n    if fontsize is None:\n        fontsize = plt.rcParams[\"font.size\"]\n    colors = [] if colors is None else list(colors)\n    if not colors:\n        colors = [None]\n    span = ax.get_xlim()[1] - ax.get_xlim()[0]\n    for i, (x, y, lab) in enumerate(zip(xs, ys, labels)):\n        c = colors[i % len(colors)]\n        ax.text(x[-1] + dx * span, y[-1], lab, color=c, va=\"center\", ha=\"left\", fontsize=fontsize)\n\n\ndef panel_crops(fig, dpi=None, pad_px=6, bbox_inches=None, pad_inches=None):\n    \"\"\"§9.2: pixel-space crop boxes for each lettered panel in the SAVED PNG.\n\n    Returns ``{letter: (x0, y0, x1, y1)}`` in image-space pixels (origin\n    top-left, matching PIL's ``Image.crop``). Panels are detected as bold\n    single-character ``Text``\n    objects placed by :func:`panel_letter`; each panel's crop is its axes'\n    tightbbox mapped into the saved file's pixel space, padded by ``pad_px``.\n    For §3.4 composites (abutting subplots sharing an axis, letter on the\n    leftmost only) the crop unions in letterless ``sharex``/``sharey`` siblings\n    on the same grid row/col so the whole composite is covered. When no axes\n    carries a panel letter (standalone plot, or a figure-composer sub-agent),\n    falls back to one crop per axes keyed by index.\n\n    ``bbox_inches`` mirrors ``Figure.savefig`` semantics: ``None`` means\n    *consult rcParams* (so under :func:`apply_figure_style` it resolves to\n    ``'tight'``); pass an explicit ``Bbox`` only if you saved with one. The\n    boxes are clamped to the saved image extent; panels with no intersection\n    are omitted.\n\n        >>> from PIL import Image\n        >>> fig.savefig(\"fig.png\")  # bbox_inches='tight' via rcParams\n        >>> with Image.open(\"fig.png\") as saved:\n        ...     for letter, box in panel_crops(fig).items():\n        ...         saved.crop(box).save(f\"panel_{letter}.png\")\n\n    Match savefig's DPI, bbox_inches and pad_inches when overriding defaults.\n    Returned boxes are pixels, never savefig bbox_inches (inches). Passing them\n    back to savefig as inches can allocate an enormous raster.\n    \"\"\"\n    import matplotlib as mpl\n    import matplotlib.text\n    from matplotlib.backends.backend_agg import FigureCanvasAgg\n    if isinstance(pad_px, bool) or not isinstance(pad_px, int) or pad_px < 0:\n        raise TypeError(\"panel_crops pad_px must be a non-negative integer\")\n    if dpi is None:\n        dpi = mpl.rcParams.get(\"savefig.dpi\", fig.dpi)\n        if dpi == \"figure\":\n            dpi = fig.dpi\n    dpi = float(dpi)\n    if bbox_inches is None:\n        bbox_inches = mpl.rcParams.get(\"savefig.bbox\")\n    # The Open-Science notebook captures and closes pyplot figures after every\n    # cell. Matplotlib 3.11 decouples a closed Figure from its backend by\n    # replacing FigureCanvasAgg with FigureCanvasBase, while the persistent\n    # notebook namespace can still retain the Figure itself. Reattach the\n    # headless canvas before asking for saved-image geometry so this helper is\n    # safe in a later QA cell as well as the producer cell.\n    if not hasattr(fig.canvas, \"get_renderer\"):\n        FigureCanvasAgg(fig)\n    fig.canvas.draw()\n    r = fig.canvas.get_renderer()\n\n    if bbox_inches == \"tight\":\n        if pad_inches is None:\n            pad_inches = mpl.rcParams.get(\"savefig.pad_inches\", 0.1)\n        tb = fig.get_tightbbox(r).padded(pad_inches)\n        ox_in, oy_in = tb.x0, tb.y0\n        W_in, H_in = tb.width, tb.height\n    elif isinstance(bbox_inches, mpl.transforms.BboxBase):\n        ox_in, oy_in = bbox_inches.x0, bbox_inches.y0\n        W_in, H_in = bbox_inches.width, bbox_inches.height\n    elif bbox_inches is None:\n        ox_in, oy_in = 0.0, 0.0\n        W_in, H_in = fig.get_size_inches()\n    else:\n        raise TypeError(\"panel_crops bbox_inches must be None, 'tight', or a matplotlib Bbox\")\n    W_px, H_px = int(round(W_in * dpi)), int(round(H_in * dpi))\n    lettered = {}\n    seen_letters = set()\n    for ax in fig.axes:\n        for t in ax.findobj(matplotlib.text.Text):\n            s = (t.get_text() or \"\").strip()\n            if t.get_gid() == \"figure-style-panel-letter\":\n                if s in seen_letters:\n                    raise ValueError(f\"Duplicate panel letter: {s!r}\")\n                seen_letters.add(s)\n                lettered[ax] = s\n                break\n\n\n\n    if not lettered:\n        lettered = {ax: str(i) for i, ax in enumerate(fig.axes)}\n    out = {}\n    for ax, letter in lettered.items():\n        bbs = [ax.get_tightbbox(r)]\n\n\n\n\n        ss = ax.get_subplotspec()\n        for sib in fig.axes:\n            if sib is ax or sib in lettered:\n                continue\n            ssib = sib.get_subplotspec()\n            same_row = ss is None or ssib is None or ss.rowspan == ssib.rowspan\n            same_col = ss is None or ssib is None or ss.colspan == ssib.colspan\n            if ((ax.get_shared_y_axes().joined(ax, sib) and same_row)\n                    or (ax.get_shared_x_axes().joined(ax, sib) and same_col)):\n                bbs.append(sib.get_tightbbox(r))\n        bb = mpl.transforms.Bbox.union(bbs)\n\n        bx0 = (bb.x0 / fig.dpi - ox_in) * dpi\n        bx1 = (bb.x1 / fig.dpi - ox_in) * dpi\n        by0 = H_px - (bb.y1 / fig.dpi - oy_in) * dpi\n        by1 = H_px - (bb.y0 / fig.dpi - oy_in) * dpi\n        box = (\n            min(max(int(bx0) - pad_px, 0), W_px),\n            min(max(int(by0) - pad_px, 0), H_px),\n            max(min(int(bx1) + pad_px, W_px), 0),\n            max(min(int(by1) + pad_px, H_px), 0),\n        )\n        if box[0] < box[2] and box[1] < box[3]:\n            out[letter] = box\n    return out\n",
          "sourceDigest": "8020d409bfb60f0c8dfa43d0dfe7bb97f7458e1a3908fefbcb6e4551e0feae4e"
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        "stdout": "\nFILE lean/OAI/Combinatorics/BooleanFunctions/Main.lean \nimport OAI.Combinatorics.BooleanFunctions.Amplification\nimport OAI.Combinatorics.BooleanFunctions.Readout\n\nnamespace OAI\n\nopen scoped BigOperators\n\nnoncomputable section\n\nnamespace SquareRootDegree\n\n/-- Every stage and the final sign readout use finitely many actual Boolean inputs. -/\ntheorem signed_violations : SignedViolations := by\n  intro C hC\n  obtain ⟨F,hF⟩ := Amplification.exists_large_ratio ((C/ReadoutCLT.constant)^2)\n  exact Observations.violation_of_observation F.map F.cost F.cost_pos F.cell_le\n    F.variance_pos C hC hF\n\n/-- Unbounded violations for ordinary real Fourier degree and uniform cube averages. -/\ntheorem main : SignedViolations ∧ ¬ BddAbove AbsoluteRatios := by\n  exact ⟨signed_violations,absolute_unbounded_of_signed signed_violations⟩\n\nend SquareRootDegree\n\nend\n\nend OAI\n\n\nFILE lean/OAI/Combinatorics/MatchingEntropy/Main.lean \nimport OAI.Combinatorics.MatchingEntropy.ExpansionSequences\n\n/-!\n# Sharp analytic bounds for perfect-matching entropy\n\nFor a finite loopless labelled multigraph on `2m` vertices, with `m ≥ 1` and a perfect\nmatching, every point of the matching polytope satisfies\n`F - (2 - 2 / m) * B ≤ H ≤ F`. Here `F` and `B` are the coordinate entropy and\ncomplement entropy with natural logarithms, and `H` is the attained supremum of entropy\nover all perfect-matching laws with the prescribed mean.\n\nThe development includes the odd-cut description of the matching polytope, sharp face\nrank bounds, minimal faces, fractional and connected-component reductions, attained\nentropy and weighted partition objectives, matching counts, degenerate cases, colored\ntriangle-expansion families, and concrete coefficient obstructions.\n-/\n\n\nFILE lean/OAI/Computability/LoopMatching/Endpoint.lean \nimport OAI.Computability.LoopMatching.LiteralChoices\nimport OAI.Computability.LoopMatching.SmallBoundary\nimport OAI.Computability.LoopMatching.Main\n\nnamespace OAI\n\nnamespace LoopMatchingReference\n\ntheorem challenge :\n    ∃ A : LoopMatching.Input → ℕ × ℕ,\n      ∃ machine : Turing.TM2ComputableInPolyTime\n          LoopMatching.encodeInput LoopMatching.encodeOutput A,\n        (∀ k, Finite (machine.tm.Γ k)) ∧\n        (∃ P : Polynomial ℕ, ∀ G,\n          (LoopMatching.encodeOutput (A G)).length ≤\n            P.eval (LoopMatching.encodeInput G).length) ∧\n        ∀ G, 0 < (A G).2 ∧\n          (A G).1 ≤ (A G).2 * LoopMatching.count G ∧\n          (A G).2 * LoopMatching.count G ≤ 2 ^ (18 * G.n) * (A G).1 ∧\n          ((A G).1 = 0 ↔ LoopMatching.count G = 0) := by\n  exact LoopMatching.fullEndpoint\n\nend LoopMatchingReference\n\nend OAI\n\n\nFILE lean/OAI/Analysis/HotSpots/Main.lean FINAL\np circleBoundary.continuous)\n    (hinj.comp circleBoundary.injective) T hr (fun U hU hne => closedTraceSpace_local_rank c hΩ hU hne)\n    g ((c.closedGradient_trace_continuous hΩ hu).comp circleBoundary.continuous) hmem\n  apply c.boundary_gradient_not_zero hΩ hu.interior hu.1.continuousOn hY he hne\n  intro x hx\n  have hh := congrArg (fun f : T => f.val (c.circleBoundaryHomeomorph.symm ⟨x,hx⟩)) h0\n  change closedGradient Ω u\n    (c.circleBoundaryHomeomorph (c.circleBoundaryHomeomorph.symm ⟨x,hx⟩)) = 0 at hh\n  simpa only [Homeomorph.apply_symm_apply] using hh\n\ninclude c\n\n\ntheorem noncritical (hΩ : AdmissibleDomain Ω)\n    {u : Plane → ℝ} (hu : InFirstNeumannEigenspace Ω u) (hne : ∃ x ∈ Ω, u x ≠ 0) :\n    ∀ x ∈ Ω, gradient u x ≠ 0 := by\n  intro x hx\n  have hp : c.G x ∈ disk := by\n    have hh := c.diskMap.map_target hx\n    change c.diskMap.symm x ∈ disk at hh\n    rwa [c.diskMap_inverse_agrees hx] at hh\n  have hh := c.noncritical_at_disk hΩ hu hne ⟨c.G x,hp⟩\n  simpa only [c.right_inverse x (subset_closure hx)] using hh\nend StrictHotSpots.Conformal.ClosedDiskChart\n\nnamespace StrictHotSpots\n\n\ntheorem main_theorem (Ω : Set Plane) (hΩ : AdmissibleDomain Ω)\n    (u : Plane → ℝ) (hu : InFirstNeumannEigenspace Ω u)\n    (hne : ∃ x ∈ Ω, u x ≠ 0) : MainConclusion Ω u := by\n  obtain ⟨c⟩ := Conformal.exists_closed_disk_chart hΩ\n  exact strict_boundary_extrema_of_noncritical hΩ.1 hΩ.2.1 hΩ.2.2.1 hu.1.continuousOn\n    (c.noncritical hΩ hu hne)\nend StrictHotSpots\nend\nend ActualNoncritical\n\n\nend OAI\n\n\nFILE lean/OAI/Analysis/RelativeGeneration/Main.lean FINAL\n:=\n  ⟨factor.nonzero, factor.weaklyClosed, factor.factor, factor.finite, factor.diffuse⟩\n\n/-- The entire relative-generator locus of an irreducible inclusion is dense Gδ in the\nnormalized trace two-topology on the original, genuine unitaries. -/\ntheorem main_theorem : MainTarget.{level} := by\n  intro Hilbert _ _ _ small large smallFactor largeFactor inclusion irreducible separablePredual\n  let smallFactor' := smallFactor.toGenerator\n  let largeFactor' := largeFactor.toGenerator\n  obtain ⟨gdelta, dense⟩ := Generator.relative_generation_of_II1_separable_predual\n    small large smallFactor' largeFactor' inclusion irreducible separablePredual\n  refine ⟨Generator.normalizedFactorTrace large largeFactor',\n    Generator.normalizedFactorTrace_isNormalizedTrace large largeFactor', ?_⟩\n  let : TopologicalSpace (unitary large) :=\n    traceTopology large (Generator.normalizedFactorTrace large largeFactor')\n  let unitaryHomeomorph := Generator.FactorTraceUnitary.traceHomeomorph large largeFactor'\n  have locus_eq : relativeGeneratorLocus small large = unitaryHomeomorph.symm ⁻¹'\n      {element : Generator.FactorTraceUnitary large largeFactor' |\n        Generator.wstar ((small : Set (Hilbert →L[ℂ] Hilbert)) ∪\n          {(element.algebra large largeFactor').val}) = large} := by\n    ext element\n    rfl\n  rw [locus_eq]\n  exact ⟨IsGδ.preimage unitaryHomeomorph.symm.continuous gdelta,\n    unitaryHomeomorph.symm.isOpenQuotientMap.dense_preimage_iff.mpr dense⟩\n\nend RelativeGeneration\n\nend\n\nend OAI\n\n\nFILE lean/OAI/MathematicalPhysics/Elasticity/Uniqueness.lean FINAL\ntain ⟨L₁,M₁,L₂,M₂,hL₁,hM₁,hL₂,hM₂,heL₁,heM₁,heL₂,heM₂,hp₁,hp₂,hexL,hexM,hd,K,hK,hout⟩ :=\n    ElasticityBoundaryPacket.admissible_common_exterior hD lam₁ mu₁ lam₂ mu₂ ha₁ ha₂ hDN\n  let l : Fin 2 → X → ℝ := ![L₁,L₂]\n  let m : Fin 2 → X → ℝ := ![M₁,M₂]\n  have hl (q : Fin 2) : ContDiff ℝ (⊤ : ℕ∞) (l q) := by fin_cases q; exact hL₁; exact hL₂\n  have hm (q : Fin 2) : ContDiff ℝ (⊤ : ℕ∞) (m q) := by fin_cases q; exact hM₁; exact hM₂\n  have hp (q : Fin 2) (x : X) : 0 < m q x ∧ 0<3*l q x+2*m q x := by\n    fin_cases q; exact hp₁ x; exact hp₂ x\n  have hcll (q : Fin 2) : ExteriorConstant (smoothOfReal (l q) (hl q)) := by\n    apply exteriorConstant_of_eq_const (hl q) hK 0\n    intro x hx\n    fin_cases q; exact (hout x hx).1; exact (hout x hx).2.2.1\n  have hclm (q : Fin 2) : ExteriorConstant (smoothOfReal (m q) (hm q)) := by\n    apply exteriorConstant_of_eq_const (hm q) hK 1\n    intro x hx\n    fin_cases q; exact (hout x hx).2.1; exact (hout x hx).2.2.2\n  have he (x) (hx : x∉Ω) : l 0 x=l 1 x ∧ m 0 x=m 1 x := ⟨(hexL hx).symm,(hexM hx).symm⟩\n  have hh := dn_global_coefficients_equal hD.1 hD.2.2.1 l m hl hm hp hcll hclm he hd\n  constructor\n  · intro x hx\n    rw [← heL₁ hx,← heL₂ hx]\n    exact congrFun hh.1 x\n  · intro x hx\n    rw [← heM₁ hx,← heM₂ hx]\n    exact congrFun hh.2 x\nend Elasticity\n\nend\nend\ntheorem Elasticity.global_uniqueness : Elasticity.MainClaim := by\n  intro Ω hD lam₁ mu₁ lam₂ mu₂ ha₁ ha₂ hDN\n  exact Elasticity.source_global_uniqueness Ω hD lam₁ mu₁ lam₂ mu₂ ha₁ ha₂ hDN\n\nend OAI\n\n",
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          "\nFILE lean/OAI/Combinatorics/BooleanFunctions/Main.lean \nimport OAI.Combinatorics.BooleanFunctions.Amplification\nimport OAI.Combinatorics.BooleanFunctions.Readout\n\nnamespace OAI\n\nopen scoped BigOperators\n\nnoncomputable section\n\nnamespace SquareRootDegree\n\n/-- Every stage and the final sign readout use finitely many actual Boolean inputs. -/\ntheorem signed_violations : SignedViolations := by\n  intro C hC\n  obtain ⟨F,hF⟩ := Amplification.exists_large_ratio ((C/ReadoutCLT.constant)^2)\n  exact Observations.violation_of_observation F.map F.cost F.cost_pos F.cell_le\n    F.variance_pos C hC hF\n\n/-- Unbounded violations for ordinary real Fourier degree and uniform cube averages. -/\ntheorem main : SignedViolations ∧ ¬ BddAbove AbsoluteRatios := by\n  exact ⟨signed_violations,absolute_unbounded_of_signed signed_violations⟩\n\nend SquareRootDegree\n\nend\n\nend OAI\n\n\nFILE lean/OAI/Combinatorics/MatchingEntropy/Main.lean \nimport OAI.Combinatorics.MatchingEntropy.ExpansionSequences\n\n/-!\n# Sharp analytic bounds for perfect-matching entropy\n\nFor a finite loopless labelled multigraph on `2m` vertices, with `m ≥ 1` and a perfect\nmatching, every point of the matching polytope satisfies\n`F - (2 - 2 / m) * B ≤ H ≤ F`. Here `F` and `B` are the coordinate entropy and\ncomplement entropy with natural logarithms, and `H` is the attained supremum of entropy\nover all perfect-matching laws with the prescribed mean.\n\nThe development includes the odd-cut description of the matching polytope, sharp face\nrank bounds, minimal faces, fractional and connected-component reductions, attained\nentropy and weighted partition objectives, matching counts, degenerate cases, colored\ntriangle-expansion families, and concrete coefficient obstructions.\n-/\n\n\nFILE lean/OAI/Computability/LoopMatching/Endpoint.lean \nimport OAI.Computability.LoopMatching.LiteralChoices\nimport OAI.Computability.LoopMatching.SmallBoundary\nimport OAI.Computability.LoopMatching.Main\n\nnamespace OAI\n\nnamespace LoopMatchingReference\n\ntheorem challenge :\n    ∃ A : LoopMatching.Input → ℕ × ℕ,\n      ∃ machine : Turing.TM2ComputableInPolyTime\n          LoopMatching.encodeInput LoopMatching.encodeOutput A,\n        (∀ k, Finite (machine.tm.Γ k)) ∧\n        (∃ P : Polynomial ℕ, ∀ G,\n          (LoopMatching.encodeOutput (A G)).length ≤\n            P.eval (LoopMatching.encodeInput G).length) ∧\n        ∀ G, 0 < (A G).2 ∧\n          (A G).1 ≤ (A G).2 * LoopMatching.count G ∧\n          (A G).2 * LoopMatching.count G ≤ 2 ^ (18 * G.n) * (A G).1 ∧\n          ((A G).1 = 0 ↔ LoopMatching.count G = 0) := by\n  exact LoopMatching.fullEndpoint\n\nend LoopMatchingReference\n\nend OAI\n\n\nFILE lean/OAI/Analysis/HotSpots/Main.lean FINAL\np circleBoundary.continuous)\n    (hinj.comp circleBoundary.injective) T hr (fun U hU hne => closedTraceSpace_local_rank c hΩ hU hne)\n    g ((c.closedGradient_trace_continuous hΩ hu).comp circleBoundary.continuous) hmem\n  apply c.boundary_gradient_not_zero hΩ hu.interior hu.1.continuousOn hY he hne\n  intro x hx\n  have hh := congrArg (fun f : T => f.val (c.circleBoundaryHomeomorph.symm ⟨x,hx⟩)) h0\n  change closedGradient Ω u\n    (c.circleBoundaryHomeomorph (c.circleBoundaryHomeomorph.symm ⟨x,hx⟩)) = 0 at hh\n  simpa only [Homeomorph.apply_symm_apply] using hh\n\ninclude c\n\n\ntheorem noncritical (hΩ : AdmissibleDomain Ω)\n    {u : Plane → ℝ} (hu : InFirstNeumannEigenspace Ω u) (hne : ∃ x ∈ Ω, u x ≠ 0) :\n    ∀ x ∈ Ω, gradient u x ≠ 0 := by\n  intro x hx\n  have hp : c.G x ∈ disk := by\n    have hh := c.diskMap.map_target hx\n    change c.diskMap.symm x ∈ disk at hh\n    rwa [c.diskMap_inverse_agrees hx] at hh\n  have hh := c.noncritical_at_disk hΩ hu hne ⟨c.G x,hp⟩\n  simpa only [c.right_inverse x (subset_closure hx)] using hh\nend StrictHotSpots.Conformal.ClosedDiskChart\n\nnamespace StrictHotSpots\n\n\ntheorem main_theorem (Ω : Set Plane) (hΩ : AdmissibleDomain Ω)\n    (u : Plane → ℝ) (hu : InFirstNeumannEigenspace Ω u)\n    (hne : ∃ x ∈ Ω, u x ≠ 0) : MainConclusion Ω u := by\n  obtain ⟨c⟩ := Conformal.exists_closed_disk_chart hΩ\n  exact strict_boundary_extrema_of_noncritical hΩ.1 hΩ.2.1 hΩ.2.2.1 hu.1.continuousOn\n    (c.noncritical hΩ hu hne)\nend StrictHotSpots\nend\nend ActualNoncritical\n\n\nend OAI\n\n\nFILE lean/OAI/Analysis/RelativeGeneration/Main.lean FINAL\n:=\n  ⟨factor.nonzero, factor.weaklyClosed, factor.factor, factor.finite, factor.diffuse⟩\n\n/-- The entire relative-generator locus of an irreducible inclusion is dense Gδ in the\nnormalized trace two-topology on the original, genuine unitaries. -/\ntheorem main_theorem : MainTarget.{level} := by\n  intro Hilbert _ _ _ small large smallFactor largeFactor inclusion irreducible separablePredual\n  let smallFactor' := smallFactor.toGenerator\n  let largeFactor' := largeFactor.toGenerator\n  obtain ⟨gdelta, dense⟩ := Generator.relative_generation_of_II1_separable_predual\n    small large smallFactor' largeFactor' inclusion irreducible separablePredual\n  refine ⟨Generator.normalizedFactorTrace large largeFactor',\n    Generator.normalizedFactorTrace_isNormalizedTrace large largeFactor', ?_⟩\n  let : TopologicalSpace (unitary large) :=\n    traceTopology large (Generator.normalizedFactorTrace large largeFactor')\n  let unitaryHomeomorph := Generator.FactorTraceUnitary.traceHomeomorph large largeFactor'\n  have locus_eq : relativeGeneratorLocus small large = unitaryHomeomorph.symm ⁻¹'\n      {element : Generator.FactorTraceUnitary large largeFactor' |\n        Generator.wstar ((small : Set (Hilbert →L[ℂ] Hilbert)) ∪\n          {(element.algebra large largeFactor').val}) = large} := by\n    ext element\n    rfl\n  rw [locus_eq]\n  exact ⟨IsGδ.preimage unitaryHomeomorph.symm.continuous gdelta,\n    unitaryHomeomorph.symm.isOpenQuotientMap.dense_preimage_iff.mpr dense⟩\n\nend RelativeGeneration\n\nend\n\nend OAI\n\n\nFILE lean/OAI/MathematicalPhysics/Elasticity/Uniqueness.lean FINAL\ntain ⟨L₁,M₁,L₂,M₂,hL₁,hM₁,hL₂,hM₂,heL₁,heM₁,heL₂,heM₂,hp₁,hp₂,hexL,hexM,hd,K,hK,hout⟩ :=\n    ElasticityBoundaryPacket.admissible_common_exterior hD lam₁ mu₁ lam₂ mu₂ ha₁ ha₂ hDN\n  let l : Fin 2 → X → ℝ := ![L₁,L₂]\n  let m : Fin 2 → X → ℝ := ![M₁,M₂]\n  have hl (q : Fin 2) : ContDiff ℝ (⊤ : ℕ∞) (l q) := by fin_cases q; exact hL₁; exact hL₂\n  have hm (q : Fin 2) : ContDiff ℝ (⊤ : ℕ∞) (m q) := by fin_cases q; exact hM₁; exact hM₂\n  have hp (q : Fin 2) (x : X) : 0 < m q x ∧ 0<3*l q x+2*m q x := by\n    fin_cases q; exact hp₁ x; exact hp₂ x\n  have hcll (q : Fin 2) : ExteriorConstant (smoothOfReal (l q) (hl q)) := by\n    apply exteriorConstant_of_eq_const (hl q) hK 0\n    intro x hx\n    fin_cases q; exact (hout x hx).1; exact (hout x hx).2.2.1\n  have hclm (q : Fin 2) : ExteriorConstant (smoothOfReal (m q) (hm q)) := by\n    apply exteriorConstant_of_eq_const (hm q) hK 1\n    intro x hx\n    fin_cases q; exact (hout x hx).2.1; exact (hout x hx).2.2.2\n  have he (x) (hx : x∉Ω) : l 0 x=l 1 x ∧ m 0 x=m 1 x := ⟨(hexL hx).symm,(hexM hx).symm⟩\n  have hh := dn_global_coefficients_equal hD.1 hD.2.2.1 l m hl hm hp hcll hclm he hd\n  constructor\n  · intro x hx\n    rw [← heL₁ hx,← heL₂ hx]\n    exact congrFun hh.1 x\n  · intro x hx\n    rw [← heM₁ hx,← heM₂ hx]\n    exact congrFun hh.2 x\nend Elasticity\n\nend\nend\ntheorem Elasticity.global_uniqueness : Elasticity.MainClaim := by\n  intro Ω hD lam₁ mu₁ lam₂ mu₂ ha₁ ha₂ hDN\n  exact Elasticity.source_global_uniqueness Ω hD lam₁ mu₁ lam₂ mu₂ ha₁ ha₂ hDN\n\nend OAI\n\n"
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          "text": "\nFILE lean/OAI/Combinatorics/BooleanFunctions/Main.lean \nimport OAI.Combinatorics.BooleanFunctions.Amplification\nimport OAI.Combinatorics.BooleanFunctions.Readout\n\nnamespace OAI\n\nopen scoped BigOperators\n\nnoncomputable section\n\nnamespace SquareRootDegree\n\n/-- Every stage and the final sign readout use finitely many actual Boolean inputs. -/\ntheorem signed_violations : SignedViolations := by\n  intro C hC\n  obtain ⟨F,hF⟩ := Amplification.exists_large_ratio ((C/ReadoutCLT.constant)^2)\n  exact Observations.violation_of_observation F.map F.cost F.cost_pos F.cell_le\n    F.variance_pos C hC hF\n\n/-- Unbounded violations for ordinary real Fourier degree and uniform cube averages. -/\ntheorem main : SignedViolations ∧ ¬ BddAbove AbsoluteRatios := by\n  exact ⟨signed_violations,absolute_unbounded_of_signed signed_violations⟩\n\nend SquareRootDegree\n\nend\n\nend OAI\n\n\nFILE lean/OAI/Combinatorics/MatchingEntropy/Main.lean \nimport OAI.Combinatorics.MatchingEntropy.ExpansionSequences\n\n/-!\n# Sharp analytic bounds for perfect-matching entropy\n\nFor a finite loopless labelled multigraph on `2m` vertices, with `m ≥ 1` and a perfect\nmatching, every point of the matching polytope satisfies\n`F - (2 - 2 / m) * B ≤ H ≤ F`. Here `F` and `B` are the coordinate entropy and\ncomplement entropy with natural logarithms, and `H` is the attained supremum of entropy\nover all perfect-matching laws with the prescribed mean.\n\nThe development includes the odd-cut description of the matching polytope, sharp face\nrank bounds, minimal faces, fractional and connected-component reductions, attained\nentropy and weighted partition objectives, matching counts, degenerate cases, colored\ntriangle-expansion families, and concrete coefficient obstructions.\n-/\n\n\nFILE lean/OAI/Computability/LoopMatching/Endpoint.lean \nimport OAI.Computability.LoopMatching.LiteralChoices\nimport OAI.Computability.LoopMatching.SmallBoundary\nimport OAI.Computability.LoopMatching.Main\n\nnamespace OAI\n\nnamespace LoopMatchingReference\n\ntheorem challenge :\n    ∃ A : LoopMatching.Input → ℕ × ℕ,\n      ∃ machine : Turing.TM2ComputableInPolyTime\n          LoopMatching.encodeInput LoopMatching.encodeOutput A,\n        (∀ k, Finite (machine.tm.Γ k)) ∧\n        (∃ P : Polynomial ℕ, ∀ G,\n          (LoopMatching.encodeOutput (A G)).length ≤\n            P.eval (LoopMatching.encodeInput G).length) ∧\n        ∀ G, 0 < (A G).2 ∧\n          (A G).1 ≤ (A G).2 * LoopMatching.count G ∧\n          (A G).2 * LoopMatching.count G ≤ 2 ^ (18 * G.n) * (A G).1 ∧\n          ((A G).1 = 0 ↔ LoopMatching.count G = 0) := by\n  exact LoopMatching.fullEndpoint\n\nend LoopMatchingReference\n\nend OAI\n\n\nFILE lean/OAI/Analysis/HotSpots/Main.lean FINAL\np circleBoundary.continuous)\n    (hinj.comp circleBoundary.injective) T hr (fun U hU hne => closedTraceSpace_local_rank c hΩ hU hne)\n    g ((c.closedGradient_trace_continuous hΩ hu).comp circleBoundary.continuous) hmem\n  apply c.boundary_gradient_not_zero hΩ hu.interior hu.1.continuousOn hY he hne\n  intro x hx\n  have hh := congrArg (fun f : T => f.val (c.circleBoundaryHomeomorph.symm ⟨x,hx⟩)) h0\n  change closedGradient Ω u\n    (c.circleBoundaryHomeomorph (c.circleBoundaryHomeomorph.symm ⟨x,hx⟩)) = 0 at hh\n  simpa only [Homeomorph.apply_symm_apply] using hh\n\ninclude c\n\n\ntheorem noncritical (hΩ : AdmissibleDomain Ω)\n    {u : Plane → ℝ} (hu : InFirstNeumannEigenspace Ω u) (hne : ∃ x ∈ Ω, u x ≠ 0) :\n    ∀ x ∈ Ω, gradient u x ≠ 0 := by\n  intro x hx\n  have hp : c.G x ∈ disk := by\n    have hh := c.diskMap.map_target hx\n    change c.diskMap.symm x ∈ disk at hh\n    rwa [c.diskMap_inverse_agrees hx] at hh\n  have hh := c.noncritical_at_disk hΩ hu hne ⟨c.G x,hp⟩\n  simpa only [c.right_inverse x (subset_closure hx)] using hh\nend StrictHotSpots.Conformal.ClosedDiskChart\n\nnamespace StrictHotSpots\n\n\ntheorem main_theorem (Ω : Set Plane) (hΩ : AdmissibleDomain Ω)\n    (u : Plane → ℝ) (hu : InFirstNeumannEigenspace Ω u)\n    (hne : ∃ x ∈ Ω, u x ≠ 0) : MainConclusion Ω u := by\n  obtain ⟨c⟩ := Conformal.exists_closed_disk_chart hΩ\n  exact strict_boundary_extrema_of_noncritical hΩ.1 hΩ.2.1 hΩ.2.2.1 hu.1.continuousOn\n    (c.noncritical hΩ hu hne)\nend StrictHotSpots\nend\nend ActualNoncritical\n\n\nend OAI\n\n\nFILE lean/OAI/Analysis/RelativeGeneration/Main.lean FINAL\n:=\n  ⟨factor.nonzero, factor.weaklyClosed, factor.factor, factor.finite, factor.diffuse⟩\n\n/-- The entire relative-generator locus of an irreducible inclusion is dense Gδ in the\nnormalized trace two-topology on the original, genuine unitaries. -/\ntheorem main_theorem : MainTarget.{level} := by\n  intro Hilbert _ _ _ small large smallFactor largeFactor inclusion irreducible separablePredual\n  let smallFactor' := smallFactor.toGenerator\n  let largeFactor' := largeFactor.toGenerator\n  obtain ⟨gdelta, dense⟩ := Generator.relative_generation_of_II1_separable_predual\n    small large smallFactor' largeFactor' inclusion irreducible separablePredual\n  refine ⟨Generator.normalizedFactorTrace large largeFactor',\n    Generator.normalizedFactorTrace_isNormalizedTrace large largeFactor', ?_⟩\n  let : TopologicalSpace (unitary large) :=\n    traceTopology large (Generator.normalizedFactorTrace large largeFactor')\n  let unitaryHomeomorph := Generator.FactorTraceUnitary.traceHomeomorph large largeFactor'\n  have locus_eq : relativeGeneratorLocus small large = unitaryHomeomorph.symm ⁻¹'\n      {element : Generator.FactorTraceUnitary large largeFactor' |\n        Generator.wstar ((small : Set (Hilbert →L[ℂ] Hilbert)) ∪\n          {(element.algebra large largeFactor').val}) = large} := by\n    ext element\n    rfl\n  rw [locus_eq]\n  exact ⟨IsGδ.preimage unitaryHomeomorph.symm.continuous gdelta,\n    unitaryHomeomorph.symm.isOpenQuotientMap.dense_preimage_iff.mpr dense⟩\n\nend RelativeGeneration\n\nend\n\nend OAI\n\n\nFILE lean/OAI/MathematicalPhysics/Elasticity/Uniqueness.lean FINAL\ntain ⟨L₁,M₁,L₂,M₂,hL₁,hM₁,hL₂,hM₂,heL₁,heM₁,heL₂,heM₂,hp₁,hp₂,hexL,hexM,hd,K,hK,hout⟩ :=\n    ElasticityBoundaryPacket.admissible_common_exterior hD lam₁ mu₁ lam₂ mu₂ ha₁ ha₂ hDN\n  let l : Fin 2 → X → ℝ := ![L₁,L₂]\n  let m : Fin 2 → X → ℝ := ![M₁,M₂]\n  have hl (q : Fin 2) : ContDiff ℝ (⊤ : ℕ∞) (l q) := by fin_cases q; exact hL₁; exact hL₂\n  have hm (q : Fin 2) : ContDiff ℝ (⊤ : ℕ∞) (m q) := by fin_cases q; exact hM₁; exact hM₂\n  have hp (q : Fin 2) (x : X) : 0 < m q x ∧ 0<3*l q x+2*m q x := by\n    fin_cases q; exact hp₁ x; exact hp₂ x\n  have hcll (q : Fin 2) : ExteriorConstant (smoothOfReal (l q) (hl q)) := by\n    apply exteriorConstant_of_eq_const (hl q) hK 0\n    intro x hx\n    fin_cases q; exact (hout x hx).1; exact (hout x hx).2.2.1\n  have hclm (q : Fin 2) : ExteriorConstant (smoothOfReal (m q) (hm q)) := by\n    apply exteriorConstant_of_eq_const (hm q) hK 1\n    intro x hx\n    fin_cases q; exact (hout x hx).2.1; exact (hout x hx).2.2.2\n  have he (x) (hx : x∉Ω) : l 0 x=l 1 x ∧ m 0 x=m 1 x := ⟨(hexL hx).symm,(hexM hx).symm⟩\n  have hh := dn_global_coefficients_equal hD.1 hD.2.2.1 l m hl hm hp hcll hclm he hd\n  constructor\n  · intro x hx\n    rw [← heL₁ hx,← heL₂ hx]\n    exact congrFun hh.1 x\n  · intro x hx\n    rw [← heM₁ hx,← heM₂ hx]\n    exact congrFun hh.2 x\nend Elasticity\n\nend\nend\ntheorem Elasticity.global_uniqueness : Elasticity.MainClaim := by\n  intro Ω hD lam₁ mu₁ lam₂ mu₂ ha₁ ha₂ hDN\n  exact Elasticity.source_global_uniqueness Ω hD lam₁ mu₁ lam₂ mu₂ ha₁ ha₂ hDN\n\nend OAI\n\n"
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          "source": "META_GREY = \"#888888\"\n\n\ndef apply_figure_style(*, frame=\"open\", font=None, sizes=(8, 7, 6), grid=False):\n    \"\"\"Set matplotlib rcParams for publication-grade output. Call once before plotting.\n\n    This sets mechanics (role-mapped font-size ladder, outward ticks, frameless\n    legends, 300-dpi save, Type-42 embedded fonts) — not a house aesthetic.\n    Frame, font and the size ladder are parameters.\n\n    frame : 'open' (bottom+left spines, default) | 'boxed' (all four) | 'none'\n    font  : sans-serif family name; None = system default sans-serif\n    sizes : (base, secondary, tick) — titles/axis-labels, legend/annotation, ticks\n    grid  : whether to draw axes.grid (default False)\n    \"\"\"\n    import matplotlib as mpl\n    if frame not in (\"open\", \"boxed\", \"none\"):\n        raise ValueError(f\"frame must be 'open'|'boxed'|'none', got {frame!r}\")\n\n    try:\n        import os, sys, glob, matplotlib.font_manager as fm\n        fdir = os.path.join(os.environ.get(\"CONDA_PREFIX\") or sys.prefix, \"fonts\")\n        if os.path.isdir(fdir):\n            known = {f.fname for f in fm.fontManager.ttflist}\n            for f in glob.glob(os.path.join(fdir, \"*.ttf\")):\n                if f not in known:\n                    fm.fontManager.addfont(f)\n    except Exception:\n        pass\n    base, secondary, tick = sizes\n    boxed = (frame == \"boxed\")\n    rc = {\n        \"font.family\": \"sans-serif\",\n        \"font.size\": base,\n        \"axes.labelsize\": base,\n        \"axes.titlesize\": base,\n        \"legend.fontsize\": secondary,\n        \"xtick.labelsize\": tick,\n        \"ytick.labelsize\": tick,\n        \"axes.linewidth\": 0.6,\n        \"xtick.direction\": \"out\", \"ytick.direction\": \"out\",\n        \"xtick.major.size\": 3, \"ytick.major.size\": 3,\n        \"xtick.major.width\": 0.6, \"ytick.major.width\": 0.6,\n        \"axes.spines.top\": boxed, \"axes.spines.right\": boxed,\n        \"axes.spines.left\": frame != \"none\", \"axes.spines.bottom\": frame != \"none\",\n        \"axes.grid\": bool(grid),\n        \"legend.frameon\": False,\n        \"figure.dpi\": 200,\n        \"savefig.dpi\": 300,\n        \"savefig.bbox\": \"tight\",\n        \"axes.titleweight\": \"normal\",\n        \"axes.titlelocation\": \"left\",\n        \"axes.labelweight\": \"normal\",\n        \"lines.linewidth\": 1.2,\n        \"patch.linewidth\": 0.6,\n        \"pdf.fonttype\": 42, \"ps.fonttype\": 42,\n    }\n    if font:\n        rc[\"font.sans-serif\"] = [font, \"DejaVu Sans\"]\n    mpl.rcParams.update(rc)\n\n\ndef set_frame(ax, style=\"open\"):\n    \"\"\"§3: set spine visibility on an existing axes. style ∈ {'open','boxed','none'}.\"\"\"\n    show = {\"open\": (False, False, True, True),\n            \"boxed\": (True, True, True, True),\n            \"none\": (False, False, False, False)}[style]\n    for side, vis in zip((\"top\", \"right\", \"bottom\", \"left\"), show):\n        ax.spines[side].set_visible(vis)\n        if vis:\n            ax.spines[side].set_linewidth(0.6)\n    ax.tick_params(direction=\"out\", length=0 if style == \"none\" else 3, width=0.6)\n\n\ndef panel_letter(ax, letter, dx=-0.18, dy=1.02, case=\"lower\", fontsize=None):\n    \"\"\"§5.7: bold panel letter outside top-left of axes. case ∈ {'lower','upper'}.\"\"\"\n    import matplotlib.pyplot as plt\n    if fontsize is None:\n        fontsize = plt.rcParams.get(\"font.size\", 8) + 1\n    s = letter.lower() if case == \"lower\" else letter.upper()\n    artist = ax.text(dx, dy, s, transform=ax.transAxes,\n                     fontweight=\"bold\", fontsize=fontsize, va=\"bottom\", ha=\"left\")\n    artist.set_gid(\"figure-style-panel-letter\")\n\n\ndef focal_palette(labels, focal, focal_color, other=\"muted\", base_colors=None):\n    \"\"\"§4.2: map labels → colors with the focal series visually dominant.\n\n    other='muted'   — desaturate base_colors (or a default cycle) toward gray\n    other='grey'    — uniform light gray for all non-focal\n    other='ordinal' — non-focal on a single light→dark gray ramp (input order)\n    \"\"\"\n    import matplotlib.colors as mcolors\n    import matplotlib.pyplot as plt\n    focal_set = {focal} if isinstance(focal, str) else set(focal)\n    n = len(labels)\n    if not focal_set & set(labels):\n        raise ValueError(f\"focal {focal!r} not found in labels\")\n    base_colors = [] if base_colors is None else list(base_colors)\n    if not base_colors:\n        base_colors = plt.rcParams[\"axes.prop_cycle\"].by_key().get(\"color\", [\"#444444\"])\n    if not base_colors:\n        base_colors = [\"#444444\"]\n    base_colors = [base_colors[i % len(base_colors)] for i in range(n)]\n    if other == \"grey\":\n        rest = [\"#BCBCBC\"] * n\n    elif other == \"ordinal\":\n        nf = max(1, n - len(focal_set))\n        ramp = [mcolors.to_hex((v, v, v)) for v in\n                ([0.55] if nf == 1 else [0.80 - 0.35 * i / (nf - 1) for i in range(nf)])]\n        rest, k = [], 0\n        for l in labels:\n            rest.append(ramp[min(k, nf - 1)]); k += (l not in focal_set)\n    elif other == \"muted\":\n        def mute(c):\n            r, g, b = mcolors.to_rgb(c)\n            m = (r + g + b) / 3\n            return mcolors.to_hex((0.3 * r + 0.7 * m, 0.3 * g + 0.7 * m, 0.3 * b + 0.7 * m))\n        rest = [mute(c) for c in base_colors]\n    else:\n        raise ValueError(\"other must be 'muted', 'grey', or 'ordinal'\")\n    return [focal_color if l in focal_set else rest[i] for i, l in enumerate(labels)]\n\n\ndef bar_with_points(ax, x, ymat, labels, colors, jitter=0.08, show_points=True,\n                    errorbar=None, point_alpha=0.5, point_size=8):\n    \"\"\"§6.1: bar = mean; optionally overlay raw points or draw an interval.\n\n    colors   : per-label color list (e.g. from focal_palette)\n    x, ymat and labels must have equal lengths; mismatches raise before drawing.\n    Each group must be a nonempty, finite, one-dimensional real numeric array.\n    Requested SD/CI intervals require at least two observations per group.\n    errorbar : None | 'sd' | 'ci95' — drawn only when show_points is False.\n               'ci95' is the t-distribution 95% CI of the mean\n               (half-width t_{0.975,n-1} · s/√n); correct at small n where the\n               z-approximation (1.96·s/√n) is markedly too narrow.\n    \"\"\"\n    import numpy as np\n    x = np.asarray(x)\n    if x.ndim != 1 or not np.issubdtype(x.dtype, np.number):\n        raise TypeError(\n            \"bar_with_points x must be one-dimensional numeric positions; \"\n            \"pass category names through labels\"\n        )\n    ymat = list(ymat)\n    labels = list(labels)\n    if len(x) != len(ymat) or len(x) != len(labels):\n        raise ValueError(\"bar_with_points x, ymat and labels must have equal lengths\")\n    if colors is not None and not isinstance(colors, str):\n        colors = list(colors)\n        if not colors:\n            colors = None\n    if x.dtype.kind not in \"iuf\" or not np.isfinite(x).all():\n        raise ValueError(\"bar_with_points x must contain finite real positions\")\n    for i, y in enumerate(ymat):\n        if np.ma.isMaskedArray(y) and np.ma.getmaskarray(y).any():\n            raise ValueError(f\"bar_with_points group {i} contains masked observations; resolve exclusions explicitly before plotting\")\n    ymat = [np.asarray(y) for y in ymat]\n    for i, y in enumerate(ymat):\n        if y.ndim != 1 or y.size == 0 or y.dtype.kind not in \"iuf\" or not np.isfinite(y).all():\n            raise ValueError(f\"bar_with_points group {i} must be nonempty, one-dimensional and finite real numeric data\")\n    if errorbar and not show_points:\n        if errorbar not in (\"sd\", \"ci95\"):\n            raise ValueError(\"errorbar must be None, 'sd', or 'ci95'\")\n        if any(y.size < 2 for y in ymat):\n            raise ValueError(\"SD/CI cannot be estimated from one observation; use raw points without an interval\")\n    means = np.array([np.mean(y) for y in ymat], float)\n    err = None\n    if errorbar and not show_points:\n        if errorbar == \"sd\":\n            err = np.array([np.std(y, ddof=1) for y in ymat])\n        elif errorbar == \"ci95\":\n            from scipy.stats import t\n            def _hw(y):\n                n = np.asarray(y).size\n                return t.ppf(0.975, n - 1) * np.std(y, ddof=1) / np.sqrt(n)\n            err = np.array([_hw(y) for y in ymat])\n    ax.bar(x, means, color=colors, width=0.7, edgecolor=\"none\",\n           yerr=err, error_kw={\"elinewidth\": 0.8, \"capsize\": 0})\n    if show_points:\n        for xi, ys in zip(x, ymat):\n            ys = np.asarray(ys)\n            if ys.ndim == 1 and ys.size > 0:\n                jit = (np.random.rand(ys.size) - 0.5) * 2 * jitter\n                ax.scatter(np.full(ys.size, xi) + jit, ys, s=point_size, color=\"black\",\n                           alpha=point_alpha, zorder=3, linewidths=0)\n    ax.set_xticks(x); ax.set_xticklabels(labels)\n    return ax\n\n\ndef strip_with_median(ax, groups, values, colors=None, jitter=0.12):\n    \"\"\"§6.1: jittered points + bold horizontal median tick per group.\n\n    groups and values must have equal lengths. Mismatches raise before drawing.\n    colors=None or an empty color sequence supplies default gray; a nonempty\n    color sequence cycles across groups.\n    Each group must be nonempty, one-dimensional and finite real numeric data;\n    resolve missing observations explicitly before plotting.\n    \"\"\"\n    import numpy as np\n    labs = list(groups)\n    values = list(values)\n    if len(labs) != len(values):\n        raise ValueError(\"strip_with_median groups and values must have equal lengths\")\n    for i, y in enumerate(values):\n        if np.ma.isMaskedArray(y) and np.ma.getmaskarray(y).any():\n            raise ValueError(f\"strip_with_median group {i} contains masked observations; resolve exclusions explicitly before plotting\")\n    values = [np.asarray(y) for y in values]\n    for i, y in enumerate(values):\n        if y.ndim != 1 or y.size == 0 or y.dtype.kind not in \"iuf\" or not np.isfinite(y).all():\n            raise ValueError(f\"strip_with_median group {i} must be nonempty, one-dimensional and finite real numeric data\")\n    colors = [] if colors is None else list(colors)\n    if not colors:\n        colors = [\"#444444\"]\n    for i, ys in enumerate(values):\n        c = colors[i % len(colors)]\n        ys = np.asarray(ys)\n        jit = (np.random.rand(ys.size) - 0.5) * 2 * jitter\n        ax.scatter(np.full(ys.size, i) + jit, ys, s=10, color=c, alpha=0.6, linewidths=0, zorder=2)\n        m = np.median(ys)\n        ax.plot([i - 0.22, i + 0.22], [m, m], color=\"black\", lw=1.6, zorder=3)\n    ax.set_xticks(range(len(labs))); ax.set_xticklabels(labs)\n    return ax\n\n\ndef goodness_arrow(ax, text=\"higher = better\", loc=\"upper left\", axis=\"y\", fontsize=None):\n    \"\"\"§3.6: small upright direction-of-goodness cue in the margin.\"\"\"\n    import matplotlib.pyplot as plt\n    if fontsize is None:\n        fontsize = plt.rcParams[\"legend.fontsize\"]\n    pos = {\"upper left\": (0.02, 0.98), \"upper right\": (0.98, 0.98),\n           \"lower left\": (0.02, 0.02), \"lower right\": (0.98, 0.02)}[loc]\n    ha = \"left\" if \"left\" in loc else \"right\"\n    va = \"top\" if \"upper\" in loc else \"bottom\"\n    arrow = \"↑ \" if axis == \"y\" else \"→ \"\n    ax.text(pos[0], pos[1], arrow + text, transform=ax.transAxes,\n            fontsize=fontsize, color=META_GREY, ha=ha, va=va)\n\n\ndef two_tier_label(name, meta):\n    \"\"\"§5: two-line label string (name / metadata). Meta line styled separately by caller.\"\"\"\n    return f\"{name}\\n{meta}\"\n\n\ndef end_of_line_labels(ax, xs, ys, labels, colors=None, dx=0.01, fontsize=None):\n    \"\"\"§6.3 / §7.3: label each line series at its right end instead of a legend box.\n\n    Each label requires a nonempty, matched x/y series. Validate all series\n    before drawing. Nonempty colors cycle; None or empty uses default text color.\n    Inner sequences must support len() and indexing; they are not copied.\n    Nonfinite endpoints are rejected rather than producing an invisible label.\n    \"\"\"\n    import matplotlib.pyplot as plt\n    import math\n    xs, ys, labels = list(xs), list(ys), list(labels)\n    if len(xs) != len(ys) or len(xs) != len(labels):\n        raise ValueError(\"end_of_line_labels xs, ys and labels must have equal lengths\")\n    if any(len(x) == 0 or len(x) != len(y) for x, y in zip(xs, ys)):\n        raise ValueError(\"end_of_line_labels each x/y series must be nonempty and have equal lengths\")\n    if any(not math.isfinite(x[-1]) or not math.isfinite(y[-1]) for x, y in zip(xs, ys)):\n        raise ValueError(\"end_of_line_labels endpoints must be finite; choose valid endpoints explicitly\")\n    if fontsize is None:\n        fontsize = plt.rcParams[\"font.size\"]\n    colors = [] if colors is None else list(colors)\n    if not colors:\n        colors = [None]\n    span = ax.get_xlim()[1] - ax.get_xlim()[0]\n    for i, (x, y, lab) in enumerate(zip(xs, ys, labels)):\n        c = colors[i % len(colors)]\n        ax.text(x[-1] + dx * span, y[-1], lab, color=c, va=\"center\", ha=\"left\", fontsize=fontsize)\n\n\ndef panel_crops(fig, dpi=None, pad_px=6, bbox_inches=None, pad_inches=None):\n    \"\"\"§9.2: pixel-space crop boxes for each lettered panel in the SAVED PNG.\n\n    Returns ``{letter: (x0, y0, x1, y1)}`` in image-space pixels (origin\n    top-left, matching PIL's ``Image.crop``). Panels are detected as bold\n    single-character ``Text``\n    objects placed by :func:`panel_letter`; each panel's crop is its axes'\n    tightbbox mapped into the saved file's pixel space, padded by ``pad_px``.\n    For §3.4 composites (abutting subplots sharing an axis, letter on the\n    leftmost only) the crop unions in letterless ``sharex``/``sharey`` siblings\n    on the same grid row/col so the whole composite is covered. When no axes\n    carries a panel letter (standalone plot, or a figure-composer sub-agent),\n    falls back to one crop per axes keyed by index.\n\n    ``bbox_inches`` mirrors ``Figure.savefig`` semantics: ``None`` means\n    *consult rcParams* (so under :func:`apply_figure_style` it resolves to\n    ``'tight'``); pass an explicit ``Bbox`` only if you saved with one. The\n    boxes are clamped to the saved image extent; panels with no intersection\n    are omitted.\n\n        >>> from PIL import Image\n        >>> fig.savefig(\"fig.png\")  # bbox_inches='tight' via rcParams\n        >>> with Image.open(\"fig.png\") as saved:\n        ...     for letter, box in panel_crops(fig).items():\n        ...         saved.crop(box).save(f\"panel_{letter}.png\")\n\n    Match savefig's DPI, bbox_inches and pad_inches when overriding defaults.\n    Returned boxes are pixels, never savefig bbox_inches (inches). Passing them\n    back to savefig as inches can allocate an enormous raster.\n    \"\"\"\n    import matplotlib as mpl\n    import matplotlib.text\n    from matplotlib.backends.backend_agg import FigureCanvasAgg\n    if isinstance(pad_px, bool) or not isinstance(pad_px, int) or pad_px < 0:\n        raise TypeError(\"panel_crops pad_px must be a non-negative integer\")\n    if dpi is None:\n        dpi = mpl.rcParams.get(\"savefig.dpi\", fig.dpi)\n        if dpi == \"figure\":\n            dpi = fig.dpi\n    dpi = float(dpi)\n    if bbox_inches is None:\n        bbox_inches = mpl.rcParams.get(\"savefig.bbox\")\n    # The Open-Science notebook captures and closes pyplot figures after every\n    # cell. Matplotlib 3.11 decouples a closed Figure from its backend by\n    # replacing FigureCanvasAgg with FigureCanvasBase, while the persistent\n    # notebook namespace can still retain the Figure itself. Reattach the\n    # headless canvas before asking for saved-image geometry so this helper is\n    # safe in a later QA cell as well as the producer cell.\n    if not hasattr(fig.canvas, \"get_renderer\"):\n        FigureCanvasAgg(fig)\n    fig.canvas.draw()\n    r = fig.canvas.get_renderer()\n\n    if bbox_inches == \"tight\":\n        if pad_inches is None:\n            pad_inches = mpl.rcParams.get(\"savefig.pad_inches\", 0.1)\n        tb = fig.get_tightbbox(r).padded(pad_inches)\n        ox_in, oy_in = tb.x0, tb.y0\n        W_in, H_in = tb.width, tb.height\n    elif isinstance(bbox_inches, mpl.transforms.BboxBase):\n        ox_in, oy_in = bbox_inches.x0, bbox_inches.y0\n        W_in, H_in = bbox_inches.width, bbox_inches.height\n    elif bbox_inches is None:\n        ox_in, oy_in = 0.0, 0.0\n        W_in, H_in = fig.get_size_inches()\n    else:\n        raise TypeError(\"panel_crops bbox_inches must be None, 'tight', or a matplotlib Bbox\")\n    W_px, H_px = int(round(W_in * dpi)), int(round(H_in * dpi))\n    lettered = {}\n    seen_letters = set()\n    for ax in fig.axes:\n        for t in ax.findobj(matplotlib.text.Text):\n            s = (t.get_text() or \"\").strip()\n            if t.get_gid() == \"figure-style-panel-letter\":\n                if s in seen_letters:\n                    raise ValueError(f\"Duplicate panel letter: {s!r}\")\n                seen_letters.add(s)\n                lettered[ax] = s\n                break\n\n\n\n    if not lettered:\n        lettered = {ax: str(i) for i, ax in enumerate(fig.axes)}\n    out = {}\n    for ax, letter in lettered.items():\n        bbs = [ax.get_tightbbox(r)]\n\n\n\n\n        ss = ax.get_subplotspec()\n        for sib in fig.axes:\n            if sib is ax or sib in lettered:\n                continue\n            ssib = sib.get_subplotspec()\n            same_row = ss is None or ssib is None or ss.rowspan == ssib.rowspan\n            same_col = ss is None or ssib is None or ss.colspan == ssib.colspan\n            if ((ax.get_shared_y_axes().joined(ax, sib) and same_row)\n                    or (ax.get_shared_x_axes().joined(ax, sib) and same_col)):\n                bbs.append(sib.get_tightbbox(r))\n        bb = mpl.transforms.Bbox.union(bbs)\n\n        bx0 = (bb.x0 / fig.dpi - ox_in) * dpi\n        bx1 = (bb.x1 / fig.dpi - ox_in) * dpi\n        by0 = H_px - (bb.y1 / fig.dpi - oy_in) * dpi\n        by1 = H_px - (bb.y0 / fig.dpi - oy_in) * dpi\n        box = (\n            min(max(int(bx0) - pad_px, 0), W_px),\n            min(max(int(by0) - pad_px, 0), H_px),\n            max(min(int(bx1) + pad_px, W_px), 0),\n            max(min(int(by1) + pad_px, H_px), 0),\n        )\n        if box[0] < box[2] and box[1] < box[3]:\n            out[letter] = box\n    return out\n",
          "sourceDigest": "8020d409bfb60f0c8dfa43d0dfe7bb97f7458e1a3908fefbcb6e4551e0feae4e"
        }
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      "helperEvidenceStatus": {
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      "environmentManifest": {
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        "captureKind": "completed-run",
        "capturedAt": "2026-10-08T10:42:58.392Z",
        "installedInventory": {
          "capturedAt": "2026-10-08T07:31:03.005Z",
          "source": "cache-reused",
          "validation": "best-effort"
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        "stdout": "\nFILE lean/OAI/NumberTheory/DukePrimeDegree/MainUnconditional.lean \nIMPORTS ['import OAI.NumberTheory.DukePrimeDegree.MainStarkDischarged', 'import OAI.NumberTheory.DukePrimeDegree.Entropy.RegularEntropyDischarge'] \nTAIL\n⟨(ν : Measure (X (d+1))), hν, ?_, htight⟩\n  exact (weakProbabilityConvergence_iff\n    (fun i => packetProbability (M i) (σ i) (hn i)\n      (hprime.odd_of_ne_two (by omega))) ν).mpr hconv\n\n/-- The complete packet measures converge weakly to any specified Haar\nprobability, in the formulation by bounded continuous test functions. -/\ntheorem prime_degree_packet_measures_tendsto_haar_unconditional\n    (d : ℕ) (hprime : Nat.Prime (d+1)) (hfive : 5 ≤ d+1)\n    (K : ℕ → Type u) [∀ i, Field (K i)] [∀ i, NumberField (K i)]\n    [∀ i, NumberField.IsTotallyReal (K i)]\n    (M : ∀ i, PrimeDegreePackets.FullLattice (d+1) (K i))\n    (σ : ∀ i, OrderedEmbeddings (d+1) (K i))\n    (hn : ∀ i, Module.finrank ℚ (K i)=d+1)\n    (hD : Tendsto (fun i => PrimeDegreePackets.multiplierDiscriminant (M i)) atTop atTop)\n    (m : ProbabilityMeasure (X (d+1)))\n    (hm : IsHaarProbability (m : Measure (X (d+1)))) :\n    WeakProbabilityConvergence (fun i => packetMeasure (M i) (σ i))\n      (m : Measure (X (d+1))) := by\n  exact (weakProbabilityConvergence_iff\n    (fun i => packetProbability (M i) (σ i) (hn i)\n      (hprime.odd_of_ne_two (by omega))) m).mpr\n    (prime_degree_packets_tendsto_haar_unconditional d hprime hfive K M σ hn hD m hm)\n\nend DukePrimeDegree\n\nend OAI\n\n\nFILE lean/OAI/NumberTheory/TotientAsymptotic/UnconditionalMain.lean \nIMPORTS ['import OAI.NumberTheory.TotientAsymptotic.FordInputs', 'import OAI.NumberTheory.TotientAsymptotic.PPTFixedBandLimit'] \nTAIL\nnus ∧ 0<cPlus ∧\n        (∀ s∈Set.Ico (0 : ℝ) 1, cMinus≤A (fk k) s ∧ A (fk k) s≤cPlus) ∧\n        Tendsto (fun x => N k x/(tupleNormalization x*A (fk k) (theta x))) atTop (nhds 1) ∧\n        ∀ᶠ x : ℝ in atTop, (cMinus/cPlus)*V x≤N k x ∧ N k x≤V x) ∧\n    ((¬∃ d : ℕ, IsTotient d ∧ k*d<ell d) →\n      (∀ x : ℝ, 0<x → N k x=0) ∧ (∀ s∈Set.Ico (0 : ℝ) 1, A (fk k) s=0)) :=\n  conditional_companion_theorem_of_structure fordScaleBounds extractedStructureInput primeNumberTheoremInput fordUnitPrimeBoxInput\n    fordRenewalInput mertensProductInput fordLemma26Input fordLemma51Input fordCoordinateConcentrationInput pptBoundedFiberPropagation k hk\n\ntheorem weighted_totient_one_two\n    :\n    ∀ k∈({1,2} : Finset ℕ), ∃ c : ℝ, 0<c ∧\n      ∀ s∈Set.Ico (0 : ℝ) 1, c≤A (fk k) s :=\n  companion_one_two_without_seed_input_of_structure fordScaleBounds extractedStructureInput primeNumberTheoremInput fordUnitPrimeBoxInput\n    fordRenewalInput mertensProductInput fordLemma26Input fordLemma51Input fordCoordinateConcentrationInput pptBoundedFiberPropagation\n\ntheorem coefficient_nonnegative (H : ℕ) {s : ℝ}\n    (hs : s ∈ Set.Ico (0 : ℝ) 1) : 0 ≤ AH H (fun _ => 1) s :=\n  AH_nonneg H hs (fun _ => 1) (fun _ _ => ⟨zero_le_one, le_rfl⟩)\n\nend TotientAsymptotic\n\nend\n\nend OAI\n\n\nFILE lean/OAI/NumberTheory/TotientAsymptotic/Main.lean \nIMPORTS ['import OAI.NumberTheory.TotientAsymptotic.FailedRowCountBound', 'import OAI.NumberTheory.TotientAsymptotic.PrimeProductValueMass', 'import OAI.NumberTheory.TotientAsymptotic.FailedRowTotalMass', 'import OAI.NumberTheory.TotientAsymptotic.CoordinateDeviationCount', 'import OAI.NumberTheory.TotientAsymptotic.ExceptionalRowMass', 'import OAI.NumberTheory.TotientAsymptotic.CofactorRowDeviation', 'import OAI.NumberTheory.TotientAsymptotic.SmoothTotientMass', 'import OAI.NumberTheory.TotientAsymptotic.PositiveGridCount', 'import OAI.NumberTheory.TotientAsymptotic.CoordinateActivePrefix', 'import OAI.NumberTheory.TotientAsymptotic.UniformRegularity', 'import OAI.NumberTheory.TotientAsymptotic.BootstrapRealCount', 'import OAI.NumberTheory.TotientAsymptotic.BandUnionCount', 'import OAI.NumberTheory.TotientAsymptotic.BootstrapResidualSize', 'import OAI.NumberTheory.TotientAsymptotic.BootstrapGoodValues', 'import OAI.NumberTheory.TotientAsymptotic.BootstrapViolation', 'import OAI.NumberTheory.TotientAsymptotic.TwoCoordinateCount', 'import OAI.NumberTheory.TotientAsymptotic.ExceptionalValueCount', 'import OAI.NumberTheory.TotientAsymptotic.TwoBandCount', 'import OAI.NumberTheory.TotientAsymptotic.PartitionedBandMass', 'import OAI.NumberTheory.TotientAsymptotic.CoarseTotientBounds', 'import OAI.NumberTheory.TotientAsymptotic.PreimageLoglogBound', 'import OAI.NumberTheory.TotientAsymptotic.PrimeExtensionCoverage', 'import OAI.NumberTheory.TotientAsymptotic.SmoothTotientCount', 'import OAI.NumberTheory.TotientAsymptotic.FordBandExponent', 'import OAI.NumberTheory.TotientAsymptotic.FordTerminalMass', 'import OAI.NumberTheory.TotientAsymptotic.FordStateTransfer', 'import OAI.NumberTheory.TotientAsymptotic.SquarefreeRoughMass', 'import OAI.NumberTheory.TotientAsymptotic.BandStateMass', 'import OAI.NumberTheory.TotientAsymptotic.SmoothTerminalBound', 'import OAI.NumberTheory.TotientAsymptotic.TerminalLogBudget', 'import OAI.NumberTheory.TotientAsymptotic.NormalTerminalFactors', 'import OAI.NumberTheory.TotientAsymptotic.CollisionBandMass', 'import OAI.NumberTheory.TotientAsymptotic.NormalBandAllocation', 'import OAI.NumberTheory.TotientAsymptotic.CommonLargestPrime', 'import OAI.NumberTheory.TotientAsymptotic.DistinctPrimeTail', 'import OAI.NumberTheory.TotientAsymptotic.PrimeFactorBands', 'import OAI.NumberTheory.TotientAsymptotic.PrimeReciprocalTails', 'import OAI.NumberTheory.TotientAsymptotic.UniformPrimeTripleBound', 'import OAI.NumberTheory.TotientAsymptotic.UniformPrimePairBound', 'import OAI.NumberTheory.TotientAsymptotic.NormalityIntervalTail', 'import OAI.NumberTheory.TotientAsymptotic.FullOmegaMoment', 'import OAI.NumberTheory.TotientAsymptotic.FordInputs', 'import OAI.NumberTheory.TotientAsymptotic.MertensProduct', 'import OAI.NumberTheory.TotientAsymptotic.PrimeNumberTheorem', 'import OAI.NumberTheory.TotientAsymptotic.SeedConsequences', 'import OAI.NumberTheory.TotientAsymptotic.SeedMinimum', 'import OAI.NumberTheory.TotientAsymptotic.ConditionalCompanion', 'import OAI.NumberTheory.TotientAsymptotic.WeightedFiber', 'import OAI.NumberTheory.TotientAsymptotic.GoodDiscard', 'import OAI.NumberTheory.TotientAsymptotic.TailBounds', 'import OAI.NumberTheory.TotientAsymptotic.Phase', 'import OAI.NumberTheory.TotientAsymptotic.Companion', 'import OAI.NumberTheory.TotientAsymptotic.PublishedFiber', 'import OAI.NumberTheory.TotientAsymptotic.Normalization', 'import OAI.NumberTheory.TotientAsymptotic.Scaling', 'import OAI.NumberTheory.TotientAsymptotic.FiniteMap', 'import OAI.NumberTheory.TotientAsymptotic.BandComparison', 'import OAI.NumberTheory.TotientAsymptotic.Mass', 'import OAI.NumberTheory.TotientAsymptotic.BandExtraction', 'import OAI.NumberTheory.TotientAsymptotic.SimplexVolume', 'import OAI.NumberTheory.TotientAsymptotic.TupleRecovery', 'import OAI.NumberTheory.TotientAsymptotic.PublishedComparison', 'import OAI.NumberTheory.TotientAsymptotic.NormalityAlignment', 'import OAI.NumberTheory.TotientAsymptotic.CollisionAlgebra', 'import OAI.NumberTheory.TotientAsymptotic.IntersectionLimit', 'import OAI.NumberTheory.TotientAsymptotic.UnionVolume', 'import OAI.NumberTheory.TotientAsymptotic.CollisionCount', 'import OAI.NumberTheory.TotientAsymptotic.CollisionSlack', 'import OAI.NumberTheory.TotientAsymptotic.UniformPrefactor', 'import OAI.NumberTheory.TotientAsymptotic.NormalitySize', 'import OAI.NumberTheory.TotientAsymptotic.CollisionCost', 'import OAI.NumberTheory.TotientAsymptotic.PrefixLower', 'import OAI.NumberTheory.TotientAsymptotic.PrimeBox', 'import OAI.NumberTheory.TotientAsymptotic.OrderedAlignment', 'import OAI.NumberTheory.TotientAsymptotic.CollisionTail', 'import OAI.NumberTheory.TotientAsymptotic.PrimeGrid', 'import OAI.NumberTheory.TotientAsymptotic.GridVolume', 'import OAI.NumberTheory.TotientAsymptotic.BandPrimeError', 'import OAI.NumberTheory.TotientAsymptotic.RenewalLower', 'import OAI.NumberTheory.TotientAsymptotic.RegionLower', 'import OAI.NumberTheory.TotientAsymptotic.CofactorMass', 'import OAI.NumberTheory.TotientAsymptotic.SimplexShell', 'import OAI.NumberTheory.TotientAsymptotic.CubeBoundary', 'import OAI.NumberTheory.TotientAsymptotic.ShellCost', 'import OAI.NumberTheory.TotientAsymptotic.CofactorEnvelope', 'import OAI.NumberTheory.TotientAsymptotic.TailCofactor', 'import OAI.NumberTheory.TotientAsymptotic.RenewalMagnitude', 'import OAI.NumberTheory.TotientAsymptotic.CofactorGrowth', 'import OAI.NumberTheory.TotientAsymptotic.CofactorShellDecay', 'import OAI.NumberTheory.TotientAsymptotic.ExtractTail', 'import OAI.NumberTheory.TotientAsymptotic.MassTailSupport', 'import OAI.NumberTheory.TotientAsymptotic.AttachTail', 'import OAI.NumberTheory.TotientAsymptotic.RemainderSplit', 'import OAI.NumberTheory.TotientAsymptotic.WitnessPrefix', 'import OAI.NumberTheory.TotientAsymptotic.PrefixRelaxation', 'import OAI.NumberTheory.TotientAsymptotic.PerturbationCost', 'import OAI.NumberTheory.TotientAsymptotic.PrimeBoxGeometry', 'import OAI.NumberTheory.TotientAsymptotic.BandGrid', 'import OAI.NumberTheory.TotientAsymptotic.GridApproximation', 'import OAI.NumberTheory.TotientAsymptotic.MassMajorant', 'import OAI.NumberTheory.TotientAsymptotic.DyadicSuffix', 'import OAI.NumberTheory.TotientAsymptotic.CollisionDyadicCount', 'import OAI.NumberTheory.TotientAsymptotic.FordSimplexEmbedding', 'import OAI.NumberTheory.TotientAsymptotic.PublishedConcentration', 'import OAI.NumberTheory.TotientAsymptotic.CenterComparison', 'import OAI.NumberTheory.TotientAsymptotic.TruncatedConcentration', 'import OAI.NumberTheory.TotientAsymptotic.ConcentrationTail', 'import OAI.NumberTheory.TotientAsymptotic.SimplexScaling', 'import OAI.NumberTheory.TotientAsymptotic.EnlargementCost', 'import OAI.NumberTheory.TotientAsymptotic.EnlargementTail', 'import OAI.NumberTheory.TotientAsymptotic.TailBoxAggregation', 'import OAI.NumberTheory.TotientAsymptotic.TailBoxCoordinates', 'import OAI.NumberTheory.TotientAsymptotic.SimplexCube', 'import OAI.NumberTheory.TotientAsymptotic.BasicCube', 'import OAI.NumberTheory.TotientAsymptotic.FullPrimeGrid', 'import OAI.NumberTheory.TotientAsymptotic.MassUpper', 'import OAI.NumberTheory.TotientAsymptotic.UnbandedLower', 'import OAI.NumberTheory.TotientAsymptotic.ScaledConcentration', 'import OAI.NumberTheory.TotientAsymptotic.PhaseBandTransfer', 'import OAI.NumberTheory.TotientAsymptotic.FullBoxConcentration', 'import OAI.NumberTheory.TotientAsymptotic.TailCutScale', 'import OAI.NumberTheory.TotientAsymptotic.NormalizedConcentration', 'import OAI.NumberTheory.TotientAsymptotic.TailArray', 'import OAI.NumberTheory.TotientAsymptotic.JoinedSimplex', 'import OAI.NumberTheory.TotientAsymptotic.PointEnclosure', 'import OAI.NumberTheory.TotientAsymptotic.ThickenedWitness', 'import OAI.NumberTheory.TotientAsymptotic.TailEncoding', 'import OAI.NumberTheory.TotientAsymptotic.WitnessAggregation', 'import OAI.NumberTheory.TotientAsymptotic.WitnessGrid', 'import OAI.NumberTheory.TotientAsymptotic.CofactorWitnessAggregation', 'import OAI.NumberTheory.TotientAsymptotic.BandRemoval', 'import OAI.NumberTheory.TotientAsymptotic.WitnessUnionLoss', 'import OAI.NumberTheory.TotientAsymptotic.AdditiveShell', 'import OAI.NumberTheory.TotientAsymptotic.AdditiveEnclosure', 'import OAI.NumberTheory.TotientAsymptotic.AdditiveCost', 'import OAI.NumberTheory.TotientAsymptotic.AdditiveShellUnion', 'import OAI.NumberTheory.TotientAsymptotic.NormalizedShell', 'import OAI.NumberTheory.TotientAsymptotic.PerturbationEnclosure', 'import OAI.NumberTheory.TotientAsymptotic.PerturbationRemoval', 'import OAI.NumberTheory.TotientAsymptotic.UnionPerturbation', 'import OAI.NumberTheory.TotientAsymptotic.VolumeComparison', 'import OAI.NumberTheory.TotientAsymptotic.MassVolumeApproximation', 'import OAI.NumberTheory.TotientAsymptotic.CollisionScale', 'import OAI.NumberTheory.TotientAsymptotic.CollisionCutoff', 'import OAI.NumberTheory.TotientAsymptotic.ResidualAlignment', 'import OAI.NumberTheory.TotientAsymptotic.SeparatedBands', 'import OAI.NumberTheory.TotientAsymptotic.SuffixFactorization', 'import OAI.NumberTheory.TotientAsymptotic.CollisionTuples', 'import OAI.NumberTheory.TotientAsymptotic.CollisionIndices', 'import OAI.NumberTheory.TotientAsymptotic.TupleCounting', 'import OAI.NumberTheory.TotientAsymptotic.LargestPrimeCount', 'import OAI.NumberTheory.TotientAsymptotic.TupleMassBound', 'import OAI.NumberTheory.TotientAsymptotic.SmoothSuffix', 'import OAI.NumberTheory.TotientAsymptotic.ResidualCutoff', 'import OAI.NumberTheory.TotientAsymptotic.WitnessCollision', 'import OAI.NumberTheory.TotientAsymptotic.TupleCoefficient', 'import OAI.NumberTheory.TotientAsymptotic.GoodTuples', 'import OAI.NumberTheory.TotientAsymptotic.StrictSlackDiscard', 'import OAI.NumberTheory.TotientAsymptotic.FordCollisionCutoffs', 'import OAI.NumberTheory.TotientAsymptotic.NormalPrimeBands', 'import OAI.NumberTheory.TotientAsymptotic.SurvivingCollision', 'import OAI.NumberTheory.TotientAsymptotic.ResidualTupleRecovery', 'import OAI.NumberTheory.TotientAsymptotic.ChosenCollision', 'import OAI.NumberTheory.TotientAsymptotic.ActualComparison', 'import OAI.NumberTheory.TotientAsymptotic.ComparisonSizes', 'import OAI.NumberTheory.TotientAsymptotic.ComparisonLargePart', 'import OAI.NumberTheory.TotientAsymptotic.CollisionSuffixSize', 'import OAI.NumberTheory.TotientAsymptotic.CollisionDyadicSize', 'import OAI.NumberTheory.TotientAsymptotic.CollisionCanceled', 'import OAI.NumberTheory.TotientAsymptotic.SurvivingComparison', 'import OAI.NumberTheory.TotientAsymptotic.CollisionGrid', 'import OAI.NumberTheory.TotientAsymptotic.CollisionLogBudget', 'import OAI.NumberTheory.TotientAsymptotic.NormalFactorBands', 'import OAI.NumberTheory.TotientAsymptotic.SurvivingBands', 'import OAI.NumberTheory.TotientAsymptotic.SurvivingAlignment', 'import OAI.NumberTheory.TotientAsymptotic.SurvivingBlockSize', 'import OAI.NumberTheory.TotientAsymptotic.CollisionGridScale', 'import OAI.NumberTheory.TotientAsymptotic.ComparisonErrorBudget', 'import OAI.NumberTheory.TotientAsymptotic.CollisionGridCount', 'import OAI.NumberTheory.TotientAsymptotic.SmoothResidualMass', 'import OAI.NumberTheory.TotientAsymptotic.CanceledPrimeMass', 'import OAI.NumberTheory.TotientAsymptotic.FordCostBudget', 'import OAI.NumberTheory.TotientAsymptotic.CollisionHeadSize', 'import OAI.NumberTheory.TotientAsymptotic.NormalHeadBand', 'import OAI.NumberTheory.TotientAsymptotic.CollisionNormalityDomain', 'import OAI.NumberTheory.TotientAsymptotic.NormalizedComparison', 'import OAI.NumberTheory.TotientAsymptotic.PairedComparisonGrid', 'import OAI.NumberTheory.TotientAsymptotic.CollisionFactorHeights', 'import OAI.NumberTheory.TotientAsymptotic.CollisionEndpointDomain', 'import OAI.NumberTheory.TotientAsymptotic.SurvivingHeights', 'import OAI.NumberTheory.TotientAsymptotic.SurvivingGridGeometry', 'import OAI.NumberTheory.TotientAsymptotic.ActualSurvivingGeometry', 'import OAI.NumberTheory.TotientAsymptotic.ComparisonGridFromHeights', 'import OAI.NumberTheory.TotientAsymptotic.HeightAlignment', 'import OAI.NumberTheory.TotientAsymptotic.ActualComparisonCutoffs', 'import OAI.NumberTheory.TotientAsymptotic.SurvivingSlack', 'import OAI.NumberTheory.TotientAsymptotic.ComparisonExponentSaving', 'import OAI.NumberTheory.TotientAsymptotic.CollisionHeadCount', 'import OAI.NumberTheory.TotientAsymptotic.AllCollisionCount', 'import OAI.NumberTheory.TotientAsymptotic.NormalityDiscard', 'import OAI.NumberTheory.TotientAsymptotic.OmegaReciprocal', 'import OAI.NumberTheory.TotientAsymptotic.OmegaDiscard', 'import OAI.NumberTheory.TotientAsymptotic.UnconditionalMain'] \nTAIL\nadSize\nimport OAI.NumberTheory.TotientAsymptotic.NormalHeadBand\nimport OAI.NumberTheory.TotientAsymptotic.CollisionNormalityDomain\nimport OAI.NumberTheory.TotientAsymptotic.NormalizedComparison\nimport OAI.NumberTheory.TotientAsymptotic.PairedComparisonGrid\nimport OAI.NumberTheory.TotientAsymptotic.CollisionFactorHeights\nimport OAI.NumberTheory.TotientAsymptotic.CollisionEndpointDomain\nimport OAI.NumberTheory.TotientAsymptotic.SurvivingHeights\nimport OAI.NumberTheory.TotientAsymptotic.SurvivingGridGeometry\nimport OAI.NumberTheory.TotientAsymptotic.ActualSurvivingGeometry\nimport OAI.NumberTheory.TotientAsymptotic.ComparisonGridFromHeights\nimport OAI.NumberTheory.TotientAsymptotic.HeightAlignment\nimport OAI.NumberTheory.TotientAsymptotic.ActualComparisonCutoffs\nimport OAI.NumberTheory.TotientAsymptotic.SurvivingSlack\nimport OAI.NumberTheory.TotientAsymptotic.ComparisonExponentSaving\n\nimport OAI.NumberTheory.TotientAsymptotic.CollisionHeadCount\n\nimport OAI.NumberTheory.TotientAsymptotic.AllCollisionCount\n\nimport OAI.NumberTheory.TotientAsymptotic.NormalityDiscard\nimport OAI.NumberTheory.TotientAsymptotic.OmegaReciprocal\nimport OAI.NumberTheory.TotientAsymptotic.OmegaDiscard\n\nimport OAI.NumberTheory.TotientAsymptotic.UnconditionalMain\n\n\nFILE lean/OAI/Combinatorics/TriangleFace/Main.lean \nIMPORTS ['import OAI.Combinatorics.TriangleFace.Faces', 'import OAI.Combinatorics.TriangleFace.MatchingImage'] \nTAIL\ne198.image_minFace\n      (localExtension G c v hc).toAffineMap (localExtension_injective G c v hc)\n      (convex_convexHull ℝ (Set.range G.indicator))\n      (convex_convexHull ℝ (Set.range (triangleExpansion G c v).indicator)) hex hx).symm\n\ntheorem triangle_expansion_face {V : Type uV} {E : Type uE}\n    [Fintype V] [Fintype E] [DecidableEq V] [DecidableEq E] (G : LooplessGraph V E) (v : V)\n    (hdeg : Fintype.card {e : E // G.Incident v e} = 3)\n    (_hmatching : Nonempty G.Matching) :\n    ∃ (c : E → Fin 3) (hc : LocalLabels G v c),\n      let H := triangleExpansion G c v\n      let f := localExtension G c v hc\n      let Q := {y | y ∈ H.polytope ∧ H.cutMass y (triangleVertices v) = 1}\n      f '' G.polytope = Q ∧\n      Convex ℝ Q ∧ IsExtreme ℝ H.polytope Q ∧\n      (∀ x, oldRestriction (f x) = x) ∧\n      (∀ y ∈ Q, f (oldRestriction y) = y) ∧\n      (∀ x e, f x (Sum.inl e) = x e) ∧\n      (∀ x i, f x (Sum.inr i) = x (localEdge G c v hc i)) ∧\n      (∀ x, (∀ e, x e = (1/3 : ℝ)) → ∀ e, f x e = (1/3 : ℝ)) ∧\n      (∀ x ∈ G.polytope, CanonicalFace198.minFace H.polytope (f x) = f '' CanonicalFace198.minFace G.polytope x) := by\n  obtain ⟨c,hc⟩ := labels_of_degree_three G v hdeg\n  exact ⟨c,hc,local_face G c v hc⟩\n\nend TriangleFace198\nend\n\nend OAI\n\n\nFILE lean/OAI/Probability/SLE/LowerPositivity.lean \nIMPORTS ['import Mathlib', 'import OAI.Probability.SLE.LowerRestartConditionalAllKappa'] \nTAIL\nexConjugate\nopen Set Filter Metric MeasureTheory Complex\nopen Set Filter MeasureTheory ProbabilityTheory Classical\nopen scoped Topology ENNReal NNReal\nopen Set Filter MeasureTheory ProbabilityTheory Metric Classical\nopen Set Filter\nopen scoped Topology NNReal ENNReal ComplexConjugate\nopen scoped ENNReal NNReal\nnoncomputable section\nopen Set MeasureTheory ProbabilityTheory Filter\nopen scoped ENNReal NNReal Topology\nsection\nopen Set Filter MeasureTheory\nopen scoped Topology ENNReal NNReal\n\nnamespace SLEExactGauge\n \ndef segment (γ : ℝ≥0 → ℂ) (s t : ℝ≥0) : Set ℂ := γ '' Icc s t\nuniverse v_lower\n \n\ndef SourceLowerMainTarget : Prop :=\n  ∀ (Ω : Type v_lower) (_ : MeasurableSpace Ω) (P : Measure Ω), IsProbabilityMeasure P →\n  ∀ κ : ℝ, 0 < κ → κ < ((8 : ℕ) : ℝ) →\n  ∀ (γ : Ω → ℝ≥0 → ℂ), IsOrdinaryChordalSLE κ γ P →\n  ∀ h : ℝ → ℝ, IsGauge h → HasSmallRadiusFormula h (hFormula κ) →\n    ∀ᵐ ω ∂P, ∀ s t : ℝ≥0, 0 < s → s < t →\n      0 < hausdorffGauge h (segment (γ ω) s t)\nend SLEExactGauge\n\nnamespace SLEExactGauge\ntheorem sourceLowerMain_proved : SourceLowerMainTarget := by\n  intro Ω m P hP κ hκ hκ8 γ hSLE h hh hf\n  let : IsProbabilityMeasure P := hP\n  exact source_lower_universe hκ hκ8 hSLE h hh hf\nend SLEExactGauge\nend\n\n \nend\nend\nend\n\nend OAI\n\n",
        "stderr": "",
        "traceback": "",
        "plain": [
          "\nFILE lean/OAI/NumberTheory/DukePrimeDegree/MainUnconditional.lean \nIMPORTS ['import OAI.NumberTheory.DukePrimeDegree.MainStarkDischarged', 'import OAI.NumberTheory.DukePrimeDegree.Entropy.RegularEntropyDischarge'] \nTAIL\n⟨(ν : Measure (X (d+1))), hν, ?_, htight⟩\n  exact (weakProbabilityConvergence_iff\n    (fun i => packetProbability (M i) (σ i) (hn i)\n      (hprime.odd_of_ne_two (by omega))) ν).mpr hconv\n\n/-- The complete packet measures converge weakly to any specified Haar\nprobability, in the formulation by bounded continuous test functions. -/\ntheorem prime_degree_packet_measures_tendsto_haar_unconditional\n    (d : ℕ) (hprime : Nat.Prime (d+1)) (hfive : 5 ≤ d+1)\n    (K : ℕ → Type u) [∀ i, Field (K i)] [∀ i, NumberField (K i)]\n    [∀ i, NumberField.IsTotallyReal (K i)]\n    (M : ∀ i, PrimeDegreePackets.FullLattice (d+1) (K i))\n    (σ : ∀ i, OrderedEmbeddings (d+1) (K i))\n    (hn : ∀ i, Module.finrank ℚ (K i)=d+1)\n    (hD : Tendsto (fun i => PrimeDegreePackets.multiplierDiscriminant (M i)) atTop atTop)\n    (m : ProbabilityMeasure (X (d+1)))\n    (hm : IsHaarProbability (m : Measure (X (d+1)))) :\n    WeakProbabilityConvergence (fun i => packetMeasure (M i) (σ i))\n      (m : Measure (X (d+1))) := by\n  exact (weakProbabilityConvergence_iff\n    (fun i => packetProbability (M i) (σ i) (hn i)\n      (hprime.odd_of_ne_two (by omega))) m).mpr\n    (prime_degree_packets_tendsto_haar_unconditional d hprime hfive K M σ hn hD m hm)\n\nend DukePrimeDegree\n\nend OAI\n\n\nFILE lean/OAI/NumberTheory/TotientAsymptotic/UnconditionalMain.lean \nIMPORTS ['import OAI.NumberTheory.TotientAsymptotic.FordInputs', 'import OAI.NumberTheory.TotientAsymptotic.PPTFixedBandLimit'] \nTAIL\nnus ∧ 0<cPlus ∧\n        (∀ s∈Set.Ico (0 : ℝ) 1, cMinus≤A (fk k) s ∧ A (fk k) s≤cPlus) ∧\n        Tendsto (fun x => N k x/(tupleNormalization x*A (fk k) (theta x))) atTop (nhds 1) ∧\n        ∀ᶠ x : ℝ in atTop, (cMinus/cPlus)*V x≤N k x ∧ N k x≤V x) ∧\n    ((¬∃ d : ℕ, IsTotient d ∧ k*d<ell d) →\n      (∀ x : ℝ, 0<x → N k x=0) ∧ (∀ s∈Set.Ico (0 : ℝ) 1, A (fk k) s=0)) :=\n  conditional_companion_theorem_of_structure fordScaleBounds extractedStructureInput primeNumberTheoremInput fordUnitPrimeBoxInput\n    fordRenewalInput mertensProductInput fordLemma26Input fordLemma51Input fordCoordinateConcentrationInput pptBoundedFiberPropagation k hk\n\ntheorem weighted_totient_one_two\n    :\n    ∀ k∈({1,2} : Finset ℕ), ∃ c : ℝ, 0<c ∧\n      ∀ s∈Set.Ico (0 : ℝ) 1, c≤A (fk k) s :=\n  companion_one_two_without_seed_input_of_structure fordScaleBounds extractedStructureInput primeNumberTheoremInput fordUnitPrimeBoxInput\n    fordRenewalInput mertensProductInput fordLemma26Input fordLemma51Input fordCoordinateConcentrationInput pptBoundedFiberPropagation\n\ntheorem coefficient_nonnegative (H : ℕ) {s : ℝ}\n    (hs : s ∈ Set.Ico (0 : ℝ) 1) : 0 ≤ AH H (fun _ => 1) s :=\n  AH_nonneg H hs (fun _ => 1) (fun _ _ => ⟨zero_le_one, le_rfl⟩)\n\nend TotientAsymptotic\n\nend\n\nend OAI\n\n\nFILE lean/OAI/NumberTheory/TotientAsymptotic/Main.lean \nIMPORTS ['import OAI.NumberTheory.TotientAsymptotic.FailedRowCountBound', 'import OAI.NumberTheory.TotientAsymptotic.PrimeProductValueMass', 'import OAI.NumberTheory.TotientAsymptotic.FailedRowTotalMass', 'import OAI.NumberTheory.TotientAsymptotic.CoordinateDeviationCount', 'import OAI.NumberTheory.TotientAsymptotic.ExceptionalRowMass', 'import OAI.NumberTheory.TotientAsymptotic.CofactorRowDeviation', 'import OAI.NumberTheory.TotientAsymptotic.SmoothTotientMass', 'import OAI.NumberTheory.TotientAsymptotic.PositiveGridCount', 'import OAI.NumberTheory.TotientAsymptotic.CoordinateActivePrefix', 'import OAI.NumberTheory.TotientAsymptotic.UniformRegularity', 'import OAI.NumberTheory.TotientAsymptotic.BootstrapRealCount', 'import OAI.NumberTheory.TotientAsymptotic.BandUnionCount', 'import OAI.NumberTheory.TotientAsymptotic.BootstrapResidualSize', 'import OAI.NumberTheory.TotientAsymptotic.BootstrapGoodValues', 'import OAI.NumberTheory.TotientAsymptotic.BootstrapViolation', 'import OAI.NumberTheory.TotientAsymptotic.TwoCoordinateCount', 'import OAI.NumberTheory.TotientAsymptotic.ExceptionalValueCount', 'import OAI.NumberTheory.TotientAsymptotic.TwoBandCount', 'import OAI.NumberTheory.TotientAsymptotic.PartitionedBandMass', 'import OAI.NumberTheory.TotientAsymptotic.CoarseTotientBounds', 'import OAI.NumberTheory.TotientAsymptotic.PreimageLoglogBound', 'import OAI.NumberTheory.TotientAsymptotic.PrimeExtensionCoverage', 'import OAI.NumberTheory.TotientAsymptotic.SmoothTotientCount', 'import OAI.NumberTheory.TotientAsymptotic.FordBandExponent', 'import OAI.NumberTheory.TotientAsymptotic.FordTerminalMass', 'import OAI.NumberTheory.TotientAsymptotic.FordStateTransfer', 'import OAI.NumberTheory.TotientAsymptotic.SquarefreeRoughMass', 'import OAI.NumberTheory.TotientAsymptotic.BandStateMass', 'import OAI.NumberTheory.TotientAsymptotic.SmoothTerminalBound', 'import OAI.NumberTheory.TotientAsymptotic.TerminalLogBudget', 'import OAI.NumberTheory.TotientAsymptotic.NormalTerminalFactors', 'import OAI.NumberTheory.TotientAsymptotic.CollisionBandMass', 'import OAI.NumberTheory.TotientAsymptotic.NormalBandAllocation', 'import OAI.NumberTheory.TotientAsymptotic.CommonLargestPrime', 'import OAI.NumberTheory.TotientAsymptotic.DistinctPrimeTail', 'import OAI.NumberTheory.TotientAsymptotic.PrimeFactorBands', 'import OAI.NumberTheory.TotientAsymptotic.PrimeReciprocalTails', 'import OAI.NumberTheory.TotientAsymptotic.UniformPrimeTripleBound', 'import OAI.NumberTheory.TotientAsymptotic.UniformPrimePairBound', 'import OAI.NumberTheory.TotientAsymptotic.NormalityIntervalTail', 'import OAI.NumberTheory.TotientAsymptotic.FullOmegaMoment', 'import OAI.NumberTheory.TotientAsymptotic.FordInputs', 'import OAI.NumberTheory.TotientAsymptotic.MertensProduct', 'import OAI.NumberTheory.TotientAsymptotic.PrimeNumberTheorem', 'import OAI.NumberTheory.TotientAsymptotic.SeedConsequences', 'import OAI.NumberTheory.TotientAsymptotic.SeedMinimum', 'import OAI.NumberTheory.TotientAsymptotic.ConditionalCompanion', 'import OAI.NumberTheory.TotientAsymptotic.WeightedFiber', 'import OAI.NumberTheory.TotientAsymptotic.GoodDiscard', 'import OAI.NumberTheory.TotientAsymptotic.TailBounds', 'import OAI.NumberTheory.TotientAsymptotic.Phase', 'import OAI.NumberTheory.TotientAsymptotic.Companion', 'import OAI.NumberTheory.TotientAsymptotic.PublishedFiber', 'import OAI.NumberTheory.TotientAsymptotic.Normalization', 'import OAI.NumberTheory.TotientAsymptotic.Scaling', 'import OAI.NumberTheory.TotientAsymptotic.FiniteMap', 'import OAI.NumberTheory.TotientAsymptotic.BandComparison', 'import OAI.NumberTheory.TotientAsymptotic.Mass', 'import OAI.NumberTheory.TotientAsymptotic.BandExtraction', 'import OAI.NumberTheory.TotientAsymptotic.SimplexVolume', 'import OAI.NumberTheory.TotientAsymptotic.TupleRecovery', 'import OAI.NumberTheory.TotientAsymptotic.PublishedComparison', 'import OAI.NumberTheory.TotientAsymptotic.NormalityAlignment', 'import OAI.NumberTheory.TotientAsymptotic.CollisionAlgebra', 'import OAI.NumberTheory.TotientAsymptotic.IntersectionLimit', 'import OAI.NumberTheory.TotientAsymptotic.UnionVolume', 'import OAI.NumberTheory.TotientAsymptotic.CollisionCount', 'import OAI.NumberTheory.TotientAsymptotic.CollisionSlack', 'import OAI.NumberTheory.TotientAsymptotic.UniformPrefactor', 'import OAI.NumberTheory.TotientAsymptotic.NormalitySize', 'import OAI.NumberTheory.TotientAsymptotic.CollisionCost', 'import OAI.NumberTheory.TotientAsymptotic.PrefixLower', 'import OAI.NumberTheory.TotientAsymptotic.PrimeBox', 'import OAI.NumberTheory.TotientAsymptotic.OrderedAlignment', 'import OAI.NumberTheory.TotientAsymptotic.CollisionTail', 'import OAI.NumberTheory.TotientAsymptotic.PrimeGrid', 'import OAI.NumberTheory.TotientAsymptotic.GridVolume', 'import OAI.NumberTheory.TotientAsymptotic.BandPrimeError', 'import OAI.NumberTheory.TotientAsymptotic.RenewalLower', 'import OAI.NumberTheory.TotientAsymptotic.RegionLower', 'import OAI.NumberTheory.TotientAsymptotic.CofactorMass', 'import OAI.NumberTheory.TotientAsymptotic.SimplexShell', 'import OAI.NumberTheory.TotientAsymptotic.CubeBoundary', 'import OAI.NumberTheory.TotientAsymptotic.ShellCost', 'import OAI.NumberTheory.TotientAsymptotic.CofactorEnvelope', 'import OAI.NumberTheory.TotientAsymptotic.TailCofactor', 'import OAI.NumberTheory.TotientAsymptotic.RenewalMagnitude', 'import OAI.NumberTheory.TotientAsymptotic.CofactorGrowth', 'import OAI.NumberTheory.TotientAsymptotic.CofactorShellDecay', 'import OAI.NumberTheory.TotientAsymptotic.ExtractTail', 'import OAI.NumberTheory.TotientAsymptotic.MassTailSupport', 'import OAI.NumberTheory.TotientAsymptotic.AttachTail', 'import OAI.NumberTheory.TotientAsymptotic.RemainderSplit', 'import OAI.NumberTheory.TotientAsymptotic.WitnessPrefix', 'import OAI.NumberTheory.TotientAsymptotic.PrefixRelaxation', 'import OAI.NumberTheory.TotientAsymptotic.PerturbationCost', 'import OAI.NumberTheory.TotientAsymptotic.PrimeBoxGeometry', 'import OAI.NumberTheory.TotientAsymptotic.BandGrid', 'import OAI.NumberTheory.TotientAsymptotic.GridApproximation', 'import OAI.NumberTheory.TotientAsymptotic.MassMajorant', 'import OAI.NumberTheory.TotientAsymptotic.DyadicSuffix', 'import OAI.NumberTheory.TotientAsymptotic.CollisionDyadicCount', 'import OAI.NumberTheory.TotientAsymptotic.FordSimplexEmbedding', 'import OAI.NumberTheory.TotientAsymptotic.PublishedConcentration', 'import OAI.NumberTheory.TotientAsymptotic.CenterComparison', 'import OAI.NumberTheory.TotientAsymptotic.TruncatedConcentration', 'import OAI.NumberTheory.TotientAsymptotic.ConcentrationTail', 'import OAI.NumberTheory.TotientAsymptotic.SimplexScaling', 'import OAI.NumberTheory.TotientAsymptotic.EnlargementCost', 'import OAI.NumberTheory.TotientAsymptotic.EnlargementTail', 'import OAI.NumberTheory.TotientAsymptotic.TailBoxAggregation', 'import OAI.NumberTheory.TotientAsymptotic.TailBoxCoordinates', 'import OAI.NumberTheory.TotientAsymptotic.SimplexCube', 'import OAI.NumberTheory.TotientAsymptotic.BasicCube', 'import OAI.NumberTheory.TotientAsymptotic.FullPrimeGrid', 'import OAI.NumberTheory.TotientAsymptotic.MassUpper', 'import OAI.NumberTheory.TotientAsymptotic.UnbandedLower', 'import OAI.NumberTheory.TotientAsymptotic.ScaledConcentration', 'import OAI.NumberTheory.TotientAsymptotic.PhaseBandTransfer', 'import OAI.NumberTheory.TotientAsymptotic.FullBoxConcentration', 'import OAI.NumberTheory.TotientAsymptotic.TailCutScale', 'import OAI.NumberTheory.TotientAsymptotic.NormalizedConcentration', 'import OAI.NumberTheory.TotientAsymptotic.TailArray', 'import OAI.NumberTheory.TotientAsymptotic.JoinedSimplex', 'import OAI.NumberTheory.TotientAsymptotic.PointEnclosure', 'import OAI.NumberTheory.TotientAsymptotic.ThickenedWitness', 'import OAI.NumberTheory.TotientAsymptotic.TailEncoding', 'import OAI.NumberTheory.TotientAsymptotic.WitnessAggregation', 'import OAI.NumberTheory.TotientAsymptotic.WitnessGrid', 'import OAI.NumberTheory.TotientAsymptotic.CofactorWitnessAggregation', 'import OAI.NumberTheory.TotientAsymptotic.BandRemoval', 'import OAI.NumberTheory.TotientAsymptotic.WitnessUnionLoss', 'import OAI.NumberTheory.TotientAsymptotic.AdditiveShell', 'import OAI.NumberTheory.TotientAsymptotic.AdditiveEnclosure', 'import OAI.NumberTheory.TotientAsymptotic.AdditiveCost', 'import OAI.NumberTheory.TotientAsymptotic.AdditiveShellUnion', 'import OAI.NumberTheory.TotientAsymptotic.NormalizedShell', 'import OAI.NumberTheory.TotientAsymptotic.PerturbationEnclosure', 'import OAI.NumberTheory.TotientAsymptotic.PerturbationRemoval', 'import OAI.NumberTheory.TotientAsymptotic.UnionPerturbation', 'import OAI.NumberTheory.TotientAsymptotic.VolumeComparison', 'import OAI.NumberTheory.TotientAsymptotic.MassVolumeApproximation', 'import OAI.NumberTheory.TotientAsymptotic.CollisionScale', 'import OAI.NumberTheory.TotientAsymptotic.CollisionCutoff', 'import OAI.NumberTheory.TotientAsymptotic.ResidualAlignment', 'import OAI.NumberTheory.TotientAsymptotic.SeparatedBands', 'import OAI.NumberTheory.TotientAsymptotic.SuffixFactorization', 'import OAI.NumberTheory.TotientAsymptotic.CollisionTuples', 'import OAI.NumberTheory.TotientAsymptotic.CollisionIndices', 'import OAI.NumberTheory.TotientAsymptotic.TupleCounting', 'import OAI.NumberTheory.TotientAsymptotic.LargestPrimeCount', 'import OAI.NumberTheory.TotientAsymptotic.TupleMassBound', 'import OAI.NumberTheory.TotientAsymptotic.SmoothSuffix', 'import OAI.NumberTheory.TotientAsymptotic.ResidualCutoff', 'import OAI.NumberTheory.TotientAsymptotic.WitnessCollision', 'import OAI.NumberTheory.TotientAsymptotic.TupleCoefficient', 'import OAI.NumberTheory.TotientAsymptotic.GoodTuples', 'import OAI.NumberTheory.TotientAsymptotic.StrictSlackDiscard', 'import OAI.NumberTheory.TotientAsymptotic.FordCollisionCutoffs', 'import OAI.NumberTheory.TotientAsymptotic.NormalPrimeBands', 'import OAI.NumberTheory.TotientAsymptotic.SurvivingCollision', 'import OAI.NumberTheory.TotientAsymptotic.ResidualTupleRecovery', 'import OAI.NumberTheory.TotientAsymptotic.ChosenCollision', 'import OAI.NumberTheory.TotientAsymptotic.ActualComparison', 'import OAI.NumberTheory.TotientAsymptotic.ComparisonSizes', 'import OAI.NumberTheory.TotientAsymptotic.ComparisonLargePart', 'import OAI.NumberTheory.TotientAsymptotic.CollisionSuffixSize', 'import OAI.NumberTheory.TotientAsymptotic.CollisionDyadicSize', 'import OAI.NumberTheory.TotientAsymptotic.CollisionCanceled', 'import OAI.NumberTheory.TotientAsymptotic.SurvivingComparison', 'import OAI.NumberTheory.TotientAsymptotic.CollisionGrid', 'import OAI.NumberTheory.TotientAsymptotic.CollisionLogBudget', 'import OAI.NumberTheory.TotientAsymptotic.NormalFactorBands', 'import OAI.NumberTheory.TotientAsymptotic.SurvivingBands', 'import OAI.NumberTheory.TotientAsymptotic.SurvivingAlignment', 'import OAI.NumberTheory.TotientAsymptotic.SurvivingBlockSize', 'import OAI.NumberTheory.TotientAsymptotic.CollisionGridScale', 'import OAI.NumberTheory.TotientAsymptotic.ComparisonErrorBudget', 'import OAI.NumberTheory.TotientAsymptotic.CollisionGridCount', 'import OAI.NumberTheory.TotientAsymptotic.SmoothResidualMass', 'import OAI.NumberTheory.TotientAsymptotic.CanceledPrimeMass', 'import OAI.NumberTheory.TotientAsymptotic.FordCostBudget', 'import OAI.NumberTheory.TotientAsymptotic.CollisionHeadSize', 'import OAI.NumberTheory.TotientAsymptotic.NormalHeadBand', 'import OAI.NumberTheory.TotientAsymptotic.CollisionNormalityDomain', 'import OAI.NumberTheory.TotientAsymptotic.NormalizedComparison', 'import OAI.NumberTheory.TotientAsymptotic.PairedComparisonGrid', 'import OAI.NumberTheory.TotientAsymptotic.CollisionFactorHeights', 'import OAI.NumberTheory.TotientAsymptotic.CollisionEndpointDomain', 'import OAI.NumberTheory.TotientAsymptotic.SurvivingHeights', 'import OAI.NumberTheory.TotientAsymptotic.SurvivingGridGeometry', 'import OAI.NumberTheory.TotientAsymptotic.ActualSurvivingGeometry', 'import OAI.NumberTheory.TotientAsymptotic.ComparisonGridFromHeights', 'import OAI.NumberTheory.TotientAsymptotic.HeightAlignment', 'import OAI.NumberTheory.TotientAsymptotic.ActualComparisonCutoffs', 'import OAI.NumberTheory.TotientAsymptotic.SurvivingSlack', 'import OAI.NumberTheory.TotientAsymptotic.ComparisonExponentSaving', 'import OAI.NumberTheory.TotientAsymptotic.CollisionHeadCount', 'import OAI.NumberTheory.TotientAsymptotic.AllCollisionCount', 'import OAI.NumberTheory.TotientAsymptotic.NormalityDiscard', 'import OAI.NumberTheory.TotientAsymptotic.OmegaReciprocal', 'import OAI.NumberTheory.TotientAsymptotic.OmegaDiscard', 'import OAI.NumberTheory.TotientAsymptotic.UnconditionalMain'] \nTAIL\nadSize\nimport OAI.NumberTheory.TotientAsymptotic.NormalHeadBand\nimport OAI.NumberTheory.TotientAsymptotic.CollisionNormalityDomain\nimport OAI.NumberTheory.TotientAsymptotic.NormalizedComparison\nimport OAI.NumberTheory.TotientAsymptotic.PairedComparisonGrid\nimport OAI.NumberTheory.TotientAsymptotic.CollisionFactorHeights\nimport OAI.NumberTheory.TotientAsymptotic.CollisionEndpointDomain\nimport OAI.NumberTheory.TotientAsymptotic.SurvivingHeights\nimport OAI.NumberTheory.TotientAsymptotic.SurvivingGridGeometry\nimport OAI.NumberTheory.TotientAsymptotic.ActualSurvivingGeometry\nimport OAI.NumberTheory.TotientAsymptotic.ComparisonGridFromHeights\nimport OAI.NumberTheory.TotientAsymptotic.HeightAlignment\nimport OAI.NumberTheory.TotientAsymptotic.ActualComparisonCutoffs\nimport OAI.NumberTheory.TotientAsymptotic.SurvivingSlack\nimport OAI.NumberTheory.TotientAsymptotic.ComparisonExponentSaving\n\nimport OAI.NumberTheory.TotientAsymptotic.CollisionHeadCount\n\nimport OAI.NumberTheory.TotientAsymptotic.AllCollisionCount\n\nimport OAI.NumberTheory.TotientAsymptotic.NormalityDiscard\nimport OAI.NumberTheory.TotientAsymptotic.OmegaReciprocal\nimport OAI.NumberTheory.TotientAsymptotic.OmegaDiscard\n\nimport OAI.NumberTheory.TotientAsymptotic.UnconditionalMain\n\n\nFILE lean/OAI/Combinatorics/TriangleFace/Main.lean \nIMPORTS ['import OAI.Combinatorics.TriangleFace.Faces', 'import OAI.Combinatorics.TriangleFace.MatchingImage'] \nTAIL\ne198.image_minFace\n      (localExtension G c v hc).toAffineMap (localExtension_injective G c v hc)\n      (convex_convexHull ℝ (Set.range G.indicator))\n      (convex_convexHull ℝ (Set.range (triangleExpansion G c v).indicator)) hex hx).symm\n\ntheorem triangle_expansion_face {V : Type uV} {E : Type uE}\n    [Fintype V] [Fintype E] [DecidableEq V] [DecidableEq E] (G : LooplessGraph V E) (v : V)\n    (hdeg : Fintype.card {e : E // G.Incident v e} = 3)\n    (_hmatching : Nonempty G.Matching) :\n    ∃ (c : E → Fin 3) (hc : LocalLabels G v c),\n      let H := triangleExpansion G c v\n      let f := localExtension G c v hc\n      let Q := {y | y ∈ H.polytope ∧ H.cutMass y (triangleVertices v) = 1}\n      f '' G.polytope = Q ∧\n      Convex ℝ Q ∧ IsExtreme ℝ H.polytope Q ∧\n      (∀ x, oldRestriction (f x) = x) ∧\n      (∀ y ∈ Q, f (oldRestriction y) = y) ∧\n      (∀ x e, f x (Sum.inl e) = x e) ∧\n      (∀ x i, f x (Sum.inr i) = x (localEdge G c v hc i)) ∧\n      (∀ x, (∀ e, x e = (1/3 : ℝ)) → ∀ e, f x e = (1/3 : ℝ)) ∧\n      (∀ x ∈ G.polytope, CanonicalFace198.minFace H.polytope (f x) = f '' CanonicalFace198.minFace G.polytope x) := by\n  obtain ⟨c,hc⟩ := labels_of_degree_three G v hdeg\n  exact ⟨c,hc,local_face G c v hc⟩\n\nend TriangleFace198\nend\n\nend OAI\n\n\nFILE lean/OAI/Probability/SLE/LowerPositivity.lean \nIMPORTS ['import Mathlib', 'import OAI.Probability.SLE.LowerRestartConditionalAllKappa'] \nTAIL\nexConjugate\nopen Set Filter Metric MeasureTheory Complex\nopen Set Filter MeasureTheory ProbabilityTheory Classical\nopen scoped Topology ENNReal NNReal\nopen Set Filter MeasureTheory ProbabilityTheory Metric Classical\nopen Set Filter\nopen scoped Topology NNReal ENNReal ComplexConjugate\nopen scoped ENNReal NNReal\nnoncomputable section\nopen Set MeasureTheory ProbabilityTheory Filter\nopen scoped ENNReal NNReal Topology\nsection\nopen Set Filter MeasureTheory\nopen scoped Topology ENNReal NNReal\n\nnamespace SLEExactGauge\n \ndef segment (γ : ℝ≥0 → ℂ) (s t : ℝ≥0) : Set ℂ := γ '' Icc s t\nuniverse v_lower\n \n\ndef SourceLowerMainTarget : Prop :=\n  ∀ (Ω : Type v_lower) (_ : MeasurableSpace Ω) (P : Measure Ω), IsProbabilityMeasure P →\n  ∀ κ : ℝ, 0 < κ → κ < ((8 : ℕ) : ℝ) →\n  ∀ (γ : Ω → ℝ≥0 → ℂ), IsOrdinaryChordalSLE κ γ P →\n  ∀ h : ℝ → ℝ, IsGauge h → HasSmallRadiusFormula h (hFormula κ) →\n    ∀ᵐ ω ∂P, ∀ s t : ℝ≥0, 0 < s → s < t →\n      0 < hausdorffGauge h (segment (γ ω) s t)\nend SLEExactGauge\n\nnamespace SLEExactGauge\ntheorem sourceLowerMain_proved : SourceLowerMainTarget := by\n  intro Ω m P hP κ hκ hκ8 γ hSLE h hh hf\n  let : IsProbabilityMeasure P := hP\n  exact source_lower_universe hκ hκ8 hSLE h hh hf\nend SLEExactGauge\nend\n\n \nend\nend\nend\n\nend OAI\n\n"
        ]
      },
      "outputs": [
        {
          "type": "stream",
          "name": "stdout",
          "text": "\nFILE lean/OAI/NumberTheory/DukePrimeDegree/MainUnconditional.lean \nIMPORTS ['import OAI.NumberTheory.DukePrimeDegree.MainStarkDischarged', 'import OAI.NumberTheory.DukePrimeDegree.Entropy.RegularEntropyDischarge'] \nTAIL\n⟨(ν : Measure (X (d+1))), hν, ?_, htight⟩\n  exact (weakProbabilityConvergence_iff\n    (fun i => packetProbability (M i) (σ i) (hn i)\n      (hprime.odd_of_ne_two (by omega))) ν).mpr hconv\n\n/-- The complete packet measures converge weakly to any specified Haar\nprobability, in the formulation by bounded continuous test functions. -/\ntheorem prime_degree_packet_measures_tendsto_haar_unconditional\n    (d : ℕ) (hprime : Nat.Prime (d+1)) (hfive : 5 ≤ d+1)\n    (K : ℕ → Type u) [∀ i, Field (K i)] [∀ i, NumberField (K i)]\n    [∀ i, NumberField.IsTotallyReal (K i)]\n    (M : ∀ i, PrimeDegreePackets.FullLattice (d+1) (K i))\n    (σ : ∀ i, OrderedEmbeddings (d+1) (K i))\n    (hn : ∀ i, Module.finrank ℚ (K i)=d+1)\n    (hD : Tendsto (fun i => PrimeDegreePackets.multiplierDiscriminant (M i)) atTop atTop)\n    (m : ProbabilityMeasure (X (d+1)))\n    (hm : IsHaarProbability (m : Measure (X (d+1)))) :\n    WeakProbabilityConvergence (fun i => packetMeasure (M i) (σ i))\n      (m : Measure (X (d+1))) := by\n  exact (weakProbabilityConvergence_iff\n    (fun i => packetProbability (M i) (σ i) (hn i)\n      (hprime.odd_of_ne_two (by omega))) m).mpr\n    (prime_degree_packets_tendsto_haar_unconditional d hprime hfive K M σ hn hD m hm)\n\nend DukePrimeDegree\n\nend OAI\n\n\nFILE lean/OAI/NumberTheory/TotientAsymptotic/UnconditionalMain.lean \nIMPORTS ['import OAI.NumberTheory.TotientAsymptotic.FordInputs', 'import OAI.NumberTheory.TotientAsymptotic.PPTFixedBandLimit'] \nTAIL\nnus ∧ 0<cPlus ∧\n        (∀ s∈Set.Ico (0 : ℝ) 1, cMinus≤A (fk k) s ∧ A (fk k) s≤cPlus) ∧\n        Tendsto (fun x => N k x/(tupleNormalization x*A (fk k) (theta x))) atTop (nhds 1) ∧\n        ∀ᶠ x : ℝ in atTop, (cMinus/cPlus)*V x≤N k x ∧ N k x≤V x) ∧\n    ((¬∃ d : ℕ, IsTotient d ∧ k*d<ell d) →\n      (∀ x : ℝ, 0<x → N k x=0) ∧ (∀ s∈Set.Ico (0 : ℝ) 1, A (fk k) s=0)) :=\n  conditional_companion_theorem_of_structure fordScaleBounds extractedStructureInput primeNumberTheoremInput fordUnitPrimeBoxInput\n    fordRenewalInput mertensProductInput fordLemma26Input fordLemma51Input fordCoordinateConcentrationInput pptBoundedFiberPropagation k hk\n\ntheorem weighted_totient_one_two\n    :\n    ∀ k∈({1,2} : Finset ℕ), ∃ c : ℝ, 0<c ∧\n      ∀ s∈Set.Ico (0 : ℝ) 1, c≤A (fk k) s :=\n  companion_one_two_without_seed_input_of_structure fordScaleBounds extractedStructureInput primeNumberTheoremInput fordUnitPrimeBoxInput\n    fordRenewalInput mertensProductInput fordLemma26Input fordLemma51Input fordCoordinateConcentrationInput pptBoundedFiberPropagation\n\ntheorem coefficient_nonnegative (H : ℕ) {s : ℝ}\n    (hs : s ∈ Set.Ico (0 : ℝ) 1) : 0 ≤ AH H (fun _ => 1) s :=\n  AH_nonneg H hs (fun _ => 1) (fun _ _ => ⟨zero_le_one, le_rfl⟩)\n\nend TotientAsymptotic\n\nend\n\nend OAI\n\n\nFILE lean/OAI/NumberTheory/TotientAsymptotic/Main.lean \nIMPORTS ['import OAI.NumberTheory.TotientAsymptotic.FailedRowCountBound', 'import OAI.NumberTheory.TotientAsymptotic.PrimeProductValueMass', 'import OAI.NumberTheory.TotientAsymptotic.FailedRowTotalMass', 'import OAI.NumberTheory.TotientAsymptotic.CoordinateDeviationCount', 'import OAI.NumberTheory.TotientAsymptotic.ExceptionalRowMass', 'import OAI.NumberTheory.TotientAsymptotic.CofactorRowDeviation', 'import OAI.NumberTheory.TotientAsymptotic.SmoothTotientMass', 'import OAI.NumberTheory.TotientAsymptotic.PositiveGridCount', 'import OAI.NumberTheory.TotientAsymptotic.CoordinateActivePrefix', 'import OAI.NumberTheory.TotientAsymptotic.UniformRegularity', 'import OAI.NumberTheory.TotientAsymptotic.BootstrapRealCount', 'import OAI.NumberTheory.TotientAsymptotic.BandUnionCount', 'import OAI.NumberTheory.TotientAsymptotic.BootstrapResidualSize', 'import OAI.NumberTheory.TotientAsymptotic.BootstrapGoodValues', 'import OAI.NumberTheory.TotientAsymptotic.BootstrapViolation', 'import OAI.NumberTheory.TotientAsymptotic.TwoCoordinateCount', 'import OAI.NumberTheory.TotientAsymptotic.ExceptionalValueCount', 'import OAI.NumberTheory.TotientAsymptotic.TwoBandCount', 'import OAI.NumberTheory.TotientAsymptotic.PartitionedBandMass', 'import OAI.NumberTheory.TotientAsymptotic.CoarseTotientBounds', 'import OAI.NumberTheory.TotientAsymptotic.PreimageLoglogBound', 'import OAI.NumberTheory.TotientAsymptotic.PrimeExtensionCoverage', 'import OAI.NumberTheory.TotientAsymptotic.SmoothTotientCount', 'import OAI.NumberTheory.TotientAsymptotic.FordBandExponent', 'import OAI.NumberTheory.TotientAsymptotic.FordTerminalMass', 'import OAI.NumberTheory.TotientAsymptotic.FordStateTransfer', 'import OAI.NumberTheory.TotientAsymptotic.SquarefreeRoughMass', 'import OAI.NumberTheory.TotientAsymptotic.BandStateMass', 'import OAI.NumberTheory.TotientAsymptotic.SmoothTerminalBound', 'import OAI.NumberTheory.TotientAsymptotic.TerminalLogBudget', 'import OAI.NumberTheory.TotientAsymptotic.NormalTerminalFactors', 'import OAI.NumberTheory.TotientAsymptotic.CollisionBandMass', 'import OAI.NumberTheory.TotientAsymptotic.NormalBandAllocation', 'import OAI.NumberTheory.TotientAsymptotic.CommonLargestPrime', 'import OAI.NumberTheory.TotientAsymptotic.DistinctPrimeTail', 'import OAI.NumberTheory.TotientAsymptotic.PrimeFactorBands', 'import OAI.NumberTheory.TotientAsymptotic.PrimeReciprocalTails', 'import OAI.NumberTheory.TotientAsymptotic.UniformPrimeTripleBound', 'import OAI.NumberTheory.TotientAsymptotic.UniformPrimePairBound', 'import OAI.NumberTheory.TotientAsymptotic.NormalityIntervalTail', 'import OAI.NumberTheory.TotientAsymptotic.FullOmegaMoment', 'import OAI.NumberTheory.TotientAsymptotic.FordInputs', 'import OAI.NumberTheory.TotientAsymptotic.MertensProduct', 'import OAI.NumberTheory.TotientAsymptotic.PrimeNumberTheorem', 'import OAI.NumberTheory.TotientAsymptotic.SeedConsequences', 'import OAI.NumberTheory.TotientAsymptotic.SeedMinimum', 'import OAI.NumberTheory.TotientAsymptotic.ConditionalCompanion', 'import OAI.NumberTheory.TotientAsymptotic.WeightedFiber', 'import OAI.NumberTheory.TotientAsymptotic.GoodDiscard', 'import OAI.NumberTheory.TotientAsymptotic.TailBounds', 'import OAI.NumberTheory.TotientAsymptotic.Phase', 'import OAI.NumberTheory.TotientAsymptotic.Companion', 'import OAI.NumberTheory.TotientAsymptotic.PublishedFiber', 'import OAI.NumberTheory.TotientAsymptotic.Normalization', 'import OAI.NumberTheory.TotientAsymptotic.Scaling', 'import OAI.NumberTheory.TotientAsymptotic.FiniteMap', 'import OAI.NumberTheory.TotientAsymptotic.BandComparison', 'import OAI.NumberTheory.TotientAsymptotic.Mass', 'import OAI.NumberTheory.TotientAsymptotic.BandExtraction', 'import OAI.NumberTheory.TotientAsymptotic.SimplexVolume', 'import OAI.NumberTheory.TotientAsymptotic.TupleRecovery', 'import OAI.NumberTheory.TotientAsymptotic.PublishedComparison', 'import OAI.NumberTheory.TotientAsymptotic.NormalityAlignment', 'import OAI.NumberTheory.TotientAsymptotic.CollisionAlgebra', 'import OAI.NumberTheory.TotientAsymptotic.IntersectionLimit', 'import OAI.NumberTheory.TotientAsymptotic.UnionVolume', 'import OAI.NumberTheory.TotientAsymptotic.CollisionCount', 'import OAI.NumberTheory.TotientAsymptotic.CollisionSlack', 'import OAI.NumberTheory.TotientAsymptotic.UniformPrefactor', 'import OAI.NumberTheory.TotientAsymptotic.NormalitySize', 'import OAI.NumberTheory.TotientAsymptotic.CollisionCost', 'import OAI.NumberTheory.TotientAsymptotic.PrefixLower', 'import OAI.NumberTheory.TotientAsymptotic.PrimeBox', 'import OAI.NumberTheory.TotientAsymptotic.OrderedAlignment', 'import OAI.NumberTheory.TotientAsymptotic.CollisionTail', 'import OAI.NumberTheory.TotientAsymptotic.PrimeGrid', 'import 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OAI.NumberTheory.TotientAsymptotic.UnconditionalMain'] \nTAIL\nadSize\nimport OAI.NumberTheory.TotientAsymptotic.NormalHeadBand\nimport OAI.NumberTheory.TotientAsymptotic.CollisionNormalityDomain\nimport OAI.NumberTheory.TotientAsymptotic.NormalizedComparison\nimport OAI.NumberTheory.TotientAsymptotic.PairedComparisonGrid\nimport OAI.NumberTheory.TotientAsymptotic.CollisionFactorHeights\nimport OAI.NumberTheory.TotientAsymptotic.CollisionEndpointDomain\nimport OAI.NumberTheory.TotientAsymptotic.SurvivingHeights\nimport OAI.NumberTheory.TotientAsymptotic.SurvivingGridGeometry\nimport OAI.NumberTheory.TotientAsymptotic.ActualSurvivingGeometry\nimport OAI.NumberTheory.TotientAsymptotic.ComparisonGridFromHeights\nimport OAI.NumberTheory.TotientAsymptotic.HeightAlignment\nimport OAI.NumberTheory.TotientAsymptotic.ActualComparisonCutoffs\nimport OAI.NumberTheory.TotientAsymptotic.SurvivingSlack\nimport OAI.NumberTheory.TotientAsymptotic.ComparisonExponentSaving\n\nimport OAI.NumberTheory.TotientAsymptotic.CollisionHeadCount\n\nimport OAI.NumberTheory.TotientAsymptotic.AllCollisionCount\n\nimport OAI.NumberTheory.TotientAsymptotic.NormalityDiscard\nimport OAI.NumberTheory.TotientAsymptotic.OmegaReciprocal\nimport OAI.NumberTheory.TotientAsymptotic.OmegaDiscard\n\nimport OAI.NumberTheory.TotientAsymptotic.UnconditionalMain\n\n\nFILE lean/OAI/Combinatorics/TriangleFace/Main.lean \nIMPORTS ['import OAI.Combinatorics.TriangleFace.Faces', 'import OAI.Combinatorics.TriangleFace.MatchingImage'] \nTAIL\ne198.image_minFace\n      (localExtension G c v hc).toAffineMap (localExtension_injective G c v hc)\n      (convex_convexHull ℝ (Set.range G.indicator))\n      (convex_convexHull ℝ (Set.range (triangleExpansion G c v).indicator)) hex hx).symm\n\ntheorem triangle_expansion_face {V : Type uV} {E : Type uE}\n    [Fintype V] [Fintype E] [DecidableEq V] [DecidableEq E] (G : LooplessGraph V E) (v : V)\n    (hdeg : Fintype.card {e : E // G.Incident v e} = 3)\n    (_hmatching : Nonempty G.Matching) :\n    ∃ (c : E → Fin 3) (hc : LocalLabels G v c),\n      let H := triangleExpansion G c v\n      let f := localExtension G c v hc\n      let Q := {y | y ∈ H.polytope ∧ H.cutMass y (triangleVertices v) = 1}\n      f '' G.polytope = Q ∧\n      Convex ℝ Q ∧ IsExtreme ℝ H.polytope Q ∧\n      (∀ x, oldRestriction (f x) = x) ∧\n      (∀ y ∈ Q, f (oldRestriction y) = y) ∧\n      (∀ x e, f x (Sum.inl e) = x e) ∧\n      (∀ x i, f x (Sum.inr i) = x (localEdge G c v hc i)) ∧\n      (∀ x, (∀ e, x e = (1/3 : ℝ)) → ∀ e, f x e = (1/3 : ℝ)) ∧\n      (∀ x ∈ G.polytope, CanonicalFace198.minFace H.polytope (f x) = f '' CanonicalFace198.minFace G.polytope x) := by\n  obtain ⟨c,hc⟩ := labels_of_degree_three G v hdeg\n  exact ⟨c,hc,local_face G c v hc⟩\n\nend TriangleFace198\nend\n\nend OAI\n\n\nFILE lean/OAI/Probability/SLE/LowerPositivity.lean \nIMPORTS ['import Mathlib', 'import OAI.Probability.SLE.LowerRestartConditionalAllKappa'] \nTAIL\nexConjugate\nopen Set Filter Metric MeasureTheory Complex\nopen Set Filter MeasureTheory ProbabilityTheory Classical\nopen scoped Topology ENNReal NNReal\nopen Set Filter MeasureTheory ProbabilityTheory Metric Classical\nopen Set Filter\nopen scoped Topology NNReal ENNReal ComplexConjugate\nopen scoped ENNReal NNReal\nnoncomputable section\nopen Set MeasureTheory ProbabilityTheory Filter\nopen scoped ENNReal NNReal Topology\nsection\nopen Set Filter MeasureTheory\nopen scoped Topology ENNReal NNReal\n\nnamespace SLEExactGauge\n \ndef segment (γ : ℝ≥0 → ℂ) (s t : ℝ≥0) : Set ℂ := γ '' Icc s t\nuniverse v_lower\n \n\ndef SourceLowerMainTarget : Prop :=\n  ∀ (Ω : Type v_lower) (_ : MeasurableSpace Ω) (P : Measure Ω), IsProbabilityMeasure P →\n  ∀ κ : ℝ, 0 < κ → κ < ((8 : ℕ) : ℝ) →\n  ∀ (γ : Ω → ℝ≥0 → ℂ), IsOrdinaryChordalSLE κ γ P →\n  ∀ h : ℝ → ℝ, IsGauge h → HasSmallRadiusFormula h (hFormula κ) →\n    ∀ᵐ ω ∂P, ∀ s t : ℝ≥0, 0 < s → s < t →\n      0 < hausdorffGauge h (segment (γ ω) s t)\nend SLEExactGauge\n\nnamespace SLEExactGauge\ntheorem sourceLowerMain_proved : SourceLowerMainTarget := by\n  intro Ω m P hP κ hκ hκ8 γ hSLE h hh hf\n  let : IsProbabilityMeasure P := hP\n  exact source_lower_universe hκ hκ8 hSLE h hh hf\nend SLEExactGauge\nend\n\n \nend\nend\nend\n\nend OAI\n\n"
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          ],
          "source": "META_GREY = \"#888888\"\n\n\ndef apply_figure_style(*, frame=\"open\", font=None, sizes=(8, 7, 6), grid=False):\n    \"\"\"Set matplotlib rcParams for publication-grade output. Call once before plotting.\n\n    This sets mechanics (role-mapped font-size ladder, outward ticks, frameless\n    legends, 300-dpi save, Type-42 embedded fonts) — not a house aesthetic.\n    Frame, font and the size ladder are parameters.\n\n    frame : 'open' (bottom+left spines, default) | 'boxed' (all four) | 'none'\n    font  : sans-serif family name; None = system default sans-serif\n    sizes : (base, secondary, tick) — titles/axis-labels, legend/annotation, ticks\n    grid  : whether to draw axes.grid (default False)\n    \"\"\"\n    import matplotlib as mpl\n    if frame not in (\"open\", \"boxed\", \"none\"):\n        raise ValueError(f\"frame must be 'open'|'boxed'|'none', got {frame!r}\")\n\n    try:\n        import os, sys, glob, matplotlib.font_manager as fm\n        fdir = os.path.join(os.environ.get(\"CONDA_PREFIX\") or sys.prefix, \"fonts\")\n        if os.path.isdir(fdir):\n            known = {f.fname for f in fm.fontManager.ttflist}\n            for f in glob.glob(os.path.join(fdir, \"*.ttf\")):\n                if f not in known:\n                    fm.fontManager.addfont(f)\n    except Exception:\n        pass\n    base, secondary, tick = sizes\n    boxed = (frame == \"boxed\")\n    rc = {\n        \"font.family\": \"sans-serif\",\n        \"font.size\": base,\n        \"axes.labelsize\": base,\n        \"axes.titlesize\": base,\n        \"legend.fontsize\": secondary,\n        \"xtick.labelsize\": tick,\n        \"ytick.labelsize\": tick,\n        \"axes.linewidth\": 0.6,\n        \"xtick.direction\": \"out\", \"ytick.direction\": \"out\",\n        \"xtick.major.size\": 3, \"ytick.major.size\": 3,\n        \"xtick.major.width\": 0.6, \"ytick.major.width\": 0.6,\n        \"axes.spines.top\": boxed, \"axes.spines.right\": boxed,\n        \"axes.spines.left\": frame != \"none\", \"axes.spines.bottom\": frame != \"none\",\n        \"axes.grid\": bool(grid),\n        \"legend.frameon\": False,\n        \"figure.dpi\": 200,\n        \"savefig.dpi\": 300,\n        \"savefig.bbox\": \"tight\",\n        \"axes.titleweight\": \"normal\",\n        \"axes.titlelocation\": \"left\",\n        \"axes.labelweight\": \"normal\",\n        \"lines.linewidth\": 1.2,\n        \"patch.linewidth\": 0.6,\n        \"pdf.fonttype\": 42, \"ps.fonttype\": 42,\n    }\n    if font:\n        rc[\"font.sans-serif\"] = [font, \"DejaVu Sans\"]\n    mpl.rcParams.update(rc)\n\n\ndef set_frame(ax, style=\"open\"):\n    \"\"\"§3: set spine visibility on an existing axes. style ∈ {'open','boxed','none'}.\"\"\"\n    show = {\"open\": (False, False, True, True),\n            \"boxed\": (True, True, True, True),\n            \"none\": (False, False, False, False)}[style]\n    for side, vis in zip((\"top\", \"right\", \"bottom\", \"left\"), show):\n        ax.spines[side].set_visible(vis)\n        if vis:\n            ax.spines[side].set_linewidth(0.6)\n    ax.tick_params(direction=\"out\", length=0 if style == \"none\" else 3, width=0.6)\n\n\ndef panel_letter(ax, letter, dx=-0.18, dy=1.02, case=\"lower\", fontsize=None):\n    \"\"\"§5.7: bold panel letter outside top-left of axes. case ∈ {'lower','upper'}.\"\"\"\n    import matplotlib.pyplot as plt\n    if fontsize is None:\n        fontsize = plt.rcParams.get(\"font.size\", 8) + 1\n    s = letter.lower() if case == \"lower\" else letter.upper()\n    artist = ax.text(dx, dy, s, transform=ax.transAxes,\n                     fontweight=\"bold\", fontsize=fontsize, va=\"bottom\", ha=\"left\")\n    artist.set_gid(\"figure-style-panel-letter\")\n\n\ndef focal_palette(labels, focal, focal_color, other=\"muted\", base_colors=None):\n    \"\"\"§4.2: map labels → colors with the focal series visually dominant.\n\n    other='muted'   — desaturate base_colors (or a default cycle) toward gray\n    other='grey'    — uniform light gray for all non-focal\n    other='ordinal' — non-focal on a single light→dark gray ramp (input order)\n    \"\"\"\n    import matplotlib.colors as mcolors\n    import matplotlib.pyplot as plt\n    focal_set = {focal} if isinstance(focal, str) else set(focal)\n    n = len(labels)\n    if not focal_set & set(labels):\n        raise ValueError(f\"focal {focal!r} not found in labels\")\n    base_colors = [] if base_colors is None else list(base_colors)\n    if not base_colors:\n        base_colors = plt.rcParams[\"axes.prop_cycle\"].by_key().get(\"color\", [\"#444444\"])\n    if not base_colors:\n        base_colors = [\"#444444\"]\n    base_colors = [base_colors[i % len(base_colors)] for i in range(n)]\n    if other == \"grey\":\n        rest = [\"#BCBCBC\"] * n\n    elif other == \"ordinal\":\n        nf = max(1, n - len(focal_set))\n        ramp = [mcolors.to_hex((v, v, v)) for v in\n                ([0.55] if nf == 1 else [0.80 - 0.35 * i / (nf - 1) for i in range(nf)])]\n        rest, k = [], 0\n        for l in labels:\n            rest.append(ramp[min(k, nf - 1)]); k += (l not in focal_set)\n    elif other == \"muted\":\n        def mute(c):\n            r, g, b = mcolors.to_rgb(c)\n            m = (r + g + b) / 3\n            return mcolors.to_hex((0.3 * r + 0.7 * m, 0.3 * g + 0.7 * m, 0.3 * b + 0.7 * m))\n        rest = [mute(c) for c in base_colors]\n    else:\n        raise ValueError(\"other must be 'muted', 'grey', or 'ordinal'\")\n    return [focal_color if l in focal_set else rest[i] for i, l in enumerate(labels)]\n\n\ndef bar_with_points(ax, x, ymat, labels, colors, jitter=0.08, show_points=True,\n                    errorbar=None, point_alpha=0.5, point_size=8):\n    \"\"\"§6.1: bar = mean; optionally overlay raw points or draw an interval.\n\n    colors   : per-label color list (e.g. from focal_palette)\n    x, ymat and labels must have equal lengths; mismatches raise before drawing.\n    Each group must be a nonempty, finite, one-dimensional real numeric array.\n    Requested SD/CI intervals require at least two observations per group.\n    errorbar : None | 'sd' | 'ci95' — drawn only when show_points is False.\n               'ci95' is the t-distribution 95% CI of the mean\n               (half-width t_{0.975,n-1} · s/√n); correct at small n where the\n               z-approximation (1.96·s/√n) is markedly too narrow.\n    \"\"\"\n    import numpy as np\n    x = np.asarray(x)\n    if x.ndim != 1 or not np.issubdtype(x.dtype, np.number):\n        raise TypeError(\n            \"bar_with_points x must be one-dimensional numeric positions; \"\n            \"pass category names through labels\"\n        )\n    ymat = list(ymat)\n    labels = list(labels)\n    if len(x) != len(ymat) or len(x) != len(labels):\n        raise ValueError(\"bar_with_points x, ymat and labels must have equal lengths\")\n    if colors is not None and not isinstance(colors, str):\n        colors = list(colors)\n        if not colors:\n            colors = None\n    if x.dtype.kind not in \"iuf\" or not np.isfinite(x).all():\n        raise ValueError(\"bar_with_points x must contain finite real positions\")\n    for i, y in enumerate(ymat):\n        if np.ma.isMaskedArray(y) and np.ma.getmaskarray(y).any():\n            raise ValueError(f\"bar_with_points group {i} contains masked observations; resolve exclusions explicitly before plotting\")\n    ymat = [np.asarray(y) for y in ymat]\n    for i, y in enumerate(ymat):\n        if y.ndim != 1 or y.size == 0 or y.dtype.kind not in \"iuf\" or not np.isfinite(y).all():\n            raise ValueError(f\"bar_with_points group {i} must be nonempty, one-dimensional and finite real numeric data\")\n    if errorbar and not show_points:\n        if errorbar not in (\"sd\", \"ci95\"):\n            raise ValueError(\"errorbar must be None, 'sd', or 'ci95'\")\n        if any(y.size < 2 for y in ymat):\n            raise ValueError(\"SD/CI cannot be estimated from one observation; use raw points without an interval\")\n    means = np.array([np.mean(y) for y in ymat], float)\n    err = None\n    if errorbar and not show_points:\n        if errorbar == \"sd\":\n            err = np.array([np.std(y, ddof=1) for y in ymat])\n        elif errorbar == \"ci95\":\n            from scipy.stats import t\n            def _hw(y):\n                n = np.asarray(y).size\n                return t.ppf(0.975, n - 1) * np.std(y, ddof=1) / np.sqrt(n)\n            err = np.array([_hw(y) for y in ymat])\n    ax.bar(x, means, color=colors, width=0.7, edgecolor=\"none\",\n           yerr=err, error_kw={\"elinewidth\": 0.8, \"capsize\": 0})\n    if show_points:\n        for xi, ys in zip(x, ymat):\n            ys = np.asarray(ys)\n            if ys.ndim == 1 and ys.size > 0:\n                jit = (np.random.rand(ys.size) - 0.5) * 2 * jitter\n                ax.scatter(np.full(ys.size, xi) + jit, ys, s=point_size, color=\"black\",\n                           alpha=point_alpha, zorder=3, linewidths=0)\n    ax.set_xticks(x); ax.set_xticklabels(labels)\n    return ax\n\n\ndef strip_with_median(ax, groups, values, colors=None, jitter=0.12):\n    \"\"\"§6.1: jittered points + bold horizontal median tick per group.\n\n    groups and values must have equal lengths. Mismatches raise before drawing.\n    colors=None or an empty color sequence supplies default gray; a nonempty\n    color sequence cycles across groups.\n    Each group must be nonempty, one-dimensional and finite real numeric data;\n    resolve missing observations explicitly before plotting.\n    \"\"\"\n    import numpy as np\n    labs = list(groups)\n    values = list(values)\n    if len(labs) != len(values):\n        raise ValueError(\"strip_with_median groups and values must have equal lengths\")\n    for i, y in enumerate(values):\n        if np.ma.isMaskedArray(y) and np.ma.getmaskarray(y).any():\n            raise ValueError(f\"strip_with_median group {i} contains masked observations; resolve exclusions explicitly before plotting\")\n    values = [np.asarray(y) for y in values]\n    for i, y in enumerate(values):\n        if y.ndim != 1 or y.size == 0 or y.dtype.kind not in \"iuf\" or not np.isfinite(y).all():\n            raise ValueError(f\"strip_with_median group {i} must be nonempty, one-dimensional and finite real numeric data\")\n    colors = [] if colors is None else list(colors)\n    if not colors:\n        colors = [\"#444444\"]\n    for i, ys in enumerate(values):\n        c = colors[i % len(colors)]\n        ys = np.asarray(ys)\n        jit = (np.random.rand(ys.size) - 0.5) * 2 * jitter\n        ax.scatter(np.full(ys.size, i) + jit, ys, s=10, color=c, alpha=0.6, linewidths=0, zorder=2)\n        m = np.median(ys)\n        ax.plot([i - 0.22, i + 0.22], [m, m], color=\"black\", lw=1.6, zorder=3)\n    ax.set_xticks(range(len(labs))); ax.set_xticklabels(labs)\n    return ax\n\n\ndef goodness_arrow(ax, text=\"higher = better\", loc=\"upper left\", axis=\"y\", fontsize=None):\n    \"\"\"§3.6: small upright direction-of-goodness cue in the margin.\"\"\"\n    import matplotlib.pyplot as plt\n    if fontsize is None:\n        fontsize = plt.rcParams[\"legend.fontsize\"]\n    pos = {\"upper left\": (0.02, 0.98), \"upper right\": (0.98, 0.98),\n           \"lower left\": (0.02, 0.02), \"lower right\": (0.98, 0.02)}[loc]\n    ha = \"left\" if \"left\" in loc else \"right\"\n    va = \"top\" if \"upper\" in loc else \"bottom\"\n    arrow = \"↑ \" if axis == \"y\" else \"→ \"\n    ax.text(pos[0], pos[1], arrow + text, transform=ax.transAxes,\n            fontsize=fontsize, color=META_GREY, ha=ha, va=va)\n\n\ndef two_tier_label(name, meta):\n    \"\"\"§5: two-line label string (name / metadata). Meta line styled separately by caller.\"\"\"\n    return f\"{name}\\n{meta}\"\n\n\ndef end_of_line_labels(ax, xs, ys, labels, colors=None, dx=0.01, fontsize=None):\n    \"\"\"§6.3 / §7.3: label each line series at its right end instead of a legend box.\n\n    Each label requires a nonempty, matched x/y series. Validate all series\n    before drawing. Nonempty colors cycle; None or empty uses default text color.\n    Inner sequences must support len() and indexing; they are not copied.\n    Nonfinite endpoints are rejected rather than producing an invisible label.\n    \"\"\"\n    import matplotlib.pyplot as plt\n    import math\n    xs, ys, labels = list(xs), list(ys), list(labels)\n    if len(xs) != len(ys) or len(xs) != len(labels):\n        raise ValueError(\"end_of_line_labels xs, ys and labels must have equal lengths\")\n    if any(len(x) == 0 or len(x) != len(y) for x, y in zip(xs, ys)):\n        raise ValueError(\"end_of_line_labels each x/y series must be nonempty and have equal lengths\")\n    if any(not math.isfinite(x[-1]) or not math.isfinite(y[-1]) for x, y in zip(xs, ys)):\n        raise ValueError(\"end_of_line_labels endpoints must be finite; choose valid endpoints explicitly\")\n    if fontsize is None:\n        fontsize = plt.rcParams[\"font.size\"]\n    colors = [] if colors is None else list(colors)\n    if not colors:\n        colors = [None]\n    span = ax.get_xlim()[1] - ax.get_xlim()[0]\n    for i, (x, y, lab) in enumerate(zip(xs, ys, labels)):\n        c = colors[i % len(colors)]\n        ax.text(x[-1] + dx * span, y[-1], lab, color=c, va=\"center\", ha=\"left\", fontsize=fontsize)\n\n\ndef panel_crops(fig, dpi=None, pad_px=6, bbox_inches=None, pad_inches=None):\n    \"\"\"§9.2: pixel-space crop boxes for each lettered panel in the SAVED PNG.\n\n    Returns ``{letter: (x0, y0, x1, y1)}`` in image-space pixels (origin\n    top-left, matching PIL's ``Image.crop``). Panels are detected as bold\n    single-character ``Text``\n    objects placed by :func:`panel_letter`; each panel's crop is its axes'\n    tightbbox mapped into the saved file's pixel space, padded by ``pad_px``.\n    For §3.4 composites (abutting subplots sharing an axis, letter on the\n    leftmost only) the crop unions in letterless ``sharex``/``sharey`` siblings\n    on the same grid row/col so the whole composite is covered. When no axes\n    carries a panel letter (standalone plot, or a figure-composer sub-agent),\n    falls back to one crop per axes keyed by index.\n\n    ``bbox_inches`` mirrors ``Figure.savefig`` semantics: ``None`` means\n    *consult rcParams* (so under :func:`apply_figure_style` it resolves to\n    ``'tight'``); pass an explicit ``Bbox`` only if you saved with one. The\n    boxes are clamped to the saved image extent; panels with no intersection\n    are omitted.\n\n        >>> from PIL import Image\n        >>> fig.savefig(\"fig.png\")  # bbox_inches='tight' via rcParams\n        >>> with Image.open(\"fig.png\") as saved:\n        ...     for letter, box in panel_crops(fig).items():\n        ...         saved.crop(box).save(f\"panel_{letter}.png\")\n\n    Match savefig's DPI, bbox_inches and pad_inches when overriding defaults.\n    Returned boxes are pixels, never savefig bbox_inches (inches). Passing them\n    back to savefig as inches can allocate an enormous raster.\n    \"\"\"\n    import matplotlib as mpl\n    import matplotlib.text\n    from matplotlib.backends.backend_agg import FigureCanvasAgg\n    if isinstance(pad_px, bool) or not isinstance(pad_px, int) or pad_px < 0:\n        raise TypeError(\"panel_crops pad_px must be a non-negative integer\")\n    if dpi is None:\n        dpi = mpl.rcParams.get(\"savefig.dpi\", fig.dpi)\n        if dpi == \"figure\":\n            dpi = fig.dpi\n    dpi = float(dpi)\n    if bbox_inches is None:\n        bbox_inches = mpl.rcParams.get(\"savefig.bbox\")\n    # The Open-Science notebook captures and closes pyplot figures after every\n    # cell. Matplotlib 3.11 decouples a closed Figure from its backend by\n    # replacing FigureCanvasAgg with FigureCanvasBase, while the persistent\n    # notebook namespace can still retain the Figure itself. Reattach the\n    # headless canvas before asking for saved-image geometry so this helper is\n    # safe in a later QA cell as well as the producer cell.\n    if not hasattr(fig.canvas, \"get_renderer\"):\n        FigureCanvasAgg(fig)\n    fig.canvas.draw()\n    r = fig.canvas.get_renderer()\n\n    if bbox_inches == \"tight\":\n        if pad_inches is None:\n            pad_inches = mpl.rcParams.get(\"savefig.pad_inches\", 0.1)\n        tb = fig.get_tightbbox(r).padded(pad_inches)\n        ox_in, oy_in = tb.x0, tb.y0\n        W_in, H_in = tb.width, tb.height\n    elif isinstance(bbox_inches, mpl.transforms.BboxBase):\n        ox_in, oy_in = bbox_inches.x0, bbox_inches.y0\n        W_in, H_in = bbox_inches.width, bbox_inches.height\n    elif bbox_inches is None:\n        ox_in, oy_in = 0.0, 0.0\n        W_in, H_in = fig.get_size_inches()\n    else:\n        raise TypeError(\"panel_crops bbox_inches must be None, 'tight', or a matplotlib Bbox\")\n    W_px, H_px = int(round(W_in * dpi)), int(round(H_in * dpi))\n    lettered = {}\n    seen_letters = set()\n    for ax in fig.axes:\n        for t in ax.findobj(matplotlib.text.Text):\n            s = (t.get_text() or \"\").strip()\n            if t.get_gid() == \"figure-style-panel-letter\":\n                if s in seen_letters:\n                    raise ValueError(f\"Duplicate panel letter: {s!r}\")\n                seen_letters.add(s)\n                lettered[ax] = s\n                break\n\n\n\n    if not lettered:\n        lettered = {ax: str(i) for i, ax in enumerate(fig.axes)}\n    out = {}\n    for ax, letter in lettered.items():\n        bbs = [ax.get_tightbbox(r)]\n\n\n\n\n        ss = ax.get_subplotspec()\n        for sib in fig.axes:\n            if sib is ax or sib in lettered:\n                continue\n            ssib = sib.get_subplotspec()\n            same_row = ss is None or ssib is None or ss.rowspan == ssib.rowspan\n            same_col = ss is None or ssib is None or ss.colspan == ssib.colspan\n            if ((ax.get_shared_y_axes().joined(ax, sib) and same_row)\n                    or (ax.get_shared_x_axes().joined(ax, sib) and same_col)):\n                bbs.append(sib.get_tightbbox(r))\n        bb = mpl.transforms.Bbox.union(bbs)\n\n        bx0 = (bb.x0 / fig.dpi - ox_in) * dpi\n        bx1 = (bb.x1 / fig.dpi - ox_in) * dpi\n        by0 = H_px - (bb.y1 / fig.dpi - oy_in) * dpi\n        by1 = H_px - (bb.y0 / fig.dpi - oy_in) * dpi\n        box = (\n            min(max(int(bx0) - pad_px, 0), W_px),\n            min(max(int(by0) - pad_px, 0), H_px),\n            max(min(int(bx1) + pad_px, W_px), 0),\n            max(min(int(by1) + pad_px, H_px), 0),\n        )\n        if box[0] < box[2] and box[1] < box[3]:\n            out[letter] = box\n    return out\n",
          "sourceDigest": "8020d409bfb60f0c8dfa43d0dfe7bb97f7458e1a3908fefbcb6e4551e0feae4e"
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        "installedInventory": {
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          "inventory-cache-best-effort"
        ]
      },
      "environmentManifestChecksum": "07186a52ee08681a2db08dcdfb28577fd86b83b896c7aa2db387a54f4790fe18"
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    {
      "runId": "notebook-run-1791456226939-106",
      "executionMode": "foreground",
      "submissionIdentity": "a1690af4-2def-4372-8ebe-170d276a4fad",
      "submissionFingerprint": "f1c9d65e60ab29ef8ecb46369a8dfd79e660930dcb49cc61a3d5b9ba95c74228",
      "admittedAt": 1791456226939,
      "kernelEpochId": "30b941a1-2823-4570-addf-fc45a3e9f366",
      "executionInvocationId": "a1690af4-2def-4372-8ebe-170d276a4fad",
      "cellId": "cell-b12ec507-29ad-493f-8005-f6801ba2700d",
      "source": "agent",
      "inputKind": "cell",
      "kernelKind": "python",
      "script": "for key in ['lean/OAI/NumberTheory/TotientAsymptotic/Main.lean','lean/OAI/Combinatorics/TriangleFace/Main.lean','lean/OAI/Probability/SLE/LowerPositivity.lean']:\n print('\\nFILE',key,'TAIL\\n'+extra[key]['content'][-1750:])\nfor k,c in configs.items():\n sol='lean/'+c['solution_module'].replace('.','/')+'.lean';v=combined[sol]['content']\n names=re.findall(r'^(?:theorem|lemma|def|abbrev)\\s+([^\\s:{(]+)',v,re.M)\n absent=[name for name in c['theorem_names'] if name.split('.')[-1] not in [n.split('.')[-1] for n in names]]\n if absent:print('NOT LOCALLY DECLARED',Path(k).name,sol,absent)",
      "status": "completed",
      "startedAt": 1791456227187,
      "cwdBefore": "$DATA/notebooks/cmuzdubm90002rxoq52hkpl0k/01a11b04-b537-7300-9a27-fde070942335/data",
      "executionCount": 54,
      "environment": "default-python",
      "rootFrameId": "root-frame-01a11b04-b537-7300-9a27-fde070942335",
      "agentFrameId": "root-frame-01a11b04-b537-7300-9a27-fde070942335",
      "messageBranchId": "message-branch-01a11b04-b537-7300-9a27-fde070942335",
      "runtimeSegmentId": "runtime-segment-1791456120066-15",
      "promptMessageId": "message-9bccc6c4-2564-41ca-9577-b2bf916389c8",
      "text": {
        "stdout": "\nFILE lean/OAI/NumberTheory/TotientAsymptotic/Main.lean TAIL\nic.SurvivingAlignment\nimport OAI.NumberTheory.TotientAsymptotic.SurvivingBlockSize\nimport OAI.NumberTheory.TotientAsymptotic.CollisionGridScale\nimport OAI.NumberTheory.TotientAsymptotic.ComparisonErrorBudget\nimport OAI.NumberTheory.TotientAsymptotic.CollisionGridCount\nimport OAI.NumberTheory.TotientAsymptotic.SmoothResidualMass\nimport OAI.NumberTheory.TotientAsymptotic.CanceledPrimeMass\nimport OAI.NumberTheory.TotientAsymptotic.FordCostBudget\nimport OAI.NumberTheory.TotientAsymptotic.CollisionHeadSize\nimport OAI.NumberTheory.TotientAsymptotic.NormalHeadBand\nimport OAI.NumberTheory.TotientAsymptotic.CollisionNormalityDomain\nimport OAI.NumberTheory.TotientAsymptotic.NormalizedComparison\nimport OAI.NumberTheory.TotientAsymptotic.PairedComparisonGrid\nimport OAI.NumberTheory.TotientAsymptotic.CollisionFactorHeights\nimport OAI.NumberTheory.TotientAsymptotic.CollisionEndpointDomain\nimport OAI.NumberTheory.TotientAsymptotic.SurvivingHeights\nimport OAI.NumberTheory.TotientAsymptotic.SurvivingGridGeometry\nimport OAI.NumberTheory.TotientAsymptotic.ActualSurvivingGeometry\nimport OAI.NumberTheory.TotientAsymptotic.ComparisonGridFromHeights\nimport OAI.NumberTheory.TotientAsymptotic.HeightAlignment\nimport OAI.NumberTheory.TotientAsymptotic.ActualComparisonCutoffs\nimport OAI.NumberTheory.TotientAsymptotic.SurvivingSlack\nimport OAI.NumberTheory.TotientAsymptotic.ComparisonExponentSaving\n\nimport OAI.NumberTheory.TotientAsymptotic.CollisionHeadCount\n\nimport OAI.NumberTheory.TotientAsymptotic.AllCollisionCount\n\nimport OAI.NumberTheory.TotientAsymptotic.NormalityDiscard\nimport OAI.NumberTheory.TotientAsymptotic.OmegaReciprocal\nimport OAI.NumberTheory.TotientAsymptotic.OmegaDiscard\n\nimport OAI.NumberTheory.TotientAsymptotic.UnconditionalMain\n\n\nFILE lean/OAI/Combinatorics/TriangleFace/Main.lean TAIL\n G c v hc\n  have hex := expansion_image_extreme G c v hc\n  refine ⟨him, ?_, ?_, restriction_extension G c v hc, ?_, ?_, ?_, ?_, ?_⟩\n  · rw [← him]\n    exact (convex_convexHull ℝ (Set.range G.indicator)).linear_image (localExtension G c v hc)\n  · rwa [← him]\n  · intro y hy\n    obtain ⟨x, hx, rfl⟩ := him.symm ▸ hy\n    rfl\n  · intro x e\n    rfl\n  · intro x i\n    rfl\n  · intro x hx e\n    rcases e with e | i\n    · exact hx e\n    · exact hx (localEdge G c v hc i)\n  · intro x hx\n    exact (CanonicalFace198.image_minFace\n      (localExtension G c v hc).toAffineMap (localExtension_injective G c v hc)\n      (convex_convexHull ℝ (Set.range G.indicator))\n      (convex_convexHull ℝ (Set.range (triangleExpansion G c v).indicator)) hex hx).symm\n\ntheorem triangle_expansion_face {V : Type uV} {E : Type uE}\n    [Fintype V] [Fintype E] [DecidableEq V] [DecidableEq E] (G : LooplessGraph V E) (v : V)\n    (hdeg : Fintype.card {e : E // G.Incident v e} = 3)\n    (_hmatching : Nonempty G.Matching) :\n    ∃ (c : E → Fin 3) (hc : LocalLabels G v c),\n      let H := triangleExpansion G c v\n      let f := localExtension G c v hc\n      let Q := {y | y ∈ H.polytope ∧ H.cutMass y (triangleVertices v) = 1}\n      f '' G.polytope = Q ∧\n      Convex ℝ Q ∧ IsExtreme ℝ H.polytope Q ∧\n      (∀ x, oldRestriction (f x) = x) ∧\n      (∀ y ∈ Q, f (oldRestriction y) = y) ∧\n      (∀ x e, f x (Sum.inl e) = x e) ∧\n      (∀ x i, f x (Sum.inr i) = x (localEdge G c v hc i)) ∧\n      (∀ x, (∀ e, x e = (1/3 : ℝ)) → ∀ e, f x e = (1/3 : ℝ)) ∧\n      (∀ x ∈ G.polytope, CanonicalFace198.minFace H.polytope (f x) = f '' CanonicalFace198.minFace G.polytope x) := by\n  obtain ⟨c,hc⟩ := labels_of_degree_three G v hdeg\n  exact ⟨c,hc,local_face G c v hc⟩\n\nend TriangleFace198\nend\n\nend OAI\n\n\nFILE lean/OAI/Probability/SLE/LowerPositivity.lean TAIL\nInnerProductSpace\nopen scoped Topology Laplacian\nopen Set Filter Metric Complex MeasureTheory\nopen Set Filter Metric Complex MeasureTheory InnerProductSpace\nopen scoped Topology Laplacian ENNReal\nopen scoped Topology ENNReal\nopen scoped Topology Laplacian RealInnerProductSpace\nopen Set Filter Metric Complex\nopen scoped Topology\nopen Set Filter Metric\nopen Set Filter MeasureTheory Metric\nopen Set Filter Complex\nopen scoped Topology ComplexConjugate NNReal\nopen scoped Topology ENNReal NNReal ComplexConjugate\nopen Set Filter Metric MeasureTheory Complex\nopen Set Filter MeasureTheory ProbabilityTheory Classical\nopen scoped Topology ENNReal NNReal\nopen Set Filter MeasureTheory ProbabilityTheory Metric Classical\nopen Set Filter\nopen scoped Topology NNReal ENNReal ComplexConjugate\nopen scoped ENNReal NNReal\nnoncomputable section\nopen Set MeasureTheory ProbabilityTheory Filter\nopen scoped ENNReal NNReal Topology\nsection\nopen Set Filter MeasureTheory\nopen scoped Topology ENNReal NNReal\n\nnamespace SLEExactGauge\n \ndef segment (γ : ℝ≥0 → ℂ) (s t : ℝ≥0) : Set ℂ := γ '' Icc s t\nuniverse v_lower\n \n\ndef SourceLowerMainTarget : Prop :=\n  ∀ (Ω : Type v_lower) (_ : MeasurableSpace Ω) (P : Measure Ω), IsProbabilityMeasure P →\n  ∀ κ : ℝ, 0 < κ → κ < ((8 : ℕ) : ℝ) →\n  ∀ (γ : Ω → ℝ≥0 → ℂ), IsOrdinaryChordalSLE κ γ P →\n  ∀ h : ℝ → ℝ, IsGauge h → HasSmallRadiusFormula h (hFormula κ) →\n    ∀ᵐ ω ∂P, ∀ s t : ℝ≥0, 0 < s → s < t →\n      0 < hausdorffGauge h (segment (γ ω) s t)\nend SLEExactGauge\n\nnamespace SLEExactGauge\ntheorem sourceLowerMain_proved : SourceLowerMainTarget := by\n  intro Ω m P hP κ hκ hκ8 γ hSLE h hh hf\n  let : IsProbabilityMeasure P := hP\n  exact source_lower_universe hκ hκ8 hSLE h hh hf\nend SLEExactGauge\nend\n\n \nend\nend\nend\n\nend OAI\n\nNOT LOCALLY DECLARED MatchingEntropyBounds.json lean/OAI/Combinatorics/MatchingEntropy/Main.lean ['OAI.MatchingEntropyBounds.Refined.pointwise_entropy']\nNOT LOCALLY DECLARED SelfSimilarCorollaries.json lean/OAI/MeasureTheory/SelfSimilar/DimensionCorollaries.lean ['OAI.EntropyRateDimension.Extensions.homogeneous_dimension_direct', 'OAI.CorSetReference.cor_set']\nNOT LOCALLY DECLARED SingletonLoopMatching.json lean/OAI/Computability/LoopMatching/Endpoint.lean ['OAI.LoopMatching.fullEndpoint']\nNOT LOCALLY DECLARED TotientCompanionZero.json lean/OAI/NumberTheory/TotientAsymptotic/Main.lean ['OAI.TotientAsymptotic.companion_zero_case']\n",
        "stderr": "",
        "traceback": "",
        "plain": [
          "\nFILE lean/OAI/NumberTheory/TotientAsymptotic/Main.lean TAIL\nic.SurvivingAlignment\nimport OAI.NumberTheory.TotientAsymptotic.SurvivingBlockSize\nimport OAI.NumberTheory.TotientAsymptotic.CollisionGridScale\nimport OAI.NumberTheory.TotientAsymptotic.ComparisonErrorBudget\nimport OAI.NumberTheory.TotientAsymptotic.CollisionGridCount\nimport OAI.NumberTheory.TotientAsymptotic.SmoothResidualMass\nimport OAI.NumberTheory.TotientAsymptotic.CanceledPrimeMass\nimport OAI.NumberTheory.TotientAsymptotic.FordCostBudget\nimport OAI.NumberTheory.TotientAsymptotic.CollisionHeadSize\nimport OAI.NumberTheory.TotientAsymptotic.NormalHeadBand\nimport OAI.NumberTheory.TotientAsymptotic.CollisionNormalityDomain\nimport OAI.NumberTheory.TotientAsymptotic.NormalizedComparison\nimport OAI.NumberTheory.TotientAsymptotic.PairedComparisonGrid\nimport OAI.NumberTheory.TotientAsymptotic.CollisionFactorHeights\nimport OAI.NumberTheory.TotientAsymptotic.CollisionEndpointDomain\nimport OAI.NumberTheory.TotientAsymptotic.SurvivingHeights\nimport OAI.NumberTheory.TotientAsymptotic.SurvivingGridGeometry\nimport OAI.NumberTheory.TotientAsymptotic.ActualSurvivingGeometry\nimport OAI.NumberTheory.TotientAsymptotic.ComparisonGridFromHeights\nimport OAI.NumberTheory.TotientAsymptotic.HeightAlignment\nimport OAI.NumberTheory.TotientAsymptotic.ActualComparisonCutoffs\nimport OAI.NumberTheory.TotientAsymptotic.SurvivingSlack\nimport OAI.NumberTheory.TotientAsymptotic.ComparisonExponentSaving\n\nimport OAI.NumberTheory.TotientAsymptotic.CollisionHeadCount\n\nimport OAI.NumberTheory.TotientAsymptotic.AllCollisionCount\n\nimport OAI.NumberTheory.TotientAsymptotic.NormalityDiscard\nimport OAI.NumberTheory.TotientAsymptotic.OmegaReciprocal\nimport OAI.NumberTheory.TotientAsymptotic.OmegaDiscard\n\nimport OAI.NumberTheory.TotientAsymptotic.UnconditionalMain\n\n\nFILE lean/OAI/Combinatorics/TriangleFace/Main.lean TAIL\n G c v hc\n  have hex := expansion_image_extreme G c v hc\n  refine ⟨him, ?_, ?_, restriction_extension G c v hc, ?_, ?_, ?_, ?_, ?_⟩\n  · rw [← him]\n    exact (convex_convexHull ℝ (Set.range G.indicator)).linear_image (localExtension G c v hc)\n  · rwa [← him]\n  · intro y hy\n    obtain ⟨x, hx, rfl⟩ := him.symm ▸ hy\n    rfl\n  · intro x e\n    rfl\n  · intro x i\n    rfl\n  · intro x hx e\n    rcases e with e | i\n    · exact hx e\n    · exact hx (localEdge G c v hc i)\n  · intro x hx\n    exact (CanonicalFace198.image_minFace\n      (localExtension G c v hc).toAffineMap (localExtension_injective G c v hc)\n      (convex_convexHull ℝ (Set.range G.indicator))\n      (convex_convexHull ℝ (Set.range (triangleExpansion G c v).indicator)) hex hx).symm\n\ntheorem triangle_expansion_face {V : Type uV} {E : Type uE}\n    [Fintype V] [Fintype E] [DecidableEq V] [DecidableEq E] (G : LooplessGraph V E) (v : V)\n    (hdeg : Fintype.card {e : E // G.Incident v e} = 3)\n    (_hmatching : Nonempty G.Matching) :\n    ∃ (c : E → Fin 3) (hc : LocalLabels G v c),\n      let H := triangleExpansion G c v\n      let f := localExtension G c v hc\n      let Q := {y | y ∈ H.polytope ∧ H.cutMass y (triangleVertices v) = 1}\n      f '' G.polytope = Q ∧\n      Convex ℝ Q ∧ IsExtreme ℝ H.polytope Q ∧\n      (∀ x, oldRestriction (f x) = x) ∧\n      (∀ y ∈ Q, f (oldRestriction y) = y) ∧\n      (∀ x e, f x (Sum.inl e) = x e) ∧\n      (∀ x i, f x (Sum.inr i) = x (localEdge G c v hc i)) ∧\n      (∀ x, (∀ e, x e = (1/3 : ℝ)) → ∀ e, f x e = (1/3 : ℝ)) ∧\n      (∀ x ∈ G.polytope, CanonicalFace198.minFace H.polytope (f x) = f '' CanonicalFace198.minFace G.polytope x) := by\n  obtain ⟨c,hc⟩ := labels_of_degree_three G v hdeg\n  exact ⟨c,hc,local_face G c v hc⟩\n\nend TriangleFace198\nend\n\nend OAI\n\n\nFILE lean/OAI/Probability/SLE/LowerPositivity.lean TAIL\nInnerProductSpace\nopen scoped Topology Laplacian\nopen Set Filter Metric Complex MeasureTheory\nopen Set Filter Metric Complex MeasureTheory InnerProductSpace\nopen scoped Topology Laplacian ENNReal\nopen scoped Topology ENNReal\nopen scoped Topology Laplacian RealInnerProductSpace\nopen Set Filter Metric Complex\nopen scoped Topology\nopen Set Filter Metric\nopen Set Filter MeasureTheory Metric\nopen Set Filter Complex\nopen scoped Topology ComplexConjugate NNReal\nopen scoped Topology ENNReal NNReal ComplexConjugate\nopen Set Filter Metric MeasureTheory Complex\nopen Set Filter MeasureTheory ProbabilityTheory Classical\nopen scoped Topology ENNReal NNReal\nopen Set Filter MeasureTheory ProbabilityTheory Metric Classical\nopen Set Filter\nopen scoped Topology NNReal ENNReal ComplexConjugate\nopen scoped ENNReal NNReal\nnoncomputable section\nopen Set MeasureTheory ProbabilityTheory Filter\nopen scoped ENNReal NNReal Topology\nsection\nopen Set Filter MeasureTheory\nopen scoped Topology ENNReal NNReal\n\nnamespace SLEExactGauge\n \ndef segment (γ : ℝ≥0 → ℂ) (s t : ℝ≥0) : Set ℂ := γ '' Icc s t\nuniverse v_lower\n \n\ndef SourceLowerMainTarget : Prop :=\n  ∀ (Ω : Type v_lower) (_ : MeasurableSpace Ω) (P : Measure Ω), IsProbabilityMeasure P →\n  ∀ κ : ℝ, 0 < κ → κ < ((8 : ℕ) : ℝ) →\n  ∀ (γ : Ω → ℝ≥0 → ℂ), IsOrdinaryChordalSLE κ γ P →\n  ∀ h : ℝ → ℝ, IsGauge h → HasSmallRadiusFormula h (hFormula κ) →\n    ∀ᵐ ω ∂P, ∀ s t : ℝ≥0, 0 < s → s < t →\n      0 < hausdorffGauge h (segment (γ ω) s t)\nend SLEExactGauge\n\nnamespace SLEExactGauge\ntheorem sourceLowerMain_proved : SourceLowerMainTarget := by\n  intro Ω m P hP κ hκ hκ8 γ hSLE h hh hf\n  let : IsProbabilityMeasure P := hP\n  exact source_lower_universe hκ hκ8 hSLE h hh hf\nend SLEExactGauge\nend\n\n \nend\nend\nend\n\nend OAI\n\nNOT LOCALLY DECLARED MatchingEntropyBounds.json lean/OAI/Combinatorics/MatchingEntropy/Main.lean ['OAI.MatchingEntropyBounds.Refined.pointwise_entropy']\nNOT LOCALLY DECLARED SelfSimilarCorollaries.json lean/OAI/MeasureTheory/SelfSimilar/DimensionCorollaries.lean ['OAI.EntropyRateDimension.Extensions.homogeneous_dimension_direct', 'OAI.CorSetReference.cor_set']\nNOT LOCALLY DECLARED SingletonLoopMatching.json lean/OAI/Computability/LoopMatching/Endpoint.lean ['OAI.LoopMatching.fullEndpoint']\nNOT LOCALLY DECLARED TotientCompanionZero.json lean/OAI/NumberTheory/TotientAsymptotic/Main.lean ['OAI.TotientAsymptotic.companion_zero_case']\n"
        ]
      },
      "outputs": [
        {
          "type": "stream",
          "name": "stdout",
          "text": "\nFILE lean/OAI/NumberTheory/TotientAsymptotic/Main.lean TAIL\nic.SurvivingAlignment\nimport OAI.NumberTheory.TotientAsymptotic.SurvivingBlockSize\nimport OAI.NumberTheory.TotientAsymptotic.CollisionGridScale\nimport OAI.NumberTheory.TotientAsymptotic.ComparisonErrorBudget\nimport OAI.NumberTheory.TotientAsymptotic.CollisionGridCount\nimport OAI.NumberTheory.TotientAsymptotic.SmoothResidualMass\nimport OAI.NumberTheory.TotientAsymptotic.CanceledPrimeMass\nimport OAI.NumberTheory.TotientAsymptotic.FordCostBudget\nimport OAI.NumberTheory.TotientAsymptotic.CollisionHeadSize\nimport OAI.NumberTheory.TotientAsymptotic.NormalHeadBand\nimport OAI.NumberTheory.TotientAsymptotic.CollisionNormalityDomain\nimport OAI.NumberTheory.TotientAsymptotic.NormalizedComparison\nimport OAI.NumberTheory.TotientAsymptotic.PairedComparisonGrid\nimport OAI.NumberTheory.TotientAsymptotic.CollisionFactorHeights\nimport OAI.NumberTheory.TotientAsymptotic.CollisionEndpointDomain\nimport OAI.NumberTheory.TotientAsymptotic.SurvivingHeights\nimport OAI.NumberTheory.TotientAsymptotic.SurvivingGridGeometry\nimport OAI.NumberTheory.TotientAsymptotic.ActualSurvivingGeometry\nimport OAI.NumberTheory.TotientAsymptotic.ComparisonGridFromHeights\nimport OAI.NumberTheory.TotientAsymptotic.HeightAlignment\nimport OAI.NumberTheory.TotientAsymptotic.ActualComparisonCutoffs\nimport OAI.NumberTheory.TotientAsymptotic.SurvivingSlack\nimport OAI.NumberTheory.TotientAsymptotic.ComparisonExponentSaving\n\nimport OAI.NumberTheory.TotientAsymptotic.CollisionHeadCount\n\nimport OAI.NumberTheory.TotientAsymptotic.AllCollisionCount\n\nimport OAI.NumberTheory.TotientAsymptotic.NormalityDiscard\nimport OAI.NumberTheory.TotientAsymptotic.OmegaReciprocal\nimport OAI.NumberTheory.TotientAsymptotic.OmegaDiscard\n\nimport OAI.NumberTheory.TotientAsymptotic.UnconditionalMain\n\n\nFILE lean/OAI/Combinatorics/TriangleFace/Main.lean TAIL\n G c v hc\n  have hex := expansion_image_extreme G c v hc\n  refine ⟨him, ?_, ?_, restriction_extension G c v hc, ?_, ?_, ?_, ?_, ?_⟩\n  · rw [← him]\n    exact (convex_convexHull ℝ (Set.range G.indicator)).linear_image (localExtension G c v hc)\n  · rwa [← him]\n  · intro y hy\n    obtain ⟨x, hx, rfl⟩ := him.symm ▸ hy\n    rfl\n  · intro x e\n    rfl\n  · intro x i\n    rfl\n  · intro x hx e\n    rcases e with e | i\n    · exact hx e\n    · exact hx (localEdge G c v hc i)\n  · intro x hx\n    exact (CanonicalFace198.image_minFace\n      (localExtension G c v hc).toAffineMap (localExtension_injective G c v hc)\n      (convex_convexHull ℝ (Set.range G.indicator))\n      (convex_convexHull ℝ (Set.range (triangleExpansion G c v).indicator)) hex hx).symm\n\ntheorem triangle_expansion_face {V : Type uV} {E : Type uE}\n    [Fintype V] [Fintype E] [DecidableEq V] [DecidableEq E] (G : LooplessGraph V E) (v : V)\n    (hdeg : Fintype.card {e : E // G.Incident v e} = 3)\n    (_hmatching : Nonempty G.Matching) :\n    ∃ (c : E → Fin 3) (hc : LocalLabels G v c),\n      let H := triangleExpansion G c v\n      let f := localExtension G c v hc\n      let Q := {y | y ∈ H.polytope ∧ H.cutMass y (triangleVertices v) = 1}\n      f '' G.polytope = Q ∧\n      Convex ℝ Q ∧ IsExtreme ℝ H.polytope Q ∧\n      (∀ x, oldRestriction (f x) = x) ∧\n      (∀ y ∈ Q, f (oldRestriction y) = y) ∧\n      (∀ x e, f x (Sum.inl e) = x e) ∧\n      (∀ x i, f x (Sum.inr i) = x (localEdge G c v hc i)) ∧\n      (∀ x, (∀ e, x e = (1/3 : ℝ)) → ∀ e, f x e = (1/3 : ℝ)) ∧\n      (∀ x ∈ G.polytope, CanonicalFace198.minFace H.polytope (f x) = f '' CanonicalFace198.minFace G.polytope x) := by\n  obtain ⟨c,hc⟩ := labels_of_degree_three G v hdeg\n  exact ⟨c,hc,local_face G c v hc⟩\n\nend TriangleFace198\nend\n\nend OAI\n\n\nFILE lean/OAI/Probability/SLE/LowerPositivity.lean TAIL\nInnerProductSpace\nopen scoped Topology Laplacian\nopen Set Filter Metric Complex MeasureTheory\nopen Set Filter Metric Complex MeasureTheory InnerProductSpace\nopen scoped Topology Laplacian ENNReal\nopen scoped Topology ENNReal\nopen scoped Topology Laplacian RealInnerProductSpace\nopen Set Filter Metric Complex\nopen scoped Topology\nopen Set Filter Metric\nopen Set Filter MeasureTheory Metric\nopen Set Filter Complex\nopen scoped Topology ComplexConjugate NNReal\nopen scoped Topology ENNReal NNReal ComplexConjugate\nopen Set Filter Metric MeasureTheory Complex\nopen Set Filter MeasureTheory ProbabilityTheory Classical\nopen scoped Topology ENNReal NNReal\nopen Set Filter MeasureTheory ProbabilityTheory Metric Classical\nopen Set Filter\nopen scoped Topology NNReal ENNReal ComplexConjugate\nopen scoped ENNReal NNReal\nnoncomputable section\nopen Set MeasureTheory ProbabilityTheory Filter\nopen scoped ENNReal NNReal Topology\nsection\nopen Set Filter MeasureTheory\nopen scoped Topology ENNReal NNReal\n\nnamespace SLEExactGauge\n \ndef segment (γ : ℝ≥0 → ℂ) (s t : ℝ≥0) : Set ℂ := γ '' Icc s t\nuniverse v_lower\n \n\ndef SourceLowerMainTarget : Prop :=\n  ∀ (Ω : Type v_lower) (_ : MeasurableSpace Ω) (P : Measure Ω), IsProbabilityMeasure P →\n  ∀ κ : ℝ, 0 < κ → κ < ((8 : ℕ) : ℝ) →\n  ∀ (γ : Ω → ℝ≥0 → ℂ), IsOrdinaryChordalSLE κ γ P →\n  ∀ h : ℝ → ℝ, IsGauge h → HasSmallRadiusFormula h (hFormula κ) →\n    ∀ᵐ ω ∂P, ∀ s t : ℝ≥0, 0 < s → s < t →\n      0 < hausdorffGauge h (segment (γ ω) s t)\nend SLEExactGauge\n\nnamespace SLEExactGauge\ntheorem sourceLowerMain_proved : SourceLowerMainTarget := by\n  intro Ω m P hP κ hκ hκ8 γ hSLE h hh hf\n  let : IsProbabilityMeasure P := hP\n  exact source_lower_universe hκ hκ8 hSLE h hh hf\nend SLEExactGauge\nend\n\n \nend\nend\nend\n\nend OAI\n\nNOT LOCALLY DECLARED MatchingEntropyBounds.json lean/OAI/Combinatorics/MatchingEntropy/Main.lean ['OAI.MatchingEntropyBounds.Refined.pointwise_entropy']\nNOT LOCALLY DECLARED SelfSimilarCorollaries.json lean/OAI/MeasureTheory/SelfSimilar/DimensionCorollaries.lean ['OAI.EntropyRateDimension.Extensions.homogeneous_dimension_direct', 'OAI.CorSetReference.cor_set']\nNOT LOCALLY DECLARED SingletonLoopMatching.json lean/OAI/Computability/LoopMatching/Endpoint.lean ['OAI.LoopMatching.fullEndpoint']\nNOT LOCALLY DECLARED TotientCompanionZero.json lean/OAI/NumberTheory/TotientAsymptotic/Main.lean ['OAI.TotientAsymptotic.companion_zero_case']\n"
        }
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          "exports": [
            "apply_figure_style",
            "set_frame",
            "panel_letter",
            "focal_palette",
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          "source": "META_GREY = \"#888888\"\n\n\ndef apply_figure_style(*, frame=\"open\", font=None, sizes=(8, 7, 6), grid=False):\n    \"\"\"Set matplotlib rcParams for publication-grade output. Call once before plotting.\n\n    This sets mechanics (role-mapped font-size ladder, outward ticks, frameless\n    legends, 300-dpi save, Type-42 embedded fonts) — not a house aesthetic.\n    Frame, font and the size ladder are parameters.\n\n    frame : 'open' (bottom+left spines, default) | 'boxed' (all four) | 'none'\n    font  : sans-serif family name; None = system default sans-serif\n    sizes : (base, secondary, tick) — titles/axis-labels, legend/annotation, ticks\n    grid  : whether to draw axes.grid (default False)\n    \"\"\"\n    import matplotlib as mpl\n    if frame not in (\"open\", \"boxed\", \"none\"):\n        raise ValueError(f\"frame must be 'open'|'boxed'|'none', got {frame!r}\")\n\n    try:\n        import os, sys, glob, matplotlib.font_manager as fm\n        fdir = os.path.join(os.environ.get(\"CONDA_PREFIX\") or sys.prefix, \"fonts\")\n        if os.path.isdir(fdir):\n            known = {f.fname for f in fm.fontManager.ttflist}\n            for f in glob.glob(os.path.join(fdir, \"*.ttf\")):\n                if f not in known:\n                    fm.fontManager.addfont(f)\n    except Exception:\n        pass\n    base, secondary, tick = sizes\n    boxed = (frame == \"boxed\")\n    rc = {\n        \"font.family\": \"sans-serif\",\n        \"font.size\": base,\n        \"axes.labelsize\": base,\n        \"axes.titlesize\": base,\n        \"legend.fontsize\": secondary,\n        \"xtick.labelsize\": tick,\n        \"ytick.labelsize\": tick,\n        \"axes.linewidth\": 0.6,\n        \"xtick.direction\": \"out\", \"ytick.direction\": \"out\",\n        \"xtick.major.size\": 3, \"ytick.major.size\": 3,\n        \"xtick.major.width\": 0.6, \"ytick.major.width\": 0.6,\n        \"axes.spines.top\": boxed, \"axes.spines.right\": boxed,\n        \"axes.spines.left\": frame != \"none\", \"axes.spines.bottom\": frame != \"none\",\n        \"axes.grid\": bool(grid),\n        \"legend.frameon\": False,\n        \"figure.dpi\": 200,\n        \"savefig.dpi\": 300,\n        \"savefig.bbox\": \"tight\",\n        \"axes.titleweight\": \"normal\",\n        \"axes.titlelocation\": \"left\",\n        \"axes.labelweight\": \"normal\",\n        \"lines.linewidth\": 1.2,\n        \"patch.linewidth\": 0.6,\n        \"pdf.fonttype\": 42, \"ps.fonttype\": 42,\n    }\n    if font:\n        rc[\"font.sans-serif\"] = [font, \"DejaVu Sans\"]\n    mpl.rcParams.update(rc)\n\n\ndef set_frame(ax, style=\"open\"):\n    \"\"\"§3: set spine visibility on an existing axes. style ∈ {'open','boxed','none'}.\"\"\"\n    show = {\"open\": (False, False, True, True),\n            \"boxed\": (True, True, True, True),\n            \"none\": (False, False, False, False)}[style]\n    for side, vis in zip((\"top\", \"right\", \"bottom\", \"left\"), show):\n        ax.spines[side].set_visible(vis)\n        if vis:\n            ax.spines[side].set_linewidth(0.6)\n    ax.tick_params(direction=\"out\", length=0 if style == \"none\" else 3, width=0.6)\n\n\ndef panel_letter(ax, letter, dx=-0.18, dy=1.02, case=\"lower\", fontsize=None):\n    \"\"\"§5.7: bold panel letter outside top-left of axes. case ∈ {'lower','upper'}.\"\"\"\n    import matplotlib.pyplot as plt\n    if fontsize is None:\n        fontsize = plt.rcParams.get(\"font.size\", 8) + 1\n    s = letter.lower() if case == \"lower\" else letter.upper()\n    artist = ax.text(dx, dy, s, transform=ax.transAxes,\n                     fontweight=\"bold\", fontsize=fontsize, va=\"bottom\", ha=\"left\")\n    artist.set_gid(\"figure-style-panel-letter\")\n\n\ndef focal_palette(labels, focal, focal_color, other=\"muted\", base_colors=None):\n    \"\"\"§4.2: map labels → colors with the focal series visually dominant.\n\n    other='muted'   — desaturate base_colors (or a default cycle) toward gray\n    other='grey'    — uniform light gray for all non-focal\n    other='ordinal' — non-focal on a single light→dark gray ramp (input order)\n    \"\"\"\n    import matplotlib.colors as mcolors\n    import matplotlib.pyplot as plt\n    focal_set = {focal} if isinstance(focal, str) else set(focal)\n    n = len(labels)\n    if not focal_set & set(labels):\n        raise ValueError(f\"focal {focal!r} not found in labels\")\n    base_colors = [] if base_colors is None else list(base_colors)\n    if not base_colors:\n        base_colors = plt.rcParams[\"axes.prop_cycle\"].by_key().get(\"color\", [\"#444444\"])\n    if not base_colors:\n        base_colors = [\"#444444\"]\n    base_colors = [base_colors[i % len(base_colors)] for i in range(n)]\n    if other == \"grey\":\n        rest = [\"#BCBCBC\"] * n\n    elif other == \"ordinal\":\n        nf = max(1, n - len(focal_set))\n        ramp = [mcolors.to_hex((v, v, v)) for v in\n                ([0.55] if nf == 1 else [0.80 - 0.35 * i / (nf - 1) for i in range(nf)])]\n        rest, k = [], 0\n        for l in labels:\n            rest.append(ramp[min(k, nf - 1)]); k += (l not in focal_set)\n    elif other == \"muted\":\n        def mute(c):\n            r, g, b = mcolors.to_rgb(c)\n            m = (r + g + b) / 3\n            return mcolors.to_hex((0.3 * r + 0.7 * m, 0.3 * g + 0.7 * m, 0.3 * b + 0.7 * m))\n        rest = [mute(c) for c in base_colors]\n    else:\n        raise ValueError(\"other must be 'muted', 'grey', or 'ordinal'\")\n    return [focal_color if l in focal_set else rest[i] for i, l in enumerate(labels)]\n\n\ndef bar_with_points(ax, x, ymat, labels, colors, jitter=0.08, show_points=True,\n                    errorbar=None, point_alpha=0.5, point_size=8):\n    \"\"\"§6.1: bar = mean; optionally overlay raw points or draw an interval.\n\n    colors   : per-label color list (e.g. from focal_palette)\n    x, ymat and labels must have equal lengths; mismatches raise before drawing.\n    Each group must be a nonempty, finite, one-dimensional real numeric array.\n    Requested SD/CI intervals require at least two observations per group.\n    errorbar : None | 'sd' | 'ci95' — drawn only when show_points is False.\n               'ci95' is the t-distribution 95% CI of the mean\n               (half-width t_{0.975,n-1} · s/√n); correct at small n where the\n               z-approximation (1.96·s/√n) is markedly too narrow.\n    \"\"\"\n    import numpy as np\n    x = np.asarray(x)\n    if x.ndim != 1 or not np.issubdtype(x.dtype, np.number):\n        raise TypeError(\n            \"bar_with_points x must be one-dimensional numeric positions; \"\n            \"pass category names through labels\"\n        )\n    ymat = list(ymat)\n    labels = list(labels)\n    if len(x) != len(ymat) or len(x) != len(labels):\n        raise ValueError(\"bar_with_points x, ymat and labels must have equal lengths\")\n    if colors is not None and not isinstance(colors, str):\n        colors = list(colors)\n        if not colors:\n            colors = None\n    if x.dtype.kind not in \"iuf\" or not np.isfinite(x).all():\n        raise ValueError(\"bar_with_points x must contain finite real positions\")\n    for i, y in enumerate(ymat):\n        if np.ma.isMaskedArray(y) and np.ma.getmaskarray(y).any():\n            raise ValueError(f\"bar_with_points group {i} contains masked observations; resolve exclusions explicitly before plotting\")\n    ymat = [np.asarray(y) for y in ymat]\n    for i, y in enumerate(ymat):\n        if y.ndim != 1 or y.size == 0 or y.dtype.kind not in \"iuf\" or not np.isfinite(y).all():\n            raise ValueError(f\"bar_with_points group {i} must be nonempty, one-dimensional and finite real numeric data\")\n    if errorbar and not show_points:\n        if errorbar not in (\"sd\", \"ci95\"):\n            raise ValueError(\"errorbar must be None, 'sd', or 'ci95'\")\n        if any(y.size < 2 for y in ymat):\n            raise ValueError(\"SD/CI cannot be estimated from one observation; use raw points without an interval\")\n    means = np.array([np.mean(y) for y in ymat], float)\n    err = None\n    if errorbar and not show_points:\n        if errorbar == \"sd\":\n            err = np.array([np.std(y, ddof=1) for y in ymat])\n        elif errorbar == \"ci95\":\n            from scipy.stats import t\n            def _hw(y):\n                n = np.asarray(y).size\n                return t.ppf(0.975, n - 1) * np.std(y, ddof=1) / np.sqrt(n)\n            err = np.array([_hw(y) for y in ymat])\n    ax.bar(x, means, color=colors, width=0.7, edgecolor=\"none\",\n           yerr=err, error_kw={\"elinewidth\": 0.8, \"capsize\": 0})\n    if show_points:\n        for xi, ys in zip(x, ymat):\n            ys = np.asarray(ys)\n            if ys.ndim == 1 and ys.size > 0:\n                jit = (np.random.rand(ys.size) - 0.5) * 2 * jitter\n                ax.scatter(np.full(ys.size, xi) + jit, ys, s=point_size, color=\"black\",\n                           alpha=point_alpha, zorder=3, linewidths=0)\n    ax.set_xticks(x); ax.set_xticklabels(labels)\n    return ax\n\n\ndef strip_with_median(ax, groups, values, colors=None, jitter=0.12):\n    \"\"\"§6.1: jittered points + bold horizontal median tick per group.\n\n    groups and values must have equal lengths. Mismatches raise before drawing.\n    colors=None or an empty color sequence supplies default gray; a nonempty\n    color sequence cycles across groups.\n    Each group must be nonempty, one-dimensional and finite real numeric data;\n    resolve missing observations explicitly before plotting.\n    \"\"\"\n    import numpy as np\n    labs = list(groups)\n    values = list(values)\n    if len(labs) != len(values):\n        raise ValueError(\"strip_with_median groups and values must have equal lengths\")\n    for i, y in enumerate(values):\n        if np.ma.isMaskedArray(y) and np.ma.getmaskarray(y).any():\n            raise ValueError(f\"strip_with_median group {i} contains masked observations; resolve exclusions explicitly before plotting\")\n    values = [np.asarray(y) for y in values]\n    for i, y in enumerate(values):\n        if y.ndim != 1 or y.size == 0 or y.dtype.kind not in \"iuf\" or not np.isfinite(y).all():\n            raise ValueError(f\"strip_with_median group {i} must be nonempty, one-dimensional and finite real numeric data\")\n    colors = [] if colors is None else list(colors)\n    if not colors:\n        colors = [\"#444444\"]\n    for i, ys in enumerate(values):\n        c = colors[i % len(colors)]\n        ys = np.asarray(ys)\n        jit = (np.random.rand(ys.size) - 0.5) * 2 * jitter\n        ax.scatter(np.full(ys.size, i) + jit, ys, s=10, color=c, alpha=0.6, linewidths=0, zorder=2)\n        m = np.median(ys)\n        ax.plot([i - 0.22, i + 0.22], [m, m], color=\"black\", lw=1.6, zorder=3)\n    ax.set_xticks(range(len(labs))); ax.set_xticklabels(labs)\n    return ax\n\n\ndef goodness_arrow(ax, text=\"higher = better\", loc=\"upper left\", axis=\"y\", fontsize=None):\n    \"\"\"§3.6: small upright direction-of-goodness cue in the margin.\"\"\"\n    import matplotlib.pyplot as plt\n    if fontsize is None:\n        fontsize = plt.rcParams[\"legend.fontsize\"]\n    pos = {\"upper left\": (0.02, 0.98), \"upper right\": (0.98, 0.98),\n           \"lower left\": (0.02, 0.02), \"lower right\": (0.98, 0.02)}[loc]\n    ha = \"left\" if \"left\" in loc else \"right\"\n    va = \"top\" if \"upper\" in loc else \"bottom\"\n    arrow = \"↑ \" if axis == \"y\" else \"→ \"\n    ax.text(pos[0], pos[1], arrow + text, transform=ax.transAxes,\n            fontsize=fontsize, color=META_GREY, ha=ha, va=va)\n\n\ndef two_tier_label(name, meta):\n    \"\"\"§5: two-line label string (name / metadata). Meta line styled separately by caller.\"\"\"\n    return f\"{name}\\n{meta}\"\n\n\ndef end_of_line_labels(ax, xs, ys, labels, colors=None, dx=0.01, fontsize=None):\n    \"\"\"§6.3 / §7.3: label each line series at its right end instead of a legend box.\n\n    Each label requires a nonempty, matched x/y series. Validate all series\n    before drawing. Nonempty colors cycle; None or empty uses default text color.\n    Inner sequences must support len() and indexing; they are not copied.\n    Nonfinite endpoints are rejected rather than producing an invisible label.\n    \"\"\"\n    import matplotlib.pyplot as plt\n    import math\n    xs, ys, labels = list(xs), list(ys), list(labels)\n    if len(xs) != len(ys) or len(xs) != len(labels):\n        raise ValueError(\"end_of_line_labels xs, ys and labels must have equal lengths\")\n    if any(len(x) == 0 or len(x) != len(y) for x, y in zip(xs, ys)):\n        raise ValueError(\"end_of_line_labels each x/y series must be nonempty and have equal lengths\")\n    if any(not math.isfinite(x[-1]) or not math.isfinite(y[-1]) for x, y in zip(xs, ys)):\n        raise ValueError(\"end_of_line_labels endpoints must be finite; choose valid endpoints explicitly\")\n    if fontsize is None:\n        fontsize = plt.rcParams[\"font.size\"]\n    colors = [] if colors is None else list(colors)\n    if not colors:\n        colors = [None]\n    span = ax.get_xlim()[1] - ax.get_xlim()[0]\n    for i, (x, y, lab) in enumerate(zip(xs, ys, labels)):\n        c = colors[i % len(colors)]\n        ax.text(x[-1] + dx * span, y[-1], lab, color=c, va=\"center\", ha=\"left\", fontsize=fontsize)\n\n\ndef panel_crops(fig, dpi=None, pad_px=6, bbox_inches=None, pad_inches=None):\n    \"\"\"§9.2: pixel-space crop boxes for each lettered panel in the SAVED PNG.\n\n    Returns ``{letter: (x0, y0, x1, y1)}`` in image-space pixels (origin\n    top-left, matching PIL's ``Image.crop``). Panels are detected as bold\n    single-character ``Text``\n    objects placed by :func:`panel_letter`; each panel's crop is its axes'\n    tightbbox mapped into the saved file's pixel space, padded by ``pad_px``.\n    For §3.4 composites (abutting subplots sharing an axis, letter on the\n    leftmost only) the crop unions in letterless ``sharex``/``sharey`` siblings\n    on the same grid row/col so the whole composite is covered. When no axes\n    carries a panel letter (standalone plot, or a figure-composer sub-agent),\n    falls back to one crop per axes keyed by index.\n\n    ``bbox_inches`` mirrors ``Figure.savefig`` semantics: ``None`` means\n    *consult rcParams* (so under :func:`apply_figure_style` it resolves to\n    ``'tight'``); pass an explicit ``Bbox`` only if you saved with one. The\n    boxes are clamped to the saved image extent; panels with no intersection\n    are omitted.\n\n        >>> from PIL import Image\n        >>> fig.savefig(\"fig.png\")  # bbox_inches='tight' via rcParams\n        >>> with Image.open(\"fig.png\") as saved:\n        ...     for letter, box in panel_crops(fig).items():\n        ...         saved.crop(box).save(f\"panel_{letter}.png\")\n\n    Match savefig's DPI, bbox_inches and pad_inches when overriding defaults.\n    Returned boxes are pixels, never savefig bbox_inches (inches). Passing them\n    back to savefig as inches can allocate an enormous raster.\n    \"\"\"\n    import matplotlib as mpl\n    import matplotlib.text\n    from matplotlib.backends.backend_agg import FigureCanvasAgg\n    if isinstance(pad_px, bool) or not isinstance(pad_px, int) or pad_px < 0:\n        raise TypeError(\"panel_crops pad_px must be a non-negative integer\")\n    if dpi is None:\n        dpi = mpl.rcParams.get(\"savefig.dpi\", fig.dpi)\n        if dpi == \"figure\":\n            dpi = fig.dpi\n    dpi = float(dpi)\n    if bbox_inches is None:\n        bbox_inches = mpl.rcParams.get(\"savefig.bbox\")\n    # The Open-Science notebook captures and closes pyplot figures after every\n    # cell. Matplotlib 3.11 decouples a closed Figure from its backend by\n    # replacing FigureCanvasAgg with FigureCanvasBase, while the persistent\n    # notebook namespace can still retain the Figure itself. Reattach the\n    # headless canvas before asking for saved-image geometry so this helper is\n    # safe in a later QA cell as well as the producer cell.\n    if not hasattr(fig.canvas, \"get_renderer\"):\n        FigureCanvasAgg(fig)\n    fig.canvas.draw()\n    r = fig.canvas.get_renderer()\n\n    if bbox_inches == \"tight\":\n        if pad_inches is None:\n            pad_inches = mpl.rcParams.get(\"savefig.pad_inches\", 0.1)\n        tb = fig.get_tightbbox(r).padded(pad_inches)\n        ox_in, oy_in = tb.x0, tb.y0\n        W_in, H_in = tb.width, tb.height\n    elif isinstance(bbox_inches, mpl.transforms.BboxBase):\n        ox_in, oy_in = bbox_inches.x0, bbox_inches.y0\n        W_in, H_in = bbox_inches.width, bbox_inches.height\n    elif bbox_inches is None:\n        ox_in, oy_in = 0.0, 0.0\n        W_in, H_in = fig.get_size_inches()\n    else:\n        raise TypeError(\"panel_crops bbox_inches must be None, 'tight', or a matplotlib Bbox\")\n    W_px, H_px = int(round(W_in * dpi)), int(round(H_in * dpi))\n    lettered = {}\n    seen_letters = set()\n    for ax in fig.axes:\n        for t in ax.findobj(matplotlib.text.Text):\n            s = (t.get_text() or \"\").strip()\n            if t.get_gid() == \"figure-style-panel-letter\":\n                if s in seen_letters:\n                    raise ValueError(f\"Duplicate panel letter: {s!r}\")\n                seen_letters.add(s)\n                lettered[ax] = s\n                break\n\n\n\n    if not lettered:\n        lettered = {ax: str(i) for i, ax in enumerate(fig.axes)}\n    out = {}\n    for ax, letter in lettered.items():\n        bbs = [ax.get_tightbbox(r)]\n\n\n\n\n        ss = ax.get_subplotspec()\n        for sib in fig.axes:\n            if sib is ax or sib in lettered:\n                continue\n            ssib = sib.get_subplotspec()\n            same_row = ss is None or ssib is None or ss.rowspan == ssib.rowspan\n            same_col = ss is None or ssib is None or ss.colspan == ssib.colspan\n            if ((ax.get_shared_y_axes().joined(ax, sib) and same_row)\n                    or (ax.get_shared_x_axes().joined(ax, sib) and same_col)):\n                bbs.append(sib.get_tightbbox(r))\n        bb = mpl.transforms.Bbox.union(bbs)\n\n        bx0 = (bb.x0 / fig.dpi - ox_in) * dpi\n        bx1 = (bb.x1 / fig.dpi - ox_in) * dpi\n        by0 = H_px - (bb.y1 / fig.dpi - oy_in) * dpi\n        by1 = H_px - (bb.y0 / fig.dpi - oy_in) * dpi\n        box = (\n            min(max(int(bx0) - pad_px, 0), W_px),\n            min(max(int(by0) - pad_px, 0), H_px),\n            max(min(int(bx1) + pad_px, W_px), 0),\n            max(min(int(by1) + pad_px, H_px), 0),\n        )\n        if box[0] < box[2] and box[1] < box[3]:\n            out[letter] = box\n    return out\n",
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      "script": "def lean_code_only(text):\n out=list(text);i=0;depth=0;string=False\n while i<len(text):\n  if depth:\n   if text.startswith('/-',i):out[i:i+2]='  ';depth+=1;i+=2;continue\n   if text.startswith('-/',i):out[i:i+2]='  ';depth-=1;i+=2;continue\n   if text[i]!='\\n':out[i]=' '\n   i+=1;continue\n  if string:\n   if text[i]=='\\\\':\n    out[i]=' ';i+=1\n    if i<len(text):out[i]=' ';i+=1\n    continue\n   if text[i]=='\"':string=False\n   if text[i]!='\\n':out[i]=' '\n   i+=1;continue\n  if text.startswith('/-',i):out[i:i+2]='  ';depth=1;i+=2;continue\n  if text.startswith('--',i):\n   j=text.find('\\n',i);j=len(text) if j<0 else j;out[i:j]=' '*(j-i);i=j;continue\n  if text[i]=='\"':string=True;out[i]=' '\n  i+=1\n return ''.join(out)\nall_impl={k:r for k,r in combined.items() if k.startswith('lean/OAI/')}\nscan={}\nfor k,r in all_impl.items():\n code=lean_code_only(r['content']);imports=re.findall(r'^import (\\S+)',code,re.M)\n scan[k]={'sha256':r['sha256'],'placeholder_tokens':[{'token':m.group(),'line':code[:m.start()].count('\\n')+1} for m in re.finditer(r'\\b(?:sorry|admit|sorryAx)\\b',code)],'axiom_declarations':[code[:m.start()].count('\\n')+1 for m in re.finditer(r'^\\s*(?:private\\s+)?axiom\\s',code,re.M)],'imports':imports,'missing_direct_OAI_imports':[x for x in imports if x.startswith('OAI.') and 'lean/'+x.replace('.','/')+'.lean' not in combined]}\nmissing_direct=sorted({x for s in scan.values() for x in s['missing_direct_OAI_imports']})\nmodule_evidence=[]\nfor cp,cfg in configs.items():\n sol='lean/'+cfg['solution_module'].replace('.','/')+'.lean';code=lean_code_only(combined[sol]['content']);decs=[]\n for target in cfg['theorem_names']:\n  name=target.rsplit('.',1)[-1];pattern=r'^(?:theorem|lemma|def|abbrev)\\s+(?:[\\w]+\\.)*'+re.escape(name)+r'(?=[\\s:{(])'\n  m=re.search(pattern,code,re.M)\n  matches=[]\n  if m:matches=[{'path':sol,'line':code[:m.start()].count('\\n')+1,'placement':'local declaration'}]\n  else:\n   for other,rr in all_impl.items():\n    oc=lean_code_only(rr['content']);om=re.search(pattern,oc,re.M)\n    if om:matches.append({'path':other,'line':oc[:om.start()].count('\\n')+1,'placement':'declaration in another supplied file; import/name resolution not compiler-checked'})\n  decs.append({'target':target,'local_declaration_found':bool(m),'declaration_locations':matches,'status':'local body present' if m else 'body in supplied import' if matches else 'imported body not supplied'})\n module_evidence.append({'configuration':cp,'solution_path':sol,'sha256':combined[sol]['sha256'],'declarations':decs,**scan[sol]})\nprint('implementation files',len(all_impl),'token findings',[(k,v['placeholder_tokens'],v['axiom_declarations']) for k,v in scan.items() if v['placeholder_tokens'] or v['axiom_declarations']])\nprint('missing distinct direct OAI imports',len(missing_direct))\nprint('declaration status',dict(collections.Counter(d['status'] for m in module_evidence for d in m['declarations'])))\nprint('nonlocal',[(m['configuration'],m['declarations']) for m in module_evidence if any(not d['local_declaration_found'] for d in m['declarations'])])",
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      "text": {
        "stdout": "implementation files 24 token findings []\nmissing distinct direct OAI imports 270\ndeclaration status {'local body present': 25, 'imported body not supplied': 4, 'body in supplied import': 1}\nnonlocal [('lean/ComparatorChallenges/MatchingEntropyBounds.json', [{'target': 'OAI.MatchingEntropyBounds.Refined.pointwise_entropy', 'local_declaration_found': False, 'declaration_locations': [], 'status': 'imported body not supplied'}]), ('lean/ComparatorChallenges/SelfSimilarCorollaries.json', [{'target': 'OAI.EntropyRateDimension.Extensions.homogeneous_dimension_direct', 'local_declaration_found': False, 'declaration_locations': [], 'status': 'imported body not supplied'}, {'target': 'OAI.CorSetReference.cor_set', 'local_declaration_found': False, 'declaration_locations': [], 'status': 'imported body not supplied'}]), ('lean/ComparatorChallenges/SingletonLoopMatching.json', [{'target': 'OAI.LoopMatching.fullEndpoint', 'local_declaration_found': False, 'declaration_locations': [{'path': 'lean/OAI/Computability/LoopMatching/Main.lean', 'line': 24, 'placement': 'declaration in another supplied file; import/name resolution not compiler-checked'}], 'status': 'body in supplied import'}]), ('lean/ComparatorChallenges/TotientCompanionZero.json', [{'target': 'OAI.TotientAsymptotic.companion_zero_case', 'local_declaration_found': False, 'declaration_locations': [], 'status': 'imported body not supplied'}])]\n",
        "stderr": "",
        "traceback": "",
        "plain": [
          "implementation files 24 token findings []\nmissing distinct direct OAI imports 270\ndeclaration status {'local body present': 25, 'imported body not supplied': 4, 'body in supplied import': 1}\nnonlocal [('lean/ComparatorChallenges/MatchingEntropyBounds.json', [{'target': 'OAI.MatchingEntropyBounds.Refined.pointwise_entropy', 'local_declaration_found': False, 'declaration_locations': [], 'status': 'imported body not supplied'}]), ('lean/ComparatorChallenges/SelfSimilarCorollaries.json', [{'target': 'OAI.EntropyRateDimension.Extensions.homogeneous_dimension_direct', 'local_declaration_found': False, 'declaration_locations': [], 'status': 'imported body not supplied'}, {'target': 'OAI.CorSetReference.cor_set', 'local_declaration_found': False, 'declaration_locations': [], 'status': 'imported body not supplied'}]), ('lean/ComparatorChallenges/SingletonLoopMatching.json', [{'target': 'OAI.LoopMatching.fullEndpoint', 'local_declaration_found': False, 'declaration_locations': [{'path': 'lean/OAI/Computability/LoopMatching/Main.lean', 'line': 24, 'placement': 'declaration in another supplied file; import/name resolution not compiler-checked'}], 'status': 'body in supplied import'}]), ('lean/ComparatorChallenges/TotientCompanionZero.json', [{'target': 'OAI.TotientAsymptotic.companion_zero_case', 'local_declaration_found': False, 'declaration_locations': [], 'status': 'imported body not supplied'}])]\n"
        ]
      },
      "outputs": [
        {
          "type": "stream",
          "name": "stdout",
          "text": "implementation files 24 token findings []\nmissing distinct direct OAI imports 270\ndeclaration status {'local body present': 25, 'imported body not supplied': 4, 'body in supplied import': 1}\nnonlocal [('lean/ComparatorChallenges/MatchingEntropyBounds.json', [{'target': 'OAI.MatchingEntropyBounds.Refined.pointwise_entropy', 'local_declaration_found': False, 'declaration_locations': [], 'status': 'imported body not supplied'}]), ('lean/ComparatorChallenges/SelfSimilarCorollaries.json', [{'target': 'OAI.EntropyRateDimension.Extensions.homogeneous_dimension_direct', 'local_declaration_found': False, 'declaration_locations': [], 'status': 'imported body not supplied'}, {'target': 'OAI.CorSetReference.cor_set', 'local_declaration_found': False, 'declaration_locations': [], 'status': 'imported body not supplied'}]), ('lean/ComparatorChallenges/SingletonLoopMatching.json', [{'target': 'OAI.LoopMatching.fullEndpoint', 'local_declaration_found': False, 'declaration_locations': [{'path': 'lean/OAI/Computability/LoopMatching/Main.lean', 'line': 24, 'placement': 'declaration in another supplied file; import/name resolution not compiler-checked'}], 'status': 'body in supplied import'}]), ('lean/ComparatorChallenges/TotientCompanionZero.json', [{'target': 'OAI.TotientAsymptotic.companion_zero_case', 'local_declaration_found': False, 'declaration_locations': [], 'status': 'imported body not supplied'}])]\n"
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          ],
          "source": "META_GREY = \"#888888\"\n\n\ndef apply_figure_style(*, frame=\"open\", font=None, sizes=(8, 7, 6), grid=False):\n    \"\"\"Set matplotlib rcParams for publication-grade output. Call once before plotting.\n\n    This sets mechanics (role-mapped font-size ladder, outward ticks, frameless\n    legends, 300-dpi save, Type-42 embedded fonts) — not a house aesthetic.\n    Frame, font and the size ladder are parameters.\n\n    frame : 'open' (bottom+left spines, default) | 'boxed' (all four) | 'none'\n    font  : sans-serif family name; None = system default sans-serif\n    sizes : (base, secondary, tick) — titles/axis-labels, legend/annotation, ticks\n    grid  : whether to draw axes.grid (default False)\n    \"\"\"\n    import matplotlib as mpl\n    if frame not in (\"open\", \"boxed\", \"none\"):\n        raise ValueError(f\"frame must be 'open'|'boxed'|'none', got {frame!r}\")\n\n    try:\n        import os, sys, glob, matplotlib.font_manager as fm\n        fdir = os.path.join(os.environ.get(\"CONDA_PREFIX\") or sys.prefix, \"fonts\")\n        if os.path.isdir(fdir):\n            known = {f.fname for f in fm.fontManager.ttflist}\n            for f in glob.glob(os.path.join(fdir, \"*.ttf\")):\n                if f not in known:\n                    fm.fontManager.addfont(f)\n    except Exception:\n        pass\n    base, secondary, tick = sizes\n    boxed = (frame == \"boxed\")\n    rc = {\n        \"font.family\": \"sans-serif\",\n        \"font.size\": base,\n        \"axes.labelsize\": base,\n        \"axes.titlesize\": base,\n        \"legend.fontsize\": secondary,\n        \"xtick.labelsize\": tick,\n        \"ytick.labelsize\": tick,\n        \"axes.linewidth\": 0.6,\n        \"xtick.direction\": \"out\", \"ytick.direction\": \"out\",\n        \"xtick.major.size\": 3, \"ytick.major.size\": 3,\n        \"xtick.major.width\": 0.6, \"ytick.major.width\": 0.6,\n        \"axes.spines.top\": boxed, \"axes.spines.right\": boxed,\n        \"axes.spines.left\": frame != \"none\", \"axes.spines.bottom\": frame != \"none\",\n        \"axes.grid\": bool(grid),\n        \"legend.frameon\": False,\n        \"figure.dpi\": 200,\n        \"savefig.dpi\": 300,\n        \"savefig.bbox\": \"tight\",\n        \"axes.titleweight\": \"normal\",\n        \"axes.titlelocation\": \"left\",\n        \"axes.labelweight\": \"normal\",\n        \"lines.linewidth\": 1.2,\n        \"patch.linewidth\": 0.6,\n        \"pdf.fonttype\": 42, \"ps.fonttype\": 42,\n    }\n    if font:\n        rc[\"font.sans-serif\"] = [font, \"DejaVu Sans\"]\n    mpl.rcParams.update(rc)\n\n\ndef set_frame(ax, style=\"open\"):\n    \"\"\"§3: set spine visibility on an existing axes. style ∈ {'open','boxed','none'}.\"\"\"\n    show = {\"open\": (False, False, True, True),\n            \"boxed\": (True, True, True, True),\n            \"none\": (False, False, False, False)}[style]\n    for side, vis in zip((\"top\", \"right\", \"bottom\", \"left\"), show):\n        ax.spines[side].set_visible(vis)\n        if vis:\n            ax.spines[side].set_linewidth(0.6)\n    ax.tick_params(direction=\"out\", length=0 if style == \"none\" else 3, width=0.6)\n\n\ndef panel_letter(ax, letter, dx=-0.18, dy=1.02, case=\"lower\", fontsize=None):\n    \"\"\"§5.7: bold panel letter outside top-left of axes. case ∈ {'lower','upper'}.\"\"\"\n    import matplotlib.pyplot as plt\n    if fontsize is None:\n        fontsize = plt.rcParams.get(\"font.size\", 8) + 1\n    s = letter.lower() if case == \"lower\" else letter.upper()\n    artist = ax.text(dx, dy, s, transform=ax.transAxes,\n                     fontweight=\"bold\", fontsize=fontsize, va=\"bottom\", ha=\"left\")\n    artist.set_gid(\"figure-style-panel-letter\")\n\n\ndef focal_palette(labels, focal, focal_color, other=\"muted\", base_colors=None):\n    \"\"\"§4.2: map labels → colors with the focal series visually dominant.\n\n    other='muted'   — desaturate base_colors (or a default cycle) toward gray\n    other='grey'    — uniform light gray for all non-focal\n    other='ordinal' — non-focal on a single light→dark gray ramp (input order)\n    \"\"\"\n    import matplotlib.colors as mcolors\n    import matplotlib.pyplot as plt\n    focal_set = {focal} if isinstance(focal, str) else set(focal)\n    n = len(labels)\n    if not focal_set & set(labels):\n        raise ValueError(f\"focal {focal!r} not found in labels\")\n    base_colors = [] if base_colors is None else list(base_colors)\n    if not base_colors:\n        base_colors = plt.rcParams[\"axes.prop_cycle\"].by_key().get(\"color\", [\"#444444\"])\n    if not base_colors:\n        base_colors = [\"#444444\"]\n    base_colors = [base_colors[i % len(base_colors)] for i in range(n)]\n    if other == \"grey\":\n        rest = [\"#BCBCBC\"] * n\n    elif other == \"ordinal\":\n        nf = max(1, n - len(focal_set))\n        ramp = [mcolors.to_hex((v, v, v)) for v in\n                ([0.55] if nf == 1 else [0.80 - 0.35 * i / (nf - 1) for i in range(nf)])]\n        rest, k = [], 0\n        for l in labels:\n            rest.append(ramp[min(k, nf - 1)]); k += (l not in focal_set)\n    elif other == \"muted\":\n        def mute(c):\n            r, g, b = mcolors.to_rgb(c)\n            m = (r + g + b) / 3\n            return mcolors.to_hex((0.3 * r + 0.7 * m, 0.3 * g + 0.7 * m, 0.3 * b + 0.7 * m))\n        rest = [mute(c) for c in base_colors]\n    else:\n        raise ValueError(\"other must be 'muted', 'grey', or 'ordinal'\")\n    return [focal_color if l in focal_set else rest[i] for i, l in enumerate(labels)]\n\n\ndef bar_with_points(ax, x, ymat, labels, colors, jitter=0.08, show_points=True,\n                    errorbar=None, point_alpha=0.5, point_size=8):\n    \"\"\"§6.1: bar = mean; optionally overlay raw points or draw an interval.\n\n    colors   : per-label color list (e.g. from focal_palette)\n    x, ymat and labels must have equal lengths; mismatches raise before drawing.\n    Each group must be a nonempty, finite, one-dimensional real numeric array.\n    Requested SD/CI intervals require at least two observations per group.\n    errorbar : None | 'sd' | 'ci95' — drawn only when show_points is False.\n               'ci95' is the t-distribution 95% CI of the mean\n               (half-width t_{0.975,n-1} · s/√n); correct at small n where the\n               z-approximation (1.96·s/√n) is markedly too narrow.\n    \"\"\"\n    import numpy as np\n    x = np.asarray(x)\n    if x.ndim != 1 or not np.issubdtype(x.dtype, np.number):\n        raise TypeError(\n            \"bar_with_points x must be one-dimensional numeric positions; \"\n            \"pass category names through labels\"\n        )\n    ymat = list(ymat)\n    labels = list(labels)\n    if len(x) != len(ymat) or len(x) != len(labels):\n        raise ValueError(\"bar_with_points x, ymat and labels must have equal lengths\")\n    if colors is not None and not isinstance(colors, str):\n        colors = list(colors)\n        if not colors:\n            colors = None\n    if x.dtype.kind not in \"iuf\" or not np.isfinite(x).all():\n        raise ValueError(\"bar_with_points x must contain finite real positions\")\n    for i, y in enumerate(ymat):\n        if np.ma.isMaskedArray(y) and np.ma.getmaskarray(y).any():\n            raise ValueError(f\"bar_with_points group {i} contains masked observations; resolve exclusions explicitly before plotting\")\n    ymat = [np.asarray(y) for y in ymat]\n    for i, y in enumerate(ymat):\n        if y.ndim != 1 or y.size == 0 or y.dtype.kind not in \"iuf\" or not np.isfinite(y).all():\n            raise ValueError(f\"bar_with_points group {i} must be nonempty, one-dimensional and finite real numeric data\")\n    if errorbar and not show_points:\n        if errorbar not in (\"sd\", \"ci95\"):\n            raise ValueError(\"errorbar must be None, 'sd', or 'ci95'\")\n        if any(y.size < 2 for y in ymat):\n            raise ValueError(\"SD/CI cannot be estimated from one observation; use raw points without an interval\")\n    means = np.array([np.mean(y) for y in ymat], float)\n    err = None\n    if errorbar and not show_points:\n        if errorbar == \"sd\":\n            err = np.array([np.std(y, ddof=1) for y in ymat])\n        elif errorbar == \"ci95\":\n            from scipy.stats import t\n            def _hw(y):\n                n = np.asarray(y).size\n                return t.ppf(0.975, n - 1) * np.std(y, ddof=1) / np.sqrt(n)\n            err = np.array([_hw(y) for y in ymat])\n    ax.bar(x, means, color=colors, width=0.7, edgecolor=\"none\",\n           yerr=err, error_kw={\"elinewidth\": 0.8, \"capsize\": 0})\n    if show_points:\n        for xi, ys in zip(x, ymat):\n            ys = np.asarray(ys)\n            if ys.ndim == 1 and ys.size > 0:\n                jit = (np.random.rand(ys.size) - 0.5) * 2 * jitter\n                ax.scatter(np.full(ys.size, xi) + jit, ys, s=point_size, color=\"black\",\n                           alpha=point_alpha, zorder=3, linewidths=0)\n    ax.set_xticks(x); ax.set_xticklabels(labels)\n    return ax\n\n\ndef strip_with_median(ax, groups, values, colors=None, jitter=0.12):\n    \"\"\"§6.1: jittered points + bold horizontal median tick per group.\n\n    groups and values must have equal lengths. Mismatches raise before drawing.\n    colors=None or an empty color sequence supplies default gray; a nonempty\n    color sequence cycles across groups.\n    Each group must be nonempty, one-dimensional and finite real numeric data;\n    resolve missing observations explicitly before plotting.\n    \"\"\"\n    import numpy as np\n    labs = list(groups)\n    values = list(values)\n    if len(labs) != len(values):\n        raise ValueError(\"strip_with_median groups and values must have equal lengths\")\n    for i, y in enumerate(values):\n        if np.ma.isMaskedArray(y) and np.ma.getmaskarray(y).any():\n            raise ValueError(f\"strip_with_median group {i} contains masked observations; resolve exclusions explicitly before plotting\")\n    values = [np.asarray(y) for y in values]\n    for i, y in enumerate(values):\n        if y.ndim != 1 or y.size == 0 or y.dtype.kind not in \"iuf\" or not np.isfinite(y).all():\n            raise ValueError(f\"strip_with_median group {i} must be nonempty, one-dimensional and finite real numeric data\")\n    colors = [] if colors is None else list(colors)\n    if not colors:\n        colors = [\"#444444\"]\n    for i, ys in enumerate(values):\n        c = colors[i % len(colors)]\n        ys = np.asarray(ys)\n        jit = (np.random.rand(ys.size) - 0.5) * 2 * jitter\n        ax.scatter(np.full(ys.size, i) + jit, ys, s=10, color=c, alpha=0.6, linewidths=0, zorder=2)\n        m = np.median(ys)\n        ax.plot([i - 0.22, i + 0.22], [m, m], color=\"black\", lw=1.6, zorder=3)\n    ax.set_xticks(range(len(labs))); ax.set_xticklabels(labs)\n    return ax\n\n\ndef goodness_arrow(ax, text=\"higher = better\", loc=\"upper left\", axis=\"y\", fontsize=None):\n    \"\"\"§3.6: small upright direction-of-goodness cue in the margin.\"\"\"\n    import matplotlib.pyplot as plt\n    if fontsize is None:\n        fontsize = plt.rcParams[\"legend.fontsize\"]\n    pos = {\"upper left\": (0.02, 0.98), \"upper right\": (0.98, 0.98),\n           \"lower left\": (0.02, 0.02), \"lower right\": (0.98, 0.02)}[loc]\n    ha = \"left\" if \"left\" in loc else \"right\"\n    va = \"top\" if \"upper\" in loc else \"bottom\"\n    arrow = \"↑ \" if axis == \"y\" else \"→ \"\n    ax.text(pos[0], pos[1], arrow + text, transform=ax.transAxes,\n            fontsize=fontsize, color=META_GREY, ha=ha, va=va)\n\n\ndef two_tier_label(name, meta):\n    \"\"\"§5: two-line label string (name / metadata). Meta line styled separately by caller.\"\"\"\n    return f\"{name}\\n{meta}\"\n\n\ndef end_of_line_labels(ax, xs, ys, labels, colors=None, dx=0.01, fontsize=None):\n    \"\"\"§6.3 / §7.3: label each line series at its right end instead of a legend box.\n\n    Each label requires a nonempty, matched x/y series. Validate all series\n    before drawing. Nonempty colors cycle; None or empty uses default text color.\n    Inner sequences must support len() and indexing; they are not copied.\n    Nonfinite endpoints are rejected rather than producing an invisible label.\n    \"\"\"\n    import matplotlib.pyplot as plt\n    import math\n    xs, ys, labels = list(xs), list(ys), list(labels)\n    if len(xs) != len(ys) or len(xs) != len(labels):\n        raise ValueError(\"end_of_line_labels xs, ys and labels must have equal lengths\")\n    if any(len(x) == 0 or len(x) != len(y) for x, y in zip(xs, ys)):\n        raise ValueError(\"end_of_line_labels each x/y series must be nonempty and have equal lengths\")\n    if any(not math.isfinite(x[-1]) or not math.isfinite(y[-1]) for x, y in zip(xs, ys)):\n        raise ValueError(\"end_of_line_labels endpoints must be finite; choose valid endpoints explicitly\")\n    if fontsize is None:\n        fontsize = plt.rcParams[\"font.size\"]\n    colors = [] if colors is None else list(colors)\n    if not colors:\n        colors = [None]\n    span = ax.get_xlim()[1] - ax.get_xlim()[0]\n    for i, (x, y, lab) in enumerate(zip(xs, ys, labels)):\n        c = colors[i % len(colors)]\n        ax.text(x[-1] + dx * span, y[-1], lab, color=c, va=\"center\", ha=\"left\", fontsize=fontsize)\n\n\ndef panel_crops(fig, dpi=None, pad_px=6, bbox_inches=None, pad_inches=None):\n    \"\"\"§9.2: pixel-space crop boxes for each lettered panel in the SAVED PNG.\n\n    Returns ``{letter: (x0, y0, x1, y1)}`` in image-space pixels (origin\n    top-left, matching PIL's ``Image.crop``). Panels are detected as bold\n    single-character ``Text``\n    objects placed by :func:`panel_letter`; each panel's crop is its axes'\n    tightbbox mapped into the saved file's pixel space, padded by ``pad_px``.\n    For §3.4 composites (abutting subplots sharing an axis, letter on the\n    leftmost only) the crop unions in letterless ``sharex``/``sharey`` siblings\n    on the same grid row/col so the whole composite is covered. When no axes\n    carries a panel letter (standalone plot, or a figure-composer sub-agent),\n    falls back to one crop per axes keyed by index.\n\n    ``bbox_inches`` mirrors ``Figure.savefig`` semantics: ``None`` means\n    *consult rcParams* (so under :func:`apply_figure_style` it resolves to\n    ``'tight'``); pass an explicit ``Bbox`` only if you saved with one. The\n    boxes are clamped to the saved image extent; panels with no intersection\n    are omitted.\n\n        >>> from PIL import Image\n        >>> fig.savefig(\"fig.png\")  # bbox_inches='tight' via rcParams\n        >>> with Image.open(\"fig.png\") as saved:\n        ...     for letter, box in panel_crops(fig).items():\n        ...         saved.crop(box).save(f\"panel_{letter}.png\")\n\n    Match savefig's DPI, bbox_inches and pad_inches when overriding defaults.\n    Returned boxes are pixels, never savefig bbox_inches (inches). Passing them\n    back to savefig as inches can allocate an enormous raster.\n    \"\"\"\n    import matplotlib as mpl\n    import matplotlib.text\n    from matplotlib.backends.backend_agg import FigureCanvasAgg\n    if isinstance(pad_px, bool) or not isinstance(pad_px, int) or pad_px < 0:\n        raise TypeError(\"panel_crops pad_px must be a non-negative integer\")\n    if dpi is None:\n        dpi = mpl.rcParams.get(\"savefig.dpi\", fig.dpi)\n        if dpi == \"figure\":\n            dpi = fig.dpi\n    dpi = float(dpi)\n    if bbox_inches is None:\n        bbox_inches = mpl.rcParams.get(\"savefig.bbox\")\n    # The Open-Science notebook captures and closes pyplot figures after every\n    # cell. Matplotlib 3.11 decouples a closed Figure from its backend by\n    # replacing FigureCanvasAgg with FigureCanvasBase, while the persistent\n    # notebook namespace can still retain the Figure itself. Reattach the\n    # headless canvas before asking for saved-image geometry so this helper is\n    # safe in a later QA cell as well as the producer cell.\n    if not hasattr(fig.canvas, \"get_renderer\"):\n        FigureCanvasAgg(fig)\n    fig.canvas.draw()\n    r = fig.canvas.get_renderer()\n\n    if bbox_inches == \"tight\":\n        if pad_inches is None:\n            pad_inches = mpl.rcParams.get(\"savefig.pad_inches\", 0.1)\n        tb = fig.get_tightbbox(r).padded(pad_inches)\n        ox_in, oy_in = tb.x0, tb.y0\n        W_in, H_in = tb.width, tb.height\n    elif isinstance(bbox_inches, mpl.transforms.BboxBase):\n        ox_in, oy_in = bbox_inches.x0, bbox_inches.y0\n        W_in, H_in = bbox_inches.width, bbox_inches.height\n    elif bbox_inches is None:\n        ox_in, oy_in = 0.0, 0.0\n        W_in, H_in = fig.get_size_inches()\n    else:\n        raise TypeError(\"panel_crops bbox_inches must be None, 'tight', or a matplotlib Bbox\")\n    W_px, H_px = int(round(W_in * dpi)), int(round(H_in * dpi))\n    lettered = {}\n    seen_letters = set()\n    for ax in fig.axes:\n        for t in ax.findobj(matplotlib.text.Text):\n            s = (t.get_text() or \"\").strip()\n            if t.get_gid() == \"figure-style-panel-letter\":\n                if s in seen_letters:\n                    raise ValueError(f\"Duplicate panel letter: {s!r}\")\n                seen_letters.add(s)\n                lettered[ax] = s\n                break\n\n\n\n    if not lettered:\n        lettered = {ax: str(i) for i, ax in enumerate(fig.axes)}\n    out = {}\n    for ax, letter in lettered.items():\n        bbs = [ax.get_tightbbox(r)]\n\n\n\n\n        ss = ax.get_subplotspec()\n        for sib in fig.axes:\n            if sib is ax or sib in lettered:\n                continue\n            ssib = sib.get_subplotspec()\n            same_row = ss is None or ssib is None or ss.rowspan == ssib.rowspan\n            same_col = ss is None or ssib is None or ss.colspan == ssib.colspan\n            if ((ax.get_shared_y_axes().joined(ax, sib) and same_row)\n                    or (ax.get_shared_x_axes().joined(ax, sib) and same_col)):\n                bbs.append(sib.get_tightbbox(r))\n        bb = mpl.transforms.Bbox.union(bbs)\n\n        bx0 = (bb.x0 / fig.dpi - ox_in) * dpi\n        bx1 = (bb.x1 / fig.dpi - ox_in) * dpi\n        by0 = H_px - (bb.y1 / fig.dpi - oy_in) * dpi\n        by1 = H_px - (bb.y0 / fig.dpi - oy_in) * dpi\n        box = (\n            min(max(int(bx0) - pad_px, 0), W_px),\n            min(max(int(by0) - pad_px, 0), H_px),\n            max(min(int(bx1) + pad_px, W_px), 0),\n            max(min(int(by1) + pad_px, H_px), 0),\n        )\n        if box[0] < box[2] and box[1] < box[3]:\n            out[letter] = box\n    return out\n",
          "sourceDigest": "8020d409bfb60f0c8dfa43d0dfe7bb97f7458e1a3908fefbcb6e4551e0feae4e"
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        "installedInventory": {
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        "stdout": "\n lean/OAI/NumberTheory/TotientAsymptotic/UnconditionalMain.lean \n theorem totient_asymptotic_formula\n    :\n    TendstoUniformlyOn (fun H => AH H (fun _ => 1)) (A (fun _ => 1))\n      atTop (Set.Ico (0 : ℝ) 1) ∧\n    (∃ cMinus cPlus : ℝ, 0 < cMinus ∧ ∀ s ∈ Set.Ico (0 : ℝ) 1,\n      cMinus ≤ A (fun _ => 1) s ∧ A (fun _ => 1) s ≤ cPlus) ∧\n    Tendsto (fun x => V x/mainTerm x) atTop (nhds 1) ∧\n    ∀ c : ℝ, 0 < c → Tendsto (fun x => V (c*x)/V x) atTop (nhds c) :=\n  conditional_main_theorem_of_structure fordScaleBounds extractedStructureInput primeNumberTheoremInput fordUnitPrimeBoxInput\n    fordRenewalInput mertensProductInput fordLemma26Input fordLemma51Input fordCoordinateConcentrationInput\n\ntheorem weighted_totient_asymptotic\n    (k : ℕ) (hk : 1≤k) :\n    TendstoUniformlyOn (fun H => AH H (fk k)) (A (fk k)) atTop (Set.Ico (0 : ℝ) 1) ∧\n    Tendsto (fun x => normalizedCount (N k) x-A (fk k) (theta x)) atTop (nhds 0) ∧\n    ((∃ d : ℕ, IsTotient d ∧ k*d<ell d) →\n      ∃ cMinus cPlus : ℝ, 0<cMinus ∧ 0<cPlus ∧\n        (∀ s∈Set.Ico (0 : ℝ) 1, cMinus≤A (fk k) s ∧ A (fk k) s≤cPlus) ∧\n        Tendsto (fun x => N k x/(tupleNormalization x*A (fk k) (theta x))) atTop (nhds 1) ∧\n        ∀ᶠ x : ℝ in atTop, (cMinus/cPlus)*V x≤N k x ∧ N k x≤V x) ∧\n    ((¬∃ d : ℕ, IsTotient d ∧ k*d<ell d) →\n      (∀ x : ℝ, 0<x → N k x=0) ∧ (∀ s∈Set.Ico (0 : ℝ) 1, A (fk k) s=0)) :=\n  conditional_companion_theorem_of_structure fordScaleBounds extractedStructureInput primeNumberTheoremInput fordUnitPrimeBoxInput\n    fordRenewalInput mertensProductInput fordLemma26Input fordLemma51Input fordCoordinateConcentrationInput pptBoundedFiberPropagation k hk\n\ntheorem weighted_totient_one_two\n    :\n    ∀ k∈({1,2} : Finset ℕ), ∃ c : ℝ, 0<c ∧\n      ∀ s∈Set.Ico (0 : ℝ) 1, c≤A (fk k) s :=\n  com\n\n lean/OAI/NumberTheory/DukePrimeDegree/MainUnconditional.lean \n theorem prime_degree_packet_measure_equidistribution_unconditional\n    (d : ℕ) (hprime : Nat.Prime (d+1)) (hfive : 5 ≤ d+1)\n    (K : ℕ → Type u) [∀ i, Field (K i)] [∀ i, NumberField (K i)]\n    [∀ i, NumberField.IsTotallyReal (K i)]\n    (M : ∀ i, PrimeDegreePackets.FullLattice (d+1) (K i))\n    (σ : ∀ i, OrderedEmbeddings (d+1) (K i))\n    (hn : ∀ i, Module.finrank ℚ (K i)=d+1)\n    (hD : Tendsto (fun i => PrimeDegreePackets.multiplierDiscriminant (M i)) atTop atTop) :\n    ∃ ν : Measure (X (d+1)), IsHaarProbability ν ∧\n      WeakProbabilityConvergence (fun i => packetMeasure (M i) (σ i)) ν ∧\n      IsTightMeasureSet (Set.range (fun i => packetMeasure (M i) (σ i))) := by\n  obtain ⟨ν, hν, hconv, htight⟩ :=\n    prime_degree_packet_equidistribution_unconditional d hprime hfive K M σ hn hD\n  refine ⟨(ν : Measure (X (d+1))), hν, ?_, htight⟩\n  exact (weakProbabilityConvergence_iff\n    (fun i => packetProbability (M i) (σ i) (hn i)\n      (hprime.odd_of_ne_two (by omega))) ν).mpr hconv\n\n/-- The complete packet measures converge weakly to any specified Haar\nprobability, in the formulation by bounded continuous test functions. -/\ntheorem prime_degree_packet_measures_tendsto_haar_unconditional\n    (d : ℕ) (hprime : Nat.Prime (d+1)) (hfive : 5 ≤ d+1)\n    (K : ℕ → Type u) [∀ i, Field (K i)] [∀ i, NumberField (K i)]\n    [∀ i, NumberField.IsTotallyReal (K i)]\n    (M : ∀ i, PrimeDegreePackets.FullLattice (d+1) (K i))\n    (σ : ∀ i, OrderedEmbeddings (d+1) (K i))\n    (hn : ∀ i, Module.finrank ℚ (K i)=d+1)\n    (hD : Tendsto (fun i => PrimeDegreePackets.multiplierDiscriminant (M i)) atTop atTop)\n    (m : ProbabilityMeasure (X (d+1)))\n    (hm : IsHaarProbability (m : Measure (X (d+1)))) :\n    WeakP\noverlap count 0\nchallenge placeholder counts Counter({1: 19, 2: 2, 3: 1, 4: 1})\n",
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          "\n lean/OAI/NumberTheory/TotientAsymptotic/UnconditionalMain.lean \n theorem totient_asymptotic_formula\n    :\n    TendstoUniformlyOn (fun H => AH H (fun _ => 1)) (A (fun _ => 1))\n      atTop (Set.Ico (0 : ℝ) 1) ∧\n    (∃ cMinus cPlus : ℝ, 0 < cMinus ∧ ∀ s ∈ Set.Ico (0 : ℝ) 1,\n      cMinus ≤ A (fun _ => 1) s ∧ A (fun _ => 1) s ≤ cPlus) ∧\n    Tendsto (fun x => V x/mainTerm x) atTop (nhds 1) ∧\n    ∀ c : ℝ, 0 < c → Tendsto (fun x => V (c*x)/V x) atTop (nhds c) :=\n  conditional_main_theorem_of_structure fordScaleBounds extractedStructureInput primeNumberTheoremInput fordUnitPrimeBoxInput\n    fordRenewalInput mertensProductInput fordLemma26Input fordLemma51Input fordCoordinateConcentrationInput\n\ntheorem weighted_totient_asymptotic\n    (k : ℕ) (hk : 1≤k) :\n    TendstoUniformlyOn (fun H => AH H (fk k)) (A (fk k)) atTop (Set.Ico (0 : ℝ) 1) ∧\n    Tendsto (fun x => normalizedCount (N k) x-A (fk k) (theta x)) atTop (nhds 0) ∧\n    ((∃ d : ℕ, IsTotient d ∧ k*d<ell d) →\n      ∃ cMinus cPlus : ℝ, 0<cMinus ∧ 0<cPlus ∧\n        (∀ s∈Set.Ico (0 : ℝ) 1, cMinus≤A (fk k) s ∧ A (fk k) s≤cPlus) ∧\n        Tendsto (fun x => N k x/(tupleNormalization x*A (fk k) (theta x))) atTop (nhds 1) ∧\n        ∀ᶠ x : ℝ in atTop, (cMinus/cPlus)*V x≤N k x ∧ N k x≤V x) ∧\n    ((¬∃ d : ℕ, IsTotient d ∧ k*d<ell d) →\n      (∀ x : ℝ, 0<x → N k x=0) ∧ (∀ s∈Set.Ico (0 : ℝ) 1, A (fk k) s=0)) :=\n  conditional_companion_theorem_of_structure fordScaleBounds extractedStructureInput primeNumberTheoremInput fordUnitPrimeBoxInput\n    fordRenewalInput mertensProductInput fordLemma26Input fordLemma51Input fordCoordinateConcentrationInput pptBoundedFiberPropagation k hk\n\ntheorem weighted_totient_one_two\n    :\n    ∀ k∈({1,2} : Finset ℕ), ∃ c : ℝ, 0<c ∧\n      ∀ s∈Set.Ico (0 : ℝ) 1, c≤A (fk k) s :=\n  com\n\n lean/OAI/NumberTheory/DukePrimeDegree/MainUnconditional.lean \n theorem prime_degree_packet_measure_equidistribution_unconditional\n    (d : ℕ) (hprime : Nat.Prime (d+1)) (hfive : 5 ≤ d+1)\n    (K : ℕ → Type u) [∀ i, Field (K i)] [∀ i, NumberField (K i)]\n    [∀ i, NumberField.IsTotallyReal (K i)]\n    (M : ∀ i, PrimeDegreePackets.FullLattice (d+1) (K i))\n    (σ : ∀ i, OrderedEmbeddings (d+1) (K i))\n    (hn : ∀ i, Module.finrank ℚ (K i)=d+1)\n    (hD : Tendsto (fun i => PrimeDegreePackets.multiplierDiscriminant (M i)) atTop atTop) :\n    ∃ ν : Measure (X (d+1)), IsHaarProbability ν ∧\n      WeakProbabilityConvergence (fun i => packetMeasure (M i) (σ i)) ν ∧\n      IsTightMeasureSet (Set.range (fun i => packetMeasure (M i) (σ i))) := by\n  obtain ⟨ν, hν, hconv, htight⟩ :=\n    prime_degree_packet_equidistribution_unconditional d hprime hfive K M σ hn hD\n  refine ⟨(ν : Measure (X (d+1))), hν, ?_, htight⟩\n  exact (weakProbabilityConvergence_iff\n    (fun i => packetProbability (M i) (σ i) (hn i)\n      (hprime.odd_of_ne_two (by omega))) ν).mpr hconv\n\n/-- The complete packet measures converge weakly to any specified Haar\nprobability, in the formulation by bounded continuous test functions. -/\ntheorem prime_degree_packet_measures_tendsto_haar_unconditional\n    (d : ℕ) (hprime : Nat.Prime (d+1)) (hfive : 5 ≤ d+1)\n    (K : ℕ → Type u) [∀ i, Field (K i)] [∀ i, NumberField (K i)]\n    [∀ i, NumberField.IsTotallyReal (K i)]\n    (M : ∀ i, PrimeDegreePackets.FullLattice (d+1) (K i))\n    (σ : ∀ i, OrderedEmbeddings (d+1) (K i))\n    (hn : ∀ i, Module.finrank ℚ (K i)=d+1)\n    (hD : Tendsto (fun i => PrimeDegreePackets.multiplierDiscriminant (M i)) atTop atTop)\n    (m : ProbabilityMeasure (X (d+1)))\n    (hm : IsHaarProbability (m : Measure (X (d+1)))) :\n    WeakP\noverlap count 0\nchallenge placeholder counts Counter({1: 19, 2: 2, 3: 1, 4: 1})\n"
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          "text": "\n lean/OAI/NumberTheory/TotientAsymptotic/UnconditionalMain.lean \n theorem totient_asymptotic_formula\n    :\n    TendstoUniformlyOn (fun H => AH H (fun _ => 1)) (A (fun _ => 1))\n      atTop (Set.Ico (0 : ℝ) 1) ∧\n    (∃ cMinus cPlus : ℝ, 0 < cMinus ∧ ∀ s ∈ Set.Ico (0 : ℝ) 1,\n      cMinus ≤ A (fun _ => 1) s ∧ A (fun _ => 1) s ≤ cPlus) ∧\n    Tendsto (fun x => V x/mainTerm x) atTop (nhds 1) ∧\n    ∀ c : ℝ, 0 < c → Tendsto (fun x => V (c*x)/V x) atTop (nhds c) :=\n  conditional_main_theorem_of_structure fordScaleBounds extractedStructureInput primeNumberTheoremInput fordUnitPrimeBoxInput\n    fordRenewalInput mertensProductInput fordLemma26Input fordLemma51Input fordCoordinateConcentrationInput\n\ntheorem weighted_totient_asymptotic\n    (k : ℕ) (hk : 1≤k) :\n    TendstoUniformlyOn (fun H => AH H (fk k)) (A (fk k)) atTop (Set.Ico (0 : ℝ) 1) ∧\n    Tendsto (fun x => normalizedCount (N k) x-A (fk k) (theta x)) atTop (nhds 0) ∧\n    ((∃ d : ℕ, IsTotient d ∧ k*d<ell d) →\n      ∃ cMinus cPlus : ℝ, 0<cMinus ∧ 0<cPlus ∧\n        (∀ s∈Set.Ico (0 : ℝ) 1, cMinus≤A (fk k) s ∧ A (fk k) s≤cPlus) ∧\n        Tendsto (fun x => N k x/(tupleNormalization x*A (fk k) (theta x))) atTop (nhds 1) ∧\n        ∀ᶠ x : ℝ in atTop, (cMinus/cPlus)*V x≤N k x ∧ N k x≤V x) ∧\n    ((¬∃ d : ℕ, IsTotient d ∧ k*d<ell d) →\n      (∀ x : ℝ, 0<x → N k x=0) ∧ (∀ s∈Set.Ico (0 : ℝ) 1, A (fk k) s=0)) :=\n  conditional_companion_theorem_of_structure fordScaleBounds extractedStructureInput primeNumberTheoremInput fordUnitPrimeBoxInput\n    fordRenewalInput mertensProductInput fordLemma26Input fordLemma51Input fordCoordinateConcentrationInput pptBoundedFiberPropagation k hk\n\ntheorem weighted_totient_one_two\n    :\n    ∀ k∈({1,2} : Finset ℕ), ∃ c : ℝ, 0<c ∧\n      ∀ s∈Set.Ico (0 : ℝ) 1, c≤A (fk k) s :=\n  com\n\n lean/OAI/NumberTheory/DukePrimeDegree/MainUnconditional.lean \n theorem prime_degree_packet_measure_equidistribution_unconditional\n    (d : ℕ) (hprime : Nat.Prime (d+1)) (hfive : 5 ≤ d+1)\n    (K : ℕ → Type u) [∀ i, Field (K i)] [∀ i, NumberField (K i)]\n    [∀ i, NumberField.IsTotallyReal (K i)]\n    (M : ∀ i, PrimeDegreePackets.FullLattice (d+1) (K i))\n    (σ : ∀ i, OrderedEmbeddings (d+1) (K i))\n    (hn : ∀ i, Module.finrank ℚ (K i)=d+1)\n    (hD : Tendsto (fun i => PrimeDegreePackets.multiplierDiscriminant (M i)) atTop atTop) :\n    ∃ ν : Measure (X (d+1)), IsHaarProbability ν ∧\n      WeakProbabilityConvergence (fun i => packetMeasure (M i) (σ i)) ν ∧\n      IsTightMeasureSet (Set.range (fun i => packetMeasure (M i) (σ i))) := by\n  obtain ⟨ν, hν, hconv, htight⟩ :=\n    prime_degree_packet_equidistribution_unconditional d hprime hfive K M σ hn hD\n  refine ⟨(ν : Measure (X (d+1))), hν, ?_, htight⟩\n  exact (weakProbabilityConvergence_iff\n    (fun i => packetProbability (M i) (σ i) (hn i)\n      (hprime.odd_of_ne_two (by omega))) ν).mpr hconv\n\n/-- The complete packet measures converge weakly to any specified Haar\nprobability, in the formulation by bounded continuous test functions. -/\ntheorem prime_degree_packet_measures_tendsto_haar_unconditional\n    (d : ℕ) (hprime : Nat.Prime (d+1)) (hfive : 5 ≤ d+1)\n    (K : ℕ → Type u) [∀ i, Field (K i)] [∀ i, NumberField (K i)]\n    [∀ i, NumberField.IsTotallyReal (K i)]\n    (M : ∀ i, PrimeDegreePackets.FullLattice (d+1) (K i))\n    (σ : ∀ i, OrderedEmbeddings (d+1) (K i))\n    (hn : ∀ i, Module.finrank ℚ (K i)=d+1)\n    (hD : Tendsto (fun i => PrimeDegreePackets.multiplierDiscriminant (M i)) atTop atTop)\n    (m : ProbabilityMeasure (X (d+1)))\n    (hm : IsHaarProbability (m : Measure (X (d+1)))) :\n    WeakP\noverlap count 0\nchallenge placeholder counts Counter({1: 19, 2: 2, 3: 1, 4: 1})\n"
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          ],
          "source": "META_GREY = \"#888888\"\n\n\ndef apply_figure_style(*, frame=\"open\", font=None, sizes=(8, 7, 6), grid=False):\n    \"\"\"Set matplotlib rcParams for publication-grade output. Call once before plotting.\n\n    This sets mechanics (role-mapped font-size ladder, outward ticks, frameless\n    legends, 300-dpi save, Type-42 embedded fonts) — not a house aesthetic.\n    Frame, font and the size ladder are parameters.\n\n    frame : 'open' (bottom+left spines, default) | 'boxed' (all four) | 'none'\n    font  : sans-serif family name; None = system default sans-serif\n    sizes : (base, secondary, tick) — titles/axis-labels, legend/annotation, ticks\n    grid  : whether to draw axes.grid (default False)\n    \"\"\"\n    import matplotlib as mpl\n    if frame not in (\"open\", \"boxed\", \"none\"):\n        raise ValueError(f\"frame must be 'open'|'boxed'|'none', got {frame!r}\")\n\n    try:\n        import os, sys, glob, matplotlib.font_manager as fm\n        fdir = os.path.join(os.environ.get(\"CONDA_PREFIX\") or sys.prefix, \"fonts\")\n        if os.path.isdir(fdir):\n            known = {f.fname for f in fm.fontManager.ttflist}\n            for f in glob.glob(os.path.join(fdir, \"*.ttf\")):\n                if f not in known:\n                    fm.fontManager.addfont(f)\n    except Exception:\n        pass\n    base, secondary, tick = sizes\n    boxed = (frame == \"boxed\")\n    rc = {\n        \"font.family\": \"sans-serif\",\n        \"font.size\": base,\n        \"axes.labelsize\": base,\n        \"axes.titlesize\": base,\n        \"legend.fontsize\": secondary,\n        \"xtick.labelsize\": tick,\n        \"ytick.labelsize\": tick,\n        \"axes.linewidth\": 0.6,\n        \"xtick.direction\": \"out\", \"ytick.direction\": \"out\",\n        \"xtick.major.size\": 3, \"ytick.major.size\": 3,\n        \"xtick.major.width\": 0.6, \"ytick.major.width\": 0.6,\n        \"axes.spines.top\": boxed, \"axes.spines.right\": boxed,\n        \"axes.spines.left\": frame != \"none\", \"axes.spines.bottom\": frame != \"none\",\n        \"axes.grid\": bool(grid),\n        \"legend.frameon\": False,\n        \"figure.dpi\": 200,\n        \"savefig.dpi\": 300,\n        \"savefig.bbox\": \"tight\",\n        \"axes.titleweight\": \"normal\",\n        \"axes.titlelocation\": \"left\",\n        \"axes.labelweight\": \"normal\",\n        \"lines.linewidth\": 1.2,\n        \"patch.linewidth\": 0.6,\n        \"pdf.fonttype\": 42, \"ps.fonttype\": 42,\n    }\n    if font:\n        rc[\"font.sans-serif\"] = [font, \"DejaVu Sans\"]\n    mpl.rcParams.update(rc)\n\n\ndef set_frame(ax, style=\"open\"):\n    \"\"\"§3: set spine visibility on an existing axes. style ∈ {'open','boxed','none'}.\"\"\"\n    show = {\"open\": (False, False, True, True),\n            \"boxed\": (True, True, True, True),\n            \"none\": (False, False, False, False)}[style]\n    for side, vis in zip((\"top\", \"right\", \"bottom\", \"left\"), show):\n        ax.spines[side].set_visible(vis)\n        if vis:\n            ax.spines[side].set_linewidth(0.6)\n    ax.tick_params(direction=\"out\", length=0 if style == \"none\" else 3, width=0.6)\n\n\ndef panel_letter(ax, letter, dx=-0.18, dy=1.02, case=\"lower\", fontsize=None):\n    \"\"\"§5.7: bold panel letter outside top-left of axes. case ∈ {'lower','upper'}.\"\"\"\n    import matplotlib.pyplot as plt\n    if fontsize is None:\n        fontsize = plt.rcParams.get(\"font.size\", 8) + 1\n    s = letter.lower() if case == \"lower\" else letter.upper()\n    artist = ax.text(dx, dy, s, transform=ax.transAxes,\n                     fontweight=\"bold\", fontsize=fontsize, va=\"bottom\", ha=\"left\")\n    artist.set_gid(\"figure-style-panel-letter\")\n\n\ndef focal_palette(labels, focal, focal_color, other=\"muted\", base_colors=None):\n    \"\"\"§4.2: map labels → colors with the focal series visually dominant.\n\n    other='muted'   — desaturate base_colors (or a default cycle) toward gray\n    other='grey'    — uniform light gray for all non-focal\n    other='ordinal' — non-focal on a single light→dark gray ramp (input order)\n    \"\"\"\n    import matplotlib.colors as mcolors\n    import matplotlib.pyplot as plt\n    focal_set = {focal} if isinstance(focal, str) else set(focal)\n    n = len(labels)\n    if not focal_set & set(labels):\n        raise ValueError(f\"focal {focal!r} not found in labels\")\n    base_colors = [] if base_colors is None else list(base_colors)\n    if not base_colors:\n        base_colors = plt.rcParams[\"axes.prop_cycle\"].by_key().get(\"color\", [\"#444444\"])\n    if not base_colors:\n        base_colors = [\"#444444\"]\n    base_colors = [base_colors[i % len(base_colors)] for i in range(n)]\n    if other == \"grey\":\n        rest = [\"#BCBCBC\"] * n\n    elif other == \"ordinal\":\n        nf = max(1, n - len(focal_set))\n        ramp = [mcolors.to_hex((v, v, v)) for v in\n                ([0.55] if nf == 1 else [0.80 - 0.35 * i / (nf - 1) for i in range(nf)])]\n        rest, k = [], 0\n        for l in labels:\n            rest.append(ramp[min(k, nf - 1)]); k += (l not in focal_set)\n    elif other == \"muted\":\n        def mute(c):\n            r, g, b = mcolors.to_rgb(c)\n            m = (r + g + b) / 3\n            return mcolors.to_hex((0.3 * r + 0.7 * m, 0.3 * g + 0.7 * m, 0.3 * b + 0.7 * m))\n        rest = [mute(c) for c in base_colors]\n    else:\n        raise ValueError(\"other must be 'muted', 'grey', or 'ordinal'\")\n    return [focal_color if l in focal_set else rest[i] for i, l in enumerate(labels)]\n\n\ndef bar_with_points(ax, x, ymat, labels, colors, jitter=0.08, show_points=True,\n                    errorbar=None, point_alpha=0.5, point_size=8):\n    \"\"\"§6.1: bar = mean; optionally overlay raw points or draw an interval.\n\n    colors   : per-label color list (e.g. from focal_palette)\n    x, ymat and labels must have equal lengths; mismatches raise before drawing.\n    Each group must be a nonempty, finite, one-dimensional real numeric array.\n    Requested SD/CI intervals require at least two observations per group.\n    errorbar : None | 'sd' | 'ci95' — drawn only when show_points is False.\n               'ci95' is the t-distribution 95% CI of the mean\n               (half-width t_{0.975,n-1} · s/√n); correct at small n where the\n               z-approximation (1.96·s/√n) is markedly too narrow.\n    \"\"\"\n    import numpy as np\n    x = np.asarray(x)\n    if x.ndim != 1 or not np.issubdtype(x.dtype, np.number):\n        raise TypeError(\n            \"bar_with_points x must be one-dimensional numeric positions; \"\n            \"pass category names through labels\"\n        )\n    ymat = list(ymat)\n    labels = list(labels)\n    if len(x) != len(ymat) or len(x) != len(labels):\n        raise ValueError(\"bar_with_points x, ymat and labels must have equal lengths\")\n    if colors is not None and not isinstance(colors, str):\n        colors = list(colors)\n        if not colors:\n            colors = None\n    if x.dtype.kind not in \"iuf\" or not np.isfinite(x).all():\n        raise ValueError(\"bar_with_points x must contain finite real positions\")\n    for i, y in enumerate(ymat):\n        if np.ma.isMaskedArray(y) and np.ma.getmaskarray(y).any():\n            raise ValueError(f\"bar_with_points group {i} contains masked observations; resolve exclusions explicitly before plotting\")\n    ymat = [np.asarray(y) for y in ymat]\n    for i, y in enumerate(ymat):\n        if y.ndim != 1 or y.size == 0 or y.dtype.kind not in \"iuf\" or not np.isfinite(y).all():\n            raise ValueError(f\"bar_with_points group {i} must be nonempty, one-dimensional and finite real numeric data\")\n    if errorbar and not show_points:\n        if errorbar not in (\"sd\", \"ci95\"):\n            raise ValueError(\"errorbar must be None, 'sd', or 'ci95'\")\n        if any(y.size < 2 for y in ymat):\n            raise ValueError(\"SD/CI cannot be estimated from one observation; use raw points without an interval\")\n    means = np.array([np.mean(y) for y in ymat], float)\n    err = None\n    if errorbar and not show_points:\n        if errorbar == \"sd\":\n            err = np.array([np.std(y, ddof=1) for y in ymat])\n        elif errorbar == \"ci95\":\n            from scipy.stats import t\n            def _hw(y):\n                n = np.asarray(y).size\n                return t.ppf(0.975, n - 1) * np.std(y, ddof=1) / np.sqrt(n)\n            err = np.array([_hw(y) for y in ymat])\n    ax.bar(x, means, color=colors, width=0.7, edgecolor=\"none\",\n           yerr=err, error_kw={\"elinewidth\": 0.8, \"capsize\": 0})\n    if show_points:\n        for xi, ys in zip(x, ymat):\n            ys = np.asarray(ys)\n            if ys.ndim == 1 and ys.size > 0:\n                jit = (np.random.rand(ys.size) - 0.5) * 2 * jitter\n                ax.scatter(np.full(ys.size, xi) + jit, ys, s=point_size, color=\"black\",\n                           alpha=point_alpha, zorder=3, linewidths=0)\n    ax.set_xticks(x); ax.set_xticklabels(labels)\n    return ax\n\n\ndef strip_with_median(ax, groups, values, colors=None, jitter=0.12):\n    \"\"\"§6.1: jittered points + bold horizontal median tick per group.\n\n    groups and values must have equal lengths. Mismatches raise before drawing.\n    colors=None or an empty color sequence supplies default gray; a nonempty\n    color sequence cycles across groups.\n    Each group must be nonempty, one-dimensional and finite real numeric data;\n    resolve missing observations explicitly before plotting.\n    \"\"\"\n    import numpy as np\n    labs = list(groups)\n    values = list(values)\n    if len(labs) != len(values):\n        raise ValueError(\"strip_with_median groups and values must have equal lengths\")\n    for i, y in enumerate(values):\n        if np.ma.isMaskedArray(y) and np.ma.getmaskarray(y).any():\n            raise ValueError(f\"strip_with_median group {i} contains masked observations; resolve exclusions explicitly before plotting\")\n    values = [np.asarray(y) for y in values]\n    for i, y in enumerate(values):\n        if y.ndim != 1 or y.size == 0 or y.dtype.kind not in \"iuf\" or not np.isfinite(y).all():\n            raise ValueError(f\"strip_with_median group {i} must be nonempty, one-dimensional and finite real numeric data\")\n    colors = [] if colors is None else list(colors)\n    if not colors:\n        colors = [\"#444444\"]\n    for i, ys in enumerate(values):\n        c = colors[i % len(colors)]\n        ys = np.asarray(ys)\n        jit = (np.random.rand(ys.size) - 0.5) * 2 * jitter\n        ax.scatter(np.full(ys.size, i) + jit, ys, s=10, color=c, alpha=0.6, linewidths=0, zorder=2)\n        m = np.median(ys)\n        ax.plot([i - 0.22, i + 0.22], [m, m], color=\"black\", lw=1.6, zorder=3)\n    ax.set_xticks(range(len(labs))); ax.set_xticklabels(labs)\n    return ax\n\n\ndef goodness_arrow(ax, text=\"higher = better\", loc=\"upper left\", axis=\"y\", fontsize=None):\n    \"\"\"§3.6: small upright direction-of-goodness cue in the margin.\"\"\"\n    import matplotlib.pyplot as plt\n    if fontsize is None:\n        fontsize = plt.rcParams[\"legend.fontsize\"]\n    pos = {\"upper left\": (0.02, 0.98), \"upper right\": (0.98, 0.98),\n           \"lower left\": (0.02, 0.02), \"lower right\": (0.98, 0.02)}[loc]\n    ha = \"left\" if \"left\" in loc else \"right\"\n    va = \"top\" if \"upper\" in loc else \"bottom\"\n    arrow = \"↑ \" if axis == \"y\" else \"→ \"\n    ax.text(pos[0], pos[1], arrow + text, transform=ax.transAxes,\n            fontsize=fontsize, color=META_GREY, ha=ha, va=va)\n\n\ndef two_tier_label(name, meta):\n    \"\"\"§5: two-line label string (name / metadata). Meta line styled separately by caller.\"\"\"\n    return f\"{name}\\n{meta}\"\n\n\ndef end_of_line_labels(ax, xs, ys, labels, colors=None, dx=0.01, fontsize=None):\n    \"\"\"§6.3 / §7.3: label each line series at its right end instead of a legend box.\n\n    Each label requires a nonempty, matched x/y series. Validate all series\n    before drawing. Nonempty colors cycle; None or empty uses default text color.\n    Inner sequences must support len() and indexing; they are not copied.\n    Nonfinite endpoints are rejected rather than producing an invisible label.\n    \"\"\"\n    import matplotlib.pyplot as plt\n    import math\n    xs, ys, labels = list(xs), list(ys), list(labels)\n    if len(xs) != len(ys) or len(xs) != len(labels):\n        raise ValueError(\"end_of_line_labels xs, ys and labels must have equal lengths\")\n    if any(len(x) == 0 or len(x) != len(y) for x, y in zip(xs, ys)):\n        raise ValueError(\"end_of_line_labels each x/y series must be nonempty and have equal lengths\")\n    if any(not math.isfinite(x[-1]) or not math.isfinite(y[-1]) for x, y in zip(xs, ys)):\n        raise ValueError(\"end_of_line_labels endpoints must be finite; choose valid endpoints explicitly\")\n    if fontsize is None:\n        fontsize = plt.rcParams[\"font.size\"]\n    colors = [] if colors is None else list(colors)\n    if not colors:\n        colors = [None]\n    span = ax.get_xlim()[1] - ax.get_xlim()[0]\n    for i, (x, y, lab) in enumerate(zip(xs, ys, labels)):\n        c = colors[i % len(colors)]\n        ax.text(x[-1] + dx * span, y[-1], lab, color=c, va=\"center\", ha=\"left\", fontsize=fontsize)\n\n\ndef panel_crops(fig, dpi=None, pad_px=6, bbox_inches=None, pad_inches=None):\n    \"\"\"§9.2: pixel-space crop boxes for each lettered panel in the SAVED PNG.\n\n    Returns ``{letter: (x0, y0, x1, y1)}`` in image-space pixels (origin\n    top-left, matching PIL's ``Image.crop``). Panels are detected as bold\n    single-character ``Text``\n    objects placed by :func:`panel_letter`; each panel's crop is its axes'\n    tightbbox mapped into the saved file's pixel space, padded by ``pad_px``.\n    For §3.4 composites (abutting subplots sharing an axis, letter on the\n    leftmost only) the crop unions in letterless ``sharex``/``sharey`` siblings\n    on the same grid row/col so the whole composite is covered. When no axes\n    carries a panel letter (standalone plot, or a figure-composer sub-agent),\n    falls back to one crop per axes keyed by index.\n\n    ``bbox_inches`` mirrors ``Figure.savefig`` semantics: ``None`` means\n    *consult rcParams* (so under :func:`apply_figure_style` it resolves to\n    ``'tight'``); pass an explicit ``Bbox`` only if you saved with one. The\n    boxes are clamped to the saved image extent; panels with no intersection\n    are omitted.\n\n        >>> from PIL import Image\n        >>> fig.savefig(\"fig.png\")  # bbox_inches='tight' via rcParams\n        >>> with Image.open(\"fig.png\") as saved:\n        ...     for letter, box in panel_crops(fig).items():\n        ...         saved.crop(box).save(f\"panel_{letter}.png\")\n\n    Match savefig's DPI, bbox_inches and pad_inches when overriding defaults.\n    Returned boxes are pixels, never savefig bbox_inches (inches). Passing them\n    back to savefig as inches can allocate an enormous raster.\n    \"\"\"\n    import matplotlib as mpl\n    import matplotlib.text\n    from matplotlib.backends.backend_agg import FigureCanvasAgg\n    if isinstance(pad_px, bool) or not isinstance(pad_px, int) or pad_px < 0:\n        raise TypeError(\"panel_crops pad_px must be a non-negative integer\")\n    if dpi is None:\n        dpi = mpl.rcParams.get(\"savefig.dpi\", fig.dpi)\n        if dpi == \"figure\":\n            dpi = fig.dpi\n    dpi = float(dpi)\n    if bbox_inches is None:\n        bbox_inches = mpl.rcParams.get(\"savefig.bbox\")\n    # The Open-Science notebook captures and closes pyplot figures after every\n    # cell. Matplotlib 3.11 decouples a closed Figure from its backend by\n    # replacing FigureCanvasAgg with FigureCanvasBase, while the persistent\n    # notebook namespace can still retain the Figure itself. Reattach the\n    # headless canvas before asking for saved-image geometry so this helper is\n    # safe in a later QA cell as well as the producer cell.\n    if not hasattr(fig.canvas, \"get_renderer\"):\n        FigureCanvasAgg(fig)\n    fig.canvas.draw()\n    r = fig.canvas.get_renderer()\n\n    if bbox_inches == \"tight\":\n        if pad_inches is None:\n            pad_inches = mpl.rcParams.get(\"savefig.pad_inches\", 0.1)\n        tb = fig.get_tightbbox(r).padded(pad_inches)\n        ox_in, oy_in = tb.x0, tb.y0\n        W_in, H_in = tb.width, tb.height\n    elif isinstance(bbox_inches, mpl.transforms.BboxBase):\n        ox_in, oy_in = bbox_inches.x0, bbox_inches.y0\n        W_in, H_in = bbox_inches.width, bbox_inches.height\n    elif bbox_inches is None:\n        ox_in, oy_in = 0.0, 0.0\n        W_in, H_in = fig.get_size_inches()\n    else:\n        raise TypeError(\"panel_crops bbox_inches must be None, 'tight', or a matplotlib Bbox\")\n    W_px, H_px = int(round(W_in * dpi)), int(round(H_in * dpi))\n    lettered = {}\n    seen_letters = set()\n    for ax in fig.axes:\n        for t in ax.findobj(matplotlib.text.Text):\n            s = (t.get_text() or \"\").strip()\n            if t.get_gid() == \"figure-style-panel-letter\":\n                if s in seen_letters:\n                    raise ValueError(f\"Duplicate panel letter: {s!r}\")\n                seen_letters.add(s)\n                lettered[ax] = s\n                break\n\n\n\n    if not lettered:\n        lettered = {ax: str(i) for i, ax in enumerate(fig.axes)}\n    out = {}\n    for ax, letter in lettered.items():\n        bbs = [ax.get_tightbbox(r)]\n\n\n\n\n        ss = ax.get_subplotspec()\n        for sib in fig.axes:\n            if sib is ax or sib in lettered:\n                continue\n            ssib = sib.get_subplotspec()\n            same_row = ss is None or ssib is None or ss.rowspan == ssib.rowspan\n            same_col = ss is None or ssib is None or ss.colspan == ssib.colspan\n            if ((ax.get_shared_y_axes().joined(ax, sib) and same_row)\n                    or (ax.get_shared_x_axes().joined(ax, sib) and same_col)):\n                bbs.append(sib.get_tightbbox(r))\n        bb = mpl.transforms.Bbox.union(bbs)\n\n        bx0 = (bb.x0 / fig.dpi - ox_in) * dpi\n        bx1 = (bb.x1 / fig.dpi - ox_in) * dpi\n        by0 = H_px - (bb.y1 / fig.dpi - oy_in) * dpi\n        by1 = H_px - (bb.y0 / fig.dpi - oy_in) * dpi\n        box = (\n            min(max(int(bx0) - pad_px, 0), W_px),\n            min(max(int(by0) - pad_px, 0), H_px),\n            max(min(int(bx1) + pad_px, W_px), 0),\n            max(min(int(by1) + pad_px, H_px), 0),\n        )\n        if box[0] < box[2] and box[1] < box[3]:\n            out[letter] = box\n    return out\n",
          "sourceDigest": "8020d409bfb60f0c8dfa43d0dfe7bb97f7458e1a3908fefbcb6e4551e0feae4e"
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        "installedInventory": {
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        "stdout": "[{\"config\": \"lean/ComparatorChallenges/BinaryMatching.json\", \"solution\": \"lean/OAI/Computability/MatchingCount/BinarySolve.lean\", \"declarations\": [{\"target\": \"OAI.BinaryMatching.deterministic_approximate_counting\", \"local_declaration_found\": true, \"declaration_locations\": [{\"path\": \"lean/OAI/Computability/MatchingCount/BinarySolve.lean\", \"line\": 129, \"placement\": \"local declaration\"}], \"status\": \"local body present\"}], \"missing_direct_count\": 1}, {\"config\": \"lean/ComparatorChallenges/ClassicalON.json\", \"solution\": \"lean/OAI/Probability/ClassicalON/Main.lean\", \"declarations\": [{\"target\": \"OAI.ClassicalON.main\", \"local_declaration_found\": true, \"declaration_locations\": [{\"path\": \"lean/OAI/Probability/ClassicalON/Main.lean\", \"line\": 19, \"placement\": \"local declaration\"}], \"status\": \"local body present\"}], \"missing_direct_count\": 1}, {\"config\": \"lean/ComparatorChallenges/ContingencyTables.json\", \"solution\": \"lean/OAI/Combinatorics/ContingencyTables/UnconditionalMain.lean\", \"declarations\": [{\"target\": \"OAI.ContingencyTables.boundedSampling\", \"local_declaration_found\": true, \"declaration_locations\": [{\"path\": \"lean/OAI/Combinatorics/ContingencyTables/UnconditionalMain.lean\", \"line\": 19, \"placement\": \"local declaration\"}], \"status\": \"local body present\"}, {\"target\": \"OAI.ContingencyTables.exactSampling\", \"local_declaration_found\": true, \"declaration_locations\": [{\"path\": \"lean/OAI/Combinatorics/ContingencyTables/UnconditionalMain.lean\", \"line\": 24, \"placement\": \"local declaration\"}], \"status\": \"local body present\"}, {\"target\": \"OAI.ContingencyTables.counting\", \"local_declaration_found\": true, \"declaration_locations\": [{\"path\": \"lean/OAI/Combinatorics/ContingencyTables/UnconditionalMain.lean\", \"line\": 29, \"placement\": \"local declaration\"}], \"status\": \"local body present\"}], \"missing_direct_count\": 6}, {\"config\": \"lean/ComparatorChallenges/DukePrimeDegree.json\", \"solution\": \"lean/OAI/NumberTheory/DukePrimeDegree/MainUnconditional.lean\", \"declarations\": [{\"target\": \"OAI.DukePrimeDegree.prime_degree_packet_measure_equidistribution_unconditional\", \"local_declaration_found\": true, \"declaration_locations\": [{\"path\": \"lean/OAI/NumberTheory/DukePrimeDegree/MainUnconditional.lean\", \"line\": 60, \"placement\": \"local declaration\"}], \"status\": \"local body present\"}, {\"target\": \"OAI.DukePrimeDegree.prime_degree_packet_measures_tendsto_haar_unconditional\", \"local_declaration_found\": true, \"declaration_locations\": [{\"path\": \"lean/OAI/NumberTheory/DukePrimeDegree/MainUnconditional.lean\", \"line\": 80, \"placement\": \"local declaration\"}], \"status\": \"local body present\"}], \"missing_direct_count\": 2}, {\"config\": \"lean/ComparatorChallenges/ElasticityUniqueness.json\", \"solution\": \"lean/OAI/MathematicalPhysics/Elasticity/Uniqueness.lean\", \"declarations\": [{\"target\": \"OAI.Elasticity.global_uniqueness\", \"local_declaration_found\": true, \"declaration_locations\": [{\"path\": \"lean/OAI/MathematicalPhysics/Elasticity/Uniqueness.lean\", \"line\": 94, \"placement\": \"local declaration\"}], \"status\": \"local body present\"}], \"missing_direct_count\": 2}]\n",
        "stderr": "",
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          "[{\"config\": \"lean/ComparatorChallenges/BinaryMatching.json\", \"solution\": \"lean/OAI/Computability/MatchingCount/BinarySolve.lean\", \"declarations\": [{\"target\": \"OAI.BinaryMatching.deterministic_approximate_counting\", \"local_declaration_found\": true, \"declaration_locations\": [{\"path\": \"lean/OAI/Computability/MatchingCount/BinarySolve.lean\", \"line\": 129, \"placement\": \"local declaration\"}], \"status\": \"local body present\"}], \"missing_direct_count\": 1}, {\"config\": \"lean/ComparatorChallenges/ClassicalON.json\", \"solution\": \"lean/OAI/Probability/ClassicalON/Main.lean\", \"declarations\": [{\"target\": \"OAI.ClassicalON.main\", \"local_declaration_found\": true, \"declaration_locations\": [{\"path\": \"lean/OAI/Probability/ClassicalON/Main.lean\", \"line\": 19, \"placement\": \"local declaration\"}], \"status\": \"local body present\"}], \"missing_direct_count\": 1}, {\"config\": \"lean/ComparatorChallenges/ContingencyTables.json\", \"solution\": \"lean/OAI/Combinatorics/ContingencyTables/UnconditionalMain.lean\", \"declarations\": [{\"target\": \"OAI.ContingencyTables.boundedSampling\", \"local_declaration_found\": true, \"declaration_locations\": [{\"path\": \"lean/OAI/Combinatorics/ContingencyTables/UnconditionalMain.lean\", \"line\": 19, \"placement\": \"local declaration\"}], \"status\": \"local body present\"}, {\"target\": \"OAI.ContingencyTables.exactSampling\", \"local_declaration_found\": true, \"declaration_locations\": [{\"path\": \"lean/OAI/Combinatorics/ContingencyTables/UnconditionalMain.lean\", \"line\": 24, \"placement\": \"local declaration\"}], \"status\": \"local body present\"}, {\"target\": \"OAI.ContingencyTables.counting\", \"local_declaration_found\": true, \"declaration_locations\": [{\"path\": \"lean/OAI/Combinatorics/ContingencyTables/UnconditionalMain.lean\", \"line\": 29, \"placement\": \"local declaration\"}], \"status\": \"local body present\"}], \"missing_direct_count\": 6}, {\"config\": \"lean/ComparatorChallenges/DukePrimeDegree.json\", \"solution\": \"lean/OAI/NumberTheory/DukePrimeDegree/MainUnconditional.lean\", \"declarations\": [{\"target\": \"OAI.DukePrimeDegree.prime_degree_packet_measure_equidistribution_unconditional\", \"local_declaration_found\": true, \"declaration_locations\": [{\"path\": \"lean/OAI/NumberTheory/DukePrimeDegree/MainUnconditional.lean\", \"line\": 60, \"placement\": \"local declaration\"}], \"status\": \"local body present\"}, {\"target\": \"OAI.DukePrimeDegree.prime_degree_packet_measures_tendsto_haar_unconditional\", \"local_declaration_found\": true, \"declaration_locations\": [{\"path\": \"lean/OAI/NumberTheory/DukePrimeDegree/MainUnconditional.lean\", \"line\": 80, \"placement\": \"local declaration\"}], \"status\": \"local body present\"}], \"missing_direct_count\": 2}, {\"config\": \"lean/ComparatorChallenges/ElasticityUniqueness.json\", \"solution\": \"lean/OAI/MathematicalPhysics/Elasticity/Uniqueness.lean\", \"declarations\": [{\"target\": \"OAI.Elasticity.global_uniqueness\", \"local_declaration_found\": true, \"declaration_locations\": [{\"path\": \"lean/OAI/MathematicalPhysics/Elasticity/Uniqueness.lean\", \"line\": 94, \"placement\": \"local declaration\"}], \"status\": \"local body present\"}], \"missing_direct_count\": 2}]\n"
        ]
      },
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        {
          "type": "stream",
          "name": "stdout",
          "text": "[{\"config\": \"lean/ComparatorChallenges/BinaryMatching.json\", \"solution\": \"lean/OAI/Computability/MatchingCount/BinarySolve.lean\", \"declarations\": [{\"target\": \"OAI.BinaryMatching.deterministic_approximate_counting\", \"local_declaration_found\": true, \"declaration_locations\": [{\"path\": \"lean/OAI/Computability/MatchingCount/BinarySolve.lean\", \"line\": 129, \"placement\": \"local declaration\"}], \"status\": \"local body present\"}], \"missing_direct_count\": 1}, {\"config\": \"lean/ComparatorChallenges/ClassicalON.json\", \"solution\": \"lean/OAI/Probability/ClassicalON/Main.lean\", \"declarations\": [{\"target\": \"OAI.ClassicalON.main\", \"local_declaration_found\": true, \"declaration_locations\": [{\"path\": \"lean/OAI/Probability/ClassicalON/Main.lean\", \"line\": 19, \"placement\": \"local declaration\"}], \"status\": \"local body present\"}], \"missing_direct_count\": 1}, {\"config\": \"lean/ComparatorChallenges/ContingencyTables.json\", \"solution\": \"lean/OAI/Combinatorics/ContingencyTables/UnconditionalMain.lean\", \"declarations\": [{\"target\": \"OAI.ContingencyTables.boundedSampling\", \"local_declaration_found\": true, \"declaration_locations\": [{\"path\": \"lean/OAI/Combinatorics/ContingencyTables/UnconditionalMain.lean\", \"line\": 19, \"placement\": \"local declaration\"}], \"status\": \"local body present\"}, {\"target\": \"OAI.ContingencyTables.exactSampling\", \"local_declaration_found\": true, \"declaration_locations\": [{\"path\": \"lean/OAI/Combinatorics/ContingencyTables/UnconditionalMain.lean\", \"line\": 24, \"placement\": \"local declaration\"}], \"status\": \"local body present\"}, {\"target\": \"OAI.ContingencyTables.counting\", \"local_declaration_found\": true, \"declaration_locations\": [{\"path\": \"lean/OAI/Combinatorics/ContingencyTables/UnconditionalMain.lean\", \"line\": 29, \"placement\": \"local declaration\"}], \"status\": \"local body present\"}], \"missing_direct_count\": 6}, {\"config\": \"lean/ComparatorChallenges/DukePrimeDegree.json\", \"solution\": \"lean/OAI/NumberTheory/DukePrimeDegree/MainUnconditional.lean\", \"declarations\": [{\"target\": \"OAI.DukePrimeDegree.prime_degree_packet_measure_equidistribution_unconditional\", \"local_declaration_found\": true, \"declaration_locations\": [{\"path\": \"lean/OAI/NumberTheory/DukePrimeDegree/MainUnconditional.lean\", \"line\": 60, \"placement\": \"local declaration\"}], \"status\": \"local body present\"}, {\"target\": \"OAI.DukePrimeDegree.prime_degree_packet_measures_tendsto_haar_unconditional\", \"local_declaration_found\": true, \"declaration_locations\": [{\"path\": \"lean/OAI/NumberTheory/DukePrimeDegree/MainUnconditional.lean\", \"line\": 80, \"placement\": \"local declaration\"}], \"status\": \"local body present\"}], \"missing_direct_count\": 2}, {\"config\": \"lean/ComparatorChallenges/ElasticityUniqueness.json\", \"solution\": \"lean/OAI/MathematicalPhysics/Elasticity/Uniqueness.lean\", \"declarations\": [{\"target\": \"OAI.Elasticity.global_uniqueness\", \"local_declaration_found\": true, \"declaration_locations\": [{\"path\": \"lean/OAI/MathematicalPhysics/Elasticity/Uniqueness.lean\", \"line\": 94, \"placement\": \"local declaration\"}], \"status\": \"local body present\"}], \"missing_direct_count\": 2}]\n"
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            "set_frame",
            "panel_letter",
            "focal_palette",
            "bar_with_points",
            "strip_with_median",
            "goodness_arrow",
            "two_tier_label",
            "end_of_line_labels",
            "panel_crops"
          ],
          "source": "META_GREY = \"#888888\"\n\n\ndef apply_figure_style(*, frame=\"open\", font=None, sizes=(8, 7, 6), grid=False):\n    \"\"\"Set matplotlib rcParams for publication-grade output. Call once before plotting.\n\n    This sets mechanics (role-mapped font-size ladder, outward ticks, frameless\n    legends, 300-dpi save, Type-42 embedded fonts) — not a house aesthetic.\n    Frame, font and the size ladder are parameters.\n\n    frame : 'open' (bottom+left spines, default) | 'boxed' (all four) | 'none'\n    font  : sans-serif family name; None = system default sans-serif\n    sizes : (base, secondary, tick) — titles/axis-labels, legend/annotation, ticks\n    grid  : whether to draw axes.grid (default False)\n    \"\"\"\n    import matplotlib as mpl\n    if frame not in (\"open\", \"boxed\", \"none\"):\n        raise ValueError(f\"frame must be 'open'|'boxed'|'none', got {frame!r}\")\n\n    try:\n        import os, sys, glob, matplotlib.font_manager as fm\n        fdir = os.path.join(os.environ.get(\"CONDA_PREFIX\") or sys.prefix, \"fonts\")\n        if os.path.isdir(fdir):\n            known = {f.fname for f in fm.fontManager.ttflist}\n            for f in glob.glob(os.path.join(fdir, \"*.ttf\")):\n                if f not in known:\n                    fm.fontManager.addfont(f)\n    except Exception:\n        pass\n    base, secondary, tick = sizes\n    boxed = (frame == \"boxed\")\n    rc = {\n        \"font.family\": \"sans-serif\",\n        \"font.size\": base,\n        \"axes.labelsize\": base,\n        \"axes.titlesize\": base,\n        \"legend.fontsize\": secondary,\n        \"xtick.labelsize\": tick,\n        \"ytick.labelsize\": tick,\n        \"axes.linewidth\": 0.6,\n        \"xtick.direction\": \"out\", \"ytick.direction\": \"out\",\n        \"xtick.major.size\": 3, \"ytick.major.size\": 3,\n        \"xtick.major.width\": 0.6, \"ytick.major.width\": 0.6,\n        \"axes.spines.top\": boxed, \"axes.spines.right\": boxed,\n        \"axes.spines.left\": frame != \"none\", \"axes.spines.bottom\": frame != \"none\",\n        \"axes.grid\": bool(grid),\n        \"legend.frameon\": False,\n        \"figure.dpi\": 200,\n        \"savefig.dpi\": 300,\n        \"savefig.bbox\": \"tight\",\n        \"axes.titleweight\": \"normal\",\n        \"axes.titlelocation\": \"left\",\n        \"axes.labelweight\": \"normal\",\n        \"lines.linewidth\": 1.2,\n        \"patch.linewidth\": 0.6,\n        \"pdf.fonttype\": 42, \"ps.fonttype\": 42,\n    }\n    if font:\n        rc[\"font.sans-serif\"] = [font, \"DejaVu Sans\"]\n    mpl.rcParams.update(rc)\n\n\ndef set_frame(ax, style=\"open\"):\n    \"\"\"§3: set spine visibility on an existing axes. style ∈ {'open','boxed','none'}.\"\"\"\n    show = {\"open\": (False, False, True, True),\n            \"boxed\": (True, True, True, True),\n            \"none\": (False, False, False, False)}[style]\n    for side, vis in zip((\"top\", \"right\", \"bottom\", \"left\"), show):\n        ax.spines[side].set_visible(vis)\n        if vis:\n            ax.spines[side].set_linewidth(0.6)\n    ax.tick_params(direction=\"out\", length=0 if style == \"none\" else 3, width=0.6)\n\n\ndef panel_letter(ax, letter, dx=-0.18, dy=1.02, case=\"lower\", fontsize=None):\n    \"\"\"§5.7: bold panel letter outside top-left of axes. case ∈ {'lower','upper'}.\"\"\"\n    import matplotlib.pyplot as plt\n    if fontsize is None:\n        fontsize = plt.rcParams.get(\"font.size\", 8) + 1\n    s = letter.lower() if case == \"lower\" else letter.upper()\n    artist = ax.text(dx, dy, s, transform=ax.transAxes,\n                     fontweight=\"bold\", fontsize=fontsize, va=\"bottom\", ha=\"left\")\n    artist.set_gid(\"figure-style-panel-letter\")\n\n\ndef focal_palette(labels, focal, focal_color, other=\"muted\", base_colors=None):\n    \"\"\"§4.2: map labels → colors with the focal series visually dominant.\n\n    other='muted'   — desaturate base_colors (or a default cycle) toward gray\n    other='grey'    — uniform light gray for all non-focal\n    other='ordinal' — non-focal on a single light→dark gray ramp (input order)\n    \"\"\"\n    import matplotlib.colors as mcolors\n    import matplotlib.pyplot as plt\n    focal_set = {focal} if isinstance(focal, str) else set(focal)\n    n = len(labels)\n    if not focal_set & set(labels):\n        raise ValueError(f\"focal {focal!r} not found in labels\")\n    base_colors = [] if base_colors is None else list(base_colors)\n    if not base_colors:\n        base_colors = plt.rcParams[\"axes.prop_cycle\"].by_key().get(\"color\", [\"#444444\"])\n    if not base_colors:\n        base_colors = [\"#444444\"]\n    base_colors = [base_colors[i % len(base_colors)] for i in range(n)]\n    if other == \"grey\":\n        rest = [\"#BCBCBC\"] * n\n    elif other == \"ordinal\":\n        nf = max(1, n - len(focal_set))\n        ramp = [mcolors.to_hex((v, v, v)) for v in\n                ([0.55] if nf == 1 else [0.80 - 0.35 * i / (nf - 1) for i in range(nf)])]\n        rest, k = [], 0\n        for l in labels:\n            rest.append(ramp[min(k, nf - 1)]); k += (l not in focal_set)\n    elif other == \"muted\":\n        def mute(c):\n            r, g, b = mcolors.to_rgb(c)\n            m = (r + g + b) / 3\n            return mcolors.to_hex((0.3 * r + 0.7 * m, 0.3 * g + 0.7 * m, 0.3 * b + 0.7 * m))\n        rest = [mute(c) for c in base_colors]\n    else:\n        raise ValueError(\"other must be 'muted', 'grey', or 'ordinal'\")\n    return [focal_color if l in focal_set else rest[i] for i, l in enumerate(labels)]\n\n\ndef bar_with_points(ax, x, ymat, labels, colors, jitter=0.08, show_points=True,\n                    errorbar=None, point_alpha=0.5, point_size=8):\n    \"\"\"§6.1: bar = mean; optionally overlay raw points or draw an interval.\n\n    colors   : per-label color list (e.g. from focal_palette)\n    x, ymat and labels must have equal lengths; mismatches raise before drawing.\n    Each group must be a nonempty, finite, one-dimensional real numeric array.\n    Requested SD/CI intervals require at least two observations per group.\n    errorbar : None | 'sd' | 'ci95' — drawn only when show_points is False.\n               'ci95' is the t-distribution 95% CI of the mean\n               (half-width t_{0.975,n-1} · s/√n); correct at small n where the\n               z-approximation (1.96·s/√n) is markedly too narrow.\n    \"\"\"\n    import numpy as np\n    x = np.asarray(x)\n    if x.ndim != 1 or not np.issubdtype(x.dtype, np.number):\n        raise TypeError(\n            \"bar_with_points x must be one-dimensional numeric positions; \"\n            \"pass category names through labels\"\n        )\n    ymat = list(ymat)\n    labels = list(labels)\n    if len(x) != len(ymat) or len(x) != len(labels):\n        raise ValueError(\"bar_with_points x, ymat and labels must have equal lengths\")\n    if colors is not None and not isinstance(colors, str):\n        colors = list(colors)\n        if not colors:\n            colors = None\n    if x.dtype.kind not in \"iuf\" or not np.isfinite(x).all():\n        raise ValueError(\"bar_with_points x must contain finite real positions\")\n    for i, y in enumerate(ymat):\n        if np.ma.isMaskedArray(y) and np.ma.getmaskarray(y).any():\n            raise ValueError(f\"bar_with_points group {i} contains masked observations; resolve exclusions explicitly before plotting\")\n    ymat = [np.asarray(y) for y in ymat]\n    for i, y in enumerate(ymat):\n        if y.ndim != 1 or y.size == 0 or y.dtype.kind not in \"iuf\" or not np.isfinite(y).all():\n            raise ValueError(f\"bar_with_points group {i} must be nonempty, one-dimensional and finite real numeric data\")\n    if errorbar and not show_points:\n        if errorbar not in (\"sd\", \"ci95\"):\n            raise ValueError(\"errorbar must be None, 'sd', or 'ci95'\")\n        if any(y.size < 2 for y in ymat):\n            raise ValueError(\"SD/CI cannot be estimated from one observation; use raw points without an interval\")\n    means = np.array([np.mean(y) for y in ymat], float)\n    err = None\n    if errorbar and not show_points:\n        if errorbar == \"sd\":\n            err = np.array([np.std(y, ddof=1) for y in ymat])\n        elif errorbar == \"ci95\":\n            from scipy.stats import t\n            def _hw(y):\n                n = np.asarray(y).size\n                return t.ppf(0.975, n - 1) * np.std(y, ddof=1) / np.sqrt(n)\n            err = np.array([_hw(y) for y in ymat])\n    ax.bar(x, means, color=colors, width=0.7, edgecolor=\"none\",\n           yerr=err, error_kw={\"elinewidth\": 0.8, \"capsize\": 0})\n    if show_points:\n        for xi, ys in zip(x, ymat):\n            ys = np.asarray(ys)\n            if ys.ndim == 1 and ys.size > 0:\n                jit = (np.random.rand(ys.size) - 0.5) * 2 * jitter\n                ax.scatter(np.full(ys.size, xi) + jit, ys, s=point_size, color=\"black\",\n                           alpha=point_alpha, zorder=3, linewidths=0)\n    ax.set_xticks(x); ax.set_xticklabels(labels)\n    return ax\n\n\ndef strip_with_median(ax, groups, values, colors=None, jitter=0.12):\n    \"\"\"§6.1: jittered points + bold horizontal median tick per group.\n\n    groups and values must have equal lengths. Mismatches raise before drawing.\n    colors=None or an empty color sequence supplies default gray; a nonempty\n    color sequence cycles across groups.\n    Each group must be nonempty, one-dimensional and finite real numeric data;\n    resolve missing observations explicitly before plotting.\n    \"\"\"\n    import numpy as np\n    labs = list(groups)\n    values = list(values)\n    if len(labs) != len(values):\n        raise ValueError(\"strip_with_median groups and values must have equal lengths\")\n    for i, y in enumerate(values):\n        if np.ma.isMaskedArray(y) and np.ma.getmaskarray(y).any():\n            raise ValueError(f\"strip_with_median group {i} contains masked observations; resolve exclusions explicitly before plotting\")\n    values = [np.asarray(y) for y in values]\n    for i, y in enumerate(values):\n        if y.ndim != 1 or y.size == 0 or y.dtype.kind not in \"iuf\" or not np.isfinite(y).all():\n            raise ValueError(f\"strip_with_median group {i} must be nonempty, one-dimensional and finite real numeric data\")\n    colors = [] if colors is None else list(colors)\n    if not colors:\n        colors = [\"#444444\"]\n    for i, ys in enumerate(values):\n        c = colors[i % len(colors)]\n        ys = np.asarray(ys)\n        jit = (np.random.rand(ys.size) - 0.5) * 2 * jitter\n        ax.scatter(np.full(ys.size, i) + jit, ys, s=10, color=c, alpha=0.6, linewidths=0, zorder=2)\n        m = np.median(ys)\n        ax.plot([i - 0.22, i + 0.22], [m, m], color=\"black\", lw=1.6, zorder=3)\n    ax.set_xticks(range(len(labs))); ax.set_xticklabels(labs)\n    return ax\n\n\ndef goodness_arrow(ax, text=\"higher = better\", loc=\"upper left\", axis=\"y\", fontsize=None):\n    \"\"\"§3.6: small upright direction-of-goodness cue in the margin.\"\"\"\n    import matplotlib.pyplot as plt\n    if fontsize is None:\n        fontsize = plt.rcParams[\"legend.fontsize\"]\n    pos = {\"upper left\": (0.02, 0.98), \"upper right\": (0.98, 0.98),\n           \"lower left\": (0.02, 0.02), \"lower right\": (0.98, 0.02)}[loc]\n    ha = \"left\" if \"left\" in loc else \"right\"\n    va = \"top\" if \"upper\" in loc else \"bottom\"\n    arrow = \"↑ \" if axis == \"y\" else \"→ \"\n    ax.text(pos[0], pos[1], arrow + text, transform=ax.transAxes,\n            fontsize=fontsize, color=META_GREY, ha=ha, va=va)\n\n\ndef two_tier_label(name, meta):\n    \"\"\"§5: two-line label string (name / metadata). Meta line styled separately by caller.\"\"\"\n    return f\"{name}\\n{meta}\"\n\n\ndef end_of_line_labels(ax, xs, ys, labels, colors=None, dx=0.01, fontsize=None):\n    \"\"\"§6.3 / §7.3: label each line series at its right end instead of a legend box.\n\n    Each label requires a nonempty, matched x/y series. Validate all series\n    before drawing. Nonempty colors cycle; None or empty uses default text color.\n    Inner sequences must support len() and indexing; they are not copied.\n    Nonfinite endpoints are rejected rather than producing an invisible label.\n    \"\"\"\n    import matplotlib.pyplot as plt\n    import math\n    xs, ys, labels = list(xs), list(ys), list(labels)\n    if len(xs) != len(ys) or len(xs) != len(labels):\n        raise ValueError(\"end_of_line_labels xs, ys and labels must have equal lengths\")\n    if any(len(x) == 0 or len(x) != len(y) for x, y in zip(xs, ys)):\n        raise ValueError(\"end_of_line_labels each x/y series must be nonempty and have equal lengths\")\n    if any(not math.isfinite(x[-1]) or not math.isfinite(y[-1]) for x, y in zip(xs, ys)):\n        raise ValueError(\"end_of_line_labels endpoints must be finite; choose valid endpoints explicitly\")\n    if fontsize is None:\n        fontsize = plt.rcParams[\"font.size\"]\n    colors = [] if colors is None else list(colors)\n    if not colors:\n        colors = [None]\n    span = ax.get_xlim()[1] - ax.get_xlim()[0]\n    for i, (x, y, lab) in enumerate(zip(xs, ys, labels)):\n        c = colors[i % len(colors)]\n        ax.text(x[-1] + dx * span, y[-1], lab, color=c, va=\"center\", ha=\"left\", fontsize=fontsize)\n\n\ndef panel_crops(fig, dpi=None, pad_px=6, bbox_inches=None, pad_inches=None):\n    \"\"\"§9.2: pixel-space crop boxes for each lettered panel in the SAVED PNG.\n\n    Returns ``{letter: (x0, y0, x1, y1)}`` in image-space pixels (origin\n    top-left, matching PIL's ``Image.crop``). Panels are detected as bold\n    single-character ``Text``\n    objects placed by :func:`panel_letter`; each panel's crop is its axes'\n    tightbbox mapped into the saved file's pixel space, padded by ``pad_px``.\n    For §3.4 composites (abutting subplots sharing an axis, letter on the\n    leftmost only) the crop unions in letterless ``sharex``/``sharey`` siblings\n    on the same grid row/col so the whole composite is covered. When no axes\n    carries a panel letter (standalone plot, or a figure-composer sub-agent),\n    falls back to one crop per axes keyed by index.\n\n    ``bbox_inches`` mirrors ``Figure.savefig`` semantics: ``None`` means\n    *consult rcParams* (so under :func:`apply_figure_style` it resolves to\n    ``'tight'``); pass an explicit ``Bbox`` only if you saved with one. The\n    boxes are clamped to the saved image extent; panels with no intersection\n    are omitted.\n\n        >>> from PIL import Image\n        >>> fig.savefig(\"fig.png\")  # bbox_inches='tight' via rcParams\n        >>> with Image.open(\"fig.png\") as saved:\n        ...     for letter, box in panel_crops(fig).items():\n        ...         saved.crop(box).save(f\"panel_{letter}.png\")\n\n    Match savefig's DPI, bbox_inches and pad_inches when overriding defaults.\n    Returned boxes are pixels, never savefig bbox_inches (inches). Passing them\n    back to savefig as inches can allocate an enormous raster.\n    \"\"\"\n    import matplotlib as mpl\n    import matplotlib.text\n    from matplotlib.backends.backend_agg import FigureCanvasAgg\n    if isinstance(pad_px, bool) or not isinstance(pad_px, int) or pad_px < 0:\n        raise TypeError(\"panel_crops pad_px must be a non-negative integer\")\n    if dpi is None:\n        dpi = mpl.rcParams.get(\"savefig.dpi\", fig.dpi)\n        if dpi == \"figure\":\n            dpi = fig.dpi\n    dpi = float(dpi)\n    if bbox_inches is None:\n        bbox_inches = mpl.rcParams.get(\"savefig.bbox\")\n    # The Open-Science notebook captures and closes pyplot figures after every\n    # cell. Matplotlib 3.11 decouples a closed Figure from its backend by\n    # replacing FigureCanvasAgg with FigureCanvasBase, while the persistent\n    # notebook namespace can still retain the Figure itself. Reattach the\n    # headless canvas before asking for saved-image geometry so this helper is\n    # safe in a later QA cell as well as the producer cell.\n    if not hasattr(fig.canvas, \"get_renderer\"):\n        FigureCanvasAgg(fig)\n    fig.canvas.draw()\n    r = fig.canvas.get_renderer()\n\n    if bbox_inches == \"tight\":\n        if pad_inches is None:\n            pad_inches = mpl.rcParams.get(\"savefig.pad_inches\", 0.1)\n        tb = fig.get_tightbbox(r).padded(pad_inches)\n        ox_in, oy_in = tb.x0, tb.y0\n        W_in, H_in = tb.width, tb.height\n    elif isinstance(bbox_inches, mpl.transforms.BboxBase):\n        ox_in, oy_in = bbox_inches.x0, bbox_inches.y0\n        W_in, H_in = bbox_inches.width, bbox_inches.height\n    elif bbox_inches is None:\n        ox_in, oy_in = 0.0, 0.0\n        W_in, H_in = fig.get_size_inches()\n    else:\n        raise TypeError(\"panel_crops bbox_inches must be None, 'tight', or a matplotlib Bbox\")\n    W_px, H_px = int(round(W_in * dpi)), int(round(H_in * dpi))\n    lettered = {}\n    seen_letters = set()\n    for ax in fig.axes:\n        for t in ax.findobj(matplotlib.text.Text):\n            s = (t.get_text() or \"\").strip()\n            if t.get_gid() == \"figure-style-panel-letter\":\n                if s in seen_letters:\n                    raise ValueError(f\"Duplicate panel letter: {s!r}\")\n                seen_letters.add(s)\n                lettered[ax] = s\n                break\n\n\n\n    if not lettered:\n        lettered = {ax: str(i) for i, ax in enumerate(fig.axes)}\n    out = {}\n    for ax, letter in lettered.items():\n        bbs = [ax.get_tightbbox(r)]\n\n\n\n\n        ss = ax.get_subplotspec()\n        for sib in fig.axes:\n            if sib is ax or sib in lettered:\n                continue\n            ssib = sib.get_subplotspec()\n            same_row = ss is None or ssib is None or ss.rowspan == ssib.rowspan\n            same_col = ss is None or ssib is None or ss.colspan == ssib.colspan\n            if ((ax.get_shared_y_axes().joined(ax, sib) and same_row)\n                    or (ax.get_shared_x_axes().joined(ax, sib) and same_col)):\n                bbs.append(sib.get_tightbbox(r))\n        bb = mpl.transforms.Bbox.union(bbs)\n\n        bx0 = (bb.x0 / fig.dpi - ox_in) * dpi\n        bx1 = (bb.x1 / fig.dpi - ox_in) * dpi\n        by0 = H_px - (bb.y1 / fig.dpi - oy_in) * dpi\n        by1 = H_px - (bb.y0 / fig.dpi - oy_in) * dpi\n        box = (\n            min(max(int(bx0) - pad_px, 0), W_px),\n            min(max(int(by0) - pad_px, 0), H_px),\n            max(min(int(bx1) + pad_px, W_px), 0),\n            max(min(int(by1) + pad_px, H_px), 0),\n        )\n        if box[0] < box[2] and box[1] < box[3]:\n            out[letter] = box\n    return out\n",
          "sourceDigest": "8020d409bfb60f0c8dfa43d0dfe7bb97f7458e1a3908fefbcb6e4551e0feae4e"
        }
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      "helperEvidenceStatus": {
        "state": "complete"
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        "captureKind": "completed-run",
        "capturedAt": "2026-10-08T10:45:26.709Z",
        "installedInventory": {
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          "source": "cache-reused",
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          "exports": [
            "apply_figure_style",
            "set_frame",
            "panel_letter",
            "focal_palette",
            "bar_with_points",
            "strip_with_median",
            "goodness_arrow",
            "two_tier_label",
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          "source": "META_GREY = \"#888888\"\n\n\ndef apply_figure_style(*, frame=\"open\", font=None, sizes=(8, 7, 6), grid=False):\n    \"\"\"Set matplotlib rcParams for publication-grade output. Call once before plotting.\n\n    This sets mechanics (role-mapped font-size ladder, outward ticks, frameless\n    legends, 300-dpi save, Type-42 embedded fonts) — not a house aesthetic.\n    Frame, font and the size ladder are parameters.\n\n    frame : 'open' (bottom+left spines, default) | 'boxed' (all four) | 'none'\n    font  : sans-serif family name; None = system default sans-serif\n    sizes : (base, secondary, tick) — titles/axis-labels, legend/annotation, ticks\n    grid  : whether to draw axes.grid (default False)\n    \"\"\"\n    import matplotlib as mpl\n    if frame not in (\"open\", \"boxed\", \"none\"):\n        raise ValueError(f\"frame must be 'open'|'boxed'|'none', got {frame!r}\")\n\n    try:\n        import os, sys, glob, matplotlib.font_manager as fm\n        fdir = os.path.join(os.environ.get(\"CONDA_PREFIX\") or sys.prefix, \"fonts\")\n        if os.path.isdir(fdir):\n            known = {f.fname for f in fm.fontManager.ttflist}\n            for f in glob.glob(os.path.join(fdir, \"*.ttf\")):\n                if f not in known:\n                    fm.fontManager.addfont(f)\n    except Exception:\n        pass\n    base, secondary, tick = sizes\n    boxed = (frame == \"boxed\")\n    rc = {\n        \"font.family\": \"sans-serif\",\n        \"font.size\": base,\n        \"axes.labelsize\": base,\n        \"axes.titlesize\": base,\n        \"legend.fontsize\": secondary,\n        \"xtick.labelsize\": tick,\n        \"ytick.labelsize\": tick,\n        \"axes.linewidth\": 0.6,\n        \"xtick.direction\": \"out\", \"ytick.direction\": \"out\",\n        \"xtick.major.size\": 3, \"ytick.major.size\": 3,\n        \"xtick.major.width\": 0.6, \"ytick.major.width\": 0.6,\n        \"axes.spines.top\": boxed, \"axes.spines.right\": boxed,\n        \"axes.spines.left\": frame != \"none\", \"axes.spines.bottom\": frame != \"none\",\n        \"axes.grid\": bool(grid),\n        \"legend.frameon\": False,\n        \"figure.dpi\": 200,\n        \"savefig.dpi\": 300,\n        \"savefig.bbox\": \"tight\",\n        \"axes.titleweight\": \"normal\",\n        \"axes.titlelocation\": \"left\",\n        \"axes.labelweight\": \"normal\",\n        \"lines.linewidth\": 1.2,\n        \"patch.linewidth\": 0.6,\n        \"pdf.fonttype\": 42, \"ps.fonttype\": 42,\n    }\n    if font:\n        rc[\"font.sans-serif\"] = [font, \"DejaVu Sans\"]\n    mpl.rcParams.update(rc)\n\n\ndef set_frame(ax, style=\"open\"):\n    \"\"\"§3: set spine visibility on an existing axes. style ∈ {'open','boxed','none'}.\"\"\"\n    show = {\"open\": (False, False, True, True),\n            \"boxed\": (True, True, True, True),\n            \"none\": (False, False, False, False)}[style]\n    for side, vis in zip((\"top\", \"right\", \"bottom\", \"left\"), show):\n        ax.spines[side].set_visible(vis)\n        if vis:\n            ax.spines[side].set_linewidth(0.6)\n    ax.tick_params(direction=\"out\", length=0 if style == \"none\" else 3, width=0.6)\n\n\ndef panel_letter(ax, letter, dx=-0.18, dy=1.02, case=\"lower\", fontsize=None):\n    \"\"\"§5.7: bold panel letter outside top-left of axes. case ∈ {'lower','upper'}.\"\"\"\n    import matplotlib.pyplot as plt\n    if fontsize is None:\n        fontsize = plt.rcParams.get(\"font.size\", 8) + 1\n    s = letter.lower() if case == \"lower\" else letter.upper()\n    artist = ax.text(dx, dy, s, transform=ax.transAxes,\n                     fontweight=\"bold\", fontsize=fontsize, va=\"bottom\", ha=\"left\")\n    artist.set_gid(\"figure-style-panel-letter\")\n\n\ndef focal_palette(labels, focal, focal_color, other=\"muted\", base_colors=None):\n    \"\"\"§4.2: map labels → colors with the focal series visually dominant.\n\n    other='muted'   — desaturate base_colors (or a default cycle) toward gray\n    other='grey'    — uniform light gray for all non-focal\n    other='ordinal' — non-focal on a single light→dark gray ramp (input order)\n    \"\"\"\n    import matplotlib.colors as mcolors\n    import matplotlib.pyplot as plt\n    focal_set = {focal} if isinstance(focal, str) else set(focal)\n    n = len(labels)\n    if not focal_set & set(labels):\n        raise ValueError(f\"focal {focal!r} not found in labels\")\n    base_colors = [] if base_colors is None else list(base_colors)\n    if not base_colors:\n        base_colors = plt.rcParams[\"axes.prop_cycle\"].by_key().get(\"color\", [\"#444444\"])\n    if not base_colors:\n        base_colors = [\"#444444\"]\n    base_colors = [base_colors[i % len(base_colors)] for i in range(n)]\n    if other == \"grey\":\n        rest = [\"#BCBCBC\"] * n\n    elif other == \"ordinal\":\n        nf = max(1, n - len(focal_set))\n        ramp = [mcolors.to_hex((v, v, v)) for v in\n                ([0.55] if nf == 1 else [0.80 - 0.35 * i / (nf - 1) for i in range(nf)])]\n        rest, k = [], 0\n        for l in labels:\n            rest.append(ramp[min(k, nf - 1)]); k += (l not in focal_set)\n    elif other == \"muted\":\n        def mute(c):\n            r, g, b = mcolors.to_rgb(c)\n            m = (r + g + b) / 3\n            return mcolors.to_hex((0.3 * r + 0.7 * m, 0.3 * g + 0.7 * m, 0.3 * b + 0.7 * m))\n        rest = [mute(c) for c in base_colors]\n    else:\n        raise ValueError(\"other must be 'muted', 'grey', or 'ordinal'\")\n    return [focal_color if l in focal_set else rest[i] for i, l in enumerate(labels)]\n\n\ndef bar_with_points(ax, x, ymat, labels, colors, jitter=0.08, show_points=True,\n                    errorbar=None, point_alpha=0.5, point_size=8):\n    \"\"\"§6.1: bar = mean; optionally overlay raw points or draw an interval.\n\n    colors   : per-label color list (e.g. from focal_palette)\n    x, ymat and labels must have equal lengths; mismatches raise before drawing.\n    Each group must be a nonempty, finite, one-dimensional real numeric array.\n    Requested SD/CI intervals require at least two observations per group.\n    errorbar : None | 'sd' | 'ci95' — drawn only when show_points is False.\n               'ci95' is the t-distribution 95% CI of the mean\n               (half-width t_{0.975,n-1} · s/√n); correct at small n where the\n               z-approximation (1.96·s/√n) is markedly too narrow.\n    \"\"\"\n    import numpy as np\n    x = np.asarray(x)\n    if x.ndim != 1 or not np.issubdtype(x.dtype, np.number):\n        raise TypeError(\n            \"bar_with_points x must be one-dimensional numeric positions; \"\n            \"pass category names through labels\"\n        )\n    ymat = list(ymat)\n    labels = list(labels)\n    if len(x) != len(ymat) or len(x) != len(labels):\n        raise ValueError(\"bar_with_points x, ymat and labels must have equal lengths\")\n    if colors is not None and not isinstance(colors, str):\n        colors = list(colors)\n        if not colors:\n            colors = None\n    if x.dtype.kind not in \"iuf\" or not np.isfinite(x).all():\n        raise ValueError(\"bar_with_points x must contain finite real positions\")\n    for i, y in enumerate(ymat):\n        if np.ma.isMaskedArray(y) and np.ma.getmaskarray(y).any():\n            raise ValueError(f\"bar_with_points group {i} contains masked observations; resolve exclusions explicitly before plotting\")\n    ymat = [np.asarray(y) for y in ymat]\n    for i, y in enumerate(ymat):\n        if y.ndim != 1 or y.size == 0 or y.dtype.kind not in \"iuf\" or not np.isfinite(y).all():\n            raise ValueError(f\"bar_with_points group {i} must be nonempty, one-dimensional and finite real numeric data\")\n    if errorbar and not show_points:\n        if errorbar not in (\"sd\", \"ci95\"):\n            raise ValueError(\"errorbar must be None, 'sd', or 'ci95'\")\n        if any(y.size < 2 for y in ymat):\n            raise ValueError(\"SD/CI cannot be estimated from one observation; use raw points without an interval\")\n    means = np.array([np.mean(y) for y in ymat], float)\n    err = None\n    if errorbar and not show_points:\n        if errorbar == \"sd\":\n            err = np.array([np.std(y, ddof=1) for y in ymat])\n        elif errorbar == \"ci95\":\n            from scipy.stats import t\n            def _hw(y):\n                n = np.asarray(y).size\n                return t.ppf(0.975, n - 1) * np.std(y, ddof=1) / np.sqrt(n)\n            err = np.array([_hw(y) for y in ymat])\n    ax.bar(x, means, color=colors, width=0.7, edgecolor=\"none\",\n           yerr=err, error_kw={\"elinewidth\": 0.8, \"capsize\": 0})\n    if show_points:\n        for xi, ys in zip(x, ymat):\n            ys = np.asarray(ys)\n            if ys.ndim == 1 and ys.size > 0:\n                jit = (np.random.rand(ys.size) - 0.5) * 2 * jitter\n                ax.scatter(np.full(ys.size, xi) + jit, ys, s=point_size, color=\"black\",\n                           alpha=point_alpha, zorder=3, linewidths=0)\n    ax.set_xticks(x); ax.set_xticklabels(labels)\n    return ax\n\n\ndef strip_with_median(ax, groups, values, colors=None, jitter=0.12):\n    \"\"\"§6.1: jittered points + bold horizontal median tick per group.\n\n    groups and values must have equal lengths. Mismatches raise before drawing.\n    colors=None or an empty color sequence supplies default gray; a nonempty\n    color sequence cycles across groups.\n    Each group must be nonempty, one-dimensional and finite real numeric data;\n    resolve missing observations explicitly before plotting.\n    \"\"\"\n    import numpy as np\n    labs = list(groups)\n    values = list(values)\n    if len(labs) != len(values):\n        raise ValueError(\"strip_with_median groups and values must have equal lengths\")\n    for i, y in enumerate(values):\n        if np.ma.isMaskedArray(y) and np.ma.getmaskarray(y).any():\n            raise ValueError(f\"strip_with_median group {i} contains masked observations; resolve exclusions explicitly before plotting\")\n    values = [np.asarray(y) for y in values]\n    for i, y in enumerate(values):\n        if y.ndim != 1 or y.size == 0 or y.dtype.kind not in \"iuf\" or not np.isfinite(y).all():\n            raise ValueError(f\"strip_with_median group {i} must be nonempty, one-dimensional and finite real numeric data\")\n    colors = [] if colors is None else list(colors)\n    if not colors:\n        colors = [\"#444444\"]\n    for i, ys in enumerate(values):\n        c = colors[i % len(colors)]\n        ys = np.asarray(ys)\n        jit = (np.random.rand(ys.size) - 0.5) * 2 * jitter\n        ax.scatter(np.full(ys.size, i) + jit, ys, s=10, color=c, alpha=0.6, linewidths=0, zorder=2)\n        m = np.median(ys)\n        ax.plot([i - 0.22, i + 0.22], [m, m], color=\"black\", lw=1.6, zorder=3)\n    ax.set_xticks(range(len(labs))); ax.set_xticklabels(labs)\n    return ax\n\n\ndef goodness_arrow(ax, text=\"higher = better\", loc=\"upper left\", axis=\"y\", fontsize=None):\n    \"\"\"§3.6: small upright direction-of-goodness cue in the margin.\"\"\"\n    import matplotlib.pyplot as plt\n    if fontsize is None:\n        fontsize = plt.rcParams[\"legend.fontsize\"]\n    pos = {\"upper left\": (0.02, 0.98), \"upper right\": (0.98, 0.98),\n           \"lower left\": (0.02, 0.02), \"lower right\": (0.98, 0.02)}[loc]\n    ha = \"left\" if \"left\" in loc else \"right\"\n    va = \"top\" if \"upper\" in loc else \"bottom\"\n    arrow = \"↑ \" if axis == \"y\" else \"→ \"\n    ax.text(pos[0], pos[1], arrow + text, transform=ax.transAxes,\n            fontsize=fontsize, color=META_GREY, ha=ha, va=va)\n\n\ndef two_tier_label(name, meta):\n    \"\"\"§5: two-line label string (name / metadata). Meta line styled separately by caller.\"\"\"\n    return f\"{name}\\n{meta}\"\n\n\ndef end_of_line_labels(ax, xs, ys, labels, colors=None, dx=0.01, fontsize=None):\n    \"\"\"§6.3 / §7.3: label each line series at its right end instead of a legend box.\n\n    Each label requires a nonempty, matched x/y series. Validate all series\n    before drawing. Nonempty colors cycle; None or empty uses default text color.\n    Inner sequences must support len() and indexing; they are not copied.\n    Nonfinite endpoints are rejected rather than producing an invisible label.\n    \"\"\"\n    import matplotlib.pyplot as plt\n    import math\n    xs, ys, labels = list(xs), list(ys), list(labels)\n    if len(xs) != len(ys) or len(xs) != len(labels):\n        raise ValueError(\"end_of_line_labels xs, ys and labels must have equal lengths\")\n    if any(len(x) == 0 or len(x) != len(y) for x, y in zip(xs, ys)):\n        raise ValueError(\"end_of_line_labels each x/y series must be nonempty and have equal lengths\")\n    if any(not math.isfinite(x[-1]) or not math.isfinite(y[-1]) for x, y in zip(xs, ys)):\n        raise ValueError(\"end_of_line_labels endpoints must be finite; choose valid endpoints explicitly\")\n    if fontsize is None:\n        fontsize = plt.rcParams[\"font.size\"]\n    colors = [] if colors is None else list(colors)\n    if not colors:\n        colors = [None]\n    span = ax.get_xlim()[1] - ax.get_xlim()[0]\n    for i, (x, y, lab) in enumerate(zip(xs, ys, labels)):\n        c = colors[i % len(colors)]\n        ax.text(x[-1] + dx * span, y[-1], lab, color=c, va=\"center\", ha=\"left\", fontsize=fontsize)\n\n\ndef panel_crops(fig, dpi=None, pad_px=6, bbox_inches=None, pad_inches=None):\n    \"\"\"§9.2: pixel-space crop boxes for each lettered panel in the SAVED PNG.\n\n    Returns ``{letter: (x0, y0, x1, y1)}`` in image-space pixels (origin\n    top-left, matching PIL's ``Image.crop``). Panels are detected as bold\n    single-character ``Text``\n    objects placed by :func:`panel_letter`; each panel's crop is its axes'\n    tightbbox mapped into the saved file's pixel space, padded by ``pad_px``.\n    For §3.4 composites (abutting subplots sharing an axis, letter on the\n    leftmost only) the crop unions in letterless ``sharex``/``sharey`` siblings\n    on the same grid row/col so the whole composite is covered. When no axes\n    carries a panel letter (standalone plot, or a figure-composer sub-agent),\n    falls back to one crop per axes keyed by index.\n\n    ``bbox_inches`` mirrors ``Figure.savefig`` semantics: ``None`` means\n    *consult rcParams* (so under :func:`apply_figure_style` it resolves to\n    ``'tight'``); pass an explicit ``Bbox`` only if you saved with one. The\n    boxes are clamped to the saved image extent; panels with no intersection\n    are omitted.\n\n        >>> from PIL import Image\n        >>> fig.savefig(\"fig.png\")  # bbox_inches='tight' via rcParams\n        >>> with Image.open(\"fig.png\") as saved:\n        ...     for letter, box in panel_crops(fig).items():\n        ...         saved.crop(box).save(f\"panel_{letter}.png\")\n\n    Match savefig's DPI, bbox_inches and pad_inches when overriding defaults.\n    Returned boxes are pixels, never savefig bbox_inches (inches). Passing them\n    back to savefig as inches can allocate an enormous raster.\n    \"\"\"\n    import matplotlib as mpl\n    import matplotlib.text\n    from matplotlib.backends.backend_agg import FigureCanvasAgg\n    if isinstance(pad_px, bool) or not isinstance(pad_px, int) or pad_px < 0:\n        raise TypeError(\"panel_crops pad_px must be a non-negative integer\")\n    if dpi is None:\n        dpi = mpl.rcParams.get(\"savefig.dpi\", fig.dpi)\n        if dpi == \"figure\":\n            dpi = fig.dpi\n    dpi = float(dpi)\n    if bbox_inches is None:\n        bbox_inches = mpl.rcParams.get(\"savefig.bbox\")\n    # The Open-Science notebook captures and closes pyplot figures after every\n    # cell. Matplotlib 3.11 decouples a closed Figure from its backend by\n    # replacing FigureCanvasAgg with FigureCanvasBase, while the persistent\n    # notebook namespace can still retain the Figure itself. Reattach the\n    # headless canvas before asking for saved-image geometry so this helper is\n    # safe in a later QA cell as well as the producer cell.\n    if not hasattr(fig.canvas, \"get_renderer\"):\n        FigureCanvasAgg(fig)\n    fig.canvas.draw()\n    r = fig.canvas.get_renderer()\n\n    if bbox_inches == \"tight\":\n        if pad_inches is None:\n            pad_inches = mpl.rcParams.get(\"savefig.pad_inches\", 0.1)\n        tb = fig.get_tightbbox(r).padded(pad_inches)\n        ox_in, oy_in = tb.x0, tb.y0\n        W_in, H_in = tb.width, tb.height\n    elif isinstance(bbox_inches, mpl.transforms.BboxBase):\n        ox_in, oy_in = bbox_inches.x0, bbox_inches.y0\n        W_in, H_in = bbox_inches.width, bbox_inches.height\n    elif bbox_inches is None:\n        ox_in, oy_in = 0.0, 0.0\n        W_in, H_in = fig.get_size_inches()\n    else:\n        raise TypeError(\"panel_crops bbox_inches must be None, 'tight', or a matplotlib Bbox\")\n    W_px, H_px = int(round(W_in * dpi)), int(round(H_in * dpi))\n    lettered = {}\n    seen_letters = set()\n    for ax in fig.axes:\n        for t in ax.findobj(matplotlib.text.Text):\n            s = (t.get_text() or \"\").strip()\n            if t.get_gid() == \"figure-style-panel-letter\":\n                if s in seen_letters:\n                    raise ValueError(f\"Duplicate panel letter: {s!r}\")\n                seen_letters.add(s)\n                lettered[ax] = s\n                break\n\n\n\n    if not lettered:\n        lettered = {ax: str(i) for i, ax in enumerate(fig.axes)}\n    out = {}\n    for ax, letter in lettered.items():\n        bbs = [ax.get_tightbbox(r)]\n\n\n\n\n        ss = ax.get_subplotspec()\n        for sib in fig.axes:\n            if sib is ax or sib in lettered:\n                continue\n            ssib = sib.get_subplotspec()\n            same_row = ss is None or ssib is None or ss.rowspan == ssib.rowspan\n            same_col = ss is None or ssib is None or ss.colspan == ssib.colspan\n            if ((ax.get_shared_y_axes().joined(ax, sib) and same_row)\n                    or (ax.get_shared_x_axes().joined(ax, sib) and same_col)):\n                bbs.append(sib.get_tightbbox(r))\n        bb = mpl.transforms.Bbox.union(bbs)\n\n        bx0 = (bb.x0 / fig.dpi - ox_in) * dpi\n        bx1 = (bb.x1 / fig.dpi - ox_in) * dpi\n        by0 = H_px - (bb.y1 / fig.dpi - oy_in) * dpi\n        by1 = H_px - (bb.y0 / fig.dpi - oy_in) * dpi\n        box = (\n            min(max(int(bx0) - pad_px, 0), W_px),\n            min(max(int(by0) - pad_px, 0), H_px),\n            max(min(int(bx1) + pad_px, W_px), 0),\n            max(min(int(by1) + pad_px, H_px), 0),\n        )\n        if box[0] < box[2] and box[1] < box[3]:\n            out[letter] = box\n    return out\n",
          "sourceDigest": "8020d409bfb60f0c8dfa43d0dfe7bb97f7458e1a3908fefbcb6e4551e0feae4e"
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        "installedInventory": {
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          "source": "cache-reused",
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            "observedVersion": "3.11.1"
          },
          {
            "reason": "package-lock-missing",
            "packageName": "numpy",
            "observedVersion": "2.5.3"
          },
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            "reason": "package-lock-missing",
            "packageName": "packaging",
            "observedVersion": "26.3"
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          {
            "reason": "package-lock-missing",
            "packageName": "pip",
            "observedVersion": "26.2.1"
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          {
            "reason": "package-lock-missing",
            "packageName": "setuptools",
            "observedVersion": "83.0.0"
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      },
      "kernelDispatched": true,
      "helperModules": [
        {
          "helperId": "figure-style",
          "skillIdentity": "figure-style",
          "packageOrigin": "builtin",
          "interfaceRevision": "1",
          "registeredGeneration": "sha256:90258343aec7bb96c2e129c39e1beb23230ee947c0db563f96913458d1795f1d",
          "exports": [
            "apply_figure_style",
            "set_frame",
            "panel_letter",
            "focal_palette",
            "bar_with_points",
            "strip_with_median",
            "goodness_arrow",
            "two_tier_label",
            "end_of_line_labels",
            "panel_crops"
          ],
          "source": "META_GREY = \"#888888\"\n\n\ndef apply_figure_style(*, frame=\"open\", font=None, sizes=(8, 7, 6), grid=False):\n    \"\"\"Set matplotlib rcParams for publication-grade output. Call once before plotting.\n\n    This sets mechanics (role-mapped font-size ladder, outward ticks, frameless\n    legends, 300-dpi save, Type-42 embedded fonts) — not a house aesthetic.\n    Frame, font and the size ladder are parameters.\n\n    frame : 'open' (bottom+left spines, default) | 'boxed' (all four) | 'none'\n    font  : sans-serif family name; None = system default sans-serif\n    sizes : (base, secondary, tick) — titles/axis-labels, legend/annotation, ticks\n    grid  : whether to draw axes.grid (default False)\n    \"\"\"\n    import matplotlib as mpl\n    if frame not in (\"open\", \"boxed\", \"none\"):\n        raise ValueError(f\"frame must be 'open'|'boxed'|'none', got {frame!r}\")\n\n    try:\n        import os, sys, glob, matplotlib.font_manager as fm\n        fdir = os.path.join(os.environ.get(\"CONDA_PREFIX\") or sys.prefix, \"fonts\")\n        if os.path.isdir(fdir):\n            known = {f.fname for f in fm.fontManager.ttflist}\n            for f in glob.glob(os.path.join(fdir, \"*.ttf\")):\n                if f not in known:\n                    fm.fontManager.addfont(f)\n    except Exception:\n        pass\n    base, secondary, tick = sizes\n    boxed = (frame == \"boxed\")\n    rc = {\n        \"font.family\": \"sans-serif\",\n        \"font.size\": base,\n        \"axes.labelsize\": base,\n        \"axes.titlesize\": base,\n        \"legend.fontsize\": secondary,\n        \"xtick.labelsize\": tick,\n        \"ytick.labelsize\": tick,\n        \"axes.linewidth\": 0.6,\n        \"xtick.direction\": \"out\", \"ytick.direction\": \"out\",\n        \"xtick.major.size\": 3, \"ytick.major.size\": 3,\n        \"xtick.major.width\": 0.6, \"ytick.major.width\": 0.6,\n        \"axes.spines.top\": boxed, \"axes.spines.right\": boxed,\n        \"axes.spines.left\": frame != \"none\", \"axes.spines.bottom\": frame != \"none\",\n        \"axes.grid\": bool(grid),\n        \"legend.frameon\": False,\n        \"figure.dpi\": 200,\n        \"savefig.dpi\": 300,\n        \"savefig.bbox\": \"tight\",\n        \"axes.titleweight\": \"normal\",\n        \"axes.titlelocation\": \"left\",\n        \"axes.labelweight\": \"normal\",\n        \"lines.linewidth\": 1.2,\n        \"patch.linewidth\": 0.6,\n        \"pdf.fonttype\": 42, \"ps.fonttype\": 42,\n    }\n    if font:\n        rc[\"font.sans-serif\"] = [font, \"DejaVu Sans\"]\n    mpl.rcParams.update(rc)\n\n\ndef set_frame(ax, style=\"open\"):\n    \"\"\"§3: set spine visibility on an existing axes. style ∈ {'open','boxed','none'}.\"\"\"\n    show = {\"open\": (False, False, True, True),\n            \"boxed\": (True, True, True, True),\n            \"none\": (False, False, False, False)}[style]\n    for side, vis in zip((\"top\", \"right\", \"bottom\", \"left\"), show):\n        ax.spines[side].set_visible(vis)\n        if vis:\n            ax.spines[side].set_linewidth(0.6)\n    ax.tick_params(direction=\"out\", length=0 if style == \"none\" else 3, width=0.6)\n\n\ndef panel_letter(ax, letter, dx=-0.18, dy=1.02, case=\"lower\", fontsize=None):\n    \"\"\"§5.7: bold panel letter outside top-left of axes. case ∈ {'lower','upper'}.\"\"\"\n    import matplotlib.pyplot as plt\n    if fontsize is None:\n        fontsize = plt.rcParams.get(\"font.size\", 8) + 1\n    s = letter.lower() if case == \"lower\" else letter.upper()\n    artist = ax.text(dx, dy, s, transform=ax.transAxes,\n                     fontweight=\"bold\", fontsize=fontsize, va=\"bottom\", ha=\"left\")\n    artist.set_gid(\"figure-style-panel-letter\")\n\n\ndef focal_palette(labels, focal, focal_color, other=\"muted\", base_colors=None):\n    \"\"\"§4.2: map labels → colors with the focal series visually dominant.\n\n    other='muted'   — desaturate base_colors (or a default cycle) toward gray\n    other='grey'    — uniform light gray for all non-focal\n    other='ordinal' — non-focal on a single light→dark gray ramp (input order)\n    \"\"\"\n    import matplotlib.colors as mcolors\n    import matplotlib.pyplot as plt\n    focal_set = {focal} if isinstance(focal, str) else set(focal)\n    n = len(labels)\n    if not focal_set & set(labels):\n        raise ValueError(f\"focal {focal!r} not found in labels\")\n    base_colors = [] if base_colors is None else list(base_colors)\n    if not base_colors:\n        base_colors = plt.rcParams[\"axes.prop_cycle\"].by_key().get(\"color\", [\"#444444\"])\n    if not base_colors:\n        base_colors = [\"#444444\"]\n    base_colors = [base_colors[i % len(base_colors)] for i in range(n)]\n    if other == \"grey\":\n        rest = [\"#BCBCBC\"] * n\n    elif other == \"ordinal\":\n        nf = max(1, n - len(focal_set))\n        ramp = [mcolors.to_hex((v, v, v)) for v in\n                ([0.55] if nf == 1 else [0.80 - 0.35 * i / (nf - 1) for i in range(nf)])]\n        rest, k = [], 0\n        for l in labels:\n            rest.append(ramp[min(k, nf - 1)]); k += (l not in focal_set)\n    elif other == \"muted\":\n        def mute(c):\n            r, g, b = mcolors.to_rgb(c)\n            m = (r + g + b) / 3\n            return mcolors.to_hex((0.3 * r + 0.7 * m, 0.3 * g + 0.7 * m, 0.3 * b + 0.7 * m))\n        rest = [mute(c) for c in base_colors]\n    else:\n        raise ValueError(\"other must be 'muted', 'grey', or 'ordinal'\")\n    return [focal_color if l in focal_set else rest[i] for i, l in enumerate(labels)]\n\n\ndef bar_with_points(ax, x, ymat, labels, colors, jitter=0.08, show_points=True,\n                    errorbar=None, point_alpha=0.5, point_size=8):\n    \"\"\"§6.1: bar = mean; optionally overlay raw points or draw an interval.\n\n    colors   : per-label color list (e.g. from focal_palette)\n    x, ymat and labels must have equal lengths; mismatches raise before drawing.\n    Each group must be a nonempty, finite, one-dimensional real numeric array.\n    Requested SD/CI intervals require at least two observations per group.\n    errorbar : None | 'sd' | 'ci95' — drawn only when show_points is False.\n               'ci95' is the t-distribution 95% CI of the mean\n               (half-width t_{0.975,n-1} · s/√n); correct at small n where the\n               z-approximation (1.96·s/√n) is markedly too narrow.\n    \"\"\"\n    import numpy as np\n    x = np.asarray(x)\n    if x.ndim != 1 or not np.issubdtype(x.dtype, np.number):\n        raise TypeError(\n            \"bar_with_points x must be one-dimensional numeric positions; \"\n            \"pass category names through labels\"\n        )\n    ymat = list(ymat)\n    labels = list(labels)\n    if len(x) != len(ymat) or len(x) != len(labels):\n        raise ValueError(\"bar_with_points x, ymat and labels must have equal lengths\")\n    if colors is not None and not isinstance(colors, str):\n        colors = list(colors)\n        if not colors:\n            colors = None\n    if x.dtype.kind not in \"iuf\" or not np.isfinite(x).all():\n        raise ValueError(\"bar_with_points x must contain finite real positions\")\n    for i, y in enumerate(ymat):\n        if np.ma.isMaskedArray(y) and np.ma.getmaskarray(y).any():\n            raise ValueError(f\"bar_with_points group {i} contains masked observations; resolve exclusions explicitly before plotting\")\n    ymat = [np.asarray(y) for y in ymat]\n    for i, y in enumerate(ymat):\n        if y.ndim != 1 or y.size == 0 or y.dtype.kind not in \"iuf\" or not np.isfinite(y).all():\n            raise ValueError(f\"bar_with_points group {i} must be nonempty, one-dimensional and finite real numeric data\")\n    if errorbar and not show_points:\n        if errorbar not in (\"sd\", \"ci95\"):\n            raise ValueError(\"errorbar must be None, 'sd', or 'ci95'\")\n        if any(y.size < 2 for y in ymat):\n            raise ValueError(\"SD/CI cannot be estimated from one observation; use raw points without an interval\")\n    means = np.array([np.mean(y) for y in ymat], float)\n    err = None\n    if errorbar and not show_points:\n        if errorbar == \"sd\":\n            err = np.array([np.std(y, ddof=1) for y in ymat])\n        elif errorbar == \"ci95\":\n            from scipy.stats import t\n            def _hw(y):\n                n = np.asarray(y).size\n                return t.ppf(0.975, n - 1) * np.std(y, ddof=1) / np.sqrt(n)\n            err = np.array([_hw(y) for y in ymat])\n    ax.bar(x, means, color=colors, width=0.7, edgecolor=\"none\",\n           yerr=err, error_kw={\"elinewidth\": 0.8, \"capsize\": 0})\n    if show_points:\n        for xi, ys in zip(x, ymat):\n            ys = np.asarray(ys)\n            if ys.ndim == 1 and ys.size > 0:\n                jit = (np.random.rand(ys.size) - 0.5) * 2 * jitter\n                ax.scatter(np.full(ys.size, xi) + jit, ys, s=point_size, color=\"black\",\n                           alpha=point_alpha, zorder=3, linewidths=0)\n    ax.set_xticks(x); ax.set_xticklabels(labels)\n    return ax\n\n\ndef strip_with_median(ax, groups, values, colors=None, jitter=0.12):\n    \"\"\"§6.1: jittered points + bold horizontal median tick per group.\n\n    groups and values must have equal lengths. Mismatches raise before drawing.\n    colors=None or an empty color sequence supplies default gray; a nonempty\n    color sequence cycles across groups.\n    Each group must be nonempty, one-dimensional and finite real numeric data;\n    resolve missing observations explicitly before plotting.\n    \"\"\"\n    import numpy as np\n    labs = list(groups)\n    values = list(values)\n    if len(labs) != len(values):\n        raise ValueError(\"strip_with_median groups and values must have equal lengths\")\n    for i, y in enumerate(values):\n        if np.ma.isMaskedArray(y) and np.ma.getmaskarray(y).any():\n            raise ValueError(f\"strip_with_median group {i} contains masked observations; resolve exclusions explicitly before plotting\")\n    values = [np.asarray(y) for y in values]\n    for i, y in enumerate(values):\n        if y.ndim != 1 or y.size == 0 or y.dtype.kind not in \"iuf\" or not np.isfinite(y).all():\n            raise ValueError(f\"strip_with_median group {i} must be nonempty, one-dimensional and finite real numeric data\")\n    colors = [] if colors is None else list(colors)\n    if not colors:\n        colors = [\"#444444\"]\n    for i, ys in enumerate(values):\n        c = colors[i % len(colors)]\n        ys = np.asarray(ys)\n        jit = (np.random.rand(ys.size) - 0.5) * 2 * jitter\n        ax.scatter(np.full(ys.size, i) + jit, ys, s=10, color=c, alpha=0.6, linewidths=0, zorder=2)\n        m = np.median(ys)\n        ax.plot([i - 0.22, i + 0.22], [m, m], color=\"black\", lw=1.6, zorder=3)\n    ax.set_xticks(range(len(labs))); ax.set_xticklabels(labs)\n    return ax\n\n\ndef goodness_arrow(ax, text=\"higher = better\", loc=\"upper left\", axis=\"y\", fontsize=None):\n    \"\"\"§3.6: small upright direction-of-goodness cue in the margin.\"\"\"\n    import matplotlib.pyplot as plt\n    if fontsize is None:\n        fontsize = plt.rcParams[\"legend.fontsize\"]\n    pos = {\"upper left\": (0.02, 0.98), \"upper right\": (0.98, 0.98),\n           \"lower left\": (0.02, 0.02), \"lower right\": (0.98, 0.02)}[loc]\n    ha = \"left\" if \"left\" in loc else \"right\"\n    va = \"top\" if \"upper\" in loc else \"bottom\"\n    arrow = \"↑ \" if axis == \"y\" else \"→ \"\n    ax.text(pos[0], pos[1], arrow + text, transform=ax.transAxes,\n            fontsize=fontsize, color=META_GREY, ha=ha, va=va)\n\n\ndef two_tier_label(name, meta):\n    \"\"\"§5: two-line label string (name / metadata). Meta line styled separately by caller.\"\"\"\n    return f\"{name}\\n{meta}\"\n\n\ndef end_of_line_labels(ax, xs, ys, labels, colors=None, dx=0.01, fontsize=None):\n    \"\"\"§6.3 / §7.3: label each line series at its right end instead of a legend box.\n\n    Each label requires a nonempty, matched x/y series. Validate all series\n    before drawing. Nonempty colors cycle; None or empty uses default text color.\n    Inner sequences must support len() and indexing; they are not copied.\n    Nonfinite endpoints are rejected rather than producing an invisible label.\n    \"\"\"\n    import matplotlib.pyplot as plt\n    import math\n    xs, ys, labels = list(xs), list(ys), list(labels)\n    if len(xs) != len(ys) or len(xs) != len(labels):\n        raise ValueError(\"end_of_line_labels xs, ys and labels must have equal lengths\")\n    if any(len(x) == 0 or len(x) != len(y) for x, y in zip(xs, ys)):\n        raise ValueError(\"end_of_line_labels each x/y series must be nonempty and have equal lengths\")\n    if any(not math.isfinite(x[-1]) or not math.isfinite(y[-1]) for x, y in zip(xs, ys)):\n        raise ValueError(\"end_of_line_labels endpoints must be finite; choose valid endpoints explicitly\")\n    if fontsize is None:\n        fontsize = plt.rcParams[\"font.size\"]\n    colors = [] if colors is None else list(colors)\n    if not colors:\n        colors = [None]\n    span = ax.get_xlim()[1] - ax.get_xlim()[0]\n    for i, (x, y, lab) in enumerate(zip(xs, ys, labels)):\n        c = colors[i % len(colors)]\n        ax.text(x[-1] + dx * span, y[-1], lab, color=c, va=\"center\", ha=\"left\", fontsize=fontsize)\n\n\ndef panel_crops(fig, dpi=None, pad_px=6, bbox_inches=None, pad_inches=None):\n    \"\"\"§9.2: pixel-space crop boxes for each lettered panel in the SAVED PNG.\n\n    Returns ``{letter: (x0, y0, x1, y1)}`` in image-space pixels (origin\n    top-left, matching PIL's ``Image.crop``). Panels are detected as bold\n    single-character ``Text``\n    objects placed by :func:`panel_letter`; each panel's crop is its axes'\n    tightbbox mapped into the saved file's pixel space, padded by ``pad_px``.\n    For §3.4 composites (abutting subplots sharing an axis, letter on the\n    leftmost only) the crop unions in letterless ``sharex``/``sharey`` siblings\n    on the same grid row/col so the whole composite is covered. When no axes\n    carries a panel letter (standalone plot, or a figure-composer sub-agent),\n    falls back to one crop per axes keyed by index.\n\n    ``bbox_inches`` mirrors ``Figure.savefig`` semantics: ``None`` means\n    *consult rcParams* (so under :func:`apply_figure_style` it resolves to\n    ``'tight'``); pass an explicit ``Bbox`` only if you saved with one. The\n    boxes are clamped to the saved image extent; panels with no intersection\n    are omitted.\n\n        >>> from PIL import Image\n        >>> fig.savefig(\"fig.png\")  # bbox_inches='tight' via rcParams\n        >>> with Image.open(\"fig.png\") as saved:\n        ...     for letter, box in panel_crops(fig).items():\n        ...         saved.crop(box).save(f\"panel_{letter}.png\")\n\n    Match savefig's DPI, bbox_inches and pad_inches when overriding defaults.\n    Returned boxes are pixels, never savefig bbox_inches (inches). Passing them\n    back to savefig as inches can allocate an enormous raster.\n    \"\"\"\n    import matplotlib as mpl\n    import matplotlib.text\n    from matplotlib.backends.backend_agg import FigureCanvasAgg\n    if isinstance(pad_px, bool) or not isinstance(pad_px, int) or pad_px < 0:\n        raise TypeError(\"panel_crops pad_px must be a non-negative integer\")\n    if dpi is None:\n        dpi = mpl.rcParams.get(\"savefig.dpi\", fig.dpi)\n        if dpi == \"figure\":\n            dpi = fig.dpi\n    dpi = float(dpi)\n    if bbox_inches is None:\n        bbox_inches = mpl.rcParams.get(\"savefig.bbox\")\n    # The Open-Science notebook captures and closes pyplot figures after every\n    # cell. Matplotlib 3.11 decouples a closed Figure from its backend by\n    # replacing FigureCanvasAgg with FigureCanvasBase, while the persistent\n    # notebook namespace can still retain the Figure itself. Reattach the\n    # headless canvas before asking for saved-image geometry so this helper is\n    # safe in a later QA cell as well as the producer cell.\n    if not hasattr(fig.canvas, \"get_renderer\"):\n        FigureCanvasAgg(fig)\n    fig.canvas.draw()\n    r = fig.canvas.get_renderer()\n\n    if bbox_inches == \"tight\":\n        if pad_inches is None:\n            pad_inches = mpl.rcParams.get(\"savefig.pad_inches\", 0.1)\n        tb = fig.get_tightbbox(r).padded(pad_inches)\n        ox_in, oy_in = tb.x0, tb.y0\n        W_in, H_in = tb.width, tb.height\n    elif isinstance(bbox_inches, mpl.transforms.BboxBase):\n        ox_in, oy_in = bbox_inches.x0, bbox_inches.y0\n        W_in, H_in = bbox_inches.width, bbox_inches.height\n    elif bbox_inches is None:\n        ox_in, oy_in = 0.0, 0.0\n        W_in, H_in = fig.get_size_inches()\n    else:\n        raise TypeError(\"panel_crops bbox_inches must be None, 'tight', or a matplotlib Bbox\")\n    W_px, H_px = int(round(W_in * dpi)), int(round(H_in * dpi))\n    lettered = {}\n    seen_letters = set()\n    for ax in fig.axes:\n        for t in ax.findobj(matplotlib.text.Text):\n            s = (t.get_text() or \"\").strip()\n            if t.get_gid() == \"figure-style-panel-letter\":\n                if s in seen_letters:\n                    raise ValueError(f\"Duplicate panel letter: {s!r}\")\n                seen_letters.add(s)\n                lettered[ax] = s\n                break\n\n\n\n    if not lettered:\n        lettered = {ax: str(i) for i, ax in enumerate(fig.axes)}\n    out = {}\n    for ax, letter in lettered.items():\n        bbs = [ax.get_tightbbox(r)]\n\n\n\n\n        ss = ax.get_subplotspec()\n        for sib in fig.axes:\n            if sib is ax or sib in lettered:\n                continue\n            ssib = sib.get_subplotspec()\n            same_row = ss is None or ssib is None or ss.rowspan == ssib.rowspan\n            same_col = ss is None or ssib is None or ss.colspan == ssib.colspan\n            if ((ax.get_shared_y_axes().joined(ax, sib) and same_row)\n                    or (ax.get_shared_x_axes().joined(ax, sib) and same_col)):\n                bbs.append(sib.get_tightbbox(r))\n        bb = mpl.transforms.Bbox.union(bbs)\n\n        bx0 = (bb.x0 / fig.dpi - ox_in) * dpi\n        bx1 = (bb.x1 / fig.dpi - ox_in) * dpi\n        by0 = H_px - (bb.y1 / fig.dpi - oy_in) * dpi\n        by1 = H_px - (bb.y0 / fig.dpi - oy_in) * dpi\n        box = (\n            min(max(int(bx0) - pad_px, 0), W_px),\n            min(max(int(by0) - pad_px, 0), H_px),\n            max(min(int(bx1) + pad_px, W_px), 0),\n            max(min(int(by1) + pad_px, H_px), 0),\n        )\n        if box[0] < box[2] and box[1] < box[3]:\n            out[letter] = box\n    return out\n",
          "sourceDigest": "8020d409bfb60f0c8dfa43d0dfe7bb97f7458e1a3908fefbcb6e4551e0feae4e"
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        "stdout": "[{\"config\": \"lean/ComparatorChallenges/SelfSimilarCorollaries.json\", \"solution\": \"lean/OAI/MeasureTheory/SelfSimilar/DimensionCorollaries.lean\", \"declarations\": [{\"target\": \"OAI.EntropyRateDimension.Extensions.homogeneous_dimension_direct\", \"local_declaration_found\": false, \"declaration_locations\": [], \"status\": \"imported body not supplied\"}, {\"target\": \"OAI.CorSetReference.cor_set\", \"local_declaration_found\": false, \"declaration_locations\": [], \"status\": \"imported body not supplied\"}], \"missing_direct_count\": 2}, {\"config\": \"lean/ComparatorChallenges/SingletonLoopMatching.json\", \"solution\": \"lean/OAI/Computability/LoopMatching/Endpoint.lean\", \"declarations\": [{\"target\": \"OAI.LoopMatching.fullEndpoint\", \"local_declaration_found\": false, \"declaration_locations\": [{\"path\": \"lean/OAI/Computability/LoopMatching/Main.lean\", \"line\": 24, \"placement\": \"declaration in another supplied file; import/name resolution not compiler-checked\"}], \"status\": \"body in supplied import\"}], \"missing_direct_count\": 2}, {\"config\": \"lean/ComparatorChallenges/SquareRootDegree.json\", \"solution\": \"lean/OAI/Combinatorics/BooleanFunctions/Main.lean\", \"declarations\": [{\"target\": \"OAI.SquareRootDegree.main\", \"local_declaration_found\": true, \"declaration_locations\": [{\"path\": \"lean/OAI/Combinatorics/BooleanFunctions/Main.lean\", \"line\": 20, \"placement\": \"local declaration\"}], \"status\": \"local body present\"}], \"missing_direct_count\": 2}, {\"config\": \"lean/ComparatorChallenges/TotientAsymptotic.json\", \"solution\": \"lean/OAI/NumberTheory/TotientAsymptotic/UnconditionalMain.lean\", \"declarations\": [{\"target\": \"OAI.TotientAsymptotic.totient_asymptotic_formula\", \"local_declaration_found\": true, \"declaration_locations\": [{\"path\": \"lean/OAI/NumberTheory/TotientAsymptotic/UnconditionalMain.lean\", \"line\": 12, \"placement\": \"local declaration\"}], \"status\": \"local body present\"}, {\"target\": \"OAI.TotientAsymptotic.weighted_totient_asymptotic\", \"local_declaration_found\": true, \"declaration_locations\": [{\"path\": \"lean/OAI/NumberTheory/TotientAsymptotic/UnconditionalMain.lean\", \"line\": 23, \"placement\": \"local declaration\"}], \"status\": \"local body present\"}, {\"target\": \"OAI.TotientAsymptotic.weighted_totient_one_two\", \"local_declaration_found\": true, \"declaration_locations\": [{\"path\": \"lean/OAI/NumberTheory/TotientAsymptotic/UnconditionalMain.lean\", \"line\": 37, \"placement\": \"local declaration\"}], \"status\": \"local body present\"}, {\"target\": \"OAI.TotientAsymptotic.coefficient_nonnegative\", \"local_declaration_found\": true, \"declaration_locations\": [{\"path\": \"lean/OAI/NumberTheory/TotientAsymptotic/UnconditionalMain.lean\", \"line\": 44, \"placement\": \"local declaration\"}], \"status\": \"local body present\"}], \"missing_direct_count\": 2}, {\"config\": \"lean/ComparatorChallenges/TotientCompanionZero.json\", \"solution\": \"lean/OAI/NumberTheory/TotientAsymptotic/Main.lean\", \"declarations\": [{\"target\": \"OAI.TotientAsymptotic.companion_zero_case\", \"local_declaration_found\": false, \"declaration_locations\": [], \"status\": \"imported body not supplied\"}], \"missing_direct_count\": 202}]\n",
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          "text": "[{\"config\": \"lean/ComparatorChallenges/SelfSimilarCorollaries.json\", \"solution\": \"lean/OAI/MeasureTheory/SelfSimilar/DimensionCorollaries.lean\", \"declarations\": [{\"target\": \"OAI.EntropyRateDimension.Extensions.homogeneous_dimension_direct\", \"local_declaration_found\": false, \"declaration_locations\": [], \"status\": \"imported body not supplied\"}, {\"target\": \"OAI.CorSetReference.cor_set\", \"local_declaration_found\": false, \"declaration_locations\": [], \"status\": \"imported body not supplied\"}], \"missing_direct_count\": 2}, {\"config\": \"lean/ComparatorChallenges/SingletonLoopMatching.json\", \"solution\": \"lean/OAI/Computability/LoopMatching/Endpoint.lean\", \"declarations\": [{\"target\": \"OAI.LoopMatching.fullEndpoint\", \"local_declaration_found\": false, \"declaration_locations\": [{\"path\": \"lean/OAI/Computability/LoopMatching/Main.lean\", \"line\": 24, \"placement\": \"declaration in another supplied file; import/name resolution not compiler-checked\"}], \"status\": \"body in supplied import\"}], \"missing_direct_count\": 2}, {\"config\": \"lean/ComparatorChallenges/SquareRootDegree.json\", \"solution\": \"lean/OAI/Combinatorics/BooleanFunctions/Main.lean\", \"declarations\": [{\"target\": \"OAI.SquareRootDegree.main\", \"local_declaration_found\": true, \"declaration_locations\": [{\"path\": \"lean/OAI/Combinatorics/BooleanFunctions/Main.lean\", \"line\": 20, \"placement\": \"local declaration\"}], \"status\": \"local body present\"}], \"missing_direct_count\": 2}, {\"config\": \"lean/ComparatorChallenges/TotientAsymptotic.json\", \"solution\": \"lean/OAI/NumberTheory/TotientAsymptotic/UnconditionalMain.lean\", \"declarations\": [{\"target\": \"OAI.TotientAsymptotic.totient_asymptotic_formula\", \"local_declaration_found\": true, \"declaration_locations\": [{\"path\": \"lean/OAI/NumberTheory/TotientAsymptotic/UnconditionalMain.lean\", \"line\": 12, \"placement\": \"local declaration\"}], \"status\": \"local body present\"}, {\"target\": \"OAI.TotientAsymptotic.weighted_totient_asymptotic\", \"local_declaration_found\": true, \"declaration_locations\": [{\"path\": \"lean/OAI/NumberTheory/TotientAsymptotic/UnconditionalMain.lean\", \"line\": 23, \"placement\": \"local declaration\"}], \"status\": \"local body present\"}, {\"target\": \"OAI.TotientAsymptotic.weighted_totient_one_two\", \"local_declaration_found\": true, \"declaration_locations\": [{\"path\": \"lean/OAI/NumberTheory/TotientAsymptotic/UnconditionalMain.lean\", \"line\": 37, \"placement\": \"local declaration\"}], \"status\": \"local body present\"}, {\"target\": \"OAI.TotientAsymptotic.coefficient_nonnegative\", \"local_declaration_found\": true, \"declaration_locations\": [{\"path\": \"lean/OAI/NumberTheory/TotientAsymptotic/UnconditionalMain.lean\", \"line\": 44, \"placement\": \"local declaration\"}], \"status\": \"local body present\"}], \"missing_direct_count\": 2}, {\"config\": \"lean/ComparatorChallenges/TotientCompanionZero.json\", \"solution\": \"lean/OAI/NumberTheory/TotientAsymptotic/Main.lean\", \"declarations\": [{\"target\": \"OAI.TotientAsymptotic.companion_zero_case\", \"local_declaration_found\": false, \"declaration_locations\": [], \"status\": \"imported body not supplied\"}], \"missing_direct_count\": 202}]\n"
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            "bar_with_points",
            "strip_with_median",
            "goodness_arrow",
            "two_tier_label",
            "end_of_line_labels",
            "panel_crops"
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          "source": "META_GREY = \"#888888\"\n\n\ndef apply_figure_style(*, frame=\"open\", font=None, sizes=(8, 7, 6), grid=False):\n    \"\"\"Set matplotlib rcParams for publication-grade output. Call once before plotting.\n\n    This sets mechanics (role-mapped font-size ladder, outward ticks, frameless\n    legends, 300-dpi save, Type-42 embedded fonts) — not a house aesthetic.\n    Frame, font and the size ladder are parameters.\n\n    frame : 'open' (bottom+left spines, default) | 'boxed' (all four) | 'none'\n    font  : sans-serif family name; None = system default sans-serif\n    sizes : (base, secondary, tick) — titles/axis-labels, legend/annotation, ticks\n    grid  : whether to draw axes.grid (default False)\n    \"\"\"\n    import matplotlib as mpl\n    if frame not in (\"open\", \"boxed\", \"none\"):\n        raise ValueError(f\"frame must be 'open'|'boxed'|'none', got {frame!r}\")\n\n    try:\n        import os, sys, glob, matplotlib.font_manager as fm\n        fdir = os.path.join(os.environ.get(\"CONDA_PREFIX\") or sys.prefix, \"fonts\")\n        if os.path.isdir(fdir):\n            known = {f.fname for f in fm.fontManager.ttflist}\n            for f in glob.glob(os.path.join(fdir, \"*.ttf\")):\n                if f not in known:\n                    fm.fontManager.addfont(f)\n    except Exception:\n        pass\n    base, secondary, tick = sizes\n    boxed = (frame == \"boxed\")\n    rc = {\n        \"font.family\": \"sans-serif\",\n        \"font.size\": base,\n        \"axes.labelsize\": base,\n        \"axes.titlesize\": base,\n        \"legend.fontsize\": secondary,\n        \"xtick.labelsize\": tick,\n        \"ytick.labelsize\": tick,\n        \"axes.linewidth\": 0.6,\n        \"xtick.direction\": \"out\", \"ytick.direction\": \"out\",\n        \"xtick.major.size\": 3, \"ytick.major.size\": 3,\n        \"xtick.major.width\": 0.6, \"ytick.major.width\": 0.6,\n        \"axes.spines.top\": boxed, \"axes.spines.right\": boxed,\n        \"axes.spines.left\": frame != \"none\", \"axes.spines.bottom\": frame != \"none\",\n        \"axes.grid\": bool(grid),\n        \"legend.frameon\": False,\n        \"figure.dpi\": 200,\n        \"savefig.dpi\": 300,\n        \"savefig.bbox\": \"tight\",\n        \"axes.titleweight\": \"normal\",\n        \"axes.titlelocation\": \"left\",\n        \"axes.labelweight\": \"normal\",\n        \"lines.linewidth\": 1.2,\n        \"patch.linewidth\": 0.6,\n        \"pdf.fonttype\": 42, \"ps.fonttype\": 42,\n    }\n    if font:\n        rc[\"font.sans-serif\"] = [font, \"DejaVu Sans\"]\n    mpl.rcParams.update(rc)\n\n\ndef set_frame(ax, style=\"open\"):\n    \"\"\"§3: set spine visibility on an existing axes. style ∈ {'open','boxed','none'}.\"\"\"\n    show = {\"open\": (False, False, True, True),\n            \"boxed\": (True, True, True, True),\n            \"none\": (False, False, False, False)}[style]\n    for side, vis in zip((\"top\", \"right\", \"bottom\", \"left\"), show):\n        ax.spines[side].set_visible(vis)\n        if vis:\n            ax.spines[side].set_linewidth(0.6)\n    ax.tick_params(direction=\"out\", length=0 if style == \"none\" else 3, width=0.6)\n\n\ndef panel_letter(ax, letter, dx=-0.18, dy=1.02, case=\"lower\", fontsize=None):\n    \"\"\"§5.7: bold panel letter outside top-left of axes. case ∈ {'lower','upper'}.\"\"\"\n    import matplotlib.pyplot as plt\n    if fontsize is None:\n        fontsize = plt.rcParams.get(\"font.size\", 8) + 1\n    s = letter.lower() if case == \"lower\" else letter.upper()\n    artist = ax.text(dx, dy, s, transform=ax.transAxes,\n                     fontweight=\"bold\", fontsize=fontsize, va=\"bottom\", ha=\"left\")\n    artist.set_gid(\"figure-style-panel-letter\")\n\n\ndef focal_palette(labels, focal, focal_color, other=\"muted\", base_colors=None):\n    \"\"\"§4.2: map labels → colors with the focal series visually dominant.\n\n    other='muted'   — desaturate base_colors (or a default cycle) toward gray\n    other='grey'    — uniform light gray for all non-focal\n    other='ordinal' — non-focal on a single light→dark gray ramp (input order)\n    \"\"\"\n    import matplotlib.colors as mcolors\n    import matplotlib.pyplot as plt\n    focal_set = {focal} if isinstance(focal, str) else set(focal)\n    n = len(labels)\n    if not focal_set & set(labels):\n        raise ValueError(f\"focal {focal!r} not found in labels\")\n    base_colors = [] if base_colors is None else list(base_colors)\n    if not base_colors:\n        base_colors = plt.rcParams[\"axes.prop_cycle\"].by_key().get(\"color\", [\"#444444\"])\n    if not base_colors:\n        base_colors = [\"#444444\"]\n    base_colors = [base_colors[i % len(base_colors)] for i in range(n)]\n    if other == \"grey\":\n        rest = [\"#BCBCBC\"] * n\n    elif other == \"ordinal\":\n        nf = max(1, n - len(focal_set))\n        ramp = [mcolors.to_hex((v, v, v)) for v in\n                ([0.55] if nf == 1 else [0.80 - 0.35 * i / (nf - 1) for i in range(nf)])]\n        rest, k = [], 0\n        for l in labels:\n            rest.append(ramp[min(k, nf - 1)]); k += (l not in focal_set)\n    elif other == \"muted\":\n        def mute(c):\n            r, g, b = mcolors.to_rgb(c)\n            m = (r + g + b) / 3\n            return mcolors.to_hex((0.3 * r + 0.7 * m, 0.3 * g + 0.7 * m, 0.3 * b + 0.7 * m))\n        rest = [mute(c) for c in base_colors]\n    else:\n        raise ValueError(\"other must be 'muted', 'grey', or 'ordinal'\")\n    return [focal_color if l in focal_set else rest[i] for i, l in enumerate(labels)]\n\n\ndef bar_with_points(ax, x, ymat, labels, colors, jitter=0.08, show_points=True,\n                    errorbar=None, point_alpha=0.5, point_size=8):\n    \"\"\"§6.1: bar = mean; optionally overlay raw points or draw an interval.\n\n    colors   : per-label color list (e.g. from focal_palette)\n    x, ymat and labels must have equal lengths; mismatches raise before drawing.\n    Each group must be a nonempty, finite, one-dimensional real numeric array.\n    Requested SD/CI intervals require at least two observations per group.\n    errorbar : None | 'sd' | 'ci95' — drawn only when show_points is False.\n               'ci95' is the t-distribution 95% CI of the mean\n               (half-width t_{0.975,n-1} · s/√n); correct at small n where the\n               z-approximation (1.96·s/√n) is markedly too narrow.\n    \"\"\"\n    import numpy as np\n    x = np.asarray(x)\n    if x.ndim != 1 or not np.issubdtype(x.dtype, np.number):\n        raise TypeError(\n            \"bar_with_points x must be one-dimensional numeric positions; \"\n            \"pass category names through labels\"\n        )\n    ymat = list(ymat)\n    labels = list(labels)\n    if len(x) != len(ymat) or len(x) != len(labels):\n        raise ValueError(\"bar_with_points x, ymat and labels must have equal lengths\")\n    if colors is not None and not isinstance(colors, str):\n        colors = list(colors)\n        if not colors:\n            colors = None\n    if x.dtype.kind not in \"iuf\" or not np.isfinite(x).all():\n        raise ValueError(\"bar_with_points x must contain finite real positions\")\n    for i, y in enumerate(ymat):\n        if np.ma.isMaskedArray(y) and np.ma.getmaskarray(y).any():\n            raise ValueError(f\"bar_with_points group {i} contains masked observations; resolve exclusions explicitly before plotting\")\n    ymat = [np.asarray(y) for y in ymat]\n    for i, y in enumerate(ymat):\n        if y.ndim != 1 or y.size == 0 or y.dtype.kind not in \"iuf\" or not np.isfinite(y).all():\n            raise ValueError(f\"bar_with_points group {i} must be nonempty, one-dimensional and finite real numeric data\")\n    if errorbar and not show_points:\n        if errorbar not in (\"sd\", \"ci95\"):\n            raise ValueError(\"errorbar must be None, 'sd', or 'ci95'\")\n        if any(y.size < 2 for y in ymat):\n            raise ValueError(\"SD/CI cannot be estimated from one observation; use raw points without an interval\")\n    means = np.array([np.mean(y) for y in ymat], float)\n    err = None\n    if errorbar and not show_points:\n        if errorbar == \"sd\":\n            err = np.array([np.std(y, ddof=1) for y in ymat])\n        elif errorbar == \"ci95\":\n            from scipy.stats import t\n            def _hw(y):\n                n = np.asarray(y).size\n                return t.ppf(0.975, n - 1) * np.std(y, ddof=1) / np.sqrt(n)\n            err = np.array([_hw(y) for y in ymat])\n    ax.bar(x, means, color=colors, width=0.7, edgecolor=\"none\",\n           yerr=err, error_kw={\"elinewidth\": 0.8, \"capsize\": 0})\n    if show_points:\n        for xi, ys in zip(x, ymat):\n            ys = np.asarray(ys)\n            if ys.ndim == 1 and ys.size > 0:\n                jit = (np.random.rand(ys.size) - 0.5) * 2 * jitter\n                ax.scatter(np.full(ys.size, xi) + jit, ys, s=point_size, color=\"black\",\n                           alpha=point_alpha, zorder=3, linewidths=0)\n    ax.set_xticks(x); ax.set_xticklabels(labels)\n    return ax\n\n\ndef strip_with_median(ax, groups, values, colors=None, jitter=0.12):\n    \"\"\"§6.1: jittered points + bold horizontal median tick per group.\n\n    groups and values must have equal lengths. Mismatches raise before drawing.\n    colors=None or an empty color sequence supplies default gray; a nonempty\n    color sequence cycles across groups.\n    Each group must be nonempty, one-dimensional and finite real numeric data;\n    resolve missing observations explicitly before plotting.\n    \"\"\"\n    import numpy as np\n    labs = list(groups)\n    values = list(values)\n    if len(labs) != len(values):\n        raise ValueError(\"strip_with_median groups and values must have equal lengths\")\n    for i, y in enumerate(values):\n        if np.ma.isMaskedArray(y) and np.ma.getmaskarray(y).any():\n            raise ValueError(f\"strip_with_median group {i} contains masked observations; resolve exclusions explicitly before plotting\")\n    values = [np.asarray(y) for y in values]\n    for i, y in enumerate(values):\n        if y.ndim != 1 or y.size == 0 or y.dtype.kind not in \"iuf\" or not np.isfinite(y).all():\n            raise ValueError(f\"strip_with_median group {i} must be nonempty, one-dimensional and finite real numeric data\")\n    colors = [] if colors is None else list(colors)\n    if not colors:\n        colors = [\"#444444\"]\n    for i, ys in enumerate(values):\n        c = colors[i % len(colors)]\n        ys = np.asarray(ys)\n        jit = (np.random.rand(ys.size) - 0.5) * 2 * jitter\n        ax.scatter(np.full(ys.size, i) + jit, ys, s=10, color=c, alpha=0.6, linewidths=0, zorder=2)\n        m = np.median(ys)\n        ax.plot([i - 0.22, i + 0.22], [m, m], color=\"black\", lw=1.6, zorder=3)\n    ax.set_xticks(range(len(labs))); ax.set_xticklabels(labs)\n    return ax\n\n\ndef goodness_arrow(ax, text=\"higher = better\", loc=\"upper left\", axis=\"y\", fontsize=None):\n    \"\"\"§3.6: small upright direction-of-goodness cue in the margin.\"\"\"\n    import matplotlib.pyplot as plt\n    if fontsize is None:\n        fontsize = plt.rcParams[\"legend.fontsize\"]\n    pos = {\"upper left\": (0.02, 0.98), \"upper right\": (0.98, 0.98),\n           \"lower left\": (0.02, 0.02), \"lower right\": (0.98, 0.02)}[loc]\n    ha = \"left\" if \"left\" in loc else \"right\"\n    va = \"top\" if \"upper\" in loc else \"bottom\"\n    arrow = \"↑ \" if axis == \"y\" else \"→ \"\n    ax.text(pos[0], pos[1], arrow + text, transform=ax.transAxes,\n            fontsize=fontsize, color=META_GREY, ha=ha, va=va)\n\n\ndef two_tier_label(name, meta):\n    \"\"\"§5: two-line label string (name / metadata). Meta line styled separately by caller.\"\"\"\n    return f\"{name}\\n{meta}\"\n\n\ndef end_of_line_labels(ax, xs, ys, labels, colors=None, dx=0.01, fontsize=None):\n    \"\"\"§6.3 / §7.3: label each line series at its right end instead of a legend box.\n\n    Each label requires a nonempty, matched x/y series. Validate all series\n    before drawing. Nonempty colors cycle; None or empty uses default text color.\n    Inner sequences must support len() and indexing; they are not copied.\n    Nonfinite endpoints are rejected rather than producing an invisible label.\n    \"\"\"\n    import matplotlib.pyplot as plt\n    import math\n    xs, ys, labels = list(xs), list(ys), list(labels)\n    if len(xs) != len(ys) or len(xs) != len(labels):\n        raise ValueError(\"end_of_line_labels xs, ys and labels must have equal lengths\")\n    if any(len(x) == 0 or len(x) != len(y) for x, y in zip(xs, ys)):\n        raise ValueError(\"end_of_line_labels each x/y series must be nonempty and have equal lengths\")\n    if any(not math.isfinite(x[-1]) or not math.isfinite(y[-1]) for x, y in zip(xs, ys)):\n        raise ValueError(\"end_of_line_labels endpoints must be finite; choose valid endpoints explicitly\")\n    if fontsize is None:\n        fontsize = plt.rcParams[\"font.size\"]\n    colors = [] if colors is None else list(colors)\n    if not colors:\n        colors = [None]\n    span = ax.get_xlim()[1] - ax.get_xlim()[0]\n    for i, (x, y, lab) in enumerate(zip(xs, ys, labels)):\n        c = colors[i % len(colors)]\n        ax.text(x[-1] + dx * span, y[-1], lab, color=c, va=\"center\", ha=\"left\", fontsize=fontsize)\n\n\ndef panel_crops(fig, dpi=None, pad_px=6, bbox_inches=None, pad_inches=None):\n    \"\"\"§9.2: pixel-space crop boxes for each lettered panel in the SAVED PNG.\n\n    Returns ``{letter: (x0, y0, x1, y1)}`` in image-space pixels (origin\n    top-left, matching PIL's ``Image.crop``). Panels are detected as bold\n    single-character ``Text``\n    objects placed by :func:`panel_letter`; each panel's crop is its axes'\n    tightbbox mapped into the saved file's pixel space, padded by ``pad_px``.\n    For §3.4 composites (abutting subplots sharing an axis, letter on the\n    leftmost only) the crop unions in letterless ``sharex``/``sharey`` siblings\n    on the same grid row/col so the whole composite is covered. When no axes\n    carries a panel letter (standalone plot, or a figure-composer sub-agent),\n    falls back to one crop per axes keyed by index.\n\n    ``bbox_inches`` mirrors ``Figure.savefig`` semantics: ``None`` means\n    *consult rcParams* (so under :func:`apply_figure_style` it resolves to\n    ``'tight'``); pass an explicit ``Bbox`` only if you saved with one. The\n    boxes are clamped to the saved image extent; panels with no intersection\n    are omitted.\n\n        >>> from PIL import Image\n        >>> fig.savefig(\"fig.png\")  # bbox_inches='tight' via rcParams\n        >>> with Image.open(\"fig.png\") as saved:\n        ...     for letter, box in panel_crops(fig).items():\n        ...         saved.crop(box).save(f\"panel_{letter}.png\")\n\n    Match savefig's DPI, bbox_inches and pad_inches when overriding defaults.\n    Returned boxes are pixels, never savefig bbox_inches (inches). Passing them\n    back to savefig as inches can allocate an enormous raster.\n    \"\"\"\n    import matplotlib as mpl\n    import matplotlib.text\n    from matplotlib.backends.backend_agg import FigureCanvasAgg\n    if isinstance(pad_px, bool) or not isinstance(pad_px, int) or pad_px < 0:\n        raise TypeError(\"panel_crops pad_px must be a non-negative integer\")\n    if dpi is None:\n        dpi = mpl.rcParams.get(\"savefig.dpi\", fig.dpi)\n        if dpi == \"figure\":\n            dpi = fig.dpi\n    dpi = float(dpi)\n    if bbox_inches is None:\n        bbox_inches = mpl.rcParams.get(\"savefig.bbox\")\n    # The Open-Science notebook captures and closes pyplot figures after every\n    # cell. Matplotlib 3.11 decouples a closed Figure from its backend by\n    # replacing FigureCanvasAgg with FigureCanvasBase, while the persistent\n    # notebook namespace can still retain the Figure itself. Reattach the\n    # headless canvas before asking for saved-image geometry so this helper is\n    # safe in a later QA cell as well as the producer cell.\n    if not hasattr(fig.canvas, \"get_renderer\"):\n        FigureCanvasAgg(fig)\n    fig.canvas.draw()\n    r = fig.canvas.get_renderer()\n\n    if bbox_inches == \"tight\":\n        if pad_inches is None:\n            pad_inches = mpl.rcParams.get(\"savefig.pad_inches\", 0.1)\n        tb = fig.get_tightbbox(r).padded(pad_inches)\n        ox_in, oy_in = tb.x0, tb.y0\n        W_in, H_in = tb.width, tb.height\n    elif isinstance(bbox_inches, mpl.transforms.BboxBase):\n        ox_in, oy_in = bbox_inches.x0, bbox_inches.y0\n        W_in, H_in = bbox_inches.width, bbox_inches.height\n    elif bbox_inches is None:\n        ox_in, oy_in = 0.0, 0.0\n        W_in, H_in = fig.get_size_inches()\n    else:\n        raise TypeError(\"panel_crops bbox_inches must be None, 'tight', or a matplotlib Bbox\")\n    W_px, H_px = int(round(W_in * dpi)), int(round(H_in * dpi))\n    lettered = {}\n    seen_letters = set()\n    for ax in fig.axes:\n        for t in ax.findobj(matplotlib.text.Text):\n            s = (t.get_text() or \"\").strip()\n            if t.get_gid() == \"figure-style-panel-letter\":\n                if s in seen_letters:\n                    raise ValueError(f\"Duplicate panel letter: {s!r}\")\n                seen_letters.add(s)\n                lettered[ax] = s\n                break\n\n\n\n    if not lettered:\n        lettered = {ax: str(i) for i, ax in enumerate(fig.axes)}\n    out = {}\n    for ax, letter in lettered.items():\n        bbs = [ax.get_tightbbox(r)]\n\n\n\n\n        ss = ax.get_subplotspec()\n        for sib in fig.axes:\n            if sib is ax or sib in lettered:\n                continue\n            ssib = sib.get_subplotspec()\n            same_row = ss is None or ssib is None or ss.rowspan == ssib.rowspan\n            same_col = ss is None or ssib is None or ss.colspan == ssib.colspan\n            if ((ax.get_shared_y_axes().joined(ax, sib) and same_row)\n                    or (ax.get_shared_x_axes().joined(ax, sib) and same_col)):\n                bbs.append(sib.get_tightbbox(r))\n        bb = mpl.transforms.Bbox.union(bbs)\n\n        bx0 = (bb.x0 / fig.dpi - ox_in) * dpi\n        bx1 = (bb.x1 / fig.dpi - ox_in) * dpi\n        by0 = H_px - (bb.y1 / fig.dpi - oy_in) * dpi\n        by1 = H_px - (bb.y0 / fig.dpi - oy_in) * dpi\n        box = (\n            min(max(int(bx0) - pad_px, 0), W_px),\n            min(max(int(by0) - pad_px, 0), H_px),\n            max(min(int(bx1) + pad_px, W_px), 0),\n            max(min(int(by1) + pad_px, H_px), 0),\n        )\n        if box[0] < box[2] and box[1] < box[3]:\n            out[letter] = box\n    return out\n",
          "sourceDigest": "8020d409bfb60f0c8dfa43d0dfe7bb97f7458e1a3908fefbcb6e4551e0feae4e"
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      "script": "print(json.dumps([{'config':m['configuration'],'solution':m['solution_path'],'declarations':m['declarations'],'missing_direct_count':len(m['missing_direct_OAI_imports'])} for m in module_evidence[20:25]],ensure_ascii=False))",
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        "stdout": "[{\"config\": \"lean/ComparatorChallenges/TriangleFace.json\", \"solution\": \"lean/OAI/Combinatorics/TriangleFace/Main.lean\", \"declarations\": [{\"target\": \"OAI.TriangleFace198.triangle_expansion_face\", \"local_declaration_found\": true, \"declaration_locations\": [{\"path\": \"lean/OAI/Combinatorics/TriangleFace/Main.lean\", \"line\": 51, \"placement\": \"local declaration\"}], \"status\": \"local body present\"}], \"missing_direct_count\": 2}, {\"config\": \"lean/ComparatorChallenges/TriangularHilbert.json\", \"solution\": \"lean/OAI/Analysis/TriangularHilbert/Main.lean\", \"declarations\": [{\"target\": \"OAI.TriangularHilbert.main_estimate\", \"local_declaration_found\": true, \"declaration_locations\": [{\"path\": \"lean/OAI/Analysis/TriangularHilbert/Main.lean\", \"line\": 38, \"placement\": \"local declaration\"}], \"status\": \"local body present\"}], \"missing_direct_count\": 1}, {\"config\": \"lean/ComparatorChallenges/TypeSystemNormalization.json\", \"solution\": \"lean/OAI/Computability/TypeSystem/Normalization.lean\", \"declarations\": [{\"target\": \"OAI.PureTypeSystem.weak_implies_strong\", \"local_declaration_found\": true, \"declaration_locations\": [{\"path\": \"lean/OAI/Computability/TypeSystem/Normalization.lean\", \"line\": 82, \"placement\": \"local declaration\"}], \"status\": \"local body present\"}], \"missing_direct_count\": 6}]\n",
        "stderr": "",
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          "[{\"config\": \"lean/ComparatorChallenges/TriangleFace.json\", \"solution\": \"lean/OAI/Combinatorics/TriangleFace/Main.lean\", \"declarations\": [{\"target\": \"OAI.TriangleFace198.triangle_expansion_face\", \"local_declaration_found\": true, \"declaration_locations\": [{\"path\": \"lean/OAI/Combinatorics/TriangleFace/Main.lean\", \"line\": 51, \"placement\": \"local declaration\"}], \"status\": \"local body present\"}], \"missing_direct_count\": 2}, {\"config\": \"lean/ComparatorChallenges/TriangularHilbert.json\", \"solution\": \"lean/OAI/Analysis/TriangularHilbert/Main.lean\", \"declarations\": [{\"target\": \"OAI.TriangularHilbert.main_estimate\", \"local_declaration_found\": true, \"declaration_locations\": [{\"path\": \"lean/OAI/Analysis/TriangularHilbert/Main.lean\", \"line\": 38, \"placement\": \"local declaration\"}], \"status\": \"local body present\"}], \"missing_direct_count\": 1}, {\"config\": \"lean/ComparatorChallenges/TypeSystemNormalization.json\", \"solution\": \"lean/OAI/Computability/TypeSystem/Normalization.lean\", \"declarations\": [{\"target\": \"OAI.PureTypeSystem.weak_implies_strong\", \"local_declaration_found\": true, \"declaration_locations\": [{\"path\": \"lean/OAI/Computability/TypeSystem/Normalization.lean\", \"line\": 82, \"placement\": \"local declaration\"}], \"status\": \"local body present\"}], \"missing_direct_count\": 6}]\n"
        ]
      },
      "outputs": [
        {
          "type": "stream",
          "name": "stdout",
          "text": "[{\"config\": \"lean/ComparatorChallenges/TriangleFace.json\", \"solution\": \"lean/OAI/Combinatorics/TriangleFace/Main.lean\", \"declarations\": [{\"target\": \"OAI.TriangleFace198.triangle_expansion_face\", \"local_declaration_found\": true, \"declaration_locations\": [{\"path\": \"lean/OAI/Combinatorics/TriangleFace/Main.lean\", \"line\": 51, \"placement\": \"local declaration\"}], \"status\": \"local body present\"}], \"missing_direct_count\": 2}, {\"config\": \"lean/ComparatorChallenges/TriangularHilbert.json\", \"solution\": \"lean/OAI/Analysis/TriangularHilbert/Main.lean\", \"declarations\": [{\"target\": \"OAI.TriangularHilbert.main_estimate\", \"local_declaration_found\": true, \"declaration_locations\": [{\"path\": \"lean/OAI/Analysis/TriangularHilbert/Main.lean\", \"line\": 38, \"placement\": \"local declaration\"}], \"status\": \"local body present\"}], \"missing_direct_count\": 1}, {\"config\": \"lean/ComparatorChallenges/TypeSystemNormalization.json\", \"solution\": \"lean/OAI/Computability/TypeSystem/Normalization.lean\", \"declarations\": [{\"target\": \"OAI.PureTypeSystem.weak_implies_strong\", \"local_declaration_found\": true, \"declaration_locations\": [{\"path\": \"lean/OAI/Computability/TypeSystem/Normalization.lean\", \"line\": 82, \"placement\": \"local declaration\"}], \"status\": \"local body present\"}], \"missing_direct_count\": 6}]\n"
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          "registeredGeneration": "sha256:90258343aec7bb96c2e129c39e1beb23230ee947c0db563f96913458d1795f1d",
          "exports": [
            "apply_figure_style",
            "set_frame",
            "panel_letter",
            "focal_palette",
            "bar_with_points",
            "strip_with_median",
            "goodness_arrow",
            "two_tier_label",
            "end_of_line_labels",
            "panel_crops"
          ],
          "source": "META_GREY = \"#888888\"\n\n\ndef apply_figure_style(*, frame=\"open\", font=None, sizes=(8, 7, 6), grid=False):\n    \"\"\"Set matplotlib rcParams for publication-grade output. Call once before plotting.\n\n    This sets mechanics (role-mapped font-size ladder, outward ticks, frameless\n    legends, 300-dpi save, Type-42 embedded fonts) — not a house aesthetic.\n    Frame, font and the size ladder are parameters.\n\n    frame : 'open' (bottom+left spines, default) | 'boxed' (all four) | 'none'\n    font  : sans-serif family name; None = system default sans-serif\n    sizes : (base, secondary, tick) — titles/axis-labels, legend/annotation, ticks\n    grid  : whether to draw axes.grid (default False)\n    \"\"\"\n    import matplotlib as mpl\n    if frame not in (\"open\", \"boxed\", \"none\"):\n        raise ValueError(f\"frame must be 'open'|'boxed'|'none', got {frame!r}\")\n\n    try:\n        import os, sys, glob, matplotlib.font_manager as fm\n        fdir = os.path.join(os.environ.get(\"CONDA_PREFIX\") or sys.prefix, \"fonts\")\n        if os.path.isdir(fdir):\n            known = {f.fname for f in fm.fontManager.ttflist}\n            for f in glob.glob(os.path.join(fdir, \"*.ttf\")):\n                if f not in known:\n                    fm.fontManager.addfont(f)\n    except Exception:\n        pass\n    base, secondary, tick = sizes\n    boxed = (frame == \"boxed\")\n    rc = {\n        \"font.family\": \"sans-serif\",\n        \"font.size\": base,\n        \"axes.labelsize\": base,\n        \"axes.titlesize\": base,\n        \"legend.fontsize\": secondary,\n        \"xtick.labelsize\": tick,\n        \"ytick.labelsize\": tick,\n        \"axes.linewidth\": 0.6,\n        \"xtick.direction\": \"out\", \"ytick.direction\": \"out\",\n        \"xtick.major.size\": 3, \"ytick.major.size\": 3,\n        \"xtick.major.width\": 0.6, \"ytick.major.width\": 0.6,\n        \"axes.spines.top\": boxed, \"axes.spines.right\": boxed,\n        \"axes.spines.left\": frame != \"none\", \"axes.spines.bottom\": frame != \"none\",\n        \"axes.grid\": bool(grid),\n        \"legend.frameon\": False,\n        \"figure.dpi\": 200,\n        \"savefig.dpi\": 300,\n        \"savefig.bbox\": \"tight\",\n        \"axes.titleweight\": \"normal\",\n        \"axes.titlelocation\": \"left\",\n        \"axes.labelweight\": \"normal\",\n        \"lines.linewidth\": 1.2,\n        \"patch.linewidth\": 0.6,\n        \"pdf.fonttype\": 42, \"ps.fonttype\": 42,\n    }\n    if font:\n        rc[\"font.sans-serif\"] = [font, \"DejaVu Sans\"]\n    mpl.rcParams.update(rc)\n\n\ndef set_frame(ax, style=\"open\"):\n    \"\"\"§3: set spine visibility on an existing axes. style ∈ {'open','boxed','none'}.\"\"\"\n    show = {\"open\": (False, False, True, True),\n            \"boxed\": (True, True, True, True),\n            \"none\": (False, False, False, False)}[style]\n    for side, vis in zip((\"top\", \"right\", \"bottom\", \"left\"), show):\n        ax.spines[side].set_visible(vis)\n        if vis:\n            ax.spines[side].set_linewidth(0.6)\n    ax.tick_params(direction=\"out\", length=0 if style == \"none\" else 3, width=0.6)\n\n\ndef panel_letter(ax, letter, dx=-0.18, dy=1.02, case=\"lower\", fontsize=None):\n    \"\"\"§5.7: bold panel letter outside top-left of axes. case ∈ {'lower','upper'}.\"\"\"\n    import matplotlib.pyplot as plt\n    if fontsize is None:\n        fontsize = plt.rcParams.get(\"font.size\", 8) + 1\n    s = letter.lower() if case == \"lower\" else letter.upper()\n    artist = ax.text(dx, dy, s, transform=ax.transAxes,\n                     fontweight=\"bold\", fontsize=fontsize, va=\"bottom\", ha=\"left\")\n    artist.set_gid(\"figure-style-panel-letter\")\n\n\ndef focal_palette(labels, focal, focal_color, other=\"muted\", base_colors=None):\n    \"\"\"§4.2: map labels → colors with the focal series visually dominant.\n\n    other='muted'   — desaturate base_colors (or a default cycle) toward gray\n    other='grey'    — uniform light gray for all non-focal\n    other='ordinal' — non-focal on a single light→dark gray ramp (input order)\n    \"\"\"\n    import matplotlib.colors as mcolors\n    import matplotlib.pyplot as plt\n    focal_set = {focal} if isinstance(focal, str) else set(focal)\n    n = len(labels)\n    if not focal_set & set(labels):\n        raise ValueError(f\"focal {focal!r} not found in labels\")\n    base_colors = [] if base_colors is None else list(base_colors)\n    if not base_colors:\n        base_colors = plt.rcParams[\"axes.prop_cycle\"].by_key().get(\"color\", [\"#444444\"])\n    if not base_colors:\n        base_colors = [\"#444444\"]\n    base_colors = [base_colors[i % len(base_colors)] for i in range(n)]\n    if other == \"grey\":\n        rest = [\"#BCBCBC\"] * n\n    elif other == \"ordinal\":\n        nf = max(1, n - len(focal_set))\n        ramp = [mcolors.to_hex((v, v, v)) for v in\n                ([0.55] if nf == 1 else [0.80 - 0.35 * i / (nf - 1) for i in range(nf)])]\n        rest, k = [], 0\n        for l in labels:\n            rest.append(ramp[min(k, nf - 1)]); k += (l not in focal_set)\n    elif other == \"muted\":\n        def mute(c):\n            r, g, b = mcolors.to_rgb(c)\n            m = (r + g + b) / 3\n            return mcolors.to_hex((0.3 * r + 0.7 * m, 0.3 * g + 0.7 * m, 0.3 * b + 0.7 * m))\n        rest = [mute(c) for c in base_colors]\n    else:\n        raise ValueError(\"other must be 'muted', 'grey', or 'ordinal'\")\n    return [focal_color if l in focal_set else rest[i] for i, l in enumerate(labels)]\n\n\ndef bar_with_points(ax, x, ymat, labels, colors, jitter=0.08, show_points=True,\n                    errorbar=None, point_alpha=0.5, point_size=8):\n    \"\"\"§6.1: bar = mean; optionally overlay raw points or draw an interval.\n\n    colors   : per-label color list (e.g. from focal_palette)\n    x, ymat and labels must have equal lengths; mismatches raise before drawing.\n    Each group must be a nonempty, finite, one-dimensional real numeric array.\n    Requested SD/CI intervals require at least two observations per group.\n    errorbar : None | 'sd' | 'ci95' — drawn only when show_points is False.\n               'ci95' is the t-distribution 95% CI of the mean\n               (half-width t_{0.975,n-1} · s/√n); correct at small n where the\n               z-approximation (1.96·s/√n) is markedly too narrow.\n    \"\"\"\n    import numpy as np\n    x = np.asarray(x)\n    if x.ndim != 1 or not np.issubdtype(x.dtype, np.number):\n        raise TypeError(\n            \"bar_with_points x must be one-dimensional numeric positions; \"\n            \"pass category names through labels\"\n        )\n    ymat = list(ymat)\n    labels = list(labels)\n    if len(x) != len(ymat) or len(x) != len(labels):\n        raise ValueError(\"bar_with_points x, ymat and labels must have equal lengths\")\n    if colors is not None and not isinstance(colors, str):\n        colors = list(colors)\n        if not colors:\n            colors = None\n    if x.dtype.kind not in \"iuf\" or not np.isfinite(x).all():\n        raise ValueError(\"bar_with_points x must contain finite real positions\")\n    for i, y in enumerate(ymat):\n        if np.ma.isMaskedArray(y) and np.ma.getmaskarray(y).any():\n            raise ValueError(f\"bar_with_points group {i} contains masked observations; resolve exclusions explicitly before plotting\")\n    ymat = [np.asarray(y) for y in ymat]\n    for i, y in enumerate(ymat):\n        if y.ndim != 1 or y.size == 0 or y.dtype.kind not in \"iuf\" or not np.isfinite(y).all():\n            raise ValueError(f\"bar_with_points group {i} must be nonempty, one-dimensional and finite real numeric data\")\n    if errorbar and not show_points:\n        if errorbar not in (\"sd\", \"ci95\"):\n            raise ValueError(\"errorbar must be None, 'sd', or 'ci95'\")\n        if any(y.size < 2 for y in ymat):\n            raise ValueError(\"SD/CI cannot be estimated from one observation; use raw points without an interval\")\n    means = np.array([np.mean(y) for y in ymat], float)\n    err = None\n    if errorbar and not show_points:\n        if errorbar == \"sd\":\n            err = np.array([np.std(y, ddof=1) for y in ymat])\n        elif errorbar == \"ci95\":\n            from scipy.stats import t\n            def _hw(y):\n                n = np.asarray(y).size\n                return t.ppf(0.975, n - 1) * np.std(y, ddof=1) / np.sqrt(n)\n            err = np.array([_hw(y) for y in ymat])\n    ax.bar(x, means, color=colors, width=0.7, edgecolor=\"none\",\n           yerr=err, error_kw={\"elinewidth\": 0.8, \"capsize\": 0})\n    if show_points:\n        for xi, ys in zip(x, ymat):\n            ys = np.asarray(ys)\n            if ys.ndim == 1 and ys.size > 0:\n                jit = (np.random.rand(ys.size) - 0.5) * 2 * jitter\n                ax.scatter(np.full(ys.size, xi) + jit, ys, s=point_size, color=\"black\",\n                           alpha=point_alpha, zorder=3, linewidths=0)\n    ax.set_xticks(x); ax.set_xticklabels(labels)\n    return ax\n\n\ndef strip_with_median(ax, groups, values, colors=None, jitter=0.12):\n    \"\"\"§6.1: jittered points + bold horizontal median tick per group.\n\n    groups and values must have equal lengths. Mismatches raise before drawing.\n    colors=None or an empty color sequence supplies default gray; a nonempty\n    color sequence cycles across groups.\n    Each group must be nonempty, one-dimensional and finite real numeric data;\n    resolve missing observations explicitly before plotting.\n    \"\"\"\n    import numpy as np\n    labs = list(groups)\n    values = list(values)\n    if len(labs) != len(values):\n        raise ValueError(\"strip_with_median groups and values must have equal lengths\")\n    for i, y in enumerate(values):\n        if np.ma.isMaskedArray(y) and np.ma.getmaskarray(y).any():\n            raise ValueError(f\"strip_with_median group {i} contains masked observations; resolve exclusions explicitly before plotting\")\n    values = [np.asarray(y) for y in values]\n    for i, y in enumerate(values):\n        if y.ndim != 1 or y.size == 0 or y.dtype.kind not in \"iuf\" or not np.isfinite(y).all():\n            raise ValueError(f\"strip_with_median group {i} must be nonempty, one-dimensional and finite real numeric data\")\n    colors = [] if colors is None else list(colors)\n    if not colors:\n        colors = [\"#444444\"]\n    for i, ys in enumerate(values):\n        c = colors[i % len(colors)]\n        ys = np.asarray(ys)\n        jit = (np.random.rand(ys.size) - 0.5) * 2 * jitter\n        ax.scatter(np.full(ys.size, i) + jit, ys, s=10, color=c, alpha=0.6, linewidths=0, zorder=2)\n        m = np.median(ys)\n        ax.plot([i - 0.22, i + 0.22], [m, m], color=\"black\", lw=1.6, zorder=3)\n    ax.set_xticks(range(len(labs))); ax.set_xticklabels(labs)\n    return ax\n\n\ndef goodness_arrow(ax, text=\"higher = better\", loc=\"upper left\", axis=\"y\", fontsize=None):\n    \"\"\"§3.6: small upright direction-of-goodness cue in the margin.\"\"\"\n    import matplotlib.pyplot as plt\n    if fontsize is None:\n        fontsize = plt.rcParams[\"legend.fontsize\"]\n    pos = {\"upper left\": (0.02, 0.98), \"upper right\": (0.98, 0.98),\n           \"lower left\": (0.02, 0.02), \"lower right\": (0.98, 0.02)}[loc]\n    ha = \"left\" if \"left\" in loc else \"right\"\n    va = \"top\" if \"upper\" in loc else \"bottom\"\n    arrow = \"↑ \" if axis == \"y\" else \"→ \"\n    ax.text(pos[0], pos[1], arrow + text, transform=ax.transAxes,\n            fontsize=fontsize, color=META_GREY, ha=ha, va=va)\n\n\ndef two_tier_label(name, meta):\n    \"\"\"§5: two-line label string (name / metadata). Meta line styled separately by caller.\"\"\"\n    return f\"{name}\\n{meta}\"\n\n\ndef end_of_line_labels(ax, xs, ys, labels, colors=None, dx=0.01, fontsize=None):\n    \"\"\"§6.3 / §7.3: label each line series at its right end instead of a legend box.\n\n    Each label requires a nonempty, matched x/y series. Validate all series\n    before drawing. Nonempty colors cycle; None or empty uses default text color.\n    Inner sequences must support len() and indexing; they are not copied.\n    Nonfinite endpoints are rejected rather than producing an invisible label.\n    \"\"\"\n    import matplotlib.pyplot as plt\n    import math\n    xs, ys, labels = list(xs), list(ys), list(labels)\n    if len(xs) != len(ys) or len(xs) != len(labels):\n        raise ValueError(\"end_of_line_labels xs, ys and labels must have equal lengths\")\n    if any(len(x) == 0 or len(x) != len(y) for x, y in zip(xs, ys)):\n        raise ValueError(\"end_of_line_labels each x/y series must be nonempty and have equal lengths\")\n    if any(not math.isfinite(x[-1]) or not math.isfinite(y[-1]) for x, y in zip(xs, ys)):\n        raise ValueError(\"end_of_line_labels endpoints must be finite; choose valid endpoints explicitly\")\n    if fontsize is None:\n        fontsize = plt.rcParams[\"font.size\"]\n    colors = [] if colors is None else list(colors)\n    if not colors:\n        colors = [None]\n    span = ax.get_xlim()[1] - ax.get_xlim()[0]\n    for i, (x, y, lab) in enumerate(zip(xs, ys, labels)):\n        c = colors[i % len(colors)]\n        ax.text(x[-1] + dx * span, y[-1], lab, color=c, va=\"center\", ha=\"left\", fontsize=fontsize)\n\n\ndef panel_crops(fig, dpi=None, pad_px=6, bbox_inches=None, pad_inches=None):\n    \"\"\"§9.2: pixel-space crop boxes for each lettered panel in the SAVED PNG.\n\n    Returns ``{letter: (x0, y0, x1, y1)}`` in image-space pixels (origin\n    top-left, matching PIL's ``Image.crop``). Panels are detected as bold\n    single-character ``Text``\n    objects placed by :func:`panel_letter`; each panel's crop is its axes'\n    tightbbox mapped into the saved file's pixel space, padded by ``pad_px``.\n    For §3.4 composites (abutting subplots sharing an axis, letter on the\n    leftmost only) the crop unions in letterless ``sharex``/``sharey`` siblings\n    on the same grid row/col so the whole composite is covered. When no axes\n    carries a panel letter (standalone plot, or a figure-composer sub-agent),\n    falls back to one crop per axes keyed by index.\n\n    ``bbox_inches`` mirrors ``Figure.savefig`` semantics: ``None`` means\n    *consult rcParams* (so under :func:`apply_figure_style` it resolves to\n    ``'tight'``); pass an explicit ``Bbox`` only if you saved with one. The\n    boxes are clamped to the saved image extent; panels with no intersection\n    are omitted.\n\n        >>> from PIL import Image\n        >>> fig.savefig(\"fig.png\")  # bbox_inches='tight' via rcParams\n        >>> with Image.open(\"fig.png\") as saved:\n        ...     for letter, box in panel_crops(fig).items():\n        ...         saved.crop(box).save(f\"panel_{letter}.png\")\n\n    Match savefig's DPI, bbox_inches and pad_inches when overriding defaults.\n    Returned boxes are pixels, never savefig bbox_inches (inches). Passing them\n    back to savefig as inches can allocate an enormous raster.\n    \"\"\"\n    import matplotlib as mpl\n    import matplotlib.text\n    from matplotlib.backends.backend_agg import FigureCanvasAgg\n    if isinstance(pad_px, bool) or not isinstance(pad_px, int) or pad_px < 0:\n        raise TypeError(\"panel_crops pad_px must be a non-negative integer\")\n    if dpi is None:\n        dpi = mpl.rcParams.get(\"savefig.dpi\", fig.dpi)\n        if dpi == \"figure\":\n            dpi = fig.dpi\n    dpi = float(dpi)\n    if bbox_inches is None:\n        bbox_inches = mpl.rcParams.get(\"savefig.bbox\")\n    # The Open-Science notebook captures and closes pyplot figures after every\n    # cell. Matplotlib 3.11 decouples a closed Figure from its backend by\n    # replacing FigureCanvasAgg with FigureCanvasBase, while the persistent\n    # notebook namespace can still retain the Figure itself. Reattach the\n    # headless canvas before asking for saved-image geometry so this helper is\n    # safe in a later QA cell as well as the producer cell.\n    if not hasattr(fig.canvas, \"get_renderer\"):\n        FigureCanvasAgg(fig)\n    fig.canvas.draw()\n    r = fig.canvas.get_renderer()\n\n    if bbox_inches == \"tight\":\n        if pad_inches is None:\n            pad_inches = mpl.rcParams.get(\"savefig.pad_inches\", 0.1)\n        tb = fig.get_tightbbox(r).padded(pad_inches)\n        ox_in, oy_in = tb.x0, tb.y0\n        W_in, H_in = tb.width, tb.height\n    elif isinstance(bbox_inches, mpl.transforms.BboxBase):\n        ox_in, oy_in = bbox_inches.x0, bbox_inches.y0\n        W_in, H_in = bbox_inches.width, bbox_inches.height\n    elif bbox_inches is None:\n        ox_in, oy_in = 0.0, 0.0\n        W_in, H_in = fig.get_size_inches()\n    else:\n        raise TypeError(\"panel_crops bbox_inches must be None, 'tight', or a matplotlib Bbox\")\n    W_px, H_px = int(round(W_in * dpi)), int(round(H_in * dpi))\n    lettered = {}\n    seen_letters = set()\n    for ax in fig.axes:\n        for t in ax.findobj(matplotlib.text.Text):\n            s = (t.get_text() or \"\").strip()\n            if t.get_gid() == \"figure-style-panel-letter\":\n                if s in seen_letters:\n                    raise ValueError(f\"Duplicate panel letter: {s!r}\")\n                seen_letters.add(s)\n                lettered[ax] = s\n                break\n\n\n\n    if not lettered:\n        lettered = {ax: str(i) for i, ax in enumerate(fig.axes)}\n    out = {}\n    for ax, letter in lettered.items():\n        bbs = [ax.get_tightbbox(r)]\n\n\n\n\n        ss = ax.get_subplotspec()\n        for sib in fig.axes:\n            if sib is ax or sib in lettered:\n                continue\n            ssib = sib.get_subplotspec()\n            same_row = ss is None or ssib is None or ss.rowspan == ssib.rowspan\n            same_col = ss is None or ssib is None or ss.colspan == ssib.colspan\n            if ((ax.get_shared_y_axes().joined(ax, sib) and same_row)\n                    or (ax.get_shared_x_axes().joined(ax, sib) and same_col)):\n                bbs.append(sib.get_tightbbox(r))\n        bb = mpl.transforms.Bbox.union(bbs)\n\n        bx0 = (bb.x0 / fig.dpi - ox_in) * dpi\n        bx1 = (bb.x1 / fig.dpi - ox_in) * dpi\n        by0 = H_px - (bb.y1 / fig.dpi - oy_in) * dpi\n        by1 = H_px - (bb.y0 / fig.dpi - oy_in) * dpi\n        box = (\n            min(max(int(bx0) - pad_px, 0), W_px),\n            min(max(int(by0) - pad_px, 0), H_px),\n            max(min(int(bx1) + pad_px, W_px), 0),\n            max(min(int(by1) + pad_px, H_px), 0),\n        )\n        if box[0] < box[2] and box[1] < box[3]:\n            out[letter] = box\n    return out\n",
          "sourceDigest": "8020d409bfb60f0c8dfa43d0dfe7bb97f7458e1a3908fefbcb6e4551e0feae4e"
        }
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        "captureKind": "completed-run",
        "capturedAt": "2026-10-08T10:45:33.848Z",
        "installedInventory": {
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          "source": "cache-reused",
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      "script": "import copy,csv\nfinal_papers=copy.deepcopy(updated)\nby_config={m['configuration']:m for m in module_evidence}\nfor paper in final_papers:\n for target in paper['targets']:\n  target['solution_file_supplied']=True\n  target['implementation_inspection']=by_config[target['configuration_path']]\nfinal_audit={'repository':pack['repository'],'commit':commit,'sampling':{'seed':20261008,'rule':'Unchanged sample of 20 families, first-listed manuscript per family','draw_order':[r['family_id'] for r in sorted(final_papers,key=lambda r:r['draw_rank'])]},'input_packages':[{'sha256':hashlib.sha256(srcpath.read_bytes()).hexdigest(),'records':len(src),'retrieved_at':pack['retrieved_at']},{'sha256':hashlib.sha256(suppath.read_bytes()).hexdigest(),'records':len(extra),'retrieved_at':supp['retrieved_at']}],'counts':{'written_proof_packages':20,'main_result_coverage':11,'supporting_or_partial_only':4,'no_linked_lean_found':5,'comparator_configurations':23,'named_solution_modules_supplied':23,'OAI_implementation_files_supplied':24,'selected_declarations':30,'selected_declarations_local':25,'selected_declarations_in_another_supplied_import':1,'selected_declarations_imported_body_not_supplied':4,'distinct_missing_direct_OAI_imports':len(missing_direct),'proof_checker_runs':0},'inspection_method':'Static comparison of paper main statements, scope notes and Comparator targets; inspection of configured entry files and selected theorem bodies. Lexical placeholder/axiom scan excludes nested Lean comments and strings; it is not Lean parsing, name resolution, type checking, dependency verification or an axiom audit.','integrity':{'new_records_size_sha256_pass':all(len(r['content'].encode())==r['bytes'] and hashlib.sha256(r['content'].encode()).hexdigest()==r['sha256'] for r in extra.values()),'previous_records_size_sha256_pass':not failures,'previous_git_blob_matches':287,'commit_matches':supp['commit']==commit},'limitations':['No compiler or Comparator execution; every paper untested.','Challenge sorry placeholders are statement specifications, not implementation-hole findings.','No sorry/admit/sorryAx tokens or axiom declarations found in supplied implementation text after comments/strings removed; imported code is not covered.','Four selected auxiliary declaration bodies are not supplied; this is an evidence gap, not a demonstrated mathematical or implementation flaw.','270 distinct direct OAI imports are absent from combined inputs; external dependencies also not audited.','Main-result coverage is static statement correspondence, not certification of correctness or all encoding equivalences.'],'source_records':[{'path':k,'sha256':rr['sha256'],'bytes':rr['bytes']} for k,rr in combined.items()],'papers':final_papers,'module_evidence':module_evidence,'implementation_scan':scan,'missing_direct_OAI_imports':missing_direct}\nPath('final-material-audit.json').write_text(json.dumps(final_audit,ensure_ascii=False,indent=2))\nfields=['family_id','paper_title','coverage','coverage_evidence','paper_statement_url','scope_url','configurations','solution_module_urls','selected_declaration_locations','imported_selected_bodies_not_supplied','implementation_placeholders_found','dependency_completeness','checker_execution']\nwith Path('final-material-evidence.csv').open('w',newline='') as fh:\n w=csv.DictWriter(fh,fieldnames=fields);w.writeheader()\n for r in final_papers:\n  ev=[t['implementation_inspection'] for t in r['targets']];decs=[d for m in ev for d in m['declarations']]\n  w.writerow({'family_id':r['family_id'],'paper_title':r['paper_title'],'coverage':r['coverage'],'coverage_evidence':r['coverage_note'],'paper_statement_url':link(r['main_statement']['path'])+'#L'+str(r['main_statement']['line']),'scope_url':link(r['scope_path']) if r['scope_path'] else '', 'configurations':' | '.join(link(m['configuration']) for m in ev),'solution_module_urls':' | '.join(link(m['solution_path']) for m in ev),'selected_declaration_locations':' | '.join(d['target']+': '+(', '.join(link(loc['path'])+'#L'+str(loc['line']) for loc in d['declaration_locations']) or d['status']) for d in decs),'imported_selected_bodies_not_supplied':' | '.join(d['target'] for d in decs if d['status']=='imported body not supplied'),'implementation_placeholders_found':'None in supplied text; dependencies not checked' if ev else 'Not applicable','dependency_completeness':'Incomplete' if ev else 'Not applicable','checker_execution':'Not run'})\nprint('saved final-material-audit.json and final-material-evidence.csv')",
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        "plain": [
          "saved final-material-audit.json and final-material-evidence.csv\n"
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          "exports": [
            "apply_figure_style",
            "set_frame",
            "panel_letter",
            "focal_palette",
            "bar_with_points",
            "strip_with_median",
            "goodness_arrow",
            "two_tier_label",
            "end_of_line_labels",
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          "source": "META_GREY = \"#888888\"\n\n\ndef apply_figure_style(*, frame=\"open\", font=None, sizes=(8, 7, 6), grid=False):\n    \"\"\"Set matplotlib rcParams for publication-grade output. Call once before plotting.\n\n    This sets mechanics (role-mapped font-size ladder, outward ticks, frameless\n    legends, 300-dpi save, Type-42 embedded fonts) — not a house aesthetic.\n    Frame, font and the size ladder are parameters.\n\n    frame : 'open' (bottom+left spines, default) | 'boxed' (all four) | 'none'\n    font  : sans-serif family name; None = system default sans-serif\n    sizes : (base, secondary, tick) — titles/axis-labels, legend/annotation, ticks\n    grid  : whether to draw axes.grid (default False)\n    \"\"\"\n    import matplotlib as mpl\n    if frame not in (\"open\", \"boxed\", \"none\"):\n        raise ValueError(f\"frame must be 'open'|'boxed'|'none', got {frame!r}\")\n\n    try:\n        import os, sys, glob, matplotlib.font_manager as fm\n        fdir = os.path.join(os.environ.get(\"CONDA_PREFIX\") or sys.prefix, \"fonts\")\n        if os.path.isdir(fdir):\n            known = {f.fname for f in fm.fontManager.ttflist}\n            for f in glob.glob(os.path.join(fdir, \"*.ttf\")):\n                if f not in known:\n                    fm.fontManager.addfont(f)\n    except Exception:\n        pass\n    base, secondary, tick = sizes\n    boxed = (frame == \"boxed\")\n    rc = {\n        \"font.family\": \"sans-serif\",\n        \"font.size\": base,\n        \"axes.labelsize\": base,\n        \"axes.titlesize\": base,\n        \"legend.fontsize\": secondary,\n        \"xtick.labelsize\": tick,\n        \"ytick.labelsize\": tick,\n        \"axes.linewidth\": 0.6,\n        \"xtick.direction\": \"out\", \"ytick.direction\": \"out\",\n        \"xtick.major.size\": 3, \"ytick.major.size\": 3,\n        \"xtick.major.width\": 0.6, \"ytick.major.width\": 0.6,\n        \"axes.spines.top\": boxed, \"axes.spines.right\": boxed,\n        \"axes.spines.left\": frame != \"none\", \"axes.spines.bottom\": frame != \"none\",\n        \"axes.grid\": bool(grid),\n        \"legend.frameon\": False,\n        \"figure.dpi\": 200,\n        \"savefig.dpi\": 300,\n        \"savefig.bbox\": \"tight\",\n        \"axes.titleweight\": \"normal\",\n        \"axes.titlelocation\": \"left\",\n        \"axes.labelweight\": \"normal\",\n        \"lines.linewidth\": 1.2,\n        \"patch.linewidth\": 0.6,\n        \"pdf.fonttype\": 42, \"ps.fonttype\": 42,\n    }\n    if font:\n        rc[\"font.sans-serif\"] = [font, \"DejaVu Sans\"]\n    mpl.rcParams.update(rc)\n\n\ndef set_frame(ax, style=\"open\"):\n    \"\"\"§3: set spine visibility on an existing axes. style ∈ {'open','boxed','none'}.\"\"\"\n    show = {\"open\": (False, False, True, True),\n            \"boxed\": (True, True, True, True),\n            \"none\": (False, False, False, False)}[style]\n    for side, vis in zip((\"top\", \"right\", \"bottom\", \"left\"), show):\n        ax.spines[side].set_visible(vis)\n        if vis:\n            ax.spines[side].set_linewidth(0.6)\n    ax.tick_params(direction=\"out\", length=0 if style == \"none\" else 3, width=0.6)\n\n\ndef panel_letter(ax, letter, dx=-0.18, dy=1.02, case=\"lower\", fontsize=None):\n    \"\"\"§5.7: bold panel letter outside top-left of axes. case ∈ {'lower','upper'}.\"\"\"\n    import matplotlib.pyplot as plt\n    if fontsize is None:\n        fontsize = plt.rcParams.get(\"font.size\", 8) + 1\n    s = letter.lower() if case == \"lower\" else letter.upper()\n    artist = ax.text(dx, dy, s, transform=ax.transAxes,\n                     fontweight=\"bold\", fontsize=fontsize, va=\"bottom\", ha=\"left\")\n    artist.set_gid(\"figure-style-panel-letter\")\n\n\ndef focal_palette(labels, focal, focal_color, other=\"muted\", base_colors=None):\n    \"\"\"§4.2: map labels → colors with the focal series visually dominant.\n\n    other='muted'   — desaturate base_colors (or a default cycle) toward gray\n    other='grey'    — uniform light gray for all non-focal\n    other='ordinal' — non-focal on a single light→dark gray ramp (input order)\n    \"\"\"\n    import matplotlib.colors as mcolors\n    import matplotlib.pyplot as plt\n    focal_set = {focal} if isinstance(focal, str) else set(focal)\n    n = len(labels)\n    if not focal_set & set(labels):\n        raise ValueError(f\"focal {focal!r} not found in labels\")\n    base_colors = [] if base_colors is None else list(base_colors)\n    if not base_colors:\n        base_colors = plt.rcParams[\"axes.prop_cycle\"].by_key().get(\"color\", [\"#444444\"])\n    if not base_colors:\n        base_colors = [\"#444444\"]\n    base_colors = [base_colors[i % len(base_colors)] for i in range(n)]\n    if other == \"grey\":\n        rest = [\"#BCBCBC\"] * n\n    elif other == \"ordinal\":\n        nf = max(1, n - len(focal_set))\n        ramp = [mcolors.to_hex((v, v, v)) for v in\n                ([0.55] if nf == 1 else [0.80 - 0.35 * i / (nf - 1) for i in range(nf)])]\n        rest, k = [], 0\n        for l in labels:\n            rest.append(ramp[min(k, nf - 1)]); k += (l not in focal_set)\n    elif other == \"muted\":\n        def mute(c):\n            r, g, b = mcolors.to_rgb(c)\n            m = (r + g + b) / 3\n            return mcolors.to_hex((0.3 * r + 0.7 * m, 0.3 * g + 0.7 * m, 0.3 * b + 0.7 * m))\n        rest = [mute(c) for c in base_colors]\n    else:\n        raise ValueError(\"other must be 'muted', 'grey', or 'ordinal'\")\n    return [focal_color if l in focal_set else rest[i] for i, l in enumerate(labels)]\n\n\ndef bar_with_points(ax, x, ymat, labels, colors, jitter=0.08, show_points=True,\n                    errorbar=None, point_alpha=0.5, point_size=8):\n    \"\"\"§6.1: bar = mean; optionally overlay raw points or draw an interval.\n\n    colors   : per-label color list (e.g. from focal_palette)\n    x, ymat and labels must have equal lengths; mismatches raise before drawing.\n    Each group must be a nonempty, finite, one-dimensional real numeric array.\n    Requested SD/CI intervals require at least two observations per group.\n    errorbar : None | 'sd' | 'ci95' — drawn only when show_points is False.\n               'ci95' is the t-distribution 95% CI of the mean\n               (half-width t_{0.975,n-1} · s/√n); correct at small n where the\n               z-approximation (1.96·s/√n) is markedly too narrow.\n    \"\"\"\n    import numpy as np\n    x = np.asarray(x)\n    if x.ndim != 1 or not np.issubdtype(x.dtype, np.number):\n        raise TypeError(\n            \"bar_with_points x must be one-dimensional numeric positions; \"\n            \"pass category names through labels\"\n        )\n    ymat = list(ymat)\n    labels = list(labels)\n    if len(x) != len(ymat) or len(x) != len(labels):\n        raise ValueError(\"bar_with_points x, ymat and labels must have equal lengths\")\n    if colors is not None and not isinstance(colors, str):\n        colors = list(colors)\n        if not colors:\n            colors = None\n    if x.dtype.kind not in \"iuf\" or not np.isfinite(x).all():\n        raise ValueError(\"bar_with_points x must contain finite real positions\")\n    for i, y in enumerate(ymat):\n        if np.ma.isMaskedArray(y) and np.ma.getmaskarray(y).any():\n            raise ValueError(f\"bar_with_points group {i} contains masked observations; resolve exclusions explicitly before plotting\")\n    ymat = [np.asarray(y) for y in ymat]\n    for i, y in enumerate(ymat):\n        if y.ndim != 1 or y.size == 0 or y.dtype.kind not in \"iuf\" or not np.isfinite(y).all():\n            raise ValueError(f\"bar_with_points group {i} must be nonempty, one-dimensional and finite real numeric data\")\n    if errorbar and not show_points:\n        if errorbar not in (\"sd\", \"ci95\"):\n            raise ValueError(\"errorbar must be None, 'sd', or 'ci95'\")\n        if any(y.size < 2 for y in ymat):\n            raise ValueError(\"SD/CI cannot be estimated from one observation; use raw points without an interval\")\n    means = np.array([np.mean(y) for y in ymat], float)\n    err = None\n    if errorbar and not show_points:\n        if errorbar == \"sd\":\n            err = np.array([np.std(y, ddof=1) for y in ymat])\n        elif errorbar == \"ci95\":\n            from scipy.stats import t\n            def _hw(y):\n                n = np.asarray(y).size\n                return t.ppf(0.975, n - 1) * np.std(y, ddof=1) / np.sqrt(n)\n            err = np.array([_hw(y) for y in ymat])\n    ax.bar(x, means, color=colors, width=0.7, edgecolor=\"none\",\n           yerr=err, error_kw={\"elinewidth\": 0.8, \"capsize\": 0})\n    if show_points:\n        for xi, ys in zip(x, ymat):\n            ys = np.asarray(ys)\n            if ys.ndim == 1 and ys.size > 0:\n                jit = (np.random.rand(ys.size) - 0.5) * 2 * jitter\n                ax.scatter(np.full(ys.size, xi) + jit, ys, s=point_size, color=\"black\",\n                           alpha=point_alpha, zorder=3, linewidths=0)\n    ax.set_xticks(x); ax.set_xticklabels(labels)\n    return ax\n\n\ndef strip_with_median(ax, groups, values, colors=None, jitter=0.12):\n    \"\"\"§6.1: jittered points + bold horizontal median tick per group.\n\n    groups and values must have equal lengths. Mismatches raise before drawing.\n    colors=None or an empty color sequence supplies default gray; a nonempty\n    color sequence cycles across groups.\n    Each group must be nonempty, one-dimensional and finite real numeric data;\n    resolve missing observations explicitly before plotting.\n    \"\"\"\n    import numpy as np\n    labs = list(groups)\n    values = list(values)\n    if len(labs) != len(values):\n        raise ValueError(\"strip_with_median groups and values must have equal lengths\")\n    for i, y in enumerate(values):\n        if np.ma.isMaskedArray(y) and np.ma.getmaskarray(y).any():\n            raise ValueError(f\"strip_with_median group {i} contains masked observations; resolve exclusions explicitly before plotting\")\n    values = [np.asarray(y) for y in values]\n    for i, y in enumerate(values):\n        if y.ndim != 1 or y.size == 0 or y.dtype.kind not in \"iuf\" or not np.isfinite(y).all():\n            raise ValueError(f\"strip_with_median group {i} must be nonempty, one-dimensional and finite real numeric data\")\n    colors = [] if colors is None else list(colors)\n    if not colors:\n        colors = [\"#444444\"]\n    for i, ys in enumerate(values):\n        c = colors[i % len(colors)]\n        ys = np.asarray(ys)\n        jit = (np.random.rand(ys.size) - 0.5) * 2 * jitter\n        ax.scatter(np.full(ys.size, i) + jit, ys, s=10, color=c, alpha=0.6, linewidths=0, zorder=2)\n        m = np.median(ys)\n        ax.plot([i - 0.22, i + 0.22], [m, m], color=\"black\", lw=1.6, zorder=3)\n    ax.set_xticks(range(len(labs))); ax.set_xticklabels(labs)\n    return ax\n\n\ndef goodness_arrow(ax, text=\"higher = better\", loc=\"upper left\", axis=\"y\", fontsize=None):\n    \"\"\"§3.6: small upright direction-of-goodness cue in the margin.\"\"\"\n    import matplotlib.pyplot as plt\n    if fontsize is None:\n        fontsize = plt.rcParams[\"legend.fontsize\"]\n    pos = {\"upper left\": (0.02, 0.98), \"upper right\": (0.98, 0.98),\n           \"lower left\": (0.02, 0.02), \"lower right\": (0.98, 0.02)}[loc]\n    ha = \"left\" if \"left\" in loc else \"right\"\n    va = \"top\" if \"upper\" in loc else \"bottom\"\n    arrow = \"↑ \" if axis == \"y\" else \"→ \"\n    ax.text(pos[0], pos[1], arrow + text, transform=ax.transAxes,\n            fontsize=fontsize, color=META_GREY, ha=ha, va=va)\n\n\ndef two_tier_label(name, meta):\n    \"\"\"§5: two-line label string (name / metadata). Meta line styled separately by caller.\"\"\"\n    return f\"{name}\\n{meta}\"\n\n\ndef end_of_line_labels(ax, xs, ys, labels, colors=None, dx=0.01, fontsize=None):\n    \"\"\"§6.3 / §7.3: label each line series at its right end instead of a legend box.\n\n    Each label requires a nonempty, matched x/y series. Validate all series\n    before drawing. Nonempty colors cycle; None or empty uses default text color.\n    Inner sequences must support len() and indexing; they are not copied.\n    Nonfinite endpoints are rejected rather than producing an invisible label.\n    \"\"\"\n    import matplotlib.pyplot as plt\n    import math\n    xs, ys, labels = list(xs), list(ys), list(labels)\n    if len(xs) != len(ys) or len(xs) != len(labels):\n        raise ValueError(\"end_of_line_labels xs, ys and labels must have equal lengths\")\n    if any(len(x) == 0 or len(x) != len(y) for x, y in zip(xs, ys)):\n        raise ValueError(\"end_of_line_labels each x/y series must be nonempty and have equal lengths\")\n    if any(not math.isfinite(x[-1]) or not math.isfinite(y[-1]) for x, y in zip(xs, ys)):\n        raise ValueError(\"end_of_line_labels endpoints must be finite; choose valid endpoints explicitly\")\n    if fontsize is None:\n        fontsize = plt.rcParams[\"font.size\"]\n    colors = [] if colors is None else list(colors)\n    if not colors:\n        colors = [None]\n    span = ax.get_xlim()[1] - ax.get_xlim()[0]\n    for i, (x, y, lab) in enumerate(zip(xs, ys, labels)):\n        c = colors[i % len(colors)]\n        ax.text(x[-1] + dx * span, y[-1], lab, color=c, va=\"center\", ha=\"left\", fontsize=fontsize)\n\n\ndef panel_crops(fig, dpi=None, pad_px=6, bbox_inches=None, pad_inches=None):\n    \"\"\"§9.2: pixel-space crop boxes for each lettered panel in the SAVED PNG.\n\n    Returns ``{letter: (x0, y0, x1, y1)}`` in image-space pixels (origin\n    top-left, matching PIL's ``Image.crop``). Panels are detected as bold\n    single-character ``Text``\n    objects placed by :func:`panel_letter`; each panel's crop is its axes'\n    tightbbox mapped into the saved file's pixel space, padded by ``pad_px``.\n    For §3.4 composites (abutting subplots sharing an axis, letter on the\n    leftmost only) the crop unions in letterless ``sharex``/``sharey`` siblings\n    on the same grid row/col so the whole composite is covered. When no axes\n    carries a panel letter (standalone plot, or a figure-composer sub-agent),\n    falls back to one crop per axes keyed by index.\n\n    ``bbox_inches`` mirrors ``Figure.savefig`` semantics: ``None`` means\n    *consult rcParams* (so under :func:`apply_figure_style` it resolves to\n    ``'tight'``); pass an explicit ``Bbox`` only if you saved with one. The\n    boxes are clamped to the saved image extent; panels with no intersection\n    are omitted.\n\n        >>> from PIL import Image\n        >>> fig.savefig(\"fig.png\")  # bbox_inches='tight' via rcParams\n        >>> with Image.open(\"fig.png\") as saved:\n        ...     for letter, box in panel_crops(fig).items():\n        ...         saved.crop(box).save(f\"panel_{letter}.png\")\n\n    Match savefig's DPI, bbox_inches and pad_inches when overriding defaults.\n    Returned boxes are pixels, never savefig bbox_inches (inches). Passing them\n    back to savefig as inches can allocate an enormous raster.\n    \"\"\"\n    import matplotlib as mpl\n    import matplotlib.text\n    from matplotlib.backends.backend_agg import FigureCanvasAgg\n    if isinstance(pad_px, bool) or not isinstance(pad_px, int) or pad_px < 0:\n        raise TypeError(\"panel_crops pad_px must be a non-negative integer\")\n    if dpi is None:\n        dpi = mpl.rcParams.get(\"savefig.dpi\", fig.dpi)\n        if dpi == \"figure\":\n            dpi = fig.dpi\n    dpi = float(dpi)\n    if bbox_inches is None:\n        bbox_inches = mpl.rcParams.get(\"savefig.bbox\")\n    # The Open-Science notebook captures and closes pyplot figures after every\n    # cell. Matplotlib 3.11 decouples a closed Figure from its backend by\n    # replacing FigureCanvasAgg with FigureCanvasBase, while the persistent\n    # notebook namespace can still retain the Figure itself. Reattach the\n    # headless canvas before asking for saved-image geometry so this helper is\n    # safe in a later QA cell as well as the producer cell.\n    if not hasattr(fig.canvas, \"get_renderer\"):\n        FigureCanvasAgg(fig)\n    fig.canvas.draw()\n    r = fig.canvas.get_renderer()\n\n    if bbox_inches == \"tight\":\n        if pad_inches is None:\n            pad_inches = mpl.rcParams.get(\"savefig.pad_inches\", 0.1)\n        tb = fig.get_tightbbox(r).padded(pad_inches)\n        ox_in, oy_in = tb.x0, tb.y0\n        W_in, H_in = tb.width, tb.height\n    elif isinstance(bbox_inches, mpl.transforms.BboxBase):\n        ox_in, oy_in = bbox_inches.x0, bbox_inches.y0\n        W_in, H_in = bbox_inches.width, bbox_inches.height\n    elif bbox_inches is None:\n        ox_in, oy_in = 0.0, 0.0\n        W_in, H_in = fig.get_size_inches()\n    else:\n        raise TypeError(\"panel_crops bbox_inches must be None, 'tight', or a matplotlib Bbox\")\n    W_px, H_px = int(round(W_in * dpi)), int(round(H_in * dpi))\n    lettered = {}\n    seen_letters = set()\n    for ax in fig.axes:\n        for t in ax.findobj(matplotlib.text.Text):\n            s = (t.get_text() or \"\").strip()\n            if t.get_gid() == \"figure-style-panel-letter\":\n                if s in seen_letters:\n                    raise ValueError(f\"Duplicate panel letter: {s!r}\")\n                seen_letters.add(s)\n                lettered[ax] = s\n                break\n\n\n\n    if not lettered:\n        lettered = {ax: str(i) for i, ax in enumerate(fig.axes)}\n    out = {}\n    for ax, letter in lettered.items():\n        bbs = [ax.get_tightbbox(r)]\n\n\n\n\n        ss = ax.get_subplotspec()\n        for sib in fig.axes:\n            if sib is ax or sib in lettered:\n                continue\n            ssib = sib.get_subplotspec()\n            same_row = ss is None or ssib is None or ss.rowspan == ssib.rowspan\n            same_col = ss is None or ssib is None or ss.colspan == ssib.colspan\n            if ((ax.get_shared_y_axes().joined(ax, sib) and same_row)\n                    or (ax.get_shared_x_axes().joined(ax, sib) and same_col)):\n                bbs.append(sib.get_tightbbox(r))\n        bb = mpl.transforms.Bbox.union(bbs)\n\n        bx0 = (bb.x0 / fig.dpi - ox_in) * dpi\n        bx1 = (bb.x1 / fig.dpi - ox_in) * dpi\n        by0 = H_px - (bb.y1 / fig.dpi - oy_in) * dpi\n        by1 = H_px - (bb.y0 / fig.dpi - oy_in) * dpi\n        box = (\n            min(max(int(bx0) - pad_px, 0), W_px),\n            min(max(int(by0) - pad_px, 0), H_px),\n            max(min(int(bx1) + pad_px, W_px), 0),\n            max(min(int(by1) + pad_px, H_px), 0),\n        )\n        if box[0] < box[2] and box[1] < box[3]:\n            out[letter] = box\n    return out\n",
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