sample_id,family_id,key,prior_verdict,verdict,evidence_basis,claim,reason,source_locator,citing_context,inspected_source_abstracts,supplementary_notes,direct_session_full_text_verified
R097,372,Eskin2001,unverifiable,unverifiable,prior_direct_evidence_only,Cekic’s holomorphic-direction parameter argument follows Eskin’s Yang–Mills analysis.,No abstract or full-text passage was available to inspect the claimed methodological relationship. Bibliographic identity alone cannot verify it.,"Eskin 2001; nearby claim also depends on Cekic 2025, which is outside this sampled source check.","{""path"": ""preprints/Global-Uniqueness-for-the-Smooth-Isotropic-Elasticity-Inverse-Problem-September-24-2026/build/sections/introduction.tex"", ""offset"": 7916, ""context"": ""theta\\binom{p}{s}\n   =M_{j,\\theta}\\binom{p}{s},\\qquad\n M_{j,\\theta}=\n \\begin{pmatrix}0&\\theta\\\\Q_{j,\\theta}\\cdot&T_{j,\\theta}\\end{pmatrix}.\n\\]\nHere $Q_{j,\\theta}\\cdot$ is the covector\n$p\\mapsto Q_{j,\\theta}\\cdot p$. The lower coefficients are smooth in $x$\nand linear in $\\theta$;\nLemma~\\ref{lem:transport-normal-form} gives their formulas.\nThe upper equation $D_\\theta p=\\theta s$ is the same for both media.\n\nTransport comparison followed by holomorphic variation of the complex\ndirection has a close precedent in Ceki\\'c's work on connection Laplacians\n\\cite{Cekic2025}, whose parameter argument follows Eskin's analysis of\nYang--Mills potentials \\cite{Eskin2001}. Ceki\\'c treats full\nconnection-system boundary data on a trivial vector bundle. The comparison\nneeded here must be obtained from physical elasticity data on the\nconstrained fibers $\\Ecal_\\theta$.\n\nOn a cylinder given by a planar disk times a transverse interval, let\n$Y_1:\\C^3\\to\\Ecal_\\theta$ be the matrix of normalized columns of one\nlocal frame realized in the first medium. Transfer its exact physical\nsolutions to the second\nmedium. The two augmented operators have the same principal part, so the\nforcing for the difference of the physical pairs has only first order.\nThe derivative bound and the supported Carleman estimate make the\ntransfer"", ""command"": ""\\cite{Eskin2001}""}",[],[],False
R056,192,ODonnellServedio,unverifiable,qualified_support_notes,supplied_source_reading_notes_with_prior_direct_metadata_or_abstract,Theorem 3 and Lemma 3 bound a Boolean readout’s signed singleton sum by the square root of the expected number of fixed coordinates.,"The December 3, 2005 author-draft notes support Lemma 3/Theorem 3 for the signed singleton Fourier sum and the square-root expected-fixed-coordinate bound. Interpreting this as total influence requires monotonicity. The final 2007 journal version and preservation of theorem numbering there are unverified.",Theorem 3; Lemma 3,"{""path"": ""preprints/Unbounded-Violations-of-the-Square-Root-Degree-Bound-September-26-2026/build/sections/00-introduction.tex"", ""offset"": 4945, ""context"": ""very output indicator \\(\\one_{\\{F=a\\}}\\) has degree at most \\(D\\). Every function of\n\\(F\\) then has degree at most \\(D\\), including\n\\(f=\\operatorname{sign}(T_F)\\). This readout has correlation\n\\(\\E[fH_N]=\\E|T_F|\\). Averaging independent copies makes its score\napproximately Gaussian, so a large ratio \\(v(F)/D\\) gives the theorem.\n\nConditional coordinate sums also appear in decision-tree inequalities.\nA subcube is obtained by fixing some coordinates. If the cells of\n\\(F\\) are subcubes, the unfixed coordinates remain independent uniform\nsigns under conditioning, so \\(v(F)\\) equals the expected number of\nfixed coordinates. O'Donnell and Servedio~\\cite[Theorem~3 and\nLemma~3]{ODonnellServedio} bound the signed singleton sum of a Boolean\nreadout by the square root of this expectation. Blais, Tan and\nWan~\\cite[Lemma~3.1 and Theorem~1]{BlaisTanWan} use the conditional\ncoordinate sum to prove square-root bounds in parity decision-tree\ndepth, where a query reveals the product of a specified set of inputs.\nOur cells need not be subcubes: their indicators are controlled by\npolynomial degree, and cancellation allows retained variance to exceed\nthat degree cost.\n\nThe main construction repeatedly increases this ratio. Revealing\n\\(m\\) independent standardized scores retains variance \\(m\\). We use\n\\(m+1\\) scores and design an observation reta"", ""command"": ""\\cite[Theorem~3 and\nLemma~3]{ODonnellServedio}""}",[],"[{""sample_id"": ""R056"", ""source_url"": ""https://www.cs.columbia.edu/~rocco/Public/os-ccc.pdf"", ""retrieved_date"": ""2026-10-08"", ""source_sha256"": ""5c303739310f8463d47603e2e9f87e4d0cab1e3e46ce6c304e5523ade70c8e21"", ""locator"": ""PDF pages 6–7 (printed pages 5–6), Lemma 3 and Theorem 3"", ""source_version"": ""Author draft 2005-12-03"", ""paraphrased_observations"": ""Lemma 3 expresses the signed sum of singleton Fourier coefficients as an expectation over a subcube partition. Theorem 3 and its proof bound this signed sum by the square root of the expected number of fixed coordinates, and then by the square root of log of partition size. The signed-sum inequality applies to Boolean functions; interpreting it as total influence requires monotonicity. The inspected source is an author draft dated December 3, 2005, not the final 2007 journal PDF."", ""evidence_mode"": ""External operator reading of original source; supplied notes, not independently retrieved by this OS session.""}]",False
R020,065,IritaniKoto,partial_support_abstract_only,qualified_support_notes,supplied_source_reading_notes_with_prior_direct_metadata_or_abstract,"For arbitrary projective bundles, Iritani–Koto construct a mirror theorem and split the quantum D-module into base-space pieces; nearby text also asserts a generic-semisimplicity equivalence and a Virasoro consequence.","The v4 notes support the mirror construction, genus-zero scope and formal generic-semisimplicity equivalence. Theorem 1.1 assumes rank at least two and a smooth projective base with globally generated dual; Remark 1.2 removes the dual-generation restriction by a line-bundle twist preserving projectivization. Formal semisimplicity does not establish convergence. The downstream Virasoro consequence remains unverified.",arXiv:2307.03696v4 abstract,"{""path"": ""preprints/Virasoro-Constraints-under-Projectivization-October-5-2026/build/sections/01-introduction.tex"", ""offset"": 4107, ""context"": ""base of a toric bundle if\nand only if they hold for its total space, for toric bundles\nconstructed from sums of line bundles. Their proof combines\nancestor localization with Brown's mirror theorem\n\\cite{Brown}; split projective bundles are among its examples.\nFor arbitrary vector bundles, Fan \\cite{Fan} proves all-genus\nreconstruction and Chern-class dependence of projective-bundle\ninvariants. His argument uses the projective completion\n$\\mathbb P_B(E\\oplus\\mathcal O)$, which is also the auxiliary\nspace used below. Reconstruction of invariants by itself\ndoes not establish compatibility with the Virasoro\ndifferential operators.\n\nIritani--Koto \\cite{IritaniKoto} construct a mirror theorem\nand a decomposition of the quantum $D$-module for arbitrary\nprojective bundles. Koto \\cite{Koto} proves a mirror theorem\nfor nonsplit toric bundles. These results concern genus zero.\nIn particular, Iritani--Koto's equivalence of generic\nsemisimplicity for a projective bundle and its base,\ncombined with Givental--Teleman, already gives the conclusion\nof Theorem~\\ref{thm:main} when the base has generically\nsemisimple quantum cohomology. The transfer proved here\nalso covers bases whose quantum cohomology is not\nsemisimple.\n\nAs in Fan's reconstruction, we use a projective completion\nof the bundle as an auxiliary space."", ""command"": ""\\cite{IritaniKoto}""}","[{""url"": ""http://arxiv.org/abs/2307.03696v4"", ""id_versioned"": ""2307.03696v4"", ""abstract"": ""We construct an I-function of the projective bundle P(V) associated with a not necessarily split vector bundle V\\to B as a Fourier transform of the S^1-equivariant J-function of the total space of V and show that it lies on the Givental Lagrangian cone of P(V). Using this result, we show that the quantum cohomology D-module of P(V) splits into the direct sum of the quantum cohomology D-modules of the base space B. This has applications to the semisimplicity of big quantum cohomology.""}]","[{""sample_id"": ""R020"", ""source_url"": ""https://arxiv.org/pdf/2307.03696v4"", ""retrieved_date"": ""2026-10-08"", ""source_sha256"": ""5139f8e0c9d46f8ccb8cb415396a0fb1fb357719b7dcfbca46234a9735b57624"", ""locator"": ""PDF pages 2 and 4; Theorem 1.1, Remarks 1.2/1.4/1.9, Corollary 1.8"", ""source_version"": ""arXiv:2307.03696v4"", ""paraphrased_observations"": ""Theorem 1.1 covers a rank at least two vector bundle over a smooth projective base with globally generated dual. Remark 1.2 removes that restriction by a line-bundle twist preserving the projectivization. Remark 1.4 concerns genus-zero invariants. Corollary 1.8 gives generic semisimplicity for the projective bundle if and only if for its base. Remark 1.9 specifies formal semisimplicity and does not establish convergence. These inspected passages do not independently establish the nearby downstream Virasoro consequence."", ""evidence_mode"": ""External operator reading of original source; supplied notes, not independently retrieved by this OS session.""}]",False
R074,230,Zhan2019Decomposition,partial_support_abstract_only,qualified_support_notes,supplied_source_reading_notes_with_prior_direct_metadata_or_abstract,"For 0<kappa<8, integrated Green-weighted two-sided radial SLE laws give natural-length-biased chordal SLE; Theorem 4.1 and specific equations implement a remaining-mass mechanism.",The arXiv v3 notes support Theorem 4.1 for 0<kappa<8 and remaining-mass/killing-measure identities in equations 4.3–4.5. The citation explicitly uses an author-hosted published article for numbering; equivalence to that edition is unverified. Proposition 2.2/equation (2.1) and subsequent past-mass derivations remain unverified.,arXiv:1509.05015 abstract; published theorem/equation locators uninspected,"{""path"": ""preprints/An-exact-Hausdorff-gauge-for-SLE-September-25-2026/build/sections/hausdorff.tex"", ""offset"": 3645, ""context"": ""raw mass\nfunctional of Proposition~\\ref{mass:construction}.\nFor example, if \\(q<T^*=\\lim_{\\ell\\to\\infty}\\tau_\\ell(z)\\), some\nfinite stopped piece extends beyond \\(q\\). Monotonicity therefore\nbounds the mass accumulated in \\(F(K)\\) before \\(q\\) by\n\\((C_*'r)^dL\\), with \\(L\\) the increasing limit in the restarted\ndisk model. No mass after \\(T^*\\) is used.\n\n\\subsection{Past mass and first visits}\n\\label{hd:palm-section}\n\nThe relation between natural-length-biased chordal SLE and integrated\nGreen-weighted two-sided radial laws was proved by\nField \\cite[Theorem~1]{Field2016} for $0<\\kappa\\le4$ in bounded\ndomains with analytic boundary, and by\nZhan \\cite[Theorem~4.1]{Zhan2019Decomposition} for $0<\\kappa<8$.\nZhan's proof connects mass strictly after an observation time to\ncentered stopped laws through the conditional remaining Green potential\n\\cite[Proposition~2.2, equation~(2.1), and equations~(4.3)--(4.5)]{Zhan2019Decomposition}.\nWe use this mechanism to derive the required past-mass identity\ndirectly for our canonical mass, from its conditional measure kernel\nand finite stopped changes of law. Retaining the strict time inequality\nkeeps each centered-law mass evaluation before its terminal time.\n\nLet \\(Q\\Subset\\HH\\) be a deterministic closed square and let\n\\(H>0\\). Write \\(M_q=\\mu_q(Q)\\), and let \\(v(z)\\) denote the first\nvis"", ""command"": ""\\cite[Theorem~4.1]{Zhan2019Decomposition}""}","[{""url"": ""http://arxiv.org/abs/1509.05015v3"", ""id_versioned"": ""1509.05015v3"", ""abstract"": ""We show that, for $κ\\in(0,8)$, the integral of the laws of two-sided radial SLE$_κ$ curves through different interior points against a measure with SLE$_κ$ Green function density is the law of a chordal SLE$_κ$ curve, biased by the path's natural length. We also show that, for $κ>0$, the integral of the laws of extended SLE$_κ(-8)$ curves through different interior points against a measure with a closed formula density restricted in a bounded set is the law of a chordal SLE$_κ$ curve, biased by the path's capacity length restricted in that set. Another result is that, for $κ\\in(4,8)$, if one integrates the laws of two-sided chordal SLE$_κ$ curves through different force points on $\\mathbb R$ against a measure with density on $\\mathbb R$, then one also gets a law that is absolutely continuous w.r.t. that of a chordal SLE$_κ$ curve. To obtain these results, we develop a framework to study stochastic processes with random lifetime, and improve the traditional Girsanov's Theorem.""}]","[{""sample_id"": ""R074"", ""source_url"": ""https://arxiv.org/pdf/1509.05015"", ""retrieved_date"": ""2026-10-08"", ""source_sha256"": ""00402d7118f4c3e21f53efacc2360c925ce7eb93b9fc25076307f94242f9dc66"", ""locator"": ""PDF page 13, Theorem 4.1 and equations 4.3–4.8"", ""source_version"": ""arXiv:1509.05015v3; document dated August 26, 2018"", ""paraphrased_observations"": ""For kappa strictly between zero and eight, Theorem 4.1 relates Green-function-weighted two-sided radial SLE laws to natural-parametrization/Minkowski-content weighted chordal SLE. Equations 4.3–4.5 on the same page provide conditional remaining-mass and killing-measure identities. These passages support the cited core identity, but do not independently prove every subsequent derivation made in the citing paper."", ""evidence_mode"": ""External operator reading of original source; supplied notes, not independently retrieved by this OS session.""}]",False
R068,215,OS75,unverifiable,unverifiable,prior_direct_evidence_only,The corrected Osterwalder–Schrader reconstruction framework uses the linear growth condition in Section IV.1.,"The cited Section IV.1 was not retrieved. Metadata verifies the paper identity, but cannot establish the correction or its precise hypotheses.",Section IV.1,"{""path"": ""preprints/The-canonical-massive-continuum-limit-of-the-two-dimensional-O3-model-October-4-2026/build/sections/method-history.tex"", ""offset"": 980, ""context"": ""elated to disagreement percolation~\\cite{VanDenBergMaes1994}.\nIn exact blocking, the connected polymer sums use the classical\ntree-graph approach to cluster expansions; a useful formulation of\nthe Penrose identity is given by Fern\\'andez and\nProcacci~\\cite{FernandezProcacci}.\n\nThe finite-volume comparison uses reflection positivity and the\nchessboard method systematized by Fr\\\""ohlich, Israel, Lieb and\nSimon~\\cite{FILS78,FILS80}.  For the final passage from Euclidean\ndistributions to a local relativistic field, we use the reconstruction\nframework of Osterwalder and Schrader~\\cite{OS73} in the corrected\nform with the linear growth condition of \\cite[Section~IV.1]{OS75}.\nReflection positivity, Euclidean covariance, clustering and the\nhierarchy of distributional bounds are consequently separate parts\nof the continuum argument.\n"", ""command"": ""\\cite[Section~IV.1]{OS75}""}",[],[],False
R004,015,Stark1974,unverifiable,unverifiable,prior_direct_evidence_only,"Stark Lemmas 3 and 8 supply a zero-free-region exception and force a quadratic subfield for a sufficiently close real zero, including nonnormal fields.","The exact lemmas were not retrieved, so constants, hypotheses, and applicability to nonnormal fields could not be checked.",Lemmas 3 and 8; nearby Equation (9) also uninspected,"{""path"": ""preprints/Equidistribution-of-Prime-Degree-Torus-Packets-with-Arbitrary-Local-Type-September-24-2026/build/analytic.tex"", ""offset"": 1009, ""context"": ""e}\nFor $D$ sufficiently large in terms of $n$,\n\\begin{equation}\\label{eq:residue}\n h\\zeta_K(1+h)\\asymp_n\\kappaK\n \\quad\\left(0<h\\le\\frac4{\\log D}\\right),\n \\qquad \\kappaK^{-1}\\ll_n\\log D.\n\\end{equation}\nFor the remaining fields, the first comparison holds uniformly for\nsufficiently small positive $h$, and $\\kappaK^{-1}\\ll_n1$.\n\\end{lemma}\n\n\\begin{proof}\nPut $L=\\log D$. Stark's Lemma~3 gives at most one zero in\n$\\operatorname{Re}s\\ge1-(4L)^{-1}$,\n$|\\operatorname{Im}s|\\le(4L)^{-1}$; such a zero is real and simple.\nHis Lemma~8, which also applies to nonnormal fields, says that a real\nzero $\\beta\\ge1-(4n!L)^{-1}$ forces a quadratic subfield of $K$\n\\cite[Lemmas~3 and~8]{Stark1974}. Since $n$ is odd, there is no such\nsubfield. Consequently, for a constant $c_n>0$, every nontrivial zero\n$\\rho$ satisfies $|1-\\rho|\\ge c_n/L$.\n\nThe real logarithmic derivative of the completed zeta function gives\n\\[\n S_K(\\sigma):=\\sum_\\rho\\operatorname{Re}\\frac1{\\sigma-\\rho}\n =\\frac1\\sigma+\\frac1{\\sigma-1}+\\frac L2\n   +\\frac{\\zeta_K'}{\\zeta_K}(\\sigma)+G_n(\\sigma),\\qquad \\sigma>1,\n\\]\nwhere zeros are counted with multiplicity and\n\\[\n G_n(\\sigma)=\\frac n2\n \\left(\\frac{\\Gamma'(\\sigma/2)}{\\Gamma(\\sigma/2)}-\\log\\pi\\right).\n\\]\nThis is the Hadamard identity in\n\\cite[Equation~(9)]{Stark1974}. The zero terms are nonnegative,\n$G_n$ is bounded on $[1,2"", ""command"": ""\\cite[Lemmas~3 and~8]{Stark1974}""}",[],[],False
R046,154,Jamneshan2023,partial_support_abstract_only,supported_attribution_notes,supplied_source_reading_notes_with_prior_direct_metadata_or_abstract,The cited results include a nonergodic Furstenberg–Zimmer structure theorem; nearby pointwise distal convergence is separately attributed to Huang–Shao–Ye and a companion proposition.,"The v4 Theorem 1.1 notes explicitly cover any group and probability algebra dynamical system without an ergodicity assumption, a transfinite relatively compact tower and weakly mixing top extension. This supports the nonergodic structure-theory attribution only; adjacent pointwise, joining and rational-time claims remain unverified.",arXiv:2103.17167v4 abstract,"{""path"": ""preprints/Pointwise-Multiple-Ergodic-Averages-for-Mixing-Transformations-October-4-2026/build/sections/inputs.tex"", ""offset"": 6328, ""context"": "" distal\nextension of the same base are relatively disjoint: every invariant\njoining identifying their bases is the conditional product over\nthat base.\n\\item For a distal transformation $R$, every $d\\ge1$, and bounded\n$g_1,\\ldots,g_d$, the averages\n\\[\n \\frac1N\\sum_{l=1}^N\\prod_{i=1}^d g_i\\circ R^{il}\n\\]\nconverge almost everywhere.\n\\end{enumerate}\nFor a flow, the maximal distal factor of time one is preserved by\nthe flow, and its action at every nonzero rational time is distal.\n\\end{proposition}\n\nThese are the conclusions of\n\\cite[Proposition~2.2]{FourfoldCompanion}. They include the\nnonergodic form of the Furstenberg--Zimmer structure theorem\n\\cite{Zimmer1976,Jamneshan2023} and the pointwise theorem for\ndistal systems \\cite[Theorem~C]{HuangShaoYe2019}. The cited proposition\ngives the component-disintegration argument needed for the latter\nin the nonergodic setting. In the flow assertion, invariance of\n$W_Y$ follows because each time commutes with time one. If\n$r=p/q\\ne0$ is rational, the $q$th power of its action is the\ndistal action at integer time $p$, so the power assertion applies.\n\nThe following short argument, from\n\\cite[Lemma~2.3]{FourfoldCompanion}, will also fix the norm-convergence\ninput explicitly.\n\\begin{lemma}[Relative norm convergence]\\label{lem:relative-norm}\nSuppose $Y\\to B$ is relatively weakly"", ""command"": ""\\cite{Zimmer1976,Jamneshan2023}""}","[{""url"": ""http://arxiv.org/abs/2103.17167v4"", ""id_versioned"": ""2103.17167v4"", ""abstract"": ""Furstenberg--Zimmer structure theory refers to the extension of the dichotomy between the compact and weakly mixing parts of a measure preserving dynamical system and the algebraic and geometric descriptions of such parts to a conditional setting, where such dichotomy is established relative to a factor and conditional analogues of those algebraic and geometric descriptions are sought. Although the unconditional dichotomy and the characterizations are known for arbitrary systems, the relative situation is understood under certain countability and separability hypotheses on the underlying groups and spaces. The aim of this article is to remove these restrictions in the relative situation and establish a Furstenberg--Zimmer structure theory in full generality. As an independent byproduct, we establish a connection between the relative analysis of systems in ergodic theory and the internal logic in certain Boolean topoi.""}]","[{""sample_id"": ""R046"", ""source_url"": ""https://arxiv.org/pdf/2103.17167v4"", ""retrieved_date"": ""2026-10-08"", ""source_sha256"": ""b3ef73623931c3bcc11b1918441cbb84610a38148a1c9577710ba47bf947e248"", ""locator"": ""PDF page 2, Theorem 1.1"", ""source_version"": ""arXiv:2103.17167v4"", ""paraphrased_observations"": ""The uncountable Furstenberg–Zimmer theorem is stated for any group and probability algebra dynamical system, with a transfinite relatively compact tower and a weakly mixing extension at the top. Ergodicity is not assumed in the statement. This supports the nonergodic structure-theory attribution, not every adjacent assertion about pointwise averages or joinings."", ""evidence_mode"": ""External operator reading of original source; supplied notes, not independently retrieved by this OS session.""}]",False
R030,113,Kasteleyn1963,unverifiable,unverifiable,prior_direct_evidence_only,Pfaffian methods give polynomial-time exact counting of perfect matchings in planar graphs.,"The source record matches, but no inspectable abstract or full-text passage was returned; the claim was not verified from this source.",Kasteleyn 1963; no theorem locator supplied,"{""path"": ""preprints/A-Fully-Polynomial-Randomized-Approximation-Scheme-for-Perfect-Matchings-in-General-Graphs-September-23-2026/build/main.tex"", ""offset"": 6908, ""context"": ""or matchings of a\nspecified size. On feasible inputs, its samplers always return a\nfeasible object and approximate the corresponding uniform law in total\nvariation.\n\n\\subsection{History and significance}\nPerfect matchings illustrate the distinction between finding a\ncombinatorial structure and counting its realizations. Edmonds gave a\npolynomial-time algorithm for finding a maximum matching in a general\ngraph~\\cite{Edmonds1965}. Valiant proved that exact counting is\n\\#P-complete even for bipartite graphs, where it is the permanent of a\nzero--one matrix~\\cite{Valiant1979}. Pfaffian methods give\npolynomial-time exact counting for planar graphs~\\cite{Kasteleyn1963}.\n\nFor approximation, three problems must be distinguished: counting all\nmatchings, counting bipartite perfect matchings, and counting perfect\nmatchings in general graphs. Jerrum and Sinclair gave an FPRAS for the\ntotal number of matchings in an arbitrary graph, summed over all\ncardinalities~\\cite[Corollary~4.5]{JerrumSinclair1989}.\nTheir perfect-matching algorithm applied when the ratio of near-perfect\nmatchings, which leave exactly two vertices unmatched, to perfect\nmatchings was bounded by a fixed polynomial~\\cite[Theorem~5.3]{JerrumSinclair1989}.\nWithout such a bound, perfect matchings can carry too little stationary\nmass for this samplin"", ""command"": ""\\cite{Kasteleyn1963}""}",[],[],False
R095,369,Miyamoto2007,unverifiable,supported_attribution_notes,supplied_source_reading_notes_with_prior_direct_metadata_or_abstract,Miyamoto Lemma 1.2 and Theorem A exclude interior critical points under a spectral-diameter condition and give nonsymmetric nearly circular convex examples.,"The RIMS1591 notes support the sampled exclusion/nonsymmetry attribution: open bounded smooth domain, nonzero second Neumann eigenfunction, strict diameter-squared/area bound near 1.378 in Theorem A, and strict sqrt(lambda_2)<j_0/diameter condition in Lemma B with Lemma 1.2. Do not replace strict inequalities with non-strict ones or infer a publication date. Proof not validated.","RIMS1591, Lemma 1.2 and Theorem A","{""path"": ""preprints/Strict-hot-spots-and-absence-of-interior-critical-points-on-smooth-simply-connected-planar-domains-September-24-2026/build/sections/01-introduction.tex"", ""offset"": 3049, ""context"": ""nNadirashvili2000}. Probabilistic coupling methods developed by Bañuelos and Burdzy \\cite{BanuelosBurdzy1999} led to further results. Pascu proved boundary-only extrema for antisymmetric eigenfunctions on $C^{1,\\alpha}$ convex planar domains with one reflection symmetry, where $0<\\alpha<1$ \\cite{Pascu2002}. Atar and Burdzy treated every first-positive eigenfunction on lip domains, a class of Lipschitz domains bounded between graphs of functions with Lipschitz constants at most one \\cite{AtarBurdzy2004}. Miyamoto obtained interior critical-point exclusion under a spectral-diameter condition, giving nonsymmetric nearly circular convex examples \\cite[Lemma~1.2 and Theorem~A]{Miyamoto2007}.\n\nThe polygonal problem has a complementary development. Siudeja proved hot-spots results for a class of acute triangles \\cite{Siudeja2015}. Judge and Mondal, with their subsequent erratum, excluded interior critical points for every first-positive Neumann eigenfunction on every Euclidean triangle \\cite{JudgeMondal2020,JudgeMondal2022}. Chen, Gui and Yao established the finer classification of nonvertex critical points \\cite{ChenGuiYao2026}. These results concern domains with corners; \\Cref{thm:main} concerns the full smooth simply connected planar class.\n\nRecent quantitative work measures how far the hot-spots conclusion can fail. For gener"", ""command"": ""\\cite[Lemma~1.2 and Theorem~A]{Miyamoto2007}""}",[],"[{""sample_id"": ""R095"", ""source_url"": ""https://www.kurims.kyoto-u.ac.jp/preprint/file/RIMS1591.pdf"", ""retrieved_date"": ""2026-10-08"", ""source_sha256"": ""f0327dfdbbd01a05e860a7cde3e5b088bbd5f28f0ae029d4c1fd8a972c20c9b5"", ""locator"": ""PDF pages 1–2, Theorem A, Lemma B, Lemma 1.2"", ""source_version"": ""RIMS1591 author preprint"", ""paraphrased_observations"": ""The title and author identify Yasuhito Miyamoto and the nearly circular planar convex domains preprint. Theorem A uses a strict diameter-squared/area bound near 1.378. Lemma B uses the strict spectral-diameter condition sqrt(lambda_2) < j_0 / diameter. Lemma 1.2 excludes interior critical points under those hypotheses. Page 1 defines an open bounded smooth domain and a nonzero second Neumann eigenfunction. Page 2 expressly does not require symmetry. The preprint URL is RIMS1591; the inspected pages do not by themselves establish the publication date."", ""evidence_mode"": ""External operator reading of original source; supplied notes, not independently retrieved by this OS session.""}]",False
R045,148,BaranyVerma2026,unverifiable,supported_attribution_notes,supplied_source_reading_notes_with_prior_direct_metadata_or_abstract,Barany–Verma Theorem 3.5 proves the entropy-rate formula under weak exponential separation while allowing exact collisions.,"The v2 notes report that page 8 permits exact overlaps under WESC, and Theorem 3.5 gives min(1, random-walk entropy/Lyapunov exponent) for general contraction ratios. This supports the sampled theorem attribution; the proof was not validated.","arXiv:2507.05835v2, Theorem 3.5 uninspected","{""path"": ""preprints/The-entropy-rate-dimension-formula-for-self-similar-measures-on-the-line-September-24-2026/build/sections/introduction.tex"", ""offset"": 6624, ""context"": ""~1.2]{Garsia1963}.\nThese results illustrate the sensitivity to overlaps and arithmetic;\nabsolute continuity and Hausdorff dimension are distinct questions.\n\nHochman showed that $\\dim_H\\mu<\\min\\{1,H(p)/\\chi\\}$\nforces superexponential concentration of cylinder maps\n\\cite[Theorem~1.1]{Hochman2014}, using inverse theorems for entropy\ngrowth under convolution.  For unequal ratios his entropy formulation\nalready keeps track of the full affine map, including its slope\n\\cite[Theorem~1.4]{Hochman2014}.\nThe entropy-rate formula under weak exponential separation follows\nfrom his work; the version allowing exact collisions is proved in\nB\\'ar\\'any--Verma \\cite[Theorem~3.5]{BaranyVerma2026}.\nFor Bernoulli convolutions, Breuillard--Varj\\'u developed approximation\nby algebraic parameters with controlled entropy\n\\cite{BreuillardVarju2019}.  Building on these and earlier entropy\nmethods, Varj\\'u proved full dimension for every transcendental parameter\nin $(1/2,1)$ \\cite[Theorem~3]{VarjuTranscendental2019}.\nRapaport proved the no-exact-overlap formula for algebraic\ncontraction ratios and arbitrary real translations, allowing signed\nunequal ratios \\cite[Theorem~2]{Rapaport2022}.\n\nRapaport--Varj\\'u extended entropy and approximation methods to\nhomogeneous three-map systems \\cite{RapaportVarju2024}.\nFor equal weights, their results inc"", ""command"": ""\\cite[Theorem~3.5]{BaranyVerma2026}""}","[{""url"": ""http://arxiv.org/abs/2507.05835v2"", ""id_versioned"": ""2507.05835v2"", ""abstract"": ""In this paper, we study the Hausdorff dimension of self-similar measures and sets on the real line, where the generating iterated function system consists of some maps that share the same fixed point. In particular, we will show that out of a Hausdorff co-dimension one exceptional set of natural parameters, such systems satisfy a weak exponential separation. This significantly strengthens the previous result of the first author and Szvák. As an application, we give the Hausdorff dimension of self-affine measures supported on the generalised 4-corner set.""}]","[{""sample_id"": ""R045"", ""source_url"": ""https://arxiv.org/pdf/2507.05835v2"", ""retrieved_date"": ""2026-10-08"", ""source_sha256"": ""b01fade7cbd6fb36052d0836df5a6dca93c584a2c8a6bb349e2d2037d77b03e8"", ""locator"": ""PDF pages 8, 10–11; discussion before Section 3 and Theorem 3.5"", ""source_version"": ""arXiv:2507.05835v2"", ""paraphrased_observations"": ""Page 8 explains that the weak exponential separation condition permits exact overlaps. Theorem 3.5 gives Hausdorff dimension of a self-similar measure under WESC as the minimum of one and random-walk entropy divided by the Lyapunov exponent. The surrounding text distinguishes this general contraction-ratio version from the earlier homogeneous case. This is a statement/attribution check, not verification of the proof."", ""evidence_mode"": ""External operator reading of original source; supplied notes, not independently retrieved by this OS session.""}]",False
