theorem,criterion_id,semantic_kind,status,frozen_criterion,response_evidence_paragraph,not_explicit_components,rationale,source_url,theorem_label,source_quotes_and_locators
T01,T01.01,assumption_domain,retained,H/D: Real r strictly greater than 2; the endpoint r=2 is outside this statement. Anchor S: “For every real $r>2$”.,"This holds for every \(r>2\), with a constant depending only on \(r\), not on the functions or the chosen cutoffs. The theorem makes no claim at \(r=2\).",,The strict r>2 range and absence of an endpoint claim are explicitly preserved.,https://raw.githubusercontent.com/openai/math/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/preprints/Annular-variation-of-the-triangular-Hilbert-transform-at-the-symmetric-point-October-5-2026/build/sections/introduction.tex,thm:variation,"preprints/Annular-variation-of-the-triangular-Hilbert-transform-at-the-symmetric-point-October-5-2026/build/sections/introduction.tex lines 27-34 (statement):
\begin{theorem}[Full annular variation]\label{thm:variation}
For every real $r>2$ there is a finite constant $C_r$ such that
\[
 \|V_r(F,G)\|_{L^{3/2}(\R^2)}
 \le C_r\|F\|_{L^3(\R^2)}\|G\|_{L^3(\R^2)}
\]
for all complex $F,G\in L^3(\R^2)$.
\end{theorem}"
T01,T01.02,assumption_domain,not_explicit,"H/D: Both inputs are complex functions in L³(R²), not arbitrary functions on any space. Anchor S: “for all complex”.","The conclusion is
\[
\|V_r(F,G)\|_{3/2}\le C_r\|F\|_3\|G\|_3.
\]
Here \(\|F\|_3=(\int_{\mathbb R^2}|F|^3)^{1/3}\), and the other norms have the corresponding meaning. If both inputs have finite \(L^3\) size, their variation has finite \(L^{3/2}\) size, bounded by the product of the input sizes. In particular, the variation is finite at almost every point.",Complex-valued inputs,The L3 domain and R2 integration are preserved; complex-valued inputs are not specified. Nothing explicitly restricts the functions to real values.,https://raw.githubusercontent.com/openai/math/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/preprints/Annular-variation-of-the-triangular-Hilbert-transform-at-the-symmetric-point-October-5-2026/build/sections/introduction.tex,thm:variation,"preprints/Annular-variation-of-the-triangular-Hilbert-transform-at-the-symmetric-point-October-5-2026/build/sections/introduction.tex lines 27-34 (statement):
\begin{theorem}[Full annular variation]\label{thm:variation}
For every real $r>2$ there is a finite constant $C_r$ such that
\[
 \|V_r(F,G)\|_{L^{3/2}(\R^2)}
 \le C_r\|F\|_{L^3(\R^2)}\|G\|_{L^3(\R^2)}
\]
for all complex $F,G\in L^3(\R^2)$.
\end{theorem}"
T01,T01.03,definition,retained,"R: The output is the annular variation V_r defined from B with shifts (x+t,y) and (x,y+t) and kernel dt/t. Anchor C: eq:bilinear-truncation.","At a point \((x,y)\), the integral samples \(F\) by moving horizontally and \(G\) by moving vertically, using the same displacement \(t\). It multiplies those values and weights them by \(1/t\). An “annular” integral includes only displacements with \(\varepsilon<|t|<R\): distances between two chosen cutoffs. Positive and negative displacements have opposite weights, allowing cancellation.",,"Horizontal/vertical shifts, common displacement, kernel and distance cutoffs preserve the operator definition.",https://raw.githubusercontent.com/openai/math/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/preprints/Annular-variation-of-the-triangular-Hilbert-transform-at-the-symmetric-point-October-5-2026/build/sections/introduction.tex,thm:variation,"preprints/Annular-variation-of-the-triangular-Hilbert-transform-at-the-symmetric-point-October-5-2026/build/sections/introduction.tex lines 1-26 (context):
\section{Introduction}\label{sec:introduction}

For complex functions $F,G\in L^3(\R^2)$ and
$0<\varepsilon<R<\infty$, define the annular triangular Hilbert transform by
\begin{equation}\label{eq:bilinear-truncation}
 B_{\varepsilon,R}(F,G)(x,y)
 =\int_{\varepsilon<|t|<R}F(x+t,y)G(x,y+t)\,\frac{\dd t}{t}.
\end{equation}
The two inputs are translated along different coordinate directions.
Every finite integral is absolutely convergent outside one common
null set; we use its usual representative there and set all quantities
to zero on the exceptional set. Lemma~\ref{lem:finite-truncations}
justifies this convention and continuity in the endpoints.

For $r>2$, its annular variation is
\begin{equation}\label{eq:variation-definition}
 V_r(F,G)(x,y)
 =\sup_{\substack{J\ge1\,,\ 0<t_0<\cdots<t_J\\t_j\in\mathbb Q}}
    \left(\sum_{j=1}^J
          |B_{t_{j-1},t_j}(F,G)(x,y)|^r\right)^{1/r}.
\end{equation}
The supremum is pointwise: the partition may depend on $(x,y)$, and
there is no restriction on the number of endpoints within a dyadic
scale interval. The use of rational endpoints makes measurability
immediate and, by endpoint continuity, does not change the supremum.

"
T01,T01.04,conclusion,retained,R: Preserve the L^(3/2) norm bound by C_r times the product of the two L³ norms. Anchor S: displayed inequality.,"The conclusion is
\[
\|V_r(F,G)\|_{3/2}\le C_r\|F\|_3\|G\|_3.
\]
Here \(\|F\|_3=(\int_{\mathbb R^2}|F|^3)^{1/3}\), and the other norms have the corresponding meaning. If both inputs have finite \(L^3\) size, their variation has finite \(L^{3/2}\) size, bounded by the product of the input sizes. In particular, the variation is finite at almost every point.",,The displayed L^(3/2) inequality and product of L3 norms preserve the result.,https://raw.githubusercontent.com/openai/math/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/preprints/Annular-variation-of-the-triangular-Hilbert-transform-at-the-symmetric-point-October-5-2026/build/sections/introduction.tex,thm:variation,"preprints/Annular-variation-of-the-triangular-Hilbert-transform-at-the-symmetric-point-October-5-2026/build/sections/introduction.tex lines 27-34 (statement):
\begin{theorem}[Full annular variation]\label{thm:variation}
For every real $r>2$ there is a finite constant $C_r$ such that
\[
 \|V_r(F,G)\|_{L^{3/2}(\R^2)}
 \le C_r\|F\|_{L^3(\R^2)}\|G\|_{L^3(\R^2)}
\]
for all complex $F,G\in L^3(\R^2)$.
\end{theorem}"
T01,T01.05,quantifier,retained,"Q: For each r there exists a finite C_r that works for all F,G; the constant is not selected separately for each pair. Anchor S: “there is a finite constant” before “for all”.","This holds for every \(r>2\), with a constant depending only on \(r\), not on the functions or the chosen cutoffs. The theorem makes no claim at \(r=2\).",,The constant is explicitly r-dependent and uniform across inputs.,https://raw.githubusercontent.com/openai/math/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/preprints/Annular-variation-of-the-triangular-Hilbert-transform-at-the-symmetric-point-October-5-2026/build/sections/introduction.tex,thm:variation,"preprints/Annular-variation-of-the-triangular-Hilbert-transform-at-the-symmetric-point-October-5-2026/build/sections/introduction.tex lines 27-34 (statement):
\begin{theorem}[Full annular variation]\label{thm:variation}
For every real $r>2$ there is a finite constant $C_r$ such that
\[
 \|V_r(F,G)\|_{L^{3/2}(\R^2)}
 \le C_r\|F\|_{L^3(\R^2)}\|G\|_{L^3(\R^2)}
\]
for all complex $F,G\in L^3(\R^2)$.
\end{theorem}"
T01,T01.06,definition_quantifier,retained,D/Q: Variation takes a pointwise supremum over all finite increasing positive rational endpoint lists; partitions may depend on location. Anchor C: eq:variation-definition and “The supremum is pointwise”.,"To measure fluctuation, choose any increasing list of cutoffs \(t_0<\cdots<t_J\). Compute the integral on each successive distance band, take its absolute value, and combine these values as
\[
\left(\sum_{j=1}^J |B_{t_{j-1},t_j}(F,G)(x,y)|^r\right)^{1/r}.
\]
The variation \(V_r(F,G)(x,y)\) is the largest value obtainable over **all such lists**. Each point may use its own list, with arbitrarily many cutoffs—even within a narrow range of scales. Thus this measures more than the size of a single truncated integral: it detects repeated changes as distance bands are added.",,"All distance-band partitions and pointwise choice are retained. Rational endpoints are not repeated, but source continuity expressly makes that choice equivalent; positivity is carried by distance cutoffs.",https://raw.githubusercontent.com/openai/math/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/preprints/Annular-variation-of-the-triangular-Hilbert-transform-at-the-symmetric-point-October-5-2026/build/sections/introduction.tex,thm:variation,"preprints/Annular-variation-of-the-triangular-Hilbert-transform-at-the-symmetric-point-October-5-2026/build/sections/introduction.tex lines 1-26 (context):
\section{Introduction}\label{sec:introduction}

For complex functions $F,G\in L^3(\R^2)$ and
$0<\varepsilon<R<\infty$, define the annular triangular Hilbert transform by
\begin{equation}\label{eq:bilinear-truncation}
 B_{\varepsilon,R}(F,G)(x,y)
 =\int_{\varepsilon<|t|<R}F(x+t,y)G(x,y+t)\,\frac{\dd t}{t}.
\end{equation}
The two inputs are translated along different coordinate directions.
Every finite integral is absolutely convergent outside one common
null set; we use its usual representative there and set all quantities
to zero on the exceptional set. Lemma~\ref{lem:finite-truncations}
justifies this convention and continuity in the endpoints.

For $r>2$, its annular variation is
\begin{equation}\label{eq:variation-definition}
 V_r(F,G)(x,y)
 =\sup_{\substack{J\ge1\,,\ 0<t_0<\cdots<t_J\\t_j\in\mathbb Q}}
    \left(\sum_{j=1}^J
          |B_{t_{j-1},t_j}(F,G)(x,y)|^r\right)^{1/r}.
\end{equation}
The supremum is pointwise: the partition may depend on $(x,y)$, and
there is no restriction on the number of endpoints within a dyadic
scale interval. The use of rational endpoints makes measurability
immediate and, by endpoint continuity, does not change the supremum.

"
T01,T01.07,technical_convention,not_explicit,E: Finite integrals use a common null-set convention; quantities are set to zero there. Anchor C: “one common null set”.,"The conclusion is
\[
\|V_r(F,G)\|_{3/2}\le C_r\|F\|_3\|G\|_3.
\]
Here \(\|F\|_3=(\int_{\mathbb R^2}|F|^3)^{1/3}\), and the other norms have the corresponding meaning. If both inputs have finite \(L^3\) size, their variation has finite \(L^{3/2}\) size, bounded by the product of the input sizes. In particular, the variation is finite at almost every point.",Common-null-set representative and zero assignment,"Almost-everywhere finiteness is stated, but the common exceptional set and assignment of zero on it are not. The omitted representative convention does not contradict the norm claim.",https://raw.githubusercontent.com/openai/math/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/preprints/Annular-variation-of-the-triangular-Hilbert-transform-at-the-symmetric-point-October-5-2026/build/sections/introduction.tex,thm:variation,"preprints/Annular-variation-of-the-triangular-Hilbert-transform-at-the-symmetric-point-October-5-2026/build/sections/introduction.tex lines 1-26 (context):
\section{Introduction}\label{sec:introduction}

For complex functions $F,G\in L^3(\R^2)$ and
$0<\varepsilon<R<\infty$, define the annular triangular Hilbert transform by
\begin{equation}\label{eq:bilinear-truncation}
 B_{\varepsilon,R}(F,G)(x,y)
 =\int_{\varepsilon<|t|<R}F(x+t,y)G(x,y+t)\,\frac{\dd t}{t}.
\end{equation}
The two inputs are translated along different coordinate directions.
Every finite integral is absolutely convergent outside one common
null set; we use its usual representative there and set all quantities
to zero on the exceptional set. Lemma~\ref{lem:finite-truncations}
justifies this convention and continuity in the endpoints.

For $r>2$, its annular variation is
\begin{equation}\label{eq:variation-definition}
 V_r(F,G)(x,y)
 =\sup_{\substack{J\ge1\,,\ 0<t_0<\cdots<t_J\\t_j\in\mathbb Q}}
    \left(\sum_{j=1}^J
          |B_{t_{j-1},t_j}(F,G)(x,y)|^r\right)^{1/r}.
\end{equation}
The supremum is pointwise: the partition may depend on $(x,y)$, and
there is no restriction on the number of endpoints within a dyadic
scale interval. The use of rational endpoints makes measurability
immediate and, by endpoint continuity, does not change the supremum.

"
T01,T01.08,scope_exception,retained,E: There is no restriction on the number of endpoints within a dyadic scale interval. Anchor C: “no restriction”.,"To measure fluctuation, choose any increasing list of cutoffs \(t_0<\cdots<t_J\). Compute the integral on each successive distance band, take its absolute value, and combine these values as
\[
\left(\sum_{j=1}^J |B_{t_{j-1},t_j}(F,G)(x,y)|^r\right)^{1/r}.
\]
The variation \(V_r(F,G)(x,y)\) is the largest value obtainable over **all such lists**. Each point may use its own list, with arbitrarily many cutoffs—even within a narrow range of scales. Thus this measures more than the size of a single truncated integral: it detects repeated changes as distance bands are added.",,Unlimited cutoffs within a scale range preserve the absence of a dyadic-count restriction.,https://raw.githubusercontent.com/openai/math/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/preprints/Annular-variation-of-the-triangular-Hilbert-transform-at-the-symmetric-point-October-5-2026/build/sections/introduction.tex,thm:variation,"preprints/Annular-variation-of-the-triangular-Hilbert-transform-at-the-symmetric-point-October-5-2026/build/sections/introduction.tex lines 1-26 (context):
\section{Introduction}\label{sec:introduction}

For complex functions $F,G\in L^3(\R^2)$ and
$0<\varepsilon<R<\infty$, define the annular triangular Hilbert transform by
\begin{equation}\label{eq:bilinear-truncation}
 B_{\varepsilon,R}(F,G)(x,y)
 =\int_{\varepsilon<|t|<R}F(x+t,y)G(x,y+t)\,\frac{\dd t}{t}.
\end{equation}
The two inputs are translated along different coordinate directions.
Every finite integral is absolutely convergent outside one common
null set; we use its usual representative there and set all quantities
to zero on the exceptional set. Lemma~\ref{lem:finite-truncations}
justifies this convention and continuity in the endpoints.

For $r>2$, its annular variation is
\begin{equation}\label{eq:variation-definition}
 V_r(F,G)(x,y)
 =\sup_{\substack{J\ge1\,,\ 0<t_0<\cdots<t_J\\t_j\in\mathbb Q}}
    \left(\sum_{j=1}^J
          |B_{t_{j-1},t_j}(F,G)(x,y)|^r\right)^{1/r}.
\end{equation}
The supremum is pointwise: the partition may depend on $(x,y)$, and
there is no restriction on the number of endpoints within a dyadic
scale interval. The use of rational endpoints makes measurability
immediate and, by endpoint continuity, does not change the supremum.

"
T02,T02.01,assumption_domain,retained,"H/D: G is finite, simple, and undirected. Anchor C: first sentence.","The theorem says that **one randomized algorithm can efficiently estimate the number of perfect matchings in any finite simple undirected graph**. A perfect matching pairs up all vertices using edges of the graph, with every vertex belonging to exactly one pair.",,All three graph restrictions are explicit.,https://raw.githubusercontent.com/openai/math/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/preprints/A-Fully-Polynomial-Randomized-Approximation-Scheme-for-Perfect-Matchings-in-General-Graphs-September-23-2026/build/main.tex,thm:main,"preprints/A-Fully-Polynomial-Randomized-Approximation-Scheme-for-Perfect-Matchings-in-General-Graphs-September-23-2026/build/main.tex lines 121-131 (context):
Let $G=(V,E)$ be a finite simple undirected graph. A perfect matching is a
set of edges incident to each vertex exactly once. Write
\[
 Z(G)=\#\{M\subseteq E:M\text{ is a perfect matching of }G\}.
\]
A fully polynomial randomized approximation scheme approximates $Z(G)$
within a prescribed relative error, with a prescribed success probability,
in time polynomial in the input length and the reciprocal error.
The confidence parameter enters through its logarithm. We prove the
following theorem, including a bound on the time of every execution.

"
T02,T02.02,definition,retained,"R: Z(G) counts perfect matchings, meaning each vertex is incident to exactly one selected edge. Anchor C: definition of Z(G).","The theorem says that **one randomized algorithm can efficiently estimate the number of perfect matchings in any finite simple undirected graph**. A perfect matching pairs up all vertices using edges of the graph, with every vertex belonging to exactly one pair.",,Perfect matching and the target count are correctly identified.,https://raw.githubusercontent.com/openai/math/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/preprints/A-Fully-Polynomial-Randomized-Approximation-Scheme-for-Perfect-Matchings-in-General-Graphs-September-23-2026/build/main.tex,thm:main,"preprints/A-Fully-Polynomial-Randomized-Approximation-Scheme-for-Perfect-Matchings-in-General-Graphs-September-23-2026/build/main.tex lines 121-131 (context):
Let $G=(V,E)$ be a finite simple undirected graph. A perfect matching is a
set of edges incident to each vertex exactly once. Write
\[
 Z(G)=\#\{M\subseteq E:M\text{ is a perfect matching of }G\}.
\]
A fully polynomial randomized approximation scheme approximates $Z(G)$
within a prescribed relative error, with a prescribed success probability,
in time polynomial in the input length and the reciprocal error.
The confidence parameter enters through its logarithm. We prove the
following theorem, including a bound on the time of every execution.

"
T02,T02.03,assumption_domain,not_explicit,H/D: Epsilon and delta are rational with 0<epsilon<1 and 0<delta<1/2. Anchor S: “rational parameters”.,You choose two parameters:,Rational parameters; 0<epsilon<1; 0<delta<1/2,"The parameter roles and admissible numeric example are retained, but rationality and the full stated epsilon/delta ranges are not. There is no explicit claim about invalid parameter choices.",https://raw.githubusercontent.com/openai/math/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/preprints/A-Fully-Polynomial-Randomized-Approximation-Scheme-for-Perfect-Matchings-in-General-Graphs-September-23-2026/build/main.tex,thm:main,"preprints/A-Fully-Polynomial-Randomized-Approximation-Scheme-for-Perfect-Matchings-in-General-Graphs-September-23-2026/build/main.tex lines 132-143 (statement):
\begin{theorem}\label{thm:main}
There is a uniform classical randomized algorithm which, given $G$ and
rational parameters $0<\eps<1$ and $0<\delta<1/2$, returns a nonnegative
rational number $\widehat Z$ such that
\[
 \Prb\bigl[(1-\eps)Z(G)\le \widehat Z\le(1+\eps)Z(G)\bigr]
 \ge 1-\delta.
\]
If $Z(G)=0$, the output is zero with certainty. The worst-case bit running
time is polynomial in the input encoding length, $\eps^{-1}$, and
$\log\delta^{-1}$.
\end{theorem}"
T02,T02.04,quantifier,retained,Q: A single uniform classical randomized algorithm accepts these inputs. Anchor S: “There is a uniform classical randomized algorithm”.,"Finally, “uniform classical” means this is a single ordinary, nonquantum algorithm that handles all valid inputs, rather than a separate procedure tailored to each graph size.",,One uniform classical randomized algorithm is correctly explained.,https://raw.githubusercontent.com/openai/math/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/preprints/A-Fully-Polynomial-Randomized-Approximation-Scheme-for-Perfect-Matchings-in-General-Graphs-September-23-2026/build/main.tex,thm:main,"preprints/A-Fully-Polynomial-Randomized-Approximation-Scheme-for-Perfect-Matchings-in-General-Graphs-September-23-2026/build/main.tex lines 132-143 (statement):
\begin{theorem}\label{thm:main}
There is a uniform classical randomized algorithm which, given $G$ and
rational parameters $0<\eps<1$ and $0<\delta<1/2$, returns a nonnegative
rational number $\widehat Z$ such that
\[
 \Prb\bigl[(1-\eps)Z(G)\le \widehat Z\le(1+\eps)Z(G)\bigr]
 \ge 1-\delta.
\]
If $Z(G)=0$, the output is zero with certainty. The worst-case bit running
time is polynomial in the input encoding length, $\eps^{-1}$, and
$\log\delta^{-1}$.
\end{theorem}"
T02,T02.05,conclusion,not_explicit,"R: Output is a nonnegative rational estimate with multiplicative relative error at most epsilon, with probability at least 1−delta. Anchor S: displayed event.","The algorithm returns an estimate \(\widehat Z\) that, with probability at least \(1-\delta\), lies between \((1-\varepsilon)Z(G)\) and \((1+\varepsilon)Z(G)\). For example, choosing \(\varepsilon=0.05\) and \(\delta=0.01\) gives an estimate within 5% of the true count with probability at least 99%. The probability is over the algorithm’s internal random choices.",Nonnegative rational output on every execution,The multiplicative guarantee and internal-randomness interpretation are retained; output nonnegativity and rational representation are not explicitly guaranteed for all executions.,https://raw.githubusercontent.com/openai/math/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/preprints/A-Fully-Polynomial-Randomized-Approximation-Scheme-for-Perfect-Matchings-in-General-Graphs-September-23-2026/build/main.tex,thm:main,"preprints/A-Fully-Polynomial-Randomized-Approximation-Scheme-for-Perfect-Matchings-in-General-Graphs-September-23-2026/build/main.tex lines 132-143 (statement):
\begin{theorem}\label{thm:main}
There is a uniform classical randomized algorithm which, given $G$ and
rational parameters $0<\eps<1$ and $0<\delta<1/2$, returns a nonnegative
rational number $\widehat Z$ such that
\[
 \Prb\bigl[(1-\eps)Z(G)\le \widehat Z\le(1+\eps)Z(G)\bigr]
 \ge 1-\delta.
\]
If $Z(G)=0$, the output is zero with certainty. The worst-case bit running
time is polynomial in the input encoding length, $\eps^{-1}$, and
$\log\delta^{-1}$.
\end{theorem}"
T02,T02.06,scope_exception,retained,"E: If Z(G)=0, output is zero with certainty, not merely high probability. Anchor S: “zero with certainty”.","If the graph has **no perfect matching**, the algorithm always returns zero.",,"The no-matching case is certain, not merely high probability.",https://raw.githubusercontent.com/openai/math/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/preprints/A-Fully-Polynomial-Randomized-Approximation-Scheme-for-Perfect-Matchings-in-General-Graphs-September-23-2026/build/main.tex,thm:main,"preprints/A-Fully-Polynomial-Randomized-Approximation-Scheme-for-Perfect-Matchings-in-General-Graphs-September-23-2026/build/main.tex lines 132-143 (statement):
\begin{theorem}\label{thm:main}
There is a uniform classical randomized algorithm which, given $G$ and
rational parameters $0<\eps<1$ and $0<\delta<1/2$, returns a nonnegative
rational number $\widehat Z$ such that
\[
 \Prb\bigl[(1-\eps)Z(G)\le \widehat Z\le(1+\eps)Z(G)\bigr]
 \ge 1-\delta.
\]
If $Z(G)=0$, the output is zero with certainty. The worst-case bit running
time is polynomial in the input encoding length, $\eps^{-1}$, and
$\log\delta^{-1}$.
\end{theorem}"
T02,T02.07,conclusion,not_explicit,"R/Q: Worst-case bit runtime is polynomial in input encoding length, epsilon inverse, and log(delta inverse). Anchor S: final sentence.","“Efficiently” has a precise meaning here: the running time is bounded by a polynomial in the encoded input size, \(1/\varepsilon\), and \(\log(1/\delta)\). Thus, requiring greater accuracy costs time, while reducing the failure probability enters the bound only logarithmically. Crucially, this time bound holds for **every execution**, even one whose estimate is inaccurate—not merely on average.",Bit-operation runtime model,"Worst-case versus expected time and the correct polynomial arguments are retained. Encoded input size suggests a bit model, but bit operations are not explicitly stated.",https://raw.githubusercontent.com/openai/math/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/preprints/A-Fully-Polynomial-Randomized-Approximation-Scheme-for-Perfect-Matchings-in-General-Graphs-September-23-2026/build/main.tex,thm:main,"preprints/A-Fully-Polynomial-Randomized-Approximation-Scheme-for-Perfect-Matchings-in-General-Graphs-September-23-2026/build/main.tex lines 132-143 (statement):
\begin{theorem}\label{thm:main}
There is a uniform classical randomized algorithm which, given $G$ and
rational parameters $0<\eps<1$ and $0<\delta<1/2$, returns a nonnegative
rational number $\widehat Z$ such that
\[
 \Prb\bigl[(1-\eps)Z(G)\le \widehat Z\le(1+\eps)Z(G)\bigr]
 \ge 1-\delta.
\]
If $Z(G)=0$, the output is zero with certainty. The worst-case bit running
time is polynomial in the input encoding length, $\eps^{-1}$, and
$\log\delta^{-1}$.
\end{theorem}"
T03,T03.01,assumption_domain,retained,H/D: Tables have nonnegative integer entries and prescribed nonnegative integer row/column margins with equal total N. Anchor C: definition of Omega; S: opening sentence.,"Here, \(\Omega(r,c)\) is the collection of all allowed matrices: row \(i\) must sum to \(r_i\), and column \(j\) must sum to \(c_j\). The two lists of sums have the same total \(N\). “Uniform” means that every distinct matrix in this collection has the same probability of being chosen.",,"Nonnegative integer matrices, prescribed row/column sums, equal totals and allowed zero margins preserve the input domain. Integer nonnegative margins follow from their role as sums of these matrices.",https://raw.githubusercontent.com/openai/math/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/preprints/Exact-Uniform-Sampling-of-Contingency-Tables-with-Arbitrary-Margins-September-24-2026/build/sections/introduction.tex,thm:main,"preprints/Exact-Uniform-Sampling-of-Contingency-Tables-with-Arbitrary-Margins-September-24-2026/build/sections/introduction.tex lines 1-17 (context):
\section{Introduction}\label{sec:introduction}

For nonnegative integer vectors $r=(r_1,\ldots,r_m)$ and
$c=(c_1,\ldots,c_n)$ with common total $N$, write
\[
 \Omega(r,c)=\left\{X\in\Z_{\ge0}^{m\times n}:
       \sum_jX_{ij}=r_i,\quad \sum_iX_{ij}=c_j\right\}.
\]
The uniform distribution on this finite set gives every table equal mass.
We study sampling when both dimensions vary and the margins are encoded
in binary. In particular, a polynomial bound in the numeric total $N$
would not, in general, give a polynomial bound in the input length.
There are no cell bounds or forbidden positions: every nonnegative integer
matrix with the prescribed margins is allowed. For probability laws on a
finite set, we use the convention
$\TV(\mu,\nu)=\tfrac12\sum_x|\mu(x)-\nu(x)|$.



preprints/Exact-Uniform-Sampling-of-Contingency-Tables-with-Arbitrary-Margins-September-24-2026/build/sections/introduction.tex lines 18-32 (statement):
\begin{theorem}\label{thm:main}
For arbitrary nonnegative integer margins of equal total, the following
algorithms exist in the model of unbiased random bits and bit operations.
\begin{enumerate}[label=\textup{(\roman*)}]
 \item Given an integer $k\ge1$, a bounded-time algorithm outputs a table
 in $\Omega(r,c)$ whose law is within $2^{-k}$ in total variation of the
 uniform law. Its running time is polynomial in
 $m,n,\log(N+1),k$.
 \item An algorithm outputs an exactly uniform table in $\Omega(r,c)$,
 terminates almost surely, and has expected running time polynomial in
 $m,n,\log(N+1)$.
\end{enumerate}
The polynomials are uniform over all margin vectors. No positivity,
sparsity, balance, or fixed-dimension hypothesis is imposed.
\end{theorem}"
T03,T03.02,assumption_domain,retained,D: All cells are allowed; there are no additional cell bounds or forbidden positions. Anchor C: “There are no cell bounds or forbidden positions”.,"These guarantees apply to arbitrary margins, including zeros and highly unequal sums, with both matrix dimensions allowed to grow. The same polynomial bounds work across all such inputs. Every cell is unrestricted apart from nonnegativity and the prescribed sums. The algorithms are explicit, but their polynomial exponents are large, so the theorem establishes theoretical efficiency without promising practical speed.",,No forbidden positions or extra cell bounds are introduced.,https://raw.githubusercontent.com/openai/math/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/preprints/Exact-Uniform-Sampling-of-Contingency-Tables-with-Arbitrary-Margins-September-24-2026/build/sections/introduction.tex,thm:main,"preprints/Exact-Uniform-Sampling-of-Contingency-Tables-with-Arbitrary-Margins-September-24-2026/build/sections/introduction.tex lines 1-17 (context):
\section{Introduction}\label{sec:introduction}

For nonnegative integer vectors $r=(r_1,\ldots,r_m)$ and
$c=(c_1,\ldots,c_n)$ with common total $N$, write
\[
 \Omega(r,c)=\left\{X\in\Z_{\ge0}^{m\times n}:
       \sum_jX_{ij}=r_i,\quad \sum_iX_{ij}=c_j\right\}.
\]
The uniform distribution on this finite set gives every table equal mass.
We study sampling when both dimensions vary and the margins are encoded
in binary. In particular, a polynomial bound in the numeric total $N$
would not, in general, give a polynomial bound in the input length.
There are no cell bounds or forbidden positions: every nonnegative integer
matrix with the prescribed margins is allowed. For probability laws on a
finite set, we use the convention
$\TV(\mu,\nu)=\tfrac12\sum_x|\mu(x)-\nu(x)|$.

"
T03,T03.03,assumption_domain,retained,"D/Q: Both dimensions m,n vary and margins are binary encoded. Anchor C: “both dimensions vary” and “encoded in binary”.","These guarantees apply to arbitrary margins, including zeros and highly unequal sums, with both matrix dimensions allowed to grow. The same polynomial bounds work across all such inputs. Every cell is unrestricted apart from nonnegativity and the prescribed sums. The algorithms are explicit, but their polynomial exponents are large, so the theorem establishes theoretical efficiency without promising practical speed.",,Both varying dimensions and binary input encoding are explicit.,https://raw.githubusercontent.com/openai/math/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/preprints/Exact-Uniform-Sampling-of-Contingency-Tables-with-Arbitrary-Margins-September-24-2026/build/sections/introduction.tex,thm:main,"preprints/Exact-Uniform-Sampling-of-Contingency-Tables-with-Arbitrary-Margins-September-24-2026/build/sections/introduction.tex lines 1-17 (context):
\section{Introduction}\label{sec:introduction}

For nonnegative integer vectors $r=(r_1,\ldots,r_m)$ and
$c=(c_1,\ldots,c_n)$ with common total $N$, write
\[
 \Omega(r,c)=\left\{X\in\Z_{\ge0}^{m\times n}:
       \sum_jX_{ij}=r_i,\quad \sum_iX_{ij}=c_j\right\}.
\]
The uniform distribution on this finite set gives every table equal mass.
We study sampling when both dimensions vary and the margins are encoded
in binary. In particular, a polynomial bound in the numeric total $N$
would not, in general, give a polynomial bound in the input length.
There are no cell bounds or forbidden positions: every nonnegative integer
matrix with the prescribed margins is allowed. For probability laws on a
finite set, we use the convention
$\TV(\mu,\nu)=\tfrac12\sum_x|\mu(x)-\nu(x)|$.

"
T03,T03.04,definition,retained,R: Uniform means equal mass for every feasible table. Anchor C: “every table equal mass”.,"Here, \(\Omega(r,c)\) is the collection of all allowed matrices: row \(i\) must sum to \(r_i\), and column \(j\) must sum to \(c_j\). The two lists of sums have the same total \(N\). “Uniform” means that every distinct matrix in this collection has the same probability of being chosen.",,Uniform law is preserved.,https://raw.githubusercontent.com/openai/math/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/preprints/Exact-Uniform-Sampling-of-Contingency-Tables-with-Arbitrary-Margins-September-24-2026/build/sections/introduction.tex,thm:main,"preprints/Exact-Uniform-Sampling-of-Contingency-Tables-with-Arbitrary-Margins-September-24-2026/build/sections/introduction.tex lines 1-17 (context):
\section{Introduction}\label{sec:introduction}

For nonnegative integer vectors $r=(r_1,\ldots,r_m)$ and
$c=(c_1,\ldots,c_n)$ with common total $N$, write
\[
 \Omega(r,c)=\left\{X\in\Z_{\ge0}^{m\times n}:
       \sum_jX_{ij}=r_i,\quad \sum_iX_{ij}=c_j\right\}.
\]
The uniform distribution on this finite set gives every table equal mass.
We study sampling when both dimensions vary and the margins are encoded
in binary. In particular, a polynomial bound in the numeric total $N$
would not, in general, give a polynomial bound in the input length.
There are no cell bounds or forbidden positions: every nonnegative integer
matrix with the prescribed margins is allowed. For probability laws on a
finite set, we use the convention
$\TV(\mu,\nu)=\tfrac12\sum_x|\mu(x)-\nu(x)|$.

"
T03,T03.05,conclusion_quantifier,retained,"H/R: For each integer k≥1, part (i) returns a feasible table within 2^(−k) total variation of uniform. Anchor S: part (i); C fixes TV as half the sum of absolute probability differences.","1. **Approximate sampling with a guaranteed runtime.** Choose any integer \(k\ge1\). The algorithm always produces a valid matrix, and its distribution differs from uniform by at most \(2^{-k}\) in total variation. Concretely, for any collection of matrices, its probability of being selected differs from the truly uniform probability by at most \(2^{-k}\). The runtime is bounded by a polynomial in \(m,n,\log(N+1)\), and \(k\). Thus, requesting error at most \(\varepsilon\) costs only a polynomial in \(\log(1/\varepsilon)\).",,"The k>=1 domain, feasibility and 2^-k TV bound are retained. Event-probability difference is a legitimate equivalent characterization of the source TV convention.",https://raw.githubusercontent.com/openai/math/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/preprints/Exact-Uniform-Sampling-of-Contingency-Tables-with-Arbitrary-Margins-September-24-2026/build/sections/introduction.tex,thm:main,"preprints/Exact-Uniform-Sampling-of-Contingency-Tables-with-Arbitrary-Margins-September-24-2026/build/sections/introduction.tex lines 1-17 (context):
\section{Introduction}\label{sec:introduction}

For nonnegative integer vectors $r=(r_1,\ldots,r_m)$ and
$c=(c_1,\ldots,c_n)$ with common total $N$, write
\[
 \Omega(r,c)=\left\{X\in\Z_{\ge0}^{m\times n}:
       \sum_jX_{ij}=r_i,\quad \sum_iX_{ij}=c_j\right\}.
\]
The uniform distribution on this finite set gives every table equal mass.
We study sampling when both dimensions vary and the margins are encoded
in binary. In particular, a polynomial bound in the numeric total $N$
would not, in general, give a polynomial bound in the input length.
There are no cell bounds or forbidden positions: every nonnegative integer
matrix with the prescribed margins is allowed. For probability laws on a
finite set, we use the convention
$\TV(\mu,\nu)=\tfrac12\sum_x|\mu(x)-\nu(x)|$.



preprints/Exact-Uniform-Sampling-of-Contingency-Tables-with-Arbitrary-Margins-September-24-2026/build/sections/introduction.tex lines 18-32 (statement):
\begin{theorem}\label{thm:main}
For arbitrary nonnegative integer margins of equal total, the following
algorithms exist in the model of unbiased random bits and bit operations.
\begin{enumerate}[label=\textup{(\roman*)}]
 \item Given an integer $k\ge1$, a bounded-time algorithm outputs a table
 in $\Omega(r,c)$ whose law is within $2^{-k}$ in total variation of the
 uniform law. Its running time is polynomial in
 $m,n,\log(N+1),k$.
 \item An algorithm outputs an exactly uniform table in $\Omega(r,c)$,
 terminates almost surely, and has expected running time polynomial in
 $m,n,\log(N+1)$.
\end{enumerate}
The polynomials are uniform over all margin vectors. No positivity,
sparsity, balance, or fixed-dimension hypothesis is imposed.
\end{theorem}"
T03,T03.06,conclusion,retained,"R: Part (i) has bounded runtime polynomial in m,n,log(N+1),k. Anchor S: part (i).","1. **Approximate sampling with a guaranteed runtime.** Choose any integer \(k\ge1\). The algorithm always produces a valid matrix, and its distribution differs from uniform by at most \(2^{-k}\) in total variation. Concretely, for any collection of matrices, its probability of being selected differs from the truly uniform probability by at most \(2^{-k}\). The runtime is bounded by a polynomial in \(m,n,\log(N+1)\), and \(k\). Thus, requesting error at most \(\varepsilon\) costs only a polynomial in \(\log(1/\varepsilon)\).",,Part (i) retains bounded time and all four complexity parameters.,https://raw.githubusercontent.com/openai/math/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/preprints/Exact-Uniform-Sampling-of-Contingency-Tables-with-Arbitrary-Margins-September-24-2026/build/sections/introduction.tex,thm:main,"preprints/Exact-Uniform-Sampling-of-Contingency-Tables-with-Arbitrary-Margins-September-24-2026/build/sections/introduction.tex lines 18-32 (statement):
\begin{theorem}\label{thm:main}
For arbitrary nonnegative integer margins of equal total, the following
algorithms exist in the model of unbiased random bits and bit operations.
\begin{enumerate}[label=\textup{(\roman*)}]
 \item Given an integer $k\ge1$, a bounded-time algorithm outputs a table
 in $\Omega(r,c)$ whose law is within $2^{-k}$ in total variation of the
 uniform law. Its running time is polynomial in
 $m,n,\log(N+1),k$.
 \item An algorithm outputs an exactly uniform table in $\Omega(r,c)$,
 terminates almost surely, and has expected running time polynomial in
 $m,n,\log(N+1)$.
\end{enumerate}
The polynomials are uniform over all margin vectors. No positivity,
sparsity, balance, or fixed-dimension hypothesis is imposed.
\end{theorem}"
T03,T03.07,conclusion_exception,retained,"R/E: Part (ii) is exactly uniform, terminates almost surely, and has expected runtime polynomial in m,n,log(N+1). Anchor S: part (ii).","2. **Exact sampling with an expected-runtime guarantee.** A second algorithm gives every valid matrix exactly the same probability. It finishes with probability one, and its average runtime is polynomial in \(m,n,\log(N+1)\). Individual runs can take much longer: an exceptionally rare branch uses exhaustive computation.",,"Exact uniformity, almost-sure termination and expected polynomial time are explicitly distinguished from a worst-case guarantee.",https://raw.githubusercontent.com/openai/math/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/preprints/Exact-Uniform-Sampling-of-Contingency-Tables-with-Arbitrary-Margins-September-24-2026/build/sections/introduction.tex,thm:main,"preprints/Exact-Uniform-Sampling-of-Contingency-Tables-with-Arbitrary-Margins-September-24-2026/build/sections/introduction.tex lines 18-32 (statement):
\begin{theorem}\label{thm:main}
For arbitrary nonnegative integer margins of equal total, the following
algorithms exist in the model of unbiased random bits and bit operations.
\begin{enumerate}[label=\textup{(\roman*)}]
 \item Given an integer $k\ge1$, a bounded-time algorithm outputs a table
 in $\Omega(r,c)$ whose law is within $2^{-k}$ in total variation of the
 uniform law. Its running time is polynomial in
 $m,n,\log(N+1),k$.
 \item An algorithm outputs an exactly uniform table in $\Omega(r,c)$,
 terminates almost surely, and has expected running time polynomial in
 $m,n,\log(N+1)$.
\end{enumerate}
The polynomials are uniform over all margin vectors. No positivity,
sparsity, balance, or fixed-dimension hypothesis is imposed.
\end{theorem}"
T03,T03.08,computational_model,not_explicit,D: Computation uses unbiased random bits and bit operations. Anchor S: “model of unbiased random bits and bit operations”.,"The dependence on **\(\log(N+1)\), rather than \(N\)**, matters because the sums are written in binary. A very large sum takes relatively few bits to specify; a runtime polynomial in \(N\) could therefore be enormous compared with the input length.",Unbiased random-bit and bit-operation model,"Binary input encoding is discussed, but unbiased random bits and bit operations as the computation model are not explicitly stated. No alternative oracle model is claimed.",https://raw.githubusercontent.com/openai/math/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/preprints/Exact-Uniform-Sampling-of-Contingency-Tables-with-Arbitrary-Margins-September-24-2026/build/sections/introduction.tex,thm:main,"preprints/Exact-Uniform-Sampling-of-Contingency-Tables-with-Arbitrary-Margins-September-24-2026/build/sections/introduction.tex lines 18-32 (statement):
\begin{theorem}\label{thm:main}
For arbitrary nonnegative integer margins of equal total, the following
algorithms exist in the model of unbiased random bits and bit operations.
\begin{enumerate}[label=\textup{(\roman*)}]
 \item Given an integer $k\ge1$, a bounded-time algorithm outputs a table
 in $\Omega(r,c)$ whose law is within $2^{-k}$ in total variation of the
 uniform law. Its running time is polynomial in
 $m,n,\log(N+1),k$.
 \item An algorithm outputs an exactly uniform table in $\Omega(r,c)$,
 terminates almost surely, and has expected running time polynomial in
 $m,n,\log(N+1)$.
\end{enumerate}
The polynomials are uniform over all margin vectors. No positivity,
sparsity, balance, or fixed-dimension hypothesis is imposed.
\end{theorem}"
T03,T03.09,quantifier_exception,retained,"Q/E: Polynomials are uniform over margins; no positivity, sparsity, balance, or fixed-dimension assumption is imposed. Anchor S: final two sentences.","These guarantees apply to arbitrary margins, including zeros and highly unequal sums, with both matrix dimensions allowed to grow. The same polynomial bounds work across all such inputs. Every cell is unrestricted apart from nonnegativity and the prescribed sums. The algorithms are explicit, but their polynomial exponents are large, so the theorem establishes theoretical efficiency without promising practical speed.",,"Arbitrary margins, zeros, imbalance and growing dimensions preserve the unrestricted family and uniform polynomial bounds.",https://raw.githubusercontent.com/openai/math/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/preprints/Exact-Uniform-Sampling-of-Contingency-Tables-with-Arbitrary-Margins-September-24-2026/build/sections/introduction.tex,thm:main,"preprints/Exact-Uniform-Sampling-of-Contingency-Tables-with-Arbitrary-Margins-September-24-2026/build/sections/introduction.tex lines 18-32 (statement):
\begin{theorem}\label{thm:main}
For arbitrary nonnegative integer margins of equal total, the following
algorithms exist in the model of unbiased random bits and bit operations.
\begin{enumerate}[label=\textup{(\roman*)}]
 \item Given an integer $k\ge1$, a bounded-time algorithm outputs a table
 in $\Omega(r,c)$ whose law is within $2^{-k}$ in total variation of the
 uniform law. Its running time is polynomial in
 $m,n,\log(N+1),k$.
 \item An algorithm outputs an exactly uniform table in $\Omega(r,c)$,
 terminates almost surely, and has expected running time polynomial in
 $m,n,\log(N+1)$.
\end{enumerate}
The polynomials are uniform over all margin vectors. No positivity,
sparsity, balance, or fixed-dimension hypothesis is imposed.
\end{theorem}"
T04,T04.01,assumption_domain,not_explicit,H/D: Integers n≥1 and 1≤d≤n. Anchor S: opening sentence.,"The bound applies to every such polynomial, with arbitrary real coefficients—even if its score is exactly zero at some inputs, which receive label \(+1\). The same constant \(8\) works for all \(n\) and \(d\). Since at most \(n\) switches can change a label, the bound improves on the automatic bound \(I(f)\le n\) when \(8d<\sqrt n\).",Explicit n>=1 and integer 1<=d<=n,"n is a count of switches and d a degree bound, but the complete declared range is absent. The sentence about all n,d is read in the supplied theorem context, not as a demonstrated claim on inadmissible inputs.",https://raw.githubusercontent.com/openai/math/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/preprints/Average-Sensitivity-of-Polynomial-Threshold-Functions-September-25-2026/build/sections/01-introduction.tex,thm:main,"preprints/Average-Sensitivity-of-Polynomial-Threshold-Functions-September-25-2026/build/sections/01-introduction.tex lines 20-28 (statement):
\begin{theorem}\label{thm:main}
Let $n\ge1$ and $1\le d\le n$ be integers. Let $p$ be a real multilinear
polynomial of degree at most $d$, and define
$f:\{-1,1\}^n\to\{-1,1\}$ by $f(x)=\sgn(p(x))$, with $\sgn(0)=1$.
Then, for average sensitivity under the uniform law,
\[
 I(f)\le 8d\sqrt n.
\]
\end{theorem}"
T04,T04.02,assumption_domain,retained,H/D: p is a real multilinear polynomial of degree at most d. Anchor S: second sentence.,"The bound applies to every such polynomial, with arbitrary real coefficients—even if its score is exactly zero at some inputs, which receive label \(+1\). The same constant \(8\) works for all \(n\) and \(d\). Since at most \(n\) switches can change a label, the bound improves on the automatic bound \(I(f)\le n\) when \(8d<\sqrt n\).",,"Real coefficients, degree at most d and a faithful explanation of multilinearity preserve this hypothesis.",https://raw.githubusercontent.com/openai/math/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/preprints/Average-Sensitivity-of-Polynomial-Threshold-Functions-September-25-2026/build/sections/01-introduction.tex,thm:main,"preprints/Average-Sensitivity-of-Polynomial-Threshold-Functions-September-25-2026/build/sections/01-introduction.tex lines 20-28 (statement):
\begin{theorem}\label{thm:main}
Let $n\ge1$ and $1\le d\le n$ be integers. Let $p$ be a real multilinear
polynomial of degree at most $d$, and define
$f:\{-1,1\}^n\to\{-1,1\}$ by $f(x)=\sgn(p(x))$, with $\sgn(0)=1$.
Then, for average sensitivity under the uniform law,
\[
 I(f)\le 8d\sqrt n.
\]
\end{theorem}"
T04,T04.03,definition,retained,D/R: f maps the ±1 Boolean cube to ±1 by the sign of p. Anchor S: definition of f.,"Imagine an input consisting of \(n\) switches, each set to either \(-1\) or \(+1\). A polynomial \(p\) gives the input a numerical score, and the function \(f\) labels it \(+1\) if the score is nonnegative and \(-1\) otherwise.",,"The Boolean cube, two labels and sign-of-polynomial rule are preserved.",https://raw.githubusercontent.com/openai/math/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/preprints/Average-Sensitivity-of-Polynomial-Threshold-Functions-September-25-2026/build/sections/01-introduction.tex,thm:main,"preprints/Average-Sensitivity-of-Polynomial-Threshold-Functions-September-25-2026/build/sections/01-introduction.tex lines 20-28 (statement):
\begin{theorem}\label{thm:main}
Let $n\ge1$ and $1\le d\le n$ be integers. Let $p$ be a real multilinear
polynomial of degree at most $d$, and define
$f:\{-1,1\}^n\to\{-1,1\}$ by $f(x)=\sgn(p(x))$, with $\sgn(0)=1$.
Then, for average sensitivity under the uniform law,
\[
 I(f)\le 8d\sqrt n.
\]
\end{theorem}"
T04,T04.04,scope_exception,retained,E: Zero is assigned sign +1; p is allowed to vanish on the cube. Anchor S: “sgn(0)=1”; N: “may vanish”.,"The bound applies to every such polynomial, with arbitrary real coefficients—even if its score is exactly zero at some inputs, which receive label \(+1\). The same constant \(8\) works for all \(n\) and \(d\). Since at most \(n\) switches can change a label, the bound improves on the automatic bound \(I(f)\le n\) when \(8d<\sqrt n\).",,Zeros are allowed and receive +1.,https://raw.githubusercontent.com/openai/math/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/preprints/Average-Sensitivity-of-Polynomial-Threshold-Functions-September-25-2026/build/sections/01-introduction.tex,thm:main,"preprints/Average-Sensitivity-of-Polynomial-Threshold-Functions-September-25-2026/build/sections/01-introduction.tex lines 20-28 (statement):
\begin{theorem}\label{thm:main}
Let $n\ge1$ and $1\le d\le n$ be integers. Let $p$ be a real multilinear
polynomial of degree at most $d$, and define
$f:\{-1,1\}^n\to\{-1,1\}$ by $f(x)=\sgn(p(x))$, with $\sgn(0)=1$.
Then, for average sensitivity under the uniform law,
\[
 I(f)\le 8d\sqrt n.
\]
\end{theorem}"
T04,T04.05,assumption_domain,retained,D: Average sensitivity is under the uniform distribution on the cube. Anchor S: “under the uniform law”; C: definition of I(f).,"Now choose an input uniformly at random. Check each switch separately: **how many switches would change the label if flipped on their own?** The expected number is the average sensitivity, \(I(f)\).",,The input distribution is explicit.,https://raw.githubusercontent.com/openai/math/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/preprints/Average-Sensitivity-of-Polynomial-Threshold-Functions-September-25-2026/build/sections/01-introduction.tex,thm:main,"preprints/Average-Sensitivity-of-Polynomial-Threshold-Functions-September-25-2026/build/sections/01-introduction.tex lines 20-28 (statement):
\begin{theorem}\label{thm:main}
Let $n\ge1$ and $1\le d\le n$ be integers. Let $p$ be a real multilinear
polynomial of degree at most $d$, and define
$f:\{-1,1\}^n\to\{-1,1\}$ by $f(x)=\sgn(p(x))$, with $\sgn(0)=1$.
Then, for average sensitivity under the uniform law,
\[
 I(f)\le 8d\sqrt n.
\]
\end{theorem}"
T04,T04.06,definition_conclusion,retained,R: I(f) is the sum of coordinate-flip disagreement probabilities and is at most 8d√n. Anchor C: eq:influence-definition; S: inequality.,"Now choose an input uniformly at random. Check each switch separately: **how many switches would change the label if flipped on their own?** The expected number is the average sensitivity, \(I(f)\).",,Counting individual coordinate flips in expectation is equivalent to summing their probabilities; the numerical bound is retained.,https://raw.githubusercontent.com/openai/math/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/preprints/Average-Sensitivity-of-Polynomial-Threshold-Functions-September-25-2026/build/sections/01-introduction.tex,thm:main,"preprints/Average-Sensitivity-of-Polynomial-Threshold-Functions-September-25-2026/build/sections/01-introduction.tex lines 1-19 (context):
\section{Introduction}\label{sec:introduction}

A polynomial threshold function assigns a sign to each vertex of the
Boolean cube by evaluating a real polynomial. On the cube, the identities
$x_i^2=1$ allow every polynomial to be replaced by a multilinear polynomial
of no larger degree without changing its values. Average sensitivity
measures the expected number of coordinate changes that reverse that sign.
For $x\in\{-1,1\}^n$, let $x^{\oplus i}$ be obtained by reversing
coordinate $i$. For $f:\{-1,1\}^n\to\{-1,1\}$, define
\begin{equation}\label{eq:influence-definition}
 I(f)=\sum_{i=1}^n\Prb\{f(X)\ne f(X^{\oplus i})\},
 \qquad X\text{ uniform on }\{-1,1\}^n.
\end{equation}
This quantity is also called the total influence of $f$. Throughout,
\[
 \sgn(t)=\begin{cases}1,&t\ge0,\\-1,&t<0.\end{cases}
\]
We prove the following estimate.



preprints/Average-Sensitivity-of-Polynomial-Threshold-Functions-September-25-2026/build/sections/01-introduction.tex lines 20-28 (statement):
\begin{theorem}\label{thm:main}
Let $n\ge1$ and $1\le d\le n$ be integers. Let $p$ be a real multilinear
polynomial of degree at most $d$, and define
$f:\{-1,1\}^n\to\{-1,1\}$ by $f(x)=\sgn(p(x))$, with $\sgn(0)=1$.
Then, for average sensitivity under the uniform law,
\[
 I(f)\le 8d\sqrt n.
\]
\end{theorem}"
T04,T04.07,quantifier_exception,retained,Q/E: The constant does not depend on d or n; d may grow with n; no regularity condition is imposed on p. Anchor N.,"The bound applies to every such polynomial, with arbitrary real coefficients—even if its score is exactly zero at some inputs, which receive label \(+1\). The same constant \(8\) works for all \(n\) and \(d\). Since at most \(n\) switches can change a label, the bound improves on the automatic bound \(I(f)\le n\) when \(8d<\sqrt n\).",,"Universal wording, arbitrary coefficients, allowed zeros and a common constant preserve lack of extra regularity and non-fixed degree. It need not repeat the phrase degree may grow.",https://raw.githubusercontent.com/openai/math/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/preprints/Average-Sensitivity-of-Polynomial-Threshold-Functions-September-25-2026/build/sections/01-introduction.tex,thm:main,"preprints/Average-Sensitivity-of-Polynomial-Threshold-Functions-September-25-2026/build/sections/01-introduction.tex lines 28-32 (scope_note):


The constant is independent of both $d$ and $n$. In particular, the degree
may grow with the dimension. The polynomial may vanish on the cube, and
no regularity condition is imposed on $p$."
T05,T05.01,assumption_domain,uncertain,"H/D: A finite, nonempty indexed family of affine maps on the real line, with real translations and 0<|r_i|<1. Anchor S and C: definition of Phi.","Start with finitely many maps that shrink the real line, possibly also reflecting and translating it. Choose maps independently using the probabilities \(p_i\). An infinite sequence of choices determines a limiting point, and \(\mu_{\Phi,p}\) describes the distribution of that point.",Affine form and explicit nonzero strict-contraction range,"In context, this can mean the supplied nondegenerate affine similarities; in isolation, shrinking maps could include nonlinear contractions or constant maps. The response later uses r_i and its logarithm but does not state the affine form or 0<|r_i|<1. Whether the abstraction preserves that class is left for expert review, not counted as a demonstrated enlargement.",https://raw.githubusercontent.com/openai/math/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/preprints/The-entropy-rate-dimension-formula-for-self-similar-measures-on-the-line-September-24-2026/build/sections/introduction.tex,thm:main,"preprints/The-entropy-rate-dimension-formula-for-self-similar-measures-on-the-line-September-24-2026/build/sections/introduction.tex lines 1-77 (context):
\section{Introduction}\label{sec:introduction}

Let $\Lambda$ be a finite nonempty alphabet, and let
\[
 \Phi=(\varphi_i)_{i\in\Lambda},\qquad
 \varphi_i(x)=r_i x+t_i,\qquad t_i\in\R,\quad 0<|r_i|<1.
\]
The family is indexed: different symbols may specify the same map.
Given a probability vector $p=(p_i)_{i\in\Lambda}$ with $p_i>0$, its
self-similar measure is the unique Borel probability measure satisfying
\[
 \mu=\sum_{i\in\Lambda}p_i(\varphi_i)_*\mu.
\]
Equivalently, $\mu$ is the law of
$\lim_{n\to\infty}\varphi_{I_1}\circ\cdots\circ\varphi_{I_n}(0)$,
where the symbols $I_j$ are independent with law $p$.
The limit exists uniformly in the address: if
$r_{\max}=\max_i|r_i|$ and $T_0=\max_i|t_i|$, all coding limits
have absolute value at most $T_0/(1-r_{\max})$.

For a word $w=i_1\cdots i_n$, write
$\varphi_w=\varphi_{i_1}\circ\cdots\circ\varphi_{i_n}$ and
$p_w=p_{i_1}\cdots p_{i_n}$.  An \emph{exact overlap} is an equality
$\varphi_u=\varphi_v$ for distinct words of the same positive length.
Equality here means equality of the complete affine maps, including
both the signed slope and the translation.  We allow all such overlaps.

The relevant entropy counts maps rather than addresses.  Set
\[
 G_n=\varphi_{I_1}\circ\cdots\circ\varphi_{I_n},\qquad
 \P(G_n=g)=\sum_{w\in\Lambda^n:\,\varphi_w=g}p_w.
\]
All logarithms below have base two.  For a finite-valued random variable
$Z$, its Shannon entropy is $H(Z)=-\sum_z\P(Z=z)\log\P(Z=z)$,
with $0\log0=0$.  Define the random-walk entropy rate and the Lyapunov
exponent by
\begin{equation}\label{eq:rate-and-lyapunov}
 h=h_{\mathrm{RW}}(\Phi,p)
   :=\lim_{n\to\infty}\frac{H(G_n)}n
    =\inf_{n\ge1}\frac{H(G_n)}n,
 \qquad
 \chi=-\sum_i p_i\log|r_i|>0.
\end{equation}
Indeed, a length-$(n+m)$ map is a function of two independent maps
with laws $G_n,G_m$, so $H(G_{n+m})\le H(G_n)+H(G_m)$; subadditivity
gives the displayed limit and infimum.  Also
$0\le h\le H(p):=-\sum_i p_i\log p_i$.  The ratio $h/\chi$ is
independent of the common logarithm base.

We use the lower Hausdorff dimension of a measure:
\[
 \dim_H\nu=\inf\{\dim_H E:E\subset\R\text{ Borel},\ \nu(E)>0\}.
\]
Feng--Hu's exact-dimensionality theorem
\cite[Theorem~2.8]{FengHu2009} applies to finite self-similar systems
without separation.  Indeed, the maps preserve a sufficiently large
compact interval, extend to contracting smooth diffeomorphisms, and have
derivative norm and least singular value both equal to $|r_i|$.
The system is therefore conformal, and its Bernoulli coding law is
ergodic.  These observations also cover negative ratios and repeated
indexed maps.  Thus, for the measure
$\mu$ above, there is a constant $d$ such that
\begin{equation}\label{eq:exact-dimensionality}
 \lim_{s\downarrow0}\frac{\log\mu(B(x,s))}{\log s}=d
 \quad\text{for $\mu$-almost every $x$}.
\end{equation}
For an exact-dimensional measure this constant equals both $\dim_H\mu$
and $\inf\{\dim_H E:\mu(E)=1\}$.
To see the equivalence, restrict to countably many sets on which the
local bounds $s^{d+\varepsilon}\le\mu(B(x,s))\le s^{d-\varepsilon}$
hold uniformly for all sufficiently small $s$.  The lower mass bound
gives a full-measure union of sets of dimension at most $d+\varepsilon$
by a covering argument; the upper mass bound gives dimension at least
$d-\varepsilon$ for every positive-mass set.  Let
$\varepsilon\downarrow0$.  Consequently either Hausdorff
measure-dimension convention gives the same statement below.



preprints/The-entropy-rate-dimension-formula-for-self-similar-measures-on-the-line-September-24-2026/build/sections/introduction.tex lines 78-88 (statement):
\begin{theorem}\label{thm:main}
For every finite nonempty indexed family
$\varphi_i(x)=r_i x+t_i$ on $\R$ with $0<|r_i|<1$, and every
strictly positive probability vector $p$, its self-similar measure satisfies
\[
 \dim_H\mu_{\Phi,p}
 =\min\left\{1,\frac{h_{\mathrm{RW}}(\Phi,p)}{\chi(\Phi,p)}\right\}.
\]
No separation assumption is required; exact overlaps and repeated
generators are allowed.
\end{theorem}"
T05,T05.02,assumption_domain,not_explicit,H/D: p is a strictly positive probability vector: each weight is positive and the weights sum to one. Anchor S: “strictly positive probability vector”.,"Start with finitely many maps that shrink the real line, possibly also reflecting and translating it. Choose maps independently using the probabilities \(p_i\). An infinite sequence of choices determines a limiting point, and \(\mu_{\Phi,p}\) describes the distribution of that point.",Every p_i>0,"Probability normalization is conveyed by choosing with probabilities, but strict positivity of each weight is absent. There is no explicit assertion for zero weights.",https://raw.githubusercontent.com/openai/math/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/preprints/The-entropy-rate-dimension-formula-for-self-similar-measures-on-the-line-September-24-2026/build/sections/introduction.tex,thm:main,"preprints/The-entropy-rate-dimension-formula-for-self-similar-measures-on-the-line-September-24-2026/build/sections/introduction.tex lines 78-88 (statement):
\begin{theorem}\label{thm:main}
For every finite nonempty indexed family
$\varphi_i(x)=r_i x+t_i$ on $\R$ with $0<|r_i|<1$, and every
strictly positive probability vector $p$, its self-similar measure satisfies
\[
 \dim_H\mu_{\Phi,p}
 =\min\left\{1,\frac{h_{\mathrm{RW}}(\Phi,p)}{\chi(\Phi,p)}\right\}.
\]
No separation assumption is required; exact overlaps and repeated
generators are allowed.
\end{theorem}"
T05,T05.03,definition,retained,R: The measure is the Borel probability measure satisfying the weighted pushforward fixed-point equation. Anchor C: definition of mu.,"Start with finitely many maps that shrink the real line, possibly also reflecting and translating it. Choose maps independently using the probabilities \(p_i\). An infinite sequence of choices determines a limiting point, and \(\mu_{\Phi,p}\) describes the distribution of that point.",,Independent infinite composition and its limiting-point law are explicitly given. This is the equivalent construction supplied in C; the fixed-point equation need not be repeated.,https://raw.githubusercontent.com/openai/math/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/preprints/The-entropy-rate-dimension-formula-for-self-similar-measures-on-the-line-September-24-2026/build/sections/introduction.tex,thm:main,"preprints/The-entropy-rate-dimension-formula-for-self-similar-measures-on-the-line-September-24-2026/build/sections/introduction.tex lines 1-77 (context):
\section{Introduction}\label{sec:introduction}

Let $\Lambda$ be a finite nonempty alphabet, and let
\[
 \Phi=(\varphi_i)_{i\in\Lambda},\qquad
 \varphi_i(x)=r_i x+t_i,\qquad t_i\in\R,\quad 0<|r_i|<1.
\]
The family is indexed: different symbols may specify the same map.
Given a probability vector $p=(p_i)_{i\in\Lambda}$ with $p_i>0$, its
self-similar measure is the unique Borel probability measure satisfying
\[
 \mu=\sum_{i\in\Lambda}p_i(\varphi_i)_*\mu.
\]
Equivalently, $\mu$ is the law of
$\lim_{n\to\infty}\varphi_{I_1}\circ\cdots\circ\varphi_{I_n}(0)$,
where the symbols $I_j$ are independent with law $p$.
The limit exists uniformly in the address: if
$r_{\max}=\max_i|r_i|$ and $T_0=\max_i|t_i|$, all coding limits
have absolute value at most $T_0/(1-r_{\max})$.

For a word $w=i_1\cdots i_n$, write
$\varphi_w=\varphi_{i_1}\circ\cdots\circ\varphi_{i_n}$ and
$p_w=p_{i_1}\cdots p_{i_n}$.  An \emph{exact overlap} is an equality
$\varphi_u=\varphi_v$ for distinct words of the same positive length.
Equality here means equality of the complete affine maps, including
both the signed slope and the translation.  We allow all such overlaps.

The relevant entropy counts maps rather than addresses.  Set
\[
 G_n=\varphi_{I_1}\circ\cdots\circ\varphi_{I_n},\qquad
 \P(G_n=g)=\sum_{w\in\Lambda^n:\,\varphi_w=g}p_w.
\]
All logarithms below have base two.  For a finite-valued random variable
$Z$, its Shannon entropy is $H(Z)=-\sum_z\P(Z=z)\log\P(Z=z)$,
with $0\log0=0$.  Define the random-walk entropy rate and the Lyapunov
exponent by
\begin{equation}\label{eq:rate-and-lyapunov}
 h=h_{\mathrm{RW}}(\Phi,p)
   :=\lim_{n\to\infty}\frac{H(G_n)}n
    =\inf_{n\ge1}\frac{H(G_n)}n,
 \qquad
 \chi=-\sum_i p_i\log|r_i|>0.
\end{equation}
Indeed, a length-$(n+m)$ map is a function of two independent maps
with laws $G_n,G_m$, so $H(G_{n+m})\le H(G_n)+H(G_m)$; subadditivity
gives the displayed limit and infimum.  Also
$0\le h\le H(p):=-\sum_i p_i\log p_i$.  The ratio $h/\chi$ is
independent of the common logarithm base.

We use the lower Hausdorff dimension of a measure:
\[
 \dim_H\nu=\inf\{\dim_H E:E\subset\R\text{ Borel},\ \nu(E)>0\}.
\]
Feng--Hu's exact-dimensionality theorem
\cite[Theorem~2.8]{FengHu2009} applies to finite self-similar systems
without separation.  Indeed, the maps preserve a sufficiently large
compact interval, extend to contracting smooth diffeomorphisms, and have
derivative norm and least singular value both equal to $|r_i|$.
The system is therefore conformal, and its Bernoulli coding law is
ergodic.  These observations also cover negative ratios and repeated
indexed maps.  Thus, for the measure
$\mu$ above, there is a constant $d$ such that
\begin{equation}\label{eq:exact-dimensionality}
 \lim_{s\downarrow0}\frac{\log\mu(B(x,s))}{\log s}=d
 \quad\text{for $\mu$-almost every $x$}.
\end{equation}
For an exact-dimensional measure this constant equals both $\dim_H\mu$
and $\inf\{\dim_H E:\mu(E)=1\}$.
To see the equivalence, restrict to countably many sets on which the
local bounds $s^{d+\varepsilon}\le\mu(B(x,s))\le s^{d-\varepsilon}$
hold uniformly for all sufficiently small $s$.  The lower mass bound
gives a full-measure union of sets of dimension at most $d+\varepsilon$
by a covering argument; the upper mass bound gives dimension at least
$d-\varepsilon$ for every positive-mass set.  Let
$\varepsilon\downarrow0$.  Consequently either Hausdorff
measure-dimension convention gives the same statement below.

"
T05,T05.04,definition,retained,R: h_RW is the limit H(G_n)/n for random composed affine maps; equal maps aggregate their word probabilities. Anchor C: law of G_n and eq:rate-and-lyapunov.,"- **The information rate \(h_{\mathrm{RW}}\).** After \(n\) choices, consider the resulting *composite map*. Different sequences can produce exactly the same map; their probabilities are added together, and they count as one outcome. The entropy of these outcomes measures how much uncertainty remains about the composite map. Its long-run growth per choice is \(h_{\mathrm{RW}}\).
- **The shrinking rate \(\chi\).** This is the average value of \(-\log_2|r_i|\). After \(n\) choices, the typical contraction factor is roughly \(2^{-n\chi}\).",,The entropy is of complete composite maps rather than words; long-run entropy growth per choice preserves the rate definition.,https://raw.githubusercontent.com/openai/math/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/preprints/The-entropy-rate-dimension-formula-for-self-similar-measures-on-the-line-September-24-2026/build/sections/introduction.tex,thm:main,"preprints/The-entropy-rate-dimension-formula-for-self-similar-measures-on-the-line-September-24-2026/build/sections/introduction.tex lines 1-77 (context):
\section{Introduction}\label{sec:introduction}

Let $\Lambda$ be a finite nonempty alphabet, and let
\[
 \Phi=(\varphi_i)_{i\in\Lambda},\qquad
 \varphi_i(x)=r_i x+t_i,\qquad t_i\in\R,\quad 0<|r_i|<1.
\]
The family is indexed: different symbols may specify the same map.
Given a probability vector $p=(p_i)_{i\in\Lambda}$ with $p_i>0$, its
self-similar measure is the unique Borel probability measure satisfying
\[
 \mu=\sum_{i\in\Lambda}p_i(\varphi_i)_*\mu.
\]
Equivalently, $\mu$ is the law of
$\lim_{n\to\infty}\varphi_{I_1}\circ\cdots\circ\varphi_{I_n}(0)$,
where the symbols $I_j$ are independent with law $p$.
The limit exists uniformly in the address: if
$r_{\max}=\max_i|r_i|$ and $T_0=\max_i|t_i|$, all coding limits
have absolute value at most $T_0/(1-r_{\max})$.

For a word $w=i_1\cdots i_n$, write
$\varphi_w=\varphi_{i_1}\circ\cdots\circ\varphi_{i_n}$ and
$p_w=p_{i_1}\cdots p_{i_n}$.  An \emph{exact overlap} is an equality
$\varphi_u=\varphi_v$ for distinct words of the same positive length.
Equality here means equality of the complete affine maps, including
both the signed slope and the translation.  We allow all such overlaps.

The relevant entropy counts maps rather than addresses.  Set
\[
 G_n=\varphi_{I_1}\circ\cdots\circ\varphi_{I_n},\qquad
 \P(G_n=g)=\sum_{w\in\Lambda^n:\,\varphi_w=g}p_w.
\]
All logarithms below have base two.  For a finite-valued random variable
$Z$, its Shannon entropy is $H(Z)=-\sum_z\P(Z=z)\log\P(Z=z)$,
with $0\log0=0$.  Define the random-walk entropy rate and the Lyapunov
exponent by
\begin{equation}\label{eq:rate-and-lyapunov}
 h=h_{\mathrm{RW}}(\Phi,p)
   :=\lim_{n\to\infty}\frac{H(G_n)}n
    =\inf_{n\ge1}\frac{H(G_n)}n,
 \qquad
 \chi=-\sum_i p_i\log|r_i|>0.
\end{equation}
Indeed, a length-$(n+m)$ map is a function of two independent maps
with laws $G_n,G_m$, so $H(G_{n+m})\le H(G_n)+H(G_m)$; subadditivity
gives the displayed limit and infimum.  Also
$0\le h\le H(p):=-\sum_i p_i\log p_i$.  The ratio $h/\chi$ is
independent of the common logarithm base.

We use the lower Hausdorff dimension of a measure:
\[
 \dim_H\nu=\inf\{\dim_H E:E\subset\R\text{ Borel},\ \nu(E)>0\}.
\]
Feng--Hu's exact-dimensionality theorem
\cite[Theorem~2.8]{FengHu2009} applies to finite self-similar systems
without separation.  Indeed, the maps preserve a sufficiently large
compact interval, extend to contracting smooth diffeomorphisms, and have
derivative norm and least singular value both equal to $|r_i|$.
The system is therefore conformal, and its Bernoulli coding law is
ergodic.  These observations also cover negative ratios and repeated
indexed maps.  Thus, for the measure
$\mu$ above, there is a constant $d$ such that
\begin{equation}\label{eq:exact-dimensionality}
 \lim_{s\downarrow0}\frac{\log\mu(B(x,s))}{\log s}=d
 \quad\text{for $\mu$-almost every $x$}.
\end{equation}
For an exact-dimensional measure this constant equals both $\dim_H\mu$
and $\inf\{\dim_H E:\mu(E)=1\}$.
To see the equivalence, restrict to countably many sets on which the
local bounds $s^{d+\varepsilon}\le\mu(B(x,s))\le s^{d-\varepsilon}$
hold uniformly for all sufficiently small $s$.  The lower mass bound
gives a full-measure union of sets of dimension at most $d+\varepsilon$
by a covering argument; the upper mass bound gives dimension at least
$d-\varepsilon$ for every positive-mass set.  Let
$\varepsilon\downarrow0$.  Consequently either Hausdorff
measure-dimension convention gives the same statement below.

"
T05,T05.05,definition,retained,"R: chi = −sum p_i log|r_i|, with a common logarithm base; it is positive. Anchor C: eq:rate-and-lyapunov.","- **The information rate \(h_{\mathrm{RW}}\).** After \(n\) choices, consider the resulting *composite map*. Different sequences can produce exactly the same map; their probabilities are added together, and they count as one outcome. The entropy of these outcomes measures how much uncertainty remains about the composite map. Its long-run growth per choice is \(h_{\mathrm{RW}}\).
- **The shrinking rate \(\chi\).** This is the average value of \(-\log_2|r_i|\). After \(n\) choices, the typical contraction factor is roughly \(2^{-n\chi}\).",,The displayed -log2|r_i| averaged using p_i preserves chi; the typical contraction expression is an intuitive consequence of the independent sampling construction.,https://raw.githubusercontent.com/openai/math/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/preprints/The-entropy-rate-dimension-formula-for-self-similar-measures-on-the-line-September-24-2026/build/sections/introduction.tex,thm:main,"preprints/The-entropy-rate-dimension-formula-for-self-similar-measures-on-the-line-September-24-2026/build/sections/introduction.tex lines 1-77 (context):
\section{Introduction}\label{sec:introduction}

Let $\Lambda$ be a finite nonempty alphabet, and let
\[
 \Phi=(\varphi_i)_{i\in\Lambda},\qquad
 \varphi_i(x)=r_i x+t_i,\qquad t_i\in\R,\quad 0<|r_i|<1.
\]
The family is indexed: different symbols may specify the same map.
Given a probability vector $p=(p_i)_{i\in\Lambda}$ with $p_i>0$, its
self-similar measure is the unique Borel probability measure satisfying
\[
 \mu=\sum_{i\in\Lambda}p_i(\varphi_i)_*\mu.
\]
Equivalently, $\mu$ is the law of
$\lim_{n\to\infty}\varphi_{I_1}\circ\cdots\circ\varphi_{I_n}(0)$,
where the symbols $I_j$ are independent with law $p$.
The limit exists uniformly in the address: if
$r_{\max}=\max_i|r_i|$ and $T_0=\max_i|t_i|$, all coding limits
have absolute value at most $T_0/(1-r_{\max})$.

For a word $w=i_1\cdots i_n$, write
$\varphi_w=\varphi_{i_1}\circ\cdots\circ\varphi_{i_n}$ and
$p_w=p_{i_1}\cdots p_{i_n}$.  An \emph{exact overlap} is an equality
$\varphi_u=\varphi_v$ for distinct words of the same positive length.
Equality here means equality of the complete affine maps, including
both the signed slope and the translation.  We allow all such overlaps.

The relevant entropy counts maps rather than addresses.  Set
\[
 G_n=\varphi_{I_1}\circ\cdots\circ\varphi_{I_n},\qquad
 \P(G_n=g)=\sum_{w\in\Lambda^n:\,\varphi_w=g}p_w.
\]
All logarithms below have base two.  For a finite-valued random variable
$Z$, its Shannon entropy is $H(Z)=-\sum_z\P(Z=z)\log\P(Z=z)$,
with $0\log0=0$.  Define the random-walk entropy rate and the Lyapunov
exponent by
\begin{equation}\label{eq:rate-and-lyapunov}
 h=h_{\mathrm{RW}}(\Phi,p)
   :=\lim_{n\to\infty}\frac{H(G_n)}n
    =\inf_{n\ge1}\frac{H(G_n)}n,
 \qquad
 \chi=-\sum_i p_i\log|r_i|>0.
\end{equation}
Indeed, a length-$(n+m)$ map is a function of two independent maps
with laws $G_n,G_m$, so $H(G_{n+m})\le H(G_n)+H(G_m)$; subadditivity
gives the displayed limit and infimum.  Also
$0\le h\le H(p):=-\sum_i p_i\log p_i$.  The ratio $h/\chi$ is
independent of the common logarithm base.

We use the lower Hausdorff dimension of a measure:
\[
 \dim_H\nu=\inf\{\dim_H E:E\subset\R\text{ Borel},\ \nu(E)>0\}.
\]
Feng--Hu's exact-dimensionality theorem
\cite[Theorem~2.8]{FengHu2009} applies to finite self-similar systems
without separation.  Indeed, the maps preserve a sufficiently large
compact interval, extend to contracting smooth diffeomorphisms, and have
derivative norm and least singular value both equal to $|r_i|$.
The system is therefore conformal, and its Bernoulli coding law is
ergodic.  These observations also cover negative ratios and repeated
indexed maps.  Thus, for the measure
$\mu$ above, there is a constant $d$ such that
\begin{equation}\label{eq:exact-dimensionality}
 \lim_{s\downarrow0}\frac{\log\mu(B(x,s))}{\log s}=d
 \quad\text{for $\mu$-almost every $x$}.
\end{equation}
For an exact-dimensional measure this constant equals both $\dim_H\mu$
and $\inf\{\dim_H E:\mu(E)=1\}$.
To see the equivalence, restrict to countably many sets on which the
local bounds $s^{d+\varepsilon}\le\mu(B(x,s))\le s^{d-\varepsilon}$
hold uniformly for all sufficiently small $s$.  The lower mass bound
gives a full-measure union of sets of dimension at most $d+\varepsilon$
by a covering argument; the upper mass bound gives dimension at least
$d-\varepsilon$ for every positive-mass set.  Let
$\varepsilon\downarrow0$.  Consequently either Hausdorff
measure-dimension convention gives the same statement below.

"
T05,T05.06,conclusion,retained,"R: Hausdorff dimension of the measure equals min{1,h_RW/chi}, not uncapped entropy divided by contraction. Anchor S: displayed equality.","The theorem says that **the dimension of this randomly generated fractal measure is the rate at which distinct information is produced, divided by the rate at which distances shrink**, capped at one because the measure lives on a line.",,The displayed dimension formula retains the ratio and ceiling.,https://raw.githubusercontent.com/openai/math/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/preprints/The-entropy-rate-dimension-formula-for-self-similar-measures-on-the-line-September-24-2026/build/sections/introduction.tex,thm:main,"preprints/The-entropy-rate-dimension-formula-for-self-similar-measures-on-the-line-September-24-2026/build/sections/introduction.tex lines 78-88 (statement):
\begin{theorem}\label{thm:main}
For every finite nonempty indexed family
$\varphi_i(x)=r_i x+t_i$ on $\R$ with $0<|r_i|<1$, and every
strictly positive probability vector $p$, its self-similar measure satisfies
\[
 \dim_H\mu_{\Phi,p}
 =\min\left\{1,\frac{h_{\mathrm{RW}}(\Phi,p)}{\chi(\Phi,p)}\right\}.
\]
No separation assumption is required; exact overlaps and repeated
generators are allowed.
\end{theorem}"
T05,T05.07,definition,retained,D: C specifies the lower Hausdorff dimension of a measure using positive-mass Borel sets and discusses exact dimensionality. Anchor C: “We use the lower Hausdorff dimension”.,"Here, the measure’s dimension also describes its typical local mass: around almost every sampled point, an interval of radius \(s\) has mass \(s^{d+o(1)}\), where \(d\) is the dimension in the formula.",,"The response uses almost-everywhere local mass scaling, rather than repeating the positive-mass-set definition. C expressly states exact dimensionality and equivalence of measure-dimension conventions, so this is acceptable, not an omission error.",https://raw.githubusercontent.com/openai/math/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/preprints/The-entropy-rate-dimension-formula-for-self-similar-measures-on-the-line-September-24-2026/build/sections/introduction.tex,thm:main,"preprints/The-entropy-rate-dimension-formula-for-self-similar-measures-on-the-line-September-24-2026/build/sections/introduction.tex lines 1-77 (context):
\section{Introduction}\label{sec:introduction}

Let $\Lambda$ be a finite nonempty alphabet, and let
\[
 \Phi=(\varphi_i)_{i\in\Lambda},\qquad
 \varphi_i(x)=r_i x+t_i,\qquad t_i\in\R,\quad 0<|r_i|<1.
\]
The family is indexed: different symbols may specify the same map.
Given a probability vector $p=(p_i)_{i\in\Lambda}$ with $p_i>0$, its
self-similar measure is the unique Borel probability measure satisfying
\[
 \mu=\sum_{i\in\Lambda}p_i(\varphi_i)_*\mu.
\]
Equivalently, $\mu$ is the law of
$\lim_{n\to\infty}\varphi_{I_1}\circ\cdots\circ\varphi_{I_n}(0)$,
where the symbols $I_j$ are independent with law $p$.
The limit exists uniformly in the address: if
$r_{\max}=\max_i|r_i|$ and $T_0=\max_i|t_i|$, all coding limits
have absolute value at most $T_0/(1-r_{\max})$.

For a word $w=i_1\cdots i_n$, write
$\varphi_w=\varphi_{i_1}\circ\cdots\circ\varphi_{i_n}$ and
$p_w=p_{i_1}\cdots p_{i_n}$.  An \emph{exact overlap} is an equality
$\varphi_u=\varphi_v$ for distinct words of the same positive length.
Equality here means equality of the complete affine maps, including
both the signed slope and the translation.  We allow all such overlaps.

The relevant entropy counts maps rather than addresses.  Set
\[
 G_n=\varphi_{I_1}\circ\cdots\circ\varphi_{I_n},\qquad
 \P(G_n=g)=\sum_{w\in\Lambda^n:\,\varphi_w=g}p_w.
\]
All logarithms below have base two.  For a finite-valued random variable
$Z$, its Shannon entropy is $H(Z)=-\sum_z\P(Z=z)\log\P(Z=z)$,
with $0\log0=0$.  Define the random-walk entropy rate and the Lyapunov
exponent by
\begin{equation}\label{eq:rate-and-lyapunov}
 h=h_{\mathrm{RW}}(\Phi,p)
   :=\lim_{n\to\infty}\frac{H(G_n)}n
    =\inf_{n\ge1}\frac{H(G_n)}n,
 \qquad
 \chi=-\sum_i p_i\log|r_i|>0.
\end{equation}
Indeed, a length-$(n+m)$ map is a function of two independent maps
with laws $G_n,G_m$, so $H(G_{n+m})\le H(G_n)+H(G_m)$; subadditivity
gives the displayed limit and infimum.  Also
$0\le h\le H(p):=-\sum_i p_i\log p_i$.  The ratio $h/\chi$ is
independent of the common logarithm base.

We use the lower Hausdorff dimension of a measure:
\[
 \dim_H\nu=\inf\{\dim_H E:E\subset\R\text{ Borel},\ \nu(E)>0\}.
\]
Feng--Hu's exact-dimensionality theorem
\cite[Theorem~2.8]{FengHu2009} applies to finite self-similar systems
without separation.  Indeed, the maps preserve a sufficiently large
compact interval, extend to contracting smooth diffeomorphisms, and have
derivative norm and least singular value both equal to $|r_i|$.
The system is therefore conformal, and its Bernoulli coding law is
ergodic.  These observations also cover negative ratios and repeated
indexed maps.  Thus, for the measure
$\mu$ above, there is a constant $d$ such that
\begin{equation}\label{eq:exact-dimensionality}
 \lim_{s\downarrow0}\frac{\log\mu(B(x,s))}{\log s}=d
 \quad\text{for $\mu$-almost every $x$}.
\end{equation}
For an exact-dimensional measure this constant equals both $\dim_H\mu$
and $\inf\{\dim_H E:\mu(E)=1\}$.
To see the equivalence, restrict to countably many sets on which the
local bounds $s^{d+\varepsilon}\le\mu(B(x,s))\le s^{d-\varepsilon}$
hold uniformly for all sufficiently small $s$.  The lower mass bound
gives a full-measure union of sets of dimension at most $d+\varepsilon$
by a covering argument; the upper mass bound gives dimension at least
$d-\varepsilon$ for every positive-mass set.  Let
$\varepsilon\downarrow0$.  Consequently either Hausdorff
measure-dimension convention gives the same statement below.

"
T05,T05.08,scope_exception,retained,E: No separation hypothesis; exact overlaps and repeated indexed generators are allowed. Anchor S: final sentence.,"The striking assertion is that **no gaps between the contracted pieces are needed**. Pieces may overlap, maps may be repeated, and different sequences may give identical composite maps. The theorem says that once those exact coincidences are accounted for in \(h_{\mathrm{RW}}\), overlaps cause no further dimension loss beyond the ceiling of one. Dimension one, however, does not by itself mean the measure has a density.",,No separation and repeated generators/exact coincidences are explicit.,https://raw.githubusercontent.com/openai/math/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/preprints/The-entropy-rate-dimension-formula-for-self-similar-measures-on-the-line-September-24-2026/build/sections/introduction.tex,thm:main,"preprints/The-entropy-rate-dimension-formula-for-self-similar-measures-on-the-line-September-24-2026/build/sections/introduction.tex lines 78-88 (statement):
\begin{theorem}\label{thm:main}
For every finite nonempty indexed family
$\varphi_i(x)=r_i x+t_i$ on $\R$ with $0<|r_i|<1$, and every
strictly positive probability vector $p$, its self-similar measure satisfies
\[
 \dim_H\mu_{\Phi,p}
 =\min\left\{1,\frac{h_{\mathrm{RW}}(\Phi,p)}{\chi(\Phi,p)}\right\}.
\]
No separation assumption is required; exact overlaps and repeated
generators are allowed.
\end{theorem}"
T05,T05.09,scope_exception,retained,E: Negative contraction ratios are allowed by the absolute-value condition. Anchor C: “negative ratios”.,"Start with finitely many maps that shrink the real line, possibly also reflecting and translating it. Choose maps independently using the probabilities \(p_i\). An infinite sequence of choices determines a limiting point, and \(\mu_{\Phi,p}\) describes the distribution of that point.",,Reflection preserves permission for negative ratios.,https://raw.githubusercontent.com/openai/math/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/preprints/The-entropy-rate-dimension-formula-for-self-similar-measures-on-the-line-September-24-2026/build/sections/introduction.tex,thm:main,"preprints/The-entropy-rate-dimension-formula-for-self-similar-measures-on-the-line-September-24-2026/build/sections/introduction.tex lines 1-77 (context):
\section{Introduction}\label{sec:introduction}

Let $\Lambda$ be a finite nonempty alphabet, and let
\[
 \Phi=(\varphi_i)_{i\in\Lambda},\qquad
 \varphi_i(x)=r_i x+t_i,\qquad t_i\in\R,\quad 0<|r_i|<1.
\]
The family is indexed: different symbols may specify the same map.
Given a probability vector $p=(p_i)_{i\in\Lambda}$ with $p_i>0$, its
self-similar measure is the unique Borel probability measure satisfying
\[
 \mu=\sum_{i\in\Lambda}p_i(\varphi_i)_*\mu.
\]
Equivalently, $\mu$ is the law of
$\lim_{n\to\infty}\varphi_{I_1}\circ\cdots\circ\varphi_{I_n}(0)$,
where the symbols $I_j$ are independent with law $p$.
The limit exists uniformly in the address: if
$r_{\max}=\max_i|r_i|$ and $T_0=\max_i|t_i|$, all coding limits
have absolute value at most $T_0/(1-r_{\max})$.

For a word $w=i_1\cdots i_n$, write
$\varphi_w=\varphi_{i_1}\circ\cdots\circ\varphi_{i_n}$ and
$p_w=p_{i_1}\cdots p_{i_n}$.  An \emph{exact overlap} is an equality
$\varphi_u=\varphi_v$ for distinct words of the same positive length.
Equality here means equality of the complete affine maps, including
both the signed slope and the translation.  We allow all such overlaps.

The relevant entropy counts maps rather than addresses.  Set
\[
 G_n=\varphi_{I_1}\circ\cdots\circ\varphi_{I_n},\qquad
 \P(G_n=g)=\sum_{w\in\Lambda^n:\,\varphi_w=g}p_w.
\]
All logarithms below have base two.  For a finite-valued random variable
$Z$, its Shannon entropy is $H(Z)=-\sum_z\P(Z=z)\log\P(Z=z)$,
with $0\log0=0$.  Define the random-walk entropy rate and the Lyapunov
exponent by
\begin{equation}\label{eq:rate-and-lyapunov}
 h=h_{\mathrm{RW}}(\Phi,p)
   :=\lim_{n\to\infty}\frac{H(G_n)}n
    =\inf_{n\ge1}\frac{H(G_n)}n,
 \qquad
 \chi=-\sum_i p_i\log|r_i|>0.
\end{equation}
Indeed, a length-$(n+m)$ map is a function of two independent maps
with laws $G_n,G_m$, so $H(G_{n+m})\le H(G_n)+H(G_m)$; subadditivity
gives the displayed limit and infimum.  Also
$0\le h\le H(p):=-\sum_i p_i\log p_i$.  The ratio $h/\chi$ is
independent of the common logarithm base.

We use the lower Hausdorff dimension of a measure:
\[
 \dim_H\nu=\inf\{\dim_H E:E\subset\R\text{ Borel},\ \nu(E)>0\}.
\]
Feng--Hu's exact-dimensionality theorem
\cite[Theorem~2.8]{FengHu2009} applies to finite self-similar systems
without separation.  Indeed, the maps preserve a sufficiently large
compact interval, extend to contracting smooth diffeomorphisms, and have
derivative norm and least singular value both equal to $|r_i|$.
The system is therefore conformal, and its Bernoulli coding law is
ergodic.  These observations also cover negative ratios and repeated
indexed maps.  Thus, for the measure
$\mu$ above, there is a constant $d$ such that
\begin{equation}\label{eq:exact-dimensionality}
 \lim_{s\downarrow0}\frac{\log\mu(B(x,s))}{\log s}=d
 \quad\text{for $\mu$-almost every $x$}.
\end{equation}
For an exact-dimensional measure this constant equals both $\dim_H\mu$
and $\inf\{\dim_H E:\mu(E)=1\}$.
To see the equivalence, restrict to countably many sets on which the
local bounds $s^{d+\varepsilon}\le\mu(B(x,s))\le s^{d-\varepsilon}$
hold uniformly for all sufficiently small $s$.  The lower mass bound
gives a full-measure union of sets of dimension at most $d+\varepsilon$
by a covering argument; the upper mass bound gives dimension at least
$d-\varepsilon$ for every positive-mass set.  Let
$\varepsilon\downarrow0$.  Consequently either Hausdorff
measure-dimension convention gives the same statement below.

"
T06,T06.01,assumption_domain,not_explicit,"H/D: A probability space; T is invertible, bimeasurable, and measure preserving. Anchor C: opening sentence.","Here, \(T\) describes one step of a probability-preserving evolution, and each bounded function \(f_j\) is a numerical measurement of the state. “Mixing” means that two events become approximately independent when separated by a sufficiently long time.",Invertibility and bimeasurability,"The probability-preserving system is retained, but invertibility and bimeasurability are not stated. The explanation refers to the supplied T, without explicitly asserting the result for noninvertible systems.",https://raw.githubusercontent.com/openai/math/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/preprints/Pointwise-Multiple-Ergodic-Averages-for-Mixing-Transformations-October-4-2026/build/sections/introduction.tex,thm:main,"preprints/Pointwise-Multiple-Ergodic-Averages-for-Mixing-Transformations-October-4-2026/build/sections/introduction.tex lines 1-19 (context):
\section{Introduction}
\label{sec:introduction}

Let $(X,\mathcal F,\mu)$ be a probability space and let $T:X\to X$
be invertible, bimeasurable, and measure preserving. We say that $T$
is \emph{mixing} if
\begin{equation}\label{eq:mixing}
 \mu(A\cap T^{-r}B)\longrightarrow\mu(A)\mu(B)
 \qquad (|r|\to\infty)
\end{equation}
for every $A,B\in\mathcal F$. For a fixed tuple of bounded measurable
functions, the consecutive multiple ergodic averages are
\[
 A_N(f_1,\ldots,f_n)(x)
 =\frac1N\sum_{k=1}^N\prod_{j=1}^n f_j(T^{jk}x).
\]
The problem is to determine their almost-everywhere behavior, not merely
their limit in norm. We prove the following result.

"
T06,T06.02,assumption,retained,"H: T is mixing in the stated sense: for every measurable A,B, mu(A intersect T^(−r)B) tends to mu(A)mu(B) as |r|→infinity. Anchor C: eq:mixing; S invokes it.","Here, \(T\) describes one step of a probability-preserving evolution, and each bounded function \(f_j\) is a numerical measurement of the state. “Mixing” means that two events become approximately independent when separated by a sufficiently long time.",,"This is an acceptable plain-language mixing description, interpreted as the limiting independence of arbitrary event pairs in the supplied definition. It does not claim exact independence at a finite time.",https://raw.githubusercontent.com/openai/math/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/preprints/Pointwise-Multiple-Ergodic-Averages-for-Mixing-Transformations-October-4-2026/build/sections/introduction.tex,thm:main,"preprints/Pointwise-Multiple-Ergodic-Averages-for-Mixing-Transformations-October-4-2026/build/sections/introduction.tex lines 1-19 (context):
\section{Introduction}
\label{sec:introduction}

Let $(X,\mathcal F,\mu)$ be a probability space and let $T:X\to X$
be invertible, bimeasurable, and measure preserving. We say that $T$
is \emph{mixing} if
\begin{equation}\label{eq:mixing}
 \mu(A\cap T^{-r}B)\longrightarrow\mu(A)\mu(B)
 \qquad (|r|\to\infty)
\end{equation}
for every $A,B\in\mathcal F$. For a fixed tuple of bounded measurable
functions, the consecutive multiple ergodic averages are
\[
 A_N(f_1,\ldots,f_n)(x)
 =\frac1N\sum_{k=1}^N\prod_{j=1}^n f_j(T^{jk}x).
\]
The problem is to determine their almost-everywhere behavior, not merely
their limit in norm. We prove the following result.



preprints/Pointwise-Multiple-Ergodic-Averages-for-Mixing-Transformations-October-4-2026/build/sections/introduction.tex lines 20-30 (statement):
\begin{theorem}\label{thm:main}
Suppose that $T$ satisfies \eqref{eq:mixing}. For every integer
$n\ge2$ and every fixed $f_1,\ldots,f_n\in L^\infty(\mu)$,
\[
 A_N(f_1,\ldots,f_n)(x)
 \longrightarrow\prod_{j=1}^n\int_X f_j\dd\mu
 \qquad\text{for $\mu$-almost every }x,
\]
as $N$ tends to infinity through all positive integers. The probability
space need not be standard, and no rate of mixing is required.
\end{theorem}"
T06,T06.03,assumption_quantifier,not_explicit,"Q/D: For every integer n≥2 and every fixed tuple f_1,...,f_n in L-infinity(mu). Anchor S.","Fix the measurements \(f_1,\ldots,f_n\). Starting from a state \(x\), for each positive integer \(k\), measure \(f_1\) after \(k\) steps, \(f_2\) after \(2k\) steps, and so on, then multiply the results. The theorem asserts that
\[
\frac1N\sum_{k=1}^N f_1(T^kx)\,f_2(T^{2k}x)\cdots f_n(T^{nk}x)
\;\longrightarrow\;
\prod_{j=1}^n\int_X f_j\,d\mu.
\]
The right-hand side is what you would obtain by multiplying the population averages of the measurements.",Explicit n>=2,Fixed bounded functions and finite n are preserved; the explicit integer lower bound n>=2 is not stated. No concrete wrong claim about a prohibited case is demonstrated.,https://raw.githubusercontent.com/openai/math/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/preprints/Pointwise-Multiple-Ergodic-Averages-for-Mixing-Transformations-October-4-2026/build/sections/introduction.tex,thm:main,"preprints/Pointwise-Multiple-Ergodic-Averages-for-Mixing-Transformations-October-4-2026/build/sections/introduction.tex lines 20-30 (statement):
\begin{theorem}\label{thm:main}
Suppose that $T$ satisfies \eqref{eq:mixing}. For every integer
$n\ge2$ and every fixed $f_1,\ldots,f_n\in L^\infty(\mu)$,
\[
 A_N(f_1,\ldots,f_n)(x)
 \longrightarrow\prod_{j=1}^n\int_X f_j\dd\mu
 \qquad\text{for $\mu$-almost every }x,
\]
as $N$ tends to infinity through all positive integers. The probability
space need not be standard, and no rate of mixing is required.
\end{theorem}"
T06,T06.04,definition,retained,"R: Averages are N inverse times the sum over k=1,...,N of the product f_j(T^(jk)x). Anchor C: definition of A_N.","Fix the measurements \(f_1,\ldots,f_n\). Starting from a state \(x\), for each positive integer \(k\), measure \(f_1\) after \(k\) steps, \(f_2\) after \(2k\) steps, and so on, then multiply the results. The theorem asserts that
\[
\frac1N\sum_{k=1}^N f_1(T^kx)\,f_2(T^{2k}x)\cdots f_n(T^{nk}x)
\;\longrightarrow\;
\prod_{j=1}^n\int_X f_j\,d\mu.
\]
The right-hand side is what you would obtain by multiplying the population averages of the measurements.",,"The formula and accompanying construction correctly use k,2k,...,nk and average the products.",https://raw.githubusercontent.com/openai/math/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/preprints/Pointwise-Multiple-Ergodic-Averages-for-Mixing-Transformations-October-4-2026/build/sections/introduction.tex,thm:main,"preprints/Pointwise-Multiple-Ergodic-Averages-for-Mixing-Transformations-October-4-2026/build/sections/introduction.tex lines 1-19 (context):
\section{Introduction}
\label{sec:introduction}

Let $(X,\mathcal F,\mu)$ be a probability space and let $T:X\to X$
be invertible, bimeasurable, and measure preserving. We say that $T$
is \emph{mixing} if
\begin{equation}\label{eq:mixing}
 \mu(A\cap T^{-r}B)\longrightarrow\mu(A)\mu(B)
 \qquad (|r|\to\infty)
\end{equation}
for every $A,B\in\mathcal F$. For a fixed tuple of bounded measurable
functions, the consecutive multiple ergodic averages are
\[
 A_N(f_1,\ldots,f_n)(x)
 =\frac1N\sum_{k=1}^N\prod_{j=1}^n f_j(T^{jk}x).
\]
The problem is to determine their almost-everywhere behavior, not merely
their limit in norm. We prove the following result.

"
T06,T06.05,conclusion,retained,"R: Limit is the product of the separate integrals, not the integral of the pointwise product. Anchor S: displayed limit.","Fix the measurements \(f_1,\ldots,f_n\). Starting from a state \(x\), for each positive integer \(k\), measure \(f_1\) after \(k\) steps, \(f_2\) after \(2k\) steps, and so on, then multiply the results. The theorem asserts that
\[
\frac1N\sum_{k=1}^N f_1(T^kx)\,f_2(T^{2k}x)\cdots f_n(T^{nk}x)
\;\longrightarrow\;
\prod_{j=1}^n\int_X f_j\,d\mu.
\]
The right-hand side is what you would obtain by multiplying the population averages of the measurements.",,The formula and indicator example preserve the product of integrals.,https://raw.githubusercontent.com/openai/math/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/preprints/Pointwise-Multiple-Ergodic-Averages-for-Mixing-Transformations-October-4-2026/build/sections/introduction.tex,thm:main,"preprints/Pointwise-Multiple-Ergodic-Averages-for-Mixing-Transformations-October-4-2026/build/sections/introduction.tex lines 20-30 (statement):
\begin{theorem}\label{thm:main}
Suppose that $T$ satisfies \eqref{eq:mixing}. For every integer
$n\ge2$ and every fixed $f_1,\ldots,f_n\in L^\infty(\mu)$,
\[
 A_N(f_1,\ldots,f_n)(x)
 \longrightarrow\prod_{j=1}^n\int_X f_j\dd\mu
 \qquad\text{for $\mu$-almost every }x,
\]
as $N$ tends to infinity through all positive integers. The probability
space need not be standard, and no rate of mixing is required.
\end{theorem}"
T06,T06.06,quantifier_exception,retained,E/Q: Convergence is mu-almost everywhere for each fixed tuple; the exceptional null set may depend on the system and tuple. Anchor S and N.,"The key strength is **convergence along individual trajectories**: it holds for every starting state except a set of probability zero, rather than merely as an averaged error over starting states. It holds as \(N\) runs through all positive integers, requires no specified speed of mixing, and imposes no “standard space” regularity assumption on the probability space. The exceptional set may depend on the chosen measurements; the theorem does not assert a single set of good starting points that works for every bounded function at once.",,"Almost-everywhere convergence for each fixed tuple is retained, with the dependency spelled out.",https://raw.githubusercontent.com/openai/math/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/preprints/Pointwise-Multiple-Ergodic-Averages-for-Mixing-Transformations-October-4-2026/build/sections/introduction.tex,thm:main,"preprints/Pointwise-Multiple-Ergodic-Averages-for-Mixing-Transformations-October-4-2026/build/sections/introduction.tex lines 20-30 (statement):
\begin{theorem}\label{thm:main}
Suppose that $T$ satisfies \eqref{eq:mixing}. For every integer
$n\ge2$ and every fixed $f_1,\ldots,f_n\in L^\infty(\mu)$,
\[
 A_N(f_1,\ldots,f_n)(x)
 \longrightarrow\prod_{j=1}^n\int_X f_j\dd\mu
 \qquad\text{for $\mu$-almost every }x,
\]
as $N$ tends to infinity through all positive integers. The probability
space need not be standard, and no rate of mixing is required.
\end{theorem}

preprints/Pointwise-Multiple-Ergodic-Averages-for-Mixing-Transformations-October-4-2026/build/sections/introduction.tex lines 30-34 (scope_note):


The exceptional null set may depend on the system and the fixed tuple.
The theorem concerns the times $k,2k,\ldots,nk$ and does not require
a common exceptional set for all bounded functions."
T06,T06.07,quantifier,retained,R/Q: N tends through all positive integers. Anchor S: final sentence.,"The key strength is **convergence along individual trajectories**: it holds for every starting state except a set of probability zero, rather than merely as an averaged error over starting states. It holds as \(N\) runs through all positive integers, requires no specified speed of mixing, and imposes no “standard space” regularity assumption on the probability space. The exceptional set may depend on the chosen measurements; the theorem does not assert a single set of good starting points that works for every bounded function at once.",,Full-sequence convergence is explicit.,https://raw.githubusercontent.com/openai/math/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/preprints/Pointwise-Multiple-Ergodic-Averages-for-Mixing-Transformations-October-4-2026/build/sections/introduction.tex,thm:main,"preprints/Pointwise-Multiple-Ergodic-Averages-for-Mixing-Transformations-October-4-2026/build/sections/introduction.tex lines 20-30 (statement):
\begin{theorem}\label{thm:main}
Suppose that $T$ satisfies \eqref{eq:mixing}. For every integer
$n\ge2$ and every fixed $f_1,\ldots,f_n\in L^\infty(\mu)$,
\[
 A_N(f_1,\ldots,f_n)(x)
 \longrightarrow\prod_{j=1}^n\int_X f_j\dd\mu
 \qquad\text{for $\mu$-almost every }x,
\]
as $N$ tends to infinity through all positive integers. The probability
space need not be standard, and no rate of mixing is required.
\end{theorem}"
T06,T06.08,scope_exception,retained,E: Space need not be standard; no mixing rate is required; no single common exceptional set for all bounded functions is required. Anchor S and N.,"The key strength is **convergence along individual trajectories**: it holds for every starting state except a set of probability zero, rather than merely as an averaged error over starting states. It holds as \(N\) runs through all positive integers, requires no specified speed of mixing, and imposes no “standard space” regularity assumption on the probability space. The exceptional set may depend on the chosen measurements; the theorem does not assert a single set of good starting points that works for every bounded function at once.",,"No rate or standard-space requirement is added, and a common exceptional set for all functions is explicitly disclaimed.",https://raw.githubusercontent.com/openai/math/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/preprints/Pointwise-Multiple-Ergodic-Averages-for-Mixing-Transformations-October-4-2026/build/sections/introduction.tex,thm:main,"preprints/Pointwise-Multiple-Ergodic-Averages-for-Mixing-Transformations-October-4-2026/build/sections/introduction.tex lines 20-30 (statement):
\begin{theorem}\label{thm:main}
Suppose that $T$ satisfies \eqref{eq:mixing}. For every integer
$n\ge2$ and every fixed $f_1,\ldots,f_n\in L^\infty(\mu)$,
\[
 A_N(f_1,\ldots,f_n)(x)
 \longrightarrow\prod_{j=1}^n\int_X f_j\dd\mu
 \qquad\text{for $\mu$-almost every }x,
\]
as $N$ tends to infinity through all positive integers. The probability
space need not be standard, and no rate of mixing is required.
\end{theorem}

preprints/Pointwise-Multiple-Ergodic-Averages-for-Mixing-Transformations-October-4-2026/build/sections/introduction.tex lines 30-34 (scope_note):


The exceptional null set may depend on the system and the fixed tuple.
The theorem concerns the times $k,2k,\ldots,nk$ and does not require
a common exceptional set for all bounded functions."
T07,T07.01,assumption_domain,retained,"H/D: Integers r,m≥1 and fixed reals R≥2,D≥1; coloring of positive integers by r colors. Anchor S; C fixes N={1,2,...}.",The theorem says that **any coloring of the positive integers with finitely many colors contains an arbitrarily large finite set whose subset sums and subset products all have the same color**.,,"Finite numbers of colors and desired finite set size convey integer counts; the later paragraph explicitly retains real R>=2,D>=1. The literal r,m>=1 notation is unnecessary in this ordinary counting context.",https://raw.githubusercontent.com/openai/math/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/preprints/Monochromatic-finite-sums-and-products-in-the-positive-integers-September-23-2026/build/sections/01_introduction.tex,thm:main,"preprints/Monochromatic-finite-sums-and-products-in-the-positive-integers-September-23-2026/build/sections/01_introduction.tex lines 13-31 (statement):
\begin{theorem}\label{thm:main}
Let $r,m\ge1$ be integers, and fix real numbers $R\ge2$ and $D\ge1$.
For every coloring $\chi:\N\to[r]$,
there are distinct positive integers $a_1<\cdots<a_m$ and a color
$c\in[r]$ such that
\[
 \chi\left(\sum_{j\in J}a_j\right)
 =\chi\left(\prod_{j\in J}a_j\right)=c
 \quad\text{for every }\varnothing\ne J\subseteq[m].
\]
Equivalently, $\FS(A)\cup\FP(A)$ is monochromatic for a set
$A\subset\N$ of cardinality $m$.
The elements can additionally be chosen to satisfy
\begin{equation}\label{eq:separated-elements}
 a_1>R,\qquad
 a_d>R\left(\sum_{k<d}a_k+\prod_{k<d}a_k\right)^D
 \quad(2\le d\le m).
\end{equation}
\end{theorem}"
T07,T07.02,quantifier,retained,Q: For every such coloring there exist a_1<...<a_m and one color c. Choices may depend on the coloring and the fixed parameters. Anchor S: “For every coloring” then “there are”.,"More precisely, you choose the number of colors \(r\) and the desired set size \(m\). No matter how the integers are colored, there are \(m\) distinct numbers \(a_1<\cdots<a_m\) such that choosing any nonempty subset and either adding or multiplying its members always produces a number of one common color. Each chosen number is used at most once in a sum or product. Since subsets with just one member count, the numbers \(a_i\) themselves also have that color.",,"For every coloring, there exists the finite set and common color after the parameters are chosen.",https://raw.githubusercontent.com/openai/math/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/preprints/Monochromatic-finite-sums-and-products-in-the-positive-integers-September-23-2026/build/sections/01_introduction.tex,thm:main,"preprints/Monochromatic-finite-sums-and-products-in-the-positive-integers-September-23-2026/build/sections/01_introduction.tex lines 13-31 (statement):
\begin{theorem}\label{thm:main}
Let $r,m\ge1$ be integers, and fix real numbers $R\ge2$ and $D\ge1$.
For every coloring $\chi:\N\to[r]$,
there are distinct positive integers $a_1<\cdots<a_m$ and a color
$c\in[r]$ such that
\[
 \chi\left(\sum_{j\in J}a_j\right)
 =\chi\left(\prod_{j\in J}a_j\right)=c
 \quad\text{for every }\varnothing\ne J\subseteq[m].
\]
Equivalently, $\FS(A)\cup\FP(A)$ is monochromatic for a set
$A\subset\N$ of cardinality $m$.
The elements can additionally be chosen to satisfy
\begin{equation}\label{eq:separated-elements}
 a_1>R,\qquad
 a_d>R\left(\sum_{k<d}a_k+\prod_{k<d}a_k\right)^D
 \quad(2\le d\le m).
\end{equation}
\end{theorem}"
T07,T07.03,assumption_domain,retained,"D: Chosen numbers are distinct positive integers, exactly m in number. Anchor S.","More precisely, you choose the number of colors \(r\) and the desired set size \(m\). No matter how the integers are colored, there are \(m\) distinct numbers \(a_1<\cdots<a_m\) such that choosing any nonempty subset and either adding or multiplying its members always produces a number of one common color. Each chosen number is used at most once in a sum or product. Since subsets with just one member count, the numbers \(a_i\) themselves also have that color.",,The specified m distinct ordered elements are positive by the opening domain.,https://raw.githubusercontent.com/openai/math/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/preprints/Monochromatic-finite-sums-and-products-in-the-positive-integers-September-23-2026/build/sections/01_introduction.tex,thm:main,"preprints/Monochromatic-finite-sums-and-products-in-the-positive-integers-September-23-2026/build/sections/01_introduction.tex lines 13-31 (statement):
\begin{theorem}\label{thm:main}
Let $r,m\ge1$ be integers, and fix real numbers $R\ge2$ and $D\ge1$.
For every coloring $\chi:\N\to[r]$,
there are distinct positive integers $a_1<\cdots<a_m$ and a color
$c\in[r]$ such that
\[
 \chi\left(\sum_{j\in J}a_j\right)
 =\chi\left(\prod_{j\in J}a_j\right)=c
 \quad\text{for every }\varnothing\ne J\subseteq[m].
\]
Equivalently, $\FS(A)\cup\FP(A)$ is monochromatic for a set
$A\subset\N$ of cardinality $m$.
The elements can additionally be chosen to satisfy
\begin{equation}\label{eq:separated-elements}
 a_1>R,\qquad
 a_d>R\left(\sum_{k<d}a_k+\prod_{k<d}a_k\right)^D
 \quad(2\le d\le m).
\end{equation}
\end{theorem}"
T07,T07.04,conclusion_quantifier,retained,"Q/R: For every nonempty subset J of [m], both its sum and its product have that same color c. Anchor S: displayed equality.","More precisely, you choose the number of colors \(r\) and the desired set size \(m\). No matter how the integers are colored, there are \(m\) distinct numbers \(a_1<\cdots<a_m\) such that choosing any nonempty subset and either adding or multiplying its members always produces a number of one common color. Each chosen number is used at most once in a sum or product. Since subsets with just one member count, the numbers \(a_i\) themselves also have that color.",,All subset sums and products share the same single color.,https://raw.githubusercontent.com/openai/math/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/preprints/Monochromatic-finite-sums-and-products-in-the-positive-integers-September-23-2026/build/sections/01_introduction.tex,thm:main,"preprints/Monochromatic-finite-sums-and-products-in-the-positive-integers-September-23-2026/build/sections/01_introduction.tex lines 13-31 (statement):
\begin{theorem}\label{thm:main}
Let $r,m\ge1$ be integers, and fix real numbers $R\ge2$ and $D\ge1$.
For every coloring $\chi:\N\to[r]$,
there are distinct positive integers $a_1<\cdots<a_m$ and a color
$c\in[r]$ such that
\[
 \chi\left(\sum_{j\in J}a_j\right)
 =\chi\left(\prod_{j\in J}a_j\right)=c
 \quad\text{for every }\varnothing\ne J\subseteq[m].
\]
Equivalently, $\FS(A)\cup\FP(A)$ is monochromatic for a set
$A\subset\N$ of cardinality $m$.
The elements can additionally be chosen to satisfy
\begin{equation}\label{eq:separated-elements}
 a_1>R,\qquad
 a_d>R\left(\sum_{k<d}a_k+\prod_{k<d}a_k\right)^D
 \quad(2\le d\le m).
\end{equation}
\end{theorem}"
T07,T07.05,definition_exception,retained,"D/E: Each selected element occurs at most once in an individual expression; singleton subsets are included, empty subsets excluded. Anchor C: “at most once” and “singleton subsets”.","More precisely, you choose the number of colors \(r\) and the desired set size \(m\). No matter how the integers are colored, there are \(m\) distinct numbers \(a_1<\cdots<a_m\) such that choosing any nonempty subset and either adding or multiplying its members always produces a number of one common color. Each chosen number is used at most once in a sum or product. Since subsets with just one member count, the numbers \(a_i\) themselves also have that color.",,No repetitions and singleton/nonempty subset conventions are explicit.,https://raw.githubusercontent.com/openai/math/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/preprints/Monochromatic-finite-sums-and-products-in-the-positive-integers-September-23-2026/build/sections/01_introduction.tex,thm:main,"preprints/Monochromatic-finite-sums-and-products-in-the-positive-integers-September-23-2026/build/sections/01_introduction.tex lines 1-12 (context):
\section{Introduction}\label{sec:introduction}

Write $[n]=\{1,\ldots,n\}$ for a positive integer $n$.
For a finite set $A\subset\N=\{1,2,\ldots\}$, write
\[
 \FS(A)=\left\{\sum_{a\in B}a:\varnothing\ne B\subseteq A\right\},
 \qquad
 \FP(A)=\left\{\prod_{a\in B}a:\varnothing\ne B\subseteq A\right\}.
\]
Thus each element of $A$ may occur at most once in an individual sum
or product, and singleton subsets are included. We prove the following.

"
T07,T07.06,conclusion,retained,R: The same selected sequence can additionally satisfy a_1>R and a_d>R(sum of previous a_k + product of previous a_k)^D for every 2≤d≤m. Anchor S: eq:separated-elements.,"The final condition strengthens the claim: **the chosen numbers can also be forced to grow extremely quickly**. You may prescribe any real \(R\ge2\) and \(D\ge1\) in advance. The first number must exceed \(R\), and every later number must satisfy
\[
a_d>R\bigl(\text{sum of the earlier numbers}
+\text{product of the earlier numbers}\bigr)^D.
\]
Even with these growth requirements, the same-color conclusion still holds.",,"The same chosen sequence satisfies the separation inequality, with arbitrary prescribed R,D; this is correctly a promised property, not an input assumption.",https://raw.githubusercontent.com/openai/math/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/preprints/Monochromatic-finite-sums-and-products-in-the-positive-integers-September-23-2026/build/sections/01_introduction.tex,thm:main,"preprints/Monochromatic-finite-sums-and-products-in-the-positive-integers-September-23-2026/build/sections/01_introduction.tex lines 13-31 (statement):
\begin{theorem}\label{thm:main}
Let $r,m\ge1$ be integers, and fix real numbers $R\ge2$ and $D\ge1$.
For every coloring $\chi:\N\to[r]$,
there are distinct positive integers $a_1<\cdots<a_m$ and a color
$c\in[r]$ such that
\[
 \chi\left(\sum_{j\in J}a_j\right)
 =\chi\left(\prod_{j\in J}a_j\right)=c
 \quad\text{for every }\varnothing\ne J\subseteq[m].
\]
Equivalently, $\FS(A)\cup\FP(A)$ is monochromatic for a set
$A\subset\N$ of cardinality $m$.
The elements can additionally be chosen to satisfy
\begin{equation}\label{eq:separated-elements}
 a_1>R,\qquad
 a_d>R\left(\sum_{k<d}a_k+\prod_{k<d}a_k\right)^D
 \quad(2\le d\le m).
\end{equation}
\end{theorem}"
T08,T08.01,assumption_domain,not_explicit,H/D: Integer b≥6 and every finite-dimensional complex vector space V. Anchor S.,"The result establishes the sixth case of Foulkes’ conjecture, for every finite dimension of \(V\). When \(b=6\), the two spaces coincide; the substantive claim covers all \(b>6\).",Complex vector space V,Integer b>=6 and finite dimension are explicit; the complex base field is not. No alternate field is positively asserted.,https://raw.githubusercontent.com/openai/math/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/preprints/Foulkes-Conjecture-for-the-Sixth-Symmetric-Power-September-25-2026/build/sections/introduction.tex,thm:main,"preprints/Foulkes-Conjecture-for-the-Sixth-Symmetric-Power-September-25-2026/build/sections/introduction.tex lines 19-26 (statement):
\begin{theorem}\label{thm:main}
For every integer \(b\ge6\) and every finite-dimensional complex vector
space \(V\), there is a \(\GL(V)\)-equivariant injection
\[
                 \Sym^6(\Sym^b V)\ \hookrightarrow\ \Sym^b(\Sym^6 V).
\]
Equivalently, \(h_b[h_6]-h_6[h_b]\) is Schur-positive for every \(b\ge6\).
\end{theorem}"
T08,T08.02,quantifier,retained,"Q: For every such b,V there exists an injection. Anchor S: quantifier order.","The symmetric power \(\Sym^r V\) is formed from products of \(r\) vectors, where changing their order makes no difference. For any integer \(b\ge6\), the theorem compares:",,The universal b and finite-dimensional V clauses together preserve existence across those inputs.,https://raw.githubusercontent.com/openai/math/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/preprints/Foulkes-Conjecture-for-the-Sixth-Symmetric-Power-September-25-2026/build/sections/introduction.tex,thm:main,"preprints/Foulkes-Conjecture-for-the-Sixth-Symmetric-Power-September-25-2026/build/sections/introduction.tex lines 19-26 (statement):
\begin{theorem}\label{thm:main}
For every integer \(b\ge6\) and every finite-dimensional complex vector
space \(V\), there is a \(\GL(V)\)-equivariant injection
\[
                 \Sym^6(\Sym^b V)\ \hookrightarrow\ \Sym^b(\Sym^6 V).
\]
Equivalently, \(h_b[h_6]-h_6[h_b]\) is Schur-positive for every \(b\ge6\).
\end{theorem}"
T08,T08.03,conclusion,retained,R: Direction is Sym^6(Sym^b V) into Sym^b(Sym^6 V). Anchor S: displayed arrow.,"It asserts that the first space embeds into the second. This embedding is an injective linear map, so it loses no information. It is also **\(\GL(V)\)-equivariant**: applying any invertible linear transformation to \(V\) and then embedding gives the same result as embedding first and then applying that transformation.",,The displayed ordered list and arrow description retain the correct direction.,https://raw.githubusercontent.com/openai/math/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/preprints/Foulkes-Conjecture-for-the-Sixth-Symmetric-Power-September-25-2026/build/sections/introduction.tex,thm:main,"preprints/Foulkes-Conjecture-for-the-Sixth-Symmetric-Power-September-25-2026/build/sections/introduction.tex lines 19-26 (statement):
\begin{theorem}\label{thm:main}
For every integer \(b\ge6\) and every finite-dimensional complex vector
space \(V\), there is a \(\GL(V)\)-equivariant injection
\[
                 \Sym^6(\Sym^b V)\ \hookrightarrow\ \Sym^b(\Sym^6 V).
\]
Equivalently, \(h_b[h_6]-h_6[h_b]\) is Schur-positive for every \(b\ge6\).
\end{theorem}"
T08,T08.04,conclusion,retained,"R: Injection is GL(V)-equivariant, preserving the group action, not merely an injection of underlying sets. Anchor S.","It asserts that the first space embeds into the second. This embedding is an injective linear map, so it loses no information. It is also **\(\GL(V)\)-equivariant**: applying any invertible linear transformation to \(V\) and then embedding gives the same result as embedding first and then applying that transformation.",,Linearity and commutation with invertible transformations preserve GL(V)-equivariance.,https://raw.githubusercontent.com/openai/math/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/preprints/Foulkes-Conjecture-for-the-Sixth-Symmetric-Power-September-25-2026/build/sections/introduction.tex,thm:main,"preprints/Foulkes-Conjecture-for-the-Sixth-Symmetric-Power-September-25-2026/build/sections/introduction.tex lines 19-26 (statement):
\begin{theorem}\label{thm:main}
For every integer \(b\ge6\) and every finite-dimensional complex vector
space \(V\), there is a \(\GL(V)\)-equivariant injection
\[
                 \Sym^6(\Sym^b V)\ \hookrightarrow\ \Sym^b(\Sym^6 V).
\]
Equivalently, \(h_b[h_6]-h_6[h_b]\) is Schur-positive for every \(b\ge6\).
\end{theorem}"
T08,T08.05,conclusion_definition,retained,R: Equivalent formulation is Schur positivity of h_b[h_6]−h_6[h_b]. Anchor S; C defines Schur positivity as nonnegative integer Schur coefficients.,"This is stronger than simply comparing dimensions. These spaces decompose into irreducible representations—the basic building blocks of their linear symmetries—and every type of building block occurs in the second space at least as many times as in the first. That is what the equivalent statement
\[
h_b[h_6]-h_6[h_b]\ \text{is Schur-positive}
\]
expresses: every coefficient in its expansion in Schur functions is nonnegative.",,Multiplicity language preserves integer counts; the written symmetric-function difference has the correct order and nonnegative coefficients.,https://raw.githubusercontent.com/openai/math/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/preprints/Foulkes-Conjecture-for-the-Sixth-Symmetric-Power-September-25-2026/build/sections/introduction.tex,thm:main,"preprints/Foulkes-Conjecture-for-the-Sixth-Symmetric-Power-September-25-2026/build/sections/introduction.tex lines 1-18 (context):
\section{The result and its context}

Foulkes' conjecture compares two ways of composing symmetric powers.
For a finite-dimensional complex vector space \(V\) and integers
\(1\le a\le b\), it asks for a \(\GL(V)\)-equivariant injection
\[
             \Sym^a(\Sym^b V)\hookrightarrow\Sym^b(\Sym^a V).
\]
Over \(\CC\), complete reducibility makes this equivalent to an
inequality between the multiplicities of every irreducible polynomial
representation. Write \(\Schur_\lambda V\) for the Schur module indexed
by a partition \(\lambda\), and \(s_\lambda\) for its character.
If \(h_r\) denotes the complete homogeneous symmetric function, then
\(h_a[h_b]\) is the stable character of \(\Sym^a(\Sym^b V)\), where
brackets denote plethysm. The conjecture says that
\(h_b[h_a]-h_a[h_b]\) is Schur-positive: all its Schur coefficients
are nonnegative integers. We prove the sixth case.



preprints/Foulkes-Conjecture-for-the-Sixth-Symmetric-Power-September-25-2026/build/sections/introduction.tex lines 19-26 (statement):
\begin{theorem}\label{thm:main}
For every integer \(b\ge6\) and every finite-dimensional complex vector
space \(V\), there is a \(\GL(V)\)-equivariant injection
\[
                 \Sym^6(\Sym^b V)\ \hookrightarrow\ \Sym^b(\Sym^6 V).
\]
Equivalently, \(h_b[h_6]-h_6[h_b]\) is Schur-positive for every \(b\ge6\).
\end{theorem}"
T08,T08.06,scope_exception,retained,E: No restriction on finite dim V beyond the stated domain; b=6 is included. Anchor N: “no restriction” and “diagonal case”.,"The result establishes the sixth case of Foulkes’ conjecture, for every finite dimension of \(V\). When \(b=6\), the two spaces coincide; the substantive claim covers all \(b>6\).",,The diagonal case and unrestricted finite dimension are explicit.,https://raw.githubusercontent.com/openai/math/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/preprints/Foulkes-Conjecture-for-the-Sixth-Symmetric-Power-September-25-2026/build/sections/introduction.tex,thm:main,"preprints/Foulkes-Conjecture-for-the-Sixth-Symmetric-Power-September-25-2026/build/sections/introduction.tex lines 26-31 (scope_note):


Theorem~\ref{thm:main} resolves this case of Foulkes' conjecture
positively, with no restriction on \(\dim V\). The diagonal case
\(b=6\) is immediate; the content is the comparison for every larger
\(b\)."
T09,T09.01,assumption_domain,retained,H/D: P subset M is an inclusion of type II_1 factors. Anchor S.,"- **Type \(\mathrm{II}_1\) factors** are infinite-dimensional operator algebras with a normalized trace, which acts like a notion of averaging, and whose central elements are just scalar multiples of the identity.
- The inclusion \(P\subset M\) is **irreducible** if the only elements of \(M\) that commute with every element of \(P\) are scalars. Thus, although \(P\) may be smaller than \(M\), it has no nontrivial commuting counterpart inside \(M\).
- **Separable predual** supplies a countability condition: the associated space \(L^2(M,\tau)\) is separable. It does not say that \(M\) is separable in operator norm.",,The response explicitly identifies type II_1 factors and the inclusion; its opening slogan is qualified by the subsequent assumptions.,https://raw.githubusercontent.com/openai/math/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/preprints/Relative-generation-and-the-generator-problem-for-finite-factors-September-23-2026/build/sections/introduction.tex,thm:relative-generation,"preprints/Relative-generation-and-the-generator-problem-for-finite-factors-September-23-2026/build/sections/introduction.tex lines 19-27 (statement):
\begin{theorem}\label{thm:relative-generation}
Let $P\subset M$ be an irreducible inclusion of type $\mathrm{II}_1$
factors, with $M$ of separable predual. The set
\[
 \{u\in\mathcal U(M):W^*(P,u)=M\}
\]
is a dense $G_\delta$ subset of the unitary group $\mathcal U(M)$
for the trace $2$-norm topology.
\end{theorem}"
T09,T09.02,assumption,retained,"H: Inclusion is irreducible, defined by P-prime intersect M = complex scalar multiples of the identity. Anchor C: “irreducible”.","- **Type \(\mathrm{II}_1\) factors** are infinite-dimensional operator algebras with a normalized trace, which acts like a notion of averaging, and whose central elements are just scalar multiples of the identity.
- The inclusion \(P\subset M\) is **irreducible** if the only elements of \(M\) that commute with every element of \(P\) are scalars. Thus, although \(P\) may be smaller than \(M\), it has no nontrivial commuting counterpart inside \(M\).
- **Separable predual** supplies a countability condition: the associated space \(L^2(M,\tau)\) is separable. It does not say that \(M\) is separable in operator norm.",,The commutant condition is faithfully restated as commuting with every element of P only for scalars.,https://raw.githubusercontent.com/openai/math/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/preprints/Relative-generation-and-the-generator-problem-for-finite-factors-September-23-2026/build/sections/introduction.tex,thm:relative-generation,"preprints/Relative-generation-and-the-generator-problem-for-finite-factors-September-23-2026/build/sections/introduction.tex lines 1-18 (context):
\section{Introduction}
\label{sec:introduction}

For a subset $S$ of a von Neumann algebra $M$, write $W^*(S)$ for the
smallest unital von Neumann subalgebra containing $S$. Thus adjoints
are included and generation is taken in the weak operator topology.
The generator problem asks whether every von Neumann algebra with
separable predual has the form $W^*(x)$ for one bounded operator $x$.
Equivalently, it asks for two self-adjoint generators: their sum
$a+ib$ recovers $a$ and $b$ as its real and imaginary parts.

We prove this assertion for type $\mathrm{II}_1$ factors by first
establishing a relative generation theorem. For a finite factor with
normalized trace $\tau$, put $\norm{x}_2=\tau(x^*x)^{1/2}$.
An inclusion $P\subset M$ is \emph{irreducible} when
$P'\cap M=\mathbb C1$. Write $\mathcal U(M)$ for the unitary group
of $M$.



preprints/Relative-generation-and-the-generator-problem-for-finite-factors-September-23-2026/build/sections/introduction.tex lines 19-27 (statement):
\begin{theorem}\label{thm:relative-generation}
Let $P\subset M$ be an irreducible inclusion of type $\mathrm{II}_1$
factors, with $M$ of separable predual. The set
\[
 \{u\in\mathcal U(M):W^*(P,u)=M\}
\]
is a dense $G_\delta$ subset of the unitary group $\mathcal U(M)$
for the trace $2$-norm topology.
\end{theorem}"
T09,T09.03,assumption_domain,retained,H/D: M has separable predual. Anchor S; N distinguishes this from operator-norm separability.,"- **Type \(\mathrm{II}_1\) factors** are infinite-dimensional operator algebras with a normalized trace, which acts like a notion of averaging, and whose central elements are just scalar multiples of the identity.
- The inclusion \(P\subset M\) is **irreducible** if the only elements of \(M\) that commute with every element of \(P\) are scalars. Thus, although \(P\) may be smaller than \(M\), it has no nontrivial commuting counterpart inside \(M\).
- **Separable predual** supplies a countability condition: the associated space \(L^2(M,\tau)\) is separable. It does not say that \(M\) is separable in operator norm.",,Separable predual is named as an assumption and correctly distinguished from norm separability. The L2 discussion explains its consequence rather than substituting an unrelated assumption.,https://raw.githubusercontent.com/openai/math/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/preprints/Relative-generation-and-the-generator-problem-for-finite-factors-September-23-2026/build/sections/introduction.tex,thm:relative-generation,"preprints/Relative-generation-and-the-generator-problem-for-finite-factors-September-23-2026/build/sections/introduction.tex lines 19-27 (statement):
\begin{theorem}\label{thm:relative-generation}
Let $P\subset M$ be an irreducible inclusion of type $\mathrm{II}_1$
factors, with $M$ of separable predual. The set
\[
 \{u\in\mathcal U(M):W^*(P,u)=M\}
\]
is a dense $G_\delta$ subset of the unitary group $\mathcal U(M)$
for the trace $2$-norm topology.
\end{theorem}

preprints/Relative-generation-and-the-generator-problem-for-finite-factors-September-23-2026/build/sections/introduction.tex lines 35-39 (scope_note):
Throughout the paper,
\emph{separable} for a finite von Neumann algebra means that its
predual is separable. This ensures separability of $L^2(M,\tau)$;
it does not require operator-norm separability of $M$.

"
T09,T09.04,definition_domain,retained,D/R: u ranges over unitary elements of M. Anchor S: U(M); C defines that notation.,"A unitary \(u\) satisfies \(u^*u=uu^*=1\). The statement
\[
W^*(P,u)=M
\]
means that starting with \(P\) and \(u\), taking algebraic combinations and adjoints, and closing in the weak operator topology produces every element of \(M\).",,The unitary domain and defining identities are explicit.,https://raw.githubusercontent.com/openai/math/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/preprints/Relative-generation-and-the-generator-problem-for-finite-factors-September-23-2026/build/sections/introduction.tex,thm:relative-generation,"preprints/Relative-generation-and-the-generator-problem-for-finite-factors-September-23-2026/build/sections/introduction.tex lines 1-18 (context):
\section{Introduction}
\label{sec:introduction}

For a subset $S$ of a von Neumann algebra $M$, write $W^*(S)$ for the
smallest unital von Neumann subalgebra containing $S$. Thus adjoints
are included and generation is taken in the weak operator topology.
The generator problem asks whether every von Neumann algebra with
separable predual has the form $W^*(x)$ for one bounded operator $x$.
Equivalently, it asks for two self-adjoint generators: their sum
$a+ib$ recovers $a$ and $b$ as its real and imaginary parts.

We prove this assertion for type $\mathrm{II}_1$ factors by first
establishing a relative generation theorem. For a finite factor with
normalized trace $\tau$, put $\norm{x}_2=\tau(x^*x)^{1/2}$.
An inclusion $P\subset M$ is \emph{irreducible} when
$P'\cap M=\mathbb C1$. Write $\mathcal U(M)$ for the unitary group
of $M$.



preprints/Relative-generation-and-the-generator-problem-for-finite-factors-September-23-2026/build/sections/introduction.tex lines 19-27 (statement):
\begin{theorem}\label{thm:relative-generation}
Let $P\subset M$ be an irreducible inclusion of type $\mathrm{II}_1$
factors, with $M$ of separable predual. The set
\[
 \{u\in\mathcal U(M):W^*(P,u)=M\}
\]
is a dense $G_\delta$ subset of the unitary group $\mathcal U(M)$
for the trace $2$-norm topology.
\end{theorem}"
T09,T09.05,definition_conclusion,retained,"R: Generation means W*(P,u)=M; adjoints and weak operator closure are included, and the existing subfactor P is retained. Anchor S and C: definition of W*.","A unitary \(u\) satisfies \(u^*u=uu^*=1\). The statement
\[
W^*(P,u)=M
\]
means that starting with \(P\) and \(u\), taking algebraic combinations and adjoints, and closing in the weak operator topology produces every element of \(M\).",,"Adjoints, closure and retention of P correctly explain generation.",https://raw.githubusercontent.com/openai/math/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/preprints/Relative-generation-and-the-generator-problem-for-finite-factors-September-23-2026/build/sections/introduction.tex,thm:relative-generation,"preprints/Relative-generation-and-the-generator-problem-for-finite-factors-September-23-2026/build/sections/introduction.tex lines 1-18 (context):
\section{Introduction}
\label{sec:introduction}

For a subset $S$ of a von Neumann algebra $M$, write $W^*(S)$ for the
smallest unital von Neumann subalgebra containing $S$. Thus adjoints
are included and generation is taken in the weak operator topology.
The generator problem asks whether every von Neumann algebra with
separable predual has the form $W^*(x)$ for one bounded operator $x$.
Equivalently, it asks for two self-adjoint generators: their sum
$a+ib$ recovers $a$ and $b$ as its real and imaginary parts.

We prove this assertion for type $\mathrm{II}_1$ factors by first
establishing a relative generation theorem. For a finite factor with
normalized trace $\tau$, put $\norm{x}_2=\tau(x^*x)^{1/2}$.
An inclusion $P\subset M$ is \emph{irreducible} when
$P'\cap M=\mathbb C1$. Write $\mathcal U(M)$ for the unitary group
of $M$.



preprints/Relative-generation-and-the-generator-problem-for-finite-factors-September-23-2026/build/sections/introduction.tex lines 19-27 (statement):
\begin{theorem}\label{thm:relative-generation}
Let $P\subset M$ be an irreducible inclusion of type $\mathrm{II}_1$
factors, with $M$ of separable predual. The set
\[
 \{u\in\mathcal U(M):W^*(P,u)=M\}
\]
is a dense $G_\delta$ subset of the unitary group $\mathcal U(M)$
for the trace $2$-norm topology.
\end{theorem}"
T09,T09.06,conclusion,retained,"R: Such unitaries form a dense G_delta subset, not a claim about every unitary. Anchor S.","- **Dense:** for every unitary \(v\) and every \(\varepsilon>0\), there is a successful \(u\) with
  \[
  \|u-v\|_2=\tau\!\left((u-v)^*(u-v)\right)^{1/2}<\varepsilon.
  \]
  This measures closeness using the trace, rather than the stronger operator norm.
- **\(G_\delta\):** the successful choices form a countable intersection of open sets. Together with density, this means they are *generic in the Baire-category sense*—a topological notion of typicality, not a probability statement.",,Density and countable intersection of open sets are both explained without asserting every unitary succeeds.,https://raw.githubusercontent.com/openai/math/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/preprints/Relative-generation-and-the-generator-problem-for-finite-factors-September-23-2026/build/sections/introduction.tex,thm:relative-generation,"preprints/Relative-generation-and-the-generator-problem-for-finite-factors-September-23-2026/build/sections/introduction.tex lines 19-27 (statement):
\begin{theorem}\label{thm:relative-generation}
Let $P\subset M$ be an irreducible inclusion of type $\mathrm{II}_1$
factors, with $M$ of separable predual. The set
\[
 \{u\in\mathcal U(M):W^*(P,u)=M\}
\]
is a dense $G_\delta$ subset of the unitary group $\mathcal U(M)$
for the trace $2$-norm topology.
\end{theorem}"
T09,T09.07,definition_topology,retained,"D: Density and G_delta refer specifically to the trace 2-norm topology, with normalized trace tau and norm tau(x*x)^(1/2). Anchor S and C.","- **Dense:** for every unitary \(v\) and every \(\varepsilon>0\), there is a successful \(u\) with
  \[
  \|u-v\|_2=\tau\!\left((u-v)^*(u-v)\right)^{1/2}<\varepsilon.
  \]
  This measures closeness using the trace, rather than the stronger operator norm.
- **\(G_\delta\):** the successful choices form a countable intersection of open sets. Together with density, this means they are *generic in the Baire-category sense*—a topological notion of typicality, not a probability statement.",,The displayed trace 2-norm is correct; norm topology is explicitly distinguished.,https://raw.githubusercontent.com/openai/math/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/preprints/Relative-generation-and-the-generator-problem-for-finite-factors-September-23-2026/build/sections/introduction.tex,thm:relative-generation,"preprints/Relative-generation-and-the-generator-problem-for-finite-factors-September-23-2026/build/sections/introduction.tex lines 1-18 (context):
\section{Introduction}
\label{sec:introduction}

For a subset $S$ of a von Neumann algebra $M$, write $W^*(S)$ for the
smallest unital von Neumann subalgebra containing $S$. Thus adjoints
are included and generation is taken in the weak operator topology.
The generator problem asks whether every von Neumann algebra with
separable predual has the form $W^*(x)$ for one bounded operator $x$.
Equivalently, it asks for two self-adjoint generators: their sum
$a+ib$ recovers $a$ and $b$ as its real and imaginary parts.

We prove this assertion for type $\mathrm{II}_1$ factors by first
establishing a relative generation theorem. For a finite factor with
normalized trace $\tau$, put $\norm{x}_2=\tau(x^*x)^{1/2}$.
An inclusion $P\subset M$ is \emph{irreducible} when
$P'\cap M=\mathbb C1$. Write $\mathcal U(M)$ for the unitary group
of $M$.



preprints/Relative-generation-and-the-generator-problem-for-finite-factors-September-23-2026/build/sections/introduction.tex lines 19-27 (statement):
\begin{theorem}\label{thm:relative-generation}
Let $P\subset M$ be an irreducible inclusion of type $\mathrm{II}_1$
factors, with $M$ of separable predual. The set
\[
 \{u\in\mathcal U(M):W^*(P,u)=M\}
\]
is a dense $G_\delta$ subset of the unitary group $\mathcal U(M)$
for the trace $2$-norm topology.
\end{theorem}"
T10,T10.01,assumption_domain,not_explicit,H/D: Omega is a bounded connected domain in R³ with C-infinity boundary. Anchor S.,The theorem says that **complete measurements on the surface of a three-dimensional elastic body uniquely determine its elastic properties everywhere inside**.,Bounded and connected domain,"Three-dimensional Euclidean physical setting and smooth boundary are conveyed, but boundedness and connectedness are not explicit. A body suggests these informally without guaranteeing them.",https://raw.githubusercontent.com/openai/math/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/preprints/Global-Uniqueness-for-the-Smooth-Isotropic-Elasticity-Inverse-Problem-September-24-2026/build/sections/introduction.tex,thm:main,"preprints/Global-Uniqueness-for-the-Smooth-Isotropic-Elasticity-Inverse-Problem-September-24-2026/build/sections/introduction.tex lines 31-44 (statement):
\begin{theorem}\label{thm:main}
Let $\Omega\subset\R^3$ be any bounded connected domain with
$C^\infty$ boundary. For $j=1,2$, let
$\lambda_j,\mu_j\in C^\infty(\overline\Omega;\R)$ satisfy
$\mu_j>0$ and $3\lambda_j+2\mu_j>0$ on $\overline\Omega$.
If
\[
 \Lambda_{\lambda_1,\mu_1}
 =\Lambda_{\lambda_2,\mu_2}
 \colon H^{1/2}(\partial\Omega;\R^3)
       \longrightarrow H^{-1/2}(\partial\Omega;\R^3),
\]
then $\lambda_1=\lambda_2$ and $\mu_1=\mu_2$ throughout $\Omega$.
\end{theorem}"
T10,T10.02,assumption_domain,not_explicit,"H/D: Both coefficient pairs lambda_j,mu_j are real and C-infinity on the closure, j=1,2. Anchor S.",The positivity assumptions say that both resistances are positive. The theorem also assumes that these functions and the body’s boundary are smooth.,Real coefficient domain; regularity on the closure,"Smooth is an acceptable synonym for C-infinity in this context, and the two material coefficients are discussed. Real-valuedness and smoothness on the entire closure are not made explicit.",https://raw.githubusercontent.com/openai/math/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/preprints/Global-Uniqueness-for-the-Smooth-Isotropic-Elasticity-Inverse-Problem-September-24-2026/build/sections/introduction.tex,thm:main,"preprints/Global-Uniqueness-for-the-Smooth-Isotropic-Elasticity-Inverse-Problem-September-24-2026/build/sections/introduction.tex lines 31-44 (statement):
\begin{theorem}\label{thm:main}
Let $\Omega\subset\R^3$ be any bounded connected domain with
$C^\infty$ boundary. For $j=1,2$, let
$\lambda_j,\mu_j\in C^\infty(\overline\Omega;\R)$ satisfy
$\mu_j>0$ and $3\lambda_j+2\mu_j>0$ on $\overline\Omega$.
If
\[
 \Lambda_{\lambda_1,\mu_1}
 =\Lambda_{\lambda_2,\mu_2}
 \colon H^{1/2}(\partial\Omega;\R^3)
       \longrightarrow H^{-1/2}(\partial\Omega;\R^3),
\]
then $\lambda_1=\lambda_2$ and $\mu_1=\mu_2$ throughout $\Omega$.
\end{theorem}"
T10,T10.03,assumption,not_explicit,H: mu_j>0 and 3lambda_j+2mu_j>0 on the entire closure. Anchor S. Strict inequalities apply to both pairs.,The positivity assumptions say that both resistances are positive. The theorem also assumes that these functions and the body’s boundary are smooth.,Positivity on the entire closure,"Positive shear and bulk modulus is mathematically equivalent to mu>0 and 3lambda+2mu>0. The requirement throughout the closure, including its boundary, is not explicit. This is not the erroneous stronger claim lambda>0.",https://raw.githubusercontent.com/openai/math/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/preprints/Global-Uniqueness-for-the-Smooth-Isotropic-Elasticity-Inverse-Problem-September-24-2026/build/sections/introduction.tex,thm:main,"preprints/Global-Uniqueness-for-the-Smooth-Isotropic-Elasticity-Inverse-Problem-September-24-2026/build/sections/introduction.tex lines 31-44 (statement):
\begin{theorem}\label{thm:main}
Let $\Omega\subset\R^3$ be any bounded connected domain with
$C^\infty$ boundary. For $j=1,2$, let
$\lambda_j,\mu_j\in C^\infty(\overline\Omega;\R)$ satisfy
$\mu_j>0$ and $3\lambda_j+2\mu_j>0$ on $\overline\Omega$.
If
\[
 \Lambda_{\lambda_1,\mu_1}
 =\Lambda_{\lambda_2,\mu_2}
 \colon H^{1/2}(\partial\Omega;\R^3)
       \longrightarrow H^{-1/2}(\partial\Omega;\R^3),
\]
then $\lambda_1=\lambda_2$ and $\mu_1=\mu_2$ throughout $\Omega$.
\end{theorem}"
T10,T10.04,definition,retained,D: Lambda is the static isotropic displacement-to-traction operator defined using the stated strain/stress equation. Anchor C: eq:physical-operator and eq:physical-dn.,"Imagine imposing a displacement on the body’s entire boundary—moving each surface point by a prescribed amount—and measuring the surface forces needed to hold it in static equilibrium. The map \(\Lambda_{\lambda,\mu}\) records this displacement-to-force response for every possible boundary displacement.",,"Static equilibrium, isotropy and displacement-to-force response preserve the operator meaning without requiring the full weak formula.",https://raw.githubusercontent.com/openai/math/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/preprints/Global-Uniqueness-for-the-Smooth-Isotropic-Elasticity-Inverse-Problem-September-24-2026/build/sections/introduction.tex,thm:main,"preprints/Global-Uniqueness-for-the-Smooth-Isotropic-Elasticity-Inverse-Problem-September-24-2026/build/sections/introduction.tex lines 1-30 (context):
\section{Introduction}\label{sec:introduction}

Let $\Omega\subset\R^3$ be a bounded connected domain with smooth boundary.
For real functions $\lambda,\mu\in C^\infty(\overline\Omega)$ satisfying
\begin{equation}\label{eq:energy-positivity}
  \mu>0,\qquad 3\lambda+2\mu>0
  \quad\text{on }\overline\Omega,
\end{equation}
define the strain, stress, and static elasticity operator by
\begin{equation}\label{eq:physical-operator}
 e(u)=\frac{\nabla u+(\nabla u)^T}{2},\qquad
 \sigma_{\lambda,\mu}(u)=\lambda(\diverg u)I+2\mu e(u),\qquad
 L_{\lambda,\mu}u=\diverg\sigma_{\lambda,\mu}(u).
\end{equation}
Here $\mu$ is the shear modulus and $\lambda+2\mu/3$ is the bulk modulus.
The inequalities in \eqref{eq:energy-positivity} give uniformly positive
elastic energy. For each $f\in H^{1/2}(\partial\Omega;\R^3)$ there is a
unique $u_f\in H^1(\Omega;\R^3)$ with $L_{\lambda,\mu}u_f=0$ and trace $f$.
The displacement-to-traction map is defined weakly by
\begin{equation}\label{eq:physical-dn}
 \langle\Lambda_{\lambda,\mu}f,g\rangle
 =\int_\Omega\left[
 \lambda(\diverg u_f)(\diverg v_g)+2\mu e(u_f):e(v_g)
 \right]\dd x,
\end{equation}
where $v_g\in H^1(\Omega;\R^3)$ has trace $g$, and $A:B=\tr(A^TB)$.
The weak equation makes this expression independent of the extension.
For smooth boundary data it is the traction
$\sigma_{\lambda,\mu}(u_f)n$, with $n$ the outward normal.

"
T10,T10.05,assumption_conclusion,retained,"H/R: Equality is of the complete maps H^(1/2)(boundary;R³)→H^(−1/2)(boundary;R³), for the same domain. Anchor S: operator equality.","Its conclusion is that if two candidate materials in the same body give **exactly the same boundary forces for every imposed boundary displacement**, their coefficients must agree at every interior point:
\[
\lambda_1=\lambda_2,\qquad \mu_1=\mu_2.
\]
Thus no two distinct materials satisfying these assumptions can be indistinguishable under complete static boundary measurements. The \(H^{1/2}\) and \(H^{-1/2}\) notation specifies the mathematical function spaces for boundary displacements and forces.",,Exact equality of the full operator for the same body is explicit; boundary displacement/force Sobolev spaces are identified. Their vector codomain is conveyed physically.,https://raw.githubusercontent.com/openai/math/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/preprints/Global-Uniqueness-for-the-Smooth-Isotropic-Elasticity-Inverse-Problem-September-24-2026/build/sections/introduction.tex,thm:main,"preprints/Global-Uniqueness-for-the-Smooth-Isotropic-Elasticity-Inverse-Problem-September-24-2026/build/sections/introduction.tex lines 31-44 (statement):
\begin{theorem}\label{thm:main}
Let $\Omega\subset\R^3$ be any bounded connected domain with
$C^\infty$ boundary. For $j=1,2$, let
$\lambda_j,\mu_j\in C^\infty(\overline\Omega;\R)$ satisfy
$\mu_j>0$ and $3\lambda_j+2\mu_j>0$ on $\overline\Omega$.
If
\[
 \Lambda_{\lambda_1,\mu_1}
 =\Lambda_{\lambda_2,\mu_2}
 \colon H^{1/2}(\partial\Omega;\R^3)
       \longrightarrow H^{-1/2}(\partial\Omega;\R^3),
\]
then $\lambda_1=\lambda_2$ and $\mu_1=\mu_2$ throughout $\Omega$.
\end{theorem}"
T10,T10.06,conclusion,retained,R: It implies both lambda_1=lambda_2 and mu_1=mu_2 throughout Omega. Anchor S: conclusion.,"Its conclusion is that if two candidate materials in the same body give **exactly the same boundary forces for every imposed boundary displacement**, their coefficients must agree at every interior point:
\[
\lambda_1=\lambda_2,\qquad \mu_1=\mu_2.
\]
Thus no two distinct materials satisfying these assumptions can be indistinguishable under complete static boundary measurements. The \(H^{1/2}\) and \(H^{-1/2}\) notation specifies the mathematical function spaces for boundary displacements and forces.",,"Both coefficients are named in the equality, with the correct interior scope.",https://raw.githubusercontent.com/openai/math/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/preprints/Global-Uniqueness-for-the-Smooth-Isotropic-Elasticity-Inverse-Problem-September-24-2026/build/sections/introduction.tex,thm:main,"preprints/Global-Uniqueness-for-the-Smooth-Isotropic-Elasticity-Inverse-Problem-September-24-2026/build/sections/introduction.tex lines 31-44 (statement):
\begin{theorem}\label{thm:main}
Let $\Omega\subset\R^3$ be any bounded connected domain with
$C^\infty$ boundary. For $j=1,2$, let
$\lambda_j,\mu_j\in C^\infty(\overline\Omega;\R)$ satisfy
$\mu_j>0$ and $3\lambda_j+2\mu_j>0$ on $\overline\Omega$.
If
\[
 \Lambda_{\lambda_1,\mu_1}
 =\Lambda_{\lambda_2,\mu_2}
 \colon H^{1/2}(\partial\Omega;\R^3)
       \longrightarrow H^{-1/2}(\partial\Omega;\R^3),
\]
then $\lambda_1=\lambda_2$ and $\mu_1=\mu_2$ throughout $\Omega$.
\end{theorem}"
T10,T10.07,data_scope,not_explicit,E: Data are the full zero-frequency boundary operator in fixed Euclidean coordinates. Anchor N.,"Its conclusion is that if two candidate materials in the same body give **exactly the same boundary forces for every imposed boundary displacement**, their coefficients must agree at every interior point:
\[
\lambda_1=\lambda_2,\qquad \mu_1=\mu_2.
\]
Thus no two distinct materials satisfying these assumptions can be indistinguishable under complete static boundary measurements. The \(H^{1/2}\) and \(H^{-1/2}\) notation specifies the mathematical function spaces for boundary displacements and forces.",Fixed Euclidean coordinates,Full static (zero-frequency) data are retained; comparison in fixed Euclidean coordinates is not expressly stated. No coordinate-gauge or partial-data claim is introduced.,https://raw.githubusercontent.com/openai/math/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/preprints/Global-Uniqueness-for-the-Smooth-Isotropic-Elasticity-Inverse-Problem-September-24-2026/build/sections/introduction.tex,thm:main,"preprints/Global-Uniqueness-for-the-Smooth-Isotropic-Elasticity-Inverse-Problem-September-24-2026/build/sections/introduction.tex lines 44-51 (scope_note):


Thus Theorem~\ref{thm:main} gives a positive resolution of the smooth
three-dimensional isotropic elastic Calder\'on uniqueness problem under
\eqref{eq:energy-positivity}. It uses the full zero-frequency boundary
operator in fixed Euclidean coordinates. The coefficients need not be
analytic or close to constants, and their agreement near the boundary is
not an additional hypothesis."
T10,T10.08,scope_exception,retained,"E: Analyticity, closeness to constants, and prior coefficient agreement near the boundary are not additional hypotheses. Anchor N.","The claim allows substantial spatial variation: the coefficients need not be nearly constant, analytic, or already known near the boundary. It is a **uniqueness statement**; by itself, it does not provide a reconstruction algorithm or say how reliably one could recover the coefficients from noisy or incomplete measurements.",,"The absent extra hypotheses are preserved in ordinary wording. Known near the boundary is a reasonable informal rendering of no required prior agreement, not a claim of noisy-data stability.",https://raw.githubusercontent.com/openai/math/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/preprints/Global-Uniqueness-for-the-Smooth-Isotropic-Elasticity-Inverse-Problem-September-24-2026/build/sections/introduction.tex,thm:main,"preprints/Global-Uniqueness-for-the-Smooth-Isotropic-Elasticity-Inverse-Problem-September-24-2026/build/sections/introduction.tex lines 44-51 (scope_note):


Thus Theorem~\ref{thm:main} gives a positive resolution of the smooth
three-dimensional isotropic elastic Calder\'on uniqueness problem under
\eqref{eq:energy-positivity}. It uses the full zero-frequency boundary
operator in fixed Euclidean coordinates. The coefficients need not be
analytic or close to constants, and their agreement near the boundary is
not an additional hypothesis."
