# Revision evidence: three selected OpenAI math papers — updated provenance

These compare older snapshots tied by the supplied initial-commit tree records to adc7f1241b42e322a6451854ab7e4b4c146bf78a against revised snapshots at fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb. Both dated editions are also retained at the later commit. The supplied archive, not a new live GitHub fetch, is the evidence source. The sample is deliberately selected and does not estimate repository-wide revision or correctness rates.

## Provenance checks

Archive SHA-256: `0223b5738ae94c6b16b1f3313f87a315c98e85b3a5195eb94462433bcd6de9e2`

All 91 manifest-listed paper files passed byte-length, SHA-256, and Git blob SHA-1 checks. The six archive-level files (LICENSE, README.md, history.md, provenance.json, repository-commits.json, repository-root-tree.json) are not entries in that manifest. Hash agreement verifies archive/manifest consistency, not independent authentication against GitHub.

## Supplied commit metadata

- [fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb](https://github.com/openai/math/commit/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb) — 2026-10-08T05:20:00Z; Merge pull request #1 from openai/codex/update-10-7  Update manuscripts and Lean formalizations; parents: adc7f1241b42e322a6451854ab7e4b4c146bf78a, 301488868beec11bfd897168433b0a64f5258559
- [301488868beec11bfd897168433b0a64f5258559](https://github.com/openai/math/commit/301488868beec11bfd897168433b0a64f5258559) — 2026-10-08T05:03:50Z; Update manuscripts and Lean formalizations; parents: adc7f1241b42e322a6451854ab7e4b4c146bf78a
- [adc7f1241b42e322a6451854ab7e4b4c146bf78a](https://github.com/openai/math/commit/adc7f1241b42e322a6451854ab7e4b4c146bf78a) — 2026-10-06T21:58:50Z; Initial commit; parents: none

The supplemental initial-commit records now tie all 45 older paper files to the initial commit by Git blob identity: Taming 12, Box Transport 19, BSD 14. Each blob hash was recomputed from the actual bytes in the original snapshots ZIP and checked against the original manifest, supplemental tree entries, and supplemental comparison rows. All sets and file sizes agree. Every recorded paper subtree hash was reconstructed, as was the initial root-tree hash, which matches the initial commit metadata in the original archive.

The older-version links below now use the initial commit; revised-version links use the later merge commit. The 45 older files remain byte-identical to their copies retained in the later archive. This strengthens historical provenance without changing the mathematical comparison.

Scope: these are offline checks against user-supplied GitHub records, not a fresh GitHub fetch or signature authentication. The full intermediate preprints-tree listing is not included; its ID is checked against the root record, but the edge linking each named paper tree to that preprints tree is supplied provenance rather than independently reconstructed.

## Supplemental provenance results

Verification ZIP SHA-256: `88a8b9eee5ee4d6eac1f7ea95bdb0331fa4960c3b25cb6dc7559b0e5ee254cd3`

| Older paper folder | Checked files | Reconstructed paper tree |
|---|---:|---|
| Exact-Birch-Swinnerton-Dyer-Formula-from-Low-Selmer-Corank-October-3-2026 | 14 | `702b0295a52d2a37ebefd9860e94c217cb39a06e` |
| Incompressible-Box-Transport-and-Finite-Computation-September-27-2026 | 19 | `01d334b9517b0eadc85ad1d8d93cd24823af33a3` |
| Taming-implies-compatibility-on-four-manifolds-September-23-2026 | 12 | `1482e92ff4ea679f0091a20f11a1d20ec0f38701` |

Initial root tree: `a8e3481a92772ee311cdc9dd6409cd7b927a3fc1`. Preprints tree ID: `272ca94583c1dca3188bbc3a1caa068cb7af4b55`.

Supplemental record hashes:

- `initial-snapshot-note.md`: SHA-256 `29a50cdb768546674069f9b8a4b5995d62b106fba5cbf2eb4a62157db502b85c`.
- `initial-commit-paper-trees.json`: SHA-256 `5c9e6e92f6342d5b88c32172bc557744b57164e96b386e50cd4a508b0787327e`.
- `initial-snapshot-comparison.json`: SHA-256 `7817f2e298cc2f26a84f999dab71868c14e76d55822b29a17efcf5d124992c8e`.

## Findings

- Taming: main existence theorem unchanged; unconditional subsidiary cone equality replaced by a conditional corollary, a cone-characterization theorem, and a strict-inclusion example. Sections 2–7 are byte-identical.
- Box Transport: headline geometric and computational theorem sources unchanged; Proposition 2.4 and its proof specify effective projection estimates and derivative-envelope requirements; the input-offset clock proof supplies a common decay envelope.
- BSD: main theorem unchanged; only an introductory citation key is deleted from the mathematical sections. Bibliography entries point to revised companion editions, which are not included here. Sections 2–10 are byte-identical.

No mathematical correctness certification, TeX rebuild, Lean execution, or validation of external companion proofs was performed. PDFs were text-extracted to cross-check the main and changed passages.

## Snapshot inventory and complete textual diffs

### Taming

Before: [Taming-implies-compatibility-on-four-manifolds-September-23-2026](https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Taming-implies-compatibility-on-four-manifolds-September-23-2026)
After: [Taming-implies-compatibility-on-four-manifolds-October-6-2026](https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/preprints/Taming-implies-compatibility-on-four-manifolds-October-6-2026)

PDF page counts: 18 → 23.

| Relative file | Status | Before bytes | After bytes |
|---|---|---:|---:|
| `README.md` | changed | 590 | 880 |
| `build/main.tex` | changed | 733 | 888 |
| `build/preamble.tex` | changed | 1599 | 1638 |
| `build/references.bib` | changed | 9744 | 9738 |
| `build/sections/01-introduction.tex` | changed | 10275 | 10307 |
| `build/sections/02-currents.tex` | identical | 6409 | 6409 |
| `build/sections/03-mass.tex` | identical | 5810 | 5810 |
| `build/sections/04-energy.tex` | identical | 9544 | 9544 |
| `build/sections/05-density.tex` | identical | 10533 | 10533 |
| `build/sections/06-splitting.tex` | identical | 10225 | 10225 |
| `build/sections/07-conclusion.tex` | identical | 2584 | 2584 |
| `build/sections/08-cones.tex` | added | — | 16943 |
| `paper.pdf` | changed | 216875 | 249763 |

#### README.md

```diff
--- adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Taming-implies-compatibility-on-four-manifolds-September-23-2026/README.md

+++ fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/preprints/Taming-implies-compatibility-on-four-manifolds-October-6-2026/README.md

@@ -2,17 +2,20 @@

 
 **Author:** OpenAI
 
-**Date:** September 23, 2026
+**Date:** October 6, 2026
 
 ## Citation
 
 ```bibtex
-@misc{OAI:Taming-implies-compatibility-on-four-manifolds-September-23-2026,
+@misc{OAI:Taming-implies-compatibility-on-four-manifolds-October-6-2026,
   author = {{OpenAI}},
   title = {{Taming implies compatibility on four-manifolds}},
   howpublished = {OpenAI Math Release preprint
-                  \href{https://github.com/openai/math/blob/main/preprints/Taming-implies-compatibility-on-four-manifolds-September-23-2026/paper.pdf}{OAI:Taming-implies-compatibility-on-four-manifolds-September-23-2026}},
+                  \href{https://github.com/openai/math/blob/main/preprints/Taming-implies-compatibility-on-four-manifolds-October-6-2026/paper.pdf}{OAI:Taming-implies-compatibility-on-four-manifolds-October-6-2026}},
   year = {2026}
 }
 ```
 
+## Version note
+
+This version characterizes the compatible cone by intersection positivity, restricts the cone-sum equality to $h_J^-=b_2^+-1$, and adds a four-torus example of strict inclusion; see the [previous version](../Taming-implies-compatibility-on-four-manifolds-September-23-2026/paper.pdf).
```

#### build/main.tex

```diff
--- adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Taming-implies-compatibility-on-four-manifolds-September-23-2026/build/main.tex

+++ fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/preprints/Taming-implies-compatibility-on-four-manifolds-October-6-2026/build/main.tex

@@ -2,13 +2,15 @@

 \input{preamble}
 \title{Taming implies compatibility on four-manifolds}
 \author{OpenAI}
-\date{September 23, 2026}
+\date{October 6, 2026}
 \begin{document}
 \maketitle
 \begin{abstract}
 We prove that every smooth almost complex structure on a closed four-manifold
 which is tamed by a symplectic form is compatible with a symplectic form.
 This gives a positive solution to Donaldson's tamed-to-compatible conjecture.
+We also characterize the compatible cone by strict intersection positivity
+on the closure of the invariant part of the taming cone.
 \end{abstract}
 \input{sections/01-introduction}
 \input{sections/02-currents}
@@ -17,6 +19,7 @@

 \input{sections/05-density}
 \input{sections/06-splitting}
 \input{sections/07-conclusion}
+\input{sections/08-cones}
 \begingroup
 \small
 \bibliographystyle{plain}
```

#### build/preamble.tex

```diff
--- adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Taming-implies-compatibility-on-four-manifolds-September-23-2026/build/preamble.tex

+++ fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/preprints/Taming-implies-compatibility-on-four-manifolds-October-6-2026/build/preamble.tex

@@ -17,6 +17,7 @@

 \newtheorem{corollary}[theorem]{Corollary}
 \theoremstyle{remark}
 \newtheorem{remark}[theorem]{Remark}
+\newtheorem{example}[theorem]{Example}
 \numberwithin{equation}{section}
 \newcommand{\R}{\mathbb R}
 \newcommand{\vol}{\,dV_g}
```

#### build/references.bib

```diff
--- adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Taming-implies-compatibility-on-four-manifolds-September-23-2026/build/references.bib

+++ fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/preprints/Taming-implies-compatibility-on-four-manifolds-October-6-2026/build/references.bib

@@ -290,6 +290,6 @@

   author = {{OpenAI}},
   title = {{Deforming hypersymplectic four-manifolds to hyperk{\"a}hler triples}},
   howpublished = {OpenAI Math Release preprint
-                  \href{https://github.com/openai/math/blob/main/preprints/Deforming-hypersymplectic-four-manifolds-to-hyperkahler-triples-September-23-2026/paper.pdf}{OAI:Deforming-hypersymplectic-four-manifolds-to-hyperkahler-triples-September-23-2026}},
+                  \href{https://github.com/openai/math/blob/main/preprints/Deforming-hypersymplectic-four-manifolds-to-hyperkahler-triples-October-7-2026/paper.pdf}{OAI:Deforming-hypersymplectic-four-manifolds-to-hyperkahler-triples-October-7-2026}},
   year = {2026}
 }
```

#### build/sections/01-introduction.tex

```diff
--- adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Taming-implies-compatibility-on-four-manifolds-September-23-2026/build/sections/01-introduction.tex

+++ fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/preprints/Taming-implies-compatibility-on-four-manifolds-October-6-2026/build/sections/01-introduction.tex

@@ -31,33 +31,50 @@

 the compatible form. In particular, it does not assert that an
 arbitrary taming class contains a compatible representative.
 
-Combining Theorem~\ref{thm:main} with Li and Zhang's cone comparison
-gives the following cohomological refinement.
-
-\begin{corollary}[Taming and compatible cones]\label{cor:taming-cones}
+There is also an intrinsic description of the compatible cone in terms
+of the taming cone and the intersection pairing.
+
+\begin{theorem}[Taming and compatible cones]\label{thm:taming-cones}
 Let \(X,J\) satisfy the hypotheses of Theorem~\ref{thm:main}, and give
 \(X\) the orientation induced by \(J\). Let
-\(\mathcal K_J^t,\mathcal K_J^c\subset H^2(X;\R)\) be the real de Rham
-classes represented by \(J\)-taming and \(J\)-compatible symplectic
-forms, respectively. Let \(H_J^-\) be the subspace represented by
-smooth closed real two-forms \(\alpha\) satisfying
-\(\alpha(Ju,Jv)=-\alpha(u,v)\). Then
-\[
- \mathcal K_J^t=\mathcal K_J^c+H_J^-,
-\]
-where the right side is a Minkowski sum. Write \(b_2^+(X)\) for
-the positive index of the intersection form. If \(b_2^+(X)=1\), then
-\(\mathcal K_J^t=\mathcal K_J^c\). In that case, every \(J\)-taming
-symplectic form \(\omega\) has a smooth \(J\)-compatible symplectic
-representative \(\eta\) with \([\eta]=[\omega]\) in \(H^2(X;\R)\).
+\(\mathcal K_J^t,\mathcal K_J^c\subset H^2(X;\R)\) be the classes
+represented by taming and compatible symplectic forms, respectively.
+Let \(H_J^-\) be the subspace represented by smooth closed real
+anti-invariant two-forms. Set
+\[
+ \mathcal V=(H_J^-)^\perp,\qquad
+ \mathcal C=\mathcal K_J^t\cap\mathcal V,
+\]
+where orthogonality is for the intersection pairing. Then
+\begin{align}
+ \mathcal K_J^t&=\mathcal C+H_J^-,\label{eq:tame-decomposition}\\
+ \mathcal K_J^c&=
+ \{a\in\mathcal C:\ a\cdot y>0
+   \text{ for every }0\ne y\in\overline{\mathcal C}\}.
+ \label{eq:compatible-duality}
+\end{align}
+The closure is taken in \(\mathcal V\), and the sum in
+\eqref{eq:tame-decomposition} is a Minkowski sum.
+\end{theorem}
+
+\begin{corollary}\label{cor:taming-cones}
+Under the same hypotheses, write \(h_J^-=\dim H_J^-\) and let
+\(b_2^+(X)\) be the positive index of the intersection form.
+If \(h_J^-=b_2^+(X)-1\), then
+\[
+ \mathcal K_J^t=\mathcal K_J^c+H_J^-.
+\]
+In particular, if \(b_2^+(X)=1\), then
+\(\mathcal K_J^t=\mathcal K_J^c\): every taming class has a smooth
+compatible symplectic representative.
 \end{corollary}
 
-\begin{proof}
-Theorem~\ref{thm:main} makes \(\mathcal K_J^c\) nonempty. Li and Zhang's
-four-dimensional cone comparison \cite[Corollary~1.1]{LiZhang2009}
-therefore gives the cone identity and, when \(b_2^+=1\), equality of
-the two cones. This equality gives the claimed representative.
-\end{proof}
+The proofs are given in Section~\ref{sec:cones}. They use separation
+in a prescribed cohomology class and the current estimates below.
+The inclusion
+\(\mathcal K_J^c+H_J^-\subseteq\mathcal K_J^t\)
+can be strict when \(h_J^-<b_2^+-1\), even if \(H_J^-=0\);
+Example~\ref{ex:strict-cones} gives an explicit four-torus.
 
 The integrable case belongs to the theory of compact complex surfaces.
 Buchdahl \cite[Theorem~11]{Buchdahl1999} and Lamari
@@ -65,10 +82,9 @@

 that even first Betti number implies the existence of a K\"ahler metric.
 Lamari's positive exact current on a surface with odd first Betti number
 also obstructs a taming form \cite[Theorem~6.1]{Lamari1999}.
-Li and Zhang proved the equivalence of taming and compatibility for
-complex surfaces and compared the two cohomology cones under the
-assumption that a compatible form already exists
-\cite[Theorem~1.2 and Corollary~1.1]{LiZhang2009}.
+Li and Zhang proved the equivalence between the existence of taming
+and compatible symplectic forms on complex surfaces
+\cite[Theorem~1.2]{LiZhang2009}.
 
 The pseudoholomorphic-curve approach has a different starting point.
 For structures on \(\mathbb{CP}^2\) tamed by the standard symplectic
@@ -77,9 +93,8 @@

 Taubes developed integration over curve moduli spaces to obtain
 compatibility for a residual subset of the smooth almost complex
 structures tamed by a fixed symplectic form when \(b_2^+=1\)
-\cite[Theorem~1]{Taubes2011}; a later correction addresses the
-\(b_1=2\) case \cite[Section~1]{Taubes2017}.
-His original Theorem~1 also states preservation of the taming
+\cite{Taubes2011,Taubes2017}.
+Theorem~1 of \cite{Taubes2011} also states preservation of the taming
 form's cohomology class when that class is rational.
 Li and Zhang extended this approach to every tamed structure on
 \(S^2\times S^2\) and
@@ -91,18 +106,11 @@

 \(\alpha(Ju,Jv)=\alpha(u,v)\); an anti-invariant one satisfies
 \(\alpha(Ju,Jv)=-\alpha(u,v)\).
 Write \(h_J^-=\dim H_J^-\), where \(H_J^-\) is the anti-invariant
-cohomology space defined in Corollary~\ref{cor:taming-cones}.
-Tan, Wang, Zhou and Zhu published an affirmative result under
-\(h_J^-=b_2^+-1\), using a strategy related to Buchdahl's
-complex-surface argument \cite[Theorem~1.1]{TWZZ2022}.
-Lin and Zhou subsequently identified difficulties in specific local
-estimates and connection formulas in that approach
-\cite[Remarks~3.4 and~3.6, Section~3.2]{LinZhou2025}; their note gives no counterexample
-to the geometric theorem.
-The same cohomological restriction remains in the compatibility
-criteria of Wang, Wang and Zhu
-\cite[Theorems~4.3 and~5.1]{WangWangZhu2023}, and in the taming
-corollary of Wang, Zhang, Zheng and Zhu
+cohomology space defined in Theorem~\ref{thm:taming-cones}.
+The condition \(h_J^-=b_2^+-1\) occurs in Buchdahl-type approaches
+\cite{TWZZ2022,LinZhou2025}, in the compatibility criteria of
+Wang, Wang and Zhu \cite[Theorems~4.3 and~5.1]{WangWangZhu2023},
+and in the taming corollary of Wang, Zhang, Zheng and Zhu
 \cite[Corollary~1.3]{WangZhangZhengZhu2025}.
 The latter paper's generalized Monge--Amp\`ere theorem starts with
 an almost K\"ahler structure.
@@ -192,7 +200,9 @@

 existence and the planar tangent description;
 Section~\ref{sec:splitting} uses them to control cutoff boundaries
 and prove closedness of \(P_E\).
-Section~\ref{sec:conclusion} completes the compatibility argument.
+Section~\ref{sec:conclusion} completes the existence argument.
+Section~\ref{sec:cones} develops the prescribed-class separation
+and proves the cone description.
 
 A triple of closed two-forms on an oriented four-manifold is
 \emph{hypersymplectic} if every nonzero real linear combination is
```

#### build/sections/08-cones.tex

```diff
--- adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Taming-implies-compatibility-on-four-manifolds-September-23-2026/build/sections/08-cones.tex

+++ fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/preprints/Taming-implies-compatibility-on-four-manifolds-October-6-2026/build/sections/08-cones.tex

@@ -0,0 +1,418 @@

+\section{The compatible cone}\label{sec:cones}
+
+The distinction between existence and realization of a specified class
+appears in the separating functional. For existence, it annihilates all
+closed invariant forms. For a specified class, it need only annihilate
+exact invariant forms. We first establish the estimates in this setting,
+then identify the classes of the resulting currents.
+
+\subsection{Invariant representatives and separation}
+
+We retain the Hermitian metric, the projection \(R\), and the closed
+lift \(B=dd^*G+H\) of Lemma~\ref{lem:lift}.
+For a closed current \(D\), let
+\(\mathfrak c(D)\in H^2(X;\R)\) denote its intersection-dual class:
+\[
+ D(\alpha)=\mathfrak c(D)\cdot[\alpha]
+ \quad\text{for every smooth closed two-form }\alpha.
+\]
+Equivalently, \(\mathfrak c(D)\) is represented by the Hodge star of
+the harmonic part of the distributional two-form associated with \(D\).
+
+\begin{lemma}\label{lem:invariant-cohomology}
+The classes represented by closed invariant forms are exactly
+\(\mathcal V=(H_J^-)^\perp\). The intersection form is positive
+definite on \(H_J^-\), and
+\[
+ H^2(X;\R)=\mathcal V\oplus H_J^-.
+\]
+The operator \(1-BR\) takes exact forms to exact invariant forms.
+\end{lemma}
+
+\begin{proof}
+Closed anti-invariant forms are self-dual, hence harmonic. Their
+intersection pairing is their \(L^2\) inner product, so it is positive
+definite on their classes. Invariant and anti-invariant forms wedge to
+zero pointwise.
+
+For any closed two-form \(\alpha\), the form
+\((1-BR)\alpha\) is closed and invariant, and
+\[
+ [\alpha]=[(1-BR)\alpha]+[HR\alpha],
+\]
+because \(BR\alpha=dd^*GR\alpha+HR\alpha\). The second summand
+is represented by a closed anti-invariant form. The two subspaces are
+orthogonal, and positive definiteness on the second makes their
+intersection zero. This proves the decomposition and the assertion
+about \(\mathcal V\).
+
+If \(\alpha\) is exact, it is \(L^2\)-orthogonal to every harmonic
+form. For a harmonic anti-invariant form \(\beta\),
+\(\ip{R\alpha}{\beta}_{L^2}=\ip{\alpha}{\beta}_{L^2}=0\).
+Thus \(HR\alpha=0\), and both \(BR\alpha\) and
+\((1-BR)\alpha\) are exact.
+\end{proof}
+
+A current is called \emph{positive} here if it annihilates
+anti-invariant forms and is nonnegative on pointwise semipositive
+invariant forms. As in Proposition~\ref{prop:separation}, it has a
+representation by a finite positive measure on \(\mathscr C\).
+Its trace measure is the projection to \(X\).
+
+\begin{lemma}[Prescribed-class separation]\label{lem:class-separation}
+Let \(a\in\mathcal V\) have no compatible representative. There
+are currents \(P,Q,T\in H^{-3}\), with \(P\) nonzero and positive,
+such that
+\begin{gather}
+ P(F)=1,\qquad P\text{ annihilates exact invariant forms},
+ \label{eq:class-exact-tests}\\
+ T=P+Q\text{ is closed},\qquad RQ=Q=*Q,
+ \label{eq:class-current}\\
+ \mathfrak c(T)\in\mathcal V,\qquad
+ \mathfrak c(T)\cdot a\le0.
+ \label{eq:class-separator-sign}
+\end{gather}
+Moreover, a class \(b\in H^2(X;\R)\) is taming if and only if
+\[
+ b\cdot\mathfrak c(N)>0
+ \quad\text{for every nonzero closed positive current }N.
+\]
+\end{lemma}
+
+\begin{proof}
+By Lemma~\ref{lem:invariant-cohomology}, the closed invariant
+representatives of \(a\) form a nonempty affine space. Separate
+this affine space from the open cone of positive invariant forms
+in the Fr\'echet space of smooth invariant forms. Hahn--Banach
+separation gives a nonzero continuous functional, nonnegative on
+the positive cone and nonpositive on the affine space. It vanishes
+on the direction space, which consists of exact invariant forms.
+Indeed, translating by any real multiple of a direction preserves
+the affine space, while positive scalings preserve the cone.
+Extend the functional by \(1-R\) and normalize it as in
+Proposition~\ref{prop:separation} to obtain \(P(F)=1\).
+The same positivity argument gives its measure representation.
+
+Define \(T(\alpha)=P((1-BR)\alpha)\) and
+\(Q(\alpha)=-P(BR\alpha)\). Lemma~\ref{lem:invariant-cohomology}
+shows that \(T\) annihilates every exact form, so it is closed.
+The projection identities give \(RQ=Q=*Q\). A closed
+anti-invariant form \(\beta\) lies in the kernel of the operator
+used to construct \(G\); hence \(B\beta=\beta\) and
+\(T(\beta)=0\). Thus \(\mathfrak c(T)\in\mathcal V\).
+On a closed invariant representative of \(a\), the currents
+\(T\) and \(P\) agree, proving \eqref{eq:class-separator-sign}.
+The Sobolev assertion follows from the measure representation and
+the order-zero lift, as in Proposition~\ref{prop:separation}.
+
+For the last statement, a taming representative evaluates strictly
+positively on every nonzero positive current. Conversely, if \(b\)
+has no taming representative, separate its affine space of closed
+representatives from the open cone of forms positive on complex
+lines. The separator annihilates all exact forms. Since arbitrary
+anti-invariant forms can be added to that cone, it also annihilates
+all anti-invariant forms. It is therefore a nonzero closed positive
+current \(N\), and \(b\cdot\mathfrak c(N)\le0\).
+\end{proof}
+
+\subsection{Pairings for currents of arbitrary class}
+
+\begin{lemma}\label{lem:cohomological-pairing}
+For closed currents \(D_1,D_2\in H^{-3}\),
+\begin{equation}\label{eq:cohomological-pairing}
+ \ip{S_rD_1}{*S_rD_2}_{L^2}
+ =\mathfrak c(D_1)\cdot\mathfrak c(D_2).
+\end{equation}
+\end{lemma}
+
+\begin{proof}
+The forms \(*S_rD_i\) lie in \(H^3\), are distributionally closed,
+and have the harmonic components representing \(\mathfrak c(D_i)\).
+Here we use commutation with \(d^*,*\), and the fact that \(S_r\)
+is the identity on harmonic forms. Their heat regularizations are
+smooth closed representatives of these classes. The intersection
+identity for smooth forms passes to the limit in \(L^2\).
+In degree two on a four-manifold,
+\(\alpha\wedge\beta=(*\alpha)\wedge(*\beta)\), giving
+\eqref{eq:cohomological-pairing}.
+\end{proof}
+
+We record the estimates that apply to the separator in
+Lemma~\ref{lem:class-separation}. In the next proposition, positive
+currents are not required to have any prescribed cohomology class.
+
+\begin{proposition}\label{prop:class-estimates}
+Suppose that \(P\ne0\) is positive, annihilates exact invariant
+forms, and \(T=P+Q\) is closed, where \(Q\in H^{-3}\) is
+anti-invariant. Write \(\lambda\) for a positive measure representing
+\(P\), and \(\mu\) for its trace. Then
+\begin{gather}
+ \mu(B_s(x))\le Cs^2,\qquad Q\in L^2,
+ \label{eq:class-growth}\\
+ \iint_{d_g(x,y)<r}|\ell-\tau_{yx}m|^2
+       \dd\lambda(x,\ell)\dd\lambda(y,m)\le Cr^4.
+ \label{eq:class-angular}
+\end{gather}
+Consequently the positive-density restriction \(P_E\) is closed,
+where \(E=\{x:\lim_{s\downarrow0}s^{-2}\mu(B_s(x))>0\}\).
+
+If \(A\) is any positive current whose trace has quadratic growth,
+then
+\begin{equation}\label{eq:class-mixed-limit}
+ \lim_{r\downarrow0}\ip{S_rA}{*S_rQ}_{L^2}=0.
+\end{equation}
+Every closed positive current has quadratic trace growth.
+\end{proposition}
+
+\begin{proof}
+Each radial test \(Du=(1-BR)dd_J^cu\) in
+Section~\ref{sec:mass} is exact by
+Lemma~\ref{lem:invariant-cohomology}. Its pointwise estimates
+therefore give the quadratic growth bound under the present
+annihilation hypothesis. The normalization \(P(F)=1\) is inessential:
+for arbitrary finite mass it only changes the constant. In particular,
+this argument applies to every closed positive current.
+
+We will use the kernel estimates for any pair \(A_1,A_2\) of positive
+currents with quadratic-growth trace measures \(\mu_1,\mu_2\).
+Write \(\lambda_1,\lambda_2\) for their representing measures.
+The proof of Lemma~\ref{lem:positive-pairing}, using
+Lemma~\ref{lem:kernel} and \eqref{eq:angular-identity}, gives
+\begin{align}
+ \|S_rA_i\|_2&\le C_i r^{-1},\label{eq:general-smoothed-mass}\\
+ \ip{S_rA_1}{*S_rA_2}_{L^2}
+ &\ge cr^{-4}\iint_{d<r}|\ell-\tau_{yx}m|^2
+            \dd\lambda_1(x,\ell)\dd\lambda_2(y,m)
+ \nonumber\\
+ &\quad-Cr^{-2}\iint(1+d/r)^{-6}
+                     \dd\mu_1(x)\dd\mu_2(y),
+ \label{eq:general-positive-pairing}
+\end{align}
+where \(d=d_g(x,y)\). These statements do not require either measure
+to be dominated by the other. The shell bound
+\eqref{eq:shell} holds separately for each \(\mu_i\), proving
+the norm bound and a uniform bound on the negative term.
+
+For any anti-invariant distribution \(W\) with \(S_rW\in L^2\),
+Lemma~\ref{lem:mixing} and \eqref{eq:general-smoothed-mass} give
+\begin{equation}\label{eq:general-mixed-bound}
+ |\ip{S_rA_i}{*S_rW}_{L^2}|
+ \le Cr\|S_rA_i\|_2\|S_rW\|_2
+ \le C_i'\|S_rW\|_2.
+\end{equation}
+Apply \eqref{eq:cohomological-pairing} to \(T,T\).
+Since \(Q\) is self-dual, expansion and these estimates yield
+\[
+ cr^{-4}\ang_r+\|S_rQ\|_2^2
+ \le C+|\mathfrak c(T)^2|+C\|S_rQ\|_2,
+\]
+where \(\ang_r\) is the angular integral in
+\eqref{eq:class-angular}. Hence \(S_rQ\) is uniformly bounded
+in \(L^2\), and the angular bound follows. Weak compactness and
+distributional convergence identify an \(L^2\) limit with \(Q\).
+
+For smooth anti-invariant \(W\),
+\[
+ \ip{S_rA}{*S_rW}=A(S_r^2W)\longrightarrow A(W)=0.
+\]
+The uniform bound \eqref{eq:general-mixed-bound}, the \(L^2\)
+contraction property of \(S_r\), and approximation by smooth
+anti-invariant forms give \eqref{eq:class-mixed-limit}.
+Finally, \eqref{eq:class-growth}--\eqref{eq:class-angular} verify
+the hypotheses of Theorem~\ref{thm:splitting}, which proves that
+\(P_E\) is closed.
+\end{proof}
+
+The next lemma permits pairing the zero-density part with positive
+currents unrelated to the original separator.
+
+\begin{lemma}[Residual pairings]\label{lem:general-residual}
+Let \(A,P'\) be positive currents whose trace measures \(\nu,\mu'\)
+have quadratic growth. Suppose
+\[
+ \lim_{s\downarrow0}s^{-2}\mu'(B_s(x))=0
+ \quad\text{for }\mu'\text{-almost every }x.
+\]
+Then
+\begin{equation}\label{eq:general-residual}
+ \liminf_{r\downarrow0}\ip{S_rA}{*S_rP'}_{L^2}\ge0.
+\end{equation}
+\end{lemma}
+
+\begin{proof}
+We first show
+\begin{equation}\label{eq:cross-small-density}
+ (\mu'\times\nu)\{(x,y):d_g(x,y)<s\}=o(s^2).
+\end{equation}
+For small \(s\), choose an \(s\)-separated maximal set of centers
+and let \(B_j\) be their radius-\(s\) balls. They cover \(X\),
+and their fixed dilates have uniformly bounded overlap, by the
+volume comparison in smooth coordinate charts. If \(x\in B_j\)
+and \(d_g(x,y)<s\), then \(y\in2B_j\). Thus
+\[
+ (\mu'\times\nu)\{d<s\}
+ \le\sum_j\mu'(B_j)\nu(2B_j).
+\]
+Bounded overlap gives
+\[
+ \sum_j\mu'(B_j)^2
+ \le C\int_X\mu'(B_{2s}(x))\dd\mu'(x)=o(s^2).
+\]
+Indeed, the integrand divided by \(s^2\) is bounded by quadratic
+growth and tends to zero at \(\mu'\)-almost every center.
+Also
+\[
+ \sum_j\nu(2B_j)^2
+ \le Cs^2\sum_j\nu(2B_j)=O(s^2).
+\]
+Cauchy--Schwarz proves \eqref{eq:cross-small-density}.
+
+In \eqref{eq:general-positive-pairing}, discard the angular term.
+The remaining negative error tends to zero: on \(d<r\), and on
+each fixed shell \(2^jr\le d<2^{j+1}r\), this follows from
+\eqref{eq:cross-small-density}. Quadratic growth bounds the shell
+contributions by \(C2^{-4j}\), allowing summation by dominated
+convergence. This proves \eqref{eq:general-residual}.
+\end{proof}
+
+\subsection{Proof of the cone description}
+
+\begin{proof}[Proof of Theorem~\ref{thm:taming-cones}]
+Adding a closed anti-invariant form does not change positivity on
+complex lines. Lemma~\ref{lem:invariant-cohomology} therefore gives
+\eqref{eq:tame-decomposition}. In particular \(\mathcal C\) is
+nonempty. It is a convex cone of positive-square classes in
+\(\mathcal V\), since its classes have closed taming representatives.
+
+A compatible form wedges strictly positively with every taming form.
+Indeed, in a unitary frame diagonalizing the compatible form, the
+wedge product is a positive linear combination of the two positive
+diagonal entries of the taming form's invariant part. Consequently
+every \(a\in\mathcal K_J^c\) lies in \(\mathcal C\) and pairs
+positively with every class in \(\mathcal C\), hence nonnegatively
+with \(\overline{\mathcal C}\).
+
+The compatible cone is open in \(\mathcal V\): add sufficiently
+small linear combinations of closed invariant representatives of a
+basis of \(\mathcal V\) to a compatible form. The intersection form
+on \(\mathcal V\) is nondegenerate by
+Lemma~\ref{lem:invariant-cohomology}. If \(a\cdot y=0\) for a
+nonzero \(y\in\overline{\mathcal C}\), a sufficiently small
+perturbation of \(a\) within the compatible cone would pair
+negatively with \(y\). Thus every compatible class satisfies the
+strict positivity in \eqref{eq:compatible-duality}.
+
+Conversely, suppose \(a\in\mathcal C\) has this strict positivity
+but no compatible representative. Apply
+Lemma~\ref{lem:class-separation} and
+Proposition~\ref{prop:class-estimates}. Let
+\[
+ P'=P-P_E,\qquad T'=P'+Q=T-P_E,\qquad
+ y=\mathfrak c(T').
+\]
+Both \(P_E\) and \(T'\) are closed. Since \(P_E\) is invariant,
+\(\mathfrak c(P_E)\in\mathcal V\), and hence \(y\in\mathcal V\).
+The trace \(\mu'\) of \(P'\) satisfies the hypotheses of
+Lemma~\ref{lem:general-residual}: it is bounded by \(\mu\) and
+is concentrated where the latter has zero two-density.
+
+We claim that \(y\in\overline{\mathcal C}\). Let \(N\) be any
+nonzero closed positive current. Its trace has quadratic growth by
+Proposition~\ref{prop:class-estimates}. By
+\eqref{eq:cohomological-pairing},
+\[
+ y\cdot\mathfrak c(N)
+ =\ip{S_rN}{*S_rP'}+\ip{S_rN}{*S_rQ}.
+\]
+The mixed term tends to zero by \eqref{eq:class-mixed-limit}, and
+Lemma~\ref{lem:general-residual} bounds the lower limit of the first
+term by zero. Thus \(y\cdot\mathfrak c(N)\ge0\). Since \(a\)
+is taming, \((y+\eps a)\cdot\mathfrak c(N)>0\) for every
+\(\eps>0\). The last part of Lemma~\ref{lem:class-separation}
+shows that \(y+\eps a\) is taming. It also lies in \(\mathcal V\),
+so \(y+\eps a\in\mathcal C\), proving the claim.
+
+It follows that \(a\cdot y\ge0\), with strict inequality if
+\(y\ne0\). If \(P'\ne0\), then \(y\ne0\). To see this,
+suppose \(y=0\) and pair \(T'\) with itself using
+\eqref{eq:cohomological-pairing}. We obtain
+\[
+ 0=\ip{S_rP'}{*S_rP'}+2\ip{S_rP'}{*S_rQ}+\|S_rQ\|_2^2.
+\]
+The residual lemma, the mixed-term limit, and strong \(L^2\)
+convergence of \(S_rQ\) force \(Q=0\). Then the nonzero positive
+current \(P'=T'\) evaluates strictly positively on a closed tamer,
+contrary to \(\mathfrak c(T')=0\).
+
+Finally, a taming representative of \(a\) gives
+\(a\cdot\mathfrak c(P_E)\ge0\), strictly if \(P_E\ne0\).
+At least one of \(P_E,P'\) is nonzero. Therefore
+\[
+ a\cdot\mathfrak c(T)
+ =a\cdot\mathfrak c(P_E)+a\cdot y>0,
+\]
+contradicting \eqref{eq:class-separator-sign}. This proves
+\eqref{eq:compatible-duality}.
+\end{proof}
+
+\begin{proof}[Proof of Corollary~\ref{cor:taming-cones}]
+The subspace \(H_J^-\) is positive definite, while its orthogonal
+complement \(\mathcal V\) contains the nonempty positive-square
+cone \(\mathcal C\). Hence \(h_J^-\le b_2^+-1\). If equality
+holds, the intersection form on \(\mathcal V\) has signature
+\((1,b_2^-)\). Convexity puts \(\mathcal C\) in one component
+of its positive-square cone. Every element of that component pairs
+strictly positively with every nonzero element of its closure,
+by Cauchy--Schwarz in coordinates of signature
+\((+,-,\ldots,-)\). Thus all \(a\in\mathcal C\) satisfy
+\eqref{eq:compatible-duality}, and
+\(\mathcal C=\mathcal K_J^c\). Equation
+\eqref{eq:tame-decomposition} gives the asserted identity.
+If \(b_2^+=1\), the inequality forces \(h_J^-=0\), proving the
+last assertion.
+\end{proof}
+
+\begin{example}[Strict inclusion on the four-torus]\label{ex:strict-cones}
+On \(X=\R^4/\mathbb Z^4\), use the Euclidean metric and standard
+orientation, and put \(e^{ij}=dx_i\wedge dx_j\). Set
+\[
+ U=e^{12}+e^{34},\qquad V_0=e^{13}-e^{24},\qquad
+ W=e^{14}+e^{23},
+\]
+\[
+ f=\tfrac14\sin(2\pi x_1),\qquad
+ k=\tfrac14\sin(2\pi x_2),\qquad
+ r=(1+f^2+k^2)^{1/2}.
+\]
+The form \(F=(U+fV_0+kW)/r\) is self-dual and has squared norm
+two, so it is the fundamental form of a smooth orthogonal almost
+complex structure \(J\) of this orientation. The form
+\[
+ \eta=U+2f e^{13}+2k e^{23}
+\]
+is closed. Its self-dual part is \(rF\), so it is invariant, and
+its anti-self-dual part is
+\[
+ f(e^{13}+e^{24})+k(e^{23}-e^{14}).
+\]
+Its Hermitian eigenvalues are \(r\pm\sqrt{f^2+k^2}>0\).
+Thus \(\eta\) is compatible. The sine terms are exact, giving
+\([\eta]=[U]\).
+
+A closed anti-invariant form is self-dual and harmonic for the flat
+metric, hence is a constant combination \(aU+bV_0+cW\).
+Orthogonality to \(F\) gives \(a+bf+ck=0\) everywhere. Varying
+\(x_1,x_2\) yields \(a=b=c=0\); thus \(H_J^-=0\).
+The closed forms \(U\pm V_0\) have invariant projections
+\((1\pm f)F/r\), so both tame \(J\). Their classes pair to zero,
+since \(U^2=V_0^2\). Neither class can have a compatible
+representative, which would wedge strictly positively with a tamer
+in the other class. Hence
+\[
+ \mathcal K_J^c+H_J^-=\mathcal K_J^c
+ \subsetneq\mathcal K_J^t.
+\]
+In Theorem~\ref{thm:taming-cones}, the class \([U+V_0]\) fails
+the strict positivity test against \([U-V_0]\in\mathcal C\),
+and conversely.
+\end{example}
```
### Box

Before: [Incompressible-Box-Transport-and-Finite-Computation-September-27-2026](https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Incompressible-Box-Transport-and-Finite-Computation-September-27-2026)
After: [Incompressible-Box-Transport-and-Finite-Computation-October-6-2026](https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/preprints/Incompressible-Box-Transport-and-Finite-Computation-October-6-2026)

PDF page counts: 76 → 78.

| Relative file | Status | Before bytes | After bytes |
|---|---|---:|---:|
| `README.md` | changed | 626 | 882 |
| `build/analysis.tex` | changed | 10806 | 13940 |
| `build/balanced.tex` | identical | 8555 | 8555 |
| `build/clocks.tex` | changed | 6530 | 7495 |
| `build/coding-conventions.tex` | identical | 11483 | 11483 |
| `build/compact-processors.tex` | identical | 9578 | 9578 |
| `build/compilers.tex` | identical | 15031 | 15031 |
| `build/geometry.tex` | identical | 16612 | 16612 |
| `build/guarded-history.tex` | identical | 33723 | 33723 |
| `build/initialization.tex` | identical | 29080 | 29080 |
| `build/introduction.tex` | identical | 10072 | 10072 |
| `build/main.tex` | changed | 2625 | 2622 |
| `build/observer-routing.tex` | identical | 47478 | 47478 |
| `build/persistent-history.tex` | identical | 26007 | 26007 |
| `build/references.bib` | identical | 7805 | 7805 |
| `build/routing-conventions.tex` | identical | 12571 | 12571 |
| `build/solid-routing.tex` | identical | 37640 | 37640 |
| `build/torus-routing.tex` | identical | 17983 | 17983 |
| `manuscript.pdf` | changed | 770717 | 767212 |

#### README.md

```diff
--- adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Incompressible-Box-Transport-and-Finite-Computation-September-27-2026/README.md

+++ fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/preprints/Incompressible-Box-Transport-and-Finite-Computation-October-6-2026/README.md

@@ -1,17 +1,20 @@

 # [Incompressible Box Transport and Finite Computation](manuscript.pdf)
 
 **Author:** OpenAI  
-**Date:** September 27, 2026
+**Date:** October 6, 2026
 
 ## Citation
 
 ```bibtex
-@misc{OAI:Incompressible-Box-Transport-and-Finite-Computation-September-27-2026,
+@misc{OAI:Incompressible-Box-Transport-and-Finite-Computation-October-6-2026,
   author = {{OpenAI}},
   title = {{Incompressible Box Transport and Finite Computation}},
   howpublished = {OpenAI Math Release preprint
-                  \href{https://github.com/openai/math/blob/main/preprints/Incompressible-Box-Transport-and-Finite-Computation-September-27-2026/manuscript.pdf}{OAI:Incompressible-Box-Transport-and-Finite-Computation-September-27-2026}},
+                  \href{https://github.com/openai/math/blob/main/preprints/Incompressible-Box-Transport-and-Finite-Computation-October-6-2026/manuscript.pdf}{OAI:Incompressible-Box-Transport-and-Finite-Computation-October-6-2026}},
   year = {2026}
 }
 ```
 
+## Version note
+
+This version supplies effective torus-projection and common-clock decay estimates, including the projected-force and pressure bounds; see the [previous version](../Incompressible-Box-Transport-and-Finite-Computation-September-27-2026/manuscript.pdf).
```

#### build/analysis.tex

```diff
--- adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Incompressible-Box-Transport-and-Finite-Computation-September-27-2026/build/analysis.tex

+++ fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/preprints/Incompressible-Box-Transport-and-Finite-Computation-October-6-2026/build/analysis.tex

@@ -162,32 +162,106 @@

 is not identified with a differently normalized viscosity.
 
 \begin{proposition}[Projection on a flat torus]\label{prop:projection-l2}
-An effective smooth mean-zero force $g$ on $\T_L^3$ has an effective
-mean-zero potential $\phi$ satisfying $\Delta\phi=\operatorname{div}g$.
-Replacing $(g,p)$ by $(g-\nabla\phi,p-\phi)$ preserves velocity and makes
-the force solenoidal. Uniform mixed-derivative bounds, temporal $L^2$
-bounds of their spatial suprema, and separately imposed time periodicity,
-eventual stationarity, or orderwise decay bounds are preserved.
+Let $g$ be a smooth force on $[0,\infty)\times\T_L^3$ with zero
+spatial mean at each time. There is a unique smooth mean-zero potential
+$\phi$ satisfying $\Delta\phi=\operatorname{div}g$. Put
+$Qg=\nabla\phi$ and $Pg=g-Qg$. Then $Pg$ is solenoidal and mean zero,
+and replacing $(g,p)$ by $(Pg,p-\phi)$ preserves the velocity and the
+pressure normalization.
+
+Use normalized spatial measure, frequencies $\xi_k=2\pi k/L$, and norms
+\[
+ \|G\|_{H^s}^2=\sum_{k\in\Z^3}(1+|\xi_k|^2)^s|\widehat G(k)|^2,
+ \qquad
+ \|G\|_{C^r}=\max_{|\beta|\le r}\|\partial_x^\beta G\|_\infty.
+\]
+For all integers $h,r\ge0$, at each time,
+\begin{equation}\label{eq:projection-estimate}
+ \begin{aligned}
+ \max\{\|\partial_t^h Pg\|_{C^r},\|\partial_t^h Qg\|_{C^r}\}
+     &\le S_L\|\partial_t^h g\|_{H^{r+2}},\\
+ \|\partial_t^h\phi\|_{C^{r+1}}
+     &\le S_Ld_L\|\partial_t^h g\|_{H^{r+2}},
+ \end{aligned}
+\end{equation}
+where $S_L=(1+52(L/(2\pi))^4)^{1/2}$ and
+$d_L=(1+(L/(2\pi))^2)^{1/2}$.
+
+Uniform bounds, temporal $L^2$ bounds on spatial suprema, and qualitative
+uniform decay of all mixed derivatives pass from $g$ to $Pg,Qg,\phi$.
+A quantitative envelope for time order $h$ and output spatial order $r$
+passes to $Pg,Qg$ through order $r$ and to $\phi$ through order $r+1$
+when the input derivatives through spatial order $r+2$ satisfy that
+envelope. The constants may depend on the derivative order.
+Periodicity and stationarity, including either property on a time tail,
+are preserved on their respective intervals.
+
+For computable $L>0$, effective mixed-derivative evaluation and effective
+derivative bounds for $g$ on finite time slabs give effective evaluation
+and such bounds for $Pg,Qg,\phi$. Corresponding supplied global input
+bounds give effective global bounds in each assertion above.
 \end{proposition}
 \begin{proof}
-In frequencies $\xi_k=2\pi k/L$, set
-$\widehat\phi(k)=-i\xi_k\cdot\widehat g(k)/|\xi_k|^2$ for $k\ne0$
-and zero for $k=0$. The multiplier from $g$ to $\nabla\phi$ is
-$\xi_k\xi_k^{\mathsf T}/|\xi_k|^2$. For a spatial derivative order $r$,
-integrate each Fourier coefficient $r+5$ times in a coordinate with largest
-$|k_i|$. There are $O(n^2)$ lattice points on the shell $\|k\|_\infty=n$,
-so the derivative series and its tail converge absolutely, with
-\[
- \|\partial_t^h\partial_x^\alpha\nabla\phi\|_\infty
- \le C_{\alpha,L}\max_i
-       \|\partial_t^h\partial_{x_i}^{|\alpha|+5}g\|_\infty.
-\]
-This bound is pointwise in time, and proves each asserted norm or weighted
-time estimate. Derivative bounds give effective Fourier tails; coefficient
-integrals can be computed by Riemann sums with derivative error estimates.
-The formula gives the Poisson identity, divergence cancellation, and the
-pressure sign by substitution. Linearity preserves the time symmetries.
-\end{proof}
+For $k\ne0$ define
+\[
+ \widehat\phi(t,k)=-\frac{i\xi_k\cdot\widehat g(t,k)}{|\xi_k|^2},
+ \qquad \widehat\phi(t,0)=0.
+\]
+The nonzero-frequency symbols of $Q$ and $P$ are complementary orthogonal
+projections. Their $H^s$ operator norms are at most one, while the
+multiplier for $\phi$ has norm at most $d_L$ from $H^s$ to $H^{s+1}$.
+For $|\beta|\le r$, Cauchy--Schwarz bounds the absolute sum of the
+Fourier series for $\partial_x^\beta G$ by
+\[
+ \left(\sum_{k\in\Z^3}(1+|\xi_k|^2)^{-2}\right)^{1/2}
+ \|G\|_{H^{r+2}}.
+\]
+The shell $|k|_\infty=n\ge1$ contains $24n^2+2\le26n^2$ points.
+Since $\sum_{n\ge1}n^{-2}\le2$, the factor is at most $S_L$.
+For the tail $|k|_\infty>N\ge1$, it is at most
+\begin{equation}\label{eq:projection-tail}
+ \left(26(L/(2\pi))^4/N\right)^{1/2}.
+\end{equation}
+Applying these estimates to the multipliers proves
+\eqref{eq:projection-estimate}, including the estimate for $\phi$ with
+one additional spatial derivative. On compact time slabs these tails
+converge uniformly for every mixed derivative. Thus the series define
+smooth fields and may be differentiated term by term. The Fourier
+identities give the Poisson equation, uniqueness of its mean-zero
+solution, and $\operatorname{div}Pg=0$. Also
+$-\nabla(p-\phi)+Pg=-\nabla p+g$.
+
+For every integer $n\ge0$, the multinomial expansion and Parseval give
+\begin{equation}\label{eq:projection-derivative-bounds}
+ \|G\|_{H^n}^2
+ \le\sum_{|\beta|\le n}
+       \frac{n!}{(n-|\beta|)!\,\beta!}
+       \|\partial_x^\beta G\|_\infty^2.
+\end{equation}
+Use $n=r+2$ and $G=\partial_t^h g$. This finite sum proves the uniform
+bounds and qualitative decay assertions. If its input suprema are
+bounded by $C_{h,\beta}a_h(t)$, it gives the same envelope $a_h$ for
+the output, with the constants specified by
+\eqref{eq:projection-estimate}--\eqref{eq:projection-derivative-bounds}.
+Integrating the squared estimates proves the temporal $L^2$ assertion
+from the corresponding input norms. The operators act at each time,
+so they preserve the stated temporal symmetries.
+
+For effective evaluation, the supplied derivative bounds and
+\eqref{eq:projection-tail} give a computable Fourier truncation on each
+finite time slab. The finitely many coefficient integrals are computed
+by Riemann sums. After scaling the torus to $[0,1]^3$, the integrand
+for $\widehat G(k)$ has Lipschitz bound
+\[
+ L\sqrt3\max_j\|\partial_{x_j}G\|_\infty
+       +2\pi|k|\|G\|_\infty.
+\]
+This gives an effective quadrature error and hence evaluation of every
+mixed derivative of all three fields. The displayed finite sums also
+compute output bounds from the supplied input bounds.
+\end{proof}
+The estimates apply to the pressure shift $\phi$; in the constructions
+with original pressure zero, the projected pressure is $-\phi$.
 Projection generally changes pressure. It supplies no compact-support
 conclusion on Euclidean space and is distinct from the direct zero-pressure
 solenoidal shear construction.
```

#### build/clocks.tex

```diff
--- adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Incompressible-Box-Transport-and-Finite-Computation-September-27-2026/build/clocks.tex

+++ fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/preprints/Incompressible-Box-Transport-and-Finite-Computation-October-6-2026/build/clocks.tex

@@ -115,8 +115,26 @@

 the dummy start. The first force is stationary after one. In the second,
 the coefficient orders are respectively $(1+t)^{-2},(1+t)^{-2}$ and
 $(1+t)^{-1}$; each time derivative gives an additional inverse power.
-This proves all the asserted bounded and temporal $L^2$ norms. The two
-profiles are separate choices. Torus projection may be applied to either
-with its corresponding changed pressure, as in
-Proposition~\ref{prop:projection-l2}.
+The spatial factors in \eqref{bc:input-offset-force} have effective
+bounds at every order. Thus for every $h,r\ge0$,
+\begin{equation}\label{bc:input-offset-common-rate}
+ \|\partial_t^h U_{\alpha_1}(t)\|_{C^r}
+ +\|\partial_t^h f_{\alpha_1}(t)\|_{C^r}
+ \le C_{h,r}(1+t)^{-1-h}.
+\end{equation}
+The constant is effective; increasing it over $[0,1]$ makes the estimate
+valid for all $t\ge0$. The exponent is common to all spatial orders,
+and the square of the envelope has integral $(1+2h)^{-1}$.
+This proves the asserted bounded and temporal $L^2$ norms.
+
+For the stationary profile, Proposition~\ref{prop:projection-l2}
+preserves stationarity after time one and every mixed-derivative bound.
+For the slow profile, \eqref{eq:projection-estimate} and
+\eqref{eq:projection-derivative-bounds} apply to
+\eqref{bc:input-offset-common-rate} through spatial order $r+2$.
+They retain the rate $(1+t)^{-1-h}$ for the projected force through
+order $r$ and for its pressure through order $r+1$, with effective
+constants and temporal $L^2$ bounds. In either projection choice the
+velocity and event are unchanged and the mean-zero pressure is
+$-\phi$. The stationary and slow profiles remain separate choices.
 \end{proof}
```

#### build/main.tex

```diff
--- adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Incompressible-Box-Transport-and-Finite-Computation-September-27-2026/build/main.tex

+++ fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/preprints/Incompressible-Box-Transport-and-Finite-Computation-October-6-2026/build/main.tex

@@ -18,7 +18,7 @@

 \hypersetup{pdftitle={Incompressible Box Transport and Finite Computation},pdfauthor={OpenAI}}
 \title{Incompressible Box Transport and Finite Computation}
 \author{OpenAI}
-\date{September 27, 2026}
+\date{October 6, 2026}
 \begin{document}\maketitle
 \begin{abstract}
 We realize finite positive diagonal affine maps of determinant one by
```
### BSD

Before: [Exact-Birch-Swinnerton-Dyer-Formula-from-Low-Selmer-Corank-October-3-2026](https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Exact-Birch-Swinnerton-Dyer-Formula-from-Low-Selmer-Corank-October-3-2026)
After: [Exact-Birch-Swinnerton-Dyer-Formula-from-Low-Selmer-Corank-October-7-2026](https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/preprints/Exact-Birch-Swinnerton-Dyer-Formula-from-Low-Selmer-Corank-October-7-2026)

PDF page counts: 94 → 94.

| Relative file | Status | Before bytes | After bytes |
|---|---|---:|---:|
| `README.md` | changed | 685 | 958 |
| `build/main.tex` | changed | 2900 | 2900 |
| `build/references.tex` | changed | 8880 | 8862 |
| `build/sections/01-introduction.tex` | changed | 13790 | 13787 |
| `build/sections/02-classical.tex` | identical | 14128 | 14128 |
| `build/sections/03-models.tex` | identical | 33118 | 33118 |
| `build/sections/04-vertical.tex` | identical | 32060 | 32060 |
| `build/sections/05-single.tex` | identical | 65191 | 65191 |
| `build/sections/06-pair.tex` | identical | 52151 | 52151 |
| `build/sections/07-theta.tex` | identical | 55696 | 55696 |
| `build/sections/08-extraction.tex` | identical | 50502 | 50502 |
| `build/sections/09-unit.tex` | identical | 21528 | 21528 |
| `build/sections/10-center.tex` | identical | 11104 | 11104 |
| `exact-bsd-low-selmer-corank.pdf` | changed | 874733 | 874353 |

#### README.md

```diff
--- adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Exact-Birch-Swinnerton-Dyer-Formula-from-Low-Selmer-Corank-October-3-2026/README.md

+++ fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/preprints/Exact-Birch-Swinnerton-Dyer-Formula-from-Low-Selmer-Corank-October-7-2026/README.md

@@ -2,17 +2,20 @@

 
 **Author:** OpenAI
 
-**Date:** October 3, 2026
+**Date:** October 7, 2026
 
 ## Citation
 
 ```bibtex
-@misc{OAI:Exact-Birch-Swinnerton-Dyer-Formula-from-Low-Selmer-Corank-October-3-2026,
+@misc{OAI:Exact-Birch-Swinnerton-Dyer-Formula-from-Low-Selmer-Corank-October-7-2026,
   author = {{OpenAI}},
   title = {{Exact Birch--Swinnerton-Dyer Formula from Low Selmer Corank}},
   howpublished = {OpenAI Math Release preprint
-                  \href{https://github.com/openai/math/blob/main/preprints/Exact-Birch-Swinnerton-Dyer-Formula-from-Low-Selmer-Corank-October-3-2026/exact-bsd-low-selmer-corank.pdf}{OAI:Exact-Birch-Swinnerton-Dyer-Formula-from-Low-Selmer-Corank-October-3-2026}},
+                  \href{https://github.com/openai/math/blob/main/preprints/Exact-Birch-Swinnerton-Dyer-Formula-from-Low-Selmer-Corank-October-7-2026/exact-bsd-low-selmer-corank.pdf}{OAI:Exact-Birch-Swinnerton-Dyer-Formula-from-Low-Selmer-Corank-October-7-2026}},
   year = {2026}
 }
 ```
 
+## Version note
+
+This version removes an obsolete supporting citation and updates citations to revised manuscripts in this collection; see the [previous version](../Exact-Birch-Swinnerton-Dyer-Formula-from-Low-Selmer-Corank-October-3-2026/exact-bsd-low-selmer-corank.pdf).
```

#### build/main.tex

```diff
--- adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Exact-Birch-Swinnerton-Dyer-Formula-from-Low-Selmer-Corank-October-3-2026/build/main.tex

+++ fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/preprints/Exact-Birch-Swinnerton-Dyer-Formula-from-Low-Selmer-Corank-October-7-2026/build/main.tex

@@ -55,7 +55,7 @@

 \displaywidowpenalty=10000
 \title{Exact Birch--Swinnerton-Dyer Formula\protect\\from Low Selmer Corank}
 \author{OpenAI}
-\date{October 3, 2026}
+\date{October 7, 2026}
 \begin{document}
 \maketitle
 \begin{abstract}
```

#### build/references.tex

```diff
--- adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Exact-Birch-Swinnerton-Dyer-Formula-from-Low-Selmer-Corank-October-3-2026/build/references.tex

+++ fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/preprints/Exact-Birch-Swinnerton-Dyer-Formula-from-Low-Selmer-Corank-October-7-2026/build/references.tex

@@ -144,21 +144,21 @@

 OpenAI,
 \emph{Goldfeld's analytic density conjecture and the $2$-converse for elliptic curves},
 OpenAI Math Release preprint
-\href{https://github.com/openai/math/blob/main/preprints/Goldfelds-analytic-density-conjecture-and-the-2-converse-for-elliptic-curves-September-23-2026/paper.pdf}{OAI:Goldfelds-analytic-density-conjecture-and-the-2-converse-for-elliptic-curves-September-23-2026},
+\href{https://github.com/openai/math/blob/main/preprints/Goldfelds-analytic-density-conjecture-and-the-2-converse-for-elliptic-curves-October-7-2026/paper.pdf}{OAI:Goldfelds-analytic-density-conjecture-and-the-2-converse-for-elliptic-curves-October-7-2026},
 2026.
 
 \bibitem{C}
 OpenAI,
 \emph{The Selmer converse for elliptic curves at every prime},
 OpenAI Math Release preprint
-\href{https://github.com/openai/math/blob/main/preprints/The-Selmer-converse-for-elliptic-curves-at-every-prime-September-24-2026/main.pdf}{OAI:The-Selmer-converse-for-elliptic-curves-at-every-prime-September-24-2026},
+\href{https://github.com/openai/math/blob/main/preprints/The-Selmer-converse-for-elliptic-curves-at-every-prime-October-7-2026/main.pdf}{OAI:The-Selmer-converse-for-elliptic-curves-at-every-prime-October-7-2026},
 2026.
 
 \bibitem{B}
 OpenAI,
 \emph{The two-primary Birch--Swinnerton-Dyer formula in Selmer corank at most one},
 OpenAI Math Release preprint
-\href{https://github.com/openai/math/blob/main/preprints/The-two-primary-Birch-Swinnerton-Dyer-formula-in-Selmer-corank-at-most-one-September-24-2026/paper.pdf}{OAI:The-two-primary-Birch-Swinnerton-Dyer-formula-in-Selmer-corank-at-most-one-September-24-2026},
+\href{https://github.com/openai/math/blob/main/preprints/The-two-primary-Birch-Swinnerton-Dyer-formula-in-Selmer-corank-at-most-one-October-6-2026/paper.pdf}{OAI:The-two-primary-Birch-Swinnerton-Dyer-formula-in-Selmer-corank-at-most-one-October-6-2026},
 2026.
 
 \bibitem{Rubin}
```

#### build/sections/01-introduction.tex

```diff
--- adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Exact-Birch-Swinnerton-Dyer-Formula-from-Low-Selmer-Corank-October-3-2026/build/sections/01-introduction.tex

+++ fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/preprints/Exact-Birch-Swinnerton-Dyer-Formula-from-Low-Selmer-Corank-October-7-2026/build/sections/01-introduction.tex

@@ -131,7 +131,7 @@

 primitivity uses the elliptic tame-stalk independence proved below
 by the Chai--Hida rigidity and monodromy method
 \cite{ChaiOrd,HidaMu}. The finite-model, switching, and theta
-methods of \cite{B,C,Dv} provide the starting constructions.
+methods of \cite{B,C} provide the starting constructions.
 The work below extends the relevant comparisons to residual
 tests and to nonaugmentation theta tests with growing \(p\)-level,
 and explains why these extensions give an exact center.
```

## Verified manifest entries

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| [preprints/Incompressible-Box-Transport-and-Finite-Computation-September-27-2026/build/solid-routing.tex](https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Incompressible-Box-Transport-and-Finite-Computation-September-27-2026/build/solid-routing.tex) | `0245e0c420f03a1684483cf11cf76b6f99f820285f2fc8e1f052444c54312883` | `921546d966504bd9479811576be27fa929d51d87` | 37640 |
| [preprints/Incompressible-Box-Transport-and-Finite-Computation-September-27-2026/build/torus-routing.tex](https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Incompressible-Box-Transport-and-Finite-Computation-September-27-2026/build/torus-routing.tex) | `fddbcd07154d9c347270d76ec2a42e6c4554b0d22f52e1c2cd82ffb50ff488b0` | `42d13725693fbe7a5b84fbbcbc335824b73162e1` | 17983 |
| [preprints/Incompressible-Box-Transport-and-Finite-Computation-September-27-2026/manuscript.pdf](https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Incompressible-Box-Transport-and-Finite-Computation-September-27-2026/manuscript.pdf) | `ef2428fde699d046033669ce92129753b4e6114617f25e24cdc871a28c3e99b6` | `c8eae2fd6cc36d60cfdae332fbcd44d58f214f53` | 770717 |
| [preprints/Taming-implies-compatibility-on-four-manifolds-October-6-2026/README.md](https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/preprints/Taming-implies-compatibility-on-four-manifolds-October-6-2026/README.md) | `45d1ab5ea22144beb58864d58bea5ad16a78b0b33929449c33ed9130213b069b` | `c6c5c6eac6e3e0706e0c23fc6a105b212a351dc4` | 880 |
| [preprints/Taming-implies-compatibility-on-four-manifolds-October-6-2026/build/main.tex](https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/preprints/Taming-implies-compatibility-on-four-manifolds-October-6-2026/build/main.tex) | `9571cf3d5ee085dcb7ad3dcd4f12b827a07cbac36a9675147613be6cf85799bf` | `b55c2c641b13e6608438001d9b051e29f5669d7b` | 888 |
| [preprints/Taming-implies-compatibility-on-four-manifolds-October-6-2026/build/preamble.tex](https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/preprints/Taming-implies-compatibility-on-four-manifolds-October-6-2026/build/preamble.tex) | `58bee60f6756a2a42c149ca0062e148b8885ef146e25b173c51510900ceaf84c` | `e3404d0db10ea2d2c62799a84fcd6db32db3d52c` | 1638 |
| [preprints/Taming-implies-compatibility-on-four-manifolds-October-6-2026/build/references.bib](https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/preprints/Taming-implies-compatibility-on-four-manifolds-October-6-2026/build/references.bib) | `1cb2ee75678baa79f3169ee0db134be84a7156a9d43639f73cae19bdb2ae9ec6` | `63206fc541744d1492b0c07da42a08d39da95557` | 9738 |
| [preprints/Taming-implies-compatibility-on-four-manifolds-October-6-2026/build/sections/01-introduction.tex](https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/preprints/Taming-implies-compatibility-on-four-manifolds-October-6-2026/build/sections/01-introduction.tex) | `214bce04a1e697417b8be7c8c9706862c7796e2c9caa3aacb596649f39e51a9d` | `d9b79cce7dd94790552a15e9d5685b24f144f917` | 10307 |
| [preprints/Taming-implies-compatibility-on-four-manifolds-October-6-2026/build/sections/02-currents.tex](https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/preprints/Taming-implies-compatibility-on-four-manifolds-October-6-2026/build/sections/02-currents.tex) | `17f02a4944c68ce917dd6e2acaae53f41ec6a25b13f15438a673e8caee10f8c3` | `6ee6282431b1483a7a5d6762803bfaf0b0fa726c` | 6409 |
| [preprints/Taming-implies-compatibility-on-four-manifolds-October-6-2026/build/sections/03-mass.tex](https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/preprints/Taming-implies-compatibility-on-four-manifolds-October-6-2026/build/sections/03-mass.tex) | `806342e2beef65fb2728975e5e5cacfc591148f983437a448afd1676095b3f93` | `a4057079cd7c95659914af62441b63a4e7b4b0b0` | 5810 |
| [preprints/Taming-implies-compatibility-on-four-manifolds-October-6-2026/build/sections/04-energy.tex](https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/preprints/Taming-implies-compatibility-on-four-manifolds-October-6-2026/build/sections/04-energy.tex) | `7fcb8945a62f12a25398ef89bcaee3c70441adb2ad2fa9e6120831ae75fbd0c6` | `42dea8fc1e2415e2675879ecfadc16630cb33e41` | 9544 |
| [preprints/Taming-implies-compatibility-on-four-manifolds-October-6-2026/build/sections/05-density.tex](https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/preprints/Taming-implies-compatibility-on-four-manifolds-October-6-2026/build/sections/05-density.tex) | `df97ab2e92e3d70dce247ac314d0e2adbb9b345018331a86dd03a3624de5fffb` | `3bca97ec278269d42b1cea42238f1b2893acbc6c` | 10533 |
| [preprints/Taming-implies-compatibility-on-four-manifolds-October-6-2026/build/sections/06-splitting.tex](https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/preprints/Taming-implies-compatibility-on-four-manifolds-October-6-2026/build/sections/06-splitting.tex) | `e256efe17b9b2b1bb4236c7ad8f8307b37d951225edc4da12278130a98649963` | `4dbdd71553fe9e86100b37fb824e506b8c5d3cd0` | 10225 |
| [preprints/Taming-implies-compatibility-on-four-manifolds-October-6-2026/build/sections/07-conclusion.tex](https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/preprints/Taming-implies-compatibility-on-four-manifolds-October-6-2026/build/sections/07-conclusion.tex) | `ad5c4074fb2716370f3ab4b8ce665ad6d616de7f714cb93b88f56986f60e7876` | `8b49ba5d5476d761bd5a5fcc235ebf5297263d8a` | 2584 |
| [preprints/Taming-implies-compatibility-on-four-manifolds-October-6-2026/build/sections/08-cones.tex](https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/preprints/Taming-implies-compatibility-on-four-manifolds-October-6-2026/build/sections/08-cones.tex) | `5b106258d563f938a4ac315a3f3b908698771d2e2d2a196f8449f7406e0974d6` | `fad0e1b6763068c4059cd51765d5baea439e0a84` | 16943 |
| [preprints/Taming-implies-compatibility-on-four-manifolds-October-6-2026/paper.pdf](https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/preprints/Taming-implies-compatibility-on-four-manifolds-October-6-2026/paper.pdf) | `cbddc627efab3dd49a69a32b270b52b6ea4528846bd3b313038a696ed3281265` | `2a910fe734003848f110acc18ff98d7ceabbf9d4` | 249763 |
| [preprints/Taming-implies-compatibility-on-four-manifolds-September-23-2026/README.md](https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Taming-implies-compatibility-on-four-manifolds-September-23-2026/README.md) | `22e6d05e847928bd34fb6dea91eab0a9385e3c9506b43168f37f46e66bcfc564` | `667a0eba3dabe21e9e1547bf8bc83ab7488bed14` | 590 |
| [preprints/Taming-implies-compatibility-on-four-manifolds-September-23-2026/build/main.tex](https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Taming-implies-compatibility-on-four-manifolds-September-23-2026/build/main.tex) | `a331a7dd4720ca319bce81ee73a4d787b10edbb24350218d8ddb971c729d7d89` | `010a6d35bfce35e59fcd5f2bb5d2baa445800c60` | 733 |
| [preprints/Taming-implies-compatibility-on-four-manifolds-September-23-2026/build/preamble.tex](https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Taming-implies-compatibility-on-four-manifolds-September-23-2026/build/preamble.tex) | `88ef7ddee1a8ce2532a7512ccd70fd8c00ff13609f4783d827f7791777680c1d` | `810c9c046a3bd7c75e70c433e65fb50849859721` | 1599 |
| [preprints/Taming-implies-compatibility-on-four-manifolds-September-23-2026/build/references.bib](https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Taming-implies-compatibility-on-four-manifolds-September-23-2026/build/references.bib) | `5eecf1f149cdeb0d0e2371a07d59b64ebc817abc6bd0f6f950a6ee91886313ba` | `fa21377dc20aa807905d5a59b47db086e9ade7af` | 9744 |
| [preprints/Taming-implies-compatibility-on-four-manifolds-September-23-2026/build/sections/01-introduction.tex](https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Taming-implies-compatibility-on-four-manifolds-September-23-2026/build/sections/01-introduction.tex) | `d55cfd6e93606acbf492acae984e38aae6181d3a0bf25935b5eea382b11ee48a` | `797ccd700e3e15d3a0b94791f0db090361490b47` | 10275 |
| [preprints/Taming-implies-compatibility-on-four-manifolds-September-23-2026/build/sections/02-currents.tex](https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Taming-implies-compatibility-on-four-manifolds-September-23-2026/build/sections/02-currents.tex) | `17f02a4944c68ce917dd6e2acaae53f41ec6a25b13f15438a673e8caee10f8c3` | `6ee6282431b1483a7a5d6762803bfaf0b0fa726c` | 6409 |
| [preprints/Taming-implies-compatibility-on-four-manifolds-September-23-2026/build/sections/03-mass.tex](https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Taming-implies-compatibility-on-four-manifolds-September-23-2026/build/sections/03-mass.tex) | `806342e2beef65fb2728975e5e5cacfc591148f983437a448afd1676095b3f93` | `a4057079cd7c95659914af62441b63a4e7b4b0b0` | 5810 |
| [preprints/Taming-implies-compatibility-on-four-manifolds-September-23-2026/build/sections/04-energy.tex](https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Taming-implies-compatibility-on-four-manifolds-September-23-2026/build/sections/04-energy.tex) | `7fcb8945a62f12a25398ef89bcaee3c70441adb2ad2fa9e6120831ae75fbd0c6` | `42dea8fc1e2415e2675879ecfadc16630cb33e41` | 9544 |
| [preprints/Taming-implies-compatibility-on-four-manifolds-September-23-2026/build/sections/05-density.tex](https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Taming-implies-compatibility-on-four-manifolds-September-23-2026/build/sections/05-density.tex) | `df97ab2e92e3d70dce247ac314d0e2adbb9b345018331a86dd03a3624de5fffb` | `3bca97ec278269d42b1cea42238f1b2893acbc6c` | 10533 |
| [preprints/Taming-implies-compatibility-on-four-manifolds-September-23-2026/build/sections/06-splitting.tex](https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Taming-implies-compatibility-on-four-manifolds-September-23-2026/build/sections/06-splitting.tex) | `e256efe17b9b2b1bb4236c7ad8f8307b37d951225edc4da12278130a98649963` | `4dbdd71553fe9e86100b37fb824e506b8c5d3cd0` | 10225 |
| [preprints/Taming-implies-compatibility-on-four-manifolds-September-23-2026/build/sections/07-conclusion.tex](https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Taming-implies-compatibility-on-four-manifolds-September-23-2026/build/sections/07-conclusion.tex) | `ad5c4074fb2716370f3ab4b8ce665ad6d616de7f714cb93b88f56986f60e7876` | `8b49ba5d5476d761bd5a5fcc235ebf5297263d8a` | 2584 |
| [preprints/Taming-implies-compatibility-on-four-manifolds-September-23-2026/paper.pdf](https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Taming-implies-compatibility-on-four-manifolds-September-23-2026/paper.pdf) | `61535cc6bd7e4f457f94c7db63f1038b35da807b6898581318ea0adcc08eff32` | `a2dca11fef1401becbc1fe3a00eb595ba73b86c5` | 216875 |

## Reproduce both provenance checks and source comparisons

Save the accompanying compare_revisions.py and run:

```sh
python compare_revisions.py paper-revision-snapshots.zip initial-commit-verification.zip > comparison.txt
```

Both filenames are arguments; substitute the actual local ZIP paths if needed. Python standard library only; no network access, packages, or archive extraction required. The script checks all 91 manifest files, all 45 older files and complete folder coverage, recorded subtree hashes, and the initial root-tree hash. It then prints per-file statuses and complete textual differences with commit-pinned paths. PDF bytes are hashed and compared but PDF text extraction is not needed for this reproduction.

The script was executed successfully on both supplied ZIPs. Its implementation is included below for a self-contained report.

```python
#!/usr/bin/env python3
"""Verify both supplied archives and print reproducible source diffs.
Usage: python compare_revisions.py paper-revision-snapshots.zip initial-commit-verification.zip
Uses only the Python standard library; reads ZIP entries without extracting files.
"""
import argparse
import difflib
import hashlib
import json
import zipfile

INITIAL = "adc7f1241b42e322a6451854ab7e4b4c146bf78a"
LATER = "fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb"
PAIRS = {
    "Taming": (
        "Taming-implies-compatibility-on-four-manifolds-September-23-2026",
        "Taming-implies-compatibility-on-four-manifolds-October-6-2026"),
    "Box Transport": (
        "Incompressible-Box-Transport-and-Finite-Computation-September-27-2026",
        "Incompressible-Box-Transport-and-Finite-Computation-October-6-2026"),
    "BSD": (
        "Exact-Birch-Swinnerton-Dyer-Formula-from-Low-Selmer-Corank-October-3-2026",
        "Exact-Birch-Swinnerton-Dyer-Formula-from-Low-Selmer-Corank-October-7-2026")
}

def check(condition, message):
    if not condition:
        raise ValueError(message)

def git_hash(kind, data):
    return hashlib.sha1(kind.encode() + b" " + str(len(data)).encode() + b"\0" + data).hexdigest()

def tree_hash(entries):
    entries = sorted(entries, key=lambda e: (
        e["path"] + ("/" if e["type"] == "tree" else "")).encode())
    data = b"".join(
        e["mode"].lstrip("0").encode() + b" " + e["path"].encode()
        + b"\0" + bytes.fromhex(e["sha"]) for e in entries)
    return git_hash("tree", data)

def load_folder(archive, folder):
    prefix = "preprints/" + folder + "/"
    return {n[len(prefix):]: archive.read(n) for n in archive.namelist()
            if n.startswith(prefix) and not n.endswith("/")}

def compare(snapshot_path, verification_path):
    with zipfile.ZipFile(snapshot_path) as z, zipfile.ZipFile(verification_path) as v:
        check(len(z.namelist()) == len(set(z.namelist())), "Duplicate snapshot ZIP paths")
        check(len(v.namelist()) == len(set(v.namelist())), "Duplicate verification ZIP paths")
        p = json.loads(z.read("provenance.json"))
        t = json.loads(v.read("initial-commit-paper-trees.json"))
        c = json.loads(v.read("initial-snapshot-comparison.json"))
        commits = json.loads(z.read("repository-commits.json"))
        check(p["commit"] == c["currentCommit"] == LATER, "Later commit mismatch")
        check(t["commit"] == c["initialCommit"] == INITIAL, "Initial commit mismatch")
        manifest = {f["path"]: f for f in p["files"]}
        check(len(manifest) == len(p["files"]) == 91, "Expected 91 unique manifest files")
        for path, f in manifest.items():
            data = z.read(path)
            check(len(data) == f["bytes"], "Length mismatch: " + path)
            check(hashlib.sha256(data).hexdigest() == f["sha256"], "SHA-256 mismatch: " + path)
            check(git_hash("blob", data) == f["gitBlobSha"], "Blob mismatch: " + path)

        comparisons = {x["path"]: x for x in c["comparisons"]}
        check(len(comparisons) == len(c["comparisons"]) == 45, "Expected 45 unique comparison rows")
        check({x["folder"] for x in t["papers"]} == {x[0] for x in PAIRS.values()},
              "Unexpected older paper folders")
        checked = []
        counts = {}
        for paper in t["papers"]:
            prefix = "preprints/" + paper["folder"] + "/"
            entries = paper["files"]
            check(len({e["path"] for e in entries}) == len(entries), "Duplicate tree entries")
            blobs = [e for e in entries if e["type"] == "blob"]
            supplied_paths = {prefix + n for n in load_folder(z, paper["folder"])}
            check(supplied_paths == {prefix + e["path"] for e in blobs}, "Incomplete older folder")
            for e in blobs:
                path = prefix + e["path"]
                data = z.read(path)
                row = comparisons[path]
                actual = git_hash("blob", data)
                check(len(data) == e["size"], "Initial size mismatch: " + path)
                check(actual == e["sha"] == row["initialGitBlobSha"]
                      == row["suppliedGitBlobSha"] == manifest[path]["gitBlobSha"],
                      "Initial blob mismatch: " + path)
                check(row["identical"] is True, "Recorded comparison mismatch: " + path)
                checked.append(path)
            directories = [("", paper["treeSha"])] + [
                (e["path"], e["sha"]) for e in entries if e["type"] == "tree"]
            for directory, expected in directories:
                children = []
                for entry in entries:
                    parent, _, name = entry["path"].rpartition("/")
                    if parent == directory:
                        children.append(dict(entry, path=name))
                check(tree_hash(children) == expected, "Subtree mismatch: " + prefix + directory)
            counts[paper["folder"]] = len(blobs)
        check(len(checked) == 45 and set(checked) == set(comparisons), "Old-file coverage mismatch")
        root = t["root"]
        check(root.get("truncated") is False, "Truncated initial root listing")
        initial_metadata = next(x for x in commits if x["sha"] == INITIAL)
        root_sha = tree_hash(root["tree"])
        check(root_sha == initial_metadata["commit"]["tree"]["sha"], "Root-tree mismatch")
        check(next(e["sha"] for e in root["tree"] if e["path"] == "preprints")
              == t["preprintsTreeSha"], "Preprints-tree identifier mismatch")

        print(json.dumps({
            "initial_commit": INITIAL,
            "later_commit": LATER,
            "manifest_files_verified": len(manifest),
            "older_files_verified": len(checked),
            "older_folder_counts": counts,
            "initial_root_tree_reconstructed": root_sha,
            "preprints_tree_id": t["preprintsTreeSha"],
            "limits": "Offline verification of supplied records. The full preprints tree listing "
                      "is not included, so the paper-tree membership edge is supplied provenance, "
                      "not independently reconstructed. No live GitHub or proof verification."
        }, indent=2))
        for title, (older, newer) in PAIRS.items():
            print("\n# " + title)
            a, b = load_folder(z, older), load_folder(z, newer)
            for name in sorted(a.keys() | b.keys()):
                status = ("added" if name not in a else "removed" if name not in b
                          else "identical" if a[name] == b[name] else "changed")
                print(status + "\t" + name)
                if status != "identical" and not name.endswith(".pdf"):
                    print("".join(difflib.unified_diff(
                        a.get(name, b"").decode().splitlines(True),
                        b.get(name, b"").decode().splitlines(True),
                        fromfile=INITIAL + "/preprints/" + older + "/" + name,
                        tofile=LATER + "/preprints/" + newer + "/" + name)), end="")

if __name__ == "__main__":
    parser = argparse.ArgumentParser(description=__doc__)
    parser.add_argument("snapshots_zip")
    parser.add_argument("initial_verification_zip")
    args = parser.parse_args()
    compare(args.snapshots_zip, args.initial_verification_zip)

```

The supplied repository license is Apache License 2.0; manuscript sources attribute authorship to OpenAI.
