# Revision evidence: three selected OpenAI math papers

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## 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 old and revised files below are both pinned at the merge commit. Earlier commit metadata is contextual; this archive does not contain a separately checked checkout of either earlier commit.

## 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/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/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
--- Taming-implies-compatibility-on-four-manifolds-September-23-2026/README.md

+++ 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
--- Taming-implies-compatibility-on-four-manifolds-September-23-2026/build/main.tex

+++ 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
--- Taming-implies-compatibility-on-four-manifolds-September-23-2026/build/preamble.tex

+++ 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
--- Taming-implies-compatibility-on-four-manifolds-September-23-2026/build/references.bib

+++ 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
--- Taming-implies-compatibility-on-four-manifolds-September-23-2026/build/sections/01-introduction.tex

+++ 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
--- Taming-implies-compatibility-on-four-manifolds-September-23-2026/build/sections/08-cones.tex

+++ 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/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/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
--- Incompressible-Box-Transport-and-Finite-Computation-September-27-2026/README.md

+++ 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
--- Incompressible-Box-Transport-and-Finite-Computation-September-27-2026/build/analysis.tex

+++ 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
--- Incompressible-Box-Transport-and-Finite-Computation-September-27-2026/build/clocks.tex

+++ 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
--- Incompressible-Box-Transport-and-Finite-Computation-September-27-2026/build/main.tex

+++ 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/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/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
--- Exact-Birch-Swinnerton-Dyer-Formula-from-Low-Selmer-Corank-October-3-2026/README.md

+++ 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
--- Exact-Birch-Swinnerton-Dyer-Formula-from-Low-Selmer-Corank-October-3-2026/build/main.tex

+++ 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
--- Exact-Birch-Swinnerton-Dyer-Formula-from-Low-Selmer-Corank-October-3-2026/build/references.tex

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

+++ 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.
```

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| [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/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/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/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/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/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/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/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/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/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/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/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/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/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/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/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/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/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/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/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/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/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/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/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/preprints/Taming-implies-compatibility-on-four-manifolds-September-23-2026/paper.pdf) | `61535cc6bd7e4f457f94c7db63f1038b35da807b6898581318ea0adcc08eff32` | `a2dca11fef1401becbc1fe3a00eb595ba73b86c5` | 216875 |

## Reproduce the hash verification and source differences

Run the following Python standard-library code with the original ZIP filename as its argument. It reads directly from ZIP entries and does not extract files.

```python
import sys, zipfile, json, hashlib, difflib
with zipfile.ZipFile(sys.argv[1]) as z:
    p = json.loads(z.read('provenance.json'))
    for f in p['files']:
        b = z.read(f['path'])
        assert len(b) == f['bytes']
        assert hashlib.sha256(b).hexdigest() == f['sha256']
        assert hashlib.sha1(b'blob ' + str(len(b)).encode() + b'\0' + b).hexdigest() == f['gitBlobSha']
    print('All manifest checks passed')
    pairs = {'Taming': ('Taming-implies-compatibility-on-four-manifolds-September-23-2026', 'Taming-implies-compatibility-on-four-manifolds-October-6-2026'), 'Box': ('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')}
    for title, (old, new) in pairs.items():
        def load(root):
            prefix = 'preprints/' + root + '/'
            return {n[len(prefix):]: z.read(n) for n in z.namelist() if n.startswith(prefix)}
        a, b = load(old), load(new)
        for name in sorted(a.keys() | b.keys()):
            if a.get(name) != b.get(name) 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=old+'/'+name, tofile=new+'/'+name)))
```

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