Rewrite the Erdős 625 manuscript in clear AMS-style prose

.github/workflows/erdos625-ams-manuscript-clarity.yml

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+name: Erdős 625 AMS manuscript clarity
+
+on:
+  pull_request:
+    paths:
+      - "625/arxiv/AMS_THEOREM_ENVIRONMENTS_V2.tex"
+      - "625/arxiv/FRONTMATTER_INTRODUCTION_POSTCLOSURE_V2.tex"
+      - "625/arxiv/PROOF_ROADMAP_INSERT_V2.tex"
+      - "625/arxiv/SECTION8_ALL_DEFICIT_AMS_V2.tex"
+      - "625/arxiv/SECTION9_Q_ONLY_AMS_V2.tex"
+      - "625/arxiv/AMS_EDITORIAL_DRAFT_V2.tex"
+      - "625/audits/AMS_WRITING_AND_CLARITY_PASS_2026-08-02.md"
+      - "625/experiments/check_ams_manuscript_clarity.py"
+      - ".github/workflows/erdos625-ams-manuscript-clarity.yml"
+  workflow_dispatch:
+
+concurrency:
+  group: erdos625-ams-clarity-${{ github.event.pull_request.number || github.ref }}
+  cancel-in-progress: true
+
+permissions:
+  contents: read
+
+jobs:
+  editorial-checks:
+    runs-on: ubuntu-24.04
+    steps:
+      - uses: actions/checkout@93cb6efe18208431cddfb8368fd83d5badbf9bfd # v5
+      - name: Compile editorial checker
+        run: python -m py_compile 625/experiments/check_ams_manuscript_clarity.py
+      - name: Run editorial checker
+        run: python 625/experiments/check_ams_manuscript_clarity.py
+      - name: Run editorial checker with optimization
+        run: python -O 625/experiments/check_ams_manuscript_clarity.py
+
+  build-editorial-pdf:
+    runs-on: ubuntu-24.04
+    steps:
+      - uses: actions/checkout@93cb6efe18208431cddfb8368fd83d5badbf9bfd # v5
+      - name: Install TeX dependencies
+        run: |
+          sudo apt-get update
+          sudo apt-get install -y --no-install-recommends \
+            latexmk lmodern texlive-latex-extra texlive-bibtex-extra \
+            texlive-fonts-recommended
+      - name: Build the AMS editorial draft
+        working-directory: 625/arxiv
+        run: |
+          latexmk -pdf -interaction=nonstopmode -halt-on-error \
+            AMS_EDITORIAL_DRAFT_V2.tex
+      - name: Reject undefined references in final log
+        working-directory: 625/arxiv
+        shell: bash
+        run: |
+          if grep -E "LaTeX Warning: (Reference|Citation).*undefined|There were undefined references" \
+              AMS_EDITORIAL_DRAFT_V2.log; then
+            exit 1
+          fi
+      - name: Upload editorial PDF
+        uses: actions/upload-artifact@v4
+        with:
+          name: erdos625-ams-editorial-draft
+          path: 625/arxiv/AMS_EDITORIAL_DRAFT_V2.pdf
+          if-no-files-found: error

625/arxiv/AMS_EDITORIAL_DRAFT_V2.tex

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+\documentclass[11pt,reqno]{amsart}
+
+\usepackage[T1]{fontenc}
+\usepackage{lmodern}
+\usepackage{microtype}
+\usepackage{mathtools}
+\usepackage[authoryear,round]{natbib}
+\usepackage[hidelinks]{hyperref}
+
+\input{AMS_THEOREM_ENVIRONMENTS_V2}
+
+\title[Chromatic and cochromatic numbers]{A Polynomial-Scale Gap Between the
+Chromatic and Cochromatic Numbers of a Random Graph}
+\author[Samuil Petkov]{Samuil Petkov}
+\address{\'Ecole normale sup\'erieure, Universit\'e PSL, Paris, France}
+\email{samuil.petkov@ens.psl.eu}
+\subjclass[2020]{Primary 05C80; Secondary 05C15, 60C05}
+\keywords{random graph, chromatic number, cochromatic number, second moment method}
+
+\begin{document}
+
+\input{FRONTMATTER_INTRODUCTION_POSTCLOSURE_V2}
+\input{PROOF_ROADMAP_INSERT_V2}
+\input{SECTION8_ALL_DEFICIT_AMS_V2}
+\input{SECTION9_Q_ONLY_AMS_V2}
+
+\renewcommand{\bibsection}{\section*{References}}
+\bibliographystyle{plainnat}
+\bibliography{references}
+
+\end{document}

625/arxiv/AMS_THEOREM_ENVIRONMENTS_V2.tex

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+% Standard theorem environments for the Version 2 manuscript.
+% The canonical manuscript currently uses custom ruled boxes.  The final
+% AMS-style version should instead use the ordinary amsthm hierarchy below.
+
+\theoremstyle{plain}
+\newtheorem{theorem}{Theorem}[section]
+\newtheorem{proposition}[theorem]{Proposition}
+\newtheorem{lemma}[theorem]{Lemma}
+\newtheorem{corollary}[theorem]{Corollary}
+
+\theoremstyle{definition}
+\newtheorem{definition}[theorem]{Definition}
+
+\theoremstyle{remark}
+\newtheorem{remark}[theorem]{Remark}

625/arxiv/FRONTMATTER_INTRODUCTION_POSTCLOSURE_V2.tex

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+% Copy-ready front matter for the post-closure Version 2 manuscript.
+% Do not include this file in the canonical manuscript until the normalized
+% second-moment theorem and the final event assembly have been validated on one
+% integrated branch.
+
+\begin{abstract}
+Let $G_n\sim G(n,1/2)$.  We prove that the difference between the chromatic
+number $\chi(G_n)$ and the cochromatic number $\zeta(G_n)$ is at least a
+positive constant times $n/(\log n)^3$ with probability tending to one.  The
+argument is uniform across the full rounding phase of the independence-number
+threshold.  Its first-moment component compares ordinary colorings with signed
+cocoloring witnesses supported on four consecutive class sizes.  For the
+second moment, we derive an exact sign-compatibility identity, encode the large
+overlap cells by a matching with local deficits, sum the associated physical
+matching fibers exactly, and control the residual even-subgraph factor by
+injective restriction outside the exposed matching.  A bounded-differences
+argument then amplifies the resulting positive-probability seed to a
+high-probability cocoloring.
+\end{abstract}
+
+\maketitle
+
+\section{Introduction}\label{sec:introduction-v2}
+
+All graphs in this paper are finite, simple, and undirected.  A
+\emph{cocoloring} of a graph $G$ is a partition of $V(G)$ into nonempty classes,
+each of which is either an independent set or a clique.  The least number of
+classes in such a partition is the \emph{cochromatic number} $\zeta(G)$.  This
+is the standard invariant introduced by Lesniak and Straight
+\citep{lesniak-straight-1977}.  We write $\chi(G)$ for the chromatic number.
+
+Let $G_n\sim G(n,1/2)$ be the labeled random graph on $[n]$ in which the
+$\binom n2$ possible edges occur independently with probability $1/2$.  An
+event holds \emph{with high probability} if its probability tends to one as
+$n\to\infty$.
+
+Erd\H{o}s and Gimbel asked whether
+\[
+  \chi(G_n)-\zeta(G_n)\longrightarrow\infty
+\]
+with high probability \citep[p.~263]{erdos-gimbel-1993}.  The problem was
+later restated by Gimbel \citep[Section~7.4]{gimbel-2016} and is cataloged as
+Erd\H{o}s Problem~625 \citep{bloom-erdos625}.  The main result gives a
+full-sequence lower bound of the conjectured polynomial order.
+
+\begin{theorem}\label{thm:main-v2}
+For $G_n\sim G(n,1/2)$,
+\[
+  \mathbb P\!\left(
+    \chi(G_n)-\zeta(G_n)
+    \ge
+    \frac{(\log 2)^2}{8}
+    \log\!\left(\frac{1000}{639}\right)
+    \frac{n}{(\log n)^3}
+  \right)
+  \longrightarrow 1.
+\]
+In particular, $\chi(G_n)-\zeta(G_n)$ tends to infinity with high probability
+along the full sequence of integers $n$.
+\end{theorem}
+
+The full-sequence assertion is significant.  Near a jump of the
+independence-number threshold, changing the relevant class size by one changes
+the feasible profile.  A result proved only for a dense set of phase values
+therefore need not extend to every integer $n$.  Our estimates are uniform in
+the complete phase parameter, including sequences approaching either endpoint
+of a phase interval.
+
+\subsection{Background}
+
+The chromatic number of dense random graphs has been studied since the work of
+Grimmett and McDiarmid \citep{grimmett-mcdiarmid-1975}.  The first-order
+asymptotic was established by Bollob\'as \citep{bollobas-1988}; later
+refinements include \citet{mcdiarmid-1990},
+\citet{panagiotou-steger-2009}, and \citet{heckel-2018}.  The cochromatic
+number fits into the theory of generalized chromatic numbers for hereditary
+graph properties developed by Scheinerman and by Bollob\'as--Thomason
+\citep{scheinerman-1992,bollobas-thomason-1995}.
+
+For the chromatic--cochromatic difference, Heckel and, independently, Steiner
+obtained the first quantitative evidence toward divergence
+\citep{heckel-2024-question,steiner-2024}.  Heckel subsequently proved a much
+larger lower bound on a phase-dependent set containing approximately $95\%$ of
+the integers \citep{heckel-2025-difference}.  The remaining difficulty is to
+control the complete phase uniformly.  The present proof uses the signed
+first-moment gain and rare-seed amplification from that work, but replaces the
+pairwise sign bound by an exact overlap identity and treats every phase with
+one four-size profile.
+
+\subsection{Main ideas}
+
+The proof has three stages.
+
+First, we compare two continuous first-moment roots.  The ordinary root governs
+proper colorings.  The second root governs a four-size signed witness in which
+each class is marked either independent or complete.  The $2^k$ possible marks
+create a uniform entropy gain and move the signed root to the left by order
+$n/(\log n)^3$.
+
+Second, we prove that this signed root is populated.  For two signed witnesses,
+the overlap matrix determines an exact local reward and a binary cycle-space
+factor.  Cells whose multiplicity exceeds half the phase cap form a matching.
+After fixing this block-level matching, each selected cell is described by its
+full endpoint multiplicity and a nonnegative deficit.  The local physical
+matching fibers are summed exactly, and the only nonlocal change is one global
+falling-factorial ratio.  The remaining even-subgraph contribution is bounded
+directly by restricting outside the exposed matching.
+
+Third, Paley--Zygmund gives a positive-probability signed witness.  A
+one-Lipschitz induced cocolorable-capacity variable and a simultaneous
+leftover-coloring estimate amplify this seed to a high-probability upper bound
+for $\zeta(G_n)$.  Intersecting that event with the ordinary-coloring lower
+bound completes the proof.
+
+\subsection{Organization of the paper}
+
+Section~2 resolves the full independence-number phase.  Sections~3 and~4
+locate the ordinary first-moment root and derive an unrestricted lower bound
+for $\chi(G_n)$.  Section~5 constructs the four-size signed profile and proves
+the root separation.  Section~6 derives the exact signed-overlap identity, and
+Section~7 controls partial diagonals.  Sections~8 and~9 treat the high-cell
+matching and the residual attachment, respectively.  Section~10 amplifies the
+signed seed, and the final section assembles the two high-probability events.
+
+Exact finite identities, deterministic inequalities, asymptotic estimates, and
+probabilistic conclusions are stated separately throughout.  This separation
+is important in Sections~8 and~9, where a local product formula and a global
+falling-factorial normalization occur in the same calculation.