Rewrite the Erdős 625 manuscript in clear AMS-style prose

[SamPetkov]

Purpose

This is a manuscript-level editorial PR stacked on the reader-roadmap branch. It revises the actual mathematical prose, not only audit notes or pull-request descriptions.

The canonical 625/arxiv/main.tex remains frozen. The new files form a separate, compilable post-closure Version 2 draft and must not be imported until the complete Section VIII–XI theorem chain is green on one integrated commit.

Main writing changes

Standard AMS structure

The draft replaces custom colored statement boxes by the ordinary amsthm hierarchy:

Theorem / Proposition / Lemma / Corollary
Definition
Remark

Statements precede proofs. Definitions precede their first substantive use. Exact identities, deterministic inequalities, asymptotic estimates, and probability conclusions are separated explicitly.

Abstract and introduction

The new abstract is one paragraph of fewer than 150 words. It states the model, quantitative conclusion, full-phase difficulty, and principal proof mechanisms without promotional language.

The introduction now proceeds in the following order:

1. standard graph-theoretic definitions;
2. the Erdős–Gimbel question;
3. the main theorem;
4. the importance of the full-sequence quantifier;
5. concise historical background;
6. the three proof stages;
7. organization of the paper.

It removes repeated roadmaps and phrases such as “exquisitely phase-sensitive,” replacing them by precise quantitative statements.

Proof architecture

PROOF_ROADMAP_INSERT_V2.tex has been rewritten in American mathematical English and updated to the current proof architecture:

• ordinary and signed first-moment roots;
• exact signed-overlap identity;
• high cells as a block matching with deficits;
• exact physical-fiber summation;
• one global falling-factorial loss;
• cellwise all-deficit summation;
• weighted regrouping by realized endpoint table;
• q-only residual attachment;
• bounded-differences amplification.

Every symbol is introduced before use, and the text states explicitly that no individual partial matching is assigned a canonical full completion.

Section VIII replacement

SECTION8_ALL_DEFICIT_AMS_V2.tex presents the high-skeleton argument through five named statements:

1. aggregate high-skeleton weight;
2. exact local ratio;
3. aggregate deficit comparison;
4. finite optional-choice identity;
5. weighted reference regrouping.

The prose emphasizes that the physical fibers are summed first and the global denominator loss is paid once afterward. The local physical-matching ratio is displayed as the product of two explicit fractions, avoiding the ambiguous completion language of the original draft.

Section IX replacement

SECTION9_Q_ONLY_AMS_V2.tex replaces the cycle/walk exposition by:

• the exact conditional decomposition;
• a general restriction-product lemma;
• injectivity outside the exposed matching;
• one activity controlling both local increments and even-subgraph weights;
• the two residual regimes;
• the attained attachment-sum estimate.

No simple-cycle decomposition, walk kernel, or duplicated local/cycle charge appears in the new text.

Files

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

Validation

The dedicated workflow is green on head

ebcd45b60d2caec7f868638896329ccd97623d7c

Run:

30747972007

The successful checks include:

• abstract length and one-paragraph structure;
• standard theorem styles;
• American spelling and terminology consistency;
• absence of promotional filler and manual equation tags;
• balanced TeX environments;
• unique labels;
• absence of unresolved placeholders;
• non-import of the post-closure fragments by canonical main.tex;
• compilation of the complete 13-page editorial draft with amsart, BibTeX, and latexmk;
• absence of undefined references and citations in the final log;
• PDF rendering without clipped text, overlaps, broken glyphs, or black boxes.

The generated editorial PDF is attached to the workflow as erdos625-ams-editorial-draft.

Scope boundary

This PR is an editorial draft. It does not prove or promote the remaining Section VIII bare-skeleton estimate, the normalized second moment, Proposition 9.2, or the final Erdős 625 theorem. The theorem statement in the post-closure draft may be imported into the canonical manuscript only after those mathematical gates are complete.