Referee-proofread and harden the Erdős 625 manuscript

[SamPetkov]

Purpose

This PR is stacked on #57. It performs a second, referee-oriented pass over the self-contained Erdős 625 manuscript. The goal is not cosmetic expansion: it reduces the amount of notation a reader must hold at once, replaces several compressed proof interfaces by exact statements, and makes the quantitative constant independently reproducible.

The theorem status remains fail-closed. This PR improves publication readiness but does not promote the full-sequence theorem until the mathematical and Lean gates in Appendix A are closed on one integrated commit.

Readability pass

• rewrites the abstract around the theorem, obstruction, method, and consequence;
• replaces the boxed repository warning with a short typographic verification note;
• presents an unnumbered main theorem rather than Theorem 0.1;
• reorganizes the introduction into previous work, four main ideas, the role of the four-size support, and a compact proof guide;
• condenses the proof-object preliminaries and removes duplicated graph definitions;
• replaces the technical proof synopsis with a reader-oriented four-stage guide;
• converts the generated legacy paragraph proofs and manual terminal squares to AMS proof environments;
• normalizes log 2 notation and several dense transitions without changing hypotheses or summation domains.

Section 8 hardening

The high-cell section now follows the valid order of operations explicitly:

1. sum the complete physical matching fiber;
2. extract exact one-cell deficit ratios;
3. pay the ambient falling-factorial loss once;
4. sum all positive deficits;
5. regroup full-containment references by endpoint table;
6. transport the table sum to the partial-diagonal weights;
7. insert the phase estimates.

The square-free endpoint comparison now has a proof. It derives the local atom ratio, the identity

g(t) / g(s) = 2^(d s + binom(d,2)),

the ambient falling-factorial inequality, and the cancellation producing the Q_ij factors. The section also isolates the reusable finite core that works for any finite endpoint alphabet with matching-supported selected cells.

Section 9 hardening

The former informal “thresholds of F” phrase is replaced by an exact function Phi_F(r'), and the cycle-space expansion is an equality before expectation.

The joint prescribed-cell estimate is derived from factorial moments. This makes explicit why demands in cells sharing rows or columns may be expanded simultaneously.

The estimate

q_ab <= C theta_ab^2

is now a named lemma with a complete proof based on log-convexity and endpoint control. The upper endpoint contributes

-U^2/24 + O(U log U)

in the exponent. The complementary residual regime now derives separately the local-reward and cycle-space bounds before combining them.

Exact constant ledger

Adds 625/experiments/check_constant_ledger_v3.py, which verifies the strengthened four-support certificate using exact rational arithmetic. It encloses 2^(1/20) by two rationals whose twentieth powers bracket 2, bounds log 2 by a rational series with an exact remainder, and checks the mean bracket, four omitted-tail bounds, and the final 139/500 ledger without floating-point decisions.

The final section now explains this certificate in the paper and uses a deterministic error sequence in the phase-resolved probability statement.

Publication-rank boundary

The accompanying audit records the distinction between presentation quality and theorem status. If the full-sequence theorem closes as stated, the contribution resolves the exceptional phase left by the existing approximately-95%-of-integers result and gives a quantitative n/(log n)^3 lower bound. The remaining rank-limiting work is therefore closure and independent replay of:

• the concrete phase/chromatic-tail package;
• the signed four-size first-moment assembly;
• the complete partial-diagonal theorem;
• the global high-skeleton/residual assembly;
• the final theorem-facing Lean root.

Validation

The dedicated workflow now:

• compiles all manuscript tools;
• runs the exact rational constant checker;
• regenerates the manuscript from the frozen canonical TeX blob;
• rejects legacy boxes, paragraph proofs, manual terminal squares, placeholders, duplicate labels, and implementation-status prose in Section 8;
• checks the exact Phi_F interface and quadratic-activity lemma;
• compiles the AMS/BibTeX manuscript;
• rejects undefined references, citations, and material overfull boxes;
• verifies full-paper length and text extractability;
• uploads representative renders of the front matter, Sections 8 and 9, and the final page.

A green workflow validates the manuscript assembly, exact constant ledger, and editorial invariants. It does not substitute for the remaining mathematical closure gates.