Document the full Erdős 625 theorem-upgrade and follow-up program

[SamPetkov]

Purpose

This draft PR records the complete post-proof research program for Erdős Problem 625 above the latest Section VIII closure branch, PR #41.

It has three goals:

1. preserve the correct public chronology of the 12 July 2026 candidate proof without converting that chronology into an unverified priority or acceptance claim;
2. state precisely which theorem improvements are already latent in the current proof architecture, conditional on closing the remaining normalized-second-moment seam;
3. formulate the most valuable next results as detailed theorem targets with mathematical proof programs, quantitative constants, and an explicit publication strategy.

The PR does not modify the canonical manuscript, the statement Erdos625Statement, or the external claim status of Problem 625.

Stack position

#34 -> #35 -> #37
                \
                 #40 -> #39 -> #41 -> this PR
                /
#36 -> #38 ----

Base branch:

agent/625-section8-decorated-reference-quotient

Head branch:

agent/625-value-upgrade-program

Review PR #39 for the theorem-by-theorem audit and PR #41 for the exact endpoint reference normalization before reviewing the research extensions in this PR.

Public chronology

The dossier records that PR #1 was created on 12 July 2026 at 19:08:31 UTC and merged at 19:29:28 UTC. Its public head was

945ed733af198fe8698e14079ddc079f2bc554d7

and it explicitly described its verdict as

Provisional internal verification: PASS.

The correct status language is therefore:

A candidate full-sequence proof was publicly deposited on GitHub on 12 July 2026. It has since undergone extensive internal computational, formal, and theorem-by-theorem auditing. External verification is still being sought, and no claim of community acceptance is made.

The public timestamp is a chronology record, not a claim about unpublished independent work.

Current proof frontier

The PR records the remaining closure theorem in exact mathematical form.

For a block matching P, endpoint multiplicities m_e, actual multiplicities

j_e = m_e - h_e,

and total multiplicities

J = sum_e j_e,
H = sum_e h_e,

the aggregate local ratio is

R_{m,d}(h)
  = C(m,h)/((d+1)...(d+h)) * 2^(-hm+h(h+1)/2).

The single global denominator ratio is

(n)_{J+H}/(n)_J = (n-J)_H <= n^H.

Hence

w(P,m-h)/w_full(P)
  <= product_e n^(h_e) R_{m_e,d_e}(h_e).

Under the high-cell condition 2h<m,

n^h R_{m,d}(h)
  <= (n m / 2^floor(2m/3))^h.

At the four-size phase this gives

rho_n = O((log n)^(7/3)/n^(1/3)) = o(1),

and the intended global estimate

BareSkeletonSum_n
  <= exp(O(n^(2/3)(log n)^(4/3)) + O(sqrt(n log n)))
  = exp(o(n/(log n)^4)).

The repaired q-only Section IX theorem then gives the normalized second moment. Thus the remaining mathematical seam is the physical-fibre/deficit reindexing, not another residual cycle decomposition.

Main-paper theorem upgrades

Phase-resolved theorem

Writing

A_4(delta) = log 2 - D_4(delta),
H_n = n/(log n)^3,

the root displacement is

r_+(n)-r_4^co(n)
  = ((log 2)^2/4 * A_4(delta_n) + o(1)) H_n.

Midpoint placement retains one half, giving the conditional high-probability theorem

chi(G_n)-zeta(G_n)
  >= ((log 2)^2/8 * A_4(delta_n) - o(1)) H_n.

Stronger certified uniform coefficient

Combining the /8 propagation with the exact certificate

A_4(delta) > log(1000/639)

gives

(log 2)^2/8 * log(1000/639)
  = 0.026896409808379...

approximately 6.68735 times the coefficient printed in the current candidate manuscript.

Immediate corollaries

The dossier gives copy-ready statements for:

• the phase-resolved gap;
• the stronger uniform constant;
• the simultaneous complement result

  min{chi(G_n),chi(complement G_n)}-zeta(G_n)
    >= c n/(log n)^3;
  

• the tunable cochromatic upper-tail family coming from the rare-seed amplifier;
• the balanced-sign rare seed.

All remain explicitly conditional on completion of the global Section VIII theorem.

New theorem target 1: remove the midpoint loss

For

k_theta = ceil(r_4^co + theta(r_+-r_4^co)),

the retained gap is

(1-theta) (log 2)^2/4 * A_4(delta_n) * n/(log n)^3.

If the proof errors requiring a positive signed-root buffer are O(n/(log n)^4), then a varying placement theta_n is admissible whenever

theta_n log n -> infinity.

The proposed first target is

theta_n = (log n)^(-1/2).

This would yield

chi(G_n)-zeta(G_n)
  >= ((log 2)^2/4 * A_4(delta_n) - o(1)) n/(log n)^3,

and the certified uniform coefficient

(log 2)^2/4 * log(1000/639)
  = 0.053792819616758...

about 13.3747 times the current displayed coefficient.

The PR lists every proof layer whose dependence on the placement buffer must be replayed.

New theorem target 2: exact four-support phase minimum

Using the envelope identity

dF_S(T)/dT = -lambda_S(T),

the phase derivative satisfies

A_4'(delta)
  = T_0'(delta)
    (lambda_infinity(T_0(delta))-lambda_4(T_0(delta))).

The proposed interval proof certifies the sign of this derivative on finitely many rational intervals and evaluates the endpoint loss rigorously.

The current numerical diagnostic is

min A_4(delta) ~= 0.520701335491,

which would correspond to coefficients approximately

0.031271565748  (midpoint),
0.062543131497  (near-root).

These numbers are deliberately labelled diagnostic until the interval proof is completed.

New theorem target 3: structural balance

For clique-class proportion rho, the sign entropy is

H(rho) = -rho log rho -(1-rho) log(1-rho).

The imbalance loss is

log 2 - H(rho),

and near rho=1/2+x it equals

2x^2 + O(x^4).

Because a Theta(n/log n) entropy loss moves the root by Theta(n/(log n)^3), the proposed stability theorem says that any cocoloring within o(n/(log n)^3) of the signed optimum must have clique proportion 1/2+o(1).

This would turn the balanced construction into a structural theorem about near-optimal cocolorings.

High-upside theorem target: slowly growing support

For supports whose width tends slowly to infinity, the dossier proposes proving

sup_delta D_{S_n}(delta) -> 0

uniformly and developing a dimension-uniform second moment. Combined with near-root placement, the target coefficient is

(log 2)^3/4
  = 0.083256162997232...

This is a separate-paper or major Version 3 target. The PR explains why another fixed five-size support is not worth rebuilding the proof around: the current diagnostic gain is only about 1.017% while the endpoint table dimension increases.

Largest follow-up target: matching upper bound

The most important separate project is

chi(G_n)-zeta(G_n)
  = O(n/(log n)^3)

with high probability, which would give the full conjectured order.

The dossier decomposes this into:

1. a first-moment lower location bound for zeta(G_n) over all signed profiles;
2. a third-order ordinary-coloring upper construction;
3. comparison of the two phase-resolved roots.

An intermediate publishable target is a two-sided O(n/(log n)^3) corridor for the cochromatic number itself.

Other follow-up directions

The PR gives detailed formulations for:

• fixed edge density p != 1/2, with one-class reward

  g_p(s)=p^C(s,2)+(1-p)^C(s,2)
  

  and an inhomogeneous Ising-type overlap factor;
• the two-independent-graph model discussed on the Erdős Problems forum, with an explicit attribution and verification boundary;
• the generic restriction-product theorem;
• the exact cycle-space factor;
• the abstract rare-seed amplifier.

Files

• 625/research/README.md
• 625/research/ERDOS625_VALUE_UPGRADE_PROGRAM_2026-07-26.md
• 625/research/ERDOS625_VALUE_UPGRADE_THEOREMS.tex
• 625/research/PUBLIC_PR_RESULT_MAP_2026-07-26.md
• 625/research/erdos625_value_upgrade_constants.py
• .github/workflows/erdos625-value-upgrade-program.yml

Exact regression layer

The standard-library coefficient script verifies under normal and optimized Python:

• canonical, /8, /4, and full-support coefficient identities;
• the exact factor-four and factor-two propagation relations;
• the criterion theta_n log n -> infinity for theta_n=(log n)^(-1/2);
• monotonicity of the formal placement coefficients;
• binary-entropy imbalance penalties.

These are deterministic arithmetic checks only. They do not prove the entropy certificate, the diagnostic phase minimum, the normalized second moment, or any random-graph theorem.

Review order

1. Verify the chronology and status language.
2. Audit the exact current proof frontier and the aggregate deficit formula.
3. Check the phase-resolved /8 coefficient and certified constant.
4. Review the near-root placement error criterion.
5. Review the balance-stability first-moment calculation.
6. Treat slowly growing support, the matching upper bound, general p, and the two-layer model as separate research programs.
7. Run the focused dossier workflow.

Scope boundary

This PR is a research and theorem-design dossier. It does not prove the physical-fibre reverse equivalence, the global all-deficit reindexing, Proposition 9.2, the near-root theorem, the exact phase minimum, balance stability, the slow-support theorem, the matching upper bound, or Erdos625Statement.

Keep the PR draft until the dossier workflow is green and its mathematical classifications have been independently reviewed.

[SamPetkov]

Validation update

The focused workflow completed successfully on the initial PR head 4d1b12206fefad3e65e6c32818fd4a0dbd2d82b6.

GitHub Actions run 30189607631 passed all gates:

• Python compilation of erdos625_value_upgrade_constants.py;
• normal execution of the coefficient, placement, and balance ledger;
• optimized execution under python -O;
• required theorem/program/status marker checks;
• lightweight TeX document and brace checks.

This validates the dossier's deterministic arithmetic and internal file structure. It does not upgrade any conditional theorem or research target to a proved random-graph result. The PR remains draft for mathematical review of the phase-resolved propagation, near-root uniformity criterion, balance-stability program, and longer-term theorem targets.

[SamPetkov]

A full section-by-section re-audit of the canonical TeX is now in stacked PR #44. The phase-resolved /8 theorem, stronger fixed coefficient, complement corollary, tunable tail, and balanced rare seed survive. Four research-program statements need narrower classifications: (1) theta_n log n -> infinity is still a proposed near-root uniformity criterion, not a proved lemma; (2) sign-entropy balance currently controls the selected four-size witness family, not every arbitrary cocoloring; (3) a slow-support estimate of the form K(m)n/(log n)^4 cannot be made o(n/(log n)^4) by requiring only K=o(log n); and (4) the current midpoint construction gives zeta <= (r_4^co+r_+)/2+o(H_n), while zeta <= r_4^co+o(H_n) is conditional on a near-root theorem. See ERDOS625_VALUE_UPGRADE_REAUDIT_CORRECTIONS_2026-07-26.md in PR #44.