2026-07-20 18:18 UTCgpt-5.6-sol · xhigh

Independent streaming certificate and whole-range validation of Xu-tame primes p=24m+1 for 0<=m<=1000000.

Internal result — value unestablished

The independent generator produced 187948 prime records: 187934 tame and 14 wild. The independent validator accepted the full 10,231,469-byte certificate, reproduced Xu's 591/5 and 7185/9 checkpoints, and confirmed the wild list 409, 577, 5569, 9601, 23929, 83449, 102001, 329617, 712321, 1134241, 1724209, 1726201, 5212561, 8813281. This does not improve the known global verification through 10^18.

Internal result — not a contribution candidate

The computation remains in the transparent ledger, but it did not establish meaningful novelty or scholarly value and is not queued for Charlie review.

Strategy and discriminator

certificate-carrying bounded enumeration

Emit one exact (a,b,t) witness for every tame prime and complete factorizations of p+a across the forced a-range for every wild prime; validate coverage and all algebra with a separate implementation.

Hypothesis: Exactly 14 primes p=24m+1 with 0<=m<=1000000 fail p=4abt-a-b for coprime integers 1<=a<=b and t>=1.

Test: A separate validator must reconstruct every progression prime, verify all tame identities, exhaust every divisor b|p+a for every wild row, and reproduce the claimed counts and list.

Rationale

Positive cases carry directly checked integer witnesses. Negative cases carry 5667 complete prime-factorization rows covering every mathematically possible a for the 14 wild primes; the validator independently regenerates every possible divisor b. Prime coverage is reconstructed separately, and three rehashed semantic corruptions were rejected for their intended mathematical reasons.

Claims requiring scrutiny

Among the 187948 primes p=24m+1 with 0<=m<=1000000, exactly 14 are Xu-wild.
The 14 wild primes are 409, 577, 5569, 9601, 23929, 83449, 102001, 329617, 712321, 1134241, 1724209, 1726201, 5212561, and 8813281.
The certificate supplies strict Erdős–Straus decompositions induced by tame parameters for every one of the other 187934 primes.
Novelty is only a candidate: searches found no longer published tame/wild classification, but priority and peer review remain unresolved.

Evidence and scope

tame_m1000000_certificate.jsonl; SHA-256 8cee70726a44e630eb87a978a5c9a5f15307c630f3e212cdef3b23f6aafc9d0f; 187950 lines; 10231469 bytes.
Internal certificate content digest d3587ce25140d76a59f002742fcf3d09033919f54220a05ce3d4bc8b6e090d65.
certify_tame_range.py; SHA-256 b7c17c85fb8ea27b20b54457abc42957384baebf3a0bdd9ff09d509429e451af.
validate_tame_certificate.py; SHA-256 78f06fc8971a9b62c29802970d3c69eedfdab42af20895fe476558a02e153c9f.
test_tame_certificate_controls.py; SHA-256 d15244cc4b224fa272ffd3c530a9b7af7410d0007914445c5951fe8ce4704fb4.
Validator definition-level search agreed with the certificate for all 325 relevant primes p<=25000.

Computational experiments

.proof-experiments/20260720-180723-e5b2b9: generator completed in 53.391 seconds, return code 0, peak child RSS 18688 KiB.
.proof-experiments/20260720-180833-c9cacb: independent validator completed in 76.245 seconds, return code 0, peak child RSS 19840 KiB.
.proof-experiments/20260720-181201-ebc70e: rehashed semantic controls completed in 8.471 seconds and rejected a false witness, missing prime, and composite factor for the expected reasons.

Independent checker

validate_tame_certificate.py imports no project code, uses a different prime enumeration, verifies all factorization products and factor primality, exhausts all wild divisors, and directly searches Xu's original numerator-summand definition through p<=25000.

Contribution gate

retroactively adjudicated

No structured gate reasons were recorded in this legacy attempt; see the adjudication ledger.

Original model outcome: candidate
Public classification: internal_result

Cross-domain transfers tested

None recorded.

Established facts

For a prime p=24m+1, Xu-tameness is equivalent to p=4abt-a-b with coprime positive a,b and t>=1.
Algebraic derivation in research_note.md, with a definition-level computational audit through p<=25000. · All prime p=24m+1 under Xu's definition. · proved
Exactly 14 relevant primes are wild through p<=24000001.
Full certificate plus successful independent validation in experiment 20260720-180833-c9cacb. · Primes p=24m+1, 0<=m<=1000000. · computed

Ruled out in this epoch

The Epoch 1 classification remained dependent on only one whole-range implementation.
The complete range 0<=m<=1000000. · A separate certificate generator and non-importing validator now agree over the full range. · .proof-experiments/20260720-180723-e5b2b9 and .proof-experiments/20260720-180833-c9cacb. · A third implementation finds a specific discrepant prime, witness, factorization, or coverage gap.
Presenting p<=24000001 as an improved global Erdős–Straus verification bound.
Full conjecture verification bounds. · The conjecture is already reported verified through 10^18; this result concerns only Xu's restricted tame classification. · https://arxiv.org/abs/2509.00128 · An independently verified full-conjecture computation beyond the current published bound.

Open leads

Isolated reconstruction in a language distinct from Python.
It directly attacks the remaining shared-session and shared-runtime risk at modest computational cost. · Write a Rust or C checker without reading the Python implementations, then validate tame_m1000000_certificate.jsonl and preserve the first discrepancy or matching counts. · high · open
Formalize the tame-equivalence lemma in Lean.
The certificate relies on a short general reduction whose hypotheses and strict-denominator conclusion are suitable for formal checking. · Create a minimal Lean statement deriving the tame denominators from p=4*a*b*t-a-b and prove positivity, distinctness, and the rational identity without sorry. · normal · open

Continuation checkpoint

Objective: Obtain an isolated, different-language reconstruction of the complete certificate and criterion.

First action: Without reading either Python implementation, specify the JSON-lines schema from the certificate itself and write validate_tame_certificate.rs to reconstruct progression-prime coverage, check tame equations, and exhaust wild divisor rows.

Stop condition: Stop with success only on counts 187948/187934/14 and the exact 14-prime list; otherwise stop immediately at and preserve the first discrepant record.

Next moves

Give only Xu's theorem, the criterion, certificate, and file hash to an isolated reviewer.
Require a checker in Rust, C, or Lean that does not inspect or reuse either Python implementation.
If it agrees, prepare a short computational note and ask Xu whether the extended classification is already known.

Citations

Tool disclosure

GPT-5 Codex produced the synthesis, Python implementations, and audit design. Web tools inspected the official source, Xu's paper, newer literature, the original Erdős PDF, and the maintained Lean statement. Python 3.12 standard-library code performed all exact computation. The experiment harness captured commands, limits, logs, hashes, versions, git state, and memory. Shell utilities, jq, and SHA-256 inspected artifacts. No CAS, SAT/SMT solver, probabilistic primality test, executed proof assistant, floating-point arithmetic, or sub-agent was used.

Duration: 1099.3s
Review state: needs Charlie review
Attempt ID: erdos-242-20260720-181822-bad8d7

Human review ledger

reject · Proof Factory contribution gate · 2026-07-20 18:38 UTC
Downgraded to an internal computational result. It extends Xu's restricted tame/wild cutoff but does not improve the known Erdős–Straus verification bound, answer a source-requested target, establish structural mathematics, or have confirmed expert relevance. Preserve the artifacts and audit trail; do not queue Charlie review or publication.