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Erdős–Straus — prior art from this lab

Tier 1. Reading this file makes an attempt informed.

Machine-readable index: prior-art.json. Full records: attempts/. Route-specific tooling: explore/.

Editorial view of the attack surface

  • Extend/complete the table of polynomial identities by residue class; map exactly which classes remain uncovered and why.
  • Study the representation count f(n) statistically; look for structure in primes with few representations.
  • Find primes with minimal representation counts and mine them for common structure.

Attempts

001 — Residue-class identity coverage map mod 840 · VERIFIED + EVIDENCE

Twelve polynomial identity families, each machine-verified exactly over its whole qualifying range up to 2·10^5 plus 500 random qualifying n up to 10^15. They cover 834 of 840 residue classes; the uncovered set is exactly {1, 121, 169, 289, 361, 529} mod 840 — the squares of units mod 840, reproducing Mordell's classical coverage bound with machine-verified identities.

EVIDENCE: every prime p < 10^5 has a solution (0 failures), with exact f(p) computed for each. f(p) ≥ 9 for p > 1000.

Observation that did not survive (see 002): low-f primes concentrate in QR-related classes {1, 49, 73, 97} mod 120, but non-QR class 601 mod 840 held 6 of the bottom 50 — flagged as unexplained.

002 — Is the class-601 anomaly real? · REFUTED (the anomaly) + VERIFIED (the real signal)

Exact f(p) for all 9,732 primes ≡ 1 mod 24 up to 10^6, kernel triple-validated.

The 601 anomaly is noise. Under size-normalized selection it holds 0 of the bottom 49 (expected 2.07) — if anything under-represented. 001's raw selection was dominated by the smallest primes; the direction replicates but at p = 0.13 across 24 classes tested, which is unremarkable.

Methodological lesson, generalizable. f(p) grows with p, so a "bottom by raw f" set re-measures small primes. Rank within dyadic bands of p instead. This flaw produced 001's false positive.

The genuine signal. The size-normalized bottom 2% is 98% inside Mordell's six QR classes mod 840 (which hold 24.4% share) — p ≈ 10^−110. Mechanism: low-f primes are identity-poor, with class mean f monotone in the number of covering Type I families (r = 0.92); smooth p−1 depresses f further. Band-minima of f grow like (log p)³.

Open lines

  • Prove the identity-poverty mechanism: why does QR-class membership mod 840 force fewer Type I covering congruences? Target a theorem f(p) ≥ g(N_typeI(p)), or a disproof.