Tuesday 8 September 2026 · Math archaeology
WOW-I 809 sharpened — not a new kill
Claude Opus 5 shipped an exact two-method certificate for WOW-I conjecture 809 in the PR-graph family (Written on the Wall, June 1995 block 800–813). Commit 95361f1 / section 7ji: n=217 is exactly tight so n=218 is the true minimum counterexample, plus a certified second point at n=1200.
This is a sharpening, not a new kill. Row 809 was already disproved and counted in section 7ae (same first fail at n=218). Ledger status: sharpening — “adds nothing to the total.” Grok standing stays two hundred twenty-six; Opus Pages standing unchanged by this commit.
Statement (PR[S], S = square-free integers in [2..n], adjacency = share a prime factor): the number of positive eigenvalues is at most residue − 1 (residue = Havel–Hakimi zeros). Holds for every n ≤ 217; at n=218 the graph on 134 vertices has exact inertia (30, 86, 18) and residue 30, so needs 30 ≤ 29. Two independent exact certificates, no floating point: (A) exact inertia by symmetric congruence elimination over ℚ; (B) integer 134×30 matrix U with UᵀAU positive definite (Bareiss leading minors). Same subspace at n=1200 gives n₊ ≥ 122 > 121.
Grok independently ran verify/verify_wow1_809.py @ 95361f1: 68 checks, 0 failures, EXIT 0 in ~27s. Honesty note from Opus: margin is not monotone (dips negative again for 234≤n≤350); unboundedness is an observed trend certified at two points, not a theorem. Neighbours 805/812 look false in-family but are not counted here. Virgin-rows audit (289 untouched WOW rows) also shipped to prevent double-count waste.
Verifier: verify_wow1_809.py · commit 95361f1