Opus 5 Kill #181: WOW Conjecture 807 False at n=345–353, and the 1988 Block Keeps Giving

Kill #181 lands with precision — exactly 9 consecutive counterexamples beyond the author's tested range. Meanwhile, the 1988 "ΣD ≤ ΣE" block yields four refutations and counting, and kill #182 (WOW 182) already has a section header.

Claude Opus 5 committed kill #181 (commit 2397f0d) with a surgically precise finding: WOW conjecture 807, which asserts λ₂ > λ₁/2 for some spectral condition, survives ≤344 but fails at exactly n=345 through 353 — and nowhere else in [20, 3000]. The verification used zero-floating-point arithmetic (Bareiss + Rayleigh-Ritz) to produce an airtight result, with two CORRECTION blocks preserving the original claims transparently. Standing: 181 kills.

The 1988 Block: A Conjecture Mine

The original Written on the Wall contains a dense block of conjectures (181–204) headed "connected graphs in which the sum of components of D is ≤ the sum of components of E" — dated July 26, 1988. This block has proven to be one of the richest veins in the graffiti corpus, now yielding four confirmed refutations in rapid succession:

Conjecture 187 (§7bw): Refuted by K₂ ∨ P_{n−2} with margin ⌈n/2⌉ − 1. Tight on exactly the three order-6 graphs that refute 188. Minimum counterexample: order 7.
Conjecture 188 (§7bu): Refuted by the same family. Laplacian mode = n, while n − μ = ⌈n/2⌉. Arithmetically extremal — the family is not merely a counterexample but the provably worst-case one.
Conjecture 189 (§7bv): Refuted under both readings of "eigenvalues" — adjacency (via K₂ ∨ P_{n−2}) and Laplacian (via C₄). Minimum counterexample: order 6 (three for 188, two for 189).
Conjecture 186 (§7bz): Favaron, Mahéo, and Saclé's own conjecture (Dec 1989). Refuted by the prism over a clique, K_a □ K₂, with minimum counterexample at order 14 (K₇ □ K₂ is one of exactly two). Deficit grows without bound.

And in what may be the next kill, WOW conjecture 182 (§7x) already has a published refutation in the README — the graph H(5,19,1) with an explicit integer certificate (99×296 = 29,304 > 25×1,125 = 28,125). No counterexample exists below order 9. An infinite family H(q,t,r) extends the margin to q−3, unbounded.

Other recent additions to the repository include §7by (conjecture 318 on triangle-free graphs, minimum counterexample order 7 with an asymptotically unbounded family), §7bs (conjecture 642, stars with triangles grafted on, smallest counterexample order 6), and §7bt (conjecture 651, ten counterexamples of order 8).

The standing count — 181 — may not survive the afternoon.