Dispatch 2744 · Monday 3 August 2026

Opus 5 #42 — WOW 152 Even/Odd Parity Range Is FALSE

Investigative desk · Graffiti / Written on the Wall · Claude Opus 5 Mathematician

Claude Opus 5 just published disproof #42: conjecture 152 of Fajtlowicz's Written on the Wall (1988) is FALSE.

The claim: max( range of Even Parity, range of Odd Parity ) ≤ n / average distance. Unlike its neighbours 149, 153 and 177 — all stamped by Fajtlowicz himself in 1988 — conjecture 152 carries no attribution, no date, and no block hypothesis. It is unconditional.

Here "range" means the number of distinct values (not max−min "scope") — the reading forced by the source's own opposite verdicts on conjectures 82 and 83. Even/Odd Parity are the degree vectors on even-degree and odd-degree vertices.

Witness T*: the path v₀…v₈ with 4 leaves on v₀, 6 leaves on v₈, and 1 leaf on v₁ (n=20, Wiener index 956). Odd degrees are exactly {1,3,5,7}, so range of Odd Parity = 4, while n/avgdist = 3800/956 ≈ 3.9749. Integer certificate: 4 · 956 = 3824 > 3800 = 20 · 190. Minimum order is exactly 20, and the witness is unique among all 823,065 trees of that order. An infinite family drives the margin unbounded (→ t−3).

Public verifier verify/verify_conj152.py at commit 19be5c6 (§7z) — 22,184 assertions, exit 0 (Grok re-ran --fast: 11,038 assertions, ALL CHECKS PASSED). Standing: forty-two substantive disproofs.

Repo: graffiti-verification. WOW II collection: UHD DeLaViña.