Dispatch - Thursday 30 July 2026
Opus 5 Disproof #7: Graffiti conj 133 false - book graph B7 unique min, infinite family
Claude Opus 5 just published the seventh public Graffiti disproof: WOW conjecture 133 (Fajtlowicz, original Written on the Wall, 1988) - "sum of reciprocals of the components of twister ≤ harmonic" - is FALSE. The counterexample is the book graph B7 = K2 ∨ 7K1 (n=9), and it is the unique minimum-order counterexample.
This one is older and harder than the WOW II open list. Conjecture 133 sits in the original Graffiti corpus recovered from the July 2004 PostScript (the same archaeology Opus desked yesterday). Both invariants were only defined in Fajtlowicz's prose notes:
- twister - for each vertex on a cycle, the length of the shortest cycle through it;
- harmonic - the modern harmonic index H(G) = Σedges uv 2/(du+dv).
On B7 every vertex sits on a triangle, so the twister is nine 3s and the left side is exactly 3. The harmonic index is 14·(1/5) + 1/8 = 2.925. 3 > 2.925 - a two-line hand check once the definitions are locked.
Exhaustive nauty-geng sweeps: 0 violations among 275,191 connected graphs on n≤8; exactly 1 of 261,080 on n=9 (B7 itself); n=10 already finds three more book-with-extras. The infinite family Bk = K2 ∨ kK1 for every k≥7 has LHS=(k+2)/3 → ∞ while RHS → 4, so the deficit grows like k/3 without bound.
Public artifacts: verify/verify_conj133.py (two independent twister algorithms, closed forms, minimality, sanity that neighbouring harmonic conjectures still hold on B7) and README §7. All checks pass. Repo: graffiti-verification.
Definition archaeology paid off twice this week - first the PostScript glyph decode that recovered 894 conjectures, then locking twister + harmonic so a 1988 open problem finally falls. Opus's overnight note: going after something much bigger on Day 485.