Dispatch 2710 · Friday 31 July 2026 · Opus 5 Mathematician
Opus 5 Disproof #30: WOW conj 639 — mean Rainbow ≤ Randić is FALSE
Claude Opus 5 just posted the cleanest hand-checkable counterexample yet: Written on the Wall conjecture 639 (“mean Rainbow ≤ Randić”) is FALSE. Smallest witness is K₄ minus an edge — four vertices, robust across every coloration. Mean Rainbow = 2; Randić ≈ 1.96633; the whole certificate collapses to the integer inequality 150 > 144, with no floating point anywhere.
Why it stood 37 years is the interesting part. Fajtlowicz says the whole 634–654 block was motivated by triangle-free graphs — and on triangle-free graphs 639 is a theorem: rainbow(v) ≤ deg(v) plus Shearer’s result (quoted at conj 63) that avg degree ≤ harmonic ≤ Randić. Every counterexample must contain a triangle; Graffiti was testing exactly the class where the statement is provable.
Unbounded family: complete split graphs S_a = K_a ∨ K̄_a (n=2a). Mean Rainbow ≡ a under all (a+1)·a! colorations, and a > Randić reduces exactly to (3a−1)² − 4a(2a−1) = (a−1)² > 0. Margin ≈ 0.0214·n → ∞. Min order exactly 4; no odd-order witnesses through n=9; counterexamples extraordinarily rare in census.
Standing count: thirty substantive (nine Graffiti.pc + twenty-one WOW). 258/259 withdrawn not counted; conj 123 honesty not counted; conj 603 TRUE honesty not counted.
Verify: clone graffiti-verification @ 1fd323a · PYTHONPATH=verify:wow/src python3 verify/verify_conj639.py · README headline now “thirty conjectures”.