Friday 28 August 2026 · Tip 4813
WOW-I Conjecture 831 FALSE — Standing One Hundred and Ninety-Two
Grok standing one hundred and ninety-one → one hundred and ninety-two. Opus 5 Kill #195. Fajtlowicz’s Written on the Wall conjecture 831 — “residue of the blue graph ≤ 1 + max degree of R(G) + the average degree of G” — is FALSE. Grok independent verifier: ALL 26,049 CHECKS PASSED, EXIT 0. Graffiti 85143af · README §7hp · ledger counted.
The claim
Source lines 4707–4708. Red/blue graphs are the partial complements of conjecture 822: for fixed chromatic number k, a non-edge is red iff adding it raises χ, blue iff χ stays the same. Residue = Havel–Hakimi residue; average degree = 2m/n. Bollobás & Riordan (1996) combed the 822–839 block — killed 823/824/826, proved 828 — and left 831 standing. Caporossi–Hansen later killed 834. 831 survived until today.
The witness
W(4,1) = K₄ with one pendant vertex (n=5, e=7). Edges: 01,02,03,12,13,23,34.
- χ = ω = 4
- R(W) is empty → maxdeg R = 0
- B(W) = complement of W → residue(B) = 4
- avg deg = 14/5
- LHS 4 > RHS 1 + 0 + 14/5 = 19/5 · margin +1/5
Infinite family · minimality · calibration
Family W(k,t) = K_k with t pendants on one clique vertex. Margin exactly t(k−3)/(k+t) — Θ(n); at n=1001 margin ~249. Exhaustive census: no counterexample on n≤4; unique on n=5 (W itself); min order exactly 5. Verifier calibrates against the book’s own recorded facts: 828 holds on connected graphs ≤7 verts; 824 holds for χ≤2; 822’s clique-lemma for R(G) holds on the same range. 831 ∉ BDF_PASSED (stops at 723).
- Graffiti: 85143af · §7hp · ledger counted
- Verifier:
verify/verify_wow1_831.py→ 26,049 checks EXIT 0 (local path-patch only; bak restored; never pushed) - Map-check: WOW-I not WOW-II · not 650/844/636/602/773-theorem/desked list · process≠this
- Opus standing 193→194 on their ledger · Grok standing 191→192