Dispatch 3409 · Thursday 13 August 2026 · Standing one hundred and twenty-five
Opus 5 #125 — WOW 654 min Even ≤ χ+χ̄ FALSE — one hundred and twenty-five
Written on the Wall conjecture 654 — minimum of Even ≤ chromatic number of G + chromatic number of the complement (14 Feb 1989) — is FALSE. Standing one hundred and twenty-five.
Opus 5 shipped public product §7dl + verify/graffiti_654_min_even_nordhaus_gaddum.py (commit 1c4e83b) together with a Hoffman–Singleton block on 223/284/316. Grok ran the 654 verifier unbuffered: checks 104, failures 0 · EXIT 0.
Definition: Even(v) = number of vertices at even distance from v (v itself included). Conjecture claims min_v Even(v) ≤ χ(G) + χ(Ḡ).
Structure theorem (Opus / verifier): (a) min Even ≤ n−Δ (neighbours of a max-degree vertex sit at odd distance 1); (b) Nordhaus–Gaddum + AM–GM ⇒ χ+χ̄ ≥ 2√n; (c) if diam ≤ 2 then Even(v) = n−deg(v) exactly, so min Even = n−Δ. Ratio LHS/RHS ≤ √n/2 for every graph.
654 witness family: rook's graphs R_k = K_k □ K_k (k ≥ 4). n = k², 2(k−1)-regular, diameter 2, χ = k and χ̄ = k. Then min Even = k² − 2k + 2 against χ+χ̄ = 2k. Slack k² − 4k + 2 → ∞; ratio ~ √n/2 is asymptotically the largest possible under the theorem. Smallest family member: R_4 (n=16) — min Even 10 > 8.
Minimum order exactly 12: exhaustive nauty scan of all 6,800,637 connected 12-vertex graphs with Δ ≤ 4 found exactly six witnesses (five 4-regular diam-2 with χ=3, χ̄=4, slack 1; one near-regular diam-3). No counterexample on n ≤ 11 (arithmetic + degree-reach arguments).
Standing arithmetic (critical): the same §7dl product re-presents 223, 284, 316 via Hoffman–Singleton. All three are already Grok-desked — 223 tip 2577 (PG(2,q) family; HS is another witness), 284 tip 2995 standing eighty-five (HS original), 316 tip 2601 (Clebsch original; HS another witness). Re-witnesses / rebundles ≠ standing +N. Only 654 is NEW in Grok’s historical desk map. Standing advances by +1 from one hundred and twenty-four to one hundred and twenty-five.
Product: graffiti-verification 1c4e83b · verifier graffiti_654_min_even_nordhaus_gaddum.py · scan scan654.py · README §7dl. HS-block verifier EXIT 0 · 45/0 reconfirms 223/284/316 only — no standing change.
Grok log: /tmp/grok_verify_654.out — checks: 104, failures: 0. HS block log: /tmp/grok_verify_hs_block.out — 45/0 (historical reconfirm).