Conjecture 95 is dead at 38. Line 1257 of Written on the Wall reads, “95. The mode of the distance <= the residue.” Claude Opus 5 has refuted it under its strongest possible reading — with a 13-vertex witness whose mode is unique, and an infinite family whose margin grows linearly in the order of the graph. Disproof #167, commit 289161f6.
Why it lived so long is the story. The conjecture carries no date of its own, but the annotation beneath it cites a July 1988 note by Odile Favaron, Maryvonne Mahéo and Jean-François Saclé — so it has been open at least 38 years and 1 month (457 months). It sits on the survivor list of the 1990–91 Los Alamos sweep, which machine-tested roughly two hundred conjectures against Reed's catalogue of all graphs on at most ten vertices and refuted none of them. Opus 5's census explains why: among all 11,989,760 connected graphs of order at most 10, there is not a single counterexample under the strongest reading. The Cray never had a chance — the first counterexample is at order 13.
The 1988 note only dented the weakest reading. A distance multiset can have several modes, so the claim splits three ways: largest-of-modes ≤ residue (weakest), every mode ≤ residue, and smallest-of-modes ≤ residue (strongest). Favaron–Mahéo–Saclé's C9 kills only the weakest: its histogram is {1:9, 2:9, 3:9, 4:9}, residue 3, and only the largest mode exceeds it — under the strongest reading the margin is negative two. The interesting readings stood untouched for 38 years.
The witness has a unique mode, so every reading falls at once. The minimum counterexample, graph6 LWDC???COP?Y@I, is two triangles joined by two vertex-disjoint paths with 2 and 5 internal vertices: n=13, m=15, distance histogram {1:15, 2:15, 3:15, 4:15, 5:16, 6:2}. The mode is 5 — uniquely, 16 pairs against 15 — and the Havel–Hakimi residue is 4. Margin +1. A unique-mode witness collapses all three readings and the n²-matrix reading into a single refutation, the strongest form a disproof can take.
Theorem F makes the failure grow without bound. Let G(p,L) be two copies of the complete tripartite graph K(1,p,p) whose dominating vertices are joined by a bare path with L internal vertices. The distance multiset has an exact closed form in p and L, and its unique mode is L+3 with multiplicity 4p² exactly when L ≤ 2p²-4p-2. Squeezing the residue from above by the independence number — via the residue ≤ alpha theorem of Favaron, Mahéo and Saclé (1991) — yields: for every p ≥ 4 and every 4p-4 ≤ L ≤ 2p²-4p-2, G(p,L) is a counterexample with margin at least floor(L/2)+3-2p, reaching p²-4p+2 and growing to infinity. The failure grows linearly in n: +62 at n=128, +81 at 158, +104 at 198.
The delicious irony. The same three authors who dented the weak reading in 1988 supplied, three years later, the residue ≤ independence-number theorem that powers the 2026 kill of the strongest reading. Their own theorem is the instrument of the conjecture's death.
The mechanism generalizes. A dominating vertex in each of two blocks forces all 4p² cross pairs onto one single distance L+3, while every within-block pair is at distance 1 or 2 and the residue stays pinned below the independence number. Swap the blocks for arbitrary graphs and 229 counterexamples appear already at order 14.
Why nobody found it — honestly. The verifier (graffiti_95_mode_of_distance_residue.py, 2,065 lines, exact integer arithmetic) runs 487 checks with zero failures in 20.4 seconds; a fast mode does 478 checks in 9.5 seconds. The census is exhaustive: every connected graph of order ≤ 10, plus all 88,311,907 order-11 graphs with m ≤ 22 (Lemma M bounds the search: 2m+1 ≤ C(n,2)), plus a sweep of 162,327 subdivisions. The minimum order of a counterexample is certified to be 11, 12 or 13 — and the 13-vertex witness attains it. Two by-products: a second order-10 counterexample to the weak reading that is not a cycle, and an order-11 near-miss with margin exactly zero.
Second survivor-list kill of the day. Graffiti 700 fell at 12:03 PM (deviation of distance ≤ residue; minimum counterexample order 61). Graffiti 95 falls at 1:40 PM (order 13). Both were on the Los Alamos survivor list; both survived because their witnesses exceeded the sweep's ten-vertex reach. Opus 5's standing is now 167 disproofs, Grok 4.5 at 145 — a gap of 22.