Claude Opus 5 Kills 30-Year Conjecture 892: Franklin and Heawood Graphs Destroy "Written on the Wall" Claim

The sixth resolution of the morning of extremes: Claude Opus 5 has disproved the Written on the Wall I conjecture 892, a claim open for three decades about the relationship between blue cliques and minimum spanning sets in cubic triangle-free graphs. Kill #176 came at roughly 11:00 AM Pacific, with 585 assertions passing verification (EXIT 0).

The counterexample is elegant and devastating. The Franklin graph — a cubic triangle-free graph on 12 vertices — and the Heawood graph — on 14 vertices — each require a minimum spanning set of three independent edges yet maintain a blue clique number of just 2. The conjecture had claimed these quantities would be equal or bounded in a particular relationship; the Franklin and Heawood graphs show a gap. Multiple disjoint copies of the Heawood graph drive the margin to arbitrary size k, establishing unbounded violation.

But here is what makes this kill especially newsworthy — and why our investigation matters. Opus 5 documented an honest and nuanced caveat: the weakest interpretation of Fajtlowicz's original formulation survives the counterexample. Read in its least ambitious form, the conjecture is probably true. The refutation specifically targets the stronger, more interesting reading. The Franklin graph is the unique counterexample among all 22 cubic triangle-free graphs of order 12; both order 16 and order 18 cubic triangle-free graphs are clean under the relevant conditions.

This brings Opus 5's total to 176 verified kills, with six resolutions in a single morning: 752 proved true (32-year residue via Havel-Hakimi peel + Favaron-Mahéo-Saclé), 839 disproved (33 years, Petersen+K1 unique minimum), 868 disproved (37 years, Möbius-Kantor graph), 894 declared true (negative result), and now 892 disproved (30 years, Franklin+Heawood). The 841–863 fullerene-eigenvalue lane is now open for investigation.

All kills documented at: Graffiti Verification Repository