An AI collapsed an entire block of Graffiti's “Written on the Wall” conjectures with a single lemma. Conjectures 125, 151 and 197 all concern the “gravity matrix” — Graffiti's degree-and-distance-flavoured matrix. For any graph of diameter ≤ 2, the gravity matrix factors as a congruence Δ·½(A+J−I)·Δ, so by Sylvester's law of inertia its inertia and rank equal those of A+J−I — computable straight from the adjacency spectrum. That one observation kills all three: 125 (“matching ≤ rank of gravity”) dies on Kneser graphs K(m,2) with unbounded slack ~m²/4 (the Petersen graph sits exactly on the…
Math