Opus 5 has shipped three more machine disproofs (#144–146), bringing the campaign's standing to 146 Graffiti conjectures refuted and five confirmed true. The mechanism this time is uniform: distance-based matrices have far smaller rank than Fajtlowicz expected — so any conjecture that assumes rank tracks graph size faithfully is in trouble.
Graffiti 134 (Randić index ≤ rank of the gravity matrix; the 'original 101,' a conjecture that survived the famous 1990 Brewster–Dinneen–Faber computer search) is false. Opus 5's inertia lemma gives rank(Gravity) = rank(Harary) + R(G) ≤ n/2, and a connected regular graph of diameter 2 refutes 134 exactly when the adjacency eigenvalue 1 has multiplicity above n/2. Kneser graphs K(m,2) have mult(1) = m(m−3)/2 and gravity-rank exactly m, so every K(m,2) with m ≥ 6 fails: K(6,2) on 15 vertices is the smallest known (Randić 7.5 vs rank 6), the Clebsch graph gives 8 > 6, and the slack m(m−1)/4 − m grows without bound. The Petersen graph is exactly tight — equality, no violation. No counterexamples among connected regular graphs of order 11–14 with degree ≤ 6.
Graffiti 282 (girth ≥ 5 ⇒ n − α ≤ rank of the distance matrix) falls to the minimum possible order — exactly 14 — and a unique witness. An exhaustive scan of all 54,283 connected girth-≥5 graphs on ≤13 vertices is clean; of 275,480 on 14 vertices exactly one hits; and all 2,045,279 on 15 vertices are clean again. The lone counterexample (graph6 M??CA?_sDOB_?wX??) is bipartite, 18 edges, girth 6, diameter 5, independence 7, distance-matrix rank 6. The Desargues graph — the bipartite double of Petersen, 20 vertices, independence 10 — posts the largest slack, with distance-matrix rank 6. GP(9,2), GP(9,4) and the dodecahedron also work.
Graffiti 604 (triangle-free ⇒ mean Even ≤ χ + χ̄) dies by blow-up: Petersen[I₂] on 20 vertices, with slack 2t − 3 for Petersen[I_t]. Hoffman–Singleton contributes +14. And an infinite family K(3k−1,k), k ≥ 3, certified by an explicit (k+1)-colouring, posts slack 125 at n = 330 and 37,036 at n = 77,520. Along the way Opus 5 proved no bipartite graph can violate 604 at all.
All three are certified by a dedicated verifier (164 checks, 0 failures; README §7dn). The notable pattern: these are conjectures a 1990 computer search had already batted at — and each fails the same way, because matrix rank was assumed to grow with the graph more faithfully than it actually does.