Opus 5's disproof #162 — the project's 165th conjecture disproved — is Graffiti 234, stamped November 1988 and the first kill inside the regular block 227:239. The printed line is "n - rank <= size / average distance", but it sits twenty-one lines below a header reading Conjectures for regular graphs (227:239) — and regularity is the whole game. Without it, the four-leaf star K₁,₄ breaks the sentence at order 5, and a one-minute "disproof" by stars would say nothing true.
For regular graphs the conjecture looks bulletproof. In exact rational arithmetic, Opus 5 checked all 19,737 connected regular graphs on at most 12 vertices and all 7,319,447 connected cubic graphs on at most 22 vertices — the last census finished twenty minutes before the commit. Zero violations; at n=22 the best cubic graph still sits 0.433 below the bound.
That census is the trap. The record slack zig-zags — −1.82 at n=12, −0.345 at n=18, jumping back up in between — because the only graphs that threaten the bound live at n ≡ 0 (mod 6). Extrapolate the right subsequence and the next step, n=24, is where the inequality breaks.
The counterexample is a necklace of four copies of K₃,₃ minus an edge: six-vertex blobs with exactly two degree-2 ports, chained into a ring by four joining edges. On 24 vertices it is cubic, with nullity 10 (rank 14) against size/average-distance = 621/73 — a margin of +109/73 — and it is minimal: no smaller regular graph fails. The family's margin grows like n/3 − 7, without bound. Two theorems certify it: the nullity of the t-ring is exactly 2t+2, and its average distance is exactly (18t² + c)/(3(6t−1)) — so the right-hand side is trapped below 9 forever while the left runs away linearly.
Why this gadget and no other: the missing edge inside K₃,₃ is what creates the kernel (a full K₃,₃ has nullity 4; the ring inherits 2 per blob), while the other natural cubic gadget, the diamond K₄−e, contributes nothing — pure diamond necklaces have nullity 0 at every length. The verifier (graffiti-verification) is a 1,261-line standard-library proof with 663 checks — Bareiss elimination, a mod-1000003 rank, and an explicit ten-vector basis for the kernel, machine-checked to t=80. Graffiti 234 was swept by the 1990–91 Cray runs, found unbroken at ten vertices, and quietly set aside for 37 years and 9 months.