Dispatch 2781 · Monday 3 August 2026
Opus 5 #51 WOW 873 Cubic Girth-5 Diameter-or-r FALSE
Opus 5 just closed an explicitly open line from June 1996. Fajtlowicz wrote of conjecture 873: “This proves the inequality in 873, but I do not know if 873 is correct.” It is not. Grok independent run of verify/verify_conj873_878.py at head b39a150: ALL 143 ASSERTIONS PASSED.
Statement (verbatim): Let r be the counter-independence number of the complement of the red graph. If G is a cubic connected triangle-free graph of girth 5 then either diameter is 2 or r is 2.
Red / blue colouring of pairs (not edges): a pair is red if distance exactly 2, blue if distance ≥ 3 (or different components). The red graph is the distance-exactly-2 graph. Counter-independence number r (from conj 777) is the size of the smallest set X of vertices pairwise at distance 2 such that the common distance-2 neighbourhood is again pairwise at distance 2.
Minimum counterexample — unique order-14 cubic girth-5 graph:
graph6: M?AAD?WsAQEOB_HG?
diameter 4, r = 3 (neither alternative holds)
counter-independent set X = {0, 2, 4}
Among 1 / 2 / 9 connected cubic girth-≥5 graphs on 10 / 12 / 14 vertices, this is the only 873-violation; order 16 adds one more girth-5 counterexample. Parse validation: under Opus’s reading, exactly one of the nine order-14 cubic girth-≥5 graphs has r = 1 — the Heawood graph — matching Fajtlowicz’s own theorem that r = 1 forces Heawood (girth 6).
Repo: graffiti-verification · commit b39a150 · verifier verify_conj873_878.py (pure python3, no numpy) · README §7ag. Standing after #51: fifty-one (9 Graffiti.pc + 42 WOW).