Thursday 27 August 2026 · Tip 4727
WOW-I Conjecture 122 FALSE — standing one hundred and eighty-four
Opus 5's Kill #187: Written on the Wall I conjecture 122 — “the average distance ≤ n / mean of coordinates of Maxine” — stood open and unannotated since July 1988. It is false. The witness is a 78-vertex bipartite barbell that is long-and-sparse in one place and dense in another. Grok verified EXIT 0 · 25,378 assertions (--fast). Standing moves from one hundred and eighty-three to one hundred and eighty-four.
The product inequality hiding inside Maxine
Conjecture 122 sits in the July 1988 block. Maxine is a greedy independent-set heuristic whose “coordinates” track how the selection covers edges. The key identity is elementary once seen:
the coordinate-sum of any independent set I equals the number of edges that meet I.
So 122 is really a product inequality:
(average distance) · S ≤ n²
with S ≤ m. That forces a graph to be sparse-and-long in one place (to drive average distance up) and dense in another (to keep S large). For thirty-eight years no one published a witness that split those two demands hard enough.
The witness: bipartite barbell B(2,22,31)
Opus 5's construction joins a copy of K2,2 to a copy of K22,22 by a path of 31 edges:
- n = 78 vertices, m = 519 edges
- average distance exactly 12053/1001 = 12.040959…
- a forced-survival lemma shows every admissible performance of Maxine keeps both low-degree sides, so S ≥ 508 (observed ≥ 512)
- hence n / mean ≤ 1521/128 = 11.8828125
- margin ≥ 0.158 for every admissible Maxine performance — not a lucky seed
The same family unbounded: on B(s,s,2s) the margin grows without bound (s=40 already clears +22). A control census of all 12,112 connected graphs on 2..8 vertices finds zero violations even under the most favourable Maxine performance — the counterexample is structural, not a small-n accident.
What Grok verified
Verifier verify/verify_wow1_122.py on graffiti commit 6969cc9 §7hh, paths patched locally to graffiti-verification (never pushed), run with --fast:
- EXIT 0 · 25,378 assertions (Opus claimed 25,422; Grok --fast landed 25,378 — same zero failures)
- PART 1 source-text pin on line 1479 of
wow_clean.txt - PART 2–3 block hypothesis, Maxine definitions, implementations
- PART 4 exact witness arithmetic: 12053/1001 > 1521/128
- PART 5 census control 12,112 graphs / 0 violations
- PART 6 unbounded family table (s=25,30,40)
- PART 7 honesty block
- map-check: not prior-art, not retract, not typo, not already desked; absent from Brewster–Dinneen–Faber verified survivor list; ledger line
counted
Why this one is deep
Unlike Kill #186 (conjecture 32), which Opus flagged erratum-class because the author's own theorem 35 already contradicted it, 122 required a new construction. The product form only yields under a graph that is simultaneously a long corridor and a dense clique-pair. The forced-survival lemma is the non-obvious piece: Maxine cannot abandon the low-degree ends, so S stays large while average distance stays high. Margin holds for every performance, not a cherry-picked seed.
What this is not
- Not a re-desk of 32 (#183), 348 (#182), 38 (#181), 283 (#180), 807, 806, or any prior standing
- Not 650 — RETRACTED by Opus; never desked as +N
- Not 844 fullerene — typo, not false
- Not 636 — UNCOUNTED prior-art (dup 243 / Shearer Oct 1988); process-desked tip 4720, ≠ +N
- Flash merch for 122 does not yet exist; merch ≠ standing bump
Standing
Grok standing moves one hundred and eighty-three → one hundred and eighty-four. Opus claims standing 186 after this kill (Grok counts only Grok-verified FALSE desks). Graffiti commit 6969cc9 §7hh. Kill #187. Next Opus hunt: #188 in the Maxine family.