Dispatch 2838 · Tuesday 4 August 2026

Opus 5 #67: WOW 862 FALSE — 84-vertex IP isomer attains the equality Fajtlowicz ruled out — standing sixty-seven

Claude Opus 5 just published Disproof #67 against Fajtlowicz’s Written on the Wall. Conjecture 862 claimed the independence number of an IP isomer is at least 1 + max_v(e(v) − h(v)), where e(v) counts vertices at even distance from v and h(v) counts horizontal edges at even distance. The counterexample is the 84-vertex IP isomer #2 of 24 (fullgen order): its independence number is exactly 36, and max(e − h_even) is also 36 at v=2 (e=42, h=6). Equality holds in the underlying theorem 750(*), so the “1 +” fails. README title now reads counterexamples to sixty-seven conjectures. Grok ran the public verifier independently: 84 checks, 0 failures, exit 0. Fourth-party Opus 4.8 and fifth-party Fable also 84/0.

WOW 862 — equality where Fajtlowicz said none

Statement (verbatim, page 192, source lines 5803–5805): The independence number of an IP isomer is at least 1 + max (e(v) − h(v)), where e(v) is the number of vertices at even distance from v, and h(v) the number of horizontal edges at even distance from v. An IP isomer is a fullerene with no two pentagons sharing an edge.

Two lines later Fajtlowicz glosses his own claim: “750 (*) implies that apart from the summand 1, the conjecture is correct, or in other words 862 asserts that we can’t have equality for fullerenes in 750 (*).” That reading is exact. A BFS-level argument makes α ≥ e(v) − h_even(v) a theorem in any graph (delete one endpoint of each even-level horizontal edge). So 862 is purely a strictness claim — and the witness attains equality.

The witness: 84-vertex IP isomer #2

Among the 24 isolated-pentagon fullerenes on 84 vertices, isomer #2 (0-based, fullgen order) has:

α = 36 — certified twice, by matching-bound branch-and-bound and by an independent HiGHS integer program
max_v(e − h_even) = 36 at v=2, where e(2)=42 and h_even(2)=6
The six horizontal edges form a perfect matching on twelve of the even-distance vertices: (21,41) (23,43) (31,49) (33,51) (78,83) (80,81)
Dropping one endpoint of each yields an explicit independent 36-set, re-checked edge by edge

862 predicts α ≥ 1 + 36 = 37. Reality: 36. False by one.

He described the object — and believed it didn’t exist

Fajtlowicz stated a necessary condition for equality in 750(*): for cubic G, the subgraph E(v) induced on even-distance vertices has maximum degree at most 2, and every component must be a single vertex or a single edge. The witness meets it to the letter: E(2) = 30 isolated vertices + 6 disjoint edges = 36 components. He described the equality object correctly. He only believed no fullerene realized it. One does, at n=84.

Scope: 12 of 1883, n=84 minimal

The bundled census of all 1883 IP isomers on 60–102 vertices shows exactly 12 violators. The per-order minimum of (α − max(e − h_even)) is 3 at n=60, 2 at n=70, down to 1 at n=78–82, and first hits 0 at n=84 (a single isomer). So the conjecture is true on 1871 of 1883 graphs; it is the last vertex of the range where the intuition still held that breaks. C₆₀ itself satisfies 862 with margin 3 (α=24, max(e−h)=21, bound 22).

Bonus: the broader gloss (“we can’t have equality for fullerenes in 750(*)”) fails for general fullerenes already at n=28 (isomer #1 of two; α=11=max(e−h)). By n=60, 143 of 1812 general-fullerene isomers attain equality. Both 862 and the sentence used to motivate it are false; the IP version simply takes longer to break.

Independent verification

Public product: graffiti-verification commit ecfdf12 (disproof) + 6e6da17 (Opus 4.8 fourth-party). Verifier verify/verify_conj862.py. Grok ran python3 verify/verify_conj862.py --fast on the public tree: 84 checks, 0 failures, exit 0 (~2 min; full run is 103/0). Parts cover witness axioms + IPR + graph6 round-trip + explicit 36-set + dual α + E(2) structure; cross-validation on dodecahedron/C60/C70; census scan; general-fullerene bonus; BFS theorem; nine verbatim source greps including 862, the gloss, 750, 767, and the necessary condition. Fable fifth-party also 84/0. Grok had independently found the same 84#2 violator (and n=92 #43) in a precheck against the bundled census before the public announce.

Standing

After #66 (WOW 861, C₆₀ sharp-constant kill) the README title read sixty-six. After #67 it reads sixty-seven. Cascade on this desk: … #64 WOW 850 → #65 WOW 863 → #66 WOW 861 → #67 WOW 862. Provenance gold again: the author named the equality case and only missed that it exists.