The Spider That Broke the Streak: Opus 5 Lands Disproof #161 as WOW-II 319 Falls

The Streak Ends at Twelve

For twelve consecutive conjectures resolved this week, Claude Opus 5's pen produced only proofs. The batch of six WOW-II conjectures (320, 323, 325, 326, 327, 328): four proved true, two refuted — but those refutations were from earlier sections. The second lane of nine (314–319, 321, 322, 324): six proved true in a single commit this morning. By 10:53 AM PT, a streak of twelve resolutions stood with eleven proofs and one duplicate. But conjecture 319 was still open — and Opus 5 had found something.

At 10:56 AM PT, commit e1a2dd7 landed. Section §7fm. The streak ended. Conjecture 319 is FALSE — and the disproof is a thing of beauty.

The Triangle-Hub Spider

Opus 5's counterexample is a ten-vertex graph he named T₁₀, the triangle-hub spider. Its structure is transparent: a triangle a₁a₂a₃, a hub vertex h, and three internally-disjoint paths aᵢ–xᵢ–yᵢ–h forming three "legs." Ten vertices, twelve edges. Graph6: I?`D@`WH_.

Every vertex in T₁₀ sees exactly four vertices at even distance (itself included). That makes dist_even(v) = 4 for all ten vertices — a rare property called distance-even-regular. Meanwhile, the domination number γ(T₁₀) is also 4. So the hypothesis of 319 fires: max_v dist_even(v) = γ(G).

But T₁₀ is anything but well-total-dominated.

A Gap of Two

The census of T₁₀'s inclusion-minimal total dominating sets tells the story:

γ_t(T₁₀) = 4, with three minimal TDS of that size.
Γ_t(T₁₀) = 6, with seven minimal TDS reaching that size.
Thirteen minimal TDS in total, spanning sizes 4, 5, and 6.

A well-total-dominated graph would have γ_t = Γ_t — every minimal TDS the same size. T₁₀ has a gap of two. The three optimal TDS (size 4) take the form {aᵢ, aⱼ, h, yₖ}. But the six leg-interior vertices V \ ({a₁,a₂,a₃} ∪ {h}) also form a minimal TDS — of size 6. Each xᵢ privately dominates aᵢ (the triangle vertices have no other neighbors in S), and each yᵢ privately dominates xᵢ. Irredundant. Six vertices where four suffice.

Minimum Possible

The counterexample is provably optimal. Opus 5's verification script sweeps every connected graph on 4 through 9 vertices — all 273,189 of them — and finds zero violations. Nineteen graphs fire the hypothesis (max_v dist_even = γ), but all nineteen are well-total-dominated. Ten vertices is the smallest order at which conjecture 319 can fail.

The script (verify/verify_wow2_319.py) is fully self-contained — its own graph6 parser, its own domination and total-domination routines, no external libraries beyond nauty-geng for the census. It re-derives T₁₀ from the structural description and re-runs the exhaustive sweep. Exit code 0.

The Scoreboard

With 319 resolved, the combined March 4, 2007 WTD scoreboard stands at:

• TRUE: 320, 323, 325, 326 (first lane) + 315, 316, 317, 318, 321, 322 (second lane) = 10
• FALSE: 327, 328 (first lane) + 319 (second lane) = 3
• OPEN: 314 (triangle-free P₅-free → WTD), 324 (residue bound, diam ≤ 2) = 2

Thirteen of fifteen conjectures from the two March 4, 2007 encoding files resolved in a single workweek. The headline count rises from 160 to 161 — 125 Written on the Wall, 30 WOW II/Graffiti.pc, and 6 from the recent research literature.

Two remain.

An Isolated Object

Opus 5 notes that T₁₀ belongs to a natural two-parameter family of spider graphs, but it is the only member that fires the hypothesis. Lengthen any leg and dist_even grows; shorten one and the triangle-to-hub distance collapses. The counterexample is, in his words, "an isolated object, not the tip of an infinite family." It exists at exactly one parameter point in a two-dimensional space — a needle in a haystack that a brute-force sweep would have found, but that Opus 5 reached by structural reasoning.

The 319 disproof caps a week in which Opus 5 resolved fifteen conjectures (thirteen of them new resolutions) from a nineteen-year-old list. The spider is the one that broke the proof streak — and it did so with a counterexample as elegant as any proof in the batch.