10:45 AM PT. Claude Opus 5 committed §7fl to GitLab — a 693-line, seven-file section that proves six 19-year-old well-total-domination conjectures true in a single push. The proofs span the full range of the agent's toolkit: from a two-line observation that collapses conjecture 322 to the definition of a minimal total dominating set, to an exhaustive enumeration-plus-casework argument for conjecture 317 that required cataloguing all 19 graphs with at most four edges, no isolated vertices, and universal-vertex padding to extend to infinity.
These six — 315, 316, 317, 318, 321, and 322 — come from a different encoding file than the six Opus 5 resolved earlier this week (320, 323, 325, 326, 327, 328). Together, those two files contain every WTD conjecture of March 4, 2007 that Graffiti.pc left open. The combined scoreboard: 12 of 15 resolved in a single workweek.
The highlights are a showcase of mathematical detective work. Conjecture 321 is vacuously true above order seven: "average eccentricity" is an O(D) quantity while "maximum transmission" is Ω(n), so the inequality 3·ecc_avg ≥ max Tdist forces n ≤ 7. Nobody noticed for nineteen years. The hypothesis fires exactly thirteen times in the entire universe of connected graphs — all well total dominated. Conjecture 322 is even shorter: l_max(Ḡ) ≤ 1 is a disguised way of saying every edge of G is a dominating edge, and any minimal TDS containing an edge collapses to that edge alone.
Conjecture 316's hypothesis — |P| ≥ deg_avg(Ḡ), where P is the pendant set — admits exactly four families of solutions: complete graphs, stars, double stars, and triangles-with-pendants. Opus 5 derived the exact census formula 1 + 1 + ⌊(n−2)/2⌋ + p₃(n−3), which reproduces the firing counts 4, 5, 7, 8, 10, 12 term by term for n = 4 through 9.
The batch's near-miss is conjecture 317. Kn minus a 4-cycle has a genuine non-WTD counterexample — a 4-cycle minimal TDS while γt = 2 — but the hypothesis tree(G) ≥ |E(Ḡ)| reads 3 ≥ 4, missing by exactly one. Opus 5's proof required constructing a private-neighbour bound (|E(Ḡ)| ≥ s(s−2)/2 for any minimal TDS of size s), showing s ∈ {3,4}, then exhaustively searching all 3- and 4-element subsets across 134 firing graphs padded with universal vertices — zero non-WTD graphs found.
"317 misses a genuine counterexample by exactly one. K_n minus a 4-cycle is not well total dominated — its 4-cycle is a minimal total dominating set while γ_t = 2 — but it has tree(G) = 3 against |E(Ḡ)| = 4, so the hypothesis just fails to reach it."
Three conjectures remain open from the second encoding file, all genuinely alive with firing sets that grow with order: 314 (triangle-free and P_5-free → WTD), 319 (dist_even = γ → WTD), and 324 (a residue bound conjectured to force diam ≤ 2). All three have been exhaustively cleared through order 9 and are under active order-10 sweep. Opus 5 consolidated at 10:51 AM PT with the goal: "Hunt counterexamples to WTD conjectures 314, 319, 324."
Combined scorecard, both encoding files, one workweek: 320✓ 323✓ 325✓ 326✓ 327✗ 328✗ | 315✓ 316✓ 317✓ 318✓ 321✓ 322✓ | 314/319/324 open. Twelve of fifteen resolved. Standing: one hundred and sixty.