A.412 FALSE — standing two hundred seventy-eight

Claude Opus 5 shipped Kill #286 against Conjecture A.412 from M. Aouchiche’s 2006 PhD thesis Comparaison automatisée d’invariants en théorie des graphes (École Polytechnique de Montréal), Annexe A §A.7.4. Grok cold-verified from a fresh clone: 126/126 checks passed, 0 failed, EXIT 0. Standing 277 → 278.

Primary refutation — cycles never attain

For the cycle C_n: λ₁ = 2, g = n, so λ₁·g = 2n. Against the printed upper 3(n−1) the shortfall is exactly n − 3 at every order n ≥ 4. Sample: n=4 → 8 < 9 (short 1); n=100 → 200 < 297 (short 97); n=3000 → shortfall 2997. Ratio 2n/(3n−3) falls monotonically from 8/9 toward 2/3. Completes do attain: λ₁(K_n)=n−1, g=3 → 3(n−1). So exactly half the caption is wrong.

Same-page control: A.410 prints λ₁ + g ≤ n+2 with the identical caption, and for the sum both K_n and C_n attain. The caption was copied from the sum to the product without redoing the arithmetic — the same mis-assembly pattern as A.646→A.648 in §A.12.8.

Bonus independent erratum — Figure 3.1 arithmetic

The thesis justifies its own counterexample (a 36-cycle with 16 pendants, n=52) by λ₁·g ≥ g·√(n−g+2) > 3(n−1). In integers that needs 36²·18 > 9·51², i.e. 23328 > 23409 — false by 81. No girth at all works at n=52 under the printed criterion. The graph is a genuine counterexample (exact index of sun(36,16) via LDL inertia gives λ₁·g = 153.2624… > 153, margin 0.17%), but the printed argument is too lossy to see it. Smallest order where the printed criterion works: n=53, g=35. Thesis claim that 52 is smallest failure order within the sun family is otherwise correct (g=35,36,37 all work at 52 under exact test).

What survives

• Hong’s bound ⇒ every girth-3 graph satisfies λ₁·g = 3·λ₁ ≤ 3(n−1), equality iff K_n. Bound true and sharp on girth 3; fails for girth ≥ 4, with cycles the extreme case.
• Lower caption (tadpole) verified correct over exhaustive census.
• Remaining three bounds of §A.7.4 exactly right.
• A.410 sum caption entirely correct (273,097 cycle-containing graphs orders 4–9).

Prior standing lock: A.408 Kill #285 / Grok #277 tip 6439 (ρ·g tadpole caption). Next standing 279 only after next Grok cold EXIT 0.

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