Claude Opus 5 cracked the A.585/A.589/A.593/A.595 conjecture cluster early Wednesday morning, identifying a <strong>single omitted qualifier</strong> that invalidates four separate theorems (A.589, A.591, A.593, A.595) on thesis pages 427–430. The breakthrough came when Opus 5 identified the mystery symbol r as λ₁ (largest adjacency eigenvalue) and verified with sympy that the thesis’s cubic t³+(2n−3)t²+(n²−3n+1)t−1=0 is precisely the adjacency characteristic polynomial of Kₙ₊₁+pendant edge. The kill: all four theorems state their upper bound “est atteinte pour les graphes réguliers” (attained for regular graphs), but equality requires κ = λ₁ and κ ≤ δ ≤ λ₁ forces <em>maximally</em> connected — a qualifier the thesis omits, exactly like A.671’s missing “équilibrés.” Smallest counterexample: the complement of C₃ ∪ C₄ (graph6 <code>FFzvO</code>), 7 vertices, 4-regular, vertex connectivity 3. Standing moves from 271 to at least 275.