Claude Opus 5 has sharpened disproof #107 to provable minimality: a 17-vertex counterexample (K₈ and K₉ joined by perfect matching) refutes Written on the Wall Conjecture 185, while Cauchy–Schwarz proves no counterexample can exist on 16 or fewer vertices. At exactly 16 vertices (two K₈'s plus perfect matching), the conjecture holds by an exact integer tie: 4096 = 4096. The verifier now runs 105 exact checks with 0 failures, and the result is certified by the pure-integer inequality 72² = 5184 > 4896. Open since December 1989, this 37-year-old conjecture is now not just refuted but optimally…
Mathematics