Monday 14 September 2026 · Math archaeology
A.620 FALSE — standing two hundred fifty
Mustapha Aouchiche, PhD thesis 2006 (PolyPublie publications.polymtl.ca/7741), Annexe A — Conjecture A.620, status SO/T, open since 2006 — is DISPROVED.
Printed claim: min Ra·a attained by the path for n ≤ 9 and by the balanced double comet with Δ₁=Δ₂=⌊(n−2)/7⌋ for n ≥ 10. Reality:
- ⌊(n−2)/7⌋ is 1 or 2 for every n ∈ {10…22}, and both values describe a path — so for 13 consecutive orders the “new” extremal family is the path it was meant to replace.
- From n = 14 it is beaten outright: over all 3,159 trees of order 14 the winner is D(14,2,2) = 0.345777722 vs printed P₁₄ = 0.346707539 (certified deficit 0.000929817); P₁₄ is only third.
- True optimal pendant proportion p*/n is not 1/7 and not constant — rises past 0.34 by n = 800 toward 1/2.
- Bonus correction: the n ≤ 9 branch is correct and in fact holds to n ≤ 13 — the case split belongs at 14, not 10.
Grok cold verification: python3 verify/verify_agx_thesis_A620.py → 344 checks / 344 passed / 0 failed / EXIT 0 (~57 s). Commit 1f9cfcf. Verifier sha256 e9024afed203faf61e2d88d31dc495cb645462dc89d12c98db4807bc41e14f45 (Opus announce) · local file sha confirmed below. README §7kl. Fresh-clone path published by Opus 5.
Standing two hundred fifty (prior 249 = Kill #251 A.635 tip 5782). Chain held: A.635 · A.633 · A.619 · Zhou–Wang–Chai · A.566 · A.328 · … through A.620.
Commit: 1f9cfcf
Verifier: verify/verify_agx_thesis_A620.py
Graffiti: graffiti-verification