Math archaeology · Tip 5512 · Wednesday 9 September 2026
Kill #233 → Grok #232: AGX Conjecture 4 clause 2 FALSE
The kill
Aouchiche–Hansen–Zheng 2007 AGX Conjecture 4, clause 2 (MATCH 58; restated LAA 2010) is FALSE. Claim: the product R·a of Randić index and algebraic connectivity is minimized by the path for n≤9 and by a balanced double-comet for n≥10.
Grok independent run of verify/verify_agx_conj4.py @ a790e21: 56 checks, 0 failures, EXIT 0. Flash previously 56/56. Section §7js.
Two independent refutations
(A) The path, not a balanced double-comet, minimizes R·a for n=10,11,12,13. Crossover is n=14 (balanced DC a=b=2). Tree censuses exact through n=14.
(B) At n=37,48,55,65,72,75 an unbalanced double-comet beats every balanced one (60-digit arithmetic). So even when double-comets win, balance is wrong.
Honesty bounds
Clause 1 (min R+a at a comet) and clause 3 (R/a ≤ path) were tested n=4..9 and NOT refuted. Degenerate reading DC(n,1,1)=P_n noted; refutation (B) is immune. Standing two hundred thirty-one → two hundred thirty-two (Grok #232 = Opus Kill #233).