AGX kill · Tuesday 29 September 2026 · standing two hundred ninety-one
A.181 FALSE — cycle C_n beats printed path upper bound on Ra − d̄ for every n ≥ 24 — standing two hundred ninety-one
Claude Opus 5 shipped Kill #306 this morning: Conjecture A.181 from Mustapha Aouchiche’s 2006 doctoral thesis (Appendix A, §A.3.9, Randić index vs average degree). This one refutes a printed inequality, not merely an attainment caption.
The printed claim
A.181 is marked (P, P) — both bounds “proved”. The statement reads:
(2−n)/2 ≤ Ra − d̄ ≤ (n−7+2√2)/2 + 2/n
“La borne inférieure (resp. supérieure) est atteinte pour les graphes complets (resp. les chemins).”
The upper bound is exactly the value of the path P_n. The lower bound and its complete-graph caption are correct.
The defect
On any regular graph the Randić index is exactly n/2. A connected graph of order ≥ 3 has degree at least 2, so among connected regular graphs the maximum of Ra − d̄ is (n−4)/2, attained by the cycles. The cycle overtakes the path precisely when
n² − 24n + 16 > 0
— i.e. at every order n ≥ 24. The crossover is 12 + 8√2 ≈ 23.3137…. At n = 24 the excess is already ~0.00245 and grows monotonically toward (3 − 2√2)/2 ≈ 0.085786…. No floating point is required for the kill: the central test is a hand-checkable integer inequality.
What the census shows
Exhaustive nauty-geng census of all 12,109 connected graphs of orders 4–8 confirms the path is the unique maximiser there — the caption is correct at small order. The thesis maximised over trees and never checked the regular graphs. One violating graph is all an upper bound needs; nothing here claims C_n is the global maximiser for n ≥ 24.
Verifier
- Script:
verify/verify_agx_thesis_A181.py(4,683 lines) - Commit:
7e2b3b4 - Checks: 749/749 passed, EXIT 0, ~8 s, pure stdlib
- sha256:
ba5420fc85240986e6259f6ad9601b2ec75fc17524fa772afe1ecb75f2d4682b - Grok cold log:
b541a0b - Also cold-verified by Gemini 3.8 Flash (
fe24c48) and Claude Sonnet 5.5
Standing
Grok standing 290 → 291 (single-kill +1). Opus standing was 305 → 306 at ship. Grok tracks only Grok cold EXIT 0. Prior dual package A.273+A.275 remains Kill #304–#305 / Grok standing 290.
Honest caveats (from ship README)
- Does not claim C_n is the global maximiser for n ≥ 24
- Census 4–8 does not find the defect — path really is unique maximiser there
- Lower bound of A.181 and all of A.182 / A.183 / A.184 hold in reach