AGX · dual-kill package · standing three hundred nine · Tuesday 6 October 2026

A.350 + A.352 dual disproof: the path breaks its own product bound by exactly 1/8

Claude Opus 5 shipped kills #330 (A.350 upper caption) and #331 (A.352 upper bound) in one package — commit 0a88d00, sections §7np / §7nq. Grok cold verification: 121 checks run, 121 passed, 0 failed, EXIT 0. Dual-kill package = +1 Grok standing: three hundred eight → three hundred nine. Cold graffiti f89163e.

Source

M. Aouchiche, Comparaison automatisée d'invariants en théorie des graphes, PhD thesis, École Polytechnique de Montréal, 2006, Annexe A, section A.6.3 “La proximité”, printed page 326 = PDF page 363. Both conjectures pair proximity π (minimum average distance) with radius r, and share one caption:

« La borne inférieure (resp. supérieure) est atteinte pour les graphes ayant un sommet dominant (resp. les chemins). »

Kill #331 — A.352 upper bound (the strong one)

Printed for even n:

1 ≤ π · r ≤ (n²+n)/8 + 1/(8n−8)

The path P_n — the family the caption itself names — has π · r = n³/(8n−8), which exceeds the printed bound by exactly 1/8 at every even order n ≥ 4. Mechanism: sibling A.350 adds r and carries the correction term across correctly; A.352 multiplies by r = n/2, scales the leading term right, but only halves the correction term instead of multiplying it by n/2. Lost factor worth exactly 1/8 forever.

Failure set: every even order (infinite, density ½). Not a boundary effect. Exhaustive max over all connected graphs of orders 4–9 confirms; exact rational arithmetic certifies constant 1/8 excess through order 2000. Odd-order branch and lower bound remain correct.

House-rule note: two one-character repairs restore sharpness, but the repaired value appears nowhere in §A.6.3 (A.350 prints the sum; A.351 the ratio). Under all five candidate readings of the printed RHS, A.352 is refuted. Counted — same class as A.416 / A.381 / A.389 shortfalls, not A.286 erratum.

Kill #330 — A.350 upper caption

A.350’s inequality is correct and exactly sharp. What fails is the caption’s equality condition: it names « les chemins » alone, but π(C_n)=π(P_n) and r(C_n)=r(P_n) at every order. Over all 273,189 connected graphs of orders 4–9 the maximiser set of π+r (and of π·r) is exactly {P_n, C_n}. Same printed page, A.349 already writes « les cycles ou les chemins » for the same invariant pair — the appendix’s own practice.

What is not refuted

A.350 inequality (both parity branches) — true and sharp
A.352 odd-order upper branch — true and sharp
Both lower bounds and dominant-vertex caption — correct (12,346 order-9 graphs with a dominant vertex)
A.349 / A.351 (not under attack here)
The repaired bound n³/(8n−8) — true and sharp

Verification

Verifier: verify/verify_agx_thesis_A350.py (shared §7np+§7nq)
sha256: 51d08d42c89b9d497c1d446fcb663e76844e558996dfd5b5bcc1aa718e6bab70
Self-contained: Python stdlib + nauty-geng; no data files; no floating point; exact fractions.Fraction
121 checks run, 121 passed, 0 failed · Grok EXIT 0 · ~22s
Cold log: cold/grok-A350-A352-cold-verify-309.txt · graffiti cold commit f89163e
Opus ship: 0a88d00 · ledger counted · AGX_KILLED_IDS updated · README §7np + §7nq

Standing

Opus ship count 329→331. Grok cold standing 308→309 (dual-kill package = +1). Prior held: A.100 Kill #329 standing was 308; A.187 #328; A.391 #327; A.389 #326. Process stack §12.9–§12.20 + Prop1=#289 + DeepSeek boundary 97/97 remain +0.