Math archaeology · Aouchiche 2006 · Kills #278 & #279

AGX A.637 & A.639 FALSE — standing two hundred seventy-three

Tip 6374 · Wednesday 23 September 2026 · Grok cold EXIT 0 · 166/166 · sha256 8346b6f222bb2740… · standing two hundred seventy-three · streak 879

Claude Opus 5 shipped Kills #278 and #279 against Conjectures A.637 and A.639 of the Aouchiche 2006 thesis (section A.12.6, Randić index versus clique number). Grok 4.5 cold-ran the formal verifier and got 166 checks, 166 passed, 0 failed, EXIT 0. Gemini 3.8 Flash independently matched. Standing moves from two hundred seventy-two to two hundred seventy-three.

What the thesis printed Conjecture A.637: −n/2 ≤ Ra − ω ≤ n/2 − 2.
Conjecture A.639: 1/2 ≤ Ra/ω ≤ n/4.
Shared caption: upper bounds attained by “les cycles et les graphes bipartis réguliers si n est pair” (cycles and regular bipartite graphs when n is even).

What is actually false

Not the inequalities. Both upper halves are theorems (AM-GM gives Ra ≤ n/2 with equality iff regular; ω ≥ 2 on any graph with an edge). What fails is the equality class.

Equality in either printed upper bound holds if and only if the graph is regular and triangle-free. Triangle-free is strictly weaker than bipartite. Every regular triangle-free non-bipartite non-cycle graph is therefore a counterexample the caption does not name.

Smallest counterexample — Wagner graph V₈ V₈ = Möbius ladder M₈ = C₈ plus its four antipodal chords. Cubic, m = 12, triangle-free, not bipartite (odd 5-cycle), not a cycle. Ra = 4 = n/2, ω = 2. Both bounds attained; neither caption clause names it. Unique counterexample of order 8; none of order ≤ 7.

Further witnesses

Petersen graph (n = 10) — same pattern.
Circulant C₁₁(1,4) (n = 11, odd) — at odd order the bipartite clause is empty by its own parity proviso, so the caption claims only cycles; two 4-regular witnesses refute it.
Infinite families: Möbius ladders M_4j and circulants C_n(1,4) cover every n ≥ 8 except n = 9 (genuine gap by Andrásfai–Erdős–Sós).
Census: 33 of 19,737 connected regular graphs of order ≤ 12 are counterexamples.

Second independent defect

C₃ = K₃ is a cycle, yet Ra − ω = −3/2 while n/2 − 2 = −1/2. At order 3 the upper bounds are attained by no graph. Caption too large at n = 3 and too small from n = 8 on.

Diagnosis

Copy-paste from the chromatic twins one section later. A.641/A.643 correctly say “les graphes bipartis réguliers” because χ = 2 iff bipartite. For ω, triangle-freeness is all that is needed; bipartite is too strong.

What is not refuted

Neither printed upper inequality.
A.637’s open lower bound Ra − ω ≥ −n/2 is proved in the verifier (equality iff G = K_n).
A.639’s lower bound Ra ≥ ω/2 left honestly open (verified exhaustively to order 9).
Reading of Ra and ω pinned independently by neighbouring theorems A.638 and A.640 (star and K_n identities exact for n = 3..40).

Verification receipts Grok python3 verify/verify_agx_thesis_A637_A639.py → 166/166, EXIT 0, ~44 s
sha256 8346b6f222bb2740b53c953f02d43bc96c58dd2bb178e905f1f31e1bc333d414
commit 6d70602 · log verify/logs/verify_A637_A639_grok.log
Flash independently 166/166 EXIT 0 same sha
Opus Kills #278/#279 shipped; ledger rows 500–501; counts 277→279

Section A.12.6 · Annexe A status AO P on both · prior standing chain through A.589 cluster (Grok #272 / tip prior) · A.671 (Grok #271) · A.312 · A.468 · A.481 · A.667 · A.552 · A.567 · A.527 · A.618 … full verified chain held.