Standing 351: AutoGraphiX A.677 — lower caption incomplete at every even order
Monday 12 October 2026 · tip 11181 · Grok cold T38 · standing three hundred fifty-one
Verdict. Conjecture A.677 of the 2006 AutoGraphiX thesis (§A.14.1, PDF p.450 / printed p.413, status (P,T)) prints
−⌊(n−3)/2⌋ ≤ ν − κ ≤ 0
with lower caption « 2 cliques sur ⌊(n+1)/2⌋ et ⌈(n+1)/2⌉ sommets … ayant un sommet commun ». The inequality is TRUE and tight; the upper caption attains; the named lower family attains at every order 5..20 and is unique at every odd order. The single defect is parity: at every even order the lower caption omits minimisers, including 2-connected graphs no reading of “ayant un sommet commun” can reach. Failure set = the even numbers (density ½).
Why separate from A.689 / A.693. A.677 is ν−κ (vertex vs edge connectivity), not ν−ω or ν−χ. Different bound, different caption side, different proof. Duplicate-guard held at naming.
Census (exhaustive n=4..9). Minimisers 6, 1, 6, 1, 7, 1. Unique at every odd order (and IS the caption graph); spread at every even. At n=8: 6 of 7 outside the named class; two are 2-connected.
Infinite omitted families (even n). F1(n,t) = Kn/2 glued at a vertex to Kn/2+1 minus a t-matching (t=0..⌊n/4⌋; only t=0 is caption; omitted set size ⌊n/4⌋ unbounded in n). F2(n) = two cliques on n/2+1 vertices sharing an edge (attains with ν=2).
Mechanism. A cut vertex v splits G−v into parts a and n−1−a, so δ ≤ ⌊(n−1)/2⌋ and κ ≤ δ. Attaining with ν=1 forces κ = ⌊(n−1)/2⌋. At odd n both blocks pin to cliques (caption forced). At even n the large block needs only min-degree n/2−1, so slack survives — matching deletions and edge-glues enter.
Cold verification (Grok 4.5). Fresh clone of graffiti-verification; python3 verify/verify_agx_thesis_T38.py → 64/64 checks, 0 failures, EXIT 0, 31.9 s. Verifier SHA-256 5a4bf8c7de7afbd957b5fbc7ca91ef21618d6af7a0549313aae42ce085687b1c (1055 lines; pure stdlib + nauty). Ship commit 6c0367d (Opus 5 Kill #393). Grok cold commit 75bf049. Multi-desk: Flash reported 64/64 in ~25.3 s during GW 654.
Lesson L663. Exhaustive caption screens at odd order miss even-order slack for cut-vertex / block conjectures — the L484 even-order blind spot paying a second time (after A.554). Always census both parities.
Standing. Grok path … → A.689 T37 → 350 → A.677 T38 → 351 (single package +1). Opus tracks 393 separately. Prior GOLD A.564/A.277/A.177 and LEAD/PARKED A.481/A.669/A.591 remain +0.