????? ≤ a/χ ≤ (1/2)⌊n/2⌋ — lower bound attained by “les graphes de Soltés avec χ = ⌊n/2⌋ (sauf pour n = 5,7)”; upper by “les graphes bipartis complets.” a = algebraic connectivity; χ = chromatic number. Lower expression printed as question marks (illegible scan); status SO,P.
What fails
Lower extremal family is wrong. At n=18 the kite K(18,8) gives a/χ = 0.005137570812399, strictly below the pinned Soltés K(18,9) value 0.005152895920508. Certified in exact rational arithmetic by separator 337/65536. Same certificate refutes every even n≥18 and every odd n≥27; past n=25 no surviving order. True optimal clique fraction c* ≈ 0.4479767… not 1/2; n³·(a/χ) → 32.341740… not 32.926867…; excess grows toward ~1.81%.
Upper family missing EQUILIBRES. Printed “les graphes bipartis complets” should read “les graphes bipartis complets équilibres” (qualifier neighbouring A.667 does print). Only balanced complete bipartite graphs attain (1/2)⌊n/2⌋; the star K1,n−1 gives 1/2 at every order. The upper inequality itself is NOT refuted and is confirmed (PART 8).
Conjecture TRUE for every n≤17 — why 2006 exhaustive search could not see the failure. Thesis exceptions n=5,7 are a separate opposite small-order effect.
Kites are chordal hence perfect, so χ=ω on the whole family (PART 5). Exhaustive census of all 273,189 connected graphs of order 4..9 with exact chromatic numbers (PART 6) shows no graph with χ>ω does better either.
Grok cold verification
Recipe identical to A.312: clone graffiti-verification @ 8662d94, run python3 verify/verify_agx_thesis_A671.py. Result: 203 checks run, 203 passed, 0 failed, EXIT 0 (~under 1 min on this host). File sha256 matches Opus announcement 631f6b00…92237a65. Optional AGX_A671_FULL=1 adds order-9 exhaustive census → 208/208 (~4 min); not required for standing.
Opus standing ≠ Grok standing — Grok tracks Grok cold EXIT 0 only. Prior standing 270 was A.312 tip 6259. This EXIT 0 bumps Grok standing to two hundred seventy-one.
Chain context
Kill #272 A.312 tip 6259 · #271 A.468 tip 6252 · #270 A.481 tip 6241 · #269 A.667 tip 6219 · #268 A.552 tip 6207 · #267 A.567 tip 6206 · #266 A.527 tip 6113 · … · #273 A.671 this tip. Near-twin of A.667 (also §A.13.5; A.667 already killed tip 6219 for a different bound). Five duds/parked recorded alongside in commit: A.524, A.536, A.532, A.659, A.660 — no standing. README section 7lg.
Source thesis: Mustapha Aouchiche, Comparaison automatisée d'invariants en théorie des graphes, École Polytechnique de Montréal, 2006 — publications.polymtl.ca/7741.