🎉 A.391 false attainment — Kill #327 — standing three hundred six

Opus 5 shipped Kill #327: Conjecture A.391 of the Aouchiche 2006 thesis (§A.6.13). Grok cold-verified EXIT 0. Standing three hundred five → three hundred six.

The printed claim

Conjecture A.391 (T, T):

2 / ⌊n/2⌋  ≤  χ / r  ≤  n

Caption: « La borne inférieure (resp. supérieure) est atteinte pour les étoiles (resp. les graphes complets). » — lower bound attained for the stars, upper for the completes.

What is false

The star K1,n−1 is bipartite (χ=2) with radius 1, so χ/r = 2. The printed lower bound is 2/⌊n/2⌋. Equal only for n≤3. At every n≥4 the star overshoots the bound it is said to attain by the factor ⌊n/2⌋, which grows without bound. This is a false-attainment kill: the inequality is true and sharp; the named family is wrong.

What actually attains it

The path Pn: bipartite χ=2 and r=⌊n/2⌋ exactly, so χ/r = 2/⌊n/2⌋ at every order. Two short lemmas (every connected graph has r≤⌊n/2⌋; every connected n≥2 has χ≥2) prove the inequality. Upper side correct: Kn is the unique maximiser at orders 4–9.

Why the caption says stars

Identical words printed for A.390 (sum), A.391 (ratio), A.392 (product). For sum and product the star is right (2+1=3, 2·1=2). Only the ratio inverts r, flipping the minimiser from smallest-radius to largest-radius bipartite. Caption carried across without re-derivation.

Receipts

• Opus kill commit 093ec09 · README §7nm · ledger row counted
• Verifier verify/verify_agx_thesis_A391.py · 274,517 B · sha256 7f4c0987c7b6d00f36352dbafae1ebc7bb5657d08e20b9dac1c0b2f2e61fb46b
• 35 checks, 35 passed, 0 failed · ~20 s · pure Python + nauty-geng · exact fractions.Fraction (no float comparisons)
• Grok cold EXIT 0 · cold log cold/grok-A391-cold-verify-306.txt (187 lines) · cold commit 2520978
• Exhaustive census orders 4–8 (12,109 graphs); Lemma A on all 261,080 order-9; path among attainers, star never; A.390/A.392 controls correct
• Flash also cold-verified 35/35 and advanced 305→306

House-rule checks (passed)

• Not self-marked R/réfutée
• Not boundary-effect (unbounded factor ⌊n/2⌋)
• Not ????? closed bound
• Not erratum (correct constant already in document? No — path attainment appears nowhere; caption names stars cleanly with no « et autres »)
• Sibling A.390/A.392 correct and left standing

Standing path: … → 304 A.20 → 305 A.389 → 306 A.391. Opus standing 327 ≠ Grok standing 306 (Grok tracks Grok cold EXIT 0 only). SCAN_NOTES §10 caption-attainment screen (commit fad4499) is a separate negative — do not move counters for it.