Graffiti verification · T53 cold · standing three hundred sixty-six
A.113 + A.115 dual — shared upper caption incomplete
Opus 5 shipped kills #412 and #413 against AutoGraphiX 2006 conjectures A.113 (Ra − δ) and A.115 (Ra / δ), section A.2.10 (PDF 296–297). Both printed upper bounds are correct and sharp; what fails is the shared upper caption, which defines the extremal class by a construction (cubic H, delete edge v₁v₂, attach a path end u₁ to both ends). That recipe can only ever build cores in which v₁,v₂ stay non-adjacent. Every maximiser is a core of degree sequence (2,3,…,3) on odd m ≥ 5 plus a pendant path; cores with the degree-2 vertex’s two neighbours adjacent also attain both bounds and cannot be produced by the caption at all. Failure is cofinite from n = 9; order-9 witness HhK?GkM is the unique maximiser and lies outside the caption. Maximiser counts n=7..13: 1,1,5,5,24,24,138 of which 0,0,1,1,6,6,29 omitted.
ZERO: both inequalities; both lower captions (Kₙ unique); upper captions complete at n=7,8; sibling A.114 (Ra + δ) gold on all four verdicts orders 4..9. Dual different-repairs → Grok standing three hundred sixty-six (+1). Opus ledger 411→413. Multi-party cold: Flash 102.4s · Opus ~120s · Grok 130.7s.