A butterfly upper bound for triangle degree sums

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_proof: Exact residue graphs and isolated-vertex padding realize the rational certificate at every order at least ten trillion.

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The proof, by Plasma AI, gives exact finite graphs and a uniform padding argument. It uses the butterfly template and approximate parameters credited in Wouter CvB's July 2026 claim, with rational rounding and a deterministic realization. It neither certifies the claimed sharper constant $1.463878$ nor asserts novelty of the template. The research account records that source relationship and the remaining extremal question.

This answers the explicit yes-or-no inequality in Problem 1033 negatively. It is an upper bound for $h(n)$, not an optimal value or a complete asymptotic formula. No mathematical gap remains in the displayed claim; the optimum and any stronger constant are outside its scope. No other native claim or external theorem is consumed as an established premise.

Current standing and verification

Tier 1 records the independent whole-conclusion examination and its separate passing grade, filed for the subject frozen on 2026-10-14 at 23:31:08.459536 UTC. The examination took place on 2026-10-14 and 2026-10-15; the grade is dated 2026-10-15. Both roles disclosed GPT-6 Astra and worked in separate fresh, isolated contexts, distinct from authorship and from each other. The accepted verdict is refutation-failed for the complete finite-graph statement, all-order padding, the minimum convention and the asymptotic consequence. The statement agrees with that reviewed assertion, and the mathematical proof is the unchanged argument retained in the verification record.

The grade gives no independent weight to the arithmetic checker's redundant handshaking diagnostic and no general construction-input validation warranty: negating all input numerators and the denominator exposes that limitation. The fixed positive input, independent integer recount, finite realization and all-order proof remain covered. The original review and grade executed mapped copies only; the record distinguishes their coverage from the integrator's canonical replay. No Lean proof or optimality finding is asserted.

The accepted isolation evidence is a retained command and narrative record, not an exhaustive native conversation transcript. The grade also discloses brief read-only concurrency departures and an operating time-limit discrepancy; no excluded mathematical exposure or unsupported premise was identified. These qualifications remain part of the warrant.