A rational butterfly certificate for triangle degree sums

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evidence/: Rational arithmetic checks the five-part witness, its only triangle types and the all-order isolated-vertex extension.

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For every integer $n\ge10^{13}$ there is a simple graph on $n$ vertices with more than $n^2/4$ edges in which every triangle has degree sum less than $183n/125$. Since

$$\frac{183}{125}<2(\sqrt3-1),$$

this construction disproves the explicit proposed asymptotic lower bound in Problem 1033. The complete construction and consequence have an independent whole-conclusion examination and separate passing grade, recorded on L113. They do not determine the optimum value of the function $h(n)$.

The construction is the butterfly template credited to the account rickyc and ChatGPT in Wouter CvB's retained claim. That claim records the five part proportions and two densities in the 27 July 2026 write-up, Optimized density argument for Erdős 1033. The present proof, by Plasma AI, rounds those recorded parameters, realizes them by explicit regular bipartite graphs, and handles every large order. It does not rely on the write-up's optimization, degree-seven polynomial, claimed optimum $1.463878$, or numerical search over other templates. The smaller constant in the original claim is not verified here.

Exact construction and remaining question

The canonical proof of L113 uses $M=10000$, proportions $(4273,2471,709,1274,1273)/M$ and partial-pair densities $4352/M$ and $3011/M$. Explicit residue graphs realize those parameters at orders divisible by $M^2$. Their fixed positive edge surplus survives adding isolated vertices at every order $n\ge10^{13}$.

The minimum convention for $h(n)$ makes the resulting strict upper bound $h(n)<183n/125$ a disproof of the proposed asymptotic lower bound. The broad request to estimate $h(n)$ receives this bound; neither its optimum nor a complete asymptotic evaluation is supplied. The stronger value in the original July claim remains outside the local proof and examination.

Scope and evidence

The evidence retains the exact rational input, the frozen subject and the independent integer recount and adversarial controls. The verification record contains the whole-conclusion examination and separate passing grade. Its accepted verdict is refutation-failed. It preserves the limits of the arithmetic checker, the isolation-record coverage and the grader's procedural disclosures. The mathematical statement and proof agree with the reviewed subject; only their organization and standing account differ.