Whole-conclusion verification of the butterfly certificate

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grade: Independent grade of the frozen E1033 whole-conclusion examination, accepting its mathematical verdict and stated execution scope while withholding independent weight from the disclosed checker limitations.

report: Independent whole-conclusion examination of the frozen E1033 construction, including finite realization, all-order padding, exact arithmetic, checker limitations and the catalog asymptotic consequence.

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The whole-conclusion report, examined on 2026-10-14 and 2026-10-15 UTC, and the independent grade, dated 2026-10-15, cover every integer $n\ge10^{13}$, finite simple undirected graphs with more than $n^2/4$ edges, and triangle degree sums below $183n/125$. They also cover the supplied minimum convention and negation of the proposed asymptotic lower bound. Both roles disclosed GPT-6 Astra and were distinct, fresh and isolated. The grade is pass and the accepted mathematical verdict is refutation-failed. It warrants L113.

The subject manifest names the exact extracts, original paths and freeze time. All mathematical chunks match the corresponding canonical text at filing. The proof is retained without mathematical changes at the native proof, and the research page is its source and scope account. The canonical checker and parameters equal the retained snapshot bytes.

Accepted scope and qualifications

The report supplies the finite realization, both directional residue counts, exact edge and triangle counts, all-order padding and fixed-gap consequence. No native claim or source theorem is consumed as an established premise. The optimum of $h(n)$, the stronger constant $1.463878$, novelty and Lean work remain outside the verdict.

The grade withholds independent weight from the tautological handshaking diagnostic and rejects a stronger interpretation of the arithmetic checker as a validator of arbitrary construction inputs. The sign-reversal witness and meaningful failing controls remain executable. Neither limitation affects the fixed positive certificate or the analytic proof.

The report and grade executed mapped copies and did not compare current canonical mathematics or run at its canonical source path. Those limits are explicit; the integrator's correspondence and replay are separate from the independent warrant. The retained isolation evidence is not an exhaustive native conversation transcript. The grade's read-only concurrency departures and operating time-limit discrepancy remain disclosed; no mathematical exposure or unsupported premise was identified.

Retained material and editorial treatment

The report and grade retain their mathematical bodies, findings, dates and verdicts without edits. Wiki metadata and navigation are added. The five reviewed extracts, their manifest, the two independent programs and both control-input files are retained unchanged. The native claim and proof state the same reviewed assertion; the research and evidence summaries explain the grade's limits. No later repair is attributed to the independent examination.

The exact reproduction commands are in the report and grade. The integer recount counts vertex and edge cardinalities separately, and the control driver checks positive and negative cases with ordinary and optimized Python. Its deliberately false specification must fail. The full graph realization and infinite-order transfer are written arguments, not conclusions of those finite executions.

Canonical correspondence and replay

On 2026-10-15 UTC the integrator compared every retained mathematical extract with the corresponding canonical text, in order, and the checker and input byte for byte. The proof's mathematical sections are preserved unchanged in L113's native proof. This correspondence check supplies no new independent review.

A clean materialization of the committed canonical checker, runtime code and retained examination package passed the canonical relative-path command, the snapshot command, integer recount and full adversarial controls. Both ordinary and optimized executions of the deliberately false specification exited one at the expected failed obligation. No live-child code or cached output supplied those results. These integration executions are distinct from the report's and grade's mapped executions and do not enlarge their stated coverage.