A computer-assisted 23/33 + ε Goldbach exceptional-set bound
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Abstract
Let E(X) denote the number of even integers<br>not exceeding X that are not a sum of two<br>primes. Building on the zero-packet framework of Zhao and the<br>exceptional-set reduction of Pintz, this manuscript claims
E(X) ≪ε X23/33+ε.
The proposed refinement keeps the fixed-class<br>R- and T-packet<br>contributions aligned during the finite maximization. Exact<br>witnesses and replay scripts accompany the argument.
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Scope and status
If the proof is validated, the exponent improves Zhao’s<br>7/10 to<br>23/33 = 0.696969…, a difference of<br>1/330.
This does not prove the binary Goldbach conjecture. The estimate may<br>still allow infinitely many exceptional even integers; Goldbach<br>asserts that there are none above the trivial small cases.
The published checkers verify the finite certificate after its<br>analytic inputs have been exported. They do not independently derive<br>those inputs from Zhao’s and Pintz’s estimates. That source-level<br>correspondence remains the principal review boundary.
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Primary sources
G. Zhao,<br>The exceptional set of Goldbach problem and Linnik’s constant,<br>arXiv:2511.05631v2 (2026).
J. Pintz,<br>A new explicit formula in the additive theory of primes with applications II,<br>arXiv:1804.09084v2 (2018).
J. Pintz,<br>A new explicit formula in the additive theory of primes with applications I,<br>Acta Arithmetica 210 (2023), 53–94.
Process note
AI-assisted drafting, coding, and internal audit were used in<br>preparing this research dossier. The mathematical claims remain the<br>author’s responsibility.