A computer-assisted 23/33 + ε bound for the exceptional set in the binary Goldbach problem
Unrefereed preprint. The finite computation is reproducible; the analytic correspondence with Zhao and Pintz requires independent specialist verification.
Abstract
Let E(X) denote the number of even integers not exceeding X that are not a sum of two primes. Building on the zero-packet framework of Zhao and the exceptional-set reduction of Pintz, this manuscript claims
The proposed refinement keeps the fixed-class R- and T-packet contributions aligned during the finite maximization. Exact witnesses and replay scripts accompany the argument.
Scope and status
If the proof is validated, the exponent improves Zhao’s 7/10 to 23/33 = 0.696969…, a difference of 1/330.
This does not prove the binary Goldbach conjecture. The estimate may still allow infinitely many exceptional even integers; Goldbach asserts that there are none above the trivial small cases.
The published checkers verify the finite certificate after its analytic inputs have been exported. They do not independently derive those inputs from Zhao’s and Pintz’s estimates. That source-level correspondence remains the principal review boundary.
Reproducibility
- Paper
- HTML manuscript
- Verification bundle
- Complete certificate archive
- Reproduction guide
- Commands and trust boundary
- Certificate manifest
- Machine-readable manifest
- Release hashes
- SHA-256 manifest
- LaTeX source
- Manuscript source
Primary sources
- G. Zhao, The exceptional set of Goldbach problem and Linnik’s constant, arXiv:2511.05631v2 (2026).
- J. Pintz, A new explicit formula in the additive theory of primes with applications II, arXiv:1804.09084v2 (2018).
- J. Pintz, A new explicit formula in the additive theory of primes with applications I, Acta Arithmetica 210 (2023), 53–94.