Building upon Helfgott's 2014 proof of the ternary prime number theorem, I reconstructed the explicit constant system for the minor-arc portion, reorganizing the explicit constants scattered across multiple inequalities into a structure based on a one-dimensional supremum problem.

Through this rewriting, the contributions of all minor-arc parts are explicitly expressed as functions, with their maximum values determining the final constant. By leveraging tail monotonicity and interval arithmetic methods, the steps that originally relied on manual estimation can be transformed into verifiable and reproducible numerical certificates.
The core goal of this work is to organize the originally complex and difficult-to-fully-verify constant estimates into a complete system that can be machine-verified, revealing the main bottleneck in lowering the threshold under fixed parameters. Read more:
A Rigorous Computational Reconstruction of the Minor-Arc Bound in Helfgott’s Proof of Ternary Goldbach
— Mirror Tang