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
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