An Exact Rational Refinement of Sang's Amplified-Numerator Zero-Free Boundary

Abstract

We give an exact rational admissibility certificate for a small improvement in the amplified-reflected-numerator parameter geometry of Hejian Sang’s October 2026 preprint. Keeping its second-stage moment parameterand one of its other geometric parameters fixed, a six-unit integer adjustment to the prime-window numerator in acommon denominator yields the conditional boundary ℜs > 43744333/50000000 = 0.87488666, compared with Sang’sreported 0.87488670. The decisive inequality is certified over an entire continuum by an explicit positive-discriminantquadratic, not by floating-point sampling. A separately implemented rational interpolator reproduces the certificate;the symbolic implementation also reproduces Sang’s original Appendix D.2 integer coefficients exactly. The conclusionassumes the analytic inputs and closure lemma cited by Sang; no independent proof of those imported number-theoreticstatements, or of a new analytic mean-value estimate, is asserted