A Phase-Uniform High-Tail Certificate for the Weil Positive-Part Operator at Support Three

This research letter establishes a computer-assisted, phase-uniformhigh-frequency bound for the Weil positive-part operator at Fouriersupport a = 3. The main result is the strict operator-norm estimate \[\left\|H_{[T_{\mathrm H},\infty)}\right\|<12,\qquadT_{\mathrm H}=3.26\times10^{22}.\] The proof combines the monotonicity of the Archimedean contribution,an explicit bound for the pole term, and a quartic polynomial majorantfor the prime-power trigonometric polynomial. The principal contribution is a rigorous reduction of theinfinite-dimensional high-tail operator estimate to a finite,computer-verifiable inequality that holds uniformly over the entiretorus of prime phases. Consequently, no Diophantine approximationor recurrence analysis of the actual arithmetic phase orbit is required. The certification uses 98 prime-power terms, 19,213 rational Fouriershifts, and an exact covering by 600 intervals. Rigorous enclosuresare obtained using Arb interval arithmetic, while the final operatorbound is verified through exact rational and integer computations.The accompanying certificate generators and independent verifiersprovide a reproducible framework for checking the result. This theorem supplies a quantitative high-frequency component forthe complementary Zero-Gram and Positive-Part approach to Weilpositivity developed in the companion paper,"Two Complementary Methods for Weil Positivity at a = 3." The result concerns the positive-part operator on a half-line atsupport exactly three. It does not establish positivity of thecomplete Weil quadratic form or prove the Riemann Hypothesis.