Critical Threshold and Double-Exponential Obstruction for Phase-Uniform Weil Certificates

Abstract

This paper studies a quantitative obstruction that arises in a specific phase-uniform proof architecture for Weil positivity associated with the Riemann zeta function. For a Paley–Wiener support parameter \(a>0\), the prime contribution is represented by the finite trigonometric polynomial $$P_{\theta,a}(u)=2\sum_{p^k 0\). Combining this threshold with the archimedean term in the corrected high-frequency reduction gives the associated phase-uniform critical frequency $$T_{\rm crit}^{\rm PU}(a)=\exp\!\left[4e^a\left(1+O(e^{-c\sqrt a})\right)\right].$$ Thus a proof architecture that uses arithmetic phase-sensitive information only up to a fixed finite frequency horizon and treats the remaining tail solely by a strict operator bound uniform over all admissible prime phases necessarily encounters a double-exponential scale. Equivalently, the maximal support parameter accessible from a finite arithmetic horizon \(T_0\) satisfies the asymptotic necessary bound $$a(T_0)\le\log\log T_0-\log 4+o(1).$$ The paper also gives an explicit onset for the analytic lower-bound mechanism, $$a_*=\frac12\log 3217,$$ supported by interval-arithmetic verification together with an analytic tail estimate, and includes a separate rigorous finite-\(a\) certificate at \(a=3\). The result is not a disproof of the Riemann Hypothesis and is not an impossibility theorem for Weil methods in general. It identifies a quantitative obstruction specifically for certificates that require the same strict Paley–Wiener operator bound to hold uniformly over the full admissible prime-phase torus.