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

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 Pθ,a(u)=2∑_pᵏ 0\). Combining this threshold with the archimedean term in the corrected high-frequency reduction gives the associated phase-uniform critical frequencyTcritPU(a)=exp\![4e^a(1+O(e-c√ a))].$$ 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(T₀)≤loglog T₀-log 4+o(1). The paper also gives an explicit onset for the analytic lower-bound mechanism, a_*=12log 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.