Critical Thresholds for Phase-Uniform Weil Certificates

Abstract

This note studies phase-uniform proof architectures for Weil positivity through Paley–Wiener compression operators. For the finite prime polynomial Pθ,aP_{\theta,a}, it introduces the operator critical threshold Ccrit(a)C_{\rm crit}(a) defined by the condition that the compressed positive-part operator remain within the 4a4a budget uniformly over all admissible prime phases. The main result is the explicit two-sided estimate (1−εa)A(a)≤Ccrit(a)≤A(a),(1-\varepsilon_a)A(a) \le C_{\rm crit}(a) \le A(a), with A(a)=2∑n<e2aΛ(n)n∼4ea,A(a)=2\sum_{n