Critical Thresholds for Phase-Uniform Weil Certificates

This note studies phase-uniform proof architectures for Weil positivity through Paley–Wiener compression operators. For the finite prime polynomial Pθ,aPθ,a, it introduces the operator critical threshold Ccrit(a)Ccrit(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)