The Range of a Finite-Scale Weil Positivity Method: Success in the First Prime Window and Termination in the Second

Abstract

We determine the exact range of validity of a split-residual Schur criterion for finite-scale positivity of the local Weil quadratic form Q_W^L. The method succeeds in the first prime window L in (1/2 log2, 1/2 log3), where it certifies lambda(7/20) > 0 (the FP-0.35 result). We show, by certified interval arithmetic and d-refined pilot computation, that the minimum eigenvalue lambda_min(L) decreases monotonically within each prime window, with infimum at the right endpoint L -> (1/2 log p)^-. At the first window's right endpoint the positive margin is already exhausted to a pilot-level knife-edge (odd sector negative for truncation dimension d<=100, marginally positive at d=120; certify-level indeterminate, as the shift-parameter enclosure width ~2.7e-4 exceeds the ~1e-9 signal by four orders of magnitude). In the second window L in (1/2 log3, log2), the double-prime cross-coupling J(tau2,tau3) -- a structure absent from the first window -- is shown to be nonzero at certify grade, but two natural conjectures about it (that it dominates positivity; that it delays margin collapse) are independently refuted; the second window loses positive definiteness at certified anchor points (odd right endpoint lambda = -0.369). The per-window right-endpoint sequence lambda((1/2 log p)^-) crosses from marginally positive to strictly negative, and an analytic extrapolation (not constructed) indicates continued descent. We conclude that the method's positive margin exhausts gradually beginning at the first window's right endpoint, terminating in the second -- a precise range-boundary result. This does NOT imply the Riemann Hypothesis, is not extrapolated past L = log2, and characterizes only this finite-scale construction. Erratum (2026-08-09): In the published PDF, §2.2 Remark 2.2 contained a broken cross-reference displayed as "??"; this has been corrected to "Remark 2.4" in the companion journal submission. No mathematical content, data, or conclusions are affected.