Sharp logarithmic coupling estimates for theta equality frames in a signed Weil-form realization

Abstract

We prove sharp two-sided estimates for the finite-frame Schur correction associated with the first m pole-killing odd theta derivatives in a weighted realization of Weil's quadratic form. For each fixed m, its norm has order beta times log(1/beta) to the power 4m+5/2 as beta tends to zero, with explicit positive constants and a full small-parameter regime. The proof combines separate half-line charge cancellation, a square-root range construction, integrated full theta-tail control and matching odd box tests. We also prove a pointwise upper bound for low-energy gamma spectral mass and an exact change of metric after equality-space elimination. Prime powers and polar terms remain in the signed Weil reduction. No uniform growing-frame estimate, complementary-form positivity or solution of the Riemann Hypothesis is asserted. The archive contains the complete standalone manuscript, compiled PDF and reproducible release packaging. The analytic theorems require no numerical certificate. Internal reconstruction, originality and external review are distinguished. Copyright © 2026 Leonard van Hemert. All rights reserved. Public access does not imply an open-content license; see NOTICE.