Beta-Logistic Weil Kernels, Spectral Localization, and Signed Short-Interval Reductions

Abstract

We study beta-logistic test kernels for the Weil explicit formula, focusing on uniform localization, signed arithmetic errors, and their relationship to zeros of the Riemann zeta function. We distinguish classical positivity and one-sided Laplace principles from kernel-specific consequences. The main reductions consist of a localized zero-sum representation, an exact diagonal/off-diagonal second-moment expansion, and a positive layer-cake transfer to ordinary Chebyshev errors in short intervals. A finite-scale sign-balance inequality is recorded with its necessary moment hypotheses. The final sufficient short-interval estimate remains unproved; our arguments do not establish the Riemann hypothesis or a new unconditional zero-free strip. Several claims, especially uniform error bounds in the spectral and arithmetic transitions, require independent line-by-line verification before submission.