Local Weil Positivity and Signed Prime-Deficit Criteria

We study a local positivity certificate and a signed arithmetic criterion associated with the

Riemann hypothesis. The computer-assisted certificate establishes QW (f) ≥ 10−40∥f∥2

2 for

complex test functions supported in [−11/10,11/10], with the complete infinite-dimensional

complement of 512 retained Legendre directions controlled. Its two arithmetic verifiers share

a frozen action assembly. For each 1/2 < σ < 1, we characterize zero exclusion in ℜs > σ

by a one-sided logarithmic bound for a smoothed prime-power deficit. The anchored explicit

formula includes all zeros; a moving spectral cutoff has uniform error less than 48. We give

an exact finite dyadic reconstruction of the same signed source, with complete auxiliary

error Oσ (Y−σ log Y). Balanced Chebyshev filters yield a centered renewal equation, and an

explicit factorial–harmonic regularization yields a Mellin numerator with finite energy on

every line 0 < ℜs < 1. These results do not bound the arithmetic response obtained by

division by ζ. Hybrid universality rules out a uniform inverse estimate for arbitrary full-

line inputs in the same positive Fourier-weighted norm; it does not exclude a selected-source

estimate. An exact past/future identity further shows why a contractive component does not

by itself control the full response. We distinguish these arithmetic results from a published

spectral route: real-zero ground-state transforms still require identification with the explicit

approximants converging to the Riemann Ξ function. The remaining arithmetic assertion

retains the prescribed source, anchor and every future scale. Neither that assertion nor

continuation of local positivity to unbounded supports is proved