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Abstract

We construct certified finite approximations to the complete negative Fourier observation of an odd Weil form in its reciprocal-theta norm. Weighted Chebyshev features on a rigorous negative-symbol cover give a positive remainder with an explicit trace bound after exact orthogonal constraints. Discrete cosine orthogonality provides a degree-independent ridge charge. At arithmetic cutoff eleven, a 1366-feature construction admits ridge parameter 10^-17 with complete trace error below 1.121 × 10^-14. A support-sensitive paired arithmetic-tail estimate yields a nonnegative addition to the localized positive reference while preserving the same negative observation. We derive the corresponding finite inverse conditions and a compact physical-to-weighted certificate transfer, retaining the complete covariance and tail costs.