Certified Real-Zero Fourier Approximations: Sharp cutoff constraints and validated finite Weil diagnostics

Abstract

We study normalized finite Fourier constructions whose entire transforms have real zeros and a prescribed lattice tail. For support parameter L and truncation index N, the reciprocal-square mass of the forced zeros controls the Gaussian factor that survives in a limit. We characterize all reachable entire limits for a prescribed sequence of cutoffs, including oscillating tail masses. When L tends to infinity, N/L² tending to infinity is necessary and sufficient for the existence of unrestricted real-zero Fourier approximants to a prescribed normalized even real-zero target of order below two. This is a specialization of classical Laguerre-Pólya convergence and does not establish convergence of Weil spectral selectors. An accompanying implementation combines degree-aware matrix compression, weighted displacement-defect bounds, exact Fourier interpolation and two validated finite-Weil moment backends. At c = 13 and N = 120, it certifies all 240 polynomial zeros as real and simple. The first-zero error is approximately 2.4363 × 10⁻⁵⁵, whereas the uncorrected function has uniform error greater than 0.0391674 on the complex disk of radius five. A target-calibrated Gaussian reduces that disk error below 5.925 × 10⁻⁶, but has error approximately 1.7653 on the disk of radius twenty. These certificates separate finite zero accuracy, functional approximation and global convergence. The contribution is an explicit application and reproducible integration of finite bounds; no proof of the Riemann hypothesis is claimed. Supplement: unchanged real-zero sieve toolkit 0.2.0, including exact source, mathematical proof notes, full interval certificates, 64 passing unit tests and reproduction instructions. The archive contains an earlier historical manuscript; ARTICLE.pdf and ARTICLE.md are the publication text. AI assistance: Generative AI, including OpenAI ChatGPT/Codex, was used extensively in derivations, software development, literature checks, internal auditing and manuscript preparation. All mathematical and software audits reported here are internal. This is a preprint and has not undergone external peer review. Licensing: The article, original figures, mathematical documentation and numerical data are licensed under CC BY 4.0. The original source code and tests, including the software wheel, are licensed under MIT. See LICENSES.txt for the exact scope and MIT notice. Third-party works and dependencies retain their own licenses. This licensing notice supersedes pending-license statements in the historical supplementary documentation.