Source-Faithful Finite-Core Reductions for a Hilbert–Pólya Positivity Criterion

This research preprint develops source-faithful finite-core reductions within a Hilbert--P\\'olya and Weil-positivity program for the Riemann hypothesis. The central objective is to identify and estimate the precise signed arithmetic balance whose nonnegative definiteness would yield the required spectral positivity. The manuscript combines functional analysis, exact prime--continuous identities, finite-dimensional Schur reductions, and computer-assisted certificates while preserving the distinction between established partial results and the remaining global arithmetic obligation. The direct analytical branch identifies the full sesquilinear form with the classical explicit formula for the completed Riemann zeta-function. It establishes the logarithmic form domain, suitable approximation cores, support consistency, and the decomposition into even and odd parity sectors. The resulting Fourier matrices represent the complete real-frequency integral on their specified input spaces: restricting the input Fourier band does not truncate the frequency integral or discard prime-power, archimedean, or mixed contributions. Historical source-channel reductions are retained with their stated dependencies and are distinguished from this direct explicit-formula branch. A central result is a quadratic-band detection criterion. For \\(L=\\log m\\) and \\(K=\\lceil L^2\\rceil\\), the manuscript shows that every hypothetical zero off the critical line would produce a negative direction in the prescribed Fourier band at every sufficiently large endpoint, in each parity sector. Consequently, nonnegative definiteness of the corresponding exact matrices along an unbounded dyadic endpoint sequence in either fixed parity is equivalent to the Riemann hypothesis. A sufficiently weak, subpower bound for the complete negative part would also suffice. These reductions specify the required quantifiers but do not establish their unbounded-family arithmetic premises. The matrix analysis retains signed prime--continuous cancellation before applying inequalities. It includes exact matrix-entry formulas, structured Schur recursions, residual-energy identities, physical-norm accounting, and the treatment of mixed couplings and zero pivots. The cumulative manuscript contains finite computer-assisted positivity certificates for both parity blocks at the eight dyadic endpoints\\[m=2^j,\\qquad10\\leq j\\leq 17.\\]These certificates concern the complete specified finite-band matrices, not all inputs at each endpoint or an unbounded endpoint family. Earlier certificates are retained as inherited content and are not rerun in the present revision. Further analytical results investigate joint arithmetic cancellation, quantitative lower bounds, and nearly null directions. Explicit theta-based vectors belong to the actual quadratic Fourier bands and have both very small signed energies and small full matrix responses. These results rule out certain proposed uniform positive power--log spectral margins without determining the sign of the remaining spectrum. A theta-weighted square-difference representation also separates positive and negative contributions to the unchanged original form and permits rigorous estimates of complete terminal portions of the signed balance. The revision of 18 September 2026 develops coefficient-uniform endpoint observation and concentration estimates for the prescribed Fourier bands. These establish positivity of the complete theta-weighted terminal piece beginning at\\[\\log m-\\frac{1}{10m},\\]for every band vector at every sufficiently large prime-power endpoint, including all actual Schur residuals. This widens the previously established all-domain collar of width\\[\\frac{\\log m}{2\\pi m^2}.\\]No numerical value of the eventual threshold is supplied. The revision also constructs an explicit unbounded dyadic sequence on which the corresponding terminal piece with width\\[\\frac{1}{2m}\\]is negative on the constant and first sine modes. These negative values concern only the separated terminal piece; they neither establish negative full Weil energies nor constitute a counterexample to the Riemann hypothesis. No optimal collar constant is claimed. The remaining interior Schur comparison is stated explicitly and remains unproved. The manuscript does not establish nonnegative definiteness of the full target matrices along an unbounded endpoint sequence or a subpower bound for their complete negative part. This deposit therefore presents a cumulative research program with partial analytic results, finite certificates, and a detailed account of unresolved proof obligations---not a completed unconditional proof of the Riemann hypothesis. Supplementary checks document selected finite identities and computations; they do not constitute a proof-assistant formalization or an independent verification of the entire analytical chain.