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

This cumulative research manuscript develops finite-core reductions and arithmetic positivity criteria related to the Riemann hypothesis within a Hilbert–Pólya/Weil-positivity framework. It brings together Schur-complement methods, weighted prime translations, signed moment representations, arithmetic cancellation estimates, and explicit accounts of unresolved proof obligations. The revision dated 9 October 2026, ``Legendre Band Core,'' localizes the complete centred arithmetic scale remainder to a moving band of Legendre moment indices. Using the established spectral allowance\[|d_k|≤ C_Lexp(κ_0 k), κ_0=1.1× 10⁻²⁵,\]Gaussian damping of the exact row weights bounds the discarded centred energy by \(C_L^2(m+1)⁻⁴\). A separate arithmetic estimate controls the discarded prime diagonal by a quantity tending to zero. These estimates preserve the complete mixed correlation with vanishing absolute error. The localization also applies to the original finite prime head, with unchanged endpoints and a vanishing correction. The retained cutoff has positive asymptotic proportion \(2κ_0\). The revision therefore does not eliminate the positive exponential allowance. The required relative bound for the retained correlation remains unproved. This deposit contains the full cumulative master, retaining earlier results, audit corrections, and the remaining open conditions. It is a research preprint, not a completed proof of the Riemann hypothesis. No complete independent audit of every inherited assertion is claimed.