Residual correction for Fourier–Schur approximation of the localized Weil form

We develop quantitative Fourier--Schur approximation for the localized Weil form and obtain explicit control of the full response omitted by finite Fourier truncation. The exact Schur matrix is defined by minimization over an infinite-dimensional Fourier complement. The associated operator is represented as a periodic logarithmic diagonal plus a perturbation bounded on weighted Fourier spaces of every order \(0<s<1/2\). At each fixed support and admissible low-dimensional space, the finite Schur matrices converge in operator norm at rate\[O\!({K⁻²ˢ}{log K}).\]A residual matrix constructed using the inverse logarithmic diagonal bounds the full Schur defect from both sides, with relative comparison factors tending to one and identical null spaces. A correction using finitely many additional Fourier rows improves the error to\[O\!({K⁻²ˢ}{(log K)^2}).\]Integration, evaluation, linear-solve, and residual-tail errors are incorporated into a computable matrix enclosure. The estimates retain the dependence of the constants on the support and provide explicit cutoff choices along growing endpoint sequences; they do not assert a uniform fixed-cutoff limit or practical computational complexity. The manuscript also investigates the arithmetic positivity problem associated with these reductions. It develops support-doubling identities, fixed moving-average criteria, centered M\"obius and Vaughan decompositions, signed prime-power recombination, and an equal-level interval representation. Known unconditional prime-number estimates are transferred to the full centered arithmetic block, with their limitations stated explicitly. The present revision adds a finite weighted representation of the retained interval average, exact coefficient identities for moving cutoffs, and a complete centered balance between two scales. An iteration lemma shows how a suitable independently established scale inequality would imply subexponential growth of the positive part of the centered block. The required signed inequality is formulated explicitly and remains unproved. This work provides approximation theorems, exact arithmetic identities, and conditional positivity criteria. It does not establish cofinal positivity, the decisive subexponential arithmetic bound, or the Riemann Hypothesis.