An Arithmetic-Fluctuation Hilbert–Pólya Mechanism: Outward-Rounded Residual-Gram Closure of the Completed 2, 3, 4 Window

This preprint develops a nonlocal arithmetic Hilbert--P\'olya mechanism based on the completed localized Weil form. Prime powers are generated as repeated returns of conservative prime delay-line colligations, while the archimedean gamma contribution and the endpoint completion are incorporated before positivity is considered. The preceding public version established unconditional continuum positivity throughout the first completed support window containing the arithmetic labels \(2\) and \(3\). The present version advances to the second prime-power window, where the label \(4=2^2\) is forced by the second return of the prime-two delay colligation with the exact coefficient\[\frac{\Lambda(4)}{\sqrt{4}}=\frac{\log 2}{2}.\]The completed Weil form is extended to a closed lower-semibounded semilocal form with compact-resolvent operator. The entire second support window is reduced to the endpoint\[R=\frac12\log 5.\]At this endpoint, exact Fourier--Galerkin formulas are obtained for the endpoint, archimedean, and prime-power sectors. Outward-rounded interval arithmetic proves strict positivity of a 165-dimensional low Fourier block and establishes the uniform high-frequency estimate\[C_{82}\geq 0.89I.\] A direct scalar substitution for the high inverse is shown rigorously to be insufficient. The proof therefore retains the operator-valued response through an inverse-free residual identity. A finite-support witness \(X_{512}\) captures the dominant inverse action, while the remaining residual Gram is divided into a finite explicit band and an analytic tail. For frequencies \(|n|\geq 100\), the residual tail is represented by an order-60 expansion containing 240 scaled Hurwitz--Lerch moment features. Exact asymptotic coefficients, Euler--Maclaurin and Abel-summation remainder estimates, binary128 assembly, and a complete a posteriori operator-error budget yield\[\left\|L_{512}-L^{\mathrm{pt}}_{512}\right\|<1.143\times 10^{-21}.\]Parity-resolved outward-rounded interval \(LDL^*\) elimination then proves\[L_{512}\geq 8\times 10^{-18}I>0.\]Consequently, the exact continuum operator-valued Schur inequality holds at \(R=\frac12\log 5\). Support monotonicity therefore proves unconditional positivity of the completed Weil form throughout\[\log 2<R\leq \frac12\log 5,\]the complete second support window containing exactly the arithmetic labels \(2,3,4\). Together with the first-window theorem, this version establishes completed positivity in the first two successive prime-power windows. The manuscript does not prove global Weil positivity or the Riemann Hypothesis. The next unresolved support window introduces the prime label \(5\) and has endpoint \(R=\frac12\log 7\).