An Arithmetic-Fluctuation Hilbert–Pólya Mechanism: Outward-Rounded Endpoint-Seven Residual Closure and a Quantitative Cofinal Temporal-Resonance Law

This preprint continues the restarted TEBAC Hilbert–Pólya program based on the completed localized Weil form. Prime powers arise as repeated returns of conservative prime delay-line colligations, while the endpoint and archimedean sectors are incorporated before positivity is considered. A Hilbert–Pólya operator is introduced only after positivity of the completed arithmetic form has been established. The preceding revisions closed, within the stated validated-numerics trust boundary, the first two completed support windows. The present revision completes the third support window\[\frac12\log 5 < R \leq \frac12\log 7,\]whose nonzero von Mangoldt labels are exactly \(2,3,4,5\). The coefficient\[\frac{\Lambda(5)}{\sqrt5}=\frac{\log 5}{\sqrt5}\]is generated as the first return of the prime-five colligation and is not introduced as an independently fitted parameter. At the closed endpoint \(L=\log 7\), the formal label \(7\) has translation length equal to the full interval length and its compressed shift vanishes. Earlier stages of the endpoint-seven analysis constructed the exact Fourier ledger for the completed \(2,3,4,5\) form and certified strict positivity of the \(165\)-dimensional low compression \(V_{82}\). They also established the complete continuum high-block estimate\[\mathcal C_{|n|\geq 83}\geq 2I,\]which gives the rigorous inverse bound \(\mathcal C_{|n|\geq 83}^{-1}\leq \tfrac12 I\). Revision v28 closes the remaining low-space residual problem. A fixed finite-support witness \(X_{512}\) is retained, and the residual Gram is decomposed into a finite band and an infinite tail. The finite and explicit residual rows are accumulated in IEEE binary128 arithmetic. For the infinite tail, the paper uses an order-60 temporal-resonance expansion with 300 scaled features and an exact operator enclosure. Outward-rounded interval \(LDL^*\) elimination proves the exact terminal estimates in both parity sectors, so that the completed endpoint form at \(R=\frac12\log 7\) is nonnegative. Support monotonicity then proves continuum positivity throughout\[\frac12\log 5 < R \leq \frac12\log 7.\]Thus, within the validated-numerics trust boundary specified in the manuscript, the first three prime-power support windows are closed. The paper also proves a quantitative cofinal temporal-resonance framework and a single cofinal residual theorem reducing all sufficiently large later support windows to uniform high-frequency coercivity, a polynomial endpoint margin, and a source-matched coefficient-decay estimate. This manuscript does not prove global Weil positivity or the Riemann Hypothesis.