An Arithmetic-Fluctuation Hilbert–Pólya Mechanism: Prime-Five Entry and an Arithmetic Temporal-Resonance Decoupling Law

This preprint continues the restarted TEBAC Hilbert–Pólya program based on the completed localized Weil form. Prime powers are generated as repeated returns of conservative prime delay-line colligations, while the endpoint and archimedean sectors are incorporated before positivity is considered. A self-adjoint Hilbert–Pólya operator is introduced only after positivity of the completed arithmetic form has been established. The preceding versions established unconditional continuum positivity, within the stated interval-arithmetic trust boundary, for the first two completed support windows. The present version begins the third support window\[\frac12\log 5 < R \leq \frac12\log 7,\]whose nonzero von Mangoldt labels are exactly \(2,3,4,5\). The new coefficient\[\frac{\Lambda(5)}{\sqrt{5}}=\frac{\log 5}{\sqrt{5}}\]is derived as the first return of the prime-five delay colligation and is not introduced as an independently fitted channel. At the closed endpoint, the formal label \(7\) has translation length equal to the full interval length, so its compressed shift vanishes. At \(L=\log 7\), the manuscript constructs the exact Fourier ledger of the completed \(2,3,4,5\) form. Outward-rounded parity certificates prove strict positivity of the \(165\)-dimensional compression \(V_{82}\), while the finite transition band and the eventual tail are certified by explicit coercive estimates. The remaining continuum problem is reduced to an exact three-block operator-valued Schur inequality, together with a temporal-resonance residual closure criterion and a finite inverse-action witness. The final source-matched residual inequality has not yet been certified. Accordingly, this version does not establish continuum positivity of the complete \(2,3,4,5\) window, global Weil positivity, or the Riemann Hypothesis.