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\12 5 < R 12 7, nonzero von Mangoldt labels are exactly \ (2, 3, 4, 5\). The new coefficient\ (5) 5= 55 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= 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 \ (V₈₂\), 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.