An Arithmetic-Fluctuation Hilbert–Pólya Mechanism: Endpoint-Seven High-Block Closure and a Quantitative Cofinal Temporal-Resonance Law

This preprint continues the restarted TEBAC Hilbert--P\'olya 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. A Hilbert--P\'olya operator is introduced only after positivity of the completed arithmetic form has been established. The first two completed support windows are already closed within the stated interval-arithmetic trust boundary. The third support window is\[\frac12\log 5 < R \leq \frac12\log 7,\]and contains exactly the nonzero von Mangoldt labels \(2,3,4,5\). 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 channel. At the closed endpoint \(L=\log 7\), the paper retains the exact Fourier ledger for the completed \(2,3,4,5\) form. The \(165\)-dimensional low compression has outward-rounded positive parity certificates, while the finite transition band satisfies coercive bounds. The principal new result of this revision is a rigorous closure of the previously unresolved endpoint-seven continuum high block: after parity reduction, the shifted high operator is divided into a finite band and an eventual tail, and the tail is shown to retain a positive coercive reserve. A fixed source-matched witness and direct residual evaluation yield validated Schur reserves in both parity sectors, so that the full high block is controlled by an exact three-block operator-valued inequality. The paper also establishes a quantitative temporal-resonance law for prime-log phases. This reduces the normalized moment problem to a finite Hilbert--Gram structure plus an explicit phase-separation error, and it yields a cofinal residual criterion for large prime-power windows. The arithmetic temporal-resonance statements are proved directly through geometric summation, Abel summation, unique factorization, and explicit phase separation; they are not assumed from an external physical hypothesis. The final source-matched residual inequality is not yet certified. Accordingly, this version does not establish continuum positivity of the complete \(2,3,4,5\) support window, global Weil positivity, or the Riemann Hypothesis.