An Arithmetic-Fluctuation Hilbert–Pólya Mechanism: Endpoint-Seven Closure, Cofinal High Coercivity, and Universal First-Order Source Annihilation

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 self-adjoint Hilbert–Pólya operator is introduced only after positivity of the completed arithmetic form has been established. Within the validated-numerics trust boundary specified in the manuscript, the preceding revisions close the first three completed prime-power support windows:\[2,3;\qquad 2,3,4;\qquad 2,3,4,5.\]In particular, the completed endpoint at\[R=\frac12\log 7\]and the full third window\[\frac12\log 5 < R \leq \frac12\log 7\]are nonnegative. The present revision re-audits the endpoint-seven high-block prime norm. The commuting labels \(222\) and \(4=2^2\) are treated jointly on their exact delay fibers, giving the combined norm\[\frac{\log 2}{4}\bigl(1+\sqrt{17}\bigr).\]Together with the labels \(333\) and \(555\), this yields the corrected prime-channel bound\[\|P_{\mathrm{pr}}\| \leq 2.241813171178084.\] After parity reduction, the shifted high block is divided into the finite band\[83\leq |n|\leq 800\]and the eventual tail\[|n|\geq 801.\]The corrected eventual-tail reserve is\[C_{|n|\geq 801}-2I\geq 0.20I.\]A fixed witness supported on \(801\leq |n|\leq 1600\), direct residual evaluation through \(|n|=20000\), and an analytic temporal-resonance tail enclosure give positive Schur reserves exceeding \(0.08020\) in the even sector and \(0.10148\) in the odd sector. Consequently,\[C_{|n|\geq 83}\geq 2I.\] The principal new analytic contribution is a universal cofinal source-compression theorem. For an endpoint \(m\), all prime sine families combine into the single weighted arithmetic phase\[F_m(n)=\sum_{q<m}\frac{\Lambda(q)}{\sqrt q}\sin\!\bigl(n\theta_q^{(m)}\bigr).\]The exact source coefficients satisfy\[A_q=\frac{\Lambda(q)}{\sqrt q}\,A_{\mathrm{osc}},\]so an order-\(J\) residual requires only two vector-polynomial coefficient families rather than \((1+r_m)^J\) independent source families. The corresponding compressed temporal-resonance moment matrix has leading block\[\begin{pmatrix}1&0\0&\kappa_m\end{pmatrix}\otimes \mathsf H_J,\]where \(\mathsf H_J\) is the Hilbert moment matrix and the nonnegative arithmetic variance \(\kappa_m\) contains the entire factor-pair resonance graph. Its normalized error satisfies\[\left\| a^{-1}\widetilde{\mathsf M}_{m,J}(a)-\widetilde{\mathsf D}_{m,J} \right\|\leq \frac{2J}{a}(1+W_m)^2\left(1+(\Delta_m^{\mathrm{nr}})^{-1}\right),\]with\[W_m\leq 2\sqrt m\log m.\] Reflection parity then produces a universal first-order source law. In the even sector the leading residual has one common moving-phase coefficient, while in the odd sector it has one fixed source coefficient. Explicit rank-one corrections of any sufficiently large finite-support witness annihilate both first-order coefficients exactly. The corrected residual rows are therefore \(O(n^{-2})\) in both parity sectors. For an explicitly computable constant \(K_m^\sharp\), their complete two-sided tail Gram satisfies\[\left\| \sum_{|n|\geq a} r_m^\sharp(n)^*r_m^\sharp(n) \right\|\leq \frac{2(K_m^\sharp)^2}{3(a-1)^3}.\]Thus the infinite source tail can be made smaller than any prescribed endpoint margin by an explicit adaptive cutoff. The revision also proves an unconditional all-endpoint high-coercivity theorem. For\[N_m=\left\lceil \max\left\{ \frac{4\log m}{\pi}, \,2\sqrt m\log m, \,\frac{\log m}{\pi}\exp\!\bigl(12+4\sqrt m\log m\bigr) \right\} \right\rceil,\]the completed high-frequency block satisfies\[C_{m,\geq N_m}\geq 2I.\]This estimate uses no zero of the Riemann zeta-function, no prime-number theorem, and no RH assumption. Revision v29 therefore discharges analytically two of the three cofinal mechanisms required by the program: uniform high-frequency invertibility at every sufficiently large endpoint and adaptive infinite-tail source decay after exact first-order annihilation. The remaining global problem is the universal finite-margin theorem: one must establish a positive lower bound for the finite transition-shortened residual matrix at all sufficiently large endpoints, together with certificates for the finitely many intermediate windows not covered by the eventual regime. The manuscript does not establish global Weil positivity or the Riemann Hypothesis. These remain open until the finite-margin theorem is proved.