An Arithmetic-Fluctuation Hilbert–Pólya Mechanism: Continuum p = 2, 3 Positivity via High-Frequency Coercivity and a 40-Moment Tail Certificate

This preprint restarts Paper A of the TEBAC Hilbert–Pólya program at the operator-mechanism level. It replaces the superseded scalar one-dimensional exponential-well construction—which can reproduce the smooth Riemann–von Mangoldt density after Weyl calibration but cannot generate the unbounded arithmetic fluctuation through a bounded or trace-class perturbation—with a nonlocal arithmetic scattering mechanism. For every prime p, the construction assigns a conservative delay-line colligation of length log p and reflection coefficient p^(−1/2−ω). Its scalar Blaschke transfer factor produces the exact prime-power phase expansion. Consequently, each prime power pᵏ arises as the k-fold return of the prime-p delay line with coefficient (log p)p^(−k/2), rather than being inserted as an independently fitted spectral channel. The prime cascade is combined with the archimedean gamma phase and the endpoint terms of the completed explicit formula. This gives the Weil distribution directly from primes and the gamma factor, independently of the locations of the zeros of the Riemann xi-function. The corresponding Weil quadratic form is localized on compact support windows and reduced to explicit finite Hermitian Gram matrices. The paper derives the exact completed p = 2, 3 sector ledger, a source-derived Kreĭn factorization, and an explicit positive three-packet Gram gate. It then extends the completed form to a closed lower-semibounded semilocal form on L²((−R,R)), whose associated operator has compact resolvent. Positivity throughout the first support window is reduced to one endpoint inequality at R = log 2 and then to a compact Birman–Schwinger problem. Exact Fourier–Galerkin formulas are obtained for all endpoint, archimedean, and prime contributions. Outward-rounded interval certificates establish positivity through Fourier level M = 82, corresponding to dimension 165. The full continuum tail is subsequently closed by combining: • a certified high-frequency coercive estimate C₈₂ ≥ 1.39I;• a finite-band evaluation of the low/high coupling;• a forty-feature moment expansion of the infinite cross Gram;• exact Hurwitz-zeta and Lerch-transcendent moment tails; and• parity-resolved interval LDL* certificates with a complete operator-error budget. The resulting continuum theorem proves unconditional nonnegativity of the completed Weil form for every support radius ½ log 3 < R ≤ log 2, that is, throughout the first completed support window containing exactly the arithmetic labels 2 and 3. The endpoint GNS completion consequently supplies canonical compatible Gram factors for every smaller support and finite-dimensional test space in this window. This result is not a proof of the Riemann Hypothesis and does not establish global Weil positivity. Test functions with larger support admit additional prime powers. The next unresolved arithmetic window is log 2 < R < ½ log 5, where the new label 4 = 2² must be derived as the second return of the prime-two delay colligation with its exact coefficient (log 2)/2. All subsequent prime-power windows must then be controlled compatibly before the global GNS and canonical-system implications can be invoked.