All-Prime Limits of Dollard-Renormalized Prolate–Jacobi Operators and Relative Fredholm Determinants

This record presents Paper II of the finite-to-all-prime program. It develops the passage from the finite-prime factor-first construction of Paper I to an all-prime renormalized prolate–Jacobi operator and its relative Fredholm determinant. For every fixed integer \(K\ge 5\) and one fixed endpoint, Dollard, finite-prefix, and real-involution convention, the manuscript proves convergence of the actual finite-prime renormalized operators along admissible cofinal prime gates. The convergence is established in the weighted Jacobi-coefficient topology, in the trace ideal \(\mathcal S_1\), and locally uniformly in norm resolvent on the common operator domain. The associated finite-prime Fredholm determinants converge locally uniformly to a holomorphic all-prime determinant. Within the fixed renormalization convention, the resulting operator, trace-class family, and determinant are independent of the admissible cofinal prime gate. The analytic core consists of an exact connected-source expansion, fixed-arity endpoint normalization, and a \(q^{5/4}\) residual gain. A deterministic finite prime core is extracted, after which the complete factor-first construction is rebased at that core. Core-normalized estimates with a core-independent arity base yield absolute all-arity summability and prime-shell Cauchy bounds. The literal source expansion is connected to the actual finite-prime objects through finite-ray Banach-holomorphic continuation. The source-level construction uses heterotyped factor and moment spaces connected by bounded reconstruction isomorphisms. Factor multiplicativity is restricted to lawful triangular products, general corners retain their exact missing-tail terms, and moment identities are transported through the factor space with exactly one reconstruction. Native physical observations and complete moving-port first jets are treated separately. The principal archived inputs are: Fixed-prime source calculus: https://doi.org/10.5281/zenodo.22701239 Finite-prime Paper I, version record: https://zenodo.org/records/22701448 Finite-prime Paper I, version DOI: https://doi.org/10.5281/zenodo.22701448 Finite-prime Paper I, all-versions DOI: https://doi.org/10.5281/zenodo.22701447 The quantification in \(K\) is pointwise: the result applies to every fixed \(K\ge 5\), but no estimate uniform in \(K\) and no canonical identification between different orders \(K\) is asserted. The manuscript also does not compare different renormalization conventions, identify the determinant with an Euler product or the completed Riemann zeta function, or claim a spectral characterization of zeta zeros. The manuscript was developed through extensive human–AI interaction. It has undergone repeated AI-assisted mathematical, symbolic, structural, and dependency audits, but has not yet received complete independent verification by a qualified human specialist. These audits are not presented as substitutes for expert review or peer review. The manuscript is therefore released as an open verification draft and candidate proof. Readers are invited to examine the arguments, reproduce the accompanying checks, and report errors or gaps. (rev. 260911)