Finite-Prime Factor-First Closure: Dollard-Renormalized Prolate–Jacobi Operators and Relative Fredholm Determinants

This record establishes the public release of version 0.1 of Paper I in the finite-to-all-prime program. The manuscript develops the finite-prime, factor-first stage of a Dollard-renormalized semilocal prolate–Jacobi construction. For every finite set of primes and every fixed integer \\(K\\ge 5\\), it constructs positive bordered-QR/Cholesky histories in heterotyped source spaces. Factor sources and moment sources are kept distinct and connected by a bounded reconstruction isomorphism, while native source observations are constructed independently and identified with the reconstructed physical columns through uniqueness. The source calculus is propagated through the Jacobi coefficients, finite Dollard subtraction, and relative prolate realization. The resulting construction provides trace-class relative operators, holomorphic Fredholm determinants, real-involution symmetry, and fixed-order comparison identities for finite prime sets. The proof records the required restriction, lawful-corner, missing-tail, first-jet, and downstream consumer compatibility conditions. The fixed-prime source calculus used as an external input is archived at Zenodo: https://doi.org/10.5281/zenodo.22701239. This manuscript does not prove an all-prime limit. It does not identify the resulting determinant with an Euler product or the completed Riemann zeta function, and it does not claim a spectral characterization of the nontrivial zeros of the zeta function. The all-prime passage is treated separately in Paper II. The manuscript was developed through extensive human–AI interaction. It has undergone repeated AI-assisted mathematical, symbolic, and structural 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)