Exact Marked Fourier Transfer Operators for Sieve Dynamics

PRIMES_SINGULARITY is a mathematical research program investigating exact structural connections between sieve dynamics, finite Fourier analysis, spatial localization, and prime-pattern counting. The principal result documented in this release is the Unified Shape–Fourier (USF) transfer framework: an exact stage-to-stage operator describing how a fully marked periodic sieve-survivor state and its spatial Fourier spectrum evolve when a new coprime sieve factor is adjoined. The all-span transfer formulation remains exact in the presence of simultaneous residue closures and requires only coprimality between the existing period and the newly adjoined factor. In a shifted Boolean-containment basis, the marked operator decomposes into scalar tuple-survival channels, leading to an exact admissibility–isometry dichotomy, fiberwise Fourier tomography, CRT-compatible composition laws, spectral-energy branching identities, and an exact Bernoulli product law for spectral prime support. The associated normalized spectral conductor is also shown to realize a classical generalized Dickman limit law. The dossier additionally records a broader investigation into prime and twin-prime localization. Exact wheel recursions, survival-window identities, CRT/Fourier decompositions, determinant-2 formulations, von Mangoldt and Möbius representations, character decompositions, and conductor–frequency reductions are developed and audited. These transformations identify the remaining obstruction as a parity-sensitive nonzero-spectrum problem involving high-conductor and Type-II/Möbius correlations. No proof of the twin-prime conjecture, no new closed-form prime generator, and no resolution of another major open problem is claimed. The USF theorem package has been internally derived and subjected to extensive finite falsification testing, including random periodic binary states, non-squarefree moduli, collision cases, and coprime composite refinements. Independent mathematical peer review and a definitive prior-art/novelty assessment remain outstanding. This deposit is intended as a persistent technical disclosure, provenance record, and self-contained handoff package for independent verification of the mathematics, the stated novelty boundary, and the open research questions.