Fused Core-Decomposition Analytic Number Theory: A Unified Framework for Additive Prime Representations

“We introduce a fused core-decomposition analytic number theory, built on the discrete recurrence (An−2)An+1=n(A n −2)A n+1 =n. The theory defines a four-dimensional state and proves its main term matches the Hardy–Littlewood singular series, with exponentially decaying error. A central square-recurrence conjecture for sieve errors is proposed; it is supported by Kloosterman sums, Bombieri–Vinogradov, and numerical evidence. Assuming this conjecture, we obtain heuristic proofs of the Goldbach, Legendre (strengthened), 3n+13n+1, and Fermat conjectures. The paper also presents a unified recurrence postulate and a novel LL-function with possible automorphic properties.”