Spectral Paradigm of Analytic Number Theory

This revised long article develops the spectral paradigm as a method in analytic number theory. It is based on the earlier consolidated manuscript Spectral Paradigm for Analytic Number Theory and on the review report supplied for that manuscript. The revision deliberately changes the balance of the work. The opening discussion of the Riemann Hypothesis and the Generalized Riemann Hypothesis is now compressed and treated as a methodological input, referring to the separate monograph Spectral-Geometric Proof of the Riemann and Grand Riemann Hypotheses for the detailed spectral-geometric closure. The main body is expanded in two directions: additive prime geometry, centered on the binary Goldbach problem, and subtractive prime geometry, centered on fixed prime differences, twin primes, and Polignac-type gaps. The central organizing principle is that arithmetic phenomena should be represented by self-adjoint Hilbert-space packets, trace-compatible explicit formulas, Fredholm determinant limits, and positive or non-escaping spectral projections. This article keeps a strict logical convention. Finite spectral-flow and distributional trace identities are stated as theorems when they follow from standard operator theory. Infinite-dimensional passages, Goldbach positivity, pair-correlation closure, and gap isolation are formulated as explicit conditional estimates or verification tasks. In this revised form, the word “proof” always means a proved implication from precisely stated spectral hypotheses; unverified analytic estimates are not used as if they were established theorems.