The Prime Anchor Sieve III: Decorrelation, Structural Collapse, and the Origin of Capacity

This preprint is the third part of a three-paper series developing the Prime Anchor Sieve (PAS), a deterministic framework for analyzing additive structure and constrained dynamical behavior within the primes. Part III introduces the Phase Machine, a model capturing how residues, modular phases, and anchor-driven interactions propagate across dyadic intervals. The central results are a Phase Decorrelation Theorem and a corresponding Structural Incoherence Principle, which together show that highly correlated or resonant configurations cannot persist at larger scales. Under standard zero-density hypotheses for Dirichlet LLL-functions, this decorrelation forces the real primes into a low-capacity regime of the form Load(J)≪x log⁡x,Load(J) √x\,log x,Load(J)≪xlogx, matching the combinatorial requirement identified in PAS I, where such a bound implies Goldbach for all sufficiently large even integers. This paper does not claim an unconditional proof of the Capacity Axiom; rather, it provides the conditional structural explanation for why capacity remains low, complementing the analytic estimates established in PAS II. Together, PAS I–III offer a complete conditional reduction of Goldbach’s problem to the underlying decorrelation and capacity mechanisms. This work is intended as the structural and dynamical component of the PAS framework; unconditional closure depends on analytic inputs that remain open in general.