This preprint is the first part of a three-paper series introducing the Prime Anchor Sieve (PAS), a deterministic framework for analyzing additive structure within the primes.Part I develops the foundations of the sieve: the definition of anchor-based elimination templates, the mechanism of redundancy collapse, and the derivation of the Bridge Theorem, which shows how PAS structure parallels key features of prime behavior inside squared intervals. The goals of this paper are: to formalize the PAS architecture in a mathematically precise way, to establish the structural rules (anchors, phases, redundancy, ledger load), to identify the combinatorial constraints any high-capacity scenario must satisfy, and to lay the groundwork for Parts II and III, where analytic tools (BDH, dispersion estimates, and phase decorrelation) are used to verify that the actual primes satisfy the structural axioms required by the sieve. This work is intended as a conceptual and structural foundation rather than a complete analytic argument. Parts II and III develop the capacity bounds and dynamical decorrelation needed to compare PAS with the real primes and to analyze consequences for additive problems such as Goldbach. This version adds an explicit Conditional Goldbach Theorem stating the analytic hypotheses under which the Capacity Axiom holds and Goldbach follows. This version is a clean, self-contained presentation of the PAS model and is intended for mathematicians interested in sieve methods, structural prime analysis, and the combinatorial aspects of additive number theory.
Elizabeth Liu (Fri,) studied this question.