The Prime Alternating Phase Framework (PAPF): A Deterministic Presieve and Structural Theory for Primes on the 6k ±1 Rails

 This document presents the Prime Alternating Phase Framework (PAPF), a deterministic,
 modular system for analyzing primes restricted to the 6k ±1 rails. PAPF is both an algorithm
 (a congruence-driven presieve with explicit activation residues and thresholds) and a structural
 theory (a 28-phase partition exposing unavoidable coverage deficits). This document proves:
 (i) per-prime capacity bounds in any 28-block; (ii) a deterministic deficit after aggregating
 primes ≤ 43; (iii) a p2 siphon/vacuum mechanism that guarantees survivors (numbers free
 of small primes) in the immediate block; and (iv) a locked-collision phenomenon with q =
 7 that lowers effective coverage in a positive density of blocks. This document also gives
 precise activation laws, correctness of the presieve, complexity bounds, worked examples, and
 tabulated activation data. Applications include twin/quadruple-prime filters (not claims of new
 unconditional infinitude within this paper), explanations of phase-patterned composite density,
 and a roadmap for higher constellations. The goal is to equip researchers with a reproducible,
 theory-backed framework that has proved practically powerful