The Prime Alternating Phase Framework (PAPF): A Deterministic Presieve and Structural Theory for Primes on the 6k ±1 Rails
Year: 2025
Type: article
Abstract: 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
Author: Stewart, Jacob
Language: English
Subfield: Algebra and Number Theory
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7081652952
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