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| Format: | Recurso digital |
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Zenodo
2025
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| Online Access: | https://doi.org/10.5281/zenodo.17127159 |
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Table of Contents:
- <p> This document presents the Prime Alternating Phase Framework (PAPF), a deterministic,<br> modular system for analyzing primes restricted to the 6k ±1 rails. PAPF is both an algorithm<br> (a congruence-driven presieve with explicit activation residues and thresholds) and a structural<br> theory (a 28-phase partition exposing unavoidable coverage deficits). This document proves:<br> (i) per-prime capacity bounds in any 28-block; (ii) a deterministic deficit after aggregating<br> primes ≤ 43; (iii) a p2 siphon/vacuum mechanism that guarantees survivors (numbers free<br> of small primes) in the immediate block; and (iv) a locked-collision phenomenon with q =<br> 7 that lowers effective coverage in a positive density of blocks. This document also gives<br> precise activation laws, correctness of the presieve, complexity bounds, worked examples, and<br> tabulated activation data. Applications include twin/quadruple-prime filters (not claims of new<br> unconditional infinitude within this paper), explanations of phase-patterned composite density,<br> and a roadmap for higher constellations. The goal is to equip researchers with a reproducible,<br> theory-backed framework that has proved practically powerful</p>