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2025
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| Online Access: | https://doi.org/10.5281/zenodo.17127159 |
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| _version_ | 1866901719364927488 |
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| author | Stewart, Jacob |
| author_facet | Stewart, Jacob |
| 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> |
| format | Recurso digital |
| id | zenodo_https___doi_org_10_5281_zenodo_17127159 |
| institution | Zenodo |
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| publishDate | 2025 |
| publisher | Zenodo |
| record_format | zenodo |
| spellingShingle | The Prime Alternating Phase Framework (PAPF): A Deterministic Presieve and Structural Theory for Primes on the 6k ±1 Rails Stewart, Jacob <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> |
| title | The Prime Alternating Phase Framework (PAPF): A Deterministic Presieve and Structural Theory for Primes on the 6k ±1 Rails |
| url | https://doi.org/10.5281/zenodo.17127159 |