From Topological Rupture to an Elegant Detour: A Definitive Logbook on Two Failed Assaults on the Collatz Conjecture
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2025
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| _version_ | 1866901387106844672 |
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| author | Peter, Romain |
| author_facet | Peter, Romain |
| contents | <p>This report provides a definitive summary of a comprehensive research program aimed at</p> <p>resolving the Collatz Conjecture. We document two distinct, sophisticated lines of attack.</p> <p>The first, a quantitative approach, reframed the problem in terms of 2-adic dynamics and</p> <p>sought a contradiction via p-adic Diophantine approximation. This route was conclusively</p> <p>shown to fail due to a ”logarithmic barrier” that prevented the application of the Subspace</p> <p>Theorem. The second, an informational approach, re-centered the problem on the arithmetic</p> <p>realizability of parity sequences, using a novel coupled 2-adic/3-adic automaton. This second</p> <p>program was a technical success, leading to a complete classification of all possible cycles in</p> <p>the profinite system. However, it ultimately proved to be an elegant detour, reducing the</p> <p>problem perfectly to the classical, unsolved question of the existence of non-trivial integer</p> <p>cycles. This document details both failed assaults, presenting their respective negative</p> <p>results as a contribution to understanding the profound difficulty of the conjecture.</p> |
| format | Recurso digital |
| id | zenodo_https___doi_org_10_5281_zenodo_17292166 |
| institution | Zenodo |
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| publishDate | 2025 |
| publisher | Zenodo |
| record_format | zenodo |
| spellingShingle | From Topological Rupture to an Elegant Detour: A Definitive Logbook on Two Failed Assaults on the Collatz Conjecture Peter, Romain <p>This report provides a definitive summary of a comprehensive research program aimed at</p> <p>resolving the Collatz Conjecture. We document two distinct, sophisticated lines of attack.</p> <p>The first, a quantitative approach, reframed the problem in terms of 2-adic dynamics and</p> <p>sought a contradiction via p-adic Diophantine approximation. This route was conclusively</p> <p>shown to fail due to a ”logarithmic barrier” that prevented the application of the Subspace</p> <p>Theorem. The second, an informational approach, re-centered the problem on the arithmetic</p> <p>realizability of parity sequences, using a novel coupled 2-adic/3-adic automaton. This second</p> <p>program was a technical success, leading to a complete classification of all possible cycles in</p> <p>the profinite system. However, it ultimately proved to be an elegant detour, reducing the</p> <p>problem perfectly to the classical, unsolved question of the existence of non-trivial integer</p> <p>cycles. This document details both failed assaults, presenting their respective negative</p> <p>results as a contribution to understanding the profound difficulty of the conjecture.</p> |
| title | From Topological Rupture to an Elegant Detour: A Definitive Logbook on Two Failed Assaults on the Collatz Conjecture |
| url | https://doi.org/10.5281/zenodo.17292166 |