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2025
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| Online Access: | https://doi.org/10.5281/zenodo.17297341 |
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| _version_ | 1866901033687449600 |
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| author | Peter, Romain |
| author_facet | Peter, Romain |
| contents | <p>The Collatz Conjecture has remained famously resistant to proof, largely due to the</p> <p>pseudo-random behavior of its trajectories. This paper documents the results of a system-</p> <p>atic, multi-stage investigation into the structural properties that underlie this difficulty. We</p> <p>first present two rigorous ”no-go” theorems, demonstrating the failure of two distinct, so-</p> <p>phisticated lines of attack. The first, a quantitative approach based on p-adic Diophantine</p> <p>approximation, is shown to fail due to a fundamental ”Logarithmic Barrier.” The second, a</p> <p>structural approach based on uniform local rules, is shown to fail due to the ”Local Freedom”</p> <p>of Collatz orbits, which allows for the realization of any finite parity prefix.</p> <p>These failures, rather than being mere dead ends, provide crucial insights into the nature</p> <p>of the problem. They strongly suggest that any viable proof must be global, non-uniform,</p> <p>and scale-dependent. Based on these findings, we propose a new and comprehensive research</p> <p>paradigm based on an analogy with the Renormalization Group (RG) in theoretical physics.</p> <p>We conjecture that the flow on the space of the system’s underlying affine transformations</p> <p>converges to a unique, universal, and contractive fixed point, and that an ”arithmetic sieve”</p> <p>prunes all pathological trajectories. This paper formalizes this research program, reducing</p> <p>the full Collatz Conjecture to two precise, open lemmas concerning the RG flow, and thus</p> <p>offering a new, promising path forward.</p> |
| format | Recurso digital |
| id | zenodo_https___doi_org_10_5281_zenodo_17297341 |
| institution | Zenodo |
| language | |
| publishDate | 2025 |
| publisher | Zenodo |
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| spellingShingle | From Structural Impasses to a Renormalization Group Program for the Collatz Conjecture Peter, Romain <p>The Collatz Conjecture has remained famously resistant to proof, largely due to the</p> <p>pseudo-random behavior of its trajectories. This paper documents the results of a system-</p> <p>atic, multi-stage investigation into the structural properties that underlie this difficulty. We</p> <p>first present two rigorous ”no-go” theorems, demonstrating the failure of two distinct, so-</p> <p>phisticated lines of attack. The first, a quantitative approach based on p-adic Diophantine</p> <p>approximation, is shown to fail due to a fundamental ”Logarithmic Barrier.” The second, a</p> <p>structural approach based on uniform local rules, is shown to fail due to the ”Local Freedom”</p> <p>of Collatz orbits, which allows for the realization of any finite parity prefix.</p> <p>These failures, rather than being mere dead ends, provide crucial insights into the nature</p> <p>of the problem. They strongly suggest that any viable proof must be global, non-uniform,</p> <p>and scale-dependent. Based on these findings, we propose a new and comprehensive research</p> <p>paradigm based on an analogy with the Renormalization Group (RG) in theoretical physics.</p> <p>We conjecture that the flow on the space of the system’s underlying affine transformations</p> <p>converges to a unique, universal, and contractive fixed point, and that an ”arithmetic sieve”</p> <p>prunes all pathological trajectories. This paper formalizes this research program, reducing</p> <p>the full Collatz Conjecture to two precise, open lemmas concerning the RG flow, and thus</p> <p>offering a new, promising path forward.</p> |
| title | From Structural Impasses to a Renormalization Group Program for the Collatz Conjecture |
| url | https://doi.org/10.5281/zenodo.17297341 |