Collatz Dynamics I: Skeleton Bound and the Elimination of Non-Trivial Cycles
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| Format: | Recurso digital |
| Langue: | anglais |
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2025
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| _version_ | 1866901143160881152 |
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| author | Moon, KyungUp |
| author_facet | Moon, KyungUp |
| contents | <p>This paper establishes the non-existence of non-trivial cycles in the Collatz map through a purely algebraic framework. The central device is the Skeleton Condition, a structural inequality that provides a strict upper bound on the correction term in the fundamental cycle equation. When juxtaposed with the necessary growth condition 2^(S(k)) > 3^k, the Skeleton Condition yields an unavoidable structural contradiction.</p> <p> </p> <p>In contrast to prior approaches—whether based on Diophantine approximation, asymptotic estimates, or large-scale computation—this argument is entirely algebraic and free of circular assumptions. The result closes the cycle subproblem of the Collatz conjecture with definitive rigor.</p> <p> </p> <p>This work constitutes the first installment of the three-paper Collatz Dynamics program. Together with the companion article Collatz Dynamics II: Drift–Compression Dynamics and Global Convergence, it delineates a unified structural pathway toward a complete deterministic resolution of the Collatz conjecture.</p> |
| format | Recurso digital |
| id | zenodo_https___doi_org_10_5281_zenodo_17266036 |
| institution | Zenodo |
| language | eng |
| publishDate | 2025 |
| publisher | Zenodo |
| record_format | zenodo |
| spellingShingle | Collatz Dynamics I: Skeleton Bound and the Elimination of Non-Trivial Cycles Moon, KyungUp Collatz conjecture 3x+1 problem cycles skeleton bound algebraic inequality dynamical systems discrete mathematics <p>This paper establishes the non-existence of non-trivial cycles in the Collatz map through a purely algebraic framework. The central device is the Skeleton Condition, a structural inequality that provides a strict upper bound on the correction term in the fundamental cycle equation. When juxtaposed with the necessary growth condition 2^(S(k)) > 3^k, the Skeleton Condition yields an unavoidable structural contradiction.</p> <p> </p> <p>In contrast to prior approaches—whether based on Diophantine approximation, asymptotic estimates, or large-scale computation—this argument is entirely algebraic and free of circular assumptions. The result closes the cycle subproblem of the Collatz conjecture with definitive rigor.</p> <p> </p> <p>This work constitutes the first installment of the three-paper Collatz Dynamics program. Together with the companion article Collatz Dynamics II: Drift–Compression Dynamics and Global Convergence, it delineates a unified structural pathway toward a complete deterministic resolution of the Collatz conjecture.</p> |
| title | Collatz Dynamics I: Skeleton Bound and the Elimination of Non-Trivial Cycles |
| topic | Collatz conjecture 3x+1 problem cycles skeleton bound algebraic inequality dynamical systems discrete mathematics |
| url | https://doi.org/10.5281/zenodo.17266036 |