Collatz Normal Form: Time as Degree-of-Freedom Elimination and the Trace-Compressed Engine
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| Format: | Recurso digital |
| Sprache: | Englisch |
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2026
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| _version_ | 1866901810572165120 |
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| author | Moon, KyungUp |
| author_facet | Moon, KyungUp |
| contents | <p>This paper introduces a trace-compressed normal form for the Collatz (3x+1) dynamics.</p> <p> </p> <p>The main contribution is structural rather than declarative:</p> <p>the Collatz iteration is rewritten via an exact change of variables that separates</p> <p>(i) multiplicative drift induced by odd steps and</p> <p>(ii) dyadic compression induced by 2-adic valuations.</p> <p> </p> <p>In the resulting normal-form coordinate, the evolution admits an exact affine increment identity driven by valuation data and a multiplicative correction cocycle.</p> <p>This representation does not modify the Collatz map, introduce auxiliary dynamics, or rely on probabilistic assumptions.</p> <p>All constructions are observational and derived directly from the original iteration.</p> <p> </p> <p>Using this framework, the paper derives an exact cocycle equation that any exact periodic numeric orbit must satisfy.</p> <p>This equation provides a necessary structural constraint linking valuation sums, odd-step counts, and an in-orbit correction product.</p> <p>The result clarifies precisely which compatibility conditions would be required for a hypothetical nontrivial cycle.</p> <p> </p> <p>Importantly, no convergence or termination claim is made.</p> <p>The paper does not assert that the cocycle constraint is unattainable, nor does it claim to resolve the Collatz conjecture.</p> <p>Rather, it isolates a compact normal-form structure and an exact constraint that any periodic scenario must obey.</p> <p> </p> <p>The trace-compressed normal form introduced here is intended as a reproducible coordinate system for analyzing long Collatz trajectories on equal footing, and as a preparatory framework for further work on cycle obstruction or non-attainability questions.</p> <p> </p> <p>A complete deterministic implementation for generating the normal-form trajectories and figures is provided in the appendix.<br><br></p> <p> </p> <p>Notes</p> <p>This upload establishes a permanent, citable record of a structural normalization of the Collatz dynamics.</p> <p>It is intended as a foundational reference rather than a proof of the conjecture.</p> <p> </p> <p>This v1.0.1 version corrects and clarifies the roles of block drift and cocycle terms,<br>and refines the cycle constraint as a necessary condition only.<br>The core normal-form construction and exact identities are unchanged.</p> |
| format | Recurso digital |
| id | zenodo_https___doi_org_10_5281_zenodo_18233316 |
| institution | Zenodo |
| language | eng |
| publishDate | 2026 |
| publisher | Zenodo |
| record_format | zenodo |
| spellingShingle | Collatz Normal Form: Time as Degree-of-Freedom Elimination and the Trace-Compressed Engine Moon, KyungUp Collatz conjecture 3x+1 problem normal form cocycle constraint valuation dynamics parity trace exact change of variables arithmetic dynamics <p>This paper introduces a trace-compressed normal form for the Collatz (3x+1) dynamics.</p> <p> </p> <p>The main contribution is structural rather than declarative:</p> <p>the Collatz iteration is rewritten via an exact change of variables that separates</p> <p>(i) multiplicative drift induced by odd steps and</p> <p>(ii) dyadic compression induced by 2-adic valuations.</p> <p> </p> <p>In the resulting normal-form coordinate, the evolution admits an exact affine increment identity driven by valuation data and a multiplicative correction cocycle.</p> <p>This representation does not modify the Collatz map, introduce auxiliary dynamics, or rely on probabilistic assumptions.</p> <p>All constructions are observational and derived directly from the original iteration.</p> <p> </p> <p>Using this framework, the paper derives an exact cocycle equation that any exact periodic numeric orbit must satisfy.</p> <p>This equation provides a necessary structural constraint linking valuation sums, odd-step counts, and an in-orbit correction product.</p> <p>The result clarifies precisely which compatibility conditions would be required for a hypothetical nontrivial cycle.</p> <p> </p> <p>Importantly, no convergence or termination claim is made.</p> <p>The paper does not assert that the cocycle constraint is unattainable, nor does it claim to resolve the Collatz conjecture.</p> <p>Rather, it isolates a compact normal-form structure and an exact constraint that any periodic scenario must obey.</p> <p> </p> <p>The trace-compressed normal form introduced here is intended as a reproducible coordinate system for analyzing long Collatz trajectories on equal footing, and as a preparatory framework for further work on cycle obstruction or non-attainability questions.</p> <p> </p> <p>A complete deterministic implementation for generating the normal-form trajectories and figures is provided in the appendix.<br><br></p> <p> </p> <p>Notes</p> <p>This upload establishes a permanent, citable record of a structural normalization of the Collatz dynamics.</p> <p>It is intended as a foundational reference rather than a proof of the conjecture.</p> <p> </p> <p>This v1.0.1 version corrects and clarifies the roles of block drift and cocycle terms,<br>and refines the cycle constraint as a necessary condition only.<br>The core normal-form construction and exact identities are unchanged.</p> |
| title | Collatz Normal Form: Time as Degree-of-Freedom Elimination and the Trace-Compressed Engine |
| topic | Collatz conjecture 3x+1 problem normal form cocycle constraint valuation dynamics parity trace exact change of variables arithmetic dynamics |
| url | https://doi.org/10.5281/zenodo.18233316 |