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| Format: | Recurso digital |
| Language: | English |
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2026
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| Online Access: | https://doi.org/10.5281/zenodo.18447604 |
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| _version_ | 1866901625709264896 |
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| author | KIM, DONGHYUK |
| author_facet | KIM, DONGHYUK |
| contents | <p>This paper investigates the validity of the Collatz conjecture through the structural properties<br>of the inverse Collatz graph from a measure-theoretic perspective. We propose a generative matrix<br>based on the interaction between the 4x + 1 chain expansion and modulo 3 arithmetic, analyzing<br>the recursive expansiveness of the inverse tree. The key results are as follows: First, by establishing<br>the modular recurrence of the 4x + 1 chain and the principle of ergodic mixing, we demonstrate<br>that the primary tree T (1) achieves an asymptotic density of 1 on the set of natural numbers.<br>Second, we prove that any inverse tree T (x) initiated from an arbitrary seed x possesses a positive<br>measure due to geometric fractal expansion. Finally, we conclude that all natural numbers must<br>connect to 1 by showing that the existence of any disjoint orbit leads to a measure-theoretic<br>contradiction—specifically, that the sum of measures would exceed 1.</p> |
| format | Recurso digital |
| id | zenodo_https___doi_org_10_5281_zenodo_18447604 |
| institution | Zenodo |
| language | eng |
| publishDate | 2026 |
| publisher | Zenodo |
| record_format | zenodo |
| spellingShingle | On the Global Connectivity of the Inverse Collatz Graph: A Measure-Theoretic Approach via Modular Fractality KIM, DONGHYUK <p>This paper investigates the validity of the Collatz conjecture through the structural properties<br>of the inverse Collatz graph from a measure-theoretic perspective. We propose a generative matrix<br>based on the interaction between the 4x + 1 chain expansion and modulo 3 arithmetic, analyzing<br>the recursive expansiveness of the inverse tree. The key results are as follows: First, by establishing<br>the modular recurrence of the 4x + 1 chain and the principle of ergodic mixing, we demonstrate<br>that the primary tree T (1) achieves an asymptotic density of 1 on the set of natural numbers.<br>Second, we prove that any inverse tree T (x) initiated from an arbitrary seed x possesses a positive<br>measure due to geometric fractal expansion. Finally, we conclude that all natural numbers must<br>connect to 1 by showing that the existence of any disjoint orbit leads to a measure-theoretic<br>contradiction—specifically, that the sum of measures would exceed 1.</p> |
| title | On the Global Connectivity of the Inverse Collatz Graph: A Measure-Theoretic Approach via Modular Fractality |
| url | https://doi.org/10.5281/zenodo.18447604 |