Proof of the Collatz Conjecture through the Inverse Expansion Graph
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| Formato: | Recurso digital |
| Lenguaje: | inglés |
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2025
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| _version_ | 1866902280127643648 |
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| author | Olgac, Enis |
| author_facet | Olgac, Enis |
| contents | <p>I published different aspects of the problem earlier, analyzed from different perspectives. This article consolidates those approaches into a single structural framework, yielding a complete proof of the Collatz conjecture. The proof is based on the inverse Collatz map. Instead of analyzing forward trajectories under the map C(n), we study the inverse expansion P, which has the same edges as the standard function graph G but with reversed interpretation. The analysis establishes four structural pillars: unique parenthood—every node in P has exactly one parent. Unicyclicity: the only directed cycle is the root cycle {1,2,4}, identified via the unique back edge (1,4). Root component: This cycle is inward closed and serves as the unique root component of P. Completeness: By closure of components and uniqueness of the back edge, no other component can exist; thus, P spans all of N⁺. Together these results prove that P is a unicyclic digraph whose root component contracts to a rooted tree covering all positive integers. Reinterpreted from end to start, all paths in P correspond exactly to orbits in G. Thus every Collatz orbit reaches 1 and then enters the cycle {1,2,4}.</p> |
| format | Recurso digital |
| id | zenodo_https___doi_org_10_5281_zenodo_17060799 |
| institution | Zenodo |
| language | eng |
| publishDate | 2025 |
| publisher | Zenodo |
| record_format | zenodo |
| spellingShingle | Proof of the Collatz Conjecture through the Inverse Expansion Graph Olgac, Enis Collatz conjecture Inverse Collatz map Unique parenthood Unicyclic digraph Rooted tree Completeness Inverse expansion Infinite ray <p>I published different aspects of the problem earlier, analyzed from different perspectives. This article consolidates those approaches into a single structural framework, yielding a complete proof of the Collatz conjecture. The proof is based on the inverse Collatz map. Instead of analyzing forward trajectories under the map C(n), we study the inverse expansion P, which has the same edges as the standard function graph G but with reversed interpretation. The analysis establishes four structural pillars: unique parenthood—every node in P has exactly one parent. Unicyclicity: the only directed cycle is the root cycle {1,2,4}, identified via the unique back edge (1,4). Root component: This cycle is inward closed and serves as the unique root component of P. Completeness: By closure of components and uniqueness of the back edge, no other component can exist; thus, P spans all of N⁺. Together these results prove that P is a unicyclic digraph whose root component contracts to a rooted tree covering all positive integers. Reinterpreted from end to start, all paths in P correspond exactly to orbits in G. Thus every Collatz orbit reaches 1 and then enters the cycle {1,2,4}.</p> |
| title | Proof of the Collatz Conjecture through the Inverse Expansion Graph |
| topic | Collatz conjecture Inverse Collatz map Unique parenthood Unicyclic digraph Rooted tree Completeness Inverse expansion Infinite ray |
| url | https://doi.org/10.5281/zenodo.17060799 |