FROM ALMOST ALL TO FINITE EXCEPTIONAL SET: A SPECTRAL COMPLETION OF TAOS COLLATZ RESULT
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2025
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| _version_ | 1866901755600568320 |
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| author | Eden, Trenton |
| author_facet | Eden, Trenton |
| contents | <p>We have shown that the adjoint Collatz operator on L2(Nn 2) is Hilbert Schmidt, hence compact. This spectral fact explains Taos almost all result: the innite-dimensional decaying subspace H<1 corresponds to typical orbits, while any exceptions must lie in the finite-dimensional subspace H≥ 1-. The Collatz conjecture thus reduces from an innite search to a finite spectral computation: verify that all eigenvalues with Lamba ≥ 1 have eigen functions supported on the known cycle {1,2,4,} This does not prove the conjecture outright, but it completes Taos result by identifying exactly where exceptions could hide and proving that hiding place is finite-dimensional.</p> |
| format | Recurso digital |
| id | zenodo_https___doi_org_10_5281_zenodo_17986838 |
| institution | Zenodo |
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| publishDate | 2025 |
| publisher | Zenodo |
| record_format | zenodo |
| spellingShingle | FROM ALMOST ALL TO FINITE EXCEPTIONAL SET: A SPECTRAL COMPLETION OF TAOS COLLATZ RESULT Eden, Trenton <p>We have shown that the adjoint Collatz operator on L2(Nn 2) is Hilbert Schmidt, hence compact. This spectral fact explains Taos almost all result: the innite-dimensional decaying subspace H<1 corresponds to typical orbits, while any exceptions must lie in the finite-dimensional subspace H≥ 1-. The Collatz conjecture thus reduces from an innite search to a finite spectral computation: verify that all eigenvalues with Lamba ≥ 1 have eigen functions supported on the known cycle {1,2,4,} This does not prove the conjecture outright, but it completes Taos result by identifying exactly where exceptions could hide and proving that hiding place is finite-dimensional.</p> |
| title | FROM ALMOST ALL TO FINITE EXCEPTIONAL SET: A SPECTRAL COMPLETION OF TAOS COLLATZ RESULT |
| url | https://doi.org/10.5281/zenodo.17986838 |