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| Format: | Preprint |
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2026
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| Online Access: | https://arxiv.org/abs/2603.25753 |
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| _version_ | 1866911547222130688 |
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| author | Chang, Edward Y. |
| author_facet | Chang, Edward Y. |
| contents | We reduce the Collatz conjecture to a fixed-modulus, one-bit orbit-mixing problem. Working with the compressed odd-to-odd Collatz map, we prove exact low-depth decomposition formulas at depths K = 3, 4, 5, reducing block-discrepancy terms to explicit run statistics. We then prove a Map Balance Theorem: among the 2^(K-3), 1 burst residues modulo 2^K that initiate gaps, the counts mapping to gap starts congruent to 3 versus congruent to 7 (mod 8) differ by exactly 1 for every K >= 5. Thus all residual bias is orbit-level, not map-level. For the dominant n congruent to 1 (mod 8) class, the gap outcome depends on a single binary variable, bit 4 of the orbit value at burst-ending times, reducing the conjecture to whether every orbit visits two residue classes modulo 32 with sufficient balance along a sparse subsequence. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_25753 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | A Structural Reduction of the Collatz Conjecture to One-Bit Orbit Mixing Chang, Edward Y. Dynamical Systems math.DS I.2.7 We reduce the Collatz conjecture to a fixed-modulus, one-bit orbit-mixing problem. Working with the compressed odd-to-odd Collatz map, we prove exact low-depth decomposition formulas at depths K = 3, 4, 5, reducing block-discrepancy terms to explicit run statistics. We then prove a Map Balance Theorem: among the 2^(K-3), 1 burst residues modulo 2^K that initiate gaps, the counts mapping to gap starts congruent to 3 versus congruent to 7 (mod 8) differ by exactly 1 for every K >= 5. Thus all residual bias is orbit-level, not map-level. For the dominant n congruent to 1 (mod 8) class, the gap outcome depends on a single binary variable, bit 4 of the orbit value at burst-ending times, reducing the conjecture to whether every orbit visits two residue classes modulo 32 with sufficient balance along a sparse subsequence. |
| title | A Structural Reduction of the Collatz Conjecture to One-Bit Orbit Mixing |
| topic | Dynamical Systems math.DS I.2.7 |
| url | https://arxiv.org/abs/2603.25753 |