{"paper":{"title":"A Coordinate System for Collatz Dynamics","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Jennifer Williams","submitted_at":"2026-07-02T05:16:36Z","abstract_excerpt":"It is well-established that every odd positive integer $n$ can be written uniquely as $n = \\lambda \\cdot 2^a \\cdot 3^b - 1$ where $\\gcd(\\lambda, 6) = 1$ and $a \\geq 1$. Building from this 3-smooth factorization, we introduce a partition of the nonnegative integers into countably many infinite triangles where each row $k$ forms a Collatz chain of alternating parity. The partition admits a coordinate system as a skeleton $\\mathcal{L}_\\lambda$ using the pair $(a, b)$ for odd positive integers within a geometric structure where row $k$ corresponds to $k = a + b$. Each position $(a, b)$ maps to $(a"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2607.01718","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2607.01718/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}