{"paper":{"title":"Logical Undefinability of the Generalized Collatz Transition Relation in B\\\"uchi Arithmetic","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["math.DS"],"primary_cat":"math.NT","authors_text":"Madhav Dhiman, Rohan Pandey","submitted_at":"2026-01-19T07:01:00Z","abstract_excerpt":"Let $q$ be an odd prime and let $d$ be an odd integer. We show that the arbitrary-step transition relation of the generalized Collatz map $T_{q,d}$ is not first-order definable in Base-2 B\\\"uchi Arithmetic ($BA_2$). We do this by demonstrating that if the transition relation were definable, the exponential set $P_q = \\{q^y : y \\in \\mathbb{N}\\}$ would also be definable in $BA_2$. Since $P_q$ is strictly non-semilinear, this yields a direct contradiction with the Cobham--Sem\\\"enov theorem. Consequently, we demonstrate that no finite automaton reading base-2 representations can recognize this tra"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2601.12772","kind":"arxiv","version":2},"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/2601.12772/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"}