{"paper":{"title":"Non-Definability of Reachability in B\\\"uchi Arithmetic for a Family of Generalized Collatz Maps","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.GM","authors_text":"Madhav Dhiman, Rohan Pandey","submitted_at":"2026-02-01T05:12:28Z","abstract_excerpt":"Let $q \\ge 3$ and $d \\ge 1$ be odd integers with $q+d$ a power of $2$. We study the generalized Collatz map $T_{q,d}$, a one-dimensional piecewise-affine map on the positive integers, and its unparameterized reachability relation $R(x,z)$, which holds when $z$ is an iterate of $x$ under $T_{q,d}$. We prove that for every such pair $(q,d)$ the relation $R$ is not first-order definable in B\\\"uchi arithmetic $\\langle \\mathbb{N}, +, V_q \\rangle$. Equivalently, no finite automaton recognizes the base-$q$ encoding of $R$. Assuming definability of $R$, we construct a first-order formula that defines "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2602.06066","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/2602.06066/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"}