{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2026:7R2SMBV5UQQT37LABYA6ALFJJH","short_pith_number":"pith:7R2SMBV5","schema_version":"1.0","canonical_sha256":"fc752606bda4213dfd600e01e02ca949faf7dc9dd7e1339b0e761b14d5cbb082","source":{"kind":"arxiv","id":"2602.06066","version":2},"attestation_state":"computed","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 "},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"2602.06066","kind":"arxiv","version":2},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.GM","submitted_at":"2026-02-01T05:12:28Z","cross_cats_sorted":[],"title_canon_sha256":"f96cd84beb77fec27b1b8001fab7f30e4cfb3732806070f607798128b4c5083a","abstract_canon_sha256":"d31fc2fc9990f3868925fd3c0fa844344d371e3ebee5d2e6af29cb274b7be83f"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-06-29T01:14:28.609295Z","signature_b64":"MrdimWCMrWqEWRKKk4qTkxYaNCzS4KVvpskhzYK2wBT+8uYzf2vEukWjZnURPEpVXaFme7e2xq8MEwR8gjVuAA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"fc752606bda4213dfd600e01e02ca949faf7dc9dd7e1339b0e761b14d5cbb082","last_reissued_at":"2026-06-29T01:14:28.608886Z","signature_status":"signed_v1","first_computed_at":"2026-06-29T01:14:28.608886Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"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"},"aliases":[{"alias_kind":"arxiv","alias_value":"2602.06066","created_at":"2026-06-29T01:14:28.608940+00:00"},{"alias_kind":"arxiv_version","alias_value":"2602.06066v2","created_at":"2026-06-29T01:14:28.608940+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2602.06066","created_at":"2026-06-29T01:14:28.608940+00:00"},{"alias_kind":"pith_short_12","alias_value":"7R2SMBV5UQQT","created_at":"2026-06-29T01:14:28.608940+00:00"},{"alias_kind":"pith_short_16","alias_value":"7R2SMBV5UQQT37LA","created_at":"2026-06-29T01:14:28.608940+00:00"},{"alias_kind":"pith_short_8","alias_value":"7R2SMBV5","created_at":"2026-06-29T01:14:28.608940+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/7R2SMBV5UQQT37LABYA6ALFJJH","json":"https://pith.science/pith/7R2SMBV5UQQT37LABYA6ALFJJH.json","graph_json":"https://pith.science/api/pith-number/7R2SMBV5UQQT37LABYA6ALFJJH/graph.json","events_json":"https://pith.science/api/pith-number/7R2SMBV5UQQT37LABYA6ALFJJH/events.json","paper":"https://pith.science/paper/7R2SMBV5"},"agent_actions":{"view_html":"https://pith.science/pith/7R2SMBV5UQQT37LABYA6ALFJJH","download_json":"https://pith.science/pith/7R2SMBV5UQQT37LABYA6ALFJJH.json","view_paper":"https://pith.science/paper/7R2SMBV5","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2602.06066&json=true","fetch_graph":"https://pith.science/api/pith-number/7R2SMBV5UQQT37LABYA6ALFJJH/graph.json","fetch_events":"https://pith.science/api/pith-number/7R2SMBV5UQQT37LABYA6ALFJJH/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/7R2SMBV5UQQT37LABYA6ALFJJH/action/timestamp_anchor","attest_storage":"https://pith.science/pith/7R2SMBV5UQQT37LABYA6ALFJJH/action/storage_attestation","attest_author":"https://pith.science/pith/7R2SMBV5UQQT37LABYA6ALFJJH/action/author_attestation","sign_citation":"https://pith.science/pith/7R2SMBV5UQQT37LABYA6ALFJJH/action/citation_signature","submit_replication":"https://pith.science/pith/7R2SMBV5UQQT37LABYA6ALFJJH/action/replication_record"}},"created_at":"2026-06-29T01:14:28.608940+00:00","updated_at":"2026-06-29T01:14:28.608940+00:00"}