{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2016:6SGT2RYPT5D7WYHPPZSKEZ6JDA","short_pith_number":"pith:6SGT2RYP","schema_version":"1.0","canonical_sha256":"f48d3d470f9f47fb60ef7e64a267c9181367eed025d7e171d9e00265c824899f","source":{"kind":"arxiv","id":"1608.08764","version":4},"attestation_state":"computed","paper":{"title":"Summand minimality and asymptotic convergence of generalized Zeckendorf decompositions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Carsten Peterson, Chi Huynh, Katherine Cordwell, Max Hlavacek, Steven J. Miller, Yen Nhi Truong Vu","submitted_at":"2016-08-31T08:23:38Z","abstract_excerpt":"Given a recurrence sequence $H$, with $H_n = c_1 H_{n-1} + \\dots + c_t H_{n-t}$ where $c_i \\in \\mathbb{N}_0$ for all $i$ and $c_1, c_t \\geq 1$, the generalized Zeckendorf decomposition (gzd) of $m \\in \\mathbb{N}_0$ is the unique representation of $m$ using $H$ composed of blocks lexicographically less than $\\sigma = (c_1, \\dots, c_t)$. We prove that the gzd of $m$ uses the fewest number of summands among all representations of $m$ using $H$, for all $m$, if and only if $\\sigma$ is weakly decreasing. We develop an algorithm for moving from any representation of $m$ to the gzd, the analysis of w"},"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":"1608.08764","kind":"arxiv","version":4},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2016-08-31T08:23:38Z","cross_cats_sorted":[],"title_canon_sha256":"b3657507c2cc2197b938debe095a2291a576a540088ddeb1840e9eca91c62955","abstract_canon_sha256":"1a916444f2ec3a6d2a4b66d5ac4d2246a510a246606bec638eeccf1ff3b0e689"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:03:25.320573Z","signature_b64":"UTsyOUq8l1Gh/5q06GKZwvRmCxrWgict+ZjteKXER1y09ALDqRgqjA72ZPw7NvdR7Ekqx6g0XKi47pcwS1Z3BA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"f48d3d470f9f47fb60ef7e64a267c9181367eed025d7e171d9e00265c824899f","last_reissued_at":"2026-05-18T00:03:25.320150Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:03:25.320150Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Summand minimality and asymptotic convergence of generalized Zeckendorf decompositions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Carsten Peterson, Chi Huynh, Katherine Cordwell, Max Hlavacek, Steven J. Miller, Yen Nhi Truong Vu","submitted_at":"2016-08-31T08:23:38Z","abstract_excerpt":"Given a recurrence sequence $H$, with $H_n = c_1 H_{n-1} + \\dots + c_t H_{n-t}$ where $c_i \\in \\mathbb{N}_0$ for all $i$ and $c_1, c_t \\geq 1$, the generalized Zeckendorf decomposition (gzd) of $m \\in \\mathbb{N}_0$ is the unique representation of $m$ using $H$ composed of blocks lexicographically less than $\\sigma = (c_1, \\dots, c_t)$. We prove that the gzd of $m$ uses the fewest number of summands among all representations of $m$ using $H$, for all $m$, if and only if $\\sigma$ is weakly decreasing. We develop an algorithm for moving from any representation of $m$ to the gzd, the analysis of w"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1608.08764","kind":"arxiv","version":4},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"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":"1608.08764","created_at":"2026-05-18T00:03:25.320218+00:00"},{"alias_kind":"arxiv_version","alias_value":"1608.08764v4","created_at":"2026-05-18T00:03:25.320218+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1608.08764","created_at":"2026-05-18T00:03:25.320218+00:00"},{"alias_kind":"pith_short_12","alias_value":"6SGT2RYPT5D7","created_at":"2026-05-18T12:30:04.600751+00:00"},{"alias_kind":"pith_short_16","alias_value":"6SGT2RYPT5D7WYHP","created_at":"2026-05-18T12:30:04.600751+00:00"},{"alias_kind":"pith_short_8","alias_value":"6SGT2RYP","created_at":"2026-05-18T12:30:04.600751+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/6SGT2RYPT5D7WYHPPZSKEZ6JDA","json":"https://pith.science/pith/6SGT2RYPT5D7WYHPPZSKEZ6JDA.json","graph_json":"https://pith.science/api/pith-number/6SGT2RYPT5D7WYHPPZSKEZ6JDA/graph.json","events_json":"https://pith.science/api/pith-number/6SGT2RYPT5D7WYHPPZSKEZ6JDA/events.json","paper":"https://pith.science/paper/6SGT2RYP"},"agent_actions":{"view_html":"https://pith.science/pith/6SGT2RYPT5D7WYHPPZSKEZ6JDA","download_json":"https://pith.science/pith/6SGT2RYPT5D7WYHPPZSKEZ6JDA.json","view_paper":"https://pith.science/paper/6SGT2RYP","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1608.08764&json=true","fetch_graph":"https://pith.science/api/pith-number/6SGT2RYPT5D7WYHPPZSKEZ6JDA/graph.json","fetch_events":"https://pith.science/api/pith-number/6SGT2RYPT5D7WYHPPZSKEZ6JDA/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/6SGT2RYPT5D7WYHPPZSKEZ6JDA/action/timestamp_anchor","attest_storage":"https://pith.science/pith/6SGT2RYPT5D7WYHPPZSKEZ6JDA/action/storage_attestation","attest_author":"https://pith.science/pith/6SGT2RYPT5D7WYHPPZSKEZ6JDA/action/author_attestation","sign_citation":"https://pith.science/pith/6SGT2RYPT5D7WYHPPZSKEZ6JDA/action/citation_signature","submit_replication":"https://pith.science/pith/6SGT2RYPT5D7WYHPPZSKEZ6JDA/action/replication_record"}},"created_at":"2026-05-18T00:03:25.320218+00:00","updated_at":"2026-05-18T00:03:25.320218+00:00"}