{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2013:OC43PPPCAC4D46V2IBP4VOMB32","short_pith_number":"pith:OC43PPPC","schema_version":"1.0","canonical_sha256":"70b9b7bde200b83e7aba405fcab981de8f3b22f78e20e5c9f5055531882cb464","source":{"kind":"arxiv","id":"1309.5600","version":3},"attestation_state":"computed","paper":{"title":"A Generalization of Fibonacci Far-Difference Representations and Gaussian Behavior","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Archit Kulkarni, Philippe Demontigny, Steven J. Miller, Thao Do, Umang Varma","submitted_at":"2013-09-22T13:43:41Z","abstract_excerpt":"A natural generalization of base B expansions is Zeckendorf's Theorem: every integer can be uniquely written as a sum of non-consecutive Fibonacci numbers $\\{F_n\\}$, with $F_{n+1} = F_n + F_{n-1}$ and $F_1=1, F_2=2$. If instead we allow the coefficients of the Fibonacci numbers in the decomposition to be zero or $\\pm 1$, the resulting expression is known as the far-difference representation. Alpert proved that a far-difference representation exists and is unique under certain restraints that generalize non-consecutiveness, specifically that two adjacent summands of the same sign must be at lea"},"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":"1309.5600","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2013-09-22T13:43:41Z","cross_cats_sorted":[],"title_canon_sha256":"02cf5b352c8a8ba2ecb8b5c5fff1b0184309fd65c623c8f4298eeb6be7f0d970","abstract_canon_sha256":"d9401187f2abfeeb451c00e30ac1e208f55c98d7fb98b001b11ac464b6aa6ebe"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:52:12.423955Z","signature_b64":"V8cYwnuzfo7iSNk4F7FUWYhKy49OneMq9QlSeXa0kBym1BeiWVJPSMvnDR4kTxpck8/7XsoS/EdEH6PPnAnsCw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"70b9b7bde200b83e7aba405fcab981de8f3b22f78e20e5c9f5055531882cb464","last_reissued_at":"2026-05-18T02:52:12.423406Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:52:12.423406Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"A Generalization of Fibonacci Far-Difference Representations and Gaussian Behavior","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Archit Kulkarni, Philippe Demontigny, Steven J. Miller, Thao Do, Umang Varma","submitted_at":"2013-09-22T13:43:41Z","abstract_excerpt":"A natural generalization of base B expansions is Zeckendorf's Theorem: every integer can be uniquely written as a sum of non-consecutive Fibonacci numbers $\\{F_n\\}$, with $F_{n+1} = F_n + F_{n-1}$ and $F_1=1, F_2=2$. If instead we allow the coefficients of the Fibonacci numbers in the decomposition to be zero or $\\pm 1$, the resulting expression is known as the far-difference representation. Alpert proved that a far-difference representation exists and is unique under certain restraints that generalize non-consecutiveness, specifically that two adjacent summands of the same sign must be at lea"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1309.5600","kind":"arxiv","version":3},"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":"1309.5600","created_at":"2026-05-18T02:52:12.423484+00:00"},{"alias_kind":"arxiv_version","alias_value":"1309.5600v3","created_at":"2026-05-18T02:52:12.423484+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1309.5600","created_at":"2026-05-18T02:52:12.423484+00:00"},{"alias_kind":"pith_short_12","alias_value":"OC43PPPCAC4D","created_at":"2026-05-18T12:27:54.935989+00:00"},{"alias_kind":"pith_short_16","alias_value":"OC43PPPCAC4D46V2","created_at":"2026-05-18T12:27:54.935989+00:00"},{"alias_kind":"pith_short_8","alias_value":"OC43PPPC","created_at":"2026-05-18T12:27:54.935989+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/OC43PPPCAC4D46V2IBP4VOMB32","json":"https://pith.science/pith/OC43PPPCAC4D46V2IBP4VOMB32.json","graph_json":"https://pith.science/api/pith-number/OC43PPPCAC4D46V2IBP4VOMB32/graph.json","events_json":"https://pith.science/api/pith-number/OC43PPPCAC4D46V2IBP4VOMB32/events.json","paper":"https://pith.science/paper/OC43PPPC"},"agent_actions":{"view_html":"https://pith.science/pith/OC43PPPCAC4D46V2IBP4VOMB32","download_json":"https://pith.science/pith/OC43PPPCAC4D46V2IBP4VOMB32.json","view_paper":"https://pith.science/paper/OC43PPPC","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1309.5600&json=true","fetch_graph":"https://pith.science/api/pith-number/OC43PPPCAC4D46V2IBP4VOMB32/graph.json","fetch_events":"https://pith.science/api/pith-number/OC43PPPCAC4D46V2IBP4VOMB32/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/OC43PPPCAC4D46V2IBP4VOMB32/action/timestamp_anchor","attest_storage":"https://pith.science/pith/OC43PPPCAC4D46V2IBP4VOMB32/action/storage_attestation","attest_author":"https://pith.science/pith/OC43PPPCAC4D46V2IBP4VOMB32/action/author_attestation","sign_citation":"https://pith.science/pith/OC43PPPCAC4D46V2IBP4VOMB32/action/citation_signature","submit_replication":"https://pith.science/pith/OC43PPPCAC4D46V2IBP4VOMB32/action/replication_record"}},"created_at":"2026-05-18T02:52:12.423484+00:00","updated_at":"2026-05-18T02:52:12.423484+00:00"}