{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2016:R4JU3DYNHMQDBV3Q5BS7UT7SHB","short_pith_number":"pith:R4JU3DYN","schema_version":"1.0","canonical_sha256":"8f134d8f0d3b2030d770e865fa4ff23862b5acf0a53e39400d9a99b4beaa7f4c","source":{"kind":"arxiv","id":"1607.01530","version":1},"attestation_state":"computed","paper":{"title":"Markoff Surfaces and Strong Approximation: 1","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.GR"],"primary_cat":"math.NT","authors_text":"Alexander Gamburd, Jean Bourgain, Peter Sarnak","submitted_at":"2016-07-06T09:20:15Z","abstract_excerpt":"We prove results pertaining to strong approximation for Markoff triples in the case of prime moduli."},"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":"1607.01530","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2016-07-06T09:20:15Z","cross_cats_sorted":["math.GR"],"title_canon_sha256":"6c6dc2bbf0f1212a59bd9f1e1d255cc8a6105efd0909b7e142246837e5ed6696","abstract_canon_sha256":"292d1152cc13f76b1108a3a2ba7402e8f42497a3b736db39a815f80904d9354a"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:11:26.212950Z","signature_b64":"7AfphzxulytwLV+H91j3IlVGbHCAUaouKr3Oe5OiCq/AuwOdyxkk2lUnVaNAzRQX18xKeNF7wSVVCtym/e3OAw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"8f134d8f0d3b2030d770e865fa4ff23862b5acf0a53e39400d9a99b4beaa7f4c","last_reissued_at":"2026-05-18T01:11:26.212344Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:11:26.212344Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Markoff Surfaces and Strong Approximation: 1","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.GR"],"primary_cat":"math.NT","authors_text":"Alexander Gamburd, Jean Bourgain, Peter Sarnak","submitted_at":"2016-07-06T09:20:15Z","abstract_excerpt":"We prove results pertaining to strong approximation for Markoff triples in the case of prime moduli."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1607.01530","kind":"arxiv","version":1},"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":"1607.01530","created_at":"2026-05-18T01:11:26.212466+00:00"},{"alias_kind":"arxiv_version","alias_value":"1607.01530v1","created_at":"2026-05-18T01:11:26.212466+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1607.01530","created_at":"2026-05-18T01:11:26.212466+00:00"},{"alias_kind":"pith_short_12","alias_value":"R4JU3DYNHMQD","created_at":"2026-05-18T12:30:41.710351+00:00"},{"alias_kind":"pith_short_16","alias_value":"R4JU3DYNHMQDBV3Q","created_at":"2026-05-18T12:30:41.710351+00:00"},{"alias_kind":"pith_short_8","alias_value":"R4JU3DYN","created_at":"2026-05-18T12:30:41.710351+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":1,"internal_anchor_count":1,"sample":[{"citing_arxiv_id":"2603.04096","citing_title":"Strong Approximation for the Character Variety of the Four-Times Punctured Sphere","ref_index":7,"is_internal_anchor":true}]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/R4JU3DYNHMQDBV3Q5BS7UT7SHB","json":"https://pith.science/pith/R4JU3DYNHMQDBV3Q5BS7UT7SHB.json","graph_json":"https://pith.science/api/pith-number/R4JU3DYNHMQDBV3Q5BS7UT7SHB/graph.json","events_json":"https://pith.science/api/pith-number/R4JU3DYNHMQDBV3Q5BS7UT7SHB/events.json","paper":"https://pith.science/paper/R4JU3DYN"},"agent_actions":{"view_html":"https://pith.science/pith/R4JU3DYNHMQDBV3Q5BS7UT7SHB","download_json":"https://pith.science/pith/R4JU3DYNHMQDBV3Q5BS7UT7SHB.json","view_paper":"https://pith.science/paper/R4JU3DYN","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1607.01530&json=true","fetch_graph":"https://pith.science/api/pith-number/R4JU3DYNHMQDBV3Q5BS7UT7SHB/graph.json","fetch_events":"https://pith.science/api/pith-number/R4JU3DYNHMQDBV3Q5BS7UT7SHB/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/R4JU3DYNHMQDBV3Q5BS7UT7SHB/action/timestamp_anchor","attest_storage":"https://pith.science/pith/R4JU3DYNHMQDBV3Q5BS7UT7SHB/action/storage_attestation","attest_author":"https://pith.science/pith/R4JU3DYNHMQDBV3Q5BS7UT7SHB/action/author_attestation","sign_citation":"https://pith.science/pith/R4JU3DYNHMQDBV3Q5BS7UT7SHB/action/citation_signature","submit_replication":"https://pith.science/pith/R4JU3DYNHMQDBV3Q5BS7UT7SHB/action/replication_record"}},"created_at":"2026-05-18T01:11:26.212466+00:00","updated_at":"2026-05-18T01:11:26.212466+00:00"}