{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2026:EKVNX2M2O35MMB6BQENRJF5G7E","short_pith_number":"pith:EKVNX2M2","schema_version":"1.0","canonical_sha256":"22aadbe99a76fac607c1811b1497a6f91dddb6199a6e54db5ed8bfef952f6de7","source":{"kind":"arxiv","id":"2606.12014","version":1},"attestation_state":"computed","paper":{"title":"Vaught's Conjecture for Unions of Products of Rooted Trees","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.LO","authors_text":"Milo\\v{s} S. Kurili\\'c","submitted_at":"2026-06-10T12:40:25Z","abstract_excerpt":"Let ${\\mathcal C} ^{\\rm rt}$ be the class of rooted trees and $\\langle {\\mathcal C} ^{\\rm rt}\\rangle _{\\dot{\\cup }\\Pi}$ its minimal closure under isomorphism, finite direct products and finite disjoint unions. Posets from that closure are isomorphic to ${\\mathbb X}= \\dot{\\bigcup} _{i<n}\\prod _{j<m_i}{\\mathbb X}_i^j$, where ${\\mathbb X}_i^j$ are rooted trees. Defining ${\\mathcal T}=\\mathop{\\rm Th} ({\\mathbb X})$, ${\\mathcal T} _i ^j=\\mathop{\\rm Th}({\\mathbb X}_i^j)$, for $i<n$ and $j<m_i$, and $\\kappa = \\prod _{i<n}\\prod _{j<m_i}I({\\mathcal T} _i^j)$, we have\n  (a) Vaught's conjecture is true f"},"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":"2606.12014","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.LO","submitted_at":"2026-06-10T12:40:25Z","cross_cats_sorted":[],"title_canon_sha256":"7db769c54ffe238a9c713660ddb8abd7707ec2612da5dd1c11c2fe80588d01c4","abstract_canon_sha256":"470be8ac01e00adf764280389440ddd10235809f626a3a30cc4fe550b4d7d0b1"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-06-11T01:10:42.841475Z","signature_b64":"RkqVcKD4s5Ofyz0MQfIuTcm8blntSbME0Mz3S429n6xLToj4U0H37sBewQsD9oYqJylhD3rLqp6tepqa1JKoAQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"22aadbe99a76fac607c1811b1497a6f91dddb6199a6e54db5ed8bfef952f6de7","last_reissued_at":"2026-06-11T01:10:42.840599Z","signature_status":"signed_v1","first_computed_at":"2026-06-11T01:10:42.840599Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Vaught's Conjecture for Unions of Products of Rooted Trees","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.LO","authors_text":"Milo\\v{s} S. Kurili\\'c","submitted_at":"2026-06-10T12:40:25Z","abstract_excerpt":"Let ${\\mathcal C} ^{\\rm rt}$ be the class of rooted trees and $\\langle {\\mathcal C} ^{\\rm rt}\\rangle _{\\dot{\\cup }\\Pi}$ its minimal closure under isomorphism, finite direct products and finite disjoint unions. Posets from that closure are isomorphic to ${\\mathbb X}= \\dot{\\bigcup} _{i<n}\\prod _{j<m_i}{\\mathbb X}_i^j$, where ${\\mathbb X}_i^j$ are rooted trees. Defining ${\\mathcal T}=\\mathop{\\rm Th} ({\\mathbb X})$, ${\\mathcal T} _i ^j=\\mathop{\\rm Th}({\\mathbb X}_i^j)$, for $i<n$ and $j<m_i$, and $\\kappa = \\prod _{i<n}\\prod _{j<m_i}I({\\mathcal T} _i^j)$, we have\n  (a) Vaught's conjecture is true f"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2606.12014","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2606.12014/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":"2606.12014","created_at":"2026-06-11T01:10:42.840756+00:00"},{"alias_kind":"arxiv_version","alias_value":"2606.12014v1","created_at":"2026-06-11T01:10:42.840756+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2606.12014","created_at":"2026-06-11T01:10:42.840756+00:00"},{"alias_kind":"pith_short_12","alias_value":"EKVNX2M2O35M","created_at":"2026-06-11T01:10:42.840756+00:00"},{"alias_kind":"pith_short_16","alias_value":"EKVNX2M2O35MMB6B","created_at":"2026-06-11T01:10:42.840756+00:00"},{"alias_kind":"pith_short_8","alias_value":"EKVNX2M2","created_at":"2026-06-11T01:10:42.840756+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/EKVNX2M2O35MMB6BQENRJF5G7E","json":"https://pith.science/pith/EKVNX2M2O35MMB6BQENRJF5G7E.json","graph_json":"https://pith.science/api/pith-number/EKVNX2M2O35MMB6BQENRJF5G7E/graph.json","events_json":"https://pith.science/api/pith-number/EKVNX2M2O35MMB6BQENRJF5G7E/events.json","paper":"https://pith.science/paper/EKVNX2M2"},"agent_actions":{"view_html":"https://pith.science/pith/EKVNX2M2O35MMB6BQENRJF5G7E","download_json":"https://pith.science/pith/EKVNX2M2O35MMB6BQENRJF5G7E.json","view_paper":"https://pith.science/paper/EKVNX2M2","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2606.12014&json=true","fetch_graph":"https://pith.science/api/pith-number/EKVNX2M2O35MMB6BQENRJF5G7E/graph.json","fetch_events":"https://pith.science/api/pith-number/EKVNX2M2O35MMB6BQENRJF5G7E/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/EKVNX2M2O35MMB6BQENRJF5G7E/action/timestamp_anchor","attest_storage":"https://pith.science/pith/EKVNX2M2O35MMB6BQENRJF5G7E/action/storage_attestation","attest_author":"https://pith.science/pith/EKVNX2M2O35MMB6BQENRJF5G7E/action/author_attestation","sign_citation":"https://pith.science/pith/EKVNX2M2O35MMB6BQENRJF5G7E/action/citation_signature","submit_replication":"https://pith.science/pith/EKVNX2M2O35MMB6BQENRJF5G7E/action/replication_record"}},"created_at":"2026-06-11T01:10:42.840756+00:00","updated_at":"2026-06-11T01:10:42.840756+00:00"}