{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2026:XH2QBGPXXXNAS7JSUVHVGKNWD5","short_pith_number":"pith:XH2QBGPX","schema_version":"1.0","canonical_sha256":"b9f50099f7bdda097d32a54f5329b61f7c078ec36ad2a85e71aeb569e5d0f1e5","source":{"kind":"arxiv","id":"2605.13608","version":1},"attestation_state":"computed","paper":{"title":"Universal homogeneous two-sorted ultrametric spaces","license":"http://creativecommons.org/licenses/by/4.0/","headline":"Treating ultrametric spaces as two-sorted structures with ordered distances yields a countable homogeneous universal space under distance-carrying embeddings.","cross_cats":[],"primary_cat":"math.LO","authors_text":"Adam Barto\\v{s}, Aleksandra Kwiatkowska, Maciej Malicki, Wies{\\l}aw Kubi\\'s","submitted_at":"2026-05-13T14:41:48Z","abstract_excerpt":"We view ultrametric spaces as two-sorted structures consisting of a set of points and of a linearly ordered set of distances. We call the appropriate notion of embeddings distance-carrying (dc for short). Those are obtained by combining isometries and linear order embeddings. We show that the class of all finite two-sorted ultrametric spaces with dc-embeddings is Fra\\\"iss\\'e, and that the limit is the countable rational Urysohn ultrametric space $\\mathbb{U}$. The space $\\mathbb{U}$ is dc-universal for all countable ultrametric spaces, and its Cauchy completion $\\overline{\\mathbb{U}}$ is dc-uni"},"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":true,"formal_links_present":true},"canonical_record":{"source":{"id":"2605.13608","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.LO","submitted_at":"2026-05-13T14:41:48Z","cross_cats_sorted":[],"title_canon_sha256":"e5389a1617b3fc928d4f66713ea083b154ccc1806c107d30ef5897bfb917ac36","abstract_canon_sha256":"2a7c8d7c931a8a5570fcf80b9dfcfe74eebe5ea6abf7288f3aa7883943c4c036"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:44:18.089907Z","signature_b64":"kt0rpV2YgRk3c1jAFobeuZzbNYnHxKnZ8GBuaPmE57QI0gOlomcEM/3oPaT7dEXDETq4LIKxpBJsYqxmi4pFDQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"b9f50099f7bdda097d32a54f5329b61f7c078ec36ad2a85e71aeb569e5d0f1e5","last_reissued_at":"2026-05-18T02:44:18.089489Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:44:18.089489Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Universal homogeneous two-sorted ultrametric spaces","license":"http://creativecommons.org/licenses/by/4.0/","headline":"Treating ultrametric spaces as two-sorted structures with ordered distances yields a countable homogeneous universal space under distance-carrying embeddings.","cross_cats":[],"primary_cat":"math.LO","authors_text":"Adam Barto\\v{s}, Aleksandra Kwiatkowska, Maciej Malicki, Wies{\\l}aw Kubi\\'s","submitted_at":"2026-05-13T14:41:48Z","abstract_excerpt":"We view ultrametric spaces as two-sorted structures consisting of a set of points and of a linearly ordered set of distances. We call the appropriate notion of embeddings distance-carrying (dc for short). Those are obtained by combining isometries and linear order embeddings. We show that the class of all finite two-sorted ultrametric spaces with dc-embeddings is Fra\\\"iss\\'e, and that the limit is the countable rational Urysohn ultrametric space $\\mathbb{U}$. The space $\\mathbb{U}$ is dc-universal for all countable ultrametric spaces, and its Cauchy completion $\\overline{\\mathbb{U}}$ is dc-uni"},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"The class of all finite two-sorted ultrametric spaces with dc-embeddings is Fraïssé, and the limit is the countable rational Urysohn ultrametric space U. The space U is dc-universal for all countable ultrametric spaces, and its Cauchy completion is dc-universal for all separable ultrametric spaces.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"That the class of finite two-sorted ultrametric spaces equipped with distance-carrying embeddings satisfies the hereditary, joint embedding, and amalgamation properties required for the Fraïssé theorem to apply.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"The countable rational Urysohn ultrametric space U is the Fraïssé limit of finite two-sorted ultrametric spaces under distance-carrying embeddings and is dc-universal for countable ultrametric spaces, with its completion universal for separable ones.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"Treating ultrametric spaces as two-sorted structures with ordered distances yields a countable homogeneous universal space under distance-carrying embeddings.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"eecf9f950813d5d6c7b95772be719592f0f4d09f8fc4ec254288e16c7574853e"},"source":{"id":"2605.13608","kind":"arxiv","version":1},"verdict":{"id":"c1734e2e-a1f6-4927-80a0-dbc179ce4924","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-14T18:51:44.248421Z","strongest_claim":"The class of all finite two-sorted ultrametric spaces with dc-embeddings is Fraïssé, and the limit is the countable rational Urysohn ultrametric space U. The space U is dc-universal for all countable ultrametric spaces, and its Cauchy completion is dc-universal for all separable ultrametric spaces.","one_line_summary":"The countable rational Urysohn ultrametric space U is the Fraïssé limit of finite two-sorted ultrametric spaces under distance-carrying embeddings and is dc-universal for countable ultrametric spaces, with its completion universal for separable ones.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"That the class of finite two-sorted ultrametric spaces equipped with distance-carrying embeddings satisfies the hereditary, joint embedding, and amalgamation properties required for the Fraïssé theorem to apply.","pith_extraction_headline":"Treating ultrametric spaces as two-sorted structures with ordered distances yields a countable homogeneous universal space under distance-carrying embeddings."},"references":{"count":33,"sample":[{"doi":"","year":2026,"title":"A. Bartoš, W. Kubiś , Hereditarily indecomposable continua as generic mathematical structures , Selecta Math. (N.S.) 32 (2026), no. 1, Paper No. 14","work_id":"6b153337-e890-493c-9cb4-65566f827110","ref_index":1,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2026,"title":"A. Bartoš, W. Kubiś, A. Kwiatkowska, M. Malicki , Generic dc-auto\\-morphisms of two-sorted ultrametric spaces , preprint, 2026","work_id":"4f380068-be15-4e42-9bd4-9a8f03b7da38","ref_index":2,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":null,"title":"R. Camerlo, A. Marcone, L. Motto Ros , Isometry groups of Polish ultrametric space , arXiv:2508.08480","work_id":"96e65b86-ae09-4cd5-9c55-ebba89541c77","ref_index":3,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":null,"title":"On homogeneous ultrametric spaces","work_id":"1d79085c-400e-4f7c-8127-0d61ec1c4ea6","ref_index":4,"cited_arxiv_id":"1509.04346","is_internal_anchor":true},{"doi":"","year":1984,"title":"F. Delon , Espaces ultram\\'etriques , J. Symbolic Logic 49 (1984), no. 2, 405--424","work_id":"2420d06a-6f5b-4e2d-8363-8d0c71c3473f","ref_index":5,"cited_arxiv_id":"","is_internal_anchor":false}],"resolved_work":33,"snapshot_sha256":"3a22d8af1fa0024415d2dc0b17bed478776ea37f6e2d7723611b08b16dbb2fa2","internal_anchors":1},"formal_canon":{"evidence_count":2,"snapshot_sha256":"2ab77da2ea35f12066d59495abff4d7f7c7b247eededd65f3c473d87fdcf5c7b"},"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":"2605.13608","created_at":"2026-05-18T02:44:18.089557+00:00"},{"alias_kind":"arxiv_version","alias_value":"2605.13608v1","created_at":"2026-05-18T02:44:18.089557+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2605.13608","created_at":"2026-05-18T02:44:18.089557+00:00"},{"alias_kind":"pith_short_12","alias_value":"XH2QBGPXXXNA","created_at":"2026-05-18T12:33:37.589309+00:00"},{"alias_kind":"pith_short_16","alias_value":"XH2QBGPXXXNAS7JS","created_at":"2026-05-18T12:33:37.589309+00:00"},{"alias_kind":"pith_short_8","alias_value":"XH2QBGPX","created_at":"2026-05-18T12:33:37.589309+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":2,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/XH2QBGPXXXNAS7JSUVHVGKNWD5","json":"https://pith.science/pith/XH2QBGPXXXNAS7JSUVHVGKNWD5.json","graph_json":"https://pith.science/api/pith-number/XH2QBGPXXXNAS7JSUVHVGKNWD5/graph.json","events_json":"https://pith.science/api/pith-number/XH2QBGPXXXNAS7JSUVHVGKNWD5/events.json","paper":"https://pith.science/paper/XH2QBGPX"},"agent_actions":{"view_html":"https://pith.science/pith/XH2QBGPXXXNAS7JSUVHVGKNWD5","download_json":"https://pith.science/pith/XH2QBGPXXXNAS7JSUVHVGKNWD5.json","view_paper":"https://pith.science/paper/XH2QBGPX","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2605.13608&json=true","fetch_graph":"https://pith.science/api/pith-number/XH2QBGPXXXNAS7JSUVHVGKNWD5/graph.json","fetch_events":"https://pith.science/api/pith-number/XH2QBGPXXXNAS7JSUVHVGKNWD5/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/XH2QBGPXXXNAS7JSUVHVGKNWD5/action/timestamp_anchor","attest_storage":"https://pith.science/pith/XH2QBGPXXXNAS7JSUVHVGKNWD5/action/storage_attestation","attest_author":"https://pith.science/pith/XH2QBGPXXXNAS7JSUVHVGKNWD5/action/author_attestation","sign_citation":"https://pith.science/pith/XH2QBGPXXXNAS7JSUVHVGKNWD5/action/citation_signature","submit_replication":"https://pith.science/pith/XH2QBGPXXXNAS7JSUVHVGKNWD5/action/replication_record"}},"created_at":"2026-05-18T02:44:18.089557+00:00","updated_at":"2026-05-18T02:44:18.089557+00:00"}