{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2026:WRCSGMLNUXKZTXKMQFUQ3R7VSE","short_pith_number":"pith:WRCSGMLN","schema_version":"1.0","canonical_sha256":"b44523316da5d599dd4c81690dc7f5913f35ac5d3dbc1983c7b1ae278167bd2c","source":{"kind":"arxiv","id":"2605.28311","version":1},"attestation_state":"computed","paper":{"title":"On metric characterizations of tree and fragmentability indices of Banach spaces","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.MG"],"primary_cat":"math.FA","authors_text":"Antonin Proch\\'azka, Estelle Basset, Gilles Lancien","submitted_at":"2026-05-27T11:13:43Z","abstract_excerpt":"We introduce two ordinal indices that are linear invariants for Banach spaces: the dyadic tree index and the sprawling tree index. We show that they are also bi-Lipschitz invariants. In fact, we characterize their values in terms of sub-Lipschitz embeddability of dyadic or countably branching diamond graphs of ordinal height. We derive applications for separable Banach spaces that are universal for complete countable metric spaces and bi-Lipschitz embeddings. We also discuss the links of these tree indices with classical fragmentability indices of Banach spaces such as the dentabilty, weak fra"},"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":"2605.28311","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2026-05-27T11:13:43Z","cross_cats_sorted":["math.MG"],"title_canon_sha256":"6e85e052c344c0b64ab46ca43240920f236c74ca13884df93c6998a27aa0fc7e","abstract_canon_sha256":"2d9a6ee2a9b8a00792c48f86e69e745614799643a24113ce594e4d81c1351edb"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-28T01:05:06.078254Z","signature_b64":"jASLIRgf/dTLCr0GEvmXZ+1Q5HfSVgu+j6+RgJw3aj8Kjga4XB4pp7Jy7MaMlT3+LDNI9nWspzO9YOebOFqhAQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"b44523316da5d599dd4c81690dc7f5913f35ac5d3dbc1983c7b1ae278167bd2c","last_reissued_at":"2026-05-28T01:05:06.077820Z","signature_status":"signed_v1","first_computed_at":"2026-05-28T01:05:06.077820Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On metric characterizations of tree and fragmentability indices of Banach spaces","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.MG"],"primary_cat":"math.FA","authors_text":"Antonin Proch\\'azka, Estelle Basset, Gilles Lancien","submitted_at":"2026-05-27T11:13:43Z","abstract_excerpt":"We introduce two ordinal indices that are linear invariants for Banach spaces: the dyadic tree index and the sprawling tree index. We show that they are also bi-Lipschitz invariants. In fact, we characterize their values in terms of sub-Lipschitz embeddability of dyadic or countably branching diamond graphs of ordinal height. We derive applications for separable Banach spaces that are universal for complete countable metric spaces and bi-Lipschitz embeddings. We also discuss the links of these tree indices with classical fragmentability indices of Banach spaces such as the dentabilty, weak fra"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2605.28311","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/2605.28311/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":"2605.28311","created_at":"2026-05-28T01:05:06.077886+00:00"},{"alias_kind":"arxiv_version","alias_value":"2605.28311v1","created_at":"2026-05-28T01:05:06.077886+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2605.28311","created_at":"2026-05-28T01:05:06.077886+00:00"},{"alias_kind":"pith_short_12","alias_value":"WRCSGMLNUXKZ","created_at":"2026-05-28T01:05:06.077886+00:00"},{"alias_kind":"pith_short_16","alias_value":"WRCSGMLNUXKZTXKM","created_at":"2026-05-28T01:05:06.077886+00:00"},{"alias_kind":"pith_short_8","alias_value":"WRCSGMLN","created_at":"2026-05-28T01:05:06.077886+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/WRCSGMLNUXKZTXKMQFUQ3R7VSE","json":"https://pith.science/pith/WRCSGMLNUXKZTXKMQFUQ3R7VSE.json","graph_json":"https://pith.science/api/pith-number/WRCSGMLNUXKZTXKMQFUQ3R7VSE/graph.json","events_json":"https://pith.science/api/pith-number/WRCSGMLNUXKZTXKMQFUQ3R7VSE/events.json","paper":"https://pith.science/paper/WRCSGMLN"},"agent_actions":{"view_html":"https://pith.science/pith/WRCSGMLNUXKZTXKMQFUQ3R7VSE","download_json":"https://pith.science/pith/WRCSGMLNUXKZTXKMQFUQ3R7VSE.json","view_paper":"https://pith.science/paper/WRCSGMLN","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2605.28311&json=true","fetch_graph":"https://pith.science/api/pith-number/WRCSGMLNUXKZTXKMQFUQ3R7VSE/graph.json","fetch_events":"https://pith.science/api/pith-number/WRCSGMLNUXKZTXKMQFUQ3R7VSE/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/WRCSGMLNUXKZTXKMQFUQ3R7VSE/action/timestamp_anchor","attest_storage":"https://pith.science/pith/WRCSGMLNUXKZTXKMQFUQ3R7VSE/action/storage_attestation","attest_author":"https://pith.science/pith/WRCSGMLNUXKZTXKMQFUQ3R7VSE/action/author_attestation","sign_citation":"https://pith.science/pith/WRCSGMLNUXKZTXKMQFUQ3R7VSE/action/citation_signature","submit_replication":"https://pith.science/pith/WRCSGMLNUXKZTXKMQFUQ3R7VSE/action/replication_record"}},"created_at":"2026-05-28T01:05:06.077886+00:00","updated_at":"2026-05-28T01:05:06.077886+00:00"}