{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2026:2BZANPDLZMB74YCELHFPFNFIDG","short_pith_number":"pith:2BZANPDL","schema_version":"1.0","canonical_sha256":"d07206bc6bcb03fe604459caf2b4a819a33dbbb36e7ae8459567f08df087b602","source":{"kind":"arxiv","id":"2606.02918","version":1},"attestation_state":"computed","paper":{"title":"Lipschitz-free spaces and purely 1-unrectifiable metric spaces","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.MG"],"primary_cat":"math.FA","authors_text":"Ram\\'on J. Aliaga","submitted_at":"2026-06-01T21:47:13Z","abstract_excerpt":"The Lipschitz-free space $\\mathcal{F}(M)$ is a canonical linearization of a complete metric space $M$ whose topological dual is the space of Lipschitz functions on $M$. We review the properties of $\\mathcal{F}(M)$ when the underlying space $M$ is purely 1-unrectifiable, that is, it contains no bi-Lipschitz copy of a subset of $\\mathbb{R}$ with positive measure. For compact $M$, this is equivalent to several Banach space properties of $\\mathcal{F}(M)$, including the Radon-Nikod\\'ym and Schur properties or admitting a predual. We shall see how the study of locally flat Lipschitz functions on $M$"},"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":"2606.02918","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2026-06-01T21:47:13Z","cross_cats_sorted":["math.MG"],"title_canon_sha256":"b049fd1980945e8beb715e040fb5de6fb6d4c45cfe67da56df24289899c07bcf","abstract_canon_sha256":"aa7262de15c4230ce30b6ded3664db9aac943d473dbe8f91665222ebac85cfa1"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-06-03T01:05:26.548241Z","signature_b64":"fWBv2pdLpRdhwbCtdUHe+EtY2OIE0rcc5xwkCTBzdevpa6ChmqYgmvoDO4P2AzZNyUKO2XwLM0fcNS7iDRzdAA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"d07206bc6bcb03fe604459caf2b4a819a33dbbb36e7ae8459567f08df087b602","last_reissued_at":"2026-06-03T01:05:26.547842Z","signature_status":"signed_v1","first_computed_at":"2026-06-03T01:05:26.547842Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Lipschitz-free spaces and purely 1-unrectifiable metric spaces","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.MG"],"primary_cat":"math.FA","authors_text":"Ram\\'on J. Aliaga","submitted_at":"2026-06-01T21:47:13Z","abstract_excerpt":"The Lipschitz-free space $\\mathcal{F}(M)$ is a canonical linearization of a complete metric space $M$ whose topological dual is the space of Lipschitz functions on $M$. We review the properties of $\\mathcal{F}(M)$ when the underlying space $M$ is purely 1-unrectifiable, that is, it contains no bi-Lipschitz copy of a subset of $\\mathbb{R}$ with positive measure. For compact $M$, this is equivalent to several Banach space properties of $\\mathcal{F}(M)$, including the Radon-Nikod\\'ym and Schur properties or admitting a predual. We shall see how the study of locally flat Lipschitz functions on $M$"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2606.02918","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/2606.02918/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":"2606.02918","created_at":"2026-06-03T01:05:26.547901+00:00"},{"alias_kind":"arxiv_version","alias_value":"2606.02918v1","created_at":"2026-06-03T01:05:26.547901+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2606.02918","created_at":"2026-06-03T01:05:26.547901+00:00"},{"alias_kind":"pith_short_12","alias_value":"2BZANPDLZMB7","created_at":"2026-06-03T01:05:26.547901+00:00"},{"alias_kind":"pith_short_16","alias_value":"2BZANPDLZMB74YCE","created_at":"2026-06-03T01:05:26.547901+00:00"},{"alias_kind":"pith_short_8","alias_value":"2BZANPDL","created_at":"2026-06-03T01:05:26.547901+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/2BZANPDLZMB74YCELHFPFNFIDG","json":"https://pith.science/pith/2BZANPDLZMB74YCELHFPFNFIDG.json","graph_json":"https://pith.science/api/pith-number/2BZANPDLZMB74YCELHFPFNFIDG/graph.json","events_json":"https://pith.science/api/pith-number/2BZANPDLZMB74YCELHFPFNFIDG/events.json","paper":"https://pith.science/paper/2BZANPDL"},"agent_actions":{"view_html":"https://pith.science/pith/2BZANPDLZMB74YCELHFPFNFIDG","download_json":"https://pith.science/pith/2BZANPDLZMB74YCELHFPFNFIDG.json","view_paper":"https://pith.science/paper/2BZANPDL","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2606.02918&json=true","fetch_graph":"https://pith.science/api/pith-number/2BZANPDLZMB74YCELHFPFNFIDG/graph.json","fetch_events":"https://pith.science/api/pith-number/2BZANPDLZMB74YCELHFPFNFIDG/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/2BZANPDLZMB74YCELHFPFNFIDG/action/timestamp_anchor","attest_storage":"https://pith.science/pith/2BZANPDLZMB74YCELHFPFNFIDG/action/storage_attestation","attest_author":"https://pith.science/pith/2BZANPDLZMB74YCELHFPFNFIDG/action/author_attestation","sign_citation":"https://pith.science/pith/2BZANPDLZMB74YCELHFPFNFIDG/action/citation_signature","submit_replication":"https://pith.science/pith/2BZANPDLZMB74YCELHFPFNFIDG/action/replication_record"}},"created_at":"2026-06-03T01:05:26.547901+00:00","updated_at":"2026-06-03T01:05:26.547901+00:00"}