{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2026:36ZOXV2MLGZBV5PU3HHXWTGW7K","short_pith_number":"pith:36ZOXV2M","schema_version":"1.0","canonical_sha256":"dfb2ebd74c59b21af5f4d9cf7b4cd6fa8806176a7324feb5346eebc016ef2599","source":{"kind":"arxiv","id":"2605.17001","version":1},"attestation_state":"computed","paper":{"title":"Strong uniqueness and rectifiability of generalized cylindrical singularities in Ricci flow","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"A Lojasiewicz inequality for the pointed W-entropy establishes strong uniqueness of generalized cylindrical tangent flows in Ricci flow.","cross_cats":[],"primary_cat":"math.DG","authors_text":"Hanbing Fang, Yu Li","submitted_at":"2026-05-16T13:54:24Z","abstract_excerpt":"In this paper, we extend the results of \\cite{fang2025strong, fang2025singular} to generalized cylinders. More precisely, we establish a Lojasiewicz inequality for the pointed $\\mathcal{W}$-entropy in Ricci flow under the assumption that the geometry near the base point is close to a generalized cylinder $\\mathbb{R}^k \\times N^{n-k}$, where $N$ is an Einstein manifold with obstruction of order three satisfying a suitable spectral condition. As an application, we prove the strong uniqueness of generalized cylindrical tangent flows. Furthermore, we show that the subset $\\mathcal{S}^k_{\\mathrm{qc"},"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":"2605.17001","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2026-05-16T13:54:24Z","cross_cats_sorted":[],"title_canon_sha256":"5da0dd1ee21e10f6a935ac6440ba5bf5abdb6ae35bbbf1c7d102fb12614c86f4","abstract_canon_sha256":"7aef4169569e4e9192d17365255e19dcbf98131f70bcd66b53ce4fe3c39a6e1e"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-20T00:03:35.377981Z","signature_b64":"SVn0j1fvjYBu1VV/od7+x9qL8gZsqMe8XBOWlHLbWvbD6eRqF4S/ZEPoE+SzqDerMIeMK7XTjX7upmHdFEnMCg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"dfb2ebd74c59b21af5f4d9cf7b4cd6fa8806176a7324feb5346eebc016ef2599","last_reissued_at":"2026-05-20T00:03:35.377161Z","signature_status":"signed_v1","first_computed_at":"2026-05-20T00:03:35.377161Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Strong uniqueness and rectifiability of generalized cylindrical singularities in Ricci flow","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"A Lojasiewicz inequality for the pointed W-entropy establishes strong uniqueness of generalized cylindrical tangent flows in Ricci flow.","cross_cats":[],"primary_cat":"math.DG","authors_text":"Hanbing Fang, Yu Li","submitted_at":"2026-05-16T13:54:24Z","abstract_excerpt":"In this paper, we extend the results of \\cite{fang2025strong, fang2025singular} to generalized cylinders. More precisely, we establish a Lojasiewicz inequality for the pointed $\\mathcal{W}$-entropy in Ricci flow under the assumption that the geometry near the base point is close to a generalized cylinder $\\mathbb{R}^k \\times N^{n-k}$, where $N$ is an Einstein manifold with obstruction of order three satisfying a suitable spectral condition. As an application, we prove the strong uniqueness of generalized cylindrical tangent flows. Furthermore, we show that the subset $\\mathcal{S}^k_{\\mathrm{qc"},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"We establish a Lojasiewicz inequality for the pointed W-entropy in Ricci flow under the assumption that the geometry near the base point is close to a generalized cylinder R^k × N^{n-k}, where N is an Einstein manifold with obstruction of order three satisfying a suitable spectral condition. As an application, we prove the strong uniqueness of generalized cylindrical tangent flows. Furthermore, we show that the subset S^k_qc(N) is horizontally parabolic k-rectifiable.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"The geometry near the base point is close to a generalized cylinder R^k × N^{n-k}, where N is an Einstein manifold with obstruction of order three satisfying a suitable spectral condition (invoked in the statement of the Lojasiewicz inequality and its applications to uniqueness and rectifiability).","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"Proves Lojasiewicz inequality for W-entropy near generalized cylinders in Ricci flow, yielding strong uniqueness of tangent flows and horizontal parabolic k-rectifiability of the corresponding singularity set.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"A Lojasiewicz inequality for the pointed W-entropy establishes strong uniqueness of generalized cylindrical tangent flows in Ricci flow.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"009a807ffe87f1d17b031f75f3b61d8804c0ea440dc4850762fa17b6563ccb56"},"source":{"id":"2605.17001","kind":"arxiv","version":1},"verdict":{"id":"876e60da-1760-4443-98f0-fc1e81ab8c37","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-19T18:42:34.010833Z","strongest_claim":"We establish a Lojasiewicz inequality for the pointed W-entropy in Ricci flow under the assumption that the geometry near the base point is close to a generalized cylinder R^k × N^{n-k}, where N is an Einstein manifold with obstruction of order three satisfying a suitable spectral condition. As an application, we prove the strong uniqueness of generalized cylindrical tangent flows. Furthermore, we show that the subset S^k_qc(N) is horizontally parabolic k-rectifiable.","one_line_summary":"Proves Lojasiewicz inequality for W-entropy near generalized cylinders in Ricci flow, yielding strong uniqueness of tangent flows and horizontal parabolic k-rectifiability of the corresponding singularity set.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"The geometry near the base point is close to a generalized cylinder R^k × N^{n-k}, where N is an Einstein manifold with obstruction of order three satisfying a suitable spectral condition (invoked in the statement of the Lojasiewicz inequality and its applications to uniqueness and rectifiability).","pith_extraction_headline":"A Lojasiewicz inequality for the pointed W-entropy establishes strong uniqueness of generalized cylindrical tangent flows in Ricci flow."},"integrity":{"clean":false,"summary":{"advisory":0,"critical":1,"by_detector":{"doi_compliance":{"total":1,"advisory":0,"critical":1,"informational":0}},"informational":0},"endpoint":"/pith/2605.17001/integrity.json","findings":[{"note":"Identifier '10.1007/s10240-025-00145-3' is syntactically valid but the DOI registry (doi.org) returned 404, and Crossref / OpenAlex / internal corpus also have no record. The cited work could not be located through any authoritative source.","detector":"doi_compliance","severity":"critical","ref_index":5,"audited_at":"2026-05-19T18:51:21.664836Z","detected_doi":"10.1007/s10240-025-00145-3","finding_type":"unresolvable_identifier","verdict_class":"cross_source","detected_arxiv_id":null}],"available":true,"detectors_run":[{"name":"citation_quote_validity","ran_at":"2026-05-19T19:49:57.215385Z","status":"completed","version":"0.1.0","findings_count":0},{"name":"cited_work_retraction","ran_at":"2026-05-19T19:23:35.600196Z","status":"completed","version":"1.0.0","findings_count":0},{"name":"doi_title_agreement","ran_at":"2026-05-19T19:01:18.832485Z","status":"completed","version":"1.0.0","findings_count":0},{"name":"doi_compliance","ran_at":"2026-05-19T18:51:21.664836Z","status":"completed","version":"1.0.0","findings_count":1},{"name":"claim_evidence","ran_at":"2026-05-19T18:41:56.198719Z","status":"completed","version":"1.0.0","findings_count":0},{"name":"ai_meta_artifact","ran_at":"2026-05-19T18:33:26.288534Z","status":"skipped","version":"1.0.0","findings_count":0}],"snapshot_sha256":"b8e5c4f583e141000abca267658065e9fab065b0462b7ae45415d4a734016271"},"references":{"count":115,"sample":[{"doi":"","year":null,"title":"Annals of Mathematics , volume=","work_id":"290eb005-54a9-430e-8275-bfd9523e3898","ref_index":1,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":null,"title":"Cheeger, J. and Tian, G. , journal=. On the cone structure at infinity of","work_id":"8be94df7-1cb7-4dbf-a6e3-85c42d8cd5c5","ref_index":2,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":null,"title":"Sesum, N. , journal=. Linear and dynamical stability of","work_id":"d0cf1bce-993c-479e-84bf-120047d4a444","ref_index":3,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2015,"title":"Colding, T. H. and Minicozzi II, W. P. , journal=. Uniqueness of blowups and. 2015 , publisher=","work_id":"52018648-5e7b-4bb6-a390-9cbd6b4f09ab","ref_index":4,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"10.1007/s10240-025-00145-3","year":null,"title":"Colding, T. H. and Minicozzi II, W. P. , title =. Publications Mathématiques de l'IHÉS , year =. doi:10.1007/s10240-025-00145-3 , url =","work_id":"8128ac8d-2823-4987-b435-4f63db7f7119","ref_index":5,"cited_arxiv_id":"","is_internal_anchor":false}],"resolved_work":115,"snapshot_sha256":"f6a22bacae997b2281267a6a5083360035faabfc4eb710670e7449bfe358505d","internal_anchors":1},"formal_canon":{"evidence_count":2,"snapshot_sha256":"5a2ce36301477f3e49ee845704e8541cc795d965d501b896b807c30ddddcf399"},"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":"2605.17001","created_at":"2026-05-20T00:03:35.377281+00:00"},{"alias_kind":"arxiv_version","alias_value":"2605.17001v1","created_at":"2026-05-20T00:03:35.377281+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2605.17001","created_at":"2026-05-20T00:03:35.377281+00:00"},{"alias_kind":"pith_short_12","alias_value":"36ZOXV2MLGZB","created_at":"2026-05-20T00:03:35.377281+00:00"},{"alias_kind":"pith_short_16","alias_value":"36ZOXV2MLGZBV5PU","created_at":"2026-05-20T00:03:35.377281+00:00"},{"alias_kind":"pith_short_8","alias_value":"36ZOXV2M","created_at":"2026-05-20T00:03:35.377281+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/36ZOXV2MLGZBV5PU3HHXWTGW7K","json":"https://pith.science/pith/36ZOXV2MLGZBV5PU3HHXWTGW7K.json","graph_json":"https://pith.science/api/pith-number/36ZOXV2MLGZBV5PU3HHXWTGW7K/graph.json","events_json":"https://pith.science/api/pith-number/36ZOXV2MLGZBV5PU3HHXWTGW7K/events.json","paper":"https://pith.science/paper/36ZOXV2M"},"agent_actions":{"view_html":"https://pith.science/pith/36ZOXV2MLGZBV5PU3HHXWTGW7K","download_json":"https://pith.science/pith/36ZOXV2MLGZBV5PU3HHXWTGW7K.json","view_paper":"https://pith.science/paper/36ZOXV2M","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2605.17001&json=true","fetch_graph":"https://pith.science/api/pith-number/36ZOXV2MLGZBV5PU3HHXWTGW7K/graph.json","fetch_events":"https://pith.science/api/pith-number/36ZOXV2MLGZBV5PU3HHXWTGW7K/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/36ZOXV2MLGZBV5PU3HHXWTGW7K/action/timestamp_anchor","attest_storage":"https://pith.science/pith/36ZOXV2MLGZBV5PU3HHXWTGW7K/action/storage_attestation","attest_author":"https://pith.science/pith/36ZOXV2MLGZBV5PU3HHXWTGW7K/action/author_attestation","sign_citation":"https://pith.science/pith/36ZOXV2MLGZBV5PU3HHXWTGW7K/action/citation_signature","submit_replication":"https://pith.science/pith/36ZOXV2MLGZBV5PU3HHXWTGW7K/action/replication_record"}},"created_at":"2026-05-20T00:03:35.377281+00:00","updated_at":"2026-05-20T00:03:35.377281+00:00"}