{"as_of":"2026-08-12T20:13:00Z","caps":{"database_statements":6,"inbound":100,"outbound":100},"context_digest":"sha256:6758b287cbc19f309824312e318a08cdadfdb52c21504d3756194dcb1b7229cf","coverage":[{"denominator":29,"lane":"reference_resolution","note":"Typed states for the displayed outbound observations.","records_observed":29,"source":"paper_references, paper_reference_links","source_observed_at":"2026-08-06T22:45:40.559134Z","state":"measured"},{"denominator":30,"lane":"standing_notices","note":"One-hop event checks from named stored sources.","records_observed":30,"source":"scholarly_work_events, retraction_status_cache","source_observed_at":"2026-08-12T06:34:41.77262+00:00","state":"measured"},{"denominator":1,"lane":"inbound_itemization","note":"Pith citing papers itemized under the disclosed page cap.","records_observed":1,"source":"paper_references, paper_reference_links","source_observed_at":"2026-08-06T22:00:37.914547Z","state":"measured"},{"denominator":1,"lane":"external_citation_measurements","note":"A source-named dated measurement, never combined with another source.","records_observed":0,"source":"pith","source_observed_at":"2026-08-06T22:00:38.284919Z","state":"measured"}],"external_citation_measurements":[],"inbound":[{"citation":{"cited_paper":{"arxiv_id":"2506.21130","last_updated":"2025-07-01T06:56:46Z","snapshot_observed_at":"2026-08-11T13:02:07.742105Z","submitted_at":"2025-06-26T10:12:30Z","title":"An Invariant for Triple-Point-Free Immersed Spheres","version":2},"cited_work":{"arxiv_id":"2506.21130","doi":null,"metadata_source":"pith","pith_arxiv_id":"2506.21130","snapshot_observed_at":"2026-08-06T22:00:38.284919Z","title":"An Invariant for Triple-Point-Free Immersed Spheres","venue":"math.GT","work_id":"be7ac7fd-eabc-4aa5-8e6a-fc7b69228bc1","year":2025},"citing_paper":{"arxiv_id":"2506.23359","last_updated":"2025-06-29T18:35:27Z","snapshot_observed_at":"2026-08-11T07:09:30.365022Z","submitted_at":"2025-06-29T18:35:27Z","title":"The Willmore Energy Landscape of Spheres and Avoidable Singularities of the Willmore Flow","version":1},"reference_index":62,"source":"arxiv_source","source_observed_at":"2026-08-06T22:00:37.914547Z"},"links":{"cited_paper":"/paper/2506.21130","citing_paper":"/paper/2506.23359"},"observation_digest":"sha256:75d707dee0e783960db58df4dec56a41440069e5f93adf3af8e0055c4fa10236","observation_id":"6b6f57ea-ffa9-4fcd-bc11-ed7256a0e842","resolution":{"observed_at":"2026-08-06T22:00:38.352454Z","resolver_source":"local_arxiv","status":"verified_exact"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-12T06:34:41.77262+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-12T06:34:41.77262+00:00","source":"crossref"},{"observed_at":"2026-08-12T06:34:36.333875+00:00","source":"retraction_watch"}],"state":"measured"}}],"links":{"evidence":"/evidence","html":"/paper/2506.21130/citation-record","integrity":"/paper/2506.21130/integrity","json":"/paper/2506.21130/citation-record.json","paper":"/paper/2506.21130"},"outbound":[{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":"raw_reference","pith_arxiv_id":null,"snapshot_observed_at":"2026-08-06T22:45:42.670190Z","title":null,"venue":null,"work_id":"f7535804-b871-4e87-8393-72211f7dc01d","year":1924},"citing_paper":{"arxiv_id":"2506.21130","last_updated":"2025-07-01T06:56:46Z","snapshot_observed_at":"2026-08-11T13:02:07.742105Z","submitted_at":"2025-06-26T10:12:30Z","title":"An Invariant for Triple-Point-Free Immersed Spheres","version":2},"reference_index":1,"source":"arxiv_source","source_observed_at":"2026-08-06T22:45:38.035877Z"},"links":{"citing_paper":"/paper/2506.21130"},"observation_digest":"sha256:7393df020ac7ac05583083de55e98329061a597d3fb41240bde0dfe226d4bdc8","observation_id":"015b23ec-de3f-4182-b13f-0ff55a34e3bb","resolution":{"observed_at":"2026-08-06T22:45:42.716190Z","resolver_source":"raw_fallback","status":"unresolved"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-12T06:34:41.77262+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-12T06:34:41.77262+00:00","source":"crossref"},{"observed_at":"2026-08-12T06:34:36.333875+00:00","source":"retraction_watch"}],"state":"measured"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":"raw_reference","pith_arxiv_id":null,"snapshot_observed_at":"2026-08-06T22:45:42.547747Z","title":"Triple points and singularities of projections of smoothly immersed surfaces","venue":null,"work_id":"b25d4e52-623c-4300-be3f-f6df6d205531","year":1974},"citing_paper":{"arxiv_id":"2506.21130","last_updated":"2025-07-01T06:56:46Z","snapshot_observed_at":"2026-08-11T13:02:07.742105Z","submitted_at":"2025-06-26T10:12:30Z","title":"An Invariant for Triple-Point-Free Immersed Spheres","version":2},"reference_index":2,"source":"arxiv_source","source_observed_at":"2026-08-06T22:45:38.153323Z"},"links":{"citing_paper":"/paper/2506.21130"},"observation_digest":"sha256:173656690aa6daac14b19a604bb53904dabeb6a2743c2d329c6705fe2c0b3282","observation_id":"d24b6af1-42aa-4bcc-9bbc-86044cd6c9ed","resolution":{"observed_at":"2026-08-06T22:45:42.609946Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-12T06:34:41.77262+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-12T06:34:41.77262+00:00","source":"crossref"},{"observed_at":"2026-08-12T06:34:36.333875+00:00","source":"retraction_watch"}],"state":"measured"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":"raw_reference","pith_arxiv_id":null,"snapshot_observed_at":"2026-08-06T22:45:42.456781Z","title":"Banchoff","venue":null,"work_id":"c1ccdc98-3841-4973-acf6-5664f8f86662","year":1974},"citing_paper":{"arxiv_id":"2506.21130","last_updated":"2025-07-01T06:56:46Z","snapshot_observed_at":"2026-08-11T13:02:07.742105Z","submitted_at":"2025-06-26T10:12:30Z","title":"An Invariant for Triple-Point-Free Immersed Spheres","version":2},"reference_index":3,"source":"arxiv_source","source_observed_at":"2026-08-06T22:45:38.263021Z"},"links":{"citing_paper":"/paper/2506.21130"},"observation_digest":"sha256:6933ed2332be0eb9d3a4ecba67f746e5cf6d4d258e115a3c3608ee2fd2cd8c95","observation_id":"74a4e1b9-b2d5-4236-911d-573d33a8178c","resolution":{"observed_at":"2026-08-06T22:45:42.482108Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-12T06:34:41.77262+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-12T06:34:41.77262+00:00","source":"crossref"},{"observed_at":"2026-08-12T06:34:36.333875+00:00","source":"retraction_watch"}],"state":"measured"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":null,"snapshot_observed_at":"2026-08-06T22:45:38.376379Z","title":"A singular example for the W illmore flow","venue":null,"work_id":null,"year":2009},"citing_paper":{"arxiv_id":"2506.21130","last_updated":"2025-07-01T06:56:46Z","snapshot_observed_at":"2026-08-11T13:02:07.742105Z","submitted_at":"2025-06-26T10:12:30Z","title":"An Invariant for Triple-Point-Free Immersed Spheres","version":2},"reference_index":4,"source":"arxiv_source","source_observed_at":"2026-08-06T22:45:38.376379Z"},"links":{"citing_paper":"/paper/2506.21130"},"observation_digest":"sha256:0e3538b7b44f846c86e9f2967bbc3961a329dc82016d2844b189ec4e768df84b","observation_id":"8736e971-e765-47f6-a572-f80f45e4604c","resolution":{"observed_at":"2026-08-06T22:45:38.376379Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":"raw_reference","pith_arxiv_id":null,"snapshot_observed_at":"2026-08-06T22:45:42.358362Z","title":"On the self-intersections of an immersed sphere","venue":null,"work_id":"134b210c-b8b9-489e-a300-4c22468a7012","year":2007},"citing_paper":{"arxiv_id":"2506.21130","last_updated":"2025-07-01T06:56:46Z","snapshot_observed_at":"2026-08-11T13:02:07.742105Z","submitted_at":"2025-06-26T10:12:30Z","title":"An Invariant for Triple-Point-Free Immersed Spheres","version":2},"reference_index":5,"source":"arxiv_source","source_observed_at":"2026-08-06T22:45:38.512333Z"},"links":{"citing_paper":"/paper/2506.21130"},"observation_digest":"sha256:41e05ad3f4954bcd16cc90461971f5ab40f56eb17b85def61b1d264fa246dabe","observation_id":"25234d25-f524-45a4-b85a-4c242bbaea38","resolution":{"observed_at":"2026-08-06T22:45:42.378423Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-12T06:34:41.77262+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-12T06:34:41.77262+00:00","source":"crossref"},{"observed_at":"2026-08-12T06:34:36.333875+00:00","source":"retraction_watch"}],"state":"measured"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":null,"snapshot_observed_at":"2026-08-06T22:45:38.655511Z","title":null,"venue":null,"work_id":null,"year":1993},"citing_paper":{"arxiv_id":"2506.21130","last_updated":"2025-07-01T06:56:46Z","snapshot_observed_at":"2026-08-11T13:02:07.742105Z","submitted_at":"2025-06-26T10:12:30Z","title":"An Invariant for Triple-Point-Free Immersed Spheres","version":2},"reference_index":6,"source":"arxiv_source","source_observed_at":"2026-08-06T22:45:38.655511Z"},"links":{"citing_paper":"/paper/2506.21130"},"observation_digest":"sha256:1c9a4f1860aec1f912e2d528154ed90990073a0a91341feaa3de18aa1e150ad1","observation_id":"4653f40f-5e48-443b-9d82-92a3c1172358","resolution":{"observed_at":"2026-08-06T22:45:38.655511Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":"raw_reference","pith_arxiv_id":null,"snapshot_observed_at":"2026-08-06T22:45:42.253238Z","title":"Chmutov, S","venue":null,"work_id":"d0f04721-01a2-493a-9e9e-c378a92f40f9","year":2012},"citing_paper":{"arxiv_id":"2506.21130","last_updated":"2025-07-01T06:56:46Z","snapshot_observed_at":"2026-08-11T13:02:07.742105Z","submitted_at":"2025-06-26T10:12:30Z","title":"An Invariant for Triple-Point-Free Immersed Spheres","version":2},"reference_index":7,"source":"arxiv_source","source_observed_at":"2026-08-06T22:45:38.800438Z"},"links":{"citing_paper":"/paper/2506.21130"},"observation_digest":"sha256:7bc63e25d78a7f23d0c4b1d205adbd759358c575ef1f300bb71e1c2fd599e9e1","observation_id":"3e66baf8-42ce-425c-8d0a-18f9ce786fbd","resolution":{"observed_at":"2026-08-06T22:45:42.294956Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-12T06:34:41.77262+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-12T06:34:41.77262+00:00","source":"crossref"},{"observed_at":"2026-08-12T06:34:36.333875+00:00","source":"retraction_watch"}],"state":"measured"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":null,"snapshot_observed_at":"2026-08-06T22:45:38.932689Z","title":"Goryunov","venue":null,"work_id":null,"year":1997},"citing_paper":{"arxiv_id":"2506.21130","last_updated":"2025-07-01T06:56:46Z","snapshot_observed_at":"2026-08-11T13:02:07.742105Z","submitted_at":"2025-06-26T10:12:30Z","title":"An Invariant for Triple-Point-Free Immersed Spheres","version":2},"reference_index":8,"source":"arxiv_source","source_observed_at":"2026-08-06T22:45:38.932689Z"},"links":{"citing_paper":"/paper/2506.21130"},"observation_digest":"sha256:421fd3e374035c214210d6aeda0a80abd97c6ae0ca67ec90e3d24e5d823993b4","observation_id":"4674b713-fd07-451e-94be-a61b1265f779","resolution":{"observed_at":"2026-08-06T22:45:38.932689Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":null,"snapshot_observed_at":"2026-08-06T22:45:39.078719Z","title":null,"venue":null,"work_id":null,"year":1976},"citing_paper":{"arxiv_id":"2506.21130","last_updated":"2025-07-01T06:56:46Z","snapshot_observed_at":"2026-08-11T13:02:07.742105Z","submitted_at":"2025-06-26T10:12:30Z","title":"An Invariant for Triple-Point-Free Immersed Spheres","version":2},"reference_index":9,"source":"arxiv_source","source_observed_at":"2026-08-06T22:45:39.078719Z"},"links":{"citing_paper":"/paper/2506.21130"},"observation_digest":"sha256:a0d8f8931b4b0fcb45b988cf9c1365791a6ced10a97278a169e7afa99229f1fb","observation_id":"e153a656-1420-402f-a949-d0e1b4c669b8","resolution":{"observed_at":"2026-08-06T22:45:39.078719Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":"raw_reference","pith_arxiv_id":null,"snapshot_observed_at":"2026-08-06T22:45:42.184321Z","title":null,"venue":null,"work_id":"2ee40777-d0df-4915-b307-5e3a50d3f286","year":2001},"citing_paper":{"arxiv_id":"2506.21130","last_updated":"2025-07-01T06:56:46Z","snapshot_observed_at":"2026-08-11T13:02:07.742105Z","submitted_at":"2025-06-26T10:12:30Z","title":"An Invariant for Triple-Point-Free Immersed Spheres","version":2},"reference_index":10,"source":"arxiv_source","source_observed_at":"2026-08-06T22:45:39.217295Z"},"links":{"citing_paper":"/paper/2506.21130"},"observation_digest":"sha256:b7d20cd01225e7bbd01b1dd260ca5111bffc826c8695195d148be5dc4bfa6380","observation_id":"21bd3824-743c-467a-9c41-bf270359ec3c","resolution":{"observed_at":"2026-08-06T22:45:42.211226Z","resolver_source":"raw_fallback","status":"unresolved"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-12T06:34:41.77262+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-12T06:34:41.77262+00:00","source":"crossref"},{"observed_at":"2026-08-12T06:34:36.333875+00:00","source":"retraction_watch"}],"state":"measured"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":null,"snapshot_observed_at":"2026-08-06T22:45:39.344061Z","title":null,"venue":null,"work_id":null,"year":1985},"citing_paper":{"arxiv_id":"2506.21130","last_updated":"2025-07-01T06:56:46Z","snapshot_observed_at":"2026-08-11T13:02:07.742105Z","submitted_at":"2025-06-26T10:12:30Z","title":"An Invariant for Triple-Point-Free Immersed Spheres","version":2},"reference_index":11,"source":"arxiv_source","source_observed_at":"2026-08-06T22:45:39.344061Z"},"links":{"citing_paper":"/paper/2506.21130"},"observation_digest":"sha256:238ba7dea08af425138311ba71d6d88a521e1cd8a8f4fd2f767d4307687e529e","observation_id":"e6abc707-bb06-4a71-a1e2-1556e551b394","resolution":{"observed_at":"2026-08-06T22:45:39.344061Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":"raw_reference","pith_arxiv_id":null,"snapshot_observed_at":"2026-08-06T22:45:42.062136Z","title":"Interlacement of double curves of immersed spheres","venue":null,"work_id":"4c3ad88e-8f44-4765-b11a-22c1bcb841b3","year":2016},"citing_paper":{"arxiv_id":"2506.21130","last_updated":"2025-07-01T06:56:46Z","snapshot_observed_at":"2026-08-11T13:02:07.742105Z","submitted_at":"2025-06-26T10:12:30Z","title":"An Invariant for Triple-Point-Free Immersed Spheres","version":2},"reference_index":12,"source":"arxiv_source","source_observed_at":"2026-08-06T22:45:39.448018Z"},"links":{"citing_paper":"/paper/2506.21130"},"observation_digest":"sha256:abc08e509968acc8171aa8f22a3e205c743cb230352b2568b45b71a122b6842e","observation_id":"d794135d-55be-4702-8161-4903dece7d9f","resolution":{"observed_at":"2026-08-06T22:45:42.110051Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-12T06:34:41.77262+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-12T06:34:41.77262+00:00","source":"crossref"},{"observed_at":"2026-08-12T06:34:36.333875+00:00","source":"retraction_watch"}],"state":"measured"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":"raw_reference","pith_arxiv_id":null,"snapshot_observed_at":"2026-08-06T22:45:41.942575Z","title":"On double points of immersions of spheres","venue":null,"work_id":"fb20b152-d45c-4424-8f35-3974199e27b9","year":2004},"citing_paper":{"arxiv_id":"2506.21130","last_updated":"2025-07-01T06:56:46Z","snapshot_observed_at":"2026-08-11T13:02:07.742105Z","submitted_at":"2025-06-26T10:12:30Z","title":"An Invariant for Triple-Point-Free Immersed Spheres","version":2},"reference_index":13,"source":"arxiv_source","source_observed_at":"2026-08-06T22:45:39.528445Z"},"links":{"citing_paper":"/paper/2506.21130"},"observation_digest":"sha256:9eeccfc3a8e88557f729ee2d2203beda7d779ee6e5e4b882d39a959fc3dac4b3","observation_id":"043508cc-4d5d-4ca8-b22d-86811016f1ea","resolution":{"observed_at":"2026-08-06T22:45:42.006713Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-12T06:34:41.77262+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-12T06:34:41.77262+00:00","source":"crossref"},{"observed_at":"2026-08-12T06:34:36.333875+00:00","source":"retraction_watch"}],"state":"measured"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":null,"snapshot_observed_at":"2026-08-06T22:45:39.614824Z","title":"A new conformal invariant and its applications to the W illmore conjecture and the first eigenvalue of compact surfaces","venue":null,"work_id":null,"year":1982},"citing_paper":{"arxiv_id":"2506.21130","last_updated":"2025-07-01T06:56:46Z","snapshot_observed_at":"2026-08-11T13:02:07.742105Z","submitted_at":"2025-06-26T10:12:30Z","title":"An Invariant for Triple-Point-Free Immersed Spheres","version":2},"reference_index":14,"source":"arxiv_source","source_observed_at":"2026-08-06T22:45:39.614824Z"},"links":{"citing_paper":"/paper/2506.21130"},"observation_digest":"sha256:8deeddfafec319e570b862e32e5fa1e48598451157afeb9ba3d1bf4e1381da1d","observation_id":"9ce6ab94-2a4f-4dd1-a17e-a84f446365b3","resolution":{"observed_at":"2026-08-06T22:45:39.614824Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":null,"snapshot_observed_at":"2026-08-06T22:45:39.718110Z","title":"Every sphere eversion has a quadruple point","venue":null,"work_id":null,"year":1980},"citing_paper":{"arxiv_id":"2506.21130","last_updated":"2025-07-01T06:56:46Z","snapshot_observed_at":"2026-08-11T13:02:07.742105Z","submitted_at":"2025-06-26T10:12:30Z","title":"An Invariant for Triple-Point-Free Immersed Spheres","version":2},"reference_index":15,"source":"arxiv_source","source_observed_at":"2026-08-06T22:45:39.718110Z"},"links":{"citing_paper":"/paper/2506.21130"},"observation_digest":"sha256:63af5b86646e03d70feb4ba1d69efc0720834875a3b86dcbe1a7e89342727e01","observation_id":"f3c93509-b5d7-41f5-88c8-fec727c3efe5","resolution":{"observed_at":"2026-08-06T22:45:39.718110Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":{"arxiv_id":"2506.23359","last_updated":"2025-06-29T18:35:27Z","snapshot_observed_at":"2026-08-11T07:09:30.365022Z","submitted_at":"2025-06-29T18:35:27Z","title":"The Willmore Energy Landscape of Spheres and Avoidable Singularities of the Willmore Flow","version":1},"cited_work":{"arxiv_id":"2506.23359","doi":null,"metadata_source":"pith","pith_arxiv_id":"2506.23359","snapshot_observed_at":"2026-08-06T22:45:40.635468Z","title":"The Willmore Energy Landscape of Spheres and Avoidable Singularities of the Willmore Flow","venue":"math.DG","work_id":"b03b1fbd-8ab3-4b60-be3e-bd149e8569c0","year":2025},"citing_paper":{"arxiv_id":"2506.21130","last_updated":"2025-07-01T06:56:46Z","snapshot_observed_at":"2026-08-11T13:02:07.742105Z","submitted_at":"2025-06-26T10:12:30Z","title":"An Invariant for Triple-Point-Free Immersed Spheres","version":2},"reference_index":16,"source":"arxiv_source","source_observed_at":"2026-08-06T22:45:39.782324Z"},"links":{"cited_paper":"/paper/2506.23359","citing_paper":"/paper/2506.21130"},"observation_digest":"sha256:ef9f634fc81876b30d392892d551f3d248ebb2711228891311d0b2ae2219bf57","observation_id":"139ec1cd-17db-409e-b512-d60a2b29885a","resolution":{"observed_at":"2026-08-06T22:45:40.691505Z","resolver_source":"local_arxiv","status":"verified_exact"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-12T06:34:41.77262+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-12T06:34:41.77262+00:00","source":"crossref"},{"observed_at":"2026-08-12T06:34:36.333875+00:00","source":"retraction_watch"}],"state":"measured"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":"raw_reference","pith_arxiv_id":null,"snapshot_observed_at":"2026-08-06T22:45:41.827082Z","title":"On the classification of germs of maps from R ^2 to R ^3","venue":null,"work_id":"71657409-53cf-44cc-ac4e-2131d08751e6","year":1985},"citing_paper":{"arxiv_id":"2506.21130","last_updated":"2025-07-01T06:56:46Z","snapshot_observed_at":"2026-08-11T13:02:07.742105Z","submitted_at":"2025-06-26T10:12:30Z","title":"An Invariant for Triple-Point-Free Immersed Spheres","version":2},"reference_index":17,"source":"arxiv_source","source_observed_at":"2026-08-06T22:45:39.857601Z"},"links":{"citing_paper":"/paper/2506.21130"},"observation_digest":"sha256:b5769f23d53eb5d6367a8240ec608444a40622faf578fe600bfb5ec9facc0a26","observation_id":"68d325ff-428c-453b-88de-8df11a3eaf8b","resolution":{"observed_at":"2026-08-06T22:45:41.889039Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-12T06:34:41.77262+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-12T06:34:41.77262+00:00","source":"crossref"},{"observed_at":"2026-08-12T06:34:36.333875+00:00","source":"retraction_watch"}],"state":"measured"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":null,"snapshot_observed_at":"2026-08-06T22:45:39.906581Z","title":"Mayer and Gieri Simonett","venue":null,"work_id":null,"year":2002},"citing_paper":{"arxiv_id":"2506.21130","last_updated":"2025-07-01T06:56:46Z","snapshot_observed_at":"2026-08-11T13:02:07.742105Z","submitted_at":"2025-06-26T10:12:30Z","title":"An Invariant for Triple-Point-Free Immersed Spheres","version":2},"reference_index":18,"source":"arxiv_source","source_observed_at":"2026-08-06T22:45:39.906581Z"},"links":{"citing_paper":"/paper/2506.21130"},"observation_digest":"sha256:9bfe8d202d137d058e729c6c9c5d9f9852b9b38b5563f238eef56e47cfbb7a0b","observation_id":"3548b5f0-52a4-4a74-9a00-ef85d5796c2b","resolution":{"observed_at":"2026-08-06T22:45:39.906581Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":"raw_reference","pith_arxiv_id":null,"snapshot_observed_at":"2026-08-06T22:45:41.714956Z","title":"Quadruple points of regular homotopies of surfaces in 3-manifolds","venue":null,"work_id":"e09a707c-cea8-4903-9924-660114b3173f","year":2000},"citing_paper":{"arxiv_id":"2506.21130","last_updated":"2025-07-01T06:56:46Z","snapshot_observed_at":"2026-08-11T13:02:07.742105Z","submitted_at":"2025-06-26T10:12:30Z","title":"An Invariant for Triple-Point-Free Immersed Spheres","version":2},"reference_index":19,"source":"arxiv_source","source_observed_at":"2026-08-06T22:45:39.963979Z"},"links":{"citing_paper":"/paper/2506.21130"},"observation_digest":"sha256:567582531b586917398cc94ed88731e7f51422c8daa9120d644adc112b574ef7","observation_id":"772d7c77-e278-4cb1-983f-e31328ae6b92","resolution":{"observed_at":"2026-08-06T22:45:41.752873Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-12T06:34:41.77262+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-12T06:34:41.77262+00:00","source":"crossref"},{"observed_at":"2026-08-12T06:34:36.333875+00:00","source":"retraction_watch"}],"state":"measured"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":"raw_reference","pith_arxiv_id":null,"snapshot_observed_at":"2026-08-06T22:45:41.593874Z","title":"Automorphisms and embeddings of surfaces and quadruple points of regular homotopies","venue":null,"work_id":"edbe93d9-6613-4432-a71b-a9445534cf65","year":2001},"citing_paper":{"arxiv_id":"2506.21130","last_updated":"2025-07-01T06:56:46Z","snapshot_observed_at":"2026-08-11T13:02:07.742105Z","submitted_at":"2025-06-26T10:12:30Z","title":"An Invariant for Triple-Point-Free Immersed Spheres","version":2},"reference_index":20,"source":"arxiv_source","source_observed_at":"2026-08-06T22:45:40.016237Z"},"links":{"citing_paper":"/paper/2506.21130"},"observation_digest":"sha256:46bf06f49f983c8053147576a2da9aad9b19fa96bbaef1bd7949d788bbd21192","observation_id":"8ed485fa-a69f-482f-bbcc-9dd9f97caf62","resolution":{"observed_at":"2026-08-06T22:45:41.652822Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-12T06:34:41.77262+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-12T06:34:41.77262+00:00","source":"crossref"},{"observed_at":"2026-08-12T06:34:36.333875+00:00","source":"retraction_watch"}],"state":"measured"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":"raw_reference","pith_arxiv_id":null,"snapshot_observed_at":"2026-08-06T22:45:41.482502Z","title":"Finite order q -invariants of immersions of surfaces into 3-space","venue":null,"work_id":"4816d895-4d03-4647-ac21-7ab53b012a6a","year":2001},"citing_paper":{"arxiv_id":"2506.21130","last_updated":"2025-07-01T06:56:46Z","snapshot_observed_at":"2026-08-11T13:02:07.742105Z","submitted_at":"2025-06-26T10:12:30Z","title":"An Invariant for Triple-Point-Free Immersed Spheres","version":2},"reference_index":21,"source":"arxiv_source","source_observed_at":"2026-08-06T22:45:40.076653Z"},"links":{"citing_paper":"/paper/2506.21130"},"observation_digest":"sha256:46e72434bd21ef2e0d3116ee380dd72fe2d50e8af40f342d29769166cff6f413","observation_id":"b3609481-fae9-4f95-bfa3-831d22d15482","resolution":{"observed_at":"2026-08-06T22:45:41.552321Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-12T06:34:41.77262+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-12T06:34:41.77262+00:00","source":"crossref"},{"observed_at":"2026-08-12T06:34:36.333875+00:00","source":"retraction_watch"}],"state":"measured"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":"raw_reference","pith_arxiv_id":null,"snapshot_observed_at":"2026-08-06T22:45:41.379668Z","title":"Order one invariants of immersions of surfaces into 3-space","venue":null,"work_id":"bbe30907-bde0-43cf-9bcd-e337c5d90a05","year":2004},"citing_paper":{"arxiv_id":"2506.21130","last_updated":"2025-07-01T06:56:46Z","snapshot_observed_at":"2026-08-11T13:02:07.742105Z","submitted_at":"2025-06-26T10:12:30Z","title":"An Invariant for Triple-Point-Free Immersed Spheres","version":2},"reference_index":22,"source":"arxiv_source","source_observed_at":"2026-08-06T22:45:40.104732Z"},"links":{"citing_paper":"/paper/2506.21130"},"observation_digest":"sha256:6056b3ac0154a9de1c64773e1ec875477351b9554aca3ae035107aadc9a79aff","observation_id":"2e6a63ff-d2a1-4b01-81dc-59f73640efa0","resolution":{"observed_at":"2026-08-06T22:45:41.413591Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-12T06:34:41.77262+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-12T06:34:41.77262+00:00","source":"crossref"},{"observed_at":"2026-08-12T06:34:36.333875+00:00","source":"retraction_watch"}],"state":"measured"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":"raw_reference","pith_arxiv_id":null,"snapshot_observed_at":"2026-08-06T22:45:41.306852Z","title":"Higher-order invariants of immersions of surfaces into 3-space","venue":null,"work_id":"0ca6d291-8d94-4687-a8ef-a2229f44f0f4","year":2006},"citing_paper":{"arxiv_id":"2506.21130","last_updated":"2025-07-01T06:56:46Z","snapshot_observed_at":"2026-08-11T13:02:07.742105Z","submitted_at":"2025-06-26T10:12:30Z","title":"An Invariant for Triple-Point-Free Immersed Spheres","version":2},"reference_index":23,"source":"arxiv_source","source_observed_at":"2026-08-06T22:45:40.154747Z"},"links":{"citing_paper":"/paper/2506.21130"},"observation_digest":"sha256:1a8bd5f47b57fd85537b1ff167c348483a3fcbecaf664742a8e76e44d83ed796","observation_id":"af811ab0-67ca-49b0-8af8-400eeda48b71","resolution":{"observed_at":"2026-08-06T22:45:41.347583Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-12T06:34:41.77262+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-12T06:34:41.77262+00:00","source":"crossref"},{"observed_at":"2026-08-12T06:34:36.333875+00:00","source":"retraction_watch"}],"state":"measured"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":"raw_reference","pith_arxiv_id":null,"snapshot_observed_at":"2026-08-06T22:45:41.186384Z","title":"Dissecting the 2-sphere by immersions","venue":null,"work_id":"02ca4e69-fee3-413f-a772-efe8f5791be2","year":2007},"citing_paper":{"arxiv_id":"2506.21130","last_updated":"2025-07-01T06:56:46Z","snapshot_observed_at":"2026-08-11T13:02:07.742105Z","submitted_at":"2025-06-26T10:12:30Z","title":"An Invariant for Triple-Point-Free Immersed Spheres","version":2},"reference_index":24,"source":"arxiv_source","source_observed_at":"2026-08-06T22:45:40.194349Z"},"links":{"citing_paper":"/paper/2506.21130"},"observation_digest":"sha256:ea991725cf485b28cff86dee575d44b1d3b82f64bdd55c73be6ee43d7cbfb015","observation_id":"1e420d64-3371-472b-9dc7-074ddc34ffca","resolution":{"observed_at":"2026-08-06T22:45:41.233946Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-12T06:34:41.77262+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-12T06:34:41.77262+00:00","source":"crossref"},{"observed_at":"2026-08-12T06:34:36.333875+00:00","source":"retraction_watch"}],"state":"measured"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":"raw_reference","pith_arxiv_id":null,"snapshot_observed_at":"2026-08-06T22:45:41.045652Z","title":"Oset Sinha and R","venue":null,"work_id":"d148f76a-9491-484f-b89f-d75389c25834","year":2018},"citing_paper":{"arxiv_id":"2506.21130","last_updated":"2025-07-01T06:56:46Z","snapshot_observed_at":"2026-08-11T13:02:07.742105Z","submitted_at":"2025-06-26T10:12:30Z","title":"An Invariant for Triple-Point-Free Immersed Spheres","version":2},"reference_index":25,"source":"arxiv_source","source_observed_at":"2026-08-06T22:45:40.250717Z"},"links":{"citing_paper":"/paper/2506.21130"},"observation_digest":"sha256:0f9e6555278fcef8c6abba69c2f697be2f862e2ed5b009dc96ef44c9a7095f22","observation_id":"5a9c192b-7707-446a-b53c-9cc7f0f8c168","resolution":{"observed_at":"2026-08-06T22:45:41.105850Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-12T06:34:41.77262+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-12T06:34:41.77262+00:00","source":"crossref"},{"observed_at":"2026-08-12T06:34:36.333875+00:00","source":"retraction_watch"}],"state":"measured"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":"raw_reference","pith_arxiv_id":null,"snapshot_observed_at":"2026-08-06T22:45:40.899514Z","title":null,"venue":null,"work_id":"f595f7af-9fb1-4e0c-99f8-db351ea829f8","year":1976},"citing_paper":{"arxiv_id":"2506.21130","last_updated":"2025-07-01T06:56:46Z","snapshot_observed_at":"2026-08-11T13:02:07.742105Z","submitted_at":"2025-06-26T10:12:30Z","title":"An Invariant for Triple-Point-Free Immersed Spheres","version":2},"reference_index":26,"source":"arxiv_source","source_observed_at":"2026-08-06T22:45:40.358437Z"},"links":{"citing_paper":"/paper/2506.21130"},"observation_digest":"sha256:a49ef8b6f6efa2d3fbbf9aa68f5358f2a0918cd52342aa9a117d8f63c3017c22","observation_id":"a0dccd8b-05db-4864-bdee-a953307c641e","resolution":{"observed_at":"2026-08-06T22:45:40.963846Z","resolver_source":"raw_fallback","status":"unresolved"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-12T06:34:41.77262+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-12T06:34:41.77262+00:00","source":"crossref"},{"observed_at":"2026-08-12T06:34:36.333875+00:00","source":"retraction_watch"}],"state":"measured"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":null,"snapshot_observed_at":"2026-08-06T22:45:40.398727Z","title":"A classification of immersions of the two-sphere","venue":null,"work_id":null,"year":1958},"citing_paper":{"arxiv_id":"2506.21130","last_updated":"2025-07-01T06:56:46Z","snapshot_observed_at":"2026-08-11T13:02:07.742105Z","submitted_at":"2025-06-26T10:12:30Z","title":"An Invariant for Triple-Point-Free Immersed Spheres","version":2},"reference_index":27,"source":"arxiv_source","source_observed_at":"2026-08-06T22:45:40.398727Z"},"links":{"citing_paper":"/paper/2506.21130"},"observation_digest":"sha256:2273ec804bc407171d6e19db9c2aff1e110e720e91758d3646a5ad23df2fcbbc","observation_id":"4910778d-7278-49ff-ae4a-20534d051e34","resolution":{"observed_at":"2026-08-06T22:45:40.398727Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":null,"snapshot_observed_at":"2026-08-06T22:45:40.458479Z","title":"Diffeomorphisms of the 2 -sphere","venue":null,"work_id":null,"year":1959},"citing_paper":{"arxiv_id":"2506.21130","last_updated":"2025-07-01T06:56:46Z","snapshot_observed_at":"2026-08-11T13:02:07.742105Z","submitted_at":"2025-06-26T10:12:30Z","title":"An Invariant for Triple-Point-Free Immersed Spheres","version":2},"reference_index":28,"source":"arxiv_source","source_observed_at":"2026-08-06T22:45:40.458479Z"},"links":{"citing_paper":"/paper/2506.21130"},"observation_digest":"sha256:2ab5206ef01ecae7de78a1cd6488459b29b6241f906927a7960be5205aec24ff","observation_id":"6995a482-653d-4b93-bb1f-edc5aaa22fe0","resolution":{"observed_at":"2026-08-06T22:45:40.458479Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":"raw_reference","pith_arxiv_id":null,"snapshot_observed_at":"2026-08-06T22:45:40.779790Z","title":null,"venue":null,"work_id":"994204c8-76b1-4f44-95fd-b1dd8bf5f4b3","year":1990},"citing_paper":{"arxiv_id":"2506.21130","last_updated":"2025-07-01T06:56:46Z","snapshot_observed_at":"2026-08-11T13:02:07.742105Z","submitted_at":"2025-06-26T10:12:30Z","title":"An Invariant for Triple-Point-Free Immersed Spheres","version":2},"reference_index":29,"source":"arxiv_source","source_observed_at":"2026-08-06T22:45:40.559134Z"},"links":{"citing_paper":"/paper/2506.21130"},"observation_digest":"sha256:e3891fa472702de6d5d298dc8fbd7942a1a2bb8cc05ec035348c155a766abe45","observation_id":"4aa40c76-4d04-4d9e-bdb6-46583a37032b","resolution":{"observed_at":"2026-08-06T22:45:40.842369Z","resolver_source":"raw_fallback","status":"unresolved"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-12T06:34:41.77262+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-12T06:34:41.77262+00:00","source":"crossref"},{"observed_at":"2026-08-12T06:34:36.333875+00:00","source":"retraction_watch"}],"state":"measured"}}],"paper":{"arxiv_id":"2506.21130","last_updated":"2025-07-01T06:56:46Z","latest_version":2,"primary_category":"math.GT","snapshot_observed_at":"2026-08-11T13:02:07.742105Z","submitted_at":"2025-06-26T10:12:30Z","title":"An Invariant for Triple-Point-Free Immersed Spheres"},"reference_resolution":{"displayed":29,"state_counts":{"malformed_identifier":0,"metadata_mismatch":0,"parse_uncertain":0,"unresolved":14,"verified_exact":1,"verified_fuzzy":14},"total_outbound_references":29},"refusal":"A citation records a reference. It does not transfer a finding from one paper to another.","schema":"pith.paper-citation-record.v1","standing_sources":[{"observed_at":"2026-08-12T06:34:41.77262+00:00","source":"crossref"},{"observed_at":"2026-08-12T06:34:36.333875+00:00","source":"retraction_watch"}],"thesis":"As of 12 August 2026, this Paper Citation Record lists 29 of 29 outbound references and 1 inbound Pith citation observation for arXiv:2506.21130."}