{"as_of":"2026-08-16T15:21:00Z","caps":{"database_statements":6,"inbound":100,"outbound":100},"context_digest":"sha256:c7d989003f050f05c704ffa74fd2f3c39dc77d9bf982e9fdccfc428d0bf67753","coverage":[{"denominator":13,"lane":"reference_resolution","note":"Typed states for the displayed outbound observations.","records_observed":13,"source":"paper_references, paper_reference_links","source_observed_at":"2026-08-14T15:12:39.909128Z","state":"measured"},{"denominator":13,"lane":"standing_notices","note":"One-hop event checks from named stored sources.","records_observed":13,"source":"scholarly_work_events, retraction_status_cache","source_observed_at":"2026-08-16T06:30:59.297886+00:00","state":"measured"},{"denominator":0,"lane":"inbound_itemization","note":"Pith citing papers itemized under the disclosed page cap.","records_observed":0,"source":"paper_references, paper_reference_links","source_observed_at":null,"state":"measured"},{"denominator":1,"lane":"external_citation_measurements","note":"A source-named dated measurement, never combined with another source.","records_observed":0,"source":"cited_works","source_observed_at":null,"state":"measured"}],"external_citation_measurements":[],"inbound":[],"links":{"evidence":"/evidence","html":"/paper/1908.02611/citation-record","integrity":"/paper/1908.02611/integrity","json":"/paper/1908.02611/citation-record.json","paper":"/paper/1908.02611"},"outbound":[{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":"raw_reference","pith_arxiv_id":null,"snapshot_observed_at":"2026-08-14T15:12:40.087776Z","title":"Berend, Multi-Invariant Sets on Tori, Transactions of the American Mathematical Society, 280 (1983), 509-532","venue":null,"work_id":"c27cf7c9-9608-4ec3-8acc-681b57dc5675","year":1983},"citing_paper":{"arxiv_id":"1908.02611","last_updated":"2021-02-18T14:07:25Z","snapshot_observed_at":"2026-08-14T14:56:06.070178Z","submitted_at":"2019-08-06T05:21:32Z","title":"Strongly Independent Matrices and Rigidity of $\\times A$-Invariant Measures on $n$-Torus","version":2},"reference_index":1,"source":"arxiv_source","source_observed_at":"2026-08-14T15:12:39.858526Z"},"links":{"citing_paper":"/paper/1908.02611"},"observation_digest":"sha256:aad4ec620b1f2025cee51c2afd32cc8b7636ef18b61796ea5b45a6bb39403132","observation_id":"78d8ef15-aba0-4f76-8e16-ea77b67e6aa3","resolution":{"observed_at":"2026-08-14T15:12:40.091775Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-16T06:30:59.297886+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-16T06:30:59.297886+00:00","source":"crossref"},{"observed_at":"2026-08-16T06:30:54.164669+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":"2026-08-14T15:12:40.077057Z","title":null,"venue":null,"work_id":"f503d052-4bf3-4371-974b-5ed6f31e8aae","year":1971},"citing_paper":{"arxiv_id":"1908.02611","last_updated":"2021-02-18T14:07:25Z","snapshot_observed_at":"2026-08-14T14:56:06.070178Z","submitted_at":"2019-08-06T05:21:32Z","title":"Strongly Independent Matrices and Rigidity of $\\times A$-Invariant Measures on $n$-Torus","version":2},"reference_index":2,"source":"arxiv_source","source_observed_at":"2026-08-14T15:12:39.863468Z"},"links":{"citing_paper":"/paper/1908.02611"},"observation_digest":"sha256:a283fe69b143efb2594394648995dfe3e08a9a98ed8075e9bf3f656236effdc4","observation_id":"ff9d3a25-52b2-498c-a9fe-af6f0bf38a24","resolution":{"observed_at":"2026-08-14T15:12:40.080966Z","resolver_source":"raw_fallback","status":"unresolved"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-16T06:30:59.297886+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-16T06:30:59.297886+00:00","source":"crossref"},{"observed_at":"2026-08-16T06:30:54.164669+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":"2026-08-14T15:12:40.066919Z","title":"Furstenberg","venue":null,"work_id":"89829bf0-9a9b-4a39-be5e-81e09f10c3f6","year":1967},"citing_paper":{"arxiv_id":"1908.02611","last_updated":"2021-02-18T14:07:25Z","snapshot_observed_at":"2026-08-14T14:56:06.070178Z","submitted_at":"2019-08-06T05:21:32Z","title":"Strongly Independent Matrices and Rigidity of $\\times A$-Invariant Measures on $n$-Torus","version":2},"reference_index":3,"source":"arxiv_source","source_observed_at":"2026-08-14T15:12:39.867452Z"},"links":{"citing_paper":"/paper/1908.02611"},"observation_digest":"sha256:0f2454cd46917900c1d3463480cd214cde3964895b331ba8a9f05827c4c108e9","observation_id":"dd635106-fcc2-40d3-863e-45e0241b232f","resolution":{"observed_at":"2026-08-14T15:12:40.070511Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-16T06:30:59.297886+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-16T06:30:59.297886+00:00","source":"crossref"},{"observed_at":"2026-08-16T06:30:54.164669+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":"2026-08-14T15:12:40.055787Z","title":null,"venue":null,"work_id":"937c2579-d9de-410e-bae1-0078cf208d71","year":2017},"citing_paper":{"arxiv_id":"1908.02611","last_updated":"2021-02-18T14:07:25Z","snapshot_observed_at":"2026-08-14T14:56:06.070178Z","submitted_at":"2019-08-06T05:21:32Z","title":"Strongly Independent Matrices and Rigidity of $\\times A$-Invariant Measures on $n$-Torus","version":2},"reference_index":4,"source":"arxiv_source","source_observed_at":"2026-08-14T15:12:39.872073Z"},"links":{"citing_paper":"/paper/1908.02611"},"observation_digest":"sha256:4307072ce6248002b47b567eb828ee8d541530233984aff75533e05a9cf83561","observation_id":"159f25c8-35e0-4e43-a0da-81262a1d99fd","resolution":{"observed_at":"2026-08-14T15:12:40.059874Z","resolver_source":"raw_fallback","status":"unresolved"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-16T06:30:59.297886+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-16T06:30:59.297886+00:00","source":"crossref"},{"observed_at":"2026-08-16T06:30:54.164669+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":"2026-08-14T15:12:40.040141Z","title":null,"venue":null,"work_id":"0da832da-2bcd-4f20-98cc-f668a6fab0d6","year":1992},"citing_paper":{"arxiv_id":"1908.02611","last_updated":"2021-02-18T14:07:25Z","snapshot_observed_at":"2026-08-14T14:56:06.070178Z","submitted_at":"2019-08-06T05:21:32Z","title":"Strongly Independent Matrices and Rigidity of $\\times A$-Invariant Measures on $n$-Torus","version":2},"reference_index":5,"source":"arxiv_source","source_observed_at":"2026-08-14T15:12:39.876213Z"},"links":{"citing_paper":"/paper/1908.02611"},"observation_digest":"sha256:f0a060123f52f66f9294cdeb5090042cd28d686f4bcaa858413e5d6c7f89e95b","observation_id":"e5d54c3a-2db8-4e86-bd51-5103d8440ce7","resolution":{"observed_at":"2026-08-14T15:12:40.046894Z","resolver_source":"raw_fallback","status":"unresolved"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-16T06:30:59.297886+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-16T06:30:59.297886+00:00","source":"crossref"},{"observed_at":"2026-08-16T06:30:54.164669+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":"2026-08-14T15:12:40.026071Z","title":"Kalinin and A","venue":null,"work_id":"76c303ea-e786-4f75-bcde-5cf1c739bac4","year":1999},"citing_paper":{"arxiv_id":"1908.02611","last_updated":"2021-02-18T14:07:25Z","snapshot_observed_at":"2026-08-14T14:56:06.070178Z","submitted_at":"2019-08-06T05:21:32Z","title":"Strongly Independent Matrices and Rigidity of $\\times A$-Invariant Measures on $n$-Torus","version":2},"reference_index":6,"source":"arxiv_source","source_observed_at":"2026-08-14T15:12:39.880723Z"},"links":{"citing_paper":"/paper/1908.02611"},"observation_digest":"sha256:f74cb51b2cb6e71cb0be22d2736b1558d8f9049e4442fe181dc87732781c1780","observation_id":"2ac6dfe4-f3c6-4184-a64f-6e09f3cc1f59","resolution":{"observed_at":"2026-08-14T15:12:40.030509Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-16T06:30:59.297886+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-16T06:30:59.297886+00:00","source":"crossref"},{"observed_at":"2026-08-16T06:30:54.164669+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":"2026-08-14T15:12:40.013992Z","title":"Katok and R","venue":null,"work_id":"36a5c0ca-1ad8-4535-a0b5-4ab8bc23f165","year":1996},"citing_paper":{"arxiv_id":"1908.02611","last_updated":"2021-02-18T14:07:25Z","snapshot_observed_at":"2026-08-14T14:56:06.070178Z","submitted_at":"2019-08-06T05:21:32Z","title":"Strongly Independent Matrices and Rigidity of $\\times A$-Invariant Measures on $n$-Torus","version":2},"reference_index":7,"source":"arxiv_source","source_observed_at":"2026-08-14T15:12:39.885959Z"},"links":{"citing_paper":"/paper/1908.02611"},"observation_digest":"sha256:f7782e95a09c70b3f01dbd1f5f774aae151e0c73ffe39ba8ab1cabdd60b8eb4d","observation_id":"26f10ae7-f9ac-456d-9d20-00dbf8cc877d","resolution":{"observed_at":"2026-08-14T15:12:40.017790Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-16T06:30:59.297886+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-16T06:30:59.297886+00:00","source":"crossref"},{"observed_at":"2026-08-16T06:30:54.164669+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":"2026-08-14T15:12:40.000323Z","title":"Katok and R","venue":null,"work_id":"da5a8d41-9cf0-40c7-89ff-fed253bf28e4","year":1998},"citing_paper":{"arxiv_id":"1908.02611","last_updated":"2021-02-18T14:07:25Z","snapshot_observed_at":"2026-08-14T14:56:06.070178Z","submitted_at":"2019-08-06T05:21:32Z","title":"Strongly Independent Matrices and Rigidity of $\\times A$-Invariant Measures on $n$-Torus","version":2},"reference_index":8,"source":"arxiv_source","source_observed_at":"2026-08-14T15:12:39.889423Z"},"links":{"citing_paper":"/paper/1908.02611"},"observation_digest":"sha256:0705a5626963f90f6adf2a30d67187b19617dc6276dbc2d6d8f6c41fe9090830","observation_id":"79c751b6-07f4-4c46-be38-8f4e77f401ff","resolution":{"observed_at":"2026-08-14T15:12:40.004791Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-16T06:30:59.297886+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-16T06:30:59.297886+00:00","source":"crossref"},{"observed_at":"2026-08-16T06:30:54.164669+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":"2026-08-14T15:12:39.988906Z","title":null,"venue":null,"work_id":"d5c75575-79aa-4699-8f40-80c642a7acda","year":2002},"citing_paper":{"arxiv_id":"1908.02611","last_updated":"2021-02-18T14:07:25Z","snapshot_observed_at":"2026-08-14T14:56:06.070178Z","submitted_at":"2019-08-06T05:21:32Z","title":"Strongly Independent Matrices and Rigidity of $\\times A$-Invariant Measures on $n$-Torus","version":2},"reference_index":9,"source":"arxiv_source","source_observed_at":"2026-08-14T15:12:39.893720Z"},"links":{"citing_paper":"/paper/1908.02611"},"observation_digest":"sha256:b48c9b71fb55b5dd80eca1465faa915c1c7b6e59c08e6f76189c0d7c54d1a1fa","observation_id":"73ef5cf9-4d72-4450-af94-1480aadc2e71","resolution":{"observed_at":"2026-08-14T15:12:39.992490Z","resolver_source":"raw_fallback","status":"unresolved"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-16T06:30:59.297886+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-16T06:30:59.297886+00:00","source":"crossref"},{"observed_at":"2026-08-16T06:30:54.164669+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":"2026-08-14T15:12:39.977070Z","title":null,"venue":null,"work_id":"b763854b-23f8-4d20-ba9e-0e3f039b0773","year":1988},"citing_paper":{"arxiv_id":"1908.02611","last_updated":"2021-02-18T14:07:25Z","snapshot_observed_at":"2026-08-14T14:56:06.070178Z","submitted_at":"2019-08-06T05:21:32Z","title":"Strongly Independent Matrices and Rigidity of $\\times A$-Invariant Measures on $n$-Torus","version":2},"reference_index":10,"source":"arxiv_source","source_observed_at":"2026-08-14T15:12:39.897687Z"},"links":{"citing_paper":"/paper/1908.02611"},"observation_digest":"sha256:6a8d23adb92a08ada450ba33cb048ea668fd6ffa6c47bf35d178fb9731c96eec","observation_id":"a8139ea9-1067-4eeb-a1ce-916fe5a0e694","resolution":{"observed_at":"2026-08-14T15:12:39.980676Z","resolver_source":"raw_fallback","status":"unresolved"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-16T06:30:59.297886+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-16T06:30:59.297886+00:00","source":"crossref"},{"observed_at":"2026-08-16T06:30:54.164669+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":"2026-08-14T15:12:39.966268Z","title":null,"venue":null,"work_id":"b4408d84-2855-4839-986c-f6bb386b77bc","year":2008},"citing_paper":{"arxiv_id":"1908.02611","last_updated":"2021-02-18T14:07:25Z","snapshot_observed_at":"2026-08-14T14:56:06.070178Z","submitted_at":"2019-08-06T05:21:32Z","title":"Strongly Independent Matrices and Rigidity of $\\times A$-Invariant Measures on $n$-Torus","version":2},"reference_index":11,"source":"arxiv_source","source_observed_at":"2026-08-14T15:12:39.901539Z"},"links":{"citing_paper":"/paper/1908.02611"},"observation_digest":"sha256:b4a26d81772bd4cf4ebc38b051e9dedc36005314d6f032c55eac951ba8d35024","observation_id":"1bb4e104-c6f4-4040-8865-4981d41a8ad5","resolution":{"observed_at":"2026-08-14T15:12:39.969901Z","resolver_source":"raw_fallback","status":"unresolved"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-16T06:30:59.297886+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-16T06:30:59.297886+00:00","source":"crossref"},{"observed_at":"2026-08-16T06:30:54.164669+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":"2026-08-14T15:12:39.953892Z","title":null,"venue":null,"work_id":"0ff0264a-5ef0-4b12-a8fe-332fa8bbf0be","year":1990},"citing_paper":{"arxiv_id":"1908.02611","last_updated":"2021-02-18T14:07:25Z","snapshot_observed_at":"2026-08-14T14:56:06.070178Z","submitted_at":"2019-08-06T05:21:32Z","title":"Strongly Independent Matrices and Rigidity of $\\times A$-Invariant Measures on $n$-Torus","version":2},"reference_index":12,"source":"arxiv_source","source_observed_at":"2026-08-14T15:12:39.905588Z"},"links":{"citing_paper":"/paper/1908.02611"},"observation_digest":"sha256:d4b9193a9ae50a0f2e8eddfbb111b2a4b80827c4f1b6eccd70d48be8ce10b1b2","observation_id":"9b050a1b-ce99-4134-82fe-d9c853c44e9d","resolution":{"observed_at":"2026-08-14T15:12:39.958312Z","resolver_source":"raw_fallback","status":"unresolved"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-16T06:30:59.297886+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-16T06:30:59.297886+00:00","source":"crossref"},{"observed_at":"2026-08-16T06:30:54.164669+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":"2026-08-14T15:12:39.939801Z","title":null,"venue":null,"work_id":"41e398c8-f509-4f25-bf8b-41c90ccd0fb3","year":1982},"citing_paper":{"arxiv_id":"1908.02611","last_updated":"2021-02-18T14:07:25Z","snapshot_observed_at":"2026-08-14T14:56:06.070178Z","submitted_at":"2019-08-06T05:21:32Z","title":"Strongly Independent Matrices and Rigidity of $\\times A$-Invariant Measures on $n$-Torus","version":2},"reference_index":13,"source":"arxiv_source","source_observed_at":"2026-08-14T15:12:39.909128Z"},"links":{"citing_paper":"/paper/1908.02611"},"observation_digest":"sha256:32239683bdb84797cddd1aed2c994defb16dd58bf53d19223add2f6673336ad0","observation_id":"9b25b297-a07e-4434-a226-2f98a1aa1d32","resolution":{"observed_at":"2026-08-14T15:12:39.945334Z","resolver_source":"raw_fallback","status":"unresolved"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-16T06:30:59.297886+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-16T06:30:59.297886+00:00","source":"crossref"},{"observed_at":"2026-08-16T06:30:54.164669+00:00","source":"retraction_watch"}],"state":"measured"}}],"paper":{"arxiv_id":"1908.02611","last_updated":"2021-02-18T14:07:25Z","latest_version":2,"primary_category":"math.DS","snapshot_observed_at":"2026-08-14T14:56:06.070178Z","submitted_at":"2019-08-06T05:21:32Z","title":"Strongly Independent Matrices and Rigidity of $\\times A$-Invariant Measures on $n$-Torus"},"reference_resolution":{"displayed":13,"state_counts":{"malformed_identifier":0,"metadata_mismatch":0,"parse_uncertain":0,"unresolved":8,"verified_exact":0,"verified_fuzzy":5},"total_outbound_references":13},"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":"2026-08-16T06:30:59.297886+00:00","source":"crossref"},{"observed_at":"2026-08-16T06:30:54.164669+00:00","source":"retraction_watch"}],"thesis":"As of 16 August 2026, this Paper Citation Record lists 13 of 13 outbound references and 0 inbound Pith citation observations for arXiv:1908.02611."}