{"as_of":"2026-08-18T22:02:00Z","caps":{"database_statements":6,"inbound":100,"outbound":100},"context_digest":"sha256:3a4443f7e571f4c5832db0f8f52507fdb0f551b92b2c931bba2b8cbcbaab4f8c","coverage":[{"denominator":53,"lane":"reference_resolution","note":"Typed states for the displayed outbound observations.","records_observed":53,"source":"paper_references, paper_reference_links","source_observed_at":"2026-08-10T23:19:40.076313Z","state":"measured"},{"denominator":53,"lane":"standing_notices","note":"One-hop event checks from named stored sources.","records_observed":53,"source":"scholarly_work_events, retraction_status_cache","source_observed_at":"2026-08-18T06:34:40.430872+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/2412.20721/citation-record","integrity":"/paper/2412.20721/integrity","json":"/paper/2412.20721/citation-record.json","paper":"/paper/2412.20721"},"outbound":[{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":"raw_reference","pith_arxiv_id":null,"snapshot_observed_at":"2026-08-10T23:19:40.810860Z","title":"The monotone circuit complexity of boolean functions","venue":null,"work_id":"044f5830-d435-4432-a617-409503d6537b","year":1987},"citing_paper":{"arxiv_id":"2412.20721","last_updated":"2024-12-30T05:41:52Z","snapshot_observed_at":"2026-08-13T11:44:17.224732Z","submitted_at":"2024-12-30T05:41:52Z","title":"Ramanujan Graphs and Interlacing Families","version":1},"reference_index":1,"source":"pdf_text","source_observed_at":"2026-08-10T23:19:39.877187Z"},"links":{"citing_paper":"/paper/2412.20721"},"observation_digest":"sha256:c35e9b404283ee63d2cfe826038e59c17b96056035604d1d91aec5940921d8be","observation_id":"5956cd60-1974-44b0-964b-b73f565b4ed6","resolution":{"observed_at":"2026-08-10T23:19:40.814160Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-18T06:34:40.430872+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-18T06:34:40.430872+00:00","source":"crossref"},{"observed_at":"2026-08-18T06:34:34.496301+00:00","source":"retraction_watch"}],"state":"measured"}},{"citation":{"cited_paper":{"arxiv_id":"1905.02264","last_updated":"2019-05-08T19:48:29Z","snapshot_observed_at":"2026-08-17T17:21:55.221619Z","submitted_at":"2019-05-06T21:14:02Z","title":"Stable multivariate generalizations of matching polynomials","version":2},"cited_work":{"arxiv_id":"1905.02264","doi":null,"metadata_source":"pith","pith_arxiv_id":"1905.02264","snapshot_observed_at":"2026-08-10T23:19:40.343720Z","title":"Stable multivariate generalizations of matching polynomials","venue":"math.CO","work_id":"f786e12d-a7ca-4bb8-b345-a87519b83a9a","year":2019},"citing_paper":{"arxiv_id":"2412.20721","last_updated":"2024-12-30T05:41:52Z","snapshot_observed_at":"2026-08-13T11:44:17.224732Z","submitted_at":"2024-12-30T05:41:52Z","title":"Ramanujan Graphs and Interlacing Families","version":1},"reference_index":2,"source":"pdf_text","source_observed_at":"2026-08-10T23:19:39.880925Z"},"links":{"cited_paper":"/paper/1905.02264","citing_paper":"/paper/2412.20721"},"observation_digest":"sha256:9d5cc37a82cd2fddc8b9d6922e806b8d59ed901f514640b5af9a58daf3a85db2","observation_id":"77d9e492-a5e6-47e1-8be7-a74bc495e599","resolution":{"observed_at":"2026-08-10T23:19:40.349016Z","resolver_source":"local_arxiv","status":"verified_exact"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-18T06:34:40.430872+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-18T06:34:40.430872+00:00","source":"crossref"},{"observed_at":"2026-08-18T06:34:34.496301+00:00","source":"retraction_watch"}],"state":"measured"}},{"citation":{"cited_paper":{"arxiv_id":"1412.1143","last_updated":"2015-07-22T18:15:39Z","snapshot_observed_at":"2026-08-14T23:09:12.903410Z","submitted_at":"2014-12-03T00:50:46Z","title":"The Kadison-Singer Problem for Strongly Rayleigh Measures and Applications to Asymmetric TSP","version":2},"cited_work":{"arxiv_id":"1412.1143","doi":null,"metadata_source":"pith","pith_arxiv_id":"1412.1143","snapshot_observed_at":"2026-08-10T23:19:40.328502Z","title":"The Kadison-Singer Problem for Strongly Rayleigh Measures and Applications to Asymmetric TSP","venue":"cs.DS","work_id":"d7bf415b-d6cb-4fdc-a997-1c08d80d510f","year":2014},"citing_paper":{"arxiv_id":"2412.20721","last_updated":"2024-12-30T05:41:52Z","snapshot_observed_at":"2026-08-13T11:44:17.224732Z","submitted_at":"2024-12-30T05:41:52Z","title":"Ramanujan Graphs and Interlacing Families","version":1},"reference_index":3,"source":"pdf_text","source_observed_at":"2026-08-10T23:19:39.884777Z"},"links":{"cited_paper":"/paper/1412.1143","citing_paper":"/paper/2412.20721"},"observation_digest":"sha256:a3d96917710c8bd9e4d9508d5a508f7ba284f3e2e1907fd340ce95cce181b0db","observation_id":"2abcd8f2-0e03-41cb-b867-e74de45c5aad","resolution":{"observed_at":"2026-08-10T23:19:40.332881Z","resolver_source":"local_arxiv","status":"verified_exact"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-18T06:34:40.430872+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-18T06:34:40.430872+00:00","source":"crossref"},{"observed_at":"2026-08-18T06:34:34.496301+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":"2026-08-10T23:19:40.801214Z","title":"Approximating the largest root and applications to interlacing families","venue":null,"work_id":"dc95825c-a5a2-446f-8189-21073a6d28ad","year":2018},"citing_paper":{"arxiv_id":"2412.20721","last_updated":"2024-12-30T05:41:52Z","snapshot_observed_at":"2026-08-13T11:44:17.224732Z","submitted_at":"2024-12-30T05:41:52Z","title":"Ramanujan Graphs and Interlacing Families","version":1},"reference_index":4,"source":"pdf_text","source_observed_at":"2026-08-10T23:19:39.888831Z"},"links":{"citing_paper":"/paper/2412.20721"},"observation_digest":"sha256:63919c1703719f0d715c9d456d9ce8dae75e49420c00cd1aa24b8da5c877198c","observation_id":"ea4a3d40-ee2e-4570-a025-550cfe1eee36","resolution":{"observed_at":"2026-08-10T23:19:40.804623Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-18T06:34:40.430872+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-18T06:34:40.430872+00:00","source":"crossref"},{"observed_at":"2026-08-18T06:34:34.496301+00:00","source":"retraction_watch"}],"state":"measured"}},{"citation":{"cited_paper":{"arxiv_id":"2408.09337","last_updated":"2025-02-05T16:59:03Z","snapshot_observed_at":"2026-08-16T13:25:47.352630Z","submitted_at":"2024-08-18T02:35:13Z","title":"$S$-transform in Finite Free Probability","version":2},"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":"2408.09337","snapshot_observed_at":"2026-08-10T23:19:39.892932Z","title":"s-transform in ﬁnite free probability","venue":null,"work_id":null,"year":2024},"citing_paper":{"arxiv_id":"2412.20721","last_updated":"2024-12-30T05:41:52Z","snapshot_observed_at":"2026-08-13T11:44:17.224732Z","submitted_at":"2024-12-30T05:41:52Z","title":"Ramanujan Graphs and Interlacing Families","version":1},"reference_index":5,"source":"pdf_text","source_observed_at":"2026-08-10T23:19:39.892932Z"},"links":{"cited_paper":"/paper/2408.09337","citing_paper":"/paper/2412.20721"},"observation_digest":"sha256:53df364b6e9276eaafd1b2f2eafba9e00312c454c22a4cbf35acdcfbf7a4c5ec","observation_id":"a4f082ca-7826-472f-974e-6fd7510f472d","resolution":{"observed_at":"2026-08-10T23:19:39.892932Z","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":"2026-08-10T23:19:40.790298Z","title":"Fin ite free cumulants: multiplicative convolutions, genus expansion and inﬁnitesimal distribu tions","venue":null,"work_id":"4df8d89c-33ed-467d-ad77-3b8e12dd7dea","year":2023},"citing_paper":{"arxiv_id":"2412.20721","last_updated":"2024-12-30T05:41:52Z","snapshot_observed_at":"2026-08-13T11:44:17.224732Z","submitted_at":"2024-12-30T05:41:52Z","title":"Ramanujan Graphs and Interlacing Families","version":1},"reference_index":6,"source":"pdf_text","source_observed_at":"2026-08-10T23:19:39.896909Z"},"links":{"citing_paper":"/paper/2412.20721"},"observation_digest":"sha256:9496f72d87162d5458c1d0d0d828f97db0fec252895530554b6c9a1069d9e622","observation_id":"1484fa07-43bc-4b13-8291-eccb1d5231e6","resolution":{"observed_at":"2026-08-10T23:19:40.794185Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-18T06:34:40.430872+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-18T06:34:40.430872+00:00","source":"crossref"},{"observed_at":"2026-08-18T06:34:34.496301+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":"2026-08-10T23:19:40.778068Z","title":"Cumulants for ﬁnite free c onvolution","venue":null,"work_id":"563c17ce-a51e-42c3-b250-875e5c9c250e","year":2018},"citing_paper":{"arxiv_id":"2412.20721","last_updated":"2024-12-30T05:41:52Z","snapshot_observed_at":"2026-08-13T11:44:17.224732Z","submitted_at":"2024-12-30T05:41:52Z","title":"Ramanujan Graphs and Interlacing Families","version":1},"reference_index":7,"source":"pdf_text","source_observed_at":"2026-08-10T23:19:39.901149Z"},"links":{"citing_paper":"/paper/2412.20721"},"observation_digest":"sha256:589c94b1f8a58d2ea180163fbeafe2479addc08a42027231235ea1a67bde66b4","observation_id":"0fc0047e-7d3c-45b2-838d-f26500e4d1e0","resolution":{"observed_at":"2026-08-10T23:19:40.782689Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-18T06:34:40.430872+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-18T06:34:40.430872+00:00","source":"crossref"},{"observed_at":"2026-08-18T06:34:34.496301+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":"2026-08-10T23:19:40.766787Z","title":"Edge rigidity and universality of random regular graphs of intermediate de gree","venue":null,"work_id":"d44fb89c-0e09-4c63-a7a8-d3cfd9084ff2","year":2020},"citing_paper":{"arxiv_id":"2412.20721","last_updated":"2024-12-30T05:41:52Z","snapshot_observed_at":"2026-08-13T11:44:17.224732Z","submitted_at":"2024-12-30T05:41:52Z","title":"Ramanujan Graphs and Interlacing Families","version":1},"reference_index":8,"source":"pdf_text","source_observed_at":"2026-08-10T23:19:39.904803Z"},"links":{"citing_paper":"/paper/2412.20721"},"observation_digest":"sha256:7198a1e71b1fce95c0e8daa66e433609f6ebaa249a028f6beecea5b79d5d07c3","observation_id":"9d073d40-6370-48b4-80ec-7b8bd8adce4d","resolution":{"observed_at":"2026-08-10T23:19:40.771078Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-18T06:34:40.430872+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-18T06:34:40.430872+00:00","source":"crossref"},{"observed_at":"2026-08-18T06:34:34.496301+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":"2026-08-10T23:19:40.755350Z","title":"Lifts, discrepancy and nearly optim al spectral gap","venue":null,"work_id":"2fd11871-1c39-402b-9933-dc8b31c5c87a","year":2006},"citing_paper":{"arxiv_id":"2412.20721","last_updated":"2024-12-30T05:41:52Z","snapshot_observed_at":"2026-08-13T11:44:17.224732Z","submitted_at":"2024-12-30T05:41:52Z","title":"Ramanujan Graphs and Interlacing Families","version":1},"reference_index":9,"source":"pdf_text","source_observed_at":"2026-08-10T23:19:39.908338Z"},"links":{"citing_paper":"/paper/2412.20721"},"observation_digest":"sha256:e7e2da76ecaf612c3cee7a8d19a4772c8b3b4e140ed8f6201abde929c6b28864","observation_id":"8a26e3f8-2300-4c80-8dcc-361a21f885f3","resolution":{"observed_at":"2026-08-10T23:19:40.759350Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-18T06:34:40.430872+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-18T06:34:40.430872+00:00","source":"crossref"},{"observed_at":"2026-08-18T06:34:34.496301+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":"2026-08-10T23:19:40.743295Z","title":"The lee-yang and p´ olya-schur programs","venue":null,"work_id":"3283e8c9-b036-493f-ad8d-402b5914cb95","year":2009},"citing_paper":{"arxiv_id":"2412.20721","last_updated":"2024-12-30T05:41:52Z","snapshot_observed_at":"2026-08-13T11:44:17.224732Z","submitted_at":"2024-12-30T05:41:52Z","title":"Ramanujan Graphs and Interlacing Families","version":1},"reference_index":10,"source":"pdf_text","source_observed_at":"2026-08-10T23:19:39.912241Z"},"links":{"citing_paper":"/paper/2412.20721"},"observation_digest":"sha256:216c4077de3f740f7ed40f0f926ec67b9f70fe70d0700243980fdb49b57ee2dc","observation_id":"e1070649-d84b-4688-a753-b58e742a519e","resolution":{"observed_at":"2026-08-10T23:19:40.747511Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-18T06:34:40.430872+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-18T06:34:40.430872+00:00","source":"crossref"},{"observed_at":"2026-08-18T06:34:34.496301+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":"2026-08-10T23:19:40.730403Z","title":"The lee-yang and p´ olya-schur programs","venue":null,"work_id":"c36147e6-631e-4590-9860-84a1ae61f318","year":2009},"citing_paper":{"arxiv_id":"2412.20721","last_updated":"2024-12-30T05:41:52Z","snapshot_observed_at":"2026-08-13T11:44:17.224732Z","submitted_at":"2024-12-30T05:41:52Z","title":"Ramanujan Graphs and Interlacing Families","version":1},"reference_index":11,"source":"pdf_text","source_observed_at":"2026-08-10T23:19:39.916272Z"},"links":{"citing_paper":"/paper/2412.20721"},"observation_digest":"sha256:1d6630cf8ca89955e31ee4b481ceda19038e56e61bb84975df29b87fbac8ace9","observation_id":"ff472b08-29af-4875-8b07-ae01b163b739","resolution":{"observed_at":"2026-08-10T23:19:40.734746Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-18T06:34:40.430872+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-18T06:34:40.430872+00:00","source":"crossref"},{"observed_at":"2026-08-18T06:34:34.496301+00:00","source":"retraction_watch"}],"state":"measured"}},{"citation":{"cited_paper":{"arxiv_id":"1502.04482","last_updated":"2019-03-05T21:20:57Z","snapshot_observed_at":"2026-08-14T22:59:54.594367Z","submitted_at":"2015-02-16T10:12:45Z","title":"A new proof of Friedman's second eigenvalue Theorem and its extension to random lifts","version":4},"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":"1502.04482","snapshot_observed_at":"2026-08-10T23:19:39.920338Z","title":"A new proof of friedman’s second eigenvalu e theorem and its ex- tension to random lifts","venue":null,"work_id":null,"year":2015},"citing_paper":{"arxiv_id":"2412.20721","last_updated":"2024-12-30T05:41:52Z","snapshot_observed_at":"2026-08-13T11:44:17.224732Z","submitted_at":"2024-12-30T05:41:52Z","title":"Ramanujan Graphs and Interlacing Families","version":1},"reference_index":12,"source":"pdf_text","source_observed_at":"2026-08-10T23:19:39.920338Z"},"links":{"cited_paper":"/paper/1502.04482","citing_paper":"/paper/2412.20721"},"observation_digest":"sha256:4b6c995ea9367fd5c982c483f57258077f87b2ae83386ee5d3f3d80360e2d0c6","observation_id":"39bdef6c-99ad-45fd-a637-77c57afd6a72","resolution":{"observed_at":"2026-08-10T23:19:39.920338Z","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":"2026-08-10T23:19:40.718697Z","title":"Eigenvalues of random lift s and polynomials of random permutation matrices","venue":null,"work_id":"c4dafaf4-547d-4b5a-9e18-1f38137013e1","year":2019},"citing_paper":{"arxiv_id":"2412.20721","last_updated":"2024-12-30T05:41:52Z","snapshot_observed_at":"2026-08-13T11:44:17.224732Z","submitted_at":"2024-12-30T05:41:52Z","title":"Ramanujan Graphs and Interlacing Families","version":1},"reference_index":13,"source":"pdf_text","source_observed_at":"2026-08-10T23:19:39.924384Z"},"links":{"citing_paper":"/paper/2412.20721"},"observation_digest":"sha256:b0884e6774cf005784721ca1afbaf72c0173e28b091751e4d3651569da2d5e6d","observation_id":"66c81abb-6e44-4868-b150-241f5ea04022","resolution":{"observed_at":"2026-08-10T23:19:40.722840Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-18T06:34:40.430872+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-18T06:34:40.430872+00:00","source":"crossref"},{"observed_at":"2026-08-18T06:34:34.496301+00:00","source":"retraction_watch"}],"state":"measured"}},{"citation":{"cited_paper":{"arxiv_id":"2303.12954","last_updated":"2023-03-22T23:33:07Z","snapshot_observed_at":"2026-08-18T16:44:08.184464Z","submitted_at":"2023-03-22T23:33:07Z","title":"On Akemann-Weaver Conjecture","version":1},"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":"2303.12954","snapshot_observed_at":"2026-08-10T23:19:39.928190Z","title":"On akemann-weaver conjecture","venue":null,"work_id":null,"year":2023},"citing_paper":{"arxiv_id":"2412.20721","last_updated":"2024-12-30T05:41:52Z","snapshot_observed_at":"2026-08-13T11:44:17.224732Z","submitted_at":"2024-12-30T05:41:52Z","title":"Ramanujan Graphs and Interlacing Families","version":1},"reference_index":14,"source":"pdf_text","source_observed_at":"2026-08-10T23:19:39.928190Z"},"links":{"cited_paper":"/paper/2303.12954","citing_paper":"/paper/2412.20721"},"observation_digest":"sha256:b0abe2894c25a1bd78e829e52d5a877ebcb8b3584877cf296f53d319b27f9600","observation_id":"855289a3-0914-4232-b88c-5aca86d7c6fb","resolution":{"observed_at":"2026-08-10T23:19:39.928190Z","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":"2026-08-10T23:19:39.932103Z","title":"A new approach to strong convergence","venue":null,"work_id":null,"year":2024},"citing_paper":{"arxiv_id":"2412.20721","last_updated":"2024-12-30T05:41:52Z","snapshot_observed_at":"2026-08-13T11:44:17.224732Z","submitted_at":"2024-12-30T05:41:52Z","title":"Ramanujan Graphs and Interlacing Families","version":1},"reference_index":15,"source":"pdf_text","source_observed_at":"2026-08-10T23:19:39.932103Z"},"links":{"citing_paper":"/paper/2412.20721"},"observation_digest":"sha256:6b6a2c45ae5840ed1465268c00f8f80285f793e69d0b9e3dabfc5a2cbb2dd621","observation_id":"c5aa375c-6e9b-4d64-8c6c-580dc8f9a7d3","resolution":{"observed_at":"2026-08-10T23:19:39.932103Z","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":"2026-08-10T23:19:40.708081Z","title":"Ramanujan graphs and shimura curves, 2006","venue":null,"work_id":"e5df2069-8d4c-421d-a5c9-48009f653729","year":2006},"citing_paper":{"arxiv_id":"2412.20721","last_updated":"2024-12-30T05:41:52Z","snapshot_observed_at":"2026-08-13T11:44:17.224732Z","submitted_at":"2024-12-30T05:41:52Z","title":"Ramanujan Graphs and Interlacing Families","version":1},"reference_index":16,"source":"pdf_text","source_observed_at":"2026-08-10T23:19:39.936589Z"},"links":{"citing_paper":"/paper/2412.20721"},"observation_digest":"sha256:c0648aa2b93ac667bf4e5fbb4f8b67bc5301d96530cd19a020f65f0f00d4ee5a","observation_id":"994994f0-7c05-49e4-9f2b-4de2928103fd","resolution":{"observed_at":"2026-08-10T23:19:40.712260Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-18T06:34:40.430872+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-18T06:34:40.430872+00:00","source":"crossref"},{"observed_at":"2026-08-18T06:34:34.496301+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":"2026-08-10T23:19:40.698055Z","title":"Ramanujan graphs in polynomial time","venue":null,"work_id":"261ab96d-0bae-44db-9d39-684829544a69","year":2016},"citing_paper":{"arxiv_id":"2412.20721","last_updated":"2024-12-30T05:41:52Z","snapshot_observed_at":"2026-08-13T11:44:17.224732Z","submitted_at":"2024-12-30T05:41:52Z","title":"Ramanujan Graphs and Interlacing Families","version":1},"reference_index":17,"source":"pdf_text","source_observed_at":"2026-08-10T23:19:39.940616Z"},"links":{"citing_paper":"/paper/2412.20721"},"observation_digest":"sha256:738d8ebf6a76229b9f2e19d90c99f15cb079ebf9e82dd6d347f1fdc3f054004f","observation_id":"bd10d6c3-cb88-467c-8c4f-fe4f299cb1cb","resolution":{"observed_at":"2026-08-10T23:19:40.701332Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-18T06:34:40.430872+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-18T06:34:40.430872+00:00","source":"crossref"},{"observed_at":"2026-08-18T06:34:34.496301+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":"2026-08-10T23:19:40.687942Z","title":"Random matrices, magic squar es and matching polynomials","venue":null,"work_id":"45f518a4-8c22-45bb-ab84-d77c7a85ef53","year":2004},"citing_paper":{"arxiv_id":"2412.20721","last_updated":"2024-12-30T05:41:52Z","snapshot_observed_at":"2026-08-13T11:44:17.224732Z","submitted_at":"2024-12-30T05:41:52Z","title":"Ramanujan Graphs and Interlacing Families","version":1},"reference_index":18,"source":"pdf_text","source_observed_at":"2026-08-10T23:19:39.944580Z"},"links":{"citing_paper":"/paper/2412.20721"},"observation_digest":"sha256:ee567c5719a62e92e5ca75a47df8cccdfa7ac8f401d7709b7071e0cc0c199773","observation_id":"6930df67-10ee-49f1-b9ab-2979081fcdf7","resolution":{"observed_at":"2026-08-10T23:19:40.691894Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-18T06:34:40.430872+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-18T06:34:40.430872+00:00","source":"crossref"},{"observed_at":"2026-08-18T06:34:34.496301+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":"2026-08-10T23:19:40.678761Z","title":"On the eigenvalues of random matrices","venue":null,"work_id":"0f94d3e8-9ddc-495c-8bc9-dbf04fc5d59f","year":1994},"citing_paper":{"arxiv_id":"2412.20721","last_updated":"2024-12-30T05:41:52Z","snapshot_observed_at":"2026-08-13T11:44:17.224732Z","submitted_at":"2024-12-30T05:41:52Z","title":"Ramanujan Graphs and Interlacing Families","version":1},"reference_index":19,"source":"pdf_text","source_observed_at":"2026-08-10T23:19:39.948763Z"},"links":{"citing_paper":"/paper/2412.20721"},"observation_digest":"sha256:ce6d6349e22da963417f5061b19facbbfde37cec0a7bb92a916d0bacb8262476","observation_id":"128bbe75-59c9-4889-878d-9913d274183a","resolution":{"observed_at":"2026-08-10T23:19:40.681736Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-18T06:34:40.430872+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-18T06:34:40.430872+00:00","source":"crossref"},{"observed_at":"2026-08-18T06:34:34.496301+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":"2026-08-10T23:19:40.667815Z","title":"Some geometric aspects of graphs and their eig enfunctions","venue":null,"work_id":"1decfc8e-3f14-42c6-a98d-eab0d450238d","year":1993},"citing_paper":{"arxiv_id":"2412.20721","last_updated":"2024-12-30T05:41:52Z","snapshot_observed_at":"2026-08-13T11:44:17.224732Z","submitted_at":"2024-12-30T05:41:52Z","title":"Ramanujan Graphs and Interlacing Families","version":1},"reference_index":20,"source":"pdf_text","source_observed_at":"2026-08-10T23:19:39.952151Z"},"links":{"citing_paper":"/paper/2412.20721"},"observation_digest":"sha256:b87e245a84f07daa197f282b7bded1482c919dd4b653d488ed35075dd74b95c6","observation_id":"9c17e973-2d63-46ee-81e3-b9aea12b967a","resolution":{"observed_at":"2026-08-10T23:19:40.672905Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-18T06:34:40.430872+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-18T06:34:40.430872+00:00","source":"crossref"},{"observed_at":"2026-08-18T06:34:34.496301+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":"2026-08-10T23:19:40.657460Z","title":"Relative expanders or weakly relatively ramanuj an graphs","venue":null,"work_id":"d32bcc6b-9493-41d4-b66c-d5afdeacb588","year":2003},"citing_paper":{"arxiv_id":"2412.20721","last_updated":"2024-12-30T05:41:52Z","snapshot_observed_at":"2026-08-13T11:44:17.224732Z","submitted_at":"2024-12-30T05:41:52Z","title":"Ramanujan Graphs and Interlacing Families","version":1},"reference_index":21,"source":"pdf_text","source_observed_at":"2026-08-10T23:19:39.955790Z"},"links":{"citing_paper":"/paper/2412.20721"},"observation_digest":"sha256:7023b7efd38eedd753b17af11f2fea702a8b65c5e4e07dba97b29f8679e1217d","observation_id":"3077e11e-b818-47d4-9cb4-ceaed9a4227e","resolution":{"observed_at":"2026-08-10T23:19:40.661417Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-18T06:34:40.430872+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-18T06:34:40.430872+00:00","source":"crossref"},{"observed_at":"2026-08-18T06:34:34.496301+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":"2026-08-10T23:19:40.645388Z","title":"A proof of Alon ’s second eigenvalue conjecture and related p roblems","venue":null,"work_id":"42be12f8-e3c7-428a-82f9-94bbb5d85c69","year":2008},"citing_paper":{"arxiv_id":"2412.20721","last_updated":"2024-12-30T05:41:52Z","snapshot_observed_at":"2026-08-13T11:44:17.224732Z","submitted_at":"2024-12-30T05:41:52Z","title":"Ramanujan Graphs and Interlacing Families","version":1},"reference_index":22,"source":"pdf_text","source_observed_at":"2026-08-10T23:19:39.959211Z"},"links":{"citing_paper":"/paper/2412.20721"},"observation_digest":"sha256:3e9514e143fdca68657a7685759b04b978037f9c63e52bfbfe26ecc186d52865","observation_id":"ec6f110b-a561-4b7b-88e3-fc939528182d","resolution":{"observed_at":"2026-08-10T23:19:40.649286Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-18T06:34:40.430872+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-18T06:34:40.430872+00:00","source":"crossref"},{"observed_at":"2026-08-18T06:34:34.496301+00:00","source":"retraction_watch"}],"state":"measured"}},{"citation":{"cited_paper":{"arxiv_id":"1911.05688","last_updated":"2019-11-13T18:11:09Z","snapshot_observed_at":"2026-08-10T17:39:16.693450Z","submitted_at":"2019-11-13T18:11:09Z","title":"On the Relativized Alon Second Eigenvalue Conjecture I: Main Theorems, Examples, and Outline of Proof","version":1},"cited_work":{"arxiv_id":"1911.05688","doi":null,"metadata_source":"pith","pith_arxiv_id":"1911.05688","snapshot_observed_at":"2026-08-10T23:19:40.164678Z","title":"On the Relativized Alon Second Eigenvalue Conjecture I: Main Theorems, Examples, and Outline of Proof","venue":"cs.DM","work_id":"98085e78-a13a-48f5-9ae2-e8ea55bf6627","year":2019},"citing_paper":{"arxiv_id":"2412.20721","last_updated":"2024-12-30T05:41:52Z","snapshot_observed_at":"2026-08-13T11:44:17.224732Z","submitted_at":"2024-12-30T05:41:52Z","title":"Ramanujan Graphs and Interlacing Families","version":1},"reference_index":23,"source":"pdf_text","source_observed_at":"2026-08-10T23:19:39.962576Z"},"links":{"cited_paper":"/paper/1911.05688","citing_paper":"/paper/2412.20721"},"observation_digest":"sha256:8f14da5a77b87f4c7eeed776e294545c5cae016ee45a02b7b34b32c5b2d5d21e","observation_id":"4ca8f3d9-6707-4365-bfd5-6af9492dea8f","resolution":{"observed_at":"2026-08-10T23:19:40.168946Z","resolver_source":"local_arxiv","status":"verified_exact"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-18T06:34:40.430872+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-18T06:34:40.430872+00:00","source":"crossref"},{"observed_at":"2026-08-18T06:34:34.496301+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":"2026-08-10T23:19:40.634588Z","title":"On the matching polynomial of a graph","venue":null,"work_id":"f7eefb70-d9b8-453f-ac0e-17097061f1d8","year":1978},"citing_paper":{"arxiv_id":"2412.20721","last_updated":"2024-12-30T05:41:52Z","snapshot_observed_at":"2026-08-13T11:44:17.224732Z","submitted_at":"2024-12-30T05:41:52Z","title":"Ramanujan Graphs and Interlacing Families","version":1},"reference_index":24,"source":"pdf_text","source_observed_at":"2026-08-10T23:19:39.966579Z"},"links":{"citing_paper":"/paper/2412.20721"},"observation_digest":"sha256:85ef98e84cf41d2b6d1361c9a7b0e4efefc7a70d360ec0139b1549d37db7a478","observation_id":"d6029a4d-7b41-4ffd-bc03-d9ba41a27cc6","resolution":{"observed_at":"2026-08-10T23:19:40.638108Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-18T06:34:40.430872+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-18T06:34:40.430872+00:00","source":"crossref"},{"observed_at":"2026-08-18T06:34:34.496301+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":"2026-08-10T23:19:40.623867Z","title":"Crystallization of random matrix o rbits","venue":null,"work_id":"ad0c7dea-478c-40b5-a204-e401fe906650","year":2020},"citing_paper":{"arxiv_id":"2412.20721","last_updated":"2024-12-30T05:41:52Z","snapshot_observed_at":"2026-08-13T11:44:17.224732Z","submitted_at":"2024-12-30T05:41:52Z","title":"Ramanujan Graphs and Interlacing Families","version":1},"reference_index":25,"source":"pdf_text","source_observed_at":"2026-08-10T23:19:39.970520Z"},"links":{"citing_paper":"/paper/2412.20721"},"observation_digest":"sha256:a30bde8c484363e7452f098bef0874b24fc68c8998a8d3597a21483ffea1ce3d","observation_id":"5e350654-5675-46bb-85d2-a04923a5578a","resolution":{"observed_at":"2026-08-10T23:19:40.627879Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-18T06:34:40.430872+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-18T06:34:40.430872+00:00","source":"crossref"},{"observed_at":"2026-08-18T06:34:34.496301+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":"2026-08-10T23:19:40.613069Z","title":"On the spectrum of graphs and their universal covering","venue":null,"work_id":"135d60bc-70da-4418-8d74-8ff2b3c1e74a","year":1995},"citing_paper":{"arxiv_id":"2412.20721","last_updated":"2024-12-30T05:41:52Z","snapshot_observed_at":"2026-08-13T11:44:17.224732Z","submitted_at":"2024-12-30T05:41:52Z","title":"Ramanujan Graphs and Interlacing Families","version":1},"reference_index":26,"source":"pdf_text","source_observed_at":"2026-08-10T23:19:39.974255Z"},"links":{"citing_paper":"/paper/2412.20721"},"observation_digest":"sha256:b90f9c590425f31a15ee42ad4967767eb98b88dc1a78211fe4e192b0c2c71e7d","observation_id":"9300f1b6-60e9-470a-8b7d-72384d4fb5b6","resolution":{"observed_at":"2026-08-10T23:19:40.616825Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-18T06:34:40.430872+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-18T06:34:40.430872+00:00","source":"crossref"},{"observed_at":"2026-08-18T06:34:34.496301+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":"2026-08-10T23:19:40.602906Z","title":"A theory of singular values for ﬁnite free prob ability","venue":null,"work_id":"90a2acc8-3357-4aa8-be72-b92e20f05ce9","year":2024},"citing_paper":{"arxiv_id":"2412.20721","last_updated":"2024-12-30T05:41:52Z","snapshot_observed_at":"2026-08-13T11:44:17.224732Z","submitted_at":"2024-12-30T05:41:52Z","title":"Ramanujan Graphs and Interlacing Families","version":1},"reference_index":27,"source":"pdf_text","source_observed_at":"2026-08-10T23:19:39.977380Z"},"links":{"citing_paper":"/paper/2412.20721"},"observation_digest":"sha256:e9df23c78522de73396fa068c6c2367d393a6d2febd0edcd065499f8cd7e5a50","observation_id":"22a4f78c-8445-44dd-950d-a40c836e140d","resolution":{"observed_at":"2026-08-10T23:19:40.606565Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-18T06:34:40.430872+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-18T06:34:40.430872+00:00","source":"crossref"},{"observed_at":"2026-08-18T06:34:34.496301+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":"2026-08-10T23:19:40.592847Z","title":"Ramanujan coverings o f graphs","venue":null,"work_id":"0e54c199-3465-4bfe-8cf2-e6a91bdfa485","year":2018},"citing_paper":{"arxiv_id":"2412.20721","last_updated":"2024-12-30T05:41:52Z","snapshot_observed_at":"2026-08-13T11:44:17.224732Z","submitted_at":"2024-12-30T05:41:52Z","title":"Ramanujan Graphs and Interlacing Families","version":1},"reference_index":28,"source":"pdf_text","source_observed_at":"2026-08-10T23:19:39.980926Z"},"links":{"citing_paper":"/paper/2412.20721"},"observation_digest":"sha256:7e44ed1b6b01e52510503ed401e2f4326ea67dc6ba1163d2300e812b398d5dd5","observation_id":"046cd92f-71b9-4631-9058-20c4f9edb9ee","resolution":{"observed_at":"2026-08-10T23:19:40.596111Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-18T06:34:40.430872+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-18T06:34:40.430872+00:00","source":"crossref"},{"observed_at":"2026-08-18T06:34:34.496301+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":"2026-08-10T23:19:40.580579Z","title":"Theory of monomer-dimer systems","venue":null,"work_id":"69076739-e64a-4c4e-97a8-4db8d602be60","year":1972},"citing_paper":{"arxiv_id":"2412.20721","last_updated":"2024-12-30T05:41:52Z","snapshot_observed_at":"2026-08-13T11:44:17.224732Z","submitted_at":"2024-12-30T05:41:52Z","title":"Ramanujan Graphs and Interlacing Families","version":1},"reference_index":29,"source":"pdf_text","source_observed_at":"2026-08-10T23:19:39.984698Z"},"links":{"citing_paper":"/paper/2412.20721"},"observation_digest":"sha256:1c8bd3e13a99beb9b89d92021b618cbb0a04ad78ef6aedbc79cd0d36f9240eb8","observation_id":"17473ba3-5763-4712-9b8b-1bfedc4bd01e","resolution":{"observed_at":"2026-08-10T23:19:40.585525Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-18T06:34:40.430872+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-18T06:34:40.430872+00:00","source":"crossref"},{"observed_at":"2026-08-18T06:34:34.496301+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":"2026-08-10T23:19:40.568998Z","title":"Near optimal spectral gaps for hyp erbolic surfaces","venue":null,"work_id":"fa76b06f-0985-43a5-b80c-eaad8c1e8657","year":2023},"citing_paper":{"arxiv_id":"2412.20721","last_updated":"2024-12-30T05:41:52Z","snapshot_observed_at":"2026-08-13T11:44:17.224732Z","submitted_at":"2024-12-30T05:41:52Z","title":"Ramanujan Graphs and Interlacing Families","version":1},"reference_index":30,"source":"pdf_text","source_observed_at":"2026-08-10T23:19:39.988028Z"},"links":{"citing_paper":"/paper/2412.20721"},"observation_digest":"sha256:ade5e9dd3877703ab2f515c039e77115ad249f604e9733ff73e3c824b3deac8c","observation_id":"da6d869a-46af-49cd-b50e-4af745ca14f2","resolution":{"observed_at":"2026-08-10T23:19:40.572877Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-18T06:34:40.430872+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-18T06:34:40.430872+00:00","source":"crossref"},{"observed_at":"2026-08-18T06:34:34.496301+00:00","source":"retraction_watch"}],"state":"measured"}},{"citation":{"cited_paper":{"arxiv_id":"2405.12161","last_updated":"2024-05-20T16:42:41Z","snapshot_observed_at":"2026-08-16T13:51:15.134990Z","submitted_at":"2024-05-20T16:42:41Z","title":"Optimal Eigenvalue Rigidity of Random Regular Graphs","version":1},"cited_work":{"arxiv_id":"2405.12161","doi":null,"metadata_source":"pith","pith_arxiv_id":"2405.12161","snapshot_observed_at":"2026-08-10T23:19:40.149367Z","title":"Optimal Eigenvalue Rigidity of Random Regular Graphs","venue":"math.PR","work_id":"eaa58ba0-8092-4c47-a739-c47c0b63400f","year":2024},"citing_paper":{"arxiv_id":"2412.20721","last_updated":"2024-12-30T05:41:52Z","snapshot_observed_at":"2026-08-13T11:44:17.224732Z","submitted_at":"2024-12-30T05:41:52Z","title":"Ramanujan Graphs and Interlacing Families","version":1},"reference_index":31,"source":"pdf_text","source_observed_at":"2026-08-10T23:19:39.991415Z"},"links":{"cited_paper":"/paper/2405.12161","citing_paper":"/paper/2412.20721"},"observation_digest":"sha256:810727aa216578c12ade9c525e21b3923453ea37c3206db5ae078f2125d19931","observation_id":"2b7821e0-609d-4b9b-a096-67c8c6f1ed4b","resolution":{"observed_at":"2026-08-10T23:19:40.153939Z","resolver_source":"local_arxiv","status":"verified_exact"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-18T06:34:40.430872+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-18T06:34:40.430872+00:00","source":"crossref"},{"observed_at":"2026-08-18T06:34:34.496301+00:00","source":"retraction_watch"}],"state":"measured"}},{"citation":{"cited_paper":{"arxiv_id":"2206.06580","last_updated":"2023-05-04T00:21:52Z","snapshot_observed_at":"2026-08-16T16:53:13.243321Z","submitted_at":"2022-06-14T03:46:00Z","title":"Edge Universality of Sparse Random Matrices","version":2},"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":"2206.06580","snapshot_observed_at":"2026-08-10T23:19:39.994865Z","title":"Edge universality of spar se random matrices","venue":null,"work_id":null,"year":2022},"citing_paper":{"arxiv_id":"2412.20721","last_updated":"2024-12-30T05:41:52Z","snapshot_observed_at":"2026-08-13T11:44:17.224732Z","submitted_at":"2024-12-30T05:41:52Z","title":"Ramanujan Graphs and Interlacing Families","version":1},"reference_index":32,"source":"pdf_text","source_observed_at":"2026-08-10T23:19:39.994865Z"},"links":{"cited_paper":"/paper/2206.06580","citing_paper":"/paper/2412.20721"},"observation_digest":"sha256:410c9598be98ae20ce4a7bc0b2cbc70c2ff9765ae1816459f58a30257c13ad86","observation_id":"c49456c2-8d24-484c-9f75-843eaa2b2c4a","resolution":{"observed_at":"2026-08-10T23:19:39.994865Z","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":"2026-08-10T23:19:40.557939Z","title":"Random matrix theory and ζ (1/2+ it)","venue":null,"work_id":"79f000bf-ff08-4240-a1dd-94bfa2cb8edb","year":2000},"citing_paper":{"arxiv_id":"2412.20721","last_updated":"2024-12-30T05:41:52Z","snapshot_observed_at":"2026-08-13T11:44:17.224732Z","submitted_at":"2024-12-30T05:41:52Z","title":"Ramanujan Graphs and Interlacing Families","version":1},"reference_index":33,"source":"pdf_text","source_observed_at":"2026-08-10T23:19:39.999358Z"},"links":{"citing_paper":"/paper/2412.20721"},"observation_digest":"sha256:4d26c7ece55656ed9a5d67f7ad718604056557bd0af6c9b97dc8cced1477b189","observation_id":"b859fb44-c723-417e-920d-d96efa8cd8bf","resolution":{"observed_at":"2026-08-10T23:19:40.561362Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-18T06:34:40.430872+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-18T06:34:40.430872+00:00","source":"crossref"},{"observed_at":"2026-08-18T06:34:34.496301+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":"2026-08-10T23:19:40.547308Z","title":"Discrete groups, expanding graphs and invariant measures , volume 125","venue":null,"work_id":"b59588d1-7b37-4c07-bf63-c1a2b559c472","year":1994},"citing_paper":{"arxiv_id":"2412.20721","last_updated":"2024-12-30T05:41:52Z","snapshot_observed_at":"2026-08-13T11:44:17.224732Z","submitted_at":"2024-12-30T05:41:52Z","title":"Ramanujan Graphs and Interlacing Families","version":1},"reference_index":34,"source":"pdf_text","source_observed_at":"2026-08-10T23:19:40.003139Z"},"links":{"citing_paper":"/paper/2412.20721"},"observation_digest":"sha256:bd43a01c44098b4775611cba27333efcb6e22ba3091632ba35d3212b3307943c","observation_id":"3c72c891-2e8e-4725-a65a-213f788da289","resolution":{"observed_at":"2026-08-10T23:19:40.550873Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-18T06:34:40.430872+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-18T06:34:40.430872+00:00","source":"crossref"},{"observed_at":"2026-08-18T06:34:34.496301+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":"2026-08-10T23:19:40.536794Z","title":"Not every uniform tree covers ramanujan graphs","venue":null,"work_id":"67458ca8-85d4-4124-898b-a6a6a3980681","year":1998},"citing_paper":{"arxiv_id":"2412.20721","last_updated":"2024-12-30T05:41:52Z","snapshot_observed_at":"2026-08-13T11:44:17.224732Z","submitted_at":"2024-12-30T05:41:52Z","title":"Ramanujan Graphs and Interlacing Families","version":1},"reference_index":35,"source":"pdf_text","source_observed_at":"2026-08-10T23:19:40.007273Z"},"links":{"citing_paper":"/paper/2412.20721"},"observation_digest":"sha256:c6ceabe48695339f3b7a5ec3aeb4f953e115e862a03f2ce0ccb6dcdaf1af2497","observation_id":"f9c20f39-0501-49ac-9eab-78d785bccda9","resolution":{"observed_at":"2026-08-10T23:19:40.540902Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-18T06:34:40.430872+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-18T06:34:40.430872+00:00","source":"crossref"},{"observed_at":"2026-08-18T06:34:34.496301+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":"2026-08-10T23:19:40.526129Z","title":"Ramanuj an graphs","venue":null,"work_id":"071ae5e8-c5e1-4b23-a698-7e4898b37248","year":1988},"citing_paper":{"arxiv_id":"2412.20721","last_updated":"2024-12-30T05:41:52Z","snapshot_observed_at":"2026-08-13T11:44:17.224732Z","submitted_at":"2024-12-30T05:41:52Z","title":"Ramanujan Graphs and Interlacing Families","version":1},"reference_index":36,"source":"pdf_text","source_observed_at":"2026-08-10T23:19:40.010601Z"},"links":{"citing_paper":"/paper/2412.20721"},"observation_digest":"sha256:8ce4c3bb6fa4c823f69829a9fc6f61706447ea67066768e68cf0468943de83a1","observation_id":"4da93a74-0e61-49ac-bd4e-f06c3ce9ffb6","resolution":{"observed_at":"2026-08-10T23:19:40.529898Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-18T06:34:40.430872+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-18T06:34:40.430872+00:00","source":"crossref"},{"observed_at":"2026-08-18T06:34:34.496301+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":"2026-08-10T23:19:40.514695Z","title":"The solution of the kadison- singer problem","venue":null,"work_id":"8e58ec64-4585-4de4-96c8-bbcf2e173058","year":2016},"citing_paper":{"arxiv_id":"2412.20721","last_updated":"2024-12-30T05:41:52Z","snapshot_observed_at":"2026-08-13T11:44:17.224732Z","submitted_at":"2024-12-30T05:41:52Z","title":"Ramanujan Graphs and Interlacing Families","version":1},"reference_index":37,"source":"pdf_text","source_observed_at":"2026-08-10T23:19:40.013954Z"},"links":{"citing_paper":"/paper/2412.20721"},"observation_digest":"sha256:136472c39a60b44cce4a200c48ba2f5b21d0d2be5a53a169553944f36a1d23de","observation_id":"af748f36-295c-4025-b3d3-dd775500b7af","resolution":{"observed_at":"2026-08-10T23:19:40.518864Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-18T06:34:40.430872+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-18T06:34:40.430872+00:00","source":"crossref"},{"observed_at":"2026-08-18T06:34:34.496301+00:00","source":"retraction_watch"}],"state":"measured"}},{"citation":{"cited_paper":{"arxiv_id":"2108.07054","last_updated":"2021-08-16T12:46:54Z","snapshot_observed_at":"2026-08-16T18:02:59.618658Z","submitted_at":"2021-08-16T12:46:54Z","title":"Polynomial convolutions and (finite) free probability","version":1},"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":"2108.07054","snapshot_observed_at":"2026-08-10T23:19:40.017114Z","title":"Polynomial convolutions and (ﬁnite) free proba bility","venue":null,"work_id":null,"year":2021},"citing_paper":{"arxiv_id":"2412.20721","last_updated":"2024-12-30T05:41:52Z","snapshot_observed_at":"2026-08-13T11:44:17.224732Z","submitted_at":"2024-12-30T05:41:52Z","title":"Ramanujan Graphs and Interlacing Families","version":1},"reference_index":38,"source":"pdf_text","source_observed_at":"2026-08-10T23:19:40.017114Z"},"links":{"cited_paper":"/paper/2108.07054","citing_paper":"/paper/2412.20721"},"observation_digest":"sha256:bca62173a4ed9e6eaece12752b878a34157e291c836ee73ff4da90d06770a89a","observation_id":"4fe76c4d-a086-491e-ab09-d51de72b742c","resolution":{"observed_at":"2026-08-10T23:19:40.017114Z","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":"2026-08-10T23:19:40.503446Z","title":"Ramanu jan graphs and the solution of the kadison-singer problem","venue":null,"work_id":"040396b1-0bb1-4541-ad24-2ea9aa83117f","year":2014},"citing_paper":{"arxiv_id":"2412.20721","last_updated":"2024-12-30T05:41:52Z","snapshot_observed_at":"2026-08-13T11:44:17.224732Z","submitted_at":"2024-12-30T05:41:52Z","title":"Ramanujan Graphs and Interlacing Families","version":1},"reference_index":39,"source":"pdf_text","source_observed_at":"2026-08-10T23:19:40.020832Z"},"links":{"citing_paper":"/paper/2412.20721"},"observation_digest":"sha256:f7cb314fe1d28d0e32f4925e444c7f1e9ca5b13960487d736e5b21d6e67cf618","observation_id":"de65a3c7-cbae-4ca2-8038-f36d75948e9a","resolution":{"observed_at":"2026-08-10T23:19:40.507203Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-18T06:34:40.430872+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-18T06:34:40.430872+00:00","source":"crossref"},{"observed_at":"2026-08-18T06:34:34.496301+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":"2026-08-10T23:19:40.491618Z","title":"Interla cing families i: Bipartite ramanujan graphs of all degrees","venue":null,"work_id":"bb7bc10b-5012-49b0-b18e-2c82c1f53aee","year":2015},"citing_paper":{"arxiv_id":"2412.20721","last_updated":"2024-12-30T05:41:52Z","snapshot_observed_at":"2026-08-13T11:44:17.224732Z","submitted_at":"2024-12-30T05:41:52Z","title":"Ramanujan Graphs and Interlacing Families","version":1},"reference_index":40,"source":"pdf_text","source_observed_at":"2026-08-10T23:19:40.024385Z"},"links":{"citing_paper":"/paper/2412.20721"},"observation_digest":"sha256:44c6b437ade710d2cfb8338b481668ae9677e1d25f195a2307c8a9bb705b7344","observation_id":"60cfc2b4-8714-4224-91aa-cb0364c9c0c3","resolution":{"observed_at":"2026-08-10T23:19:40.496067Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-18T06:34:40.430872+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-18T06:34:40.430872+00:00","source":"crossref"},{"observed_at":"2026-08-18T06:34:34.496301+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":"2026-08-10T23:19:40.480026Z","title":"Interla cing families ii: Mixed characteristic polynomials and the kadison—singer problem","venue":null,"work_id":"0c8157ac-f624-4c9a-b6c8-9270aeab7d3f","year":2015},"citing_paper":{"arxiv_id":"2412.20721","last_updated":"2024-12-30T05:41:52Z","snapshot_observed_at":"2026-08-13T11:44:17.224732Z","submitted_at":"2024-12-30T05:41:52Z","title":"Ramanujan Graphs and Interlacing Families","version":1},"reference_index":41,"source":"pdf_text","source_observed_at":"2026-08-10T23:19:40.027753Z"},"links":{"citing_paper":"/paper/2412.20721"},"observation_digest":"sha256:5e6dc30683ea40ba993399cc5906faea40845ab64526f765bb8fb4685008a67f","observation_id":"50be58b5-8586-49f6-a796-86a02fe69e14","resolution":{"observed_at":"2026-08-10T23:19:40.484040Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-18T06:34:40.430872+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-18T06:34:40.430872+00:00","source":"crossref"},{"observed_at":"2026-08-18T06:34:34.496301+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":"2026-08-10T23:19:40.467963Z","title":"Interla cing families iv: Bipartite ramanujan graphs of all sizes","venue":null,"work_id":"1604b0d6-defb-4c90-9324-3272f45198d4","year":2018},"citing_paper":{"arxiv_id":"2412.20721","last_updated":"2024-12-30T05:41:52Z","snapshot_observed_at":"2026-08-13T11:44:17.224732Z","submitted_at":"2024-12-30T05:41:52Z","title":"Ramanujan Graphs and Interlacing Families","version":1},"reference_index":42,"source":"pdf_text","source_observed_at":"2026-08-10T23:19:40.031278Z"},"links":{"citing_paper":"/paper/2412.20721"},"observation_digest":"sha256:decb112cea11435ab1c6f646dc8c7af4ac9344bf1ebada5a655d0f0f631f6f35","observation_id":"90df3ddc-bcb9-4d9e-bcb0-a25cfba338c4","resolution":{"observed_at":"2026-08-10T23:19:40.472375Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-18T06:34:40.430872+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-18T06:34:40.430872+00:00","source":"crossref"},{"observed_at":"2026-08-18T06:34:34.496301+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":"2026-08-10T23:19:40.456806Z","title":"Finite fr ee convolutions of polynomials","venue":null,"work_id":"2234e0f8-7277-4eba-89d7-f649bea57507","year":2022},"citing_paper":{"arxiv_id":"2412.20721","last_updated":"2024-12-30T05:41:52Z","snapshot_observed_at":"2026-08-13T11:44:17.224732Z","submitted_at":"2024-12-30T05:41:52Z","title":"Ramanujan Graphs and Interlacing Families","version":1},"reference_index":43,"source":"pdf_text","source_observed_at":"2026-08-10T23:19:40.035721Z"},"links":{"citing_paper":"/paper/2412.20721"},"observation_digest":"sha256:fe040c98e4b564d0a1012b3622222fda5094d0e0e5bd1009221509587f5b898d","observation_id":"85b497b5-d91c-4b3e-b673-2751bf8809eb","resolution":{"observed_at":"2026-08-10T23:19:40.460541Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-18T06:34:40.430872+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-18T06:34:40.430872+00:00","source":"crossref"},{"observed_at":"2026-08-18T06:34:34.496301+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":"2026-08-10T23:19:40.445979Z","title":"Explicit group-theoretical co nstructions of combi- natorial schemes and their application to the design of expanders a nd concentrators","venue":null,"work_id":"3616bc49-f91e-43ad-990a-c9b48a973757","year":1988},"citing_paper":{"arxiv_id":"2412.20721","last_updated":"2024-12-30T05:41:52Z","snapshot_observed_at":"2026-08-13T11:44:17.224732Z","submitted_at":"2024-12-30T05:41:52Z","title":"Ramanujan Graphs and Interlacing Families","version":1},"reference_index":44,"source":"pdf_text","source_observed_at":"2026-08-10T23:19:40.039430Z"},"links":{"citing_paper":"/paper/2412.20721"},"observation_digest":"sha256:87c97cba0f4104927000748eba66c5286f1e279ee6b2d3eec9e57078550e9353","observation_id":"f45ab26f-fac6-4fe7-9645-753c3b67b2cb","resolution":{"observed_at":"2026-08-10T23:19:40.450001Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-18T06:34:40.430872+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-18T06:34:40.430872+00:00","source":"crossref"},{"observed_at":"2026-08-18T06:34:34.496301+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":"2026-08-10T23:19:40.434871Z","title":"The expected eigenvalue distribution of a larg e regular graph","venue":null,"work_id":"6ed1889d-a161-4175-a673-657e51bd7b3a","year":1981},"citing_paper":{"arxiv_id":"2412.20721","last_updated":"2024-12-30T05:41:52Z","snapshot_observed_at":"2026-08-13T11:44:17.224732Z","submitted_at":"2024-12-30T05:41:52Z","title":"Ramanujan Graphs and Interlacing Families","version":1},"reference_index":45,"source":"pdf_text","source_observed_at":"2026-08-10T23:19:40.043877Z"},"links":{"citing_paper":"/paper/2412.20721"},"observation_digest":"sha256:c7a0d6844cd815e81c5be99847363e49badf87f6d42f82e1c6bff9c43b488090","observation_id":"83d8308b-8e38-436a-9a26-72000aa8a08a","resolution":{"observed_at":"2026-08-10T23:19:40.438578Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-18T06:34:40.430872+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-18T06:34:40.430872+00:00","source":"crossref"},{"observed_at":"2026-08-18T06:34:34.496301+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":"2026-08-10T23:19:40.425245Z","title":"The distribution o f the largest nontrivial eigenvalues in families of random regular graphs","venue":null,"work_id":"21f12ce6-f58e-4eb3-ac6f-88b8b598fc39","year":2008},"citing_paper":{"arxiv_id":"2412.20721","last_updated":"2024-12-30T05:41:52Z","snapshot_observed_at":"2026-08-13T11:44:17.224732Z","submitted_at":"2024-12-30T05:41:52Z","title":"Ramanujan Graphs and Interlacing Families","version":1},"reference_index":46,"source":"pdf_text","source_observed_at":"2026-08-10T23:19:40.047985Z"},"links":{"citing_paper":"/paper/2412.20721"},"observation_digest":"sha256:abf6c1758517532e0cc38d0fe02bddb93e73fc04bcb7a29b29631004e9ef829f","observation_id":"e57a55d2-fc1b-40c3-b883-6f33afeed942","resolution":{"observed_at":"2026-08-10T23:19:40.428528Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-18T06:34:40.430872+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-18T06:34:40.430872+00:00","source":"crossref"},{"observed_at":"2026-08-18T06:34:34.496301+00:00","source":"retraction_watch"}],"state":"measured"}},{"citation":{"cited_paper":{"arxiv_id":"1904.03500","last_updated":"2019-04-10T17:04:45Z","snapshot_observed_at":"2026-08-14T16:50:38.897545Z","submitted_at":"2019-04-06T17:36:33Z","title":"$X$-Ramanujan Graphs","version":2},"cited_work":{"arxiv_id":"1904.03500","doi":null,"metadata_source":"pith","pith_arxiv_id":"1904.03500","snapshot_observed_at":"2026-08-10T23:19:40.109444Z","title":"$X$-Ramanujan Graphs","venue":"math.CO","work_id":"298283d4-367d-4a08-b844-fda445e5aea5","year":2019},"citing_paper":{"arxiv_id":"2412.20721","last_updated":"2024-12-30T05:41:52Z","snapshot_observed_at":"2026-08-13T11:44:17.224732Z","submitted_at":"2024-12-30T05:41:52Z","title":"Ramanujan Graphs and Interlacing Families","version":1},"reference_index":47,"source":"pdf_text","source_observed_at":"2026-08-10T23:19:40.051998Z"},"links":{"cited_paper":"/paper/1904.03500","citing_paper":"/paper/2412.20721"},"observation_digest":"sha256:e49137f7a426fe90a9de48d297827cd4578b2e7712982c9e2696ef3e0041bd7a","observation_id":"575744be-97b6-414e-a50e-c40c101d5b91","resolution":{"observed_at":"2026-08-10T23:19:40.114972Z","resolver_source":"local_arxiv","status":"verified_exact"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-18T06:34:40.430872+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-18T06:34:40.430872+00:00","source":"crossref"},{"observed_at":"2026-08-18T06:34:34.496301+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":"2026-08-10T23:19:40.415602Z","title":"Expansion of random graphs: New proofs, new r esults","venue":null,"work_id":"d12285a2-f9e5-4f79-b2ad-666b46c1245b","year":2015},"citing_paper":{"arxiv_id":"2412.20721","last_updated":"2024-12-30T05:41:52Z","snapshot_observed_at":"2026-08-13T11:44:17.224732Z","submitted_at":"2024-12-30T05:41:52Z","title":"Ramanujan Graphs and Interlacing Families","version":1},"reference_index":48,"source":"pdf_text","source_observed_at":"2026-08-10T23:19:40.056321Z"},"links":{"citing_paper":"/paper/2412.20721"},"observation_digest":"sha256:58b7282eabde32f4e46416122a49e66b64242e8d7575fc80fa5650714d605f64","observation_id":"13b3b0e7-6b6d-4333-8709-d208f03ce0a1","resolution":{"observed_at":"2026-08-10T23:19:40.419043Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-18T06:34:40.430872+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-18T06:34:40.430872+00:00","source":"crossref"},{"observed_at":"2026-08-18T06:34:34.496301+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":"2026-08-10T23:19:40.405423Z","title":"Mixed determinants and the kadison–singer problem","venue":null,"work_id":"dac8910e-3a86-489b-afc6-dcc2771970f4","year":2020},"citing_paper":{"arxiv_id":"2412.20721","last_updated":"2024-12-30T05:41:52Z","snapshot_observed_at":"2026-08-13T11:44:17.224732Z","submitted_at":"2024-12-30T05:41:52Z","title":"Ramanujan Graphs and Interlacing Families","version":1},"reference_index":49,"source":"pdf_text","source_observed_at":"2026-08-10T23:19:40.060353Z"},"links":{"citing_paper":"/paper/2412.20721"},"observation_digest":"sha256:9ea4af10e8f796438f3b38c146fc26643a52146d8bb80ad96208a766bfcfe6ce","observation_id":"4008a32e-1182-4032-b65c-c6a000a8b7c4","resolution":{"observed_at":"2026-08-10T23:19:40.409038Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-18T06:34:40.430872+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-18T06:34:40.430872+00:00","source":"crossref"},{"observed_at":"2026-08-18T06:34:34.496301+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":"2026-08-10T23:19:40.394118Z","title":"Colloque de combinatoire ´ enum´ erative","venue":null,"work_id":"8cae6225-bffc-4e0b-901d-a707ab1d301f","year":1985},"citing_paper":{"arxiv_id":"2412.20721","last_updated":"2024-12-30T05:41:52Z","snapshot_observed_at":"2026-08-13T11:44:17.224732Z","submitted_at":"2024-12-30T05:41:52Z","title":"Ramanujan Graphs and Interlacing Families","version":1},"reference_index":50,"source":"pdf_text","source_observed_at":"2026-08-10T23:19:40.064036Z"},"links":{"citing_paper":"/paper/2412.20721"},"observation_digest":"sha256:9913f6e2f8fa4dd45a74de91cb733e31461b5e15009508110bee3f115931de80","observation_id":"f9b6d051-185c-4ec1-a747-651b9e0d300a","resolution":{"observed_at":"2026-08-10T23:19:40.398364Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-18T06:34:40.430872+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-18T06:34:40.430872+00:00","source":"crossref"},{"observed_at":"2026-08-18T06:34:34.496301+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":"2026-08-10T23:19:40.382294Z","title":"Limit laws for random matrices and free product s","venue":null,"work_id":"7cf7189f-8688-42a5-997a-410f2ab4281c","year":1991},"citing_paper":{"arxiv_id":"2412.20721","last_updated":"2024-12-30T05:41:52Z","snapshot_observed_at":"2026-08-13T11:44:17.224732Z","submitted_at":"2024-12-30T05:41:52Z","title":"Ramanujan Graphs and Interlacing Families","version":1},"reference_index":51,"source":"pdf_text","source_observed_at":"2026-08-10T23:19:40.068469Z"},"links":{"citing_paper":"/paper/2412.20721"},"observation_digest":"sha256:33146f85a137e44c2e923073e5b2cf654a4245c94bf4ecf65aecb57ccf979a90","observation_id":"df55988e-718e-4bfa-96bc-e81e9e91a193","resolution":{"observed_at":"2026-08-10T23:19:40.386142Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-18T06:34:40.430872+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-18T06:34:40.430872+00:00","source":"crossref"},{"observed_at":"2026-08-18T06:34:34.496301+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":"2026-08-10T23:19:40.370247Z","title":"On the location of the roots of certain types of polynomials","venue":null,"work_id":"5db43700-6f7c-45b1-9292-28402de0728c","year":1922},"citing_paper":{"arxiv_id":"2412.20721","last_updated":"2024-12-30T05:41:52Z","snapshot_observed_at":"2026-08-13T11:44:17.224732Z","submitted_at":"2024-12-30T05:41:52Z","title":"Ramanujan Graphs and Interlacing Families","version":1},"reference_index":52,"source":"pdf_text","source_observed_at":"2026-08-10T23:19:40.072656Z"},"links":{"citing_paper":"/paper/2412.20721"},"observation_digest":"sha256:f0115306fc343fdddef9ca20b8715bf3c206fcf812a644b8c459f3ad3c5fccd2","observation_id":"be8c7e2d-960e-48de-ae36-9da77be8695e","resolution":{"observed_at":"2026-08-10T23:19:40.374749Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-18T06:34:40.430872+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-18T06:34:40.430872+00:00","source":"crossref"},{"observed_at":"2026-08-18T06:34:34.496301+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":"2026-08-10T23:19:40.357887Z","title":"Models of random regular graphs","venue":null,"work_id":"555fc3a9-1584-4749-bdbf-1ed41932cfab","year":1999},"citing_paper":{"arxiv_id":"2412.20721","last_updated":"2024-12-30T05:41:52Z","snapshot_observed_at":"2026-08-13T11:44:17.224732Z","submitted_at":"2024-12-30T05:41:52Z","title":"Ramanujan Graphs and Interlacing Families","version":1},"reference_index":53,"source":"pdf_text","source_observed_at":"2026-08-10T23:19:40.076313Z"},"links":{"citing_paper":"/paper/2412.20721"},"observation_digest":"sha256:3399ea6b75df29d69b8e5fd3c44a08d0fa5e324c9997729d73650eb2f76e8c14","observation_id":"d0e5d22d-0ee7-4fd4-98c2-5f5faf30f9ad","resolution":{"observed_at":"2026-08-10T23:19:40.361976Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-18T06:34:40.430872+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-18T06:34:40.430872+00:00","source":"crossref"},{"observed_at":"2026-08-18T06:34:34.496301+00:00","source":"retraction_watch"}],"state":"measured"}}],"paper":{"arxiv_id":"2412.20721","last_updated":"2024-12-30T05:41:52Z","latest_version":1,"primary_category":"math.CO","snapshot_observed_at":"2026-08-13T11:44:17.224732Z","submitted_at":"2024-12-30T05:41:52Z","title":"Ramanujan Graphs and Interlacing Families"},"reference_resolution":{"displayed":53,"state_counts":{"malformed_identifier":0,"metadata_mismatch":0,"parse_uncertain":0,"unresolved":6,"verified_exact":5,"verified_fuzzy":42},"total_outbound_references":53},"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":"2026-08-18T06:34:40.430872+00:00","source":"crossref"},{"observed_at":"2026-08-18T06:34:34.496301+00:00","source":"retraction_watch"}],"thesis":"As of 18 August 2026, this Paper Citation Record lists 53 of 53 outbound references and 0 inbound Pith citation observations for arXiv:2412.20721."}