{"as_of":"2026-08-12T16:06:00Z","caps":{"database_statements":6,"inbound":100,"outbound":100},"context_digest":"sha256:c9bd8c6954518e4a7b6e009f9d4698795d0a4d48879808b3e264bd547bb07388","coverage":[{"denominator":69,"lane":"reference_resolution","note":"Typed states for the displayed outbound observations.","records_observed":69,"source":"paper_references, paper_reference_links","source_observed_at":"2026-08-11T22:02:08.245459Z","state":"measured"},{"denominator":70,"lane":"standing_notices","note":"One-hop event checks from named stored sources.","records_observed":70,"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":"2026-08-10T20:37:39.747038Z","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":"2026-08-10T20:37:39.908986Z","state":"measured"}],"external_citation_measurements":[],"inbound":[{"citation":{"cited_paper":{"arxiv_id":"2412.03997","last_updated":"2025-01-29T09:31:52Z","snapshot_observed_at":"2026-08-11T21:50:24.929338Z","submitted_at":"2024-12-05T09:18:57Z","title":"Free energy minimizers with radial densities: classification and quantitative stability","version":2},"cited_work":{"arxiv_id":"2412.03997","doi":null,"metadata_source":"pith","pith_arxiv_id":"2412.03997","snapshot_observed_at":"2026-08-10T20:37:39.908986Z","title":"Free energy minimizers with radial densities: classification and quantitative stability","venue":"math.AP","work_id":"09fc9454-59cb-4ea9-8f46-436e69f8ed51","year":2024},"citing_paper":{"arxiv_id":"2501.07900","last_updated":"2025-01-14T07:30:54Z","snapshot_observed_at":"2026-08-10T20:27:38.439191Z","submitted_at":"2025-01-14T07:30:54Z","title":"The one-dimensional equilibrium shape of a crystal","version":1},"reference_index":1,"source":"arxiv_source","source_observed_at":"2026-08-10T20:37:39.747038Z"},"links":{"cited_paper":"/paper/2412.03997","citing_paper":"/paper/2501.07900"},"observation_digest":"sha256:251cc55b2f5ff79cbd3b83347b85eae82f90c608da8afbc7e82b923c6cbbe1c0","observation_id":"b9322bd9-39fb-4655-babd-04033520be94","resolution":{"observed_at":"2026-08-10T20:37:39.913538Z","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/2412.03997/citation-record","integrity":"/paper/2412.03997/integrity","json":"/paper/2412.03997/citation-record.json","paper":"/paper/2412.03997"},"outbound":[{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":null,"snapshot_observed_at":"2026-08-11T22:02:07.576379Z","title":"Ambrosio, N","venue":null,"work_id":null,"year":2000},"citing_paper":{"arxiv_id":"2412.03997","last_updated":"2025-01-29T09:31:52Z","snapshot_observed_at":"2026-08-11T21:50:24.929338Z","submitted_at":"2024-12-05T09:18:57Z","title":"Free energy minimizers with radial densities: classification and quantitative stability","version":2},"reference_index":1,"source":"pdf_text","source_observed_at":"2026-08-11T22:02:07.576379Z"},"links":{"citing_paper":"/paper/2412.03997"},"observation_digest":"sha256:85128b9aec0037ddc24b74d1e456a435592dc18ac1997f18d2d6babdea947e3b","observation_id":"dbc9e99f-9816-4e41-8cba-b2b3032802fb","resolution":{"observed_at":"2026-08-11T22:02:07.576379Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":{"arxiv_id":"2405.08918","last_updated":"2026-07-11T09:25:50Z","snapshot_observed_at":"2026-08-07T13:50:24.758187Z","submitted_at":"2024-05-14T19:15:17Z","title":"New spectral Bishop-Gromov and Bonnet-Myers theorems and applications to isoperimetry","version":4},"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":"2405.08918","snapshot_observed_at":"2026-08-11T22:02:07.586103Z","title":"New spectral Bishop–Gromov and B onnet–Myers theorems and applications to isoperimetry","venue":null,"work_id":null,"year":2024},"citing_paper":{"arxiv_id":"2412.03997","last_updated":"2025-01-29T09:31:52Z","snapshot_observed_at":"2026-08-11T21:50:24.929338Z","submitted_at":"2024-12-05T09:18:57Z","title":"Free energy minimizers with radial densities: classification and quantitative stability","version":2},"reference_index":2,"source":"pdf_text","source_observed_at":"2026-08-11T22:02:07.586103Z"},"links":{"cited_paper":"/paper/2405.08918","citing_paper":"/paper/2412.03997"},"observation_digest":"sha256:e8ffa9bc5b87ac8e5d40a3154d724895ae52d4b3944b02262d61d04e5d9ad1b3","observation_id":"60e85712-f053-4180-91c4-6d2ea3c938fe","resolution":{"observed_at":"2026-08-11T22:02:07.586103Z","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":"2026-08-11T22:02:10.443273Z","title":"Equilibrium shapes of cr ystals in a gravitational field: Crystals on a table","venue":null,"work_id":"0401420c-de93-49c8-8ce9-a1cf21d24b92","year":1983},"citing_paper":{"arxiv_id":"2412.03997","last_updated":"2025-01-29T09:31:52Z","snapshot_observed_at":"2026-08-11T21:50:24.929338Z","submitted_at":"2024-12-05T09:18:57Z","title":"Free energy minimizers with radial densities: classification and quantitative stability","version":2},"reference_index":3,"source":"pdf_text","source_observed_at":"2026-08-11T22:02:07.595641Z"},"links":{"citing_paper":"/paper/2412.03997"},"observation_digest":"sha256:ce2b2cbf10638dcb3574d60b9e83f6d3270648deaecf3c1f591d1d6d70879b1a","observation_id":"c657a0d2-c848-4cb2-99b9-3e1dd5bbe405","resolution":{"observed_at":"2026-08-11T22:02:10.450170Z","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":"2026-08-11T22:02:10.419653Z","title":"Diffusions hypercontractives","venue":null,"work_id":"2851b713-ed42-4b1a-8999-a97b22f7c567","year":1983},"citing_paper":{"arxiv_id":"2412.03997","last_updated":"2025-01-29T09:31:52Z","snapshot_observed_at":"2026-08-11T21:50:24.929338Z","submitted_at":"2024-12-05T09:18:57Z","title":"Free energy minimizers with radial densities: classification and quantitative stability","version":2},"reference_index":4,"source":"pdf_text","source_observed_at":"2026-08-11T22:02:07.604167Z"},"links":{"citing_paper":"/paper/2412.03997"},"observation_digest":"sha256:8205cdc955451bc0e5d72d30050d4cee80617744ca139d925a427c169f4bd2b6","observation_id":"fb05c55a-27ee-4d13-804f-0e60f4215163","resolution":{"observed_at":"2026-08-11T22:02:10.426266Z","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":"2026-08-11T22:02:10.390179Z","title":"Lévy–Gromov’s isoperimetric in equality for an infinite di- mensional diffusion generator","venue":null,"work_id":"88e2ac73-b213-4ac2-8397-a79881217f75","year":1996},"citing_paper":{"arxiv_id":"2412.03997","last_updated":"2025-01-29T09:31:52Z","snapshot_observed_at":"2026-08-11T21:50:24.929338Z","submitted_at":"2024-12-05T09:18:57Z","title":"Free energy minimizers with radial densities: classification and quantitative stability","version":2},"reference_index":5,"source":"pdf_text","source_observed_at":"2026-08-11T22:02:07.612762Z"},"links":{"citing_paper":"/paper/2412.03997"},"observation_digest":"sha256:2e18fe36ae1fef694fa8995aa9dba8605bb0895431de17ea759ba9cbd79a6850","observation_id":"8d77b5d1-b138-478c-aaea-9b4fb746c4bd","resolution":{"observed_at":"2026-08-11T22:02:10.396604Z","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":"2026-08-11T22:02:10.348380Z","title":"Stability of hypersurface s with constant mean curvature","venue":null,"work_id":"b625d28f-3416-4978-b04e-70124a303b21","year":2012},"citing_paper":{"arxiv_id":"2412.03997","last_updated":"2025-01-29T09:31:52Z","snapshot_observed_at":"2026-08-11T21:50:24.929338Z","submitted_at":"2024-12-05T09:18:57Z","title":"Free energy minimizers with radial densities: classification and quantitative stability","version":2},"reference_index":6,"source":"pdf_text","source_observed_at":"2026-08-11T22:02:07.621666Z"},"links":{"citing_paper":"/paper/2412.03997"},"observation_digest":"sha256:ee7483e49f90fffc9669914fbfccf29486c9ee291d2e80a6520892385c08e0e1","observation_id":"91be3404-0ddd-4efa-9919-f8f819556988","resolution":{"observed_at":"2026-08-11T22:02:10.354724Z","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":"2026-08-11T22:02:10.326942Z","title":"Propriétés de concavité du profil isopérimétr ique et applications","venue":null,"work_id":"b28ecf45-3fef-4c58-9ae2-3e28796bdaa8","year":2003},"citing_paper":{"arxiv_id":"2412.03997","last_updated":"2025-01-29T09:31:52Z","snapshot_observed_at":"2026-08-11T21:50:24.929338Z","submitted_at":"2024-12-05T09:18:57Z","title":"Free energy minimizers with radial densities: classification and quantitative stability","version":2},"reference_index":7,"source":"pdf_text","source_observed_at":"2026-08-11T22:02:07.629866Z"},"links":{"citing_paper":"/paper/2412.03997"},"observation_digest":"sha256:53fc1211700961276877ebade2b95e8af118ea63c0b54a8f35d6255a6929cff4","observation_id":"943d0311-c87b-4908-a2c5-96158adfa4fa","resolution":{"observed_at":"2026-08-11T22:02:10.334249Z","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":"2026-08-11T22:02:10.277568Z","title":"A quantitative iso perimetric inequality on the sphere","venue":null,"work_id":"b0ae48d3-7fe7-415d-bb1a-6000505d455c","year":2017},"citing_paper":{"arxiv_id":"2412.03997","last_updated":"2025-01-29T09:31:52Z","snapshot_observed_at":"2026-08-11T21:50:24.929338Z","submitted_at":"2024-12-05T09:18:57Z","title":"Free energy minimizers with radial densities: classification and quantitative stability","version":2},"reference_index":8,"source":"pdf_text","source_observed_at":"2026-08-11T22:02:07.638368Z"},"links":{"citing_paper":"/paper/2412.03997"},"observation_digest":"sha256:fce9d5c66e4a3d5e05bc03d9e8af96a37108300ac8f22de81499d0bf85f2e8f6","observation_id":"f466f649-6cb8-422a-be28-c9884363b2a7","resolution":{"observed_at":"2026-08-11T22:02:10.287831Z","resolver_source":"raw_fallback","status":"malformed_identifier"},"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":"2026-08-11T22:02:10.250579Z","title":"A sharp quantita tive isoperimetric in- equality in hyperbolic n-space","venue":null,"work_id":"12df5749-71db-40ed-9410-d02610bb737d","year":2015},"citing_paper":{"arxiv_id":"2412.03997","last_updated":"2025-01-29T09:31:52Z","snapshot_observed_at":"2026-08-11T21:50:24.929338Z","submitted_at":"2024-12-05T09:18:57Z","title":"Free energy minimizers with radial densities: classification and quantitative stability","version":2},"reference_index":9,"source":"pdf_text","source_observed_at":"2026-08-11T22:02:07.647899Z"},"links":{"citing_paper":"/paper/2412.03997"},"observation_digest":"sha256:6ac1851c89211f1af4a53f556c5eb8d4f2f06d2f6cef428656e82f06c62d8351","observation_id":"da296e1e-9f60-439b-966d-99202efef81d","resolution":{"observed_at":"2026-08-11T22:02:10.258946Z","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":"2026-08-11T22:02:10.220338Z","title":"Isoperimetry in sur- faces of revolution with density","venue":null,"work_id":"4159f03f-bced-4937-bebf-13715d77ccb8","year":2018},"citing_paper":{"arxiv_id":"2412.03997","last_updated":"2025-01-29T09:31:52Z","snapshot_observed_at":"2026-08-11T21:50:24.929338Z","submitted_at":"2024-12-05T09:18:57Z","title":"Free energy minimizers with radial densities: classification and quantitative stability","version":2},"reference_index":10,"source":"pdf_text","source_observed_at":"2026-08-11T22:02:07.655603Z"},"links":{"citing_paper":"/paper/2412.03997"},"observation_digest":"sha256:320503f156fc95442e5f56ee6d929e36f6353ddfc1e2811144fe084849fa019a","observation_id":"e5f52f7e-19f4-40c0-9016-7d53b0f59f77","resolution":{"observed_at":"2026-08-11T22:02:10.226664Z","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":"2026-08-11T22:02:10.192489Z","title":"Isoperimetric Regions in Rn with Density rp","venue":null,"work_id":"2702f562-9d52-4a94-bf0b-a5fd650a7fa4","year":2016},"citing_paper":{"arxiv_id":"2412.03997","last_updated":"2025-01-29T09:31:52Z","snapshot_observed_at":"2026-08-11T21:50:24.929338Z","submitted_at":"2024-12-05T09:18:57Z","title":"Free energy minimizers with radial densities: classification and quantitative stability","version":2},"reference_index":11,"source":"pdf_text","source_observed_at":"2026-08-11T22:02:07.664405Z"},"links":{"citing_paper":"/paper/2412.03997"},"observation_digest":"sha256:067d199448c06e38ab6d1f76d14782fbb32bba36ff978986819fb44a84d04562","observation_id":"fc4ace88-e264-48b0-9de6-378e78d03b4d","resolution":{"observed_at":"2026-08-11T22:02:10.202611Z","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":"2026-08-11T22:02:10.167574Z","title":null,"venue":null,"work_id":"e20b53f1-6e9f-4e91-a403-bc2d3e61beb3","year":2013},"citing_paper":{"arxiv_id":"2412.03997","last_updated":"2025-01-29T09:31:52Z","snapshot_observed_at":"2026-08-11T21:50:24.929338Z","submitted_at":"2024-12-05T09:18:57Z","title":"Free energy minimizers with radial densities: classification and quantitative stability","version":2},"reference_index":12,"source":"pdf_text","source_observed_at":"2026-08-11T22:02:07.671818Z"},"links":{"citing_paper":"/paper/2412.03997"},"observation_digest":"sha256:b61e16da00acff98a4c9d3dc27bdf2b65850360171fb5e24f7e962055eb0195c","observation_id":"133e4925-a4cc-4b75-96b7-0b0c9e84b3a7","resolution":{"observed_at":"2026-08-11T22:02:10.175720Z","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":"2026-08-11T22:02:10.140741Z","title":"Stabilité isopérimétrique","venue":null,"work_id":"8d499f54-d544-4510-84d6-fff9bd5e8d58","year":1996},"citing_paper":{"arxiv_id":"2412.03997","last_updated":"2025-01-29T09:31:52Z","snapshot_observed_at":"2026-08-11T21:50:24.929338Z","submitted_at":"2024-12-05T09:18:57Z","title":"Free energy minimizers with radial densities: classification and quantitative stability","version":2},"reference_index":13,"source":"pdf_text","source_observed_at":"2026-08-11T22:02:07.679851Z"},"links":{"citing_paper":"/paper/2412.03997"},"observation_digest":"sha256:bf7d33eb7e4b448419acf2653606b57da4e4d4d2a6c5ce3c52f358fe3d980168","observation_id":"2eb22d53-ade8-49d7-abe5-400d86e53a00","resolution":{"observed_at":"2026-08-11T22:02:10.148991Z","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":"2026-08-11T22:02:10.121700Z","title":"Free boundary stable hypersu rfaces in manifolds with density and rigidity results","venue":null,"work_id":"76735d60-f7bd-4a46-85f5-aa16a4843678","year":2014},"citing_paper":{"arxiv_id":"2412.03997","last_updated":"2025-01-29T09:31:52Z","snapshot_observed_at":"2026-08-11T21:50:24.929338Z","submitted_at":"2024-12-05T09:18:57Z","title":"Free energy minimizers with radial densities: classification and quantitative stability","version":2},"reference_index":14,"source":"pdf_text","source_observed_at":"2026-08-11T22:02:07.687354Z"},"links":{"citing_paper":"/paper/2412.03997"},"observation_digest":"sha256:edada5382c194a8a765a1cd6f8c146b8f503d3280545ab9137d1ad6d57ced562","observation_id":"e18f0ba3-3796-443c-b28a-e90dc990aae5","resolution":{"observed_at":"2026-08-11T22:02:10.127808Z","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":"2026-08-11T22:02:10.100080Z","title":"Proof of the log-convex density conjec ture","venue":null,"work_id":"bd18dab5-b5b8-4177-aade-403223a88814","year":2019},"citing_paper":{"arxiv_id":"2412.03997","last_updated":"2025-01-29T09:31:52Z","snapshot_observed_at":"2026-08-11T21:50:24.929338Z","submitted_at":"2024-12-05T09:18:57Z","title":"Free energy minimizers with radial densities: classification and quantitative stability","version":2},"reference_index":15,"source":"pdf_text","source_observed_at":"2026-08-11T22:02:07.697924Z"},"links":{"citing_paper":"/paper/2412.03997"},"observation_digest":"sha256:a8efdfe49f269da994f4e8e820607ea991458943a1eb951d50cf881f30c9c51f","observation_id":"2f444c72-e5cf-4778-9c24-389de4669833","resolution":{"observed_at":"2026-08-11T22:02:10.106602Z","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":"2026-08-11T22:02:10.076370Z","title":"The Riemann ian quantitative isoperi- metric inequality","venue":null,"work_id":"43b108b3-796d-4e4d-9c15-f5246b17e26a","year":2022},"citing_paper":{"arxiv_id":"2412.03997","last_updated":"2025-01-29T09:31:52Z","snapshot_observed_at":"2026-08-11T21:50:24.929338Z","submitted_at":"2024-12-05T09:18:57Z","title":"Free energy minimizers with radial densities: classification and quantitative stability","version":2},"reference_index":16,"source":"pdf_text","source_observed_at":"2026-08-11T22:02:07.705848Z"},"links":{"citing_paper":"/paper/2412.03997"},"observation_digest":"sha256:fabfae9d5f8c701fe6140fa585a0e72e678587dab9f3231033d1a6ba917803ca","observation_id":"73566fbc-d3b0-4c91-b6cb-4108c5264712","resolution":{"observed_at":"2026-08-11T22:02:10.084852Z","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":"2026-08-11T22:02:10.050987Z","title":"On the i soperimetric deficit in Gauss space","venue":null,"work_id":"cdd1096a-4f47-4679-af7d-6f4f420ad544","year":2011},"citing_paper":{"arxiv_id":"2412.03997","last_updated":"2025-01-29T09:31:52Z","snapshot_observed_at":"2026-08-11T21:50:24.929338Z","submitted_at":"2024-12-05T09:18:57Z","title":"Free energy minimizers with radial densities: classification and quantitative stability","version":2},"reference_index":17,"source":"pdf_text","source_observed_at":"2026-08-11T22:02:07.712850Z"},"links":{"citing_paper":"/paper/2412.03997"},"observation_digest":"sha256:5ce4d4fdb5275d4f2ba3c9c7e2c9c4b786fcb5b8aacd88b19ad42960fbdfe25a","observation_id":"9a3ba58e-e4f3-4603-95e8-ffa2e6bf79d9","resolution":{"observed_at":"2026-08-11T22:02:10.057198Z","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":"2026-08-11T22:02:10.023608Z","title":"A selection principle f or the sharp quantitative isoperimetric inequality","venue":null,"work_id":"74a8e7ab-4ce6-46e3-bda6-8ce899d61eb0","year":2012},"citing_paper":{"arxiv_id":"2412.03997","last_updated":"2025-01-29T09:31:52Z","snapshot_observed_at":"2026-08-11T21:50:24.929338Z","submitted_at":"2024-12-05T09:18:57Z","title":"Free energy minimizers with radial densities: classification and quantitative stability","version":2},"reference_index":18,"source":"pdf_text","source_observed_at":"2026-08-11T22:02:07.720522Z"},"links":{"citing_paper":"/paper/2412.03997"},"observation_digest":"sha256:74905a7fed7b9dbc707b14fbab262e03fbd834bcbdcfd87ca73d9efba3a03dd5","observation_id":"566638e3-f630-4b01-b1ea-d4ef13b87dbc","resolution":{"observed_at":"2026-08-11T22:02:10.032310Z","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":"2026-08-11T22:02:10.001900Z","title":"Best constants for the i soperimetric inequality in quantitative form","venue":null,"work_id":"bc4ab2bd-1ca5-4187-9679-c353ddc529f2","year":2013},"citing_paper":{"arxiv_id":"2412.03997","last_updated":"2025-01-29T09:31:52Z","snapshot_observed_at":"2026-08-11T21:50:24.929338Z","submitted_at":"2024-12-05T09:18:57Z","title":"Free energy minimizers with radial densities: classification and quantitative stability","version":2},"reference_index":19,"source":"pdf_text","source_observed_at":"2026-08-11T22:02:07.729209Z"},"links":{"citing_paper":"/paper/2412.03997"},"observation_digest":"sha256:7425544c21e412359f5a197be2b0a78e599e84cc4945b111f745044176bf7213","observation_id":"9338e73d-30c0-4540-a338-dc2c0f706326","resolution":{"observed_at":"2026-08-11T22:02:10.009605Z","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":"2026-08-11T22:02:09.960479Z","title":"Sharp quantitative stability for isoperimetric inequalities with homogeneou s weights","venue":null,"work_id":"ddf744d2-fac0-4b3f-9d45-1a4c2aff30c5","year":2022},"citing_paper":{"arxiv_id":"2412.03997","last_updated":"2025-01-29T09:31:52Z","snapshot_observed_at":"2026-08-11T21:50:24.929338Z","submitted_at":"2024-12-05T09:18:57Z","title":"Free energy minimizers with radial densities: classification and quantitative stability","version":2},"reference_index":20,"source":"pdf_text","source_observed_at":"2026-08-11T22:02:07.735903Z"},"links":{"citing_paper":"/paper/2412.03997"},"observation_digest":"sha256:facc6b4ba23082140f36e5b2fd1e24f1169075b36af88d2293fec7768b60e0eb","observation_id":"1ef4db32-f3c1-4a59-b7ea-9dffd8d52500","resolution":{"observed_at":"2026-08-11T22:02:09.982215Z","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":"2026-08-11T22:02:09.936423Z","title":"Differential geometry of manifolds with den sity","venue":null,"work_id":"c63559b5-5c79-454a-959d-3591685a3f71","year":2006},"citing_paper":{"arxiv_id":"2412.03997","last_updated":"2025-01-29T09:31:52Z","snapshot_observed_at":"2026-08-11T21:50:24.929338Z","submitted_at":"2024-12-05T09:18:57Z","title":"Free energy minimizers with radial densities: classification and quantitative stability","version":2},"reference_index":21,"source":"pdf_text","source_observed_at":"2026-08-11T22:02:07.742683Z"},"links":{"citing_paper":"/paper/2412.03997"},"observation_digest":"sha256:7adac171266501474ad2d23f65fc753a2964b2407b54f00dbc2778e77e363ce9","observation_id":"a3320915-5a43-473d-9f12-5016670b626a","resolution":{"observed_at":"2026-08-11T22:02:09.946590Z","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":"2026-08-11T22:02:09.897788Z","title":"A two-point function ap proach to connectedness of drops in convex potentials","venue":null,"work_id":"323e7139-689f-42b0-8d3d-a8eb42b9a09d","year":2022},"citing_paper":{"arxiv_id":"2412.03997","last_updated":"2025-01-29T09:31:52Z","snapshot_observed_at":"2026-08-11T21:50:24.929338Z","submitted_at":"2024-12-05T09:18:57Z","title":"Free energy minimizers with radial densities: classification and quantitative stability","version":2},"reference_index":22,"source":"pdf_text","source_observed_at":"2026-08-11T22:02:07.748829Z"},"links":{"citing_paper":"/paper/2412.03997"},"observation_digest":"sha256:3c75929167360afdff6c6c4ab6a204dfb3a9e73acd57594371f4b545faae3ea4","observation_id":"d3e062db-7902-499f-b50e-7408f8031888","resolution":{"observed_at":"2026-08-11T22:02:09.908943Z","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":"2026-08-11T22:02:09.853342Z","title":"A quantitativ e version of the isoperimetric inequality: the anisotropic case","venue":null,"work_id":"b6b5c515-1e74-4eba-9f48-526ddeaa080d","year":2005},"citing_paper":{"arxiv_id":"2412.03997","last_updated":"2025-01-29T09:31:52Z","snapshot_observed_at":"2026-08-11T21:50:24.929338Z","submitted_at":"2024-12-05T09:18:57Z","title":"Free energy minimizers with radial densities: classification and quantitative stability","version":2},"reference_index":23,"source":"pdf_text","source_observed_at":"2026-08-11T22:02:07.755394Z"},"links":{"citing_paper":"/paper/2412.03997"},"observation_digest":"sha256:e0a58bec4b116514273f327bc64cdd590aa971b3f0f286ced0ffed2c00d03bdf","observation_id":"7add7d2d-1713-46df-8575-a20815258b70","resolution":{"observed_at":"2026-08-11T22:02:09.861271Z","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":"2026-08-11T22:02:09.817673Z","title":"A sharp stability result for t he relative isoperimetric inequality inside convex cones","venue":null,"work_id":"faf85df5-3da8-4313-bd6b-9f671733d511","year":2013},"citing_paper":{"arxiv_id":"2412.03997","last_updated":"2025-01-29T09:31:52Z","snapshot_observed_at":"2026-08-11T21:50:24.929338Z","submitted_at":"2024-12-05T09:18:57Z","title":"Free energy minimizers with radial densities: classification and quantitative stability","version":2},"reference_index":24,"source":"pdf_text","source_observed_at":"2026-08-11T22:02:07.763118Z"},"links":{"citing_paper":"/paper/2412.03997"},"observation_digest":"sha256:0734c0d4d4349bd7b22a81e2db6dc34830f5808bd0127a6b150487a5b7fd7467","observation_id":"bbf9904b-8918-43bf-b908-a5c4e19e86b8","resolution":{"observed_at":"2026-08-11T22:02:09.826366Z","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":"2026-08-11T22:02:09.774277Z","title":"On the isoperimetric problem f or radial log-convex den- sities","venue":null,"work_id":"8bfae5ea-9ad3-453a-a25c-f9aac1db0e7c","year":2013},"citing_paper":{"arxiv_id":"2412.03997","last_updated":"2025-01-29T09:31:52Z","snapshot_observed_at":"2026-08-11T21:50:24.929338Z","submitted_at":"2024-12-05T09:18:57Z","title":"Free energy minimizers with radial densities: classification and quantitative stability","version":2},"reference_index":25,"source":"pdf_text","source_observed_at":"2026-08-11T22:02:07.770549Z"},"links":{"citing_paper":"/paper/2412.03997"},"observation_digest":"sha256:18ebb1080728c648f73c868543493355d8950fd2085412641047163c05521308","observation_id":"e6c07bc1-fdbb-41d9-acf0-eae10f0d7702","resolution":{"observed_at":"2026-08-11T22:02:09.782511Z","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":"2026-08-11T22:02:09.744091Z","title":"On the shape of liquid drops and crystals in the small mass regime","venue":null,"work_id":"e4c32cf0-a65f-4e75-b83b-9b1e27b10d9e","year":2011},"citing_paper":{"arxiv_id":"2412.03997","last_updated":"2025-01-29T09:31:52Z","snapshot_observed_at":"2026-08-11T21:50:24.929338Z","submitted_at":"2024-12-05T09:18:57Z","title":"Free energy minimizers with radial densities: classification and quantitative stability","version":2},"reference_index":26,"source":"pdf_text","source_observed_at":"2026-08-11T22:02:07.778212Z"},"links":{"citing_paper":"/paper/2412.03997"},"observation_digest":"sha256:a43ee3bfdf2115c624adffa49d91bd5f17f92e38e42d12487e9f6eb99ac183af","observation_id":"89d489ae-ac8f-4134-9f24-198d67655fff","resolution":{"observed_at":"2026-08-11T22:02:09.754924Z","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":"2026-08-11T22:02:09.703710Z","title":"A mass transport ation approach to quantita- tive isoperimetric inequalities","venue":null,"work_id":"5f8a6c46-23bd-4816-8a41-1dcbc026495e","year":2010},"citing_paper":{"arxiv_id":"2412.03997","last_updated":"2025-01-29T09:31:52Z","snapshot_observed_at":"2026-08-11T21:50:24.929338Z","submitted_at":"2024-12-05T09:18:57Z","title":"Free energy minimizers with radial densities: classification and quantitative stability","version":2},"reference_index":27,"source":"pdf_text","source_observed_at":"2026-08-11T22:02:07.786820Z"},"links":{"citing_paper":"/paper/2412.03997"},"observation_digest":"sha256:acf1c21b81c3d6aa0b89c6fc686fdf04006fc20f02237eebc596bc28bb9f3007","observation_id":"0c868f4a-b208-4d2a-875e-416932e2bcf1","resolution":{"observed_at":"2026-08-11T22:02:09.718055Z","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":"2026-08-11T22:02:09.653861Z","title":null,"venue":null,"work_id":"4c6fb590-9699-4549-bfb0-b541c67744d9","year":2012},"citing_paper":{"arxiv_id":"2412.03997","last_updated":"2025-01-29T09:31:52Z","snapshot_observed_at":"2026-08-11T21:50:24.929338Z","submitted_at":"2024-12-05T09:18:57Z","title":"Free energy minimizers with radial densities: classification and quantitative stability","version":2},"reference_index":28,"source":"pdf_text","source_observed_at":"2026-08-11T22:02:07.799073Z"},"links":{"citing_paper":"/paper/2412.03997"},"observation_digest":"sha256:8f0065616c7edf80873eba7fcdbd10e7f11f4f6e4b609f34b8a5af40e58a9882","observation_id":"15f5728b-02d9-473b-987e-4cec195051d9","resolution":{"observed_at":"2026-08-11T22:02:09.661748Z","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":"2026-08-11T22:02:09.627055Z","title":"The sessile liquid drop. I. Symmetric case","venue":null,"work_id":"6d72b175-46d9-4359-a31c-5255ec4c9368","year":1980},"citing_paper":{"arxiv_id":"2412.03997","last_updated":"2025-01-29T09:31:52Z","snapshot_observed_at":"2026-08-11T21:50:24.929338Z","submitted_at":"2024-12-05T09:18:57Z","title":"Free energy minimizers with radial densities: classification and quantitative stability","version":2},"reference_index":29,"source":"pdf_text","source_observed_at":"2026-08-11T22:02:07.812486Z"},"links":{"citing_paper":"/paper/2412.03997"},"observation_digest":"sha256:849e740eda8b5ec8504daa860e581be81520d05f974d393541d0f9aa0b5c243d","observation_id":"05d6a1c6-e057-4971-a82f-da2bef8c0681","resolution":{"observed_at":"2026-08-11T22:02:09.634892Z","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":"2026-08-11T22:02:09.605680Z","title":"Stability in the isoperimetric problem fo r convex or nearly spherical do- mains in Rn","venue":null,"work_id":"846a9201-1f31-4c75-a4f4-f6cd6aa8711c","year":1989},"citing_paper":{"arxiv_id":"2412.03997","last_updated":"2025-01-29T09:31:52Z","snapshot_observed_at":"2026-08-11T21:50:24.929338Z","submitted_at":"2024-12-05T09:18:57Z","title":"Free energy minimizers with radial densities: classification and quantitative stability","version":2},"reference_index":30,"source":"pdf_text","source_observed_at":"2026-08-11T22:02:07.822994Z"},"links":{"citing_paper":"/paper/2412.03997"},"observation_digest":"sha256:a145fb65bfe8ef1a0024d51f83b50273a9f02bc7d973bd19c6f0bf95e5e1ce11","observation_id":"695397a1-8933-4376-b19f-a942c3baaa93","resolution":{"observed_at":"2026-08-11T22:02:09.611585Z","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":"2026-08-11T22:02:09.577867Z","title":"The quantitative isoperimetric inequality and related topics","venue":null,"work_id":"f59d189a-b43b-4462-ab57-fa70db1e65a7","year":2015},"citing_paper":{"arxiv_id":"2412.03997","last_updated":"2025-01-29T09:31:52Z","snapshot_observed_at":"2026-08-11T21:50:24.929338Z","submitted_at":"2024-12-05T09:18:57Z","title":"Free energy minimizers with radial densities: classification and quantitative stability","version":2},"reference_index":31,"source":"pdf_text","source_observed_at":"2026-08-11T22:02:07.831695Z"},"links":{"citing_paper":"/paper/2412.03997"},"observation_digest":"sha256:d9d3c56f66537d49c580c7324cbb46f171d20ad11c3b4b2b5bb0321571504dcc","observation_id":"140227cf-c30f-4321-a59e-9c7cd9f35f51","resolution":{"observed_at":"2026-08-11T22:02:09.588742Z","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":"2026-08-11T22:02:09.541031Z","title":"A strong form of the quantitative isoperimetric inequality","venue":null,"work_id":"1d46a171-d51f-40b6-9811-1212308b1f4f","year":2014},"citing_paper":{"arxiv_id":"2412.03997","last_updated":"2025-01-29T09:31:52Z","snapshot_observed_at":"2026-08-11T21:50:24.929338Z","submitted_at":"2024-12-05T09:18:57Z","title":"Free energy minimizers with radial densities: classification and quantitative stability","version":2},"reference_index":32,"source":"pdf_text","source_observed_at":"2026-08-11T22:02:07.844914Z"},"links":{"citing_paper":"/paper/2412.03997"},"observation_digest":"sha256:f2a7104f2339fa33912b8b40731b5243eee16a4584c53d70b7bf31774ad4a825","observation_id":"b7f91faf-eec2-43d7-b46d-8fec8cd7e3a3","resolution":{"observed_at":"2026-08-11T22:02:09.551450Z","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":"2026-08-11T22:02:09.512769Z","title":"Some weighted isoperimetri c inequalities in quan- titative form","venue":null,"work_id":"f568e090-4125-4356-b3ec-c4dd8a48a125","year":2023},"citing_paper":{"arxiv_id":"2412.03997","last_updated":"2025-01-29T09:31:52Z","snapshot_observed_at":"2026-08-11T21:50:24.929338Z","submitted_at":"2024-12-05T09:18:57Z","title":"Free energy minimizers with radial densities: classification and quantitative stability","version":2},"reference_index":33,"source":"pdf_text","source_observed_at":"2026-08-11T22:02:07.853120Z"},"links":{"citing_paper":"/paper/2412.03997"},"observation_digest":"sha256:06cecb0fbf620cc051dcfb73ad84bc1bb35b5f812f014ad1841af3b09481891f","observation_id":"c22bd5cb-cfe1-408a-a41f-7823137d8edf","resolution":{"observed_at":"2026-08-11T22:02:09.521786Z","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":"2026-08-11T22:02:09.485224Z","title":"The sharp quantita tive isoperimetric inequal- ity","venue":null,"work_id":"c6ca9d4c-85a1-4f8a-98ca-858fc37d526d","year":2008},"citing_paper":{"arxiv_id":"2412.03997","last_updated":"2025-01-29T09:31:52Z","snapshot_observed_at":"2026-08-11T21:50:24.929338Z","submitted_at":"2024-12-05T09:18:57Z","title":"Free energy minimizers with radial densities: classification and quantitative stability","version":2},"reference_index":34,"source":"pdf_text","source_observed_at":"2026-08-11T22:02:07.858467Z"},"links":{"citing_paper":"/paper/2412.03997"},"observation_digest":"sha256:694ff6a683e10aa427c2cabadbce716c9310ac369c612da1844348eb4ffce5d1","observation_id":"a8a34fa5-0094-484b-80ca-2aeb7a98b5f9","resolution":{"observed_at":"2026-08-11T22:02:09.493768Z","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":"2026-08-11T22:02:09.456004Z","title":"Existence and regularity for the problem of a pendent liquid drop","venue":null,"work_id":"78cdefff-ef12-4b40-945a-e1818e17d40d","year":1980},"citing_paper":{"arxiv_id":"2412.03997","last_updated":"2025-01-29T09:31:52Z","snapshot_observed_at":"2026-08-11T21:50:24.929338Z","submitted_at":"2024-12-05T09:18:57Z","title":"Free energy minimizers with radial densities: classification and quantitative stability","version":2},"reference_index":35,"source":"pdf_text","source_observed_at":"2026-08-11T22:02:07.880371Z"},"links":{"citing_paper":"/paper/2412.03997"},"observation_digest":"sha256:eaf009f423ae60172e4d5d17eee61b0b0f5212c90cdcaf92e698c05e38bc4b85","observation_id":"e756b663-c90b-4de8-9203-2e096fbea40f","resolution":{"observed_at":"2026-08-11T22:02:09.463031Z","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":"2026-08-11T22:02:09.427105Z","title":"Isoperimetric inequalities in Riemannian manifolds","venue":null,"work_id":"59285e89-c807-4d43-974d-e3b014355190","year":1986},"citing_paper":{"arxiv_id":"2412.03997","last_updated":"2025-01-29T09:31:52Z","snapshot_observed_at":"2026-08-11T21:50:24.929338Z","submitted_at":"2024-12-05T09:18:57Z","title":"Free energy minimizers with radial densities: classification and quantitative stability","version":2},"reference_index":36,"source":"pdf_text","source_observed_at":"2026-08-11T22:02:07.895115Z"},"links":{"citing_paper":"/paper/2412.03997"},"observation_digest":"sha256:40e893c7f13fa450db56615e5f3bec0c087c2524415a63282f80ebef365bd25b","observation_id":"4dba083b-e7bf-4821-81a4-c8c99217dcae","resolution":{"observed_at":"2026-08-11T22:02:09.439942Z","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":"2026-08-11T22:02:09.393504Z","title":"A quantitative isoperimetric inequality in n -dimensional space","venue":null,"work_id":"705bad40-0e5b-47c2-8e21-4a777ce91073","year":1992},"citing_paper":{"arxiv_id":"2412.03997","last_updated":"2025-01-29T09:31:52Z","snapshot_observed_at":"2026-08-11T21:50:24.929338Z","submitted_at":"2024-12-05T09:18:57Z","title":"Free energy minimizers with radial densities: classification and quantitative stability","version":2},"reference_index":37,"source":"pdf_text","source_observed_at":"2026-08-11T22:02:07.906449Z"},"links":{"citing_paper":"/paper/2412.03997"},"observation_digest":"sha256:643faeb9f45cf55c9a2703ab0dcba046a69262b823222a5487f057f0f3b6322d","observation_id":"91af780a-ad3a-4cd9-95d4-6cb5fff4b5ef","resolution":{"observed_at":"2026-08-11T22:02:09.400271Z","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":"2026-08-11T22:02:09.366455Z","title":"Some theorems on the free energies of cryst al surfaces","venue":null,"work_id":"221ee893-37d2-44b2-a44a-4f6685cf5e4a","year":1951},"citing_paper":{"arxiv_id":"2412.03997","last_updated":"2025-01-29T09:31:52Z","snapshot_observed_at":"2026-08-11T21:50:24.929338Z","submitted_at":"2024-12-05T09:18:57Z","title":"Free energy minimizers with radial densities: classification and quantitative stability","version":2},"reference_index":38,"source":"pdf_text","source_observed_at":"2026-08-11T22:02:07.914006Z"},"links":{"citing_paper":"/paper/2412.03997"},"observation_digest":"sha256:90838748abd7cbc383c7ad1dd5dec344ab13c7b1b4b56791168a05077f5795e9","observation_id":"f565d358-787c-40c6-b65b-ff8c46511343","resolution":{"observed_at":"2026-08-11T22:02:09.375419Z","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":"2026-08-11T22:02:09.340808Z","title":"The log-convex density conjecture and vertic al surface area in warped products","venue":null,"work_id":"8f75041a-1cc6-4b12-976b-1723e5aec6a8","year":2015},"citing_paper":{"arxiv_id":"2412.03997","last_updated":"2025-01-29T09:31:52Z","snapshot_observed_at":"2026-08-11T21:50:24.929338Z","submitted_at":"2024-12-05T09:18:57Z","title":"Free energy minimizers with radial densities: classification and quantitative stability","version":2},"reference_index":39,"source":"pdf_text","source_observed_at":"2026-08-11T22:02:07.922583Z"},"links":{"citing_paper":"/paper/2412.03997"},"observation_digest":"sha256:935feb604cd9365205ae8d7ddeecba217e8248f41fd683515559b38826d3a4c6","observation_id":"3f32e8b3-c8ac-46e6-aa4b-673501b67891","resolution":{"observed_at":"2026-08-11T22:02:09.351442Z","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":"2026-08-11T22:02:09.309845Z","title":"A weighted relative isoperimetric inequal ity in convex cones","venue":null,"work_id":"79574ff8-e58f-4456-9157-bb09fa148547","year":2021},"citing_paper":{"arxiv_id":"2412.03997","last_updated":"2025-01-29T09:31:52Z","snapshot_observed_at":"2026-08-11T21:50:24.929338Z","submitted_at":"2024-12-05T09:18:57Z","title":"Free energy minimizers with radial densities: classification and quantitative stability","version":2},"reference_index":40,"source":"pdf_text","source_observed_at":"2026-08-11T22:02:07.932542Z"},"links":{"citing_paper":"/paper/2412.03997"},"observation_digest":"sha256:6a852fa8efbe462c5e6ddbddc3edcfe69221af97ffa451702b86f61018aa7b27","observation_id":"4768f7c4-820b-4cd4-b1e6-465b508078da","resolution":{"observed_at":"2026-08-11T22:02:09.321946Z","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":"2026-08-11T22:02:09.276844Z","title":"On the equilibrium shape of a crystal","venue":null,"work_id":"2f22ab58-2625-46db-9c88-d539747a574d","year":2024},"citing_paper":{"arxiv_id":"2412.03997","last_updated":"2025-01-29T09:31:52Z","snapshot_observed_at":"2026-08-11T21:50:24.929338Z","submitted_at":"2024-12-05T09:18:57Z","title":"Free energy minimizers with radial densities: classification and quantitative stability","version":2},"reference_index":41,"source":"pdf_text","source_observed_at":"2026-08-11T22:02:07.940464Z"},"links":{"citing_paper":"/paper/2412.03997"},"observation_digest":"sha256:1c96ec21feb1470850ccd0ab8dd214ea46a62d873b2103b7c70562cc9459986f","observation_id":"e6e0ce60-7205-4ca0-bad6-c1aebc50cfdb","resolution":{"observed_at":"2026-08-11T22:02:09.285815Z","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":{"arxiv_id":"2501.07900","last_updated":"2025-01-14T07:30:54Z","snapshot_observed_at":"2026-08-10T20:27:38.439191Z","submitted_at":"2025-01-14T07:30:54Z","title":"The one-dimensional equilibrium shape of a crystal","version":1},"cited_work":{"arxiv_id":"2501.07900","doi":null,"metadata_source":"pith","pith_arxiv_id":"2501.07900","snapshot_observed_at":"2026-08-11T22:02:08.529224Z","title":"The one-dimensional equilibrium shape of a crystal","venue":"math-ph","work_id":"e91fe9d8-9d4c-43fa-87da-6be1a7c924c0","year":2025},"citing_paper":{"arxiv_id":"2412.03997","last_updated":"2025-01-29T09:31:52Z","snapshot_observed_at":"2026-08-11T21:50:24.929338Z","submitted_at":"2024-12-05T09:18:57Z","title":"Free energy minimizers with radial densities: classification and quantitative stability","version":2},"reference_index":42,"source":"pdf_text","source_observed_at":"2026-08-11T22:02:07.950733Z"},"links":{"cited_paper":"/paper/2501.07900","citing_paper":"/paper/2412.03997"},"observation_digest":"sha256:fa720a9ffccfc60daae170310a7e8f80352d4940e4275e4a0e7bbac3b7f806c5","observation_id":"e6ba1a55-265c-43af-8d5b-e3e680049019","resolution":{"observed_at":"2026-08-11T22:02:08.560187Z","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":{"arxiv_id":"2304.01866","last_updated":"2023-04-04T15:17:47Z","snapshot_observed_at":"2026-08-03T02:40:19.959762Z","submitted_at":"2023-04-04T15:17:47Z","title":"Minimizing the free energy","version":1},"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":"2304.01866","snapshot_observed_at":"2026-08-11T22:02:07.957095Z","title":"Indrei and A","venue":null,"work_id":null,"year":2023},"citing_paper":{"arxiv_id":"2412.03997","last_updated":"2025-01-29T09:31:52Z","snapshot_observed_at":"2026-08-11T21:50:24.929338Z","submitted_at":"2024-12-05T09:18:57Z","title":"Free energy minimizers with radial densities: classification and quantitative stability","version":2},"reference_index":43,"source":"pdf_text","source_observed_at":"2026-08-11T22:02:07.957095Z"},"links":{"cited_paper":"/paper/2304.01866","citing_paper":"/paper/2412.03997"},"observation_digest":"sha256:943c3826b4b93bb6fc1d96ec2f321fc02b4a6d87b40cfa057368cb8b1f5f0358","observation_id":"4247f6c1-d651-43d5-b258-a1293c6522c9","resolution":{"observed_at":"2026-08-11T22:02:07.957095Z","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":"2026-08-11T22:02:09.251286Z","title":"On the isoperimetric nature of a rearrangem ent inequality and its consequences for some variational problems","venue":null,"work_id":"f0a09aa6-64d2-40cf-8adb-2d8cdc5cb293","year":1986},"citing_paper":{"arxiv_id":"2412.03997","last_updated":"2025-01-29T09:31:52Z","snapshot_observed_at":"2026-08-11T21:50:24.929338Z","submitted_at":"2024-12-05T09:18:57Z","title":"Free energy minimizers with radial densities: classification and quantitative stability","version":2},"reference_index":44,"source":"pdf_text","source_observed_at":"2026-08-11T22:02:07.967150Z"},"links":{"citing_paper":"/paper/2412.03997"},"observation_digest":"sha256:1d388e8078c9c478297be853e3081d537d40e01350873c612f6f4c623f0e4099","observation_id":"dc20bf18-97ad-4054-91f9-3512c373f984","resolution":{"observed_at":"2026-08-11T22:02:09.258996Z","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":"2026-08-11T22:02:09.217655Z","title":"On isoperimetric se ts of radially symmetric measures","venue":null,"work_id":"b4622332-28d3-418b-878b-66d143d6e474","year":2011},"citing_paper":{"arxiv_id":"2412.03997","last_updated":"2025-01-29T09:31:52Z","snapshot_observed_at":"2026-08-11T21:50:24.929338Z","submitted_at":"2024-12-05T09:18:57Z","title":"Free energy minimizers with radial densities: classification and quantitative stability","version":2},"reference_index":45,"source":"pdf_text","source_observed_at":"2026-08-11T22:02:07.978057Z"},"links":{"citing_paper":"/paper/2412.03997"},"observation_digest":"sha256:f7075b09dd1e2172d654e8ba952eab1c29777aa5718f98505833dbe784375be5","observation_id":"20499b43-e599-48e7-93ae-f72c9fe51eb5","resolution":{"observed_at":"2026-08-11T22:02:09.226718Z","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":"2026-08-11T22:02:09.195433Z","title":"A class of weighted isoperimetric inequ alities in hyperbolic space","venue":null,"work_id":"c94a2f6e-15a9-49f4-9a57-788164bf8926","year":2023},"citing_paper":{"arxiv_id":"2412.03997","last_updated":"2025-01-29T09:31:52Z","snapshot_observed_at":"2026-08-11T21:50:24.929338Z","submitted_at":"2024-12-05T09:18:57Z","title":"Free energy minimizers with radial densities: classification and quantitative stability","version":2},"reference_index":46,"source":"pdf_text","source_observed_at":"2026-08-11T22:02:07.984915Z"},"links":{"citing_paper":"/paper/2412.03997"},"observation_digest":"sha256:9e018d98fa27929c7445c5ce2983dc1879ae35024d2a0b3084d4dec71172c20d","observation_id":"c38886d0-d211-487e-9b82-d2ef0dbf77a5","resolution":{"observed_at":"2026-08-11T22:02:09.201630Z","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":"2026-08-11T22:02:09.170626Z","title":null,"venue":null,"work_id":"d6afde86-c8ad-4076-ab9e-ff3e7eba60da","year":2012},"citing_paper":{"arxiv_id":"2412.03997","last_updated":"2025-01-29T09:31:52Z","snapshot_observed_at":"2026-08-11T21:50:24.929338Z","submitted_at":"2024-12-05T09:18:57Z","title":"Free energy minimizers with radial densities: classification and quantitative stability","version":2},"reference_index":47,"source":"pdf_text","source_observed_at":"2026-08-11T22:02:07.991077Z"},"links":{"citing_paper":"/paper/2412.03997"},"observation_digest":"sha256:8400dbe4745da82e6124329d8c5a4b1e0486793bd27172ca7477d7f731eca2e2","observation_id":"dbbf492b-5824-4a34-800c-939ad2833853","resolution":{"observed_at":"2026-08-11T22:02:09.177605Z","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":"2026-08-11T22:02:09.147279Z","title":null,"venue":null,"work_id":"33a4e4e2-dab4-4652-b7c8-371f806403de","year":2012},"citing_paper":{"arxiv_id":"2412.03997","last_updated":"2025-01-29T09:31:52Z","snapshot_observed_at":"2026-08-11T21:50:24.929338Z","submitted_at":"2024-12-05T09:18:57Z","title":"Free energy minimizers with radial densities: classification and quantitative stability","version":2},"reference_index":48,"source":"pdf_text","source_observed_at":"2026-08-11T22:02:07.998203Z"},"links":{"citing_paper":"/paper/2412.03997"},"observation_digest":"sha256:382d65f2931f5c0be7bdba0f13fb0f165e0e5c653b06b602d0ff582aa735647a","observation_id":"9af523e7-6add-4c4d-9fd1-88bb573622d5","resolution":{"observed_at":"2026-08-11T22:02:09.154179Z","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":{"arxiv_id":"2405.14676","last_updated":"2024-05-23T15:11:53Z","snapshot_observed_at":"2026-07-06T18:18:43.328203Z","submitted_at":"2024-05-23T15:11:53Z","title":"Stable minimal hypersurfaces in $\\mathbb R^6$","version":1},"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":"2405.14676","snapshot_observed_at":"2026-08-11T22:02:08.007552Z","title":"Stable minimal hypersurfaces in R6","venue":null,"work_id":null,"year":2024},"citing_paper":{"arxiv_id":"2412.03997","last_updated":"2025-01-29T09:31:52Z","snapshot_observed_at":"2026-08-11T21:50:24.929338Z","submitted_at":"2024-12-05T09:18:57Z","title":"Free energy minimizers with radial densities: classification and quantitative stability","version":2},"reference_index":49,"source":"pdf_text","source_observed_at":"2026-08-11T22:02:08.007552Z"},"links":{"cited_paper":"/paper/2405.14676","citing_paper":"/paper/2412.03997"},"observation_digest":"sha256:6625759bcbfe6dfa16c645a4b450e878b95fb6c83cd078983872f9a1ca8066ea","observation_id":"bc2e25e7-50d9-4b1a-87b2-27561f52c710","resolution":{"observed_at":"2026-08-11T22:02:08.007552Z","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":"2026-08-11T22:02:09.126006Z","title":"Equilibrium shapes for planar crystals i n an external field","venue":null,"work_id":"808bc0ff-94ac-4dea-b5d5-cf7d5fd4848d","year":1998},"citing_paper":{"arxiv_id":"2412.03997","last_updated":"2025-01-29T09:31:52Z","snapshot_observed_at":"2026-08-11T21:50:24.929338Z","submitted_at":"2024-12-05T09:18:57Z","title":"Free energy minimizers with radial densities: classification and quantitative stability","version":2},"reference_index":50,"source":"pdf_text","source_observed_at":"2026-08-11T22:02:08.016945Z"},"links":{"citing_paper":"/paper/2412.03997"},"observation_digest":"sha256:6d8c113127d2353cfc3a84369e6dfcbc547851811f2a0e018af8a8b46f7aa662","observation_id":"27e80527-8754-4b8f-b435-b7385e81394f","resolution":{"observed_at":"2026-08-11T22:02:09.134484Z","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":{"arxiv_id":"1712.07690","last_updated":"2017-12-19T13:47:56Z","snapshot_observed_at":"2026-07-06T06:15:24.909952Z","submitted_at":"2017-12-19T13:47:56Z","title":"A weighted isoperimetric inequality on the hyperbolic plane","version":1},"cited_work":{"arxiv_id":"1712.07690","doi":null,"metadata_source":"pith","pith_arxiv_id":"1712.07690","snapshot_observed_at":"2026-08-11T22:02:08.362643Z","title":"A weighted isoperimetric inequality on the hyperbolic plane","venue":"math.AP","work_id":"8600f1b5-6947-401f-a089-fd11bc05bc6a","year":2017},"citing_paper":{"arxiv_id":"2412.03997","last_updated":"2025-01-29T09:31:52Z","snapshot_observed_at":"2026-08-11T21:50:24.929338Z","submitted_at":"2024-12-05T09:18:57Z","title":"Free energy minimizers with radial densities: classification and quantitative stability","version":2},"reference_index":51,"source":"pdf_text","source_observed_at":"2026-08-11T22:02:08.023982Z"},"links":{"cited_paper":"/paper/1712.07690","citing_paper":"/paper/2412.03997"},"observation_digest":"sha256:e441a2dcf369e92b7d02b725a5744a8020a70e51c51c142b67c8e5f80c316bfa","observation_id":"477eea70-8163-479d-88cb-22384add76b8","resolution":{"observed_at":"2026-08-11T22:02:08.374846Z","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":"2026-08-11T22:02:09.086552Z","title":"An isoperimetric inequality in the pl ane with a log-convex density","venue":null,"work_id":"72d19b00-d936-4d27-a962-ebf2a756a233","year":2018},"citing_paper":{"arxiv_id":"2412.03997","last_updated":"2025-01-29T09:31:52Z","snapshot_observed_at":"2026-08-11T21:50:24.929338Z","submitted_at":"2024-12-05T09:18:57Z","title":"Free energy minimizers with radial densities: classification and quantitative stability","version":2},"reference_index":52,"source":"pdf_text","source_observed_at":"2026-08-11T22:02:08.032228Z"},"links":{"citing_paper":"/paper/2412.03997"},"observation_digest":"sha256:c4ee870cd333dbd036cebfc9cb985e7a3c5749859a4093f4f2832d040cde9a6f","observation_id":"8a093251-9f3b-4892-85f3-bb6195288c0b","resolution":{"observed_at":"2026-08-11T22:02:09.098093Z","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":"2026-08-11T22:02:09.057036Z","title":null,"venue":null,"work_id":"dbf81c99-27c1-48b7-9d56-2b5d2cf3cf23","year":2016},"citing_paper":{"arxiv_id":"2412.03997","last_updated":"2025-01-29T09:31:52Z","snapshot_observed_at":"2026-08-11T21:50:24.929338Z","submitted_at":"2024-12-05T09:18:57Z","title":"Free energy minimizers with radial densities: classification and quantitative stability","version":2},"reference_index":53,"source":"pdf_text","source_observed_at":"2026-08-11T22:02:08.049145Z"},"links":{"citing_paper":"/paper/2412.03997"},"observation_digest":"sha256:4ff34aefb5dde22a2fb7ec9bbd9b4c4be004fffdc112f10f2de69a06ea09d739","observation_id":"d141401e-8f47-4a38-839f-9a3e8492d8bc","resolution":{"observed_at":"2026-08-11T22:02:09.068013Z","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":"2026-08-11T22:02:09.031165Z","title":"Manifolds with density","venue":null,"work_id":"d1363965-9c08-4359-bb0f-d8c02d6c10ce","year":2005},"citing_paper":{"arxiv_id":"2412.03997","last_updated":"2025-01-29T09:31:52Z","snapshot_observed_at":"2026-08-11T21:50:24.929338Z","submitted_at":"2024-12-05T09:18:57Z","title":"Free energy minimizers with radial densities: classification and quantitative stability","version":2},"reference_index":54,"source":"pdf_text","source_observed_at":"2026-08-11T22:02:08.064659Z"},"links":{"citing_paper":"/paper/2412.03997"},"observation_digest":"sha256:a5fd8d7c7b68d6583238611bf2118a80feb6fbdd722fd12ef6bacf8eada36e9a","observation_id":"70c5c413-05ea-4192-9ff4-30ea8572879f","resolution":{"observed_at":"2026-08-11T22:02:09.036940Z","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":"2026-08-11T22:02:09.010110Z","title":"Regularity of isoperimetric hypersurface s in Riemannian manifolds","venue":null,"work_id":"836b3a3f-a483-4b26-bb1f-ffc4873fb91e","year":2003},"citing_paper":{"arxiv_id":"2412.03997","last_updated":"2025-01-29T09:31:52Z","snapshot_observed_at":"2026-08-11T21:50:24.929338Z","submitted_at":"2024-12-05T09:18:57Z","title":"Free energy minimizers with radial densities: classification and quantitative stability","version":2},"reference_index":55,"source":"pdf_text","source_observed_at":"2026-08-11T22:02:08.070060Z"},"links":{"citing_paper":"/paper/2412.03997"},"observation_digest":"sha256:88ee8559f0137a9031372372b17190e457662427dd22899f98fad76df58d46d3","observation_id":"49424e0f-1cb9-4243-bf75-d5b666dda01a","resolution":{"observed_at":"2026-08-11T22:02:09.015861Z","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":"2026-08-11T22:02:08.987984Z","title":"Steiner and Schwarz s ymmetrization in warped products and fiber bundles with density","venue":null,"work_id":"10d9f1cc-252e-43a8-9937-8e12056b3559","year":2011},"citing_paper":{"arxiv_id":"2412.03997","last_updated":"2025-01-29T09:31:52Z","snapshot_observed_at":"2026-08-11T21:50:24.929338Z","submitted_at":"2024-12-05T09:18:57Z","title":"Free energy minimizers with radial densities: classification and quantitative stability","version":2},"reference_index":56,"source":"pdf_text","source_observed_at":"2026-08-11T22:02:08.078900Z"},"links":{"citing_paper":"/paper/2412.03997"},"observation_digest":"sha256:5c1e16d618991436122c201b4af8228f73eb9ea8a6940d9cbe0e0c8ca89002ec","observation_id":"d549179e-01a7-4a28-9f1b-8623ac131797","resolution":{"observed_at":"2026-08-11T22:02:08.993928Z","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":"2026-08-11T22:02:08.960182Z","title":"Existence of isoperimetric regions in Rn with density","venue":null,"work_id":"9818c530-f2ea-4385-91b5-b3ac45f7d58a","year":2013},"citing_paper":{"arxiv_id":"2412.03997","last_updated":"2025-01-29T09:31:52Z","snapshot_observed_at":"2026-08-11T21:50:24.929338Z","submitted_at":"2024-12-05T09:18:57Z","title":"Free energy minimizers with radial densities: classification and quantitative stability","version":2},"reference_index":57,"source":"pdf_text","source_observed_at":"2026-08-11T22:02:08.084724Z"},"links":{"citing_paper":"/paper/2412.03997"},"observation_digest":"sha256:119ad4249b98268cc28f7351814c26b3df08ed19c37148828e0f11ab3b053352","observation_id":"2787dab5-3d93-40a1-bc33-40cf5b5457c5","resolution":{"observed_at":"2026-08-11T22:02:08.967071Z","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":"2026-08-11T22:02:08.934722Z","title":"A strong form of the quantitative Wulff ine quality","venue":null,"work_id":"f149b303-6f05-4556-ab98-840dfd649cb3","year":2016},"citing_paper":{"arxiv_id":"2412.03997","last_updated":"2025-01-29T09:31:52Z","snapshot_observed_at":"2026-08-11T21:50:24.929338Z","submitted_at":"2024-12-05T09:18:57Z","title":"Free energy minimizers with radial densities: classification and quantitative stability","version":2},"reference_index":58,"source":"pdf_text","source_observed_at":"2026-08-11T22:02:08.091828Z"},"links":{"citing_paper":"/paper/2412.03997"},"observation_digest":"sha256:aed3766f7c18517af9b6e82db27b39c5d4f450ad4df0cd2fb8ab42390642f45f","observation_id":"1c26f0de-3924-40a7-838f-987863d4b883","resolution":{"observed_at":"2026-08-11T22:02:08.940557Z","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":"2026-08-11T22:02:08.901695Z","title":"Stability of the Wulff shape","venue":null,"work_id":"5360915a-85f3-4790-b246-7155712ca775","year":1998},"citing_paper":{"arxiv_id":"2412.03997","last_updated":"2025-01-29T09:31:52Z","snapshot_observed_at":"2026-08-11T21:50:24.929338Z","submitted_at":"2024-12-05T09:18:57Z","title":"Free energy minimizers with radial densities: classification and quantitative stability","version":2},"reference_index":59,"source":"pdf_text","source_observed_at":"2026-08-11T22:02:08.102813Z"},"links":{"citing_paper":"/paper/2412.03997"},"observation_digest":"sha256:08b9823195fdf42796e9dc131d9e83b5812797a5dddcee56a50d223a94a8f6dd","observation_id":"a021dfbf-ed79-448a-a00e-86511bfad07f","resolution":{"observed_at":"2026-08-11T22:02:08.912727Z","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":"2026-08-11T22:02:08.866116Z","title":"Regularity of free bounda ries in anisotropic capillar- ity problems and the validity of Young’s law","venue":null,"work_id":"9c777aaf-24b4-4440-83e8-8feee84e3775","year":2015},"citing_paper":{"arxiv_id":"2412.03997","last_updated":"2025-01-29T09:31:52Z","snapshot_observed_at":"2026-08-11T21:50:24.929338Z","submitted_at":"2024-12-05T09:18:57Z","title":"Free energy minimizers with radial densities: classification and quantitative stability","version":2},"reference_index":60,"source":"pdf_text","source_observed_at":"2026-08-11T22:02:08.122084Z"},"links":{"citing_paper":"/paper/2412.03997"},"observation_digest":"sha256:9d22c70f727eecd60a6ca821b31c0be9c0aaa66fb2b7ae38fa4011210d578310","observation_id":"d24a13ac-a6cb-4a77-9a67-261c700e823d","resolution":{"observed_at":"2026-08-11T22:02:08.880583Z","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":"2026-08-11T22:02:08.839697Z","title":"Stable and isoperimetric regions in some w eighted manifolds with boundary","venue":null,"work_id":"5607e2a0-8446-46f1-bd4f-9bac1a00a9c2","year":2021},"citing_paper":{"arxiv_id":"2412.03997","last_updated":"2025-01-29T09:31:52Z","snapshot_observed_at":"2026-08-11T21:50:24.929338Z","submitted_at":"2024-12-05T09:18:57Z","title":"Free energy minimizers with radial densities: classification and quantitative stability","version":2},"reference_index":61,"source":"pdf_text","source_observed_at":"2026-08-11T22:02:08.152836Z"},"links":{"citing_paper":"/paper/2412.03997"},"observation_digest":"sha256:913ac7652bd390caa767bb5d9c5febeae9bb620a276bb4abee9347b6074d6e81","observation_id":"e1a6e46d-27cc-4c3c-9ece-a3a7f306adb2","resolution":{"observed_at":"2026-08-11T22:02:08.847985Z","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":"2026-08-11T22:02:08.809890Z","title":"On the is operimetric problem in Euclidean space with density","venue":null,"work_id":"691471b3-a5c5-4457-91b0-95793a01c32e","year":2008},"citing_paper":{"arxiv_id":"2412.03997","last_updated":"2025-01-29T09:31:52Z","snapshot_observed_at":"2026-08-11T21:50:24.929338Z","submitted_at":"2024-12-05T09:18:57Z","title":"Free energy minimizers with radial densities: classification and quantitative stability","version":2},"reference_index":62,"source":"pdf_text","source_observed_at":"2026-08-11T22:02:08.159329Z"},"links":{"citing_paper":"/paper/2412.03997"},"observation_digest":"sha256:6e05d4d59b314bad8ac2f3879538b33e33e2cdb07b1ae8697b3cd148b27fbc43","observation_id":"8052b423-7e77-4561-b939-88da01dffab3","resolution":{"observed_at":"2026-08-11T22:02:08.816914Z","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":"2026-08-11T22:02:08.783806Z","title":"Locally constrained inverse cur vature flows","venue":null,"work_id":"66d34282-4121-4edc-ae22-887ee30ebcdb","year":2019},"citing_paper":{"arxiv_id":"2412.03997","last_updated":"2025-01-29T09:31:52Z","snapshot_observed_at":"2026-08-11T21:50:24.929338Z","submitted_at":"2024-12-05T09:18:57Z","title":"Free energy minimizers with radial densities: classification and quantitative stability","version":2},"reference_index":63,"source":"pdf_text","source_observed_at":"2026-08-11T22:02:08.166866Z"},"links":{"citing_paper":"/paper/2412.03997"},"observation_digest":"sha256:2ba61551a6806108e975abff9299553a15744d4b83a728869aeca0c4f6f4c44d","observation_id":"b2f29eda-b8d4-462b-af19-0b554e73dc3a","resolution":{"observed_at":"2026-08-11T22:02:08.792768Z","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":"2026-08-11T22:02:08.756439Z","title":"Approaching the isoperimetric problem in H C m via the hyperbolic log- convex density conjecture","venue":null,"work_id":"62184b46-de2c-4fac-99ae-e1387833e46e","year":2024},"citing_paper":{"arxiv_id":"2412.03997","last_updated":"2025-01-29T09:31:52Z","snapshot_observed_at":"2026-08-11T21:50:24.929338Z","submitted_at":"2024-12-05T09:18:57Z","title":"Free energy minimizers with radial densities: classification and quantitative stability","version":2},"reference_index":64,"source":"pdf_text","source_observed_at":"2026-08-11T22:02:08.175195Z"},"links":{"citing_paper":"/paper/2412.03997"},"observation_digest":"sha256:ed112ea9d9f8ba81a5dc2f9ff0e0cb73c22e99678a83d49cd8fba14510b9ddbb","observation_id":"326d9e78-9089-4ab1-85cf-f75a06f976d7","resolution":{"observed_at":"2026-08-11T22:02:08.762195Z","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":"2026-08-11T22:02:08.723555Z","title":"Quantitative C 1-stability of spheres in ra nk one symmetric spaces of non-compact type","venue":null,"work_id":"ee7a13ae-00a6-42a6-8000-a776feff6a8c","year":2024},"citing_paper":{"arxiv_id":"2412.03997","last_updated":"2025-01-29T09:31:52Z","snapshot_observed_at":"2026-08-11T21:50:24.929338Z","submitted_at":"2024-12-05T09:18:57Z","title":"Free energy minimizers with radial densities: classification and quantitative stability","version":2},"reference_index":65,"source":"pdf_text","source_observed_at":"2026-08-11T22:02:08.183585Z"},"links":{"citing_paper":"/paper/2412.03997"},"observation_digest":"sha256:805e40a2ea535983e3ffb5385a09e47413a3409a4c5a1970a02d3d844ea48f4c","observation_id":"91e97efd-86e6-488d-bd44-a6ff63121b67","resolution":{"observed_at":"2026-08-11T22:02:08.730449Z","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":"2026-08-11T22:02:08.687151Z","title":"Einfache Beweise der isoperimetrischen H auptsätze","venue":null,"work_id":"1234c696-0339-49b2-82df-ea0a3c525ab4","year":null},"citing_paper":{"arxiv_id":"2412.03997","last_updated":"2025-01-29T09:31:52Z","snapshot_observed_at":"2026-08-11T21:50:24.929338Z","submitted_at":"2024-12-05T09:18:57Z","title":"Free energy minimizers with radial densities: classification and quantitative stability","version":2},"reference_index":66,"source":"pdf_text","source_observed_at":"2026-08-11T22:02:08.211062Z"},"links":{"citing_paper":"/paper/2412.03997"},"observation_digest":"sha256:c2b9dda92f9b5fa706e6ceba956bc0d9999deeca1104b96638339986a8721ab7","observation_id":"c024524d-fe3f-4ca8-a830-cc84a90f3fcb","resolution":{"observed_at":"2026-08-11T22:02:08.698148Z","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":"2026-08-11T22:02:08.660781Z","title":"On the sphericity of liquid droplets","venue":null,"work_id":"b479cbd9-e953-4527-b237-4138e1a8f3ca","year":1984},"citing_paper":{"arxiv_id":"2412.03997","last_updated":"2025-01-29T09:31:52Z","snapshot_observed_at":"2026-08-11T21:50:24.929338Z","submitted_at":"2024-12-05T09:18:57Z","title":"Free energy minimizers with radial densities: classification and quantitative stability","version":2},"reference_index":67,"source":"pdf_text","source_observed_at":"2026-08-11T22:02:08.218685Z"},"links":{"citing_paper":"/paper/2412.03997"},"observation_digest":"sha256:2ea4772b1c51d03c444307bd66e95d6cb79eca0ee0255b1711cbf96a8863954c","observation_id":"14d397c4-3112-4af0-93c4-f9c8d26486ff","resolution":{"observed_at":"2026-08-11T22:02:08.669567Z","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":"2026-08-11T22:02:08.632286Z","title":"The spaces BV and quasilinear equation s","venue":null,"work_id":"1afdf394-9772-4a12-9f67-e38482af07d3","year":1967},"citing_paper":{"arxiv_id":"2412.03997","last_updated":"2025-01-29T09:31:52Z","snapshot_observed_at":"2026-08-11T21:50:24.929338Z","submitted_at":"2024-12-05T09:18:57Z","title":"Free energy minimizers with radial densities: classification and quantitative stability","version":2},"reference_index":68,"source":"pdf_text","source_observed_at":"2026-08-11T22:02:08.237182Z"},"links":{"citing_paper":"/paper/2412.03997"},"observation_digest":"sha256:a0f4dfabae4d08574360045def08e1ee7acdf31156279fcc3352a515be6cb441","observation_id":"dcc74a6e-19ee-4b29-bedd-3073dc707421","resolution":{"observed_at":"2026-08-11T22:02:08.639391Z","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":"2026-08-11T22:02:08.606711Z","title":"The symmetry of sessile and pendent drops","venue":null,"work_id":"edc5077a-592b-49b0-bc6f-fb00de0a06e7","year":1980},"citing_paper":{"arxiv_id":"2412.03997","last_updated":"2025-01-29T09:31:52Z","snapshot_observed_at":"2026-08-11T21:50:24.929338Z","submitted_at":"2024-12-05T09:18:57Z","title":"Free energy minimizers with radial densities: classification and quantitative stability","version":2},"reference_index":69,"source":"pdf_text","source_observed_at":"2026-08-11T22:02:08.245459Z"},"links":{"citing_paper":"/paper/2412.03997"},"observation_digest":"sha256:6dbaf9baee7e677df316d2b7128bcccdd68abf0c3216737c4f20b74d22ea689b","observation_id":"70502dc1-6ed0-4e05-8124-531f7a4e1fdd","resolution":{"observed_at":"2026-08-11T22:02:08.614898Z","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"}}],"paper":{"arxiv_id":"2412.03997","last_updated":"2025-01-29T09:31:52Z","latest_version":2,"primary_category":"math.AP","snapshot_observed_at":"2026-08-11T21:50:24.929338Z","submitted_at":"2024-12-05T09:18:57Z","title":"Free energy minimizers with radial densities: classification and quantitative stability"},"reference_resolution":{"displayed":69,"state_counts":{"malformed_identifier":1,"metadata_mismatch":0,"parse_uncertain":0,"unresolved":9,"verified_exact":2,"verified_fuzzy":57},"total_outbound_references":69},"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 69 of 69 outbound references and 1 inbound Pith citation observation for arXiv:2412.03997."}