{"as_of":"2026-08-08T23:53:00Z","caps":{"database_statements":6,"inbound":100,"outbound":100},"context_digest":"sha256:5bce1c5470d6d86d036289359d265da652aa6735727949b2864e4a26e39932c2","coverage":[{"denominator":35,"lane":"reference_resolution","note":"Typed states for the displayed outbound observations.","records_observed":35,"source":"paper_references, paper_reference_links","source_observed_at":"2026-08-07T10:46:08.092561Z","state":"measured"},{"denominator":35,"lane":"standing_notices","note":"One-hop event checks from named stored sources.","records_observed":35,"source":"scholarly_work_events, retraction_status_cache","source_observed_at":"2026-08-08T06:32:00.761636+00:00","state":"measured"},{"denominator":0,"lane":"inbound_itemization","note":"Pith citing papers itemized under the disclosed page cap.","records_observed":0,"source":"paper_references, paper_reference_links","source_observed_at":null,"state":"measured"},{"denominator":1,"lane":"external_citation_measurements","note":"A source-named dated measurement, never combined with another source.","records_observed":0,"source":"cited_works","source_observed_at":null,"state":"measured"}],"external_citation_measurements":[],"inbound":[],"links":{"evidence":"/evidence","html":"/paper/2506.04937/citation-record","integrity":"/paper/2506.04937/integrity","json":"/paper/2506.04937/citation-record.json","paper":"/paper/2506.04937"},"outbound":[{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":"raw_reference","pith_arxiv_id":null,"snapshot_observed_at":"2026-08-07T10:46:08.547050Z","title":"Dirichlet’s problem for multiple valued functions and the regularity of mass minimizing integral currents","venue":null,"work_id":"534345fd-4dda-4949-acb1-1cea7c3d950d","year":1979},"citing_paper":{"arxiv_id":"2506.04937","last_updated":"2025-06-05T12:11:00Z","snapshot_observed_at":"2026-08-07T10:27:52.481856Z","submitted_at":"2025-06-05T12:11:00Z","title":"Gradient estimates and parabolic frequency monotonicity for positive solutions of the heat equation under generalized Ricci flow","version":1},"reference_index":1,"source":"pdf_text","source_observed_at":"2026-08-07T10:46:07.971851Z"},"links":{"citing_paper":"/paper/2506.04937"},"observation_digest":"sha256:e5dc816e5cea5302ba368e8fed23140a6af42f6496926bd96afe6d91ccdeefec","observation_id":"3fe9149f-084c-42ed-806d-ccdf4b9f7438","resolution":{"observed_at":"2026-08-07T10:46:08.551956Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-08T06:32:00.761636+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-08T06:32:00.761636+00:00","source":"crossref"},{"observed_at":"2026-08-08T06:31:55.24221+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":"2026-08-07T10:46:08.531339Z","title":"Gradient estimates for the heat equation under the Ricci-harmonic map flow","venue":null,"work_id":"34fbb204-84be-49ff-bb04-ccc3a4bf5b38","year":2015},"citing_paper":{"arxiv_id":"2506.04937","last_updated":"2025-06-05T12:11:00Z","snapshot_observed_at":"2026-08-07T10:27:52.481856Z","submitted_at":"2025-06-05T12:11:00Z","title":"Gradient estimates and parabolic frequency monotonicity for positive solutions of the heat equation under generalized Ricci flow","version":1},"reference_index":2,"source":"pdf_text","source_observed_at":"2026-08-07T10:46:07.976484Z"},"links":{"citing_paper":"/paper/2506.04937"},"observation_digest":"sha256:09767209806609510ee9b621708eaa8a77f5ce55a12fc8c8ddec323badffc6a9","observation_id":"b57afc3e-7e0b-453f-a55d-e09e00fc1fe7","resolution":{"observed_at":"2026-08-07T10:46:08.538415Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-08T06:32:00.761636+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-08T06:32:00.761636+00:00","source":"crossref"},{"observed_at":"2026-08-08T06:31:55.24221+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":"2026-08-07T10:46:08.514236Z","title":"Gradient estimates for the heat equation under the Ricci flow","venue":null,"work_id":"7567f906-2000-43e8-8f52-cbf35f1ad121","year":2010},"citing_paper":{"arxiv_id":"2506.04937","last_updated":"2025-06-05T12:11:00Z","snapshot_observed_at":"2026-08-07T10:27:52.481856Z","submitted_at":"2025-06-05T12:11:00Z","title":"Gradient estimates and parabolic frequency monotonicity for positive solutions of the heat equation under generalized Ricci flow","version":1},"reference_index":3,"source":"pdf_text","source_observed_at":"2026-08-07T10:46:07.980043Z"},"links":{"citing_paper":"/paper/2506.04937"},"observation_digest":"sha256:6a267f18938546ea482f2b79377a2713df687fd6e7bf2fec8d25f2235862a097","observation_id":"61cdf08a-def5-4194-802d-9b1749c143ea","resolution":{"observed_at":"2026-08-07T10:46:08.521208Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-08T06:32:00.761636+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-08T06:32:00.761636+00:00","source":"crossref"},{"observed_at":"2026-08-08T06:31:55.24221+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":"2026-08-07T10:46:08.498037Z","title":"Parabolic frequency on Ricci flows","venue":null,"work_id":"ddb420c2-d334-4544-adab-dc0ab4230414","year":2023},"citing_paper":{"arxiv_id":"2506.04937","last_updated":"2025-06-05T12:11:00Z","snapshot_observed_at":"2026-08-07T10:27:52.481856Z","submitted_at":"2025-06-05T12:11:00Z","title":"Gradient estimates and parabolic frequency monotonicity for positive solutions of the heat equation under generalized Ricci flow","version":1},"reference_index":4,"source":"pdf_text","source_observed_at":"2026-08-07T10:46:07.983631Z"},"links":{"citing_paper":"/paper/2506.04937"},"observation_digest":"sha256:5ae21cf024d96ba745fc07eb716c5169f27f6772d3a3e355c3a81d9ca3c256e6","observation_id":"86792135-f5b8-4e73-9668-ff26c66911c2","resolution":{"observed_at":"2026-08-07T10:46:08.503473Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-08T06:32:00.761636+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-08T06:32:00.761636+00:00","source":"crossref"},{"observed_at":"2026-08-08T06:31:55.24221+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":"2026-08-07T10:46:08.481029Z","title":"Parabolic frequency for the mean curvature flow","venue":null,"work_id":"2d03adb3-ce32-49a3-9827-38809a0e038f","year":2024},"citing_paper":{"arxiv_id":"2506.04937","last_updated":"2025-06-05T12:11:00Z","snapshot_observed_at":"2026-08-07T10:27:52.481856Z","submitted_at":"2025-06-05T12:11:00Z","title":"Gradient estimates and parabolic frequency monotonicity for positive solutions of the heat equation under generalized Ricci flow","version":1},"reference_index":5,"source":"pdf_text","source_observed_at":"2026-08-07T10:46:07.987384Z"},"links":{"citing_paper":"/paper/2506.04937"},"observation_digest":"sha256:e2b3d78f5a130b2cbda4233845c4fc96ee0cfeb7fc93631f73d780d280f2a992","observation_id":"165541cd-17e5-4106-951d-44ee0ca015ad","resolution":{"observed_at":"2026-08-07T10:46:08.487614Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-08T06:32:00.761636+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-08T06:32:00.761636+00:00","source":"crossref"},{"observed_at":"2026-08-08T06:31:55.24221+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":"2026-08-07T10:46:08.467313Z","title":"Lu, and L","venue":null,"work_id":"a9fdc097-5db0-43f4-91f3-32e705241382","year":2006},"citing_paper":{"arxiv_id":"2506.04937","last_updated":"2025-06-05T12:11:00Z","snapshot_observed_at":"2026-08-07T10:27:52.481856Z","submitted_at":"2025-06-05T12:11:00Z","title":"Gradient estimates and parabolic frequency monotonicity for positive solutions of the heat equation under generalized Ricci flow","version":1},"reference_index":6,"source":"pdf_text","source_observed_at":"2026-08-07T10:46:07.991472Z"},"links":{"citing_paper":"/paper/2506.04937"},"observation_digest":"sha256:95f43df209d52e771c90c8240dfd0e9f1b7f486eaf1b0ee90849ee7f1b53d1f4","observation_id":"a94719c4-da28-45d7-85e1-774a3c2a14aa","resolution":{"observed_at":"2026-08-07T10:46:08.473070Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-08T06:32:00.761636+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-08T06:32:00.761636+00:00","source":"crossref"},{"observed_at":"2026-08-08T06:31:55.24221+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":"2026-08-07T10:46:08.452774Z","title":"Harmonic functions with polynomial growth","venue":null,"work_id":"5616be77-90a8-4b0b-adce-5f55520d76e3","year":1997},"citing_paper":{"arxiv_id":"2506.04937","last_updated":"2025-06-05T12:11:00Z","snapshot_observed_at":"2026-08-07T10:27:52.481856Z","submitted_at":"2025-06-05T12:11:00Z","title":"Gradient estimates and parabolic frequency monotonicity for positive solutions of the heat equation under generalized Ricci flow","version":1},"reference_index":7,"source":"pdf_text","source_observed_at":"2026-08-07T10:46:07.995723Z"},"links":{"citing_paper":"/paper/2506.04937"},"observation_digest":"sha256:ecbc5b003c623e2e2a6dfa87aef788609f13d61b0e13608ce272d3cebf8628bb","observation_id":"c02a947d-68d0-4de4-94a7-b684e1b3afa7","resolution":{"observed_at":"2026-08-07T10:46:08.457278Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-08T06:32:00.761636+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-08T06:32:00.761636+00:00","source":"crossref"},{"observed_at":"2026-08-08T06:31:55.24221+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":"2026-08-07T10:46:08.437324Z","title":"Parabolic frequency on manifolds","venue":null,"work_id":"db47eca6-2825-43ff-84d0-d9e27148b565","year":2022},"citing_paper":{"arxiv_id":"2506.04937","last_updated":"2025-06-05T12:11:00Z","snapshot_observed_at":"2026-08-07T10:27:52.481856Z","submitted_at":"2025-06-05T12:11:00Z","title":"Gradient estimates and parabolic frequency monotonicity for positive solutions of the heat equation under generalized Ricci flow","version":1},"reference_index":8,"source":"pdf_text","source_observed_at":"2026-08-07T10:46:07.999179Z"},"links":{"citing_paper":"/paper/2506.04937"},"observation_digest":"sha256:ce4562eca415355f85c078c1ddc94433fa42c250731aaecc1e4fe24275fd0e22","observation_id":"34edf084-d5b4-4f0a-a18f-a1da8b3b74e5","resolution":{"observed_at":"2026-08-07T10:46:08.442483Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-08T06:32:00.761636+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-08T06:32:00.761636+00:00","source":"crossref"},{"observed_at":"2026-08-08T06:31:55.24221+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":"2026-08-07T10:46:08.421636Z","title":"Ricci flow, Killing spinors, and T-duality in generalized geometry","venue":null,"work_id":"5419a4e0-5983-425f-a79a-9abb4ea3344a","year":2019},"citing_paper":{"arxiv_id":"2506.04937","last_updated":"2025-06-05T12:11:00Z","snapshot_observed_at":"2026-08-07T10:27:52.481856Z","submitted_at":"2025-06-05T12:11:00Z","title":"Gradient estimates and parabolic frequency monotonicity for positive solutions of the heat equation under generalized Ricci flow","version":1},"reference_index":9,"source":"pdf_text","source_observed_at":"2026-08-07T10:46:08.002872Z"},"links":{"citing_paper":"/paper/2506.04937"},"observation_digest":"sha256:3403b0456644853693f760bd8be6301ce7f6ff296f4523524838d4c3b35f16a5","observation_id":"6fae6501-64b0-4259-b431-4501d7eb89ca","resolution":{"observed_at":"2026-08-07T10:46:08.426434Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-08T06:32:00.761636+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-08T06:32:00.761636+00:00","source":"crossref"},{"observed_at":"2026-08-08T06:31:55.24221+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":"2026-08-07T10:46:08.408223Z","title":"Streets.Generalized Ricci flow","venue":null,"work_id":"ebaa49c2-7d9d-437e-82ec-85316c9b120d","year":2021},"citing_paper":{"arxiv_id":"2506.04937","last_updated":"2025-06-05T12:11:00Z","snapshot_observed_at":"2026-08-07T10:27:52.481856Z","submitted_at":"2025-06-05T12:11:00Z","title":"Gradient estimates and parabolic frequency monotonicity for positive solutions of the heat equation under generalized Ricci flow","version":1},"reference_index":10,"source":"pdf_text","source_observed_at":"2026-08-07T10:46:08.006338Z"},"links":{"citing_paper":"/paper/2506.04937"},"observation_digest":"sha256:c4dc5c7406db6cad05f031ae9e3f9ef8041b2dea564b06cc4f5c908cf26cb123","observation_id":"4dda47b2-09a1-44c0-b25a-09688e8ffbd8","resolution":{"observed_at":"2026-08-07T10:46:08.412943Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-08T06:32:00.761636+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-08T06:32:00.761636+00:00","source":"crossref"},{"observed_at":"2026-08-08T06:31:55.24221+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":"2026-08-07T10:46:08.392547Z","title":"Monotonicity properties of variational integrals,A p weights and unique continuation","venue":null,"work_id":"dd141b2e-17bf-4bda-b69d-e7ea52d95ed3","year":1986},"citing_paper":{"arxiv_id":"2506.04937","last_updated":"2025-06-05T12:11:00Z","snapshot_observed_at":"2026-08-07T10:27:52.481856Z","submitted_at":"2025-06-05T12:11:00Z","title":"Gradient estimates and parabolic frequency monotonicity for positive solutions of the heat equation under generalized Ricci flow","version":1},"reference_index":11,"source":"pdf_text","source_observed_at":"2026-08-07T10:46:08.009726Z"},"links":{"citing_paper":"/paper/2506.04937"},"observation_digest":"sha256:d60f6ee538f4cd985b843053f0eba89cafd312ab10e9a5291b770b42297346d5","observation_id":"f6616e36-979e-4dc8-bb40-9609fc0a2af0","resolution":{"observed_at":"2026-08-07T10:46:08.397497Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-08T06:32:00.761636+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-08T06:32:00.761636+00:00","source":"crossref"},{"observed_at":"2026-08-08T06:31:55.24221+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":"2026-08-07T10:46:08.379022Z","title":"Unique continuation for elliptic operators: a geometric- variational approach","venue":null,"work_id":"6859d049-1460-4c99-866b-8947565f99c7","year":1987},"citing_paper":{"arxiv_id":"2506.04937","last_updated":"2025-06-05T12:11:00Z","snapshot_observed_at":"2026-08-07T10:27:52.481856Z","submitted_at":"2025-06-05T12:11:00Z","title":"Gradient estimates and parabolic frequency monotonicity for positive solutions of the heat equation under generalized Ricci flow","version":1},"reference_index":12,"source":"pdf_text","source_observed_at":"2026-08-07T10:46:08.013335Z"},"links":{"citing_paper":"/paper/2506.04937"},"observation_digest":"sha256:9f32166f0da36e6363e589c1919a36a78df16aab24cf78a732d5be0f3a570175","observation_id":"7c79f139-bdef-4f6a-8030-da4573c85e92","resolution":{"observed_at":"2026-08-07T10:46:08.384592Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-08T06:32:00.761636+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-08T06:32:00.761636+00:00","source":"crossref"},{"observed_at":"2026-08-08T06:31:55.24221+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":"2026-08-07T10:46:08.363924Z","title":"A matrix Harnack estimate for the heat equation","venue":null,"work_id":"455829f7-92ad-4a1f-8627-3af54dc91b91","year":1993},"citing_paper":{"arxiv_id":"2506.04937","last_updated":"2025-06-05T12:11:00Z","snapshot_observed_at":"2026-08-07T10:27:52.481856Z","submitted_at":"2025-06-05T12:11:00Z","title":"Gradient estimates and parabolic frequency monotonicity for positive solutions of the heat equation under generalized Ricci flow","version":1},"reference_index":13,"source":"pdf_text","source_observed_at":"2026-08-07T10:46:08.017821Z"},"links":{"citing_paper":"/paper/2506.04937"},"observation_digest":"sha256:5191e62ae78f5d0b38410b55d239a66fb7c3b12311e24ebd7cc76655e7f5f59c","observation_id":"3803d451-efe2-4050-9a23-484239c5ab0c","resolution":{"observed_at":"2026-08-07T10:46:08.369538Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-08T06:32:00.761636+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-08T06:32:00.761636+00:00","source":"crossref"},{"observed_at":"2026-08-08T06:31:55.24221+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":"2026-08-07T10:46:08.349248Z","title":"Geometric measure of singular sets of elliptic equations","venue":null,"work_id":"5463e93d-a678-402d-ba8f-91fc88fe2144","year":1998},"citing_paper":{"arxiv_id":"2506.04937","last_updated":"2025-06-05T12:11:00Z","snapshot_observed_at":"2026-08-07T10:27:52.481856Z","submitted_at":"2025-06-05T12:11:00Z","title":"Gradient estimates and parabolic frequency monotonicity for positive solutions of the heat equation under generalized Ricci flow","version":1},"reference_index":14,"source":"pdf_text","source_observed_at":"2026-08-07T10:46:08.021178Z"},"links":{"citing_paper":"/paper/2506.04937"},"observation_digest":"sha256:e3026c348333bfc85badaf1f71662e3e7979b8ff41c5e39ca143ddb973da2de2","observation_id":"f19b3053-39b0-4fdc-89b8-16cde4c7caea","resolution":{"observed_at":"2026-08-07T10:46:08.354107Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-08T06:32:00.761636+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-08T06:32:00.761636+00:00","source":"crossref"},{"observed_at":"2026-08-08T06:31:55.24221+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":"2026-08-07T10:46:08.335611Z","title":"On the shrinking entropy functional for generalized Ricci flow","venue":null,"work_id":"dd4983f8-7120-4657-9464-742140db8385","year":2025},"citing_paper":{"arxiv_id":"2506.04937","last_updated":"2025-06-05T12:11:00Z","snapshot_observed_at":"2026-08-07T10:27:52.481856Z","submitted_at":"2025-06-05T12:11:00Z","title":"Gradient estimates and parabolic frequency monotonicity for positive solutions of the heat equation under generalized Ricci flow","version":1},"reference_index":15,"source":"pdf_text","source_observed_at":"2026-08-07T10:46:08.024599Z"},"links":{"citing_paper":"/paper/2506.04937"},"observation_digest":"sha256:fb01a1ccd10b5d05af355e7d50955753c3ed07893546577cf996f67210b36227","observation_id":"a6dfdc96-4daa-4947-a2b9-117710c44f71","resolution":{"observed_at":"2026-08-07T10:46:08.341442Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-08T06:32:00.761636+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-08T06:32:00.761636+00:00","source":"crossref"},{"observed_at":"2026-08-08T06:31:55.24221+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":"2026-08-07T10:46:08.325460Z","title":"Geometric measure of singular sets of elliptic equations","venue":null,"work_id":"c2b7c15a-ca4a-4ee2-8584-b13630c40260","year":2023},"citing_paper":{"arxiv_id":"2506.04937","last_updated":"2025-06-05T12:11:00Z","snapshot_observed_at":"2026-08-07T10:27:52.481856Z","submitted_at":"2025-06-05T12:11:00Z","title":"Gradient estimates and parabolic frequency monotonicity for positive solutions of the heat equation under generalized Ricci flow","version":1},"reference_index":16,"source":"pdf_text","source_observed_at":"2026-08-07T10:46:08.028581Z"},"links":{"citing_paper":"/paper/2506.04937"},"observation_digest":"sha256:cd31f0677bd4156aa4803d1052a9d1197d0fa15c461cfbeac2537520942a6cce","observation_id":"7528c1d2-bc22-43b9-8552-c5b53c117417","resolution":{"observed_at":"2026-08-07T10:46:08.328877Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-08T06:32:00.761636+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-08T06:32:00.761636+00:00","source":"crossref"},{"observed_at":"2026-08-08T06:31:55.24221+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":"2026-08-07T10:46:08.314982Z","title":"Parabolic frequency monotonicity on Ricci flow and Ricci- harmonic flow with bounded curvatures","venue":null,"work_id":"0bf95e65-4f67-439d-875a-d5e6d7191989","year":2023},"citing_paper":{"arxiv_id":"2506.04937","last_updated":"2025-06-05T12:11:00Z","snapshot_observed_at":"2026-08-07T10:27:52.481856Z","submitted_at":"2025-06-05T12:11:00Z","title":"Gradient estimates and parabolic frequency monotonicity for positive solutions of the heat equation under generalized Ricci flow","version":1},"reference_index":17,"source":"pdf_text","source_observed_at":"2026-08-07T10:46:08.031906Z"},"links":{"citing_paper":"/paper/2506.04937"},"observation_digest":"sha256:616baf7ec382dd2b775b96e4deedb8fe1698e82bd48d5b25282c2a79cd0869fc","observation_id":"e8222dec-dab6-413d-8945-a74e200ec514","resolution":{"observed_at":"2026-08-07T10:46:08.318673Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-08T06:32:00.761636+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-08T06:32:00.761636+00:00","source":"crossref"},{"observed_at":"2026-08-08T06:31:55.24221+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":"2026-08-07T10:46:08.304302Z","title":"Gradient estimates and parabolic frequency under the LaplacianG 2 flow","venue":null,"work_id":"828a22c4-2921-41d7-979c-1b65840b8522","year":2025},"citing_paper":{"arxiv_id":"2506.04937","last_updated":"2025-06-05T12:11:00Z","snapshot_observed_at":"2026-08-07T10:27:52.481856Z","submitted_at":"2025-06-05T12:11:00Z","title":"Gradient estimates and parabolic frequency monotonicity for positive solutions of the heat equation under generalized Ricci flow","version":1},"reference_index":18,"source":"pdf_text","source_observed_at":"2026-08-07T10:46:08.035230Z"},"links":{"citing_paper":"/paper/2506.04937"},"observation_digest":"sha256:1e162015f1b04acf957798e7c665352e94f9586d2e99195f43bdf5eb324044bb","observation_id":"f6587948-15ec-4e26-9550-94061a48dd02","resolution":{"observed_at":"2026-08-07T10:46:08.307802Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-08T06:32:00.761636+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-08T06:32:00.761636+00:00","source":"crossref"},{"observed_at":"2026-08-08T06:31:55.24221+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":"2026-08-07T10:46:08.293617Z","title":"On the parabolic kernel of the Schr¨ odinger operator","venue":null,"work_id":"23ed678f-d3f3-4363-b00d-079245f52c36","year":1986},"citing_paper":{"arxiv_id":"2506.04937","last_updated":"2025-06-05T12:11:00Z","snapshot_observed_at":"2026-08-07T10:27:52.481856Z","submitted_at":"2025-06-05T12:11:00Z","title":"Gradient estimates and parabolic frequency monotonicity for positive solutions of the heat equation under generalized Ricci flow","version":1},"reference_index":19,"source":"pdf_text","source_observed_at":"2026-08-07T10:46:08.038658Z"},"links":{"citing_paper":"/paper/2506.04937"},"observation_digest":"sha256:a2ce6351cb259feb29d6e493d0afe4f23563ba6b6e32fbbde9144edabfd906ee","observation_id":"976e5e2c-b0d2-4c57-8111-241484b4fe9c","resolution":{"observed_at":"2026-08-07T10:46:08.297110Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-08T06:32:00.761636+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-08T06:32:00.761636+00:00","source":"crossref"},{"observed_at":"2026-08-08T06:31:55.24221+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":"2026-08-07T10:46:08.282275Z","title":"Parabolic frequency monotonicity on compact manifolds","venue":null,"work_id":"e36cd83e-e021-4daa-83fa-4a85f946206e","year":2019},"citing_paper":{"arxiv_id":"2506.04937","last_updated":"2025-06-05T12:11:00Z","snapshot_observed_at":"2026-08-07T10:27:52.481856Z","submitted_at":"2025-06-05T12:11:00Z","title":"Gradient estimates and parabolic frequency monotonicity for positive solutions of the heat equation under generalized Ricci flow","version":1},"reference_index":20,"source":"pdf_text","source_observed_at":"2026-08-07T10:46:08.041902Z"},"links":{"citing_paper":"/paper/2506.04937"},"observation_digest":"sha256:3d8ced46a5dbca4d4ca2541233f4ce2f2d3452443dc262e208942e7a514c145a","observation_id":"8042c5dc-73f3-499c-a656-f32493ce04ec","resolution":{"observed_at":"2026-08-07T10:46:08.286586Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-08T06:32:00.761636+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-08T06:32:00.761636+00:00","source":"crossref"},{"observed_at":"2026-08-08T06:31:55.24221+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":"2026-08-07T10:46:08.272280Z","title":"Matrix Li-Yau-Hamilton estimates under Ricci flow and parabolic frequency","venue":null,"work_id":"ec444721-9280-4ada-b01a-0a6e9b383b67","year":2024},"citing_paper":{"arxiv_id":"2506.04937","last_updated":"2025-06-05T12:11:00Z","snapshot_observed_at":"2026-08-07T10:27:52.481856Z","submitted_at":"2025-06-05T12:11:00Z","title":"Gradient estimates and parabolic frequency monotonicity for positive solutions of the heat equation under generalized Ricci flow","version":1},"reference_index":21,"source":"pdf_text","source_observed_at":"2026-08-07T10:46:08.045257Z"},"links":{"citing_paper":"/paper/2506.04937"},"observation_digest":"sha256:e71a1ebee5853a7975a6357b59efd65bb6eb9bd0eb33763eaf4c9c818b525f15","observation_id":"d9922dc4-9e4a-4a43-ae65-40834d6d5e5a","resolution":{"observed_at":"2026-08-07T10:46:08.275986Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-08T06:32:00.761636+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-08T06:32:00.761636+00:00","source":"crossref"},{"observed_at":"2026-08-08T06:31:55.24221+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":"2026-08-07T10:46:08.261779Z","title":"Entropy and heat kernel on generalized Ricci flow","venue":null,"work_id":"da2fba2f-08fe-497c-9a57-d6e2994be2fe","year":2024},"citing_paper":{"arxiv_id":"2506.04937","last_updated":"2025-06-05T12:11:00Z","snapshot_observed_at":"2026-08-07T10:27:52.481856Z","submitted_at":"2025-06-05T12:11:00Z","title":"Gradient estimates and parabolic frequency monotonicity for positive solutions of the heat equation under generalized Ricci flow","version":1},"reference_index":22,"source":"pdf_text","source_observed_at":"2026-08-07T10:46:08.048557Z"},"links":{"citing_paper":"/paper/2506.04937"},"observation_digest":"sha256:a108270c57e66d85a42ee44c111926827e8a123a71fbbc0b89fa331f8abe5056","observation_id":"013c66fd-e272-4a5f-afbe-bb8c64087d81","resolution":{"observed_at":"2026-08-07T10:46:08.265602Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-08T06:32:00.761636+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-08T06:32:00.761636+00:00","source":"crossref"},{"observed_at":"2026-08-08T06:31:55.24221+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":"2026-08-07T10:46:08.250854Z","title":"Nodal sets of solutions of elliptic and parabolic equations","venue":null,"work_id":"dafb9a95-dec0-4748-be3c-a39231d45b79","year":1991},"citing_paper":{"arxiv_id":"2506.04937","last_updated":"2025-06-05T12:11:00Z","snapshot_observed_at":"2026-08-07T10:27:52.481856Z","submitted_at":"2025-06-05T12:11:00Z","title":"Gradient estimates and parabolic frequency monotonicity for positive solutions of the heat equation under generalized Ricci flow","version":1},"reference_index":23,"source":"pdf_text","source_observed_at":"2026-08-07T10:46:08.051857Z"},"links":{"citing_paper":"/paper/2506.04937"},"observation_digest":"sha256:dfeb5ed2e4e8e9cec3470072ae6c5b65d46704a5a943c791a9ef5957085242ed","observation_id":"a265a1db-2f91-4e1a-9044-95e0d2a8e671","resolution":{"observed_at":"2026-08-07T10:46:08.254400Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-08T06:32:00.761636+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-08T06:32:00.761636+00:00","source":"crossref"},{"observed_at":"2026-08-08T06:31:55.24221+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":"2026-08-07T10:46:08.239081Z","title":"A note on parabolic frequency and a theorem of Hardy-P´ olya-Szeg¨ o","venue":null,"work_id":"21344ad2-4f3a-4738-bde1-52bd2dcec534","year":2022},"citing_paper":{"arxiv_id":"2506.04937","last_updated":"2025-06-05T12:11:00Z","snapshot_observed_at":"2026-08-07T10:27:52.481856Z","submitted_at":"2025-06-05T12:11:00Z","title":"Gradient estimates and parabolic frequency monotonicity for positive solutions of the heat equation under generalized Ricci flow","version":1},"reference_index":24,"source":"pdf_text","source_observed_at":"2026-08-07T10:46:08.055757Z"},"links":{"citing_paper":"/paper/2506.04937"},"observation_digest":"sha256:cf7908754f7ac996c0309fec63252648ab88f83e36e62af105cfbaf028836abb","observation_id":"08eea182-cd20-4253-9e7c-b6683149798d","resolution":{"observed_at":"2026-08-07T10:46:08.243818Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-08T06:32:00.761636+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-08T06:32:00.761636+00:00","source":"crossref"},{"observed_at":"2026-08-08T06:31:55.24221+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":"2026-08-07T10:46:08.228514Z","title":"Gradient estimates for solutions of the heat equation under Ricci flow","venue":null,"work_id":"96b7628d-aa71-4694-b5ff-9187bb0f2c69","year":2009},"citing_paper":{"arxiv_id":"2506.04937","last_updated":"2025-06-05T12:11:00Z","snapshot_observed_at":"2026-08-07T10:27:52.481856Z","submitted_at":"2025-06-05T12:11:00Z","title":"Gradient estimates and parabolic frequency monotonicity for positive solutions of the heat equation under generalized Ricci flow","version":1},"reference_index":25,"source":"pdf_text","source_observed_at":"2026-08-07T10:46:08.059126Z"},"links":{"citing_paper":"/paper/2506.04937"},"observation_digest":"sha256:026df14ca449e571efad186a5114664438779c8a2f738f035d9aa92a5c0d7638","observation_id":"3dce8e63-6149-4549-bf8a-3e304f1ca170","resolution":{"observed_at":"2026-08-07T10:46:08.232141Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-08T06:32:00.761636+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-08T06:32:00.761636+00:00","source":"crossref"},{"observed_at":"2026-08-08T06:31:55.24221+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":"2026-08-07T10:46:08.218560Z","title":"A Harnack inequality for parabolic differential equations","venue":null,"work_id":"fd566760-1c6e-4336-ad2f-c6d781d3dbbc","year":1964},"citing_paper":{"arxiv_id":"2506.04937","last_updated":"2025-06-05T12:11:00Z","snapshot_observed_at":"2026-08-07T10:27:52.481856Z","submitted_at":"2025-06-05T12:11:00Z","title":"Gradient estimates and parabolic frequency monotonicity for positive solutions of the heat equation under generalized Ricci flow","version":1},"reference_index":26,"source":"pdf_text","source_observed_at":"2026-08-07T10:46:08.062545Z"},"links":{"citing_paper":"/paper/2506.04937"},"observation_digest":"sha256:9a7a18e292ee9ba4c95c90e188d90e1e454511c3e13f550d3de0973dcbf81cd7","observation_id":"34a048ff-44ae-42d3-8152-094c0ea904b2","resolution":{"observed_at":"2026-08-07T10:46:08.221937Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-08T06:32:00.761636+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-08T06:32:00.761636+00:00","source":"crossref"},{"observed_at":"2026-08-08T06:31:55.24221+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":"2026-08-07T10:46:08.208130Z","title":"In: Analysis, complexgeometry, and mathematical physics: in honor of Duong H","venue":null,"work_id":"21b37767-f1fa-44a8-bbe1-2b9587717774","year":2015},"citing_paper":{"arxiv_id":"2506.04937","last_updated":"2025-06-05T12:11:00Z","snapshot_observed_at":"2026-08-07T10:27:52.481856Z","submitted_at":"2025-06-05T12:11:00Z","title":"Gradient estimates and parabolic frequency monotonicity for positive solutions of the heat equation under generalized Ricci flow","version":1},"reference_index":27,"source":"pdf_text","source_observed_at":"2026-08-07T10:46:08.065908Z"},"links":{"citing_paper":"/paper/2506.04937"},"observation_digest":"sha256:24a3030317a1e1277732d2da860057540652fc6a6c4f361b85ef9ec14f85cd10","observation_id":"f349aee7-a91f-4214-b654-d70ff062ac99","resolution":{"observed_at":"2026-08-07T10:46:08.211507Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-08T06:32:00.761636+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-08T06:32:00.761636+00:00","source":"crossref"},{"observed_at":"2026-08-08T06:31:55.24221+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":"2026-08-07T10:46:08.197776Z","title":"A gradient flow for worldsheet nonlinear sigma models","venue":null,"work_id":"04374aa2-f958-44fb-9ff7-be1710b41f8f","year":2006},"citing_paper":{"arxiv_id":"2506.04937","last_updated":"2025-06-05T12:11:00Z","snapshot_observed_at":"2026-08-07T10:27:52.481856Z","submitted_at":"2025-06-05T12:11:00Z","title":"Gradient estimates and parabolic frequency monotonicity for positive solutions of the heat equation under generalized Ricci flow","version":1},"reference_index":28,"source":"pdf_text","source_observed_at":"2026-08-07T10:46:08.069349Z"},"links":{"citing_paper":"/paper/2506.04937"},"observation_digest":"sha256:7404076a9c94d9e260dd24851c7217d9dd2b3fe96ac0e84144b54f53b8ffcd68","observation_id":"67eb3fc4-3761-4da2-94af-f5b85b04b772","resolution":{"observed_at":"2026-08-07T10:46:08.201174Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-08T06:32:00.761636+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-08T06:32:00.761636+00:00","source":"crossref"},{"observed_at":"2026-08-08T06:31:55.24221+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":"2026-08-07T10:46:08.187702Z","title":"Unique continuation for parabolic equations","venue":null,"work_id":"1b6ee7ad-3fbe-46c1-951a-79ae2308dc79","year":1996},"citing_paper":{"arxiv_id":"2506.04937","last_updated":"2025-06-05T12:11:00Z","snapshot_observed_at":"2026-08-07T10:27:52.481856Z","submitted_at":"2025-06-05T12:11:00Z","title":"Gradient estimates and parabolic frequency monotonicity for positive solutions of the heat equation under generalized Ricci flow","version":1},"reference_index":29,"source":"pdf_text","source_observed_at":"2026-08-07T10:46:08.072853Z"},"links":{"citing_paper":"/paper/2506.04937"},"observation_digest":"sha256:9eca80bc409ce38745a8b974af6ff9f06244bebfaaefd6d09ebd6ffd0ccd9ccb","observation_id":"59cca460-7402-4fdc-b9b3-fdc544bebf1b","resolution":{"observed_at":"2026-08-07T10:46:08.191241Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-08T06:32:00.761636+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-08T06:32:00.761636+00:00","source":"crossref"},{"observed_at":"2026-08-08T06:31:55.24221+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":"2026-08-07T10:46:08.176550Z","title":"Generalized geometry, T-duality, and renormalization group flow","venue":null,"work_id":"cb16b5bf-39f6-4a33-b26e-ff383b4f0a82","year":2017},"citing_paper":{"arxiv_id":"2506.04937","last_updated":"2025-06-05T12:11:00Z","snapshot_observed_at":"2026-08-07T10:27:52.481856Z","submitted_at":"2025-06-05T12:11:00Z","title":"Gradient estimates and parabolic frequency monotonicity for positive solutions of the heat equation under generalized Ricci flow","version":1},"reference_index":30,"source":"pdf_text","source_observed_at":"2026-08-07T10:46:08.076284Z"},"links":{"citing_paper":"/paper/2506.04937"},"observation_digest":"sha256:1fbd42a32efc28a5fafbdc4487aaa47082bf3b7e275688132ac76a932e455a85","observation_id":"87611d5d-41d8-44d3-b615-5ebced6b7f0a","resolution":{"observed_at":"2026-08-07T10:46:08.180741Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-08T06:32:00.761636+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-08T06:32:00.761636+00:00","source":"crossref"},{"observed_at":"2026-08-08T06:31:55.24221+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":"2026-08-07T10:46:08.164568Z","title":"Scalar curvature, entropy, and generalized Ricci flow","venue":null,"work_id":"b50fd754-159b-4924-9c6b-81cf193771ff","year":2023},"citing_paper":{"arxiv_id":"2506.04937","last_updated":"2025-06-05T12:11:00Z","snapshot_observed_at":"2026-08-07T10:27:52.481856Z","submitted_at":"2025-06-05T12:11:00Z","title":"Gradient estimates and parabolic frequency monotonicity for positive solutions of the heat equation under generalized Ricci flow","version":1},"reference_index":31,"source":"pdf_text","source_observed_at":"2026-08-07T10:46:08.079647Z"},"links":{"citing_paper":"/paper/2506.04937"},"observation_digest":"sha256:4009b1f91fdd29a1aa8afc3675a731178681fae61cb6853158f1c13521702f35","observation_id":"e0b60fc2-d3ae-4fa0-b9ff-4bc70a907906","resolution":{"observed_at":"2026-08-07T10:46:08.168422Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-08T06:32:00.761636+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-08T06:32:00.761636+00:00","source":"crossref"},{"observed_at":"2026-08-08T06:31:55.24221+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":"2026-08-07T10:46:08.153589Z","title":"A parabolic flow of pluriclosed metrics","venue":null,"work_id":"b849f992-8391-4a31-8dbb-8c41ece0c6ca","year":2010},"citing_paper":{"arxiv_id":"2506.04937","last_updated":"2025-06-05T12:11:00Z","snapshot_observed_at":"2026-08-07T10:27:52.481856Z","submitted_at":"2025-06-05T12:11:00Z","title":"Gradient estimates and parabolic frequency monotonicity for positive solutions of the heat equation under generalized Ricci flow","version":1},"reference_index":32,"source":"pdf_text","source_observed_at":"2026-08-07T10:46:08.082894Z"},"links":{"citing_paper":"/paper/2506.04937"},"observation_digest":"sha256:a09034b7c95a0e6601b1e111765b4ec61c37d3bfc0798316f8cdd7802c26b831","observation_id":"4a97b0e9-f6df-4bf2-b6f2-65b0c25b862e","resolution":{"observed_at":"2026-08-07T10:46:08.157508Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-08T06:32:00.761636+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-08T06:32:00.761636+00:00","source":"crossref"},{"observed_at":"2026-08-08T06:31:55.24221+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":"2026-08-07T10:46:08.142420Z","title":"Generalized K¨ ahler geometry and the pluriclosed flow","venue":null,"work_id":"142ec2bd-d220-4f49-b3d9-77439e62f3ac","year":2012},"citing_paper":{"arxiv_id":"2506.04937","last_updated":"2025-06-05T12:11:00Z","snapshot_observed_at":"2026-08-07T10:27:52.481856Z","submitted_at":"2025-06-05T12:11:00Z","title":"Gradient estimates and parabolic frequency monotonicity for positive solutions of the heat equation under generalized Ricci flow","version":1},"reference_index":33,"source":"pdf_text","source_observed_at":"2026-08-07T10:46:08.086259Z"},"links":{"citing_paper":"/paper/2506.04937"},"observation_digest":"sha256:f125f9b9c4cbe7ca6c328b08f60386f4780aae3e5eff6c800459f6ae3332ba04","observation_id":"619f21b4-252b-485f-9e5d-3cda21bc5b6a","resolution":{"observed_at":"2026-08-07T10:46:08.146098Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-08T06:32:00.761636+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-08T06:32:00.761636+00:00","source":"crossref"},{"observed_at":"2026-08-08T06:31:55.24221+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":"2026-08-07T10:46:08.131989Z","title":"Regularity results for pluriclosed flow","venue":null,"work_id":"976416bd-4b37-4c20-b302-df5df0216be5","year":2013},"citing_paper":{"arxiv_id":"2506.04937","last_updated":"2025-06-05T12:11:00Z","snapshot_observed_at":"2026-08-07T10:27:52.481856Z","submitted_at":"2025-06-05T12:11:00Z","title":"Gradient estimates and parabolic frequency monotonicity for positive solutions of the heat equation under generalized Ricci flow","version":1},"reference_index":34,"source":"pdf_text","source_observed_at":"2026-08-07T10:46:08.089563Z"},"links":{"citing_paper":"/paper/2506.04937"},"observation_digest":"sha256:0b52d8003e7ae1d0d91a91c587a46106e6499fc9cd5b18e8e60eac76769be643","observation_id":"c5a7bc90-dc14-4da8-8ea7-64bde9b4ad51","resolution":{"observed_at":"2026-08-07T10:46:08.135583Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-08T06:32:00.761636+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-08T06:32:00.761636+00:00","source":"crossref"},{"observed_at":"2026-08-08T06:31:55.24221+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":"2026-08-07T10:46:08.118371Z","title":"Some gradient estimates for the heat equation on domains and for an equation by Perelman","venue":null,"work_id":"38126624-117a-4561-af7a-b6436d2dfa0d","year":2006},"citing_paper":{"arxiv_id":"2506.04937","last_updated":"2025-06-05T12:11:00Z","snapshot_observed_at":"2026-08-07T10:27:52.481856Z","submitted_at":"2025-06-05T12:11:00Z","title":"Gradient estimates and parabolic frequency monotonicity for positive solutions of the heat equation under generalized Ricci flow","version":1},"reference_index":35,"source":"pdf_text","source_observed_at":"2026-08-07T10:46:08.092561Z"},"links":{"citing_paper":"/paper/2506.04937"},"observation_digest":"sha256:c54d7bc3161e1412245b509c8e6f7f9e7994c4ab9606df51abe0c226a6f1b32b","observation_id":"c3ec6f29-24f8-46b9-821b-da5e4111c293","resolution":{"observed_at":"2026-08-07T10:46:08.124374Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-08T06:32:00.761636+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-08T06:32:00.761636+00:00","source":"crossref"},{"observed_at":"2026-08-08T06:31:55.24221+00:00","source":"retraction_watch"}],"state":"measured"}}],"paper":{"arxiv_id":"2506.04937","last_updated":"2025-06-05T12:11:00Z","latest_version":1,"primary_category":"math.DG","snapshot_observed_at":"2026-08-07T10:27:52.481856Z","submitted_at":"2025-06-05T12:11:00Z","title":"Gradient estimates and parabolic frequency monotonicity for positive solutions of the heat equation under generalized Ricci flow"},"reference_resolution":{"displayed":35,"state_counts":{"malformed_identifier":0,"metadata_mismatch":0,"parse_uncertain":0,"unresolved":0,"verified_exact":0,"verified_fuzzy":35},"total_outbound_references":35},"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":"2026-08-08T06:32:00.761636+00:00","source":"crossref"},{"observed_at":"2026-08-08T06:31:55.24221+00:00","source":"retraction_watch"}],"thesis":"As of 8 August 2026, this Paper Citation Record lists 35 of 35 outbound references and 0 inbound Pith citation observations for arXiv:2506.04937."}