{"as_of":"2026-08-14T09:26:00Z","caps":{"database_statements":6,"inbound":100,"outbound":100},"context_digest":"sha256:0eb670ae38a67427b3e1877c7f74d18bcf914430927a734ec4c3e30ee4cd9e15","coverage":[{"denominator":42,"lane":"reference_resolution","note":"Typed states for the displayed outbound observations.","records_observed":42,"source":"paper_references, paper_reference_links","source_observed_at":"2026-08-05T10:20:42.481207Z","state":"measured"},{"denominator":42,"lane":"standing_notices","note":"One-hop event checks from named stored sources.","records_observed":42,"source":"scholarly_work_events, retraction_status_cache","source_observed_at":"2026-08-14T06:32:32.682623+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/2509.04265/citation-record","integrity":"/paper/2509.04265/integrity","json":"/paper/2509.04265/citation-record.json","paper":"/paper/2509.04265"},"outbound":[{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":"raw_reference","pith_arxiv_id":null,"snapshot_observed_at":"2026-08-05T10:20:43.318588Z","title":"Arnold.Mathematical methods of classical mechanics, volume 60","venue":null,"work_id":"f3ff1e38-e409-4d5d-b870-3ff2103c5ff9","year":2013},"citing_paper":{"arxiv_id":"2509.04265","last_updated":"2026-06-05T15:03:49Z","snapshot_observed_at":"2026-08-13T12:35:19.818364Z","submitted_at":"2025-09-04T14:42:22Z","title":"Reinforcement-Learning-Guided Data-Driven Estimation of Spectral Properties of Stochastic Koopman Semigroups","version":3},"reference_index":1,"source":"pdf_text","source_observed_at":"2026-08-05T10:20:42.296454Z"},"links":{"citing_paper":"/paper/2509.04265"},"observation_digest":"sha256:d7317a4aff5f6d41015088ddcfcd073f2b5202c49b0728fa1fd1e602716d89d4","observation_id":"69d793a7-34e8-4659-8eef-aa0ceafe4fcb","resolution":{"observed_at":"2026-08-05T10:20:43.323326Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-14T06:32:32.682623+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-14T06:32:32.682623+00:00","source":"crossref"},{"observed_at":"2026-08-14T06:32:18.44784+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":"2026-08-05T10:20:43.303576Z","title":"Batchelor.An introduction to fluid dynamics","venue":null,"work_id":"70255a51-1448-4e3a-9a92-fa9b80c0f74e","year":2000},"citing_paper":{"arxiv_id":"2509.04265","last_updated":"2026-06-05T15:03:49Z","snapshot_observed_at":"2026-08-13T12:35:19.818364Z","submitted_at":"2025-09-04T14:42:22Z","title":"Reinforcement-Learning-Guided Data-Driven Estimation of Spectral Properties of Stochastic Koopman Semigroups","version":3},"reference_index":2,"source":"pdf_text","source_observed_at":"2026-08-05T10:20:42.301155Z"},"links":{"citing_paper":"/paper/2509.04265"},"observation_digest":"sha256:67766997e46b7801124c12293ffec3a661eff0562183b6bed6835f2ed4bbad5c","observation_id":"4dc3f410-db54-4aa5-b261-65babfc4f64c","resolution":{"observed_at":"2026-08-05T10:20:43.308068Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-14T06:32:32.682623+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-14T06:32:32.682623+00:00","source":"crossref"},{"observed_at":"2026-08-14T06:32:18.44784+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":"2026-08-05T10:20:43.289313Z","title":"Athena scientific, 2012","venue":null,"work_id":"767c44d5-8bdf-4e29-af0f-b758741d6e62","year":2012},"citing_paper":{"arxiv_id":"2509.04265","last_updated":"2026-06-05T15:03:49Z","snapshot_observed_at":"2026-08-13T12:35:19.818364Z","submitted_at":"2025-09-04T14:42:22Z","title":"Reinforcement-Learning-Guided Data-Driven Estimation of Spectral Properties of Stochastic Koopman Semigroups","version":3},"reference_index":3,"source":"pdf_text","source_observed_at":"2026-08-05T10:20:42.305994Z"},"links":{"citing_paper":"/paper/2509.04265"},"observation_digest":"sha256:bf8e19ecd1d68ad955b8bc9201d8ddec15f26e17d27614bc423d90d1dd8fa83c","observation_id":"ae31addb-1591-4133-bdfb-72d9f28f4489","resolution":{"observed_at":"2026-08-05T10:20:43.293814Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-14T06:32:32.682623+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-14T06:32:32.682623+00:00","source":"crossref"},{"observed_at":"2026-08-14T06:32:18.44784+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":"2026-08-05T10:20:43.274342Z","title":"Athena Scientific, 2008","venue":null,"work_id":"2a7ff4a9-451a-4b29-b82e-80502eebf8f7","year":2008},"citing_paper":{"arxiv_id":"2509.04265","last_updated":"2026-06-05T15:03:49Z","snapshot_observed_at":"2026-08-13T12:35:19.818364Z","submitted_at":"2025-09-04T14:42:22Z","title":"Reinforcement-Learning-Guided Data-Driven Estimation of Spectral Properties of Stochastic Koopman Semigroups","version":3},"reference_index":4,"source":"pdf_text","source_observed_at":"2026-08-05T10:20:42.311233Z"},"links":{"citing_paper":"/paper/2509.04265"},"observation_digest":"sha256:7c719777d7b79d303ef4e4b3201b7b18c5d1de0abc18dcee373a4e51cac42bf2","observation_id":"27e6547c-8d4a-4304-8ede-13b07b7da21a","resolution":{"observed_at":"2026-08-05T10:20:43.279245Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-14T06:32:32.682623+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-14T06:32:32.682623+00:00","source":"crossref"},{"observed_at":"2026-08-14T06:32:18.44784+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":"2026-08-05T10:20:43.259720Z","title":"Brunton and Nathan J","venue":null,"work_id":"8fea9bd9-00b4-4373-ae4f-45b540fb9194","year":2022},"citing_paper":{"arxiv_id":"2509.04265","last_updated":"2026-06-05T15:03:49Z","snapshot_observed_at":"2026-08-13T12:35:19.818364Z","submitted_at":"2025-09-04T14:42:22Z","title":"Reinforcement-Learning-Guided Data-Driven Estimation of Spectral Properties of Stochastic Koopman Semigroups","version":3},"reference_index":5,"source":"pdf_text","source_observed_at":"2026-08-05T10:20:42.316679Z"},"links":{"citing_paper":"/paper/2509.04265"},"observation_digest":"sha256:cc8622b44f8f4b31b8c500d6036579707a9a58a3d396b41c69cb6170cbc824bd","observation_id":"4c8a7947-b53d-47c8-a4a4-018b936756f3","resolution":{"observed_at":"2026-08-05T10:20:43.264440Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-14T06:32:32.682623+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-14T06:32:32.682623+00:00","source":"crossref"},{"observed_at":"2026-08-14T06:32:18.44784+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":"2026-08-05T10:20:43.244924Z","title":"Brunton, Joshua L","venue":null,"work_id":"3ffb053c-f7ac-46d4-a0f4-4ef2554862bf","year":2016},"citing_paper":{"arxiv_id":"2509.04265","last_updated":"2026-06-05T15:03:49Z","snapshot_observed_at":"2026-08-13T12:35:19.818364Z","submitted_at":"2025-09-04T14:42:22Z","title":"Reinforcement-Learning-Guided Data-Driven Estimation of Spectral Properties of Stochastic Koopman Semigroups","version":3},"reference_index":6,"source":"pdf_text","source_observed_at":"2026-08-05T10:20:42.321170Z"},"links":{"citing_paper":"/paper/2509.04265"},"observation_digest":"sha256:b14343a364a5a98b0cbf150358d1bedf914d8e6e264a76b2ca0208e69e565312","observation_id":"439b3e8f-0ff4-4598-b173-0f60523da9cc","resolution":{"observed_at":"2026-08-05T10:20:43.249625Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-14T06:32:32.682623+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-14T06:32:32.682623+00:00","source":"crossref"},{"observed_at":"2026-08-14T06:32:18.44784+00:00","source":"retraction_watch"}],"state":"measured"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":null,"snapshot_observed_at":"2026-08-05T10:20:42.326058Z","title":"Colbrook and Alex Townsend","venue":null,"work_id":null,"year":2024},"citing_paper":{"arxiv_id":"2509.04265","last_updated":"2026-06-05T15:03:49Z","snapshot_observed_at":"2026-08-13T12:35:19.818364Z","submitted_at":"2025-09-04T14:42:22Z","title":"Reinforcement-Learning-Guided Data-Driven Estimation of Spectral Properties of Stochastic Koopman Semigroups","version":3},"reference_index":7,"source":"pdf_text","source_observed_at":"2026-08-05T10:20:42.326058Z"},"links":{"citing_paper":"/paper/2509.04265"},"observation_digest":"sha256:50a8924a87fee6ce81a2620db43e326fdb9e11876f340666140d2d7c1f5d197a","observation_id":"3bdd2999-b42c-480b-808a-1e6b811e3f82","resolution":{"observed_at":"2026-08-05T10:20:42.326058Z","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":"2006.87158","doi":null,"metadata_source":"raw_reference","pith_arxiv_id":null,"snapshot_observed_at":"2026-08-05T10:20:42.861761Z","title":null,"venue":null,"work_id":"3de57f01-d794-4a0c-881c-5cd92c2ebfa3","year":2006},"citing_paper":{"arxiv_id":"2509.04265","last_updated":"2026-06-05T15:03:49Z","snapshot_observed_at":"2026-08-13T12:35:19.818364Z","submitted_at":"2025-09-04T14:42:22Z","title":"Reinforcement-Learning-Guided Data-Driven Estimation of Spectral Properties of Stochastic Koopman Semigroups","version":3},"reference_index":8,"source":"pdf_text","source_observed_at":"2026-08-05T10:20:42.330041Z"},"links":{"citing_paper":"/paper/2509.04265"},"observation_digest":"sha256:8bcc9d7f095dbb0bce03c984f309602531b0f502d39e74d3a92daa23b0344814","observation_id":"81385606-811b-4e36-a5fe-67b2099b68dc","resolution":{"observed_at":"2026-08-05T10:20:42.869563Z","resolver_source":"raw_fallback","status":"metadata_mismatch"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-14T06:32:32.682623+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-14T06:32:32.682623+00:00","source":"crossref"},{"observed_at":"2026-08-14T06:32:18.44784+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":"2026-08-05T10:20:43.220871Z","title":"Impulses and physiological states in theoretical models of nerve membrane.Biophysical journal, 1(6):445–466, 1961","venue":null,"work_id":"497ac6c9-8642-43b7-b382-48f9d481a6c3","year":1961},"citing_paper":{"arxiv_id":"2509.04265","last_updated":"2026-06-05T15:03:49Z","snapshot_observed_at":"2026-08-13T12:35:19.818364Z","submitted_at":"2025-09-04T14:42:22Z","title":"Reinforcement-Learning-Guided Data-Driven Estimation of Spectral Properties of Stochastic Koopman Semigroups","version":3},"reference_index":9,"source":"pdf_text","source_observed_at":"2026-08-05T10:20:42.334770Z"},"links":{"citing_paper":"/paper/2509.04265"},"observation_digest":"sha256:c63179dbda5c6b29c06b47367f0d02ae37561397b8a54b31ba01c04d80cf94f6","observation_id":"46b421d0-0501-422c-a72e-8ac4de69bc0d","resolution":{"observed_at":"2026-08-05T10:20:43.225485Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-14T06:32:32.682623+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-14T06:32:32.682623+00:00","source":"crossref"},{"observed_at":"2026-08-14T06:32:18.44784+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":"2026-08-05T10:20:43.207219Z","title":"Springer Science & Business Media, 2013","venue":null,"work_id":"89c57f3f-9c85-47e9-82ce-9680e41e26b3","year":2013},"citing_paper":{"arxiv_id":"2509.04265","last_updated":"2026-06-05T15:03:49Z","snapshot_observed_at":"2026-08-13T12:35:19.818364Z","submitted_at":"2025-09-04T14:42:22Z","title":"Reinforcement-Learning-Guided Data-Driven Estimation of Spectral Properties of Stochastic Koopman Semigroups","version":3},"reference_index":10,"source":"pdf_text","source_observed_at":"2026-08-05T10:20:42.339862Z"},"links":{"citing_paper":"/paper/2509.04265"},"observation_digest":"sha256:e26e59b5f8cdc9ecf175783dfd6e9898af95b9063a0aa926fe7b5465bff07564","observation_id":"3762be78-92cd-4577-aeff-68ed29315b03","resolution":{"observed_at":"2026-08-05T10:20:43.211615Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-14T06:32:32.682623+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-14T06:32:32.682623+00:00","source":"crossref"},{"observed_at":"2026-08-14T06:32:18.44784+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":"2026-08-05T10:20:43.193149Z","title":"Hale.Ordinary differential equations","venue":null,"work_id":"ee2bb8fc-ae41-4cc3-b650-e061ae60e1ed","year":2009},"citing_paper":{"arxiv_id":"2509.04265","last_updated":"2026-06-05T15:03:49Z","snapshot_observed_at":"2026-08-13T12:35:19.818364Z","submitted_at":"2025-09-04T14:42:22Z","title":"Reinforcement-Learning-Guided Data-Driven Estimation of Spectral Properties of Stochastic Koopman Semigroups","version":3},"reference_index":11,"source":"pdf_text","source_observed_at":"2026-08-05T10:20:42.344425Z"},"links":{"citing_paper":"/paper/2509.04265"},"observation_digest":"sha256:52d20360be5cca994a13029831b216647ba477c338796ee769b854e63a4666d7","observation_id":"83cc52e0-6805-4b79-b0cb-b2a339882c8e","resolution":{"observed_at":"2026-08-05T10:20:43.197692Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-14T06:32:32.682623+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-14T06:32:32.682623+00:00","source":"crossref"},{"observed_at":"2026-08-14T06:32:18.44784+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":"2026-08-05T10:20:43.177899Z","title":"Hirsch, Stephen Smale, and Robert L","venue":null,"work_id":"81124120-5e54-4d97-978d-8828c06b12ae","year":2013},"citing_paper":{"arxiv_id":"2509.04265","last_updated":"2026-06-05T15:03:49Z","snapshot_observed_at":"2026-08-13T12:35:19.818364Z","submitted_at":"2025-09-04T14:42:22Z","title":"Reinforcement-Learning-Guided Data-Driven Estimation of Spectral Properties of Stochastic Koopman Semigroups","version":3},"reference_index":12,"source":"pdf_text","source_observed_at":"2026-08-05T10:20:42.348793Z"},"links":{"citing_paper":"/paper/2509.04265"},"observation_digest":"sha256:94e771824ec57a56b0210d032c8a4c543aab934af03716d74f757ee8ac10318d","observation_id":"4036d147-a313-4dbe-8d52-45d0d767d55c","resolution":{"observed_at":"2026-08-05T10:20:43.182557Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-14T06:32:32.682623+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-14T06:32:32.682623+00:00","source":"crossref"},{"observed_at":"2026-08-14T06:32:18.44784+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":"2026-08-05T10:20:43.162498Z","title":"Cambridge university press, 2012","venue":null,"work_id":"2ca36b0e-3389-4480-ba74-3bd432757f30","year":2012},"citing_paper":{"arxiv_id":"2509.04265","last_updated":"2026-06-05T15:03:49Z","snapshot_observed_at":"2026-08-13T12:35:19.818364Z","submitted_at":"2025-09-04T14:42:22Z","title":"Reinforcement-Learning-Guided Data-Driven Estimation of Spectral Properties of Stochastic Koopman Semigroups","version":3},"reference_index":13,"source":"pdf_text","source_observed_at":"2026-08-05T10:20:42.353265Z"},"links":{"citing_paper":"/paper/2509.04265"},"observation_digest":"sha256:b4052227046cd546682da82a301560c47a5456de7582a0903ade36c5c16a41a3","observation_id":"9dd1d92c-85ff-4702-a606-df352c109c81","resolution":{"observed_at":"2026-08-05T10:20:43.167485Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-14T06:32:32.682623+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-14T06:32:32.682623+00:00","source":"crossref"},{"observed_at":"2026-08-14T06:32:18.44784+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":"2026-08-05T10:20:43.148104Z","title":null,"venue":null,"work_id":"73ac0cbf-de38-40b3-98fa-800145351b4a","year":1992},"citing_paper":{"arxiv_id":"2509.04265","last_updated":"2026-06-05T15:03:49Z","snapshot_observed_at":"2026-08-13T12:35:19.818364Z","submitted_at":"2025-09-04T14:42:22Z","title":"Reinforcement-Learning-Guided Data-Driven Estimation of Spectral Properties of Stochastic Koopman Semigroups","version":3},"reference_index":14,"source":"pdf_text","source_observed_at":"2026-08-05T10:20:42.357486Z"},"links":{"citing_paper":"/paper/2509.04265"},"observation_digest":"sha256:da0e7ae455f08796059b04235347aed85b3d16ef3c7003bc602d8f63b3f22841","observation_id":"e8e760ae-f6e4-41c6-b7e5-c369f3dfff80","resolution":{"observed_at":"2026-08-05T10:20:43.152771Z","resolver_source":"raw_fallback","status":"unresolved"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-14T06:32:32.682623+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-14T06:32:32.682623+00:00","source":"crossref"},{"observed_at":"2026-08-14T06:32:18.44784+00:00","source":"retraction_watch"}],"state":"measured"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":null,"snapshot_observed_at":"2026-08-05T10:20:42.361609Z","title":"Koopman operators with intrinsic observables in rigged reproducing kernel hilbert spaces.arXiv preprint arXiv:2403.02524, 2024","venue":null,"work_id":null,"year":2024},"citing_paper":{"arxiv_id":"2509.04265","last_updated":"2026-06-05T15:03:49Z","snapshot_observed_at":"2026-08-13T12:35:19.818364Z","submitted_at":"2025-09-04T14:42:22Z","title":"Reinforcement-Learning-Guided Data-Driven Estimation of Spectral Properties of Stochastic Koopman Semigroups","version":3},"reference_index":15,"source":"pdf_text","source_observed_at":"2026-08-05T10:20:42.361609Z"},"links":{"citing_paper":"/paper/2509.04265"},"observation_digest":"sha256:846227b48393b55b321b316e6f908b3bd55f775cd823b2be130cca37f7d6863a","observation_id":"71b32b6b-402a-4370-a50b-0cede0375069","resolution":{"observed_at":"2026-08-05T10:20:42.361609Z","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":"2026-08-05T10:20:43.133091Z","title":"Asymptotic phase and amplitude for classical and semiclassical stochastic oscillators via koopman operator theory.Mathematics, 9(18):2188, 2021","venue":null,"work_id":"1921aa61-ecf0-4f5c-b054-544cc3802dc1","year":2021},"citing_paper":{"arxiv_id":"2509.04265","last_updated":"2026-06-05T15:03:49Z","snapshot_observed_at":"2026-08-13T12:35:19.818364Z","submitted_at":"2025-09-04T14:42:22Z","title":"Reinforcement-Learning-Guided Data-Driven Estimation of Spectral Properties of Stochastic Koopman Semigroups","version":3},"reference_index":16,"source":"pdf_text","source_observed_at":"2026-08-05T10:20:42.365870Z"},"links":{"citing_paper":"/paper/2509.04265"},"observation_digest":"sha256:b84078adf70acc81458fbbadce394abeff2080a8027156e906c88e64b74dd369","observation_id":"3ee23459-678e-4674-bf1f-15470d9c3b77","resolution":{"observed_at":"2026-08-05T10:20:43.137961Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-14T06:32:32.682623+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-14T06:32:32.682623+00:00","source":"crossref"},{"observed_at":"2026-08-14T06:32:18.44784+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":"2026-08-05T10:20:43.116673Z","title":"Near-optimal reinforcement learning in polynomial time.Machine learning, 49(2):209–232, 2002","venue":null,"work_id":"19e70636-1444-4345-9e52-efcc655373c0","year":2002},"citing_paper":{"arxiv_id":"2509.04265","last_updated":"2026-06-05T15:03:49Z","snapshot_observed_at":"2026-08-13T12:35:19.818364Z","submitted_at":"2025-09-04T14:42:22Z","title":"Reinforcement-Learning-Guided Data-Driven Estimation of Spectral Properties of Stochastic Koopman Semigroups","version":3},"reference_index":17,"source":"pdf_text","source_observed_at":"2026-08-05T10:20:42.371025Z"},"links":{"citing_paper":"/paper/2509.04265"},"observation_digest":"sha256:e25a8c2572452838b62eb7b96ad1953f90a1ccf5fb303b229abf6abc432529bc","observation_id":"09aad655-2a31-4a27-be12-08d45d38ce26","resolution":{"observed_at":"2026-08-05T10:20:43.121786Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-14T06:32:32.682623+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-14T06:32:32.682623+00:00","source":"crossref"},{"observed_at":"2026-08-14T06:32:18.44784+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":"2026-08-05T10:20:43.101785Z","title":"Data- driven approximation of the koopman generator: Model reduction, system identification, and control.Physica D: Nonlinear Phenomena, 406:132416, 2020","venue":null,"work_id":"a92f45b1-319d-4078-b3d2-59f7ab80e62b","year":2020},"citing_paper":{"arxiv_id":"2509.04265","last_updated":"2026-06-05T15:03:49Z","snapshot_observed_at":"2026-08-13T12:35:19.818364Z","submitted_at":"2025-09-04T14:42:22Z","title":"Reinforcement-Learning-Guided Data-Driven Estimation of Spectral Properties of Stochastic Koopman Semigroups","version":3},"reference_index":18,"source":"pdf_text","source_observed_at":"2026-08-05T10:20:42.375829Z"},"links":{"citing_paper":"/paper/2509.04265"},"observation_digest":"sha256:e0a6eb2721d763390db8f275c5a28e5a1ab68c934067d875813ab16b34990867","observation_id":"6720a8d7-4644-4f36-be27-f222e49d302c","resolution":{"observed_at":"2026-08-05T10:20:43.106445Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-14T06:32:32.682623+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-14T06:32:32.682623+00:00","source":"crossref"},{"observed_at":"2026-08-14T06:32:18.44784+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":"2026-08-05T10:20:43.086968Z","title":null,"venue":null,"work_id":"f1ea03af-f69b-4cb7-b13c-ec2ea4b8ec41","year":1931},"citing_paper":{"arxiv_id":"2509.04265","last_updated":"2026-06-05T15:03:49Z","snapshot_observed_at":"2026-08-13T12:35:19.818364Z","submitted_at":"2025-09-04T14:42:22Z","title":"Reinforcement-Learning-Guided Data-Driven Estimation of Spectral Properties of Stochastic Koopman Semigroups","version":3},"reference_index":19,"source":"pdf_text","source_observed_at":"2026-08-05T10:20:42.380308Z"},"links":{"citing_paper":"/paper/2509.04265"},"observation_digest":"sha256:21a60c8896b4583d5eb1b20a09e5bd77a19b83dac25bb8673fcb479926067426","observation_id":"c7b609b5-7ecd-435b-b348-5908f3b7f3eb","resolution":{"observed_at":"2026-08-05T10:20:43.092494Z","resolver_source":"raw_fallback","status":"unresolved"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-14T06:32:32.682623+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-14T06:32:32.682623+00:00","source":"crossref"},{"observed_at":"2026-08-14T06:32:18.44784+00:00","source":"retraction_watch"}],"state":"measured"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":null,"snapshot_observed_at":"2026-08-05T10:20:42.384766Z","title":"Koopman and John von Neumann","venue":null,"work_id":null,"year":1932},"citing_paper":{"arxiv_id":"2509.04265","last_updated":"2026-06-05T15:03:49Z","snapshot_observed_at":"2026-08-13T12:35:19.818364Z","submitted_at":"2025-09-04T14:42:22Z","title":"Reinforcement-Learning-Guided Data-Driven Estimation of Spectral Properties of Stochastic Koopman Semigroups","version":3},"reference_index":20,"source":"pdf_text","source_observed_at":"2026-08-05T10:20:42.384766Z"},"links":{"citing_paper":"/paper/2509.04265"},"observation_digest":"sha256:07cbc6b4523b799f3ba4ec9ca6d8df16835f677ac686ac09eedcff50984d712f","observation_id":"dfa64a37-efde-404c-9526-06a27db0cf5e","resolution":{"observed_at":"2026-08-05T10:20:42.384766Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":{"arxiv_id":"2508.20339","last_updated":"2025-08-28T01:21:17Z","snapshot_observed_at":"2026-08-05T15:03:26.784758Z","submitted_at":"2025-08-28T01:21:17Z","title":"Artificial neural network solver for Fokker-Planck and Koopman eigenfunctions","version":1},"cited_work":{"arxiv_id":"2508.20339","doi":null,"metadata_source":"pith","pith_arxiv_id":"2508.20339","snapshot_observed_at":"2026-08-05T10:20:42.608102Z","title":"Artificial neural network solver for Fokker-Planck and Koopman eigenfunctions","venue":"math.NA","work_id":"bf062e72-3b8a-4a25-935b-c3ec8f9c990e","year":2025},"citing_paper":{"arxiv_id":"2509.04265","last_updated":"2026-06-05T15:03:49Z","snapshot_observed_at":"2026-08-13T12:35:19.818364Z","submitted_at":"2025-09-04T14:42:22Z","title":"Reinforcement-Learning-Guided Data-Driven Estimation of Spectral Properties of Stochastic Koopman Semigroups","version":3},"reference_index":21,"source":"pdf_text","source_observed_at":"2026-08-05T10:20:42.389239Z"},"links":{"cited_paper":"/paper/2508.20339","citing_paper":"/paper/2509.04265"},"observation_digest":"sha256:49e4378a52ccf897efc122a9ff84097b5944c712a27005d03a483115b5cd2f04","observation_id":"127c7162-524a-4898-b3bf-73e574c0385f","resolution":{"observed_at":"2026-08-05T10:20:42.612809Z","resolver_source":"local_arxiv","status":"verified_exact"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-14T06:32:32.682623+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-14T06:32:32.682623+00:00","source":"crossref"},{"observed_at":"2026-08-14T06:32:18.44784+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":"2026-08-05T10:20:43.072070Z","title":"Kutz, Steven L","venue":null,"work_id":"81bf9864-71ad-4e66-a00d-d6dd8d29e158","year":2016},"citing_paper":{"arxiv_id":"2509.04265","last_updated":"2026-06-05T15:03:49Z","snapshot_observed_at":"2026-08-13T12:35:19.818364Z","submitted_at":"2025-09-04T14:42:22Z","title":"Reinforcement-Learning-Guided Data-Driven Estimation of Spectral Properties of Stochastic Koopman Semigroups","version":3},"reference_index":22,"source":"pdf_text","source_observed_at":"2026-08-05T10:20:42.393720Z"},"links":{"citing_paper":"/paper/2509.04265"},"observation_digest":"sha256:d8f7c65ad710aa97d0d579acc1d8fb861ced2f845ca8bd8446212438129509b6","observation_id":"a3bc76ce-d682-4463-abc9-9ba42b91f8b7","resolution":{"observed_at":"2026-08-05T10:20:43.076677Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-14T06:32:32.682623+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-14T06:32:32.682623+00:00","source":"crossref"},{"observed_at":"2026-08-14T06:32:18.44784+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":"2026-08-05T10:20:43.058081Z","title":null,"venue":null,"work_id":"0b330320-66ae-4276-9c23-4e9a9e137511","year":2017},"citing_paper":{"arxiv_id":"2509.04265","last_updated":"2026-06-05T15:03:49Z","snapshot_observed_at":"2026-08-13T12:35:19.818364Z","submitted_at":"2025-09-04T14:42:22Z","title":"Reinforcement-Learning-Guided Data-Driven Estimation of Spectral Properties of Stochastic Koopman Semigroups","version":3},"reference_index":23,"source":"pdf_text","source_observed_at":"2026-08-05T10:20:42.397877Z"},"links":{"citing_paper":"/paper/2509.04265"},"observation_digest":"sha256:68007dad7dbf838ca3770ef8ad06e1095563d458fd627ecbf01276a913e15ae2","observation_id":"d3aa6ed0-dc4f-418e-b405-438fda8991bc","resolution":{"observed_at":"2026-08-05T10:20:43.062296Z","resolver_source":"raw_fallback","status":"unresolved"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-14T06:32:32.682623+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-14T06:32:32.682623+00:00","source":"crossref"},{"observed_at":"2026-08-14T06:32:18.44784+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":"2026-08-05T10:20:43.043882Z","title":"Spectral properties of dynamical systems, model reduction and decompositions.Nonlinear Dynamics, 41:309–325, 2005","venue":null,"work_id":"605ec3e3-fe2f-468e-8b75-87da848776fe","year":2005},"citing_paper":{"arxiv_id":"2509.04265","last_updated":"2026-06-05T15:03:49Z","snapshot_observed_at":"2026-08-13T12:35:19.818364Z","submitted_at":"2025-09-04T14:42:22Z","title":"Reinforcement-Learning-Guided Data-Driven Estimation of Spectral Properties of Stochastic Koopman Semigroups","version":3},"reference_index":24,"source":"pdf_text","source_observed_at":"2026-08-05T10:20:42.401940Z"},"links":{"citing_paper":"/paper/2509.04265"},"observation_digest":"sha256:176aff6ba489ae678dcf53b75c114e425acc5a3a046f977583e6e2e6e4cee08b","observation_id":"8e7af0de-23f9-46fb-bff3-ad004a891b63","resolution":{"observed_at":"2026-08-05T10:20:43.048706Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-14T06:32:32.682623+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-14T06:32:32.682623+00:00","source":"crossref"},{"observed_at":"2026-08-14T06:32:18.44784+00:00","source":"retraction_watch"}],"state":"measured"}},{"citation":{"cited_paper":{"arxiv_id":"1312.5602","last_updated":"2013-12-19T16:00:08Z","snapshot_observed_at":"2026-08-14T03:19:40.736445Z","submitted_at":"2013-12-19T16:00:08Z","title":"Playing Atari with Deep Reinforcement Learning","version":1},"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":"1312.5602","snapshot_observed_at":"2026-08-05T10:20:42.406349Z","title":"Playing atari with deep reinforcement learning.arXiv preprint arXiv:1312.5602, 2013","venue":null,"work_id":null,"year":2013},"citing_paper":{"arxiv_id":"2509.04265","last_updated":"2026-06-05T15:03:49Z","snapshot_observed_at":"2026-08-13T12:35:19.818364Z","submitted_at":"2025-09-04T14:42:22Z","title":"Reinforcement-Learning-Guided Data-Driven Estimation of Spectral Properties of Stochastic Koopman Semigroups","version":3},"reference_index":25,"source":"pdf_text","source_observed_at":"2026-08-05T10:20:42.406349Z"},"links":{"cited_paper":"/paper/1312.5602","citing_paper":"/paper/2509.04265"},"observation_digest":"sha256:e352d693c9e0b7de4ceabcf29218ef4d96f08126ad5999099cb6adf168b0a6e7","observation_id":"16609e9c-7558-4799-9c88-f4f5123c80b7","resolution":{"observed_at":"2026-08-05T10:20:42.406349Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":null,"snapshot_observed_at":"2026-08-05T10:20:42.410752Z","title":"Error bounds for approximate value iteration","venue":null,"work_id":null,"year":1999},"citing_paper":{"arxiv_id":"2509.04265","last_updated":"2026-06-05T15:03:49Z","snapshot_observed_at":"2026-08-13T12:35:19.818364Z","submitted_at":"2025-09-04T14:42:22Z","title":"Reinforcement-Learning-Guided Data-Driven Estimation of Spectral Properties of Stochastic Koopman Semigroups","version":3},"reference_index":26,"source":"pdf_text","source_observed_at":"2026-08-05T10:20:42.410752Z"},"links":{"citing_paper":"/paper/2509.04265"},"observation_digest":"sha256:23dec8fc445b6e45c53499d74685ceaecfcea6bb8fb61b63b2bf1c6ed376af55","observation_id":"d07faefa-fb63-4e3a-ae97-bb8ff127756c","resolution":{"observed_at":"2026-08-05T10:20:42.410752Z","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":"2026-08-05T10:20:43.020887Z","title":"Murray.Mathematical biology: I","venue":null,"work_id":"bccc0dcc-07c9-428a-8c17-cd70a22b06ce","year":2007},"citing_paper":{"arxiv_id":"2509.04265","last_updated":"2026-06-05T15:03:49Z","snapshot_observed_at":"2026-08-13T12:35:19.818364Z","submitted_at":"2025-09-04T14:42:22Z","title":"Reinforcement-Learning-Guided Data-Driven Estimation of Spectral Properties of Stochastic Koopman Semigroups","version":3},"reference_index":27,"source":"pdf_text","source_observed_at":"2026-08-05T10:20:42.414825Z"},"links":{"citing_paper":"/paper/2509.04265"},"observation_digest":"sha256:ecd277d07ea7813aae217ce42cd1f572b84ccf74fd36d5c17829b98b55bef804","observation_id":"e0f58c7a-f659-4879-aa9c-f7fb5612cb63","resolution":{"observed_at":"2026-08-05T10:20:43.024952Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-14T06:32:32.682623+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-14T06:32:32.682623+00:00","source":"crossref"},{"observed_at":"2026-08-14T06:32:18.44784+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":"2026-08-05T10:20:43.006778Z","title":"An active pulse transmission line simulating nerve axon","venue":null,"work_id":"bb699797-f148-41f0-a1c6-f84dd3b17c06","year":2061},"citing_paper":{"arxiv_id":"2509.04265","last_updated":"2026-06-05T15:03:49Z","snapshot_observed_at":"2026-08-13T12:35:19.818364Z","submitted_at":"2025-09-04T14:42:22Z","title":"Reinforcement-Learning-Guided Data-Driven Estimation of Spectral Properties of Stochastic Koopman Semigroups","version":3},"reference_index":28,"source":"pdf_text","source_observed_at":"2026-08-05T10:20:42.418939Z"},"links":{"citing_paper":"/paper/2509.04265"},"observation_digest":"sha256:fa5428bc091cfacb844572978992a10a89d331059d4c9bbfc73c93f5fc0fcd7f","observation_id":"8cbc7749-8a24-4ff0-9fa5-df4d1b34e650","resolution":{"observed_at":"2026-08-05T10:20:43.011899Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-14T06:32:32.682623+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-14T06:32:32.682623+00:00","source":"crossref"},{"observed_at":"2026-08-14T06:32:18.44784+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":"2026-08-05T10:20:42.993262Z","title":"Pavliotis.Stochastic Processes and Applications: Diffusion Processes, the Fokker-Planck and Langevin Equations","venue":null,"work_id":"f1e233d9-7c9b-43cf-9cc3-b3b751401496","year":2016},"citing_paper":{"arxiv_id":"2509.04265","last_updated":"2026-06-05T15:03:49Z","snapshot_observed_at":"2026-08-13T12:35:19.818364Z","submitted_at":"2025-09-04T14:42:22Z","title":"Reinforcement-Learning-Guided Data-Driven Estimation of Spectral Properties of Stochastic Koopman Semigroups","version":3},"reference_index":29,"source":"pdf_text","source_observed_at":"2026-08-05T10:20:42.423501Z"},"links":{"citing_paper":"/paper/2509.04265"},"observation_digest":"sha256:addf497be90b73da5477d3d64fae3d69bd27fc67d8845949340000049ebc9153","observation_id":"9eb2602a-02df-4d2f-81b7-60028975a00e","resolution":{"observed_at":"2026-08-05T10:20:42.997777Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-14T06:32:32.682623+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-14T06:32:32.682623+00:00","source":"crossref"},{"observed_at":"2026-08-14T06:32:18.44784+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":"2026-08-05T10:20:42.979835Z","title":"Applied Mathematical Sciences","venue":null,"work_id":"31440056-f389-4976-93ea-80c9c31cc356","year":2012},"citing_paper":{"arxiv_id":"2509.04265","last_updated":"2026-06-05T15:03:49Z","snapshot_observed_at":"2026-08-13T12:35:19.818364Z","submitted_at":"2025-09-04T14:42:22Z","title":"Reinforcement-Learning-Guided Data-Driven Estimation of Spectral Properties of Stochastic Koopman Semigroups","version":3},"reference_index":30,"source":"pdf_text","source_observed_at":"2026-08-05T10:20:42.427410Z"},"links":{"citing_paper":"/paper/2509.04265"},"observation_digest":"sha256:4a64da45a52e257f3d2ffc7c74849df789a615961d71cd75de3cd877e2c05d08","observation_id":"da00f610-dc6c-4713-bc1c-395a3f018e43","resolution":{"observed_at":"2026-08-05T10:20:42.983866Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-14T06:32:32.682623+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-14T06:32:32.682623+00:00","source":"crossref"},{"observed_at":"2026-08-14T06:32:18.44784+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":"2026-08-05T10:20:42.965614Z","title":"Rowley, Igor Mezi´c, Shervin Bagheri, Philipp Schlatter, and Dan S","venue":null,"work_id":"2d7b6efe-2e0e-47a5-8f9f-0fd19abb42f1","year":2009},"citing_paper":{"arxiv_id":"2509.04265","last_updated":"2026-06-05T15:03:49Z","snapshot_observed_at":"2026-08-13T12:35:19.818364Z","submitted_at":"2025-09-04T14:42:22Z","title":"Reinforcement-Learning-Guided Data-Driven Estimation of Spectral Properties of Stochastic Koopman Semigroups","version":3},"reference_index":31,"source":"pdf_text","source_observed_at":"2026-08-05T10:20:42.431416Z"},"links":{"citing_paper":"/paper/2509.04265"},"observation_digest":"sha256:5d8a720fd1fbb07b1dc86754968e8fbae90900f2879312bf2f5a4a1eedfbed96","observation_id":"619c67ef-a741-4398-82c0-daebe33814a2","resolution":{"observed_at":"2026-08-05T10:20:42.970193Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-14T06:32:32.682623+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-14T06:32:32.682623+00:00","source":"crossref"},{"observed_at":"2026-08-14T06:32:18.44784+00:00","source":"retraction_watch"}],"state":"measured"}},{"citation":{"cited_paper":{"arxiv_id":"1707.06347","last_updated":"2017-08-28T09:20:06Z","snapshot_observed_at":"2026-07-06T02:11:23.670680Z","submitted_at":"2017-07-20T02:32:33Z","title":"Proximal Policy Optimization Algorithms","version":2},"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":"1707.06347","snapshot_observed_at":"2026-08-05T10:20:42.436545Z","title":"Proximal policy optimization algorithms.arXiv preprint arXiv:1707.06347, 2017","venue":null,"work_id":null,"year":2017},"citing_paper":{"arxiv_id":"2509.04265","last_updated":"2026-06-05T15:03:49Z","snapshot_observed_at":"2026-08-13T12:35:19.818364Z","submitted_at":"2025-09-04T14:42:22Z","title":"Reinforcement-Learning-Guided Data-Driven Estimation of Spectral Properties of Stochastic Koopman Semigroups","version":3},"reference_index":32,"source":"pdf_text","source_observed_at":"2026-08-05T10:20:42.436545Z"},"links":{"cited_paper":"/paper/1707.06347","citing_paper":"/paper/2509.04265"},"observation_digest":"sha256:231bd8d42c5e93d7b5b1fe1c2dc4ebc304f4674241d699da144d86f6d39e5668","observation_id":"c845461a-a967-4890-8154-42617d713cba","resolution":{"observed_at":"2026-08-05T10:20:42.436545Z","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":"2026-08-05T10:20:42.950354Z","title":"Introduction to multi-armed bandits.Foundations and Trends® in Machine Learning, 12 (1-2):1–286, 2019","venue":null,"work_id":"d6937fed-5fdb-4ff3-bc9d-1f4f005d8910","year":2019},"citing_paper":{"arxiv_id":"2509.04265","last_updated":"2026-06-05T15:03:49Z","snapshot_observed_at":"2026-08-13T12:35:19.818364Z","submitted_at":"2025-09-04T14:42:22Z","title":"Reinforcement-Learning-Guided Data-Driven Estimation of Spectral Properties of Stochastic Koopman Semigroups","version":3},"reference_index":33,"source":"pdf_text","source_observed_at":"2026-08-05T10:20:42.441187Z"},"links":{"citing_paper":"/paper/2509.04265"},"observation_digest":"sha256:de81be947bc12302413b118628beb7c9c8faf6de1cc8beb2b849d7a48b958495","observation_id":"a40c1e36-31d0-4eca-b9b1-9c6a3f42e9f7","resolution":{"observed_at":"2026-08-05T10:20:42.955026Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-14T06:32:32.682623+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-14T06:32:32.682623+00:00","source":"crossref"},{"observed_at":"2026-08-14T06:32:18.44784+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":"2026-08-05T10:20:42.934763Z","title":"Strogatz.Nonlinear dynamics and chaos: with applications to physics, biology, chemistry, and engineering (studies in nonlinearity), volume 1","venue":null,"work_id":"59eeb426-709d-4039-a1dd-80ac17723a51","year":2001},"citing_paper":{"arxiv_id":"2509.04265","last_updated":"2026-06-05T15:03:49Z","snapshot_observed_at":"2026-08-13T12:35:19.818364Z","submitted_at":"2025-09-04T14:42:22Z","title":"Reinforcement-Learning-Guided Data-Driven Estimation of Spectral Properties of Stochastic Koopman Semigroups","version":3},"reference_index":34,"source":"pdf_text","source_observed_at":"2026-08-05T10:20:42.445326Z"},"links":{"citing_paper":"/paper/2509.04265"},"observation_digest":"sha256:9771e3b953086c6cba3507bce249bfe566a7d332930644ce2f0ed0a475a08ddf","observation_id":"06f0cfc7-c623-457f-9e4b-c894ce5d0a95","resolution":{"observed_at":"2026-08-05T10:20:42.940534Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-14T06:32:32.682623+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-14T06:32:32.682623+00:00","source":"crossref"},{"observed_at":"2026-08-14T06:32:18.44784+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":"2026-08-05T10:20:42.920390Z","title":"Sutton and Andrew G","venue":null,"work_id":"778a5038-2e0d-4be0-bd3d-ba55579f00b3","year":1998},"citing_paper":{"arxiv_id":"2509.04265","last_updated":"2026-06-05T15:03:49Z","snapshot_observed_at":"2026-08-13T12:35:19.818364Z","submitted_at":"2025-09-04T14:42:22Z","title":"Reinforcement-Learning-Guided Data-Driven Estimation of Spectral Properties of Stochastic Koopman Semigroups","version":3},"reference_index":35,"source":"pdf_text","source_observed_at":"2026-08-05T10:20:42.449703Z"},"links":{"citing_paper":"/paper/2509.04265"},"observation_digest":"sha256:5b25e824d52759cb389712d09659ed8210f217d8b0d9e9ad98a4be09216675bf","observation_id":"1bc379e5-b7d4-4a20-8529-8b6d18853fcb","resolution":{"observed_at":"2026-08-05T10:20:42.924924Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-14T06:32:32.682623+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-14T06:32:32.682623+00:00","source":"crossref"},{"observed_at":"2026-08-14T06:32:18.44784+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":"2026-08-05T10:20:42.906903Z","title":"Definition and data-driven reconstruction of asymptotic phase and amplitudes of stochastic oscillators via koopman operator theory","venue":null,"work_id":"81e203ba-db2f-4e2a-91c1-d22c69b56687","year":2023},"citing_paper":{"arxiv_id":"2509.04265","last_updated":"2026-06-05T15:03:49Z","snapshot_observed_at":"2026-08-13T12:35:19.818364Z","submitted_at":"2025-09-04T14:42:22Z","title":"Reinforcement-Learning-Guided Data-Driven Estimation of Spectral Properties of Stochastic Koopman Semigroups","version":3},"reference_index":36,"source":"pdf_text","source_observed_at":"2026-08-05T10:20:42.453845Z"},"links":{"citing_paper":"/paper/2509.04265"},"observation_digest":"sha256:f74a497816fe9fe135b428d493a2755d0fecf1b0efa953401f64f2d6f48c28cb","observation_id":"4e88e091-cda1-486c-96ef-78b97c9bb858","resolution":{"observed_at":"2026-08-05T10:20:42.911204Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-14T06:32:32.682623+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-14T06:32:32.682623+00:00","source":"crossref"},{"observed_at":"2026-08-14T06:32:18.44784+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":"2026-08-05T10:20:42.892548Z","title":"Tu.Dynamic mode decomposition: Theory and applications","venue":null,"work_id":"9149f358-5cc7-4e84-9eec-e6c063add6a7","year":2013},"citing_paper":{"arxiv_id":"2509.04265","last_updated":"2026-06-05T15:03:49Z","snapshot_observed_at":"2026-08-13T12:35:19.818364Z","submitted_at":"2025-09-04T14:42:22Z","title":"Reinforcement-Learning-Guided Data-Driven Estimation of Spectral Properties of Stochastic Koopman Semigroups","version":3},"reference_index":37,"source":"pdf_text","source_observed_at":"2026-08-05T10:20:42.458612Z"},"links":{"citing_paper":"/paper/2509.04265"},"observation_digest":"sha256:1ab8fbe0f5bfe93e2c2211f1a2767e6a52e09ee9a04d4885e3b8fd2f665456b0","observation_id":"2e89519c-a407-4da8-98c9-19a11f3936b5","resolution":{"observed_at":"2026-08-05T10:20:42.896929Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-14T06:32:32.682623+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-14T06:32:32.682623+00:00","source":"crossref"},{"observed_at":"2026-08-14T06:32:18.44784+00:00","source":"retraction_watch"}],"state":"measured"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":null,"snapshot_observed_at":"2026-08-05T10:20:42.463145Z","title":"Texts in Applied Mathematics","venue":null,"work_id":null,"year":2003},"citing_paper":{"arxiv_id":"2509.04265","last_updated":"2026-06-05T15:03:49Z","snapshot_observed_at":"2026-08-13T12:35:19.818364Z","submitted_at":"2025-09-04T14:42:22Z","title":"Reinforcement-Learning-Guided Data-Driven Estimation of Spectral Properties of Stochastic Koopman Semigroups","version":3},"reference_index":38,"source":"pdf_text","source_observed_at":"2026-08-05T10:20:42.463145Z"},"links":{"citing_paper":"/paper/2509.04265"},"observation_digest":"sha256:5ef3f1d5f1d64cc0af09628e4c3acbee01e063b66dc04e7c61972261d5e942b9","observation_id":"21a9ee02-0488-4f4d-845e-3bbb8dc88241","resolution":{"observed_at":"2026-08-05T10:20:42.463145Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":null,"snapshot_observed_at":"2026-08-05T10:20:42.467525Z","title":"Williams, Ioannis G","venue":null,"work_id":null,"year":2015},"citing_paper":{"arxiv_id":"2509.04265","last_updated":"2026-06-05T15:03:49Z","snapshot_observed_at":"2026-08-13T12:35:19.818364Z","submitted_at":"2025-09-04T14:42:22Z","title":"Reinforcement-Learning-Guided Data-Driven Estimation of Spectral Properties of Stochastic Koopman Semigroups","version":3},"reference_index":39,"source":"pdf_text","source_observed_at":"2026-08-05T10:20:42.467525Z"},"links":{"citing_paper":"/paper/2509.04265"},"observation_digest":"sha256:4ad3a02cf6e65e4ab40e7a5f3c229393904984fed5df97847096a1f39bea88fc","observation_id":"5a4c0d27-39ae-44dd-9846-966ebd500361","resolution":{"observed_at":"2026-08-05T10:20:42.467525Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":null,"snapshot_observed_at":"2026-08-05T10:20:42.472179Z","title":"Williams, Clarence W","venue":null,"work_id":null,"year":2015},"citing_paper":{"arxiv_id":"2509.04265","last_updated":"2026-06-05T15:03:49Z","snapshot_observed_at":"2026-08-13T12:35:19.818364Z","submitted_at":"2025-09-04T14:42:22Z","title":"Reinforcement-Learning-Guided Data-Driven Estimation of Spectral Properties of Stochastic Koopman Semigroups","version":3},"reference_index":40,"source":"pdf_text","source_observed_at":"2026-08-05T10:20:42.472179Z"},"links":{"citing_paper":"/paper/2509.04265"},"observation_digest":"sha256:769c7d6a280313fb743a14130a712e58ea3b400cc22e95de3cf8ae813a920c20","observation_id":"c03011dd-af52-4ac7-a471-ed5f096ef6a6","resolution":{"observed_at":"2026-08-05T10:20:42.472179Z","resolver_source":null,"status":"malformed_identifier"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":{"arxiv_id":"2501.13301","last_updated":"2025-08-19T04:09:52Z","snapshot_observed_at":"2026-08-10T16:14:44.214456Z","submitted_at":"2025-01-23T01:02:52Z","title":"A Data-Driven Framework for Koopman Semigroup Estimation in Stochastic Dynamical Systems","version":4},"cited_work":{"arxiv_id":"2501.13301","doi":null,"metadata_source":"pith","pith_arxiv_id":"2501.13301","snapshot_observed_at":"2026-08-05T10:20:42.555567Z","title":"A Data-Driven Framework for Koopman Semigroup Estimation in Stochastic Dynamical Systems","venue":"math.DS","work_id":"8434a335-b1dc-473d-ab41-6757e7657ad4","year":2025},"citing_paper":{"arxiv_id":"2509.04265","last_updated":"2026-06-05T15:03:49Z","snapshot_observed_at":"2026-08-13T12:35:19.818364Z","submitted_at":"2025-09-04T14:42:22Z","title":"Reinforcement-Learning-Guided Data-Driven Estimation of Spectral Properties of Stochastic Koopman Semigroups","version":3},"reference_index":41,"source":"pdf_text","source_observed_at":"2026-08-05T10:20:42.476624Z"},"links":{"cited_paper":"/paper/2501.13301","citing_paper":"/paper/2509.04265"},"observation_digest":"sha256:7c7544c6fdc11c3abc1af581e3c7d7b3325e4df3a7840b2bbf940e836dea7934","observation_id":"00f462e8-193f-4bfb-9a06-e6002f79162c","resolution":{"observed_at":"2026-08-05T10:20:42.562421Z","resolver_source":"local_arxiv","status":"verified_exact"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-14T06:32:32.682623+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-14T06:32:32.682623+00:00","source":"crossref"},{"observed_at":"2026-08-14T06:32:18.44784+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":"2026-08-05T10:20:42.878935Z","title":"Reskoopnet: Learning koopman representations for complex dynamics with spectral residuals","venue":null,"work_id":"89259088-24a7-464e-a23f-154e5d18af97","year":2025},"citing_paper":{"arxiv_id":"2509.04265","last_updated":"2026-06-05T15:03:49Z","snapshot_observed_at":"2026-08-13T12:35:19.818364Z","submitted_at":"2025-09-04T14:42:22Z","title":"Reinforcement-Learning-Guided Data-Driven Estimation of Spectral Properties of Stochastic Koopman Semigroups","version":3},"reference_index":42,"source":"pdf_text","source_observed_at":"2026-08-05T10:20:42.481207Z"},"links":{"citing_paper":"/paper/2509.04265"},"observation_digest":"sha256:6304a050f04eba1e79359fed4e639d40e8b130adc85ea6596eff7b572ada8183","observation_id":"6f15877f-c4fb-47eb-a497-0d096af7602d","resolution":{"observed_at":"2026-08-05T10:20:42.883382Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-14T06:32:32.682623+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-14T06:32:32.682623+00:00","source":"crossref"},{"observed_at":"2026-08-14T06:32:18.44784+00:00","source":"retraction_watch"}],"state":"measured"}}],"paper":{"arxiv_id":"2509.04265","last_updated":"2026-06-05T15:03:49Z","latest_version":3,"primary_category":"math.DS","snapshot_observed_at":"2026-08-13T12:35:19.818364Z","submitted_at":"2025-09-04T14:42:22Z","title":"Reinforcement-Learning-Guided Data-Driven Estimation of Spectral Properties of Stochastic Koopman Semigroups"},"reference_resolution":{"displayed":42,"state_counts":{"malformed_identifier":1,"metadata_mismatch":1,"parse_uncertain":0,"unresolved":11,"verified_exact":2,"verified_fuzzy":27},"total_outbound_references":42},"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":"2026-08-14T06:32:32.682623+00:00","source":"crossref"},{"observed_at":"2026-08-14T06:32:18.44784+00:00","source":"retraction_watch"}],"thesis":"As of 14 August 2026, this Paper Citation Record lists 42 of 42 outbound references and 0 inbound Pith citation observations for arXiv:2509.04265."}