{"as_of":"2026-08-10T06:26:00Z","caps":{"database_statements":6,"inbound":100,"outbound":100},"context_digest":"sha256:75ffeaf27576f3d259703739e6b39cf5eaaa343c36ec94ea4c1a9ef8f45f013e","coverage":[{"denominator":78,"lane":"reference_resolution","note":"Typed states for the displayed outbound observations.","records_observed":78,"source":"paper_references, paper_reference_links","source_observed_at":"2026-07-13T07:17:37.978825Z","state":"measured"},{"denominator":78,"lane":"standing_notices","note":"One-hop event checks from named stored sources.","records_observed":78,"source":"scholarly_work_events, retraction_status_cache","source_observed_at":"2026-08-09T06:31:02.800959+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/2607.08788/citation-record","integrity":"/paper/2607.08788/integrity","json":"/paper/2607.08788/citation-record.json","paper":"/paper/2607.08788"},"outbound":[{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":null,"snapshot_observed_at":"2026-07-13T07:17:37.978825Z","title":"Natarajan, and K","venue":null,"work_id":null,"year":1974},"citing_paper":{"arxiv_id":"2607.08788","last_updated":"2026-06-26T22:21:40Z","snapshot_observed_at":"2026-08-09T14:09:25.924672Z","submitted_at":"2026-06-26T22:21:40Z","title":"Matched Generators for the Karhunen--Lo\\`eve Transform: A Double-Commutator Eigenvalue Theory","version":1},"reference_index":1,"source":"pdf_text","source_observed_at":"2026-07-13T07:17:37.978825Z"},"links":{"citing_paper":"/paper/2607.08788"},"observation_digest":"sha256:086e62c319029d68109e5bb6eb8855c0cfa0679162703b5492ab3888f35d6bcd","observation_id":"07473623-f301-48f5-8678-1986f1748695","resolution":{"observed_at":"2026-07-13T07:17:37.978825Z","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":"2026-07-13T07:17:37.978825Z","title":"Graph isomorphism in quasipolynomial time","venue":null,"work_id":null,"year":2016},"citing_paper":{"arxiv_id":"2607.08788","last_updated":"2026-06-26T22:21:40Z","snapshot_observed_at":"2026-08-09T14:09:25.924672Z","submitted_at":"2026-06-26T22:21:40Z","title":"Matched Generators for the Karhunen--Lo\\`eve Transform: A Double-Commutator Eigenvalue Theory","version":1},"reference_index":2,"source":"pdf_text","source_observed_at":"2026-07-13T07:17:37.978825Z"},"links":{"citing_paper":"/paper/2607.08788"},"observation_digest":"sha256:d4553a0f940fc88b193db5cf471da17d897fc2640e050ffe92acbca63c7c4e42","observation_id":"f8e7ed95-aa4d-4e16-947f-6c8d7ec87596","resolution":{"observed_at":"2026-07-13T07:17:37.978825Z","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":"2026-07-13T07:17:37.978825Z","title":"Cambridge University Press, 2nd edition, 1993","venue":null,"work_id":null,"year":1993},"citing_paper":{"arxiv_id":"2607.08788","last_updated":"2026-06-26T22:21:40Z","snapshot_observed_at":"2026-08-09T14:09:25.924672Z","submitted_at":"2026-06-26T22:21:40Z","title":"Matched Generators for the Karhunen--Lo\\`eve Transform: A Double-Commutator Eigenvalue Theory","version":1},"reference_index":3,"source":"pdf_text","source_observed_at":"2026-07-13T07:17:37.978825Z"},"links":{"citing_paper":"/paper/2607.08788"},"observation_digest":"sha256:4ffcb02f53da99bcdef4b215a984cb725aa7b3babaddead137fc6fed470b9877","observation_id":"3c5a5cb1-8d2f-4f86-9e9d-d92261de1071","resolution":{"observed_at":"2026-07-13T07:17:37.978825Z","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":"2026-07-13T07:17:37.978825Z","title":"Brockett","venue":null,"work_id":null,"year":1991},"citing_paper":{"arxiv_id":"2607.08788","last_updated":"2026-06-26T22:21:40Z","snapshot_observed_at":"2026-08-09T14:09:25.924672Z","submitted_at":"2026-06-26T22:21:40Z","title":"Matched Generators for the Karhunen--Lo\\`eve Transform: A Double-Commutator Eigenvalue Theory","version":1},"reference_index":4,"source":"pdf_text","source_observed_at":"2026-07-13T07:17:37.978825Z"},"links":{"citing_paper":"/paper/2607.08788"},"observation_digest":"sha256:ac09dc321c1b4ba50a82176c68518ea188864b4b694f753272cb8d4e4648c4b4","observation_id":"88c99feb-fd29-4074-a571-4506514ac835","resolution":{"observed_at":"2026-07-13T07:17:37.978825Z","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":"2026-07-13T07:17:37.978825Z","title":"Jacobi angles for simultaneous diagonalization","venue":null,"work_id":null,"year":1996},"citing_paper":{"arxiv_id":"2607.08788","last_updated":"2026-06-26T22:21:40Z","snapshot_observed_at":"2026-08-09T14:09:25.924672Z","submitted_at":"2026-06-26T22:21:40Z","title":"Matched Generators for the Karhunen--Lo\\`eve Transform: A Double-Commutator Eigenvalue Theory","version":1},"reference_index":5,"source":"pdf_text","source_observed_at":"2026-07-13T07:17:37.978825Z"},"links":{"citing_paper":"/paper/2607.08788"},"observation_digest":"sha256:a32cd67ca8f8c25b04e9c65e465f3244cfea0748f69432aef4aa5d3da909c9ec","observation_id":"4a210ef4-823a-4e94-a80a-6abba6186631","resolution":{"observed_at":"2026-07-13T07:17:37.978825Z","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":"2026-07-13T07:17:37.978825Z","title":"Riley Casper, F","venue":null,"work_id":null,"year":2023},"citing_paper":{"arxiv_id":"2607.08788","last_updated":"2026-06-26T22:21:40Z","snapshot_observed_at":"2026-08-09T14:09:25.924672Z","submitted_at":"2026-06-26T22:21:40Z","title":"Matched Generators for the Karhunen--Lo\\`eve Transform: A Double-Commutator Eigenvalue Theory","version":1},"reference_index":6,"source":"pdf_text","source_observed_at":"2026-07-13T07:17:37.978825Z"},"links":{"citing_paper":"/paper/2607.08788"},"observation_digest":"sha256:c3494513c2b36c1665d359f9b68d768bb94716f4ad803301c0ffbb805f9a6ffe","observation_id":"7cca100f-4a2f-471d-b292-3f050bc9dafb","resolution":{"observed_at":"2026-07-13T07:17:37.978825Z","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":"2026-07-13T07:17:37.978825Z","title":"Riley Casper and Milen Yakimov","venue":null,"work_id":null,"year":2022},"citing_paper":{"arxiv_id":"2607.08788","last_updated":"2026-06-26T22:21:40Z","snapshot_observed_at":"2026-08-09T14:09:25.924672Z","submitted_at":"2026-06-26T22:21:40Z","title":"Matched Generators for the Karhunen--Lo\\`eve Transform: A Double-Commutator Eigenvalue Theory","version":1},"reference_index":7,"source":"pdf_text","source_observed_at":"2026-07-13T07:17:37.978825Z"},"links":{"citing_paper":"/paper/2607.08788"},"observation_digest":"sha256:c5ef4f69ad1fed8597ebd212d895149fc3cd4b9ad50bacbb7522e32ad0df5f2b","observation_id":"29d1d83d-ccc3-4651-aba7-d7edafe2333e","resolution":{"observed_at":"2026-07-13T07:17:37.978825Z","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":"2026-07-13T07:17:37.978825Z","title":null,"venue":null,"work_id":null,"year":2008},"citing_paper":{"arxiv_id":"2607.08788","last_updated":"2026-06-26T22:21:40Z","snapshot_observed_at":"2026-08-09T14:09:25.924672Z","submitted_at":"2026-06-26T22:21:40Z","title":"Matched Generators for the Karhunen--Lo\\`eve Transform: A Double-Commutator Eigenvalue Theory","version":1},"reference_index":8,"source":"pdf_text","source_observed_at":"2026-07-13T07:17:37.978825Z"},"links":{"citing_paper":"/paper/2607.08788"},"observation_digest":"sha256:5f206cf76081d99e088983c4e7a0e16aebedb3215931ef7562d6395f402ae4b1","observation_id":"da74a1d6-56e0-4d2c-92ce-cfec890314ad","resolution":{"observed_at":"2026-07-13T07:17:37.978825Z","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":"2026-07-13T07:17:37.978825Z","title":"Chu and Kenneth R","venue":null,"work_id":null,"year":1990},"citing_paper":{"arxiv_id":"2607.08788","last_updated":"2026-06-26T22:21:40Z","snapshot_observed_at":"2026-08-09T14:09:25.924672Z","submitted_at":"2026-06-26T22:21:40Z","title":"Matched Generators for the Karhunen--Lo\\`eve Transform: A Double-Commutator Eigenvalue Theory","version":1},"reference_index":9,"source":"pdf_text","source_observed_at":"2026-07-13T07:17:37.978825Z"},"links":{"citing_paper":"/paper/2607.08788"},"observation_digest":"sha256:79cffc5ad27cfd816b4fce5ff3c8eace2eb02bbc0860031e12c5f505e1d5b251","observation_id":"3de091a0-b2b9-404d-8a12-c2b678be7686","resolution":{"observed_at":"2026-07-13T07:17:37.978825Z","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":"2026-07-13T07:17:37.978825Z","title":"Interscience, 1953","venue":null,"work_id":null,"year":1953},"citing_paper":{"arxiv_id":"2607.08788","last_updated":"2026-06-26T22:21:40Z","snapshot_observed_at":"2026-08-09T14:09:25.924672Z","submitted_at":"2026-06-26T22:21:40Z","title":"Matched Generators for the Karhunen--Lo\\`eve Transform: A Double-Commutator Eigenvalue Theory","version":1},"reference_index":10,"source":"pdf_text","source_observed_at":"2026-07-13T07:17:37.978825Z"},"links":{"citing_paper":"/paper/2607.08788"},"observation_digest":"sha256:0362fb511d4b4fed9a33a68357a6a23880aa686fd95f44a3894e262323ad6275","observation_id":"72e9978e-50c7-4af5-ba1b-ef85df598243","resolution":{"observed_at":"2026-07-13T07:17:37.978825Z","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":"2026-07-13T07:17:37.978825Z","title":"Cover and Joy A","venue":null,"work_id":null,"year":2006},"citing_paper":{"arxiv_id":"2607.08788","last_updated":"2026-06-26T22:21:40Z","snapshot_observed_at":"2026-08-09T14:09:25.924672Z","submitted_at":"2026-06-26T22:21:40Z","title":"Matched Generators for the Karhunen--Lo\\`eve Transform: A Double-Commutator Eigenvalue Theory","version":1},"reference_index":11,"source":"pdf_text","source_observed_at":"2026-07-13T07:17:37.978825Z"},"links":{"citing_paper":"/paper/2607.08788"},"observation_digest":"sha256:d1dbb60524956213fb2ca38969f157549b331419e865a02e7e272ac1a2f49dfc","observation_id":"fc0eb227-99ec-47de-abad-c692bacfd6b9","resolution":{"observed_at":"2026-07-13T07:17:37.978825Z","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":"2026-07-13T07:17:37.978825Z","title":"Society for Industrial and Applied Mathematics, Philadelphia, 1992","venue":null,"work_id":null,"year":1992},"citing_paper":{"arxiv_id":"2607.08788","last_updated":"2026-06-26T22:21:40Z","snapshot_observed_at":"2026-08-09T14:09:25.924672Z","submitted_at":"2026-06-26T22:21:40Z","title":"Matched Generators for the Karhunen--Lo\\`eve Transform: A Double-Commutator Eigenvalue Theory","version":1},"reference_index":12,"source":"pdf_text","source_observed_at":"2026-07-13T07:17:37.978825Z"},"links":{"citing_paper":"/paper/2607.08788"},"observation_digest":"sha256:75ec17744db11e466024bb78f6329045d4bb7d9eb5e2824ef31a0779069dbafe","observation_id":"b6e61be6-7133-4b5c-93a1-bb326ac33f21","resolution":{"observed_at":"2026-07-13T07:17:37.978825Z","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":"2026-07-13T07:17:37.978825Z","title":null,"venue":null,"work_id":null,"year":1993},"citing_paper":{"arxiv_id":"2607.08788","last_updated":"2026-06-26T22:21:40Z","snapshot_observed_at":"2026-08-09T14:09:25.924672Z","submitted_at":"2026-06-26T22:21:40Z","title":"Matched Generators for the Karhunen--Lo\\`eve Transform: A Double-Commutator Eigenvalue Theory","version":1},"reference_index":13,"source":"pdf_text","source_observed_at":"2026-07-13T07:17:37.978825Z"},"links":{"citing_paper":"/paper/2607.08788"},"observation_digest":"sha256:d555332f102ecce7249771ba2fb280b45b6293d2dc810892c2406a7fe750f978","observation_id":"8f7a2df3-3290-401e-ac66-f3d06ca06a33","resolution":{"observed_at":"2026-07-13T07:17:37.978825Z","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":"2026-07-13T07:17:37.978825Z","title":"Institute of Mathematical Statistics, Hayward, CA, 1988","venue":null,"work_id":null,"year":1988},"citing_paper":{"arxiv_id":"2607.08788","last_updated":"2026-06-26T22:21:40Z","snapshot_observed_at":"2026-08-09T14:09:25.924672Z","submitted_at":"2026-06-26T22:21:40Z","title":"Matched Generators for the Karhunen--Lo\\`eve Transform: A Double-Commutator Eigenvalue Theory","version":1},"reference_index":14,"source":"pdf_text","source_observed_at":"2026-07-13T07:17:37.978825Z"},"links":{"citing_paper":"/paper/2607.08788"},"observation_digest":"sha256:314d0825e8114b943f50fe6ac84f25ab4944c353fe08ef0926c964e9facb9ad3","observation_id":"7ebf452d-92b6-4f7c-8b9b-71159f5ff0cc","resolution":{"observed_at":"2026-07-13T07:17:37.978825Z","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":"2026-07-13T07:17:37.978825Z","title":"What can we do in a symmetry-constrained perspective? the importance of the total charge’s status in quantum reference frame frameworks.Quantum, 10:2126, 2026","venue":null,"work_id":null,"year":2026},"citing_paper":{"arxiv_id":"2607.08788","last_updated":"2026-06-26T22:21:40Z","snapshot_observed_at":"2026-08-09T14:09:25.924672Z","submitted_at":"2026-06-26T22:21:40Z","title":"Matched Generators for the Karhunen--Lo\\`eve Transform: A Double-Commutator Eigenvalue Theory","version":1},"reference_index":15,"source":"pdf_text","source_observed_at":"2026-07-13T07:17:37.978825Z"},"links":{"citing_paper":"/paper/2607.08788"},"observation_digest":"sha256:0618e456e1de156901e9c0f31753ba60a4ab7947edc296256474eccc0d28c1a5","observation_id":"c9d8f0a3-4206-4d2a-b959-81597c9004d1","resolution":{"observed_at":"2026-07-13T07:17:37.978825Z","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":"2026-07-13T07:17:37.978825Z","title":null,"venue":null,"work_id":null,"year":1986},"citing_paper":{"arxiv_id":"2607.08788","last_updated":"2026-06-26T22:21:40Z","snapshot_observed_at":"2026-08-09T14:09:25.924672Z","submitted_at":"2026-06-26T22:21:40Z","title":"Matched Generators for the Karhunen--Lo\\`eve Transform: A Double-Commutator Eigenvalue Theory","version":1},"reference_index":16,"source":"pdf_text","source_observed_at":"2026-07-13T07:17:37.978825Z"},"links":{"citing_paper":"/paper/2607.08788"},"observation_digest":"sha256:28bef5a0b9b7e27618f52be244b29317e1df6e107a245bb960f3f0583c4f6643","observation_id":"28f3044c-69a8-4b9e-8296-3cf776e96c2c","resolution":{"observed_at":"2026-07-13T07:17:37.978825Z","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":"2026-07-13T07:17:37.978825Z","title":null,"venue":null,"work_id":null,"year":1989},"citing_paper":{"arxiv_id":"2607.08788","last_updated":"2026-06-26T22:21:40Z","snapshot_observed_at":"2026-08-09T14:09:25.924672Z","submitted_at":"2026-06-26T22:21:40Z","title":"Matched Generators for the Karhunen--Lo\\`eve Transform: A Double-Commutator Eigenvalue Theory","version":1},"reference_index":17,"source":"pdf_text","source_observed_at":"2026-07-13T07:17:37.978825Z"},"links":{"citing_paper":"/paper/2607.08788"},"observation_digest":"sha256:ec8c550959c7a7079ec6fa4fe54025a612a7383724c7be4ab16c25cfd1d9158d","observation_id":"4bf9da3a-00cf-41df-ab90-e1db794e951d","resolution":{"observed_at":"2026-07-13T07:17:37.978825Z","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":"2026-07-13T07:17:37.978825Z","title":"Dur ´an and F","venue":null,"work_id":null,"year":2004},"citing_paper":{"arxiv_id":"2607.08788","last_updated":"2026-06-26T22:21:40Z","snapshot_observed_at":"2026-08-09T14:09:25.924672Z","submitted_at":"2026-06-26T22:21:40Z","title":"Matched Generators for the Karhunen--Lo\\`eve Transform: A Double-Commutator Eigenvalue Theory","version":1},"reference_index":18,"source":"pdf_text","source_observed_at":"2026-07-13T07:17:37.978825Z"},"links":{"citing_paper":"/paper/2607.08788"},"observation_digest":"sha256:9f83dff98ef6a895c25907a5398ffb35d825d6534e77379ebf8913fe04ed913f","observation_id":"5a45741d-a502-48f9-97ea-281a76ef5446","resolution":{"observed_at":"2026-07-13T07:17:37.978825Z","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":"2026-07-13T07:17:37.978825Z","title":"Suboptimality of the Karhunen–Lo `eve transform for transform coding.IEEE Transactions on Information Theory, 50(8):1605–1619, 2004","venue":null,"work_id":null,"year":2004},"citing_paper":{"arxiv_id":"2607.08788","last_updated":"2026-06-26T22:21:40Z","snapshot_observed_at":"2026-08-09T14:09:25.924672Z","submitted_at":"2026-06-26T22:21:40Z","title":"Matched Generators for the Karhunen--Lo\\`eve Transform: A Double-Commutator Eigenvalue Theory","version":1},"reference_index":19,"source":"pdf_text","source_observed_at":"2026-07-13T07:17:37.978825Z"},"links":{"citing_paper":"/paper/2607.08788"},"observation_digest":"sha256:87e1b19373c6f2fc089c170d6068df824da8311fab38f1319f44c17b5ee28c12","observation_id":"f091c1a4-4b0f-440c-b05d-b52d782d1238","resolution":{"observed_at":"2026-07-13T07:17:37.978825Z","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":"2026-07-13T07:17:37.978825Z","title":"On a theorem of Weyl concerning eigenvalues of linear transformations","venue":null,"work_id":null,"year":1949},"citing_paper":{"arxiv_id":"2607.08788","last_updated":"2026-06-26T22:21:40Z","snapshot_observed_at":"2026-08-09T14:09:25.924672Z","submitted_at":"2026-06-26T22:21:40Z","title":"Matched Generators for the Karhunen--Lo\\`eve Transform: A Double-Commutator Eigenvalue Theory","version":1},"reference_index":20,"source":"pdf_text","source_observed_at":"2026-07-13T07:17:37.978825Z"},"links":{"citing_paper":"/paper/2607.08788"},"observation_digest":"sha256:7d7b225407b2f28f7a12c7095b3e34ce5bf7f6f809ee74121a5a93bdc558ed2e","observation_id":"20f6840a-c8f7-49e9-a08c-f7457ee11814","resolution":{"observed_at":"2026-07-13T07:17:37.978825Z","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":"2026-07-13T07:17:37.978825Z","title":"Resection of high frequency oscillations predicts seizure outcome in the individual patient.Scientific Reports, 7(1):13836, 2017","venue":null,"work_id":null,"year":2017},"citing_paper":{"arxiv_id":"2607.08788","last_updated":"2026-06-26T22:21:40Z","snapshot_observed_at":"2026-08-09T14:09:25.924672Z","submitted_at":"2026-06-26T22:21:40Z","title":"Matched Generators for the Karhunen--Lo\\`eve Transform: A Double-Commutator Eigenvalue Theory","version":1},"reference_index":21,"source":"pdf_text","source_observed_at":"2026-07-13T07:17:37.978825Z"},"links":{"citing_paper":"/paper/2607.08788"},"observation_digest":"sha256:4bbf86504a0aa4f00579537b4ac1efb56236a52af1d7daf2d564437614238f23","observation_id":"fde706d5-3a91-4bb2-a834-0ded6cb761c6","resolution":{"observed_at":"2026-07-13T07:17:37.978825Z","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":"2026-07-13T07:17:37.978825Z","title":"Interictal iEEG during slow-wave sleep with HFO markings","venue":null,"work_id":null,"year":2021},"citing_paper":{"arxiv_id":"2607.08788","last_updated":"2026-06-26T22:21:40Z","snapshot_observed_at":"2026-08-09T14:09:25.924672Z","submitted_at":"2026-06-26T22:21:40Z","title":"Matched Generators for the Karhunen--Lo\\`eve Transform: A Double-Commutator Eigenvalue Theory","version":1},"reference_index":22,"source":"pdf_text","source_observed_at":"2026-07-13T07:17:37.978825Z"},"links":{"citing_paper":"/paper/2607.08788"},"observation_digest":"sha256:93b8b3c8cd7e4809703e4ef487b82ffe65409df50a9de32bc98d611c7da93c7a","observation_id":"b5ff764f-3065-4506-b822-a72b1ab57515","resolution":{"observed_at":"2026-07-13T07:17:37.978825Z","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":"2026-07-13T07:17:37.978825Z","title":null,"venue":null,"work_id":null,"year":1984},"citing_paper":{"arxiv_id":"2607.08788","last_updated":"2026-06-26T22:21:40Z","snapshot_observed_at":"2026-08-09T14:09:25.924672Z","submitted_at":"2026-06-26T22:21:40Z","title":"Matched Generators for the Karhunen--Lo\\`eve Transform: A Double-Commutator Eigenvalue Theory","version":1},"reference_index":23,"source":"pdf_text","source_observed_at":"2026-07-13T07:17:37.978825Z"},"links":{"citing_paper":"/paper/2607.08788"},"observation_digest":"sha256:26e3e4b075824568f0de7759a7f76b0d10b694ac4bd06956b4c1b553e9e34483","observation_id":"1dae45a2-8491-4548-8b1e-605998b9713d","resolution":{"observed_at":"2026-07-13T07:17:37.978825Z","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":"2026-07-13T07:17:37.978825Z","title":"Folland.A Course in Abstract Harmonic Analysis","venue":null,"work_id":null,"year":1995},"citing_paper":{"arxiv_id":"2607.08788","last_updated":"2026-06-26T22:21:40Z","snapshot_observed_at":"2026-08-09T14:09:25.924672Z","submitted_at":"2026-06-26T22:21:40Z","title":"Matched Generators for the Karhunen--Lo\\`eve Transform: A Double-Commutator Eigenvalue Theory","version":1},"reference_index":24,"source":"pdf_text","source_observed_at":"2026-07-13T07:17:37.978825Z"},"links":{"citing_paper":"/paper/2607.08788"},"observation_digest":"sha256:8bf228cb799d60a04f0c1c7eee4547cf68b6693818ed51040b4776c2cf5b63a8","observation_id":"3f628f05-e334-419c-bd40-541cd3e5d9f3","resolution":{"observed_at":"2026-07-13T07:17:37.978825Z","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":"2026-07-13T07:17:37.978825Z","title":"Springer, 2001","venue":null,"work_id":null,"year":2001},"citing_paper":{"arxiv_id":"2607.08788","last_updated":"2026-06-26T22:21:40Z","snapshot_observed_at":"2026-08-09T14:09:25.924672Z","submitted_at":"2026-06-26T22:21:40Z","title":"Matched Generators for the Karhunen--Lo\\`eve Transform: A Double-Commutator Eigenvalue Theory","version":1},"reference_index":25,"source":"pdf_text","source_observed_at":"2026-07-13T07:17:37.978825Z"},"links":{"citing_paper":"/paper/2607.08788"},"observation_digest":"sha256:17c506a39bde45286109f5071e216e6251740ce2ed7a79c3318b7c93c82835b1","observation_id":"9d46c596-cb0f-41e2-8ee2-f6a0428cbfd6","resolution":{"observed_at":"2026-07-13T07:17:37.978825Z","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":"2026-07-13T07:17:37.978825Z","title":null,"venue":null,"work_id":null,"year":2001},"citing_paper":{"arxiv_id":"2607.08788","last_updated":"2026-06-26T22:21:40Z","snapshot_observed_at":"2026-08-09T14:09:25.924672Z","submitted_at":"2026-06-26T22:21:40Z","title":"Matched Generators for the Karhunen--Lo\\`eve Transform: A Double-Commutator Eigenvalue Theory","version":1},"reference_index":26,"source":"pdf_text","source_observed_at":"2026-07-13T07:17:37.978825Z"},"links":{"citing_paper":"/paper/2607.08788"},"observation_digest":"sha256:3ec59c9c875e9091aa2dc950bfcb203af594b5539ac90e702b5b920848ea480f","observation_id":"2ac1701c-fd0b-4117-a640-782cbc035b8e","resolution":{"observed_at":"2026-07-13T07:17:37.978825Z","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":"2026-07-13T07:17:37.978825Z","title":null,"venue":null,"work_id":null,"year":2006},"citing_paper":{"arxiv_id":"2607.08788","last_updated":"2026-06-26T22:21:40Z","snapshot_observed_at":"2026-08-09T14:09:25.924672Z","submitted_at":"2026-06-26T22:21:40Z","title":"Matched Generators for the Karhunen--Lo\\`eve Transform: A Double-Commutator Eigenvalue Theory","version":1},"reference_index":27,"source":"pdf_text","source_observed_at":"2026-07-13T07:17:37.978825Z"},"links":{"citing_paper":"/paper/2607.08788"},"observation_digest":"sha256:562dfcdaeae9fc0f971a6cff828d1dbd2f671ec19430321d92e987e85dfbba65","observation_id":"79d62700-ac09-4124-8b93-68e49c347b32","resolution":{"observed_at":"2026-07-13T07:17:37.978825Z","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":"2026-07-13T07:17:37.978825Z","title":"Alberto Gr ¨unbaum and Luc Haine","venue":null,"work_id":null,"year":1997},"citing_paper":{"arxiv_id":"2607.08788","last_updated":"2026-06-26T22:21:40Z","snapshot_observed_at":"2026-08-09T14:09:25.924672Z","submitted_at":"2026-06-26T22:21:40Z","title":"Matched Generators for the Karhunen--Lo\\`eve Transform: A Double-Commutator Eigenvalue Theory","version":1},"reference_index":28,"source":"pdf_text","source_observed_at":"2026-07-13T07:17:37.978825Z"},"links":{"citing_paper":"/paper/2607.08788"},"observation_digest":"sha256:e0e0757a04324d304f9b1e3a67f807b81737d1c3136b3b04d4dabac81de6672a","observation_id":"4af6be1b-9861-4a31-87b0-1824a63234fa","resolution":{"observed_at":"2026-07-13T07:17:37.978825Z","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":"2026-07-13T07:17:37.978825Z","title":"Alberto Gr ¨unbaum, Luigi Longhi, and Marvin Perlstadt","venue":null,"work_id":null,"year":1982},"citing_paper":{"arxiv_id":"2607.08788","last_updated":"2026-06-26T22:21:40Z","snapshot_observed_at":"2026-08-09T14:09:25.924672Z","submitted_at":"2026-06-26T22:21:40Z","title":"Matched Generators for the Karhunen--Lo\\`eve Transform: A Double-Commutator Eigenvalue Theory","version":1},"reference_index":29,"source":"pdf_text","source_observed_at":"2026-07-13T07:17:37.978825Z"},"links":{"citing_paper":"/paper/2607.08788"},"observation_digest":"sha256:7728124e50a799242c1093ed23d1af521d39df2600de3eea01e9f68657c12adf","observation_id":"7e0f28e9-8da9-4776-bd71-99936a6c5f62","resolution":{"observed_at":"2026-07-13T07:17:37.978825Z","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":"2026-07-13T07:17:37.978825Z","title":"Alberto Gr ¨unbaum, In ´es Pacharoni, and Ignacio Zurri ´an","venue":null,"work_id":null,"year":2020},"citing_paper":{"arxiv_id":"2607.08788","last_updated":"2026-06-26T22:21:40Z","snapshot_observed_at":"2026-08-09T14:09:25.924672Z","submitted_at":"2026-06-26T22:21:40Z","title":"Matched Generators for the Karhunen--Lo\\`eve Transform: A Double-Commutator Eigenvalue Theory","version":1},"reference_index":30,"source":"pdf_text","source_observed_at":"2026-07-13T07:17:37.978825Z"},"links":{"citing_paper":"/paper/2607.08788"},"observation_digest":"sha256:64f2e5a73fb7726e29e6ee5ae542ce85717aefd8c76ec6b232b3c426620df905","observation_id":"825a3c67-5a48-460c-a6ad-9bf087bc3d37","resolution":{"observed_at":"2026-07-13T07:17:37.978825Z","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":"2026-07-13T07:17:37.978825Z","title":"Alberto Gr ¨unbaum, Luc Vinet, and Alexei Zhedanov","venue":null,"work_id":null,"year":2018},"citing_paper":{"arxiv_id":"2607.08788","last_updated":"2026-06-26T22:21:40Z","snapshot_observed_at":"2026-08-09T14:09:25.924672Z","submitted_at":"2026-06-26T22:21:40Z","title":"Matched Generators for the Karhunen--Lo\\`eve Transform: A Double-Commutator Eigenvalue Theory","version":1},"reference_index":31,"source":"pdf_text","source_observed_at":"2026-07-13T07:17:37.978825Z"},"links":{"citing_paper":"/paper/2607.08788"},"observation_digest":"sha256:2e57c37aa60a32bf7df61747414610557994e9be12837fe1fcbcbae1e74914e5","observation_id":"73d4e3e4-81b2-4fae-9cb4-96b554b9f6ba","resolution":{"observed_at":"2026-07-13T07:17:37.978825Z","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":"2026-07-13T07:17:37.978825Z","title":"Alberto Gr ¨unbaum and Milen Yakimov","venue":null,"work_id":null,"year":2002},"citing_paper":{"arxiv_id":"2607.08788","last_updated":"2026-06-26T22:21:40Z","snapshot_observed_at":"2026-08-09T14:09:25.924672Z","submitted_at":"2026-06-26T22:21:40Z","title":"Matched Generators for the Karhunen--Lo\\`eve Transform: A Double-Commutator Eigenvalue Theory","version":1},"reference_index":32,"source":"pdf_text","source_observed_at":"2026-07-13T07:17:37.978825Z"},"links":{"citing_paper":"/paper/2607.08788"},"observation_digest":"sha256:5ae7511b5a9bdf24d2a082f87f4309a0067cd0a194394e065c1fdbb3fc1bf2ed","observation_id":"66a228c8-8943-42f7-ba8f-cf10f714bc99","resolution":{"observed_at":"2026-07-13T07:17:37.978825Z","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":"2026-07-13T07:17:37.978825Z","title":"Alberto Gr ¨unbaum and Milen Yakimov","venue":null,"work_id":null,"year":2004},"citing_paper":{"arxiv_id":"2607.08788","last_updated":"2026-06-26T22:21:40Z","snapshot_observed_at":"2026-08-09T14:09:25.924672Z","submitted_at":"2026-06-26T22:21:40Z","title":"Matched Generators for the Karhunen--Lo\\`eve Transform: A Double-Commutator Eigenvalue Theory","version":1},"reference_index":33,"source":"pdf_text","source_observed_at":"2026-07-13T07:17:37.978825Z"},"links":{"citing_paper":"/paper/2607.08788"},"observation_digest":"sha256:ab59d147d7e14ff7228acd7bcdc08b8941c8fcc4076049fa85580faf964cdbe2","observation_id":"42323e24-75b4-4a80-9446-dc9a59fd487e","resolution":{"observed_at":"2026-07-13T07:17:37.978825Z","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":"2026-07-13T07:17:37.978825Z","title":"Zur theorie der orthogonalen funktionensysteme.Mathematische Annalen, 69(3):331– 371, 1910","venue":null,"work_id":null,"year":1910},"citing_paper":{"arxiv_id":"2607.08788","last_updated":"2026-06-26T22:21:40Z","snapshot_observed_at":"2026-08-09T14:09:25.924672Z","submitted_at":"2026-06-26T22:21:40Z","title":"Matched Generators for the Karhunen--Lo\\`eve Transform: A Double-Commutator Eigenvalue Theory","version":1},"reference_index":34,"source":"pdf_text","source_observed_at":"2026-07-13T07:17:37.978825Z"},"links":{"citing_paper":"/paper/2607.08788"},"observation_digest":"sha256:29a2404a1b2b368767a9b995e28cb2885626ab59a19721f6ad40066cac9cd7ff","observation_id":"d8c60f6b-ed00-4d51-bdf3-cc98e63a31c5","resolution":{"observed_at":"2026-07-13T07:17:37.978825Z","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":"2026-07-13T07:17:37.978825Z","title":"Commutative rings of difference operators and an adelic flag manifold","venue":null,"work_id":null,"year":2000},"citing_paper":{"arxiv_id":"2607.08788","last_updated":"2026-06-26T22:21:40Z","snapshot_observed_at":"2026-08-09T14:09:25.924672Z","submitted_at":"2026-06-26T22:21:40Z","title":"Matched Generators for the Karhunen--Lo\\`eve Transform: A Double-Commutator Eigenvalue Theory","version":1},"reference_index":35,"source":"pdf_text","source_observed_at":"2026-07-13T07:17:37.978825Z"},"links":{"citing_paper":"/paper/2607.08788"},"observation_digest":"sha256:466203a92583698ce4323c61628f167b2fcd497512d4d56236ae35e4c412603c","observation_id":"c76ff1c8-b4fb-4cc4-b97a-da291210bb52","resolution":{"observed_at":"2026-07-13T07:17:37.978825Z","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":"2026-07-13T07:17:37.978825Z","title":"Moore.Optimization and Dynamical Systems","venue":null,"work_id":null,"year":1994},"citing_paper":{"arxiv_id":"2607.08788","last_updated":"2026-06-26T22:21:40Z","snapshot_observed_at":"2026-08-09T14:09:25.924672Z","submitted_at":"2026-06-26T22:21:40Z","title":"Matched Generators for the Karhunen--Lo\\`eve Transform: A Double-Commutator Eigenvalue Theory","version":1},"reference_index":36,"source":"pdf_text","source_observed_at":"2026-07-13T07:17:37.978825Z"},"links":{"citing_paper":"/paper/2607.08788"},"observation_digest":"sha256:cace75ec03bc9cdda16777d3d6fc91c593a1d341f0d1ea9a14546445babbff47","observation_id":"acd455df-22ab-4e6c-ab3a-1e79c7eccf65","resolution":{"observed_at":"2026-07-13T07:17:37.978825Z","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":"2026-07-13T07:17:37.978825Z","title":"Analysis of a complex of statistical variables into principal components.Journal of Educational Psychology, 24:417–441, 498–520, 1933","venue":null,"work_id":null,"year":1933},"citing_paper":{"arxiv_id":"2607.08788","last_updated":"2026-06-26T22:21:40Z","snapshot_observed_at":"2026-08-09T14:09:25.924672Z","submitted_at":"2026-06-26T22:21:40Z","title":"Matched Generators for the Karhunen--Lo\\`eve Transform: A Double-Commutator Eigenvalue Theory","version":1},"reference_index":37,"source":"pdf_text","source_observed_at":"2026-07-13T07:17:37.978825Z"},"links":{"citing_paper":"/paper/2607.08788"},"observation_digest":"sha256:b4aae0843cdd6904d9e5e7e234fde40fc8e40064be798b121a5ac762e2642104","observation_id":"9c58cb8b-ef52-4b4f-924d-229da45dbff3","resolution":{"observed_at":"2026-07-13T07:17:37.978825Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":{"arxiv_id":"2605.20440","last_updated":"2026-07-29T04:24:05Z","snapshot_observed_at":"2026-08-02T13:46:37.105111Z","submitted_at":"2026-05-19T19:47:40Z","title":"Exact Symmetry as Algebra: A Machine-Verified Tensor Calculus that Enforces Physical Selection Rules","version":2},"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":"2605.20440","snapshot_observed_at":"2026-07-13T07:17:37.978825Z","title":"Clarkson, Misha Kilmer, Haim Avron, and Lior Horesh","venue":null,"work_id":null,"year":2026},"citing_paper":{"arxiv_id":"2607.08788","last_updated":"2026-06-26T22:21:40Z","snapshot_observed_at":"2026-08-09T14:09:25.924672Z","submitted_at":"2026-06-26T22:21:40Z","title":"Matched Generators for the Karhunen--Lo\\`eve Transform: A Double-Commutator Eigenvalue Theory","version":1},"reference_index":38,"source":"pdf_text","source_observed_at":"2026-07-13T07:17:37.978825Z"},"links":{"cited_paper":"/paper/2605.20440","citing_paper":"/paper/2607.08788"},"observation_digest":"sha256:a3388a02ddc6990b00c139308d7a3bdcec2ef31d8bde48fc255ff3d48067558f","observation_id":"324dc4fe-b1c8-40a7-81bf-599afc9fe623","resolution":{"observed_at":"2026-07-13T07:17:37.978825Z","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":"2026-07-13T07:17:37.978825Z","title":"Huang and P","venue":null,"work_id":null,"year":1963},"citing_paper":{"arxiv_id":"2607.08788","last_updated":"2026-06-26T22:21:40Z","snapshot_observed_at":"2026-08-09T14:09:25.924672Z","submitted_at":"2026-06-26T22:21:40Z","title":"Matched Generators for the Karhunen--Lo\\`eve Transform: A Double-Commutator Eigenvalue Theory","version":1},"reference_index":39,"source":"pdf_text","source_observed_at":"2026-07-13T07:17:37.978825Z"},"links":{"citing_paper":"/paper/2607.08788"},"observation_digest":"sha256:be1d713bf4abdcc6c4bfbd62fed70ef182715002cb9f975c04a9a9efb14ccffd","observation_id":"57829dc5-7b89-4402-9780-b6f3ae39fb39","resolution":{"observed_at":"2026-07-13T07:17:37.978825Z","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":"2026-07-13T07:17:37.978825Z","title":null,"venue":null,"work_id":null,"year":1979},"citing_paper":{"arxiv_id":"2607.08788","last_updated":"2026-06-26T22:21:40Z","snapshot_observed_at":"2026-08-09T14:09:25.924672Z","submitted_at":"2026-06-26T22:21:40Z","title":"Matched Generators for the Karhunen--Lo\\`eve Transform: A Double-Commutator Eigenvalue Theory","version":1},"reference_index":40,"source":"pdf_text","source_observed_at":"2026-07-13T07:17:37.978825Z"},"links":{"citing_paper":"/paper/2607.08788"},"observation_digest":"sha256:90ecb2acffbc03f000072bcf6ad366b9347f2f3f1ec04975af561067f0961d2b","observation_id":"fcab7dfb-4371-49c5-a4f2-5f9a1d77ac0d","resolution":{"observed_at":"2026-07-13T07:17:37.978825Z","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":"2026-07-13T07:17:37.978825Z","title":"¨Uber lineare methoden in der wahrscheinlichkeitsrechnung.Annales Academiae Scientiarum Fennicae, Ser","venue":null,"work_id":null,"year":1947},"citing_paper":{"arxiv_id":"2607.08788","last_updated":"2026-06-26T22:21:40Z","snapshot_observed_at":"2026-08-09T14:09:25.924672Z","submitted_at":"2026-06-26T22:21:40Z","title":"Matched Generators for the Karhunen--Lo\\`eve Transform: A Double-Commutator Eigenvalue Theory","version":1},"reference_index":41,"source":"pdf_text","source_observed_at":"2026-07-13T07:17:37.978825Z"},"links":{"citing_paper":"/paper/2607.08788"},"observation_digest":"sha256:0114221be79500b5715990a0988b23a55a2b6bd21da59cd6e02ede99e8af8b1e","observation_id":"55ec2a5f-f71c-49d8-ad44-eb54c0b68936","resolution":{"observed_at":"2026-07-13T07:17:37.978825Z","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":"2026-07-13T07:17:37.978825Z","title":"Lesky, and Ren ´e F","venue":null,"work_id":null,"year":2010},"citing_paper":{"arxiv_id":"2607.08788","last_updated":"2026-06-26T22:21:40Z","snapshot_observed_at":"2026-08-09T14:09:25.924672Z","submitted_at":"2026-06-26T22:21:40Z","title":"Matched Generators for the Karhunen--Lo\\`eve Transform: A Double-Commutator Eigenvalue Theory","version":1},"reference_index":42,"source":"pdf_text","source_observed_at":"2026-07-13T07:17:37.978825Z"},"links":{"citing_paper":"/paper/2607.08788"},"observation_digest":"sha256:64230242ff0208ff33efffb72ca32e55546f898fab6356685f80677d98fda337","observation_id":"e33b4421-d431-4e0a-9803-34fb37eca2bd","resolution":{"observed_at":"2026-07-13T07:17:37.978825Z","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":"2026-07-13T07:17:37.978825Z","title":null,"venue":null,"work_id":null,"year":1943},"citing_paper":{"arxiv_id":"2607.08788","last_updated":"2026-06-26T22:21:40Z","snapshot_observed_at":"2026-08-09T14:09:25.924672Z","submitted_at":"2026-06-26T22:21:40Z","title":"Matched Generators for the Karhunen--Lo\\`eve Transform: A Double-Commutator Eigenvalue Theory","version":1},"reference_index":43,"source":"pdf_text","source_observed_at":"2026-07-13T07:17:37.978825Z"},"links":{"citing_paper":"/paper/2607.08788"},"observation_digest":"sha256:2073c4a6e24295f72d45a48458192233449eb74a4415ab9b12f12fbcdc4b7d4d","observation_id":"b949e167-a382-40ba-b890-d9e7441dcf9c","resolution":{"observed_at":"2026-07-13T07:17:37.978825Z","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":"2026-07-13T07:17:37.978825Z","title":null,"venue":null,"work_id":null,"year":2002},"citing_paper":{"arxiv_id":"2607.08788","last_updated":"2026-06-26T22:21:40Z","snapshot_observed_at":"2026-08-09T14:09:25.924672Z","submitted_at":"2026-06-26T22:21:40Z","title":"Matched Generators for the Karhunen--Lo\\`eve Transform: A Double-Commutator Eigenvalue Theory","version":1},"reference_index":44,"source":"pdf_text","source_observed_at":"2026-07-13T07:17:37.978825Z"},"links":{"citing_paper":"/paper/2607.08788"},"observation_digest":"sha256:387a76b833501877cf8daba6cfc9de71994e4c520954de8a366d7584af93ea2f","observation_id":"48ef271b-b982-4945-b21b-826de56798d9","resolution":{"observed_at":"2026-07-13T07:17:37.978825Z","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":"2026-07-13T07:17:37.978825Z","title":null,"venue":null,"work_id":null,"year":1955},"citing_paper":{"arxiv_id":"2607.08788","last_updated":"2026-06-26T22:21:40Z","snapshot_observed_at":"2026-08-09T14:09:25.924672Z","submitted_at":"2026-06-26T22:21:40Z","title":"Matched Generators for the Karhunen--Lo\\`eve Transform: A Double-Commutator Eigenvalue Theory","version":1},"reference_index":45,"source":"pdf_text","source_observed_at":"2026-07-13T07:17:37.978825Z"},"links":{"citing_paper":"/paper/2607.08788"},"observation_digest":"sha256:8ee62307c85bd8307fb7155e170b52131ab9fc0bf5d0424a75fd6b0dc7aadf44","observation_id":"4ebe9989-a0b3-446a-8da4-c23c23111e38","resolution":{"observed_at":"2026-07-13T07:17:37.978825Z","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":"2026-07-13T07:17:37.978825Z","title":"Landau and Henry O","venue":null,"work_id":null,"year":1961},"citing_paper":{"arxiv_id":"2607.08788","last_updated":"2026-06-26T22:21:40Z","snapshot_observed_at":"2026-08-09T14:09:25.924672Z","submitted_at":"2026-06-26T22:21:40Z","title":"Matched Generators for the Karhunen--Lo\\`eve Transform: A Double-Commutator Eigenvalue Theory","version":1},"reference_index":46,"source":"pdf_text","source_observed_at":"2026-07-13T07:17:37.978825Z"},"links":{"citing_paper":"/paper/2607.08788"},"observation_digest":"sha256:58918af491a6ff4b19eb7e5c05a493257121f11ed5c0094656fb4547e8145466","observation_id":"ff724e55-82f7-432e-8400-26549916a5a5","resolution":{"observed_at":"2026-07-13T07:17:37.978825Z","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":"2026-07-13T07:17:37.978825Z","title":"Springer, 4th edition, 1978","venue":null,"work_id":null,"year":1978},"citing_paper":{"arxiv_id":"2607.08788","last_updated":"2026-06-26T22:21:40Z","snapshot_observed_at":"2026-08-09T14:09:25.924672Z","submitted_at":"2026-06-26T22:21:40Z","title":"Matched Generators for the Karhunen--Lo\\`eve Transform: A Double-Commutator Eigenvalue Theory","version":1},"reference_index":47,"source":"pdf_text","source_observed_at":"2026-07-13T07:17:37.978825Z"},"links":{"citing_paper":"/paper/2607.08788"},"observation_digest":"sha256:69c5eafc76896fa4b47fed0ff57b2e5ec8c4aae5f3826cf3c2cd8dac88d66e4d","observation_id":"a9fea912-2475-49a2-b1cc-0308536a9db5","resolution":{"observed_at":"2026-07-13T07:17:37.978825Z","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":"2026-07-13T07:17:37.978825Z","title":null,"venue":null,"work_id":null,"year":1989},"citing_paper":{"arxiv_id":"2607.08788","last_updated":"2026-06-26T22:21:40Z","snapshot_observed_at":"2026-08-09T14:09:25.924672Z","submitted_at":"2026-06-26T22:21:40Z","title":"Matched Generators for the Karhunen--Lo\\`eve Transform: A Double-Commutator Eigenvalue Theory","version":1},"reference_index":48,"source":"pdf_text","source_observed_at":"2026-07-13T07:17:37.978825Z"},"links":{"citing_paper":"/paper/2607.08788"},"observation_digest":"sha256:f7d9d73e0ce8ce6c3c9b53d992f27642d1ea8550ab2954076033898d71fe7ca0","observation_id":"03fd5956-44ba-42b5-8941-9e95565397ab","resolution":{"observed_at":"2026-07-13T07:17:37.978825Z","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":"2026-07-13T07:17:37.978825Z","title":"Functions of positive and negative type, and their connection with the theory of integral equations.Philosophical Transactions of the Royal Society of London A, 209:415–446, 1909","venue":null,"work_id":null,"year":1909},"citing_paper":{"arxiv_id":"2607.08788","last_updated":"2026-06-26T22:21:40Z","snapshot_observed_at":"2026-08-09T14:09:25.924672Z","submitted_at":"2026-06-26T22:21:40Z","title":"Matched Generators for the Karhunen--Lo\\`eve Transform: A Double-Commutator Eigenvalue Theory","version":1},"reference_index":49,"source":"pdf_text","source_observed_at":"2026-07-13T07:17:37.978825Z"},"links":{"citing_paper":"/paper/2607.08788"},"observation_digest":"sha256:7d6dc370bd4ae35582099d6d864863f00585efc8be64b0bc32437d9278894042","observation_id":"a677271c-3a15-4c87-8b7c-8326dd0626e8","resolution":{"observed_at":"2026-07-13T07:17:37.978825Z","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":"2026-07-13T07:17:37.978825Z","title":"Moody and Roger G","venue":null,"work_id":null,"year":2001},"citing_paper":{"arxiv_id":"2607.08788","last_updated":"2026-06-26T22:21:40Z","snapshot_observed_at":"2026-08-09T14:09:25.924672Z","submitted_at":"2026-06-26T22:21:40Z","title":"Matched Generators for the Karhunen--Lo\\`eve Transform: A Double-Commutator Eigenvalue Theory","version":1},"reference_index":50,"source":"pdf_text","source_observed_at":"2026-07-13T07:17:37.978825Z"},"links":{"citing_paper":"/paper/2607.08788"},"observation_digest":"sha256:bb1e450975cce3d74514ef3bda3f8c720618d3d5526bd0815a1171be01fb82ef","observation_id":"25c0835a-7ef7-4429-b8a7-33f7ea263413","resolution":{"observed_at":"2026-07-13T07:17:37.978825Z","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":"2026-07-13T07:17:37.978825Z","title":"The fractional order fourier transform and its application to quantum mechanics","venue":null,"work_id":null,"year":1980},"citing_paper":{"arxiv_id":"2607.08788","last_updated":"2026-06-26T22:21:40Z","snapshot_observed_at":"2026-08-09T14:09:25.924672Z","submitted_at":"2026-06-26T22:21:40Z","title":"Matched Generators for the Karhunen--Lo\\`eve Transform: A Double-Commutator Eigenvalue Theory","version":1},"reference_index":51,"source":"pdf_text","source_observed_at":"2026-07-13T07:17:37.978825Z"},"links":{"citing_paper":"/paper/2607.08788"},"observation_digest":"sha256:231ccae055f90064f93900bddaf444dca6d0246526c4273c40a0bf0ad732f811","observation_id":"a56cf0fe-22e5-4e04-acaa-a0e103242fc7","resolution":{"observed_at":"2026-07-13T07:17:37.978825Z","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":"2026-07-13T07:17:37.978825Z","title":"On lines and planes of closest fit to systems of points in space.Philosophical Magazine, Series 6, 2(11):559–572, 1901","venue":null,"work_id":null,"year":1901},"citing_paper":{"arxiv_id":"2607.08788","last_updated":"2026-06-26T22:21:40Z","snapshot_observed_at":"2026-08-09T14:09:25.924672Z","submitted_at":"2026-06-26T22:21:40Z","title":"Matched Generators for the Karhunen--Lo\\`eve Transform: A Double-Commutator Eigenvalue Theory","version":1},"reference_index":52,"source":"pdf_text","source_observed_at":"2026-07-13T07:17:37.978825Z"},"links":{"citing_paper":"/paper/2607.08788"},"observation_digest":"sha256:62083a6910ec6969a08260b215991284032457accabe2374663c5e01042909b9","observation_id":"b21fd085-ef5d-4d67-8e94-abadfa03584b","resolution":{"observed_at":"2026-07-13T07:17:37.978825Z","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":"2026-07-13T07:17:37.978825Z","title":"Percival and Andrew T","venue":null,"work_id":null,"year":1993},"citing_paper":{"arxiv_id":"2607.08788","last_updated":"2026-06-26T22:21:40Z","snapshot_observed_at":"2026-08-09T14:09:25.924672Z","submitted_at":"2026-06-26T22:21:40Z","title":"Matched Generators for the Karhunen--Lo\\`eve Transform: A Double-Commutator Eigenvalue Theory","version":1},"reference_index":53,"source":"pdf_text","source_observed_at":"2026-07-13T07:17:37.978825Z"},"links":{"citing_paper":"/paper/2607.08788"},"observation_digest":"sha256:1c9bace945fdfb1fa92ddd470a3152964180fce579af993dd3fc730fe00bbe38","observation_id":"86c1643c-c47b-4dec-bef5-8e10802afbb7","resolution":{"observed_at":"2026-07-13T07:17:37.978825Z","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":"2026-07-13T07:17:37.978825Z","title":"Joint approximate diagonalization of positive definite Hermitian matrices.SIAM Journal on Matrix Analysis and Applications, 22(4):1136–1152, 2001","venue":null,"work_id":null,"year":2001},"citing_paper":{"arxiv_id":"2607.08788","last_updated":"2026-06-26T22:21:40Z","snapshot_observed_at":"2026-08-09T14:09:25.924672Z","submitted_at":"2026-06-26T22:21:40Z","title":"Matched Generators for the Karhunen--Lo\\`eve Transform: A Double-Commutator Eigenvalue Theory","version":1},"reference_index":54,"source":"pdf_text","source_observed_at":"2026-07-13T07:17:37.978825Z"},"links":{"citing_paper":"/paper/2607.08788"},"observation_digest":"sha256:12b566942dc0261d9861ff1831841d5cea49e08db037f7e7ec134ad734043d38","observation_id":"990208ef-94fe-499b-a693-00f4fc40ac8f","resolution":{"observed_at":"2026-07-13T07:17:37.978825Z","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":"2026-07-13T07:17:37.978825Z","title":null,"venue":null,"work_id":null,"year":2008},"citing_paper":{"arxiv_id":"2607.08788","last_updated":"2026-06-26T22:21:40Z","snapshot_observed_at":"2026-08-09T14:09:25.924672Z","submitted_at":"2026-06-26T22:21:40Z","title":"Matched Generators for the Karhunen--Lo\\`eve Transform: A Double-Commutator Eigenvalue Theory","version":1},"reference_index":55,"source":"pdf_text","source_observed_at":"2026-07-13T07:17:37.978825Z"},"links":{"citing_paper":"/paper/2607.08788"},"observation_digest":"sha256:2f88977c936e55ecd3828457ff78a4884281165bd3f11c4ecdd6b41f7b867dc1","observation_id":"6136e430-e832-4b40-a0e8-d730ea528d69","resolution":{"observed_at":"2026-07-13T07:17:37.978825Z","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":"2026-07-13T07:17:37.978825Z","title":null,"venue":null,"work_id":null,"year":2008},"citing_paper":{"arxiv_id":"2607.08788","last_updated":"2026-06-26T22:21:40Z","snapshot_observed_at":"2026-08-09T14:09:25.924672Z","submitted_at":"2026-06-26T22:21:40Z","title":"Matched Generators for the Karhunen--Lo\\`eve Transform: A Double-Commutator Eigenvalue Theory","version":1},"reference_index":56,"source":"pdf_text","source_observed_at":"2026-07-13T07:17:37.978825Z"},"links":{"citing_paper":"/paper/2607.08788"},"observation_digest":"sha256:655f7120ced572a8c01f88d5670a22fb06109452dd7905762e198510f3031f94","observation_id":"0be385ad-33af-4791-aec9-5fe7f4fb1ad0","resolution":{"observed_at":"2026-07-13T07:17:37.978825Z","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":"2026-07-13T07:17:37.978825Z","title":null,"venue":null,"work_id":null,"year":2008},"citing_paper":{"arxiv_id":"2607.08788","last_updated":"2026-06-26T22:21:40Z","snapshot_observed_at":"2026-08-09T14:09:25.924672Z","submitted_at":"2026-06-26T22:21:40Z","title":"Matched Generators for the Karhunen--Lo\\`eve Transform: A Double-Commutator Eigenvalue Theory","version":1},"reference_index":57,"source":"pdf_text","source_observed_at":"2026-07-13T07:17:37.978825Z"},"links":{"citing_paper":"/paper/2607.08788"},"observation_digest":"sha256:a0f312f78dd9e249e0c6a3500e3ae57f7bee9811d12e4d09d31ccf44d3ef8395","observation_id":"f54293b8-d219-4566-9596-6c1d2bb9ea80","resolution":{"observed_at":"2026-07-13T07:17:37.978825Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":{"arxiv_id":"2210.07980","last_updated":"2023-02-07T18:31:27Z","snapshot_observed_at":"2026-07-06T14:05:43.023213Z","submitted_at":"2022-10-14T17:25:36Z","title":"Representation Theory for Geometric Quantum Machine Learning","version":2},"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":"2210.07980","snapshot_observed_at":"2026-07-13T07:17:37.978825Z","title":"Nguyen, Louis Schatzki, Patrick J","venue":null,"work_id":null,"year":2023},"citing_paper":{"arxiv_id":"2607.08788","last_updated":"2026-06-26T22:21:40Z","snapshot_observed_at":"2026-08-09T14:09:25.924672Z","submitted_at":"2026-06-26T22:21:40Z","title":"Matched Generators for the Karhunen--Lo\\`eve Transform: A Double-Commutator Eigenvalue Theory","version":1},"reference_index":58,"source":"pdf_text","source_observed_at":"2026-07-13T07:17:37.978825Z"},"links":{"cited_paper":"/paper/2210.07980","citing_paper":"/paper/2607.08788"},"observation_digest":"sha256:e75de1e40d4ef746b1e5bd5d209f0a956c3b7859cad68fce88bf9f14f8a606ec","observation_id":"6547d939-629b-4992-9512-25ec5e617448","resolution":{"observed_at":"2026-07-13T07:17:37.978825Z","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":"2026-07-13T07:17:37.978825Z","title":null,"venue":null,"work_id":null,"year":2013},"citing_paper":{"arxiv_id":"2607.08788","last_updated":"2026-06-26T22:21:40Z","snapshot_observed_at":"2026-08-09T14:09:25.924672Z","submitted_at":"2026-06-26T22:21:40Z","title":"Matched Generators for the Karhunen--Lo\\`eve Transform: A Double-Commutator Eigenvalue Theory","version":1},"reference_index":59,"source":"pdf_text","source_observed_at":"2026-07-13T07:17:37.978825Z"},"links":{"citing_paper":"/paper/2607.08788"},"observation_digest":"sha256:7924875cf056a4a1d1845d0e322c469891e1db8bf4989cd56991a43663ebfa45","observation_id":"bcd4ff74-c82e-4d46-9e41-c82a7a832c26","resolution":{"observed_at":"2026-07-13T07:17:37.978825Z","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":"2026-07-13T07:17:37.978825Z","title":null,"venue":null,"work_id":null,"year":2010},"citing_paper":{"arxiv_id":"2607.08788","last_updated":"2026-06-26T22:21:40Z","snapshot_observed_at":"2026-08-09T14:09:25.924672Z","submitted_at":"2026-06-26T22:21:40Z","title":"Matched Generators for the Karhunen--Lo\\`eve Transform: A Double-Commutator Eigenvalue Theory","version":1},"reference_index":60,"source":"pdf_text","source_observed_at":"2026-07-13T07:17:37.978825Z"},"links":{"citing_paper":"/paper/2607.08788"},"observation_digest":"sha256:526392979278281f75239c11de2a8216c5f7982eed60d739ed55abaec2fc622e","observation_id":"5212a97c-da16-4163-8e31-7679838701f8","resolution":{"observed_at":"2026-07-13T07:17:37.978825Z","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":"2026-07-13T07:17:37.978825Z","title":"Springer, New York, 1977","venue":null,"work_id":null,"year":1977},"citing_paper":{"arxiv_id":"2607.08788","last_updated":"2026-06-26T22:21:40Z","snapshot_observed_at":"2026-08-09T14:09:25.924672Z","submitted_at":"2026-06-26T22:21:40Z","title":"Matched Generators for the Karhunen--Lo\\`eve Transform: A Double-Commutator Eigenvalue Theory","version":1},"reference_index":61,"source":"pdf_text","source_observed_at":"2026-07-13T07:17:37.978825Z"},"links":{"citing_paper":"/paper/2607.08788"},"observation_digest":"sha256:1619b9703d0c1de9617faac93fb51b518696c4636b423eea2c44e3ffc1df6156","observation_id":"864d6e07-c38e-4881-ab47-32ae726c9b36","resolution":{"observed_at":"2026-07-13T07:17:37.978825Z","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":"2026-07-13T07:17:37.978825Z","title":"Shuman, Sunil K","venue":null,"work_id":null,"year":2013},"citing_paper":{"arxiv_id":"2607.08788","last_updated":"2026-06-26T22:21:40Z","snapshot_observed_at":"2026-08-09T14:09:25.924672Z","submitted_at":"2026-06-26T22:21:40Z","title":"Matched Generators for the Karhunen--Lo\\`eve Transform: A Double-Commutator Eigenvalue Theory","version":1},"reference_index":62,"source":"pdf_text","source_observed_at":"2026-07-13T07:17:37.978825Z"},"links":{"citing_paper":"/paper/2607.08788"},"observation_digest":"sha256:452e22401ff851b6dcce6849a5df4bac4c8e0c78da91251f6db27ef6ab3f123d","observation_id":"5653d9ea-2b5c-4483-bf9c-34a4c2033acc","resolution":{"observed_at":"2026-07-13T07:17:37.978825Z","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":"2026-07-13T07:17:37.978825Z","title":"Prolate spheroidal wave functions, Fourier analysis and uncertainty—V: The discrete case.Bell System Technical Journal, 57(5):1371–1430, 1978","venue":null,"work_id":null,"year":1978},"citing_paper":{"arxiv_id":"2607.08788","last_updated":"2026-06-26T22:21:40Z","snapshot_observed_at":"2026-08-09T14:09:25.924672Z","submitted_at":"2026-06-26T22:21:40Z","title":"Matched Generators for the Karhunen--Lo\\`eve Transform: A Double-Commutator Eigenvalue Theory","version":1},"reference_index":63,"source":"pdf_text","source_observed_at":"2026-07-13T07:17:37.978825Z"},"links":{"citing_paper":"/paper/2607.08788"},"observation_digest":"sha256:085fc1c42778e9e763843e0174e3cb7c6c471e29ca39e52cdd9c20333db7a8eb","observation_id":"12b4b853-5292-43bc-84e8-87ef8f2c4ea5","resolution":{"observed_at":"2026-07-13T07:17:37.978825Z","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":"2026-07-13T07:17:37.978825Z","title":null,"venue":null,"work_id":null,"year":1961},"citing_paper":{"arxiv_id":"2607.08788","last_updated":"2026-06-26T22:21:40Z","snapshot_observed_at":"2026-08-09T14:09:25.924672Z","submitted_at":"2026-06-26T22:21:40Z","title":"Matched Generators for the Karhunen--Lo\\`eve Transform: A Double-Commutator Eigenvalue Theory","version":1},"reference_index":64,"source":"pdf_text","source_observed_at":"2026-07-13T07:17:37.978825Z"},"links":{"citing_paper":"/paper/2607.08788"},"observation_digest":"sha256:b8470972eebd0a1871df7e10e753c5eb2a7bc51c4ae87107b53a9e4a4b3a443d","observation_id":"277fa492-5a52-4395-b4ef-0eb755bf2fcc","resolution":{"observed_at":"2026-07-13T07:17:37.978825Z","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":"2026-07-13T07:17:37.978825Z","title":"Smith and Robert B","venue":null,"work_id":null,"year":2015},"citing_paper":{"arxiv_id":"2607.08788","last_updated":"2026-06-26T22:21:40Z","snapshot_observed_at":"2026-08-09T14:09:25.924672Z","submitted_at":"2026-06-26T22:21:40Z","title":"Matched Generators for the Karhunen--Lo\\`eve Transform: A Double-Commutator Eigenvalue Theory","version":1},"reference_index":65,"source":"pdf_text","source_observed_at":"2026-07-13T07:17:37.978825Z"},"links":{"citing_paper":"/paper/2607.08788"},"observation_digest":"sha256:232b4af8d94472c4481ecbaaf8e8cd3568986aaae0100b8736bcf33ac12cc306","observation_id":"881f955d-5e82-4d21-b8aa-337cf79a1a40","resolution":{"observed_at":"2026-07-13T07:17:37.978825Z","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":"2026-07-13T07:17:37.978825Z","title":"Cambridge University Press, Cambridge, 1999","venue":null,"work_id":null,"year":1999},"citing_paper":{"arxiv_id":"2607.08788","last_updated":"2026-06-26T22:21:40Z","snapshot_observed_at":"2026-08-09T14:09:25.924672Z","submitted_at":"2026-06-26T22:21:40Z","title":"Matched Generators for the Karhunen--Lo\\`eve Transform: A Double-Commutator Eigenvalue Theory","version":1},"reference_index":66,"source":"pdf_text","source_observed_at":"2026-07-13T07:17:37.978825Z"},"links":{"citing_paper":"/paper/2607.08788"},"observation_digest":"sha256:fc2466fe7cabebfc921cc5a8187a4ce9661625ff9ad7898cb8a73545a96d30c0","observation_id":"148f2b55-f1ae-4779-b484-d5d36b2d16ef","resolution":{"observed_at":"2026-07-13T07:17:37.978825Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":{"arxiv_id":"2604.03634","last_updated":"2026-06-29T03:46:46Z","snapshot_observed_at":"2026-07-13T12:43:52.957328Z","submitted_at":"2026-04-04T08:08:34Z","title":"Algebraic Diversity: Group-Theoretic Spectral Estimation from Single Observations","version":6},"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":"2604.03634","snapshot_observed_at":"2026-07-13T07:17:37.978825Z","title":"Thornton","venue":null,"work_id":null,"year":2026},"citing_paper":{"arxiv_id":"2607.08788","last_updated":"2026-06-26T22:21:40Z","snapshot_observed_at":"2026-08-09T14:09:25.924672Z","submitted_at":"2026-06-26T22:21:40Z","title":"Matched Generators for the Karhunen--Lo\\`eve Transform: A Double-Commutator Eigenvalue Theory","version":1},"reference_index":67,"source":"pdf_text","source_observed_at":"2026-07-13T07:17:37.978825Z"},"links":{"cited_paper":"/paper/2604.03634","citing_paper":"/paper/2607.08788"},"observation_digest":"sha256:d32c1451a0424d4a63ff7d20ef3a411e1593c26906e35a51d47f33bb49021904","observation_id":"565fc109-9cca-4bd0-bd87-9cc06e42a012","resolution":{"observed_at":"2026-07-13T07:17:37.978825Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":{"arxiv_id":"2604.19983","last_updated":"2026-06-03T23:12:35Z","snapshot_observed_at":"2026-07-06T23:06:31.110859Z","submitted_at":"2026-04-21T20:47:14Z","title":"Algebraic Diversity: Principles of a Group-Theoretic Approach to Signal Processing","version":5},"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":"2604.19983","snapshot_observed_at":"2026-07-13T07:17:37.978825Z","title":"Thornton","venue":null,"work_id":null,"year":2026},"citing_paper":{"arxiv_id":"2607.08788","last_updated":"2026-06-26T22:21:40Z","snapshot_observed_at":"2026-08-09T14:09:25.924672Z","submitted_at":"2026-06-26T22:21:40Z","title":"Matched Generators for the Karhunen--Lo\\`eve Transform: A Double-Commutator Eigenvalue Theory","version":1},"reference_index":68,"source":"pdf_text","source_observed_at":"2026-07-13T07:17:37.978825Z"},"links":{"cited_paper":"/paper/2604.19983","citing_paper":"/paper/2607.08788"},"observation_digest":"sha256:e67990298f612caabcb2ca8b3bfaaac80a08a88b37377e5a7c5554e8672f59dd","observation_id":"042fd19b-5660-4f39-a5ef-635e9d5b1554","resolution":{"observed_at":"2026-07-13T07:17:37.978825Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":{"arxiv_id":"2605.00848","last_updated":"2026-04-15T06:26:51Z","snapshot_observed_at":"2026-08-02T12:30:32.892094Z","submitted_at":"2026-04-15T06:26:51Z","title":"Continuous Algebraic Diversity: Unifying Spectral, Wavelet, and Time-Frequency Analysis via Lie Group Actions","version":1},"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":"2605.00848","snapshot_observed_at":"2026-07-13T07:17:37.978825Z","title":"Thornton","venue":null,"work_id":null,"year":2026},"citing_paper":{"arxiv_id":"2607.08788","last_updated":"2026-06-26T22:21:40Z","snapshot_observed_at":"2026-08-09T14:09:25.924672Z","submitted_at":"2026-06-26T22:21:40Z","title":"Matched Generators for the Karhunen--Lo\\`eve Transform: A Double-Commutator Eigenvalue Theory","version":1},"reference_index":69,"source":"pdf_text","source_observed_at":"2026-07-13T07:17:37.978825Z"},"links":{"cited_paper":"/paper/2605.00848","citing_paper":"/paper/2607.08788"},"observation_digest":"sha256:0aede7d8faf757d59f39dc873f1e4b5308bbea061586e6d5b2f8e6bc5a00cb61","observation_id":"a28c98eb-2550-4b6e-876a-2145a24aa66c","resolution":{"observed_at":"2026-07-13T07:17:37.978825Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":{"arxiv_id":"2605.00834","last_updated":"2026-05-08T00:02:42Z","snapshot_observed_at":"2026-07-06T23:14:11.211164Z","submitted_at":"2026-04-04T09:02:16Z","title":"Polynomial-Time Optimal Group Selection via the Double-Commutator Eigenvalue Problem","version":2},"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":"2605.00834","snapshot_observed_at":"2026-07-13T07:17:37.978825Z","title":"Thornton","venue":null,"work_id":null,"year":2026},"citing_paper":{"arxiv_id":"2607.08788","last_updated":"2026-06-26T22:21:40Z","snapshot_observed_at":"2026-08-09T14:09:25.924672Z","submitted_at":"2026-06-26T22:21:40Z","title":"Matched Generators for the Karhunen--Lo\\`eve Transform: A Double-Commutator Eigenvalue Theory","version":1},"reference_index":70,"source":"pdf_text","source_observed_at":"2026-07-13T07:17:37.978825Z"},"links":{"cited_paper":"/paper/2605.00834","citing_paper":"/paper/2607.08788"},"observation_digest":"sha256:87c6bcf427489873cd55c855e7bfb869a486850cf0337863dee31a69afdd0758","observation_id":"88a507cf-bf54-4c6b-9d0c-767973a4d491","resolution":{"observed_at":"2026-07-13T07:17:37.978825Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":{"arxiv_id":"2605.11589","last_updated":"2026-06-26T22:32:48Z","snapshot_observed_at":"2026-08-02T01:46:40.454629Z","submitted_at":"2026-05-12T06:15:31Z","title":"Unification of Signal Transform Theory","version":4},"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":"2605.11589","snapshot_observed_at":"2026-07-13T07:17:37.978825Z","title":"Thornton","venue":null,"work_id":null,"year":2026},"citing_paper":{"arxiv_id":"2607.08788","last_updated":"2026-06-26T22:21:40Z","snapshot_observed_at":"2026-08-09T14:09:25.924672Z","submitted_at":"2026-06-26T22:21:40Z","title":"Matched Generators for the Karhunen--Lo\\`eve Transform: A Double-Commutator Eigenvalue Theory","version":1},"reference_index":71,"source":"pdf_text","source_observed_at":"2026-07-13T07:17:37.978825Z"},"links":{"cited_paper":"/paper/2605.11589","citing_paper":"/paper/2607.08788"},"observation_digest":"sha256:7fa07c58fdeb56933f15728f533bde25b0579d96879e37f954370a261f2c3982","observation_id":"4fd6aa6b-88c3-42ab-a8a7-571c99fa6ca1","resolution":{"observed_at":"2026-07-13T07:17:37.978825Z","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":"2026-07-13T07:17:37.978825Z","title":"Tracy and Harold Widom","venue":null,"work_id":null,"year":1994},"citing_paper":{"arxiv_id":"2607.08788","last_updated":"2026-06-26T22:21:40Z","snapshot_observed_at":"2026-08-09T14:09:25.924672Z","submitted_at":"2026-06-26T22:21:40Z","title":"Matched Generators for the Karhunen--Lo\\`eve Transform: A Double-Commutator Eigenvalue Theory","version":1},"reference_index":72,"source":"pdf_text","source_observed_at":"2026-07-13T07:17:37.978825Z"},"links":{"citing_paper":"/paper/2607.08788"},"observation_digest":"sha256:6f6bf452451dbcac04be23be5477f9702a9405f373f22e7cb262f848121e7096","observation_id":"258c22c4-9016-41ec-9eb3-1d2f60229556","resolution":{"observed_at":"2026-07-13T07:17:37.978825Z","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":"2026-07-13T07:17:37.978825Z","title":"Tracy and Harold Widom","venue":null,"work_id":null,"year":1994},"citing_paper":{"arxiv_id":"2607.08788","last_updated":"2026-06-26T22:21:40Z","snapshot_observed_at":"2026-08-09T14:09:25.924672Z","submitted_at":"2026-06-26T22:21:40Z","title":"Matched Generators for the Karhunen--Lo\\`eve Transform: A Double-Commutator Eigenvalue Theory","version":1},"reference_index":73,"source":"pdf_text","source_observed_at":"2026-07-13T07:17:37.978825Z"},"links":{"citing_paper":"/paper/2607.08788"},"observation_digest":"sha256:92b5f0550e221d31672740976cc2aa08d37db5a0bc1a348143b0885b5c23b19a","observation_id":"6bcf1042-40ad-4bac-aeca-121ecb2bbecd","resolution":{"observed_at":"2026-07-13T07:17:37.978825Z","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":"2026-07-13T07:17:37.978825Z","title":"Sch ¨onberger, Juan Nunez-Iglesias, Franc ¸ois Boulogne, Joshua D","venue":null,"work_id":null,"year":2014},"citing_paper":{"arxiv_id":"2607.08788","last_updated":"2026-06-26T22:21:40Z","snapshot_observed_at":"2026-08-09T14:09:25.924672Z","submitted_at":"2026-06-26T22:21:40Z","title":"Matched Generators for the Karhunen--Lo\\`eve Transform: A Double-Commutator Eigenvalue Theory","version":1},"reference_index":74,"source":"pdf_text","source_observed_at":"2026-07-13T07:17:37.978825Z"},"links":{"citing_paper":"/paper/2607.08788"},"observation_digest":"sha256:ea011b830dba8acf5084add5e2b2eb008fe9c035ef6186d062afc6b8c5c44140","observation_id":"e1ca5fc0-df7e-41d2-a13f-03bdf78b49db","resolution":{"observed_at":"2026-07-13T07:17:37.978825Z","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":"2026-07-13T07:17:37.978825Z","title":"Cambridge University Press, 2018","venue":null,"work_id":null,"year":2018},"citing_paper":{"arxiv_id":"2607.08788","last_updated":"2026-06-26T22:21:40Z","snapshot_observed_at":"2026-08-09T14:09:25.924672Z","submitted_at":"2026-06-26T22:21:40Z","title":"Matched Generators for the Karhunen--Lo\\`eve Transform: A Double-Commutator Eigenvalue Theory","version":1},"reference_index":75,"source":"pdf_text","source_observed_at":"2026-07-13T07:17:37.978825Z"},"links":{"citing_paper":"/paper/2607.08788"},"observation_digest":"sha256:c1598e49d8f3a3700f6ef890929128e940ce96f73e57c34e73e679a0b2779316","observation_id":"04ed0c66-1b3a-42ea-a939-3f74d01401c1","resolution":{"observed_at":"2026-07-13T07:17:37.978825Z","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":"2026-07-13T07:17:37.978825Z","title":"A reduction of a graph to a canonical form and an algebra arising during this reduction.Nauchno-Technicheskaya Informatsia, Ser","venue":null,"work_id":null,"year":1968},"citing_paper":{"arxiv_id":"2607.08788","last_updated":"2026-06-26T22:21:40Z","snapshot_observed_at":"2026-08-09T14:09:25.924672Z","submitted_at":"2026-06-26T22:21:40Z","title":"Matched Generators for the Karhunen--Lo\\`eve Transform: A Double-Commutator Eigenvalue Theory","version":1},"reference_index":76,"source":"pdf_text","source_observed_at":"2026-07-13T07:17:37.978825Z"},"links":{"citing_paper":"/paper/2607.08788"},"observation_digest":"sha256:118162c4a2f96863d915d8a55b79e7741abe6b33e1bb9855a3f60dd581823e66","observation_id":"26c5cd3a-cf00-4df4-a7bd-0faa7c03ca20","resolution":{"observed_at":"2026-07-13T07:17:37.978825Z","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":"2026-07-13T07:17:37.978825Z","title":"Williams, Ioannis G","venue":null,"work_id":null,"year":2015},"citing_paper":{"arxiv_id":"2607.08788","last_updated":"2026-06-26T22:21:40Z","snapshot_observed_at":"2026-08-09T14:09:25.924672Z","submitted_at":"2026-06-26T22:21:40Z","title":"Matched Generators for the Karhunen--Lo\\`eve Transform: A Double-Commutator Eigenvalue Theory","version":1},"reference_index":77,"source":"pdf_text","source_observed_at":"2026-07-13T07:17:37.978825Z"},"links":{"citing_paper":"/paper/2607.08788"},"observation_digest":"sha256:3c5c457626a2f67f4fff4105f2249387069352b1aad723fa87b6c415564ed832","observation_id":"430260a5-c8fe-4924-a192-3339a1c7e62e","resolution":{"observed_at":"2026-07-13T07:17:37.978825Z","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":"2026-07-13T07:17:37.978825Z","title":null,"venue":null,"work_id":null,"year":1993},"citing_paper":{"arxiv_id":"2607.08788","last_updated":"2026-06-26T22:21:40Z","snapshot_observed_at":"2026-08-09T14:09:25.924672Z","submitted_at":"2026-06-26T22:21:40Z","title":"Matched Generators for the Karhunen--Lo\\`eve Transform: A Double-Commutator Eigenvalue Theory","version":1},"reference_index":78,"source":"pdf_text","source_observed_at":"2026-07-13T07:17:37.978825Z"},"links":{"citing_paper":"/paper/2607.08788"},"observation_digest":"sha256:6943693b65bc206cc74c8e7708b093fbaf1ccfe56dbc05ae469d1c3f9090136e","observation_id":"5c78db78-17e4-4ee8-b38a-25f45bcab07a","resolution":{"observed_at":"2026-07-13T07:17:37.978825Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}}],"paper":{"arxiv_id":"2607.08788","last_updated":"2026-06-26T22:21:40Z","latest_version":1,"primary_category":"eess.SP","snapshot_observed_at":"2026-08-09T14:09:25.924672Z","submitted_at":"2026-06-26T22:21:40Z","title":"Matched Generators for the Karhunen--Lo\\`eve Transform: A Double-Commutator Eigenvalue Theory"},"reference_resolution":{"displayed":78,"state_counts":{"malformed_identifier":0,"metadata_mismatch":0,"parse_uncertain":0,"unresolved":78,"verified_exact":0,"verified_fuzzy":0},"total_outbound_references":78},"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":"2026-08-09T06:31:02.800959+00:00","source":"crossref"},{"observed_at":"2026-08-09T06:30:57.326959+00:00","source":"retraction_watch"}],"thesis":"As of 10 August 2026, this Paper Citation Record lists 78 of 78 outbound references and 0 inbound Pith citation observations for arXiv:2607.08788."}