{"as_of":"2026-08-16T22:20:00Z","caps":{"database_statements":6,"inbound":100,"outbound":100},"context_digest":"sha256:34470da27f5d74c7cc95f87f86ee0ddc9d63477eb6516a8b65a48e75a9c75691","coverage":[{"denominator":40,"lane":"reference_resolution","note":"Typed states for the displayed outbound observations.","records_observed":40,"source":"paper_references, paper_reference_links","source_observed_at":"2026-08-15T16:39:36.629805Z","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":"2026-08-16T06:30:59.297886+00:00","state":"measured"},{"denominator":2,"lane":"inbound_itemization","note":"Pith citing papers itemized under the disclosed page cap.","records_observed":2,"source":"paper_references, paper_reference_links","source_observed_at":"2026-08-15T16:39:36.446181Z","state":"measured"},{"denominator":1,"lane":"external_citation_measurements","note":"A source-named dated measurement, never combined with another source.","records_observed":0,"source":"pith","source_observed_at":"2026-07-01T16:25:49.575020Z","state":"measured"}],"external_citation_measurements":[],"inbound":[{"citation":{"cited_paper":{"arxiv_id":"2509.04581","last_updated":"2026-06-02T07:23:48Z","snapshot_observed_at":"2026-08-16T22:09:08.791003Z","submitted_at":"2025-09-04T18:03:57Z","title":"Symmetric entanglers for non-invertible SPT phases","version":3},"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":"2509.04581","snapshot_observed_at":"2026-08-15T16:39:36.446181Z","title":"Dirichlet","venue":null,"work_id":null,"year":2025},"citing_paper":{"arxiv_id":"2509.04581","last_updated":"2026-06-02T07:23:48Z","snapshot_observed_at":"2026-08-16T22:09:08.791003Z","submitted_at":"2025-09-04T18:03:57Z","title":"Symmetric entanglers for non-invertible SPT phases","version":3},"reference_index":3,"source":"pdf_text","source_observed_at":"2026-08-15T16:39:36.446181Z"},"links":{"cited_paper":"/paper/2509.04581","citing_paper":"/paper/2509.04581"},"observation_digest":"sha256:48293b758ffeb339d88697410d7ddc268118407c36facbbc2c86ddecfd02598a","observation_id":"2f5fa32b-ca58-442a-8d1f-f813189e6526","resolution":{"observed_at":"2026-08-15T16:39:36.446181Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":{"arxiv_id":"2509.04581","last_updated":"2026-06-02T07:23:48Z","snapshot_observed_at":"2026-08-16T22:09:08.791003Z","submitted_at":"2025-09-04T18:03:57Z","title":"Symmetric entanglers for non-invertible SPT phases","version":3},"cited_work":{"arxiv_id":"2509.04581","doi":null,"metadata_source":"pith","pith_arxiv_id":"2509.04581","snapshot_observed_at":"2026-07-01T16:25:49.575020Z","title":"Symmetric entanglers for non-invertible SPT phases","venue":"cond-mat.str-el","work_id":"ffb3f02e-ad85-4512-b1aa-da48bd53b7d1","year":2025},"citing_paper":{"arxiv_id":"2605.27193","last_updated":"2026-05-26T15:45:35Z","snapshot_observed_at":"2026-08-08T05:59:46.124891Z","submitted_at":"2026-05-26T15:45:35Z","title":"Algebras of order parameters in one-dimensional spin systems","version":1},"reference_index":141,"source":"arxiv_source","source_observed_at":"2026-07-01T16:24:07.943937Z"},"links":{"cited_paper":"/paper/2509.04581","citing_paper":"/paper/2605.27193"},"observation_digest":"sha256:956a0835e9a6bfdbf797b13f0a09f762e8f42b3e4087fbe0aa6d3d8c52c73abc","observation_id":"fdc8b4da-3ead-4067-a648-4ab08251367c","resolution":{"observed_at":"2026-07-01T16:25:49.576378Z","resolver_source":"local_arxiv","status":"verified_exact"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-16T06:30:59.297886+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-16T06:30:59.297886+00:00","source":"crossref"},{"observed_at":"2026-08-16T06:30:54.164669+00:00","source":"retraction_watch"}],"state":"measured"}}],"links":{"evidence":"/evidence","html":"/paper/2509.04581/citation-record","integrity":"/paper/2509.04581/integrity","json":"/paper/2509.04581/citation-record.json","paper":"/paper/2509.04581"},"outbound":[{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":"raw_reference","pith_arxiv_id":null,"snapshot_observed_at":"2026-08-15T16:39:37.314991Z","title":null,"venue":null,"work_id":"5a692240-764f-4f0f-97cb-646606f3f406","year":null},"citing_paper":{"arxiv_id":"2509.04581","last_updated":"2026-06-02T07:23:48Z","snapshot_observed_at":"2026-08-16T22:09:08.791003Z","submitted_at":"2025-09-04T18:03:57Z","title":"Symmetric entanglers for non-invertible SPT phases","version":3},"reference_index":1,"source":"pdf_text","source_observed_at":"2026-08-15T16:39:36.440231Z"},"links":{"citing_paper":"/paper/2509.04581"},"observation_digest":"sha256:7f8cc5c036cea092c140571eecf41461f009346be79d3df3e2499d75db61fa9a","observation_id":"577d3bae-8e1f-4433-953a-907ad52b4882","resolution":{"observed_at":"2026-08-15T16:39:37.319966Z","resolver_source":"raw_fallback","status":"unresolved"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-16T06:30:59.297886+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-16T06:30:59.297886+00:00","source":"crossref"},{"observed_at":"2026-08-16T06:30:54.164669+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":"2026-08-15T16:39:37.286815Z","title":"trivial phase,","venue":null,"work_id":"bf119f1f-f9e5-4700-b3ae-f9c87023072d","year":null},"citing_paper":{"arxiv_id":"2509.04581","last_updated":"2026-06-02T07:23:48Z","snapshot_observed_at":"2026-08-16T22:09:08.791003Z","submitted_at":"2025-09-04T18:03:57Z","title":"Symmetric entanglers for non-invertible SPT phases","version":3},"reference_index":2,"source":"pdf_text","source_observed_at":"2026-08-15T16:39:36.457520Z"},"links":{"citing_paper":"/paper/2509.04581"},"observation_digest":"sha256:624548d02a67a16e0d7e00255d96ea05b13b8ee83706b45fb27af11701928455","observation_id":"f1d1d365-c7b2-4b88-ae93-23ad068fa1a2","resolution":{"observed_at":"2026-08-15T16:39:37.291709Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-16T06:30:59.297886+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-16T06:30:59.297886+00:00","source":"crossref"},{"observed_at":"2026-08-16T06:30:54.164669+00:00","source":"retraction_watch"}],"state":"measured"}},{"citation":{"cited_paper":{"arxiv_id":"2509.04581","last_updated":"2026-06-02T07:23:48Z","snapshot_observed_at":"2026-08-16T22:09:08.791003Z","submitted_at":"2025-09-04T18:03:57Z","title":"Symmetric entanglers for non-invertible SPT phases","version":3},"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":"2509.04581","snapshot_observed_at":"2026-08-15T16:39:36.446181Z","title":"Dirichlet","venue":null,"work_id":null,"year":2025},"citing_paper":{"arxiv_id":"2509.04581","last_updated":"2026-06-02T07:23:48Z","snapshot_observed_at":"2026-08-16T22:09:08.791003Z","submitted_at":"2025-09-04T18:03:57Z","title":"Symmetric entanglers for non-invertible SPT phases","version":3},"reference_index":3,"source":"pdf_text","source_observed_at":"2026-08-15T16:39:36.446181Z"},"links":{"cited_paper":"/paper/2509.04581","citing_paper":"/paper/2509.04581"},"observation_digest":"sha256:48293b758ffeb339d88697410d7ddc268118407c36facbbc2c86ddecfd02598a","observation_id":"2f5fa32b-ca58-442a-8d1f-f813189e6526","resolution":{"observed_at":"2026-08-15T16:39:36.446181Z","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":"2026-08-15T16:39:37.300737Z","title":"overall group element","venue":null,"work_id":"3c1d0be7-909b-44ff-8257-4babcf676537","year":null},"citing_paper":{"arxiv_id":"2509.04581","last_updated":"2026-06-02T07:23:48Z","snapshot_observed_at":"2026-08-16T22:09:08.791003Z","submitted_at":"2025-09-04T18:03:57Z","title":"Symmetric entanglers for non-invertible SPT phases","version":3},"reference_index":4,"source":"pdf_text","source_observed_at":"2026-08-15T16:39:36.452901Z"},"links":{"citing_paper":"/paper/2509.04581"},"observation_digest":"sha256:cc20cb8efab532794b644ab15cc1614fcde1755300b689979f9ef2193d64ce77","observation_id":"8241d5d0-4c6e-4621-8ddf-faf7ec5aee28","resolution":{"observed_at":"2026-08-15T16:39:37.306318Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-16T06:30:59.297886+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-16T06:30:59.297886+00:00","source":"crossref"},{"observed_at":"2026-08-16T06:30:54.164669+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":"2026-08-15T16:39:37.273272Z","title":"internal","venue":null,"work_id":"3b580da0-9e29-4890-bdf9-f0e52a2499c5","year":null},"citing_paper":{"arxiv_id":"2509.04581","last_updated":"2026-06-02T07:23:48Z","snapshot_observed_at":"2026-08-16T22:09:08.791003Z","submitted_at":"2025-09-04T18:03:57Z","title":"Symmetric entanglers for non-invertible SPT phases","version":3},"reference_index":5,"source":"pdf_text","source_observed_at":"2026-08-15T16:39:36.462235Z"},"links":{"citing_paper":"/paper/2509.04581"},"observation_digest":"sha256:53c64c1aaaa58c86662b6a2f7b40dd945e6bbd9db1e0f56826602e3a0876e111","observation_id":"e941a3e7-5d25-4b47-b225-b71101ca705b","resolution":{"observed_at":"2026-08-15T16:39:37.278078Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-16T06:30:59.297886+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-16T06:30:59.297886+00:00","source":"crossref"},{"observed_at":"2026-08-16T06:30:54.164669+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":"2026-08-15T16:39:36.468442Z","title":"Thorngren and Y","venue":null,"work_id":null,"year":2024},"citing_paper":{"arxiv_id":"2509.04581","last_updated":"2026-06-02T07:23:48Z","snapshot_observed_at":"2026-08-16T22:09:08.791003Z","submitted_at":"2025-09-04T18:03:57Z","title":"Symmetric entanglers for non-invertible SPT phases","version":3},"reference_index":6,"source":"pdf_text","source_observed_at":"2026-08-15T16:39:36.468442Z"},"links":{"citing_paper":"/paper/2509.04581"},"observation_digest":"sha256:1782c876c4eda37e30b015938894ba1fe9022aa47b8470a52ccf7ef1f9f9a9c2","observation_id":"758d9f02-02e2-4bc1-ae55-5358ebf22b5d","resolution":{"observed_at":"2026-08-15T16:39:36.468442Z","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":"2026-08-15T16:39:37.248904Z","title":"Zhang and C","venue":null,"work_id":"9e6072cf-53fa-4f1b-95c9-510e5f512a1b","year":2024},"citing_paper":{"arxiv_id":"2509.04581","last_updated":"2026-06-02T07:23:48Z","snapshot_observed_at":"2026-08-16T22:09:08.791003Z","submitted_at":"2025-09-04T18:03:57Z","title":"Symmetric entanglers for non-invertible SPT phases","version":3},"reference_index":7,"source":"pdf_text","source_observed_at":"2026-08-15T16:39:36.473233Z"},"links":{"citing_paper":"/paper/2509.04581"},"observation_digest":"sha256:7fb3dd5e1326a42eb6769e3056672340d34dde43fbf866ccc6bf4b4a08eaef2b","observation_id":"e4eb13f2-c351-4f54-a0da-740aeb7785a3","resolution":{"observed_at":"2026-08-15T16:39:37.253043Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-16T06:30:59.297886+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-16T06:30:59.297886+00:00","source":"crossref"},{"observed_at":"2026-08-16T06:30:54.164669+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":"2026-08-15T16:39:36.478768Z","title":"Bhardwaj, L","venue":null,"work_id":null,"year":2025},"citing_paper":{"arxiv_id":"2509.04581","last_updated":"2026-06-02T07:23:48Z","snapshot_observed_at":"2026-08-16T22:09:08.791003Z","submitted_at":"2025-09-04T18:03:57Z","title":"Symmetric entanglers for non-invertible SPT phases","version":3},"reference_index":8,"source":"pdf_text","source_observed_at":"2026-08-15T16:39:36.478768Z"},"links":{"citing_paper":"/paper/2509.04581"},"observation_digest":"sha256:5b97ad98c712d8a295894a78022666bca1873592736462d514c720abee59e01b","observation_id":"89d1e244-40ac-49f4-9f38-fc5d5314827d","resolution":{"observed_at":"2026-08-15T16:39:36.478768Z","resolver_source":null,"status":"malformed_identifier"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":"10.1007/jhep05(2021)204","metadata_source":"doi_reference","pith_arxiv_id":null,"snapshot_observed_at":"2026-08-15T16:39:36.768443Z","title":"Inamura, Topological field theories and symme- try protected topological phases with fusion category symmetries, Journal of High Energy Physics2021, 10.1007/jhep05(2021)204 (2021)","venue":null,"work_id":"57c27948-1d68-4913-9e4a-f2a2dcd0b2d9","year":2021},"citing_paper":{"arxiv_id":"2509.04581","last_updated":"2026-06-02T07:23:48Z","snapshot_observed_at":"2026-08-16T22:09:08.791003Z","submitted_at":"2025-09-04T18:03:57Z","title":"Symmetric entanglers for non-invertible SPT phases","version":3},"reference_index":9,"source":"pdf_text","source_observed_at":"2026-08-15T16:39:36.484119Z"},"links":{"citing_paper":"/paper/2509.04581"},"observation_digest":"sha256:290549f371beb17be0ed6cade0545daca6893f3f000cb5a3cf0d99a9579f040b","observation_id":"33bb36ad-f584-41b8-939e-2db9cb955d10","resolution":{"observed_at":"2026-08-15T16:39:36.774800Z","resolver_source":"doi","status":"verified_exact"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-16T06:30:59.297886+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-16T06:30:59.297886+00:00","source":"crossref"},{"observed_at":"2026-08-16T06:30:54.164669+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":"2026-08-15T16:39:36.488628Z","title":"Inamura and S","venue":null,"work_id":null,"year":2024},"citing_paper":{"arxiv_id":"2509.04581","last_updated":"2026-06-02T07:23:48Z","snapshot_observed_at":"2026-08-16T22:09:08.791003Z","submitted_at":"2025-09-04T18:03:57Z","title":"Symmetric entanglers for non-invertible SPT phases","version":3},"reference_index":10,"source":"pdf_text","source_observed_at":"2026-08-15T16:39:36.488628Z"},"links":{"citing_paper":"/paper/2509.04581"},"observation_digest":"sha256:38b931ec74a84b2bbfecc760f8c239430377e75fe34fa4d9ddf9e95a5faab4e4","observation_id":"87c6f07e-5e34-433a-bb95-702d5bcdbcfd","resolution":{"observed_at":"2026-08-15T16:39:36.488628Z","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":"2026-08-15T16:39:36.493819Z","title":"Fechisin, N","venue":null,"work_id":null,"year":2025},"citing_paper":{"arxiv_id":"2509.04581","last_updated":"2026-06-02T07:23:48Z","snapshot_observed_at":"2026-08-16T22:09:08.791003Z","submitted_at":"2025-09-04T18:03:57Z","title":"Symmetric entanglers for non-invertible SPT phases","version":3},"reference_index":11,"source":"pdf_text","source_observed_at":"2026-08-15T16:39:36.493819Z"},"links":{"citing_paper":"/paper/2509.04581"},"observation_digest":"sha256:d06d8aa1e0723820aee7a24b017eb4690b8553795e0449061c766569d11d5571","observation_id":"2925e3c7-5639-4878-a39a-0bcde72b231e","resolution":{"observed_at":"2026-08-15T16:39:36.493819Z","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":"2026-08-15T16:39:37.233622Z","title":"Seifnashri and S.-H","venue":null,"work_id":"81845a46-165b-4d5e-bd88-418ccf749e57","year":2024},"citing_paper":{"arxiv_id":"2509.04581","last_updated":"2026-06-02T07:23:48Z","snapshot_observed_at":"2026-08-16T22:09:08.791003Z","submitted_at":"2025-09-04T18:03:57Z","title":"Symmetric entanglers for non-invertible SPT phases","version":3},"reference_index":12,"source":"pdf_text","source_observed_at":"2026-08-15T16:39:36.497979Z"},"links":{"citing_paper":"/paper/2509.04581"},"observation_digest":"sha256:b9ef1fe6229684c314c5110fd465cd04562e934a852c5d07ed3e08ac6278f2a5","observation_id":"aa199b15-1abd-4470-86ed-d3a67cbb08fb","resolution":{"observed_at":"2026-08-15T16:39:37.239172Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-16T06:30:59.297886+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-16T06:30:59.297886+00:00","source":"crossref"},{"observed_at":"2026-08-16T06:30:54.164669+00:00","source":"retraction_watch"}],"state":"measured"}},{"citation":{"cited_paper":{"arxiv_id":"2412.20546","last_updated":"2025-03-31T16:03:21Z","snapshot_observed_at":"2026-08-15T15:48:13.737670Z","submitted_at":"2024-12-29T18:34:34Z","title":"Non-invertible SPTs: an on-site realization of (1+1)d anomaly-free fusion category symmetry","version":2},"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":"2412.20546","snapshot_observed_at":"2026-08-15T16:39:36.501791Z","title":null,"venue":null,"work_id":null,"year":2025},"citing_paper":{"arxiv_id":"2509.04581","last_updated":"2026-06-02T07:23:48Z","snapshot_observed_at":"2026-08-16T22:09:08.791003Z","submitted_at":"2025-09-04T18:03:57Z","title":"Symmetric entanglers for non-invertible SPT phases","version":3},"reference_index":13,"source":"pdf_text","source_observed_at":"2026-08-15T16:39:36.501791Z"},"links":{"cited_paper":"/paper/2412.20546","citing_paper":"/paper/2509.04581"},"observation_digest":"sha256:c96c2195d12b55d4e9fbdc4abeb2e9a58282b2a5c35e03feef4f3306352f8cce","observation_id":"f3452de6-667a-4268-b775-fcccbd5cc6f6","resolution":{"observed_at":"2026-08-15T16:39:36.501791Z","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":"2026-08-15T16:39:36.507112Z","title":"Chen, Z.-C","venue":null,"work_id":null,"year":2013},"citing_paper":{"arxiv_id":"2509.04581","last_updated":"2026-06-02T07:23:48Z","snapshot_observed_at":"2026-08-16T22:09:08.791003Z","submitted_at":"2025-09-04T18:03:57Z","title":"Symmetric entanglers for non-invertible SPT phases","version":3},"reference_index":14,"source":"pdf_text","source_observed_at":"2026-08-15T16:39:36.507112Z"},"links":{"citing_paper":"/paper/2509.04581"},"observation_digest":"sha256:e36c57768a7d7a2451a0e43168c12e80dc3d1083422b4d13f4020be5d15679b5","observation_id":"74424239-c45a-400e-a1e9-44369f9c0231","resolution":{"observed_at":"2026-08-15T16:39:36.507112Z","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":"10.1103/physrevb.98.125101","metadata_source":"doi_reference","pith_arxiv_id":null,"snapshot_observed_at":"2026-08-15T16:39:36.739316Z","title":"Kapustin, A","venue":null,"work_id":"8dcf3c06-3e30-4744-a717-f8a271f8f3ad","year":2018},"citing_paper":{"arxiv_id":"2509.04581","last_updated":"2026-06-02T07:23:48Z","snapshot_observed_at":"2026-08-16T22:09:08.791003Z","submitted_at":"2025-09-04T18:03:57Z","title":"Symmetric entanglers for non-invertible SPT phases","version":3},"reference_index":15,"source":"pdf_text","source_observed_at":"2026-08-15T16:39:36.512438Z"},"links":{"citing_paper":"/paper/2509.04581"},"observation_digest":"sha256:62280ce7ba132cbcd68030293d8f8e6d3baed73475a5a81dcf3b5fa82e1d0468","observation_id":"94b23000-8b3d-4ba4-b14d-a3c8b76149d2","resolution":{"observed_at":"2026-08-15T16:39:36.744675Z","resolver_source":"doi","status":"verified_exact"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-16T06:30:59.297886+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-16T06:30:59.297886+00:00","source":"crossref"},{"observed_at":"2026-08-16T06:30:54.164669+00:00","source":"retraction_watch"}],"state":"measured"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":"10.1103/physrevb.99.035103","metadata_source":"doi_reference","pith_arxiv_id":null,"snapshot_observed_at":"2026-08-15T16:39:36.726251Z","title":"Turzillo and M","venue":null,"work_id":"a4d301ec-ada1-4853-a7ad-3ea67870b566","year":2019},"citing_paper":{"arxiv_id":"2509.04581","last_updated":"2026-06-02T07:23:48Z","snapshot_observed_at":"2026-08-16T22:09:08.791003Z","submitted_at":"2025-09-04T18:03:57Z","title":"Symmetric entanglers for non-invertible SPT phases","version":3},"reference_index":16,"source":"pdf_text","source_observed_at":"2026-08-15T16:39:36.517218Z"},"links":{"citing_paper":"/paper/2509.04581"},"observation_digest":"sha256:d54e3527bee2626f8341233ff7f01fa13f7ce658888cb61ff5dc2c03ab306117","observation_id":"34d024c0-ad92-4440-b77a-d08cf7e98f0d","resolution":{"observed_at":"2026-08-15T16:39:36.730447Z","resolver_source":"doi","status":"verified_exact"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-16T06:30:59.297886+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-16T06:30:59.297886+00:00","source":"crossref"},{"observed_at":"2026-08-16T06:30:54.164669+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":"2026-08-15T16:39:36.523030Z","title":"Bultinck, D","venue":null,"work_id":null,"year":2017},"citing_paper":{"arxiv_id":"2509.04581","last_updated":"2026-06-02T07:23:48Z","snapshot_observed_at":"2026-08-16T22:09:08.791003Z","submitted_at":"2025-09-04T18:03:57Z","title":"Symmetric entanglers for non-invertible SPT phases","version":3},"reference_index":17,"source":"pdf_text","source_observed_at":"2026-08-15T16:39:36.523030Z"},"links":{"citing_paper":"/paper/2509.04581"},"observation_digest":"sha256:af0f4b274b95e733a049cd6fb0be6be75805278a4e70424d728c53976cfb3e55","observation_id":"a13b2439-6e3f-45b8-a655-f66b477ab636","resolution":{"observed_at":"2026-08-15T16:39:36.523030Z","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":"2026-08-15T16:39:37.219103Z","title":"Ren, S.-Q","venue":null,"work_id":"f68f831e-4955-4f17-b2cb-ccb74b129881","year":2024},"citing_paper":{"arxiv_id":"2509.04581","last_updated":"2026-06-02T07:23:48Z","snapshot_observed_at":"2026-08-16T22:09:08.791003Z","submitted_at":"2025-09-04T18:03:57Z","title":"Symmetric entanglers for non-invertible SPT phases","version":3},"reference_index":18,"source":"pdf_text","source_observed_at":"2026-08-15T16:39:36.527394Z"},"links":{"citing_paper":"/paper/2509.04581"},"observation_digest":"sha256:d1e03c1b033499c01c7b994e615a38a06eb52bcc29b679832797d4a53228af39","observation_id":"ef78f3fb-5b3f-470d-93d2-32209deabdb1","resolution":{"observed_at":"2026-08-15T16:39:37.223701Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-16T06:30:59.297886+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-16T06:30:59.297886+00:00","source":"crossref"},{"observed_at":"2026-08-16T06:30:54.164669+00:00","source":"retraction_watch"}],"state":"measured"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":"10.1007/jhep10(2024)034","metadata_source":"doi_reference","pith_arxiv_id":null,"snapshot_observed_at":"2026-08-15T16:39:36.705559Z","title":"Turzillo and M","venue":null,"work_id":"0ebb771c-db52-4d02-83df-f2a1760b1c8f","year":2024},"citing_paper":{"arxiv_id":"2509.04581","last_updated":"2026-06-02T07:23:48Z","snapshot_observed_at":"2026-08-16T22:09:08.791003Z","submitted_at":"2025-09-04T18:03:57Z","title":"Symmetric entanglers for non-invertible SPT phases","version":3},"reference_index":19,"source":"pdf_text","source_observed_at":"2026-08-15T16:39:36.534179Z"},"links":{"citing_paper":"/paper/2509.04581"},"observation_digest":"sha256:e97d798e6070502b74320eb7a3ebde530473c5e9b99465221ee190f67c76d911","observation_id":"d765d721-c010-41b8-8c12-82939e4e5525","resolution":{"observed_at":"2026-08-15T16:39:36.710529Z","resolver_source":"doi","status":"verified_exact"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-16T06:30:59.297886+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-16T06:30:59.297886+00:00","source":"crossref"},{"observed_at":"2026-08-16T06:30:54.164669+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":"2026-08-15T16:39:37.205561Z","title":"Chen, Z.-C","venue":null,"work_id":"ae399059-4c25-4156-b38e-ca5d26596b26","year":2012},"citing_paper":{"arxiv_id":"2509.04581","last_updated":"2026-06-02T07:23:48Z","snapshot_observed_at":"2026-08-16T22:09:08.791003Z","submitted_at":"2025-09-04T18:03:57Z","title":"Symmetric entanglers for non-invertible SPT phases","version":3},"reference_index":20,"source":"pdf_text","source_observed_at":"2026-08-15T16:39:36.538500Z"},"links":{"citing_paper":"/paper/2509.04581"},"observation_digest":"sha256:551917d5a58cbe0a970435da8757ebd04d1564c47933d7aeb8bfebc8d38cb80f","observation_id":"28ce9aee-9b14-4a7a-97ed-7438ce44e6cc","resolution":{"observed_at":"2026-08-15T16:39:37.210220Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-16T06:30:59.297886+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-16T06:30:59.297886+00:00","source":"crossref"},{"observed_at":"2026-08-16T06:30:54.164669+00:00","source":"retraction_watch"}],"state":"measured"}},{"citation":{"cited_paper":{"arxiv_id":"1508.02595","last_updated":"2018-02-23T19:00:29Z","snapshot_observed_at":"2026-08-14T22:36:56.408916Z","submitted_at":"2015-08-11T13:51:47Z","title":"Quantum Information Meets Quantum Matter -- From Quantum Entanglement to Topological Phase in Many-Body Systems","version":4},"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":"1508.02595","snapshot_observed_at":"2026-08-15T16:39:36.542869Z","title":null,"venue":null,"work_id":null,"year":2018},"citing_paper":{"arxiv_id":"2509.04581","last_updated":"2026-06-02T07:23:48Z","snapshot_observed_at":"2026-08-16T22:09:08.791003Z","submitted_at":"2025-09-04T18:03:57Z","title":"Symmetric entanglers for non-invertible SPT phases","version":3},"reference_index":21,"source":"pdf_text","source_observed_at":"2026-08-15T16:39:36.542869Z"},"links":{"cited_paper":"/paper/1508.02595","citing_paper":"/paper/2509.04581"},"observation_digest":"sha256:bf753b1e95f160a989b6edef69d3e46f24df3264f6c76e5537efc4ece5113605","observation_id":"cecaf81e-e97f-46e0-ae34-bc53b3f1c4f8","resolution":{"observed_at":"2026-08-15T16:39:36.542869Z","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":"2026-08-15T16:39:37.190059Z","title":"Tantivasadakarn, R","venue":null,"work_id":"ab149a1e-a66b-42c7-8521-7f354e23014f","year":2023},"citing_paper":{"arxiv_id":"2509.04581","last_updated":"2026-06-02T07:23:48Z","snapshot_observed_at":"2026-08-16T22:09:08.791003Z","submitted_at":"2025-09-04T18:03:57Z","title":"Symmetric entanglers for non-invertible SPT phases","version":3},"reference_index":22,"source":"pdf_text","source_observed_at":"2026-08-15T16:39:36.548393Z"},"links":{"citing_paper":"/paper/2509.04581"},"observation_digest":"sha256:4ae25222a39c48380808171b4ee8bd6f039303232811ed07b1c4c8d769fd17d2","observation_id":"6bd07c0e-7cd9-4012-981a-dff538aa43c0","resolution":{"observed_at":"2026-08-15T16:39:37.195579Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-16T06:30:59.297886+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-16T06:30:59.297886+00:00","source":"crossref"},{"observed_at":"2026-08-16T06:30:54.164669+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":"2026-08-15T16:39:37.174033Z","title":"Zhang, Topological invariants for symmetry-protected topological phase entanglers, Physical Review B107, 235104 (2023)","venue":null,"work_id":"0cf8b590-b949-4516-934c-00e6d264a107","year":2023},"citing_paper":{"arxiv_id":"2509.04581","last_updated":"2026-06-02T07:23:48Z","snapshot_observed_at":"2026-08-16T22:09:08.791003Z","submitted_at":"2025-09-04T18:03:57Z","title":"Symmetric entanglers for non-invertible SPT phases","version":3},"reference_index":23,"source":"pdf_text","source_observed_at":"2026-08-15T16:39:36.554393Z"},"links":{"citing_paper":"/paper/2509.04581"},"observation_digest":"sha256:107a67977869a9f5dbd2b6e6b107555e7a8c323df1fe31b55b17b4c890650a3c","observation_id":"d2e1e25a-ac9c-4a7f-8634-d5ca71aaf2ca","resolution":{"observed_at":"2026-08-15T16:39:37.179219Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-16T06:30:59.297886+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-16T06:30:59.297886+00:00","source":"crossref"},{"observed_at":"2026-08-16T06:30:54.164669+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":"2026-08-15T16:39:36.558769Z","title":"Tantivasadakarn, R","venue":null,"work_id":null,"year":2024},"citing_paper":{"arxiv_id":"2509.04581","last_updated":"2026-06-02T07:23:48Z","snapshot_observed_at":"2026-08-16T22:09:08.791003Z","submitted_at":"2025-09-04T18:03:57Z","title":"Symmetric entanglers for non-invertible SPT phases","version":3},"reference_index":24,"source":"pdf_text","source_observed_at":"2026-08-15T16:39:36.558769Z"},"links":{"citing_paper":"/paper/2509.04581"},"observation_digest":"sha256:de1d70dfc0bad999c39fe64a7d7795fec7129255b4338ebb3c3256b666fecec1","observation_id":"ecbc0fbc-cb43-44b4-ba84-28a71c98741f","resolution":{"observed_at":"2026-08-15T16:39:36.558769Z","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":"2026-08-15T16:39:36.563587Z","title":"Gaiotto and J","venue":null,"work_id":null,"year":2021},"citing_paper":{"arxiv_id":"2509.04581","last_updated":"2026-06-02T07:23:48Z","snapshot_observed_at":"2026-08-16T22:09:08.791003Z","submitted_at":"2025-09-04T18:03:57Z","title":"Symmetric entanglers for non-invertible SPT phases","version":3},"reference_index":25,"source":"pdf_text","source_observed_at":"2026-08-15T16:39:36.563587Z"},"links":{"citing_paper":"/paper/2509.04581"},"observation_digest":"sha256:3e34c182fc17956f28266f44d0fe7b9851dd77ad026e46680be175fd8a0d1703","observation_id":"f60004fd-ac88-4f24-ad12-e0c2c936e742","resolution":{"observed_at":"2026-08-15T16:39:36.563587Z","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":"2026-08-15T16:39:37.160287Z","title":"Moradi, S","venue":null,"work_id":"9f67a9e1-2e80-4c71-8de6-490a74d7a03d","year":2023},"citing_paper":{"arxiv_id":"2509.04581","last_updated":"2026-06-02T07:23:48Z","snapshot_observed_at":"2026-08-16T22:09:08.791003Z","submitted_at":"2025-09-04T18:03:57Z","title":"Symmetric entanglers for non-invertible SPT phases","version":3},"reference_index":26,"source":"pdf_text","source_observed_at":"2026-08-15T16:39:36.568764Z"},"links":{"citing_paper":"/paper/2509.04581"},"observation_digest":"sha256:017488e0f076194b8417251efbbb679c9048a12d57426f59e3a4184b0846ee2d","observation_id":"6dcb22b3-386b-4fdd-af70-ed53558921c7","resolution":{"observed_at":"2026-08-15T16:39:37.165095Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-16T06:30:59.297886+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-16T06:30:59.297886+00:00","source":"crossref"},{"observed_at":"2026-08-16T06:30:54.164669+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":"2026-08-15T16:39:36.572724Z","title":"Bhardwaj and S","venue":null,"work_id":null,"year":2023},"citing_paper":{"arxiv_id":"2509.04581","last_updated":"2026-06-02T07:23:48Z","snapshot_observed_at":"2026-08-16T22:09:08.791003Z","submitted_at":"2025-09-04T18:03:57Z","title":"Symmetric entanglers for non-invertible SPT phases","version":3},"reference_index":27,"source":"pdf_text","source_observed_at":"2026-08-15T16:39:36.572724Z"},"links":{"citing_paper":"/paper/2509.04581"},"observation_digest":"sha256:ac4799e066327b269cc416ae162ee95a8c122e0bafc834ff943b317241f75b42","observation_id":"962289f8-e74f-4ff6-8958-d614073fda7b","resolution":{"observed_at":"2026-08-15T16:39:36.572724Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":{"arxiv_id":"0909.3140","last_updated":"2009-12-29T02:10:39Z","snapshot_observed_at":"2026-08-16T22:05:19.183889Z","submitted_at":"2009-09-17T02:17:33Z","title":"Fusion categories and homotopy theory","version":2},"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":"0909.3140","snapshot_observed_at":"2026-08-15T16:39:36.576160Z","title":"Etingof, D","venue":null,"work_id":null,"year":2009},"citing_paper":{"arxiv_id":"2509.04581","last_updated":"2026-06-02T07:23:48Z","snapshot_observed_at":"2026-08-16T22:09:08.791003Z","submitted_at":"2025-09-04T18:03:57Z","title":"Symmetric entanglers for non-invertible SPT phases","version":3},"reference_index":28,"source":"pdf_text","source_observed_at":"2026-08-15T16:39:36.576160Z"},"links":{"cited_paper":"/paper/0909.3140","citing_paper":"/paper/2509.04581"},"observation_digest":"sha256:26bdcfbaa1ba52b531e8b8b7ca96d2a2262e884f500677e4a23347fa6184cb95","observation_id":"33feffee-79b9-4f06-aef3-8e85dcff0f26","resolution":{"observed_at":"2026-08-15T16:39:36.576160Z","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":"2026-08-15T16:39:36.580967Z","title":"Barkeshli, P","venue":null,"work_id":null,"year":2019},"citing_paper":{"arxiv_id":"2509.04581","last_updated":"2026-06-02T07:23:48Z","snapshot_observed_at":"2026-08-16T22:09:08.791003Z","submitted_at":"2025-09-04T18:03:57Z","title":"Symmetric entanglers for non-invertible SPT phases","version":3},"reference_index":29,"source":"pdf_text","source_observed_at":"2026-08-15T16:39:36.580967Z"},"links":{"citing_paper":"/paper/2509.04581"},"observation_digest":"sha256:51cfc5cca821a705092684da68c7dc5b6b72107cb83ddd6da9b0f85d86667c84","observation_id":"20101cf3-136f-44ac-baea-9dc2d4e5b4f9","resolution":{"observed_at":"2026-08-15T16:39:36.580967Z","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":"2026-08-15T16:39:36.584954Z","title":"Lootens, C","venue":null,"work_id":null,"year":2023},"citing_paper":{"arxiv_id":"2509.04581","last_updated":"2026-06-02T07:23:48Z","snapshot_observed_at":"2026-08-16T22:09:08.791003Z","submitted_at":"2025-09-04T18:03:57Z","title":"Symmetric entanglers for non-invertible SPT phases","version":3},"reference_index":30,"source":"pdf_text","source_observed_at":"2026-08-15T16:39:36.584954Z"},"links":{"citing_paper":"/paper/2509.04581"},"observation_digest":"sha256:e2e47971cd77cb4fcba8b4d7dd90872e63929fc42eb8d73635e39679d2f62650","observation_id":"6c5a59b7-f3bd-4189-99ef-42efb7e42b25","resolution":{"observed_at":"2026-08-15T16:39:36.584954Z","resolver_source":null,"status":"malformed_identifier"},"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":"2026-08-15T16:39:36.590279Z","title":"Lootens, C","venue":null,"work_id":null,"year":2024},"citing_paper":{"arxiv_id":"2509.04581","last_updated":"2026-06-02T07:23:48Z","snapshot_observed_at":"2026-08-16T22:09:08.791003Z","submitted_at":"2025-09-04T18:03:57Z","title":"Symmetric entanglers for non-invertible SPT phases","version":3},"reference_index":31,"source":"pdf_text","source_observed_at":"2026-08-15T16:39:36.590279Z"},"links":{"citing_paper":"/paper/2509.04581"},"observation_digest":"sha256:3d77e241777614b6ba7e77b19af98f1c11f7bcdf39895335c707420f0e552f63","observation_id":"7f051b55-5209-4ec6-9570-90eabb4ab206","resolution":{"observed_at":"2026-08-15T16:39:36.590279Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":{"arxiv_id":"2503.21764","last_updated":"2025-03-28T17:28:20Z","snapshot_observed_at":"2026-08-16T12:46:11.683310Z","submitted_at":"2025-03-27T17:58:33Z","title":"Phases with non-invertible symmetries in 1+1D $\\unicode{x2013}$ symmetry protected topological orders as duality automorphisms","version":2},"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":"2503.21764","snapshot_observed_at":"2026-08-15T16:39:36.594427Z","title":null,"venue":null,"work_id":null,"year":2025},"citing_paper":{"arxiv_id":"2509.04581","last_updated":"2026-06-02T07:23:48Z","snapshot_observed_at":"2026-08-16T22:09:08.791003Z","submitted_at":"2025-09-04T18:03:57Z","title":"Symmetric entanglers for non-invertible SPT phases","version":3},"reference_index":32,"source":"pdf_text","source_observed_at":"2026-08-15T16:39:36.594427Z"},"links":{"cited_paper":"/paper/2503.21764","citing_paper":"/paper/2509.04581"},"observation_digest":"sha256:476048796cc911c05dc51f2e431a279cfcbe6722baeb10f7e86c35b036cfb9ae","observation_id":"b44caac8-2fd2-42ba-9019-36879cc5cc06","resolution":{"observed_at":"2026-08-15T16:39:36.594427Z","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":"2026-08-15T16:39:36.599820Z","title":null,"venue":null,"work_id":null,"year":2023},"citing_paper":{"arxiv_id":"2509.04581","last_updated":"2026-06-02T07:23:48Z","snapshot_observed_at":"2026-08-16T22:09:08.791003Z","submitted_at":"2025-09-04T18:03:57Z","title":"Symmetric entanglers for non-invertible SPT phases","version":3},"reference_index":33,"source":"pdf_text","source_observed_at":"2026-08-15T16:39:36.599820Z"},"links":{"citing_paper":"/paper/2509.04581"},"observation_digest":"sha256:0a59ffbf7889c5f32eabce60f272945eb6b5a4ede3cceab59aabab5e9f30a194","observation_id":"5f40f00a-350e-4eec-8037-6d62a940ee45","resolution":{"observed_at":"2026-08-15T16:39:36.599820Z","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":"2026-08-15T16:39:36.604035Z","title":null,"venue":null,"work_id":null,"year":2024},"citing_paper":{"arxiv_id":"2509.04581","last_updated":"2026-06-02T07:23:48Z","snapshot_observed_at":"2026-08-16T22:09:08.791003Z","submitted_at":"2025-09-04T18:03:57Z","title":"Symmetric entanglers for non-invertible SPT phases","version":3},"reference_index":34,"source":"pdf_text","source_observed_at":"2026-08-15T16:39:36.604035Z"},"links":{"citing_paper":"/paper/2509.04581"},"observation_digest":"sha256:f8455ed654e8ea2c466edbfd209ba046c89be2516c1b9bd51daf9fdcd341451a","observation_id":"dfe36873-60bb-4a02-b297-4854da254910","resolution":{"observed_at":"2026-08-15T16:39:36.604035Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":{"arxiv_id":"2405.04619","last_updated":"2024-05-07T19:05:07Z","snapshot_observed_at":"2026-08-16T13:54:33.923439Z","submitted_at":"2024-05-07T19:05:07Z","title":"Non-anomalous non-invertible symmetries in 1+1D from gapped boundaries of SymTFTs","version":1},"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":"2405.04619","snapshot_observed_at":"2026-08-15T16:39:36.608522Z","title":"Putrov and R","venue":null,"work_id":null,"year":2024},"citing_paper":{"arxiv_id":"2509.04581","last_updated":"2026-06-02T07:23:48Z","snapshot_observed_at":"2026-08-16T22:09:08.791003Z","submitted_at":"2025-09-04T18:03:57Z","title":"Symmetric entanglers for non-invertible SPT phases","version":3},"reference_index":35,"source":"pdf_text","source_observed_at":"2026-08-15T16:39:36.608522Z"},"links":{"cited_paper":"/paper/2405.04619","citing_paper":"/paper/2509.04581"},"observation_digest":"sha256:4b28a8da279a789490546b375249ac9db4f4d5f5f05490b2548aad9db6f6c0d8","observation_id":"2dbcecdc-c3a9-4e8c-b500-6054623dd0e7","resolution":{"observed_at":"2026-08-15T16:39:36.608522Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":{"arxiv_id":"2506.11288","last_updated":"2025-06-12T20:46:36Z","snapshot_observed_at":"2026-08-07T11:58:55.653710Z","submitted_at":"2025-06-12T20:46:36Z","title":"Bulk Excitations of Invertible Phases","version":1},"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":"2506.11288","snapshot_observed_at":"2026-08-15T16:39:36.613187Z","title":null,"venue":null,"work_id":null,"year":2025},"citing_paper":{"arxiv_id":"2509.04581","last_updated":"2026-06-02T07:23:48Z","snapshot_observed_at":"2026-08-16T22:09:08.791003Z","submitted_at":"2025-09-04T18:03:57Z","title":"Symmetric entanglers for non-invertible SPT phases","version":3},"reference_index":36,"source":"pdf_text","source_observed_at":"2026-08-15T16:39:36.613187Z"},"links":{"cited_paper":"/paper/2506.11288","citing_paper":"/paper/2509.04581"},"observation_digest":"sha256:d02678fc039c306a377375dc1e6e9457480efc049d500a58db1c7ca5a729b58c","observation_id":"795ce8ab-8ca7-4f66-8e6f-3ee7fc460eed","resolution":{"observed_at":"2026-08-15T16:39:36.613187Z","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":"2026-08-15T16:39:37.145824Z","title":"Etingof, S","venue":null,"work_id":"badd17ed-fa8f-4a18-953a-86a9df093226","year":2016},"citing_paper":{"arxiv_id":"2509.04581","last_updated":"2026-06-02T07:23:48Z","snapshot_observed_at":"2026-08-16T22:09:08.791003Z","submitted_at":"2025-09-04T18:03:57Z","title":"Symmetric entanglers for non-invertible SPT phases","version":3},"reference_index":37,"source":"pdf_text","source_observed_at":"2026-08-15T16:39:36.617419Z"},"links":{"citing_paper":"/paper/2509.04581"},"observation_digest":"sha256:ec4f18e10b25b3cff0d954974623a4cddbe28c7e8611ead948331694d1bc62ba","observation_id":"a8f03d6a-22e8-48b6-82f8-7bb41652d7a3","resolution":{"observed_at":"2026-08-15T16:39:37.151303Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-16T06:30:59.297886+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-16T06:30:59.297886+00:00","source":"crossref"},{"observed_at":"2026-08-16T06:30:54.164669+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":"2026-08-15T16:39:36.621849Z","title":"Garre-Rubio, L","venue":null,"work_id":null,"year":2023},"citing_paper":{"arxiv_id":"2509.04581","last_updated":"2026-06-02T07:23:48Z","snapshot_observed_at":"2026-08-16T22:09:08.791003Z","submitted_at":"2025-09-04T18:03:57Z","title":"Symmetric entanglers for non-invertible SPT phases","version":3},"reference_index":38,"source":"pdf_text","source_observed_at":"2026-08-15T16:39:36.621849Z"},"links":{"citing_paper":"/paper/2509.04581"},"observation_digest":"sha256:591896989ade0de8009a9004ef6fb4e0285969a47f1a79570b7436ab34ea0239","observation_id":"c225e010-bc1f-4f63-98c2-e4de6b104b23","resolution":{"observed_at":"2026-08-15T16:39:36.621849Z","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":"2026-08-15T16:39:36.625895Z","title":null,"venue":null,"work_id":null,"year":2025},"citing_paper":{"arxiv_id":"2509.04581","last_updated":"2026-06-02T07:23:48Z","snapshot_observed_at":"2026-08-16T22:09:08.791003Z","submitted_at":"2025-09-04T18:03:57Z","title":"Symmetric entanglers for non-invertible SPT phases","version":3},"reference_index":39,"source":"pdf_text","source_observed_at":"2026-08-15T16:39:36.625895Z"},"links":{"citing_paper":"/paper/2509.04581"},"observation_digest":"sha256:0ff787382f167d04815c2ef62eb07acf6ee2d7872b16e6902a33f971a5025ef7","observation_id":"6139c839-4215-4441-bcbe-73243d1fbad5","resolution":{"observed_at":"2026-08-15T16:39:36.625895Z","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":"2026-08-15T16:39:36.629805Z","title":"Ignacio Cirac, D","venue":null,"work_id":null,"year":2017},"citing_paper":{"arxiv_id":"2509.04581","last_updated":"2026-06-02T07:23:48Z","snapshot_observed_at":"2026-08-16T22:09:08.791003Z","submitted_at":"2025-09-04T18:03:57Z","title":"Symmetric entanglers for non-invertible SPT phases","version":3},"reference_index":40,"source":"pdf_text","source_observed_at":"2026-08-15T16:39:36.629805Z"},"links":{"citing_paper":"/paper/2509.04581"},"observation_digest":"sha256:32b1227f382a1fbd19a735dcfb3b2c86d697ebf57aed5d66fbd89be52a788ad8","observation_id":"69bde5be-8e2b-4616-84fa-975ec7cd24a1","resolution":{"observed_at":"2026-08-15T16:39:36.629805Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}}],"paper":{"arxiv_id":"2509.04581","last_updated":"2026-06-02T07:23:48Z","latest_version":3,"primary_category":"cond-mat.str-el","snapshot_observed_at":"2026-08-16T22:09:08.791003Z","submitted_at":"2025-09-04T18:03:57Z","title":"Symmetric entanglers for non-invertible SPT phases"},"reference_resolution":{"displayed":40,"state_counts":{"malformed_identifier":2,"metadata_mismatch":0,"parse_uncertain":0,"unresolved":23,"verified_exact":4,"verified_fuzzy":11},"total_outbound_references":40},"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":"2026-08-16T06:30:59.297886+00:00","source":"crossref"},{"observed_at":"2026-08-16T06:30:54.164669+00:00","source":"retraction_watch"}],"thesis":"As of 16 August 2026, this Paper Citation Record lists 40 of 40 outbound references and 2 inbound Pith citation observations for arXiv:2509.04581."}