{"as_of":"2026-08-14T10:07:00Z","caps":{"database_statements":6,"inbound":100,"outbound":100},"context_digest":"sha256:a0fb364f3a36f372e5c4590acab24846b3396a4f6e1c51845e66402107bc6c0d","coverage":[{"denominator":38,"lane":"reference_resolution","note":"Typed states for the displayed outbound observations.","records_observed":38,"source":"paper_references, paper_reference_links","source_observed_at":"2026-08-04T16:41:06.488506Z","state":"measured"},{"denominator":38,"lane":"standing_notices","note":"One-hop event checks from named stored sources.","records_observed":38,"source":"scholarly_work_events, retraction_status_cache","source_observed_at":"2026-08-14T06:32:32.682623+00:00","state":"measured"},{"denominator":0,"lane":"inbound_itemization","note":"Pith citing papers itemized under the disclosed page cap.","records_observed":0,"source":"paper_references, paper_reference_links","source_observed_at":null,"state":"measured"},{"denominator":1,"lane":"external_citation_measurements","note":"A source-named dated measurement, never combined with another source.","records_observed":0,"source":"cited_works","source_observed_at":null,"state":"measured"}],"external_citation_measurements":[],"inbound":[],"links":{"evidence":"/evidence","html":"/paper/2509.13228/citation-record","integrity":"/paper/2509.13228/integrity","json":"/paper/2509.13228/citation-record.json","paper":"/paper/2509.13228"},"outbound":[{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":null,"snapshot_observed_at":"2026-08-04T16:41:04.632789Z","title":"``Neumann Domains on Graphs and Manifolds\"","venue":null,"work_id":null,"year":2020},"citing_paper":{"arxiv_id":"2509.13228","last_updated":"2026-07-19T21:19:46Z","snapshot_observed_at":"2026-08-08T04:26:07.234835Z","submitted_at":"2025-09-16T16:37:31Z","title":"On Courant-type bounds and spectral partitioning via Neumann domains on quantum graphs","version":2},"reference_index":1,"source":"arxiv_source","source_observed_at":"2026-08-04T16:41:04.632789Z"},"links":{"citing_paper":"/paper/2509.13228"},"observation_digest":"sha256:75cf14eac4d60499b71f802de457d93484c72731017da35ce3b4d4fb63de4c2f","observation_id":"b6ccc658-c0dd-47f5-a07d-4d8ab602049f","resolution":{"observed_at":"2026-08-04T16:41:04.632789Z","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":"2026-08-04T16:41:04.703517Z","title":null,"venue":null,"work_id":null,"year":2018},"citing_paper":{"arxiv_id":"2509.13228","last_updated":"2026-07-19T21:19:46Z","snapshot_observed_at":"2026-08-08T04:26:07.234835Z","submitted_at":"2025-09-16T16:37:31Z","title":"On Courant-type bounds and spectral partitioning via Neumann domains on quantum graphs","version":2},"reference_index":2,"source":"arxiv_source","source_observed_at":"2026-08-04T16:41:04.703517Z"},"links":{"citing_paper":"/paper/2509.13228"},"observation_digest":"sha256:3073759c562b3d2f071168b2d8678c1a89d710cc1b658fdf2fee7f5bfaa9313f","observation_id":"8e668c87-2f57-4d99-bc10-55d8f016a501","resolution":{"observed_at":"2026-08-04T16:41:04.703517Z","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":"2026-08-04T16:41:04.785927Z","title":"Alon and R","venue":null,"work_id":null,"year":2021},"citing_paper":{"arxiv_id":"2509.13228","last_updated":"2026-07-19T21:19:46Z","snapshot_observed_at":"2026-08-08T04:26:07.234835Z","submitted_at":"2025-09-16T16:37:31Z","title":"On Courant-type bounds and spectral partitioning via Neumann domains on quantum graphs","version":2},"reference_index":3,"source":"arxiv_source","source_observed_at":"2026-08-04T16:41:04.785927Z"},"links":{"citing_paper":"/paper/2509.13228"},"observation_digest":"sha256:a99d7ecabed5f0932f1def3ebb6f3fc6d2e18a2593427d9d11d658b6171a8b83","observation_id":"b57b2be7-0ce9-41a5-8eab-17293ebe53c4","resolution":{"observed_at":"2026-08-04T16:41:04.785927Z","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":"2026-08-04T16:41:04.869139Z","title":null,"venue":null,"work_id":null,"year":2012},"citing_paper":{"arxiv_id":"2509.13228","last_updated":"2026-07-19T21:19:46Z","snapshot_observed_at":"2026-08-08T04:26:07.234835Z","submitted_at":"2025-09-16T16:37:31Z","title":"On Courant-type bounds and spectral partitioning via Neumann domains on quantum graphs","version":2},"reference_index":4,"source":"arxiv_source","source_observed_at":"2026-08-04T16:41:04.869139Z"},"links":{"citing_paper":"/paper/2509.13228"},"observation_digest":"sha256:cc284c23a213fe0c1ca3171ab44017faac11e01fc5247f362a9fe3c2dc2b38f6","observation_id":"fa05699f-52fd-40e8-8795-bbfcb0c08820","resolution":{"observed_at":"2026-08-04T16:41:04.869139Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":{"arxiv_id":"1609.07471","last_updated":"2016-09-23T19:34:05Z","snapshot_observed_at":"2026-07-06T05:12:02.236230Z","submitted_at":"2016-09-23T19:34:05Z","title":"Eigenvalue estimates on quantum graphs","version":1},"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":"1609.07471","snapshot_observed_at":"2026-08-04T16:41:04.958435Z","title":null,"venue":null,"work_id":null,"year":2016},"citing_paper":{"arxiv_id":"2509.13228","last_updated":"2026-07-19T21:19:46Z","snapshot_observed_at":"2026-08-08T04:26:07.234835Z","submitted_at":"2025-09-16T16:37:31Z","title":"On Courant-type bounds and spectral partitioning via Neumann domains on quantum graphs","version":2},"reference_index":5,"source":"arxiv_source","source_observed_at":"2026-08-04T16:41:04.958435Z"},"links":{"cited_paper":"/paper/1609.07471","citing_paper":"/paper/2509.13228"},"observation_digest":"sha256:48c73678645385cecf8900547a56adf8aef8f1d36f92457b8b387abba3784486","observation_id":"991cbf6d-57d2-4643-be82-8a5f97d96b69","resolution":{"observed_at":"2026-08-04T16:41:04.958435Z","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":"2026-08-04T16:41:05.009669Z","title":null,"venue":null,"work_id":null,"year":2017},"citing_paper":{"arxiv_id":"2509.13228","last_updated":"2026-07-19T21:19:46Z","snapshot_observed_at":"2026-08-08T04:26:07.234835Z","submitted_at":"2025-09-16T16:37:31Z","title":"On Courant-type bounds and spectral partitioning via Neumann domains on quantum graphs","version":2},"reference_index":6,"source":"arxiv_source","source_observed_at":"2026-08-04T16:41:05.009669Z"},"links":{"citing_paper":"/paper/2509.13228"},"observation_digest":"sha256:a83d10586a2a2af5832468be69cc72cbc1fa3c7084dce5a09a6e225cc331d0c8","observation_id":"6603ba14-fd67-4643-9932-39b563f99972","resolution":{"observed_at":"2026-08-04T16:41:05.009669Z","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":"2026-08-04T16:41:05.065174Z","title":null,"venue":null,"work_id":null,"year":2012},"citing_paper":{"arxiv_id":"2509.13228","last_updated":"2026-07-19T21:19:46Z","snapshot_observed_at":"2026-08-08T04:26:07.234835Z","submitted_at":"2025-09-16T16:37:31Z","title":"On Courant-type bounds and spectral partitioning via Neumann domains on quantum graphs","version":2},"reference_index":7,"source":"arxiv_source","source_observed_at":"2026-08-04T16:41:05.065174Z"},"links":{"citing_paper":"/paper/2509.13228"},"observation_digest":"sha256:0fcee86493f6d3bb1296ef0a0fbd6b95e311bbfb75ffadde53ad3bf712eac8e3","observation_id":"2057281e-3340-4f2d-b6d5-55121576ca58","resolution":{"observed_at":"2026-08-04T16:41:05.065174Z","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":"2026-08-04T16:41:05.109017Z","title":null,"venue":null,"work_id":null,"year":2015},"citing_paper":{"arxiv_id":"2509.13228","last_updated":"2026-07-19T21:19:46Z","snapshot_observed_at":"2026-08-08T04:26:07.234835Z","submitted_at":"2025-09-16T16:37:31Z","title":"On Courant-type bounds and spectral partitioning via Neumann domains on quantum graphs","version":2},"reference_index":8,"source":"arxiv_source","source_observed_at":"2026-08-04T16:41:05.109017Z"},"links":{"citing_paper":"/paper/2509.13228"},"observation_digest":"sha256:ba1966418a33f246d17435f0ab7bf22194ff5a427510a2bca7c51c0dcc289f77","observation_id":"ec242423-6285-4595-9ee4-fa90a72947a8","resolution":{"observed_at":"2026-08-04T16:41:05.109017Z","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":"2026-08-04T16:41:05.158782Z","title":null,"venue":null,"work_id":null,"year":2018},"citing_paper":{"arxiv_id":"2509.13228","last_updated":"2026-07-19T21:19:46Z","snapshot_observed_at":"2026-08-08T04:26:07.234835Z","submitted_at":"2025-09-16T16:37:31Z","title":"On Courant-type bounds and spectral partitioning via Neumann domains on quantum graphs","version":2},"reference_index":9,"source":"arxiv_source","source_observed_at":"2026-08-04T16:41:05.158782Z"},"links":{"citing_paper":"/paper/2509.13228"},"observation_digest":"sha256:e1c67d75b60e87860fd62aa8c478d8640a2097f173db53fa89de94c90de41edc","observation_id":"1fc352c2-61b5-433f-99f2-dab42f121848","resolution":{"observed_at":"2026-08-04T16:41:05.158782Z","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":"2026-08-04T16:41:05.216855Z","title":null,"venue":null,"work_id":null,"year":2013},"citing_paper":{"arxiv_id":"2509.13228","last_updated":"2026-07-19T21:19:46Z","snapshot_observed_at":"2026-08-08T04:26:07.234835Z","submitted_at":"2025-09-16T16:37:31Z","title":"On Courant-type bounds and spectral partitioning via Neumann domains on quantum graphs","version":2},"reference_index":10,"source":"arxiv_source","source_observed_at":"2026-08-04T16:41:05.216855Z"},"links":{"citing_paper":"/paper/2509.13228"},"observation_digest":"sha256:9056b9489e67253fd13f56db70a4f8d0d4a2fbb7f3fedaf71c1e48788c9e7894","observation_id":"c69dcdc4-c22d-4d86-a046-04b46ac5f65c","resolution":{"observed_at":"2026-08-04T16:41:05.216855Z","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":"2026-08-04T16:41:05.271099Z","title":null,"venue":null,"work_id":null,"year":2008},"citing_paper":{"arxiv_id":"2509.13228","last_updated":"2026-07-19T21:19:46Z","snapshot_observed_at":"2026-08-08T04:26:07.234835Z","submitted_at":"2025-09-16T16:37:31Z","title":"On Courant-type bounds and spectral partitioning via Neumann domains on quantum graphs","version":2},"reference_index":11,"source":"arxiv_source","source_observed_at":"2026-08-04T16:41:05.271099Z"},"links":{"citing_paper":"/paper/2509.13228"},"observation_digest":"sha256:72480813686280446d51081f4ad2f97ea76500eb7c7318786f07651d61aacb80","observation_id":"88cb1f2c-1e54-4054-af07-c3c19c14a18f","resolution":{"observed_at":"2026-08-04T16:41:05.271099Z","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":"2026-08-04T16:41:05.354923Z","title":"B., Kurasov, P., Mugnolo, D., Edge connectivity and the spectral gap of combinatorial and quantum graphs, J.\\ Phys.\\ A: Math.\\ Theor.\\ 50 (2017), 365201, 29pp","venue":null,"work_id":null,"year":2017},"citing_paper":{"arxiv_id":"2509.13228","last_updated":"2026-07-19T21:19:46Z","snapshot_observed_at":"2026-08-08T04:26:07.234835Z","submitted_at":"2025-09-16T16:37:31Z","title":"On Courant-type bounds and spectral partitioning via Neumann domains on quantum graphs","version":2},"reference_index":12,"source":"arxiv_source","source_observed_at":"2026-08-04T16:41:05.354923Z"},"links":{"citing_paper":"/paper/2509.13228"},"observation_digest":"sha256:f41ad25b8c3fcab8b46ccb5c74bd9ea498750187c097395d4c678f2261f11ec2","observation_id":"ad146daf-0197-4014-9e37-6c7e330cff82","resolution":{"observed_at":"2026-08-04T16:41:05.354923Z","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":"2026-08-04T16:41:05.441253Z","title":"B., Kurasov, P., Mugnolo, D., Surgery principles for the spectral analysis of quantum graphs, Trans.\\ Amer.\\ Math.\\ Soc.\\ 372 (2019), 5153--5197","venue":null,"work_id":null,"year":2019},"citing_paper":{"arxiv_id":"2509.13228","last_updated":"2026-07-19T21:19:46Z","snapshot_observed_at":"2026-08-08T04:26:07.234835Z","submitted_at":"2025-09-16T16:37:31Z","title":"On Courant-type bounds and spectral partitioning via Neumann domains on quantum graphs","version":2},"reference_index":13,"source":"arxiv_source","source_observed_at":"2026-08-04T16:41:05.441253Z"},"links":{"citing_paper":"/paper/2509.13228"},"observation_digest":"sha256:0f59e54371967a0541087323c68abfa75cbd29e3aaac52fc1e67b88be60a297d","observation_id":"eb782403-f9ff-4e12-b3a6-20483cd84774","resolution":{"observed_at":"2026-08-04T16:41:05.441253Z","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":"2026-08-04T16:41:05.519018Z","title":"(ed.), Shape optimization and spectral theory, De Gruyter Open, Warsaw-Berlin, 2017","venue":null,"work_id":null,"year":2016},"citing_paper":{"arxiv_id":"2509.13228","last_updated":"2026-07-19T21:19:46Z","snapshot_observed_at":"2026-08-08T04:26:07.234835Z","submitted_at":"2025-09-16T16:37:31Z","title":"On Courant-type bounds and spectral partitioning via Neumann domains on quantum graphs","version":2},"reference_index":14,"source":"arxiv_source","source_observed_at":"2026-08-04T16:41:05.519018Z"},"links":{"citing_paper":"/paper/2509.13228"},"observation_digest":"sha256:c8d4cc5d30cc048c6bad3b0fb4c7704dccaaf1ba9c5af1a18179609b7b4d0644","observation_id":"b6c3aac0-e5ce-45c5-82ef-8b11204687aa","resolution":{"observed_at":"2026-08-04T16:41:05.519018Z","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":"2026-08-04T16:41:05.584309Z","title":null,"venue":null,"work_id":null,"year":2017},"citing_paper":{"arxiv_id":"2509.13228","last_updated":"2026-07-19T21:19:46Z","snapshot_observed_at":"2026-08-08T04:26:07.234835Z","submitted_at":"2025-09-16T16:37:31Z","title":"On Courant-type bounds and spectral partitioning via Neumann domains on quantum graphs","version":2},"reference_index":15,"source":"arxiv_source","source_observed_at":"2026-08-04T16:41:05.584309Z"},"links":{"citing_paper":"/paper/2509.13228"},"observation_digest":"sha256:71055663aa2963048f7ee56e2cf76260884eb69da573fb8a70656f6e5561a9e2","observation_id":"a94aabd9-639f-4c10-a4d4-59d16c062462","resolution":{"observed_at":"2026-08-04T16:41:05.584309Z","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":"2026-08-04T16:41:05.603502Z","title":null,"venue":null,"work_id":null,"year":1923},"citing_paper":{"arxiv_id":"2509.13228","last_updated":"2026-07-19T21:19:46Z","snapshot_observed_at":"2026-08-08T04:26:07.234835Z","submitted_at":"2025-09-16T16:37:31Z","title":"On Courant-type bounds and spectral partitioning via Neumann domains on quantum graphs","version":2},"reference_index":16,"source":"arxiv_source","source_observed_at":"2026-08-04T16:41:05.603502Z"},"links":{"citing_paper":"/paper/2509.13228"},"observation_digest":"sha256:1674216b83b04cc09998f14e7e07388b4cf428d87cc6f53c5665fc42312dfe56","observation_id":"574f641f-4265-4a25-bb41-3b5fe03d37dc","resolution":{"observed_at":"2026-08-04T16:41:05.603502Z","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":"2026-08-04T16:41:05.688631Z","title":null,"venue":null,"work_id":null,"year":2005},"citing_paper":{"arxiv_id":"2509.13228","last_updated":"2026-07-19T21:19:46Z","snapshot_observed_at":"2026-08-08T04:26:07.234835Z","submitted_at":"2025-09-16T16:37:31Z","title":"On Courant-type bounds and spectral partitioning via Neumann domains on quantum graphs","version":2},"reference_index":17,"source":"arxiv_source","source_observed_at":"2026-08-04T16:41:05.688631Z"},"links":{"citing_paper":"/paper/2509.13228"},"observation_digest":"sha256:38f98a8fd179598bb4621d37d74993e4740786f65a31bd9e3c033df321820245","observation_id":"5ae0cdb9-2045-4867-b449-9d384de42033","resolution":{"observed_at":"2026-08-04T16:41:05.688631Z","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":"2026-08-04T16:41:05.857848Z","title":null,"venue":null,"work_id":null,"year":2005},"citing_paper":{"arxiv_id":"2509.13228","last_updated":"2026-07-19T21:19:46Z","snapshot_observed_at":"2026-08-08T04:26:07.234835Z","submitted_at":"2025-09-16T16:37:31Z","title":"On Courant-type bounds and spectral partitioning via Neumann domains on quantum graphs","version":2},"reference_index":18,"source":"arxiv_source","source_observed_at":"2026-08-04T16:41:05.857848Z"},"links":{"citing_paper":"/paper/2509.13228"},"observation_digest":"sha256:9b839c23cc877558f226ebd95ece93a2925839eca9500257a1c46f85c1fdd12f","observation_id":"4dcdbbe7-9888-43cd-8eff-748fe822444b","resolution":{"observed_at":"2026-08-04T16:41:05.857848Z","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":"2026-08-04T16:41:05.980082Z","title":null,"venue":null,"work_id":null,"year":1991},"citing_paper":{"arxiv_id":"2509.13228","last_updated":"2026-07-19T21:19:46Z","snapshot_observed_at":"2026-08-08T04:26:07.234835Z","submitted_at":"2025-09-16T16:37:31Z","title":"On Courant-type bounds and spectral partitioning via Neumann domains on quantum graphs","version":2},"reference_index":19,"source":"arxiv_source","source_observed_at":"2026-08-04T16:41:05.980082Z"},"links":{"citing_paper":"/paper/2509.13228"},"observation_digest":"sha256:ca085c782897e33168e178d202d46bc8709b9bf0ae4920bfdb2a530fbe1bc143","observation_id":"9397694d-3aea-4796-89fa-6885a80fa013","resolution":{"observed_at":"2026-08-04T16:41:05.980082Z","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":"2026-08-04T16:41:06.061135Z","title":null,"venue":null,"work_id":null,"year":2005},"citing_paper":{"arxiv_id":"2509.13228","last_updated":"2026-07-19T21:19:46Z","snapshot_observed_at":"2026-08-08T04:26:07.234835Z","submitted_at":"2025-09-16T16:37:31Z","title":"On Courant-type bounds and spectral partitioning via Neumann domains on quantum graphs","version":2},"reference_index":20,"source":"arxiv_source","source_observed_at":"2026-08-04T16:41:06.061135Z"},"links":{"citing_paper":"/paper/2509.13228"},"observation_digest":"sha256:8877a5e12c5aa74e0ab77a12ffe00144352a1737b2d618b35b4e707d2c423720","observation_id":"1d883d7b-e214-47d4-97a1-b376b7a8e2fc","resolution":{"observed_at":"2026-08-04T16:41:06.061135Z","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":"2026-08-04T16:41:06.261518Z","title":null,"venue":null,"work_id":null,"year":2009},"citing_paper":{"arxiv_id":"2509.13228","last_updated":"2026-07-19T21:19:46Z","snapshot_observed_at":"2026-08-08T04:26:07.234835Z","submitted_at":"2025-09-16T16:37:31Z","title":"On Courant-type bounds and spectral partitioning via Neumann domains on quantum graphs","version":2},"reference_index":21,"source":"arxiv_source","source_observed_at":"2026-08-04T16:41:06.261518Z"},"links":{"citing_paper":"/paper/2509.13228"},"observation_digest":"sha256:dc24628082464cada2bbce5a82e024bb89ca8d9159e2264b7a6d47f0a77a5dc7","observation_id":"8304cad7-b718-4aa5-9694-6c56c0326e54","resolution":{"observed_at":"2026-08-04T16:41:06.261518Z","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":"2026-08-04T16:41:06.368255Z","title":"B., Mugnolo, D., Plümer, M., Asymptotics and Estimates for Spectral Minimal Partitions of Metric Graphs, Int.\\ Equ.\\ Oper.\\ Theory 93 (2021), 26","venue":null,"work_id":null,"year":2021},"citing_paper":{"arxiv_id":"2509.13228","last_updated":"2026-07-19T21:19:46Z","snapshot_observed_at":"2026-08-08T04:26:07.234835Z","submitted_at":"2025-09-16T16:37:31Z","title":"On Courant-type bounds and spectral partitioning via Neumann domains on quantum graphs","version":2},"reference_index":22,"source":"arxiv_source","source_observed_at":"2026-08-04T16:41:06.368255Z"},"links":{"citing_paper":"/paper/2509.13228"},"observation_digest":"sha256:c3fb1133957c811d346d9f4e5a330cb052b87044f25c52a0ecaaaefb67b8907a","observation_id":"a7db5b63-f55e-4127-9242-7831ca6208e9","resolution":{"observed_at":"2026-08-04T16:41:06.368255Z","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":"2026-08-04T16:41:06.372843Z","title":"B., Mugnolo, D., Plümer, M., On Pleijel’s Nodal Domain Theorem for Quantum Graphs","venue":null,"work_id":null,"year":2021},"citing_paper":{"arxiv_id":"2509.13228","last_updated":"2026-07-19T21:19:46Z","snapshot_observed_at":"2026-08-08T04:26:07.234835Z","submitted_at":"2025-09-16T16:37:31Z","title":"On Courant-type bounds and spectral partitioning via Neumann domains on quantum graphs","version":2},"reference_index":23,"source":"arxiv_source","source_observed_at":"2026-08-04T16:41:06.372843Z"},"links":{"citing_paper":"/paper/2509.13228"},"observation_digest":"sha256:2486b46b356c579dc5c20c76b09ee49c7fc6e99a14de6a9587c7028d57375bbf","observation_id":"91016afd-bece-4eaa-9439-28339f16a692","resolution":{"observed_at":"2026-08-04T16:41:06.372843Z","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":"2026-08-04T16:41:06.383275Z","title":"B., Interlacing and Friedlander-type inequalities for spectral minimal partitions of metric graphs, Lett","venue":null,"work_id":null,"year":2021},"citing_paper":{"arxiv_id":"2509.13228","last_updated":"2026-07-19T21:19:46Z","snapshot_observed_at":"2026-08-08T04:26:07.234835Z","submitted_at":"2025-09-16T16:37:31Z","title":"On Courant-type bounds and spectral partitioning via Neumann domains on quantum graphs","version":2},"reference_index":24,"source":"arxiv_source","source_observed_at":"2026-08-04T16:41:06.383275Z"},"links":{"citing_paper":"/paper/2509.13228"},"observation_digest":"sha256:da87e5536504844465ba67a05a25ae5a7eb3970c2576e6683b02c646213359d9","observation_id":"3a2cb971-3a44-4295-94d1-3e545a475132","resolution":{"observed_at":"2026-08-04T16:41:06.383275Z","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":"2026-08-04T16:41:06.389225Z","title":"B., Geometric spectral theory of quantum graphs, Commun.\\ Math.\\ 32 (2024), no","venue":null,"work_id":null,"year":2024},"citing_paper":{"arxiv_id":"2509.13228","last_updated":"2026-07-19T21:19:46Z","snapshot_observed_at":"2026-08-08T04:26:07.234835Z","submitted_at":"2025-09-16T16:37:31Z","title":"On Courant-type bounds and spectral partitioning via Neumann domains on quantum graphs","version":2},"reference_index":25,"source":"arxiv_source","source_observed_at":"2026-08-04T16:41:06.389225Z"},"links":{"citing_paper":"/paper/2509.13228"},"observation_digest":"sha256:a2df5cd68227da44ecbda2896c0b885f6c12a27331e86ffaed30670ebcb4fff5","observation_id":"585cfcf9-c6f8-4a24-a6c8-438616a55428","resolution":{"observed_at":"2026-08-04T16:41:06.389225Z","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":"2026-08-04T16:41:06.395623Z","title":"B., Kurasov, P., L\\'ena, C., Mugnolo, D., A theory of spectral partitions of metric graphs, Calc.\\ Var.\\ Partial Differential Equations 60 (2021), 61","venue":null,"work_id":null,"year":2021},"citing_paper":{"arxiv_id":"2509.13228","last_updated":"2026-07-19T21:19:46Z","snapshot_observed_at":"2026-08-08T04:26:07.234835Z","submitted_at":"2025-09-16T16:37:31Z","title":"On Courant-type bounds and spectral partitioning via Neumann domains on quantum graphs","version":2},"reference_index":26,"source":"arxiv_source","source_observed_at":"2026-08-04T16:41:06.395623Z"},"links":{"citing_paper":"/paper/2509.13228"},"observation_digest":"sha256:b135383dc0f5cf3c51f724ff3830e17dd4c0ae947c1aa3c2526ed3a227ee1298","observation_id":"cbb16652-3ba7-4d19-9fac-a93d05e91cdc","resolution":{"observed_at":"2026-08-04T16:41:06.395623Z","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":"2026-08-04T16:41:06.401274Z","title":"B., Kurasov, P., Malenov\\'a, G., Mugnolo, D., On the spectral gap of a quantum graph, Ann.\\ Henri Poincar\\'e 17 (2016), 2439--2473","venue":null,"work_id":null,"year":2016},"citing_paper":{"arxiv_id":"2509.13228","last_updated":"2026-07-19T21:19:46Z","snapshot_observed_at":"2026-08-08T04:26:07.234835Z","submitted_at":"2025-09-16T16:37:31Z","title":"On Courant-type bounds and spectral partitioning via Neumann domains on quantum graphs","version":2},"reference_index":27,"source":"arxiv_source","source_observed_at":"2026-08-04T16:41:06.401274Z"},"links":{"citing_paper":"/paper/2509.13228"},"observation_digest":"sha256:1efb6f14ff789de06b8cdd373cfba809542cbeb4b9e0ea8a30effb6c5d2cac87","observation_id":"3c0493ea-7ee5-47ef-b139-51c7b30db4a7","resolution":{"observed_at":"2026-08-04T16:41:06.401274Z","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":"2026-08-04T16:41:06.406634Z","title":"and Smilansky, U., Periodic orbit theory and spectral statistics for quantum graphs, Ann","venue":null,"work_id":null,"year":1999},"citing_paper":{"arxiv_id":"2509.13228","last_updated":"2026-07-19T21:19:46Z","snapshot_observed_at":"2026-08-08T04:26:07.234835Z","submitted_at":"2025-09-16T16:37:31Z","title":"On Courant-type bounds and spectral partitioning via Neumann domains on quantum graphs","version":2},"reference_index":28,"source":"arxiv_source","source_observed_at":"2026-08-04T16:41:06.406634Z"},"links":{"citing_paper":"/paper/2509.13228"},"observation_digest":"sha256:bad865bbf9df83add99191282e59b06c47a1f241b4cb3e6b616d6a36ead53c42","observation_id":"58767ef8-fb31-48ab-a4d5-1bb5c5c10333","resolution":{"observed_at":"2026-08-04T16:41:06.406634Z","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":"2026-08-04T16:41:06.424345Z","title":"Rep.\\ Math.\\ Phys.\\ 82 (2018), 63--80","venue":null,"work_id":null,"year":2018},"citing_paper":{"arxiv_id":"2509.13228","last_updated":"2026-07-19T21:19:46Z","snapshot_observed_at":"2026-08-08T04:26:07.234835Z","submitted_at":"2025-09-16T16:37:31Z","title":"On Courant-type bounds and spectral partitioning via Neumann domains on quantum graphs","version":2},"reference_index":29,"source":"arxiv_source","source_observed_at":"2026-08-04T16:41:06.424345Z"},"links":{"citing_paper":"/paper/2509.13228"},"observation_digest":"sha256:4d5f0cc43fc8ce8942ea7cfd445ec4690c602ac8fa469db7c900093b513ee971","observation_id":"ebc13694-445c-4063-900a-a76c80d16ecf","resolution":{"observed_at":"2026-08-04T16:41:06.424345Z","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":"2026-08-04T16:41:06.431744Z","title":"Lett Math Phys 109, 2491–2512 (2019)","venue":null,"work_id":null,"year":2019},"citing_paper":{"arxiv_id":"2509.13228","last_updated":"2026-07-19T21:19:46Z","snapshot_observed_at":"2026-08-08T04:26:07.234835Z","submitted_at":"2025-09-16T16:37:31Z","title":"On Courant-type bounds and spectral partitioning via Neumann domains on quantum graphs","version":2},"reference_index":30,"source":"arxiv_source","source_observed_at":"2026-08-04T16:41:06.431744Z"},"links":{"citing_paper":"/paper/2509.13228"},"observation_digest":"sha256:30cf0778a86642a8f44925811f1e3c7d8bed5026cb3264040d9a1a7d93889fb8","observation_id":"6693b1a8-8873-4e9b-a4f7-10325022e7ce","resolution":{"observed_at":"2026-08-04T16:41:06.431744Z","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":"2026-08-04T16:41:06.441708Z","title":null,"venue":null,"work_id":null,"year":2024},"citing_paper":{"arxiv_id":"2509.13228","last_updated":"2026-07-19T21:19:46Z","snapshot_observed_at":"2026-08-08T04:26:07.234835Z","submitted_at":"2025-09-16T16:37:31Z","title":"On Courant-type bounds and spectral partitioning via Neumann domains on quantum graphs","version":2},"reference_index":31,"source":"arxiv_source","source_observed_at":"2026-08-04T16:41:06.441708Z"},"links":{"citing_paper":"/paper/2509.13228"},"observation_digest":"sha256:8cc98d5ae8a265cb3e07690b52f5190c3a28a85a5555d11223c9e6238ca91093","observation_id":"d2749100-ea8d-420f-9911-a6b3881ed18f","resolution":{"observed_at":"2026-08-04T16:41:06.441708Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":{"arxiv_id":"1912.07549","last_updated":"2021-03-15T20:24:17Z","snapshot_observed_at":"2026-08-14T01:29:08.382747Z","submitted_at":"2019-12-16T17:54:29Z","title":"What is actually a metric graph?","version":2},"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":"1912.07549","snapshot_observed_at":"2026-08-04T16:41:06.451513Z","title":null,"venue":null,"work_id":null,"year":2019},"citing_paper":{"arxiv_id":"2509.13228","last_updated":"2026-07-19T21:19:46Z","snapshot_observed_at":"2026-08-08T04:26:07.234835Z","submitted_at":"2025-09-16T16:37:31Z","title":"On Courant-type bounds and spectral partitioning via Neumann domains on quantum graphs","version":2},"reference_index":32,"source":"arxiv_source","source_observed_at":"2026-08-04T16:41:06.451513Z"},"links":{"cited_paper":"/paper/1912.07549","citing_paper":"/paper/2509.13228"},"observation_digest":"sha256:af17389263965adc45d6ccec6c18c03219ad9619dd4f3d3dd731e9ea0f576b19","observation_id":"bce37f29-36d1-4376-83d2-bac7c694a826","resolution":{"observed_at":"2026-08-04T16:41:06.451513Z","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":"2026-08-04T16:41:06.458698Z","title":null,"venue":null,"work_id":null,"year":1987},"citing_paper":{"arxiv_id":"2509.13228","last_updated":"2026-07-19T21:19:46Z","snapshot_observed_at":"2026-08-08T04:26:07.234835Z","submitted_at":"2025-09-16T16:37:31Z","title":"On Courant-type bounds and spectral partitioning via Neumann domains on quantum graphs","version":2},"reference_index":33,"source":"arxiv_source","source_observed_at":"2026-08-04T16:41:06.458698Z"},"links":{"citing_paper":"/paper/2509.13228"},"observation_digest":"sha256:f9734e132d6d7f47c2ac04eff5b0f7999717e65a381eca942933d6110724b954","observation_id":"ed2988c5-fe10-4390-9a91-314de7ff72b7","resolution":{"observed_at":"2026-08-04T16:41:06.458698Z","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":"2026-08-04T16:41:06.463085Z","title":"Comm.\\ Pure Appl.\\ Math., 9 (1956), 543–550","venue":null,"work_id":null,"year":1956},"citing_paper":{"arxiv_id":"2509.13228","last_updated":"2026-07-19T21:19:46Z","snapshot_observed_at":"2026-08-08T04:26:07.234835Z","submitted_at":"2025-09-16T16:37:31Z","title":"On Courant-type bounds and spectral partitioning via Neumann domains on quantum graphs","version":2},"reference_index":34,"source":"arxiv_source","source_observed_at":"2026-08-04T16:41:06.463085Z"},"links":{"citing_paper":"/paper/2509.13228"},"observation_digest":"sha256:ea66f242b7c4ce8866672042cc7385a7dd4c28b8c3386600a6131ec19a1e5587","observation_id":"e0b09254-fee5-4fab-997d-020b573c09b8","resolution":{"observed_at":"2026-08-04T16:41:06.463085Z","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":"2026-08-04T16:41:06.469821Z","title":null,"venue":null,"work_id":null,"year":2017},"citing_paper":{"arxiv_id":"2509.13228","last_updated":"2026-07-19T21:19:46Z","snapshot_observed_at":"2026-08-08T04:26:07.234835Z","submitted_at":"2025-09-16T16:37:31Z","title":"On Courant-type bounds and spectral partitioning via Neumann domains on quantum graphs","version":2},"reference_index":35,"source":"arxiv_source","source_observed_at":"2026-08-04T16:41:06.469821Z"},"links":{"citing_paper":"/paper/2509.13228"},"observation_digest":"sha256:047f4f811adfd1ff153057a9142561250895f759000940076ba0695156e08dcf","observation_id":"08f61173-4a07-4619-a87f-1918a8b7ff02","resolution":{"observed_at":"2026-08-04T16:41:06.469821Z","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":"2026-08-04T16:41:06.476586Z","title":"3, 167--178","venue":null,"work_id":null,"year":2006},"citing_paper":{"arxiv_id":"2509.13228","last_updated":"2026-07-19T21:19:46Z","snapshot_observed_at":"2026-08-08T04:26:07.234835Z","submitted_at":"2025-09-16T16:37:31Z","title":"On Courant-type bounds and spectral partitioning via Neumann domains on quantum graphs","version":2},"reference_index":36,"source":"arxiv_source","source_observed_at":"2026-08-04T16:41:06.476586Z"},"links":{"citing_paper":"/paper/2509.13228"},"observation_digest":"sha256:6031f8755bf6ea7f6cfd132e22ec9d6cee9b9e940845a88425e694529f669330","observation_id":"100d65be-2eba-42f6-8031-403fbdd5b075","resolution":{"observed_at":"2026-08-04T16:41:06.476586Z","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":"2026-08-04T16:41:06.483632Z","title":"J.\\ Phys.\\ A: Math.\\ Theor.\\ 54 (2021), 015202, 14 pp","venue":null,"work_id":null,"year":2021},"citing_paper":{"arxiv_id":"2509.13228","last_updated":"2026-07-19T21:19:46Z","snapshot_observed_at":"2026-08-08T04:26:07.234835Z","submitted_at":"2025-09-16T16:37:31Z","title":"On Courant-type bounds and spectral partitioning via Neumann domains on quantum graphs","version":2},"reference_index":37,"source":"arxiv_source","source_observed_at":"2026-08-04T16:41:06.483632Z"},"links":{"citing_paper":"/paper/2509.13228"},"observation_digest":"sha256:52c7a1332ef8da00f7c826fdbfbe3fe6926d5db9ed857be2b25b46d53487db15","observation_id":"ab075762-75b6-4c20-b5cc-0952f0c9eabd","resolution":{"observed_at":"2026-08-04T16:41:06.483632Z","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":"2026-08-04T16:41:06.488506Z","title":"Journal de Mathématiques Pures et Appliquées (1836), 106–186","venue":null,"work_id":null,"year":null},"citing_paper":{"arxiv_id":"2509.13228","last_updated":"2026-07-19T21:19:46Z","snapshot_observed_at":"2026-08-08T04:26:07.234835Z","submitted_at":"2025-09-16T16:37:31Z","title":"On Courant-type bounds and spectral partitioning via Neumann domains on quantum graphs","version":2},"reference_index":38,"source":"arxiv_source","source_observed_at":"2026-08-04T16:41:06.488506Z"},"links":{"citing_paper":"/paper/2509.13228"},"observation_digest":"sha256:41363a295929e3e06b67f5c790bae0cc4d62d5c10eea42e7f4e7677e4307d327","observation_id":"f8bdc092-54bd-4d28-8a52-1533c3051555","resolution":{"observed_at":"2026-08-04T16:41:06.488506Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}}],"paper":{"arxiv_id":"2509.13228","last_updated":"2026-07-19T21:19:46Z","latest_version":2,"primary_category":"math.SP","snapshot_observed_at":"2026-08-08T04:26:07.234835Z","submitted_at":"2025-09-16T16:37:31Z","title":"On Courant-type bounds and spectral partitioning via Neumann domains on quantum graphs"},"reference_resolution":{"displayed":38,"state_counts":{"malformed_identifier":0,"metadata_mismatch":0,"parse_uncertain":0,"unresolved":38,"verified_exact":0,"verified_fuzzy":0},"total_outbound_references":38},"refusal":"A citation records a reference. It does not transfer a finding from one paper to another.","schema":"pith.paper-citation-record.v1","standing_sources":[{"observed_at":"2026-08-14T06:32:32.682623+00:00","source":"crossref"},{"observed_at":"2026-08-14T06:32:18.44784+00:00","source":"retraction_watch"}],"thesis":"As of 14 August 2026, this Paper Citation Record lists 38 of 38 outbound references and 0 inbound Pith citation observations for arXiv:2509.13228."}