{"as_of":"2026-08-13T19:46:00Z","caps":{"database_statements":6,"inbound":100,"outbound":100},"context_digest":"sha256:3aea03911fe0057c62039ce2d2b2ad5404bc92cae748cf59abea1c5347ada5f9","coverage":[{"denominator":49,"lane":"reference_resolution","note":"Typed states for the displayed outbound observations.","records_observed":49,"source":"paper_references, paper_reference_links","source_observed_at":"2026-08-05T00:54:17.047306Z","state":"measured"},{"denominator":49,"lane":"standing_notices","note":"One-hop event checks from named stored sources.","records_observed":49,"source":"scholarly_work_events, retraction_status_cache","source_observed_at":"2026-08-13T06:32:02.005865+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/2608.00556/citation-record","integrity":"/paper/2608.00556/integrity","json":"/paper/2608.00556/citation-record.json","paper":"/paper/2608.00556"},"outbound":[{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":null,"snapshot_observed_at":"2026-08-05T00:54:13.797850Z","title":"Banyaga and D","venue":null,"work_id":null,"year":2004},"citing_paper":{"arxiv_id":"2608.00556","last_updated":"2026-08-01T09:21:36Z","snapshot_observed_at":"2026-08-10T00:06:11.727871Z","submitted_at":"2026-08-01T09:21:36Z","title":"Fundamental examples of height functions on closed manifolds and their 1st derivatives","version":1},"reference_index":1,"source":"pdf_text","source_observed_at":"2026-08-05T00:54:13.797850Z"},"links":{"citing_paper":"/paper/2608.00556"},"observation_digest":"sha256:c6379f0ccb71a0e6961e212e442d3656a8659bcf11f5f23349a3dbe2d74a7dca","observation_id":"e8584070-7ed4-4784-9972-5c83fedde0f0","resolution":{"observed_at":"2026-08-05T00:54:13.797850Z","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":"2026-08-05T00:54:17.873352Z","title":"Banchoﬀ and F","venue":null,"work_id":"578e251a-28c4-4f53-998f-bc3c19eb1771","year":1975},"citing_paper":{"arxiv_id":"2608.00556","last_updated":"2026-08-01T09:21:36Z","snapshot_observed_at":"2026-08-10T00:06:11.727871Z","submitted_at":"2026-08-01T09:21:36Z","title":"Fundamental examples of height functions on closed manifolds and their 1st derivatives","version":1},"reference_index":2,"source":"pdf_text","source_observed_at":"2026-08-05T00:54:13.952707Z"},"links":{"citing_paper":"/paper/2608.00556"},"observation_digest":"sha256:80593d559f33943261cd0fde10cb637dfbdd3beeef44e0a1ecf14691e5f5faef","observation_id":"5d3a06ec-39cf-4590-8f7a-c81786a61041","resolution":{"observed_at":"2026-08-05T00:54:17.877662Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-13T06:32:02.005865+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-13T06:32:02.005865+00:00","source":"crossref"},{"observed_at":"2026-08-13T06:31:53.387327+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":"2026-08-05T00:54:17.863382Z","title":"Burlet and V","venue":null,"work_id":"8e92752f-a51d-4f6f-bf98-142210e096ee","year":1985},"citing_paper":{"arxiv_id":"2608.00556","last_updated":"2026-08-01T09:21:36Z","snapshot_observed_at":"2026-08-10T00:06:11.727871Z","submitted_at":"2026-08-01T09:21:36Z","title":"Fundamental examples of height functions on closed manifolds and their 1st derivatives","version":1},"reference_index":3,"source":"pdf_text","source_observed_at":"2026-08-05T00:54:14.100225Z"},"links":{"citing_paper":"/paper/2608.00556"},"observation_digest":"sha256:ca5199042c653e9a7fccee1756f7bfb5174f0d05451c0d39b2486608ec018073","observation_id":"0147e878-477c-4ca4-83a2-760e1a53c07f","resolution":{"observed_at":"2026-08-05T00:54:17.866606Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-13T06:32:02.005865+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-13T06:32:02.005865+00:00","source":"crossref"},{"observed_at":"2026-08-13T06:31:53.387327+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":"2026-08-05T00:54:14.253125Z","title":"Bott, Nondegenerate critical manifolds , Ann","venue":null,"work_id":null,"year":1954},"citing_paper":{"arxiv_id":"2608.00556","last_updated":"2026-08-01T09:21:36Z","snapshot_observed_at":"2026-08-10T00:06:11.727871Z","submitted_at":"2026-08-01T09:21:36Z","title":"Fundamental examples of height functions on closed manifolds and their 1st derivatives","version":1},"reference_index":4,"source":"pdf_text","source_observed_at":"2026-08-05T00:54:14.253125Z"},"links":{"citing_paper":"/paper/2608.00556"},"observation_digest":"sha256:c307274e04b06e579988520a8e60d16d37712d198734fa654f648cf3697e380f","observation_id":"b79e3abb-cee9-4bb6-82c2-a7293056e89f","resolution":{"observed_at":"2026-08-05T00:54:14.253125Z","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":"2026-08-05T00:54:17.847401Z","title":null,"venue":null,"work_id":"e22a3d9f-4e31-4c75-8615-3e6e5188cadb","year":null},"citing_paper":{"arxiv_id":"2608.00556","last_updated":"2026-08-01T09:21:36Z","snapshot_observed_at":"2026-08-10T00:06:11.727871Z","submitted_at":"2026-08-01T09:21:36Z","title":"Fundamental examples of height functions on closed manifolds and their 1st derivatives","version":1},"reference_index":5,"source":"pdf_text","source_observed_at":"2026-08-05T00:54:14.412967Z"},"links":{"citing_paper":"/paper/2608.00556"},"observation_digest":"sha256:c6be01cc3701ea37a608a4123a62a95b9504abb0c20c8e70dc37bcdaf46ec2cb","observation_id":"ca835993-5969-4a38-bcd9-993e9490208e","resolution":{"observed_at":"2026-08-05T00:54:17.850778Z","resolver_source":"raw_fallback","status":"unresolved"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-13T06:32:02.005865+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-13T06:32:02.005865+00:00","source":"crossref"},{"observed_at":"2026-08-13T06:31:53.387327+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":"2026-08-05T00:54:17.838226Z","title":"Burlet and G","venue":null,"work_id":"58bb69d0-4eb8-4830-92bf-e4f781946d21","year":1974},"citing_paper":{"arxiv_id":"2608.00556","last_updated":"2026-08-01T09:21:36Z","snapshot_observed_at":"2026-08-10T00:06:11.727871Z","submitted_at":"2026-08-01T09:21:36Z","title":"Fundamental examples of height functions on closed manifolds and their 1st derivatives","version":1},"reference_index":6,"source":"pdf_text","source_observed_at":"2026-08-05T00:54:14.583214Z"},"links":{"citing_paper":"/paper/2608.00556"},"observation_digest":"sha256:82df99d04f9836fc226072a1f3d6ccab4f1b9cc95e20825151f3c624e2041ec5","observation_id":"5e19cef3-8121-484c-b883-06f674dc27e7","resolution":{"observed_at":"2026-08-05T00:54:17.841261Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-13T06:32:02.005865+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-13T06:32:02.005865+00:00","source":"crossref"},{"observed_at":"2026-08-13T06:31:53.387327+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":"2026-08-05T00:54:17.828528Z","title":null,"venue":null,"work_id":"12c2b60e-d0c5-4a04-b169-1e8c55316cd1","year":1990},"citing_paper":{"arxiv_id":"2608.00556","last_updated":"2026-08-01T09:21:36Z","snapshot_observed_at":"2026-08-10T00:06:11.727871Z","submitted_at":"2026-08-01T09:21:36Z","title":"Fundamental examples of height functions on closed manifolds and their 1st derivatives","version":1},"reference_index":7,"source":"pdf_text","source_observed_at":"2026-08-05T00:54:14.703640Z"},"links":{"citing_paper":"/paper/2608.00556"},"observation_digest":"sha256:0ddd1348d2aefe41e839c2d304f643a05edac277144baea448398d81fdfb8b85","observation_id":"89778b40-be1f-4efd-b6c9-a32ce320eaa7","resolution":{"observed_at":"2026-08-05T00:54:17.831654Z","resolver_source":"raw_fallback","status":"unresolved"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-13T06:32:02.005865+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-13T06:32:02.005865+00:00","source":"crossref"},{"observed_at":"2026-08-13T06:31:53.387327+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":"2026-08-05T00:54:14.888024Z","title":"Gelbukh, A ﬁnite graph is homeomorphic to the Reeb graph of a Morse-Bot t function , Mathematica Slovaca, 71 (3), 757–772, 2021; doi: 10.1515/m s-2021-0018","venue":null,"work_id":null,"year":2021},"citing_paper":{"arxiv_id":"2608.00556","last_updated":"2026-08-01T09:21:36Z","snapshot_observed_at":"2026-08-10T00:06:11.727871Z","submitted_at":"2026-08-01T09:21:36Z","title":"Fundamental examples of height functions on closed manifolds and their 1st derivatives","version":1},"reference_index":8,"source":"pdf_text","source_observed_at":"2026-08-05T00:54:14.888024Z"},"links":{"citing_paper":"/paper/2608.00556"},"observation_digest":"sha256:4e1fb0f54c0ebabda6719894fb49e303565ecc923b9d9eda686a0653348486ee","observation_id":"43fbbb40-07ea-42d4-8b33-2ff92042d557","resolution":{"observed_at":"2026-08-05T00:54:14.888024Z","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":"2026-08-05T00:54:15.017586Z","title":"Gelbukh, Morse-Bott functions with two critical values on a surface , Czechoslovak Mathe- matical Journal, 71 (3), 865–880, 2021; doi: 10.21136/CMJ","venue":null,"work_id":null,"year":2021},"citing_paper":{"arxiv_id":"2608.00556","last_updated":"2026-08-01T09:21:36Z","snapshot_observed_at":"2026-08-10T00:06:11.727871Z","submitted_at":"2026-08-01T09:21:36Z","title":"Fundamental examples of height functions on closed manifolds and their 1st derivatives","version":1},"reference_index":9,"source":"pdf_text","source_observed_at":"2026-08-05T00:54:15.017586Z"},"links":{"citing_paper":"/paper/2608.00556"},"observation_digest":"sha256:e783f50f21adac4f8b5a8957b3f067bc546bf14b40d56f0d6d35c3b2b4d70e33","observation_id":"a4ed3d6c-44ce-487f-87f4-ac9242d20a1a","resolution":{"observed_at":"2026-08-05T00:54:15.017586Z","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":"raw_reference","pith_arxiv_id":null,"snapshot_observed_at":"2026-08-05T00:54:17.818173Z","title":"Gelbukh, Realization of a graph as the Reeb graph of a height function o n an embedded surface, Topol","venue":null,"work_id":"960f06ec-22da-4605-8edd-a1e6b0c635f8","year":2023},"citing_paper":{"arxiv_id":"2608.00556","last_updated":"2026-08-01T09:21:36Z","snapshot_observed_at":"2026-08-10T00:06:11.727871Z","submitted_at":"2026-08-01T09:21:36Z","title":"Fundamental examples of height functions on closed manifolds and their 1st derivatives","version":1},"reference_index":10,"source":"pdf_text","source_observed_at":"2026-08-05T00:54:15.155921Z"},"links":{"citing_paper":"/paper/2608.00556"},"observation_digest":"sha256:cba246d511af85e387d534ff44d0e285043a418c19972898c6ea8b1192ef9a52","observation_id":"963b031d-7408-44aa-8790-ad88b791b275","resolution":{"observed_at":"2026-08-05T00:54:17.821315Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-13T06:32:02.005865+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-13T06:32:02.005865+00:00","source":"crossref"},{"observed_at":"2026-08-13T06:31:53.387327+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":"2026-08-05T00:54:17.808185Z","title":"Gelbukh, On the topology of the Reeb graph , Publicationes Mathematicae Debrecen 104(3–","venue":null,"work_id":"87430683-cd28-43fe-86d5-939cca79f97a","year":null},"citing_paper":{"arxiv_id":"2608.00556","last_updated":"2026-08-01T09:21:36Z","snapshot_observed_at":"2026-08-10T00:06:11.727871Z","submitted_at":"2026-08-01T09:21:36Z","title":"Fundamental examples of height functions on closed manifolds and their 1st derivatives","version":1},"reference_index":11,"source":"pdf_text","source_observed_at":"2026-08-05T00:54:15.359148Z"},"links":{"citing_paper":"/paper/2608.00556"},"observation_digest":"sha256:c01411d3344ac302f62be8da761fdfb4413f4b738bfb939849eaa67d79e7d6f1","observation_id":"69a85b87-1620-4d8b-95af-2a6d4a6d1e7d","resolution":{"observed_at":"2026-08-05T00:54:17.811586Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-13T06:32:02.005865+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-13T06:32:02.005865+00:00","source":"crossref"},{"observed_at":"2026-08-13T06:31:53.387327+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":"2026-08-05T00:54:15.547172Z","title":"Gelbukh, Realization of a digraph as the Reeb graph of a Morse-Bott fun ction on a given surface, Topology and its Applications, 2024","venue":null,"work_id":null,"year":2024},"citing_paper":{"arxiv_id":"2608.00556","last_updated":"2026-08-01T09:21:36Z","snapshot_observed_at":"2026-08-10T00:06:11.727871Z","submitted_at":"2026-08-01T09:21:36Z","title":"Fundamental examples of height functions on closed manifolds and their 1st derivatives","version":1},"reference_index":12,"source":"pdf_text","source_observed_at":"2026-08-05T00:54:15.547172Z"},"links":{"citing_paper":"/paper/2608.00556"},"observation_digest":"sha256:316db76bf36c7a7e3d4be00283acb174fbca972223c95c4d578a44ac14e29568","observation_id":"e2251e1b-0f81-4bbb-a04a-e2db61f16658","resolution":{"observed_at":"2026-08-05T00:54:15.547172Z","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":"2026-08-05T00:54:15.713254Z","title":"Gelbukh, Reeb Graphs of Morse-Bott Functions on a Given Surface , Bulletin of the Iranian Mathematical Society, Volume 50 Article number 84, 2024, 1– 17","venue":null,"work_id":null,"year":2024},"citing_paper":{"arxiv_id":"2608.00556","last_updated":"2026-08-01T09:21:36Z","snapshot_observed_at":"2026-08-10T00:06:11.727871Z","submitted_at":"2026-08-01T09:21:36Z","title":"Fundamental examples of height functions on closed manifolds and their 1st derivatives","version":1},"reference_index":13,"source":"pdf_text","source_observed_at":"2026-08-05T00:54:15.713254Z"},"links":{"citing_paper":"/paper/2608.00556"},"observation_digest":"sha256:28527b4fadebefa6a4256b3da395eb04361fa57e3552b56775adf9e26e2d051c","observation_id":"563387ff-6070-4d5f-bbd6-479d755ea6ea","resolution":{"observed_at":"2026-08-05T00:54:15.713254Z","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":"publication/4035013","doi":null,"metadata_source":"raw_reference","pith_arxiv_id":null,"snapshot_observed_at":"2026-08-05T00:54:17.544670Z","title":null,"venue":null,"work_id":"b55ed66b-3e68-46f1-8836-feae1d8730ca","year":null},"citing_paper":{"arxiv_id":"2608.00556","last_updated":"2026-08-01T09:21:36Z","snapshot_observed_at":"2026-08-10T00:06:11.727871Z","submitted_at":"2026-08-01T09:21:36Z","title":"Fundamental examples of height functions on closed manifolds and their 1st derivatives","version":1},"reference_index":14,"source":"pdf_text","source_observed_at":"2026-08-05T00:54:15.854542Z"},"links":{"citing_paper":"/paper/2608.00556"},"observation_digest":"sha256:aad4c741206aa74f790557a3138313e35580fafa3ed04d3bd1c4817f27dc0d01","observation_id":"8d55b1e2-28f1-4454-b316-612ab3fcf19f","resolution":{"observed_at":"2026-08-05T00:54:17.549962Z","resolver_source":"raw_fallback","status":"verified_exact"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-13T06:32:02.005865+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-13T06:32:02.005865+00:00","source":"crossref"},{"observed_at":"2026-08-13T06:31:53.387327+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":"2026-08-05T00:54:15.961604Z","title":"Golubitsky and V","venue":null,"work_id":null,"year":1974},"citing_paper":{"arxiv_id":"2608.00556","last_updated":"2026-08-01T09:21:36Z","snapshot_observed_at":"2026-08-10T00:06:11.727871Z","submitted_at":"2026-08-01T09:21:36Z","title":"Fundamental examples of height functions on closed manifolds and their 1st derivatives","version":1},"reference_index":15,"source":"pdf_text","source_observed_at":"2026-08-05T00:54:15.961604Z"},"links":{"citing_paper":"/paper/2608.00556"},"observation_digest":"sha256:686b954cff1894c542dff9ea40c3fb2b77a78ce9b1ce82032aad4acba7f89b52","observation_id":"82605845-16aa-4db4-89b9-d5527f3f0c37","resolution":{"observed_at":"2026-08-05T00:54:15.961604Z","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":"2026-08-05T00:54:16.089899Z","title":null,"venue":null,"work_id":null,"year":1988},"citing_paper":{"arxiv_id":"2608.00556","last_updated":"2026-08-01T09:21:36Z","snapshot_observed_at":"2026-08-10T00:06:11.727871Z","submitted_at":"2026-08-01T09:21:36Z","title":"Fundamental examples of height functions on closed manifolds and their 1st derivatives","version":1},"reference_index":16,"source":"pdf_text","source_observed_at":"2026-08-05T00:54:16.089899Z"},"links":{"citing_paper":"/paper/2608.00556"},"observation_digest":"sha256:2028dd01662884f673fb568289ef8eb87eedb9b0580a1d3a169823522352dd07","observation_id":"a5c30dc0-a66d-4ab1-af7f-2777d73f0177","resolution":{"observed_at":"2026-08-05T00:54:16.089899Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":{"arxiv_id":"1902.08841","last_updated":"2021-07-27T09:55:56Z","snapshot_observed_at":"2026-08-13T02:51:22.471449Z","submitted_at":"2019-02-23T19:55:38Z","title":"On Reeb graphs induced from smooth functions on 3-dimensional closed orientable manifolds with finitely many singular values","version":5},"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":"1902.08841","snapshot_observed_at":"2026-08-05T00:54:16.202466Z","title":"Kitazawa, On Reeb graphs induced from smooth functions on 3-dimensional closed ori- entable manifolds with ﬁnitely many singular values , Topol","venue":null,"work_id":null,"year":2022},"citing_paper":{"arxiv_id":"2608.00556","last_updated":"2026-08-01T09:21:36Z","snapshot_observed_at":"2026-08-10T00:06:11.727871Z","submitted_at":"2026-08-01T09:21:36Z","title":"Fundamental examples of height functions on closed manifolds and their 1st derivatives","version":1},"reference_index":17,"source":"pdf_text","source_observed_at":"2026-08-05T00:54:16.202466Z"},"links":{"cited_paper":"/paper/1902.08841","citing_paper":"/paper/2608.00556"},"observation_digest":"sha256:c099506a375c80793aa0cbb2650fb56807a4aa105bbba6c247bad7c63d22959b","observation_id":"25dea9b2-7d08-407b-baa3-58ab36094df2","resolution":{"observed_at":"2026-08-05T00:54:16.202466Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":{"arxiv_id":"1908.04340","last_updated":"2022-03-25T10:19:39Z","snapshot_observed_at":"2026-08-13T13:30:04.387820Z","submitted_at":"2019-08-12T19:11:23Z","title":"On Reeb graphs induced from smooth functions on closed or open manifolds","version":8},"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":"1908.04340","snapshot_observed_at":"2026-08-05T00:54:16.362099Z","title":"Kitazawa, On Reeb graphs induced from smooth functions on closed or ope n sur- faces, Methods of Functional Analysis and Topology Vol","venue":null,"work_id":null,"year":1908},"citing_paper":{"arxiv_id":"2608.00556","last_updated":"2026-08-01T09:21:36Z","snapshot_observed_at":"2026-08-10T00:06:11.727871Z","submitted_at":"2026-08-01T09:21:36Z","title":"Fundamental examples of height functions on closed manifolds and their 1st derivatives","version":1},"reference_index":18,"source":"pdf_text","source_observed_at":"2026-08-05T00:54:16.362099Z"},"links":{"cited_paper":"/paper/1908.04340","citing_paper":"/paper/2608.00556"},"observation_digest":"sha256:7b69242c0cdf41fcb91c4f1571360cc748fdf73928934f41406cf51a94acc9d3","observation_id":"d4a3b6e0-0d81-40bf-ae2f-ad35a2329ec2","resolution":{"observed_at":"2026-08-05T00:54:16.362099Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":{"arxiv_id":"2302.02339","last_updated":"2023-02-19T13:58:00Z","snapshot_observed_at":"2026-08-13T12:50:45.488191Z","submitted_at":"2023-02-05T09:01:31Z","title":"Real algebraic functions on closed manifolds whose Reeb spaces are given graphs","version":3},"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":"2302.02339","snapshot_observed_at":"2026-08-05T00:54:16.552763Z","title":"Kitazawa, Real algebraic functions on closed manifolds whose Reeb gra phs are given graphs, Methods of Functional Analysis and Topology Vol","venue":null,"work_id":null,"year":2023},"citing_paper":{"arxiv_id":"2608.00556","last_updated":"2026-08-01T09:21:36Z","snapshot_observed_at":"2026-08-10T00:06:11.727871Z","submitted_at":"2026-08-01T09:21:36Z","title":"Fundamental examples of height functions on closed manifolds and their 1st derivatives","version":1},"reference_index":19,"source":"pdf_text","source_observed_at":"2026-08-05T00:54:16.552763Z"},"links":{"cited_paper":"/paper/2302.02339","citing_paper":"/paper/2608.00556"},"observation_digest":"sha256:b6de35cf62a8ccdc5c6cd652ffcc68adf9fe8e7f5dda5094971a1bf1daaf0ba3","observation_id":"0f8ead0f-b7bb-4df8-a540-ae1b4d4afbb4","resolution":{"observed_at":"2026-08-05T00:54:16.552763Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":{"arxiv_id":"2108.06913","last_updated":"2023-06-15T13:04:46Z","snapshot_observed_at":"2026-08-13T18:28:19.665615Z","submitted_at":"2021-08-16T06:17:58Z","title":"Realization problems of graphs as Reeb graphs of Morse functions with prescribed preimages","version":4},"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":"2108.06913","snapshot_observed_at":"2026-08-05T00:54:16.725125Z","title":"Kitazawa, Constructing Morse functions with given Reeb graphs and lev el sets , accepted for publication in Topol","venue":null,"work_id":null,"year":2025},"citing_paper":{"arxiv_id":"2608.00556","last_updated":"2026-08-01T09:21:36Z","snapshot_observed_at":"2026-08-10T00:06:11.727871Z","submitted_at":"2026-08-01T09:21:36Z","title":"Fundamental examples of height functions on closed manifolds and their 1st derivatives","version":1},"reference_index":20,"source":"pdf_text","source_observed_at":"2026-08-05T00:54:16.725125Z"},"links":{"cited_paper":"/paper/2108.06913","citing_paper":"/paper/2608.00556"},"observation_digest":"sha256:7db5c038e6f0eb1b519963d31f55b80b20e583dc057c7f4e17a47344e05cfe6b","observation_id":"e167c158-d63a-4da3-8877-28e5efa14e5a","resolution":{"observed_at":"2026-08-05T00:54:16.725125Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":{"arxiv_id":"2303.10723","last_updated":"2024-09-15T06:55:00Z","snapshot_observed_at":"2026-08-13T12:21:51.108545Z","submitted_at":"2023-03-19T17:40:33Z","title":"Reconstructing real algebraic maps locally like moment-maps with prescribed images and compositions with the canonical projections to the $1$-dimensional real affine space","version":8},"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":"2303.10723","snapshot_observed_at":"2026-08-05T00:54:16.889552Z","title":null,"venue":null,"work_id":null,"year":null},"citing_paper":{"arxiv_id":"2608.00556","last_updated":"2026-08-01T09:21:36Z","snapshot_observed_at":"2026-08-10T00:06:11.727871Z","submitted_at":"2026-08-01T09:21:36Z","title":"Fundamental examples of height functions on closed manifolds and their 1st derivatives","version":1},"reference_index":21,"source":"pdf_text","source_observed_at":"2026-08-05T00:54:16.889552Z"},"links":{"cited_paper":"/paper/2303.10723","citing_paper":"/paper/2608.00556"},"observation_digest":"sha256:3f10329c5840c2995529c344485e7477575ad10f3231e5c72b87320774580ac2","observation_id":"6ec27fd1-de96-4a0b-9da2-606714af954b","resolution":{"observed_at":"2026-08-05T00:54:16.889552Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":{"arxiv_id":"2312.10646","last_updated":"2025-03-27T06:41:16Z","snapshot_observed_at":"2026-08-13T05:00:09.431995Z","submitted_at":"2023-12-17T08:12:55Z","title":"Some remark on real algebraic maps which are topologically special generic maps and generalize the canonical projections of the unit spheres","version":6},"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":"2312.10646","snapshot_observed_at":"2026-08-05T00:54:16.949117Z","title":"Kitazawa, Some remark on real algebraic maps which are topologically s pecial generic maps and generalize the canonical projections of the unit sp heres, arXiv:2312.10646, 2024","venue":null,"work_id":null,"year":2024},"citing_paper":{"arxiv_id":"2608.00556","last_updated":"2026-08-01T09:21:36Z","snapshot_observed_at":"2026-08-10T00:06:11.727871Z","submitted_at":"2026-08-01T09:21:36Z","title":"Fundamental examples of height functions on closed manifolds and their 1st derivatives","version":1},"reference_index":22,"source":"pdf_text","source_observed_at":"2026-08-05T00:54:16.949117Z"},"links":{"cited_paper":"/paper/2312.10646","citing_paper":"/paper/2608.00556"},"observation_digest":"sha256:e82bdf8ce3342454b4a0e2ba2037b8f519119ed8f390d3d0d56be2dd13110975","observation_id":"40703d07-9ce5-4642-bacf-c546e69c5d52","resolution":{"observed_at":"2026-08-05T00:54:16.949117Z","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":"2026-08-05T00:54:16.968854Z","title":null,"venue":null,"work_id":null,"year":2026},"citing_paper":{"arxiv_id":"2608.00556","last_updated":"2026-08-01T09:21:36Z","snapshot_observed_at":"2026-08-10T00:06:11.727871Z","submitted_at":"2026-08-01T09:21:36Z","title":"Fundamental examples of height functions on closed manifolds and their 1st derivatives","version":1},"reference_index":23,"source":"pdf_text","source_observed_at":"2026-08-05T00:54:16.968854Z"},"links":{"citing_paper":"/paper/2608.00556"},"observation_digest":"sha256:d91887d8c185f0344eaf0aa9d4fc548caa382f50150f6fce50f609432c1486be","observation_id":"64fb4e93-a10e-4c0f-bd7e-bcd2450267a4","resolution":{"observed_at":"2026-08-05T00:54:16.968854Z","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":"2026-08-05T00:54:16.972320Z","title":"Kitazawa, Fundamental examples of Reeb spaces of smooth functions deﬁ ned from two graphs of smooth functions with same asymptotic behaviors , arXiv:2602.17014, 2026/2","venue":null,"work_id":null,"year":2026},"citing_paper":{"arxiv_id":"2608.00556","last_updated":"2026-08-01T09:21:36Z","snapshot_observed_at":"2026-08-10T00:06:11.727871Z","submitted_at":"2026-08-01T09:21:36Z","title":"Fundamental examples of height functions on closed manifolds and their 1st derivatives","version":1},"reference_index":24,"source":"pdf_text","source_observed_at":"2026-08-05T00:54:16.972320Z"},"links":{"citing_paper":"/paper/2608.00556"},"observation_digest":"sha256:3765aacfff3ef97e3aab60f5ca7eff0de01001b4657bb4bde52b3e51ce03042e","observation_id":"3fb814f6-cb3f-411c-9291-d460e7c39494","resolution":{"observed_at":"2026-08-05T00:54:16.972320Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":{"arxiv_id":"2607.17180","last_updated":"2026-07-27T08:31:17Z","snapshot_observed_at":"2026-08-05T19:57:55.637664Z","submitted_at":"2026-07-19T10:35:38Z","title":"Regions represented as foliated forms and natural smooth maps onto them","version":2},"cited_work":{"arxiv_id":"2607.17180","doi":null,"metadata_source":"pith","pith_arxiv_id":"2607.17180","snapshot_observed_at":"2026-08-05T00:54:17.255244Z","title":"Regions represented as foliated forms and natural smooth maps onto them","venue":"math.AG","work_id":"d7b64f80-3cd5-4bcf-b1d6-d6db296f4a60","year":2026},"citing_paper":{"arxiv_id":"2608.00556","last_updated":"2026-08-01T09:21:36Z","snapshot_observed_at":"2026-08-10T00:06:11.727871Z","submitted_at":"2026-08-01T09:21:36Z","title":"Fundamental examples of height functions on closed manifolds and their 1st derivatives","version":1},"reference_index":25,"source":"pdf_text","source_observed_at":"2026-08-05T00:54:16.975440Z"},"links":{"cited_paper":"/paper/2607.17180","citing_paper":"/paper/2608.00556"},"observation_digest":"sha256:85054b533f2240f06e5a716cc0642e787fefba97d9a6c14b2b23d9a84124aaec","observation_id":"c3ca4fb5-96b6-464e-9652-3f411e29625f","resolution":{"observed_at":"2026-08-05T00:54:17.259564Z","resolver_source":"local_arxiv","status":"verified_exact"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-13T06:32:02.005865+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-13T06:32:02.005865+00:00","source":"crossref"},{"observed_at":"2026-08-13T06:31:53.387327+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":"2026-08-05T00:54:17.775019Z","title":null,"venue":null,"work_id":"d48eb8fa-c734-4998-b934-753a86181590","year":1999},"citing_paper":{"arxiv_id":"2608.00556","last_updated":"2026-08-01T09:21:36Z","snapshot_observed_at":"2026-08-10T00:06:11.727871Z","submitted_at":"2026-08-01T09:21:36Z","title":"Fundamental examples of height functions on closed manifolds and their 1st derivatives","version":1},"reference_index":26,"source":"pdf_text","source_observed_at":"2026-08-05T00:54:16.978966Z"},"links":{"citing_paper":"/paper/2608.00556"},"observation_digest":"sha256:130fe5db1438bdc67534782c095335b35854d0b789466d2573ba8a5912ecd1b3","observation_id":"c14033b5-e6dd-4074-8dae-9c2f6d178c22","resolution":{"observed_at":"2026-08-05T00:54:17.778289Z","resolver_source":"raw_fallback","status":"unresolved"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-13T06:32:02.005865+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-13T06:32:02.005865+00:00","source":"crossref"},{"observed_at":"2026-08-13T06:31:53.387327+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":"2026-08-05T00:54:16.981839Z","title":"Martinez-Alfaro, I","venue":null,"work_id":null,"year":2016},"citing_paper":{"arxiv_id":"2608.00556","last_updated":"2026-08-01T09:21:36Z","snapshot_observed_at":"2026-08-10T00:06:11.727871Z","submitted_at":"2026-08-01T09:21:36Z","title":"Fundamental examples of height functions on closed manifolds and their 1st derivatives","version":1},"reference_index":27,"source":"pdf_text","source_observed_at":"2026-08-05T00:54:16.981839Z"},"links":{"citing_paper":"/paper/2608.00556"},"observation_digest":"sha256:f0517d2c6779473fabccabb3c14dc2d2350b62602c219bece1a51621acdcad9b","observation_id":"202d5e83-e071-4a28-a670-28eab3a446c7","resolution":{"observed_at":"2026-08-05T00:54:16.981839Z","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":"2026-08-05T00:54:16.985400Z","title":"Masumoto and O","venue":null,"work_id":null,"year":2011},"citing_paper":{"arxiv_id":"2608.00556","last_updated":"2026-08-01T09:21:36Z","snapshot_observed_at":"2026-08-10T00:06:11.727871Z","submitted_at":"2026-08-01T09:21:36Z","title":"Fundamental examples of height functions on closed manifolds and their 1st derivatives","version":1},"reference_index":28,"source":"pdf_text","source_observed_at":"2026-08-05T00:54:16.985400Z"},"links":{"citing_paper":"/paper/2608.00556"},"observation_digest":"sha256:bb3e60eaf9551d7f84e17b49e3526b889bcaa9b6c777418673015943b7f8ccb3","observation_id":"37ef1885-c803-462a-b8d5-fc5faf56922e","resolution":{"observed_at":"2026-08-05T00:54:16.985400Z","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":"2026-08-05T00:54:17.753509Z","title":"N Mather, Stability of C∞ mappings","venue":null,"work_id":"b025a8b7-50f0-4d6f-b616-519f9ef1b1ca","year":1968},"citing_paper":{"arxiv_id":"2608.00556","last_updated":"2026-08-01T09:21:36Z","snapshot_observed_at":"2026-08-10T00:06:11.727871Z","submitted_at":"2026-08-01T09:21:36Z","title":"Fundamental examples of height functions on closed manifolds and their 1st derivatives","version":1},"reference_index":29,"source":"pdf_text","source_observed_at":"2026-08-05T00:54:16.988501Z"},"links":{"citing_paper":"/paper/2608.00556"},"observation_digest":"sha256:44d2182350430ccf1c5942707ced5fad92fa41813a6c4dc85f68508f37bed31c","observation_id":"c4253306-5c6d-43ad-810d-8e85c3b41d59","resolution":{"observed_at":"2026-08-05T00:54:17.756624Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-13T06:32:02.005865+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-13T06:32:02.005865+00:00","source":"crossref"},{"observed_at":"2026-08-13T06:31:53.387327+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":"2026-08-05T00:54:17.744128Z","title":null,"venue":null,"work_id":"84b189fb-abc6-4e5a-b73a-77ca671aec76","year":1968},"citing_paper":{"arxiv_id":"2608.00556","last_updated":"2026-08-01T09:21:36Z","snapshot_observed_at":"2026-08-10T00:06:11.727871Z","submitted_at":"2026-08-01T09:21:36Z","title":"Fundamental examples of height functions on closed manifolds and their 1st derivatives","version":1},"reference_index":30,"source":"pdf_text","source_observed_at":"2026-08-05T00:54:16.991252Z"},"links":{"citing_paper":"/paper/2608.00556"},"observation_digest":"sha256:9ac219a4d336c236bcf029e8f7bc4dfcbceb4a355256d01c1d862eac9f62a82a","observation_id":"9b511d86-c346-4bfd-98de-684683188e4c","resolution":{"observed_at":"2026-08-05T00:54:17.747173Z","resolver_source":"raw_fallback","status":"unresolved"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-13T06:32:02.005865+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-13T06:32:02.005865+00:00","source":"crossref"},{"observed_at":"2026-08-13T06:31:53.387327+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":"2026-08-05T00:54:17.733355Z","title":null,"venue":null,"work_id":"09deef3d-ac4e-4001-95b0-711f14327219","year":1968},"citing_paper":{"arxiv_id":"2608.00556","last_updated":"2026-08-01T09:21:36Z","snapshot_observed_at":"2026-08-10T00:06:11.727871Z","submitted_at":"2026-08-01T09:21:36Z","title":"Fundamental examples of height functions on closed manifolds and their 1st derivatives","version":1},"reference_index":31,"source":"pdf_text","source_observed_at":"2026-08-05T00:54:16.994262Z"},"links":{"citing_paper":"/paper/2608.00556"},"observation_digest":"sha256:5324aec8cc6e7ff980cb3dd1858b72c3ce7c0a630758ba8945f578da1988d57a","observation_id":"b9c72295-e211-4335-bc9d-b57615126db9","resolution":{"observed_at":"2026-08-05T00:54:17.737117Z","resolver_source":"raw_fallback","status":"unresolved"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-13T06:32:02.005865+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-13T06:32:02.005865+00:00","source":"crossref"},{"observed_at":"2026-08-13T06:31:53.387327+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":"2026-08-05T00:54:17.723420Z","title":null,"venue":null,"work_id":"cf3faf42-2bd7-4324-87a4-040e36ee17d4","year":1969},"citing_paper":{"arxiv_id":"2608.00556","last_updated":"2026-08-01T09:21:36Z","snapshot_observed_at":"2026-08-10T00:06:11.727871Z","submitted_at":"2026-08-01T09:21:36Z","title":"Fundamental examples of height functions on closed manifolds and their 1st derivatives","version":1},"reference_index":32,"source":"pdf_text","source_observed_at":"2026-08-05T00:54:16.997115Z"},"links":{"citing_paper":"/paper/2608.00556"},"observation_digest":"sha256:808db535a0a2c79260fb83b30eafc2f643947d1f41fc07fad0a26bed326735db","observation_id":"1a3a6a9d-2ef2-435a-891d-50826283d9ab","resolution":{"observed_at":"2026-08-05T00:54:17.726540Z","resolver_source":"raw_fallback","status":"unresolved"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-13T06:32:02.005865+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-13T06:32:02.005865+00:00","source":"crossref"},{"observed_at":"2026-08-13T06:31:53.387327+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":"2026-08-05T00:54:17.713709Z","title":null,"venue":null,"work_id":"33f207be-a0d7-4d51-8288-5f6a8292f4af","year":1969},"citing_paper":{"arxiv_id":"2608.00556","last_updated":"2026-08-01T09:21:36Z","snapshot_observed_at":"2026-08-10T00:06:11.727871Z","submitted_at":"2026-08-01T09:21:36Z","title":"Fundamental examples of height functions on closed manifolds and their 1st derivatives","version":1},"reference_index":33,"source":"pdf_text","source_observed_at":"2026-08-05T00:54:17.000472Z"},"links":{"citing_paper":"/paper/2608.00556"},"observation_digest":"sha256:4a8cc15afcc6a685323d639c79674c5f711ee8484c25af32d815e3555c626236","observation_id":"24a45bc1-332f-49cb-9df1-bf4c023b5965","resolution":{"observed_at":"2026-08-05T00:54:17.717098Z","resolver_source":"raw_fallback","status":"unresolved"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-13T06:32:02.005865+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-13T06:32:02.005865+00:00","source":"crossref"},{"observed_at":"2026-08-13T06:31:53.387327+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":"2026-08-05T00:54:17.704118Z","title":null,"venue":null,"work_id":"cddd51b4-3646-449c-bfd4-802767d30cbd","year":1970},"citing_paper":{"arxiv_id":"2608.00556","last_updated":"2026-08-01T09:21:36Z","snapshot_observed_at":"2026-08-10T00:06:11.727871Z","submitted_at":"2026-08-01T09:21:36Z","title":"Fundamental examples of height functions on closed manifolds and their 1st derivatives","version":1},"reference_index":34,"source":"pdf_text","source_observed_at":"2026-08-05T00:54:17.003327Z"},"links":{"citing_paper":"/paper/2608.00556"},"observation_digest":"sha256:4f1af5b476e048d491ea7871907221753057918e8bb67b2b73309231a4eee09d","observation_id":"1c2b0c48-01f8-4a69-a000-a0cf0b780c29","resolution":{"observed_at":"2026-08-05T00:54:17.707407Z","resolver_source":"raw_fallback","status":"unresolved"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-13T06:32:02.005865+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-13T06:32:02.005865+00:00","source":"crossref"},{"observed_at":"2026-08-13T06:31:53.387327+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":"2026-08-05T00:54:17.694658Z","title":null,"venue":null,"work_id":"99cc3736-1605-4eab-b830-5f6795cc60e5","year":1969},"citing_paper":{"arxiv_id":"2608.00556","last_updated":"2026-08-01T09:21:36Z","snapshot_observed_at":"2026-08-10T00:06:11.727871Z","submitted_at":"2026-08-01T09:21:36Z","title":"Fundamental examples of height functions on closed manifolds and their 1st derivatives","version":1},"reference_index":35,"source":"pdf_text","source_observed_at":"2026-08-05T00:54:17.006103Z"},"links":{"citing_paper":"/paper/2608.00556"},"observation_digest":"sha256:55123d6180a9c13b6c96648c3da1a264135bd0776cbd245a99333a08f48e292e","observation_id":"29433566-c28b-4599-ae78-c712cbb5070a","resolution":{"observed_at":"2026-08-05T00:54:17.697800Z","resolver_source":"raw_fallback","status":"unresolved"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-13T06:32:02.005865+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-13T06:32:02.005865+00:00","source":"crossref"},{"observed_at":"2026-08-13T06:31:53.387327+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":"2026-08-05T00:54:17.684504Z","title":"Mather, Notes on topological stability , Bull","venue":null,"work_id":"2202776f-e5cf-4e0f-a8cf-9f31129a7c80","year":2012},"citing_paper":{"arxiv_id":"2608.00556","last_updated":"2026-08-01T09:21:36Z","snapshot_observed_at":"2026-08-10T00:06:11.727871Z","submitted_at":"2026-08-01T09:21:36Z","title":"Fundamental examples of height functions on closed manifolds and their 1st derivatives","version":1},"reference_index":36,"source":"pdf_text","source_observed_at":"2026-08-05T00:54:17.009122Z"},"links":{"citing_paper":"/paper/2608.00556"},"observation_digest":"sha256:77759f9dbd8d16de1f71224abb1e74f814a5275a650baac3d1240b939f000fb2","observation_id":"f6080977-5c33-4fe4-b0a8-b3495cc054c6","resolution":{"observed_at":"2026-08-05T00:54:17.687647Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-13T06:32:02.005865+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-13T06:32:02.005865+00:00","source":"crossref"},{"observed_at":"2026-08-13T06:31:53.387327+00:00","source":"retraction_watch"}],"state":"measured"}},{"citation":{"cited_paper":{"arxiv_id":"1805.06727","last_updated":"2018-09-17T09:56:21Z","snapshot_observed_at":"2026-08-02T14:25:22.727238Z","submitted_at":"2018-05-17T12:36:45Z","title":"Realization of a graph as the Reeb graph of a Morse function on a manifold","version":2},"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":"1805.06727","snapshot_observed_at":"2026-08-05T00:54:17.011798Z","title":null,"venue":null,"work_id":null,"year":2018},"citing_paper":{"arxiv_id":"2608.00556","last_updated":"2026-08-01T09:21:36Z","snapshot_observed_at":"2026-08-10T00:06:11.727871Z","submitted_at":"2026-08-01T09:21:36Z","title":"Fundamental examples of height functions on closed manifolds and their 1st derivatives","version":1},"reference_index":37,"source":"pdf_text","source_observed_at":"2026-08-05T00:54:17.011798Z"},"links":{"cited_paper":"/paper/1805.06727","citing_paper":"/paper/2608.00556"},"observation_digest":"sha256:85a47e8e12727d535666c2975dd5b7b2489ca61629d1929f32dfdff315ccf44f","observation_id":"719236fb-8068-4ec7-86be-65597ec51c18","resolution":{"observed_at":"2026-08-05T00:54:17.011798Z","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":"2026-08-05T00:54:17.014734Z","title":"Milnor, Morse Theory , Annals of Mathematic Studies AM-51, Princeton University Press; 1st Edition (1963.5.1)","venue":null,"work_id":null,"year":1963},"citing_paper":{"arxiv_id":"2608.00556","last_updated":"2026-08-01T09:21:36Z","snapshot_observed_at":"2026-08-10T00:06:11.727871Z","submitted_at":"2026-08-01T09:21:36Z","title":"Fundamental examples of height functions on closed manifolds and their 1st derivatives","version":1},"reference_index":38,"source":"pdf_text","source_observed_at":"2026-08-05T00:54:17.014734Z"},"links":{"citing_paper":"/paper/2608.00556"},"observation_digest":"sha256:622918dcc5b21c46c26d250b7a86b6359b3db1aa273b075474b54ab02efe9a25","observation_id":"da1fbe2a-7b74-420f-a84d-400b9e309c74","resolution":{"observed_at":"2026-08-05T00:54:17.014734Z","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":"2026-08-05T00:54:17.668761Z","title":"Milnor, Lectures on the h-cobordism theorem , Math","venue":null,"work_id":"7be24752-83d9-4505-9151-7f3a9cb87b40","year":1965},"citing_paper":{"arxiv_id":"2608.00556","last_updated":"2026-08-01T09:21:36Z","snapshot_observed_at":"2026-08-10T00:06:11.727871Z","submitted_at":"2026-08-01T09:21:36Z","title":"Fundamental examples of height functions on closed manifolds and their 1st derivatives","version":1},"reference_index":39,"source":"pdf_text","source_observed_at":"2026-08-05T00:54:17.017497Z"},"links":{"citing_paper":"/paper/2608.00556"},"observation_digest":"sha256:8fc781b8f5b78f66b3393f53f811aeb24c22589fa49d12f48c1511f2714d3638","observation_id":"8f2c1365-65b5-4431-b5fc-74e069db44c5","resolution":{"observed_at":"2026-08-05T00:54:17.672063Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-13T06:32:02.005865+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-13T06:32:02.005865+00:00","source":"crossref"},{"observed_at":"2026-08-13T06:31:53.387327+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":"2026-08-05T00:54:17.658874Z","title":"Morishita and O","venue":null,"work_id":"be5f30ed-d6e0-44bd-8f11-dd2417b7bf33","year":null},"citing_paper":{"arxiv_id":"2608.00556","last_updated":"2026-08-01T09:21:36Z","snapshot_observed_at":"2026-08-10T00:06:11.727871Z","submitted_at":"2026-08-01T09:21:36Z","title":"Fundamental examples of height functions on closed manifolds and their 1st derivatives","version":1},"reference_index":40,"source":"pdf_text","source_observed_at":"2026-08-05T00:54:17.020272Z"},"links":{"citing_paper":"/paper/2608.00556"},"observation_digest":"sha256:a193f207eb4c206c0588a47fbef8ffb909459106f4f35d96aa06bc800aefbd08","observation_id":"c8932ef7-b193-4fee-8297-12fefe3896aa","resolution":{"observed_at":"2026-08-05T00:54:17.662043Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-13T06:32:02.005865+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-13T06:32:02.005865+00:00","source":"crossref"},{"observed_at":"2026-08-13T06:31:53.387327+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":"2026-08-05T00:54:17.023546Z","title":null,"venue":null,"work_id":null,"year":1946},"citing_paper":{"arxiv_id":"2608.00556","last_updated":"2026-08-01T09:21:36Z","snapshot_observed_at":"2026-08-10T00:06:11.727871Z","submitted_at":"2026-08-01T09:21:36Z","title":"Fundamental examples of height functions on closed manifolds and their 1st derivatives","version":1},"reference_index":41,"source":"pdf_text","source_observed_at":"2026-08-05T00:54:17.023546Z"},"links":{"citing_paper":"/paper/2608.00556"},"observation_digest":"sha256:8310b3113c5934b27c25f70f97c5451ddf313b39ff03f2d08c8f4e3dc4906f7b","observation_id":"5fc54445-424e-491d-b92e-fe8431715269","resolution":{"observed_at":"2026-08-05T00:54:17.023546Z","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":"2026-08-05T00:54:17.026457Z","title":"Saeki, Notes on the topology of folds , J","venue":null,"work_id":null,"year":1992},"citing_paper":{"arxiv_id":"2608.00556","last_updated":"2026-08-01T09:21:36Z","snapshot_observed_at":"2026-08-10T00:06:11.727871Z","submitted_at":"2026-08-01T09:21:36Z","title":"Fundamental examples of height functions on closed manifolds and their 1st derivatives","version":1},"reference_index":42,"source":"pdf_text","source_observed_at":"2026-08-05T00:54:17.026457Z"},"links":{"citing_paper":"/paper/2608.00556"},"observation_digest":"sha256:9ab44c64f088643f354dd9ad793c9f7dd78c59bb796c3eedfdeeb35fcf0b4d32","observation_id":"4e420f73-c8ba-47b4-8b0a-e2c9aa22c61b","resolution":{"observed_at":"2026-08-05T00:54:17.026457Z","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":"2026-08-05T00:54:17.029470Z","title":"Saeki, Topology of special generic maps of manifolds into Euclidea n spaces, Topology Appl","venue":null,"work_id":null,"year":1993},"citing_paper":{"arxiv_id":"2608.00556","last_updated":"2026-08-01T09:21:36Z","snapshot_observed_at":"2026-08-10T00:06:11.727871Z","submitted_at":"2026-08-01T09:21:36Z","title":"Fundamental examples of height functions on closed manifolds and their 1st derivatives","version":1},"reference_index":43,"source":"pdf_text","source_observed_at":"2026-08-05T00:54:17.029470Z"},"links":{"citing_paper":"/paper/2608.00556"},"observation_digest":"sha256:b7483aca45bba3cba588ea79ee87ef8bf3f62f8b538e235eae2799e33b82bdfa","observation_id":"52c98521-acc9-4762-ad0f-c948e8543e3b","resolution":{"observed_at":"2026-08-05T00:54:17.029470Z","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":"2026-08-05T00:54:17.032535Z","title":"Saeki, Topology of singular ﬁbers of diﬀerentiable maps , Lecture Notes in Math., Vol","venue":null,"work_id":null,"year":2004},"citing_paper":{"arxiv_id":"2608.00556","last_updated":"2026-08-01T09:21:36Z","snapshot_observed_at":"2026-08-10T00:06:11.727871Z","submitted_at":"2026-08-01T09:21:36Z","title":"Fundamental examples of height functions on closed manifolds and their 1st derivatives","version":1},"reference_index":44,"source":"pdf_text","source_observed_at":"2026-08-05T00:54:17.032535Z"},"links":{"citing_paper":"/paper/2608.00556"},"observation_digest":"sha256:37c34dc28b64f27443c451742e7539e1e79611964df8eec1ba047de8fefd12da","observation_id":"c9482f1b-ed65-4c49-848b-5146fcad1bb3","resolution":{"observed_at":"2026-08-05T00:54:17.032535Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":"2006.0168","doi":null,"metadata_source":"raw_reference","pith_arxiv_id":null,"snapshot_observed_at":"2026-08-05T00:54:17.170515Z","title":null,"venue":null,"work_id":"52f420f2-2d14-497f-8fe7-8d298a6ba52a","year":2022},"citing_paper":{"arxiv_id":"2608.00556","last_updated":"2026-08-01T09:21:36Z","snapshot_observed_at":"2026-08-10T00:06:11.727871Z","submitted_at":"2026-08-01T09:21:36Z","title":"Fundamental examples of height functions on closed manifolds and their 1st derivatives","version":1},"reference_index":45,"source":"pdf_text","source_observed_at":"2026-08-05T00:54:17.035269Z"},"links":{"citing_paper":"/paper/2608.00556"},"observation_digest":"sha256:736f51415ae663beae4624ec0dc5af2dcfb40aef310da9cddd7d74cb8ae8e526","observation_id":"b8a750d7-7be9-41b8-b7fc-4df79261a713","resolution":{"observed_at":"2026-08-05T00:54:17.176363Z","resolver_source":"raw_fallback","status":"verified_exact"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-13T06:32:02.005865+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-13T06:32:02.005865+00:00","source":"crossref"},{"observed_at":"2026-08-13T06:31:53.387327+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":"2026-08-05T00:54:17.038102Z","title":"Saeki, Reeb spaces of smooth functions on manifolds II , Res","venue":null,"work_id":null,"year":2024},"citing_paper":{"arxiv_id":"2608.00556","last_updated":"2026-08-01T09:21:36Z","snapshot_observed_at":"2026-08-10T00:06:11.727871Z","submitted_at":"2026-08-01T09:21:36Z","title":"Fundamental examples of height functions on closed manifolds and their 1st derivatives","version":1},"reference_index":46,"source":"pdf_text","source_observed_at":"2026-08-05T00:54:17.038102Z"},"links":{"citing_paper":"/paper/2608.00556"},"observation_digest":"sha256:f57bfe86ddf581c377a7d1509271307d51215b9f8d667d2ed28a85289a5feb22","observation_id":"a7c7dd44-499a-441e-9014-8c9191a96e44","resolution":{"observed_at":"2026-08-05T00:54:17.038102Z","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":"2026-08-05T00:54:17.041081Z","title":"Sharko, About Kronrod-Reeb graph of a function on a manifold , Methods of Functional Analysis and Topology 12 (2006), 389–396","venue":null,"work_id":null,"year":2006},"citing_paper":{"arxiv_id":"2608.00556","last_updated":"2026-08-01T09:21:36Z","snapshot_observed_at":"2026-08-10T00:06:11.727871Z","submitted_at":"2026-08-01T09:21:36Z","title":"Fundamental examples of height functions on closed manifolds and their 1st derivatives","version":1},"reference_index":47,"source":"pdf_text","source_observed_at":"2026-08-05T00:54:17.041081Z"},"links":{"citing_paper":"/paper/2608.00556"},"observation_digest":"sha256:0ebbe1bc31f5d6cc42113fb8021dd43422fe4c579f10cf22335ef66926565a4f","observation_id":"9fbb6d04-a343-413d-b967-69b9c20dd7eb","resolution":{"observed_at":"2026-08-05T00:54:17.041081Z","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":"2026-08-05T00:54:17.044071Z","title":"Thom, Les singularites des applications diﬀerentiables , Ann","venue":null,"work_id":null,"year":1955},"citing_paper":{"arxiv_id":"2608.00556","last_updated":"2026-08-01T09:21:36Z","snapshot_observed_at":"2026-08-10T00:06:11.727871Z","submitted_at":"2026-08-01T09:21:36Z","title":"Fundamental examples of height functions on closed manifolds and their 1st derivatives","version":1},"reference_index":48,"source":"pdf_text","source_observed_at":"2026-08-05T00:54:17.044071Z"},"links":{"citing_paper":"/paper/2608.00556"},"observation_digest":"sha256:609b28cf956cdfedea0247fd7df68aba86155dcd29eba0c75dca411a8d0cf6e9","observation_id":"27f67d91-8329-41aa-b1e3-13ad555b6f8e","resolution":{"observed_at":"2026-08-05T00:54:17.044071Z","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":"2026-08-05T00:54:17.047306Z","title":"Whitney, On singularities of mappings of Euclidean spaces: I, mappin gs of the plane into the plane , Ann","venue":null,"work_id":null,"year":1955},"citing_paper":{"arxiv_id":"2608.00556","last_updated":"2026-08-01T09:21:36Z","snapshot_observed_at":"2026-08-10T00:06:11.727871Z","submitted_at":"2026-08-01T09:21:36Z","title":"Fundamental examples of height functions on closed manifolds and their 1st derivatives","version":1},"reference_index":49,"source":"pdf_text","source_observed_at":"2026-08-05T00:54:17.047306Z"},"links":{"citing_paper":"/paper/2608.00556"},"observation_digest":"sha256:de2a0b89fe9bbd43d35b235e723c98c1ba67adcb1223bec9dd5182db896806ed","observation_id":"1ad9c0c5-39f0-49a3-8baf-5546be47fc18","resolution":{"observed_at":"2026-08-05T00:54:17.047306Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}}],"paper":{"arxiv_id":"2608.00556","last_updated":"2026-08-01T09:21:36Z","latest_version":1,"primary_category":"math.DG","snapshot_observed_at":"2026-08-10T00:06:11.727871Z","submitted_at":"2026-08-01T09:21:36Z","title":"Fundamental examples of height functions on closed manifolds and their 1st derivatives"},"reference_resolution":{"displayed":49,"state_counts":{"malformed_identifier":3,"metadata_mismatch":0,"parse_uncertain":0,"unresolved":34,"verified_exact":3,"verified_fuzzy":9},"total_outbound_references":49},"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":"2026-08-13T06:32:02.005865+00:00","source":"crossref"},{"observed_at":"2026-08-13T06:31:53.387327+00:00","source":"retraction_watch"}],"thesis":"As of 13 August 2026, this Paper Citation Record lists 49 of 49 outbound references and 0 inbound Pith citation observations for arXiv:2608.00556."}