{"as_of":"2026-08-05T09:10:00Z","caps":{"database_statements":6,"inbound":100,"outbound":100},"context_digest":"sha256:350340879c13759ef86a55f580771789660351ae82a999e10150711d4b37ab00","coverage":[{"denominator":25,"lane":"reference_resolution","note":"Typed states for the displayed outbound observations.","records_observed":25,"source":"paper_references, paper_reference_links","source_observed_at":"2026-05-10T06:00:03.812477Z","state":"measured"},{"denominator":25,"lane":"standing_notices","note":"One-hop event checks from named stored sources.","records_observed":25,"source":"scholarly_work_events, retraction_status_cache","source_observed_at":"2026-08-05T06:32:48.257954+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/2604.17445/citation-record","integrity":"/paper/2604.17445/integrity","json":"/paper/2604.17445/citation-record.json","paper":"/paper/2604.17445"},"outbound":[{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":"raw_reference","pith_arxiv_id":null,"snapshot_observed_at":"2026-06-05T21:23:00.469572Z","title":"Aigner.Markov’s theorem and 100 years of the uniqueness conjecture","venue":null,"work_id":"d726b8d0-549b-4528-911f-e7e84631f079","year":2013},"citing_paper":{"arxiv_id":"2604.17445","last_updated":"2026-04-19T13:52:56Z","snapshot_observed_at":"2026-07-06T23:04:32.734175Z","submitted_at":"2026-04-19T13:52:56Z","title":"Orderings of Generalized k-Markov Numbers","version":1},"reference_index":1,"source":"pdf_text","source_observed_at":"2026-05-10T06:00:03.812477Z"},"links":{"citing_paper":"/paper/2604.17445"},"observation_digest":"sha256:527cea5d7cef0335361afdc87c3a64cd9e3396d7a9d6c3089dd396f4c845e586","observation_id":"1cd92cce-6fa4-468c-b046-813f7a63783b","resolution":{"observed_at":"2026-05-21T18:15:28.443838Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-05T06:32:48.257954+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-05T06:32:48.257954+00:00","source":"crossref"},{"observed_at":"2026-08-05T06:32:44.755628+00:00","source":"retraction_watch"}],"state":"measured"}},{"citation":{"cited_paper":{"arxiv_id":"2512.04026","last_updated":"2026-04-13T09:13:48Z","snapshot_observed_at":"2026-07-06T22:37:41.555068Z","submitted_at":"2025-12-03T18:02:48Z","title":"Orderings of k-Markov Numbers","version":2},"cited_work":{"arxiv_id":"2512.04026","doi":null,"metadata_source":"pith","pith_arxiv_id":"2512.04026","snapshot_observed_at":"2026-06-29T16:03:33.999944Z","title":"Orderings of k-Markov Numbers","venue":"math.NT","work_id":"4134eb10-4a4a-4a07-b49d-fb4610f40f68","year":2025},"citing_paper":{"arxiv_id":"2604.17445","last_updated":"2026-04-19T13:52:56Z","snapshot_observed_at":"2026-07-06T23:04:32.734175Z","submitted_at":"2026-04-19T13:52:56Z","title":"Orderings of Generalized k-Markov Numbers","version":1},"reference_index":2,"source":"pdf_text","source_observed_at":"2026-05-10T06:00:03.812477Z"},"links":{"cited_paper":"/paper/2512.04026","citing_paper":"/paper/2604.17445"},"observation_digest":"sha256:5f8c08720d298cb19ca52766a61e5b84874fdb361e16b83f6949725da2509649","observation_id":"0ff200d1-e95d-4c4c-8b6f-2858255f634d","resolution":{"observed_at":"2026-05-10T06:01:13.283956Z","resolver_source":"local_arxiv","status":"verified_exact"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-05T06:32:48.257954+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-05T06:32:48.257954+00:00","source":"crossref"},{"observed_at":"2026-08-05T06:32:44.755628+00:00","source":"retraction_watch"}],"state":"measured"}},{"citation":{"cited_paper":{"arxiv_id":"2507.06900","last_updated":"2025-07-22T15:38:20Z","snapshot_observed_at":"2026-08-04T02:32:42.426599Z","submitted_at":"2025-07-09T14:36:31Z","title":"Cluster algebraic interpretation of generalized Markov numbers and their matrixizations","version":2},"cited_work":{"arxiv_id":"2507.06900","doi":null,"metadata_source":"arxiv_reference","pith_arxiv_id":"2507.06900","snapshot_observed_at":"2026-06-30T23:25:07.104696Z","title":"Banaian and Y","venue":null,"work_id":"beb45872-1952-4f6b-ba47-8618501b6f3e","year":2025},"citing_paper":{"arxiv_id":"2604.17445","last_updated":"2026-04-19T13:52:56Z","snapshot_observed_at":"2026-07-06T23:04:32.734175Z","submitted_at":"2026-04-19T13:52:56Z","title":"Orderings of Generalized k-Markov Numbers","version":1},"reference_index":3,"source":"pdf_text","source_observed_at":"2026-05-10T06:00:03.812477Z"},"links":{"cited_paper":"/paper/2507.06900","citing_paper":"/paper/2604.17445"},"observation_digest":"sha256:0c026ce816e5d968f9f69071a952f2ee3fd63ab1babde338c9e4228a35987224","observation_id":"f3e0c6c6-1901-4c2c-be6c-1aa965228f8f","resolution":{"observed_at":"2026-05-10T06:01:13.289096Z","resolver_source":"arxiv_id","status":"verified_exact"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-05T06:32:48.257954+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-05T06:32:48.257954+00:00","source":"crossref"},{"observed_at":"2026-08-05T06:32:44.755628+00:00","source":"retraction_watch"}],"state":"measured"}},{"citation":{"cited_paper":{"arxiv_id":"2409.04957","last_updated":"2024-11-19T23:33:35Z","snapshot_observed_at":"2026-07-06T19:12:02.473662Z","submitted_at":"2024-09-08T03:09:12Z","title":"Skein relations for punctured surfaces","version":2},"cited_work":{"arxiv_id":"2409.04957","doi":null,"metadata_source":"arxiv_reference","pith_arxiv_id":"2409.04957","snapshot_observed_at":"2026-07-04T08:09:41.739450Z","title":"Skein relations for punctured surfaces","venue":null,"work_id":"d491b17e-a3cd-44f8-9d3b-c73ad608037c","year":2024},"citing_paper":{"arxiv_id":"2604.17445","last_updated":"2026-04-19T13:52:56Z","snapshot_observed_at":"2026-07-06T23:04:32.734175Z","submitted_at":"2026-04-19T13:52:56Z","title":"Orderings of Generalized k-Markov Numbers","version":1},"reference_index":4,"source":"pdf_text","source_observed_at":"2026-05-10T06:00:03.812477Z"},"links":{"cited_paper":"/paper/2409.04957","citing_paper":"/paper/2604.17445"},"observation_digest":"sha256:412d58007f801a75ae933dcf44b57c8bf45264d59ead1cbfae6bf9abd0702ebf","observation_id":"abc22947-2620-4612-b6c5-9eeae8625ff5","resolution":{"observed_at":"2026-05-10T06:01:13.281655Z","resolver_source":"arxiv_id","status":"verified_exact"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-05T06:32:48.257954+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-05T06:32:48.257954+00:00","source":"crossref"},{"observed_at":"2026-08-05T06:32:44.755628+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":"2026-06-05T21:23:00.469572Z","title":"Banaian, E","venue":null,"work_id":"e8b553fa-f73e-469c-b680-9f8aaaf29aeb","year":2020},"citing_paper":{"arxiv_id":"2604.17445","last_updated":"2026-04-19T13:52:56Z","snapshot_observed_at":"2026-07-06T23:04:32.734175Z","submitted_at":"2026-04-19T13:52:56Z","title":"Orderings of Generalized k-Markov Numbers","version":1},"reference_index":5,"source":"pdf_text","source_observed_at":"2026-05-10T06:00:03.812477Z"},"links":{"citing_paper":"/paper/2604.17445"},"observation_digest":"sha256:22485c6a14ede4f9ee69a9c2c1c3ab7aa9797faf84c474c2857e87645305cd2c","observation_id":"37558946-ff9e-4ad5-9a68-0f23995d04d6","resolution":{"observed_at":"2026-05-21T18:15:28.437609Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-05T06:32:48.257954+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-05T06:32:48.257954+00:00","source":"crossref"},{"observed_at":"2026-08-05T06:32:44.755628+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":"2026-06-05T21:23:00.469572Z","title":"Banaian and A","venue":null,"work_id":"10efd5dd-3df3-4e89-81cb-e48bfa1778e0","year":2024},"citing_paper":{"arxiv_id":"2604.17445","last_updated":"2026-04-19T13:52:56Z","snapshot_observed_at":"2026-07-06T23:04:32.734175Z","submitted_at":"2026-04-19T13:52:56Z","title":"Orderings of Generalized k-Markov Numbers","version":1},"reference_index":6,"source":"pdf_text","source_observed_at":"2026-05-10T06:00:03.812477Z"},"links":{"citing_paper":"/paper/2604.17445"},"observation_digest":"sha256:20e83671e964a35f0530445bf924d8bd7ca6f54f0a4b756f038d8c4f41503088","observation_id":"5ddad0cd-4e1c-41f1-98cc-ea79170c9ac4","resolution":{"observed_at":"2026-05-21T18:15:28.447043Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-05T06:32:48.257954+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-05T06:32:48.257954+00:00","source":"crossref"},{"observed_at":"2026-08-05T06:32:44.755628+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":"2026-06-05T21:23:00.469572Z","title":"Beineke, T","venue":null,"work_id":"6d9191d6-6e7b-4fcd-8cc6-51a49dda00cf","year":2011},"citing_paper":{"arxiv_id":"2604.17445","last_updated":"2026-04-19T13:52:56Z","snapshot_observed_at":"2026-07-06T23:04:32.734175Z","submitted_at":"2026-04-19T13:52:56Z","title":"Orderings of Generalized k-Markov Numbers","version":1},"reference_index":7,"source":"pdf_text","source_observed_at":"2026-05-10T06:00:03.812477Z"},"links":{"citing_paper":"/paper/2604.17445"},"observation_digest":"sha256:07a1f939fbf5cf28539b232b1862183185d9e3f69ad25541bfccffc8d3318f73","observation_id":"1047461f-e173-4784-bff6-504fb3b7225f","resolution":{"observed_at":"2026-05-21T18:15:28.414618Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-05T06:32:48.257954+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-05T06:32:48.257954+00:00","source":"crossref"},{"observed_at":"2026-08-05T06:32:44.755628+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":"2026-06-05T21:23:00.469572Z","title":"C ¸ anak¸ cı and R","venue":null,"work_id":"6e48fb7a-ae51-46e5-bdd7-909e47cc7102","year":2018},"citing_paper":{"arxiv_id":"2604.17445","last_updated":"2026-04-19T13:52:56Z","snapshot_observed_at":"2026-07-06T23:04:32.734175Z","submitted_at":"2026-04-19T13:52:56Z","title":"Orderings of Generalized k-Markov Numbers","version":1},"reference_index":8,"source":"pdf_text","source_observed_at":"2026-05-10T06:00:03.812477Z"},"links":{"citing_paper":"/paper/2604.17445"},"observation_digest":"sha256:93ee40b6d2a117991027cb330e85b583d897d997b7854f84fd5ca3354f282f3a","observation_id":"2e53a8eb-7224-4610-be72-4cba4446800b","resolution":{"observed_at":"2026-05-21T18:15:28.404760Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-05T06:32:48.257954+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-05T06:32:48.257954+00:00","source":"crossref"},{"observed_at":"2026-08-05T06:32:44.755628+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":"2026-06-05T21:23:00.469572Z","title":"Canakci and R","venue":null,"work_id":"35395aa5-d4e4-4307-8e85-ead439036a55","year":2013},"citing_paper":{"arxiv_id":"2604.17445","last_updated":"2026-04-19T13:52:56Z","snapshot_observed_at":"2026-07-06T23:04:32.734175Z","submitted_at":"2026-04-19T13:52:56Z","title":"Orderings of Generalized k-Markov Numbers","version":1},"reference_index":9,"source":"pdf_text","source_observed_at":"2026-05-10T06:00:03.812477Z"},"links":{"citing_paper":"/paper/2604.17445"},"observation_digest":"sha256:87fd7559328296b58aa7b2a4d5d6b88680db9e325c78580ecde8c3b3df6a95a4","observation_id":"b0721764-8bc1-44b1-9449-d3e07b5f72c5","resolution":{"observed_at":"2026-05-21T18:15:28.427842Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-05T06:32:48.257954+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-05T06:32:48.257954+00:00","source":"crossref"},{"observed_at":"2026-08-05T06:32:44.755628+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":"2026-06-05T21:23:00.469572Z","title":"Chekhov and M","venue":null,"work_id":"5ad31808-148b-4b8f-a99d-aac85ddb016d","year":2014},"citing_paper":{"arxiv_id":"2604.17445","last_updated":"2026-04-19T13:52:56Z","snapshot_observed_at":"2026-07-06T23:04:32.734175Z","submitted_at":"2026-04-19T13:52:56Z","title":"Orderings of Generalized k-Markov Numbers","version":1},"reference_index":10,"source":"pdf_text","source_observed_at":"2026-05-10T06:00:03.812477Z"},"links":{"citing_paper":"/paper/2604.17445"},"observation_digest":"sha256:faec2eff0d0c2c6973cfdd57b4465db38cf6c48b3260a12e29336d52761bbb49","observation_id":"d33eb6bc-9a95-4a6c-b2e4-8194014c46a2","resolution":{"observed_at":"2026-05-21T18:15:28.407858Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-05T06:32:48.257954+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-05T06:32:48.257954+00:00","source":"crossref"},{"observed_at":"2026-08-05T06:32:44.755628+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":"2026-06-05T21:23:00.469572Z","title":null,"venue":null,"work_id":"2ecc9c59-fe52-4c2b-b9ae-b7195ff6175b","year":2022},"citing_paper":{"arxiv_id":"2604.17445","last_updated":"2026-04-19T13:52:56Z","snapshot_observed_at":"2026-07-06T23:04:32.734175Z","submitted_at":"2026-04-19T13:52:56Z","title":"Orderings of Generalized k-Markov Numbers","version":1},"reference_index":11,"source":"pdf_text","source_observed_at":"2026-05-10T06:00:03.812477Z"},"links":{"citing_paper":"/paper/2604.17445"},"observation_digest":"sha256:3bee573e3c2dac4d39990f083251902a6835ada4962522a55f2f742be7b74c1e","observation_id":"3f7b1fa7-cc40-440c-a826-b12b286f1337","resolution":{"observed_at":"2026-05-21T18:15:28.440693Z","resolver_source":"raw_fallback","status":"unresolved"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-05T06:32:48.257954+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-05T06:32:48.257954+00:00","source":"crossref"},{"observed_at":"2026-08-05T06:32:44.755628+00:00","source":"retraction_watch"}],"state":"measured"}},{"citation":{"cited_paper":{"arxiv_id":"2109.09639","last_updated":"2022-09-17T05:41:15Z","snapshot_observed_at":"2026-07-06T11:49:35.126128Z","submitted_at":"2021-09-20T15:46:47Z","title":"Positive integer solutions to $(x+y)^2+(y+z)^2+(z+x)^2=12xyz$","version":4},"cited_work":{"arxiv_id":"2109.09639","doi":null,"metadata_source":"arxiv_reference","pith_arxiv_id":"2109.09639","snapshot_observed_at":"2026-06-05T21:23:00.469572Z","title":"Gyoda,Positive integer solutions to(x+y) 2 + (y+z) 2 + (z+x) 2 = 12xyz","venue":null,"work_id":"8a927dd1-7731-42f5-a312-74befdee2878","year":2021},"citing_paper":{"arxiv_id":"2604.17445","last_updated":"2026-04-19T13:52:56Z","snapshot_observed_at":"2026-07-06T23:04:32.734175Z","submitted_at":"2026-04-19T13:52:56Z","title":"Orderings of Generalized k-Markov Numbers","version":1},"reference_index":12,"source":"pdf_text","source_observed_at":"2026-05-10T06:00:03.812477Z"},"links":{"cited_paper":"/paper/2109.09639","citing_paper":"/paper/2604.17445"},"observation_digest":"sha256:5d0f6a8e65af25350e0784b118cc3383d75c659bb364865cde30f3c51e10d954","observation_id":"e8b4df7d-582b-4234-ab19-55c38125c773","resolution":{"observed_at":"2026-05-10T06:01:13.286557Z","resolver_source":"arxiv_id","status":"verified_exact"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-05T06:32:48.257954+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-05T06:32:48.257954+00:00","source":"crossref"},{"observed_at":"2026-08-05T06:32:44.755628+00:00","source":"retraction_watch"}],"state":"measured"}},{"citation":{"cited_paper":{"arxiv_id":"2312.07329","last_updated":"2024-07-11T13:36:44Z","snapshot_observed_at":"2026-07-06T17:00:29.200522Z","submitted_at":"2023-12-12T14:45:12Z","title":"Uniqueness theorem of generalized Markov numbers that are prime powers","version":2},"cited_work":{"arxiv_id":"2312.07329","doi":null,"metadata_source":"arxiv_reference","pith_arxiv_id":"2312.07329","snapshot_observed_at":"2026-06-29T16:03:33.992106Z","title":"Gyoda and S","venue":null,"work_id":"7457a2e2-3e86-42b1-a874-787d3188343e","year":2023},"citing_paper":{"arxiv_id":"2604.17445","last_updated":"2026-04-19T13:52:56Z","snapshot_observed_at":"2026-07-06T23:04:32.734175Z","submitted_at":"2026-04-19T13:52:56Z","title":"Orderings of Generalized k-Markov Numbers","version":1},"reference_index":13,"source":"pdf_text","source_observed_at":"2026-05-10T06:00:03.812477Z"},"links":{"cited_paper":"/paper/2312.07329","citing_paper":"/paper/2604.17445"},"observation_digest":"sha256:fef45ee48d2974f9e2007cfe4cc7f0e718b4ebe380399178896f64a124b1935f","observation_id":"4f1495be-925e-49ae-b5df-daea68572ce2","resolution":{"observed_at":"2026-05-10T06:01:13.294126Z","resolver_source":"arxiv_id","status":"verified_exact"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-05T06:32:48.257954+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-05T06:32:48.257954+00:00","source":"crossref"},{"observed_at":"2026-08-05T06:32:44.755628+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":"2026-06-05T21:23:00.469572Z","title":"Gyoda and K","venue":null,"work_id":"73f9365c-d8a1-4cee-85a3-ddc94a8c0a1c","year":2023},"citing_paper":{"arxiv_id":"2604.17445","last_updated":"2026-04-19T13:52:56Z","snapshot_observed_at":"2026-07-06T23:04:32.734175Z","submitted_at":"2026-04-19T13:52:56Z","title":"Orderings of Generalized k-Markov Numbers","version":1},"reference_index":14,"source":"pdf_text","source_observed_at":"2026-05-10T06:00:03.812477Z"},"links":{"citing_paper":"/paper/2604.17445"},"observation_digest":"sha256:b1a12f8a34f3c91867e166b05284f3f727a884026fe4a25b01665559fb3bb82d","observation_id":"08644b7c-c5cd-4675-8839-07ba1b468fe1","resolution":{"observed_at":"2026-05-21T18:15:28.398444Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-05T06:32:48.257954+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-05T06:32:48.257954+00:00","source":"crossref"},{"observed_at":"2026-08-05T06:32:44.755628+00:00","source":"retraction_watch"}],"state":"measured"}},{"citation":{"cited_paper":{"arxiv_id":"2204.11443","last_updated":"2022-04-26T12:20:01Z","snapshot_observed_at":"2026-07-06T13:03:22.436518Z","submitted_at":"2022-04-25T05:28:54Z","title":"On the monotonicity of the generalized Markov numbers","version":2},"cited_work":{"arxiv_id":"2204.11443","doi":null,"metadata_source":"arxiv_reference","pith_arxiv_id":"2204.11443","snapshot_observed_at":"2026-06-05T21:23:00.469572Z","title":null,"venue":null,"work_id":"b52728b6-d21d-46b9-96f3-cbd2ca0f9de4","year":2022},"citing_paper":{"arxiv_id":"2604.17445","last_updated":"2026-04-19T13:52:56Z","snapshot_observed_at":"2026-07-06T23:04:32.734175Z","submitted_at":"2026-04-19T13:52:56Z","title":"Orderings of Generalized k-Markov Numbers","version":1},"reference_index":15,"source":"pdf_text","source_observed_at":"2026-05-10T06:00:03.812477Z"},"links":{"cited_paper":"/paper/2204.11443","citing_paper":"/paper/2604.17445"},"observation_digest":"sha256:acbe576489c612e3538b2963fe63f464f014284d2848941c2ad47ff26e7e0a57","observation_id":"cd4545fa-0aa4-415d-837c-622b369717a5","resolution":{"observed_at":"2026-05-10T06:01:13.291721Z","resolver_source":"arxiv_id","status":"verified_exact"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-05T06:32:48.257954+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-05T06:32:48.257954+00:00","source":"crossref"},{"observed_at":"2026-08-05T06:32:44.755628+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":"2026-06-05T21:23:00.469572Z","title":"Kantarcı O˘ guz","venue":null,"work_id":"ce7c19f5-b9ca-4025-9b43-1d5a544ba193","year":2025},"citing_paper":{"arxiv_id":"2604.17445","last_updated":"2026-04-19T13:52:56Z","snapshot_observed_at":"2026-07-06T23:04:32.734175Z","submitted_at":"2026-04-19T13:52:56Z","title":"Orderings of Generalized k-Markov Numbers","version":1},"reference_index":16,"source":"pdf_text","source_observed_at":"2026-05-10T06:00:03.812477Z"},"links":{"citing_paper":"/paper/2604.17445"},"observation_digest":"sha256:6571c50a2b64e3f5b84814dfb6e909fde59c3b846c46b11b7582161299a97a3d","observation_id":"ce6dc8d3-8c64-4966-a29a-4aa0ccf70859","resolution":{"observed_at":"2026-05-21T18:15:28.417372Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-05T06:32:48.257954+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-05T06:32:48.257954+00:00","source":"crossref"},{"observed_at":"2026-08-05T06:32:44.755628+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":"2026-06-05T21:23:00.469572Z","title":"Lagisquet, E","venue":null,"work_id":"93114aa1-163a-4255-9d49-e481f485a365","year":2021},"citing_paper":{"arxiv_id":"2604.17445","last_updated":"2026-04-19T13:52:56Z","snapshot_observed_at":"2026-07-06T23:04:32.734175Z","submitted_at":"2026-04-19T13:52:56Z","title":"Orderings of Generalized k-Markov Numbers","version":1},"reference_index":17,"source":"pdf_text","source_observed_at":"2026-05-10T06:00:03.812477Z"},"links":{"citing_paper":"/paper/2604.17445"},"observation_digest":"sha256:e027eaf8c5c378e73da87e4b2b04270e88c57df8097d79cc0620f566330efc77","observation_id":"25ff9ffd-c49d-46bc-96c8-816096c1311f","resolution":{"observed_at":"2026-05-21T18:15:28.434601Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-05T06:32:48.257954+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-05T06:32:48.257954+00:00","source":"crossref"},{"observed_at":"2026-08-05T06:32:44.755628+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":"2026-06-05T21:23:00.469572Z","title":null,"venue":null,"work_id":"a066fd58-4960-4fac-ac1a-2eaa42ba94f4","year":2023},"citing_paper":{"arxiv_id":"2604.17445","last_updated":"2026-04-19T13:52:56Z","snapshot_observed_at":"2026-07-06T23:04:32.734175Z","submitted_at":"2026-04-19T13:52:56Z","title":"Orderings of Generalized k-Markov Numbers","version":1},"reference_index":18,"source":"pdf_text","source_observed_at":"2026-05-10T06:00:03.812477Z"},"links":{"citing_paper":"/paper/2604.17445"},"observation_digest":"sha256:f1a8966c7d880339eed51c29aa4bb62a4c503d785e90bab5bb11cd0df5186566","observation_id":"4b2fec98-7ebe-412c-a246-8400921ee715","resolution":{"observed_at":"2026-05-21T18:15:28.393065Z","resolver_source":"raw_fallback","status":"unresolved"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-05T06:32:48.257954+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-05T06:32:48.257954+00:00","source":"crossref"},{"observed_at":"2026-08-05T06:32:44.755628+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":"2026-06-05T21:23:00.469572Z","title":null,"venue":null,"work_id":"1c769ee2-a63a-47b7-bf3f-213cfec32cf2","year":null},"citing_paper":{"arxiv_id":"2604.17445","last_updated":"2026-04-19T13:52:56Z","snapshot_observed_at":"2026-07-06T23:04:32.734175Z","submitted_at":"2026-04-19T13:52:56Z","title":"Orderings of Generalized k-Markov Numbers","version":1},"reference_index":19,"source":"pdf_text","source_observed_at":"2026-05-10T06:00:03.812477Z"},"links":{"citing_paper":"/paper/2604.17445"},"observation_digest":"sha256:00f785b30b22b0c2d98b6331ab63b00bd98f6feebc6632f797fa5734bacd69bc","observation_id":"e93f2caf-2a06-446f-8c65-b7707b7546a5","resolution":{"observed_at":"2026-05-21T18:15:28.431334Z","resolver_source":"raw_fallback","status":"unresolved"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-05T06:32:48.257954+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-05T06:32:48.257954+00:00","source":"crossref"},{"observed_at":"2026-08-05T06:32:44.755628+00:00","source":"retraction_watch"}],"state":"measured"}},{"citation":{"cited_paper":{"arxiv_id":"2101.03316","last_updated":"2021-01-09T08:17:15Z","snapshot_observed_at":"2026-07-06T10:31:09.519606Z","submitted_at":"2021-01-09T08:17:15Z","title":"Convexity and Aigner's Conjectures","version":1},"cited_work":{"arxiv_id":"2101.03316","doi":null,"metadata_source":"arxiv_reference","pith_arxiv_id":"2101.03316","snapshot_observed_at":"2026-06-05T21:23:00.469572Z","title":"McShane,Convexity and Aigner’s Conjectures","venue":null,"work_id":"062d76ac-c200-486b-86b7-ad6199df3edd","year":2021},"citing_paper":{"arxiv_id":"2604.17445","last_updated":"2026-04-19T13:52:56Z","snapshot_observed_at":"2026-07-06T23:04:32.734175Z","submitted_at":"2026-04-19T13:52:56Z","title":"Orderings of Generalized k-Markov Numbers","version":1},"reference_index":20,"source":"pdf_text","source_observed_at":"2026-05-10T06:00:03.812477Z"},"links":{"cited_paper":"/paper/2101.03316","citing_paper":"/paper/2604.17445"},"observation_digest":"sha256:e5a3b3f8f2929c871b3eeab77d1b91ca2df4da8cd5239f1626e69661d3f2a3c8","observation_id":"54ee0c31-32fb-4907-bbe8-956d2d7b0c7b","resolution":{"observed_at":"2026-05-10T06:01:13.296598Z","resolver_source":"arxiv_id","status":"verified_exact"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-05T06:32:48.257954+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-05T06:32:48.257954+00:00","source":"crossref"},{"observed_at":"2026-08-05T06:32:44.755628+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":"2026-06-05T21:23:00.469572Z","title":"Musiker and R","venue":null,"work_id":"a8f8be88-4311-413b-8910-68431b27d48d","year":2010},"citing_paper":{"arxiv_id":"2604.17445","last_updated":"2026-04-19T13:52:56Z","snapshot_observed_at":"2026-07-06T23:04:32.734175Z","submitted_at":"2026-04-19T13:52:56Z","title":"Orderings of Generalized k-Markov Numbers","version":1},"reference_index":21,"source":"pdf_text","source_observed_at":"2026-05-10T06:00:03.812477Z"},"links":{"citing_paper":"/paper/2604.17445"},"observation_digest":"sha256:17cd690def20ecf1b436fbe5d5c98bc8105fe421c7927c21feefabac495b28b1","observation_id":"5f53a7f8-e773-43b2-ba6f-aee5d8f92204","resolution":{"observed_at":"2026-05-21T18:15:28.424597Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-05T06:32:48.257954+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-05T06:32:48.257954+00:00","source":"crossref"},{"observed_at":"2026-08-05T06:32:44.755628+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":"2026-06-05T21:23:00.469572Z","title":"Musiker, R","venue":null,"work_id":"240172ae-5d16-4d60-8980-6ef43d2799d9","year":2013},"citing_paper":{"arxiv_id":"2604.17445","last_updated":"2026-04-19T13:52:56Z","snapshot_observed_at":"2026-07-06T23:04:32.734175Z","submitted_at":"2026-04-19T13:52:56Z","title":"Orderings of Generalized k-Markov Numbers","version":1},"reference_index":22,"source":"pdf_text","source_observed_at":"2026-05-10T06:00:03.812477Z"},"links":{"citing_paper":"/paper/2604.17445"},"observation_digest":"sha256:6c954eef91f10cbf9f893380cbb7b4c979c07d94345d7dc844ed9c1da8ae839f","observation_id":"c723f2c4-3b40-4bf1-b660-ac366d5cb203","resolution":{"observed_at":"2026-05-21T18:15:28.411041Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-05T06:32:48.257954+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-05T06:32:48.257954+00:00","source":"crossref"},{"observed_at":"2026-08-05T06:32:44.755628+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":"2026-06-05T21:23:00.469572Z","title":null,"venue":null,"work_id":"e3f9c64c-d024-405f-9f83-217b6312f438","year":2020},"citing_paper":{"arxiv_id":"2604.17445","last_updated":"2026-04-19T13:52:56Z","snapshot_observed_at":"2026-07-06T23:04:32.734175Z","submitted_at":"2026-04-19T13:52:56Z","title":"Orderings of Generalized k-Markov Numbers","version":1},"reference_index":23,"source":"pdf_text","source_observed_at":"2026-05-10T06:00:03.812477Z"},"links":{"citing_paper":"/paper/2604.17445"},"observation_digest":"sha256:916bc94d097b629787d1f8abbe70be22148e3ab6df5b622b72bb911fbcf536f5","observation_id":"b60410c1-e2e8-4829-a715-e2386003acd7","resolution":{"observed_at":"2026-05-21T18:15:28.401549Z","resolver_source":"raw_fallback","status":"unresolved"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-05T06:32:48.257954+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-05T06:32:48.257954+00:00","source":"crossref"},{"observed_at":"2026-08-05T06:32:44.755628+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":"2026-06-05T21:23:00.469572Z","title":"Rabideau and R","venue":null,"work_id":"413f3e06-9a7d-434b-b3a8-03575825a3d8","year":2020},"citing_paper":{"arxiv_id":"2604.17445","last_updated":"2026-04-19T13:52:56Z","snapshot_observed_at":"2026-07-06T23:04:32.734175Z","submitted_at":"2026-04-19T13:52:56Z","title":"Orderings of Generalized k-Markov Numbers","version":1},"reference_index":24,"source":"pdf_text","source_observed_at":"2026-05-10T06:00:03.812477Z"},"links":{"citing_paper":"/paper/2604.17445"},"observation_digest":"sha256:4cf2550cf144f80fb59f4f0120a3214f3f28d4a6f75608ab98e1be2a373a01b3","observation_id":"55553d87-a75f-4811-9703-8a774f5cd97b","resolution":{"observed_at":"2026-05-21T18:15:28.420608Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-05T06:32:48.257954+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-05T06:32:48.257954+00:00","source":"crossref"},{"observed_at":"2026-08-05T06:32:44.755628+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":"2026-06-05T21:23:00.469572Z","title":"Reutenauer.From Christoffel words to Markoff numbers","venue":null,"work_id":"c428d4fe-4350-4995-9cd6-af8092fdd64e","year":2019},"citing_paper":{"arxiv_id":"2604.17445","last_updated":"2026-04-19T13:52:56Z","snapshot_observed_at":"2026-07-06T23:04:32.734175Z","submitted_at":"2026-04-19T13:52:56Z","title":"Orderings of Generalized k-Markov Numbers","version":1},"reference_index":25,"source":"pdf_text","source_observed_at":"2026-05-10T06:00:03.812477Z"},"links":{"citing_paper":"/paper/2604.17445"},"observation_digest":"sha256:34cb23df3792783d88ab0e3a7e9ad0d5e1808b8a00f9cbf2b77660e89e7e9249","observation_id":"d251fac5-46f7-4a2c-b017-08c131ac45f5","resolution":{"observed_at":"2026-05-21T18:15:28.395757Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-05T06:32:48.257954+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-05T06:32:48.257954+00:00","source":"crossref"},{"observed_at":"2026-08-05T06:32:44.755628+00:00","source":"retraction_watch"}],"state":"measured"}}],"paper":{"arxiv_id":"2604.17445","last_updated":"2026-04-19T13:52:56Z","latest_version":1,"primary_category":"math.NT","snapshot_observed_at":"2026-07-06T23:04:32.734175Z","submitted_at":"2026-04-19T13:52:56Z","title":"Orderings of Generalized k-Markov Numbers"},"reference_resolution":{"displayed":25,"state_counts":{"malformed_identifier":0,"metadata_mismatch":0,"parse_uncertain":0,"unresolved":4,"verified_exact":7,"verified_fuzzy":14},"total_outbound_references":25},"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":"2026-08-05T06:32:48.257954+00:00","source":"crossref"},{"observed_at":"2026-08-05T06:32:44.755628+00:00","source":"retraction_watch"}],"thesis":"As of 5 August 2026, this Paper Citation Record lists 25 of 25 outbound references and 0 inbound Pith citation observations for arXiv:2604.17445."}