{"as_of":"2026-08-19T12:05:00Z","caps":{"database_statements":6,"inbound":100,"outbound":100},"context_digest":"sha256:8b597b776ad56c0c3796cd2c2479158725e3f150976d36ed16759ee585588f83","coverage":[{"denominator":26,"lane":"reference_resolution","note":"Typed states for the displayed outbound observations.","records_observed":26,"source":"paper_references, paper_reference_links","source_observed_at":"2026-08-06T22:10:20.888746Z","state":"measured"},{"denominator":26,"lane":"standing_notices","note":"One-hop event checks from named stored sources.","records_observed":26,"source":"scholarly_work_events, retraction_status_cache","source_observed_at":"2026-08-19T06:32:44.657259+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/2506.22730/citation-record","integrity":"/paper/2506.22730/integrity","json":"/paper/2506.22730/citation-record.json","paper":"/paper/2506.22730"},"outbound":[{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":"raw_reference","pith_arxiv_id":null,"snapshot_observed_at":"2026-08-06T22:10:25.138098Z","title":"Birkhoff, Lattice Theory (3rd ed.), Amer","venue":null,"work_id":"e3e8f3b0-d858-4d15-b791-d5c1f9b87391","year":null},"citing_paper":{"arxiv_id":"2506.22730","last_updated":"2025-06-28T02:23:27Z","snapshot_observed_at":"2026-08-16T03:27:27.880724Z","submitted_at":"2025-06-28T02:23:27Z","title":"Invariants of toric double determinantal rings","version":1},"reference_index":1,"source":"pdf_text","source_observed_at":"2026-08-06T22:10:19.433504Z"},"links":{"citing_paper":"/paper/2506.22730"},"observation_digest":"sha256:2763ea9cebc5bb91c1f9d548bea84dd5cf8420e47871beca1279be00f37029e1","observation_id":"ef62baaa-bc14-4e83-8af9-7ff14ecac1fd","resolution":{"observed_at":"2026-08-06T22:10:25.203630Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-19T06:32:44.657259+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-19T06:32:44.657259+00:00","source":"crossref"},{"observed_at":"2026-08-19T06:32:39.956319+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":"2026-08-06T22:10:25.012680Z","title":"Blose, P","venue":null,"work_id":"8c0296a6-d4a0-47e2-b4bd-53368985c771","year":2021},"citing_paper":{"arxiv_id":"2506.22730","last_updated":"2025-06-28T02:23:27Z","snapshot_observed_at":"2026-08-16T03:27:27.880724Z","submitted_at":"2025-06-28T02:23:27Z","title":"Invariants of toric double determinantal rings","version":1},"reference_index":2,"source":"pdf_text","source_observed_at":"2026-08-06T22:10:19.466437Z"},"links":{"citing_paper":"/paper/2506.22730"},"observation_digest":"sha256:a9fe18711af31b13d60e528acd9402a03b0d032ccee4f9a48c221f157e3dfcf1","observation_id":"85da4c85-ec5c-4475-83e9-d08b2edb1bf5","resolution":{"observed_at":"2026-08-06T22:10:25.073179Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-19T06:32:44.657259+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-19T06:32:44.657259+00:00","source":"crossref"},{"observed_at":"2026-08-19T06:32:39.956319+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":"2026-08-06T22:10:24.874517Z","title":"Bruns, A","venue":null,"work_id":"1932b403-4dd1-49a3-b353-5cf2d4641bc4","year":2022},"citing_paper":{"arxiv_id":"2506.22730","last_updated":"2025-06-28T02:23:27Z","snapshot_observed_at":"2026-08-16T03:27:27.880724Z","submitted_at":"2025-06-28T02:23:27Z","title":"Invariants of toric double determinantal rings","version":1},"reference_index":3,"source":"pdf_text","source_observed_at":"2026-08-06T22:10:19.493162Z"},"links":{"citing_paper":"/paper/2506.22730"},"observation_digest":"sha256:fe4f5c7acf83c8f5ab243ead390828d5848cab9e4dd40c4be7d21ff9f27b3fb6","observation_id":"9bf9cc0b-0174-4c4b-bbd5-b8d8909d4108","resolution":{"observed_at":"2026-08-06T22:10:24.929607Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-19T06:32:44.657259+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-19T06:32:44.657259+00:00","source":"crossref"},{"observed_at":"2026-08-19T06:32:39.956319+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":"2026-08-06T22:10:24.767781Z","title":"Cohen-Macaulay rings, Cambridge Stud","venue":null,"work_id":"b400c6c2-467e-4239-a487-635240874d83","year":1993},"citing_paper":{"arxiv_id":"2506.22730","last_updated":"2025-06-28T02:23:27Z","snapshot_observed_at":"2026-08-16T03:27:27.880724Z","submitted_at":"2025-06-28T02:23:27Z","title":"Invariants of toric double determinantal rings","version":1},"reference_index":4,"source":"pdf_text","source_observed_at":"2026-08-06T22:10:19.499837Z"},"links":{"citing_paper":"/paper/2506.22730"},"observation_digest":"sha256:35782810b476ca014565f159cf8baa5007c92cc523c6f5c35141cf88d04736fd","observation_id":"a5e14e98-d84b-41d2-8c29-4e5a2dcca4ec","resolution":{"observed_at":"2026-08-06T22:10:24.812638Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-19T06:32:44.657259+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-19T06:32:44.657259+00:00","source":"crossref"},{"observed_at":"2026-08-19T06:32:39.956319+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":"2026-08-06T22:10:24.650952Z","title":"Carlitz, V","venue":null,"work_id":"555148ee-be45-482e-945c-a74cf27e306c","year":1978},"citing_paper":{"arxiv_id":"2506.22730","last_updated":"2025-06-28T02:23:27Z","snapshot_observed_at":"2026-08-16T03:27:27.880724Z","submitted_at":"2025-06-28T02:23:27Z","title":"Invariants of toric double determinantal rings","version":1},"reference_index":5,"source":"pdf_text","source_observed_at":"2026-08-06T22:10:19.505328Z"},"links":{"citing_paper":"/paper/2506.22730"},"observation_digest":"sha256:ed73e7d09b74bc60d0767e5e190b851fbd52515b7495c4e997206d7cdc282a46","observation_id":"5f75b034-da23-43a8-ac0f-6c6845698de7","resolution":{"observed_at":"2026-08-06T22:10:24.715969Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-19T06:32:44.657259+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-19T06:32:44.657259+00:00","source":"crossref"},{"observed_at":"2026-08-19T06:32:39.956319+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":"2026-08-06T22:10:24.532948Z","title":"Conca, Ladder determinantal rings, J","venue":null,"work_id":"6f2e237a-2ba1-43e4-8619-261c063840ab","year":1995},"citing_paper":{"arxiv_id":"2506.22730","last_updated":"2025-06-28T02:23:27Z","snapshot_observed_at":"2026-08-16T03:27:27.880724Z","submitted_at":"2025-06-28T02:23:27Z","title":"Invariants of toric double determinantal rings","version":1},"reference_index":6,"source":"pdf_text","source_observed_at":"2026-08-06T22:10:19.547968Z"},"links":{"citing_paper":"/paper/2506.22730"},"observation_digest":"sha256:1c4d23a3871315ed83ca24b988b7385a871346938c925b0c314fa2b3c726187c","observation_id":"93ca9e0a-526c-4614-9c0b-bcc50759dee6","resolution":{"observed_at":"2026-08-06T22:10:24.575391Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-19T06:32:44.657259+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-19T06:32:44.657259+00:00","source":"crossref"},{"observed_at":"2026-08-19T06:32:39.956319+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":"2026-08-06T22:10:24.420433Z","title":"Conca, E","venue":null,"work_id":"517bde57-2115-41f4-9794-fa74efadb6c3","year":2020},"citing_paper":{"arxiv_id":"2506.22730","last_updated":"2025-06-28T02:23:27Z","snapshot_observed_at":"2026-08-16T03:27:27.880724Z","submitted_at":"2025-06-28T02:23:27Z","title":"Invariants of toric double determinantal rings","version":1},"reference_index":7,"source":"pdf_text","source_observed_at":"2026-08-06T22:10:19.614966Z"},"links":{"citing_paper":"/paper/2506.22730"},"observation_digest":"sha256:f06ccbdd6d0606e45682fbbbddb82fdb4289efa0105b6928534e9654fd2d820b","observation_id":"8267f613-c4e1-4d67-86fb-e712b2580a57","resolution":{"observed_at":"2026-08-06T22:10:24.482659Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-19T06:32:44.657259+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-19T06:32:44.657259+00:00","source":"crossref"},{"observed_at":"2026-08-19T06:32:39.956319+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":"2026-08-06T22:10:24.249808Z","title":"Syzygies of Hibi Rings","venue":null,"work_id":"c49c5fa7-e9ef-4325-baa3-75657c367f35","year":2015},"citing_paper":{"arxiv_id":"2506.22730","last_updated":"2025-06-28T02:23:27Z","snapshot_observed_at":"2026-08-16T03:27:27.880724Z","submitted_at":"2025-06-28T02:23:27Z","title":"Invariants of toric double determinantal rings","version":1},"reference_index":8,"source":"pdf_text","source_observed_at":"2026-08-06T22:10:19.712886Z"},"links":{"citing_paper":"/paper/2506.22730"},"observation_digest":"sha256:0dc9d699311cb2f38132ddde978230b043ada731be631c75855277f2300b4758","observation_id":"2f237e2a-de4f-476e-a752-782ba9b57274","resolution":{"observed_at":"2026-08-06T22:10:24.333785Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-19T06:32:44.657259+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-19T06:32:44.657259+00:00","source":"crossref"},{"observed_at":"2026-08-19T06:32:39.956319+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":"2026-08-06T22:10:24.039059Z","title":null,"venue":null,"work_id":"57e668a7-ee28-4d65-9999-b1b967fd8d39","year":2015},"citing_paper":{"arxiv_id":"2506.22730","last_updated":"2025-06-28T02:23:27Z","snapshot_observed_at":"2026-08-16T03:27:27.880724Z","submitted_at":"2025-06-28T02:23:27Z","title":"Invariants of toric double determinantal rings","version":1},"reference_index":9,"source":"pdf_text","source_observed_at":"2026-08-06T22:10:19.840390Z"},"links":{"citing_paper":"/paper/2506.22730"},"observation_digest":"sha256:03418f1b04a6cbf6a19fbac3b720b78e9d919c6e3af7e82c7257f6a7ec186801","observation_id":"a5ff1bce-6b76-4994-b4f4-c82a7e49b116","resolution":{"observed_at":"2026-08-06T22:10:24.153084Z","resolver_source":"raw_fallback","status":"unresolved"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-19T06:32:44.657259+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-19T06:32:44.657259+00:00","source":"crossref"},{"observed_at":"2026-08-19T06:32:39.956319+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":"2026-08-06T22:10:23.871698Z","title":"Gasanova, L","venue":null,"work_id":"56bf9af2-3b78-4afe-b507-d56f90a332fe","year":2024},"citing_paper":{"arxiv_id":"2506.22730","last_updated":"2025-06-28T02:23:27Z","snapshot_observed_at":"2026-08-16T03:27:27.880724Z","submitted_at":"2025-06-28T02:23:27Z","title":"Invariants of toric double determinantal rings","version":1},"reference_index":10,"source":"pdf_text","source_observed_at":"2026-08-06T22:10:19.955205Z"},"links":{"citing_paper":"/paper/2506.22730"},"observation_digest":"sha256:baf5fcef8493743893d429fbfda5ae7ee8264c7843edeb7a05e0a668b6db424e","observation_id":"ac26ba7d-79b6-42ad-9aea-92310242d14a","resolution":{"observed_at":"2026-08-06T22:10:23.939732Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-19T06:32:44.657259+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-19T06:32:44.657259+00:00","source":"crossref"},{"observed_at":"2026-08-19T06:32:39.956319+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":"2026-08-06T22:10:23.711088Z","title":"Gonciulea and C","venue":null,"work_id":"5b127618-af64-401c-b358-67c187e3d938","year":2000},"citing_paper":{"arxiv_id":"2506.22730","last_updated":"2025-06-28T02:23:27Z","snapshot_observed_at":"2026-08-16T03:27:27.880724Z","submitted_at":"2025-06-28T02:23:27Z","title":"Invariants of toric double determinantal rings","version":1},"reference_index":11,"source":"pdf_text","source_observed_at":"2026-08-06T22:10:20.024914Z"},"links":{"citing_paper":"/paper/2506.22730"},"observation_digest":"sha256:83444176792989ad34a620bfa7237c3a88c767a601fc305d6915cb65eeab6640","observation_id":"e87f1be6-2964-4af6-a08f-a537421c9364","resolution":{"observed_at":"2026-08-06T22:10:23.800001Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-19T06:32:44.657259+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-19T06:32:44.657259+00:00","source":"crossref"},{"observed_at":"2026-08-19T06:32:39.956319+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":"2026-08-06T22:10:23.557845Z","title":"Gorla, Mixed ladder determinantal varieties from two-sided ladders, J","venue":null,"work_id":"1abc3963-59ad-4797-83bc-41e323182d04","year":2007},"citing_paper":{"arxiv_id":"2506.22730","last_updated":"2025-06-28T02:23:27Z","snapshot_observed_at":"2026-08-16T03:27:27.880724Z","submitted_at":"2025-06-28T02:23:27Z","title":"Invariants of toric double determinantal rings","version":1},"reference_index":12,"source":"pdf_text","source_observed_at":"2026-08-06T22:10:20.104265Z"},"links":{"citing_paper":"/paper/2506.22730"},"observation_digest":"sha256:60bce71c0603edd400611ca443536cacfe9a77399b5aa2f3cdbf6136f5fe7c5d","observation_id":"912b63b4-c1c3-42c1-99ff-2346949e3a33","resolution":{"observed_at":"2026-08-06T22:10:23.638350Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-19T06:32:44.657259+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-19T06:32:44.657259+00:00","source":"crossref"},{"observed_at":"2026-08-19T06:32:39.956319+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":"2026-08-06T22:10:23.351624Z","title":"Herzog, T","venue":null,"work_id":"b240e284-7db5-4be2-87f9-b9d040eaee9f","year":2011},"citing_paper":{"arxiv_id":"2506.22730","last_updated":"2025-06-28T02:23:27Z","snapshot_observed_at":"2026-08-16T03:27:27.880724Z","submitted_at":"2025-06-28T02:23:27Z","title":"Invariants of toric double determinantal rings","version":1},"reference_index":13,"source":"pdf_text","source_observed_at":"2026-08-06T22:10:20.161570Z"},"links":{"citing_paper":"/paper/2506.22730"},"observation_digest":"sha256:8b8bd6e98e08d19eaa9931db0bf1df64d7dec9d57ca6e18b4494f691233b1e0a","observation_id":"11520746-ceb9-41e7-be73-017c120128bf","resolution":{"observed_at":"2026-08-06T22:10:23.444249Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-19T06:32:44.657259+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-19T06:32:44.657259+00:00","source":"crossref"},{"observed_at":"2026-08-19T06:32:39.956319+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":"2026-08-06T22:10:23.128811Z","title":"Com- mutative Algebra and Combinatorics","venue":null,"work_id":"275efe79-f71d-4c11-845a-b59c3b28991b","year":1987},"citing_paper":{"arxiv_id":"2506.22730","last_updated":"2025-06-28T02:23:27Z","snapshot_observed_at":"2026-08-16T03:27:27.880724Z","submitted_at":"2025-06-28T02:23:27Z","title":"Invariants of toric double determinantal rings","version":1},"reference_index":14,"source":"pdf_text","source_observed_at":"2026-08-06T22:10:20.209110Z"},"links":{"citing_paper":"/paper/2506.22730"},"observation_digest":"sha256:4ef2b2829921480ae94f4881cc70c14f54eb113b68b0508f93fadee4763ac3a8","observation_id":"71336742-55c8-4d64-a0a0-d632993738b5","resolution":{"observed_at":"2026-08-06T22:10:23.225224Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-19T06:32:44.657259+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-19T06:32:44.657259+00:00","source":"crossref"},{"observed_at":"2026-08-19T06:32:39.956319+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":"2026-08-06T22:10:22.938300Z","title":"Hibi, Hilbert functions of Cohen-Macaulay integral domains and chain conditions of finite partially ordered sets, J","venue":null,"work_id":"94a6b5be-bf23-455d-8cdc-5476162a212d","year":1991},"citing_paper":{"arxiv_id":"2506.22730","last_updated":"2025-06-28T02:23:27Z","snapshot_observed_at":"2026-08-16T03:27:27.880724Z","submitted_at":"2025-06-28T02:23:27Z","title":"Invariants of toric double determinantal rings","version":1},"reference_index":15,"source":"pdf_text","source_observed_at":"2026-08-06T22:10:20.227138Z"},"links":{"citing_paper":"/paper/2506.22730"},"observation_digest":"sha256:433694e401bc8b31b1d1fa8cfaf0c2534414ec8d73ddd22bcae68dcdf1414271","observation_id":"a43b76a8-836c-4c25-8415-2c867f98d109","resolution":{"observed_at":"2026-08-06T22:10:23.007665Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-19T06:32:44.657259+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-19T06:32:44.657259+00:00","source":"crossref"},{"observed_at":"2026-08-19T06:32:39.956319+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":"2026-08-06T22:10:22.763217Z","title":"Illian, L","venue":null,"work_id":"87c14078-3c20-4c8f-8d2a-75c073a6a013","year":2025},"citing_paper":{"arxiv_id":"2506.22730","last_updated":"2025-06-28T02:23:27Z","snapshot_observed_at":"2026-08-16T03:27:27.880724Z","submitted_at":"2025-06-28T02:23:27Z","title":"Invariants of toric double determinantal rings","version":1},"reference_index":16,"source":"pdf_text","source_observed_at":"2026-08-06T22:10:20.249696Z"},"links":{"citing_paper":"/paper/2506.22730"},"observation_digest":"sha256:7f595ddb8cea0d49cc96eeaac8205072882cd5be8d5120d5020704cd7e3e6b24","observation_id":"25d073f2-2918-4f07-b3b7-f8c9596ffbb0","resolution":{"observed_at":"2026-08-06T22:10:22.861711Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-19T06:32:44.657259+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-19T06:32:44.657259+00:00","source":"crossref"},{"observed_at":"2026-08-19T06:32:39.956319+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":"2026-08-06T22:10:22.606430Z","title":"Klein, N","venue":null,"work_id":"381c9c18-9a17-446b-bc4b-ebeb7d7e46bb","year":2020},"citing_paper":{"arxiv_id":"2506.22730","last_updated":"2025-06-28T02:23:27Z","snapshot_observed_at":"2026-08-16T03:27:27.880724Z","submitted_at":"2025-06-28T02:23:27Z","title":"Invariants of toric double determinantal rings","version":1},"reference_index":17,"source":"pdf_text","source_observed_at":"2026-08-06T22:10:20.266120Z"},"links":{"citing_paper":"/paper/2506.22730"},"observation_digest":"sha256:0882517538375e00375d7fa17618e27cf75be9f3dc63a06c56cf3836c18a47ec","observation_id":"5a75141b-d846-4643-9b82-b4fad510768c","resolution":{"observed_at":"2026-08-06T22:10:22.677070Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-19T06:32:44.657259+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-19T06:32:44.657259+00:00","source":"crossref"},{"observed_at":"2026-08-19T06:32:39.956319+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":"2026-08-06T22:10:22.428949Z","title":"Li, Nakajima’s quiver varieties and triangular bases of rank-2 cluster algebras, J","venue":null,"work_id":"26ad59ba-8437-4782-a694-27b2afd3d9aa","year":2023},"citing_paper":{"arxiv_id":"2506.22730","last_updated":"2025-06-28T02:23:27Z","snapshot_observed_at":"2026-08-16T03:27:27.880724Z","submitted_at":"2025-06-28T02:23:27Z","title":"Invariants of toric double determinantal rings","version":1},"reference_index":18,"source":"pdf_text","source_observed_at":"2026-08-06T22:10:20.283793Z"},"links":{"citing_paper":"/paper/2506.22730"},"observation_digest":"sha256:dc284ee1f6cb515e3e9afb79e570533563cd66a8492242158e221af0fe1138c8","observation_id":"9441c206-b7e9-49ac-b36e-725d73410cd0","resolution":{"observed_at":"2026-08-06T22:10:22.510603Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-19T06:32:44.657259+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-19T06:32:44.657259+00:00","source":"crossref"},{"observed_at":"2026-08-19T06:32:39.956319+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":"2026-08-06T22:10:22.268945Z","title":"Li, Bipartite determinantal ideals and concurrent vertex maps, J","venue":null,"work_id":"bd0835d3-b332-4dd9-a562-79086e05c682","year":2024},"citing_paper":{"arxiv_id":"2506.22730","last_updated":"2025-06-28T02:23:27Z","snapshot_observed_at":"2026-08-16T03:27:27.880724Z","submitted_at":"2025-06-28T02:23:27Z","title":"Invariants of toric double determinantal rings","version":1},"reference_index":19,"source":"pdf_text","source_observed_at":"2026-08-06T22:10:20.302917Z"},"links":{"citing_paper":"/paper/2506.22730"},"observation_digest":"sha256:40f582ff5111a35e3075c67be52e4a77ff02b13cf469a44ac70dc9304c257bd1","observation_id":"309d0b32-c6a9-49f8-b32b-d7caafc84093","resolution":{"observed_at":"2026-08-06T22:10:22.338515Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-19T06:32:44.657259+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-19T06:32:44.657259+00:00","source":"crossref"},{"observed_at":"2026-08-19T06:32:39.956319+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":"2026-08-06T22:10:22.128639Z","title":null,"venue":null,"work_id":"cc1f1b69-0768-4a02-a4a0-d24cadbd2d52","year":2022},"citing_paper":{"arxiv_id":"2506.22730","last_updated":"2025-06-28T02:23:27Z","snapshot_observed_at":"2026-08-16T03:27:27.880724Z","submitted_at":"2025-06-28T02:23:27Z","title":"Invariants of toric double determinantal rings","version":1},"reference_index":20,"source":"pdf_text","source_observed_at":"2026-08-06T22:10:20.350719Z"},"links":{"citing_paper":"/paper/2506.22730"},"observation_digest":"sha256:7126ace34abad5bca76095ac826d669a140d8b64ac201f786bc839c0a556ded6","observation_id":"b002075f-18e4-4381-bec1-d71aa4ab3032","resolution":{"observed_at":"2026-08-06T22:10:22.193171Z","resolver_source":"raw_fallback","status":"unresolved"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-19T06:32:44.657259+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-19T06:32:44.657259+00:00","source":"crossref"},{"observed_at":"2026-08-19T06:32:39.956319+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":"2026-08-06T22:10:21.980506Z","title":"MacMahon, Combinatory Analysis, Two Volumes (Bound As One), Chelsea Publishing Co., New York, 1960","venue":null,"work_id":"a45631ac-e904-470c-ba2b-70eb08e4562e","year":1960},"citing_paper":{"arxiv_id":"2506.22730","last_updated":"2025-06-28T02:23:27Z","snapshot_observed_at":"2026-08-16T03:27:27.880724Z","submitted_at":"2025-06-28T02:23:27Z","title":"Invariants of toric double determinantal rings","version":1},"reference_index":21,"source":"pdf_text","source_observed_at":"2026-08-06T22:10:20.379634Z"},"links":{"citing_paper":"/paper/2506.22730"},"observation_digest":"sha256:c26ba8bfb03966c0b940ec5016243bf72c40f719fbe430331f0d7b5f17d9e521","observation_id":"d8c37475-13ab-41dc-8533-a145cb239e2d","resolution":{"observed_at":"2026-08-06T22:10:22.056841Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-19T06:32:44.657259+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-19T06:32:44.657259+00:00","source":"crossref"},{"observed_at":"2026-08-19T06:32:39.956319+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":"2026-08-06T22:10:21.758791Z","title":null,"venue":null,"work_id":"8d0c0de1-ecf3-447a-a7e6-64bef3869fbc","year":2023},"citing_paper":{"arxiv_id":"2506.22730","last_updated":"2025-06-28T02:23:27Z","snapshot_observed_at":"2026-08-16T03:27:27.880724Z","submitted_at":"2025-06-28T02:23:27Z","title":"Invariants of toric double determinantal rings","version":1},"reference_index":22,"source":"pdf_text","source_observed_at":"2026-08-06T22:10:20.417696Z"},"links":{"citing_paper":"/paper/2506.22730"},"observation_digest":"sha256:025cab6c57cfe96b0ff918337c25ee62ea6a793852da6f757995eea1a54d6531","observation_id":"b6f097a7-96de-4e21-9261-fb55f226d1c4","resolution":{"observed_at":"2026-08-06T22:10:21.876390Z","resolver_source":"raw_fallback","status":"unresolved"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-19T06:32:44.657259+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-19T06:32:44.657259+00:00","source":"crossref"},{"observed_at":"2026-08-19T06:32:39.956319+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":"2026-08-06T22:10:21.588142Z","title":"Grayson, M","venue":null,"work_id":"d2e71b18-7352-4f88-a24d-d49deafadd4e","year":null},"citing_paper":{"arxiv_id":"2506.22730","last_updated":"2025-06-28T02:23:27Z","snapshot_observed_at":"2026-08-16T03:27:27.880724Z","submitted_at":"2025-06-28T02:23:27Z","title":"Invariants of toric double determinantal rings","version":1},"reference_index":23,"source":"pdf_text","source_observed_at":"2026-08-06T22:10:20.487009Z"},"links":{"citing_paper":"/paper/2506.22730"},"observation_digest":"sha256:28237b4012a9b7de83f729700545d2c5e86caf4901309fa860389a77ac2946a0","observation_id":"ce5bfe9d-84f7-43aa-8d20-368070fcd78b","resolution":{"observed_at":"2026-08-06T22:10:21.676705Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-19T06:32:44.657259+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-19T06:32:44.657259+00:00","source":"crossref"},{"observed_at":"2026-08-19T06:32:39.956319+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":"2026-08-06T22:10:21.360029Z","title":null,"venue":null,"work_id":"5e03db09-4e57-4ab8-8ccb-2d0cc9ad50ff","year":1996},"citing_paper":{"arxiv_id":"2506.22730","last_updated":"2025-06-28T02:23:27Z","snapshot_observed_at":"2026-08-16T03:27:27.880724Z","submitted_at":"2025-06-28T02:23:27Z","title":"Invariants of toric double determinantal rings","version":1},"reference_index":24,"source":"pdf_text","source_observed_at":"2026-08-06T22:10:20.635401Z"},"links":{"citing_paper":"/paper/2506.22730"},"observation_digest":"sha256:6273da502870186ed0993627374f7e481454eb5ac971b33a3f11371e21fdc565","observation_id":"ff4947c6-bb83-4d4e-91f1-2ccaac631ff9","resolution":{"observed_at":"2026-08-06T22:10:21.484701Z","resolver_source":"raw_fallback","status":"unresolved"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-19T06:32:44.657259+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-19T06:32:44.657259+00:00","source":"crossref"},{"observed_at":"2026-08-19T06:32:39.956319+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":"2026-08-06T22:10:21.124566Z","title":"Sturmfels, Gr¨ obner bases and convex polytopes, University Lecture Series, 9, American Mathe- matical Soc., 1996","venue":null,"work_id":"8e898463-eca3-4e45-bd09-c322557404ee","year":1996},"citing_paper":{"arxiv_id":"2506.22730","last_updated":"2025-06-28T02:23:27Z","snapshot_observed_at":"2026-08-16T03:27:27.880724Z","submitted_at":"2025-06-28T02:23:27Z","title":"Invariants of toric double determinantal rings","version":1},"reference_index":25,"source":"pdf_text","source_observed_at":"2026-08-06T22:10:20.769544Z"},"links":{"citing_paper":"/paper/2506.22730"},"observation_digest":"sha256:35b8ad3a10229f73cb4d78b0a8683ab74e2b863333d793a0cb34c5fe9efd4b24","observation_id":"0525a2cc-0d3f-40cd-afe9-ea441dad03c4","resolution":{"observed_at":"2026-08-06T22:10:21.256042Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-19T06:32:44.657259+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-19T06:32:44.657259+00:00","source":"crossref"},{"observed_at":"2026-08-19T06:32:39.956319+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":"2026-08-06T22:10:20.958839Z","title":"Svanes, Coherent cohomology on schubert subschemes of flag schemes and applications, Advances in Math","venue":null,"work_id":"520049a4-2765-469c-93d7-f3a6b3bebd63","year":1974},"citing_paper":{"arxiv_id":"2506.22730","last_updated":"2025-06-28T02:23:27Z","snapshot_observed_at":"2026-08-16T03:27:27.880724Z","submitted_at":"2025-06-28T02:23:27Z","title":"Invariants of toric double determinantal rings","version":1},"reference_index":26,"source":"pdf_text","source_observed_at":"2026-08-06T22:10:20.888746Z"},"links":{"citing_paper":"/paper/2506.22730"},"observation_digest":"sha256:d6e06d23d1a134307485b850c36ad4ffd3eecfae4ac26ab475c01b9ab1ce1e94","observation_id":"27927ae4-12f2-4a80-bfe7-7b2040e1561f","resolution":{"observed_at":"2026-08-06T22:10:21.045762Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-19T06:32:44.657259+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-19T06:32:44.657259+00:00","source":"crossref"},{"observed_at":"2026-08-19T06:32:39.956319+00:00","source":"retraction_watch"}],"state":"measured"}}],"paper":{"arxiv_id":"2506.22730","last_updated":"2025-06-28T02:23:27Z","latest_version":1,"primary_category":"math.AC","snapshot_observed_at":"2026-08-16T03:27:27.880724Z","submitted_at":"2025-06-28T02:23:27Z","title":"Invariants of toric double determinantal rings"},"reference_resolution":{"displayed":26,"state_counts":{"malformed_identifier":0,"metadata_mismatch":0,"parse_uncertain":0,"unresolved":4,"verified_exact":0,"verified_fuzzy":22},"total_outbound_references":26},"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":"2026-08-19T06:32:44.657259+00:00","source":"crossref"},{"observed_at":"2026-08-19T06:32:39.956319+00:00","source":"retraction_watch"}],"thesis":"As of 19 August 2026, this Paper Citation Record lists 26 of 26 outbound references and 0 inbound Pith citation observations for arXiv:2506.22730."}