{"as_of":"2026-08-10T07:29:00Z","caps":{"database_statements":6,"inbound":100,"outbound":100},"context_digest":"sha256:13521065b9ae88e0ca0d85dfe18720ff5ccddecf9d2abce04f3769ffe47c7ee8","coverage":[{"denominator":29,"lane":"reference_resolution","note":"Typed states for the displayed outbound observations.","records_observed":29,"source":"paper_references, paper_reference_links","source_observed_at":"2026-05-10T00:40:37.969330Z","state":"measured"},{"denominator":30,"lane":"standing_notices","note":"One-hop event checks from named stored sources.","records_observed":30,"source":"scholarly_work_events, retraction_status_cache","source_observed_at":"2026-08-10T06:31:04.303077+00:00","state":"measured"},{"denominator":1,"lane":"inbound_itemization","note":"Pith citing papers itemized under the disclosed page cap.","records_observed":1,"source":"paper_references, paper_reference_links","source_observed_at":"2026-08-03T14:55:05.012043Z","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":[{"citation":{"cited_paper":{"arxiv_id":"2604.20233","last_updated":"2026-04-29T16:10:16Z","snapshot_observed_at":"2026-08-08T03:04:25.215031Z","submitted_at":"2026-04-22T06:32:37Z","title":"Entropy lower bounds and sum-product phenomena","version":2},"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":"2604.20233","snapshot_observed_at":"2026-08-03T14:55:05.012043Z","title":"Gavalakis, M","venue":null,"work_id":null,"year":2026},"citing_paper":{"arxiv_id":"2607.29042","last_updated":"2026-07-31T05:43:56Z","snapshot_observed_at":"2026-08-09T00:27:15.435269Z","submitted_at":"2026-07-31T05:43:56Z","title":"The Entropic Sum-Product Phenomenon","version":1},"reference_index":9,"source":"pdf_text","source_observed_at":"2026-08-03T14:55:05.012043Z"},"links":{"cited_paper":"/paper/2604.20233","citing_paper":"/paper/2607.29042"},"observation_digest":"sha256:c03774c4cc2ff92089f54aae3ce9be6d028835537edb345f8160d9b6a326f3cc","observation_id":"c83b36f6-b798-4fc7-96c7-618d7a2b2673","resolution":{"observed_at":"2026-08-03T14:55:05.012043Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}}],"links":{"evidence":"/evidence","html":"/paper/2604.20233/citation-record","integrity":"/paper/2604.20233/integrity","json":"/paper/2604.20233/citation-record.json","paper":"/paper/2604.20233"},"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":"Growth estimates in positive characteristic via collisions","venue":null,"work_id":"0246f595-f67a-425f-8fe3-9ead17e2c4f9","year":2017},"citing_paper":{"arxiv_id":"2604.20233","last_updated":"2026-04-29T16:10:16Z","snapshot_observed_at":"2026-08-08T03:04:25.215031Z","submitted_at":"2026-04-22T06:32:37Z","title":"Entropy lower bounds and sum-product phenomena","version":2},"reference_index":1,"source":"arxiv_source","source_observed_at":"2026-05-10T00:40:37.969330Z"},"links":{"citing_paper":"/paper/2604.20233"},"observation_digest":"sha256:0d619da8099640e0cbae157643d1893086a4b42c064cfe4a1b4f7b269d2aae5e","observation_id":"7f014c36-1ea0-43bd-853a-4c1ed7696186","resolution":{"observed_at":"2026-05-23T10:52:53.212286Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-10T06:31:04.303077+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-10T06:31:04.303077+00:00","source":"crossref"},{"observed_at":"2026-08-10T06:30:57.382061+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":"Extracting randomness using few independent sources","venue":null,"work_id":"c37fd07e-d99d-47cf-9289-e55bafc68115","year":2006},"citing_paper":{"arxiv_id":"2604.20233","last_updated":"2026-04-29T16:10:16Z","snapshot_observed_at":"2026-08-08T03:04:25.215031Z","submitted_at":"2026-04-22T06:32:37Z","title":"Entropy lower bounds and sum-product phenomena","version":2},"reference_index":2,"source":"arxiv_source","source_observed_at":"2026-05-10T00:40:37.969330Z"},"links":{"citing_paper":"/paper/2604.20233"},"observation_digest":"sha256:beaa75c517a05f6bf43ef44c1ad1bf4c071c4b99fac6cb391a9780931b9d268a","observation_id":"1ced7992-78e9-49c3-a7a6-8c0a9434436b","resolution":{"observed_at":"2026-05-23T10:52:53.220527Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-10T06:31:04.303077+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-10T06:31:04.303077+00:00","source":"crossref"},{"observed_at":"2026-08-10T06:30:57.382061+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":"An improved sum-product bound for quaternions","venue":null,"work_id":"4f10db17-d137-405d-b95c-3650cdd19cb3","year":2019},"citing_paper":{"arxiv_id":"2604.20233","last_updated":"2026-04-29T16:10:16Z","snapshot_observed_at":"2026-08-08T03:04:25.215031Z","submitted_at":"2026-04-22T06:32:37Z","title":"Entropy lower bounds and sum-product phenomena","version":2},"reference_index":3,"source":"arxiv_source","source_observed_at":"2026-05-10T00:40:37.969330Z"},"links":{"citing_paper":"/paper/2604.20233"},"observation_digest":"sha256:3e502e672d8d369a9784f94e323967015f16fd77d41952bea0c99901951c21de","observation_id":"b381a8cb-916f-4530-a3be-b42be5a86874","resolution":{"observed_at":"2026-05-23T10:52:53.207471Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-10T06:31:04.303077+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-10T06:31:04.303077+00:00","source":"crossref"},{"observed_at":"2026-08-10T06:30:57.382061+00:00","source":"retraction_watch"}],"state":"measured"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":"2501.29470","doi":null,"metadata_source":"arxiv_reference","pith_arxiv_id":null,"snapshot_observed_at":"2026-06-05T21:23:00.469572Z","title":"Control and its applications in additive combinatorics","venue":null,"work_id":"f6b2747a-6cd2-4e08-a8ea-508004dbfe41","year":2025},"citing_paper":{"arxiv_id":"2604.20233","last_updated":"2026-04-29T16:10:16Z","snapshot_observed_at":"2026-08-08T03:04:25.215031Z","submitted_at":"2026-04-22T06:32:37Z","title":"Entropy lower bounds and sum-product phenomena","version":2},"reference_index":4,"source":"arxiv_source","source_observed_at":"2026-05-10T00:40:37.969330Z"},"links":{"citing_paper":"/paper/2604.20233"},"observation_digest":"sha256:7e54994ba08ca6859cb8038c6d6097ba6efd222c1f0d24949e347dcf33ffdfd0","observation_id":"81c834ab-c1b8-41c2-9a1e-eac3418f6e48","resolution":{"observed_at":"2026-05-11T13:46:03.575641Z","resolver_source":"arxiv_id","status":"verified_exact"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-10T06:31:04.303077+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-10T06:31:04.303077+00:00","source":"crossref"},{"observed_at":"2026-08-10T06:30:57.382061+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":"Recherches sur les nombres","venue":null,"work_id":"209257d1-33b3-4038-a850-76d43d016761","year":null},"citing_paper":{"arxiv_id":"2604.20233","last_updated":"2026-04-29T16:10:16Z","snapshot_observed_at":"2026-08-08T03:04:25.215031Z","submitted_at":"2026-04-22T06:32:37Z","title":"Entropy lower bounds and sum-product phenomena","version":2},"reference_index":5,"source":"arxiv_source","source_observed_at":"2026-05-10T00:40:37.969330Z"},"links":{"citing_paper":"/paper/2604.20233"},"observation_digest":"sha256:649d651a9e9be5d2245d717e6405eae86b32fb809d6836d031f0ef3d3cf12276","observation_id":"ca528028-12b8-4be6-b055-0b350aad1681","resolution":{"observed_at":"2026-05-23T10:52:53.217380Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-10T06:31:04.303077+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-10T06:31:04.303077+00:00","source":"crossref"},{"observed_at":"2026-08-10T06:30:57.382061+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":"Elements of Information Theory (New York: John Wiley and Sons, 2nd edition, 2006)","venue":null,"work_id":"9c47db07-8cfe-4560-985d-e40f44b9b399","year":2006},"citing_paper":{"arxiv_id":"2604.20233","last_updated":"2026-04-29T16:10:16Z","snapshot_observed_at":"2026-08-08T03:04:25.215031Z","submitted_at":"2026-04-22T06:32:37Z","title":"Entropy lower bounds and sum-product phenomena","version":2},"reference_index":6,"source":"arxiv_source","source_observed_at":"2026-05-10T00:40:37.969330Z"},"links":{"citing_paper":"/paper/2604.20233"},"observation_digest":"sha256:49305e5f541d9b4d4f5b0d47b8d6d159324f9ca861242f1d7ac89b5073997fa2","observation_id":"f6337866-5b08-4b68-bde1-1fbdaccb2379","resolution":{"observed_at":"2026-05-23T10:52:53.186048Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-10T06:31:04.303077+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-10T06:31:04.303077+00:00","source":"crossref"},{"observed_at":"2026-08-10T06:30:57.382061+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":"On the addition of residue classes","venue":null,"work_id":"b005ab96-722e-4f14-b706-459f6e0e0570","year":1935},"citing_paper":{"arxiv_id":"2604.20233","last_updated":"2026-04-29T16:10:16Z","snapshot_observed_at":"2026-08-08T03:04:25.215031Z","submitted_at":"2026-04-22T06:32:37Z","title":"Entropy lower bounds and sum-product phenomena","version":2},"reference_index":7,"source":"arxiv_source","source_observed_at":"2026-05-10T00:40:37.969330Z"},"links":{"citing_paper":"/paper/2604.20233"},"observation_digest":"sha256:d9f44234e1feb8b557244bbb4e3b1a1c76a1bcc17db012d0d1b4ebd111151d4e","observation_id":"f6d82d3b-1291-4c94-ae74-f374a546bab2","resolution":{"observed_at":"2026-05-23T10:52:53.174770Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-10T06:31:04.303077+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-10T06:31:04.303077+00:00","source":"crossref"},{"observed_at":"2026-08-10T06:30:57.382061+00:00","source":"retraction_watch"}],"state":"measured"}},{"citation":{"cited_paper":{"arxiv_id":"1612.02719","last_updated":"2016-12-08T16:38:20Z","snapshot_observed_at":"2026-08-08T19:41:07.147749Z","submitted_at":"2016-12-08T16:38:20Z","title":"A short proof of Rudnev's point-plane incidence bound","version":1},"cited_work":{"arxiv_id":"1612.02719","doi":null,"metadata_source":"pith","pith_arxiv_id":"1612.02719","snapshot_observed_at":"2026-07-02T05:56:40.851047Z","title":"A short proof of R udnev’s point-plane incidence bound","venue":"math.CO","work_id":"a6980b5d-cd88-41d9-be49-55463014d326","year":2016},"citing_paper":{"arxiv_id":"2604.20233","last_updated":"2026-04-29T16:10:16Z","snapshot_observed_at":"2026-08-08T03:04:25.215031Z","submitted_at":"2026-04-22T06:32:37Z","title":"Entropy lower bounds and sum-product phenomena","version":2},"reference_index":8,"source":"arxiv_source","source_observed_at":"2026-05-10T00:40:37.969330Z"},"links":{"cited_paper":"/paper/1612.02719","citing_paper":"/paper/2604.20233"},"observation_digest":"sha256:78e185a6bc38dbb9214744a4f088c34e0b652d50330acd40d7e017f022310934","observation_id":"142a5003-0713-4e2a-b3cb-b36c78efaada","resolution":{"observed_at":"2026-05-11T13:46:03.677743Z","resolver_source":"arxiv_id","status":"verified_exact"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-10T06:31:04.303077+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-10T06:31:04.303077+00:00","source":"crossref"},{"observed_at":"2026-08-10T06:30:57.382061+00:00","source":"retraction_watch"}],"state":"measured"}},{"citation":{"cited_paper":{"arxiv_id":"2404.02244","last_updated":"2024-05-20T20:12:42Z","snapshot_observed_at":"2026-08-07T02:57:47.594209Z","submitted_at":"2024-04-02T19:03:51Z","title":"Marton's Conjecture in abelian groups with bounded torsion","version":2},"cited_work":{"arxiv_id":"2404.02244","doi":null,"metadata_source":"pith","pith_arxiv_id":"2404.02244","snapshot_observed_at":"2026-07-09T18:46:26.487030Z","title":"Marton's conjecture in abelian groups with bounded torsion","venue":"math.NT","work_id":"b5e7d8d0-6fbb-48f9-a276-91ce678dbca9","year":2024},"citing_paper":{"arxiv_id":"2604.20233","last_updated":"2026-04-29T16:10:16Z","snapshot_observed_at":"2026-08-08T03:04:25.215031Z","submitted_at":"2026-04-22T06:32:37Z","title":"Entropy lower bounds and sum-product phenomena","version":2},"reference_index":9,"source":"arxiv_source","source_observed_at":"2026-05-10T00:40:37.969330Z"},"links":{"cited_paper":"/paper/2404.02244","citing_paper":"/paper/2604.20233"},"observation_digest":"sha256:ddc2b075376b5e2290024d5e5c0c733cef94c4860e2bc9fc66efa7857a126219","observation_id":"1d134952-51cc-49ad-a84c-73ae5b06c06c","resolution":{"observed_at":"2026-05-11T13:46:03.635992Z","resolver_source":"arxiv_id","status":"verified_exact"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-10T06:31:04.303077+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-10T06:31:04.303077+00:00","source":"crossref"},{"observed_at":"2026-08-10T06:30:57.382061+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":"On a conjecture of Marton","venue":null,"work_id":"60a326ab-8b8e-48e5-83e4-e5fe9bd8b829","year":2025},"citing_paper":{"arxiv_id":"2604.20233","last_updated":"2026-04-29T16:10:16Z","snapshot_observed_at":"2026-08-08T03:04:25.215031Z","submitted_at":"2026-04-22T06:32:37Z","title":"Entropy lower bounds and sum-product phenomena","version":2},"reference_index":10,"source":"arxiv_source","source_observed_at":"2026-05-10T00:40:37.969330Z"},"links":{"citing_paper":"/paper/2604.20233"},"observation_digest":"sha256:30dc73cb9bc6b38a72d331e6c76d310312ee72e11a86b160daa78127d733864f","observation_id":"0b41e6e9-2610-4323-8892-cf67bde38f55","resolution":{"observed_at":"2026-05-23T10:52:53.167080Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-10T06:31:04.303077+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-10T06:31:04.303077+00:00","source":"crossref"},{"observed_at":"2026-08-10T06:30:57.382061+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":"Entropy and the discrete central limit theorem","venue":null,"work_id":"611db320-3f6e-446a-95fd-5847692d5eaf","year":2024},"citing_paper":{"arxiv_id":"2604.20233","last_updated":"2026-04-29T16:10:16Z","snapshot_observed_at":"2026-08-08T03:04:25.215031Z","submitted_at":"2026-04-22T06:32:37Z","title":"Entropy lower bounds and sum-product phenomena","version":2},"reference_index":11,"source":"arxiv_source","source_observed_at":"2026-05-10T00:40:37.969330Z"},"links":{"citing_paper":"/paper/2604.20233"},"observation_digest":"sha256:bbbfc22d87d15a14948bfae78499d3c90d2fdd74ea5f3d1181975ef77d72bd30","observation_id":"fa8b17ea-6c4f-48b8-aac6-73b3898b6f98","resolution":{"observed_at":"2026-05-23T10:52:53.190751Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-10T06:31:04.303077+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-10T06:31:04.303077+00:00","source":"crossref"},{"observed_at":"2026-08-10T06:30:57.382061+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":"Sumsets and entropy revisited","venue":null,"work_id":"2ded97e3-4b0c-4c13-84ae-f09d8fffa8d9","year":2025},"citing_paper":{"arxiv_id":"2604.20233","last_updated":"2026-04-29T16:10:16Z","snapshot_observed_at":"2026-08-08T03:04:25.215031Z","submitted_at":"2026-04-22T06:32:37Z","title":"Entropy lower bounds and sum-product phenomena","version":2},"reference_index":12,"source":"arxiv_source","source_observed_at":"2026-05-10T00:40:37.969330Z"},"links":{"citing_paper":"/paper/2604.20233"},"observation_digest":"sha256:56076622d29fe6cf1893af4c821e962d584ba463c88f9b4339ab9700dc2cd065","observation_id":"6ec7c561-49aa-4e15-a47c-c85b5842eb5b","resolution":{"observed_at":"2026-05-23T10:52:53.161675Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-10T06:31:04.303077+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-10T06:31:04.303077+00:00","source":"crossref"},{"observed_at":"2026-08-10T06:30:57.382061+00:00","source":"retraction_watch"}],"state":"measured"}},{"citation":{"cited_paper":{"arxiv_id":"2406.18798","last_updated":"2026-03-31T21:24:26Z","snapshot_observed_at":"2026-07-06T18:37:39.282367Z","submitted_at":"2024-06-27T00:08:17Z","title":"On an entropic analogue of additive energy","version":4},"cited_work":{"arxiv_id":"2406.18798","doi":null,"metadata_source":"arxiv_reference","pith_arxiv_id":"2406.18798","snapshot_observed_at":"2026-06-05T21:23:00.469572Z","title":"On an entropic analogue of additive energy","venue":null,"work_id":"d3c36667-8932-4668-a931-856ad3dd853e","year":2024},"citing_paper":{"arxiv_id":"2604.20233","last_updated":"2026-04-29T16:10:16Z","snapshot_observed_at":"2026-08-08T03:04:25.215031Z","submitted_at":"2026-04-22T06:32:37Z","title":"Entropy lower bounds and sum-product phenomena","version":2},"reference_index":13,"source":"arxiv_source","source_observed_at":"2026-05-10T00:40:37.969330Z"},"links":{"cited_paper":"/paper/2406.18798","citing_paper":"/paper/2604.20233"},"observation_digest":"sha256:d1fec22fc57c5900e4275a3d51ce179a4d9b9428fc889c0d56544efd5b39015e","observation_id":"ebd6defd-4cc4-4476-b0e0-692e55a32d17","resolution":{"observed_at":"2026-05-27T15:06:19.979776Z","resolver_source":"arxiv_id","status":"verified_exact"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-10T06:31:04.303077+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-10T06:31:04.303077+00:00","source":"crossref"},{"observed_at":"2026-08-10T06:30:57.382061+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":"A new entropy power inequality for integer-valued random variables","venue":null,"work_id":"02f0c4e8-33c2-45c7-a56a-2ae53b648d5f","year":2014},"citing_paper":{"arxiv_id":"2604.20233","last_updated":"2026-04-29T16:10:16Z","snapshot_observed_at":"2026-08-08T03:04:25.215031Z","submitted_at":"2026-04-22T06:32:37Z","title":"Entropy lower bounds and sum-product phenomena","version":2},"reference_index":14,"source":"arxiv_source","source_observed_at":"2026-05-10T00:40:37.969330Z"},"links":{"citing_paper":"/paper/2604.20233"},"observation_digest":"sha256:55cc70c6611631fa6f2e7be845794b72e273ca54d599e30517514dba89269e6c","observation_id":"68c9a086-9969-4f76-9c11-39e5f91e7dd5","resolution":{"observed_at":"2026-05-23T10:52:53.170570Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-10T06:31:04.303077+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-10T06:31:04.303077+00:00","source":"crossref"},{"observed_at":"2026-08-10T06:30:57.382061+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":"Sumset and inverse sumset inequalities for differential entropy and mutual information","venue":null,"work_id":"eec8be9b-fa57-4815-b6b4-55ebff09a004","year":2014},"citing_paper":{"arxiv_id":"2604.20233","last_updated":"2026-04-29T16:10:16Z","snapshot_observed_at":"2026-08-08T03:04:25.215031Z","submitted_at":"2026-04-22T06:32:37Z","title":"Entropy lower bounds and sum-product phenomena","version":2},"reference_index":15,"source":"arxiv_source","source_observed_at":"2026-05-10T00:40:37.969330Z"},"links":{"citing_paper":"/paper/2604.20233"},"observation_digest":"sha256:ad9376fbb5ea55f63a87827609daf1dc64bd4d068c3d3452569539dc4996a3a8","observation_id":"b64353a3-8a0c-4b28-a6cb-adfdab151839","resolution":{"observed_at":"2026-05-23T10:52:53.153029Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-10T06:31:04.303077+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-10T06:31:04.303077+00:00","source":"crossref"},{"observed_at":"2026-08-10T06:30:57.382061+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":"Exponential sum estimates over prime fields","venue":null,"work_id":"bff1dfd5-027b-4598-8288-4b6510a443ac","year":2020},"citing_paper":{"arxiv_id":"2604.20233","last_updated":"2026-04-29T16:10:16Z","snapshot_observed_at":"2026-08-08T03:04:25.215031Z","submitted_at":"2026-04-22T06:32:37Z","title":"Entropy lower bounds and sum-product phenomena","version":2},"reference_index":16,"source":"arxiv_source","source_observed_at":"2026-05-10T00:40:37.969330Z"},"links":{"citing_paper":"/paper/2604.20233"},"observation_digest":"sha256:78a17bf32d30d8c9aa1daea2b0685c4e529b3fa867705c960397b0ce3574f53c","observation_id":"420b92dd-39dc-494e-b358-1ed13055dfb9","resolution":{"observed_at":"2026-05-23T10:52:53.158185Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-10T06:31:04.303077+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-10T06:31:04.303077+00:00","source":"crossref"},{"observed_at":"2026-08-10T06:30:57.382061+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":"S zemerédi– T rotter-type theorems in dimension 3","venue":null,"work_id":"052df51b-cdcc-4949-9618-34c85d4a84f8","year":2015},"citing_paper":{"arxiv_id":"2604.20233","last_updated":"2026-04-29T16:10:16Z","snapshot_observed_at":"2026-08-08T03:04:25.215031Z","submitted_at":"2026-04-22T06:32:37Z","title":"Entropy lower bounds and sum-product phenomena","version":2},"reference_index":17,"source":"arxiv_source","source_observed_at":"2026-05-10T00:40:37.969330Z"},"links":{"citing_paper":"/paper/2604.20233"},"observation_digest":"sha256:a30fc7882bdd08b77c14936d5211f916011d33462af2854c9a8f3f3d63e513fb","observation_id":"80e703cf-7aa7-442b-8d81-51b1c1e1b276","resolution":{"observed_at":"2026-05-23T10:52:53.149430Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-10T06:31:04.303077+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-10T06:31:04.303077+00:00","source":"crossref"},{"observed_at":"2026-08-10T06:30:57.382061+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":"Entropic additive energy and entropy inequalities for sums and products","venue":null,"work_id":"7db0d38e-869a-4c4b-b691-69714c94e239","year":2026},"citing_paper":{"arxiv_id":"2604.20233","last_updated":"2026-04-29T16:10:16Z","snapshot_observed_at":"2026-08-08T03:04:25.215031Z","submitted_at":"2026-04-22T06:32:37Z","title":"Entropy lower bounds and sum-product phenomena","version":2},"reference_index":18,"source":"arxiv_source","source_observed_at":"2026-05-10T00:40:37.969330Z"},"links":{"citing_paper":"/paper/2604.20233"},"observation_digest":"sha256:8ae00f9a5759234302dd3f703360fc7fa12dfaf6afe32f83af6d4730417348d2","observation_id":"c1cca4f0-ee82-49bc-b52b-8c33ced865d8","resolution":{"observed_at":"2026-05-23T10:52:53.177950Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-10T06:31:04.303077+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-10T06:31:04.303077+00:00","source":"crossref"},{"observed_at":"2026-08-10T06:30:57.382061+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":"An improved sum-product estimate for general finite fields","venue":null,"work_id":"d18704fb-366b-4450-891f-3376630806ef","year":2011},"citing_paper":{"arxiv_id":"2604.20233","last_updated":"2026-04-29T16:10:16Z","snapshot_observed_at":"2026-08-08T03:04:25.215031Z","submitted_at":"2026-04-22T06:32:37Z","title":"Entropy lower bounds and sum-product phenomena","version":2},"reference_index":19,"source":"arxiv_source","source_observed_at":"2026-05-10T00:40:37.969330Z"},"links":{"citing_paper":"/paper/2604.20233"},"observation_digest":"sha256:bd821d67b66b4faf3acc725910569105f4f44b50ffdc07700518c021eeb308d0","observation_id":"abe95c0a-00c6-4e06-aed0-0afb5dc5bf48","resolution":{"observed_at":"2026-05-23T10:52:53.193763Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-10T06:31:04.303077+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-10T06:31:04.303077+00:00","source":"crossref"},{"observed_at":"2026-08-10T06:30:57.382061+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":"Discretised sum-product theorems by Shannon-type inequalities","venue":null,"work_id":"704ed5e2-1ecc-495f-ad16-a11df806f74f","year":2025},"citing_paper":{"arxiv_id":"2604.20233","last_updated":"2026-04-29T16:10:16Z","snapshot_observed_at":"2026-08-08T03:04:25.215031Z","submitted_at":"2026-04-22T06:32:37Z","title":"Entropy lower bounds and sum-product phenomena","version":2},"reference_index":20,"source":"arxiv_source","source_observed_at":"2026-05-10T00:40:37.969330Z"},"links":{"citing_paper":"/paper/2604.20233"},"observation_digest":"sha256:30eb215b30aefefc23bce7ff1d6e926a388b97f21848e2ef01169ffa09a25f2d","observation_id":"a47244ec-6216-41b5-b660-8a602c5c85bc","resolution":{"observed_at":"2026-05-23T10:52:53.201391Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-10T06:31:04.303077+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-10T06:31:04.303077+00:00","source":"crossref"},{"observed_at":"2026-08-10T06:30:57.382061+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":"``Sum-product in finite fields via entropy.'' Blog post at https://pohoatza.wordpress.com/2026/01/12/sum-product-in-finite -fields-via-entropy , January 12 (2026)","venue":null,"work_id":"7725dc03-ff0b-4d9a-957c-0126383eac54","year":2026},"citing_paper":{"arxiv_id":"2604.20233","last_updated":"2026-04-29T16:10:16Z","snapshot_observed_at":"2026-08-08T03:04:25.215031Z","submitted_at":"2026-04-22T06:32:37Z","title":"Entropy lower bounds and sum-product phenomena","version":2},"reference_index":21,"source":"arxiv_source","source_observed_at":"2026-05-10T00:40:37.969330Z"},"links":{"citing_paper":"/paper/2604.20233"},"observation_digest":"sha256:686eb4186e444bc17159c43caa3b25e7d4142a0d6e033e6042095d9bd9f4062e","observation_id":"5b3a3866-f4d0-4501-bc31-edf58ebe7fe3","resolution":{"observed_at":"2026-05-23T10:52:53.141564Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-10T06:31:04.303077+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-10T06:31:04.303077+00:00","source":"crossref"},{"observed_at":"2026-08-10T06:30:57.382061+00:00","source":"retraction_watch"}],"state":"measured"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":"2512.11217","doi":null,"metadata_source":"arxiv_reference","pith_arxiv_id":null,"snapshot_observed_at":"2026-06-05T21:23:00.469572Z","title":"Raghavan,Improved Bounds for the Freiman-Ruzsa Theorem","venue":null,"work_id":"92362552-96d3-44b5-bec1-d85e579f70bd","year":2025},"citing_paper":{"arxiv_id":"2604.20233","last_updated":"2026-04-29T16:10:16Z","snapshot_observed_at":"2026-08-08T03:04:25.215031Z","submitted_at":"2026-04-22T06:32:37Z","title":"Entropy lower bounds and sum-product phenomena","version":2},"reference_index":22,"source":"arxiv_source","source_observed_at":"2026-05-10T00:40:37.969330Z"},"links":{"citing_paper":"/paper/2604.20233"},"observation_digest":"sha256:8fd170c0e24289f78d4d5f0f5ae5c575a78ba9e094436cd1a7926c0d50490902","observation_id":"714576f3-5f13-40d6-aab0-60cea7852c97","resolution":{"observed_at":"2026-05-11T13:46:03.641963Z","resolver_source":"arxiv_id","status":"verified_exact"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-10T06:31:04.303077+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-10T06:31:04.303077+00:00","source":"crossref"},{"observed_at":"2026-08-10T06:30:57.382061+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":"On the M inkowski distances and products of sum sets","venue":null,"work_id":"c0b5ef73-0161-47a5-b7b7-9a69768d6c12","year":2015},"citing_paper":{"arxiv_id":"2604.20233","last_updated":"2026-04-29T16:10:16Z","snapshot_observed_at":"2026-08-08T03:04:25.215031Z","submitted_at":"2026-04-22T06:32:37Z","title":"Entropy lower bounds and sum-product phenomena","version":2},"reference_index":23,"source":"arxiv_source","source_observed_at":"2026-05-10T00:40:37.969330Z"},"links":{"citing_paper":"/paper/2604.20233"},"observation_digest":"sha256:2f723a3b4cfe331cb4eb6afe39539576c9d7fa8a0302a0059414f1ad070f448c","observation_id":"df0b7ddf-72fc-4202-b7c4-67fa06bd2579","resolution":{"observed_at":"2026-05-23T10:52:53.137017Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-10T06:31:04.303077+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-10T06:31:04.303077+00:00","source":"crossref"},{"observed_at":"2026-08-10T06:30:57.382061+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":"Sumsets and entropy","venue":null,"work_id":"dc97a73a-ce81-4b41-a62a-5847f543a910","year":2009},"citing_paper":{"arxiv_id":"2604.20233","last_updated":"2026-04-29T16:10:16Z","snapshot_observed_at":"2026-08-08T03:04:25.215031Z","submitted_at":"2026-04-22T06:32:37Z","title":"Entropy lower bounds and sum-product phenomena","version":2},"reference_index":24,"source":"arxiv_source","source_observed_at":"2026-05-10T00:40:37.969330Z"},"links":{"citing_paper":"/paper/2604.20233"},"observation_digest":"sha256:3b5c60c08957d932ca2af28ef355ab4aff6baa3f0e9777cb008035c2247aa7c1","observation_id":"6ee0e7e7-be06-49ba-adb1-686652ff3470","resolution":{"observed_at":"2026-05-23T10:52:53.182566Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-10T06:31:04.303077+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-10T06:31:04.303077+00:00","source":"crossref"},{"observed_at":"2026-08-10T06:30:57.382061+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":"A mathematical theory of communication","venue":null,"work_id":"4570ebba-2003-40d9-b480-604ec835bb60","year":1948},"citing_paper":{"arxiv_id":"2604.20233","last_updated":"2026-04-29T16:10:16Z","snapshot_observed_at":"2026-08-08T03:04:25.215031Z","submitted_at":"2026-04-22T06:32:37Z","title":"Entropy lower bounds and sum-product phenomena","version":2},"reference_index":25,"source":"arxiv_source","source_observed_at":"2026-05-10T00:40:37.969330Z"},"links":{"citing_paper":"/paper/2604.20233"},"observation_digest":"sha256:97b066ebadf7a4b88af54eb65d172c256a4f5c9a4fb7b5893291f81a8f7eb2e9","observation_id":"63a0d57a-f351-4e8e-8c2e-05f68916482a","resolution":{"observed_at":"2026-05-23T10:52:53.133345Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-10T06:31:04.303077+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-10T06:31:04.303077+00:00","source":"crossref"},{"observed_at":"2026-08-10T06:30:57.382061+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":"Some inequalities satisfied by the quantities of information of F isher and S hannon","venue":null,"work_id":"5972bac5-4257-4692-8fd8-3b2945e6a4c6","year":1959},"citing_paper":{"arxiv_id":"2604.20233","last_updated":"2026-04-29T16:10:16Z","snapshot_observed_at":"2026-08-08T03:04:25.215031Z","submitted_at":"2026-04-22T06:32:37Z","title":"Entropy lower bounds and sum-product phenomena","version":2},"reference_index":26,"source":"arxiv_source","source_observed_at":"2026-05-10T00:40:37.969330Z"},"links":{"citing_paper":"/paper/2604.20233"},"observation_digest":"sha256:5f10c82faaae3d0caf8398a1cc2309c108627b744662980409c7400c4ad693dc","observation_id":"eb40d3fb-a28a-4bae-9802-0bd4c67c20b2","resolution":{"observed_at":"2026-05-23T10:52:53.123263Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-10T06:31:04.303077+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-10T06:31:04.303077+00:00","source":"crossref"},{"observed_at":"2026-08-10T06:30:57.382061+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":"Sumset and inverse sumset theory for S hannon entropy","venue":null,"work_id":"6de1206c-9e07-46bf-bc03-7d34de285d56","year":2010},"citing_paper":{"arxiv_id":"2604.20233","last_updated":"2026-04-29T16:10:16Z","snapshot_observed_at":"2026-08-08T03:04:25.215031Z","submitted_at":"2026-04-22T06:32:37Z","title":"Entropy lower bounds and sum-product phenomena","version":2},"reference_index":27,"source":"arxiv_source","source_observed_at":"2026-05-10T00:40:37.969330Z"},"links":{"citing_paper":"/paper/2604.20233"},"observation_digest":"sha256:b5fdec0b0dc6c06743aeac2adae392c8ec0c6546c690f29d46083b48d0ef76bb","observation_id":"ed5efdc3-eb2e-4c31-b5f1-3418d12c1968","resolution":{"observed_at":"2026-05-23T10:52:53.144950Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-10T06:31:04.303077+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-10T06:31:04.303077+00:00","source":"crossref"},{"observed_at":"2026-08-10T06:30:57.382061+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":"John-type theorems for generalized arithmetic progressions and iterated sumsets","venue":null,"work_id":"021453d9-57df-42a4-ba85-8fd6c64392fc","year":2008},"citing_paper":{"arxiv_id":"2604.20233","last_updated":"2026-04-29T16:10:16Z","snapshot_observed_at":"2026-08-08T03:04:25.215031Z","submitted_at":"2026-04-22T06:32:37Z","title":"Entropy lower bounds and sum-product phenomena","version":2},"reference_index":28,"source":"arxiv_source","source_observed_at":"2026-05-10T00:40:37.969330Z"},"links":{"citing_paper":"/paper/2604.20233"},"observation_digest":"sha256:628ed2a3d21a9fac484da6d7f7818d1694b62dea37674a7a3545f7552123f193","observation_id":"c1f5b123-8001-4966-a9e3-7c1bad5c4867","resolution":{"observed_at":"2026-05-23T10:52:53.128610Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-10T06:31:04.303077+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-10T06:31:04.303077+00:00","source":"crossref"},{"observed_at":"2026-08-10T06:30:57.382061+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":"Pseudorandomness","venue":null,"work_id":"38464e85-a23d-4839-b61f-dcb986aa6b3d","year":2012},"citing_paper":{"arxiv_id":"2604.20233","last_updated":"2026-04-29T16:10:16Z","snapshot_observed_at":"2026-08-08T03:04:25.215031Z","submitted_at":"2026-04-22T06:32:37Z","title":"Entropy lower bounds and sum-product phenomena","version":2},"reference_index":29,"source":"arxiv_source","source_observed_at":"2026-05-10T00:40:37.969330Z"},"links":{"citing_paper":"/paper/2604.20233"},"observation_digest":"sha256:73f2d31d71ee9a3b306fb9055d4dab8a4a757aec5ea11845317b6da9e514ddfd","observation_id":"0d35d617-43e0-4a39-9c76-94c888c66007","resolution":{"observed_at":"2026-05-23T10:52:53.198207Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-10T06:31:04.303077+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-10T06:31:04.303077+00:00","source":"crossref"},{"observed_at":"2026-08-10T06:30:57.382061+00:00","source":"retraction_watch"}],"state":"measured"}}],"paper":{"arxiv_id":"2604.20233","last_updated":"2026-04-29T16:10:16Z","latest_version":2,"primary_category":"math.CO","snapshot_observed_at":"2026-08-08T03:04:25.215031Z","submitted_at":"2026-04-22T06:32:37Z","title":"Entropy lower bounds and sum-product phenomena"},"reference_resolution":{"displayed":29,"state_counts":{"malformed_identifier":0,"metadata_mismatch":0,"parse_uncertain":0,"unresolved":0,"verified_exact":5,"verified_fuzzy":24},"total_outbound_references":29},"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":"2026-08-10T06:31:04.303077+00:00","source":"crossref"},{"observed_at":"2026-08-10T06:30:57.382061+00:00","source":"retraction_watch"}],"thesis":"As of 10 August 2026, this Paper Citation Record lists 29 of 29 outbound references and 1 inbound Pith citation observation for arXiv:2604.20233."}