{"as_of":"2026-08-19T14:48:00Z","caps":{"database_statements":6,"inbound":100,"outbound":100},"context_digest":"sha256:1dca7770939801d6b4b63e757fdd948bc2e4fc7556394617ebbdd6a21dad7c4f","coverage":[{"denominator":28,"lane":"reference_resolution","note":"Typed states for the displayed outbound observations.","records_observed":28,"source":"paper_references, paper_reference_links","source_observed_at":"2026-08-16T12:14:06.566011Z","state":"measured"},{"denominator":28,"lane":"standing_notices","note":"One-hop event checks from named stored sources.","records_observed":28,"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/2504.13454/citation-record","integrity":"/paper/2504.13454/integrity","json":"/paper/2504.13454/citation-record.json","paper":"/paper/2504.13454"},"outbound":[{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":"raw_reference","pith_arxiv_id":null,"snapshot_observed_at":"2026-08-16T12:14:06.899894Z","title":"Abe, Strong semimodular lattices and Frankl’s conjec ture, Algebra Universalis","venue":null,"work_id":"bd39cf39-1935-4b5a-a27c-f1bfa13caf3e","year":2000},"citing_paper":{"arxiv_id":"2504.13454","last_updated":"2025-04-18T04:12:43Z","snapshot_observed_at":"2026-08-18T16:43:14.830124Z","submitted_at":"2025-04-18T04:12:43Z","title":"On the Averaging Problem of Ideal Families Related to Frankl's Conjecture with Formal Proof by Lean 4","version":1},"reference_index":1,"source":"pdf_text","source_observed_at":"2026-08-16T12:14:06.463834Z"},"links":{"citing_paper":"/paper/2504.13454"},"observation_digest":"sha256:c539b9ed4758a39f6e2929aea9028b1c59f7ff62dede18b04ecf7a411aece729","observation_id":"e210d94b-970c-4d7c-8fc3-9d857248332f","resolution":{"observed_at":"2026-08-16T12:14:06.904614Z","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":{"arxiv_id":"2211.11731","last_updated":"2024-07-08T07:42:36Z","snapshot_observed_at":"2026-08-16T16:14:26.176186Z","submitted_at":"2022-11-21T18:53:06Z","title":"Improved Lower Bound for Frankl's Union-Closed Sets Conjecture","version":4},"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":"2211.11731","snapshot_observed_at":"2026-08-16T12:14:06.468314Z","title":"Alweiss, B","venue":null,"work_id":null,"year":2022},"citing_paper":{"arxiv_id":"2504.13454","last_updated":"2025-04-18T04:12:43Z","snapshot_observed_at":"2026-08-18T16:43:14.830124Z","submitted_at":"2025-04-18T04:12:43Z","title":"On the Averaging Problem of Ideal Families Related to Frankl's Conjecture with Formal Proof by Lean 4","version":1},"reference_index":2,"source":"pdf_text","source_observed_at":"2026-08-16T12:14:06.468314Z"},"links":{"cited_paper":"/paper/2211.11731","citing_paper":"/paper/2504.13454"},"observation_digest":"sha256:e398bbd892b22166a700e844e79f35cc21b3cdd1527fc050b5870f2de15071dc","observation_id":"49884886-a2e2-4a19-a6e8-c64ce069a4bc","resolution":{"observed_at":"2026-08-16T12:14:06.468314Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":"raw_reference","pith_arxiv_id":null,"snapshot_observed_at":"2026-08-16T12:14:06.887876Z","title":"Boˇ snjak and P","venue":null,"work_id":"c4296c8d-3e0a-440e-9f54-72f3bc448842","year":2008},"citing_paper":{"arxiv_id":"2504.13454","last_updated":"2025-04-18T04:12:43Z","snapshot_observed_at":"2026-08-18T16:43:14.830124Z","submitted_at":"2025-04-18T04:12:43Z","title":"On the Averaging Problem of Ideal Families Related to Frankl's Conjecture with Formal Proof by Lean 4","version":1},"reference_index":3,"source":"pdf_text","source_observed_at":"2026-08-16T12:14:06.472583Z"},"links":{"citing_paper":"/paper/2504.13454"},"observation_digest":"sha256:df50177b2f2748e0e490f3bf4f8312aeffeecd324a5b4c2ab29c3792234946ed","observation_id":"89567db4-65e2-418b-b68f-d6e07fafa698","resolution":{"observed_at":"2026-08-16T12:14:06.892169Z","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":"2026-08-16T12:14:06.877114Z","title":"Bruhn and O","venue":null,"work_id":"0c1f96ed-a4c2-4e90-ad36-f55920ee2e54","year":2015},"citing_paper":{"arxiv_id":"2504.13454","last_updated":"2025-04-18T04:12:43Z","snapshot_observed_at":"2026-08-18T16:43:14.830124Z","submitted_at":"2025-04-18T04:12:43Z","title":"On the Averaging Problem of Ideal Families Related to Frankl's Conjecture with Formal Proof by Lean 4","version":1},"reference_index":4,"source":"pdf_text","source_observed_at":"2026-08-16T12:14:06.476641Z"},"links":{"citing_paper":"/paper/2504.13454"},"observation_digest":"sha256:360635734d56f6c5beb90422788038bcd52eb7215d70e6ed7fa960c63b0a2e63","observation_id":"f25899fa-d740-4d9d-8733-9794e0784391","resolution":{"observed_at":"2026-08-16T12:14:06.881045Z","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":{"arxiv_id":"2212.12500","last_updated":"2025-02-16T14:42:27Z","snapshot_observed_at":"2026-08-17T20:54:37.628834Z","submitted_at":"2022-12-23T17:48:48Z","title":"Better bounds for the union-closed sets conjecture using the entropy approach","version":2},"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":"2212.12500","snapshot_observed_at":"2026-08-16T12:14:06.480791Z","title":"Cambie, Better bounds for the union-closed sets conje cture using the entropy approach (2022)","venue":null,"work_id":null,"year":2022},"citing_paper":{"arxiv_id":"2504.13454","last_updated":"2025-04-18T04:12:43Z","snapshot_observed_at":"2026-08-18T16:43:14.830124Z","submitted_at":"2025-04-18T04:12:43Z","title":"On the Averaging Problem of Ideal Families Related to Frankl's Conjecture with Formal Proof by Lean 4","version":1},"reference_index":5,"source":"pdf_text","source_observed_at":"2026-08-16T12:14:06.480791Z"},"links":{"cited_paper":"/paper/2212.12500","citing_paper":"/paper/2504.13454"},"observation_digest":"sha256:c4ac86ca8d26c2418d9bd2cce8746b309c780b7e8eb829e6365f720817b2130f","observation_id":"af1ddc52-044a-439a-8241-fc51c2e3f4b8","resolution":{"observed_at":"2026-08-16T12:14:06.480791Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":{"arxiv_id":"2211.11689","last_updated":"2022-11-21T18:04:53Z","snapshot_observed_at":"2026-08-18T03:37:49.599998Z","submitted_at":"2022-11-21T18:04:53Z","title":"Approximate union closed conjecture","version":1},"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":"2211.11689","snapshot_observed_at":"2026-08-16T12:14:06.485125Z","title":"Chase, S","venue":null,"work_id":null,"year":2022},"citing_paper":{"arxiv_id":"2504.13454","last_updated":"2025-04-18T04:12:43Z","snapshot_observed_at":"2026-08-18T16:43:14.830124Z","submitted_at":"2025-04-18T04:12:43Z","title":"On the Averaging Problem of Ideal Families Related to Frankl's Conjecture with Formal Proof by Lean 4","version":1},"reference_index":6,"source":"pdf_text","source_observed_at":"2026-08-16T12:14:06.485125Z"},"links":{"cited_paper":"/paper/2211.11689","citing_paper":"/paper/2504.13454"},"observation_digest":"sha256:6e67f8ea2a8987f24afdb63c921a6cdd066351d97dbf8762cf0659fa4680abf8","observation_id":"95dfe79f-62e8-40d8-8cb6-4ea5cf65d8e4","resolution":{"observed_at":"2026-08-16T12:14:06.485125Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":"raw_reference","pith_arxiv_id":null,"snapshot_observed_at":"2026-08-16T12:14:06.865103Z","title":"Cz´ edli, M","venue":null,"work_id":"eb69119f-155a-4538-ad98-72c62d593ec4","year":2009},"citing_paper":{"arxiv_id":"2504.13454","last_updated":"2025-04-18T04:12:43Z","snapshot_observed_at":"2026-08-18T16:43:14.830124Z","submitted_at":"2025-04-18T04:12:43Z","title":"On the Averaging Problem of Ideal Families Related to Frankl's Conjecture with Formal Proof by Lean 4","version":1},"reference_index":7,"source":"pdf_text","source_observed_at":"2026-08-16T12:14:06.489433Z"},"links":{"citing_paper":"/paper/2504.13454"},"observation_digest":"sha256:3d98de243bfcf43cd52caf7caffd53c0c8b500fba146f8902db83bf7758f7e3b","observation_id":"49eb1e8c-436d-40bd-acb2-c46444a5924c","resolution":{"observed_at":"2026-08-16T12:14:06.868829Z","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":"2026-08-16T12:14:06.854470Z","title":"Frankl, On the Union-Closed Sets Conjecture, Unpubli shed manuscript, 1979","venue":null,"work_id":"cde94810-933e-4162-be1f-58a126e2bf3b","year":1979},"citing_paper":{"arxiv_id":"2504.13454","last_updated":"2025-04-18T04:12:43Z","snapshot_observed_at":"2026-08-18T16:43:14.830124Z","submitted_at":"2025-04-18T04:12:43Z","title":"On the Averaging Problem of Ideal Families Related to Frankl's Conjecture with Formal Proof by Lean 4","version":1},"reference_index":8,"source":"pdf_text","source_observed_at":"2026-08-16T12:14:06.492965Z"},"links":{"citing_paper":"/paper/2504.13454"},"observation_digest":"sha256:1b15f42583ad13a4fc417e2b34cdd39c225e7d8422d04814585a6abae982cf46","observation_id":"d5618e60-119c-420c-b2c2-26b8b9113e7c","resolution":{"observed_at":"2026-08-16T12:14:06.858255Z","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":{"arxiv_id":"2202.10087","last_updated":"2022-02-21T09:58:45Z","snapshot_observed_at":"2026-08-16T23:10:20.470346Z","submitted_at":"2022-02-21T09:58:45Z","title":"Finite groups admitting a coprime automorphism satisfying an additional polynomial identity","version":1},"cited_work":{"arxiv_id":"2202.10087","doi":null,"metadata_source":"pith","pith_arxiv_id":"2202.10087","snapshot_observed_at":"2026-08-16T12:14:06.659782Z","title":"Finite groups admitting a coprime automorphism satisfying an additional polynomial identity","venue":"math.GR","work_id":"f6c5568f-ee6d-4fba-93f7-c9fd8f00e15f","year":2022},"citing_paper":{"arxiv_id":"2504.13454","last_updated":"2025-04-18T04:12:43Z","snapshot_observed_at":"2026-08-18T16:43:14.830124Z","submitted_at":"2025-04-18T04:12:43Z","title":"On the Averaging Problem of Ideal Families Related to Frankl's Conjecture with Formal Proof by Lean 4","version":1},"reference_index":9,"source":"pdf_text","source_observed_at":"2026-08-16T12:14:06.496623Z"},"links":{"cited_paper":"/paper/2202.10087","citing_paper":"/paper/2504.13454"},"observation_digest":"sha256:ce76c77f8880c9f901255b557bcb0b6150a8e15fa6c75242b10887ec693bccd6","observation_id":"7899b308-d289-47be-a756-1eb95e25bcf6","resolution":{"observed_at":"2026-08-16T12:14:06.663865Z","resolver_source":"local_arxiv","status":"metadata_mismatch"},"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":"2026-08-16T12:14:06.843979Z","title":"https://github.com/features/copilot","venue":null,"work_id":"b5e4db7f-19ed-4b6c-839f-536dc2701578","year":null},"citing_paper":{"arxiv_id":"2504.13454","last_updated":"2025-04-18T04:12:43Z","snapshot_observed_at":"2026-08-18T16:43:14.830124Z","submitted_at":"2025-04-18T04:12:43Z","title":"On the Averaging Problem of Ideal Families Related to Frankl's Conjecture with Formal Proof by Lean 4","version":1},"reference_index":10,"source":"pdf_text","source_observed_at":"2026-08-16T12:14:06.500614Z"},"links":{"citing_paper":"/paper/2504.13454"},"observation_digest":"sha256:949ba8fa5f2206a78fce01dee72f706134901c5dd5779762c0db1d7dfe7b7f40","observation_id":"b374fddf-0b14-468a-a14c-2b87e5e7adc6","resolution":{"observed_at":"2026-08-16T12:14:06.847552Z","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":"2026-08-16T12:14:06.833719Z","title":"Gusakov, Formalizing the excluded minor characteri zation of binary ma- troids in the Lean theorem prover, Master Thesis, Universit y of Waterloo, 2024","venue":null,"work_id":"0ca364d2-89e3-4a98-937b-d55fe4135a9e","year":2024},"citing_paper":{"arxiv_id":"2504.13454","last_updated":"2025-04-18T04:12:43Z","snapshot_observed_at":"2026-08-18T16:43:14.830124Z","submitted_at":"2025-04-18T04:12:43Z","title":"On the Averaging Problem of Ideal Families Related to Frankl's Conjecture with Formal Proof by Lean 4","version":1},"reference_index":11,"source":"pdf_text","source_observed_at":"2026-08-16T12:14:06.504302Z"},"links":{"citing_paper":"/paper/2504.13454"},"observation_digest":"sha256:eeccd8f611977bfa726012e20750246d97f9ac5bee3c2f959c195abe0ed6f87c","observation_id":"2fa72bc7-56c8-4f7f-baec-540d3357bc23","resolution":{"observed_at":"2026-08-16T12:14:06.837490Z","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":"2026-08-16T12:14:06.823798Z","title":"Hachimori and K","venue":null,"work_id":"bbaac978-fd78-4ccd-92b2-92992dce24f9","year":2024},"citing_paper":{"arxiv_id":"2504.13454","last_updated":"2025-04-18T04:12:43Z","snapshot_observed_at":"2026-08-18T16:43:14.830124Z","submitted_at":"2025-04-18T04:12:43Z","title":"On the Averaging Problem of Ideal Families Related to Frankl's Conjecture with Formal Proof by Lean 4","version":1},"reference_index":12,"source":"pdf_text","source_observed_at":"2026-08-16T12:14:06.507724Z"},"links":{"citing_paper":"/paper/2504.13454"},"observation_digest":"sha256:dc5c6b04b13836b56056840efb534ee3101708a5d451bcd953b98ad77452db2a","observation_id":"09346e27-cfb0-46b9-8d35-775738f2f610","resolution":{"observed_at":"2026-08-16T12:14:06.827526Z","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":"2026-08-16T12:14:06.813364Z","title":"Kashiwabara, Formal proof of a problem of ideal famil ies by Lean 4, GitHub","venue":null,"work_id":"043b54d3-6170-4776-b877-78fe4652dfcf","year":null},"citing_paper":{"arxiv_id":"2504.13454","last_updated":"2025-04-18T04:12:43Z","snapshot_observed_at":"2026-08-18T16:43:14.830124Z","submitted_at":"2025-04-18T04:12:43Z","title":"On the Averaging Problem of Ideal Families Related to Frankl's Conjecture with Formal Proof by Lean 4","version":1},"reference_index":13,"source":"pdf_text","source_observed_at":"2026-08-16T12:14:06.511338Z"},"links":{"citing_paper":"/paper/2504.13454"},"observation_digest":"sha256:778163922078ab164a3381994ee8e1c99d9208bf14d9f462d62788bc1d262712","observation_id":"c507f94c-da0f-452a-aef2-e09470238247","resolution":{"observed_at":"2026-08-16T12:14:06.817134Z","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":"2026-08-16T12:14:06.803097Z","title":"Lean Copilot: Assisting formal pro of development in Lean,","venue":null,"work_id":"9014f3de-e433-4c01-822a-139ba4097d4a","year":null},"citing_paper":{"arxiv_id":"2504.13454","last_updated":"2025-04-18T04:12:43Z","snapshot_observed_at":"2026-08-18T16:43:14.830124Z","submitted_at":"2025-04-18T04:12:43Z","title":"On the Averaging Problem of Ideal Families Related to Frankl's Conjecture with Formal Proof by Lean 4","version":1},"reference_index":14,"source":"pdf_text","source_observed_at":"2026-08-16T12:14:06.514759Z"},"links":{"citing_paper":"/paper/2504.13454"},"observation_digest":"sha256:ad1f34ee8666eec4729ce4336e44ba282fbc5d15105758b7e5f98dc410037237","observation_id":"fd85a6ff-3230-4a17-9099-ef651d68bd5e","resolution":{"observed_at":"2026-08-16T12:14:06.806728Z","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":"2026-08-16T12:14:06.792464Z","title":null,"venue":null,"work_id":"c069c886-77fe-45e6-8f03-d5e55537edab","year":null},"citing_paper":{"arxiv_id":"2504.13454","last_updated":"2025-04-18T04:12:43Z","snapshot_observed_at":"2026-08-18T16:43:14.830124Z","submitted_at":"2025-04-18T04:12:43Z","title":"On the Averaging Problem of Ideal Families Related to Frankl's Conjecture with Formal Proof by Lean 4","version":1},"reference_index":15,"source":"pdf_text","source_observed_at":"2026-08-16T12:14:06.518312Z"},"links":{"citing_paper":"/paper/2504.13454"},"observation_digest":"sha256:7f27d9fbbd190eec07adedf1a2e99675e90ebf1a000b88d441ae7b2f72a6d2fd","observation_id":"3f3a26e2-1de4-469e-b83d-fdcfa541aed5","resolution":{"observed_at":"2026-08-16T12:14:06.796184Z","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":"2026-08-16T12:14:06.780814Z","title":"Liu, Improving the lower bound for the union-closed s ets conjecture via conditionally IID coupling, 58th Annual Conference on Information Scienc es and Systems (CISS), IEEE, 2024","venue":null,"work_id":"0a5ed741-b746-4282-9bd6-0edd38695ba5","year":2024},"citing_paper":{"arxiv_id":"2504.13454","last_updated":"2025-04-18T04:12:43Z","snapshot_observed_at":"2026-08-18T16:43:14.830124Z","submitted_at":"2025-04-18T04:12:43Z","title":"On the Averaging Problem of Ideal Families Related to Frankl's Conjecture with Formal Proof by Lean 4","version":1},"reference_index":16,"source":"pdf_text","source_observed_at":"2026-08-16T12:14:06.522278Z"},"links":{"citing_paper":"/paper/2504.13454"},"observation_digest":"sha256:9866e1838630bc581e7b5b0844f6c69075e615679ce97af03975112319f5ad08","observation_id":"794a08ef-792b-4449-8bd0-9586644e0136","resolution":{"observed_at":"2026-08-16T12:14:06.785214Z","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":{"arxiv_id":"1207.3604","last_updated":"2012-07-16T08:49:10Z","snapshot_observed_at":"2026-08-15T00:53:10.415199Z","submitted_at":"2012-07-16T08:49:10Z","title":"Formalizing Frankl's Conjecture: FC-families","version":1},"cited_work":{"arxiv_id":"1207.3604","doi":null,"metadata_source":"pith","pith_arxiv_id":"1207.3604","snapshot_observed_at":"2026-08-16T12:14:06.644094Z","title":"Formalizing Frankl's Conjecture: FC-families","venue":"cs.DM","work_id":"c1190616-e5ad-4f46-9d8d-41a596ba71c8","year":2012},"citing_paper":{"arxiv_id":"2504.13454","last_updated":"2025-04-18T04:12:43Z","snapshot_observed_at":"2026-08-18T16:43:14.830124Z","submitted_at":"2025-04-18T04:12:43Z","title":"On the Averaging Problem of Ideal Families Related to Frankl's Conjecture with Formal Proof by Lean 4","version":1},"reference_index":17,"source":"pdf_text","source_observed_at":"2026-08-16T12:14:06.525746Z"},"links":{"cited_paper":"/paper/1207.3604","citing_paper":"/paper/2504.13454"},"observation_digest":"sha256:907287e1be15a27607760508a1c08e28b0ff6d1803a78dfbae90bd62f3820bd4","observation_id":"fbe974af-2bff-46d0-9ee9-c48f790177b7","resolution":{"observed_at":"2026-08-16T12:14:06.648226Z","resolver_source":"local_arxiv","status":"verified_exact"},"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":{"arxiv_id":"2309.01704","last_updated":"2023-09-09T21:42:27Z","snapshot_observed_at":"2026-08-16T15:03:26.385887Z","submitted_at":"2023-09-04T16:29:20Z","title":"A Note on the Union-closed Sets Conjecture","version":3},"cited_work":{"arxiv_id":"2309.01704","doi":null,"metadata_source":"pith","pith_arxiv_id":"2309.01704","snapshot_observed_at":"2026-08-16T12:14:06.626299Z","title":"A Note on the Union-closed Sets Conjecture","venue":"math.CO","work_id":"639b97e8-76d1-4521-8498-d31d77173abd","year":2023},"citing_paper":{"arxiv_id":"2504.13454","last_updated":"2025-04-18T04:12:43Z","snapshot_observed_at":"2026-08-18T16:43:14.830124Z","submitted_at":"2025-04-18T04:12:43Z","title":"On the Averaging Problem of Ideal Families Related to Frankl's Conjecture with Formal Proof by Lean 4","version":1},"reference_index":18,"source":"pdf_text","source_observed_at":"2026-08-16T12:14:06.529504Z"},"links":{"cited_paper":"/paper/2309.01704","citing_paper":"/paper/2504.13454"},"observation_digest":"sha256:5e4d5ea9e9ef142701f28595948ad19c7b8b79e3b9a5c78df9afeb3d3b3dc79d","observation_id":"80d2fa7d-3fde-4a11-9e02-b23b58b48eb7","resolution":{"observed_at":"2026-08-16T12:14:06.632127Z","resolver_source":"local_arxiv","status":"verified_exact"},"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":"2026-08-16T12:14:06.769143Z","title":"Morris, FC-families and improved bounds for Frankl’ s conjecture, European Journal of Combinatorics 27 (2006), 269–282","venue":null,"work_id":"f9df7725-171e-4c1a-974e-a6df5257fb0c","year":2006},"citing_paper":{"arxiv_id":"2504.13454","last_updated":"2025-04-18T04:12:43Z","snapshot_observed_at":"2026-08-18T16:43:14.830124Z","submitted_at":"2025-04-18T04:12:43Z","title":"On the Averaging Problem of Ideal Families Related to Frankl's Conjecture with Formal Proof by Lean 4","version":1},"reference_index":19,"source":"pdf_text","source_observed_at":"2026-08-16T12:14:06.533432Z"},"links":{"citing_paper":"/paper/2504.13454"},"observation_digest":"sha256:dd91f4556b5b3a4a222a54f2a6d1778c471ec3fc3499030a83cfd8f39c9429af","observation_id":"e5ca7b8f-0795-4f1b-b136-dcfb8e14ae57","resolution":{"observed_at":"2026-08-16T12:14:06.773262Z","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":"2026-08-16T12:14:06.757845Z","title":"de Moura et al., The Lean 4 theorem prover and programm ing language, Proceedings of the 13th International Conference on Interactive Theore m Proving (ITP 2022), 2021","venue":null,"work_id":"c353d8c5-856d-4cdf-af7b-9e7a7340e942","year":2022},"citing_paper":{"arxiv_id":"2504.13454","last_updated":"2025-04-18T04:12:43Z","snapshot_observed_at":"2026-08-18T16:43:14.830124Z","submitted_at":"2025-04-18T04:12:43Z","title":"On the Averaging Problem of Ideal Families Related to Frankl's Conjecture with Formal Proof by Lean 4","version":1},"reference_index":20,"source":"pdf_text","source_observed_at":"2026-08-16T12:14:06.536787Z"},"links":{"citing_paper":"/paper/2504.13454"},"observation_digest":"sha256:786396b097db96fc64bce20b4daa785821be5617e3f4fe468c8772b0f16920b8","observation_id":"73bb5a95-e7b8-4007-87db-1bfb159f581c","resolution":{"observed_at":"2026-08-16T12:14:06.761609Z","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":{"arxiv_id":"2208.03803","last_updated":"2023-04-04T07:30:31Z","snapshot_observed_at":"2026-08-18T02:28:33.252760Z","submitted_at":"2022-08-07T20:29:19Z","title":"Notes on the Union Closed Sets Conjecture","version":2},"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":"2208.03803","snapshot_observed_at":"2026-08-16T12:14:06.540855Z","title":"Nagel, Notes on the Union Closed Sets Conjecture (202 2)","venue":null,"work_id":null,"year":null},"citing_paper":{"arxiv_id":"2504.13454","last_updated":"2025-04-18T04:12:43Z","snapshot_observed_at":"2026-08-18T16:43:14.830124Z","submitted_at":"2025-04-18T04:12:43Z","title":"On the Averaging Problem of Ideal Families Related to Frankl's Conjecture with Formal Proof by Lean 4","version":1},"reference_index":21,"source":"pdf_text","source_observed_at":"2026-08-16T12:14:06.540855Z"},"links":{"cited_paper":"/paper/2208.03803","citing_paper":"/paper/2504.13454"},"observation_digest":"sha256:1d04e6cf9b8090aa60a17ea1c858548370f7938b4c19a636c83a2951eba0e681","observation_id":"b6dd28ab-f458-40e3-bdce-00c311bce93c","resolution":{"observed_at":"2026-08-16T12:14:06.540855Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":"raw_reference","pith_arxiv_id":null,"snapshot_observed_at":"2026-08-16T12:14:06.745520Z","title":"ChatGPT, https://openai.com","venue":null,"work_id":"f1d0a264-e8f2-4a39-bbec-8bc94155794f","year":null},"citing_paper":{"arxiv_id":"2504.13454","last_updated":"2025-04-18T04:12:43Z","snapshot_observed_at":"2026-08-18T16:43:14.830124Z","submitted_at":"2025-04-18T04:12:43Z","title":"On the Averaging Problem of Ideal Families Related to Frankl's Conjecture with Formal Proof by Lean 4","version":1},"reference_index":22,"source":"pdf_text","source_observed_at":"2026-08-16T12:14:06.544756Z"},"links":{"citing_paper":"/paper/2504.13454"},"observation_digest":"sha256:29efa790fc6924e0819474a8d451c40b727947f3fa1b36f088b732fb3ae8af1b","observation_id":"6b1ca10f-5551-4284-8192-028e6eb1c7d5","resolution":{"observed_at":"2026-08-16T12:14:06.749475Z","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":{"arxiv_id":"2211.13139","last_updated":"2022-11-23T17:12:29Z","snapshot_observed_at":"2026-08-16T16:13:48.516987Z","submitted_at":"2022-11-23T17:12:29Z","title":"Extension of a Method of Gilmer","version":1},"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":"2211.13139","snapshot_observed_at":"2026-08-16T12:14:06.548564Z","title":"Pebody, Extension of a method of Gilmer (2022)","venue":null,"work_id":null,"year":2022},"citing_paper":{"arxiv_id":"2504.13454","last_updated":"2025-04-18T04:12:43Z","snapshot_observed_at":"2026-08-18T16:43:14.830124Z","submitted_at":"2025-04-18T04:12:43Z","title":"On the Averaging Problem of Ideal Families Related to Frankl's Conjecture with Formal Proof by Lean 4","version":1},"reference_index":23,"source":"pdf_text","source_observed_at":"2026-08-16T12:14:06.548564Z"},"links":{"cited_paper":"/paper/2211.13139","citing_paper":"/paper/2504.13454"},"observation_digest":"sha256:4ea170d7aa3a4739c0e953f6bece01d51b9ee3cba596f0523587675bc85d8112","observation_id":"d4e9d2a1-4a1e-4f63-8649-c991ad68cc0e","resolution":{"observed_at":"2026-08-16T12:14:06.548564Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":"raw_reference","pith_arxiv_id":null,"snapshot_observed_at":"2026-08-16T12:14:06.734214Z","title":"Poonen, Union-closed families, Journal of Combinat orial Theory, Series A 59 (1992), 253–268","venue":null,"work_id":"7a2122c8-4571-4ae6-a9d8-d4a1472ce2e1","year":1992},"citing_paper":{"arxiv_id":"2504.13454","last_updated":"2025-04-18T04:12:43Z","snapshot_observed_at":"2026-08-18T16:43:14.830124Z","submitted_at":"2025-04-18T04:12:43Z","title":"On the Averaging Problem of Ideal Families Related to Frankl's Conjecture with Formal Proof by Lean 4","version":1},"reference_index":24,"source":"pdf_text","source_observed_at":"2026-08-16T12:14:06.552132Z"},"links":{"citing_paper":"/paper/2504.13454"},"observation_digest":"sha256:8ea96a202a5d97c45648431d8b63aa37d7548eaa8bca19dda04c152e95822f8e","observation_id":"ea996ed0-a583-4d27-b1fa-776f2663046c","resolution":{"observed_at":"2026-08-16T12:14:06.738124Z","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":"2026-08-16T12:14:06.723121Z","title":"Reinhold, Frankl’s conjecture is true for lower semi modular lattices, Graphs and Com- binatorics, 16 (2000), 115–116","venue":null,"work_id":"7217b73b-db43-4f7c-9b4f-6bb1a666171d","year":2000},"citing_paper":{"arxiv_id":"2504.13454","last_updated":"2025-04-18T04:12:43Z","snapshot_observed_at":"2026-08-18T16:43:14.830124Z","submitted_at":"2025-04-18T04:12:43Z","title":"On the Averaging Problem of Ideal Families Related to Frankl's Conjecture with Formal Proof by Lean 4","version":1},"reference_index":25,"source":"pdf_text","source_observed_at":"2026-08-16T12:14:06.555779Z"},"links":{"citing_paper":"/paper/2504.13454"},"observation_digest":"sha256:e7a396545de27b3282d6027fc7aa9de6ae50661d9e1e1bbaf65886404fc1585d","observation_id":"b3ae7706-0240-4f7d-aa01-f0d007aa8b5d","resolution":{"observed_at":"2026-08-16T12:14:06.727267Z","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":"2026-08-16T12:14:06.712074Z","title":"Roberts and J","venue":null,"work_id":"e0a06e6a-bcf7-4ef6-bc74-cb4725f42f12","year":2010},"citing_paper":{"arxiv_id":"2504.13454","last_updated":"2025-04-18T04:12:43Z","snapshot_observed_at":"2026-08-18T16:43:14.830124Z","submitted_at":"2025-04-18T04:12:43Z","title":"On the Averaging Problem of Ideal Families Related to Frankl's Conjecture with Formal Proof by Lean 4","version":1},"reference_index":26,"source":"pdf_text","source_observed_at":"2026-08-16T12:14:06.559116Z"},"links":{"citing_paper":"/paper/2504.13454"},"observation_digest":"sha256:7c2fef795a1e959a9aa5bfb4fc1c98dc9805deefc896dd50f02ce2e39b46361b","observation_id":"48c5db4b-9359-4328-bb8c-1437aaa0c8b6","resolution":{"observed_at":"2026-08-16T12:14:06.715855Z","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":{"arxiv_id":"2211.11504","last_updated":"2023-06-19T11:25:34Z","snapshot_observed_at":"2026-08-16T16:14:31.047773Z","submitted_at":"2022-11-21T14:42:37Z","title":"An improved lower bound for the union-closed set conjecture","version":3},"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":"2211.11504","snapshot_observed_at":"2026-08-16T12:14:06.562430Z","title":"Sawin, An improved lower bound for the union-closed s et conjecture (2022)","venue":null,"work_id":null,"year":2022},"citing_paper":{"arxiv_id":"2504.13454","last_updated":"2025-04-18T04:12:43Z","snapshot_observed_at":"2026-08-18T16:43:14.830124Z","submitted_at":"2025-04-18T04:12:43Z","title":"On the Averaging Problem of Ideal Families Related to Frankl's Conjecture with Formal Proof by Lean 4","version":1},"reference_index":27,"source":"pdf_text","source_observed_at":"2026-08-16T12:14:06.562430Z"},"links":{"cited_paper":"/paper/2211.11504","citing_paper":"/paper/2504.13454"},"observation_digest":"sha256:aeec3dabaa607d358b2d8fb74af5c64846e810bbf90fafa8f90822514669da71","observation_id":"0169e471-40e3-4a95-a464-57af04bafa08","resolution":{"observed_at":"2026-08-16T12:14:06.562430Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":"raw_reference","pith_arxiv_id":null,"snapshot_observed_at":"2026-08-16T12:14:06.700279Z","title":"Yu, Dimension-free bounds for the union-closed sets conjecture","venue":null,"work_id":"89f4a484-66ed-4af2-ade3-05575b09da1b","year":2023},"citing_paper":{"arxiv_id":"2504.13454","last_updated":"2025-04-18T04:12:43Z","snapshot_observed_at":"2026-08-18T16:43:14.830124Z","submitted_at":"2025-04-18T04:12:43Z","title":"On the Averaging Problem of Ideal Families Related to Frankl's Conjecture with Formal Proof by Lean 4","version":1},"reference_index":28,"source":"pdf_text","source_observed_at":"2026-08-16T12:14:06.566011Z"},"links":{"citing_paper":"/paper/2504.13454"},"observation_digest":"sha256:eaee2f013035ecbe9879d93d15ae1c9f6ee5761ff4a104fd872cd5977be473f7","observation_id":"775e441c-d057-4a45-989e-aa578b586e1a","resolution":{"observed_at":"2026-08-16T12:14:06.704240Z","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":"2504.13454","last_updated":"2025-04-18T04:12:43Z","latest_version":1,"primary_category":"math.CO","snapshot_observed_at":"2026-08-18T16:43:14.830124Z","submitted_at":"2025-04-18T04:12:43Z","title":"On the Averaging Problem of Ideal Families Related to Frankl's Conjecture with Formal Proof by Lean 4"},"reference_resolution":{"displayed":28,"state_counts":{"malformed_identifier":0,"metadata_mismatch":1,"parse_uncertain":0,"unresolved":7,"verified_exact":2,"verified_fuzzy":18},"total_outbound_references":28},"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 28 of 28 outbound references and 0 inbound Pith citation observations for arXiv:2504.13454."}