{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2026:TJPUHTZOJR42GAJLSJWLJFEIKN","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"791cfa8ac26c8d2d69fd18b08ec07da1ea773ded8b751a0c4d528f379e90a8f8","cross_cats_sorted":["cs.IT","math.IT"],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CO","submitted_at":"2026-07-31T17:57:08Z","title_canon_sha256":"c643ea48d21076c25b2392807e5230529a9f089dfeccdbd1a455f1465d356312"},"schema_version":"1.0","source":{"id":"2607.29681","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2607.29681","created_at":"2026-08-03T01:41:44Z"},{"alias_kind":"arxiv_version","alias_value":"2607.29681v1","created_at":"2026-08-03T01:41:44Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2607.29681","created_at":"2026-08-03T01:41:44Z"},{"alias_kind":"pith_short_12","alias_value":"TJPUHTZOJR42","created_at":"2026-08-03T01:41:44Z"},{"alias_kind":"pith_short_16","alias_value":"TJPUHTZOJR42GAJL","created_at":"2026-08-03T01:41:44Z"},{"alias_kind":"pith_short_8","alias_value":"TJPUHTZO","created_at":"2026-08-03T01:41:44Z"}],"graph_snapshots":[{"event_id":"sha256:59d73afdfcea93d1ddfedf438984873b081251e008d10036e66ea4a2c23c393e","target":"graph","created_at":"2026-08-03T01:41:44Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"integrity":{"available":true,"clean":true,"detectors_run":[],"endpoint":"/pith/2607.29681/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"We give new lower bounds for the Shannon capacities of small odd cycles: $\\Theta(C_7)\\geq3.258805369885\\ldots$, $\\Theta(C_{11})\\geq5.294502522149\\ldots$, $\\Theta(C_{13})\\geq6.302455083464\\ldots$, $\\Theta(C_{15})\\geq7.301600534487\\ldots$, $\\Theta(C_{19})\\geq9.357192705918\\ldots$, $\\Theta(C_{21})\\geq10.342455853338\\ldots$, and $\\Theta(C_{23})\\geq11.328224257774\\ldots$. The bounds are obtained by an iterative procedure due to Gao (2026) which is based on a method by Itty, Rosin, Carstensen and Reichman (2026). The bounds are fully formalised in Lean.","authors_text":"Jeroen Zuiddam, Pjotr Buys, Sven Polak","cross_cats":["cs.IT","math.IT"],"headline":"","license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CO","submitted_at":"2026-07-31T17:57:08Z","title":"Lean-verified lower bounds for the Shannon capacity of odd cycles"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2607.29681","kind":"arxiv","version":1},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:3b189a35eae30475f6f8849f9363e4808f1afb31a8891a0edd1dce480584917d","target":"record","created_at":"2026-08-03T01:41:44Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"791cfa8ac26c8d2d69fd18b08ec07da1ea773ded8b751a0c4d528f379e90a8f8","cross_cats_sorted":["cs.IT","math.IT"],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CO","submitted_at":"2026-07-31T17:57:08Z","title_canon_sha256":"c643ea48d21076c25b2392807e5230529a9f089dfeccdbd1a455f1465d356312"},"schema_version":"1.0","source":{"id":"2607.29681","kind":"arxiv","version":1}},"canonical_sha256":"9a5f43cf2e4c79a3012b926cb494885360e2ec445d300e1f8a671618dea2c3eb","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"9a5f43cf2e4c79a3012b926cb494885360e2ec445d300e1f8a671618dea2c3eb","first_computed_at":"2026-08-03T01:41:44.217966Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-08-03T01:41:44.217966Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"3JpOM0D4z/5v+bEiW1AcB3DmmOJESljY1+P6JhJv90RqC3AHNXL8j5py2WYyt7rvlaTqpXRnWtFphhGTRqsTAw==","signature_status":"signed_v1","signed_at":"2026-08-03T01:41:44.219844Z","signed_message":"canonical_sha256_bytes"},"source_id":"2607.29681","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:3b189a35eae30475f6f8849f9363e4808f1afb31a8891a0edd1dce480584917d","sha256:59d73afdfcea93d1ddfedf438984873b081251e008d10036e66ea4a2c23c393e"],"state_sha256":"ec963c1bd37f09dcc95e72d032f49f4465a4ec94759aa43937a5dc419ee69e32"}