{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2018:TYAEGFEXTLYR5TF6ZBYNDPMJHL","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":"9f1020d7a9336ba55b250dac883ada173ee1660adabaaf4fee641ad0b4b4f00b","cross_cats_sorted":["math.CO"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RA","submitted_at":"2018-07-22T23:34:58Z","title_canon_sha256":"6d95933e71c5a45b086270f01577d5c4b5026845171c8dc6d4150bfc15a4e5d5"},"schema_version":"1.0","source":{"id":"1807.08384","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1807.08384","created_at":"2026-05-18T00:04:44Z"},{"alias_kind":"arxiv_version","alias_value":"1807.08384v1","created_at":"2026-05-18T00:04:44Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1807.08384","created_at":"2026-05-18T00:04:44Z"},{"alias_kind":"pith_short_12","alias_value":"TYAEGFEXTLYR","created_at":"2026-05-18T12:32:56Z"},{"alias_kind":"pith_short_16","alias_value":"TYAEGFEXTLYR5TF6","created_at":"2026-05-18T12:32:56Z"},{"alias_kind":"pith_short_8","alias_value":"TYAEGFEX","created_at":"2026-05-18T12:32:56Z"}],"graph_snapshots":[{"event_id":"sha256:93d1d8862b250632e3f6e8b7376bd97bd427cc0fbca79ddb45e75a9e8d4889c0","target":"graph","created_at":"2026-05-18T00:04:44Z","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"},"paper":{"abstract_excerpt":"Let $L$ be an $n$-element finite lattice. We prove that if $L$ has strictly more than $2^{n-5}$ congruences, then $L$ is planar. This result is sharp, since for each natural number $n\\geq 8$, there exists a non-planar lattice with exactly $2^{n-5}$ congruences.","authors_text":"G\\'abor Cz\\'edli","cross_cats":["math.CO"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RA","submitted_at":"2018-07-22T23:34:58Z","title":"Lattices with many congruences are planar"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1807.08384","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":"sha256:1e03a314988217d3c319b36f49711d91c87863d6bde3868b0495932da10738c4","target":"record","created_at":"2026-05-18T00:04:44Z","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":"9f1020d7a9336ba55b250dac883ada173ee1660adabaaf4fee641ad0b4b4f00b","cross_cats_sorted":["math.CO"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RA","submitted_at":"2018-07-22T23:34:58Z","title_canon_sha256":"6d95933e71c5a45b086270f01577d5c4b5026845171c8dc6d4150bfc15a4e5d5"},"schema_version":"1.0","source":{"id":"1807.08384","kind":"arxiv","version":1}},"canonical_sha256":"9e004314979af11eccbec870d1bd893af689c593e880887b9602483229cbf543","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"9e004314979af11eccbec870d1bd893af689c593e880887b9602483229cbf543","first_computed_at":"2026-05-18T00:04:44.555340Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:04:44.555340Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"r6A8NWLtSVMWcf83iidaTfoXYHRRmzJmswXI+J9qjI30ds72jdU5RiSITSHeqP+G72KbeTQWEA0wjDW2FCgkCw==","signature_status":"signed_v1","signed_at":"2026-05-18T00:04:44.555983Z","signed_message":"canonical_sha256_bytes"},"source_id":"1807.08384","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:1e03a314988217d3c319b36f49711d91c87863d6bde3868b0495932da10738c4","sha256:93d1d8862b250632e3f6e8b7376bd97bd427cc0fbca79ddb45e75a9e8d4889c0"],"state_sha256":"02defe8d67f03263c7b73b64a6209f31a2528e17a75469f947df6119b27e5f8b"}