{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2010:7PICQKF62HBITJBKXZELZ25KCJ","short_pith_number":"pith:7PICQKF6","canonical_record":{"source":{"id":"1003.5742","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.LO","submitted_at":"2010-03-30T06:47:13Z","cross_cats_sorted":["math.CT"],"title_canon_sha256":"173831b99767b18409c5a7b59a871ff30a1ba87d6d58254cbcb578132d2ae447","abstract_canon_sha256":"995c27f1501ea81d43b5b5230752eb7e49d69b79a4f6ea609d16e21b6b64d0ac"},"schema_version":"1.0"},"canonical_sha256":"fbd02828bed1c289a42abe48bcebaa125afff3b207e29a7a18c1b254e7267a56","source":{"kind":"arxiv","id":"1003.5742","version":3},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1003.5742","created_at":"2026-05-18T02:55:57Z"},{"alias_kind":"arxiv_version","alias_value":"1003.5742v3","created_at":"2026-05-18T02:55:57Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1003.5742","created_at":"2026-05-18T02:55:57Z"},{"alias_kind":"pith_short_12","alias_value":"7PICQKF62HBI","created_at":"2026-05-18T12:26:05Z"},{"alias_kind":"pith_short_16","alias_value":"7PICQKF62HBITJBK","created_at":"2026-05-18T12:26:05Z"},{"alias_kind":"pith_short_8","alias_value":"7PICQKF6","created_at":"2026-05-18T12:26:05Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2010:7PICQKF62HBITJBKXZELZ25KCJ","target":"record","payload":{"canonical_record":{"source":{"id":"1003.5742","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.LO","submitted_at":"2010-03-30T06:47:13Z","cross_cats_sorted":["math.CT"],"title_canon_sha256":"173831b99767b18409c5a7b59a871ff30a1ba87d6d58254cbcb578132d2ae447","abstract_canon_sha256":"995c27f1501ea81d43b5b5230752eb7e49d69b79a4f6ea609d16e21b6b64d0ac"},"schema_version":"1.0"},"canonical_sha256":"fbd02828bed1c289a42abe48bcebaa125afff3b207e29a7a18c1b254e7267a56","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:55:57.787378Z","signature_b64":"4rt879eoL1vusdhuCBX4GYxGUL1HB7Pq513ajWn8BrrfV0c7oHZMY9FcQ/fupEyfw7fdY2Z5IfCSC7IiXyHzCw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"fbd02828bed1c289a42abe48bcebaa125afff3b207e29a7a18c1b254e7267a56","last_reissued_at":"2026-05-18T02:55:57.786828Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:55:57.786828Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1003.5742","source_version":3,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T02:55:57Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"uwNcWs5AJFR/3gCysinoW6MG6fxd5xwBgoAORSHHd/+HHfiIAXMgF/KexTfepz8BMvKuGJ2GjQE93b7fldEkDQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-23T12:56:26.040919Z"},"content_sha256":"55eddf7c73472cbd157bf5530497ef7209c3fd25002855346ab42808a1859653","schema_version":"1.0","event_id":"sha256:55eddf7c73472cbd157bf5530497ef7209c3fd25002855346ab42808a1859653"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2010:7PICQKF62HBITJBKXZELZ25KCJ","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"The possible values of critical points between varieties of lattices","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CT"],"primary_cat":"math.LO","authors_text":"Pierre Gillibert (LMNO)","submitted_at":"2010-03-30T06:47:13Z","abstract_excerpt":"We denote by Conc(L) the semilattice of all finitely generated congruences of a lattice L. For varieties (i.e., equational classes) V and W of lattices such that V is contained neither in W nor its dual, and such that every simple member of W contains a prime interval, we prove that there exists a bounded lattice A in V with at most aleph 2 elements such that Conc(A) is not isomorphic to Conc(B) for any B in W. The bound aleph 2 is optimal. As a corollary of our results, there are continuum many congruence classes of locally finite varieties of (bounded) modular lattices."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1003.5742","kind":"arxiv","version":3},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T02:55:57Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"RmDpI22AFMY144huGrmlGu1gZRaUJqA/FxMexpUhf2j4l16uRMf5r7z1k1J7smLwSd1xDDAkSP9RoTwmXfgABg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-23T12:56:26.041252Z"},"content_sha256":"4808c47360b334ec2e1392b14da1e212a83b0d2f24dcef06afa29134959f0957","schema_version":"1.0","event_id":"sha256:4808c47360b334ec2e1392b14da1e212a83b0d2f24dcef06afa29134959f0957"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/7PICQKF62HBITJBKXZELZ25KCJ/bundle.json","state_url":"https://pith.science/pith/7PICQKF62HBITJBKXZELZ25KCJ/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/7PICQKF62HBITJBKXZELZ25KCJ/bundle.json","status":"primary"}],"public_keys":[{"key_id":"pith-v1-2026-05","algorithm":"ed25519","format":"raw","public_key_b64":"stVStoiQhXFxp4s2pdzPNoqVNBMojDU/fJ2db5S3CbM=","public_key_hex":"b2d552b68890857171a78b36a5dccf368a953413288c353f7c9d9d6f94b709b3","fingerprint_sha256_b32_first128bits":"RVFV5Z2OI2J3ZUO7ERDEBCYNKS","fingerprint_sha256_hex":"8d4b5ee74e4693bcd1df2446408b0d54","rotates_at":null,"url":"https://pith.science/pith-signing-key.json","notes":"Pith uses this Ed25519 key to sign canonical record SHA-256 digests. Verify with: ed25519_verify(public_key, message=canonical_sha256_bytes, signature=base64decode(signature_b64))."}],"merge_version":"pith-open-graph-merge-v1","built_at":"2026-06-23T12:56:26Z","links":{"resolver":"https://pith.science/pith/7PICQKF62HBITJBKXZELZ25KCJ","bundle":"https://pith.science/pith/7PICQKF62HBITJBKXZELZ25KCJ/bundle.json","state":"https://pith.science/pith/7PICQKF62HBITJBKXZELZ25KCJ/state.json","well_known_bundle":"https://pith.science/.well-known/pith/7PICQKF62HBITJBKXZELZ25KCJ/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2010:7PICQKF62HBITJBKXZELZ25KCJ","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":"995c27f1501ea81d43b5b5230752eb7e49d69b79a4f6ea609d16e21b6b64d0ac","cross_cats_sorted":["math.CT"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.LO","submitted_at":"2010-03-30T06:47:13Z","title_canon_sha256":"173831b99767b18409c5a7b59a871ff30a1ba87d6d58254cbcb578132d2ae447"},"schema_version":"1.0","source":{"id":"1003.5742","kind":"arxiv","version":3}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1003.5742","created_at":"2026-05-18T02:55:57Z"},{"alias_kind":"arxiv_version","alias_value":"1003.5742v3","created_at":"2026-05-18T02:55:57Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1003.5742","created_at":"2026-05-18T02:55:57Z"},{"alias_kind":"pith_short_12","alias_value":"7PICQKF62HBI","created_at":"2026-05-18T12:26:05Z"},{"alias_kind":"pith_short_16","alias_value":"7PICQKF62HBITJBK","created_at":"2026-05-18T12:26:05Z"},{"alias_kind":"pith_short_8","alias_value":"7PICQKF6","created_at":"2026-05-18T12:26:05Z"}],"graph_snapshots":[{"event_id":"sha256:4808c47360b334ec2e1392b14da1e212a83b0d2f24dcef06afa29134959f0957","target":"graph","created_at":"2026-05-18T02:55:57Z","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":"We denote by Conc(L) the semilattice of all finitely generated congruences of a lattice L. For varieties (i.e., equational classes) V and W of lattices such that V is contained neither in W nor its dual, and such that every simple member of W contains a prime interval, we prove that there exists a bounded lattice A in V with at most aleph 2 elements such that Conc(A) is not isomorphic to Conc(B) for any B in W. The bound aleph 2 is optimal. As a corollary of our results, there are continuum many congruence classes of locally finite varieties of (bounded) modular lattices.","authors_text":"Pierre Gillibert (LMNO)","cross_cats":["math.CT"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.LO","submitted_at":"2010-03-30T06:47:13Z","title":"The possible values of critical points between varieties of lattices"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1003.5742","kind":"arxiv","version":3},"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":"sha256:55eddf7c73472cbd157bf5530497ef7209c3fd25002855346ab42808a1859653","target":"record","created_at":"2026-05-18T02:55:57Z","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":"995c27f1501ea81d43b5b5230752eb7e49d69b79a4f6ea609d16e21b6b64d0ac","cross_cats_sorted":["math.CT"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.LO","submitted_at":"2010-03-30T06:47:13Z","title_canon_sha256":"173831b99767b18409c5a7b59a871ff30a1ba87d6d58254cbcb578132d2ae447"},"schema_version":"1.0","source":{"id":"1003.5742","kind":"arxiv","version":3}},"canonical_sha256":"fbd02828bed1c289a42abe48bcebaa125afff3b207e29a7a18c1b254e7267a56","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"fbd02828bed1c289a42abe48bcebaa125afff3b207e29a7a18c1b254e7267a56","first_computed_at":"2026-05-18T02:55:57.786828Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T02:55:57.786828Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"4rt879eoL1vusdhuCBX4GYxGUL1HB7Pq513ajWn8BrrfV0c7oHZMY9FcQ/fupEyfw7fdY2Z5IfCSC7IiXyHzCw==","signature_status":"signed_v1","signed_at":"2026-05-18T02:55:57.787378Z","signed_message":"canonical_sha256_bytes"},"source_id":"1003.5742","source_kind":"arxiv","source_version":3}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:55eddf7c73472cbd157bf5530497ef7209c3fd25002855346ab42808a1859653","sha256:4808c47360b334ec2e1392b14da1e212a83b0d2f24dcef06afa29134959f0957"],"state_sha256":"2c0a953b2b9094678fc428be710670660a7c9c90451c6827ea59cd24438a50a0"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"GpQSXq3eZw2Z+EV0vH44B/oMg8x3P53wArUqnFqXHiWg0gq1cu/H/Bq7QhQY0/JnrYhfsH5MYQfegIJF9D16AQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-23T12:56:26.043078Z","bundle_sha256":"23a366de13369ce88c6927959a9e8c0398189edff7d464866ef99d082ea58d55"}}