{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2010:5LBEZRVHGFQKXVIRTRW5FXBF65","short_pith_number":"pith:5LBEZRVH","canonical_record":{"source":{"id":"1009.0949","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RA","submitted_at":"2010-08-16T14:36:43Z","cross_cats_sorted":[],"title_canon_sha256":"0fee4a0e6f533cc7b941c0cf58caee31aaf3ac9df75d1819f2f1288b4f4da173","abstract_canon_sha256":"9e5589ca08c38de73acd0ae0791365ee3a6f86f84793f1e0fade07f1b1728374"},"schema_version":"1.0"},"canonical_sha256":"eac24cc6a73160abd5119c6dd2dc25f779bc0a2c1008f3a6eb177d4a6b409b9c","source":{"kind":"arxiv","id":"1009.0949","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1009.0949","created_at":"2026-05-18T04:27:56Z"},{"alias_kind":"arxiv_version","alias_value":"1009.0949v2","created_at":"2026-05-18T04:27:56Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1009.0949","created_at":"2026-05-18T04:27:56Z"},{"alias_kind":"pith_short_12","alias_value":"5LBEZRVHGFQK","created_at":"2026-05-18T12:26:04Z"},{"alias_kind":"pith_short_16","alias_value":"5LBEZRVHGFQKXVIR","created_at":"2026-05-18T12:26:04Z"},{"alias_kind":"pith_short_8","alias_value":"5LBEZRVH","created_at":"2026-05-18T12:26:04Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2010:5LBEZRVHGFQKXVIRTRW5FXBF65","target":"record","payload":{"canonical_record":{"source":{"id":"1009.0949","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RA","submitted_at":"2010-08-16T14:36:43Z","cross_cats_sorted":[],"title_canon_sha256":"0fee4a0e6f533cc7b941c0cf58caee31aaf3ac9df75d1819f2f1288b4f4da173","abstract_canon_sha256":"9e5589ca08c38de73acd0ae0791365ee3a6f86f84793f1e0fade07f1b1728374"},"schema_version":"1.0"},"canonical_sha256":"eac24cc6a73160abd5119c6dd2dc25f779bc0a2c1008f3a6eb177d4a6b409b9c","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:27:56.684888Z","signature_b64":"IHjU0W5yHNXKYtgnOu/iwwwBer9RLHLoYKTpZwkJopNcJhJ6IX4TbkBi9smh/rBUANBe8jJAwmmNWYi0wPv2AQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"eac24cc6a73160abd5119c6dd2dc25f779bc0a2c1008f3a6eb177d4a6b409b9c","last_reissued_at":"2026-05-18T04:27:56.684357Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:27:56.684357Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1009.0949","source_version":2,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T04:27:56Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Qv1wcEp9rpH0QZDJbC8xEze33On6dNlpSF6qtL40WqJ1PI682EpPO2NVuHP++UHzxrrOY9ufSTGBTFEB8oEUAA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-26T19:08:46.743838Z"},"content_sha256":"27a5052546b4584a7d8f4d4682ff07c3573402c4f80fc9e43808ae659632ca71","schema_version":"1.0","event_id":"sha256:27a5052546b4584a7d8f4d4682ff07c3573402c4f80fc9e43808ae659632ca71"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2010:5LBEZRVHGFQKXVIRTRW5FXBF65","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Infinite combinatorial issues raised by lifting problems in universal algebra","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.RA","authors_text":"Friedrich Wehrung (LMNO)","submitted_at":"2010-08-16T14:36:43Z","abstract_excerpt":"The critical point between varieties A and B of algebras is defined as the least cardinality of the semilattice of compact congruences of a member of A but of no member of B, if it exists. The study of critical points gives rise to a whole array of problems, often involving lifting problems of either diagrams or objects, with respect to functors. These, in turn, involve problems that belong to infinite combinatorics. We survey some of the combinatorial problems and results thus encountered. The corresponding problematic is articulated around the notion of a k-ladder (for proving that a critica"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1009.0949","kind":"arxiv","version":2},"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":"2026-05-18T04:27:56Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"rdqwNdo/dpIZ7EU363BiIq1k5DoQuqlOrD8pT7usNH1pS8Lg/PpLPh3l/plkgvNfziH7Z/eDn6ZpfxgcTqoNCw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-26T19:08:46.744514Z"},"content_sha256":"066df0fa602ab516e88df65926091be32e4129b4557a8e059d9bee4f6aff2ab1","schema_version":"1.0","event_id":"sha256:066df0fa602ab516e88df65926091be32e4129b4557a8e059d9bee4f6aff2ab1"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/5LBEZRVHGFQKXVIRTRW5FXBF65/bundle.json","state_url":"https://pith.science/pith/5LBEZRVHGFQKXVIRTRW5FXBF65/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/5LBEZRVHGFQKXVIRTRW5FXBF65/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":"2026-05-26T19:08:46Z","links":{"resolver":"https://pith.science/pith/5LBEZRVHGFQKXVIRTRW5FXBF65","bundle":"https://pith.science/pith/5LBEZRVHGFQKXVIRTRW5FXBF65/bundle.json","state":"https://pith.science/pith/5LBEZRVHGFQKXVIRTRW5FXBF65/state.json","well_known_bundle":"https://pith.science/.well-known/pith/5LBEZRVHGFQKXVIRTRW5FXBF65/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2010:5LBEZRVHGFQKXVIRTRW5FXBF65","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":"9e5589ca08c38de73acd0ae0791365ee3a6f86f84793f1e0fade07f1b1728374","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RA","submitted_at":"2010-08-16T14:36:43Z","title_canon_sha256":"0fee4a0e6f533cc7b941c0cf58caee31aaf3ac9df75d1819f2f1288b4f4da173"},"schema_version":"1.0","source":{"id":"1009.0949","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1009.0949","created_at":"2026-05-18T04:27:56Z"},{"alias_kind":"arxiv_version","alias_value":"1009.0949v2","created_at":"2026-05-18T04:27:56Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1009.0949","created_at":"2026-05-18T04:27:56Z"},{"alias_kind":"pith_short_12","alias_value":"5LBEZRVHGFQK","created_at":"2026-05-18T12:26:04Z"},{"alias_kind":"pith_short_16","alias_value":"5LBEZRVHGFQKXVIR","created_at":"2026-05-18T12:26:04Z"},{"alias_kind":"pith_short_8","alias_value":"5LBEZRVH","created_at":"2026-05-18T12:26:04Z"}],"graph_snapshots":[{"event_id":"sha256:066df0fa602ab516e88df65926091be32e4129b4557a8e059d9bee4f6aff2ab1","target":"graph","created_at":"2026-05-18T04:27:56Z","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":"The critical point between varieties A and B of algebras is defined as the least cardinality of the semilattice of compact congruences of a member of A but of no member of B, if it exists. The study of critical points gives rise to a whole array of problems, often involving lifting problems of either diagrams or objects, with respect to functors. These, in turn, involve problems that belong to infinite combinatorics. We survey some of the combinatorial problems and results thus encountered. The corresponding problematic is articulated around the notion of a k-ladder (for proving that a critica","authors_text":"Friedrich Wehrung (LMNO)","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RA","submitted_at":"2010-08-16T14:36:43Z","title":"Infinite combinatorial issues raised by lifting problems in universal algebra"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1009.0949","kind":"arxiv","version":2},"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":"sha256:27a5052546b4584a7d8f4d4682ff07c3573402c4f80fc9e43808ae659632ca71","target":"record","created_at":"2026-05-18T04:27:56Z","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":"9e5589ca08c38de73acd0ae0791365ee3a6f86f84793f1e0fade07f1b1728374","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RA","submitted_at":"2010-08-16T14:36:43Z","title_canon_sha256":"0fee4a0e6f533cc7b941c0cf58caee31aaf3ac9df75d1819f2f1288b4f4da173"},"schema_version":"1.0","source":{"id":"1009.0949","kind":"arxiv","version":2}},"canonical_sha256":"eac24cc6a73160abd5119c6dd2dc25f779bc0a2c1008f3a6eb177d4a6b409b9c","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"eac24cc6a73160abd5119c6dd2dc25f779bc0a2c1008f3a6eb177d4a6b409b9c","first_computed_at":"2026-05-18T04:27:56.684357Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T04:27:56.684357Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"IHjU0W5yHNXKYtgnOu/iwwwBer9RLHLoYKTpZwkJopNcJhJ6IX4TbkBi9smh/rBUANBe8jJAwmmNWYi0wPv2AQ==","signature_status":"signed_v1","signed_at":"2026-05-18T04:27:56.684888Z","signed_message":"canonical_sha256_bytes"},"source_id":"1009.0949","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:27a5052546b4584a7d8f4d4682ff07c3573402c4f80fc9e43808ae659632ca71","sha256:066df0fa602ab516e88df65926091be32e4129b4557a8e059d9bee4f6aff2ab1"],"state_sha256":"dc5f5ae3664b3e4b3d0f38baaa26aa37913658b52ad7a720e397657b9d302b3c"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"AwCnaFVG1VmEiSzMy/2hyTJy79R+gtlPeIpegvtL5T2O4wEfd+uEsm5xWE9MwHeltvzPY0SrBBABR7Z10rsaCQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-26T19:08:46.748183Z","bundle_sha256":"10d374d48b50ce6e4e9804c4cff89d0b7de8d88d2d286b7ab7e4c56a5d192819"}}