{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2024:P6RORCZVDV6CAF267HC6VD6GVC","short_pith_number":"pith:P6RORCZV","canonical_record":{"source":{"id":"2411.15992","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CO","submitted_at":"2024-11-24T22:02:58Z","cross_cats_sorted":[],"title_canon_sha256":"9c7e3d115fa1a6d39f002aca43e0df7dcd4153571d413e3f303d2bb2eb70bea0","abstract_canon_sha256":"3578b2d43749c63556240c019b5949848141937968dfdc37dcb406d12e3cf122"},"schema_version":"1.0"},"canonical_sha256":"7fa2e88b351d7c20175ef9c5ea8fc6a8acb1caad41892588d5eaeab4ee51b74e","source":{"kind":"arxiv","id":"2411.15992","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2411.15992","created_at":"2026-07-05T09:40:02Z"},{"alias_kind":"arxiv_version","alias_value":"2411.15992v1","created_at":"2026-07-05T09:40:02Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2411.15992","created_at":"2026-07-05T09:40:02Z"},{"alias_kind":"pith_short_12","alias_value":"P6RORCZVDV6C","created_at":"2026-07-05T09:40:02Z"},{"alias_kind":"pith_short_16","alias_value":"P6RORCZVDV6CAF26","created_at":"2026-07-05T09:40:02Z"},{"alias_kind":"pith_short_8","alias_value":"P6RORCZV","created_at":"2026-07-05T09:40:02Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2024:P6RORCZVDV6CAF267HC6VD6GVC","target":"record","payload":{"canonical_record":{"source":{"id":"2411.15992","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CO","submitted_at":"2024-11-24T22:02:58Z","cross_cats_sorted":[],"title_canon_sha256":"9c7e3d115fa1a6d39f002aca43e0df7dcd4153571d413e3f303d2bb2eb70bea0","abstract_canon_sha256":"3578b2d43749c63556240c019b5949848141937968dfdc37dcb406d12e3cf122"},"schema_version":"1.0"},"canonical_sha256":"7fa2e88b351d7c20175ef9c5ea8fc6a8acb1caad41892588d5eaeab4ee51b74e","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T09:40:02.096524Z","signature_b64":"CM4Fee3z1Ooad7IdhepbzMaAK1OfgHDUoUfwGD+MDuYBLxF3S3g5aOXVj2WnuxI6j9l+keavVqisaD1Y99GeCg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"7fa2e88b351d7c20175ef9c5ea8fc6a8acb1caad41892588d5eaeab4ee51b74e","last_reissued_at":"2026-07-05T09:40:02.096064Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T09:40:02.096064Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2411.15992","source_version":1,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-07-05T09:40:02Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"uFzJXo9IZVnMo51MekLgYs5BRP6DxvAokmQhdLVYsdmo5L9pnQCgXgwL2PuxEoGVMtCxYxuN86fLFhv1aGVxBQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-13T03:11:26.093335Z"},"content_sha256":"b18e642af7e1470d8a56c9e051e7ada6b62d69c6a4f627336c2b342aab303569","schema_version":"1.0","event_id":"sha256:b18e642af7e1470d8a56c9e051e7ada6b62d69c6a4f627336c2b342aab303569"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2024:P6RORCZVDV6CAF267HC6VD6GVC","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"The Heawood approach to Tait colorings and defining vertex sets","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"E. Yu. Lerner","submitted_at":"2024-11-24T22:02:58Z","abstract_excerpt":"Given a simple biconnected planar cubic graph, we associate each its vertex among $2n$ ones with the so-called spin, i.e., a variable which takes on values $\\pm 1$. P. J. Heawood has proved that a Tait coloring, accurate to the choice of a color for one edge, is equivalent to the choice of spin values so as to make the sum of these value at vertices of any face be a multiple of~3. We treat faces, which satisfy this condition, as {\\it proper}. The condition that guarantee the propriety of faces define a system of linear equations (SLE) with respect to variables, which take on nonzero values in "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2411.15992","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2411.15992/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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":"2026-07-05T09:40:02Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"C58TIQ5gBU4aSgxD31Z62qYoRpKtXKjSPLjEzjE0+S8rNdtOHiJHJ3WvO4HqvpW2zuXbHtUipF9s6Rr7KfFZDA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-13T03:11:26.094321Z"},"content_sha256":"f1ea62a7a693a68d0e17f835da0d7fda0b1b63af592e3a7b3689fe710c3ebf50","schema_version":"1.0","event_id":"sha256:f1ea62a7a693a68d0e17f835da0d7fda0b1b63af592e3a7b3689fe710c3ebf50"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/P6RORCZVDV6CAF267HC6VD6GVC/bundle.json","state_url":"https://pith.science/pith/P6RORCZVDV6CAF267HC6VD6GVC/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/P6RORCZVDV6CAF267HC6VD6GVC/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":"2026-08-13T03:11:26Z","links":{"resolver":"https://pith.science/pith/P6RORCZVDV6CAF267HC6VD6GVC","bundle":"https://pith.science/pith/P6RORCZVDV6CAF267HC6VD6GVC/bundle.json","state":"https://pith.science/pith/P6RORCZVDV6CAF267HC6VD6GVC/state.json","well_known_bundle":"https://pith.science/.well-known/pith/P6RORCZVDV6CAF267HC6VD6GVC/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2024:P6RORCZVDV6CAF267HC6VD6GVC","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":"3578b2d43749c63556240c019b5949848141937968dfdc37dcb406d12e3cf122","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CO","submitted_at":"2024-11-24T22:02:58Z","title_canon_sha256":"9c7e3d115fa1a6d39f002aca43e0df7dcd4153571d413e3f303d2bb2eb70bea0"},"schema_version":"1.0","source":{"id":"2411.15992","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2411.15992","created_at":"2026-07-05T09:40:02Z"},{"alias_kind":"arxiv_version","alias_value":"2411.15992v1","created_at":"2026-07-05T09:40:02Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2411.15992","created_at":"2026-07-05T09:40:02Z"},{"alias_kind":"pith_short_12","alias_value":"P6RORCZVDV6C","created_at":"2026-07-05T09:40:02Z"},{"alias_kind":"pith_short_16","alias_value":"P6RORCZVDV6CAF26","created_at":"2026-07-05T09:40:02Z"},{"alias_kind":"pith_short_8","alias_value":"P6RORCZV","created_at":"2026-07-05T09:40:02Z"}],"graph_snapshots":[{"event_id":"sha256:f1ea62a7a693a68d0e17f835da0d7fda0b1b63af592e3a7b3689fe710c3ebf50","target":"graph","created_at":"2026-07-05T09:40:02Z","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/2411.15992/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"Given a simple biconnected planar cubic graph, we associate each its vertex among $2n$ ones with the so-called spin, i.e., a variable which takes on values $\\pm 1$. P. J. Heawood has proved that a Tait coloring, accurate to the choice of a color for one edge, is equivalent to the choice of spin values so as to make the sum of these value at vertices of any face be a multiple of~3. We treat faces, which satisfy this condition, as {\\it proper}. The condition that guarantee the propriety of faces define a system of linear equations (SLE) with respect to variables, which take on nonzero values in ","authors_text":"E. Yu. Lerner","cross_cats":[],"headline":"","license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CO","submitted_at":"2024-11-24T22:02:58Z","title":"The Heawood approach to Tait colorings and defining vertex sets"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2411.15992","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:b18e642af7e1470d8a56c9e051e7ada6b62d69c6a4f627336c2b342aab303569","target":"record","created_at":"2026-07-05T09:40:02Z","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":"3578b2d43749c63556240c019b5949848141937968dfdc37dcb406d12e3cf122","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CO","submitted_at":"2024-11-24T22:02:58Z","title_canon_sha256":"9c7e3d115fa1a6d39f002aca43e0df7dcd4153571d413e3f303d2bb2eb70bea0"},"schema_version":"1.0","source":{"id":"2411.15992","kind":"arxiv","version":1}},"canonical_sha256":"7fa2e88b351d7c20175ef9c5ea8fc6a8acb1caad41892588d5eaeab4ee51b74e","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"7fa2e88b351d7c20175ef9c5ea8fc6a8acb1caad41892588d5eaeab4ee51b74e","first_computed_at":"2026-07-05T09:40:02.096064Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T09:40:02.096064Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"CM4Fee3z1Ooad7IdhepbzMaAK1OfgHDUoUfwGD+MDuYBLxF3S3g5aOXVj2WnuxI6j9l+keavVqisaD1Y99GeCg==","signature_status":"signed_v1","signed_at":"2026-07-05T09:40:02.096524Z","signed_message":"canonical_sha256_bytes"},"source_id":"2411.15992","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:b18e642af7e1470d8a56c9e051e7ada6b62d69c6a4f627336c2b342aab303569","sha256:f1ea62a7a693a68d0e17f835da0d7fda0b1b63af592e3a7b3689fe710c3ebf50"],"state_sha256":"c4bdf3a0c47d7a04ceb89f1efebd3de0d5f37aed2c8cd4b2458fb000447e1ead"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"rWawEjwJ25nClOjWMRiS1C+8iFoYV3ll17YC5cFtDiZGLtEtv8TpI74TFh10t3hsI7F7s0FuCypxobR3cxuqBQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-13T03:11:26.099091Z","bundle_sha256":"de99a6f565d293997ff7c7d8527c8ae95cb567b8754f7e17d8cd3c3513a17172"}}