{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2015:KVTYQZQT7BAQKUAZGLN2BN77BO","short_pith_number":"pith:KVTYQZQT","canonical_record":{"source":{"id":"1507.04109","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2015-07-15T07:49:52Z","cross_cats_sorted":["math-ph","math.CO","math.MP"],"title_canon_sha256":"948965905c8f0a6f613cc11055ebff427f5e9466d3e57f56b183a72544f6989f","abstract_canon_sha256":"94b473e458addec35145f838f1f2dca77f5f99674e0d36af361d1fe0234ec9f5"},"schema_version":"1.0"},"canonical_sha256":"5567886613f84105501932dba0b7ff0b80b32b578d679aa7da36793fd59cc2ed","source":{"kind":"arxiv","id":"1507.04109","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1507.04109","created_at":"2026-05-18T01:33:42Z"},{"alias_kind":"arxiv_version","alias_value":"1507.04109v2","created_at":"2026-05-18T01:33:42Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1507.04109","created_at":"2026-05-18T01:33:42Z"},{"alias_kind":"pith_short_12","alias_value":"KVTYQZQT7BAQ","created_at":"2026-05-18T12:29:29Z"},{"alias_kind":"pith_short_16","alias_value":"KVTYQZQT7BAQKUAZ","created_at":"2026-05-18T12:29:29Z"},{"alias_kind":"pith_short_8","alias_value":"KVTYQZQT","created_at":"2026-05-18T12:29:29Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2015:KVTYQZQT7BAQKUAZGLN2BN77BO","target":"record","payload":{"canonical_record":{"source":{"id":"1507.04109","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2015-07-15T07:49:52Z","cross_cats_sorted":["math-ph","math.CO","math.MP"],"title_canon_sha256":"948965905c8f0a6f613cc11055ebff427f5e9466d3e57f56b183a72544f6989f","abstract_canon_sha256":"94b473e458addec35145f838f1f2dca77f5f99674e0d36af361d1fe0234ec9f5"},"schema_version":"1.0"},"canonical_sha256":"5567886613f84105501932dba0b7ff0b80b32b578d679aa7da36793fd59cc2ed","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:33:42.491342Z","signature_b64":"gAavv0khQ79kxyZHRC68Vs2C1StJD4j6QEnj0W9Q5CzPv0ksMLPraK8NNLnblKiqSCaxMSqojm7EyEo6F0RqAg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"5567886613f84105501932dba0b7ff0b80b32b578d679aa7da36793fd59cc2ed","last_reissued_at":"2026-05-18T01:33:42.490892Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:33:42.490892Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1507.04109","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":"2026-05-18T01:33:42Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"vUznHFMpSWaitU5S1SdEWRWdT8SVAPHZUc8zffF/5INLZzKng8l0Zqi7C4gnBgJ9WjuiNQXC3hCe3YSgF3syCQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-28T19:13:10.155754Z"},"content_sha256":"6dcd43187142a51d811cd2216fa0e3186c5ee5047360d03223155ae947ae239d","schema_version":"1.0","event_id":"sha256:6dcd43187142a51d811cd2216fa0e3186c5ee5047360d03223155ae947ae239d"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2015:KVTYQZQT7BAQKUAZGLN2BN77BO","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"The 1-2 model","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math-ph","math.CO","math.MP"],"primary_cat":"math.PR","authors_text":"Geoffrey R. Grimmett, Zhongyang Li","submitted_at":"2015-07-15T07:49:52Z","abstract_excerpt":"The current paper is a short review of rigorous results for the 1-2 model. The 1-2 model on the hexagonal lattice is a model of statistical mechanics in which each vertex is constrained to have degree either 1 or 2. It was proposed in a study by Schwartz and Bruck of constrained coding systems, and is strongly connected to the dimer model on a decoration of the lattice, and to an enhanced Ising model and an associated polygon model on the graph derived from the hexagonal lattice by adding a further vertex in the middle of each edge.\n  The general 1-2 model possesses three parameters $a$, $b$, "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1507.04109","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":"2026-05-18T01:33:42Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Zw9t09Xk3uiFZtEZQQW4BuSeeLFrkkfLIYOlg0kly1HpB35O1PxyH+cmIELO7b1bCoKB3uLmBM6FVwg7jKNSCQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-28T19:13:10.156447Z"},"content_sha256":"50220a89818d86d98edb854964004c688563803f0d24b9c64241e87e241cdab3","schema_version":"1.0","event_id":"sha256:50220a89818d86d98edb854964004c688563803f0d24b9c64241e87e241cdab3"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/KVTYQZQT7BAQKUAZGLN2BN77BO/bundle.json","state_url":"https://pith.science/pith/KVTYQZQT7BAQKUAZGLN2BN77BO/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/KVTYQZQT7BAQKUAZGLN2BN77BO/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":"2026-05-28T19:13:10Z","links":{"resolver":"https://pith.science/pith/KVTYQZQT7BAQKUAZGLN2BN77BO","bundle":"https://pith.science/pith/KVTYQZQT7BAQKUAZGLN2BN77BO/bundle.json","state":"https://pith.science/pith/KVTYQZQT7BAQKUAZGLN2BN77BO/state.json","well_known_bundle":"https://pith.science/.well-known/pith/KVTYQZQT7BAQKUAZGLN2BN77BO/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2015:KVTYQZQT7BAQKUAZGLN2BN77BO","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":"94b473e458addec35145f838f1f2dca77f5f99674e0d36af361d1fe0234ec9f5","cross_cats_sorted":["math-ph","math.CO","math.MP"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2015-07-15T07:49:52Z","title_canon_sha256":"948965905c8f0a6f613cc11055ebff427f5e9466d3e57f56b183a72544f6989f"},"schema_version":"1.0","source":{"id":"1507.04109","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1507.04109","created_at":"2026-05-18T01:33:42Z"},{"alias_kind":"arxiv_version","alias_value":"1507.04109v2","created_at":"2026-05-18T01:33:42Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1507.04109","created_at":"2026-05-18T01:33:42Z"},{"alias_kind":"pith_short_12","alias_value":"KVTYQZQT7BAQ","created_at":"2026-05-18T12:29:29Z"},{"alias_kind":"pith_short_16","alias_value":"KVTYQZQT7BAQKUAZ","created_at":"2026-05-18T12:29:29Z"},{"alias_kind":"pith_short_8","alias_value":"KVTYQZQT","created_at":"2026-05-18T12:29:29Z"}],"graph_snapshots":[{"event_id":"sha256:50220a89818d86d98edb854964004c688563803f0d24b9c64241e87e241cdab3","target":"graph","created_at":"2026-05-18T01:33:42Z","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 current paper is a short review of rigorous results for the 1-2 model. The 1-2 model on the hexagonal lattice is a model of statistical mechanics in which each vertex is constrained to have degree either 1 or 2. It was proposed in a study by Schwartz and Bruck of constrained coding systems, and is strongly connected to the dimer model on a decoration of the lattice, and to an enhanced Ising model and an associated polygon model on the graph derived from the hexagonal lattice by adding a further vertex in the middle of each edge.\n  The general 1-2 model possesses three parameters $a$, $b$, ","authors_text":"Geoffrey R. Grimmett, Zhongyang Li","cross_cats":["math-ph","math.CO","math.MP"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2015-07-15T07:49:52Z","title":"The 1-2 model"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1507.04109","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":"sha256:6dcd43187142a51d811cd2216fa0e3186c5ee5047360d03223155ae947ae239d","target":"record","created_at":"2026-05-18T01:33:42Z","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":"94b473e458addec35145f838f1f2dca77f5f99674e0d36af361d1fe0234ec9f5","cross_cats_sorted":["math-ph","math.CO","math.MP"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2015-07-15T07:49:52Z","title_canon_sha256":"948965905c8f0a6f613cc11055ebff427f5e9466d3e57f56b183a72544f6989f"},"schema_version":"1.0","source":{"id":"1507.04109","kind":"arxiv","version":2}},"canonical_sha256":"5567886613f84105501932dba0b7ff0b80b32b578d679aa7da36793fd59cc2ed","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"5567886613f84105501932dba0b7ff0b80b32b578d679aa7da36793fd59cc2ed","first_computed_at":"2026-05-18T01:33:42.490892Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:33:42.490892Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"gAavv0khQ79kxyZHRC68Vs2C1StJD4j6QEnj0W9Q5CzPv0ksMLPraK8NNLnblKiqSCaxMSqojm7EyEo6F0RqAg==","signature_status":"signed_v1","signed_at":"2026-05-18T01:33:42.491342Z","signed_message":"canonical_sha256_bytes"},"source_id":"1507.04109","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:6dcd43187142a51d811cd2216fa0e3186c5ee5047360d03223155ae947ae239d","sha256:50220a89818d86d98edb854964004c688563803f0d24b9c64241e87e241cdab3"],"state_sha256":"8c3d9d55fc79e793656090fc2c23c6de05947b2c9279e46345db4af71899dc22"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"dRj7SumzTdPbZIhAzRyh/Za+MyNm3rzmme7e05KfkhTg1wTZHmlW8BvvAS+fckQmhMy3F3LY5x12rAI+hzC5Dg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-28T19:13:10.159964Z","bundle_sha256":"55f470b0ff9e8ac9f2c6e6773111f16458f7620dc80023a4aa1634500434d849"}}