{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2014:KP3XFN74RTPBMXFECMRIYTWIBQ","short_pith_number":"pith:KP3XFN74","canonical_record":{"source":{"id":"1406.5936","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AC","submitted_at":"2014-06-23T15:05:42Z","cross_cats_sorted":["math.ST","stat.TH"],"title_canon_sha256":"e4997904384b81c9e312f5c17e5411db35e4c7d99f8e1e6a49aeca5698e321e2","abstract_canon_sha256":"b8f8724685eff853f823d860b79c194736994ba7790f8fdbe6111d9e6b6b8c52"},"schema_version":"1.0"},"canonical_sha256":"53f772b7fc8cde165ca413228c4ec80c1d5232ec5c3e0c01997de6cc37c89048","source":{"kind":"arxiv","id":"1406.5936","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1406.5936","created_at":"2026-05-18T02:48:56Z"},{"alias_kind":"arxiv_version","alias_value":"1406.5936v2","created_at":"2026-05-18T02:48:56Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1406.5936","created_at":"2026-05-18T02:48:56Z"},{"alias_kind":"pith_short_12","alias_value":"KP3XFN74RTPB","created_at":"2026-05-18T12:28:35Z"},{"alias_kind":"pith_short_16","alias_value":"KP3XFN74RTPBMXFE","created_at":"2026-05-18T12:28:35Z"},{"alias_kind":"pith_short_8","alias_value":"KP3XFN74","created_at":"2026-05-18T12:28:35Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2014:KP3XFN74RTPBMXFECMRIYTWIBQ","target":"record","payload":{"canonical_record":{"source":{"id":"1406.5936","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AC","submitted_at":"2014-06-23T15:05:42Z","cross_cats_sorted":["math.ST","stat.TH"],"title_canon_sha256":"e4997904384b81c9e312f5c17e5411db35e4c7d99f8e1e6a49aeca5698e321e2","abstract_canon_sha256":"b8f8724685eff853f823d860b79c194736994ba7790f8fdbe6111d9e6b6b8c52"},"schema_version":"1.0"},"canonical_sha256":"53f772b7fc8cde165ca413228c4ec80c1d5232ec5c3e0c01997de6cc37c89048","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:48:56.145553Z","signature_b64":"KljGxQzGY3f62RTy8l9+zDxiY8iGhuLGV9MctJuoycN2aiLiZ2gxepf7a4ujuf1f5sLIeifgyDcI0uEqVPKlBg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"53f772b7fc8cde165ca413228c4ec80c1d5232ec5c3e0c01997de6cc37c89048","last_reissued_at":"2026-05-18T02:48:56.145019Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:48:56.145019Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1406.5936","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":"2026-05-18T02:48:56Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"JSbo0OTDJhbXcoyQrwerdM7xk4VsQpxAF3VYTuHerc7NlO++UYF6Pumo7bAC8MGvxSI3zBYedB+HWD86uw6cAQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-03T05:33:25.901553Z"},"content_sha256":"9d14efba54969bbae4b18d7d8cc6a5a4d95fc90ba5a08bdb6f1c66c6b514b113","schema_version":"1.0","event_id":"sha256:9d14efba54969bbae4b18d7d8cc6a5a4d95fc90ba5a08bdb6f1c66c6b514b113"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2014:KP3XFN74RTPBMXFECMRIYTWIBQ","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"The Markov basis of $K_{3,N}$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.ST","stat.TH"],"primary_cat":"math.AC","authors_text":"Johannes Rauh, Seth Sullivant","submitted_at":"2014-06-23T15:05:42Z","abstract_excerpt":"This document explains how to obtain a Markov basis of the graphical model of the complete bipartite graph $K_{3,N}$ with binary nodes. The computations illustrate the theory developed in arXiv:1404.6392 that explains how to compute Markov bases of toric fiber products."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1406.5936","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":"2026-05-18T02:48:56Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"ylHgtnXdeROY92VvukNgK9k/xhwr93fARA4lchem2WDbhWs6XqnAh3/nNCrgvgHO6Oxj+OWaF6RUULYwv22OAQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-03T05:33:25.901895Z"},"content_sha256":"1b6e2f487b781c666a04d52700aeafda36fdc01d23c7f830f8039a0babc12c7b","schema_version":"1.0","event_id":"sha256:1b6e2f487b781c666a04d52700aeafda36fdc01d23c7f830f8039a0babc12c7b"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/KP3XFN74RTPBMXFECMRIYTWIBQ/bundle.json","state_url":"https://pith.science/pith/KP3XFN74RTPBMXFECMRIYTWIBQ/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/KP3XFN74RTPBMXFECMRIYTWIBQ/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":"2026-06-03T05:33:25Z","links":{"resolver":"https://pith.science/pith/KP3XFN74RTPBMXFECMRIYTWIBQ","bundle":"https://pith.science/pith/KP3XFN74RTPBMXFECMRIYTWIBQ/bundle.json","state":"https://pith.science/pith/KP3XFN74RTPBMXFECMRIYTWIBQ/state.json","well_known_bundle":"https://pith.science/.well-known/pith/KP3XFN74RTPBMXFECMRIYTWIBQ/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2014:KP3XFN74RTPBMXFECMRIYTWIBQ","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":"b8f8724685eff853f823d860b79c194736994ba7790f8fdbe6111d9e6b6b8c52","cross_cats_sorted":["math.ST","stat.TH"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AC","submitted_at":"2014-06-23T15:05:42Z","title_canon_sha256":"e4997904384b81c9e312f5c17e5411db35e4c7d99f8e1e6a49aeca5698e321e2"},"schema_version":"1.0","source":{"id":"1406.5936","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1406.5936","created_at":"2026-05-18T02:48:56Z"},{"alias_kind":"arxiv_version","alias_value":"1406.5936v2","created_at":"2026-05-18T02:48:56Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1406.5936","created_at":"2026-05-18T02:48:56Z"},{"alias_kind":"pith_short_12","alias_value":"KP3XFN74RTPB","created_at":"2026-05-18T12:28:35Z"},{"alias_kind":"pith_short_16","alias_value":"KP3XFN74RTPBMXFE","created_at":"2026-05-18T12:28:35Z"},{"alias_kind":"pith_short_8","alias_value":"KP3XFN74","created_at":"2026-05-18T12:28:35Z"}],"graph_snapshots":[{"event_id":"sha256:1b6e2f487b781c666a04d52700aeafda36fdc01d23c7f830f8039a0babc12c7b","target":"graph","created_at":"2026-05-18T02:48:56Z","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":"This document explains how to obtain a Markov basis of the graphical model of the complete bipartite graph $K_{3,N}$ with binary nodes. The computations illustrate the theory developed in arXiv:1404.6392 that explains how to compute Markov bases of toric fiber products.","authors_text":"Johannes Rauh, Seth Sullivant","cross_cats":["math.ST","stat.TH"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AC","submitted_at":"2014-06-23T15:05:42Z","title":"The Markov basis of $K_{3,N}$"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1406.5936","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":"sha256:9d14efba54969bbae4b18d7d8cc6a5a4d95fc90ba5a08bdb6f1c66c6b514b113","target":"record","created_at":"2026-05-18T02:48:56Z","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":"b8f8724685eff853f823d860b79c194736994ba7790f8fdbe6111d9e6b6b8c52","cross_cats_sorted":["math.ST","stat.TH"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AC","submitted_at":"2014-06-23T15:05:42Z","title_canon_sha256":"e4997904384b81c9e312f5c17e5411db35e4c7d99f8e1e6a49aeca5698e321e2"},"schema_version":"1.0","source":{"id":"1406.5936","kind":"arxiv","version":2}},"canonical_sha256":"53f772b7fc8cde165ca413228c4ec80c1d5232ec5c3e0c01997de6cc37c89048","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"53f772b7fc8cde165ca413228c4ec80c1d5232ec5c3e0c01997de6cc37c89048","first_computed_at":"2026-05-18T02:48:56.145019Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T02:48:56.145019Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"KljGxQzGY3f62RTy8l9+zDxiY8iGhuLGV9MctJuoycN2aiLiZ2gxepf7a4ujuf1f5sLIeifgyDcI0uEqVPKlBg==","signature_status":"signed_v1","signed_at":"2026-05-18T02:48:56.145553Z","signed_message":"canonical_sha256_bytes"},"source_id":"1406.5936","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:9d14efba54969bbae4b18d7d8cc6a5a4d95fc90ba5a08bdb6f1c66c6b514b113","sha256:1b6e2f487b781c666a04d52700aeafda36fdc01d23c7f830f8039a0babc12c7b"],"state_sha256":"38eb577c5c392239ff8da5d5349835e92b0f7037751717e48fd58ce6e9499b28"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"E3A4XXha0iIjkFUDXHBvqAobCMITU2ci562d7RthcJFDH9fEOwMkIPi2VMUCEUHMMnqKYxCT4+1Us1iQcfqtBA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-03T05:33:25.903784Z","bundle_sha256":"98cb6ef390b1460a2abb5ebaa2636705fa215bfc90dde13b2feff5c97d2f3d88"}}