{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2016:X5Q54T3F2KA7QJ2TYKVQJDKSE3","short_pith_number":"pith:X5Q54T3F","canonical_record":{"source":{"id":"1611.00492","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2016-11-02T07:40:12Z","cross_cats_sorted":["math.MG"],"title_canon_sha256":"dc8a52cd1e43ba6f5ba16b606fae8909ec7c162fe27e06ba6b328fb9e0fa3715","abstract_canon_sha256":"03c95fbae5b154910b01c327212944ed72fa3c2df31a3d02f4be8b79d0238350"},"schema_version":"1.0"},"canonical_sha256":"bf61de4f65d281f82753c2ab048d5226c90d088f2e6b5004c4ceca49f932269c","source":{"kind":"arxiv","id":"1611.00492","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1611.00492","created_at":"2026-05-17T23:49:28Z"},{"alias_kind":"arxiv_version","alias_value":"1611.00492v2","created_at":"2026-05-17T23:49:28Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1611.00492","created_at":"2026-05-17T23:49:28Z"},{"alias_kind":"pith_short_12","alias_value":"X5Q54T3F2KA7","created_at":"2026-05-18T12:30:51Z"},{"alias_kind":"pith_short_16","alias_value":"X5Q54T3F2KA7QJ2T","created_at":"2026-05-18T12:30:51Z"},{"alias_kind":"pith_short_8","alias_value":"X5Q54T3F","created_at":"2026-05-18T12:30:51Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2016:X5Q54T3F2KA7QJ2TYKVQJDKSE3","target":"record","payload":{"canonical_record":{"source":{"id":"1611.00492","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2016-11-02T07:40:12Z","cross_cats_sorted":["math.MG"],"title_canon_sha256":"dc8a52cd1e43ba6f5ba16b606fae8909ec7c162fe27e06ba6b328fb9e0fa3715","abstract_canon_sha256":"03c95fbae5b154910b01c327212944ed72fa3c2df31a3d02f4be8b79d0238350"},"schema_version":"1.0"},"canonical_sha256":"bf61de4f65d281f82753c2ab048d5226c90d088f2e6b5004c4ceca49f932269c","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:49:28.192440Z","signature_b64":"j4ixOJ2ZJZDttBuo2+IP2rFXlHeZ35NPeSzkEDC39I0T3apByYeozRtszh08lsYHzWyx3A/9baZqJ66MS+9tBg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"bf61de4f65d281f82753c2ab048d5226c90d088f2e6b5004c4ceca49f932269c","last_reissued_at":"2026-05-17T23:49:28.191722Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:49:28.191722Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1611.00492","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":"2026-05-17T23:49:28Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"GxSSIz3k6IPs/eADaT3DKQ9A8IWGl5hzcJNMnKuelxex9Dqj5xUedezQEEH9cOU+Z2yDmeq3VQiQEqRZvWovCg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-30T21:32:50.845230Z"},"content_sha256":"d5329bd14c87343a27d8d9b3a11966af53ec6b1fbf01d398a3509c0854246153","schema_version":"1.0","event_id":"sha256:d5329bd14c87343a27d8d9b3a11966af53ec6b1fbf01d398a3509c0854246153"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2016:X5Q54T3F2KA7QJ2TYKVQJDKSE3","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Plabic graphs and zonotopal tilings","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.MG"],"primary_cat":"math.CO","authors_text":"Pavel Galashin","submitted_at":"2016-11-02T07:40:12Z","abstract_excerpt":"We say that two sets $S,T\\subset\\{1,2,\\dots,n\\}$ are chord separated if there does not exist a cyclically ordered quadruple $a,b,c,d$ of integers satisfying $a,c\\in S-T$ and $b,d\\in T-S$. This is a weaker version of Leclerc and Zelevinsky's weak separation. We show that every maximal by inclusion collection of pairwise chord separated sets is also maximal by size. Moreover, we prove that such collections are precisely vertex label collections of fine zonotopal tilings of the three-dimensional cyclic zonotope. In our construction, plabic graphs and square moves appear naturally as horizontal se"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1611.00492","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":"2026-05-17T23:49:28Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"9628wpzhnfWxIvK516MqFcI1GPRbITlz5t+9CngUt0JdTdkTDTJAdLz8Nr8qDQlnglpnMLfRE7l5bqj7mozABA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-30T21:32:50.845630Z"},"content_sha256":"f0cf84e236458bb6ed7be53ef1f070e5a01529dcdb9fd744ad208f467b8a4db7","schema_version":"1.0","event_id":"sha256:f0cf84e236458bb6ed7be53ef1f070e5a01529dcdb9fd744ad208f467b8a4db7"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/X5Q54T3F2KA7QJ2TYKVQJDKSE3/bundle.json","state_url":"https://pith.science/pith/X5Q54T3F2KA7QJ2TYKVQJDKSE3/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/X5Q54T3F2KA7QJ2TYKVQJDKSE3/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":"2026-06-30T21:32:50Z","links":{"resolver":"https://pith.science/pith/X5Q54T3F2KA7QJ2TYKVQJDKSE3","bundle":"https://pith.science/pith/X5Q54T3F2KA7QJ2TYKVQJDKSE3/bundle.json","state":"https://pith.science/pith/X5Q54T3F2KA7QJ2TYKVQJDKSE3/state.json","well_known_bundle":"https://pith.science/.well-known/pith/X5Q54T3F2KA7QJ2TYKVQJDKSE3/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2016:X5Q54T3F2KA7QJ2TYKVQJDKSE3","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":"03c95fbae5b154910b01c327212944ed72fa3c2df31a3d02f4be8b79d0238350","cross_cats_sorted":["math.MG"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2016-11-02T07:40:12Z","title_canon_sha256":"dc8a52cd1e43ba6f5ba16b606fae8909ec7c162fe27e06ba6b328fb9e0fa3715"},"schema_version":"1.0","source":{"id":"1611.00492","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1611.00492","created_at":"2026-05-17T23:49:28Z"},{"alias_kind":"arxiv_version","alias_value":"1611.00492v2","created_at":"2026-05-17T23:49:28Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1611.00492","created_at":"2026-05-17T23:49:28Z"},{"alias_kind":"pith_short_12","alias_value":"X5Q54T3F2KA7","created_at":"2026-05-18T12:30:51Z"},{"alias_kind":"pith_short_16","alias_value":"X5Q54T3F2KA7QJ2T","created_at":"2026-05-18T12:30:51Z"},{"alias_kind":"pith_short_8","alias_value":"X5Q54T3F","created_at":"2026-05-18T12:30:51Z"}],"graph_snapshots":[{"event_id":"sha256:f0cf84e236458bb6ed7be53ef1f070e5a01529dcdb9fd744ad208f467b8a4db7","target":"graph","created_at":"2026-05-17T23:49:28Z","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 say that two sets $S,T\\subset\\{1,2,\\dots,n\\}$ are chord separated if there does not exist a cyclically ordered quadruple $a,b,c,d$ of integers satisfying $a,c\\in S-T$ and $b,d\\in T-S$. This is a weaker version of Leclerc and Zelevinsky's weak separation. We show that every maximal by inclusion collection of pairwise chord separated sets is also maximal by size. Moreover, we prove that such collections are precisely vertex label collections of fine zonotopal tilings of the three-dimensional cyclic zonotope. In our construction, plabic graphs and square moves appear naturally as horizontal se","authors_text":"Pavel Galashin","cross_cats":["math.MG"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2016-11-02T07:40:12Z","title":"Plabic graphs and zonotopal tilings"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1611.00492","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:d5329bd14c87343a27d8d9b3a11966af53ec6b1fbf01d398a3509c0854246153","target":"record","created_at":"2026-05-17T23:49:28Z","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":"03c95fbae5b154910b01c327212944ed72fa3c2df31a3d02f4be8b79d0238350","cross_cats_sorted":["math.MG"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2016-11-02T07:40:12Z","title_canon_sha256":"dc8a52cd1e43ba6f5ba16b606fae8909ec7c162fe27e06ba6b328fb9e0fa3715"},"schema_version":"1.0","source":{"id":"1611.00492","kind":"arxiv","version":2}},"canonical_sha256":"bf61de4f65d281f82753c2ab048d5226c90d088f2e6b5004c4ceca49f932269c","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"bf61de4f65d281f82753c2ab048d5226c90d088f2e6b5004c4ceca49f932269c","first_computed_at":"2026-05-17T23:49:28.191722Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-17T23:49:28.191722Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"j4ixOJ2ZJZDttBuo2+IP2rFXlHeZ35NPeSzkEDC39I0T3apByYeozRtszh08lsYHzWyx3A/9baZqJ66MS+9tBg==","signature_status":"signed_v1","signed_at":"2026-05-17T23:49:28.192440Z","signed_message":"canonical_sha256_bytes"},"source_id":"1611.00492","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:d5329bd14c87343a27d8d9b3a11966af53ec6b1fbf01d398a3509c0854246153","sha256:f0cf84e236458bb6ed7be53ef1f070e5a01529dcdb9fd744ad208f467b8a4db7"],"state_sha256":"3013cbb31826cecf4b5d0344e6a397f799165950ec82edb644979289fb878a01"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"vcJVg8R+I0QjrfGCPHUrLHIWwEY0WDKITOlmgEJE+uco+1oh0/jZzcq/tEAWTsWVDFc2BmfBzObGWpAWx561Aw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-30T21:32:50.847784Z","bundle_sha256":"81e8bbd4a09217747cf75710825df1088d3e38cc86dea1b2ae87b8f3542a6959"}}