{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2021:HOZ7SX7UTQ4UKXNYU35YW6HQPJ","short_pith_number":"pith:HOZ7SX7U","canonical_record":{"source":{"id":"2104.01382","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CO","submitted_at":"2021-04-03T11:53:56Z","cross_cats_sorted":[],"title_canon_sha256":"9e53f1b141de6f825360d1a15d1e720a2fc7e4051384d22b6035e9b81370e19a","abstract_canon_sha256":"c17e5354b95dfbc90b13cba47730bf7dddb7bccc43c5836e9f2bceac7d6f0ddd"},"schema_version":"1.0"},"canonical_sha256":"3bb3f95ff49c39455db8a6fb8b78f07a4af5d58da69a176de61cf64c0f80f528","source":{"kind":"arxiv","id":"2104.01382","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2104.01382","created_at":"2026-07-05T02:28:59Z"},{"alias_kind":"arxiv_version","alias_value":"2104.01382v1","created_at":"2026-07-05T02:28:59Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2104.01382","created_at":"2026-07-05T02:28:59Z"},{"alias_kind":"pith_short_12","alias_value":"HOZ7SX7UTQ4U","created_at":"2026-07-05T02:28:59Z"},{"alias_kind":"pith_short_16","alias_value":"HOZ7SX7UTQ4UKXNY","created_at":"2026-07-05T02:28:59Z"},{"alias_kind":"pith_short_8","alias_value":"HOZ7SX7U","created_at":"2026-07-05T02:28:59Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2021:HOZ7SX7UTQ4UKXNYU35YW6HQPJ","target":"record","payload":{"canonical_record":{"source":{"id":"2104.01382","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CO","submitted_at":"2021-04-03T11:53:56Z","cross_cats_sorted":[],"title_canon_sha256":"9e53f1b141de6f825360d1a15d1e720a2fc7e4051384d22b6035e9b81370e19a","abstract_canon_sha256":"c17e5354b95dfbc90b13cba47730bf7dddb7bccc43c5836e9f2bceac7d6f0ddd"},"schema_version":"1.0"},"canonical_sha256":"3bb3f95ff49c39455db8a6fb8b78f07a4af5d58da69a176de61cf64c0f80f528","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T02:28:59.404244Z","signature_b64":"uY0EERu4+J1SZqiffaRxmX2BRpqu3hob0jHrYUYa6Dd/JnZMdZ61PKIXXrS6tvzglSJGulNL+PMzST8jZYDkAQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"3bb3f95ff49c39455db8a6fb8b78f07a4af5d58da69a176de61cf64c0f80f528","last_reissued_at":"2026-07-05T02:28:59.403707Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T02:28:59.403707Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2104.01382","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":"2026-07-05T02:28:59Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"krle15ErS2l2prN/fsdo3X9CZbprPQyv3DpAxRuRIcfGCfvnOZiBF/6sZ60ayvFRCfzL1vTo1ATr+w3XsgEuBA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-15T06:05:55.769573Z"},"content_sha256":"09db9cb8c04cfa085a6106ac225b12a9e8eee7f0ef559db6c6865a0eb1fae999","schema_version":"1.0","event_id":"sha256:09db9cb8c04cfa085a6106ac225b12a9e8eee7f0ef559db6c6865a0eb1fae999"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2021:HOZ7SX7UTQ4UKXNYU35YW6HQPJ","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"A note on cycle lengths in graphs of chromatic number five and six","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Qingyi Huo","submitted_at":"2021-04-03T11:53:56Z","abstract_excerpt":"In this note, we prove that every non-complete $(k+1)$-critical graph contains cycles of all lengths modulo $k$, where $k=4,5$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2104.01382","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/2104.01382/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":"2026-07-05T02:28:59Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"XHAh5Ixa+/jzxP6zVA6Q6SYGqfJkUDJ3rfdcziPV2KQo+BxQRATzuq9sw1oM+V8yrqkmlK9JDgnDNLY8VCpvBw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-15T06:05:55.770109Z"},"content_sha256":"21940172387af64374d5114f25d7f677404825915da962130d77b0b98c36ec63","schema_version":"1.0","event_id":"sha256:21940172387af64374d5114f25d7f677404825915da962130d77b0b98c36ec63"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/HOZ7SX7UTQ4UKXNYU35YW6HQPJ/bundle.json","state_url":"https://pith.science/pith/HOZ7SX7UTQ4UKXNYU35YW6HQPJ/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/HOZ7SX7UTQ4UKXNYU35YW6HQPJ/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":"2026-08-15T06:05:55Z","links":{"resolver":"https://pith.science/pith/HOZ7SX7UTQ4UKXNYU35YW6HQPJ","bundle":"https://pith.science/pith/HOZ7SX7UTQ4UKXNYU35YW6HQPJ/bundle.json","state":"https://pith.science/pith/HOZ7SX7UTQ4UKXNYU35YW6HQPJ/state.json","well_known_bundle":"https://pith.science/.well-known/pith/HOZ7SX7UTQ4UKXNYU35YW6HQPJ/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2021:HOZ7SX7UTQ4UKXNYU35YW6HQPJ","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":"c17e5354b95dfbc90b13cba47730bf7dddb7bccc43c5836e9f2bceac7d6f0ddd","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CO","submitted_at":"2021-04-03T11:53:56Z","title_canon_sha256":"9e53f1b141de6f825360d1a15d1e720a2fc7e4051384d22b6035e9b81370e19a"},"schema_version":"1.0","source":{"id":"2104.01382","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2104.01382","created_at":"2026-07-05T02:28:59Z"},{"alias_kind":"arxiv_version","alias_value":"2104.01382v1","created_at":"2026-07-05T02:28:59Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2104.01382","created_at":"2026-07-05T02:28:59Z"},{"alias_kind":"pith_short_12","alias_value":"HOZ7SX7UTQ4U","created_at":"2026-07-05T02:28:59Z"},{"alias_kind":"pith_short_16","alias_value":"HOZ7SX7UTQ4UKXNY","created_at":"2026-07-05T02:28:59Z"},{"alias_kind":"pith_short_8","alias_value":"HOZ7SX7U","created_at":"2026-07-05T02:28:59Z"}],"graph_snapshots":[{"event_id":"sha256:21940172387af64374d5114f25d7f677404825915da962130d77b0b98c36ec63","target":"graph","created_at":"2026-07-05T02:28:59Z","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/2104.01382/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"In this note, we prove that every non-complete $(k+1)$-critical graph contains cycles of all lengths modulo $k$, where $k=4,5$.","authors_text":"Qingyi Huo","cross_cats":[],"headline":"","license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CO","submitted_at":"2021-04-03T11:53:56Z","title":"A note on cycle lengths in graphs of chromatic number five and six"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2104.01382","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":"sha256:09db9cb8c04cfa085a6106ac225b12a9e8eee7f0ef559db6c6865a0eb1fae999","target":"record","created_at":"2026-07-05T02:28:59Z","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":"c17e5354b95dfbc90b13cba47730bf7dddb7bccc43c5836e9f2bceac7d6f0ddd","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CO","submitted_at":"2021-04-03T11:53:56Z","title_canon_sha256":"9e53f1b141de6f825360d1a15d1e720a2fc7e4051384d22b6035e9b81370e19a"},"schema_version":"1.0","source":{"id":"2104.01382","kind":"arxiv","version":1}},"canonical_sha256":"3bb3f95ff49c39455db8a6fb8b78f07a4af5d58da69a176de61cf64c0f80f528","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"3bb3f95ff49c39455db8a6fb8b78f07a4af5d58da69a176de61cf64c0f80f528","first_computed_at":"2026-07-05T02:28:59.403707Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T02:28:59.403707Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"uY0EERu4+J1SZqiffaRxmX2BRpqu3hob0jHrYUYa6Dd/JnZMdZ61PKIXXrS6tvzglSJGulNL+PMzST8jZYDkAQ==","signature_status":"signed_v1","signed_at":"2026-07-05T02:28:59.404244Z","signed_message":"canonical_sha256_bytes"},"source_id":"2104.01382","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:09db9cb8c04cfa085a6106ac225b12a9e8eee7f0ef559db6c6865a0eb1fae999","sha256:21940172387af64374d5114f25d7f677404825915da962130d77b0b98c36ec63"],"state_sha256":"bea95afdd54b3b3c2e02c0332db4277ba6cf4e853c0c013068f7cecf17ba5709"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"BUB6byrfhuI6zGyM+OWU+8oUP8V/HuI+0zIaSm1yxTh6OsMuiycPprNBs4gJhStvBCyvjWx/X1nPcJ3oYFmQCA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-15T06:05:55.788600Z","bundle_sha256":"f8c37d65c1e940611b7429288e5d4a719c4aefbbf54b30828dba5e338154fdf6"}}