{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2022:3FBGZNAJPQ7RPI6SQ4MBEAYKVS","short_pith_number":"pith:3FBGZNAJ","canonical_record":{"source":{"id":"2208.14063","kind":"arxiv","version":2},"metadata":{"license":"http://creativecommons.org/licenses/by-nc-sa/4.0/","primary_cat":"math.AT","submitted_at":"2022-08-30T08:29:20Z","cross_cats_sorted":["math.CO"],"title_canon_sha256":"b318bdfc9223c383a370ed503bc8aa9a57c594f1ade813dc6863bdcc057af0d6","abstract_canon_sha256":"327284ecbb14e0fa4b4db9d0cf07b474f2e0f74a0abb55f90e52cefe9844b433"},"schema_version":"1.0"},"canonical_sha256":"d9426cb4097c3f17a3d2871812030aac840786e70cd76fc65ec2673eec3b4bd8","source":{"kind":"arxiv","id":"2208.14063","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2208.14063","created_at":"2026-07-05T11:06:59Z"},{"alias_kind":"arxiv_version","alias_value":"2208.14063v2","created_at":"2026-07-05T11:06:59Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2208.14063","created_at":"2026-07-05T11:06:59Z"},{"alias_kind":"pith_short_12","alias_value":"3FBGZNAJPQ7R","created_at":"2026-07-05T11:06:59Z"},{"alias_kind":"pith_short_16","alias_value":"3FBGZNAJPQ7RPI6S","created_at":"2026-07-05T11:06:59Z"},{"alias_kind":"pith_short_8","alias_value":"3FBGZNAJ","created_at":"2026-07-05T11:06:59Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2022:3FBGZNAJPQ7RPI6SQ4MBEAYKVS","target":"record","payload":{"canonical_record":{"source":{"id":"2208.14063","kind":"arxiv","version":2},"metadata":{"license":"http://creativecommons.org/licenses/by-nc-sa/4.0/","primary_cat":"math.AT","submitted_at":"2022-08-30T08:29:20Z","cross_cats_sorted":["math.CO"],"title_canon_sha256":"b318bdfc9223c383a370ed503bc8aa9a57c594f1ade813dc6863bdcc057af0d6","abstract_canon_sha256":"327284ecbb14e0fa4b4db9d0cf07b474f2e0f74a0abb55f90e52cefe9844b433"},"schema_version":"1.0"},"canonical_sha256":"d9426cb4097c3f17a3d2871812030aac840786e70cd76fc65ec2673eec3b4bd8","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T11:06:59.236024Z","signature_b64":"YsYeQ/MdJXBEeJVtTqvsuDgNpACDuCwWxZ2ymDzV8d9cOR3uQqjM9vipxRTdcOi29Cut2Ck1FYRvkDFNg4nLBQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"d9426cb4097c3f17a3d2871812030aac840786e70cd76fc65ec2673eec3b4bd8","last_reissued_at":"2026-07-05T11:06:59.235586Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T11:06:59.235586Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2208.14063","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":"2026-07-05T11:06:59Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Qr9UpjmYbCjtzZSHOAGypxWl2mWyxk4ljK03uF02D2EDSTk8MHLNFLkFGV7jHuoBbNlJCL18HrXAZaPKMS70AA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-19T18:54:22.817616Z"},"content_sha256":"82da0d2dc715bca13232f3f532d456bd76df263760d96b89f495407d9b77aa58","schema_version":"1.0","event_id":"sha256:82da0d2dc715bca13232f3f532d456bd76df263760d96b89f495407d9b77aa58"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2022:3FBGZNAJPQ7RPI6SQ4MBEAYKVS","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Minimal Path and Acyclic Models in the Path Complex","license":"http://creativecommons.org/licenses/by-nc-sa/4.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"math.AT","authors_text":"Shing-Tung Yau, Xinxing Tang","submitted_at":"2022-08-30T08:29:20Z","abstract_excerpt":"In this paper, firstly, we will study the structure of the path complex $(\\Omega_*(G;\\Z),\\partial)$ of a digraph $G$ via the $\\Z$-generators of $\\Omega_*(G,\\Z)$ under strongly regular condition, which is called the minimal path in \\cite{HY}. In particular, we will study various examples of the minimal $3$-paths. Secondly, we will show that the supporting sub-digraph of minimal path has acyclic path homologies. Thirdly, we will consider the applications of such an acyclic model."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2208.14063","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2208.14063/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":"2026-07-05T11:06:59Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"V0KCDzz4HrAz9+rg2hTLjuaPqucPn7PUoKchg8yEOxfJWfmj6RgiOMlGi8767NAbUTBd95z8p5wFCQnOdZSaDQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-19T18:54:22.818492Z"},"content_sha256":"ce2e52264fdbbb91ac27090f511262fd2c2e7b03908b99f1a3c694756198621a","schema_version":"1.0","event_id":"sha256:ce2e52264fdbbb91ac27090f511262fd2c2e7b03908b99f1a3c694756198621a"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/3FBGZNAJPQ7RPI6SQ4MBEAYKVS/bundle.json","state_url":"https://pith.science/pith/3FBGZNAJPQ7RPI6SQ4MBEAYKVS/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/3FBGZNAJPQ7RPI6SQ4MBEAYKVS/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":"2026-08-19T18:54:22Z","links":{"resolver":"https://pith.science/pith/3FBGZNAJPQ7RPI6SQ4MBEAYKVS","bundle":"https://pith.science/pith/3FBGZNAJPQ7RPI6SQ4MBEAYKVS/bundle.json","state":"https://pith.science/pith/3FBGZNAJPQ7RPI6SQ4MBEAYKVS/state.json","well_known_bundle":"https://pith.science/.well-known/pith/3FBGZNAJPQ7RPI6SQ4MBEAYKVS/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2022:3FBGZNAJPQ7RPI6SQ4MBEAYKVS","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":"327284ecbb14e0fa4b4db9d0cf07b474f2e0f74a0abb55f90e52cefe9844b433","cross_cats_sorted":["math.CO"],"license":"http://creativecommons.org/licenses/by-nc-sa/4.0/","primary_cat":"math.AT","submitted_at":"2022-08-30T08:29:20Z","title_canon_sha256":"b318bdfc9223c383a370ed503bc8aa9a57c594f1ade813dc6863bdcc057af0d6"},"schema_version":"1.0","source":{"id":"2208.14063","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2208.14063","created_at":"2026-07-05T11:06:59Z"},{"alias_kind":"arxiv_version","alias_value":"2208.14063v2","created_at":"2026-07-05T11:06:59Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2208.14063","created_at":"2026-07-05T11:06:59Z"},{"alias_kind":"pith_short_12","alias_value":"3FBGZNAJPQ7R","created_at":"2026-07-05T11:06:59Z"},{"alias_kind":"pith_short_16","alias_value":"3FBGZNAJPQ7RPI6S","created_at":"2026-07-05T11:06:59Z"},{"alias_kind":"pith_short_8","alias_value":"3FBGZNAJ","created_at":"2026-07-05T11:06:59Z"}],"graph_snapshots":[{"event_id":"sha256:ce2e52264fdbbb91ac27090f511262fd2c2e7b03908b99f1a3c694756198621a","target":"graph","created_at":"2026-07-05T11:06:59Z","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/2208.14063/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"In this paper, firstly, we will study the structure of the path complex $(\\Omega_*(G;\\Z),\\partial)$ of a digraph $G$ via the $\\Z$-generators of $\\Omega_*(G,\\Z)$ under strongly regular condition, which is called the minimal path in \\cite{HY}. In particular, we will study various examples of the minimal $3$-paths. Secondly, we will show that the supporting sub-digraph of minimal path has acyclic path homologies. Thirdly, we will consider the applications of such an acyclic model.","authors_text":"Shing-Tung Yau, Xinxing Tang","cross_cats":["math.CO"],"headline":"","license":"http://creativecommons.org/licenses/by-nc-sa/4.0/","primary_cat":"math.AT","submitted_at":"2022-08-30T08:29:20Z","title":"Minimal Path and Acyclic Models in the Path Complex"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2208.14063","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":"sha256:82da0d2dc715bca13232f3f532d456bd76df263760d96b89f495407d9b77aa58","target":"record","created_at":"2026-07-05T11:06:59Z","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":"327284ecbb14e0fa4b4db9d0cf07b474f2e0f74a0abb55f90e52cefe9844b433","cross_cats_sorted":["math.CO"],"license":"http://creativecommons.org/licenses/by-nc-sa/4.0/","primary_cat":"math.AT","submitted_at":"2022-08-30T08:29:20Z","title_canon_sha256":"b318bdfc9223c383a370ed503bc8aa9a57c594f1ade813dc6863bdcc057af0d6"},"schema_version":"1.0","source":{"id":"2208.14063","kind":"arxiv","version":2}},"canonical_sha256":"d9426cb4097c3f17a3d2871812030aac840786e70cd76fc65ec2673eec3b4bd8","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"d9426cb4097c3f17a3d2871812030aac840786e70cd76fc65ec2673eec3b4bd8","first_computed_at":"2026-07-05T11:06:59.235586Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T11:06:59.235586Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"YsYeQ/MdJXBEeJVtTqvsuDgNpACDuCwWxZ2ymDzV8d9cOR3uQqjM9vipxRTdcOi29Cut2Ck1FYRvkDFNg4nLBQ==","signature_status":"signed_v1","signed_at":"2026-07-05T11:06:59.236024Z","signed_message":"canonical_sha256_bytes"},"source_id":"2208.14063","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:82da0d2dc715bca13232f3f532d456bd76df263760d96b89f495407d9b77aa58","sha256:ce2e52264fdbbb91ac27090f511262fd2c2e7b03908b99f1a3c694756198621a"],"state_sha256":"289c3a0e610b3ca64ecb9952d742fde9e5fcbf796fcde73ec1621bfc2baaa6fa"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"/yv/jqUvvvv/sIKS11sqxj5a5GhKBQVEgrM3GnJLA62UOHWYc8qK+O1W4HceR3op/MUHnOG2XymksPdv//NyBQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-19T18:54:22.826505Z","bundle_sha256":"951c789cea7d65a64cb8141164d2693cc5c146c6a178ba6e9ade6c59817b226f"}}