{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2025:K3PI2HE5IR6DYSEJQZSYF6KA6P","short_pith_number":"pith:K3PI2HE5","canonical_record":{"source":{"id":"2507.04109","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2025-07-05T17:39:25Z","cross_cats_sorted":[],"title_canon_sha256":"72ba524e09687331078c0f452204fb1a9842404725ea17ed3521757fad0fa162","abstract_canon_sha256":"8f728a8114c3936ced9e234992679fee9bfd0ad68eccd5019a065ee64b8efcbf"},"schema_version":"1.0"},"canonical_sha256":"56de8d1c9d447c3c4889866582f940f3ce82962eb5a63ab244ae7b758160a2e0","source":{"kind":"arxiv","id":"2507.04109","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2507.04109","created_at":"2026-07-05T11:32:42Z"},{"alias_kind":"arxiv_version","alias_value":"2507.04109v1","created_at":"2026-07-05T11:32:42Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2507.04109","created_at":"2026-07-05T11:32:42Z"},{"alias_kind":"pith_short_12","alias_value":"K3PI2HE5IR6D","created_at":"2026-07-05T11:32:42Z"},{"alias_kind":"pith_short_16","alias_value":"K3PI2HE5IR6DYSEJ","created_at":"2026-07-05T11:32:42Z"},{"alias_kind":"pith_short_8","alias_value":"K3PI2HE5","created_at":"2026-07-05T11:32:42Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2025:K3PI2HE5IR6DYSEJQZSYF6KA6P","target":"record","payload":{"canonical_record":{"source":{"id":"2507.04109","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2025-07-05T17:39:25Z","cross_cats_sorted":[],"title_canon_sha256":"72ba524e09687331078c0f452204fb1a9842404725ea17ed3521757fad0fa162","abstract_canon_sha256":"8f728a8114c3936ced9e234992679fee9bfd0ad68eccd5019a065ee64b8efcbf"},"schema_version":"1.0"},"canonical_sha256":"56de8d1c9d447c3c4889866582f940f3ce82962eb5a63ab244ae7b758160a2e0","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T11:32:42.089116Z","signature_b64":"qzIaT945+MXuhi9CNu80K2dqKs4LiY1ojl26lIzfxQn+35O63/xLlmkSfME1BBzEY5fnYevRrRZYLWzIwLqXDg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"56de8d1c9d447c3c4889866582f940f3ce82962eb5a63ab244ae7b758160a2e0","last_reissued_at":"2026-07-05T11:32:42.088726Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T11:32:42.088726Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2507.04109","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":"2026-07-05T11:32:42Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"v18776FNvnx//0GAzhbTCkc5T7KNut2qI3kKv+5xbI2DEdVTL6C1exQebgorIrUHU6bCXFO091BROC7pozSSBg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-09T04:28:39.413156Z"},"content_sha256":"7df40359dbf5ec33fcde48315686ebe876685147ba984f83c9c70d5c37649a5d","schema_version":"1.0","event_id":"sha256:7df40359dbf5ec33fcde48315686ebe876685147ba984f83c9c70d5c37649a5d"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2025:K3PI2HE5IR6DYSEJQZSYF6KA6P","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Lin-Lu-Yau Ricci curvature on hypergraphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.DG","authors_text":"Liang Zhao, Yulu Tian","submitted_at":"2025-07-05T17:39:25Z","abstract_excerpt":"In this paper, we introduce a unified framework for defining Lin-Lu-Yau (LLY) Ricci curvature on both undirected and directed hypergraphs. By establishing upper bounds and monotonicity properties for the parameterized curvature $\\kappa_\\alpha$, we justify the well-posedness and compatibility of our definitions. Furthermore, we prove a Bonnet-Myers-type theorem for hypergraphs, which highlights the potential of LLY Ricci curvature in hypergraph analysis, particularly in studying geometric and structural properties. Our results extends the foundational definitions of graph Ricci curvature by Oll"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2507.04109","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/2507.04109/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":"2026-07-05T11:32:42Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"MlCm4XoC3EwNsyEGNgTYf9EsHAbmYyNJAM0NX2NP/YZmPJXs6cUbdrH+0fSfSCzJx+Nsopmv+g86OsG0Ex+oCw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-09T04:28:39.413999Z"},"content_sha256":"5218073113c1c890809d05571ae90fea0c0c312f7425fefe32425ecb119b29d8","schema_version":"1.0","event_id":"sha256:5218073113c1c890809d05571ae90fea0c0c312f7425fefe32425ecb119b29d8"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/K3PI2HE5IR6DYSEJQZSYF6KA6P/bundle.json","state_url":"https://pith.science/pith/K3PI2HE5IR6DYSEJQZSYF6KA6P/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/K3PI2HE5IR6DYSEJQZSYF6KA6P/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":"2026-08-09T04:28:39Z","links":{"resolver":"https://pith.science/pith/K3PI2HE5IR6DYSEJQZSYF6KA6P","bundle":"https://pith.science/pith/K3PI2HE5IR6DYSEJQZSYF6KA6P/bundle.json","state":"https://pith.science/pith/K3PI2HE5IR6DYSEJQZSYF6KA6P/state.json","well_known_bundle":"https://pith.science/.well-known/pith/K3PI2HE5IR6DYSEJQZSYF6KA6P/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2025:K3PI2HE5IR6DYSEJQZSYF6KA6P","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":"8f728a8114c3936ced9e234992679fee9bfd0ad68eccd5019a065ee64b8efcbf","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2025-07-05T17:39:25Z","title_canon_sha256":"72ba524e09687331078c0f452204fb1a9842404725ea17ed3521757fad0fa162"},"schema_version":"1.0","source":{"id":"2507.04109","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2507.04109","created_at":"2026-07-05T11:32:42Z"},{"alias_kind":"arxiv_version","alias_value":"2507.04109v1","created_at":"2026-07-05T11:32:42Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2507.04109","created_at":"2026-07-05T11:32:42Z"},{"alias_kind":"pith_short_12","alias_value":"K3PI2HE5IR6D","created_at":"2026-07-05T11:32:42Z"},{"alias_kind":"pith_short_16","alias_value":"K3PI2HE5IR6DYSEJ","created_at":"2026-07-05T11:32:42Z"},{"alias_kind":"pith_short_8","alias_value":"K3PI2HE5","created_at":"2026-07-05T11:32:42Z"}],"graph_snapshots":[{"event_id":"sha256:5218073113c1c890809d05571ae90fea0c0c312f7425fefe32425ecb119b29d8","target":"graph","created_at":"2026-07-05T11:32:42Z","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"},"integrity":{"available":true,"clean":true,"detectors_run":[],"endpoint":"/pith/2507.04109/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"In this paper, we introduce a unified framework for defining Lin-Lu-Yau (LLY) Ricci curvature on both undirected and directed hypergraphs. By establishing upper bounds and monotonicity properties for the parameterized curvature $\\kappa_\\alpha$, we justify the well-posedness and compatibility of our definitions. Furthermore, we prove a Bonnet-Myers-type theorem for hypergraphs, which highlights the potential of LLY Ricci curvature in hypergraph analysis, particularly in studying geometric and structural properties. Our results extends the foundational definitions of graph Ricci curvature by Oll","authors_text":"Liang Zhao, Yulu Tian","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2025-07-05T17:39:25Z","title":"Lin-Lu-Yau Ricci curvature on hypergraphs"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2507.04109","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":"sha256:7df40359dbf5ec33fcde48315686ebe876685147ba984f83c9c70d5c37649a5d","target":"record","created_at":"2026-07-05T11:32:42Z","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":"8f728a8114c3936ced9e234992679fee9bfd0ad68eccd5019a065ee64b8efcbf","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2025-07-05T17:39:25Z","title_canon_sha256":"72ba524e09687331078c0f452204fb1a9842404725ea17ed3521757fad0fa162"},"schema_version":"1.0","source":{"id":"2507.04109","kind":"arxiv","version":1}},"canonical_sha256":"56de8d1c9d447c3c4889866582f940f3ce82962eb5a63ab244ae7b758160a2e0","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"56de8d1c9d447c3c4889866582f940f3ce82962eb5a63ab244ae7b758160a2e0","first_computed_at":"2026-07-05T11:32:42.088726Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T11:32:42.088726Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"qzIaT945+MXuhi9CNu80K2dqKs4LiY1ojl26lIzfxQn+35O63/xLlmkSfME1BBzEY5fnYevRrRZYLWzIwLqXDg==","signature_status":"signed_v1","signed_at":"2026-07-05T11:32:42.089116Z","signed_message":"canonical_sha256_bytes"},"source_id":"2507.04109","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:7df40359dbf5ec33fcde48315686ebe876685147ba984f83c9c70d5c37649a5d","sha256:5218073113c1c890809d05571ae90fea0c0c312f7425fefe32425ecb119b29d8"],"state_sha256":"3b5aaad66951b80689d348a3d86eb1f1049ee622d32f7fa7eda7a0303a015691"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"k7h2T449WqqaRB8XITE8y1YJO5xkqJe/i3z2JGOdHu7wJeIAU4XF+kspj4JD6EQ4zdpaMJfoVVL3f35mt8PwAA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-09T04:28:39.419893Z","bundle_sha256":"7559629a8fbdb0fdf8bd6f84adc2515806e1bf2d956c58ed0f249166f55e80d7"}}