{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2016:RLAU2MSSLQKYCQSNM35CCBYNPC","short_pith_number":"pith:RLAU2MSS","canonical_record":{"source":{"id":"1605.05432","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2016-05-18T04:05:35Z","cross_cats_sorted":[],"title_canon_sha256":"0093e781d3017688768d54551a3388ec2f8e7e4ec86429da9f86030d6fb2576f","abstract_canon_sha256":"e027608d5cb9efce470e95915f4180991bd0bd9a15543980e6497df9d44152bb"},"schema_version":"1.0"},"canonical_sha256":"8ac14d32525c1581424d66fa21070d7892ac772450207393430dfd08dbcf88a6","source":{"kind":"arxiv","id":"1605.05432","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1605.05432","created_at":"2026-05-18T00:09:53Z"},{"alias_kind":"arxiv_version","alias_value":"1605.05432v2","created_at":"2026-05-18T00:09:53Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1605.05432","created_at":"2026-05-18T00:09:53Z"},{"alias_kind":"pith_short_12","alias_value":"RLAU2MSSLQKY","created_at":"2026-05-18T12:30:41Z"},{"alias_kind":"pith_short_16","alias_value":"RLAU2MSSLQKYCQSN","created_at":"2026-05-18T12:30:41Z"},{"alias_kind":"pith_short_8","alias_value":"RLAU2MSS","created_at":"2026-05-18T12:30:41Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2016:RLAU2MSSLQKYCQSNM35CCBYNPC","target":"record","payload":{"canonical_record":{"source":{"id":"1605.05432","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2016-05-18T04:05:35Z","cross_cats_sorted":[],"title_canon_sha256":"0093e781d3017688768d54551a3388ec2f8e7e4ec86429da9f86030d6fb2576f","abstract_canon_sha256":"e027608d5cb9efce470e95915f4180991bd0bd9a15543980e6497df9d44152bb"},"schema_version":"1.0"},"canonical_sha256":"8ac14d32525c1581424d66fa21070d7892ac772450207393430dfd08dbcf88a6","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:09:53.395986Z","signature_b64":"KRCz0MoPUPRF+Va5rBFRZk/a2f9jbIiMCqInj8RUqPfleApDQBvwl0EB5WuAkQNFvq+2VhIHKqmroE00aJFmCA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"8ac14d32525c1581424d66fa21070d7892ac772450207393430dfd08dbcf88a6","last_reissued_at":"2026-05-18T00:09:53.395253Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:09:53.395253Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1605.05432","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":"2026-05-18T00:09:53Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"idld65p7Pd3Fge8nRO43pIRBN5t3j8HdDXtVHn0V//73YtvxDVQ9NzAbuGrxyifEGldAQPFMZn3KXG2joXoHCQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-01T02:34:25.376412Z"},"content_sha256":"2abb9d0dd86cc922d1b735d4290f5bfbb220dd7418993250839aa284baebdee7","schema_version":"1.0","event_id":"sha256:2abb9d0dd86cc922d1b735d4290f5bfbb220dd7418993250839aa284baebdee7"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2016:RLAU2MSSLQKYCQSNM35CCBYNPC","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Global Poincar\\'e inequality on Graphs via Conical Curvature-Dimension Conditions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.DG","authors_text":"Sajjad Lakzian, Zachary McGuirk","submitted_at":"2016-05-18T04:05:35Z","abstract_excerpt":"We introduce and study the conical curvature-dimension condition, $CCD(K,N)$, for graphs. We show that $CCD(K,N)$ provides necessary and sufficient conditions for the underlying graph to satisfy a sharp global Poincar\\'e inequality which in turn translates to a sharp lower bound for the first eigenvalues of these graphs. Another application of the conical curvature-dimension analysis is finding a sharp estimate on the curvature of complete graphs."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1605.05432","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":"2026-05-18T00:09:53Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"+Q8UMDi8rzkB3NwbqrSHRsNNwUv/Z21SaraRnFqqBh9yxjngwLvr3svHHHZPX+VcTlAxNQP8HerMYKIAAlBuDg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-01T02:34:25.377177Z"},"content_sha256":"cabc075f436e02c12131a1c1ff6d8de80d26fc41cd0cf8b9add08f5b2b320acb","schema_version":"1.0","event_id":"sha256:cabc075f436e02c12131a1c1ff6d8de80d26fc41cd0cf8b9add08f5b2b320acb"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/RLAU2MSSLQKYCQSNM35CCBYNPC/bundle.json","state_url":"https://pith.science/pith/RLAU2MSSLQKYCQSNM35CCBYNPC/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/RLAU2MSSLQKYCQSNM35CCBYNPC/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":"2026-06-01T02:34:25Z","links":{"resolver":"https://pith.science/pith/RLAU2MSSLQKYCQSNM35CCBYNPC","bundle":"https://pith.science/pith/RLAU2MSSLQKYCQSNM35CCBYNPC/bundle.json","state":"https://pith.science/pith/RLAU2MSSLQKYCQSNM35CCBYNPC/state.json","well_known_bundle":"https://pith.science/.well-known/pith/RLAU2MSSLQKYCQSNM35CCBYNPC/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2016:RLAU2MSSLQKYCQSNM35CCBYNPC","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":"e027608d5cb9efce470e95915f4180991bd0bd9a15543980e6497df9d44152bb","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2016-05-18T04:05:35Z","title_canon_sha256":"0093e781d3017688768d54551a3388ec2f8e7e4ec86429da9f86030d6fb2576f"},"schema_version":"1.0","source":{"id":"1605.05432","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1605.05432","created_at":"2026-05-18T00:09:53Z"},{"alias_kind":"arxiv_version","alias_value":"1605.05432v2","created_at":"2026-05-18T00:09:53Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1605.05432","created_at":"2026-05-18T00:09:53Z"},{"alias_kind":"pith_short_12","alias_value":"RLAU2MSSLQKY","created_at":"2026-05-18T12:30:41Z"},{"alias_kind":"pith_short_16","alias_value":"RLAU2MSSLQKYCQSN","created_at":"2026-05-18T12:30:41Z"},{"alias_kind":"pith_short_8","alias_value":"RLAU2MSS","created_at":"2026-05-18T12:30:41Z"}],"graph_snapshots":[{"event_id":"sha256:cabc075f436e02c12131a1c1ff6d8de80d26fc41cd0cf8b9add08f5b2b320acb","target":"graph","created_at":"2026-05-18T00:09:53Z","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 introduce and study the conical curvature-dimension condition, $CCD(K,N)$, for graphs. We show that $CCD(K,N)$ provides necessary and sufficient conditions for the underlying graph to satisfy a sharp global Poincar\\'e inequality which in turn translates to a sharp lower bound for the first eigenvalues of these graphs. Another application of the conical curvature-dimension analysis is finding a sharp estimate on the curvature of complete graphs.","authors_text":"Sajjad Lakzian, Zachary McGuirk","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2016-05-18T04:05:35Z","title":"Global Poincar\\'e inequality on Graphs via Conical Curvature-Dimension Conditions"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1605.05432","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":"sha256:2abb9d0dd86cc922d1b735d4290f5bfbb220dd7418993250839aa284baebdee7","target":"record","created_at":"2026-05-18T00:09:53Z","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":"e027608d5cb9efce470e95915f4180991bd0bd9a15543980e6497df9d44152bb","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2016-05-18T04:05:35Z","title_canon_sha256":"0093e781d3017688768d54551a3388ec2f8e7e4ec86429da9f86030d6fb2576f"},"schema_version":"1.0","source":{"id":"1605.05432","kind":"arxiv","version":2}},"canonical_sha256":"8ac14d32525c1581424d66fa21070d7892ac772450207393430dfd08dbcf88a6","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"8ac14d32525c1581424d66fa21070d7892ac772450207393430dfd08dbcf88a6","first_computed_at":"2026-05-18T00:09:53.395253Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:09:53.395253Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"KRCz0MoPUPRF+Va5rBFRZk/a2f9jbIiMCqInj8RUqPfleApDQBvwl0EB5WuAkQNFvq+2VhIHKqmroE00aJFmCA==","signature_status":"signed_v1","signed_at":"2026-05-18T00:09:53.395986Z","signed_message":"canonical_sha256_bytes"},"source_id":"1605.05432","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:2abb9d0dd86cc922d1b735d4290f5bfbb220dd7418993250839aa284baebdee7","sha256:cabc075f436e02c12131a1c1ff6d8de80d26fc41cd0cf8b9add08f5b2b320acb"],"state_sha256":"c7162057586b28efde3eb22d9a221aba4bdd24544383a32c788f1beee93a8d56"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"2Qdw05vL+c4/qX89POiLv9vieicIxd1Ab+mVZcly6RVzXfnw5fvRDMfc5u1tVcy2V2zjgOA1JmPLGlIbfDcyBw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-01T02:34:25.381285Z","bundle_sha256":"b223f6ff6ba525645dbf0a570adf87ef70891075f04cb74f81199c826401b012"}}