{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2008:VTLEPFEF4EQVWAMDJC76NHVVAL","short_pith_number":"pith:VTLEPFEF","canonical_record":{"source":{"id":"0807.2233","kind":"arxiv","version":4},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2008-07-14T19:40:57Z","cross_cats_sorted":["math.MG"],"title_canon_sha256":"01aefe2cc6ce436b1c70669dd11f4284b4e4977d27de78b8513e44cd97d04324","abstract_canon_sha256":"0330a83564e490f939a7c63e77fce5d1648928588e22f1b778cb00e0a5ebe823"},"schema_version":"1.0"},"canonical_sha256":"acd6479485e1215b018348bfe69eb502e68c8aa5d4e31d82be9808f40c17c3e1","source":{"kind":"arxiv","id":"0807.2233","version":4},"source_aliases":[{"alias_kind":"arxiv","alias_value":"0807.2233","created_at":"2026-05-18T02:26:23Z"},{"alias_kind":"arxiv_version","alias_value":"0807.2233v4","created_at":"2026-05-18T02:26:23Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.0807.2233","created_at":"2026-05-18T02:26:23Z"},{"alias_kind":"pith_short_12","alias_value":"VTLEPFEF4EQV","created_at":"2026-05-18T12:25:58Z"},{"alias_kind":"pith_short_16","alias_value":"VTLEPFEF4EQVWAMD","created_at":"2026-05-18T12:25:58Z"},{"alias_kind":"pith_short_8","alias_value":"VTLEPFEF","created_at":"2026-05-18T12:25:58Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2008:VTLEPFEF4EQVWAMDJC76NHVVAL","target":"record","payload":{"canonical_record":{"source":{"id":"0807.2233","kind":"arxiv","version":4},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2008-07-14T19:40:57Z","cross_cats_sorted":["math.MG"],"title_canon_sha256":"01aefe2cc6ce436b1c70669dd11f4284b4e4977d27de78b8513e44cd97d04324","abstract_canon_sha256":"0330a83564e490f939a7c63e77fce5d1648928588e22f1b778cb00e0a5ebe823"},"schema_version":"1.0"},"canonical_sha256":"acd6479485e1215b018348bfe69eb502e68c8aa5d4e31d82be9808f40c17c3e1","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:26:23.968503Z","signature_b64":"zF7JSzKWJBPzKIWGfiPDQDNg+OWwgx2chU9CmYovdcS6jlfSsFegqYASZpquRBf7QnfEMkiHBB6BqF4cQXLCBA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"acd6479485e1215b018348bfe69eb502e68c8aa5d4e31d82be9808f40c17c3e1","last_reissued_at":"2026-05-18T02:26:23.967763Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:26:23.967763Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"0807.2233","source_version":4,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T02:26:23Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"ixP7CS2hi7svOD6WbjcT6aQYqmyDY8sieapvAo8/9NTAaqp0Y1xiDx4irSQvRYBMh4sjdaMk7+uhrlveRdFgBA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-27T09:17:52.748658Z"},"content_sha256":"ea1c3c2b33e2c8b3430c7dfa1c064ad141a4bdac4f16f68866b2e261d0c3c787","schema_version":"1.0","event_id":"sha256:ea1c3c2b33e2c8b3430c7dfa1c064ad141a4bdac4f16f68866b2e261d0c3c787"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2008:VTLEPFEF4EQVWAMDJC76NHVVAL","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Aspects of area formulas by way of Luzin, Rad\\'o, and Reichelderfer on metric measure spaces","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.MG"],"primary_cat":"math.CA","authors_text":"Niko Marola, William P. Ziemer","submitted_at":"2008-07-14T19:40:57Z","abstract_excerpt":"We consider some measure-theoretic properties of functions belonging to a Sobolev-type class on metric measure spaces that admit a Poincar\\'e inequality and are equipped with a doubling measure. The properties we have selected to study are those that are related to area formulas."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0807.2233","kind":"arxiv","version":4},"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":"2026-05-18T02:26:23Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"c/hfOlNZa8+3q7UpnT/J0AysS6rOYEykrlWi9Xte8X9UBEPJrv40BY0T6MAkIRM9fXp7/e6IYwt9I4yeIDtoBA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-27T09:17:52.749336Z"},"content_sha256":"5e331dd9d7cfa95bb7822428f8412765e7cda88bd2d0cf8608a3cb1a67f4fd1f","schema_version":"1.0","event_id":"sha256:5e331dd9d7cfa95bb7822428f8412765e7cda88bd2d0cf8608a3cb1a67f4fd1f"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/VTLEPFEF4EQVWAMDJC76NHVVAL/bundle.json","state_url":"https://pith.science/pith/VTLEPFEF4EQVWAMDJC76NHVVAL/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/VTLEPFEF4EQVWAMDJC76NHVVAL/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":"2026-05-27T09:17:52Z","links":{"resolver":"https://pith.science/pith/VTLEPFEF4EQVWAMDJC76NHVVAL","bundle":"https://pith.science/pith/VTLEPFEF4EQVWAMDJC76NHVVAL/bundle.json","state":"https://pith.science/pith/VTLEPFEF4EQVWAMDJC76NHVVAL/state.json","well_known_bundle":"https://pith.science/.well-known/pith/VTLEPFEF4EQVWAMDJC76NHVVAL/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2008:VTLEPFEF4EQVWAMDJC76NHVVAL","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":"0330a83564e490f939a7c63e77fce5d1648928588e22f1b778cb00e0a5ebe823","cross_cats_sorted":["math.MG"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2008-07-14T19:40:57Z","title_canon_sha256":"01aefe2cc6ce436b1c70669dd11f4284b4e4977d27de78b8513e44cd97d04324"},"schema_version":"1.0","source":{"id":"0807.2233","kind":"arxiv","version":4}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"0807.2233","created_at":"2026-05-18T02:26:23Z"},{"alias_kind":"arxiv_version","alias_value":"0807.2233v4","created_at":"2026-05-18T02:26:23Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.0807.2233","created_at":"2026-05-18T02:26:23Z"},{"alias_kind":"pith_short_12","alias_value":"VTLEPFEF4EQV","created_at":"2026-05-18T12:25:58Z"},{"alias_kind":"pith_short_16","alias_value":"VTLEPFEF4EQVWAMD","created_at":"2026-05-18T12:25:58Z"},{"alias_kind":"pith_short_8","alias_value":"VTLEPFEF","created_at":"2026-05-18T12:25:58Z"}],"graph_snapshots":[{"event_id":"sha256:5e331dd9d7cfa95bb7822428f8412765e7cda88bd2d0cf8608a3cb1a67f4fd1f","target":"graph","created_at":"2026-05-18T02:26:23Z","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 consider some measure-theoretic properties of functions belonging to a Sobolev-type class on metric measure spaces that admit a Poincar\\'e inequality and are equipped with a doubling measure. The properties we have selected to study are those that are related to area formulas.","authors_text":"Niko Marola, William P. Ziemer","cross_cats":["math.MG"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2008-07-14T19:40:57Z","title":"Aspects of area formulas by way of Luzin, Rad\\'o, and Reichelderfer on metric measure spaces"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0807.2233","kind":"arxiv","version":4},"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:ea1c3c2b33e2c8b3430c7dfa1c064ad141a4bdac4f16f68866b2e261d0c3c787","target":"record","created_at":"2026-05-18T02:26:23Z","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":"0330a83564e490f939a7c63e77fce5d1648928588e22f1b778cb00e0a5ebe823","cross_cats_sorted":["math.MG"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2008-07-14T19:40:57Z","title_canon_sha256":"01aefe2cc6ce436b1c70669dd11f4284b4e4977d27de78b8513e44cd97d04324"},"schema_version":"1.0","source":{"id":"0807.2233","kind":"arxiv","version":4}},"canonical_sha256":"acd6479485e1215b018348bfe69eb502e68c8aa5d4e31d82be9808f40c17c3e1","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"acd6479485e1215b018348bfe69eb502e68c8aa5d4e31d82be9808f40c17c3e1","first_computed_at":"2026-05-18T02:26:23.967763Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T02:26:23.967763Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"zF7JSzKWJBPzKIWGfiPDQDNg+OWwgx2chU9CmYovdcS6jlfSsFegqYASZpquRBf7QnfEMkiHBB6BqF4cQXLCBA==","signature_status":"signed_v1","signed_at":"2026-05-18T02:26:23.968503Z","signed_message":"canonical_sha256_bytes"},"source_id":"0807.2233","source_kind":"arxiv","source_version":4}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:ea1c3c2b33e2c8b3430c7dfa1c064ad141a4bdac4f16f68866b2e261d0c3c787","sha256:5e331dd9d7cfa95bb7822428f8412765e7cda88bd2d0cf8608a3cb1a67f4fd1f"],"state_sha256":"1fb5881ba56872695143a689f4d6049d3427b5f3493d708fee1b5615b3822e12"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"q6g+mXiNaDyux0X1f03XgGj50KMmKZ+w7tJtf1i3LxoVVAsmNg0HdklQpoLeNvk4WgeiY5ZfEqt2/Qg9+s4aDQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-27T09:17:52.753026Z","bundle_sha256":"9ba2b3a2d58b7e2a7e164d2134f4dbf2c5f4c34199ac5881d46880a7097dbae6"}}