{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2015:GJDW3DNZA4VD4IGS747RQDCEM4","short_pith_number":"pith:GJDW3DNZ","canonical_record":{"source":{"id":"1507.08931","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2015-07-31T16:21:04Z","cross_cats_sorted":[],"title_canon_sha256":"1e7c5e492405cd94d09099a1b8a74446265846a0c07fb976d3aed92533de7956","abstract_canon_sha256":"d3b6188fae14050942a6a8d3e7090d6e011d0540142eb7774e60e388c32467c8"},"schema_version":"1.0"},"canonical_sha256":"32476d8db9072a3e20d2ff3f180c44670536d4bf01611c183951072f0f32a83c","source":{"kind":"arxiv","id":"1507.08931","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1507.08931","created_at":"2026-05-18T01:17:08Z"},{"alias_kind":"arxiv_version","alias_value":"1507.08931v2","created_at":"2026-05-18T01:17:08Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1507.08931","created_at":"2026-05-18T01:17:08Z"},{"alias_kind":"pith_short_12","alias_value":"GJDW3DNZA4VD","created_at":"2026-05-18T12:29:22Z"},{"alias_kind":"pith_short_16","alias_value":"GJDW3DNZA4VD4IGS","created_at":"2026-05-18T12:29:22Z"},{"alias_kind":"pith_short_8","alias_value":"GJDW3DNZ","created_at":"2026-05-18T12:29:22Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2015:GJDW3DNZA4VD4IGS747RQDCEM4","target":"record","payload":{"canonical_record":{"source":{"id":"1507.08931","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2015-07-31T16:21:04Z","cross_cats_sorted":[],"title_canon_sha256":"1e7c5e492405cd94d09099a1b8a74446265846a0c07fb976d3aed92533de7956","abstract_canon_sha256":"d3b6188fae14050942a6a8d3e7090d6e011d0540142eb7774e60e388c32467c8"},"schema_version":"1.0"},"canonical_sha256":"32476d8db9072a3e20d2ff3f180c44670536d4bf01611c183951072f0f32a83c","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:17:08.061932Z","signature_b64":"y6DOjeeSP/DsJRjlMg+2c1QFLzcDQT3u0EPhw019BD8p0j861HACmqhjeFueGGSdHccYEZyTdfpL7y6w6H6JDw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"32476d8db9072a3e20d2ff3f180c44670536d4bf01611c183951072f0f32a83c","last_reissued_at":"2026-05-18T01:17:08.059872Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:17:08.059872Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1507.08931","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":"2026-05-18T01:17:08Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"5jg7vewdxLKrB3LUPQRJCoQ3kV4t9KAcqudL/tYr3/ZvEI1THJgh2yZUH10qN9D20mki7OIo3iipTiAejgODAw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-10T07:27:34.036370Z"},"content_sha256":"4688f98965633de38eea4504ec21c9eff1286325a6e5c78db5e8f527ce3b8c42","schema_version":"1.0","event_id":"sha256:4688f98965633de38eea4504ec21c9eff1286325a6e5c78db5e8f527ce3b8c42"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2015:GJDW3DNZA4VD4IGS747RQDCEM4","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Volume comparison for $C^{1,1}$ metrics","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.DG","authors_text":"Melanie Graf","submitted_at":"2015-07-31T16:21:04Z","abstract_excerpt":"We establish volume comparison results for balls in Riemannian manifolds with $C^{1,1}$-metrics with a lower bound on the Ricci tensor and for the evolution of spacelike, acausal, causally complete hypersurfaces with an upper bound on the mean curvature in spacetimes with $C^{1,1}$-metrics with a lower bound on the timelike Ricci curvature. These results are then used to give proofs of Myers' theorem and of Hawking's singularity theorem in this regularity."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1507.08931","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":"2026-05-18T01:17:08Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"M/in7IBgqKXf8nG4z7KK7Ze60CGeeKvQ+ac/Z9QUPyDaf6B0VQiai2BvNusZXztPP1PS4lWTBFJHc6BhOoM0Bw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-10T07:27:34.037079Z"},"content_sha256":"96377ad262ba0452d28b5a1b0748acb01c637e909dbf019b338b6949bed34a77","schema_version":"1.0","event_id":"sha256:96377ad262ba0452d28b5a1b0748acb01c637e909dbf019b338b6949bed34a77"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/GJDW3DNZA4VD4IGS747RQDCEM4/bundle.json","state_url":"https://pith.science/pith/GJDW3DNZA4VD4IGS747RQDCEM4/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/GJDW3DNZA4VD4IGS747RQDCEM4/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":"2026-06-10T07:27:34Z","links":{"resolver":"https://pith.science/pith/GJDW3DNZA4VD4IGS747RQDCEM4","bundle":"https://pith.science/pith/GJDW3DNZA4VD4IGS747RQDCEM4/bundle.json","state":"https://pith.science/pith/GJDW3DNZA4VD4IGS747RQDCEM4/state.json","well_known_bundle":"https://pith.science/.well-known/pith/GJDW3DNZA4VD4IGS747RQDCEM4/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2015:GJDW3DNZA4VD4IGS747RQDCEM4","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":"d3b6188fae14050942a6a8d3e7090d6e011d0540142eb7774e60e388c32467c8","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2015-07-31T16:21:04Z","title_canon_sha256":"1e7c5e492405cd94d09099a1b8a74446265846a0c07fb976d3aed92533de7956"},"schema_version":"1.0","source":{"id":"1507.08931","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1507.08931","created_at":"2026-05-18T01:17:08Z"},{"alias_kind":"arxiv_version","alias_value":"1507.08931v2","created_at":"2026-05-18T01:17:08Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1507.08931","created_at":"2026-05-18T01:17:08Z"},{"alias_kind":"pith_short_12","alias_value":"GJDW3DNZA4VD","created_at":"2026-05-18T12:29:22Z"},{"alias_kind":"pith_short_16","alias_value":"GJDW3DNZA4VD4IGS","created_at":"2026-05-18T12:29:22Z"},{"alias_kind":"pith_short_8","alias_value":"GJDW3DNZ","created_at":"2026-05-18T12:29:22Z"}],"graph_snapshots":[{"event_id":"sha256:96377ad262ba0452d28b5a1b0748acb01c637e909dbf019b338b6949bed34a77","target":"graph","created_at":"2026-05-18T01:17:08Z","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 establish volume comparison results for balls in Riemannian manifolds with $C^{1,1}$-metrics with a lower bound on the Ricci tensor and for the evolution of spacelike, acausal, causally complete hypersurfaces with an upper bound on the mean curvature in spacetimes with $C^{1,1}$-metrics with a lower bound on the timelike Ricci curvature. These results are then used to give proofs of Myers' theorem and of Hawking's singularity theorem in this regularity.","authors_text":"Melanie Graf","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2015-07-31T16:21:04Z","title":"Volume comparison for $C^{1,1}$ metrics"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1507.08931","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":"sha256:4688f98965633de38eea4504ec21c9eff1286325a6e5c78db5e8f527ce3b8c42","target":"record","created_at":"2026-05-18T01:17:08Z","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":"d3b6188fae14050942a6a8d3e7090d6e011d0540142eb7774e60e388c32467c8","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2015-07-31T16:21:04Z","title_canon_sha256":"1e7c5e492405cd94d09099a1b8a74446265846a0c07fb976d3aed92533de7956"},"schema_version":"1.0","source":{"id":"1507.08931","kind":"arxiv","version":2}},"canonical_sha256":"32476d8db9072a3e20d2ff3f180c44670536d4bf01611c183951072f0f32a83c","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"32476d8db9072a3e20d2ff3f180c44670536d4bf01611c183951072f0f32a83c","first_computed_at":"2026-05-18T01:17:08.059872Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:17:08.059872Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"y6DOjeeSP/DsJRjlMg+2c1QFLzcDQT3u0EPhw019BD8p0j861HACmqhjeFueGGSdHccYEZyTdfpL7y6w6H6JDw==","signature_status":"signed_v1","signed_at":"2026-05-18T01:17:08.061932Z","signed_message":"canonical_sha256_bytes"},"source_id":"1507.08931","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:4688f98965633de38eea4504ec21c9eff1286325a6e5c78db5e8f527ce3b8c42","sha256:96377ad262ba0452d28b5a1b0748acb01c637e909dbf019b338b6949bed34a77"],"state_sha256":"643cbc880d799593d7692f1ccadb0becbcb5477629bfe1a38c9aff0462351750"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"C8XGzCmAqb9zwrPFgENFVuWr/tEwztt8sOHIrOmu55+d++rDpV/OzEXsWAN/9XjHCthJ7xY141fGBnzITkpNAQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-10T07:27:34.040875Z","bundle_sha256":"7cd658dae17fe11cb4998f1a673eea6d5de9b9f89702d9dfb849ee2967cf3c2a"}}