{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2019:P3KORDFZXFZPBSJDJISVYMTBX6","short_pith_number":"pith:P3KORDFZ","canonical_record":{"source":{"id":"1911.00120","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2019-10-31T21:44:19Z","cross_cats_sorted":["math.MG"],"title_canon_sha256":"d4cada8b968c0c9d9a5201b9d73af67b6d0ace1fc92df6d00ed9969a852a1aec","abstract_canon_sha256":"ebe081a1be3c8488dd577d7878f78defaf38343d1f4abbc52cb64cb3b4be5107"},"schema_version":"1.0"},"canonical_sha256":"7ed4e88cb9b972f0c9234a255c3261bf837980061b1c99961ee13fc545f9f409","source":{"kind":"arxiv","id":"1911.00120","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1911.00120","created_at":"2026-07-05T00:16:23Z"},{"alias_kind":"arxiv_version","alias_value":"1911.00120v1","created_at":"2026-07-05T00:16:23Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1911.00120","created_at":"2026-07-05T00:16:23Z"},{"alias_kind":"pith_short_12","alias_value":"P3KORDFZXFZP","created_at":"2026-07-05T00:16:23Z"},{"alias_kind":"pith_short_16","alias_value":"P3KORDFZXFZPBSJD","created_at":"2026-07-05T00:16:23Z"},{"alias_kind":"pith_short_8","alias_value":"P3KORDFZ","created_at":"2026-07-05T00:16:23Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2019:P3KORDFZXFZPBSJDJISVYMTBX6","target":"record","payload":{"canonical_record":{"source":{"id":"1911.00120","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2019-10-31T21:44:19Z","cross_cats_sorted":["math.MG"],"title_canon_sha256":"d4cada8b968c0c9d9a5201b9d73af67b6d0ace1fc92df6d00ed9969a852a1aec","abstract_canon_sha256":"ebe081a1be3c8488dd577d7878f78defaf38343d1f4abbc52cb64cb3b4be5107"},"schema_version":"1.0"},"canonical_sha256":"7ed4e88cb9b972f0c9234a255c3261bf837980061b1c99961ee13fc545f9f409","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T00:16:23.716426Z","signature_b64":"cSZVECv6u8VsSOfS299uRsrwb3plGkYLNy05sy2OLgXWq7jvj7AUMwOCSmRFWKHeJk3v3V2T8AIwK+T28tpIDQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"7ed4e88cb9b972f0c9234a255c3261bf837980061b1c99961ee13fc545f9f409","last_reissued_at":"2026-07-05T00:16:23.715943Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T00:16:23.715943Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1911.00120","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":"2026-07-05T00:16:23Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"nROYZGbyZ4Ow9kDP5yztZfxxbqMVrmSUN+jS+i6A+gpbVhV3jjEmZSfyI4e99Iwbuv7zjO91q69QJrdgdEOyAA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-06T08:27:49.005500Z"},"content_sha256":"5783cdf7688b19026526469ae4845ab655f3fc7e02bfc13f73e2dbfbcebd85eb","schema_version":"1.0","event_id":"sha256:5783cdf7688b19026526469ae4845ab655f3fc7e02bfc13f73e2dbfbcebd85eb"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2019:P3KORDFZXFZPBSJDJISVYMTBX6","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Lipschitz-Volume Rigidity and Globalization","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.MG"],"primary_cat":"math.DG","authors_text":"Nan Li","submitted_at":"2019-10-31T21:44:19Z","abstract_excerpt":"Let $X$ and $Y$ be length metric spaces. Let $\\mathcal H^n$ denote the $n$-dimensional Hausdorff measure. The Lipschitz-Volume Rigidity is a property that if there exists a 1-Lipschitz map $f\\colon X\\to Y$ and $0<\\mathcal H^n(X)=\\mathcal H^n(f(X))<\\infty$, then $f$ preserves the length of path. This property holds for smooth manifolds but doesn't hold for all singular spaces. We survey the Lipschitz-Volume Rigidity Theorems on singular spaces with lower curvature bounds and discuss some related open problems."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1911.00120","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/1911.00120/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":"2026-07-05T00:16:23Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"LT5yXNDHUO0YvMH3OP868eGLxVfLTXYRVXPJrTnGkF/cK3QCNotjnevSwIYfKpjpmc+runJCtXCLDRlJaxW5Bg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-06T08:27:49.006053Z"},"content_sha256":"6b78d856a60767dada20445124f354f6db3e49f1c3e3b5d5bc08137a0e84b83e","schema_version":"1.0","event_id":"sha256:6b78d856a60767dada20445124f354f6db3e49f1c3e3b5d5bc08137a0e84b83e"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/P3KORDFZXFZPBSJDJISVYMTBX6/bundle.json","state_url":"https://pith.science/pith/P3KORDFZXFZPBSJDJISVYMTBX6/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/P3KORDFZXFZPBSJDJISVYMTBX6/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":"2026-08-06T08:27:49Z","links":{"resolver":"https://pith.science/pith/P3KORDFZXFZPBSJDJISVYMTBX6","bundle":"https://pith.science/pith/P3KORDFZXFZPBSJDJISVYMTBX6/bundle.json","state":"https://pith.science/pith/P3KORDFZXFZPBSJDJISVYMTBX6/state.json","well_known_bundle":"https://pith.science/.well-known/pith/P3KORDFZXFZPBSJDJISVYMTBX6/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2019:P3KORDFZXFZPBSJDJISVYMTBX6","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":"ebe081a1be3c8488dd577d7878f78defaf38343d1f4abbc52cb64cb3b4be5107","cross_cats_sorted":["math.MG"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2019-10-31T21:44:19Z","title_canon_sha256":"d4cada8b968c0c9d9a5201b9d73af67b6d0ace1fc92df6d00ed9969a852a1aec"},"schema_version":"1.0","source":{"id":"1911.00120","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1911.00120","created_at":"2026-07-05T00:16:23Z"},{"alias_kind":"arxiv_version","alias_value":"1911.00120v1","created_at":"2026-07-05T00:16:23Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1911.00120","created_at":"2026-07-05T00:16:23Z"},{"alias_kind":"pith_short_12","alias_value":"P3KORDFZXFZP","created_at":"2026-07-05T00:16:23Z"},{"alias_kind":"pith_short_16","alias_value":"P3KORDFZXFZPBSJD","created_at":"2026-07-05T00:16:23Z"},{"alias_kind":"pith_short_8","alias_value":"P3KORDFZ","created_at":"2026-07-05T00:16:23Z"}],"graph_snapshots":[{"event_id":"sha256:6b78d856a60767dada20445124f354f6db3e49f1c3e3b5d5bc08137a0e84b83e","target":"graph","created_at":"2026-07-05T00:16:23Z","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"},"integrity":{"available":true,"clean":true,"detectors_run":[],"endpoint":"/pith/1911.00120/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"Let $X$ and $Y$ be length metric spaces. Let $\\mathcal H^n$ denote the $n$-dimensional Hausdorff measure. The Lipschitz-Volume Rigidity is a property that if there exists a 1-Lipschitz map $f\\colon X\\to Y$ and $0<\\mathcal H^n(X)=\\mathcal H^n(f(X))<\\infty$, then $f$ preserves the length of path. This property holds for smooth manifolds but doesn't hold for all singular spaces. We survey the Lipschitz-Volume Rigidity Theorems on singular spaces with lower curvature bounds and discuss some related open problems.","authors_text":"Nan Li","cross_cats":["math.MG"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2019-10-31T21:44:19Z","title":"Lipschitz-Volume Rigidity and Globalization"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1911.00120","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":"sha256:5783cdf7688b19026526469ae4845ab655f3fc7e02bfc13f73e2dbfbcebd85eb","target":"record","created_at":"2026-07-05T00:16:23Z","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":"ebe081a1be3c8488dd577d7878f78defaf38343d1f4abbc52cb64cb3b4be5107","cross_cats_sorted":["math.MG"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2019-10-31T21:44:19Z","title_canon_sha256":"d4cada8b968c0c9d9a5201b9d73af67b6d0ace1fc92df6d00ed9969a852a1aec"},"schema_version":"1.0","source":{"id":"1911.00120","kind":"arxiv","version":1}},"canonical_sha256":"7ed4e88cb9b972f0c9234a255c3261bf837980061b1c99961ee13fc545f9f409","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"7ed4e88cb9b972f0c9234a255c3261bf837980061b1c99961ee13fc545f9f409","first_computed_at":"2026-07-05T00:16:23.715943Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T00:16:23.715943Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"cSZVECv6u8VsSOfS299uRsrwb3plGkYLNy05sy2OLgXWq7jvj7AUMwOCSmRFWKHeJk3v3V2T8AIwK+T28tpIDQ==","signature_status":"signed_v1","signed_at":"2026-07-05T00:16:23.716426Z","signed_message":"canonical_sha256_bytes"},"source_id":"1911.00120","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:5783cdf7688b19026526469ae4845ab655f3fc7e02bfc13f73e2dbfbcebd85eb","sha256:6b78d856a60767dada20445124f354f6db3e49f1c3e3b5d5bc08137a0e84b83e"],"state_sha256":"37c282ef057f740354e81bbe8ae57b7a2984ec6440065d85bd554e9628deccdd"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"IMgEhl1vxkE3ki5vpOVGSpb4NjOCizx1xf2GEvxPE481L6I2oAU+EKhLR5H0mO4g6mAgEWGe66OFcGmSECkoAg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-06T08:27:49.010902Z","bundle_sha256":"ed01a6e6a2f6e8aa26f58ebd4ac3deb443f5d6945f1d0363e38642bd318e2f1b"}}