{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2011:BUJI26HXU6O2GABUPKIFKI7XCC","short_pith_number":"pith:BUJI26HX","canonical_record":{"source":{"id":"1111.2184","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2011-11-09T12:09:19Z","cross_cats_sorted":[],"title_canon_sha256":"d4a0455821022823f2ab9dc6330a310480e92aaaa5e548b620c601e24376a7a6","abstract_canon_sha256":"be7cd14216522ac20d3ad6beab2e6d524fbfbb9a7cc7403f49a4f8e74e3909e8"},"schema_version":"1.0"},"canonical_sha256":"0d128d78f7a79da300347a905523f710bdd4a584e516b8b99fc82cbaffe48f0b","source":{"kind":"arxiv","id":"1111.2184","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1111.2184","created_at":"2026-05-18T04:08:43Z"},{"alias_kind":"arxiv_version","alias_value":"1111.2184v1","created_at":"2026-05-18T04:08:43Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1111.2184","created_at":"2026-05-18T04:08:43Z"},{"alias_kind":"pith_short_12","alias_value":"BUJI26HXU6O2","created_at":"2026-05-18T12:26:24Z"},{"alias_kind":"pith_short_16","alias_value":"BUJI26HXU6O2GABU","created_at":"2026-05-18T12:26:24Z"},{"alias_kind":"pith_short_8","alias_value":"BUJI26HX","created_at":"2026-05-18T12:26:24Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2011:BUJI26HXU6O2GABUPKIFKI7XCC","target":"record","payload":{"canonical_record":{"source":{"id":"1111.2184","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2011-11-09T12:09:19Z","cross_cats_sorted":[],"title_canon_sha256":"d4a0455821022823f2ab9dc6330a310480e92aaaa5e548b620c601e24376a7a6","abstract_canon_sha256":"be7cd14216522ac20d3ad6beab2e6d524fbfbb9a7cc7403f49a4f8e74e3909e8"},"schema_version":"1.0"},"canonical_sha256":"0d128d78f7a79da300347a905523f710bdd4a584e516b8b99fc82cbaffe48f0b","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:08:43.959169Z","signature_b64":"+cUtdKyO0ZoPObUTvE9oAonitATEDaRSlgIQ+xWR/LMcMTxfnylS/Y2UzD1+6buGdcyssks4ua5VIk4rN0DYBA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"0d128d78f7a79da300347a905523f710bdd4a584e516b8b99fc82cbaffe48f0b","last_reissued_at":"2026-05-18T04:08:43.958559Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:08:43.958559Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1111.2184","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":"2026-05-18T04:08:43Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"57GwcYFnzYEKVTwTNK74SSY5PMW7q0f2vt01f6f1yR5P0rPaeOp05+lPsiFeVC2WBTQWlxkDUxJZT6qFBOL7Ag==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-21T00:16:38.182032Z"},"content_sha256":"f4482e0ee928cc1dc6dc84e198c357285eeb474216a1c2302dfb180775cba699","schema_version":"1.0","event_id":"sha256:f4482e0ee928cc1dc6dc84e198c357285eeb474216a1c2302dfb180775cba699"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2011:BUJI26HXU6O2GABUPKIFKI7XCC","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Lower Ricci Curvature, Branching, and Bi-Lipschitz Structure of Uniform Reifenberg Spaces","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.DG","authors_text":"Aaron Naber, Tobias Holck Colding","submitted_at":"2011-11-09T12:09:19Z","abstract_excerpt":"We study here limit spaces $(M_\\alpha,g_\\alpha,p_\\alpha)\\stackrel{GH}{\\rightarrow} (Y,d_Y,p)$, where the $M_\\alpha$ have a lower Ricci curvature bound and are volume noncollapsed. Such limits $Y$ may be quite singular, however it is known that there is a subset of full measure $\\cR(Y)\\subseteq Y$, called {\\it regular} points, along with coverings by the almost regular points $\\cap_\\epsilon \\cup_r\\cR_{\\epsilon,r}(Y)=\\cR(Y)$ such that each of the {\\it Reifenberg sets} $\\cR_{\\epsilon,r}(Y)$ is bi-H\\\"older homeomorphic to a manifold. It has been an ongoing question as to the bi-Lipschitz regularit"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1111.2184","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":""},"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":"2026-05-18T04:08:43Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"IGK5KsRz8ygwHUfM0c7UdYZXnDAx08VKlcrNbm5R8o1vBWXPpTk9wS/CeY6nccHtWksvPDYbNZEkp9bYMjbQDg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-21T00:16:38.182398Z"},"content_sha256":"ba74c0009e0d5a43deb1dcf8b18dfe11be68c86561c5cc441b78efe3416058b8","schema_version":"1.0","event_id":"sha256:ba74c0009e0d5a43deb1dcf8b18dfe11be68c86561c5cc441b78efe3416058b8"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/BUJI26HXU6O2GABUPKIFKI7XCC/bundle.json","state_url":"https://pith.science/pith/BUJI26HXU6O2GABUPKIFKI7XCC/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/BUJI26HXU6O2GABUPKIFKI7XCC/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":"2026-06-21T00:16:38Z","links":{"resolver":"https://pith.science/pith/BUJI26HXU6O2GABUPKIFKI7XCC","bundle":"https://pith.science/pith/BUJI26HXU6O2GABUPKIFKI7XCC/bundle.json","state":"https://pith.science/pith/BUJI26HXU6O2GABUPKIFKI7XCC/state.json","well_known_bundle":"https://pith.science/.well-known/pith/BUJI26HXU6O2GABUPKIFKI7XCC/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2011:BUJI26HXU6O2GABUPKIFKI7XCC","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":"be7cd14216522ac20d3ad6beab2e6d524fbfbb9a7cc7403f49a4f8e74e3909e8","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2011-11-09T12:09:19Z","title_canon_sha256":"d4a0455821022823f2ab9dc6330a310480e92aaaa5e548b620c601e24376a7a6"},"schema_version":"1.0","source":{"id":"1111.2184","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1111.2184","created_at":"2026-05-18T04:08:43Z"},{"alias_kind":"arxiv_version","alias_value":"1111.2184v1","created_at":"2026-05-18T04:08:43Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1111.2184","created_at":"2026-05-18T04:08:43Z"},{"alias_kind":"pith_short_12","alias_value":"BUJI26HXU6O2","created_at":"2026-05-18T12:26:24Z"},{"alias_kind":"pith_short_16","alias_value":"BUJI26HXU6O2GABU","created_at":"2026-05-18T12:26:24Z"},{"alias_kind":"pith_short_8","alias_value":"BUJI26HX","created_at":"2026-05-18T12:26:24Z"}],"graph_snapshots":[{"event_id":"sha256:ba74c0009e0d5a43deb1dcf8b18dfe11be68c86561c5cc441b78efe3416058b8","target":"graph","created_at":"2026-05-18T04:08:43Z","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 study here limit spaces $(M_\\alpha,g_\\alpha,p_\\alpha)\\stackrel{GH}{\\rightarrow} (Y,d_Y,p)$, where the $M_\\alpha$ have a lower Ricci curvature bound and are volume noncollapsed. Such limits $Y$ may be quite singular, however it is known that there is a subset of full measure $\\cR(Y)\\subseteq Y$, called {\\it regular} points, along with coverings by the almost regular points $\\cap_\\epsilon \\cup_r\\cR_{\\epsilon,r}(Y)=\\cR(Y)$ such that each of the {\\it Reifenberg sets} $\\cR_{\\epsilon,r}(Y)$ is bi-H\\\"older homeomorphic to a manifold. It has been an ongoing question as to the bi-Lipschitz regularit","authors_text":"Aaron Naber, Tobias Holck Colding","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2011-11-09T12:09:19Z","title":"Lower Ricci Curvature, Branching, and Bi-Lipschitz Structure of Uniform Reifenberg Spaces"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1111.2184","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:f4482e0ee928cc1dc6dc84e198c357285eeb474216a1c2302dfb180775cba699","target":"record","created_at":"2026-05-18T04:08:43Z","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":"be7cd14216522ac20d3ad6beab2e6d524fbfbb9a7cc7403f49a4f8e74e3909e8","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2011-11-09T12:09:19Z","title_canon_sha256":"d4a0455821022823f2ab9dc6330a310480e92aaaa5e548b620c601e24376a7a6"},"schema_version":"1.0","source":{"id":"1111.2184","kind":"arxiv","version":1}},"canonical_sha256":"0d128d78f7a79da300347a905523f710bdd4a584e516b8b99fc82cbaffe48f0b","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"0d128d78f7a79da300347a905523f710bdd4a584e516b8b99fc82cbaffe48f0b","first_computed_at":"2026-05-18T04:08:43.958559Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T04:08:43.958559Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"+cUtdKyO0ZoPObUTvE9oAonitATEDaRSlgIQ+xWR/LMcMTxfnylS/Y2UzD1+6buGdcyssks4ua5VIk4rN0DYBA==","signature_status":"signed_v1","signed_at":"2026-05-18T04:08:43.959169Z","signed_message":"canonical_sha256_bytes"},"source_id":"1111.2184","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:f4482e0ee928cc1dc6dc84e198c357285eeb474216a1c2302dfb180775cba699","sha256:ba74c0009e0d5a43deb1dcf8b18dfe11be68c86561c5cc441b78efe3416058b8"],"state_sha256":"6efbb2c26c3ac344257db594e08ed3b2b6fda0d47c59b590ee6199c6ee4a1838"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"+/Y0R02KL7FQDi09WgCAGQujITnsJPkkC6AV/nC0KUM7kVXjlWHEkKXHhIm30vTZR7kwStpuRoSn4LA3OCqrBA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-21T00:16:38.184386Z","bundle_sha256":"6d675e0327c5ea85140e712e7cb93d8dae8cc1632da2d33b6ade4088cc260110"}}