{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2019:SLNOIM5WWZWA2YBSJZDZ4TDU4C","short_pith_number":"pith:SLNOIM5W","canonical_record":{"source":{"id":"1912.03615","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2019-12-08T04:51:47Z","cross_cats_sorted":["math.MG"],"title_canon_sha256":"52c7f7dcf9a4890ca9f33d602eddb8b026f1a5a5992540523423c797905c131a","abstract_canon_sha256":"78c7936f76e365f3e1b3f485e9442f759e52659b79c4fe5b81be5dcfd8ea3a4b"},"schema_version":"1.0"},"canonical_sha256":"92dae433b6b66c0d60324e479e4c74e090a9313856bdef91442c8d89b6f32818","source":{"kind":"arxiv","id":"1912.03615","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1912.03615","created_at":"2026-07-05T00:24:47Z"},{"alias_kind":"arxiv_version","alias_value":"1912.03615v1","created_at":"2026-07-05T00:24:47Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1912.03615","created_at":"2026-07-05T00:24:47Z"},{"alias_kind":"pith_short_12","alias_value":"SLNOIM5WWZWA","created_at":"2026-07-05T00:24:47Z"},{"alias_kind":"pith_short_16","alias_value":"SLNOIM5WWZWA2YBS","created_at":"2026-07-05T00:24:47Z"},{"alias_kind":"pith_short_8","alias_value":"SLNOIM5W","created_at":"2026-07-05T00:24:47Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2019:SLNOIM5WWZWA2YBSJZDZ4TDU4C","target":"record","payload":{"canonical_record":{"source":{"id":"1912.03615","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2019-12-08T04:51:47Z","cross_cats_sorted":["math.MG"],"title_canon_sha256":"52c7f7dcf9a4890ca9f33d602eddb8b026f1a5a5992540523423c797905c131a","abstract_canon_sha256":"78c7936f76e365f3e1b3f485e9442f759e52659b79c4fe5b81be5dcfd8ea3a4b"},"schema_version":"1.0"},"canonical_sha256":"92dae433b6b66c0d60324e479e4c74e090a9313856bdef91442c8d89b6f32818","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T00:24:47.117978Z","signature_b64":"oc4S8Pfy3thqu5udTS/nG/crCjbzKBIwG8WU6hirdUTm1S1++Bq81O/+VKwA27Z3lGjU0IuQmkZ9yWNAfO1FBA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"92dae433b6b66c0d60324e479e4c74e090a9313856bdef91442c8d89b6f32818","last_reissued_at":"2026-07-05T00:24:47.117466Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T00:24:47.117466Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1912.03615","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":"2026-07-05T00:24:47Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"JD88wamu2uoRsRqRehu2WlxXweUGE7vNBXrwc8YQ7FajlVV1wdFlfp1OHtZA1nzMl21SfyEUSwDMUezPkeKvCw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-06T06:58:07.410617Z"},"content_sha256":"0045f5cb58bb255ed32f78e76ce278ec798f4f73e8ce884c311df76e7464addb","schema_version":"1.0","event_id":"sha256:0045f5cb58bb255ed32f78e76ce278ec798f4f73e8ce884c311df76e7464addb"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2019:SLNOIM5WWZWA2YBSJZDZ4TDU4C","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Quantitative Estimates on the Singular Sets of Alexandrov Spaces","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.MG"],"primary_cat":"math.DG","authors_text":"Aaron Naber, Nan Li","submitted_at":"2019-12-08T04:51:47Z","abstract_excerpt":"Let $X\\in\\text{Alex}\\,^n(-1)$ be an $n$-dimensional Alexandrov space with curvature $\\ge -1$. Let the $r$-scale $(k,\\epsilon)$-singular set $\\mathcal S^k_{\\epsilon,\\,r}(X)$ be the collection of $x\\in X$ so that $B_r(x)$ is not $\\epsilon r$-close to a ball in any splitting space $\\mathbb R^{k+1}\\times Z$. We show that there exists $C(n,\\epsilon)>0$ and $\\beta(n,\\epsilon)>0$, independent of the volume, so that for any disjoint collection $\\big\\{B_{r_i}(x_i):x_i\\in \\mathcal S_{\\epsilon,\\,\\beta r_i}^k(X)\\cap B_1, \\,r_i\\le 1\\big\\}$, the packing estimate $\\sum r_i^k\\le C$ holds. Consequently, we obt"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1912.03615","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/1912.03615/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":"2026-07-05T00:24:47Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"IMKvcWWuXfuCdjZoDNA/+wbfTpCor9BteEFNI2HcyZwmkOkBVAFTOoA8QxHZpy0gwZZLtcQgCWOerLddXJ4KCw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-06T06:58:07.411006Z"},"content_sha256":"9b70d5ce2b43fafcf49696e1dd3d2ab76fa12275d486a1232936797df1ec773a","schema_version":"1.0","event_id":"sha256:9b70d5ce2b43fafcf49696e1dd3d2ab76fa12275d486a1232936797df1ec773a"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/SLNOIM5WWZWA2YBSJZDZ4TDU4C/bundle.json","state_url":"https://pith.science/pith/SLNOIM5WWZWA2YBSJZDZ4TDU4C/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/SLNOIM5WWZWA2YBSJZDZ4TDU4C/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":"2026-08-06T06:58:07Z","links":{"resolver":"https://pith.science/pith/SLNOIM5WWZWA2YBSJZDZ4TDU4C","bundle":"https://pith.science/pith/SLNOIM5WWZWA2YBSJZDZ4TDU4C/bundle.json","state":"https://pith.science/pith/SLNOIM5WWZWA2YBSJZDZ4TDU4C/state.json","well_known_bundle":"https://pith.science/.well-known/pith/SLNOIM5WWZWA2YBSJZDZ4TDU4C/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2019:SLNOIM5WWZWA2YBSJZDZ4TDU4C","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":"78c7936f76e365f3e1b3f485e9442f759e52659b79c4fe5b81be5dcfd8ea3a4b","cross_cats_sorted":["math.MG"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2019-12-08T04:51:47Z","title_canon_sha256":"52c7f7dcf9a4890ca9f33d602eddb8b026f1a5a5992540523423c797905c131a"},"schema_version":"1.0","source":{"id":"1912.03615","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1912.03615","created_at":"2026-07-05T00:24:47Z"},{"alias_kind":"arxiv_version","alias_value":"1912.03615v1","created_at":"2026-07-05T00:24:47Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1912.03615","created_at":"2026-07-05T00:24:47Z"},{"alias_kind":"pith_short_12","alias_value":"SLNOIM5WWZWA","created_at":"2026-07-05T00:24:47Z"},{"alias_kind":"pith_short_16","alias_value":"SLNOIM5WWZWA2YBS","created_at":"2026-07-05T00:24:47Z"},{"alias_kind":"pith_short_8","alias_value":"SLNOIM5W","created_at":"2026-07-05T00:24:47Z"}],"graph_snapshots":[{"event_id":"sha256:9b70d5ce2b43fafcf49696e1dd3d2ab76fa12275d486a1232936797df1ec773a","target":"graph","created_at":"2026-07-05T00:24:47Z","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/1912.03615/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"Let $X\\in\\text{Alex}\\,^n(-1)$ be an $n$-dimensional Alexandrov space with curvature $\\ge -1$. Let the $r$-scale $(k,\\epsilon)$-singular set $\\mathcal S^k_{\\epsilon,\\,r}(X)$ be the collection of $x\\in X$ so that $B_r(x)$ is not $\\epsilon r$-close to a ball in any splitting space $\\mathbb R^{k+1}\\times Z$. We show that there exists $C(n,\\epsilon)>0$ and $\\beta(n,\\epsilon)>0$, independent of the volume, so that for any disjoint collection $\\big\\{B_{r_i}(x_i):x_i\\in \\mathcal S_{\\epsilon,\\,\\beta r_i}^k(X)\\cap B_1, \\,r_i\\le 1\\big\\}$, the packing estimate $\\sum r_i^k\\le C$ holds. Consequently, we obt","authors_text":"Aaron Naber, Nan Li","cross_cats":["math.MG"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2019-12-08T04:51:47Z","title":"Quantitative Estimates on the Singular Sets of Alexandrov Spaces"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1912.03615","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":"sha256:0045f5cb58bb255ed32f78e76ce278ec798f4f73e8ce884c311df76e7464addb","target":"record","created_at":"2026-07-05T00:24:47Z","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":"78c7936f76e365f3e1b3f485e9442f759e52659b79c4fe5b81be5dcfd8ea3a4b","cross_cats_sorted":["math.MG"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2019-12-08T04:51:47Z","title_canon_sha256":"52c7f7dcf9a4890ca9f33d602eddb8b026f1a5a5992540523423c797905c131a"},"schema_version":"1.0","source":{"id":"1912.03615","kind":"arxiv","version":1}},"canonical_sha256":"92dae433b6b66c0d60324e479e4c74e090a9313856bdef91442c8d89b6f32818","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"92dae433b6b66c0d60324e479e4c74e090a9313856bdef91442c8d89b6f32818","first_computed_at":"2026-07-05T00:24:47.117466Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T00:24:47.117466Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"oc4S8Pfy3thqu5udTS/nG/crCjbzKBIwG8WU6hirdUTm1S1++Bq81O/+VKwA27Z3lGjU0IuQmkZ9yWNAfO1FBA==","signature_status":"signed_v1","signed_at":"2026-07-05T00:24:47.117978Z","signed_message":"canonical_sha256_bytes"},"source_id":"1912.03615","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:0045f5cb58bb255ed32f78e76ce278ec798f4f73e8ce884c311df76e7464addb","sha256:9b70d5ce2b43fafcf49696e1dd3d2ab76fa12275d486a1232936797df1ec773a"],"state_sha256":"0681f55557f6c15d0d86b472c116ffdd730abc07191b376fda97849532cac292"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"datvUEve9ERVlcC7Qv4hrybcOOD5PV/KkWDJiAmtfA+fTWIQJqN3TFYusXauU94Two3wuwyp/m5K2r4HsEAyDQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-06T06:58:07.414591Z","bundle_sha256":"d7e86e6f2269296a8f7120e96eda40a89fdf081ce050ea1f9fa87abcbe85cd30"}}