{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2013:BU4ML7SWWYU65YITWSI5YM5KVR","short_pith_number":"pith:BU4ML7SW","canonical_record":{"source":{"id":"1305.2810","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2013-05-13T15:24:50Z","cross_cats_sorted":["math.AP"],"title_canon_sha256":"48b993d252ee7f71d7965ec697d803746751d5d1cf4bfbc66ec5d1cb6c9d67f8","abstract_canon_sha256":"b9765fd0746d79a8d621cfd48ba82fa6df020755db6e5d23bbfedeecd57cea83"},"schema_version":"1.0"},"canonical_sha256":"0d38c5fe56b629eee113b491dc33aaac693e9402f4df12ae9e72a9f71ebf6900","source":{"kind":"arxiv","id":"1305.2810","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1305.2810","created_at":"2026-05-18T00:36:41Z"},{"alias_kind":"arxiv_version","alias_value":"1305.2810v1","created_at":"2026-05-18T00:36:41Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1305.2810","created_at":"2026-05-18T00:36:41Z"},{"alias_kind":"pith_short_12","alias_value":"BU4ML7SWWYU6","created_at":"2026-05-18T12:27:40Z"},{"alias_kind":"pith_short_16","alias_value":"BU4ML7SWWYU65YIT","created_at":"2026-05-18T12:27:40Z"},{"alias_kind":"pith_short_8","alias_value":"BU4ML7SW","created_at":"2026-05-18T12:27:40Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2013:BU4ML7SWWYU65YITWSI5YM5KVR","target":"record","payload":{"canonical_record":{"source":{"id":"1305.2810","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2013-05-13T15:24:50Z","cross_cats_sorted":["math.AP"],"title_canon_sha256":"48b993d252ee7f71d7965ec697d803746751d5d1cf4bfbc66ec5d1cb6c9d67f8","abstract_canon_sha256":"b9765fd0746d79a8d621cfd48ba82fa6df020755db6e5d23bbfedeecd57cea83"},"schema_version":"1.0"},"canonical_sha256":"0d38c5fe56b629eee113b491dc33aaac693e9402f4df12ae9e72a9f71ebf6900","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:36:41.865624Z","signature_b64":"hicpGCL2j7I99wQR5kN8TyhWuPTPYQWr6I/2u1DahbuercZmtWy7IhZDwjz11VmUnVbUo2gQ4EiVBzJsWuqcDg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"0d38c5fe56b629eee113b491dc33aaac693e9402f4df12ae9e72a9f71ebf6900","last_reissued_at":"2026-05-18T00:36:41.864870Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:36:41.864870Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1305.2810","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":"2026-05-18T00:36:41Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"FZrnBfAUJcfCTTyEJEJib0cs/nah80810c4KFGOOIVGH5wO8NUqYdLVsHCPazW8kRiDgyvjs0ZBvYN+x9JTJAw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-09T01:01:26.114843Z"},"content_sha256":"658ba9a01ddf0a38187cd1c92cde0132b088a4d814224ece591818278b4d920a","schema_version":"1.0","event_id":"sha256:658ba9a01ddf0a38187cd1c92cde0132b088a4d814224ece591818278b4d920a"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2013:BU4ML7SWWYU65YITWSI5YM5KVR","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"On the characterization of some classes of proximally smooth sets","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AP"],"primary_cat":"math.CA","authors_text":"Graziano Crasta, Ilaria Fragal\\`a","submitted_at":"2013-05-13T15:24:50Z","abstract_excerpt":"We provide a complete characterization of closed sets with empty interior and positive reach in $\\mathbb{R}^2$. As a consequence, we characterize open bounded domains in $\\mathbb{R}^2$ whose high ridge and cut locus agree, and hence $C^1$ planar domains whose normal distance to the cut locus is constant along the boundary. The latter results extends to convex domains in $\\mathbb{R}^n$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1305.2810","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":"2026-05-18T00:36:41Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"bV1X/zprbdFtkVO0YEEhkordMuqkUI29nu0T4qjPW+EAd+vNByKZd4ec3x7dCq7iVWMdwKPGzmRNu4zyWefHAw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-09T01:01:26.115187Z"},"content_sha256":"54706484b5920c4d0952c5ed6ecc621b22d0370f06a3f5f85350b8e9c53b0ba7","schema_version":"1.0","event_id":"sha256:54706484b5920c4d0952c5ed6ecc621b22d0370f06a3f5f85350b8e9c53b0ba7"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/BU4ML7SWWYU65YITWSI5YM5KVR/bundle.json","state_url":"https://pith.science/pith/BU4ML7SWWYU65YITWSI5YM5KVR/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/BU4ML7SWWYU65YITWSI5YM5KVR/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":"2026-06-09T01:01:26Z","links":{"resolver":"https://pith.science/pith/BU4ML7SWWYU65YITWSI5YM5KVR","bundle":"https://pith.science/pith/BU4ML7SWWYU65YITWSI5YM5KVR/bundle.json","state":"https://pith.science/pith/BU4ML7SWWYU65YITWSI5YM5KVR/state.json","well_known_bundle":"https://pith.science/.well-known/pith/BU4ML7SWWYU65YITWSI5YM5KVR/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2013:BU4ML7SWWYU65YITWSI5YM5KVR","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":"b9765fd0746d79a8d621cfd48ba82fa6df020755db6e5d23bbfedeecd57cea83","cross_cats_sorted":["math.AP"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2013-05-13T15:24:50Z","title_canon_sha256":"48b993d252ee7f71d7965ec697d803746751d5d1cf4bfbc66ec5d1cb6c9d67f8"},"schema_version":"1.0","source":{"id":"1305.2810","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1305.2810","created_at":"2026-05-18T00:36:41Z"},{"alias_kind":"arxiv_version","alias_value":"1305.2810v1","created_at":"2026-05-18T00:36:41Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1305.2810","created_at":"2026-05-18T00:36:41Z"},{"alias_kind":"pith_short_12","alias_value":"BU4ML7SWWYU6","created_at":"2026-05-18T12:27:40Z"},{"alias_kind":"pith_short_16","alias_value":"BU4ML7SWWYU65YIT","created_at":"2026-05-18T12:27:40Z"},{"alias_kind":"pith_short_8","alias_value":"BU4ML7SW","created_at":"2026-05-18T12:27:40Z"}],"graph_snapshots":[{"event_id":"sha256:54706484b5920c4d0952c5ed6ecc621b22d0370f06a3f5f85350b8e9c53b0ba7","target":"graph","created_at":"2026-05-18T00:36:41Z","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 provide a complete characterization of closed sets with empty interior and positive reach in $\\mathbb{R}^2$. As a consequence, we characterize open bounded domains in $\\mathbb{R}^2$ whose high ridge and cut locus agree, and hence $C^1$ planar domains whose normal distance to the cut locus is constant along the boundary. The latter results extends to convex domains in $\\mathbb{R}^n$.","authors_text":"Graziano Crasta, Ilaria Fragal\\`a","cross_cats":["math.AP"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2013-05-13T15:24:50Z","title":"On the characterization of some classes of proximally smooth sets"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1305.2810","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":"sha256:658ba9a01ddf0a38187cd1c92cde0132b088a4d814224ece591818278b4d920a","target":"record","created_at":"2026-05-18T00:36:41Z","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":"b9765fd0746d79a8d621cfd48ba82fa6df020755db6e5d23bbfedeecd57cea83","cross_cats_sorted":["math.AP"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2013-05-13T15:24:50Z","title_canon_sha256":"48b993d252ee7f71d7965ec697d803746751d5d1cf4bfbc66ec5d1cb6c9d67f8"},"schema_version":"1.0","source":{"id":"1305.2810","kind":"arxiv","version":1}},"canonical_sha256":"0d38c5fe56b629eee113b491dc33aaac693e9402f4df12ae9e72a9f71ebf6900","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"0d38c5fe56b629eee113b491dc33aaac693e9402f4df12ae9e72a9f71ebf6900","first_computed_at":"2026-05-18T00:36:41.864870Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:36:41.864870Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"hicpGCL2j7I99wQR5kN8TyhWuPTPYQWr6I/2u1DahbuercZmtWy7IhZDwjz11VmUnVbUo2gQ4EiVBzJsWuqcDg==","signature_status":"signed_v1","signed_at":"2026-05-18T00:36:41.865624Z","signed_message":"canonical_sha256_bytes"},"source_id":"1305.2810","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:658ba9a01ddf0a38187cd1c92cde0132b088a4d814224ece591818278b4d920a","sha256:54706484b5920c4d0952c5ed6ecc621b22d0370f06a3f5f85350b8e9c53b0ba7"],"state_sha256":"95069cab232d7f635bec09ba46e5f1e86784675deebc72a6cb6daabb894ccefb"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"yksa7BkZ+05ae3HRW2mrAA77rM7OIob9qdKCy2JFl6cKTTqUjWkzaoNUCSe242mf3SJell5lPKSkkMkd1Ay5Cg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-09T01:01:26.117208Z","bundle_sha256":"5c59ba09b82a110b925c4f3b5327f8702c60e6254590479795ca653b87e668c1"}}