{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2018:DAFPA4HQAXK23DAS63GSBMTKBT","short_pith_number":"pith:DAFPA4HQ","canonical_record":{"source":{"id":"1803.08484","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.MG","submitted_at":"2018-03-22T17:34:21Z","cross_cats_sorted":["math.PR"],"title_canon_sha256":"170eb20de1238a6f14268254a44819091e1137676c74fe198a61b41a3f0e0e09","abstract_canon_sha256":"c3f2e5027590a7f96dd5ad17d05655f2db9ac758f2c75c8ac123a2b187a315f5"},"schema_version":"1.0"},"canonical_sha256":"180af070f005d5ad8c12f6cd20b26a0cca53f0838f5b75f61e3e15c7a90cb5d5","source":{"kind":"arxiv","id":"1803.08484","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1803.08484","created_at":"2026-05-18T00:20:22Z"},{"alias_kind":"arxiv_version","alias_value":"1803.08484v1","created_at":"2026-05-18T00:20:22Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1803.08484","created_at":"2026-05-18T00:20:22Z"},{"alias_kind":"pith_short_12","alias_value":"DAFPA4HQAXK2","created_at":"2026-05-18T12:32:19Z"},{"alias_kind":"pith_short_16","alias_value":"DAFPA4HQAXK23DAS","created_at":"2026-05-18T12:32:19Z"},{"alias_kind":"pith_short_8","alias_value":"DAFPA4HQ","created_at":"2026-05-18T12:32:19Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2018:DAFPA4HQAXK23DAS63GSBMTKBT","target":"record","payload":{"canonical_record":{"source":{"id":"1803.08484","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.MG","submitted_at":"2018-03-22T17:34:21Z","cross_cats_sorted":["math.PR"],"title_canon_sha256":"170eb20de1238a6f14268254a44819091e1137676c74fe198a61b41a3f0e0e09","abstract_canon_sha256":"c3f2e5027590a7f96dd5ad17d05655f2db9ac758f2c75c8ac123a2b187a315f5"},"schema_version":"1.0"},"canonical_sha256":"180af070f005d5ad8c12f6cd20b26a0cca53f0838f5b75f61e3e15c7a90cb5d5","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:20:22.731396Z","signature_b64":"kpuVTUoKxOQuQ1I7mJma4zK0l6iXfZN43PeYIjOxenxOqWgkvj/4+oW7lB3E8gvRh65+4zhltaA7KkMHuRzHCA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"180af070f005d5ad8c12f6cd20b26a0cca53f0838f5b75f61e3e15c7a90cb5d5","last_reissued_at":"2026-05-18T00:20:22.730769Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:20:22.730769Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1803.08484","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":"2026-05-18T00:20:22Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"tcPGYkPDWoh21mcltQ+rVbTsV6HkAd60SIBsiDNlfoynMz3BmGnTZ9oTuS9ti95TB9rWmVyx3zUBND2rfXwJDQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-06T21:27:23.518433Z"},"content_sha256":"52037df98681aa8ab34c4dd8edc1750965673ae1cc1f0190e85362ee58c700e1","schema_version":"1.0","event_id":"sha256:52037df98681aa8ab34c4dd8edc1750965673ae1cc1f0190e85362ee58c700e1"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2018:DAFPA4HQAXK23DAS63GSBMTKBT","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Circumspheres of sets of n+1 random points in the d-dimensional Euclidean unit ball (0<n<d+1)","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.PR"],"primary_cat":"math.MG","authors_text":"G\\'erard Le Ca\\\"er","submitted_at":"2018-03-22T17:34:21Z","abstract_excerpt":"In the d dimensional Euclidean space, any set of n+1 independent random points, uniformly distributed in the interior of a unit ball of center O, determines almost surely a circumsphere of center C and of radius R, with n positive and less than d+1, and a n flat when n is positive and less than d. The projection of O on the n flat is named O'. The focus is set on circumspheres which are contained in this unit ball. For any d larger than 1 and any n positive and less than d, the joint probability density function of the distance D = O'C and of R has a simple closed form expression. Their margin"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1803.08484","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":"2026-05-18T00:20:22Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"SHxRxDNaaMNfD7N670oLspafRiGLnUlfXzgQK4/lprS3+Vk52Xp7fNT+l5knN1NViQ7sNwqws2k2uW3oyYaKCQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-06T21:27:23.519071Z"},"content_sha256":"68ba3562df1df958a21a7a24c5e268cd37a4008a05e61a01703194d9e1f49cca","schema_version":"1.0","event_id":"sha256:68ba3562df1df958a21a7a24c5e268cd37a4008a05e61a01703194d9e1f49cca"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/DAFPA4HQAXK23DAS63GSBMTKBT/bundle.json","state_url":"https://pith.science/pith/DAFPA4HQAXK23DAS63GSBMTKBT/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/DAFPA4HQAXK23DAS63GSBMTKBT/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":"2026-06-06T21:27:23Z","links":{"resolver":"https://pith.science/pith/DAFPA4HQAXK23DAS63GSBMTKBT","bundle":"https://pith.science/pith/DAFPA4HQAXK23DAS63GSBMTKBT/bundle.json","state":"https://pith.science/pith/DAFPA4HQAXK23DAS63GSBMTKBT/state.json","well_known_bundle":"https://pith.science/.well-known/pith/DAFPA4HQAXK23DAS63GSBMTKBT/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2018:DAFPA4HQAXK23DAS63GSBMTKBT","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":"c3f2e5027590a7f96dd5ad17d05655f2db9ac758f2c75c8ac123a2b187a315f5","cross_cats_sorted":["math.PR"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.MG","submitted_at":"2018-03-22T17:34:21Z","title_canon_sha256":"170eb20de1238a6f14268254a44819091e1137676c74fe198a61b41a3f0e0e09"},"schema_version":"1.0","source":{"id":"1803.08484","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1803.08484","created_at":"2026-05-18T00:20:22Z"},{"alias_kind":"arxiv_version","alias_value":"1803.08484v1","created_at":"2026-05-18T00:20:22Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1803.08484","created_at":"2026-05-18T00:20:22Z"},{"alias_kind":"pith_short_12","alias_value":"DAFPA4HQAXK2","created_at":"2026-05-18T12:32:19Z"},{"alias_kind":"pith_short_16","alias_value":"DAFPA4HQAXK23DAS","created_at":"2026-05-18T12:32:19Z"},{"alias_kind":"pith_short_8","alias_value":"DAFPA4HQ","created_at":"2026-05-18T12:32:19Z"}],"graph_snapshots":[{"event_id":"sha256:68ba3562df1df958a21a7a24c5e268cd37a4008a05e61a01703194d9e1f49cca","target":"graph","created_at":"2026-05-18T00:20:22Z","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":"In the d dimensional Euclidean space, any set of n+1 independent random points, uniformly distributed in the interior of a unit ball of center O, determines almost surely a circumsphere of center C and of radius R, with n positive and less than d+1, and a n flat when n is positive and less than d. The projection of O on the n flat is named O'. The focus is set on circumspheres which are contained in this unit ball. For any d larger than 1 and any n positive and less than d, the joint probability density function of the distance D = O'C and of R has a simple closed form expression. Their margin","authors_text":"G\\'erard Le Ca\\\"er","cross_cats":["math.PR"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.MG","submitted_at":"2018-03-22T17:34:21Z","title":"Circumspheres of sets of n+1 random points in the d-dimensional Euclidean unit ball (0<n<d+1)"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1803.08484","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":"sha256:52037df98681aa8ab34c4dd8edc1750965673ae1cc1f0190e85362ee58c700e1","target":"record","created_at":"2026-05-18T00:20:22Z","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":"c3f2e5027590a7f96dd5ad17d05655f2db9ac758f2c75c8ac123a2b187a315f5","cross_cats_sorted":["math.PR"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.MG","submitted_at":"2018-03-22T17:34:21Z","title_canon_sha256":"170eb20de1238a6f14268254a44819091e1137676c74fe198a61b41a3f0e0e09"},"schema_version":"1.0","source":{"id":"1803.08484","kind":"arxiv","version":1}},"canonical_sha256":"180af070f005d5ad8c12f6cd20b26a0cca53f0838f5b75f61e3e15c7a90cb5d5","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"180af070f005d5ad8c12f6cd20b26a0cca53f0838f5b75f61e3e15c7a90cb5d5","first_computed_at":"2026-05-18T00:20:22.730769Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:20:22.730769Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"kpuVTUoKxOQuQ1I7mJma4zK0l6iXfZN43PeYIjOxenxOqWgkvj/4+oW7lB3E8gvRh65+4zhltaA7KkMHuRzHCA==","signature_status":"signed_v1","signed_at":"2026-05-18T00:20:22.731396Z","signed_message":"canonical_sha256_bytes"},"source_id":"1803.08484","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:52037df98681aa8ab34c4dd8edc1750965673ae1cc1f0190e85362ee58c700e1","sha256:68ba3562df1df958a21a7a24c5e268cd37a4008a05e61a01703194d9e1f49cca"],"state_sha256":"246f7a2e7c009cab6a56ff7851b937d895a135906ac06e7aa96aab24296ab865"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"wT0ywJ4gQpj1k7+bTFFWaoM2NbLxvsDpEdNgJCdrJXSvmeqzSaZI0pVYVNgsooCYA9WG2e6TIcoIBzDuAvwwAw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-06T21:27:23.522243Z","bundle_sha256":"e87b8e7b25555982f0dd4a396a13c7515df13d55824369feed673806130f0e16"}}