{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2004:OBKY44UBQPN65PV6ASLVBWPPTW","short_pith_number":"pith:OBKY44UB","canonical_record":{"source":{"id":"math/0410168","kind":"arxiv","version":1},"metadata":{"license":"","primary_cat":"math.PR","submitted_at":"2004-10-06T15:37:52Z","cross_cats_sorted":[],"title_canon_sha256":"0ba2300c52808ec55c41f6cd6c83415c8b8b6bed2d1fdee544257dc05b0b3a42","abstract_canon_sha256":"61bff6a634bf4e74fd4c9173fa864441693dd9c34e2352d2c9a55216050c4561"},"schema_version":"1.0"},"canonical_sha256":"70558e728183dbeebebe049750d9ef9da2b0b9485163e8d67d65fd8f001a49ac","source":{"kind":"arxiv","id":"math/0410168","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"math/0410168","created_at":"2026-05-18T01:05:25Z"},{"alias_kind":"arxiv_version","alias_value":"math/0410168v1","created_at":"2026-05-18T01:05:25Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.math/0410168","created_at":"2026-05-18T01:05:25Z"},{"alias_kind":"pith_short_12","alias_value":"OBKY44UBQPN6","created_at":"2026-05-18T12:25:52Z"},{"alias_kind":"pith_short_16","alias_value":"OBKY44UBQPN65PV6","created_at":"2026-05-18T12:25:52Z"},{"alias_kind":"pith_short_8","alias_value":"OBKY44UB","created_at":"2026-05-18T12:25:52Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2004:OBKY44UBQPN65PV6ASLVBWPPTW","target":"record","payload":{"canonical_record":{"source":{"id":"math/0410168","kind":"arxiv","version":1},"metadata":{"license":"","primary_cat":"math.PR","submitted_at":"2004-10-06T15:37:52Z","cross_cats_sorted":[],"title_canon_sha256":"0ba2300c52808ec55c41f6cd6c83415c8b8b6bed2d1fdee544257dc05b0b3a42","abstract_canon_sha256":"61bff6a634bf4e74fd4c9173fa864441693dd9c34e2352d2c9a55216050c4561"},"schema_version":"1.0"},"canonical_sha256":"70558e728183dbeebebe049750d9ef9da2b0b9485163e8d67d65fd8f001a49ac","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:05:25.782106Z","signature_b64":"O3Lf9bPDJC1XVccyECATP7oerxZvNxTu1Ox0cQ4Yy69joHrE3lsS4uJdI7hQhLnSib4zJBqPMKSUieOTKIGCCw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"70558e728183dbeebebe049750d9ef9da2b0b9485163e8d67d65fd8f001a49ac","last_reissued_at":"2026-05-18T01:05:25.781435Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:05:25.781435Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"math/0410168","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":"2026-05-18T01:05:25Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"gclHkNz85P/RaUumS0mEQO+w8Ijf5F5jP2s5zLUNM5unZpu4gQnZgwemej1z0hWzFk1eTT7Lo5Q7lZyyKVQ7AA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-07-05T03:21:44.220182Z"},"content_sha256":"2497e2db65300f0c60cd864ef5fec271710496a5a6fadf4b6083e0035fba9243","schema_version":"1.0","event_id":"sha256:2497e2db65300f0c60cd864ef5fec271710496a5a6fadf4b6083e0035fba9243"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2004:OBKY44UBQPN65PV6ASLVBWPPTW","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Measure concentration for Euclidean distance in the case of dependent random variables","license":"","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Katalin Marton","submitted_at":"2004-10-06T15:37:52Z","abstract_excerpt":"Let q^n be a continuous density function in n-dimensional Euclidean space.\n We think of q^n as the density function of some random sequence X^n with values in \\BbbR^n. For I\\subset[1,n], let X_I denote the collection of coordinates X_i, i\\in I, and let \\bar X_I denote the collection of coordinates\n X_i, i\\notin I. We denote by Q_I(x_I|\\bar x_I) the joint conditional density function of X_I, given \\bar X_I. We prove measure concentration for q^n in the case when, for an appropriate class of sets I, (i) the conditional densities Q_I(x_I|\\bar x_I), as functions of x_I, uniformly satisfy a logarit"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math/0410168","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":"2026-05-18T01:05:25Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"XQDSaXtiX8/fs0qWIbSLd09+5yjNj0kpW7JyYRNo/7wZOi0WNiu35HWtRyywVRu5No8oG7zalL+9B4uInPNuAg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-07-05T03:21:44.220540Z"},"content_sha256":"f89d09b08702829a34ad5246a26b11b651c055fe4272badca7e00fe71892dc70","schema_version":"1.0","event_id":"sha256:f89d09b08702829a34ad5246a26b11b651c055fe4272badca7e00fe71892dc70"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/OBKY44UBQPN65PV6ASLVBWPPTW/bundle.json","state_url":"https://pith.science/pith/OBKY44UBQPN65PV6ASLVBWPPTW/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/OBKY44UBQPN65PV6ASLVBWPPTW/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":"2026-07-05T03:21:44Z","links":{"resolver":"https://pith.science/pith/OBKY44UBQPN65PV6ASLVBWPPTW","bundle":"https://pith.science/pith/OBKY44UBQPN65PV6ASLVBWPPTW/bundle.json","state":"https://pith.science/pith/OBKY44UBQPN65PV6ASLVBWPPTW/state.json","well_known_bundle":"https://pith.science/.well-known/pith/OBKY44UBQPN65PV6ASLVBWPPTW/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2004:OBKY44UBQPN65PV6ASLVBWPPTW","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":"61bff6a634bf4e74fd4c9173fa864441693dd9c34e2352d2c9a55216050c4561","cross_cats_sorted":[],"license":"","primary_cat":"math.PR","submitted_at":"2004-10-06T15:37:52Z","title_canon_sha256":"0ba2300c52808ec55c41f6cd6c83415c8b8b6bed2d1fdee544257dc05b0b3a42"},"schema_version":"1.0","source":{"id":"math/0410168","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"math/0410168","created_at":"2026-05-18T01:05:25Z"},{"alias_kind":"arxiv_version","alias_value":"math/0410168v1","created_at":"2026-05-18T01:05:25Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.math/0410168","created_at":"2026-05-18T01:05:25Z"},{"alias_kind":"pith_short_12","alias_value":"OBKY44UBQPN6","created_at":"2026-05-18T12:25:52Z"},{"alias_kind":"pith_short_16","alias_value":"OBKY44UBQPN65PV6","created_at":"2026-05-18T12:25:52Z"},{"alias_kind":"pith_short_8","alias_value":"OBKY44UB","created_at":"2026-05-18T12:25:52Z"}],"graph_snapshots":[{"event_id":"sha256:f89d09b08702829a34ad5246a26b11b651c055fe4272badca7e00fe71892dc70","target":"graph","created_at":"2026-05-18T01:05:25Z","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":"Let q^n be a continuous density function in n-dimensional Euclidean space.\n We think of q^n as the density function of some random sequence X^n with values in \\BbbR^n. For I\\subset[1,n], let X_I denote the collection of coordinates X_i, i\\in I, and let \\bar X_I denote the collection of coordinates\n X_i, i\\notin I. We denote by Q_I(x_I|\\bar x_I) the joint conditional density function of X_I, given \\bar X_I. We prove measure concentration for q^n in the case when, for an appropriate class of sets I, (i) the conditional densities Q_I(x_I|\\bar x_I), as functions of x_I, uniformly satisfy a logarit","authors_text":"Katalin Marton","cross_cats":[],"headline":"","license":"","primary_cat":"math.PR","submitted_at":"2004-10-06T15:37:52Z","title":"Measure concentration for Euclidean distance in the case of dependent random variables"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math/0410168","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":"sha256:2497e2db65300f0c60cd864ef5fec271710496a5a6fadf4b6083e0035fba9243","target":"record","created_at":"2026-05-18T01:05:25Z","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":"61bff6a634bf4e74fd4c9173fa864441693dd9c34e2352d2c9a55216050c4561","cross_cats_sorted":[],"license":"","primary_cat":"math.PR","submitted_at":"2004-10-06T15:37:52Z","title_canon_sha256":"0ba2300c52808ec55c41f6cd6c83415c8b8b6bed2d1fdee544257dc05b0b3a42"},"schema_version":"1.0","source":{"id":"math/0410168","kind":"arxiv","version":1}},"canonical_sha256":"70558e728183dbeebebe049750d9ef9da2b0b9485163e8d67d65fd8f001a49ac","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"70558e728183dbeebebe049750d9ef9da2b0b9485163e8d67d65fd8f001a49ac","first_computed_at":"2026-05-18T01:05:25.781435Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:05:25.781435Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"O3Lf9bPDJC1XVccyECATP7oerxZvNxTu1Ox0cQ4Yy69joHrE3lsS4uJdI7hQhLnSib4zJBqPMKSUieOTKIGCCw==","signature_status":"signed_v1","signed_at":"2026-05-18T01:05:25.782106Z","signed_message":"canonical_sha256_bytes"},"source_id":"math/0410168","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:2497e2db65300f0c60cd864ef5fec271710496a5a6fadf4b6083e0035fba9243","sha256:f89d09b08702829a34ad5246a26b11b651c055fe4272badca7e00fe71892dc70"],"state_sha256":"90e2f7a1bcb96128662a2e62a12111c67ae2e36b63a4355de0d31d498f41059c"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"8w3sBZVO1H9r/TNfMAq6FNlM1i8GKcnrCg5fSGbNT7rUuuAMi7qj9T/+OOD8WKJWz8BtDcx9MwLky4PViwGGAg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-07-05T03:21:44.222426Z","bundle_sha256":"10d024c0d4bb986459590bc19af5cbc2a83f311c8e2274fb9eb3fa738dea7c8d"}}