{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2013:6TYZCJELXIOPYDB76DRY7RBQNW","short_pith_number":"pith:6TYZCJEL","canonical_record":{"source":{"id":"1302.5828","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2013-02-23T17:30:11Z","cross_cats_sorted":["math.CO","math.OC","math.PR"],"title_canon_sha256":"7eb16f4315fc6d9253467dc823a0493075b39f60003f5e88e5ce83c14c6a8065","abstract_canon_sha256":"1600bddf664f3df6dda46c83249571823fa3ac65e97f54a677774f776cac41c9"},"schema_version":"1.0"},"canonical_sha256":"f4f191248bba1cfc0c3ff0e38fc4306dbee400e257a42175304863991c8969d6","source":{"kind":"arxiv","id":"1302.5828","version":3},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1302.5828","created_at":"2026-05-18T03:04:24Z"},{"alias_kind":"arxiv_version","alias_value":"1302.5828v3","created_at":"2026-05-18T03:04:24Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1302.5828","created_at":"2026-05-18T03:04:24Z"},{"alias_kind":"pith_short_12","alias_value":"6TYZCJELXIOP","created_at":"2026-05-18T12:27:36Z"},{"alias_kind":"pith_short_16","alias_value":"6TYZCJELXIOPYDB7","created_at":"2026-05-18T12:27:36Z"},{"alias_kind":"pith_short_8","alias_value":"6TYZCJEL","created_at":"2026-05-18T12:27:36Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2013:6TYZCJELXIOPYDB76DRY7RBQNW","target":"record","payload":{"canonical_record":{"source":{"id":"1302.5828","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2013-02-23T17:30:11Z","cross_cats_sorted":["math.CO","math.OC","math.PR"],"title_canon_sha256":"7eb16f4315fc6d9253467dc823a0493075b39f60003f5e88e5ce83c14c6a8065","abstract_canon_sha256":"1600bddf664f3df6dda46c83249571823fa3ac65e97f54a677774f776cac41c9"},"schema_version":"1.0"},"canonical_sha256":"f4f191248bba1cfc0c3ff0e38fc4306dbee400e257a42175304863991c8969d6","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:04:24.243823Z","signature_b64":"Y/0hnNTM53YAo0o8M+1AjLviq1j2R/fA0TbPQ5ECvYCk9Y0HnD8aKWZsP2TxE9a5AHpWONkLUaUMlWEEk0KoAw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"f4f191248bba1cfc0c3ff0e38fc4306dbee400e257a42175304863991c8969d6","last_reissued_at":"2026-05-18T03:04:24.243484Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:04:24.243484Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1302.5828","source_version":3,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T03:04:24Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"K2z0DwTHS6GMYMafCr/TP0kBep4rYSto6W5Lg2qpKKsN4hnt+8ProOhQ7p79ijF+hgD9Unz3Ss8kUUn8XG1DBA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-23T12:23:26.346838Z"},"content_sha256":"6d36b3ea4440d5bd1cc9159460406a6835844f5030856514f2de4113a41503e3","schema_version":"1.0","event_id":"sha256:6d36b3ea4440d5bd1cc9159460406a6835844f5030856514f2de4113a41503e3"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2013:6TYZCJELXIOPYDB76DRY7RBQNW","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"A Hamilton-Jacobi equation for the continuum limit of non-dominated sorting","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO","math.OC","math.PR"],"primary_cat":"math.AP","authors_text":"Alfred O. Hero, Jeff Calder, Selim Esedo\\=glu","submitted_at":"2013-02-23T17:30:11Z","abstract_excerpt":"We show that non-dominated sorting of a sequence of i.i.d. random variables in Euclidean space has a continuum limit that corresponds to solving a Hamilton-Jacobi equation involving the probability density function of the random variables. Non-dominated sorting is a fundamental problem in multi-objective optimization, and is equivalent to finding the canonical antichain partition and to problems involving the longest chain among Euclidean points. As an application of this result, we show that non-dominated sorting is asymptotically stable under random perturbations in the data. We give a numer"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1302.5828","kind":"arxiv","version":3},"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":"2026-05-18T03:04:24Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"zDHTa+/LuBgUPdugRPukv+ddgt/kMcDzunjgr/zC3gWKIK4DBsW8DjCdRk5R2aZqBgfLEgL7zaN3f+anCaeZDw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-23T12:23:26.347187Z"},"content_sha256":"1da8a1c92b0fb8544fd21964d36641ee11f06882d55e241b3d65d8a069cf79a4","schema_version":"1.0","event_id":"sha256:1da8a1c92b0fb8544fd21964d36641ee11f06882d55e241b3d65d8a069cf79a4"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/6TYZCJELXIOPYDB76DRY7RBQNW/bundle.json","state_url":"https://pith.science/pith/6TYZCJELXIOPYDB76DRY7RBQNW/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/6TYZCJELXIOPYDB76DRY7RBQNW/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":"2026-06-23T12:23:26Z","links":{"resolver":"https://pith.science/pith/6TYZCJELXIOPYDB76DRY7RBQNW","bundle":"https://pith.science/pith/6TYZCJELXIOPYDB76DRY7RBQNW/bundle.json","state":"https://pith.science/pith/6TYZCJELXIOPYDB76DRY7RBQNW/state.json","well_known_bundle":"https://pith.science/.well-known/pith/6TYZCJELXIOPYDB76DRY7RBQNW/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2013:6TYZCJELXIOPYDB76DRY7RBQNW","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":"1600bddf664f3df6dda46c83249571823fa3ac65e97f54a677774f776cac41c9","cross_cats_sorted":["math.CO","math.OC","math.PR"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2013-02-23T17:30:11Z","title_canon_sha256":"7eb16f4315fc6d9253467dc823a0493075b39f60003f5e88e5ce83c14c6a8065"},"schema_version":"1.0","source":{"id":"1302.5828","kind":"arxiv","version":3}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1302.5828","created_at":"2026-05-18T03:04:24Z"},{"alias_kind":"arxiv_version","alias_value":"1302.5828v3","created_at":"2026-05-18T03:04:24Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1302.5828","created_at":"2026-05-18T03:04:24Z"},{"alias_kind":"pith_short_12","alias_value":"6TYZCJELXIOP","created_at":"2026-05-18T12:27:36Z"},{"alias_kind":"pith_short_16","alias_value":"6TYZCJELXIOPYDB7","created_at":"2026-05-18T12:27:36Z"},{"alias_kind":"pith_short_8","alias_value":"6TYZCJEL","created_at":"2026-05-18T12:27:36Z"}],"graph_snapshots":[{"event_id":"sha256:1da8a1c92b0fb8544fd21964d36641ee11f06882d55e241b3d65d8a069cf79a4","target":"graph","created_at":"2026-05-18T03:04:24Z","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 show that non-dominated sorting of a sequence of i.i.d. random variables in Euclidean space has a continuum limit that corresponds to solving a Hamilton-Jacobi equation involving the probability density function of the random variables. Non-dominated sorting is a fundamental problem in multi-objective optimization, and is equivalent to finding the canonical antichain partition and to problems involving the longest chain among Euclidean points. As an application of this result, we show that non-dominated sorting is asymptotically stable under random perturbations in the data. We give a numer","authors_text":"Alfred O. Hero, Jeff Calder, Selim Esedo\\=glu","cross_cats":["math.CO","math.OC","math.PR"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2013-02-23T17:30:11Z","title":"A Hamilton-Jacobi equation for the continuum limit of non-dominated sorting"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1302.5828","kind":"arxiv","version":3},"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":"sha256:6d36b3ea4440d5bd1cc9159460406a6835844f5030856514f2de4113a41503e3","target":"record","created_at":"2026-05-18T03:04:24Z","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":"1600bddf664f3df6dda46c83249571823fa3ac65e97f54a677774f776cac41c9","cross_cats_sorted":["math.CO","math.OC","math.PR"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2013-02-23T17:30:11Z","title_canon_sha256":"7eb16f4315fc6d9253467dc823a0493075b39f60003f5e88e5ce83c14c6a8065"},"schema_version":"1.0","source":{"id":"1302.5828","kind":"arxiv","version":3}},"canonical_sha256":"f4f191248bba1cfc0c3ff0e38fc4306dbee400e257a42175304863991c8969d6","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"f4f191248bba1cfc0c3ff0e38fc4306dbee400e257a42175304863991c8969d6","first_computed_at":"2026-05-18T03:04:24.243484Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:04:24.243484Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"Y/0hnNTM53YAo0o8M+1AjLviq1j2R/fA0TbPQ5ECvYCk9Y0HnD8aKWZsP2TxE9a5AHpWONkLUaUMlWEEk0KoAw==","signature_status":"signed_v1","signed_at":"2026-05-18T03:04:24.243823Z","signed_message":"canonical_sha256_bytes"},"source_id":"1302.5828","source_kind":"arxiv","source_version":3}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:6d36b3ea4440d5bd1cc9159460406a6835844f5030856514f2de4113a41503e3","sha256:1da8a1c92b0fb8544fd21964d36641ee11f06882d55e241b3d65d8a069cf79a4"],"state_sha256":"4ae65d2335c95f8e1ef4ceb479b950f4de02514692051daa47a4ad5db526540a"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"LRC4Vnjn/q7pccKOzGWYWIcG54zYCLTTdc9CTRSwB9ksjnT5tQen0Hsm02MK9qpJ5jIX6nVYHg3NjHtU2O2OBg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-23T12:23:26.349227Z","bundle_sha256":"121476ee6aa7ebc7d8a7e2dd9caecca7918412c3bd146d0ba4aad0703988f250"}}