{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2018:YC2K5WZ6WBMDQMZDYWDVNBBBCP","short_pith_number":"pith:YC2K5WZ6","canonical_record":{"source":{"id":"1803.10370","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2018-03-28T01:12:03Z","cross_cats_sorted":[],"title_canon_sha256":"6da7930f8199ed63022c94c0a6031bffcf79b8df41d0d78783a0c1a32f8d7ac9","abstract_canon_sha256":"8f260e6c13680f869b6b3dbf5cc491f7fbf44216e2107f1f6df8897bd4ae4585"},"schema_version":"1.0"},"canonical_sha256":"c0b4aedb3eb058383323c58756842113deca6e1adb62da6db797367bd3884146","source":{"kind":"arxiv","id":"1803.10370","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1803.10370","created_at":"2026-05-18T00:19:55Z"},{"alias_kind":"arxiv_version","alias_value":"1803.10370v1","created_at":"2026-05-18T00:19:55Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1803.10370","created_at":"2026-05-18T00:19:55Z"},{"alias_kind":"pith_short_12","alias_value":"YC2K5WZ6WBMD","created_at":"2026-05-18T12:33:04Z"},{"alias_kind":"pith_short_16","alias_value":"YC2K5WZ6WBMDQMZD","created_at":"2026-05-18T12:33:04Z"},{"alias_kind":"pith_short_8","alias_value":"YC2K5WZ6","created_at":"2026-05-18T12:33:04Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2018:YC2K5WZ6WBMDQMZDYWDVNBBBCP","target":"record","payload":{"canonical_record":{"source":{"id":"1803.10370","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2018-03-28T01:12:03Z","cross_cats_sorted":[],"title_canon_sha256":"6da7930f8199ed63022c94c0a6031bffcf79b8df41d0d78783a0c1a32f8d7ac9","abstract_canon_sha256":"8f260e6c13680f869b6b3dbf5cc491f7fbf44216e2107f1f6df8897bd4ae4585"},"schema_version":"1.0"},"canonical_sha256":"c0b4aedb3eb058383323c58756842113deca6e1adb62da6db797367bd3884146","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:19:55.661349Z","signature_b64":"Ij2MjI7eGahZ2+bKrgYJ1901qEDukwLMj8G9E7otJ3cGsX3jgjjrNgs9vWPMDa1q/OlbobesmKOQcxZ99xeZCw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"c0b4aedb3eb058383323c58756842113deca6e1adb62da6db797367bd3884146","last_reissued_at":"2026-05-18T00:19:55.660615Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:19:55.660615Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1803.10370","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":"2026-05-18T00:19:55Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"pcyF4KYOyIWycg5E9japqozUk2tBb4mZYJUxbgl2Img0h289j5aQQqerydfo68SZ9JPqqCY3wy+lf5G/wqSQCg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-09T05:04:07.941124Z"},"content_sha256":"cb579af5ed559e8de127b5bede935ed8c0a1585ac2f1e6662a7c90448d3765cf","schema_version":"1.0","event_id":"sha256:cb579af5ed559e8de127b5bede935ed8c0a1585ac2f1e6662a7c90448d3765cf"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2018:YC2K5WZ6WBMDQMZDYWDVNBBBCP","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Best finite approximations of Benford's Law","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Arno Berger, Chuang Xu","submitted_at":"2018-03-28T01:12:03Z","abstract_excerpt":"For arbitrary Borel probability measures with compact support on the real line, characterizations are established of the best finitely supported approximations, relative to three familiar probability metrics (Levy, Kantorovich, and Kolmogorov), given any number of atoms, and allowing for additional constraints regarding weights or positions of atoms. As an application, best (constrained or unconstrained) approximations are identified for Benford's Law (logarithmic distribution of significands) and other familiar distributions. The results complement and extend known facts in the literature; th"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1803.10370","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":"2026-05-18T00:19:55Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"L8jB+9s2vf69+YN9WFPXtwmy05rkTftruLaQO1w0nQrpzf/n5fw5qw38jwV9pyyn4cepQuKNjNg4dpZjWBKVAg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-09T05:04:07.941849Z"},"content_sha256":"4bdd92d2a9b08bd2617d779d1d5005c3583e7d8b674d869745c12f5aacf199ef","schema_version":"1.0","event_id":"sha256:4bdd92d2a9b08bd2617d779d1d5005c3583e7d8b674d869745c12f5aacf199ef"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/YC2K5WZ6WBMDQMZDYWDVNBBBCP/bundle.json","state_url":"https://pith.science/pith/YC2K5WZ6WBMDQMZDYWDVNBBBCP/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/YC2K5WZ6WBMDQMZDYWDVNBBBCP/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":"2026-06-09T05:04:07Z","links":{"resolver":"https://pith.science/pith/YC2K5WZ6WBMDQMZDYWDVNBBBCP","bundle":"https://pith.science/pith/YC2K5WZ6WBMDQMZDYWDVNBBBCP/bundle.json","state":"https://pith.science/pith/YC2K5WZ6WBMDQMZDYWDVNBBBCP/state.json","well_known_bundle":"https://pith.science/.well-known/pith/YC2K5WZ6WBMDQMZDYWDVNBBBCP/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2018:YC2K5WZ6WBMDQMZDYWDVNBBBCP","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":"8f260e6c13680f869b6b3dbf5cc491f7fbf44216e2107f1f6df8897bd4ae4585","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2018-03-28T01:12:03Z","title_canon_sha256":"6da7930f8199ed63022c94c0a6031bffcf79b8df41d0d78783a0c1a32f8d7ac9"},"schema_version":"1.0","source":{"id":"1803.10370","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1803.10370","created_at":"2026-05-18T00:19:55Z"},{"alias_kind":"arxiv_version","alias_value":"1803.10370v1","created_at":"2026-05-18T00:19:55Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1803.10370","created_at":"2026-05-18T00:19:55Z"},{"alias_kind":"pith_short_12","alias_value":"YC2K5WZ6WBMD","created_at":"2026-05-18T12:33:04Z"},{"alias_kind":"pith_short_16","alias_value":"YC2K5WZ6WBMDQMZD","created_at":"2026-05-18T12:33:04Z"},{"alias_kind":"pith_short_8","alias_value":"YC2K5WZ6","created_at":"2026-05-18T12:33:04Z"}],"graph_snapshots":[{"event_id":"sha256:4bdd92d2a9b08bd2617d779d1d5005c3583e7d8b674d869745c12f5aacf199ef","target":"graph","created_at":"2026-05-18T00:19:55Z","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":"For arbitrary Borel probability measures with compact support on the real line, characterizations are established of the best finitely supported approximations, relative to three familiar probability metrics (Levy, Kantorovich, and Kolmogorov), given any number of atoms, and allowing for additional constraints regarding weights or positions of atoms. As an application, best (constrained or unconstrained) approximations are identified for Benford's Law (logarithmic distribution of significands) and other familiar distributions. The results complement and extend known facts in the literature; th","authors_text":"Arno Berger, Chuang Xu","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2018-03-28T01:12:03Z","title":"Best finite approximations of Benford's Law"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1803.10370","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:cb579af5ed559e8de127b5bede935ed8c0a1585ac2f1e6662a7c90448d3765cf","target":"record","created_at":"2026-05-18T00:19:55Z","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":"8f260e6c13680f869b6b3dbf5cc491f7fbf44216e2107f1f6df8897bd4ae4585","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2018-03-28T01:12:03Z","title_canon_sha256":"6da7930f8199ed63022c94c0a6031bffcf79b8df41d0d78783a0c1a32f8d7ac9"},"schema_version":"1.0","source":{"id":"1803.10370","kind":"arxiv","version":1}},"canonical_sha256":"c0b4aedb3eb058383323c58756842113deca6e1adb62da6db797367bd3884146","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"c0b4aedb3eb058383323c58756842113deca6e1adb62da6db797367bd3884146","first_computed_at":"2026-05-18T00:19:55.660615Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:19:55.660615Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"Ij2MjI7eGahZ2+bKrgYJ1901qEDukwLMj8G9E7otJ3cGsX3jgjjrNgs9vWPMDa1q/OlbobesmKOQcxZ99xeZCw==","signature_status":"signed_v1","signed_at":"2026-05-18T00:19:55.661349Z","signed_message":"canonical_sha256_bytes"},"source_id":"1803.10370","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:cb579af5ed559e8de127b5bede935ed8c0a1585ac2f1e6662a7c90448d3765cf","sha256:4bdd92d2a9b08bd2617d779d1d5005c3583e7d8b674d869745c12f5aacf199ef"],"state_sha256":"3dc316d871ad302ad1d9197789ce6fe8e44e079d1d79bf85df4b64a57a583104"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"kFn11Yweh3UVqDrI1UFO26F2rivuqi/zzUOxf5pHKjhGxlyDZsPlkB0Gq6Kxvk8tUCAXfpt7Au77kpHKspF8CQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-09T05:04:07.949654Z","bundle_sha256":"57400255c36a4f79c97566c253e49761fa29bc0c55a66cdae23ef576fda8ec50"}}