{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2018:L3CXGFG22EY673N7EINYN5HHWP","short_pith_number":"pith:L3CXGFG2","canonical_record":{"source":{"id":"1805.01053","kind":"arxiv","version":4},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2018-05-02T23:28:59Z","cross_cats_sorted":[],"title_canon_sha256":"57372096e69f9cbdb86bdd02fddbd8698e8b3bf5011b1c3441c04526ed9f49e4","abstract_canon_sha256":"e6934f9daf7ac1e64b6f65f7e488e41579e6083418c2dd5f83c2af6887de87f0"},"schema_version":"1.0"},"canonical_sha256":"5ec57314dad131efedbf221b86f4e7b3fd5e220b154a931f5906b9ec4c2c3838","source":{"kind":"arxiv","id":"1805.01053","version":4},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1805.01053","created_at":"2026-07-05T00:18:04Z"},{"alias_kind":"arxiv_version","alias_value":"1805.01053v4","created_at":"2026-07-05T00:18:04Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1805.01053","created_at":"2026-07-05T00:18:04Z"},{"alias_kind":"pith_short_12","alias_value":"L3CXGFG22EY6","created_at":"2026-07-05T00:18:04Z"},{"alias_kind":"pith_short_16","alias_value":"L3CXGFG22EY673N7","created_at":"2026-07-05T00:18:04Z"},{"alias_kind":"pith_short_8","alias_value":"L3CXGFG2","created_at":"2026-07-05T00:18:04Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2018:L3CXGFG22EY673N7EINYN5HHWP","target":"record","payload":{"canonical_record":{"source":{"id":"1805.01053","kind":"arxiv","version":4},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2018-05-02T23:28:59Z","cross_cats_sorted":[],"title_canon_sha256":"57372096e69f9cbdb86bdd02fddbd8698e8b3bf5011b1c3441c04526ed9f49e4","abstract_canon_sha256":"e6934f9daf7ac1e64b6f65f7e488e41579e6083418c2dd5f83c2af6887de87f0"},"schema_version":"1.0"},"canonical_sha256":"5ec57314dad131efedbf221b86f4e7b3fd5e220b154a931f5906b9ec4c2c3838","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T00:18:04.406039Z","signature_b64":"SVDokVs3mXLOcLToy5lgdTULgm9ClnRSrccfoWkVEQUXS9swKWCxFM6cvtzB2D9CjHo5/actKIK72jXG7hLACQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"5ec57314dad131efedbf221b86f4e7b3fd5e220b154a931f5906b9ec4c2c3838","last_reissued_at":"2026-07-05T00:18:04.405594Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T00:18:04.405594Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1805.01053","source_version":4,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-07-05T00:18:04Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"0AsfmH+0DC5tRTvLjcfcZ7o+FYaVdmWUIHfFDHKaGBcbCcfBorltZ6VTziGU/BX5qWaI8hjDJTM1BDjTXTTOAg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-07-05T13:32:03.890688Z"},"content_sha256":"f0f856cea42d54cdcda444964f1df7b6c231df8dd3fe2255ea65c63dec7ba1b3","schema_version":"1.0","event_id":"sha256:f0f856cea42d54cdcda444964f1df7b6c231df8dd3fe2255ea65c63dec7ba1b3"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2018:L3CXGFG22EY673N7EINYN5HHWP","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Mean Field Analysis of Neural Networks: A Law of Large Numbers","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Justin Sirignano, Konstantinos Spiliopoulos","submitted_at":"2018-05-02T23:28:59Z","abstract_excerpt":"Machine learning, and in particular neural network models, have revolutionized fields such as image, text, and speech recognition. Today, many important real-world applications in these areas are driven by neural networks. There are also growing applications in engineering, robotics, medicine, and finance. Despite their immense success in practice, there is limited mathematical understanding of neural networks. This paper illustrates how neural networks can be studied via stochastic analysis, and develops approaches for addressing some of the technical challenges which arise. We analyze one-la"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1805.01053","kind":"arxiv","version":4},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/1805.01053/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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":"2026-07-05T00:18:04Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"qPygEg1SLqPw/EIoHOOrsWubF6UAoChtNzKBj4orK8CU3fK8SXul3ItwWHB8WVV9yNwrE8xwl/2vorlNIfjMAQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-07-05T13:32:03.891081Z"},"content_sha256":"5cf97af38ba4b217f678d93f925d30fc69b4581140dc1b31a31f3dd9b1cb1d94","schema_version":"1.0","event_id":"sha256:5cf97af38ba4b217f678d93f925d30fc69b4581140dc1b31a31f3dd9b1cb1d94"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/L3CXGFG22EY673N7EINYN5HHWP/bundle.json","state_url":"https://pith.science/pith/L3CXGFG22EY673N7EINYN5HHWP/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/L3CXGFG22EY673N7EINYN5HHWP/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":"2026-07-05T13:32:03Z","links":{"resolver":"https://pith.science/pith/L3CXGFG22EY673N7EINYN5HHWP","bundle":"https://pith.science/pith/L3CXGFG22EY673N7EINYN5HHWP/bundle.json","state":"https://pith.science/pith/L3CXGFG22EY673N7EINYN5HHWP/state.json","well_known_bundle":"https://pith.science/.well-known/pith/L3CXGFG22EY673N7EINYN5HHWP/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2018:L3CXGFG22EY673N7EINYN5HHWP","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":"e6934f9daf7ac1e64b6f65f7e488e41579e6083418c2dd5f83c2af6887de87f0","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2018-05-02T23:28:59Z","title_canon_sha256":"57372096e69f9cbdb86bdd02fddbd8698e8b3bf5011b1c3441c04526ed9f49e4"},"schema_version":"1.0","source":{"id":"1805.01053","kind":"arxiv","version":4}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1805.01053","created_at":"2026-07-05T00:18:04Z"},{"alias_kind":"arxiv_version","alias_value":"1805.01053v4","created_at":"2026-07-05T00:18:04Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1805.01053","created_at":"2026-07-05T00:18:04Z"},{"alias_kind":"pith_short_12","alias_value":"L3CXGFG22EY6","created_at":"2026-07-05T00:18:04Z"},{"alias_kind":"pith_short_16","alias_value":"L3CXGFG22EY673N7","created_at":"2026-07-05T00:18:04Z"},{"alias_kind":"pith_short_8","alias_value":"L3CXGFG2","created_at":"2026-07-05T00:18:04Z"}],"graph_snapshots":[{"event_id":"sha256:5cf97af38ba4b217f678d93f925d30fc69b4581140dc1b31a31f3dd9b1cb1d94","target":"graph","created_at":"2026-07-05T00:18:04Z","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"},"integrity":{"available":true,"clean":true,"detectors_run":[],"endpoint":"/pith/1805.01053/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"Machine learning, and in particular neural network models, have revolutionized fields such as image, text, and speech recognition. Today, many important real-world applications in these areas are driven by neural networks. There are also growing applications in engineering, robotics, medicine, and finance. Despite their immense success in practice, there is limited mathematical understanding of neural networks. This paper illustrates how neural networks can be studied via stochastic analysis, and develops approaches for addressing some of the technical challenges which arise. We analyze one-la","authors_text":"Justin Sirignano, Konstantinos Spiliopoulos","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2018-05-02T23:28:59Z","title":"Mean Field Analysis of Neural Networks: A Law of Large Numbers"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1805.01053","kind":"arxiv","version":4},"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:f0f856cea42d54cdcda444964f1df7b6c231df8dd3fe2255ea65c63dec7ba1b3","target":"record","created_at":"2026-07-05T00:18:04Z","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":"e6934f9daf7ac1e64b6f65f7e488e41579e6083418c2dd5f83c2af6887de87f0","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2018-05-02T23:28:59Z","title_canon_sha256":"57372096e69f9cbdb86bdd02fddbd8698e8b3bf5011b1c3441c04526ed9f49e4"},"schema_version":"1.0","source":{"id":"1805.01053","kind":"arxiv","version":4}},"canonical_sha256":"5ec57314dad131efedbf221b86f4e7b3fd5e220b154a931f5906b9ec4c2c3838","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"5ec57314dad131efedbf221b86f4e7b3fd5e220b154a931f5906b9ec4c2c3838","first_computed_at":"2026-07-05T00:18:04.405594Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T00:18:04.405594Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"SVDokVs3mXLOcLToy5lgdTULgm9ClnRSrccfoWkVEQUXS9swKWCxFM6cvtzB2D9CjHo5/actKIK72jXG7hLACQ==","signature_status":"signed_v1","signed_at":"2026-07-05T00:18:04.406039Z","signed_message":"canonical_sha256_bytes"},"source_id":"1805.01053","source_kind":"arxiv","source_version":4}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:f0f856cea42d54cdcda444964f1df7b6c231df8dd3fe2255ea65c63dec7ba1b3","sha256:5cf97af38ba4b217f678d93f925d30fc69b4581140dc1b31a31f3dd9b1cb1d94"],"state_sha256":"9cef7ddea629c100fc95deb5435fb3ab5cb7d6e0ddd5ab4a50ecf9da3f2ca0cd"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"IB6Bus6sNHBCmrwFv5tgi6wBpbjJsPkNrCdpwNy3yYjczAN69JUWdmK5nFLZzpekO/YxgU2sgA7//AHQl4PuDQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-07-05T13:32:03.894040Z","bundle_sha256":"d28f119d518070356e47b1274f601f940d9fc40e0faa991b3e08a2687ca0c107"}}