{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2020:GM5T3E6YI4QGCCTY5SBRAK6EH3","short_pith_number":"pith:GM5T3E6Y","canonical_record":{"source":{"id":"2009.09431","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.ST","submitted_at":"2020-09-20T14:03:13Z","cross_cats_sorted":["cs.IT","hep-th","math.IT","math.OC","stat.ML","stat.TH"],"title_canon_sha256":"4381000fef3380ee1e8d156e288646c20575d3a3ab9419ce143cb3d454bdd8ff","abstract_canon_sha256":"f5e28f2612aa6eb6af18b7d4c054dd030aa7c257c9279d7f7f3e4bbfb3712aae"},"schema_version":"1.0"},"canonical_sha256":"333b3d93d84720610a78ec83102bc43ecaf15c494c4f1ee00e58be8c5aad4e2f","source":{"kind":"arxiv","id":"2009.09431","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2009.09431","created_at":"2026-07-05T05:02:22Z"},{"alias_kind":"arxiv_version","alias_value":"2009.09431v2","created_at":"2026-07-05T05:02:22Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2009.09431","created_at":"2026-07-05T05:02:22Z"},{"alias_kind":"pith_short_12","alias_value":"GM5T3E6YI4QG","created_at":"2026-07-05T05:02:22Z"},{"alias_kind":"pith_short_16","alias_value":"GM5T3E6YI4QGCCTY","created_at":"2026-07-05T05:02:22Z"},{"alias_kind":"pith_short_8","alias_value":"GM5T3E6Y","created_at":"2026-07-05T05:02:22Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2020:GM5T3E6YI4QGCCTY5SBRAK6EH3","target":"record","payload":{"canonical_record":{"source":{"id":"2009.09431","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.ST","submitted_at":"2020-09-20T14:03:13Z","cross_cats_sorted":["cs.IT","hep-th","math.IT","math.OC","stat.ML","stat.TH"],"title_canon_sha256":"4381000fef3380ee1e8d156e288646c20575d3a3ab9419ce143cb3d454bdd8ff","abstract_canon_sha256":"f5e28f2612aa6eb6af18b7d4c054dd030aa7c257c9279d7f7f3e4bbfb3712aae"},"schema_version":"1.0"},"canonical_sha256":"333b3d93d84720610a78ec83102bc43ecaf15c494c4f1ee00e58be8c5aad4e2f","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T05:02:22.500590Z","signature_b64":"aoLHCttA9AlGvlBEqTx5c+sh23mtj4eu2nqWZNyT0tvMNeHFQwvVygV19pXvZxuGIlwHXRFtTVCfUsNyiwvHAQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"333b3d93d84720610a78ec83102bc43ecaf15c494c4f1ee00e58be8c5aad4e2f","last_reissued_at":"2026-07-05T05:02:22.500168Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T05:02:22.500168Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2009.09431","source_version":2,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-07-05T05:02:22Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Z2yFQefR7AOXEjxOlRogwCjec5cdpRNXEv8eFKHmxRF0tDrVZLKRyzWoC34dUjRAfEQJOpZELUScp/PWoxCtAg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-07-06T09:15:33.139615Z"},"content_sha256":"61558b5aa0c966cffd6d57984501040aaba3e6260509488dd719433c5858079e","schema_version":"1.0","event_id":"sha256:61558b5aa0c966cffd6d57984501040aaba3e6260509488dd719433c5858079e"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2020:GM5T3E6YI4QGCCTY5SBRAK6EH3","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Lagrangian and Hamiltonian Mechanics for Probabilities on the Statistical Manifold","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.IT","hep-th","math.IT","math.OC","stat.ML","stat.TH"],"primary_cat":"math.ST","authors_text":"Giovanni Pistone, Goffredo Chirco, Luigi Malag\\`o","submitted_at":"2020-09-20T14:03:13Z","abstract_excerpt":"We provide an Information-Geometric formulation of Classical Mechanics on the Riemannian manifold of probability distributions, which is an affine manifold endowed with a dually-flat connection. In a non-parametric formalism, we consider the full set of positive probability functions on a finite sample space, and we provide a specific expression for the tangent and cotangent spaces over the statistical manifold, in terms of a Hilbert bundle structure that we call the Statistical Bundle. In this setting, we compute velocities and accelerations of a one-dimensional statistical model using the ca"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2009.09431","kind":"arxiv","version":2},"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/2009.09431/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":"2026-07-05T05:02:22Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"NJLLiAjACw+EQZGQ1wIiEeCS4eyqevuYCWN+yQzTXucN83ZTbLMUtlBDFByLnsKStXuCisFr+4t8eh+Pw4c2Cg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-07-06T09:15:33.139987Z"},"content_sha256":"aa5c0dbbec9af56eb8c12c9e0377067045a21cd19887ce290eeeda586ad8acae","schema_version":"1.0","event_id":"sha256:aa5c0dbbec9af56eb8c12c9e0377067045a21cd19887ce290eeeda586ad8acae"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/GM5T3E6YI4QGCCTY5SBRAK6EH3/bundle.json","state_url":"https://pith.science/pith/GM5T3E6YI4QGCCTY5SBRAK6EH3/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/GM5T3E6YI4QGCCTY5SBRAK6EH3/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":"2026-07-06T09:15:33Z","links":{"resolver":"https://pith.science/pith/GM5T3E6YI4QGCCTY5SBRAK6EH3","bundle":"https://pith.science/pith/GM5T3E6YI4QGCCTY5SBRAK6EH3/bundle.json","state":"https://pith.science/pith/GM5T3E6YI4QGCCTY5SBRAK6EH3/state.json","well_known_bundle":"https://pith.science/.well-known/pith/GM5T3E6YI4QGCCTY5SBRAK6EH3/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2020:GM5T3E6YI4QGCCTY5SBRAK6EH3","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":"f5e28f2612aa6eb6af18b7d4c054dd030aa7c257c9279d7f7f3e4bbfb3712aae","cross_cats_sorted":["cs.IT","hep-th","math.IT","math.OC","stat.ML","stat.TH"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.ST","submitted_at":"2020-09-20T14:03:13Z","title_canon_sha256":"4381000fef3380ee1e8d156e288646c20575d3a3ab9419ce143cb3d454bdd8ff"},"schema_version":"1.0","source":{"id":"2009.09431","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2009.09431","created_at":"2026-07-05T05:02:22Z"},{"alias_kind":"arxiv_version","alias_value":"2009.09431v2","created_at":"2026-07-05T05:02:22Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2009.09431","created_at":"2026-07-05T05:02:22Z"},{"alias_kind":"pith_short_12","alias_value":"GM5T3E6YI4QG","created_at":"2026-07-05T05:02:22Z"},{"alias_kind":"pith_short_16","alias_value":"GM5T3E6YI4QGCCTY","created_at":"2026-07-05T05:02:22Z"},{"alias_kind":"pith_short_8","alias_value":"GM5T3E6Y","created_at":"2026-07-05T05:02:22Z"}],"graph_snapshots":[{"event_id":"sha256:aa5c0dbbec9af56eb8c12c9e0377067045a21cd19887ce290eeeda586ad8acae","target":"graph","created_at":"2026-07-05T05:02:22Z","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"},"integrity":{"available":true,"clean":true,"detectors_run":[],"endpoint":"/pith/2009.09431/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"We provide an Information-Geometric formulation of Classical Mechanics on the Riemannian manifold of probability distributions, which is an affine manifold endowed with a dually-flat connection. In a non-parametric formalism, we consider the full set of positive probability functions on a finite sample space, and we provide a specific expression for the tangent and cotangent spaces over the statistical manifold, in terms of a Hilbert bundle structure that we call the Statistical Bundle. In this setting, we compute velocities and accelerations of a one-dimensional statistical model using the ca","authors_text":"Giovanni Pistone, Goffredo Chirco, Luigi Malag\\`o","cross_cats":["cs.IT","hep-th","math.IT","math.OC","stat.ML","stat.TH"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.ST","submitted_at":"2020-09-20T14:03:13Z","title":"Lagrangian and Hamiltonian Mechanics for Probabilities on the Statistical Manifold"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2009.09431","kind":"arxiv","version":2},"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":"sha256:61558b5aa0c966cffd6d57984501040aaba3e6260509488dd719433c5858079e","target":"record","created_at":"2026-07-05T05:02:22Z","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":"f5e28f2612aa6eb6af18b7d4c054dd030aa7c257c9279d7f7f3e4bbfb3712aae","cross_cats_sorted":["cs.IT","hep-th","math.IT","math.OC","stat.ML","stat.TH"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.ST","submitted_at":"2020-09-20T14:03:13Z","title_canon_sha256":"4381000fef3380ee1e8d156e288646c20575d3a3ab9419ce143cb3d454bdd8ff"},"schema_version":"1.0","source":{"id":"2009.09431","kind":"arxiv","version":2}},"canonical_sha256":"333b3d93d84720610a78ec83102bc43ecaf15c494c4f1ee00e58be8c5aad4e2f","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"333b3d93d84720610a78ec83102bc43ecaf15c494c4f1ee00e58be8c5aad4e2f","first_computed_at":"2026-07-05T05:02:22.500168Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T05:02:22.500168Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"aoLHCttA9AlGvlBEqTx5c+sh23mtj4eu2nqWZNyT0tvMNeHFQwvVygV19pXvZxuGIlwHXRFtTVCfUsNyiwvHAQ==","signature_status":"signed_v1","signed_at":"2026-07-05T05:02:22.500590Z","signed_message":"canonical_sha256_bytes"},"source_id":"2009.09431","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:61558b5aa0c966cffd6d57984501040aaba3e6260509488dd719433c5858079e","sha256:aa5c0dbbec9af56eb8c12c9e0377067045a21cd19887ce290eeeda586ad8acae"],"state_sha256":"fed9b2bcd7dd1dbf5594168cec8da763e9cb7cb03b9db83a578802f0d61d684d"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"jDTmMZt29RZOjUx3D3hucLWJV+Cdw7BMJ8eRgeAjOTms/GWGUuH2XG10RGZiehrmKHlmllo/Mo5G86LY7IjIAA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-07-06T09:15:33.141969Z","bundle_sha256":"6098c77dc457ef57eb38f765d8197da535791140718b928ead302a327e39f42e"}}