{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2009:YBS5GNMG6URC4U74ZC7ZEKQYFB","short_pith_number":"pith:YBS5GNMG","canonical_record":{"source":{"id":"0910.4514","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math-ph","submitted_at":"2009-10-23T14:07:46Z","cross_cats_sorted":["math.MP"],"title_canon_sha256":"a91c17ca81d62dc5e74951c906da1fd54deabd82590deeb12de2ba644263a694","abstract_canon_sha256":"3e446375fa5322eed70f3bbc79fc0563b908b9153f18b289d0c6bf6e7b3c87c6"},"schema_version":"1.0"},"canonical_sha256":"c065d33586f5222e53fcc8bf922a18286739d0bc74f3cf64cf246d2bf716fe9c","source":{"kind":"arxiv","id":"0910.4514","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"0910.4514","created_at":"2026-05-18T02:11:12Z"},{"alias_kind":"arxiv_version","alias_value":"0910.4514v2","created_at":"2026-05-18T02:11:12Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.0910.4514","created_at":"2026-05-18T02:11:12Z"},{"alias_kind":"pith_short_12","alias_value":"YBS5GNMG6URC","created_at":"2026-05-18T12:26:02Z"},{"alias_kind":"pith_short_16","alias_value":"YBS5GNMG6URC4U74","created_at":"2026-05-18T12:26:02Z"},{"alias_kind":"pith_short_8","alias_value":"YBS5GNMG","created_at":"2026-05-18T12:26:02Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2009:YBS5GNMG6URC4U74ZC7ZEKQYFB","target":"record","payload":{"canonical_record":{"source":{"id":"0910.4514","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math-ph","submitted_at":"2009-10-23T14:07:46Z","cross_cats_sorted":["math.MP"],"title_canon_sha256":"a91c17ca81d62dc5e74951c906da1fd54deabd82590deeb12de2ba644263a694","abstract_canon_sha256":"3e446375fa5322eed70f3bbc79fc0563b908b9153f18b289d0c6bf6e7b3c87c6"},"schema_version":"1.0"},"canonical_sha256":"c065d33586f5222e53fcc8bf922a18286739d0bc74f3cf64cf246d2bf716fe9c","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:11:12.158152Z","signature_b64":"vF9YIpeCJkNUPak5lzTbduXT9VhlCqMmyqNI6jsiKHcO4Lhn/6JlEChKXcWjCSktA3qP0jfDPurTuo9PN7iDAg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"c065d33586f5222e53fcc8bf922a18286739d0bc74f3cf64cf246d2bf716fe9c","last_reissued_at":"2026-05-18T02:11:12.157394Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:11:12.157394Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"0910.4514","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":"2026-05-18T02:11:12Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"oRxKnvpMmKBS+h7M1wk9HbLJEl459Vu54FvuIBZhp0BEDBLMHKIsMG/eVimvqqcO5zg7O8I21RfUVC8NRtOHDA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-07-02T22:57:29.404975Z"},"content_sha256":"51031f630c0ed20b44b2f2521a90432153b2fbdbf81c37abd86528a87a95cf79","schema_version":"1.0","event_id":"sha256:51031f630c0ed20b44b2f2521a90432153b2fbdbf81c37abd86528a87a95cf79"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2009:YBS5GNMG6URC4U74ZC7ZEKQYFB","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Granger causality and transfer entropy are equivalent for Gaussian variables","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.MP"],"primary_cat":"math-ph","authors_text":"Adam B Barrett, Anil K. Seth, Lionel Barnett","submitted_at":"2009-10-23T14:07:46Z","abstract_excerpt":"Granger causality is a statistical notion of causal influence based on prediction via vector autoregression. Developed originally in the field of econometrics, it has since found application in a broader arena, particularly in neuroscience. More recently transfer entropy, an information-theoretic measure of time-directed information transfer between jointly dependent processes, has gained traction in a similarly wide field. While it has been recognized that the two concepts must be related, the exact relationship has until now not been formally described. Here we show that for Gaussian variabl"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0910.4514","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":""},"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":"2026-05-18T02:11:12Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"HsdPTGADx+vUIYwEhTxoG185Q2z48uYQFb+xshV+lNf84/R3yaBF2dKQHQeT6ohgeFsXzqVjnpfFcK0o5p7HCQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-07-02T22:57:29.405340Z"},"content_sha256":"59fbe0f3bbea0cf993ebf436e04950fe106a01bdd1dc4581564bb0d306285de7","schema_version":"1.0","event_id":"sha256:59fbe0f3bbea0cf993ebf436e04950fe106a01bdd1dc4581564bb0d306285de7"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/YBS5GNMG6URC4U74ZC7ZEKQYFB/bundle.json","state_url":"https://pith.science/pith/YBS5GNMG6URC4U74ZC7ZEKQYFB/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/YBS5GNMG6URC4U74ZC7ZEKQYFB/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":"2026-07-02T22:57:29Z","links":{"resolver":"https://pith.science/pith/YBS5GNMG6URC4U74ZC7ZEKQYFB","bundle":"https://pith.science/pith/YBS5GNMG6URC4U74ZC7ZEKQYFB/bundle.json","state":"https://pith.science/pith/YBS5GNMG6URC4U74ZC7ZEKQYFB/state.json","well_known_bundle":"https://pith.science/.well-known/pith/YBS5GNMG6URC4U74ZC7ZEKQYFB/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2009:YBS5GNMG6URC4U74ZC7ZEKQYFB","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":"3e446375fa5322eed70f3bbc79fc0563b908b9153f18b289d0c6bf6e7b3c87c6","cross_cats_sorted":["math.MP"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math-ph","submitted_at":"2009-10-23T14:07:46Z","title_canon_sha256":"a91c17ca81d62dc5e74951c906da1fd54deabd82590deeb12de2ba644263a694"},"schema_version":"1.0","source":{"id":"0910.4514","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"0910.4514","created_at":"2026-05-18T02:11:12Z"},{"alias_kind":"arxiv_version","alias_value":"0910.4514v2","created_at":"2026-05-18T02:11:12Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.0910.4514","created_at":"2026-05-18T02:11:12Z"},{"alias_kind":"pith_short_12","alias_value":"YBS5GNMG6URC","created_at":"2026-05-18T12:26:02Z"},{"alias_kind":"pith_short_16","alias_value":"YBS5GNMG6URC4U74","created_at":"2026-05-18T12:26:02Z"},{"alias_kind":"pith_short_8","alias_value":"YBS5GNMG","created_at":"2026-05-18T12:26:02Z"}],"graph_snapshots":[{"event_id":"sha256:59fbe0f3bbea0cf993ebf436e04950fe106a01bdd1dc4581564bb0d306285de7","target":"graph","created_at":"2026-05-18T02:11:12Z","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":"Granger causality is a statistical notion of causal influence based on prediction via vector autoregression. Developed originally in the field of econometrics, it has since found application in a broader arena, particularly in neuroscience. More recently transfer entropy, an information-theoretic measure of time-directed information transfer between jointly dependent processes, has gained traction in a similarly wide field. While it has been recognized that the two concepts must be related, the exact relationship has until now not been formally described. Here we show that for Gaussian variabl","authors_text":"Adam B Barrett, Anil K. Seth, Lionel Barnett","cross_cats":["math.MP"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math-ph","submitted_at":"2009-10-23T14:07:46Z","title":"Granger causality and transfer entropy are equivalent for Gaussian variables"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0910.4514","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":"sha256:51031f630c0ed20b44b2f2521a90432153b2fbdbf81c37abd86528a87a95cf79","target":"record","created_at":"2026-05-18T02:11:12Z","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":"3e446375fa5322eed70f3bbc79fc0563b908b9153f18b289d0c6bf6e7b3c87c6","cross_cats_sorted":["math.MP"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math-ph","submitted_at":"2009-10-23T14:07:46Z","title_canon_sha256":"a91c17ca81d62dc5e74951c906da1fd54deabd82590deeb12de2ba644263a694"},"schema_version":"1.0","source":{"id":"0910.4514","kind":"arxiv","version":2}},"canonical_sha256":"c065d33586f5222e53fcc8bf922a18286739d0bc74f3cf64cf246d2bf716fe9c","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"c065d33586f5222e53fcc8bf922a18286739d0bc74f3cf64cf246d2bf716fe9c","first_computed_at":"2026-05-18T02:11:12.157394Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T02:11:12.157394Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"vF9YIpeCJkNUPak5lzTbduXT9VhlCqMmyqNI6jsiKHcO4Lhn/6JlEChKXcWjCSktA3qP0jfDPurTuo9PN7iDAg==","signature_status":"signed_v1","signed_at":"2026-05-18T02:11:12.158152Z","signed_message":"canonical_sha256_bytes"},"source_id":"0910.4514","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:51031f630c0ed20b44b2f2521a90432153b2fbdbf81c37abd86528a87a95cf79","sha256:59fbe0f3bbea0cf993ebf436e04950fe106a01bdd1dc4581564bb0d306285de7"],"state_sha256":"8ae397c4f3e43978d42c3ea4e04e6b2351bb4913f34f01846cd889747673c900"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"XF6SOAEP3soL/iKy8vMO3wJ+cxOUH/Tb7pzSHfLZ6BJw5slIZUNBhxYtmQQHMfuIpBsUzz2AKKtXN460bydcCw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-07-02T22:57:29.407304Z","bundle_sha256":"0540379522ef8369925cdb465b9642ec41da4592994b9ca6aa347c80a9626e64"}}