{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2014:ERUTVSBTADV7FH6TEKHKJD5K4B","short_pith_number":"pith:ERUTVSBT","canonical_record":{"source":{"id":"1407.6784","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AG","submitted_at":"2014-07-25T04:11:36Z","cross_cats_sorted":["math-ph","math.MP"],"title_canon_sha256":"0b0a642489bbdb655db5b9b538c8743bb06bcf035f01881914ca8be4aa859b52","abstract_canon_sha256":"d31e016173939646dbc40f122e59231e933e594068de3c371a45830e5060d689"},"schema_version":"1.0"},"canonical_sha256":"24693ac83300ebf29fd3228ea48faae05fca8eb1b4cbce991bd7a57f0b42b08e","source":{"kind":"arxiv","id":"1407.6784","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1407.6784","created_at":"2026-05-18T02:46:32Z"},{"alias_kind":"arxiv_version","alias_value":"1407.6784v1","created_at":"2026-05-18T02:46:32Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1407.6784","created_at":"2026-05-18T02:46:32Z"},{"alias_kind":"pith_short_12","alias_value":"ERUTVSBTADV7","created_at":"2026-05-18T12:28:28Z"},{"alias_kind":"pith_short_16","alias_value":"ERUTVSBTADV7FH6T","created_at":"2026-05-18T12:28:28Z"},{"alias_kind":"pith_short_8","alias_value":"ERUTVSBT","created_at":"2026-05-18T12:28:28Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2014:ERUTVSBTADV7FH6TEKHKJD5K4B","target":"record","payload":{"canonical_record":{"source":{"id":"1407.6784","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AG","submitted_at":"2014-07-25T04:11:36Z","cross_cats_sorted":["math-ph","math.MP"],"title_canon_sha256":"0b0a642489bbdb655db5b9b538c8743bb06bcf035f01881914ca8be4aa859b52","abstract_canon_sha256":"d31e016173939646dbc40f122e59231e933e594068de3c371a45830e5060d689"},"schema_version":"1.0"},"canonical_sha256":"24693ac83300ebf29fd3228ea48faae05fca8eb1b4cbce991bd7a57f0b42b08e","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:46:32.248600Z","signature_b64":"mxsMNr7Qqwyp3NDBIJEvQo65Va8MkfWL9M78HauB25qpwpY2EZUMsFuI8DElDB2MLOQ77KWAo2A+/XWLlTz3Bg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"24693ac83300ebf29fd3228ea48faae05fca8eb1b4cbce991bd7a57f0b42b08e","last_reissued_at":"2026-05-18T02:46:32.248130Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:46:32.248130Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1407.6784","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":"2026-05-18T02:46:32Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"677rpqZJhgqm97sv2N1LWott1t420+j6sDENumxaGg8wXt02ICyntQYVxLEqSTu7b9SGn8HDsQ1AoWeZZPNdAw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-12T09:49:21.651411Z"},"content_sha256":"98c09f3c8d0d28747b27402d0457b88a4726e7eebf1e803903914ff452ec85aa","schema_version":"1.0","event_id":"sha256:98c09f3c8d0d28747b27402d0457b88a4726e7eebf1e803903914ff452ec85aa"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2014:ERUTVSBTADV7FH6TEKHKJD5K4B","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Algebraic Stochastic Calculus","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math-ph","math.MP"],"primary_cat":"math.AG","authors_text":"Renaud Gauthier","submitted_at":"2014-07-25T04:11:36Z","abstract_excerpt":"We develop the foundations of Algebraic Stochastic Calculus, with an aim to replacing what is typically referred to as Stochastic Calculus by a purely categorical version thereof. We first give a sheaf theoretic reinterpretation of Probability Theory. We regard probability spaces (X, F, P) as Grothendieck sites (F, J_P) on which Brownian motions are defined via sheaves in symmetric monoidal infinity-categories. Due to the complex nature of such a formalism we are naturally led to considering a purely categorical, time independent formalism in which stochastic differential equations are replace"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1407.6784","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":"2026-05-18T02:46:32Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Dzwm3FPjhR3sOC4jabWfd8hVPqqyx0DlOzqLeLQv0IlL1j8li3FlJWRppQgz61/uJfe+585HpsZEt3tZSMOODg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-12T09:49:21.651763Z"},"content_sha256":"93acc658c5a998fa8bbf962ddffbdf117f8fb0c5ace827dfbfe957df0fc083bc","schema_version":"1.0","event_id":"sha256:93acc658c5a998fa8bbf962ddffbdf117f8fb0c5ace827dfbfe957df0fc083bc"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/ERUTVSBTADV7FH6TEKHKJD5K4B/bundle.json","state_url":"https://pith.science/pith/ERUTVSBTADV7FH6TEKHKJD5K4B/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/ERUTVSBTADV7FH6TEKHKJD5K4B/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":"2026-06-12T09:49:21Z","links":{"resolver":"https://pith.science/pith/ERUTVSBTADV7FH6TEKHKJD5K4B","bundle":"https://pith.science/pith/ERUTVSBTADV7FH6TEKHKJD5K4B/bundle.json","state":"https://pith.science/pith/ERUTVSBTADV7FH6TEKHKJD5K4B/state.json","well_known_bundle":"https://pith.science/.well-known/pith/ERUTVSBTADV7FH6TEKHKJD5K4B/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2014:ERUTVSBTADV7FH6TEKHKJD5K4B","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":"d31e016173939646dbc40f122e59231e933e594068de3c371a45830e5060d689","cross_cats_sorted":["math-ph","math.MP"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AG","submitted_at":"2014-07-25T04:11:36Z","title_canon_sha256":"0b0a642489bbdb655db5b9b538c8743bb06bcf035f01881914ca8be4aa859b52"},"schema_version":"1.0","source":{"id":"1407.6784","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1407.6784","created_at":"2026-05-18T02:46:32Z"},{"alias_kind":"arxiv_version","alias_value":"1407.6784v1","created_at":"2026-05-18T02:46:32Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1407.6784","created_at":"2026-05-18T02:46:32Z"},{"alias_kind":"pith_short_12","alias_value":"ERUTVSBTADV7","created_at":"2026-05-18T12:28:28Z"},{"alias_kind":"pith_short_16","alias_value":"ERUTVSBTADV7FH6T","created_at":"2026-05-18T12:28:28Z"},{"alias_kind":"pith_short_8","alias_value":"ERUTVSBT","created_at":"2026-05-18T12:28:28Z"}],"graph_snapshots":[{"event_id":"sha256:93acc658c5a998fa8bbf962ddffbdf117f8fb0c5ace827dfbfe957df0fc083bc","target":"graph","created_at":"2026-05-18T02:46:32Z","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 develop the foundations of Algebraic Stochastic Calculus, with an aim to replacing what is typically referred to as Stochastic Calculus by a purely categorical version thereof. We first give a sheaf theoretic reinterpretation of Probability Theory. We regard probability spaces (X, F, P) as Grothendieck sites (F, J_P) on which Brownian motions are defined via sheaves in symmetric monoidal infinity-categories. Due to the complex nature of such a formalism we are naturally led to considering a purely categorical, time independent formalism in which stochastic differential equations are replace","authors_text":"Renaud Gauthier","cross_cats":["math-ph","math.MP"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AG","submitted_at":"2014-07-25T04:11:36Z","title":"Algebraic Stochastic Calculus"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1407.6784","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":"sha256:98c09f3c8d0d28747b27402d0457b88a4726e7eebf1e803903914ff452ec85aa","target":"record","created_at":"2026-05-18T02:46:32Z","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":"d31e016173939646dbc40f122e59231e933e594068de3c371a45830e5060d689","cross_cats_sorted":["math-ph","math.MP"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AG","submitted_at":"2014-07-25T04:11:36Z","title_canon_sha256":"0b0a642489bbdb655db5b9b538c8743bb06bcf035f01881914ca8be4aa859b52"},"schema_version":"1.0","source":{"id":"1407.6784","kind":"arxiv","version":1}},"canonical_sha256":"24693ac83300ebf29fd3228ea48faae05fca8eb1b4cbce991bd7a57f0b42b08e","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"24693ac83300ebf29fd3228ea48faae05fca8eb1b4cbce991bd7a57f0b42b08e","first_computed_at":"2026-05-18T02:46:32.248130Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T02:46:32.248130Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"mxsMNr7Qqwyp3NDBIJEvQo65Va8MkfWL9M78HauB25qpwpY2EZUMsFuI8DElDB2MLOQ77KWAo2A+/XWLlTz3Bg==","signature_status":"signed_v1","signed_at":"2026-05-18T02:46:32.248600Z","signed_message":"canonical_sha256_bytes"},"source_id":"1407.6784","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:98c09f3c8d0d28747b27402d0457b88a4726e7eebf1e803903914ff452ec85aa","sha256:93acc658c5a998fa8bbf962ddffbdf117f8fb0c5ace827dfbfe957df0fc083bc"],"state_sha256":"c406e31b89d340035af4bd0b0f375a787087f465e49b685f69c89e264aade120"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"nGOBuFy7k9nMUEI5UZFOIyksPqEzsXnW0gRYMr8i8uMdkSSIDb8e6ZfXmsh5QRzfBM81DNClAu3SaeO/3BmXCA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-12T09:49:21.653646Z","bundle_sha256":"fabd34e7916f49ddd89aed619fd200febb64d3931b444b722c57895b88460c63"}}