{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2012:2SHUGAOU2G63MCPPRORPBR65HQ","short_pith_number":"pith:2SHUGAOU","canonical_record":{"source":{"id":"1206.0378","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2012-06-02T13:14:06Z","cross_cats_sorted":["math.OA","quant-ph"],"title_canon_sha256":"1df03c9971de6ff3e03fb3a050748fdfa2992a98e89598ce8b70320c9d11347e","abstract_canon_sha256":"f8e66b8a612032169e3f23dfb8ddc4bf2b68051c8a3656a921f857e99a744fab"},"schema_version":"1.0"},"canonical_sha256":"d48f4301d4d1bdb609ef8ba2f0c7dd3c38d8a1fbd4c88d4caf1e003938b383c6","source":{"kind":"arxiv","id":"1206.0378","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1206.0378","created_at":"2026-05-18T02:03:45Z"},{"alias_kind":"arxiv_version","alias_value":"1206.0378v2","created_at":"2026-05-18T02:03:45Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1206.0378","created_at":"2026-05-18T02:03:45Z"},{"alias_kind":"pith_short_12","alias_value":"2SHUGAOU2G63","created_at":"2026-05-18T12:26:50Z"},{"alias_kind":"pith_short_16","alias_value":"2SHUGAOU2G63MCPP","created_at":"2026-05-18T12:26:50Z"},{"alias_kind":"pith_short_8","alias_value":"2SHUGAOU","created_at":"2026-05-18T12:26:50Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2012:2SHUGAOU2G63MCPPRORPBR65HQ","target":"record","payload":{"canonical_record":{"source":{"id":"1206.0378","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2012-06-02T13:14:06Z","cross_cats_sorted":["math.OA","quant-ph"],"title_canon_sha256":"1df03c9971de6ff3e03fb3a050748fdfa2992a98e89598ce8b70320c9d11347e","abstract_canon_sha256":"f8e66b8a612032169e3f23dfb8ddc4bf2b68051c8a3656a921f857e99a744fab"},"schema_version":"1.0"},"canonical_sha256":"d48f4301d4d1bdb609ef8ba2f0c7dd3c38d8a1fbd4c88d4caf1e003938b383c6","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:03:45.552574Z","signature_b64":"ybxIU8BNgN6BHJ6SwTfIsetyshmggOAyjjZNRFj6DsYvIX7g6cSY6T+YQ1JRsMj2ELWQIKJCPinC8juXS6qOBw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"d48f4301d4d1bdb609ef8ba2f0c7dd3c38d8a1fbd4c88d4caf1e003938b383c6","last_reissued_at":"2026-05-18T02:03:45.552209Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:03:45.552209Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1206.0378","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":"2026-05-18T02:03:45Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"OuSyrrPkVUxsZXFwoQakf7tlG9cLPohvItQWbkepp197ETvk7/H8686snb1xx3cHZChcJw/XP3JL7xzFyBG+AA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-27T13:44:08.596304Z"},"content_sha256":"aac14034ff7218134e612b5a38a1ed4d44c869e355ed27ebb1783d161eac437b","schema_version":"1.0","event_id":"sha256:aac14034ff7218134e612b5a38a1ed4d44c869e355ed27ebb1783d161eac437b"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2012:2SHUGAOU2G63MCPPRORPBR65HQ","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Weak Markov Processes as Linear Systems","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.OA","quant-ph"],"primary_cat":"math.FA","authors_text":"Rolf Gohm","submitted_at":"2012-06-02T13:14:06Z","abstract_excerpt":"A noncommutative Fornasini-Marchesini system (a multi-variable version of a linear system) can be realized within a weak Markov process (a model for quantum evolution). For a discrete time parameter the resulting structure is worked out systematically and some quantum mechanical interpretations are given. We introduce subprocesses and quotient processes and then the notion of a $\\gamma$-extension for processes which leads to a complete classification of all the ways in which processes can be built from subprocesses and quotient processes. We show that within a $\\gamma$-extension we have a casc"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1206.0378","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":"2026-05-18T02:03:45Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"8o8bwydYS4Ig5S9+hj/o5E34KUVS71QWWYG/gfqyP+A5vpQdp17sAvVauV8ZjxhzONYjJMnafYehZjTwNn3xDA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-27T13:44:08.596982Z"},"content_sha256":"52ccb260e439cd9930cedcf6be8204807365856653271d47a2884c1155ad8568","schema_version":"1.0","event_id":"sha256:52ccb260e439cd9930cedcf6be8204807365856653271d47a2884c1155ad8568"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/2SHUGAOU2G63MCPPRORPBR65HQ/bundle.json","state_url":"https://pith.science/pith/2SHUGAOU2G63MCPPRORPBR65HQ/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/2SHUGAOU2G63MCPPRORPBR65HQ/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":"2026-05-27T13:44:08Z","links":{"resolver":"https://pith.science/pith/2SHUGAOU2G63MCPPRORPBR65HQ","bundle":"https://pith.science/pith/2SHUGAOU2G63MCPPRORPBR65HQ/bundle.json","state":"https://pith.science/pith/2SHUGAOU2G63MCPPRORPBR65HQ/state.json","well_known_bundle":"https://pith.science/.well-known/pith/2SHUGAOU2G63MCPPRORPBR65HQ/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2012:2SHUGAOU2G63MCPPRORPBR65HQ","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":"f8e66b8a612032169e3f23dfb8ddc4bf2b68051c8a3656a921f857e99a744fab","cross_cats_sorted":["math.OA","quant-ph"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2012-06-02T13:14:06Z","title_canon_sha256":"1df03c9971de6ff3e03fb3a050748fdfa2992a98e89598ce8b70320c9d11347e"},"schema_version":"1.0","source":{"id":"1206.0378","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1206.0378","created_at":"2026-05-18T02:03:45Z"},{"alias_kind":"arxiv_version","alias_value":"1206.0378v2","created_at":"2026-05-18T02:03:45Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1206.0378","created_at":"2026-05-18T02:03:45Z"},{"alias_kind":"pith_short_12","alias_value":"2SHUGAOU2G63","created_at":"2026-05-18T12:26:50Z"},{"alias_kind":"pith_short_16","alias_value":"2SHUGAOU2G63MCPP","created_at":"2026-05-18T12:26:50Z"},{"alias_kind":"pith_short_8","alias_value":"2SHUGAOU","created_at":"2026-05-18T12:26:50Z"}],"graph_snapshots":[{"event_id":"sha256:52ccb260e439cd9930cedcf6be8204807365856653271d47a2884c1155ad8568","target":"graph","created_at":"2026-05-18T02:03:45Z","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":"A noncommutative Fornasini-Marchesini system (a multi-variable version of a linear system) can be realized within a weak Markov process (a model for quantum evolution). For a discrete time parameter the resulting structure is worked out systematically and some quantum mechanical interpretations are given. We introduce subprocesses and quotient processes and then the notion of a $\\gamma$-extension for processes which leads to a complete classification of all the ways in which processes can be built from subprocesses and quotient processes. We show that within a $\\gamma$-extension we have a casc","authors_text":"Rolf Gohm","cross_cats":["math.OA","quant-ph"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2012-06-02T13:14:06Z","title":"Weak Markov Processes as Linear Systems"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1206.0378","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:aac14034ff7218134e612b5a38a1ed4d44c869e355ed27ebb1783d161eac437b","target":"record","created_at":"2026-05-18T02:03:45Z","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":"f8e66b8a612032169e3f23dfb8ddc4bf2b68051c8a3656a921f857e99a744fab","cross_cats_sorted":["math.OA","quant-ph"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2012-06-02T13:14:06Z","title_canon_sha256":"1df03c9971de6ff3e03fb3a050748fdfa2992a98e89598ce8b70320c9d11347e"},"schema_version":"1.0","source":{"id":"1206.0378","kind":"arxiv","version":2}},"canonical_sha256":"d48f4301d4d1bdb609ef8ba2f0c7dd3c38d8a1fbd4c88d4caf1e003938b383c6","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"d48f4301d4d1bdb609ef8ba2f0c7dd3c38d8a1fbd4c88d4caf1e003938b383c6","first_computed_at":"2026-05-18T02:03:45.552209Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T02:03:45.552209Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"ybxIU8BNgN6BHJ6SwTfIsetyshmggOAyjjZNRFj6DsYvIX7g6cSY6T+YQ1JRsMj2ELWQIKJCPinC8juXS6qOBw==","signature_status":"signed_v1","signed_at":"2026-05-18T02:03:45.552574Z","signed_message":"canonical_sha256_bytes"},"source_id":"1206.0378","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:aac14034ff7218134e612b5a38a1ed4d44c869e355ed27ebb1783d161eac437b","sha256:52ccb260e439cd9930cedcf6be8204807365856653271d47a2884c1155ad8568"],"state_sha256":"7a93234ac22a9a75f62148969294df9816b28b561e55ee9d9ed7043252f93fb7"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"xeGvVRQh3OdhikHRFg3S1CYoCtTW+0VgxSZNcieiFsSebMWb+w2wTNF5b6OQKdp2yMWScaSAFtVkANHfxEYSAw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-27T13:44:08.600412Z","bundle_sha256":"f922d2a89c2cfaf9a59a76cef8b240435e544ec1cec56653445f0aecb6952920"}}