{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2015:QIJ72G5H4UNOMTRV55ULM53OIQ","short_pith_number":"pith:QIJ72G5H","canonical_record":{"source":{"id":"1508.06448","kind":"arxiv","version":4},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math-ph","submitted_at":"2015-08-26T11:31:07Z","cross_cats_sorted":["math.CT","math.MP","math.PR"],"title_canon_sha256":"050ada523a58f183a0b543aea961e4e0a2d43f4705e67f93444340dc8dbd731c","abstract_canon_sha256":"66a4eafcde56014acac8a0a69e1ebec7efd4e64d9b2dd7e801cf7cb08108718e"},"schema_version":"1.0"},"canonical_sha256":"8213fd1ba7e51ae64e35ef68b6776e442ff290cdef0d4f099e0c1b2659cd0c9c","source":{"kind":"arxiv","id":"1508.06448","version":4},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1508.06448","created_at":"2026-05-18T00:37:48Z"},{"alias_kind":"arxiv_version","alias_value":"1508.06448v4","created_at":"2026-05-18T00:37:48Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1508.06448","created_at":"2026-05-18T00:37:48Z"},{"alias_kind":"pith_short_12","alias_value":"QIJ72G5H4UNO","created_at":"2026-05-18T12:29:37Z"},{"alias_kind":"pith_short_16","alias_value":"QIJ72G5H4UNOMTRV","created_at":"2026-05-18T12:29:37Z"},{"alias_kind":"pith_short_8","alias_value":"QIJ72G5H","created_at":"2026-05-18T12:29:37Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2015:QIJ72G5H4UNOMTRV55ULM53OIQ","target":"record","payload":{"canonical_record":{"source":{"id":"1508.06448","kind":"arxiv","version":4},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math-ph","submitted_at":"2015-08-26T11:31:07Z","cross_cats_sorted":["math.CT","math.MP","math.PR"],"title_canon_sha256":"050ada523a58f183a0b543aea961e4e0a2d43f4705e67f93444340dc8dbd731c","abstract_canon_sha256":"66a4eafcde56014acac8a0a69e1ebec7efd4e64d9b2dd7e801cf7cb08108718e"},"schema_version":"1.0"},"canonical_sha256":"8213fd1ba7e51ae64e35ef68b6776e442ff290cdef0d4f099e0c1b2659cd0c9c","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:37:48.874797Z","signature_b64":"1VurlsL4RC4BHnoBr9aPTt+9R59TAp8qUAC3g1jd+2UOesxZq3K+KDWepXLL3Pf2/Ne1smtMU2KFPl0RPo4lAQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"8213fd1ba7e51ae64e35ef68b6776e442ff290cdef0d4f099e0c1b2659cd0c9c","last_reissued_at":"2026-05-18T00:37:48.874137Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:37:48.874137Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1508.06448","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":"2026-05-18T00:37:48Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"WrK6oglsE56qsPxZqFQc6ZEyA6+9tFJQB07iZSoPwR70WhAESFLlvIuOvSXN+A8xnRd80KTm1ArVTIXSY0oZAw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-20T13:56:21.603763Z"},"content_sha256":"0370e8afc71cc77039bb3894409235152dca8fa5b98eaacf18246d9df181d2f2","schema_version":"1.0","event_id":"sha256:0370e8afc71cc77039bb3894409235152dca8fa5b98eaacf18246d9df181d2f2"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2015:QIJ72G5H4UNOMTRV55ULM53OIQ","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"A Compositional Framework for Markov Processes","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CT","math.MP","math.PR"],"primary_cat":"math-ph","authors_text":"Blake S. Pollard, Brendan Fong, John C. Baez","submitted_at":"2015-08-26T11:31:07Z","abstract_excerpt":"We define the concept of an \"open\" Markov process, or more precisely, continuous-time Markov chain, which is one where probability can flow in or out of certain states called \"inputs\" and \"outputs\". One can build up a Markov process from smaller open pieces. This process is formalized by making open Markov processes into the morphisms of a dagger compact category. We show that the behavior of a detailed balanced open Markov process is determined by a principle of minimum dissipation, closely related to Prigogine's principle of minimum entropy production. Using this fact, we set up a functor ma"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1508.06448","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":""},"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":"2026-05-18T00:37:48Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"ShFpAbwXHNLgwqMGRXqOf3F8E1DRvZNMogdTUa4RHJGgZpKzxoEMqU+ZITRr9p5Ojtj90/4EZ3NXlUDnAUUnAQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-20T13:56:21.604402Z"},"content_sha256":"812843d575158ec7d5024e235b550c2345462458bf22eb22df5e3aa461973574","schema_version":"1.0","event_id":"sha256:812843d575158ec7d5024e235b550c2345462458bf22eb22df5e3aa461973574"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/QIJ72G5H4UNOMTRV55ULM53OIQ/bundle.json","state_url":"https://pith.science/pith/QIJ72G5H4UNOMTRV55ULM53OIQ/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/QIJ72G5H4UNOMTRV55ULM53OIQ/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":"2026-08-20T13:56:21Z","links":{"resolver":"https://pith.science/pith/QIJ72G5H4UNOMTRV55ULM53OIQ","bundle":"https://pith.science/pith/QIJ72G5H4UNOMTRV55ULM53OIQ/bundle.json","state":"https://pith.science/pith/QIJ72G5H4UNOMTRV55ULM53OIQ/state.json","well_known_bundle":"https://pith.science/.well-known/pith/QIJ72G5H4UNOMTRV55ULM53OIQ/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2015:QIJ72G5H4UNOMTRV55ULM53OIQ","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":"66a4eafcde56014acac8a0a69e1ebec7efd4e64d9b2dd7e801cf7cb08108718e","cross_cats_sorted":["math.CT","math.MP","math.PR"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math-ph","submitted_at":"2015-08-26T11:31:07Z","title_canon_sha256":"050ada523a58f183a0b543aea961e4e0a2d43f4705e67f93444340dc8dbd731c"},"schema_version":"1.0","source":{"id":"1508.06448","kind":"arxiv","version":4}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1508.06448","created_at":"2026-05-18T00:37:48Z"},{"alias_kind":"arxiv_version","alias_value":"1508.06448v4","created_at":"2026-05-18T00:37:48Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1508.06448","created_at":"2026-05-18T00:37:48Z"},{"alias_kind":"pith_short_12","alias_value":"QIJ72G5H4UNO","created_at":"2026-05-18T12:29:37Z"},{"alias_kind":"pith_short_16","alias_value":"QIJ72G5H4UNOMTRV","created_at":"2026-05-18T12:29:37Z"},{"alias_kind":"pith_short_8","alias_value":"QIJ72G5H","created_at":"2026-05-18T12:29:37Z"}],"graph_snapshots":[{"event_id":"sha256:812843d575158ec7d5024e235b550c2345462458bf22eb22df5e3aa461973574","target":"graph","created_at":"2026-05-18T00:37:48Z","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 define the concept of an \"open\" Markov process, or more precisely, continuous-time Markov chain, which is one where probability can flow in or out of certain states called \"inputs\" and \"outputs\". One can build up a Markov process from smaller open pieces. This process is formalized by making open Markov processes into the morphisms of a dagger compact category. We show that the behavior of a detailed balanced open Markov process is determined by a principle of minimum dissipation, closely related to Prigogine's principle of minimum entropy production. Using this fact, we set up a functor ma","authors_text":"Blake S. Pollard, Brendan Fong, John C. Baez","cross_cats":["math.CT","math.MP","math.PR"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math-ph","submitted_at":"2015-08-26T11:31:07Z","title":"A Compositional Framework for Markov Processes"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1508.06448","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":"sha256:0370e8afc71cc77039bb3894409235152dca8fa5b98eaacf18246d9df181d2f2","target":"record","created_at":"2026-05-18T00:37:48Z","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":"66a4eafcde56014acac8a0a69e1ebec7efd4e64d9b2dd7e801cf7cb08108718e","cross_cats_sorted":["math.CT","math.MP","math.PR"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math-ph","submitted_at":"2015-08-26T11:31:07Z","title_canon_sha256":"050ada523a58f183a0b543aea961e4e0a2d43f4705e67f93444340dc8dbd731c"},"schema_version":"1.0","source":{"id":"1508.06448","kind":"arxiv","version":4}},"canonical_sha256":"8213fd1ba7e51ae64e35ef68b6776e442ff290cdef0d4f099e0c1b2659cd0c9c","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"8213fd1ba7e51ae64e35ef68b6776e442ff290cdef0d4f099e0c1b2659cd0c9c","first_computed_at":"2026-05-18T00:37:48.874137Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:37:48.874137Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"1VurlsL4RC4BHnoBr9aPTt+9R59TAp8qUAC3g1jd+2UOesxZq3K+KDWepXLL3Pf2/Ne1smtMU2KFPl0RPo4lAQ==","signature_status":"signed_v1","signed_at":"2026-05-18T00:37:48.874797Z","signed_message":"canonical_sha256_bytes"},"source_id":"1508.06448","source_kind":"arxiv","source_version":4}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:0370e8afc71cc77039bb3894409235152dca8fa5b98eaacf18246d9df181d2f2","sha256:812843d575158ec7d5024e235b550c2345462458bf22eb22df5e3aa461973574"],"state_sha256":"537487816c008a713b45b836bf85ff970bcd280495c394a2099aa6b86ffc66b9"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"a7CfqHJxnzyxGt8T8z4r3hROdE0NfEd8IpFgw9DqGInCCpWnEZMpe9h/KKQAoQzE5bXUQhcNo0utdDYGwpR3Cg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-20T13:56:21.611542Z","bundle_sha256":"da1dfa2b6045e038498f849a9352f1329f359beb2cc7c8d11750298dd1c72df4"}}