{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2025:UWPT7YT4INTX4ZBUMSG27BPHF7","short_pith_number":"pith:UWPT7YT4","canonical_record":{"source":{"id":"2502.15276","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2025-02-21T08:04:57Z","cross_cats_sorted":["cs.SY","eess.SY","math.CT"],"title_canon_sha256":"c556a60cca075dbb0e59a42b6ac1bbaad2061c87b796b75b5b8b6f45528b7d8c","abstract_canon_sha256":"ec014b3c73f81d6dde2100ef97e1ac23841929d71debe335cc956bd5a2057f89"},"schema_version":"1.0"},"canonical_sha256":"a59f3fe27c43677e6434648daf85e72ffd10d0921c05bd9526ae66f09ff988d8","source":{"kind":"arxiv","id":"2502.15276","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2502.15276","created_at":"2026-07-05T11:45:55Z"},{"alias_kind":"arxiv_version","alias_value":"2502.15276v2","created_at":"2026-07-05T11:45:55Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2502.15276","created_at":"2026-07-05T11:45:55Z"},{"alias_kind":"pith_short_12","alias_value":"UWPT7YT4INTX","created_at":"2026-07-05T11:45:55Z"},{"alias_kind":"pith_short_16","alias_value":"UWPT7YT4INTX4ZBU","created_at":"2026-07-05T11:45:55Z"},{"alias_kind":"pith_short_8","alias_value":"UWPT7YT4","created_at":"2026-07-05T11:45:55Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2025:UWPT7YT4INTX4ZBUMSG27BPHF7","target":"record","payload":{"canonical_record":{"source":{"id":"2502.15276","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2025-02-21T08:04:57Z","cross_cats_sorted":["cs.SY","eess.SY","math.CT"],"title_canon_sha256":"c556a60cca075dbb0e59a42b6ac1bbaad2061c87b796b75b5b8b6f45528b7d8c","abstract_canon_sha256":"ec014b3c73f81d6dde2100ef97e1ac23841929d71debe335cc956bd5a2057f89"},"schema_version":"1.0"},"canonical_sha256":"a59f3fe27c43677e6434648daf85e72ffd10d0921c05bd9526ae66f09ff988d8","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T11:45:55.972238Z","signature_b64":"ImdVV7Vde11EaoOw47cMoxslIXT+2+Slk3Qboa8LxNamosnXPoinvzu0a+wJEelkUa60Bt0G4IeCF/6unWbOBg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"a59f3fe27c43677e6434648daf85e72ffd10d0921c05bd9526ae66f09ff988d8","last_reissued_at":"2026-07-05T11:45:55.971711Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T11:45:55.971711Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2502.15276","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":"2026-07-05T11:45:55Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"7yKMM07qetXM+0gHKOObkFn7ZnZMQb/QFHonASKBGHO76APwBEWR3NrFnvM9BVvXvaxjX2Hga/9fKSteuJMhAg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-08T08:05:05.351740Z"},"content_sha256":"24fc61a8bb29b4445e2a3991cb528dadc025214c81a9aaec1824e705da68d4ca","schema_version":"1.0","event_id":"sha256:24fc61a8bb29b4445e2a3991cb528dadc025214c81a9aaec1824e705da68d4ca"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2025:UWPT7YT4INTX4ZBUMSG27BPHF7","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Categorical Lyapunov Theory I: Stability of Flows","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.SY","eess.SY","math.CT"],"primary_cat":"math.DS","authors_text":"Aaron D. Ames, Joe Moeller, Paulo Tabuada","submitted_at":"2025-02-21T08:04:57Z","abstract_excerpt":"Lyapunov's theorem provides a fundamental characterization of the stability of dynamical systems. This paper presents a categorical framework for Lyapunov theory, generalizing stability analysis with Lyapunov functions categorically. Core to our approach is the set of axioms underlying a setting for stability, which give the necessary ingredients for ``doing Lyapunov theory'' in a category of interest. With these minimal assumptions, we define the stability of equilibria, formulate Lyapunov morphisms, and demonstrate that the existence of Lyapunov morphisms is necessary and sufficient for esta"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2502.15276","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/2502.15276/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":"2026-07-05T11:45:55Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"t1YP6AGVE2EUEvn8nBRqGfzRgLqe/9wb5ByusQsHTNNF2GJqv4vNs9KyFwzh0rB4U4Upp7fxvA2YcmhUCKhTDg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-08T08:05:05.352337Z"},"content_sha256":"e2d48a05bcbed12e5ec013ef298c3eeb911bf4c05af50adcbe62b98284ca9d1e","schema_version":"1.0","event_id":"sha256:e2d48a05bcbed12e5ec013ef298c3eeb911bf4c05af50adcbe62b98284ca9d1e"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/UWPT7YT4INTX4ZBUMSG27BPHF7/bundle.json","state_url":"https://pith.science/pith/UWPT7YT4INTX4ZBUMSG27BPHF7/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/UWPT7YT4INTX4ZBUMSG27BPHF7/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":"2026-08-08T08:05:05Z","links":{"resolver":"https://pith.science/pith/UWPT7YT4INTX4ZBUMSG27BPHF7","bundle":"https://pith.science/pith/UWPT7YT4INTX4ZBUMSG27BPHF7/bundle.json","state":"https://pith.science/pith/UWPT7YT4INTX4ZBUMSG27BPHF7/state.json","well_known_bundle":"https://pith.science/.well-known/pith/UWPT7YT4INTX4ZBUMSG27BPHF7/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2025:UWPT7YT4INTX4ZBUMSG27BPHF7","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":"ec014b3c73f81d6dde2100ef97e1ac23841929d71debe335cc956bd5a2057f89","cross_cats_sorted":["cs.SY","eess.SY","math.CT"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2025-02-21T08:04:57Z","title_canon_sha256":"c556a60cca075dbb0e59a42b6ac1bbaad2061c87b796b75b5b8b6f45528b7d8c"},"schema_version":"1.0","source":{"id":"2502.15276","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2502.15276","created_at":"2026-07-05T11:45:55Z"},{"alias_kind":"arxiv_version","alias_value":"2502.15276v2","created_at":"2026-07-05T11:45:55Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2502.15276","created_at":"2026-07-05T11:45:55Z"},{"alias_kind":"pith_short_12","alias_value":"UWPT7YT4INTX","created_at":"2026-07-05T11:45:55Z"},{"alias_kind":"pith_short_16","alias_value":"UWPT7YT4INTX4ZBU","created_at":"2026-07-05T11:45:55Z"},{"alias_kind":"pith_short_8","alias_value":"UWPT7YT4","created_at":"2026-07-05T11:45:55Z"}],"graph_snapshots":[{"event_id":"sha256:e2d48a05bcbed12e5ec013ef298c3eeb911bf4c05af50adcbe62b98284ca9d1e","target":"graph","created_at":"2026-07-05T11:45:55Z","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/2502.15276/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"Lyapunov's theorem provides a fundamental characterization of the stability of dynamical systems. This paper presents a categorical framework for Lyapunov theory, generalizing stability analysis with Lyapunov functions categorically. Core to our approach is the set of axioms underlying a setting for stability, which give the necessary ingredients for ``doing Lyapunov theory'' in a category of interest. With these minimal assumptions, we define the stability of equilibria, formulate Lyapunov morphisms, and demonstrate that the existence of Lyapunov morphisms is necessary and sufficient for esta","authors_text":"Aaron D. Ames, Joe Moeller, Paulo Tabuada","cross_cats":["cs.SY","eess.SY","math.CT"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2025-02-21T08:04:57Z","title":"Categorical Lyapunov Theory I: Stability of Flows"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2502.15276","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":"sha256:24fc61a8bb29b4445e2a3991cb528dadc025214c81a9aaec1824e705da68d4ca","target":"record","created_at":"2026-07-05T11:45:55Z","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":"ec014b3c73f81d6dde2100ef97e1ac23841929d71debe335cc956bd5a2057f89","cross_cats_sorted":["cs.SY","eess.SY","math.CT"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2025-02-21T08:04:57Z","title_canon_sha256":"c556a60cca075dbb0e59a42b6ac1bbaad2061c87b796b75b5b8b6f45528b7d8c"},"schema_version":"1.0","source":{"id":"2502.15276","kind":"arxiv","version":2}},"canonical_sha256":"a59f3fe27c43677e6434648daf85e72ffd10d0921c05bd9526ae66f09ff988d8","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"a59f3fe27c43677e6434648daf85e72ffd10d0921c05bd9526ae66f09ff988d8","first_computed_at":"2026-07-05T11:45:55.971711Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T11:45:55.971711Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"ImdVV7Vde11EaoOw47cMoxslIXT+2+Slk3Qboa8LxNamosnXPoinvzu0a+wJEelkUa60Bt0G4IeCF/6unWbOBg==","signature_status":"signed_v1","signed_at":"2026-07-05T11:45:55.972238Z","signed_message":"canonical_sha256_bytes"},"source_id":"2502.15276","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:24fc61a8bb29b4445e2a3991cb528dadc025214c81a9aaec1824e705da68d4ca","sha256:e2d48a05bcbed12e5ec013ef298c3eeb911bf4c05af50adcbe62b98284ca9d1e"],"state_sha256":"85dfde1f207d2628548be5425e1b843b0d37e98c9be273c03c7fae70e9be7ac6"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"kZGvuo3gF9LnpGm+B+VIfobKv02blKtWKDs1G7ERRhV3C1qUs8q6ljhejIK9jt94c/c8rOsNXPoQuvo62x9xBQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-08T08:05:05.357911Z","bundle_sha256":"0db5a53e0d8842b0823ddd5818476d4e7dea73478c7bb7d20baa237c9e5cebf5"}}