{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2026:7HAHO3X56XGZBOWCVCJHLHG6FC","short_pith_number":"pith:7HAHO3X5","canonical_record":{"source":{"id":"2607.05097","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.PR","submitted_at":"2026-07-06T13:55:34Z","cross_cats_sorted":["math.DS"],"title_canon_sha256":"43fbfa3fb54be2828c80e4e0e254f2eba277a09d1b48fde46544fc17e06e7b22","abstract_canon_sha256":"efb663096cd1992a277bbd3bedb1815879b14f0bfd7365d3bd9fb52d8651a5c3"},"schema_version":"1.0"},"canonical_sha256":"f9c0776efdf5cd90bac2a892759cde28908e05d96bd20959086cbfa3f8907b05","source":{"kind":"arxiv","id":"2607.05097","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2607.05097","created_at":"2026-07-07T03:19:15Z"},{"alias_kind":"arxiv_version","alias_value":"2607.05097v1","created_at":"2026-07-07T03:19:15Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2607.05097","created_at":"2026-07-07T03:19:15Z"},{"alias_kind":"pith_short_12","alias_value":"7HAHO3X56XGZ","created_at":"2026-07-07T03:19:15Z"},{"alias_kind":"pith_short_16","alias_value":"7HAHO3X56XGZBOWC","created_at":"2026-07-07T03:19:15Z"},{"alias_kind":"pith_short_8","alias_value":"7HAHO3X5","created_at":"2026-07-07T03:19:15Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2026:7HAHO3X56XGZBOWCVCJHLHG6FC","target":"record","payload":{"canonical_record":{"source":{"id":"2607.05097","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.PR","submitted_at":"2026-07-06T13:55:34Z","cross_cats_sorted":["math.DS"],"title_canon_sha256":"43fbfa3fb54be2828c80e4e0e254f2eba277a09d1b48fde46544fc17e06e7b22","abstract_canon_sha256":"efb663096cd1992a277bbd3bedb1815879b14f0bfd7365d3bd9fb52d8651a5c3"},"schema_version":"1.0"},"canonical_sha256":"f9c0776efdf5cd90bac2a892759cde28908e05d96bd20959086cbfa3f8907b05","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-07T03:19:15.929785Z","signature_b64":"+33Dg4fJcn+DbTfAefJWC/L3Wtn4xjyQ+vuZ0COKkuBMdBbplzLqUf4ozS3hiAIQQf/213Qa5mn3/U7XZLdLCA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"f9c0776efdf5cd90bac2a892759cde28908e05d96bd20959086cbfa3f8907b05","last_reissued_at":"2026-07-07T03:19:15.929373Z","signature_status":"signed_v1","first_computed_at":"2026-07-07T03:19:15.929373Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2607.05097","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":"2026-07-07T03:19:15Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"WmPvbfUGQUJ7tr7HfbwTjx7QgXMVPF0Pj6la5DmNxSF3hQY9R3UjaSygSibLIRCUCXhL1uNf6xZXG2+4OvwZBw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-08T09:14:09.717251Z"},"content_sha256":"84c48f386029d78800d2bcf00da38f0428994b6dc08e9bbb33ffa0844f3b666d","schema_version":"1.0","event_id":"sha256:84c48f386029d78800d2bcf00da38f0428994b6dc08e9bbb33ffa0844f3b666d"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2026:7HAHO3X56XGZBOWCVCJHLHG6FC","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Asymptotics of Lyapunov Exponents and Phase Transitions for Fluids with Degenerate Forcing","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["math.DS"],"primary_cat":"math.PR","authors_text":"Declan Stacy, Martin Hairer","submitted_at":"2026-07-06T13:55:34Z","abstract_excerpt":"In this paper we analyze the Lyapunov exponents of a slow-fast system where the slow component is an Ornstein-Uhlenbeck process which perturbs the linear evolution of a fast variable through a bilinear form. These naturally arise in many finite-dimensional models for turbulence such as Galerkin truncation of 2D Navier-Stokes, the Lorenz 96 system, and the Lorenz 63 system. Using our general results about the slow-fast system, we are able to prove phase transitions in the ergodicity of each of these models when degenerate stochastic forcing is applied: as a parameter (e.g. noise strength or vis"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2607.05097","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2607.05097/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":"2026-07-07T03:19:15Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"/sUwVZ4+yg0UKXEOPV43ruyMK3z6gz6GiKCLHmWb0kj3P8aBUVwuNfvt6zU4/KUsRr1Ybc9aUzLOKh0TYaz+Bw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-08T09:14:09.718151Z"},"content_sha256":"a0316ec012e2351aae1565288026cb8f14d576485e5200d38abe7c95fcd267b9","schema_version":"1.0","event_id":"sha256:a0316ec012e2351aae1565288026cb8f14d576485e5200d38abe7c95fcd267b9"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/7HAHO3X56XGZBOWCVCJHLHG6FC/bundle.json","state_url":"https://pith.science/pith/7HAHO3X56XGZBOWCVCJHLHG6FC/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/7HAHO3X56XGZBOWCVCJHLHG6FC/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":"2026-08-08T09:14:09Z","links":{"resolver":"https://pith.science/pith/7HAHO3X56XGZBOWCVCJHLHG6FC","bundle":"https://pith.science/pith/7HAHO3X56XGZBOWCVCJHLHG6FC/bundle.json","state":"https://pith.science/pith/7HAHO3X56XGZBOWCVCJHLHG6FC/state.json","well_known_bundle":"https://pith.science/.well-known/pith/7HAHO3X56XGZBOWCVCJHLHG6FC/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2026:7HAHO3X56XGZBOWCVCJHLHG6FC","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":"efb663096cd1992a277bbd3bedb1815879b14f0bfd7365d3bd9fb52d8651a5c3","cross_cats_sorted":["math.DS"],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.PR","submitted_at":"2026-07-06T13:55:34Z","title_canon_sha256":"43fbfa3fb54be2828c80e4e0e254f2eba277a09d1b48fde46544fc17e06e7b22"},"schema_version":"1.0","source":{"id":"2607.05097","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2607.05097","created_at":"2026-07-07T03:19:15Z"},{"alias_kind":"arxiv_version","alias_value":"2607.05097v1","created_at":"2026-07-07T03:19:15Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2607.05097","created_at":"2026-07-07T03:19:15Z"},{"alias_kind":"pith_short_12","alias_value":"7HAHO3X56XGZ","created_at":"2026-07-07T03:19:15Z"},{"alias_kind":"pith_short_16","alias_value":"7HAHO3X56XGZBOWC","created_at":"2026-07-07T03:19:15Z"},{"alias_kind":"pith_short_8","alias_value":"7HAHO3X5","created_at":"2026-07-07T03:19:15Z"}],"graph_snapshots":[{"event_id":"sha256:a0316ec012e2351aae1565288026cb8f14d576485e5200d38abe7c95fcd267b9","target":"graph","created_at":"2026-07-07T03:19:15Z","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/2607.05097/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"In this paper we analyze the Lyapunov exponents of a slow-fast system where the slow component is an Ornstein-Uhlenbeck process which perturbs the linear evolution of a fast variable through a bilinear form. These naturally arise in many finite-dimensional models for turbulence such as Galerkin truncation of 2D Navier-Stokes, the Lorenz 96 system, and the Lorenz 63 system. Using our general results about the slow-fast system, we are able to prove phase transitions in the ergodicity of each of these models when degenerate stochastic forcing is applied: as a parameter (e.g. noise strength or vis","authors_text":"Declan Stacy, Martin Hairer","cross_cats":["math.DS"],"headline":"","license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.PR","submitted_at":"2026-07-06T13:55:34Z","title":"Asymptotics of Lyapunov Exponents and Phase Transitions for Fluids with Degenerate Forcing"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2607.05097","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":"sha256:84c48f386029d78800d2bcf00da38f0428994b6dc08e9bbb33ffa0844f3b666d","target":"record","created_at":"2026-07-07T03:19:15Z","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":"efb663096cd1992a277bbd3bedb1815879b14f0bfd7365d3bd9fb52d8651a5c3","cross_cats_sorted":["math.DS"],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.PR","submitted_at":"2026-07-06T13:55:34Z","title_canon_sha256":"43fbfa3fb54be2828c80e4e0e254f2eba277a09d1b48fde46544fc17e06e7b22"},"schema_version":"1.0","source":{"id":"2607.05097","kind":"arxiv","version":1}},"canonical_sha256":"f9c0776efdf5cd90bac2a892759cde28908e05d96bd20959086cbfa3f8907b05","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"f9c0776efdf5cd90bac2a892759cde28908e05d96bd20959086cbfa3f8907b05","first_computed_at":"2026-07-07T03:19:15.929373Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-07T03:19:15.929373Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"+33Dg4fJcn+DbTfAefJWC/L3Wtn4xjyQ+vuZ0COKkuBMdBbplzLqUf4ozS3hiAIQQf/213Qa5mn3/U7XZLdLCA==","signature_status":"signed_v1","signed_at":"2026-07-07T03:19:15.929785Z","signed_message":"canonical_sha256_bytes"},"source_id":"2607.05097","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:84c48f386029d78800d2bcf00da38f0428994b6dc08e9bbb33ffa0844f3b666d","sha256:a0316ec012e2351aae1565288026cb8f14d576485e5200d38abe7c95fcd267b9"],"state_sha256":"675bb296ecea07319667393fb035215e5788a7d6a7b3364c9d4b1a4406998ed5"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"uSb1pnoEUYnJdwTVojbQQK3uyEYmgq+Tsj2EhSsDD9jtJIIjMLugecJM6qmaPxRFJHXrE2E5ocyA2ReKv0iyDA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-08T09:14:09.724416Z","bundle_sha256":"5b34741ad1454725b357abe10495d6a4626b0df7afe8f95ae978e6568460ac20"}}