{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2026:C6JH6NCBZNYX7ASH26GT2WO3TS","short_pith_number":"pith:C6JH6NCB","canonical_record":{"source":{"id":"2605.12883","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.AP","submitted_at":"2026-05-13T01:57:33Z","cross_cats_sorted":[],"title_canon_sha256":"9fa5f56df8d8a9019e80c6ccc3d6c1e92388a991f4d5535ffe2372e37465c774","abstract_canon_sha256":"002c0e91c278888340a876b02fce17c06c8521b744ccf821f17216111d17af59"},"schema_version":"1.0"},"canonical_sha256":"17927f3441cb717f8247d78d3d59db9cbb3443339426cf3f99ddef603145fff5","source":{"kind":"arxiv","id":"2605.12883","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2605.12883","created_at":"2026-05-18T03:09:11Z"},{"alias_kind":"arxiv_version","alias_value":"2605.12883v1","created_at":"2026-05-18T03:09:11Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2605.12883","created_at":"2026-05-18T03:09:11Z"},{"alias_kind":"pith_short_12","alias_value":"C6JH6NCBZNYX","created_at":"2026-05-18T12:33:37Z"},{"alias_kind":"pith_short_16","alias_value":"C6JH6NCBZNYX7ASH","created_at":"2026-05-18T12:33:37Z"},{"alias_kind":"pith_short_8","alias_value":"C6JH6NCB","created_at":"2026-05-18T12:33:37Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2026:C6JH6NCBZNYX7ASH26GT2WO3TS","target":"record","payload":{"canonical_record":{"source":{"id":"2605.12883","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.AP","submitted_at":"2026-05-13T01:57:33Z","cross_cats_sorted":[],"title_canon_sha256":"9fa5f56df8d8a9019e80c6ccc3d6c1e92388a991f4d5535ffe2372e37465c774","abstract_canon_sha256":"002c0e91c278888340a876b02fce17c06c8521b744ccf821f17216111d17af59"},"schema_version":"1.0"},"canonical_sha256":"17927f3441cb717f8247d78d3d59db9cbb3443339426cf3f99ddef603145fff5","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:09:11.034512Z","signature_b64":"vQ52aCEWnXn8I3rpWaRxFoKESdTaDCqb4NpWnd01pRHQ6OW0t6Y9b3rr3iEcuUCmjpDeEwvMUMuaWOS+5CYMDw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"17927f3441cb717f8247d78d3d59db9cbb3443339426cf3f99ddef603145fff5","last_reissued_at":"2026-05-18T03:09:11.033578Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:09:11.033578Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2605.12883","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":"2026-05-18T03:09:11Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"+hKNqLEu7bOj6qFEXwaKtLuTjeLHpv2n+0Z49V84XsfFXkCW5SJzwcCNhMCqPIZrb2k8Zkgm/ph6CFIo99eNDw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-31T08:20:58.062625Z"},"content_sha256":"da561560586e15a35a8dbfc516668de0e2c504adad3ee5f972ed89b01902cadc","schema_version":"1.0","event_id":"sha256:da561560586e15a35a8dbfc516668de0e2c504adad3ee5f972ed89b01902cadc"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2026:C6JH6NCBZNYX7ASH26GT2WO3TS","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Mixing and Small-Scale Formation in a Passive Divergence-Free Vector Field","license":"http://creativecommons.org/licenses/by/4.0/","headline":"Numerical simulations indicate that passive divergence-free vector fields mix at least exponentially when the advecting field is chosen at each instant to maximize instantaneous decay of the negative Sobolev norm.","cross_cats":[],"primary_cat":"math.AP","authors_text":"Anuj Kumar, Franziska Weber","submitted_at":"2026-05-13T01:57:33Z","abstract_excerpt":"We study mixing for a divergence-free passive vector field $u$ transported by another divergence-free vector field $U$, where $u$ evolves according to $ \\partial_t u + (U \\cdot \\nabla) u + \\nabla p = 0.$ In recent years, a lot of attention has been given to the question of optimal mixing in the scalar case, where there is a Sobolev constraint on the advecting velocity. In the vector setting considered here, however, the pressure term introduces substantial difficulties, since the simple Lagrangian perspective available in the scalar case is no longer applicable.\n  In this paper, we investigate"},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"Numerical simulations provide evidence that the optimal mixing rate is at least exponential in time.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"That a divergence-free U with bounded W^{1,q} norm exists at each instant which instantaneously maximizes the decay of the H^{-α} norm of u, and that the resulting evolution remains well-defined.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"Derives lower bounds on mixing rates for passive divergence-free vector fields under W^{1,q} constraints and provides numerical evidence for at least exponential optimal mixing via H^{-α} norm decay.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"Numerical simulations indicate that passive divergence-free vector fields mix at least exponentially when the advecting field is chosen at each instant to maximize instantaneous decay of the negative Sobolev norm.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"53592b250fc30bf57f5de4a43b95a794140d8002b5d6daa8a89b1679f2e7c7c2"},"source":{"id":"2605.12883","kind":"arxiv","version":1},"verdict":{"id":"b96ace55-e8ae-48ca-9e14-86b91990ccfa","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-14T19:01:50.766280Z","strongest_claim":"Numerical simulations provide evidence that the optimal mixing rate is at least exponential in time.","one_line_summary":"Derives lower bounds on mixing rates for passive divergence-free vector fields under W^{1,q} constraints and provides numerical evidence for at least exponential optimal mixing via H^{-α} norm decay.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"That a divergence-free U with bounded W^{1,q} norm exists at each instant which instantaneously maximizes the decay of the H^{-α} norm of u, and that the resulting evolution remains well-defined.","pith_extraction_headline":"Numerical simulations indicate that passive divergence-free vector fields mix at least exponentially when the advecting field is chosen at each instant to maximize instantaneous decay of the negative Sobolev norm."},"references":{"count":34,"sample":[{"doi":"","year":2019,"title":"G. Alberti, G. Crippa, and A. L. Mazzucato. Exponential self-similar mixing by incompressible flows.J. Amer. Math. Soc., 32(2):445–490, 2019","work_id":"24354c68-bf6a-4f80-9757-acc993d34dae","ref_index":1,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2003,"title":"A. Bressan. A lemma and a conjecture on the cost of rearrangements.Rendiconti del Seminario Matematico della Universita di Padova, 110:97–102, 2003","work_id":"97e98849-dc5a-4222-99af-49a2eb2560a6","ref_index":2,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2018,"title":"H. Brezis and P. Mironescu. Gagliardo-Nirenberg inequalities and non-inequalities: the full story. Ann. Inst. H. Poincar´ e C Anal. Non Lin´ eaire, 35(5):1355–1376, 2018. 24 Mixing and Small-Scale For","work_id":"223ff261-0efa-4f8d-974d-7d4834292a1c","ref_index":3,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2019,"title":"H. Brezis and P. Mironescu. Where Sobolev interacts with Gagliardo-Nirenberg.J. Funct. Anal., 277(8):2839–2864, 2019","work_id":"7466ed71-4cf7-4261-a212-89eb5b611134","ref_index":4,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2024,"title":"E. Bru` e, M. Colombo, and A. Kumar. Sharp Nonuniqueness in the Transport Equation with Sobolev Velocity Field.arXiv preprint arXiv:2405.01670, 2024","work_id":"98c9c402-c853-4c6a-bb80-bd65c826b09d","ref_index":5,"cited_arxiv_id":"","is_internal_anchor":false}],"resolved_work":34,"snapshot_sha256":"7b7b6f80381aac2578a09939379e421c1144804784aaeabf1b2e47f78f890ada","internal_anchors":0},"formal_canon":{"evidence_count":2,"snapshot_sha256":"32c9b3cb00f02d03abe922af88fab66cae5506d1fbe72f19a13f379424b7e683"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"},"verdict_id":"b96ace55-e8ae-48ca-9e14-86b91990ccfa"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T03:09:11Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"xTxH/lw3Rh/62UnU/7RmV8agCmbGVceA8/siHwY62TXDixN71CB1knIyr7jSgI1tPohNwpH/9JEn0rgKY/UXBg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-31T08:20:58.063697Z"},"content_sha256":"64f8a12e0deb54645ced8c958f7996df4dba7d94a58fc3d617944744a55495fe","schema_version":"1.0","event_id":"sha256:64f8a12e0deb54645ced8c958f7996df4dba7d94a58fc3d617944744a55495fe"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/C6JH6NCBZNYX7ASH26GT2WO3TS/bundle.json","state_url":"https://pith.science/pith/C6JH6NCBZNYX7ASH26GT2WO3TS/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/C6JH6NCBZNYX7ASH26GT2WO3TS/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":"2026-05-31T08:20:58Z","links":{"resolver":"https://pith.science/pith/C6JH6NCBZNYX7ASH26GT2WO3TS","bundle":"https://pith.science/pith/C6JH6NCBZNYX7ASH26GT2WO3TS/bundle.json","state":"https://pith.science/pith/C6JH6NCBZNYX7ASH26GT2WO3TS/state.json","well_known_bundle":"https://pith.science/.well-known/pith/C6JH6NCBZNYX7ASH26GT2WO3TS/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2026:C6JH6NCBZNYX7ASH26GT2WO3TS","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":"002c0e91c278888340a876b02fce17c06c8521b744ccf821f17216111d17af59","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.AP","submitted_at":"2026-05-13T01:57:33Z","title_canon_sha256":"9fa5f56df8d8a9019e80c6ccc3d6c1e92388a991f4d5535ffe2372e37465c774"},"schema_version":"1.0","source":{"id":"2605.12883","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2605.12883","created_at":"2026-05-18T03:09:11Z"},{"alias_kind":"arxiv_version","alias_value":"2605.12883v1","created_at":"2026-05-18T03:09:11Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2605.12883","created_at":"2026-05-18T03:09:11Z"},{"alias_kind":"pith_short_12","alias_value":"C6JH6NCBZNYX","created_at":"2026-05-18T12:33:37Z"},{"alias_kind":"pith_short_16","alias_value":"C6JH6NCBZNYX7ASH","created_at":"2026-05-18T12:33:37Z"},{"alias_kind":"pith_short_8","alias_value":"C6JH6NCB","created_at":"2026-05-18T12:33:37Z"}],"graph_snapshots":[{"event_id":"sha256:64f8a12e0deb54645ced8c958f7996df4dba7d94a58fc3d617944744a55495fe","target":"graph","created_at":"2026-05-18T03:09:11Z","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":4,"items":[{"attestation":"unclaimed","claim_id":"C1","kind":"strongest_claim","source":"verdict.strongest_claim","status":"machine_extracted","text":"Numerical simulations provide evidence that the optimal mixing rate is at least exponential in time."},{"attestation":"unclaimed","claim_id":"C2","kind":"weakest_assumption","source":"verdict.weakest_assumption","status":"machine_extracted","text":"That a divergence-free U with bounded W^{1,q} norm exists at each instant which instantaneously maximizes the decay of the H^{-α} norm of u, and that the resulting evolution remains well-defined."},{"attestation":"unclaimed","claim_id":"C3","kind":"one_line_summary","source":"verdict.one_line_summary","status":"machine_extracted","text":"Derives lower bounds on mixing rates for passive divergence-free vector fields under W^{1,q} constraints and provides numerical evidence for at least exponential optimal mixing via H^{-α} norm decay."},{"attestation":"unclaimed","claim_id":"C4","kind":"headline","source":"verdict.pith_extraction.headline","status":"machine_extracted","text":"Numerical simulations indicate that passive divergence-free vector fields mix at least exponentially when the advecting field is chosen at each instant to maximize instantaneous decay of the negative Sobolev norm."}],"snapshot_sha256":"53592b250fc30bf57f5de4a43b95a794140d8002b5d6daa8a89b1679f2e7c7c2"},"formal_canon":{"evidence_count":2,"snapshot_sha256":"32c9b3cb00f02d03abe922af88fab66cae5506d1fbe72f19a13f379424b7e683"},"paper":{"abstract_excerpt":"We study mixing for a divergence-free passive vector field $u$ transported by another divergence-free vector field $U$, where $u$ evolves according to $ \\partial_t u + (U \\cdot \\nabla) u + \\nabla p = 0.$ In recent years, a lot of attention has been given to the question of optimal mixing in the scalar case, where there is a Sobolev constraint on the advecting velocity. In the vector setting considered here, however, the pressure term introduces substantial difficulties, since the simple Lagrangian perspective available in the scalar case is no longer applicable.\n  In this paper, we investigate","authors_text":"Anuj Kumar, Franziska Weber","cross_cats":[],"headline":"Numerical simulations indicate that passive divergence-free vector fields mix at least exponentially when the advecting field is chosen at each instant to maximize instantaneous decay of the negative Sobolev norm.","license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.AP","submitted_at":"2026-05-13T01:57:33Z","title":"Mixing and Small-Scale Formation in a Passive Divergence-Free Vector Field"},"references":{"count":34,"internal_anchors":0,"resolved_work":34,"sample":[{"cited_arxiv_id":"","doi":"","is_internal_anchor":false,"ref_index":1,"title":"G. Alberti, G. Crippa, and A. L. Mazzucato. Exponential self-similar mixing by incompressible flows.J. Amer. Math. Soc., 32(2):445–490, 2019","work_id":"24354c68-bf6a-4f80-9757-acc993d34dae","year":2019},{"cited_arxiv_id":"","doi":"","is_internal_anchor":false,"ref_index":2,"title":"A. Bressan. A lemma and a conjecture on the cost of rearrangements.Rendiconti del Seminario Matematico della Universita di Padova, 110:97–102, 2003","work_id":"97e98849-dc5a-4222-99af-49a2eb2560a6","year":2003},{"cited_arxiv_id":"","doi":"","is_internal_anchor":false,"ref_index":3,"title":"H. Brezis and P. Mironescu. Gagliardo-Nirenberg inequalities and non-inequalities: the full story. Ann. Inst. H. Poincar´ e C Anal. Non Lin´ eaire, 35(5):1355–1376, 2018. 24 Mixing and Small-Scale For","work_id":"223ff261-0efa-4f8d-974d-7d4834292a1c","year":2018},{"cited_arxiv_id":"","doi":"","is_internal_anchor":false,"ref_index":4,"title":"H. Brezis and P. Mironescu. Where Sobolev interacts with Gagliardo-Nirenberg.J. Funct. Anal., 277(8):2839–2864, 2019","work_id":"7466ed71-4cf7-4261-a212-89eb5b611134","year":2019},{"cited_arxiv_id":"","doi":"","is_internal_anchor":false,"ref_index":5,"title":"E. Bru` e, M. Colombo, and A. Kumar. Sharp Nonuniqueness in the Transport Equation with Sobolev Velocity Field.arXiv preprint arXiv:2405.01670, 2024","work_id":"98c9c402-c853-4c6a-bb80-bd65c826b09d","year":2024}],"snapshot_sha256":"7b7b6f80381aac2578a09939379e421c1144804784aaeabf1b2e47f78f890ada"},"source":{"id":"2605.12883","kind":"arxiv","version":1},"verdict":{"created_at":"2026-05-14T19:01:50.766280Z","id":"b96ace55-e8ae-48ca-9e14-86b91990ccfa","model_set":{"reader":"grok-4.3"},"one_line_summary":"Derives lower bounds on mixing rates for passive divergence-free vector fields under W^{1,q} constraints and provides numerical evidence for at least exponential optimal mixing via H^{-α} norm decay.","pipeline_version":"pith-pipeline@v0.9.0","pith_extraction_headline":"Numerical simulations indicate that passive divergence-free vector fields mix at least exponentially when the advecting field is chosen at each instant to maximize instantaneous decay of the negative Sobolev norm.","strongest_claim":"Numerical simulations provide evidence that the optimal mixing rate is at least exponential in time.","weakest_assumption":"That a divergence-free U with bounded W^{1,q} norm exists at each instant which instantaneously maximizes the decay of the H^{-α} norm of u, and that the resulting evolution remains well-defined."}},"verdict_id":"b96ace55-e8ae-48ca-9e14-86b91990ccfa"}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:da561560586e15a35a8dbfc516668de0e2c504adad3ee5f972ed89b01902cadc","target":"record","created_at":"2026-05-18T03:09:11Z","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":"002c0e91c278888340a876b02fce17c06c8521b744ccf821f17216111d17af59","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.AP","submitted_at":"2026-05-13T01:57:33Z","title_canon_sha256":"9fa5f56df8d8a9019e80c6ccc3d6c1e92388a991f4d5535ffe2372e37465c774"},"schema_version":"1.0","source":{"id":"2605.12883","kind":"arxiv","version":1}},"canonical_sha256":"17927f3441cb717f8247d78d3d59db9cbb3443339426cf3f99ddef603145fff5","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"17927f3441cb717f8247d78d3d59db9cbb3443339426cf3f99ddef603145fff5","first_computed_at":"2026-05-18T03:09:11.033578Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:09:11.033578Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"vQ52aCEWnXn8I3rpWaRxFoKESdTaDCqb4NpWnd01pRHQ6OW0t6Y9b3rr3iEcuUCmjpDeEwvMUMuaWOS+5CYMDw==","signature_status":"signed_v1","signed_at":"2026-05-18T03:09:11.034512Z","signed_message":"canonical_sha256_bytes"},"source_id":"2605.12883","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:da561560586e15a35a8dbfc516668de0e2c504adad3ee5f972ed89b01902cadc","sha256:64f8a12e0deb54645ced8c958f7996df4dba7d94a58fc3d617944744a55495fe"],"state_sha256":"abbf086607f2bdc286e41c684b2eee4cee3e91d2dc84948c180941a04295477a"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"IKIaMglZeTR+H52qurKPgco8ZkejFGucx5CG3NdzjbE3BxdbSr78gF4nrI5diqGBtuNOZnX/+E31yc0htd7VCQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-31T08:20:58.068089Z","bundle_sha256":"2940f278a48c1bc2c9094a8559d4fa19628b4b582f4c8f002bac578e2dfc03d9"}}