{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2025:ALS7KGBYZBU5Z3X4GSBZOPWTJ5","short_pith_number":"pith:ALS7KGBY","canonical_record":{"source":{"id":"2509.08168","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.AP","submitted_at":"2025-09-09T22:11:10Z","cross_cats_sorted":["math.FA"],"title_canon_sha256":"aa0d6b761745fc6605c118d210a5771d82a55b3ba7d12729f155c0fced9115c6","abstract_canon_sha256":"b96e9904339309ef231b19ab337006d990557f37cd72468a019d0671c9d22070"},"schema_version":"1.0"},"canonical_sha256":"02e5f51838c869dceefc3483973ed34f6f2b5ea39ded586c3ef7557514997c57","source":{"kind":"arxiv","id":"2509.08168","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2509.08168","created_at":"2026-07-05T12:08:24Z"},{"alias_kind":"arxiv_version","alias_value":"2509.08168v1","created_at":"2026-07-05T12:08:24Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2509.08168","created_at":"2026-07-05T12:08:24Z"},{"alias_kind":"pith_short_12","alias_value":"ALS7KGBYZBU5","created_at":"2026-07-05T12:08:24Z"},{"alias_kind":"pith_short_16","alias_value":"ALS7KGBYZBU5Z3X4","created_at":"2026-07-05T12:08:24Z"},{"alias_kind":"pith_short_8","alias_value":"ALS7KGBY","created_at":"2026-07-05T12:08:24Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2025:ALS7KGBYZBU5Z3X4GSBZOPWTJ5","target":"record","payload":{"canonical_record":{"source":{"id":"2509.08168","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.AP","submitted_at":"2025-09-09T22:11:10Z","cross_cats_sorted":["math.FA"],"title_canon_sha256":"aa0d6b761745fc6605c118d210a5771d82a55b3ba7d12729f155c0fced9115c6","abstract_canon_sha256":"b96e9904339309ef231b19ab337006d990557f37cd72468a019d0671c9d22070"},"schema_version":"1.0"},"canonical_sha256":"02e5f51838c869dceefc3483973ed34f6f2b5ea39ded586c3ef7557514997c57","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T12:08:24.211486Z","signature_b64":"KSfxXtpiADa6e5uWzkZcL1Nf6xHnJYXAXpt7hF1EEhAxRPQ869jhSINHPJ+T5DQeHyXpjkg+FUPqglGYOX03BA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"02e5f51838c869dceefc3483973ed34f6f2b5ea39ded586c3ef7557514997c57","last_reissued_at":"2026-07-05T12:08:24.210985Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T12:08:24.210985Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2509.08168","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":"2026-07-05T12:08:24Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"+/fo9J5JxcQbGxDeQ9Fh+CaakvwyPUF4AeU+9EOF7B0fMKXE/oiLBwqhh5SsPlVo9E3KyXBptnsqa3g5BbZOAQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-09T07:23:08.963054Z"},"content_sha256":"d0e4449c2c16573eb210c1dd3d958890ce9ed47f82cd6ea92806f5b76fbb4c99","schema_version":"1.0","event_id":"sha256:d0e4449c2c16573eb210c1dd3d958890ce9ed47f82cd6ea92806f5b76fbb4c99"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2025:ALS7KGBYZBU5Z3X4GSBZOPWTJ5","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Pathological solutions of Navier-Stokes equations on $\\mathbb{T}^2$ with gradients in Hardy spaces","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["math.FA"],"primary_cat":"math.AP","authors_text":"Antonio Hidalgo-Torn\\'e, Jan Burczak","submitted_at":"2025-09-09T22:11:10Z","abstract_excerpt":"For an arbitrary smooth initial datum, we construct multiple nonzero solutions to the $2$d Navier-Stokes equations, with their gradients in the Hardy space $\\mathcal{H}^p$ with any $p \\in (0,1)$. Thus, in terms of the path space $C(\\mathcal{H}^p)$ for vorticity, $p=1$ is the threshold value distinguishing between non-uniqueness and uniqueness regimes. In order to obtain our result, we develop the needed theory of Hardy spaces on periodic domains."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2509.08168","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/2509.08168/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":"2026-07-05T12:08:24Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"EZsCkFpZIR0skiXpHI3hdixvipSbRdic10yzOX38BF+kVUF+ZExWUa7YJEgBpA+zDjLi9PJpe5JMsGrIHPGXBg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-09T07:23:08.963568Z"},"content_sha256":"772d348d02e98484baae644774f4d34da6d483462a69f069c83ba567aaa41c93","schema_version":"1.0","event_id":"sha256:772d348d02e98484baae644774f4d34da6d483462a69f069c83ba567aaa41c93"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/ALS7KGBYZBU5Z3X4GSBZOPWTJ5/bundle.json","state_url":"https://pith.science/pith/ALS7KGBYZBU5Z3X4GSBZOPWTJ5/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/ALS7KGBYZBU5Z3X4GSBZOPWTJ5/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":"2026-08-09T07:23:08Z","links":{"resolver":"https://pith.science/pith/ALS7KGBYZBU5Z3X4GSBZOPWTJ5","bundle":"https://pith.science/pith/ALS7KGBYZBU5Z3X4GSBZOPWTJ5/bundle.json","state":"https://pith.science/pith/ALS7KGBYZBU5Z3X4GSBZOPWTJ5/state.json","well_known_bundle":"https://pith.science/.well-known/pith/ALS7KGBYZBU5Z3X4GSBZOPWTJ5/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2025:ALS7KGBYZBU5Z3X4GSBZOPWTJ5","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":"b96e9904339309ef231b19ab337006d990557f37cd72468a019d0671c9d22070","cross_cats_sorted":["math.FA"],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.AP","submitted_at":"2025-09-09T22:11:10Z","title_canon_sha256":"aa0d6b761745fc6605c118d210a5771d82a55b3ba7d12729f155c0fced9115c6"},"schema_version":"1.0","source":{"id":"2509.08168","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2509.08168","created_at":"2026-07-05T12:08:24Z"},{"alias_kind":"arxiv_version","alias_value":"2509.08168v1","created_at":"2026-07-05T12:08:24Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2509.08168","created_at":"2026-07-05T12:08:24Z"},{"alias_kind":"pith_short_12","alias_value":"ALS7KGBYZBU5","created_at":"2026-07-05T12:08:24Z"},{"alias_kind":"pith_short_16","alias_value":"ALS7KGBYZBU5Z3X4","created_at":"2026-07-05T12:08:24Z"},{"alias_kind":"pith_short_8","alias_value":"ALS7KGBY","created_at":"2026-07-05T12:08:24Z"}],"graph_snapshots":[{"event_id":"sha256:772d348d02e98484baae644774f4d34da6d483462a69f069c83ba567aaa41c93","target":"graph","created_at":"2026-07-05T12:08:24Z","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/2509.08168/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"For an arbitrary smooth initial datum, we construct multiple nonzero solutions to the $2$d Navier-Stokes equations, with their gradients in the Hardy space $\\mathcal{H}^p$ with any $p \\in (0,1)$. Thus, in terms of the path space $C(\\mathcal{H}^p)$ for vorticity, $p=1$ is the threshold value distinguishing between non-uniqueness and uniqueness regimes. In order to obtain our result, we develop the needed theory of Hardy spaces on periodic domains.","authors_text":"Antonio Hidalgo-Torn\\'e, Jan Burczak","cross_cats":["math.FA"],"headline":"","license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.AP","submitted_at":"2025-09-09T22:11:10Z","title":"Pathological solutions of Navier-Stokes equations on $\\mathbb{T}^2$ with gradients in Hardy spaces"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2509.08168","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:d0e4449c2c16573eb210c1dd3d958890ce9ed47f82cd6ea92806f5b76fbb4c99","target":"record","created_at":"2026-07-05T12:08:24Z","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":"b96e9904339309ef231b19ab337006d990557f37cd72468a019d0671c9d22070","cross_cats_sorted":["math.FA"],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.AP","submitted_at":"2025-09-09T22:11:10Z","title_canon_sha256":"aa0d6b761745fc6605c118d210a5771d82a55b3ba7d12729f155c0fced9115c6"},"schema_version":"1.0","source":{"id":"2509.08168","kind":"arxiv","version":1}},"canonical_sha256":"02e5f51838c869dceefc3483973ed34f6f2b5ea39ded586c3ef7557514997c57","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"02e5f51838c869dceefc3483973ed34f6f2b5ea39ded586c3ef7557514997c57","first_computed_at":"2026-07-05T12:08:24.210985Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T12:08:24.210985Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"KSfxXtpiADa6e5uWzkZcL1Nf6xHnJYXAXpt7hF1EEhAxRPQ869jhSINHPJ+T5DQeHyXpjkg+FUPqglGYOX03BA==","signature_status":"signed_v1","signed_at":"2026-07-05T12:08:24.211486Z","signed_message":"canonical_sha256_bytes"},"source_id":"2509.08168","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:d0e4449c2c16573eb210c1dd3d958890ce9ed47f82cd6ea92806f5b76fbb4c99","sha256:772d348d02e98484baae644774f4d34da6d483462a69f069c83ba567aaa41c93"],"state_sha256":"5a2baad06b06eaec0860afa70f690c3737a019d7b5e869dfc6e54b211bcc353a"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"TEX/qt/YueiXjClLLcWnnzQxiUpIo5UFgqo8IprLmytiMit+ZZwXKBfavFaQDa02wFixhjNTjLwe9FWE0G6oDg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-09T07:23:08.968821Z","bundle_sha256":"ad01aee959873315b1ac4e9fdfc1183325521af36b4db1240bb16a49aed23764"}}