{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2026:Z3EM57ANUKG76O7DM7YJE7OI7L","short_pith_number":"pith:Z3EM57AN","canonical_record":{"source":{"id":"2606.12192","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2026-06-10T15:15:51Z","cross_cats_sorted":[],"title_canon_sha256":"7a74f61838903384b2f99e22094dc0474591feb80de34181f8a5a85560172780","abstract_canon_sha256":"4b976d3ed91487069fae278272b648507e03841271d40ec8f4ed1865d2c8055e"},"schema_version":"1.0"},"canonical_sha256":"cec8cefc0da28dff3be367f0927dc8fac727c5ecdd8ce09cc6ca39633bd9f99f","source":{"kind":"arxiv","id":"2606.12192","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2606.12192","created_at":"2026-06-11T01:10:53Z"},{"alias_kind":"arxiv_version","alias_value":"2606.12192v1","created_at":"2026-06-11T01:10:53Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2606.12192","created_at":"2026-06-11T01:10:53Z"},{"alias_kind":"pith_short_12","alias_value":"Z3EM57ANUKG7","created_at":"2026-06-11T01:10:53Z"},{"alias_kind":"pith_short_16","alias_value":"Z3EM57ANUKG76O7D","created_at":"2026-06-11T01:10:53Z"},{"alias_kind":"pith_short_8","alias_value":"Z3EM57AN","created_at":"2026-06-11T01:10:53Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2026:Z3EM57ANUKG76O7DM7YJE7OI7L","target":"record","payload":{"canonical_record":{"source":{"id":"2606.12192","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2026-06-10T15:15:51Z","cross_cats_sorted":[],"title_canon_sha256":"7a74f61838903384b2f99e22094dc0474591feb80de34181f8a5a85560172780","abstract_canon_sha256":"4b976d3ed91487069fae278272b648507e03841271d40ec8f4ed1865d2c8055e"},"schema_version":"1.0"},"canonical_sha256":"cec8cefc0da28dff3be367f0927dc8fac727c5ecdd8ce09cc6ca39633bd9f99f","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-06-11T01:10:53.347199Z","signature_b64":"t01xzxMWnumAJ/UfPQIxGEeWTq4qBmvjzv9CBRpe71cVFs3DYAsuaYNecpEoSNZhpeVecforlfydYG//DYXmDg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"cec8cefc0da28dff3be367f0927dc8fac727c5ecdd8ce09cc6ca39633bd9f99f","last_reissued_at":"2026-06-11T01:10:53.346388Z","signature_status":"signed_v1","first_computed_at":"2026-06-11T01:10:53.346388Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2606.12192","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":"2026-06-11T01:10:53Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"vuWNOZHPLbILBuJqq76T6CjnwzEoMaD1PffNs3krczqoECJWqcDb2MKqYSvj6JStS7D2bKWry3EneRz6clpkBQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-19T08:44:45.927694Z"},"content_sha256":"ec5ccaaad84d258b1a13fbe333e8b4085e20516d374c95065ff74691e40812df","schema_version":"1.0","event_id":"sha256:ec5ccaaad84d258b1a13fbe333e8b4085e20516d374c95065ff74691e40812df"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2026:Z3EM57ANUKG76O7DM7YJE7OI7L","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"A Generalized Framework for $L^r$ Convex Integration and its Application to Geophysical Models","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Daniel W. Boutros, Edriss S. Titi, Simon Markfelder","submitted_at":"2026-06-10T15:15:51Z","abstract_excerpt":"In this paper a general framework for convex integration is developed, in order to construct weak solutions to the Cauchy problem, by building on ideas from [C. De Lellis and L. Sz\\'ekelyhidi, Arch. Ration. Mech. Anal., 195 (2010)] and [S. Markfelder, Nonlinearity, 37 (2024)]. This framework may be applied to a large family of partial differential equations in order to construct weak solutions in $L^\\infty ((0,T) \\times \\Omega)$ (for a bounded domain $\\Omega)$ which are weakly continuous in time with respect to the weak topology of $L^r (\\Omega)$ for some $r \\in (1,\\infty)$. This allows us to "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2606.12192","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/2606.12192/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":"2026-06-11T01:10:53Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"8xuyXFT4mFQw9pTilxbWMk0/EHGKIQkEc8nowGfqcNxWXGv/j9AAAO5HC7n/gEN/mpiHGjHukBrFRAcH8rKYBQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-19T08:44:45.928219Z"},"content_sha256":"121b281b615c2cb46fff4c6f3b02c99a69bc6da3131b8e0e842a00fbe1fdc9b0","schema_version":"1.0","event_id":"sha256:121b281b615c2cb46fff4c6f3b02c99a69bc6da3131b8e0e842a00fbe1fdc9b0"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/Z3EM57ANUKG76O7DM7YJE7OI7L/bundle.json","state_url":"https://pith.science/pith/Z3EM57ANUKG76O7DM7YJE7OI7L/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/Z3EM57ANUKG76O7DM7YJE7OI7L/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":"2026-08-19T08:44:45Z","links":{"resolver":"https://pith.science/pith/Z3EM57ANUKG76O7DM7YJE7OI7L","bundle":"https://pith.science/pith/Z3EM57ANUKG76O7DM7YJE7OI7L/bundle.json","state":"https://pith.science/pith/Z3EM57ANUKG76O7DM7YJE7OI7L/state.json","well_known_bundle":"https://pith.science/.well-known/pith/Z3EM57ANUKG76O7DM7YJE7OI7L/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2026:Z3EM57ANUKG76O7DM7YJE7OI7L","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":"4b976d3ed91487069fae278272b648507e03841271d40ec8f4ed1865d2c8055e","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2026-06-10T15:15:51Z","title_canon_sha256":"7a74f61838903384b2f99e22094dc0474591feb80de34181f8a5a85560172780"},"schema_version":"1.0","source":{"id":"2606.12192","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2606.12192","created_at":"2026-06-11T01:10:53Z"},{"alias_kind":"arxiv_version","alias_value":"2606.12192v1","created_at":"2026-06-11T01:10:53Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2606.12192","created_at":"2026-06-11T01:10:53Z"},{"alias_kind":"pith_short_12","alias_value":"Z3EM57ANUKG7","created_at":"2026-06-11T01:10:53Z"},{"alias_kind":"pith_short_16","alias_value":"Z3EM57ANUKG76O7D","created_at":"2026-06-11T01:10:53Z"},{"alias_kind":"pith_short_8","alias_value":"Z3EM57AN","created_at":"2026-06-11T01:10:53Z"}],"graph_snapshots":[{"event_id":"sha256:121b281b615c2cb46fff4c6f3b02c99a69bc6da3131b8e0e842a00fbe1fdc9b0","target":"graph","created_at":"2026-06-11T01:10:53Z","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/2606.12192/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"In this paper a general framework for convex integration is developed, in order to construct weak solutions to the Cauchy problem, by building on ideas from [C. De Lellis and L. Sz\\'ekelyhidi, Arch. Ration. Mech. Anal., 195 (2010)] and [S. Markfelder, Nonlinearity, 37 (2024)]. This framework may be applied to a large family of partial differential equations in order to construct weak solutions in $L^\\infty ((0,T) \\times \\Omega)$ (for a bounded domain $\\Omega)$ which are weakly continuous in time with respect to the weak topology of $L^r (\\Omega)$ for some $r \\in (1,\\infty)$. This allows us to ","authors_text":"Daniel W. Boutros, Edriss S. Titi, Simon Markfelder","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2026-06-10T15:15:51Z","title":"A Generalized Framework for $L^r$ Convex Integration and its Application to Geophysical Models"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2606.12192","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:ec5ccaaad84d258b1a13fbe333e8b4085e20516d374c95065ff74691e40812df","target":"record","created_at":"2026-06-11T01:10:53Z","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":"4b976d3ed91487069fae278272b648507e03841271d40ec8f4ed1865d2c8055e","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2026-06-10T15:15:51Z","title_canon_sha256":"7a74f61838903384b2f99e22094dc0474591feb80de34181f8a5a85560172780"},"schema_version":"1.0","source":{"id":"2606.12192","kind":"arxiv","version":1}},"canonical_sha256":"cec8cefc0da28dff3be367f0927dc8fac727c5ecdd8ce09cc6ca39633bd9f99f","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"cec8cefc0da28dff3be367f0927dc8fac727c5ecdd8ce09cc6ca39633bd9f99f","first_computed_at":"2026-06-11T01:10:53.346388Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-06-11T01:10:53.346388Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"t01xzxMWnumAJ/UfPQIxGEeWTq4qBmvjzv9CBRpe71cVFs3DYAsuaYNecpEoSNZhpeVecforlfydYG//DYXmDg==","signature_status":"signed_v1","signed_at":"2026-06-11T01:10:53.347199Z","signed_message":"canonical_sha256_bytes"},"source_id":"2606.12192","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:ec5ccaaad84d258b1a13fbe333e8b4085e20516d374c95065ff74691e40812df","sha256:121b281b615c2cb46fff4c6f3b02c99a69bc6da3131b8e0e842a00fbe1fdc9b0"],"state_sha256":"adaf3a653d363944794a57b049a7ae38f34b0391c07762aa12cd9cfd906aab4b"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"E7PyUJXFcs5xuoNWiQQfexCFOg7/USLMWcUHpCI5KpjDYjoaSCCRnYKiV9lDJxGHUdma1yCfdoPjJiaAA+P9Cw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-19T08:44:45.933164Z","bundle_sha256":"043a9f67436295ee46efc6b3456298c3fb33e9b6590d2db6407a347bf1138e4c"}}