{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2024:GIUXVA3H32OAYPEXHVI7UG2FUQ","short_pith_number":"pith:GIUXVA3H","canonical_record":{"source":{"id":"2410.15750","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.AP","submitted_at":"2024-10-21T08:11:00Z","cross_cats_sorted":[],"title_canon_sha256":"357e3e4fa6af78bb5d7d17f928bddc4173b3fea0859e80a412f28bf695677af1","abstract_canon_sha256":"d3d145ebad99f2da2c50710b98d9eea22cf719d7535f4cc293c7b3305639e0f7"},"schema_version":"1.0"},"canonical_sha256":"32297a8367de9c0c3c973d51fa1b45a41f30e2a58c2b6465b02a177aef74a48f","source":{"kind":"arxiv","id":"2410.15750","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2410.15750","created_at":"2026-07-05T09:23:24Z"},{"alias_kind":"arxiv_version","alias_value":"2410.15750v1","created_at":"2026-07-05T09:23:24Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2410.15750","created_at":"2026-07-05T09:23:24Z"},{"alias_kind":"pith_short_12","alias_value":"GIUXVA3H32OA","created_at":"2026-07-05T09:23:24Z"},{"alias_kind":"pith_short_16","alias_value":"GIUXVA3H32OAYPEX","created_at":"2026-07-05T09:23:24Z"},{"alias_kind":"pith_short_8","alias_value":"GIUXVA3H","created_at":"2026-07-05T09:23:24Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2024:GIUXVA3H32OAYPEXHVI7UG2FUQ","target":"record","payload":{"canonical_record":{"source":{"id":"2410.15750","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.AP","submitted_at":"2024-10-21T08:11:00Z","cross_cats_sorted":[],"title_canon_sha256":"357e3e4fa6af78bb5d7d17f928bddc4173b3fea0859e80a412f28bf695677af1","abstract_canon_sha256":"d3d145ebad99f2da2c50710b98d9eea22cf719d7535f4cc293c7b3305639e0f7"},"schema_version":"1.0"},"canonical_sha256":"32297a8367de9c0c3c973d51fa1b45a41f30e2a58c2b6465b02a177aef74a48f","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T09:23:24.162545Z","signature_b64":"9gHCRaHS6dnhe9oFcFPsDpufE2hvPJlejlweM0Q2ERe+9b0uznKZJ5vamKndLE86cXulN1AExfPYP2ERWLI6Dg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"32297a8367de9c0c3c973d51fa1b45a41f30e2a58c2b6465b02a177aef74a48f","last_reissued_at":"2026-07-05T09:23:24.162094Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T09:23:24.162094Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2410.15750","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":"2026-07-05T09:23:24Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"ncjohZA6kJFB0LBJD5ev2anqjfa+qt2WdLEpwNB4AtzvPZ/WZ3hBZTcsBNYtG3kQiMi0K5HxhVHZy4NuN3H/Aw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-14T14:47:41.620105Z"},"content_sha256":"3bcce7a8d0da651b2b058446324341c1b269e07cb04e6ef7f0d2336a2f82445f","schema_version":"1.0","event_id":"sha256:3bcce7a8d0da651b2b058446324341c1b269e07cb04e6ef7f0d2336a2f82445f"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2024:GIUXVA3H32OAYPEXHVI7UG2FUQ","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Normalized solutions for a class of Sobolev critical Schrodinger systems","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Houwang Li, Tianhao Liu, Wenming Zou","submitted_at":"2024-10-21T08:11:00Z","abstract_excerpt":"This paper focuses on the existence and multiplicity of normalized solutions for the coupled Schrodinger system with Sobolev critical coupling term. We present several existence and multiplicity results under some explicit conditions. Furthermore, we present a non-existence result for the defocusing case. This paper, together with the paper [T. Bartsch, H. W. Li and W. M. Zou. Calc. Var. Partial Differential Equations 62 (2023) ], provides a more comprehensive understanding of normalized solutions for Sobolev critical systems. We believe our methods can also address the open problem of the mul"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2410.15750","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/2410.15750/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":"2026-07-05T09:23:24Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"BPfRojVVky79+kjVrk5J6hciaihYDtwWFYHIHzK/sc2DuGVNRxnF1KFKrMYYrBtIRgtMicrOPvVAVrW9zvzZAw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-14T14:47:41.621252Z"},"content_sha256":"2a61d01f56154ad9b41f8167e242881d4c62cd855c44b2cd53e59ad046da6d09","schema_version":"1.0","event_id":"sha256:2a61d01f56154ad9b41f8167e242881d4c62cd855c44b2cd53e59ad046da6d09"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/GIUXVA3H32OAYPEXHVI7UG2FUQ/bundle.json","state_url":"https://pith.science/pith/GIUXVA3H32OAYPEXHVI7UG2FUQ/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/GIUXVA3H32OAYPEXHVI7UG2FUQ/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":"2026-08-14T14:47:41Z","links":{"resolver":"https://pith.science/pith/GIUXVA3H32OAYPEXHVI7UG2FUQ","bundle":"https://pith.science/pith/GIUXVA3H32OAYPEXHVI7UG2FUQ/bundle.json","state":"https://pith.science/pith/GIUXVA3H32OAYPEXHVI7UG2FUQ/state.json","well_known_bundle":"https://pith.science/.well-known/pith/GIUXVA3H32OAYPEXHVI7UG2FUQ/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2024:GIUXVA3H32OAYPEXHVI7UG2FUQ","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":"d3d145ebad99f2da2c50710b98d9eea22cf719d7535f4cc293c7b3305639e0f7","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.AP","submitted_at":"2024-10-21T08:11:00Z","title_canon_sha256":"357e3e4fa6af78bb5d7d17f928bddc4173b3fea0859e80a412f28bf695677af1"},"schema_version":"1.0","source":{"id":"2410.15750","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2410.15750","created_at":"2026-07-05T09:23:24Z"},{"alias_kind":"arxiv_version","alias_value":"2410.15750v1","created_at":"2026-07-05T09:23:24Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2410.15750","created_at":"2026-07-05T09:23:24Z"},{"alias_kind":"pith_short_12","alias_value":"GIUXVA3H32OA","created_at":"2026-07-05T09:23:24Z"},{"alias_kind":"pith_short_16","alias_value":"GIUXVA3H32OAYPEX","created_at":"2026-07-05T09:23:24Z"},{"alias_kind":"pith_short_8","alias_value":"GIUXVA3H","created_at":"2026-07-05T09:23:24Z"}],"graph_snapshots":[{"event_id":"sha256:2a61d01f56154ad9b41f8167e242881d4c62cd855c44b2cd53e59ad046da6d09","target":"graph","created_at":"2026-07-05T09:23:24Z","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/2410.15750/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"This paper focuses on the existence and multiplicity of normalized solutions for the coupled Schrodinger system with Sobolev critical coupling term. We present several existence and multiplicity results under some explicit conditions. Furthermore, we present a non-existence result for the defocusing case. This paper, together with the paper [T. Bartsch, H. W. Li and W. M. Zou. Calc. Var. Partial Differential Equations 62 (2023) ], provides a more comprehensive understanding of normalized solutions for Sobolev critical systems. We believe our methods can also address the open problem of the mul","authors_text":"Houwang Li, Tianhao Liu, Wenming Zou","cross_cats":[],"headline":"","license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.AP","submitted_at":"2024-10-21T08:11:00Z","title":"Normalized solutions for a class of Sobolev critical Schrodinger systems"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2410.15750","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":"sha256:3bcce7a8d0da651b2b058446324341c1b269e07cb04e6ef7f0d2336a2f82445f","target":"record","created_at":"2026-07-05T09:23:24Z","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":"d3d145ebad99f2da2c50710b98d9eea22cf719d7535f4cc293c7b3305639e0f7","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.AP","submitted_at":"2024-10-21T08:11:00Z","title_canon_sha256":"357e3e4fa6af78bb5d7d17f928bddc4173b3fea0859e80a412f28bf695677af1"},"schema_version":"1.0","source":{"id":"2410.15750","kind":"arxiv","version":1}},"canonical_sha256":"32297a8367de9c0c3c973d51fa1b45a41f30e2a58c2b6465b02a177aef74a48f","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"32297a8367de9c0c3c973d51fa1b45a41f30e2a58c2b6465b02a177aef74a48f","first_computed_at":"2026-07-05T09:23:24.162094Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T09:23:24.162094Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"9gHCRaHS6dnhe9oFcFPsDpufE2hvPJlejlweM0Q2ERe+9b0uznKZJ5vamKndLE86cXulN1AExfPYP2ERWLI6Dg==","signature_status":"signed_v1","signed_at":"2026-07-05T09:23:24.162545Z","signed_message":"canonical_sha256_bytes"},"source_id":"2410.15750","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:3bcce7a8d0da651b2b058446324341c1b269e07cb04e6ef7f0d2336a2f82445f","sha256:2a61d01f56154ad9b41f8167e242881d4c62cd855c44b2cd53e59ad046da6d09"],"state_sha256":"9bf0cd230317d059c7ce9477deb650b27143af28e26d4390de79ecb8c5ea138d"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"IXFmhYCtzuEFeON0jslW/3RUbqExMyFDqXoPBOCCcN0mbatWirvmfsvgX778LlDJQRMky/7YfehswJNWrpkxBg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-14T14:47:41.629574Z","bundle_sha256":"b7de0645b144c232e5b8ef49e9888ae2e70bf198d010e9ba5cee3098e41fdf0b"}}