{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2009:MK4RFQYGWCTIHGBKV6MMP7TR4Y","short_pith_number":"pith:MK4RFQYG","canonical_record":{"source":{"id":"0911.4534","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2009-11-24T03:45:13Z","cross_cats_sorted":[],"title_canon_sha256":"dc6ff3e662546077f29546dcd37ce4cad60530b07629a6cba2fc7decd7122c3e","abstract_canon_sha256":"e14e0ed30fced1840e1d7175ac978fed7e9caadbd827e6a8266f86a999521d9a"},"schema_version":"1.0"},"canonical_sha256":"62b912c306b0a683982aaf98c7fe71e62852d570fafe6fd3f5a4bdc2a0d59eee","source":{"kind":"arxiv","id":"0911.4534","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"0911.4534","created_at":"2026-05-18T01:35:33Z"},{"alias_kind":"arxiv_version","alias_value":"0911.4534v2","created_at":"2026-05-18T01:35:33Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.0911.4534","created_at":"2026-05-18T01:35:33Z"},{"alias_kind":"pith_short_12","alias_value":"MK4RFQYGWCTI","created_at":"2026-05-18T12:26:00Z"},{"alias_kind":"pith_short_16","alias_value":"MK4RFQYGWCTIHGBK","created_at":"2026-05-18T12:26:00Z"},{"alias_kind":"pith_short_8","alias_value":"MK4RFQYG","created_at":"2026-05-18T12:26:00Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2009:MK4RFQYGWCTIHGBKV6MMP7TR4Y","target":"record","payload":{"canonical_record":{"source":{"id":"0911.4534","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2009-11-24T03:45:13Z","cross_cats_sorted":[],"title_canon_sha256":"dc6ff3e662546077f29546dcd37ce4cad60530b07629a6cba2fc7decd7122c3e","abstract_canon_sha256":"e14e0ed30fced1840e1d7175ac978fed7e9caadbd827e6a8266f86a999521d9a"},"schema_version":"1.0"},"canonical_sha256":"62b912c306b0a683982aaf98c7fe71e62852d570fafe6fd3f5a4bdc2a0d59eee","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:35:33.347665Z","signature_b64":"xaRga8UmM0r8Y6Hd8/3eunwsPjpwg2VsoXsULhT4+4HiBAU+AzcSPh3gJ7cWDqlT4AhGNlSSc5WG6fLExfZzBw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"62b912c306b0a683982aaf98c7fe71e62852d570fafe6fd3f5a4bdc2a0d59eee","last_reissued_at":"2026-05-18T01:35:33.347008Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:35:33.347008Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"0911.4534","source_version":2,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T01:35:33Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"XKT9evZ041HzHanGKP1Q5L1GQlQSHplU6aoEu164gZNiSDpfHsDOOkOw39dTjhK3Pv6GEF1OX3l6IsfElTb7Ag==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-01T10:11:48.517183Z"},"content_sha256":"bf58ced7dd0246118fcb1b0b8d441f7a6110a262fd674ab8b36afde34e41ed5b","schema_version":"1.0","event_id":"sha256:bf58ced7dd0246118fcb1b0b8d441f7a6110a262fd674ab8b36afde34e41ed5b"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2009:MK4RFQYGWCTIHGBKV6MMP7TR4Y","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Stability and Unconditional Uniqueness of Solutions for Energy Critical Wave Equations in High Dimensions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Aynur Bulut, Dong Li, Magdalena Czubak, Nata\\v{s}a Pavlovi\\'c, Xiaoyi Zhang","submitted_at":"2009-11-24T03:45:13Z","abstract_excerpt":"In this paper we establish a complete local theory for the energy-critical nonlinear wave equation (NLW) in high dimensions ${\\mathbb R} \\times {\\mathbb R}^d$ with $d \\geq 6$. We prove the stability of solutions under the weak condition that the perturbation of the linear flow is small in certain space-time norms. As a by-product of our stability analysis, we also prove local well-posedness of solutions for which we only assume the smallness of the linear evolution. These results provide essential technical tools that can be applied towards obtaining the extension to high dimensions of the ana"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0911.4534","kind":"arxiv","version":2},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"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":"2026-05-18T01:35:33Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"irZBSY7MPdKpuTNECTvjkSBQU4pFh942qGXYWtFZEngrjKoySO3vG7zehixCGeJu7bEuWW1JzJ6Wba8wGuT1Cg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-01T10:11:48.517553Z"},"content_sha256":"992be14ed0a988187a78a62f823240aee28391e4e768145e8d355553c1086a73","schema_version":"1.0","event_id":"sha256:992be14ed0a988187a78a62f823240aee28391e4e768145e8d355553c1086a73"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/MK4RFQYGWCTIHGBKV6MMP7TR4Y/bundle.json","state_url":"https://pith.science/pith/MK4RFQYGWCTIHGBKV6MMP7TR4Y/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/MK4RFQYGWCTIHGBKV6MMP7TR4Y/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":"2026-06-01T10:11:48Z","links":{"resolver":"https://pith.science/pith/MK4RFQYGWCTIHGBKV6MMP7TR4Y","bundle":"https://pith.science/pith/MK4RFQYGWCTIHGBKV6MMP7TR4Y/bundle.json","state":"https://pith.science/pith/MK4RFQYGWCTIHGBKV6MMP7TR4Y/state.json","well_known_bundle":"https://pith.science/.well-known/pith/MK4RFQYGWCTIHGBKV6MMP7TR4Y/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2009:MK4RFQYGWCTIHGBKV6MMP7TR4Y","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":"e14e0ed30fced1840e1d7175ac978fed7e9caadbd827e6a8266f86a999521d9a","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2009-11-24T03:45:13Z","title_canon_sha256":"dc6ff3e662546077f29546dcd37ce4cad60530b07629a6cba2fc7decd7122c3e"},"schema_version":"1.0","source":{"id":"0911.4534","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"0911.4534","created_at":"2026-05-18T01:35:33Z"},{"alias_kind":"arxiv_version","alias_value":"0911.4534v2","created_at":"2026-05-18T01:35:33Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.0911.4534","created_at":"2026-05-18T01:35:33Z"},{"alias_kind":"pith_short_12","alias_value":"MK4RFQYGWCTI","created_at":"2026-05-18T12:26:00Z"},{"alias_kind":"pith_short_16","alias_value":"MK4RFQYGWCTIHGBK","created_at":"2026-05-18T12:26:00Z"},{"alias_kind":"pith_short_8","alias_value":"MK4RFQYG","created_at":"2026-05-18T12:26:00Z"}],"graph_snapshots":[{"event_id":"sha256:992be14ed0a988187a78a62f823240aee28391e4e768145e8d355553c1086a73","target":"graph","created_at":"2026-05-18T01:35:33Z","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"},"paper":{"abstract_excerpt":"In this paper we establish a complete local theory for the energy-critical nonlinear wave equation (NLW) in high dimensions ${\\mathbb R} \\times {\\mathbb R}^d$ with $d \\geq 6$. We prove the stability of solutions under the weak condition that the perturbation of the linear flow is small in certain space-time norms. As a by-product of our stability analysis, we also prove local well-posedness of solutions for which we only assume the smallness of the linear evolution. These results provide essential technical tools that can be applied towards obtaining the extension to high dimensions of the ana","authors_text":"Aynur Bulut, Dong Li, Magdalena Czubak, Nata\\v{s}a Pavlovi\\'c, Xiaoyi Zhang","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2009-11-24T03:45:13Z","title":"Stability and Unconditional Uniqueness of Solutions for Energy Critical Wave Equations in High Dimensions"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0911.4534","kind":"arxiv","version":2},"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:bf58ced7dd0246118fcb1b0b8d441f7a6110a262fd674ab8b36afde34e41ed5b","target":"record","created_at":"2026-05-18T01:35:33Z","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":"e14e0ed30fced1840e1d7175ac978fed7e9caadbd827e6a8266f86a999521d9a","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2009-11-24T03:45:13Z","title_canon_sha256":"dc6ff3e662546077f29546dcd37ce4cad60530b07629a6cba2fc7decd7122c3e"},"schema_version":"1.0","source":{"id":"0911.4534","kind":"arxiv","version":2}},"canonical_sha256":"62b912c306b0a683982aaf98c7fe71e62852d570fafe6fd3f5a4bdc2a0d59eee","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"62b912c306b0a683982aaf98c7fe71e62852d570fafe6fd3f5a4bdc2a0d59eee","first_computed_at":"2026-05-18T01:35:33.347008Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:35:33.347008Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"xaRga8UmM0r8Y6Hd8/3eunwsPjpwg2VsoXsULhT4+4HiBAU+AzcSPh3gJ7cWDqlT4AhGNlSSc5WG6fLExfZzBw==","signature_status":"signed_v1","signed_at":"2026-05-18T01:35:33.347665Z","signed_message":"canonical_sha256_bytes"},"source_id":"0911.4534","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:bf58ced7dd0246118fcb1b0b8d441f7a6110a262fd674ab8b36afde34e41ed5b","sha256:992be14ed0a988187a78a62f823240aee28391e4e768145e8d355553c1086a73"],"state_sha256":"8b29e9e1a8e8aa612bb49a86d35d900756038084a6a0e86b74241351d6490e4f"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"WVO4C/OV06G3PXQ9hXtRS8VOb7DZkUtSA1OCy2Q5vFuhp6Ih2/psoKiHMkXBAMci9LJCql8Ssiyakpdx6C8xAQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-01T10:11:48.519499Z","bundle_sha256":"64f9c0ed48ca996ef5d86e8c1241c2da2c64efd3bea3159a30fba92e120622d7"}}