{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2000:FN6M2BNEH7HLKXDSSUIXVQL4VI","short_pith_number":"pith:FN6M2BNE","canonical_record":{"source":{"id":"math/0010068","kind":"arxiv","version":3},"metadata":{"license":"","primary_cat":"math.AP","submitted_at":"2000-10-07T03:11:59Z","cross_cats_sorted":[],"title_canon_sha256":"56409cc4393c51e2f20cb99c48dc16b3b54813484c89f6a3934f7c28cd91bbe6","abstract_canon_sha256":"81bb734f727381a9ec9741b1b01d09ff88a209cf33512aa2b24f5fe2036f7e01"},"schema_version":"1.0"},"canonical_sha256":"2b7ccd05a43fceb55c7295117ac17caa17a495fc1415b155039833ab7a137873","source":{"kind":"arxiv","id":"math/0010068","version":3},"source_aliases":[{"alias_kind":"arxiv","alias_value":"math/0010068","created_at":"2026-07-04T14:34:51Z"},{"alias_kind":"arxiv_version","alias_value":"math/0010068v3","created_at":"2026-07-04T14:34:51Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.math/0010068","created_at":"2026-07-04T14:34:51Z"},{"alias_kind":"pith_short_12","alias_value":"FN6M2BNEH7HL","created_at":"2026-07-04T14:34:51Z"},{"alias_kind":"pith_short_16","alias_value":"FN6M2BNEH7HLKXDS","created_at":"2026-07-04T14:34:51Z"},{"alias_kind":"pith_short_8","alias_value":"FN6M2BNE","created_at":"2026-07-04T14:34:51Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2000:FN6M2BNEH7HLKXDSSUIXVQL4VI","target":"record","payload":{"canonical_record":{"source":{"id":"math/0010068","kind":"arxiv","version":3},"metadata":{"license":"","primary_cat":"math.AP","submitted_at":"2000-10-07T03:11:59Z","cross_cats_sorted":[],"title_canon_sha256":"56409cc4393c51e2f20cb99c48dc16b3b54813484c89f6a3934f7c28cd91bbe6","abstract_canon_sha256":"81bb734f727381a9ec9741b1b01d09ff88a209cf33512aa2b24f5fe2036f7e01"},"schema_version":"1.0"},"canonical_sha256":"2b7ccd05a43fceb55c7295117ac17caa17a495fc1415b155039833ab7a137873","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-04T14:34:51.268736Z","signature_b64":"ztEPyWQtcItx7SG9meaIb5Jlhucw6KKdhKWdZOiODa6z6HItpGpEpgebdWlQW+dcK6Z3gZMPExcQE0CJEIhNAg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"2b7ccd05a43fceb55c7295117ac17caa17a495fc1415b155039833ab7a137873","last_reissued_at":"2026-07-04T14:34:51.267869Z","signature_status":"signed_v1","first_computed_at":"2026-07-04T14:34:51.267869Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"math/0010068","source_version":3,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-07-04T14:34:51Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"KVCUdqxab9ScFZxOJ6JQxiAE/wKKVIMUOU95nFEkbtxWrUPk/WGh3VyncIRxpx7DaF8p3FMlQnQ8hNC2oc+uCw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-13T03:59:39.710364Z"},"content_sha256":"6eacc0345c8b8eeb874d7175485779060fa297068abbf1268bf770a050928074","schema_version":"1.0","event_id":"sha256:6eacc0345c8b8eeb874d7175485779060fa297068abbf1268bf770a050928074"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2000:FN6M2BNEH7HLKXDSSUIXVQL4VI","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Global regularity of wave maps I. Small critical Sobolev norm in high dimension","license":"","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Terence Tao","submitted_at":"2000-10-07T03:11:59Z","abstract_excerpt":"We show that wave maps from Minkowski space $R^{1+n}$ to a sphere are globally smooth if the initial data is smooth and has small norm in the critical Sobolev space $\\dot H^{n/2}$ in the high dimensional case $n \\geq 5$. A major difficulty, not present in the earlier results, is that the $\\dot H^{n/2}$ norm barely fails to control $L^\\infty$, potentially causing a logarithmic divergence in the nonlinearity; however, this can be overcome by using co-ordinate frames adapted to the wave map by approximate parallel transport. In the sequel of this paper we address the more interesting two-dimensio"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math/0010068","kind":"arxiv","version":3},"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/math/0010068/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":"2026-07-04T14:34:51Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"ScLN952pk8PR0i6LfFnrqk83M9meL9d2sJD8ThyKxdqgOUcak0q76MjOMrrFlfNDLFA9hueQ0eJ7Xk2Z1fNWBA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-13T03:59:39.710865Z"},"content_sha256":"da0865b0ca45571a917ef5e9779f7b109e7e992cdcbad169f592000ac74dab34","schema_version":"1.0","event_id":"sha256:da0865b0ca45571a917ef5e9779f7b109e7e992cdcbad169f592000ac74dab34"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/FN6M2BNEH7HLKXDSSUIXVQL4VI/bundle.json","state_url":"https://pith.science/pith/FN6M2BNEH7HLKXDSSUIXVQL4VI/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/FN6M2BNEH7HLKXDSSUIXVQL4VI/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":"2026-08-13T03:59:39Z","links":{"resolver":"https://pith.science/pith/FN6M2BNEH7HLKXDSSUIXVQL4VI","bundle":"https://pith.science/pith/FN6M2BNEH7HLKXDSSUIXVQL4VI/bundle.json","state":"https://pith.science/pith/FN6M2BNEH7HLKXDSSUIXVQL4VI/state.json","well_known_bundle":"https://pith.science/.well-known/pith/FN6M2BNEH7HLKXDSSUIXVQL4VI/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2000:FN6M2BNEH7HLKXDSSUIXVQL4VI","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":"81bb734f727381a9ec9741b1b01d09ff88a209cf33512aa2b24f5fe2036f7e01","cross_cats_sorted":[],"license":"","primary_cat":"math.AP","submitted_at":"2000-10-07T03:11:59Z","title_canon_sha256":"56409cc4393c51e2f20cb99c48dc16b3b54813484c89f6a3934f7c28cd91bbe6"},"schema_version":"1.0","source":{"id":"math/0010068","kind":"arxiv","version":3}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"math/0010068","created_at":"2026-07-04T14:34:51Z"},{"alias_kind":"arxiv_version","alias_value":"math/0010068v3","created_at":"2026-07-04T14:34:51Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.math/0010068","created_at":"2026-07-04T14:34:51Z"},{"alias_kind":"pith_short_12","alias_value":"FN6M2BNEH7HL","created_at":"2026-07-04T14:34:51Z"},{"alias_kind":"pith_short_16","alias_value":"FN6M2BNEH7HLKXDS","created_at":"2026-07-04T14:34:51Z"},{"alias_kind":"pith_short_8","alias_value":"FN6M2BNE","created_at":"2026-07-04T14:34:51Z"}],"graph_snapshots":[{"event_id":"sha256:da0865b0ca45571a917ef5e9779f7b109e7e992cdcbad169f592000ac74dab34","target":"graph","created_at":"2026-07-04T14:34:51Z","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/math/0010068/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"We show that wave maps from Minkowski space $R^{1+n}$ to a sphere are globally smooth if the initial data is smooth and has small norm in the critical Sobolev space $\\dot H^{n/2}$ in the high dimensional case $n \\geq 5$. A major difficulty, not present in the earlier results, is that the $\\dot H^{n/2}$ norm barely fails to control $L^\\infty$, potentially causing a logarithmic divergence in the nonlinearity; however, this can be overcome by using co-ordinate frames adapted to the wave map by approximate parallel transport. In the sequel of this paper we address the more interesting two-dimensio","authors_text":"Terence Tao","cross_cats":[],"headline":"","license":"","primary_cat":"math.AP","submitted_at":"2000-10-07T03:11:59Z","title":"Global regularity of wave maps I. Small critical Sobolev norm in high dimension"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math/0010068","kind":"arxiv","version":3},"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":"sha256:6eacc0345c8b8eeb874d7175485779060fa297068abbf1268bf770a050928074","target":"record","created_at":"2026-07-04T14:34:51Z","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":"81bb734f727381a9ec9741b1b01d09ff88a209cf33512aa2b24f5fe2036f7e01","cross_cats_sorted":[],"license":"","primary_cat":"math.AP","submitted_at":"2000-10-07T03:11:59Z","title_canon_sha256":"56409cc4393c51e2f20cb99c48dc16b3b54813484c89f6a3934f7c28cd91bbe6"},"schema_version":"1.0","source":{"id":"math/0010068","kind":"arxiv","version":3}},"canonical_sha256":"2b7ccd05a43fceb55c7295117ac17caa17a495fc1415b155039833ab7a137873","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"2b7ccd05a43fceb55c7295117ac17caa17a495fc1415b155039833ab7a137873","first_computed_at":"2026-07-04T14:34:51.267869Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-04T14:34:51.267869Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"ztEPyWQtcItx7SG9meaIb5Jlhucw6KKdhKWdZOiODa6z6HItpGpEpgebdWlQW+dcK6Z3gZMPExcQE0CJEIhNAg==","signature_status":"signed_v1","signed_at":"2026-07-04T14:34:51.268736Z","signed_message":"canonical_sha256_bytes"},"source_id":"math/0010068","source_kind":"arxiv","source_version":3}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:6eacc0345c8b8eeb874d7175485779060fa297068abbf1268bf770a050928074","sha256:da0865b0ca45571a917ef5e9779f7b109e7e992cdcbad169f592000ac74dab34"],"state_sha256":"d68c18312e06f74f67a52932dfb1e3e0afdef745b58e3c582b3f853c92cd83fe"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"slqdY8b4yo4ONaViiiuSDUq9NnG0HLnTEZSWEcgu2tpT/kUGXzAbpb4YzU7/G/+FYCFTa4WN6gjea0cV7VyXBg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-13T03:59:39.715635Z","bundle_sha256":"22aca6b7653e8f24d177a26001e068259497c51fd7b4f8af091a02acd4880080"}}