{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2015:76PCV57RINI7Q4TEISBRRA3G4T","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":"375501af5e75c95565c8357f1b1eef9a86b896c84dc102a6b69d35f13812ed65","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2015-10-07T20:30:23Z","title_canon_sha256":"a355747250f170a3a5c5c745df4c35101d2df726130f7435b0b344438a5cf76a"},"schema_version":"1.0","source":{"id":"1510.02119","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1510.02119","created_at":"2026-05-18T01:03:31Z"},{"alias_kind":"arxiv_version","alias_value":"1510.02119v2","created_at":"2026-05-18T01:03:31Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1510.02119","created_at":"2026-05-18T01:03:31Z"},{"alias_kind":"pith_short_12","alias_value":"76PCV57RINI7","created_at":"2026-05-18T12:29:07Z"},{"alias_kind":"pith_short_16","alias_value":"76PCV57RINI7Q4TE","created_at":"2026-05-18T12:29:07Z"},{"alias_kind":"pith_short_8","alias_value":"76PCV57R","created_at":"2026-05-18T12:29:07Z"}],"graph_snapshots":[{"event_id":"sha256:94ed68ca9fe13814956c4d69c6a3cc6cebbca49f8f7217248a8a12ba1279b524","target":"graph","created_at":"2026-05-18T01:03:31Z","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":"We prove a strong form of the quantitative Sobolev inequality in $\\mathbb{R}^n$ for $p\\geq 2$, where the deficit of a function $u\\in \\dot W^{1,p} $ controls $\\| \\nabla u -\\nabla v\\|_{L^p}$ for an extremal function $v$ in the Sobolev inequality.","authors_text":"Alessio Figalli, Robin Neumayer","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2015-10-07T20:30:23Z","title":"Gradient stability for the Sobolev inequality: the case $p\\geq 2$"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1510.02119","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":"sha256:96a3da4b87e841fbd328b2418461cf3ad9ea552cf9479f28131c5ad3890f3317","target":"record","created_at":"2026-05-18T01:03:31Z","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":"375501af5e75c95565c8357f1b1eef9a86b896c84dc102a6b69d35f13812ed65","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2015-10-07T20:30:23Z","title_canon_sha256":"a355747250f170a3a5c5c745df4c35101d2df726130f7435b0b344438a5cf76a"},"schema_version":"1.0","source":{"id":"1510.02119","kind":"arxiv","version":2}},"canonical_sha256":"ff9e2af7f14351f872644483188366e4de5412e33a96f24341606408708f2dfc","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"ff9e2af7f14351f872644483188366e4de5412e33a96f24341606408708f2dfc","first_computed_at":"2026-05-18T01:03:31.045060Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:03:31.045060Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"NePCTp7qPsEVe414IdrSyAlil3y46w2LYCGwiHDHhIHU5iYa7gltibUb0a1/XRFxTOnHl6+TWF6Yg0gLA2uXCA==","signature_status":"signed_v1","signed_at":"2026-05-18T01:03:31.045605Z","signed_message":"canonical_sha256_bytes"},"source_id":"1510.02119","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:96a3da4b87e841fbd328b2418461cf3ad9ea552cf9479f28131c5ad3890f3317","sha256:94ed68ca9fe13814956c4d69c6a3cc6cebbca49f8f7217248a8a12ba1279b524"],"state_sha256":"a2c21b82570ce57ceede1834e062db8d40f2bc002e122673a1675ec4501a618f"}