{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2011:GWL4OODURET64SXMG3RXCFKE6P","short_pith_number":"pith:GWL4OODU","canonical_record":{"source":{"id":"1110.6124","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2011-10-27T15:49:41Z","cross_cats_sorted":["math.AP"],"title_canon_sha256":"9de0455411de5a750aef21e5f3ff753144abe284bb82798c62697adf2e0c068f","abstract_canon_sha256":"cb3b1513f9c208d59dfd3e5293c9b2762105022917903b64b3821c084e964e3a"},"schema_version":"1.0"},"canonical_sha256":"3597c738748927ee4aec36e3711544f3efb47843f338f9f2bff2eb8201069725","source":{"kind":"arxiv","id":"1110.6124","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1110.6124","created_at":"2026-05-18T04:10:08Z"},{"alias_kind":"arxiv_version","alias_value":"1110.6124v2","created_at":"2026-05-18T04:10:08Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1110.6124","created_at":"2026-05-18T04:10:08Z"},{"alias_kind":"pith_short_12","alias_value":"GWL4OODURET6","created_at":"2026-05-18T12:26:30Z"},{"alias_kind":"pith_short_16","alias_value":"GWL4OODURET64SXM","created_at":"2026-05-18T12:26:30Z"},{"alias_kind":"pith_short_8","alias_value":"GWL4OODU","created_at":"2026-05-18T12:26:30Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2011:GWL4OODURET64SXMG3RXCFKE6P","target":"record","payload":{"canonical_record":{"source":{"id":"1110.6124","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2011-10-27T15:49:41Z","cross_cats_sorted":["math.AP"],"title_canon_sha256":"9de0455411de5a750aef21e5f3ff753144abe284bb82798c62697adf2e0c068f","abstract_canon_sha256":"cb3b1513f9c208d59dfd3e5293c9b2762105022917903b64b3821c084e964e3a"},"schema_version":"1.0"},"canonical_sha256":"3597c738748927ee4aec36e3711544f3efb47843f338f9f2bff2eb8201069725","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:10:08.297649Z","signature_b64":"r101hAOCiWQf+d16jvCWplrD/iXqZ4RVW9IQK0vD++PgGWpo7v5gpN0MQFsVG1eBLNndi9pzTizJ+LLGoaWTBw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"3597c738748927ee4aec36e3711544f3efb47843f338f9f2bff2eb8201069725","last_reissued_at":"2026-05-18T04:10:08.296725Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:10:08.296725Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1110.6124","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":"2026-05-18T04:10:08Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"MfFrLGUlmPc7RwW0xmDXc2HXEjvhqp2Zf6yInVYNem5HdBbwecZak0IdBHcYGDQmyJkNccTFQr7/0khLw3Q0Aw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-04T03:55:31.817337Z"},"content_sha256":"1c2c59cf1f7fceeee38f2d2dc9b4244ba2977d6df25f37e1ecae64a8a955f17c","schema_version":"1.0","event_id":"sha256:1c2c59cf1f7fceeee38f2d2dc9b4244ba2977d6df25f37e1ecae64a8a955f17c"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2011:GWL4OODURET64SXMG3RXCFKE6P","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"A planar bi-Lipschitz extension Theorem","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AP"],"primary_cat":"math.FA","authors_text":"Aldo Pratelli, Sara Daneri","submitted_at":"2011-10-27T15:49:41Z","abstract_excerpt":"We prove that, given a planar bi-Lipschitz homeomorphism $u$ defined on the boundary of the unit square, it is possible to extend it to a function $v$ of the whole square, in such a way that $v$ is still bi-Lipschitz. In particular, denoting by $L$ and $\\widetilde L$ the bi-Lipschitz constants of $u$ and $v$, with our construction one has $\\widetilde L \\leq C L^4$ (being $C$ an explicit geometrical constant). The same result was proved in 1980 by Tukia (see \\cite{Tukia}), using a completely different argument, but without any estimate on the constant $\\widetilde L$. In particular, the function"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1110.6124","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":"2026-05-18T04:10:08Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"muTwXe/9gOTkrO3Omly/SbAhQ/GrK+i3/QikU6j8EJ1N37I0AzT9A6ttPO65B7T4VXyNsd2D707bey8nqDQmBg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-04T03:55:31.817682Z"},"content_sha256":"25c0744a1ed03d1371348aa267dc0a553a05bde0295b021b32aecee1816840a6","schema_version":"1.0","event_id":"sha256:25c0744a1ed03d1371348aa267dc0a553a05bde0295b021b32aecee1816840a6"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/GWL4OODURET64SXMG3RXCFKE6P/bundle.json","state_url":"https://pith.science/pith/GWL4OODURET64SXMG3RXCFKE6P/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/GWL4OODURET64SXMG3RXCFKE6P/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":"2026-06-04T03:55:31Z","links":{"resolver":"https://pith.science/pith/GWL4OODURET64SXMG3RXCFKE6P","bundle":"https://pith.science/pith/GWL4OODURET64SXMG3RXCFKE6P/bundle.json","state":"https://pith.science/pith/GWL4OODURET64SXMG3RXCFKE6P/state.json","well_known_bundle":"https://pith.science/.well-known/pith/GWL4OODURET64SXMG3RXCFKE6P/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2011:GWL4OODURET64SXMG3RXCFKE6P","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":"cb3b1513f9c208d59dfd3e5293c9b2762105022917903b64b3821c084e964e3a","cross_cats_sorted":["math.AP"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2011-10-27T15:49:41Z","title_canon_sha256":"9de0455411de5a750aef21e5f3ff753144abe284bb82798c62697adf2e0c068f"},"schema_version":"1.0","source":{"id":"1110.6124","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1110.6124","created_at":"2026-05-18T04:10:08Z"},{"alias_kind":"arxiv_version","alias_value":"1110.6124v2","created_at":"2026-05-18T04:10:08Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1110.6124","created_at":"2026-05-18T04:10:08Z"},{"alias_kind":"pith_short_12","alias_value":"GWL4OODURET6","created_at":"2026-05-18T12:26:30Z"},{"alias_kind":"pith_short_16","alias_value":"GWL4OODURET64SXM","created_at":"2026-05-18T12:26:30Z"},{"alias_kind":"pith_short_8","alias_value":"GWL4OODU","created_at":"2026-05-18T12:26:30Z"}],"graph_snapshots":[{"event_id":"sha256:25c0744a1ed03d1371348aa267dc0a553a05bde0295b021b32aecee1816840a6","target":"graph","created_at":"2026-05-18T04:10:08Z","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 that, given a planar bi-Lipschitz homeomorphism $u$ defined on the boundary of the unit square, it is possible to extend it to a function $v$ of the whole square, in such a way that $v$ is still bi-Lipschitz. In particular, denoting by $L$ and $\\widetilde L$ the bi-Lipschitz constants of $u$ and $v$, with our construction one has $\\widetilde L \\leq C L^4$ (being $C$ an explicit geometrical constant). The same result was proved in 1980 by Tukia (see \\cite{Tukia}), using a completely different argument, but without any estimate on the constant $\\widetilde L$. In particular, the function","authors_text":"Aldo Pratelli, Sara Daneri","cross_cats":["math.AP"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2011-10-27T15:49:41Z","title":"A planar bi-Lipschitz extension Theorem"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1110.6124","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":"sha256:1c2c59cf1f7fceeee38f2d2dc9b4244ba2977d6df25f37e1ecae64a8a955f17c","target":"record","created_at":"2026-05-18T04:10:08Z","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":"cb3b1513f9c208d59dfd3e5293c9b2762105022917903b64b3821c084e964e3a","cross_cats_sorted":["math.AP"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2011-10-27T15:49:41Z","title_canon_sha256":"9de0455411de5a750aef21e5f3ff753144abe284bb82798c62697adf2e0c068f"},"schema_version":"1.0","source":{"id":"1110.6124","kind":"arxiv","version":2}},"canonical_sha256":"3597c738748927ee4aec36e3711544f3efb47843f338f9f2bff2eb8201069725","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"3597c738748927ee4aec36e3711544f3efb47843f338f9f2bff2eb8201069725","first_computed_at":"2026-05-18T04:10:08.296725Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T04:10:08.296725Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"r101hAOCiWQf+d16jvCWplrD/iXqZ4RVW9IQK0vD++PgGWpo7v5gpN0MQFsVG1eBLNndi9pzTizJ+LLGoaWTBw==","signature_status":"signed_v1","signed_at":"2026-05-18T04:10:08.297649Z","signed_message":"canonical_sha256_bytes"},"source_id":"1110.6124","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:1c2c59cf1f7fceeee38f2d2dc9b4244ba2977d6df25f37e1ecae64a8a955f17c","sha256:25c0744a1ed03d1371348aa267dc0a553a05bde0295b021b32aecee1816840a6"],"state_sha256":"5eadc5017134aad74286afc4b9fc3943bb9b9900392d196cd4de5cfb7bd3c92d"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"eGd+6arTjtUM4ppQVmPzypZ/RGQvfDIhNyJhW40AZ2K85GGE0KajhCi7WKFa8QGBmqkbZFT191BK948slpEZCg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-04T03:55:31.819569Z","bundle_sha256":"9f92b25bff59e0b813756f745966b6ca0343b66e6ecbbe1be2fe9b8af660ef9b"}}