{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2012:VHF2ZZLQQKBQ5D5XRYQHAOIZRS","short_pith_number":"pith:VHF2ZZLQ","canonical_record":{"source":{"id":"1204.3441","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.MG","submitted_at":"2012-04-16T11:15:25Z","cross_cats_sorted":["math.FA"],"title_canon_sha256":"118c11ff436385f0e235480925bb39d93cfcbb635f07bfc09f00e9296aace525","abstract_canon_sha256":"df2b89c6c0a9bd98021d52dae7f01c3750d3a50d08ac34f3ae343aa0a80fd26b"},"schema_version":"1.0"},"canonical_sha256":"a9cbace57082830e8fb78e207039198c8f125f0704d29e2a3f1aee767c406c88","source":{"kind":"arxiv","id":"1204.3441","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1204.3441","created_at":"2026-05-18T03:57:51Z"},{"alias_kind":"arxiv_version","alias_value":"1204.3441v1","created_at":"2026-05-18T03:57:51Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1204.3441","created_at":"2026-05-18T03:57:51Z"},{"alias_kind":"pith_short_12","alias_value":"VHF2ZZLQQKBQ","created_at":"2026-05-18T12:27:25Z"},{"alias_kind":"pith_short_16","alias_value":"VHF2ZZLQQKBQ5D5X","created_at":"2026-05-18T12:27:25Z"},{"alias_kind":"pith_short_8","alias_value":"VHF2ZZLQ","created_at":"2026-05-18T12:27:25Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2012:VHF2ZZLQQKBQ5D5XRYQHAOIZRS","target":"record","payload":{"canonical_record":{"source":{"id":"1204.3441","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.MG","submitted_at":"2012-04-16T11:15:25Z","cross_cats_sorted":["math.FA"],"title_canon_sha256":"118c11ff436385f0e235480925bb39d93cfcbb635f07bfc09f00e9296aace525","abstract_canon_sha256":"df2b89c6c0a9bd98021d52dae7f01c3750d3a50d08ac34f3ae343aa0a80fd26b"},"schema_version":"1.0"},"canonical_sha256":"a9cbace57082830e8fb78e207039198c8f125f0704d29e2a3f1aee767c406c88","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:57:51.530695Z","signature_b64":"04dx4/jrW9Ro8DbRMLl+YB+s2rk+8tFP7+U1bJ+ZsZLGMYhwnYUwI4JoCRAx42TobFaWI0QdDFD/VeSLdlMgDQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"a9cbace57082830e8fb78e207039198c8f125f0704d29e2a3f1aee767c406c88","last_reissued_at":"2026-05-18T03:57:51.530069Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:57:51.530069Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1204.3441","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":"2026-05-18T03:57:51Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"0AD6nhFwRtWNvA8HfM0b55y+0QZrURLZMHNkM2ycsRDqa2r8LlR40o8VF/AnCdKM0yejq+rHTcLzHE/4XB5sBQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-27T11:06:59.221925Z"},"content_sha256":"cee419648c928185314b4a4d40d23a6a0707394125753289dafa7f4c495cc402","schema_version":"1.0","event_id":"sha256:cee419648c928185314b4a4d40d23a6a0707394125753289dafa7f4c495cc402"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2012:VHF2ZZLQQKBQ5D5XRYQHAOIZRS","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Sharp geometric rigidity of isometries on Heisenberg groups","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.FA"],"primary_cat":"math.MG","authors_text":"D. V. Isangulova, S. K. Vodopyanov","submitted_at":"2012-04-16T11:15:25Z","abstract_excerpt":"We prove sharp geometric rigidity estimates for isometries on Heisenberg groups. Our main result asserts that every $(1+\\varepsilon)$-quasi-isometry on a John domain of the Heisenberg group $\\mathbb{H}^n$, $n>1$, is close to some isometry up to proximity order $\\sqrt{\\varepsilon}+\\varepsilon$ in the uniform norm, and up to proximity order $\\varepsilon$ in the $L_p^1$-norm. We give examples showing the asymptotic sharpness of our results."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1204.3441","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":""},"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":"2026-05-18T03:57:51Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"nMX/vlqzIgQldMEwcDLpTx1b0tqvlVycTAJgvg7tsl2Z6OMlS/PegChkNgw1xlxKx9e7p+qgopfBEsmJLYUrDA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-27T11:06:59.222261Z"},"content_sha256":"d0d50eabfaf42aa00eea697f36e4c5b74a3e080a57b4d747a864d1f0d9a04b38","schema_version":"1.0","event_id":"sha256:d0d50eabfaf42aa00eea697f36e4c5b74a3e080a57b4d747a864d1f0d9a04b38"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/VHF2ZZLQQKBQ5D5XRYQHAOIZRS/bundle.json","state_url":"https://pith.science/pith/VHF2ZZLQQKBQ5D5XRYQHAOIZRS/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/VHF2ZZLQQKBQ5D5XRYQHAOIZRS/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":"2026-06-27T11:06:59Z","links":{"resolver":"https://pith.science/pith/VHF2ZZLQQKBQ5D5XRYQHAOIZRS","bundle":"https://pith.science/pith/VHF2ZZLQQKBQ5D5XRYQHAOIZRS/bundle.json","state":"https://pith.science/pith/VHF2ZZLQQKBQ5D5XRYQHAOIZRS/state.json","well_known_bundle":"https://pith.science/.well-known/pith/VHF2ZZLQQKBQ5D5XRYQHAOIZRS/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2012:VHF2ZZLQQKBQ5D5XRYQHAOIZRS","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":"df2b89c6c0a9bd98021d52dae7f01c3750d3a50d08ac34f3ae343aa0a80fd26b","cross_cats_sorted":["math.FA"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.MG","submitted_at":"2012-04-16T11:15:25Z","title_canon_sha256":"118c11ff436385f0e235480925bb39d93cfcbb635f07bfc09f00e9296aace525"},"schema_version":"1.0","source":{"id":"1204.3441","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1204.3441","created_at":"2026-05-18T03:57:51Z"},{"alias_kind":"arxiv_version","alias_value":"1204.3441v1","created_at":"2026-05-18T03:57:51Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1204.3441","created_at":"2026-05-18T03:57:51Z"},{"alias_kind":"pith_short_12","alias_value":"VHF2ZZLQQKBQ","created_at":"2026-05-18T12:27:25Z"},{"alias_kind":"pith_short_16","alias_value":"VHF2ZZLQQKBQ5D5X","created_at":"2026-05-18T12:27:25Z"},{"alias_kind":"pith_short_8","alias_value":"VHF2ZZLQ","created_at":"2026-05-18T12:27:25Z"}],"graph_snapshots":[{"event_id":"sha256:d0d50eabfaf42aa00eea697f36e4c5b74a3e080a57b4d747a864d1f0d9a04b38","target":"graph","created_at":"2026-05-18T03:57:51Z","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"},"paper":{"abstract_excerpt":"We prove sharp geometric rigidity estimates for isometries on Heisenberg groups. Our main result asserts that every $(1+\\varepsilon)$-quasi-isometry on a John domain of the Heisenberg group $\\mathbb{H}^n$, $n>1$, is close to some isometry up to proximity order $\\sqrt{\\varepsilon}+\\varepsilon$ in the uniform norm, and up to proximity order $\\varepsilon$ in the $L_p^1$-norm. We give examples showing the asymptotic sharpness of our results.","authors_text":"D. V. Isangulova, S. K. Vodopyanov","cross_cats":["math.FA"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.MG","submitted_at":"2012-04-16T11:15:25Z","title":"Sharp geometric rigidity of isometries on Heisenberg groups"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1204.3441","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:cee419648c928185314b4a4d40d23a6a0707394125753289dafa7f4c495cc402","target":"record","created_at":"2026-05-18T03:57:51Z","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":"df2b89c6c0a9bd98021d52dae7f01c3750d3a50d08ac34f3ae343aa0a80fd26b","cross_cats_sorted":["math.FA"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.MG","submitted_at":"2012-04-16T11:15:25Z","title_canon_sha256":"118c11ff436385f0e235480925bb39d93cfcbb635f07bfc09f00e9296aace525"},"schema_version":"1.0","source":{"id":"1204.3441","kind":"arxiv","version":1}},"canonical_sha256":"a9cbace57082830e8fb78e207039198c8f125f0704d29e2a3f1aee767c406c88","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"a9cbace57082830e8fb78e207039198c8f125f0704d29e2a3f1aee767c406c88","first_computed_at":"2026-05-18T03:57:51.530069Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:57:51.530069Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"04dx4/jrW9Ro8DbRMLl+YB+s2rk+8tFP7+U1bJ+ZsZLGMYhwnYUwI4JoCRAx42TobFaWI0QdDFD/VeSLdlMgDQ==","signature_status":"signed_v1","signed_at":"2026-05-18T03:57:51.530695Z","signed_message":"canonical_sha256_bytes"},"source_id":"1204.3441","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:cee419648c928185314b4a4d40d23a6a0707394125753289dafa7f4c495cc402","sha256:d0d50eabfaf42aa00eea697f36e4c5b74a3e080a57b4d747a864d1f0d9a04b38"],"state_sha256":"67c8ea6d33c3d72f74861646de560ba9427f9ec4d7666d408c7292097ca4f32c"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"iBbkO57C2myzDCh5jC5VDU2Ur7a1I4DE+/KuF7/h5MPqLDMDL9iFDv7HV0kcdnjiUoqINtlcHhmyCefTggTSBw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-27T11:06:59.224079Z","bundle_sha256":"1404f01a455157e8a00e61e06c841957c62521f554e188285cc6d97dd081801e"}}