{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2015:X5H5LUKBT66H5UHN6SIQX3LZOT","short_pith_number":"pith:X5H5LUKB","canonical_record":{"source":{"id":"1512.05859","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2015-12-18T06:27:38Z","cross_cats_sorted":[],"title_canon_sha256":"1c760f150dbc636ea92f40315ca8cbbb123a490bfa1e8950419ad7fb0561d764","abstract_canon_sha256":"c8d42ae66d7af404821e5194fd96a0e5c91e00def7499dea2e96e50d97281892"},"schema_version":"1.0"},"canonical_sha256":"bf4fd5d1419fbc7ed0edf4910bed7974cec431c00b13bb41d63da4224cc51c67","source":{"kind":"arxiv","id":"1512.05859","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1512.05859","created_at":"2026-05-18T01:24:06Z"},{"alias_kind":"arxiv_version","alias_value":"1512.05859v1","created_at":"2026-05-18T01:24:06Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1512.05859","created_at":"2026-05-18T01:24:06Z"},{"alias_kind":"pith_short_12","alias_value":"X5H5LUKBT66H","created_at":"2026-05-18T12:29:47Z"},{"alias_kind":"pith_short_16","alias_value":"X5H5LUKBT66H5UHN","created_at":"2026-05-18T12:29:47Z"},{"alias_kind":"pith_short_8","alias_value":"X5H5LUKB","created_at":"2026-05-18T12:29:47Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2015:X5H5LUKBT66H5UHN6SIQX3LZOT","target":"record","payload":{"canonical_record":{"source":{"id":"1512.05859","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2015-12-18T06:27:38Z","cross_cats_sorted":[],"title_canon_sha256":"1c760f150dbc636ea92f40315ca8cbbb123a490bfa1e8950419ad7fb0561d764","abstract_canon_sha256":"c8d42ae66d7af404821e5194fd96a0e5c91e00def7499dea2e96e50d97281892"},"schema_version":"1.0"},"canonical_sha256":"bf4fd5d1419fbc7ed0edf4910bed7974cec431c00b13bb41d63da4224cc51c67","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:24:06.047811Z","signature_b64":"2dXCZWFx4jY4aY1zH1vuEoN8QKGezdKN9MF5ZsvteQM0W7rK6u2bpB4SUdAfm3mcvYf75tCbgeNxHAGRkYuNDQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"bf4fd5d1419fbc7ed0edf4910bed7974cec431c00b13bb41d63da4224cc51c67","last_reissued_at":"2026-05-18T01:24:06.047188Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:24:06.047188Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1512.05859","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":"2026-05-18T01:24:06Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"yBla/u0MkFqk71RFThWWqhs3DhCzKuHv+bfz6mEXhE56fpcoeJAFcyWI0iTBtdddF6O2eTAiaslqnRvllCaNDw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-23T21:30:52.859558Z"},"content_sha256":"29757f0a646bf7635ed9a82ff36971e5262defbf9296c816deddfba4ac8ee6b0","schema_version":"1.0","event_id":"sha256:29757f0a646bf7635ed9a82ff36971e5262defbf9296c816deddfba4ac8ee6b0"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2015:X5H5LUKBT66H5UHN6SIQX3LZOT","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Umbilic hypersurfaces of constant sigma-k curvature in the Heisenberg group","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.DG","authors_text":"Hung-Lin Chiu, Jenn-Fang Hwang, Jih-Hsin Cheng, Paul Yang","submitted_at":"2015-12-18T06:27:38Z","abstract_excerpt":"We study immersed, connected, umbilic hypersurfaces in the Heisenberg group $H_{n}$ with $n$ $\\geq $ $2.$ We show that such a hypersurface, if closed, must be rotationally invariant up to a Heisenberg translation. Moreover, we prove that, among others, Pansu spheres are the only such spheres with positive constant sigma-k curvature up to Heisenberg translations."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1512.05859","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":"2026-05-18T01:24:06Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"0lSLVKbBs0aFLXf7Dqx1gbVdOyCpPOopsNhiRexFDMonuf2q9fyvx9zElDXgwOhI6De+E/ukdDGYlNAglxjSAw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-23T21:30:52.859904Z"},"content_sha256":"c89f76038fb349adb3579c2b5651c966ef423200536b7b5e7edad7e056401a14","schema_version":"1.0","event_id":"sha256:c89f76038fb349adb3579c2b5651c966ef423200536b7b5e7edad7e056401a14"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/X5H5LUKBT66H5UHN6SIQX3LZOT/bundle.json","state_url":"https://pith.science/pith/X5H5LUKBT66H5UHN6SIQX3LZOT/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/X5H5LUKBT66H5UHN6SIQX3LZOT/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":"2026-06-23T21:30:52Z","links":{"resolver":"https://pith.science/pith/X5H5LUKBT66H5UHN6SIQX3LZOT","bundle":"https://pith.science/pith/X5H5LUKBT66H5UHN6SIQX3LZOT/bundle.json","state":"https://pith.science/pith/X5H5LUKBT66H5UHN6SIQX3LZOT/state.json","well_known_bundle":"https://pith.science/.well-known/pith/X5H5LUKBT66H5UHN6SIQX3LZOT/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2015:X5H5LUKBT66H5UHN6SIQX3LZOT","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":"c8d42ae66d7af404821e5194fd96a0e5c91e00def7499dea2e96e50d97281892","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2015-12-18T06:27:38Z","title_canon_sha256":"1c760f150dbc636ea92f40315ca8cbbb123a490bfa1e8950419ad7fb0561d764"},"schema_version":"1.0","source":{"id":"1512.05859","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1512.05859","created_at":"2026-05-18T01:24:06Z"},{"alias_kind":"arxiv_version","alias_value":"1512.05859v1","created_at":"2026-05-18T01:24:06Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1512.05859","created_at":"2026-05-18T01:24:06Z"},{"alias_kind":"pith_short_12","alias_value":"X5H5LUKBT66H","created_at":"2026-05-18T12:29:47Z"},{"alias_kind":"pith_short_16","alias_value":"X5H5LUKBT66H5UHN","created_at":"2026-05-18T12:29:47Z"},{"alias_kind":"pith_short_8","alias_value":"X5H5LUKB","created_at":"2026-05-18T12:29:47Z"}],"graph_snapshots":[{"event_id":"sha256:c89f76038fb349adb3579c2b5651c966ef423200536b7b5e7edad7e056401a14","target":"graph","created_at":"2026-05-18T01:24:06Z","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 study immersed, connected, umbilic hypersurfaces in the Heisenberg group $H_{n}$ with $n$ $\\geq $ $2.$ We show that such a hypersurface, if closed, must be rotationally invariant up to a Heisenberg translation. Moreover, we prove that, among others, Pansu spheres are the only such spheres with positive constant sigma-k curvature up to Heisenberg translations.","authors_text":"Hung-Lin Chiu, Jenn-Fang Hwang, Jih-Hsin Cheng, Paul Yang","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2015-12-18T06:27:38Z","title":"Umbilic hypersurfaces of constant sigma-k curvature in the Heisenberg group"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1512.05859","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":"sha256:29757f0a646bf7635ed9a82ff36971e5262defbf9296c816deddfba4ac8ee6b0","target":"record","created_at":"2026-05-18T01:24:06Z","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":"c8d42ae66d7af404821e5194fd96a0e5c91e00def7499dea2e96e50d97281892","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2015-12-18T06:27:38Z","title_canon_sha256":"1c760f150dbc636ea92f40315ca8cbbb123a490bfa1e8950419ad7fb0561d764"},"schema_version":"1.0","source":{"id":"1512.05859","kind":"arxiv","version":1}},"canonical_sha256":"bf4fd5d1419fbc7ed0edf4910bed7974cec431c00b13bb41d63da4224cc51c67","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"bf4fd5d1419fbc7ed0edf4910bed7974cec431c00b13bb41d63da4224cc51c67","first_computed_at":"2026-05-18T01:24:06.047188Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:24:06.047188Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"2dXCZWFx4jY4aY1zH1vuEoN8QKGezdKN9MF5ZsvteQM0W7rK6u2bpB4SUdAfm3mcvYf75tCbgeNxHAGRkYuNDQ==","signature_status":"signed_v1","signed_at":"2026-05-18T01:24:06.047811Z","signed_message":"canonical_sha256_bytes"},"source_id":"1512.05859","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:29757f0a646bf7635ed9a82ff36971e5262defbf9296c816deddfba4ac8ee6b0","sha256:c89f76038fb349adb3579c2b5651c966ef423200536b7b5e7edad7e056401a14"],"state_sha256":"ef21f5abb7a2852e2544255e3123a721450c928b18b8982d2ea644f13a5068c4"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"2qo3fLH4JPlzJDmePdxiCRb1mR9YeqGLcmAtXE5a0PHUcgfUjdOxHSWEp0f89ALS5CjF62wJNrX5LmlgrMhqBA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-23T21:30:52.861808Z","bundle_sha256":"94ad007c3de0c2a638e60f4b649600abcdc3b52bbdbe0fa4eb6c35da3d52f5bf"}}