{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2010:B7TRPTXYPPTI7WKIYDFYXNYUEU","short_pith_number":"pith:B7TRPTXY","canonical_record":{"source":{"id":"1003.0355","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2010-03-01T14:19:18Z","cross_cats_sorted":["math.GT"],"title_canon_sha256":"25c091bd0dd6f7d0182e5a3e6e8d5270c7bfce395d4365fb670e5a639851befd","abstract_canon_sha256":"10e5ca673f3b370e931d37eb18743461519eb87563016d889ba94bac29cba3d5"},"schema_version":"1.0"},"canonical_sha256":"0fe717cef87be68fd948c0cb8bb7142503c7695badf4b29943b81d3be4d609f2","source":{"kind":"arxiv","id":"1003.0355","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1003.0355","created_at":"2026-05-18T04:04:43Z"},{"alias_kind":"arxiv_version","alias_value":"1003.0355v2","created_at":"2026-05-18T04:04:43Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1003.0355","created_at":"2026-05-18T04:04:43Z"},{"alias_kind":"pith_short_12","alias_value":"B7TRPTXYPPTI","created_at":"2026-05-18T12:26:05Z"},{"alias_kind":"pith_short_16","alias_value":"B7TRPTXYPPTI7WKI","created_at":"2026-05-18T12:26:05Z"},{"alias_kind":"pith_short_8","alias_value":"B7TRPTXY","created_at":"2026-05-18T12:26:05Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2010:B7TRPTXYPPTI7WKIYDFYXNYUEU","target":"record","payload":{"canonical_record":{"source":{"id":"1003.0355","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2010-03-01T14:19:18Z","cross_cats_sorted":["math.GT"],"title_canon_sha256":"25c091bd0dd6f7d0182e5a3e6e8d5270c7bfce395d4365fb670e5a639851befd","abstract_canon_sha256":"10e5ca673f3b370e931d37eb18743461519eb87563016d889ba94bac29cba3d5"},"schema_version":"1.0"},"canonical_sha256":"0fe717cef87be68fd948c0cb8bb7142503c7695badf4b29943b81d3be4d609f2","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:04:43.836173Z","signature_b64":"89QXEEgIsT3sV9toDcoBSsAbFR4KtfH3MfLq0pew1jafirZwWK7FISRQP9Zxr/h1q2mtBqCasHoyB+5UXb6oBg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"0fe717cef87be68fd948c0cb8bb7142503c7695badf4b29943b81d3be4d609f2","last_reissued_at":"2026-05-18T04:04:43.835384Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:04:43.835384Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1003.0355","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":"2026-05-18T04:04:43Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"jjaTrP/q7VufUc9mQWwPQ6j6bg2mV5Y7X9OUKUs12GOeOx8KzfM5IbUnsK37R7AAfogNqBaIwgNwDjpEWJ4ODg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-02T13:31:00.009619Z"},"content_sha256":"704a4e015a62bf833fe09d0d805f5c8a5ee2f87c7da12cbcb6f5a2a7ca024278","schema_version":"1.0","event_id":"sha256:704a4e015a62bf833fe09d0d805f5c8a5ee2f87c7da12cbcb6f5a2a7ca024278"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2010:B7TRPTXYPPTI7WKIYDFYXNYUEU","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Isoparametric functions and exotic spheres","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.GT"],"primary_cat":"math.DG","authors_text":"Jianquan Ge, Zizhou Tang","submitted_at":"2010-03-01T14:19:18Z","abstract_excerpt":"The first part of the paper is to improve the fundamental theory of isoparametric functions on general Riemannian manifolds. Next we focus our attention on exotic spheres, especially on \"exotic\" 4-spheres (if exist) and the Gromoll-Meyer sphere. In particular, as one of main results we prove: there exists no properly transnormal function on any exotic 4-sphere if it exists. Furthermore, by projecting an $S^3$-invariant isoparametric function on $Sp(2)$, we construct a properly transnormal but not an isoparametric function on the Gromoll-Meyer sphere with two points as the focal varieties."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1003.0355","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":"2026-05-18T04:04:43Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"KnqeX8/fwOKYP/69tlbvOMA9KAZR42KMUSZWVp1cHg2L563+F3HUwL0N8+f2RGm7B26ORH2q0svlOKj2inQxCw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-02T13:31:00.009985Z"},"content_sha256":"d9e3408bf662d900dff0b2d6d868b2e4fa596a5d06290e8b10f4efcf521b359e","schema_version":"1.0","event_id":"sha256:d9e3408bf662d900dff0b2d6d868b2e4fa596a5d06290e8b10f4efcf521b359e"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/B7TRPTXYPPTI7WKIYDFYXNYUEU/bundle.json","state_url":"https://pith.science/pith/B7TRPTXYPPTI7WKIYDFYXNYUEU/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/B7TRPTXYPPTI7WKIYDFYXNYUEU/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":"2026-06-02T13:31:00Z","links":{"resolver":"https://pith.science/pith/B7TRPTXYPPTI7WKIYDFYXNYUEU","bundle":"https://pith.science/pith/B7TRPTXYPPTI7WKIYDFYXNYUEU/bundle.json","state":"https://pith.science/pith/B7TRPTXYPPTI7WKIYDFYXNYUEU/state.json","well_known_bundle":"https://pith.science/.well-known/pith/B7TRPTXYPPTI7WKIYDFYXNYUEU/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2010:B7TRPTXYPPTI7WKIYDFYXNYUEU","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":"10e5ca673f3b370e931d37eb18743461519eb87563016d889ba94bac29cba3d5","cross_cats_sorted":["math.GT"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2010-03-01T14:19:18Z","title_canon_sha256":"25c091bd0dd6f7d0182e5a3e6e8d5270c7bfce395d4365fb670e5a639851befd"},"schema_version":"1.0","source":{"id":"1003.0355","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1003.0355","created_at":"2026-05-18T04:04:43Z"},{"alias_kind":"arxiv_version","alias_value":"1003.0355v2","created_at":"2026-05-18T04:04:43Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1003.0355","created_at":"2026-05-18T04:04:43Z"},{"alias_kind":"pith_short_12","alias_value":"B7TRPTXYPPTI","created_at":"2026-05-18T12:26:05Z"},{"alias_kind":"pith_short_16","alias_value":"B7TRPTXYPPTI7WKI","created_at":"2026-05-18T12:26:05Z"},{"alias_kind":"pith_short_8","alias_value":"B7TRPTXY","created_at":"2026-05-18T12:26:05Z"}],"graph_snapshots":[{"event_id":"sha256:d9e3408bf662d900dff0b2d6d868b2e4fa596a5d06290e8b10f4efcf521b359e","target":"graph","created_at":"2026-05-18T04:04:43Z","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":"The first part of the paper is to improve the fundamental theory of isoparametric functions on general Riemannian manifolds. Next we focus our attention on exotic spheres, especially on \"exotic\" 4-spheres (if exist) and the Gromoll-Meyer sphere. In particular, as one of main results we prove: there exists no properly transnormal function on any exotic 4-sphere if it exists. Furthermore, by projecting an $S^3$-invariant isoparametric function on $Sp(2)$, we construct a properly transnormal but not an isoparametric function on the Gromoll-Meyer sphere with two points as the focal varieties.","authors_text":"Jianquan Ge, Zizhou Tang","cross_cats":["math.GT"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2010-03-01T14:19:18Z","title":"Isoparametric functions and exotic spheres"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1003.0355","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":"sha256:704a4e015a62bf833fe09d0d805f5c8a5ee2f87c7da12cbcb6f5a2a7ca024278","target":"record","created_at":"2026-05-18T04:04:43Z","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":"10e5ca673f3b370e931d37eb18743461519eb87563016d889ba94bac29cba3d5","cross_cats_sorted":["math.GT"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2010-03-01T14:19:18Z","title_canon_sha256":"25c091bd0dd6f7d0182e5a3e6e8d5270c7bfce395d4365fb670e5a639851befd"},"schema_version":"1.0","source":{"id":"1003.0355","kind":"arxiv","version":2}},"canonical_sha256":"0fe717cef87be68fd948c0cb8bb7142503c7695badf4b29943b81d3be4d609f2","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"0fe717cef87be68fd948c0cb8bb7142503c7695badf4b29943b81d3be4d609f2","first_computed_at":"2026-05-18T04:04:43.835384Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T04:04:43.835384Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"89QXEEgIsT3sV9toDcoBSsAbFR4KtfH3MfLq0pew1jafirZwWK7FISRQP9Zxr/h1q2mtBqCasHoyB+5UXb6oBg==","signature_status":"signed_v1","signed_at":"2026-05-18T04:04:43.836173Z","signed_message":"canonical_sha256_bytes"},"source_id":"1003.0355","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:704a4e015a62bf833fe09d0d805f5c8a5ee2f87c7da12cbcb6f5a2a7ca024278","sha256:d9e3408bf662d900dff0b2d6d868b2e4fa596a5d06290e8b10f4efcf521b359e"],"state_sha256":"602cfbc20356029931afb35f7bea0e379c06cbeed0862eee2c61d7f2b3315e31"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"nkiQCd8/mLsMdml7/MWo8puLTXCCIbhXDheUP0fIX6tD9cbLHsUtrNlxv85k62/Xppf9UAIA/lz8XN5Vri+VDA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-02T13:31:00.012521Z","bundle_sha256":"df946e811e6c373a676ccfa5eb4120fedfcd7d1e598d3ffa4985be70f35cdaf6"}}