{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2015:RQTZU3MUJ4SKGHFHA7JUGNB7VQ","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":"8a095e7da31268e6d314616ba7db789e6768642e57185bd6e64937bd63c3556f","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GT","submitted_at":"2015-07-19T18:47:21Z","title_canon_sha256":"3ee2457d589b857fa684c932c72b4774bc315d3af2fe74416be61ea61699bc71"},"schema_version":"1.0","source":{"id":"1507.05319","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1507.05319","created_at":"2026-05-18T01:20:22Z"},{"alias_kind":"arxiv_version","alias_value":"1507.05319v2","created_at":"2026-05-18T01:20:22Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1507.05319","created_at":"2026-05-18T01:20:22Z"},{"alias_kind":"pith_short_12","alias_value":"RQTZU3MUJ4SK","created_at":"2026-05-18T12:29:39Z"},{"alias_kind":"pith_short_16","alias_value":"RQTZU3MUJ4SKGHFH","created_at":"2026-05-18T12:29:39Z"},{"alias_kind":"pith_short_8","alias_value":"RQTZU3MU","created_at":"2026-05-18T12:29:39Z"}],"graph_snapshots":[{"event_id":"sha256:b69dbfb9acbd5f8651472a71cb3ca07ab927aec3e66ededffc1348775b543cab","target":"graph","created_at":"2026-05-18T01:20:22Z","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 construct a large class of pathological $n$-dimensional topological spheres in ${\\mathbb R}^{n+1}$ by showing that for any Cantor set $C\\subset {\\mathbb R}^{n+1}$ there is a topological embedding $f:{\\mathbb S}^n\\to{\\mathbb R}^{n+1}$ of the Sobolev class $W^{1,n}$ whose image contains the Cantor set $C$.","authors_text":"Piotr Haj{\\l}asz, Xiaodan Zhou","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GT","submitted_at":"2015-07-19T18:47:21Z","title":"Sobolev Embedding of a Sphere Containing An Arbitrary Cantor Set in the image"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1507.05319","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:ec9b6163a594727bae6a354f30b5d9459cfb7b6d25ebf8cf344b3b536e4ca804","target":"record","created_at":"2026-05-18T01:20:22Z","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":"8a095e7da31268e6d314616ba7db789e6768642e57185bd6e64937bd63c3556f","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GT","submitted_at":"2015-07-19T18:47:21Z","title_canon_sha256":"3ee2457d589b857fa684c932c72b4774bc315d3af2fe74416be61ea61699bc71"},"schema_version":"1.0","source":{"id":"1507.05319","kind":"arxiv","version":2}},"canonical_sha256":"8c279a6d944f24a31ca707d343343fac36c85dfb21b9f3c66516afa0bbbb3fce","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"8c279a6d944f24a31ca707d343343fac36c85dfb21b9f3c66516afa0bbbb3fce","first_computed_at":"2026-05-18T01:20:22.197645Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:20:22.197645Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"QWpeLuKeEGCnAeA9hI0VBUz8Vk65Ueev3jX8mwbagNFMkllAxa4IUGZwOGN3vGfvJXKu+mUxeZ9HiDZoafqbBA==","signature_status":"signed_v1","signed_at":"2026-05-18T01:20:22.198265Z","signed_message":"canonical_sha256_bytes"},"source_id":"1507.05319","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:ec9b6163a594727bae6a354f30b5d9459cfb7b6d25ebf8cf344b3b536e4ca804","sha256:b69dbfb9acbd5f8651472a71cb3ca07ab927aec3e66ededffc1348775b543cab"],"state_sha256":"eb1df120c2058d9a64c6489366b90ebc00c1ff7e5c1907d8b87649effef012b3"}