{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2022:PX4AME6PKPEU3H32HX6ADKMPDN","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":"7b6f841ed2c538f259fa340149a1b82a6446afa3591d92a3007ca8f48f637c9b","cross_cats_sorted":["math.DG"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2022-01-25T09:41:47Z","title_canon_sha256":"ca24926342c4e0aae6d4bba9bb9d7758934bbe14ea80a620e2ab6ac723e8ba88"},"schema_version":"1.0","source":{"id":"2201.10204","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2201.10204","created_at":"2026-07-05T03:54:04Z"},{"alias_kind":"arxiv_version","alias_value":"2201.10204v2","created_at":"2026-07-05T03:54:04Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2201.10204","created_at":"2026-07-05T03:54:04Z"},{"alias_kind":"pith_short_12","alias_value":"PX4AME6PKPEU","created_at":"2026-07-05T03:54:04Z"},{"alias_kind":"pith_short_16","alias_value":"PX4AME6PKPEU3H32","created_at":"2026-07-05T03:54:04Z"},{"alias_kind":"pith_short_8","alias_value":"PX4AME6P","created_at":"2026-07-05T03:54:04Z"}],"graph_snapshots":[{"event_id":"sha256:e4a1598ef8c6a46bc2a7f9715b0228ace591895654dae7d35640843b5a9673d6","target":"graph","created_at":"2026-07-05T03:54:04Z","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"},"integrity":{"available":true,"clean":true,"detectors_run":[],"endpoint":"/pith/2201.10204/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"Consider an area minimizing current modulo $p$ of dimension $m$ in a smooth Riemannian manifold of dimension $m+1$. We prove that its interior singular set is, up to a relatively closed set of dimension at most $m-2$, a $C^{1,\\alpha}$ submanifold of dimension $m-1$ at which, locally, $N\\leq p$ regular sheets of the current join transversally, each sheet counted with a positive multiplicity $k_i$ so that $\\sum_i k_i = p$. This completes the analysis of the structure of the singular set of area minimizing hypersurfaces modulo $p$, initiated by J. Taylor for $m=2$ and $p=3$ and extended by the au","authors_text":"Andrea Marchese, Camillo De Lellis, Jonas Hirsch, Luca Spolaor, Salvatore Stuvard","cross_cats":["math.DG"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2022-01-25T09:41:47Z","title":"Fine structure of the singular set of area minimizing hypersurfaces modulo $p$"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2201.10204","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":"sha256:49d8c98956133da6e820d0e296719fec909c456b4a6b2154e7a6b06e8d728246","target":"record","created_at":"2026-07-05T03:54:04Z","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":"7b6f841ed2c538f259fa340149a1b82a6446afa3591d92a3007ca8f48f637c9b","cross_cats_sorted":["math.DG"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2022-01-25T09:41:47Z","title_canon_sha256":"ca24926342c4e0aae6d4bba9bb9d7758934bbe14ea80a620e2ab6ac723e8ba88"},"schema_version":"1.0","source":{"id":"2201.10204","kind":"arxiv","version":2}},"canonical_sha256":"7df80613cf53c94d9f7a3dfc01a98f1b77f4677a388b6cdb60b4e20fead4a526","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"7df80613cf53c94d9f7a3dfc01a98f1b77f4677a388b6cdb60b4e20fead4a526","first_computed_at":"2026-07-05T03:54:04.655392Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T03:54:04.655392Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"LMtwi7meh8HsKFOR3pO3Js2/J/Yf1qX6vAT4ZxkRRmoP7OSgbFC8WINY73hw+nD5yHYY3egONPzCZxRmPNatDQ==","signature_status":"signed_v1","signed_at":"2026-07-05T03:54:04.655905Z","signed_message":"canonical_sha256_bytes"},"source_id":"2201.10204","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:49d8c98956133da6e820d0e296719fec909c456b4a6b2154e7a6b06e8d728246","sha256:e4a1598ef8c6a46bc2a7f9715b0228ace591895654dae7d35640843b5a9673d6"],"state_sha256":"147f10467387c07c086e99aaa9ba3841d2da97c67a59bd20e887fd10b89afc16"}