{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2019:BFRNJTTKCYUHMQQB4AC2QW3SHE","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":"f4a19766212298f78fe30c5ef728a2a66e5f10ec512cda028c0280a9e3d2e92d","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2019-11-20T20:50:49Z","title_canon_sha256":"00d20ace63aff4a79b8c8b659a0d9c424db1f22e0489ae4d94b475d3965d8757"},"schema_version":"1.0","source":{"id":"1911.09166","kind":"arxiv","version":3}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1911.09166","created_at":"2026-07-05T04:50:29Z"},{"alias_kind":"arxiv_version","alias_value":"1911.09166v3","created_at":"2026-07-05T04:50:29Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1911.09166","created_at":"2026-07-05T04:50:29Z"},{"alias_kind":"pith_short_12","alias_value":"BFRNJTTKCYUH","created_at":"2026-07-05T04:50:29Z"},{"alias_kind":"pith_short_16","alias_value":"BFRNJTTKCYUHMQQB","created_at":"2026-07-05T04:50:29Z"},{"alias_kind":"pith_short_8","alias_value":"BFRNJTTK","created_at":"2026-07-05T04:50:29Z"}],"graph_snapshots":[{"event_id":"sha256:75a46ec91c3b1b17fee8810fe9f0ba252fdf33b8275cbd1f96e08a0b43088e43","target":"graph","created_at":"2026-07-05T04:50:29Z","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/1911.09166/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"We introduce a combinatorial argument to study closed minimal hypersurfaces of bounded area and high Morse index. Let $(M^{n+1},g)$ be a closed Riemannian manifold and $\\Sigma\\subset M$ be a closed embedded minimal hypersurface with area at most $A>0$ and with a singular set of Hausdorff dimension at most $n-7$. We show the following bounds: there is $C_A>0$ depending only on $n$, $g$, and $A$ so that $$\\sum_{i=0}^n b^i(\\Sigma) \\leq C_A \\big(1+index(\\Sigma)\\big) \\quad \\text{ if $3\\leq n+1\\leq 7$},$$ $$\\mathcal{H}^{n-7}\\big(Sing(\\Sigma)\\big) \\leq C_A \\big(1+index(\\Sigma)\\big)^{7/n} \\quad \\text{","authors_text":"Antoine Song","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2019-11-20T20:50:49Z","title":"Morse index, Betti numbers and singular set of bounded area minimal hypersurfaces"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1911.09166","kind":"arxiv","version":3},"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":"sha256:2bd92cb762c7bd2a7df03fac9c139ce2d114749680470d8dd4f59c6efcb6adc4","target":"record","created_at":"2026-07-05T04:50:29Z","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":"f4a19766212298f78fe30c5ef728a2a66e5f10ec512cda028c0280a9e3d2e92d","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2019-11-20T20:50:49Z","title_canon_sha256":"00d20ace63aff4a79b8c8b659a0d9c424db1f22e0489ae4d94b475d3965d8757"},"schema_version":"1.0","source":{"id":"1911.09166","kind":"arxiv","version":3}},"canonical_sha256":"0962d4ce6a1628764201e005a85b72393883d42699f5eba30cee3a429b2d01cf","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"0962d4ce6a1628764201e005a85b72393883d42699f5eba30cee3a429b2d01cf","first_computed_at":"2026-07-05T04:50:29.984970Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T04:50:29.984970Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"gAK0nIex2AqEEu7anGkB+UkMcuLQDVDzaAJ3sTldcbyggU1Lr+Zt4Bejg7vYoEgj9WsgfafCzjxVGPuBJx3kDA==","signature_status":"signed_v1","signed_at":"2026-07-05T04:50:29.985379Z","signed_message":"canonical_sha256_bytes"},"source_id":"1911.09166","source_kind":"arxiv","source_version":3}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:2bd92cb762c7bd2a7df03fac9c139ce2d114749680470d8dd4f59c6efcb6adc4","sha256:75a46ec91c3b1b17fee8810fe9f0ba252fdf33b8275cbd1f96e08a0b43088e43"],"state_sha256":"6ed31f4a312c0882c51665ccc506e4d621909b29f6a181beb9f0804a58811cb8"}