{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2026:WP3DTEIPTTUDXANMW6YBYGN2QJ","merge_version":"pith-open-graph-merge-v1","event_count":4,"valid_event_count":4,"invalid_event_count":0,"equivocation_count":1,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"4a7ca490ee250c2c8a1f1cb94628fe78fe6862e4f616d81c487f5f91429b966b","cross_cats_sorted":["math.AT","math.PR","math.ST","stat.TH"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2026-05-21T10:09:14Z","title_canon_sha256":"910f12038a0843767644604def07bac7c4f2dd603ef1d0d914bfadc5f69f0921"},"schema_version":"1.0","source":{"id":"2605.22265","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2605.22265","created_at":"2026-05-22T01:04:35Z"},{"alias_kind":"arxiv_version","alias_value":"2605.22265v1","created_at":"2026-05-22T01:04:35Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2605.22265","created_at":"2026-05-22T01:04:35Z"},{"alias_kind":"pith_short_12","alias_value":"WP3DTEIPTTUD","created_at":"2026-05-22T01:04:35Z"},{"alias_kind":"pith_short_16","alias_value":"WP3DTEIPTTUDXANM","created_at":"2026-05-22T01:04:35Z"},{"alias_kind":"pith_short_8","alias_value":"WP3DTEIP","created_at":"2026-05-22T01:04:35Z"}],"graph_snapshots":[{"event_id":"sha256:322eebde04ba29d82cdaa4359f96f6e459f68ddc1af8f3804acd193f2422b3cc","target":"graph","created_at":"2026-05-22T01:04:35Z","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/2605.22265/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"Let $M^n$ be a compact orientable Riemannian smooth submanifold of dimension $n \\ge 2$ in $\\mathbf R^d$. We construct a family of deformed Hodge Laplacians $\\Delta ^*_t, t \\in \\mathbf R_{+},$ acting on differential forms using the extrinsic geometry of $M^n$ and prove their uniform convergence to the Hodge Laplacian $\\Delta^*$ as $t \\to 0^+$. Given a point cloud $S_m \\subset M^n$, we define symmetrized empirical operators $\\Delta^*_{sym, t, S_m}$ and establish their spectral convergence in probability to $\\Delta^*$, as $t \\to 0^+$, under suitable scaling regimes. This extends the scalar framew","authors_text":"H\\^ong V\\^an L\\^e","cross_cats":["math.AT","math.PR","math.ST","stat.TH"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2026-05-21T10:09:14Z","title":"Empirical Hodge Laplacians, Cohomology Ring, and Manifold Learning"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2605.22265","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":"sha256:8df4cd385bc0fed8849a87541b268ee575e4d6d5eddb7b60e185266829d48609","target":"record","created_at":"2026-05-22T01:04:35Z","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":"4a7ca490ee250c2c8a1f1cb94628fe78fe6862e4f616d81c487f5f91429b966b","cross_cats_sorted":["math.AT","math.PR","math.ST","stat.TH"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2026-05-21T10:09:14Z","title_canon_sha256":"910f12038a0843767644604def07bac7c4f2dd603ef1d0d914bfadc5f69f0921"},"schema_version":"1.0","source":{"id":"2605.22265","kind":"arxiv","version":1}},"canonical_sha256":"b3f639910f9ce83b81acb7b01c19ba8240acae296420623e2d250eca2070a92d","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"b3f639910f9ce83b81acb7b01c19ba8240acae296420623e2d250eca2070a92d","first_computed_at":"2026-05-22T01:04:35.171875Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-22T01:04:35.171875Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"4sob3yRFN5XAA0VIqrIWj1bhIRH4NQQC+t9l1f0dd+Xnemv3LchcROiCqRfF5Q//JKmkDeBE58FQrXmO+wrPAg==","signature_status":"signed_v1","signed_at":"2026-05-22T01:04:35.172653Z","signed_message":"canonical_sha256_bytes"},"source_id":"2605.22265","source_kind":"arxiv","source_version":1}}},"equivocations":[{"signer_id":"pith.science","event_type":"integrity_finding","target":"integrity","event_ids":["sha256:5555d9c0ab13e2aeb7286449a64465aa5374e07c649cc9a3f8f0196fecb2aebb","sha256:d57feed6d8b1c8bce33661719f479850a7524128f6cc0b9b22a916601bffaa40"]}],"invalid_events":[],"applied_event_ids":["sha256:8df4cd385bc0fed8849a87541b268ee575e4d6d5eddb7b60e185266829d48609","sha256:322eebde04ba29d82cdaa4359f96f6e459f68ddc1af8f3804acd193f2422b3cc"],"state_sha256":"9f48ad7dae91fa22d17abd59a261bcf832324cf6eb4d686a7af6e3758718e04c"}