{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2012:D2OHRRGFJPJFFBYRDZZYROCIE2","short_pith_number":"pith:D2OHRRGF","schema_version":"1.0","canonical_sha256":"1e9c78c4c54bd25287111e7388b848268cac04ef08a706dc057e49399d015313","source":{"kind":"arxiv","id":"1209.0639","version":3},"attestation_state":"computed","paper":{"title":"Random Morse functions and spectral geometry","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AP","math.PR"],"primary_cat":"math.DG","authors_text":"Liviu I. Nicolaescu","submitted_at":"2012-09-04T13:29:31Z","abstract_excerpt":"We study random Morse functions on a Riemann manifold $(M^m,g)$ defined as a random Gaussian weighted superpositions of eigenfunctions of the Laplacian of the metric $g$. The randomness is determined by a fixed Schwartz function $w$ and a small parameter $\\varepsilon>0$. We first prove that as $\\varepsilon\\to 0$ the expected distribution of critical values of this random function approaches a universal measure on $\\mathbb{R}$, independent of $g$, that can be explicitly described in terms the expected distribution of eigenvalues of the Gaussian Wigner ensemble of random $(m+1)\\times (m+1)$ symm"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1209.0639","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2012-09-04T13:29:31Z","cross_cats_sorted":["math.AP","math.PR"],"title_canon_sha256":"c12fff9a732fc7bc249cd483950dc6727d8fd16aae18d45f0b44685cbe1962dc","abstract_canon_sha256":"cc34f716213ee0a52d3d1ebc271461591136e94372af7263ae8a3482c133f5f3"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:56:44.498565Z","signature_b64":"yNx6YbfpyNc7IrL8odTDPuUf5ZH5uGx1rmTCHksrg+Xbetg1yI4YQdFQ3IrA3jdxyJ+HM2p0GNAxn18llfElAQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"1e9c78c4c54bd25287111e7388b848268cac04ef08a706dc057e49399d015313","last_reissued_at":"2026-05-18T02:56:44.497843Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:56:44.497843Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Random Morse functions and spectral geometry","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AP","math.PR"],"primary_cat":"math.DG","authors_text":"Liviu I. Nicolaescu","submitted_at":"2012-09-04T13:29:31Z","abstract_excerpt":"We study random Morse functions on a Riemann manifold $(M^m,g)$ defined as a random Gaussian weighted superpositions of eigenfunctions of the Laplacian of the metric $g$. The randomness is determined by a fixed Schwartz function $w$ and a small parameter $\\varepsilon>0$. We first prove that as $\\varepsilon\\to 0$ the expected distribution of critical values of this random function approaches a universal measure on $\\mathbb{R}$, independent of $g$, that can be explicitly described in terms the expected distribution of eigenvalues of the Gaussian Wigner ensemble of random $(m+1)\\times (m+1)$ symm"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1209.0639","kind":"arxiv","version":3},"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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1209.0639","created_at":"2026-05-18T02:56:44.497972+00:00"},{"alias_kind":"arxiv_version","alias_value":"1209.0639v3","created_at":"2026-05-18T02:56:44.497972+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1209.0639","created_at":"2026-05-18T02:56:44.497972+00:00"},{"alias_kind":"pith_short_12","alias_value":"D2OHRRGFJPJF","created_at":"2026-05-18T12:27:01.376967+00:00"},{"alias_kind":"pith_short_16","alias_value":"D2OHRRGFJPJFFBYR","created_at":"2026-05-18T12:27:01.376967+00:00"},{"alias_kind":"pith_short_8","alias_value":"D2OHRRGF","created_at":"2026-05-18T12:27:01.376967+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/D2OHRRGFJPJFFBYRDZZYROCIE2","json":"https://pith.science/pith/D2OHRRGFJPJFFBYRDZZYROCIE2.json","graph_json":"https://pith.science/api/pith-number/D2OHRRGFJPJFFBYRDZZYROCIE2/graph.json","events_json":"https://pith.science/api/pith-number/D2OHRRGFJPJFFBYRDZZYROCIE2/events.json","paper":"https://pith.science/paper/D2OHRRGF"},"agent_actions":{"view_html":"https://pith.science/pith/D2OHRRGFJPJFFBYRDZZYROCIE2","download_json":"https://pith.science/pith/D2OHRRGFJPJFFBYRDZZYROCIE2.json","view_paper":"https://pith.science/paper/D2OHRRGF","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1209.0639&json=true","fetch_graph":"https://pith.science/api/pith-number/D2OHRRGFJPJFFBYRDZZYROCIE2/graph.json","fetch_events":"https://pith.science/api/pith-number/D2OHRRGFJPJFFBYRDZZYROCIE2/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/D2OHRRGFJPJFFBYRDZZYROCIE2/action/timestamp_anchor","attest_storage":"https://pith.science/pith/D2OHRRGFJPJFFBYRDZZYROCIE2/action/storage_attestation","attest_author":"https://pith.science/pith/D2OHRRGFJPJFFBYRDZZYROCIE2/action/author_attestation","sign_citation":"https://pith.science/pith/D2OHRRGFJPJFFBYRDZZYROCIE2/action/citation_signature","submit_replication":"https://pith.science/pith/D2OHRRGFJPJFFBYRDZZYROCIE2/action/replication_record"}},"created_at":"2026-05-18T02:56:44.497972+00:00","updated_at":"2026-05-18T02:56:44.497972+00:00"}