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This result is then generalized to singular values of rectangular random matrices with i.i.d. entries.\n  We also prove that for two fixed real numbers $\\lambda_1,\\lambda_2$ with a sufficient lower bound on $|\\lambda_1-\\lambda_2|$, we have a joint singular value small ball estimate for any $\\epsilon>0$ $$ \\mathbb{P}(\\sigma_{min}(A-\\lambda_1I_n)\\l"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"2502.13819","kind":"arxiv","version":2},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.PR","submitted_at":"2025-02-19T15:31:35Z","cross_cats_sorted":[],"title_canon_sha256":"a2f53cd0731d15fc85dd2a10f7106d984522a6b66a17b0dfb57baeefe32a818f","abstract_canon_sha256":"fa1206a2b26cb25895859e36acfd798ac3f76cee106a124833d73feaa7e38017"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T10:22:13.856709Z","signature_b64":"VI79iqOAgIrbwJAu1UWjD0ECKP3Wzr4ATmzslNY5+xE7gZGA8xbwBInH+2v71Ueiu+w/BHi/MozJ/HMca8ZUBg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"b85b1ef4c75cfdacf443adfaa619dbd9aed729e657d333bf617e7e3c1faff2bd","last_reissued_at":"2026-07-05T10:22:13.856008Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T10:22:13.856008Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Simplicity of singular value spectrum of random matrices and two-point quantitative invertibility","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Yi Han","submitted_at":"2025-02-19T15:31:35Z","abstract_excerpt":"Let $A$ be an $n\\times n$ random matrix with independent, identically distributed mean 0, variance 1 subgaussian entries. We prove that $$ \\mathbb{P}(A\\text{ has distinct singular values})\\geq 1-e^{-cn} $$ for some $c>0$, confirming a conjecture of Vu. This result is then generalized to singular values of rectangular random matrices with i.i.d. entries.\n  We also prove that for two fixed real numbers $\\lambda_1,\\lambda_2$ with a sufficient lower bound on $|\\lambda_1-\\lambda_2|$, we have a joint singular value small ball estimate for any $\\epsilon>0$ $$ \\mathbb{P}(\\sigma_{min}(A-\\lambda_1I_n)\\l"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2502.13819","kind":"arxiv","version":2},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2502.13819/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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":"2502.13819","created_at":"2026-07-05T10:22:13.856099+00:00"},{"alias_kind":"arxiv_version","alias_value":"2502.13819v2","created_at":"2026-07-05T10:22:13.856099+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2502.13819","created_at":"2026-07-05T10:22:13.856099+00:00"},{"alias_kind":"pith_short_12","alias_value":"XBNR55GHLT62","created_at":"2026-07-05T10:22:13.856099+00:00"},{"alias_kind":"pith_short_16","alias_value":"XBNR55GHLT62Z5CD","created_at":"2026-07-05T10:22:13.856099+00:00"},{"alias_kind":"pith_short_8","alias_value":"XBNR55GH","created_at":"2026-07-05T10:22:13.856099+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":1,"internal_anchor_count":0,"sample":[{"citing_arxiv_id":"2503.08139","citing_title":"The eigenvalue gap of inhomogeneous symmetric discrete random matrix","ref_index":14,"is_internal_anchor":false}]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/XBNR55GHLT62Z5CDVX5KMGO33G","json":"https://pith.science/pith/XBNR55GHLT62Z5CDVX5KMGO33G.json","graph_json":"https://pith.science/api/pith-number/XBNR55GHLT62Z5CDVX5KMGO33G/graph.json","events_json":"https://pith.science/api/pith-number/XBNR55GHLT62Z5CDVX5KMGO33G/events.json","paper":"https://pith.science/paper/XBNR55GH"},"agent_actions":{"view_html":"https://pith.science/pith/XBNR55GHLT62Z5CDVX5KMGO33G","download_json":"https://pith.science/pith/XBNR55GHLT62Z5CDVX5KMGO33G.json","view_paper":"https://pith.science/paper/XBNR55GH","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2502.13819&json=true","fetch_graph":"https://pith.science/api/pith-number/XBNR55GHLT62Z5CDVX5KMGO33G/graph.json","fetch_events":"https://pith.science/api/pith-number/XBNR55GHLT62Z5CDVX5KMGO33G/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/XBNR55GHLT62Z5CDVX5KMGO33G/action/timestamp_anchor","attest_storage":"https://pith.science/pith/XBNR55GHLT62Z5CDVX5KMGO33G/action/storage_attestation","attest_author":"https://pith.science/pith/XBNR55GHLT62Z5CDVX5KMGO33G/action/author_attestation","sign_citation":"https://pith.science/pith/XBNR55GHLT62Z5CDVX5KMGO33G/action/citation_signature","submit_replication":"https://pith.science/pith/XBNR55GHLT62Z5CDVX5KMGO33G/action/replication_record"}},"created_at":"2026-07-05T10:22:13.856099+00:00","updated_at":"2026-07-05T10:22:13.856099+00:00"}