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We prove that, for any $m \\geq 2$, $n \\geq 2$ and $x \\geq |n-m|+1$, $\\kappa_2(G_{m \\times n})$ satisfies $\n  \\frac{1}{\\sqrt{2\\pi}} ({c}/{x})^{|n-m|+1} < P(\\frac{\\kappa_2(G_{m \\times n})} {{n}/{(|n-m|+1)}}> x) <\n  \\frac{1}{\\sqrt{2\\pi}} ({C}/{x})^{|n-m|+1}, $ where $0.245 \\leq c \\leq 2.000$ and $ 5.013 \\leq C \\leq 6.414$ are universal positive constants indepen"},"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":"0810.0800","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.NA","submitted_at":"2008-10-05T04:18:54Z","cross_cats_sorted":["cs.NA"],"title_canon_sha256":"71580427fc49b439e3bd8340d0c85dca09c706b9a270728cbca29c975fcc2b12","abstract_canon_sha256":"79dec084bf2142293d972be301219dda97bef6669a597dbf3ce2b06650716731"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-06-03T22:06:10.495970Z","signature_b64":"YPfcTadNluTX700tYIoTid6lCZ8Ce0crwfXq2ANYwokUfDbBZpoU671BWy3MS+K4rTkmXuKithDYy0LoiiGvDA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"52a1c6491f2bf2c70d97cac0e1a69002f3e2e76bd372c0e2aa98199f59eb56d1","last_reissued_at":"2026-06-03T22:06:10.495383Z","signature_status":"signed_v1","first_computed_at":"2026-06-03T22:06:10.495383Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Condition Numbers of Gaussian Random Matrices","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.NA"],"primary_cat":"cs.NA","authors_text":"Jack Dongarra, Zizhong Chen","submitted_at":"2008-10-05T04:18:54Z","abstract_excerpt":"Let $G_{m \\times n}$ be an $m \\times n$ real random matrix whose elements are independent and identically distributed standard normal random variables, and let $\\kappa_2(G_{m \\times n})$ be the 2-norm condition number of $G_{m \\times n}$. We prove that, for any $m \\geq 2$, $n \\geq 2$ and $x \\geq |n-m|+1$, $\\kappa_2(G_{m \\times n})$ satisfies $\n  \\frac{1}{\\sqrt{2\\pi}} ({c}/{x})^{|n-m|+1} < P(\\frac{\\kappa_2(G_{m \\times n})} {{n}/{(|n-m|+1)}}> x) <\n  \\frac{1}{\\sqrt{2\\pi}} ({C}/{x})^{|n-m|+1}, $ where $0.245 \\leq c \\leq 2.000$ and $ 5.013 \\leq C \\leq 6.414$ are universal positive constants indepen"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0810.0800","kind":"arxiv","version":1},"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/0810.0800/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":"0810.0800","created_at":"2026-06-03T22:06:10.495461+00:00"},{"alias_kind":"arxiv_version","alias_value":"0810.0800v1","created_at":"2026-06-03T22:06:10.495461+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.0810.0800","created_at":"2026-06-03T22:06:10.495461+00:00"},{"alias_kind":"pith_short_12","alias_value":"KKQ4MSI7FPZM","created_at":"2026-06-03T22:06:10.495461+00:00"},{"alias_kind":"pith_short_16","alias_value":"KKQ4MSI7FPZMODMX","created_at":"2026-06-03T22:06:10.495461+00:00"},{"alias_kind":"pith_short_8","alias_value":"KKQ4MSI7","created_at":"2026-06-03T22:06:10.495461+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/KKQ4MSI7FPZMODMXZLAODJUQAL","json":"https://pith.science/pith/KKQ4MSI7FPZMODMXZLAODJUQAL.json","graph_json":"https://pith.science/api/pith-number/KKQ4MSI7FPZMODMXZLAODJUQAL/graph.json","events_json":"https://pith.science/api/pith-number/KKQ4MSI7FPZMODMXZLAODJUQAL/events.json","paper":"https://pith.science/paper/KKQ4MSI7"},"agent_actions":{"view_html":"https://pith.science/pith/KKQ4MSI7FPZMODMXZLAODJUQAL","download_json":"https://pith.science/pith/KKQ4MSI7FPZMODMXZLAODJUQAL.json","view_paper":"https://pith.science/paper/KKQ4MSI7","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=0810.0800&json=true","fetch_graph":"https://pith.science/api/pith-number/KKQ4MSI7FPZMODMXZLAODJUQAL/graph.json","fetch_events":"https://pith.science/api/pith-number/KKQ4MSI7FPZMODMXZLAODJUQAL/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/KKQ4MSI7FPZMODMXZLAODJUQAL/action/timestamp_anchor","attest_storage":"https://pith.science/pith/KKQ4MSI7FPZMODMXZLAODJUQAL/action/storage_attestation","attest_author":"https://pith.science/pith/KKQ4MSI7FPZMODMXZLAODJUQAL/action/author_attestation","sign_citation":"https://pith.science/pith/KKQ4MSI7FPZMODMXZLAODJUQAL/action/citation_signature","submit_replication":"https://pith.science/pith/KKQ4MSI7FPZMODMXZLAODJUQAL/action/replication_record"}},"created_at":"2026-06-03T22:06:10.495461+00:00","updated_at":"2026-06-03T22:06:10.495461+00:00"}