{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2014:WXA6R6MRLZAFLBOOLYU4BBXPW7","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":"13d0346f4f44d670474493cf65ef48ad455d57bf34aa4264bafd42c0fe16a33f","cross_cats_sorted":["cond-mat.stat-mech","hep-lat","math.MP"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math-ph","submitted_at":"2014-09-01T11:00:10Z","title_canon_sha256":"6256bca14feefbaa6a6e91d682fa6485219a2e61cdb398690258eafbb71bd969"},"schema_version":"1.0","source":{"id":"1409.0360","kind":"arxiv","version":3}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1409.0360","created_at":"2026-05-18T02:25:26Z"},{"alias_kind":"arxiv_version","alias_value":"1409.0360v3","created_at":"2026-05-18T02:25:26Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1409.0360","created_at":"2026-05-18T02:25:26Z"},{"alias_kind":"pith_short_12","alias_value":"WXA6R6MRLZAF","created_at":"2026-05-18T12:28:54Z"},{"alias_kind":"pith_short_16","alias_value":"WXA6R6MRLZAFLBOO","created_at":"2026-05-18T12:28:54Z"},{"alias_kind":"pith_short_8","alias_value":"WXA6R6MR","created_at":"2026-05-18T12:28:54Z"}],"graph_snapshots":[{"event_id":"sha256:098d9f5f27813e4a2c025b0f62c8a885cda9cbbe856325d6a99582d03808f899","target":"graph","created_at":"2026-05-18T02:25:26Z","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"},"paper":{"abstract_excerpt":"Rectangular real $N \\times (N + \\nu)$ matrices $W$ with a Gaussian distribution appear very frequently in data analysis, condensed matter physics and quantum field theory. A central question concerns the correlations encoded in the spectral statistics of $WW^T$. The extreme eigenvalues of $W W^T$ are of particular interest. We explicitly compute the distribution and the gap probability of the smallest non-zero eigenvalue in this ensemble, both for arbitrary fixed $N$ and $\\nu$, and in the universal large $N$ limit with $\\nu$ fixed. We uncover an integrable Pfaffian structure valid for all even","authors_text":"G. Akemann, M. Kieburg, R. Wegner, T. Guhr, T. Wirtz","cross_cats":["cond-mat.stat-mech","hep-lat","math.MP"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math-ph","submitted_at":"2014-09-01T11:00:10Z","title":"Completing the picture for the smallest eigenvalue of real Wishart matrices"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1409.0360","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":"sha256:3021c92f6c3c959bf61e0a896d6f245704ea3039e83b9f47d527a7ad6a86d60f","target":"record","created_at":"2026-05-18T02:25:26Z","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":"13d0346f4f44d670474493cf65ef48ad455d57bf34aa4264bafd42c0fe16a33f","cross_cats_sorted":["cond-mat.stat-mech","hep-lat","math.MP"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math-ph","submitted_at":"2014-09-01T11:00:10Z","title_canon_sha256":"6256bca14feefbaa6a6e91d682fa6485219a2e61cdb398690258eafbb71bd969"},"schema_version":"1.0","source":{"id":"1409.0360","kind":"arxiv","version":3}},"canonical_sha256":"b5c1e8f9915e405585ce5e29c086efb7c518bb4f07a5b96ccdeec2172202f575","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"b5c1e8f9915e405585ce5e29c086efb7c518bb4f07a5b96ccdeec2172202f575","first_computed_at":"2026-05-18T02:25:26.356167Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T02:25:26.356167Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"NkfChPucfApKe7ycicwi+u3S9F9LDL5elSkE8X8lw7tt2gYM0R4I4rzgd35gAh1IdJU/EhXqSkUxVuMJSjrYAw==","signature_status":"signed_v1","signed_at":"2026-05-18T02:25:26.356555Z","signed_message":"canonical_sha256_bytes"},"source_id":"1409.0360","source_kind":"arxiv","source_version":3}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:3021c92f6c3c959bf61e0a896d6f245704ea3039e83b9f47d527a7ad6a86d60f","sha256:098d9f5f27813e4a2c025b0f62c8a885cda9cbbe856325d6a99582d03808f899"],"state_sha256":"dbf0cbf12fe16124d5c1f5631da484c5c069630fba45221b6b28ef8baeba45ad"}