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We study the probability \\begin{equation*} p_{2n}^{\\left(\\ell\\right)} = \\mathbb{P}\\left[M_{2n} \\, \\mbox{has no real eigenvalues}\\right], \\end{equation*} where $M_{2n}$ is the $2n\\times 2n$ left top minor of a $(2n+\\ell)\\times(2n+\\ell)$ orthogonal matrix. We prove that this probability is given in terms of a determinant identity minus a weighted Hankel matrix of size $n\\times n$ that depends on the truncation "},"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":"1905.03154","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2019-05-08T15:32:24Z","cross_cats_sorted":["math-ph","math.FA","math.MP"],"title_canon_sha256":"50e2ffdcc0ab1570f23996cd87528b90131aac107e662e44263024c5472f6a53","abstract_canon_sha256":"34327ab022d1c0c331bf0ad141a568d38661d4c14fea8a2b3662d0e8c25a395e"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:46:42.714167Z","signature_b64":"XOy9PbkfE1OnPd4LY6sB+UDmtsUn6IpUJmn7qBj/tIV5WAhkLjvJzYSVC3lBnVgcXbD3hD6+sM1EYXry/iO0Aw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"cba7ad65a80dbeb63d67e48803efb87f1781b6399661a7fd93712ca7bf8dfcb6","last_reissued_at":"2026-05-17T23:46:42.713420Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:46:42.713420Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On pure complex spectrum for truncations of random orthogonal matrices and Kac polynomials","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math-ph","math.FA","math.MP"],"primary_cat":"math.PR","authors_text":"Martin Gebert, Mihail Poplavskyi","submitted_at":"2019-05-08T15:32:24Z","abstract_excerpt":"Let $O(2n+\\ell)$ be the group of orthogonal matrices of size $\\left(2n+\\ell\\right)\\times \\left(2n+\\ell\\right)$ equipped with the probability distribution given by normalized Haar measure. 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