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In this paper we express the Cullis' determinant of a matrix $X$ as the Pfaffian of the matrix obtained from $X$ by matrix multiplication and transposition.\n  Relying on this result, we present an efficient polynomial-time algorithm for calculating the Cullis' determinant of given matrix."},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"we express the Cullis' determinant of a matrix X as the Pfaffian of the matrix obtained from X by matrix multiplication and transposition","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"The specific matrix construction via multiplication and transposition produces a skew-symmetric matrix whose Pfaffian exactly matches the alternating sum of maximal minors for arbitrary rectangular X.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"The Cullis' determinant of rectangular matrix X equals the Pfaffian of a matrix obtained from X by multiplication and transposition, enabling an efficient polynomial-time algorithm.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"The Cullis determinant of a rectangular matrix equals the Pfaffian of a skew-symmetric matrix obtained from it by multiplication and transposition.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"bd0009cedd5354e35d3e2cdbda6862291fb7830fc6348b5536b9b7a449588067"},"source":{"id":"2605.14010","kind":"arxiv","version":1},"verdict":{"id":"8768bc25-71be-4f8a-857e-c8b0977d8fed","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-15T02:49:03.614993Z","strongest_claim":"we express the Cullis' determinant of a matrix X as the Pfaffian of the matrix obtained from X by matrix multiplication and transposition","one_line_summary":"The Cullis' determinant of rectangular matrix X equals the Pfaffian of a matrix obtained from X by multiplication and transposition, enabling an efficient polynomial-time algorithm.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"The specific matrix construction via multiplication and transposition produces a skew-symmetric matrix whose Pfaffian exactly matches the alternating sum of maximal minors for arbitrary rectangular X.","pith_extraction_headline":"The Cullis determinant of a rectangular matrix equals the Pfaffian of a skew-symmetric matrix obtained from it by multiplication and transposition."},"references":{"count":20,"sample":[{"doi":"","year":2010,"title":"A. 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