Proper Transposed Sesqui Arrays at All Sylvester--Hadamard Powers

Abstract

For every power of two t = 2k, k ≥ 2, we construct a proper transposed sesqui array SAT(4t-2,t,-,t-1,t:(2t-1)×2t). The columns form the classical Sylvester--Hadamard trace design. We reduce the cell-ordering problem to a two-to-one finite-field map. Odd field dimensions are handled by an explicit inverse-pair map. In even dimension a Subiaco map and its zero fibre reduce the problem to two fixed algebraic curves. Geometric irreducibility, Artin--Schreier trace covers, and a Hasse--Weil estimate cover all even dimensions at least 34; fifteen exact polynomial-gcd certificates close the remaining even dimensions. A fixed resultant certificate proves properness, and an explicit array closes t = 4. Status: Public Beta v0.3; internally verified candidate proof; external mathematical review pending.