Paley Two-Edge Switching for Proper Transposed Sesqui Arrays

Abstract

For every odd prime power q congruent to 3 modulo 4 with q ≥ 11, we construct an exact-parameter proper transposed sesqui array. The construction crosses two edges of a classical Paley matching. Five quadratic-character conditions preserve the required concurrences, while a nonzero second-moment defect forces properness. Existence follows from exact character counts and the Hasse bound for a family of elliptic curves, with an explicit witness for q = 11. Status: Public Beta v0.3; internally verified candidate proof; external mathematical review pending.