Digitwise Structure and Reverse Stability in Magic Squares: Sharp Thresholds and Emirp Constructions

This paper studies reverse stability in magic squares under fixed-width digit reversal in positional bases. The main result is an exact threshold theorem. For a declared line family of length n in base b, if n≤b, then reverse stability is equivalent to digitwise magic. The proof proceeds through a coefficient-bound argument for signature-difference polynomials and yields the exact sharp threshold b2−b+1. A matching all-base construction shows that n=b+1 is the first line length at which unequal digit signatures become arithmetically admissible, establishing sharpness at the line-signature level. The paper also develops consequences for strict-emirp magic squares, including order-3 centre-offset structure, the 6ℤ offset lattice, reversible-complement conditions, and a practical verification framework separating additive structure from primality, uniqueness and reverse-disjointness requirements. A computed audit of supplied strict-emirp archives covering orders 3×3 through 9×9 is reported. The audit is presented as a project computation rather than an independent external replication. Empirical observations, search diagnostics, construction ecology and open problems are explicitly separated from proved results through a status-discipline framework. The work aims to distinguish clearly between theorem, computation, observation and conjecture while providing a foundation for future research into reverse-stable emirp-prime magic squares, nested constructions and higher-dimensional extensions.