Dimension-independent stability of the real Böttcher–Wenzel inequality

The Böttcher–Wenzel inequality bounds the squared Frobenius norm of the commutator of two unit matrices by two. We prove that every almost maximizing pair of real matrices lies within \(C√δ\) of an exact maximizing pair, where \(δ\) is the deficit and \(C\) is independent of the matrix size. The distance exponent \(1/2\) is optimal. The proof combines the known equality classification and Audenaert’s singular-value refinement with two further steps. A calculation of the normal Hessian gives quadratic growth of the deficit away from the smooth family of equality pairs. Simultaneous compression to a space of dimension at most eight preserves the order of the deficit, so that fixed-dimensional quadratic growth yields a uniform estimate. The constant is existential, and neither matrix is required to be symmetric or normal.