Dimension-independent stability of the Böttcher–Wenzel inequality

For complex matrices A and B of unit Frobenius norm in every dimension n >= 2, define the Böttcher-Wenzel deficit by D(A,B) = 2 - ||AB-BA||_F^2. We prove that the squared distance to the full set of normalized equality pairs is at most C D(A,B), where C is finite and independent of the matrix size. Both matrices therefore concentrate, with total squared error O(D), on one two-dimensional subspace supporting a trace-free equality pair. The proof compresses both matrices to a common space of dimension at most eight while preserving the order of the deficit and gives a uniform lower bound for the normal quadratic form at every equality pair. The optimal near-equality constant for the squared distance is 1 uniformly over all dimensions; it is also 1 in each fixed dimension at least three and is 1/2 in dimension two. Explicit families attain these limits and prove sharpness of the square-root distance exponent. The exact optimal global constant remains open.