A generic counterexample to the smallest-absolute-multiplier rule

For real data $U$, consider the real stationary pairs $(X,c)$ with $|\det X|=1$ and $X^{T}(U-X)=cI$, and the rule that selects a pair with smallest $|c|$ as the nearest such matrix in Frobenius norm. The rule fails for every diagonal $U=\operatorname{diag}(s_1,s_2,s_3)$ with $7/4<s_1<s_2<s_3<44/25$. Every stationary matrix is then diagonal, no stationary pair has $0\le c\le 13/25$, and a monotonicity comparison of the root patterns shows that the unique multiplier of least absolute value is negative, with exactly one negative entry in its matrix. Replacing that entry by its absolute value preserves $|\det X|=1$ and strictly reduces the distance. Orthogonal invariance extends the failure to every real $3\times3$ matrix with distinct singular values in $(7/4,44/25)$, a nonempty open set that a proper real algebraic exceptional set cannot cover.