Order of dynamical and control systems on maximal compact subgroups

The paper rigorously proves an extended Bruhat order on the Weyl–Tits group U (Theorem 1.1), its coset version for a hyperbolic H (Theorem 3.12), and a partial algebraic characterization of the order of control sets (Theorem 4.8). The model’s solution reaches the same statements but contains a key mistake in part (a): it asserts S(u) = π^{-1}(cl(B(π(u)))), which is false because π^{-1}(cl(B(w))) is a union of Schubert closures S(v) over all fiber choices, not a single S(u). The model also omits necessary structural assumptions (e.g., finite center, existence of a complexification to ensure C is abelian) used in the paper. Hence, while the conclusions match, the model’s proof is not correct.