A counterexample to R-superlinear convergence of cyclic steepest descent

Cyclic steepest descent recomputes an exact steepest-descent stepsize once per cycle and reuses it for m updates. Dai's ICM 2022 survey states that the method is likely to converge R-superlinearly on n-dimensional convex quadratics when m is at least (n+1)/2. This preprint disproves the corresponding universal assertion by an exact threshold counterexample. For n = m = 2, A = diag(1,3), b = 0, and x0 = (1,1/3)^T, the complete nonterminating orbit is x_k = 2^{-k}(1,(-1)^k/3)^T, and its Euclidean error norm has ratio exactly 1/2 at every iteration. Hence the convergence is exactly R-linear and not R-superlinear. The same calculation identifies the two-eigendirection invariant mechanism explaining why cyclic reuse preserves this linear trajectory.