Exact counterexamples to R-superlinear convergence of cyclic steepest descent

Cyclic steepest descent (CSD) recomputes the exact steepest-descent stepsize once per cycle and reuses it for m consecutive updates. An ICM 2022 survey describes CSD as likely to converge R-superlinearly on n-dimensional convex quadratics when m is at least the ceiling of (n+1)/2. This record gives two closed-form counterexamples at that threshold. First, for n=m=2, A=diag(1,3), b=0, and x0=(1,1/3)^T, every recomputed stepsize equals 1/2 and the complete nonterminating orbit is xk=2^(-k)(1,(-1)^k/3)^T, so successive error norms have ratio 1/2. Second, for n=3, m=2, and A=diag(1,2,3), an explicit full-support, nonresonant projective period-two orbit satisfies g_(k+4)=g_k/49 and e_(k+4)=e_k/49. Thus the universal R-superlinear convergence assertion fails through both a balanced two-dimensional zig-zag mechanism and a genuinely three-dimensional non-zig-zag mechanism. These are exact exceptional witnesses; no claim of generic or positive-measure failure is made.