Concave matrix-function error transfer can fail for exact Nyström approximations

We give exact Nyström counterexamples to two proposed concave matrix-function
error-transfer principles. A rational positive-definite $6\times6$ matrix has a
rank-three Nyström approximation that is exactly optimal in spectral norm, yet
applying $f(x)=\min(x,1)$ destroys optimality. A rational lower bound certifies
the strict violation at a fixed nonzero perturbation. Separately, a
growing-dimensional Nyström construction rules out every dimension-independent
nuclear-norm relative-excess factor $1+C\varepsilon$, even when the input
excess is required to be arbitrarily small. Both examples use the same fixed
nonnegative nondecreasing concave function and satisfy positive-semidefinite
order. They do not contradict positive transfer theorems for operator-monotone
functions.