  We introduce notions of a constraint metric approximation and of a constraint
stability of a metric approximation. This is done in the language of group
equations with coefficients. We give an example of a group which is not
constraintly sofic. In building it, we find a sofic representation of free
group with trivial commutant among extreme points of the convex structure on
the space of sofic representations.
  We consider the centralizer equation in permutations as an instance of this
new general setting. We characterize permutations $p\in S_k$ whose centralizer
is stable in permutations with respect to the normalized Hamming distance, that
is, a permutation which almost centralizes $p$ is near a centralizing
permutation. This answers a question of Gorenstein-Sandler-Mills (1962).
