Constraint metric approximations and equations in groups

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).