This paper concentrates on the fragility margins of discrete-time Piecewise Affine (PWA) closed-loop dynamics. Starting from the case where the nominal trajectories are controlled by a PWA controller using a positioning mechanism within a binary search tree (BST), we are interested in preserving the properties of the nominal dynamics, particularly the positive invariance in the presence of perturbations in the control law representation. As the main result, we define (and effectively construct) the fragility margin for a hyperplane defining the partition of the nominal PWA controller. This hyperplane will be identified through a node in a BST, and the proposed margin characterizes the degrees of freedom in the perturbation of its coefficients as it might result from a quantization operation. From the mathematical standpoint, the fragility margin is a set in the coefficients' space, and the properties of this set are formally described.