  We quantify the rigidity of branching microstructures in shape memory alloys
undergoing cubic-to-tetragonal transformations in the geometrically linearized
theory by making use of Tartar's H-measures. The main result is a
$B^{2/3}_{1,\infty}$-estimate for the characteristic functions of twins, which
heuristically suggests that the larger-scale interfaces can cluster on a set of
Hausdorff-dimension $3-\frac{2}{3}$. We provide evidence indicating that the
dimension is optimal. Furthermore, we get an essentially local lower bound for
the blow-up behavior of the limiting energy density close to a habit plane.
