  In this article we derive an (almost) optimal scaling law for a singular
perturbation problem associated with the Tartar square. As in \cite{W97, C99},
our upper bound quantifies the well-known construction which is used in the
literature to prove flexibility of the Tartar square in the sense of
flexibility of approximate solutions to the differential inclusion. The main
novelty of our article is the derivation of an (up to logarithmic powers
matching) ansatz free lower bound which relies on a bootstrap argument in
Fourier space and is related to a quantification of the interaction of a
nonlinearity and a negative Sobolev space in the form of "a chain rule in a
negative Sobolev space". Both the lower and the upper bound arguments give
evidence of the involved "infinite order of lamination".
