  We prove a scale-free, quantitative unique continuation principle for
functions in the range of the spectral projector $\chi_{(-\infty,E]}(H_L)$ of a
Schr\"odinger operator $H_L$ on a cube of side $L\in \mathbb{N}$, with bounded
potential. Such estimates are also called, depending on the context,
uncertainty principles, observability estimates, or spectral inequalities. We
apply it to (i) prove a Wegner estimate for random Schr\"odinger operators with
non-linear parameter-dependence and to (ii) exhibit the dependence of the
control cost on geometric model parameters for the heat equation in a
multi-scale domain.
