Null-controllable data for cascade heat equations: almost sharp spectral spaces and modal critical times

We study one-dimensional cascades of heat equations controlled through their last component. For a fixed time $T>0$, we consider the Hilbert space $\mathcal{C}_T$ of all initial data that can be steered exactly to zero at time $T$, endowed with the norm of the minimal $L^2$ control. This space can b