  We classify thick subcategories in a Paquette-Y\i ld\i r\i m completion
$\overline{\mathcal{C}}$ of a discrete cluster category of Dynkin type
$A_{\infty}$. To do this we introduce the notion of homologically connected
objects, and the hc (=homologically connected) decomposition of an object into
homologically connected objects in a $\mathrm{Hom}$-finite, Krull-Schmidt
triangulated category. We show that any object in a $\overline{\mathcal{C}}$
has a hc decomposition, and that the hc decomposition determines the thick
closure of an object. Moreover, we use this result to classify the classical
generators of $\overline{\mathcal{C}}$ as homologically connected objects
satisfying a maximality condition.
  Every homologically connected object has an invariant, known as the
homological length, and we show that in $\overline{\mathcal{C}}$ this
homological length is an upper bound for the generation time of a classical
generator. This allows us to provide an upper bound for the Orlov spectrum of
$\overline{\mathcal{C}}$, as well as giving the Rouquier dimension.
