  We provide a closed-form expression for the motivic Kontsevich-Soibelman
invariant for M-theory in the background of the toric Calabi-Yau threefold
$K_{\mathbb{F}_0}$. This encodes the refined BPS spectrum of $SU(2)$ 5d ${\cal
N}=1$ Yang-Mills theory on $S^1\times \mathbb{R}^4$, corresponding to rank-zero
Donaldson-Thomas invariants for $K_{\mathbb{F}_0}$, anywhere on the Coulomb
branch.
