  We study BPS states of 5d $\mathcal{N}=1$ $SU(2)$ Yang-Mills theory on
$S^1\times \mathbb{R}^4$. Geometric engineering relates these to enumerative
invariants for the local Hirzebruch surface $\mathbb{F}_0$. We illustrate
computations of Vafa-Witten invariants via exponential networks, verifying
fiber-base symmetry of the spectrum at certain points in moduli space, and
matching with mirror descriptions based on quivers and exceptional collections.
Albeit infinite, parts of the spectrum organize in families described by simple
algebraic equations. Varying the radius of the M-theory circle interpolates
smoothly with the spectrum of 4d $\mathcal{N}=2$ Seiberg-Witten theory,
recovering spectral networks in the limit.
