  In this paper we apply the idea of Higgs branch localization to 5d
supersymmetric theories of vector multiplet and hypermultiplets, obtained as
the rigid limit of $\mathcal{N} = 1$ supergravity with all auxiliary fields. On
supersymmetric K-contact/Sasakian background, the Higgs branch BPS equations
can be interpreted as 5d generalizations of the Seiberg-Witten equations. We
discuss the properties and local behavior of the solutions near closed Reeb
orbits. For $U(1)$ gauge theories, we show the suppression of the deformed
Coulomb branch, and the partition function is dominated by 5d Seiberg-Witten
solutions at large $\zeta$-limit. For squashed $S^5$ and $Y^{pq}$ manifolds, we
show the matching between poles in the perturbative Coulomb branch matrix
model, and the bound on local winding numbers of the BPS solutions.
