  We study the infrared physics of 5d $\cal N=1$ Yang-Mills theories
compactified on $\mathbb{S}^1$, with a view toward 4d and 5d limits. Global
structures of the simplest Coulombic moduli spaces are outlined, with an
emphasis on how multiple planar 4d Seiberg-Witten geometries are embedded in
the cigar geometry of a single 5d theory on $\mathbb{S}^1$. The Coulomb phase
boundaries in the decompactification limit are given particular attention and
related to how the wall-crossings by 5d BPS particles turn off. On the other
hand, the elliptic genera of magnetic BPS strings do wall-cross and retain the
memory of 4d wall-crossings, which we review with the example of dP$_2$ theory.
Along the way, we also offer a general field theory proof of the odd shift of
electric charge on Sp$(k)_\pi$ instanton solitons, previously observed via
geometric engineering for low-rank supersymmetric theories.
