  We address a number of outstanding questions associated with the analytic
properties of the universal equation of state of the $\phi^4$ theory, which
describes the critical behavior of the Ising model and ubiquitous critical
points of the liquid-gas type. We focus on the relation between spinodal points
that limit the domain of metastability for temperatures below the critical
temperature, i.e., $T < T_{\rm c}$, and Lee-Yang edge singularities that
restrict the domain of analyticity around the point of zero magnetic field $H$
for $T > T_{\rm c}$. The extended analyticity conjecture (due to Fonseca and
Zamolodchikov) posits that, for $T < T_{\rm c}$, the Lee-Yang edge
singularities are the closest singularities to the real $H$ axis. This has
interesting implications, in particular, that the spinodal singularities must
lie off the real $H$ axis for $d < 4$, in contrast to the commonly known result
of the mean-field approximation. We find that the parametric representation of
the Ising equation of state obtained in the $\varepsilon = 4-d$ expansion, as
well as the equation of state of the ${\rm O}(N)$-symmetric $\phi^4$ theory at
large $N$, are both nontrivially consistent with the conjecture. We analyze the
reason for the difficulty of addressing this issue using the $\varepsilon$
expansion. It is related to the long-standing paradox associated with the fact
that the vicinity of the Lee-Yang edge singularity is described by Fisher's
$\phi^3$ theory, which remains nonperturbative even for $d\to 4$, where the
equation of state of the $\phi^4$ theory is expected to approach the mean-field
result. We resolve this paradox by deriving the Ginzburg criterion that
determines the size of the region around the Lee-Yang edge singularity where
mean-field theory no longer applies.
