  Identifying flux vacua in string theory with stabilized complex structure
moduli presents a significant challenge, necessitating the minimization of a
scalar potential complicated by infinitely many exponential corrections. In
order to obtain exact results we connect three central topics:
transcendentality or algebraicity of coupling functions, emergent symmetries,
and the distribution of vacua. Beginning with explicit examples, we determine
the first exact landscape of flux vacua with a vanishing superpotential within
F-theory compactifications on a genuine Calabi-Yau fourfold. We find that along
certain symmetry loci in moduli space the generically transcendental vacuum
conditions become algebraic and can be described using the periods of a K3
surface. On such loci the vacua become dense when we do not bound the flux
tadpole, while imposing the tadpole bound yields a small finite landscape of
distinct vacua. Away from these symmetry loci, the transcendentality of the
fourfold periods ensures that there are only a finite number of vacua with a
vanishing superpotential, even when the tadpole constraint is removed. These
observations exemplify the general patterns emerging in the bulk of moduli
space that we expose in this work. They are deeply tied to the arithmetic
structure underlying flux vacua and generalize the finiteness claims about
rational CFTs and rank-two attractors. From a mathematical perspective, our
study is linked with the recent landmark results by Baldi, Klingler, and Ullmo
about the Hodge locus that arose from connecting tame geometry and Hodge
theory.
