  We construct new six-functor formalisms capturing cohomological invariants of
varieties with potentials. Starting from any six-functor formalism $C$, encoded
as a coefficient system, we associate a new six-functor formalism
$C_{\text{exp}}$. This requires in particular constructing the convolution
product symmetric monoidal structure at the $\infty$-categorical level. We
study $C_{\text{exp}}$ and how it relates to $C$. We also define motives in
$C_{\text{exp}}$ attached to varieties with potential and study their
properties.
