A Unified Reduction Framework for Laplace-type Integrals

We introduce a unified reduction framework for Laplace-type integrals
\[
\int_\Gamma f(x)\,e^{-\phi(x)}\,dx,
\]
centered on a canonical reduction operator acting on amplitudes. This operator
provides an intrinsic algebraic formulation of the exactness underlying
integration by parts for exponential integrals, identifying precisely when two
amplitudes yield the same integral up to boundary contributions. 
The resulting reduction quotient furnishes a universal space encoding all reduction-generated identities for Laplace-type integrals with phase 
$\phi$ and admits a canonical realization as twisted de Rham cohomology.

Within this structure we establish canonical reduction normal forms,
characterize finite-dimensional reduction modules, and construct natural
duality pairings between reduced amplitudes and admissible contours. These
results reveal a unified mechanism underlying reduction, cohomological
structure, and differential closure: in particular, finite reduction is shown to
be equivalent to holonomic behavior of the associated Laplace-type integrals.
The framework provides a conceptual foundation for a broad class of phenomena
in exponential integral theory, including recurrence structures, Pfaffian and
higher-order differential systems, and finite-dimensional invariance. Further
developments concerning parameter dependence, creative telescoping, and Stokes
structure are outlined for subsequent work.
\end{abstract}