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 operatorprovides an intrinsic algebraic formulation of the exactness underlyingintegration by parts for exponential integrals, identifying precisely when twoamplitudes 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 naturalduality pairings between reduced amplitudes and admissible contours. Theseresults reveal a unified mechanism underlying reduction, cohomologicalstructure, and differential closure: in particular, finite reduction is shown tobe equivalent to holonomic behavior of the associated Laplace-type integrals.The framework provides a conceptual foundation for a broad class of phenomenain exponential integral theory, including recurrence structures, Pfaffian andhigher-order differential systems, and finite-dimensional invariance. Furtherdevelopments concerning parameter dependence, creative telescoping, and Stokesstructure are outlined for subsequent work.\end{abstract}