Exact Nonperturbative Vacuum Structure of 0+1‑Dimensional ϕ⁴ Theory and a Spectral‑Lifting Framework

Perturbative expansions in interacting quantum field theories are generically asymptotic and divergent, even in models that are mathematically stable. A controlled understanding of the exact analytic structure underlying such divergences is therefore essential. In this work, the interacting vacuum of scalar ϕ⁴ theory in 0+1 Euclidean dimensions—equivalently, the quantum anharmonic oscillator—is analyzed exactly in closed form. The ultra‑local vacuum functional is derived explicitly and shown to be governed by a modified Bessel function kernel rather than by polynomial dependence on the coupling. The weak‑coupling expansion is demonstrated to be Gevrey‑1 with zero radius of convergence, while the exact expression is proven to be Borel summable and uniquely fixed by the Nevanlinna–Sokal theorem for positive coupling. Strong‑coupling asymptotics are obtained directly from the exact kernel, establishing finiteness and analytic control across the full coupling range. The structural lessons of the exactly solvable model are then reformulated in a spectral language suitable for higher‑dimensional field theory. A mode‑resolved kernel motivated by the exact solution is introduced, and its ultraviolet behavior is analyzed rigorously. It is shown that a naive replacement of the mass scale by a momentum variable does not by itself lead to ultraviolet suppression. This observation motivates the introduction of a spectral‑lifting framework: if the interacting vacuum in higher dimensions is governed by a lifted kernel with sufficiently decaying high‑energy tail, then polynomial ultraviolet divergences in vacuum energy integrals are replaced by spectral saturation. Within this framework, a conditional ultraviolet finiteness theorem is proven under explicit assumptions on the lifted kernel. An effective complementary‑error‑function realization is presented as a continuum representative of the required nonpolynomial decay. Exact results are confined to 0+1 dimensions, while higher‑dimensional conclusions are rigorous given the stated lifting assumptions. The remaining open problem is the first‑principles derivation of the lifted kernel from fully interacting quantum field theory. This preprint provides an exact nonperturbative reference point for understanding the divergence of perturbation theory, the role of Borel summation, and the spectral organization of interacting vacuum fluctuations.