A Convergent Kinetic-Term Perturbation Expansion for $\phi^4$ Theory

We revisit scalar $\phi^4$ theory and construct a reorganized perturbative expansion in which the kinetic operator, rather than the quartic interaction, is treated as the perturbation. Starting from the exactly solvable $0$-dimensional model, we show that the resulting series is convergent for positive coupling and can be written as an expansion in negative powers of the quartic coupling $\lambda$. We extend the construction to higher-dimensional field theory using an auxiliary field, and we formulate a discrete lattice version in which multi-site contributions are systematically organized. We explicitly compute the leading terms in the expansion, study their continuum limit, and compare against brute-force numerical evaluations of the partition function. We discuss the relation of this expansion to standard weak-coupling perturbation theory, strong-coupling expansions, and resummation techniques, and we outline possible applications to nonperturbative studies of scalar field theories.