When Independent Gaussian Models Break Down: Characterizing Regime-Dependent Modeling Failures in $\phi^4$ Theory

In practical physical systems, modeling assumptions of Gaussianity and basis independence break down due to self-interactions. We study a specific instance of one-dimensional $\phi^4$ theory on a lattice, analyzing how the interaction strength and system size jointly affect the marginal and joint distributions of frequency-based representation of the field (i.e., Fourier modes). We find that models relying on Gaussian and independent Fourier modes fail primarily from structured dependencies rather than marginal non-Gaussianity, since individual modes become approximately Gaussian despite mode coupling growing with size. Based on this, we identify three distinct regimes that delineate where traditional methods remain effective and where more expressive models are needed. Our results provide a computationally simple diagnostic to establish when Gaussian models are insufficient, and establish a concrete design criterion that future nonlinear models must satisfy.