Area Scaling of Dynamical Degrees of Freedom in Regularised Scalar Field Theory

How many canonical degrees of freedom does a quantum field theory actually use during its Hamiltonian evolution? For a UV/IR-regularised classical scalar field, we address this question directly at the level of phase-space dynamics by identifying the minimal symplectic dimension required to reproduce a single trajectory by an autonomous Hamiltonian system. Using symplectic model order reduction as a structure-preserving diagnostic, we show that for the free scalar field this minimal dimension is controlled not by the volume-extensive number of discretised field variables, but by the much smaller number of distinct normal-mode frequencies below the ultraviolet cutoff. In flat space, this leads to an area-type scaling with the size of the region, up to slowly varying corrections. On geodesic balls in maximally symmetric curved spaces, positive curvature induces mild super-area growth, while negative curvature suppresses the scaling, with the flat result recovered smoothly in the small-curvature limit. Numerical experiments further indicate that this behaviour persists in weakly interacting $\lambda {\phi }^4$ theory over quasi-integrable time scales. Beyond counting, the reduced dynamics exhibits a distinctive internal structure: it decomposes into independent oscillator blocks, while linear combinations of these blocks generate a larger family of apparent field modes whose Poisson brackets are governed by a projector rather than the identity. This reveals a purely classical and dynamical mechanism by which overlapping degrees of freedom arise, without modifying canonical structures by hand. Our results provide a controlled field-theoretic setting in which area-type scaling and overlap phenomena can be studied prior to quantisation, helping to identify which aspects of such structures–often discussed in holographic contexts–can already arise from classical Hamiltonian dynamics.