Additive Order and Composition in Classical Dynamics

We provide a physically grounded instantiation of a structural result: any empirically co-herent description of processes admitting composition, refinement stability, and context-stablecomparability must support an additive ordering functional. While such a functional is oftenidentified with time in physical theories, its necessity is rarely made explicit. We demonstratethis necessity in a fully classical setting by analyzing heat diffusion under conserved bound-ary conditions. The result shows that additive temporal structure is not a modeling choicebut a prerequisite for the mathematical and empirical coherence of classical partial differentialequations.