Information Limits for Stable Reduction of Coupled Diffusion Systems

Eliminating an unresolved diffusion field from a coupled parabolic system produces an operator-valued memory term and a contribution from the hidden initial state. We study whether a reduced prediction of the retained field meets a prescribed tolerance. For known hidden initial data, the discarded response, damped by diffusion and transmitted through the coupling, gives an upper error bound; an auxiliary diffusion problem supplies a lower bound. Rigorous discretisation bounds then yield tests that accept, reject or remain undecided. If the discarded initial state is known only through a norm bound, the minimum worst-case error equals that bound multiplied by the time-uniform gain from the missing state to the retained trajectory. Our main result encloses this infinite-dimensional gain using a finite spectral Galerkin evolution, its residual in the full equations and an explicit diffusion tail. The enclosure converges without compactness or smallness of the coupling. Strict comparisons of the optimal radius with a tolerance have finite certificates if all endpoint quantities admit arbitrarily accurate rigorous enclosures. Multiplication-coupling examples with infinitely many dynamically coupled modes illustrate both settings, including explicit finite-element error budgets.