RelationFirst: What Composition Requires from Learned Representations

Accurate local predictions do not establish whether a learned representation will remain reliable within a larger computation. The open question is what information the consuming computation needs and how errors in that information affect its result. RelationFirst makes both requirements precise. Composition sufficiency identifies the distinctions a component must preserve across contexts, while propagation geometry determines how local approximation errors affect the assembled result. Stationary elliptic PDEs make these requirements testable. Classical elimination exposes an exact interface for the discrete problem, and the assembled equilibrium reveals how response errors propagate. We train models on isolated components, freeze them, and reuse them in held-out assemblies. With the output decoder and composer fixed, changing only the supplied interaction shifts the global equilibrium. A single propagation law predicts these effects across distinct nonlinear systems, including cases where a learned response helps and where it harms. Perhaps counter-intuitively, in a controlled intervention, even a local error with only half the magnitude yields a larger global error across all 72 tested assemblies, driven purely by the shift in error direction. These results show why reusable representations must be evaluated through the behavior they preserve under composition and the consequences of their errors in use. These criteria can guide the design and evaluation of learned components in numerical solvers and coupled simulators.