Prove necessity of exactness for out-of-distribution generalization

Prove that exactness of the representation consulted at inference is necessary for out-of-distribution generalization, rather than merely establishing that exactness predicts observed successes and failures better than architecture, capacity, data, or scale.

Background

The paper proposes a representational criterion according to which a system generalizes out of distribution only when the representation used at inference is structurally equivalent to the underlying data-generating mechanism. The authors apply this criterion to piecewise-affine extrapolation, relational binding, hybrid architectures, neuro-symbolic systems, and epistemic uncertainty.

However, the authors explicitly characterize the criterion as an argument rather than a theorem. Establishing necessity would require a formal result ruling out out-of-distribution generalization by a fitted, structurally nonequivalent approximation under the relevant assumptions, rather than relying on the empirical and conceptual evidence presented in the paper.