We prove that any genuinely emergent physical description is necessarily non-injective and intrinsically compressive. The argument is purely structural: given a surjective projection: O whose observables are entirely defined by, if the effective level is not structurally isomorphic to the fundamental level, then cannot be injective, and the associated projection entropy S_ > 0. The proof uses only surjectivity, the definition of observables by projection, and the non-isomorphism condition; it is independent of any specific physical framework. As a corollary, descriptive redundancy, non-injectivity of, and S_ > 0 are co-extensive in any projective emergent framework, and any non-trivial gauge structure requires these properties. The result reframes non-injectivity from a modelling choice to a structural necessity: a fully injective emergent description would reduce to a mere reparametrisation of the fundamental level. We discuss how this universal constraint is explicitly realised in the Cosmochrony relational framework, where non-injectivity of the projection underlies the emergence of gauge structure, spacetime geometry, and quantum correlations.
Jérôme Beau (Thu,) studied this question.