We prove that any genuinely emergent (i. e. \ non-isomorphic effective) physical description is necessarily non-injective and intrinsically compressive. The argument is purely structural: given a surjective projection: O such that O carries no independent microstate-resolving structure beyond what provides, and such that 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, informational completeness of for observables, and the non-isomorphism condition; it is independent of any specific physical framework. The result constitutes a no-go theorem for faithful emergence: a surjective projection that is both informationally faithful and genuinely emergent cannot exist. As a corollary, descriptive redundancy, non-injectivity of, and S_ > 0 are co-extensive in any projective emergent framework; any non-trivial gauge structure requires these properties; and any theory admitting a regime where its own description ceases to be valid — such as standard cosmology at t tPlanck — is structurally consistent with the presence of a non-injective projection. We provide a categorical reformulation (genuine emergence corresponds to a non-faithful functor) and an informational one (the Kullback–Leibler divergence of any state distribution with respect to its projection lift is strictly positive). The result is illustrated by a concrete spin model and discussed in relation to the projective incompleteness of CDM and the Standard Model, whose fundamental constants are identified as invariants of an unspecified projection not accessible from the effective level alone. A further corollary establishes that non-injectivity is recursive: any tower of genuinely distinct emergent levels carries one independent non-injective projection per stratum, with entropy accumulating across levels. This recursive structure implies that colour confinement — the non-observability of free quarks — is a structural necessity of any framework in which colour-neutral hadrons arise as genuinely emergent from coloured degrees of freedom.
Jérôme Beau (Sun,) studied this question.