Non-Injectivity as a Structural Necessity of Genuine Emergence: A No-Go Theorem for Faithful Emergent Descriptions

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 $\Pi : \Omega \to \mathcal{O}$ such that $\mathcal{O}$ carries no independent microstate-resolving structure beyond what $\Pi$ provides, and such that the effective level is not structurally isomorphic to the fundamental level, then $\Pi$ cannot be injective and the associated projection entropy $S_\Pi > 0$. The proof uses only surjectivity, informational completeness of $\Pi$ 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 $\Pi$, and $S_\Pi > 0$ 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 $t \lesssim t_\mathrm{Planck}$ — 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 $\Lambda$CDM and the Standard Model, whose fundamental constants are identified as invariants of an unspecified projection $\Pi$ 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.