That genuinely emergent descriptions are many-to-one — that distinct microconfigurations share one effective state — is a familiar, at times definitional, feature of emergence, not a new observation. We do not restate it; we prove a chain of previously unstated implications that turn it from a stipulation into a structural necessity. Our central result is a no-go theorem: for a surjective projection: O whose observables carry no resolving structure beyond what supplies, faithfulness (injectivity) is equivalent to the absence of emergence (structural isomorphism O) ; hence no genuinely emergent projection can be information-preserving, and its projection entropy is strictly positive, S_ > 0. From this we derive, as further results: (i) descriptive redundancy, non-injectivity, and S_ > 0 are co-extensive, and any non-trivial gauge structure requires them; (ii) the compression is recursive and exactly additive across a tower of emergent levels; (iii) wherever colour-neutral states are genuinely emergent from coloured degrees of freedom, the colour projection is non-injective, making the non-observability of free colour structural rather than dynamical. The theorem admits a categorical form (genuine emergence is the non-faithfulness of a functor) and an informational one (a strict, not merely non-negative, data-processing loss). It clarifies apparent counterexamples: exact holographic dualities, when fully faithful, are isomorphisms rather than genuine emergent projections in this sense, while causal-emergence constructions in which ``macro beats micro'' remain many-to-one and lossy in S_. The argument assumes no metric, Hilbert space, Lagrangian, or dynamics, and applies uniformly to renormalisation-group, coarse-grained, causal-set, spin-foam, and pre-geometric projections.
Jérôme Beau (2026) studied this question.