Non-Injective Observable Maps Do Not Explain Bell Violations: A Redundant-Extension No-Go Theorem

Non-injective maps occur in coarse-grained descriptions, but their relation to Bell non-factorisability requires a separate compatibility theorem. We prove that non-injectivity alone has no Bell-discriminating power. Any Bell-local model can be made arbitrarily non-injective by adjoining erased independent labels, without changing any observable probability or Bell functional. More strongly, every finite empirical model, Bell-local or not, admits a setting-independent non-injective pushforward realisation. Fibre multiplicity and information discarded inside fibres therefore cannot by themselves bound the CHSH value. We give an explicit Bell-local model whose four observable maps have fibres of size eight, and show that output Shannon entropy does not measure non-injectivity. We also isolate the correct obstruction in the binary CHSH scenario: the PR parity family has compatible one-site marginals but no setting-independent global coupling over \((A_0,A_1,B_0,B_1)\). The obstruction is cross-context incompatibility, not many-to-one behaviour within any individual context. Interpretive status: non-injective projection may remain part of an effective architecture, but it cannot explain Bell violation unless an additional global-coupling obstruction is supplied or derived.