Proposition: Presentation descent and residual coverage

The response descends to reconstructed counterterms iff

A zero tested coefficient residual implies the asserted functional identity when the coefficient/test map is injective on a known subspace containing all possible tested residuals. Without separation, this implication can fail.

Proof. Equal reconstructions differ by a reconstruction-kernel vector; their responses agree exactly when $M$ annihilates it. The coverage assertion is injectivity. A nonzero element in the test map's kernel supplies the counterexample without it.