Corollary: A strict finite-dimensional obstruction

If the complete complement is nonconstant, then ${\epsilon }_{\mathrm{rec}}(\mathcal{N})>0$.

Proof. Decoder Choi matrices are positive, have a fixed partial trace, and have fixed total trace. They form a compact set in finite dimensions. The decoder error is continuous, so its infimum is attained. If that minimum were zero, Theorem thm:exact would force a constant complement.