Capacity and target compatibility

Let ${\mathfrak{I}}_K$ be the permitted implementations at capacity $K$, with ${\mathfrak{I}}_K\subseteq {\mathfrak{I}}_{K^{\prime }}$ for $K\le K^{\prime }$. For a fixed task error, put

Then ${\epsilon }_{K^{\prime }}\le {\epsilon }_K$, simply because the infimum is taken over a larger set. There is no implication that ${\epsilon }_K\to 0$.

Feasibility, approximation of a specified target, and compatibility of two ideal targets are separate questions. The last depends on the named observation algebra. The residual obstruction in thm:residual-obstruction cannot be cured merely by allocating more work bits.