Theorem: Exact batch-encoding finite-capacity formulation

For arbitrary batch encoding with the complete $K$-state boundary, the classical optimum is

If $K\ge 2^n$, the optimum is one.

For a more restricted streaming architecture, the right-hand side remains an upper bound, but attainment is not automatic.

Proof. The deterministic-table reduction applies to the finite $n$-bit task. Every decoder assigns an $n$-bit answer word to each memory value. Its score on input $x$ is

For fixed words, nearest-word encoding attains the best batch score. Optimize the word set and average over uniform inputs.

A restricted streaming encoder is a subset of the batch encoding class and therefore cannot exceed its optimum.