Remark: A surviving quantum memory changes the hypothesis

The theorem does not apply to a decoder that returns an unmeasured $C_{\mathrm{out}}$ while destructively reading only a syndrome. In the exact normal form of Theorem thm:exact, the syndrome state is independent of the code. Reading and forgetting that syndrome can leave the returned code untouched.

Likewise, a coherent register retained alongside a classical record invalidates the record-only premise. Unrestricted classical storage does not by itself invalidate it.