Corollary: A dimension constraint, not an area law

If the outputs are finite dimensional and exact recovery holds with $\mathrm{rank}\sigma =r$, then

This is a Hilbert-space dimension bound, not a statement about geometric area.

Proof. The proof of Theorem thm:exact constructs the $dr$ orthonormal vectors $|r_{i\alpha }⟩$ in $R$. No geometric variable enters their counting.