Corollary: Residual-rank dimension bound

Exact recovery implies

If ${\sigma }_B$ is pure, $\mathcal{N}$ is an isometric encoding. Equation (eq:dimension-bound) is not an area law.

Proof. The $dr$ vectors $r_{i\alpha }$ constructed in the proof of Theorem thm:exact are orthonormal in $R$. If $r=1$, (eq:normal-form) reduces to conjugation by an isometry $C\to R$. No geometric area or gravitational dynamics enters this count.