Proposition: A non-Gaussian occupation-constrained alternative

If only $⟨N_h+N_g⟩\le E_{\mathrm{pl}}$ is known, then for every integer $D\ge 0$ the same mode comparison obeys

Proof. Project onto total occupation at most $D$. Its discarded probability is at most $E_{\mathrm{pl}}/(D+1)$. The distance to the normalized projected state is at most the square root of that probability, as follows by purification. On this invariant sector, $\Vert J\Vert \le D$, so the unitary comparison is at most $D|\theta |$. Two projection comparisons and contractivity give the result.