Theorem: Exact coarse-record instrument gap

The two coarse instruments satisfy

The input ${\rho }_{+}$ already attains this distance without a reference.

Proof. Write $Z=P_0-P_1=-Q$ and ${\kappa }_0=1/\sqrt{2}$. The actual middle map is

This follows by adding the two phase conjugations in (eq:path-table); off-diagonal entries acquire $\cos (\pi /4)={\kappa }_0$. Only the middle recorded block differs. The diamond norm of $\mathrm{id}-{\mathrm{Ad}}_Z$ is two: the channel-difference upper bound is two, and ${\rho }_{+}$ gives orthogonal outputs. Multiplying by $(8/25)(1-{\kappa }_0)/2$ and using half-diamond normalization proves (eq:exact-gap).