Theorem: Exact sequential agreement

For every compiled tower,

For every $M\in {\mathcal{A}}_{\mathrm{inv}}$,

The assertion holds at every circuit depth.

Proof. By Lemma lem:covariance, each root pinching can be moved past all later gates. Pinchings on different roots commute. Their final composition is pinching on all roots, which equals ${\Delta }_{\mathcal{N}}\rho$ by anchoring.

For invariant $M$, self-adjointness gives