A common plateau, and a controlled defect outside it

Exact gauge equality (eq:v-gauge) holds for every allowed switch because all share the same plateau $U$.

For a real smooth compact $\xi$ supported to the future of ${\Sigma }_0$, but not necessarily in that plateau, the exact identity is instead

Thus the represented defect direction is

If the supports of such parameters lie in a common compact source region, the same scalar-response estimate gives

where the last seminorm includes the tensor components.

In the displaced state, the difference of the represented observables has the state-relative root-mean-square bound

Indeed, displacement transforms this difference to $2\mathrm{Re}⟨d_{\xi },z⟩+\sqrt{ϵ}\Phi (d_{\xi })$, and the triangle inequality in the state's quadratic norm proves the estimate.

A small right-hand side is an error allowance, not exact invariance. There is no operator-norm bound for the unbounded field difference uniform over all states. Changes of initial surface or coordinate Cauchy data are excluded from the present stability theorem.