Remark: Reading of the defect

For a gauge parameter in the edge zone the first-order observable changes, under $(h,{\varphi }_0)\mapsto (h+K\xi ,{\varphi }_0+\kappa X_{\xi }{\varphi }_0)$, by $\kappa \int f{{\Delta }^{(1)}}_{\epsilon ,\xi }$ at order $\kappa$ (Section sec:expansion). By the order argument of VI and VII-A, no coefficient beginning at ${\kappa }^2$ can cancel it. The regulator preserves exactly the gauge directions supported where it equals one; it does not preserve directions supported where it ramps. The frozen algebra (eq:ga) was chosen accordingly before any calculation.