Switch, regulator, plateau, gauge group

For $\kappa$-dependent objects $\mathsf{T}={\sum }_j{\kappa }^j{\mathsf{T}}_j$ the formal pullback is applied coefficientwise and truncated; this is the Taylor expansion of the genuine pullback of the genuine $\kappa$-dependent field by $F_{\kappa }$. Explicitly,

all modulo ${\kappa }^3$. Equation (eq:metricpull) is the complete second-order metric law. Setting $\zeta =0$ and reading the $\lambda$-linear part of the one-parameter family $\exp (\lambda \kappa {\mathcal{L}}_{\xi })$ gives the infinitesimal law ${\delta }_{\xi }h=K\xi$, ${\delta }_{\xi }k={\mathcal{L}}_{\xi }h$, equivalently, in the notation $q=\eta +\kappa h$ with $\kappa$-dependent $h$, ${\delta }_{\xi }h=K\xi +\kappa {\mathcal{L}}_{\xi }h$.