Theorem: Variation of the cubic-completed coefficient

For every smooth $\gamma$, every free ${\varphi }_0$, and every $\xi \in {\mathfrak{g}}_U$,

Proof. (eq:F3first) is (eq:XAv) at first order. For (eq:F3), $\mathcal{T}$ cancels, and we split the difference in two steps. First, with $\gamma +K\xi$ fixed in the first slot, (eq:affine) and Theorem thm:u3law give

Second, apply (eq:XAv) with $(h,k)=(\gamma ,u_3[\gamma ])$, $F$ the flow of $\kappa \xi$ ($\zeta =0$), for which $k^{\prime }=u_3[\gamma ]+s_{\xi }$: ${{\mathcal{A}}_2}^{\mathrm{sc}}[\gamma +K\xi ,u_3[\gamma ]+s_{\xi }]={{\mathcal{A}}_2}^{\mathrm{sc}}[\gamma ,u_3[\gamma ]]$. By (eq:affine),

Adding the two steps gives (eq:F3). ($u_3[\gamma ]$ is smooth on the closed slab and extends smoothly across its boundary; X-A's identities use only values on $M_T$.)