Virasoro modes give an Aut O action with field covariance

For a vector of weight $d$, every $L_M(n)$ with $n>d-d_0$ kills it by [F2]. Thus only finitely many $n$ and finitely many powers of $\sum c_n{L}_M(n)$ act nontrivially on that vector. On the jet modulo $t^{r+1}$, the coefficient of degree $k+1$ in $\exp (\sum c_n{\mathrm{\ell }}_n)(t)$ is $-c_k$ plus an expression in $c_1,\dots ,c_{k-1}$; recursive solution gives unique $c_1,\dots ,c_r$. The Virasoro central term in [F4] vanishes for positive indices. The assignment ${\mathrm{\ell }}_n\mapsto -L_M(n)$ is anti-Lie for the right action, since $[-L_M(m),-L_M(n)]=(m-n)L_M(m+n)=-(-L_M([{\mathrm{\ell }}_m,{\mathrm{\ell }}_n]))$. The finite Baker–Campbell–Hausdorff formula on each nilpotent jet quotient therefore gives compatible right actions; $[L_M(0),L_M(n)]=-n{L}_M(n)$ supplies the scaling compatibility. Integer weights in [F2] make $a^{L_M(0)}$ single-valued. More explicitly, for $A$ as in the Statement and a unit $b\in A^{\times }$, $b^{L_M(0)}$ acts on $M_d\otimes A$ by $b^d$, which is defined for every integer $d$. For a fixed $m\in M_d$, ${\sum }_{n\ge 1}{c}_n{L}_M(n)m$ is finite because $L_M(n)m=0$ for $n>d-d_0$, and every sufficiently high power of this degree-lowering sum kills $m$; hence its exponential is a finite sum even when $c_n\in A$. The finite-jet factorization recursion and the Baker–Campbell–Hausdorff identities use only ring operations, rational scalars, and inversion of the unit leading coefficient, so they hold over $A$ and give the asserted functorial right action. On $V$, [F4] gives $[L_V(0),L_V(n)]=-n{L}_V(n)$, so $L_V(n)V_d\subseteq V_{d-n}$; the same argument applies using the nonnegative integral grading and finite-dimensional stages in [F3].

Taking the conformal-vector residues in [F5] gives, for every integer $q\ge 0$, $[L_M(q),Y_M(v,z)]={\sum }_{j=0}^{q+1}\left(\genfrac{}{}{0px}{}{q+1}{j}\right)z^{q+1-j}{Y}_M(L(j-1)v,z)\mathrm{.}$ The $j=0$ term is $z^{q+1}{Y}_M(Tv,z)=z^{q+1}{\mathrm{\partial }}_z{Y}_M(v,z)$, by $L(-1)=T$ in [F3] and the translation identity obtained from the module Jacobi identity. Hence $[-L_M(q),Y_M(v,z)]=-z^{q+1}{\mathrm{\partial }}_z{Y}_M(v,z)-{\sum }_{j=1}^{q+1}\left(\genfrac{}{}{0px}{}{q+1}{j}\right)z^{q+1-j}{Y}_M(L(j-1)v,z)\mathrm{.}$ This is the infinitesimal covariance formula for the coordinate vector field ${\mathrm{\ell }}_q=-z^{q+1}{\mathrm{\partial }}_z$: its base-point velocity is $-z^{q+1}$ and its relative-coordinate velocity is $-(z+u{)}^{q+1}+z^{q+1}=-{\sum }_{j=1}^{q+1}\left(\genfrac{}{}{0px}{}{q+1}{j}\right)z^{q+1-j}{u}^j$. Under $R_V$, the latter acts on $v$ by the state term in the displayed commutator.

On every finite jet, the covariance identity derived in step 1.2 and composed below uses only polynomial operations and inversion of the leading coefficient. The same finite-jet computation therefore holds over any commutative $\mathbb{C}$-algebra $A$. For ${\rho }_z$, take $A=\mathbb{C}\mathrm{⟦}z\mathrm{⟧}$; its leading coefficient ${\rho }^{\mathrm{\prime }}(z)$ is a unit, so step 1.1 applies. For a homogeneous $v\in V_d$, positive modes lower weight and the diagonal factor preserves weight; consequently $R_{V,A}({\rho }_z)v$ has coefficients in $V_{\le d}$. Since $V_{\le d}$ is finite dimensional, lower truncation of $Y_M(e,w)m$ is uniform over a basis $e$ of this stage for each fixed $m\in M$. Writing $\rho (z)=zq(z)$ with $q(0)=a_1\mathrm{\ne }0$, each integer power $\rho (z{)}^n$ has order $n$; the uniform lower bound on the field powers and the nonnegative $z$-orders of the state coefficients therefore make every coefficient of the substituted field a finite sum. Thus the covariance identity is well-defined coefficientwise. For the formal flow of each ${\mathrm{\ell }}_q$, differentiate both sides of the claimed covariance identity with respect to the flow parameter. The derivative of the conjugated left side is the commutator in step 1.2; the chain rule on the right gives the base-point and relative-coordinate velocities computed there. At parameter zero the two sides coincide, so their coefficientwise formal differential equation has equal initial values and equal recursively determined coefficients. This proves covariance for each elementary flow. For $\rho (t)=at$, the $q=0$ commutator in step 1.2 integrates to $a^{L_M(0)}{Y}_M(v,z)a^{-L_M(0)}=Y_M(a^{L_V(0)}v,az)$; integer weights remove any logarithm choice. Every finite jet of a general $\rho$ is a product of such a scaling and the elementary positive-mode flows by the recursion in step 1.1. Composing their covariance identities using the right-action law in step 1.1 proves the formula on every finite jet and therefore coefficientwise on the formal series.