\documentclass[11pt]{article} \usepackage[letterpaper,top=0.89in,bottom=0.91in,left=0.96in,right=0.96in]{geometry} \usepackage[T1]{fontenc} \usepackage{lmodern,microtype,amsmath,amssymb,amsthm,mathtools} \usepackage[hidelinks]{hyperref} \usepackage{bookmark} \allowdisplaybreaks[1] \setlength{\parskip}{1.5pt} \setlength{\parindent}{1.15em} \setlength{\emergencystretch}{1.5em} \newcommand{\C}{\mathbb C} \newcommand{\ket}[1]{\lvert #1\rangle} \newcommand{\Span}{\operatorname{span}} \newcommand{\wt}{\operatorname{wt}} \newcommand{\gr}{\operatorname{gr}} \newcommand{\Sym}{\operatorname{Sym}} \newcommand{\Supp}{\operatorname{Supp}} \newcommand{\Ann}{\operatorname{Ann}} \newcommand{\id}{\operatorname{id}} \newcommand{\m}{\mathfrak m} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{proposition}[theorem]{Proposition} \newtheorem{corollary}[theorem]{Corollary} \newtheorem*{restatedcutoff}{Proposition~\ref{prop:cutoff}} \newtheorem*{restatedmain}{Theorem~\ref{thm:main}} \newtheorem*{restatedsfg}{Theorem~\ref{thm:sfg}} \theoremstyle{remark} \newtheorem{remark}[theorem]{Remark} \hypersetup{pdftitle={C2-Cofiniteness of Finite-Group Orbifolds}, pdfauthor={Andrew Riesen}} \title{$C_2$-Cofiniteness of Finite-Group Orbifolds} \author{Andrew Riesen} \date{29 October 2026} \begin{document} \maketitle \begin{abstract} Let $V$ be a simple vertex algebra with a connected nonnegative integer grading, and let $G$ be a finite group of grading-preserving automorphisms. We prove that $C_2$-cofiniteness of $V$ implies $C_2$-cofiniteness of $V^G$. The proof bounds the length and height of monomials in strong generators representing classes in a relative $C_2$ quotient. The length bound follows from a good filtration and finite-group invariant theory. The height bound follows from differential equations for products of fields, together with identities for invariants and vacuum coefficients. The argument also shows that finite strong generation passes to $V^G$, without assuming $C_2$-cofiniteness. For strongly rational vertex operator algebras, the cofiniteness result and McRae's orbifold theorem imply strong rationality of every finite-group orbifold. \end{abstract} \section{Introduction}\label{sec:intro} Taking the fixed-point subalgebra of a vertex algebra under a finite group of automorphisms, or orbifolding, is a basic operation in vertex operator algebra theory and conformal field theory \cite{DVVV,DLM,FLM,GR}. A longstanding conjecture predicts that finite-group orbifolds of strongly rational vertex operator algebras are again strongly rational. The $C_2$ condition, introduced by Zhu in his work on modular invariance \cite{Zhu}, is central to this conjecture. For a vertex algebra $A$, write \[ C_2(A)=\Span_{\C}\{a_{-2}b:a,b\in A\},\qquad R_A=A/C_2(A). \] The quotient $R_A$ is a commutative algebra with product $[a][b]=[a_{-1}b]$, and $A$ is \emph{$C_2$-cofinite} when $R_A$ is finite dimensional. For vertex operator algebras, this condition implies that Zhu's algebra is finite dimensional \cite[Proposition 3.6]{DLM}, and hence that there are only finitely many irreducible admissible modules up to isomorphism \cite[Theorem 3.4(b)]{DLM}. Miyamoto proved $C_2$-cofiniteness for cyclic orbifolds of simple $C_2$-cofinite vertex operator algebras of CFT type, and hence for finite solvable groups \cite{Miyamoto}. In subsequent work, Carnahan and Miyamoto established regularity of solvable fixed points for simple nonnegatively graded regular vertex operator algebras with a nondegenerate invariant bilinear form \cite{CM}. For arbitrary finite groups, McRae proved that a fixed-point algebra of a strongly rational vertex operator algebra is strongly rational as soon as it is $C_2$-cofinite \cite[Corollary 4.23]{McRae}. This reduces the strong-rationality problem to $C_2$-cofiniteness. We prove that $C_2$-cofiniteness passes to fixed points for every finite group. More precisely, let $V$ be a simple vertex algebra over $\mathbb{C}$ with a \emph{connected nonnegative integer grading}, meaning \[ V=\bigoplus_{n\ge0}V_n,\qquad V_0=\C\ket{0},\qquad a_mV_k\subseteq V_{h+k-m-1}\quad(a\in V_h), \] with $V_n=0$ for $n<0$. Let $G$ be a finite group of grading-preserving automorphisms. \begin{theorem}\label{thm:main} If $V$ is $C_2$-cofinite, then $V^G$ is $C_2$-cofinite. \end{theorem} In particular, the theorem applies to simple $C_2$-cofinite vertex operator algebras of CFT type. McRae's theorem then implies that every finite-group orbifold of a strongly rational vertex operator algebra is strongly rational (Corollary~\ref{cor:rational}). We also prove that finite strong generation passes to fixed points, without assuming $C_2$-cofiniteness. \begin{theorem}\label{thm:sfg} Let $V$ be a simple, strongly finitely generated vertex algebra with a connected nonnegative integer grading. For every finite group $G$ of grading-preserving automorphisms, $V^G$ is strongly finitely generated. \end{theorem} Under the connected-grading assumption above, Theorem~\ref{thm:sfg} resolves the finite-group case of the vertex algebra Hilbert problem formulated in \cite[Conjecture~1.1]{LianLinshaw}. \paragraph{Notation.} We write tuples of formal variables and multi-indices in bold, and their individual components in italics. For $\mathbf{z}=(z_1,\ldots,z_m)$ and $\boldsymbol{\alpha}=(\alpha_1,\ldots,\alpha_m)\in\mathbb Z_{\ge0}^m$, put \[ \mathbf{z}^{\boldsymbol{\alpha}}=\prod_{i=1}^m z_i^{\alpha_i}, \qquad |\boldsymbol{\alpha}|=\sum_{i=1}^m\alpha_i, \qquad \boldsymbol{\alpha}!=\prod_{i=1}^m\alpha_i!, \qquad \partial_{\mathbf{z}}^{\boldsymbol{\alpha}} =\prod_{i=1}^m\partial_{z_i}^{\alpha_i}. \] For a formal series with vector coefficients, $[\mathbf{z}^{\boldsymbol{\alpha}}]$ denotes coefficient extraction, leaving unextracted variables unchanged. Inequalities between multi-indices are componentwise. Write $[n]=\{1,\ldots,n\}$, $I^c=[n]\setminus I$, and $\mathbf{z}_I=(z_i)_{i\in I}$, with indices in increasing order. All tensor products are over $\C$, and linear maps act coefficientwise on formal series. \paragraph{Proof Strategy for Theorem~\ref{thm:main}.} The argument has three steps. \begin{enumerate} \item \emph{The Relative Quotient.} Set \[ Q=V/\Span_{\C}\{b_{-2}v:b\in V^G,\ v\in V\}. \] Taking invariants gives $Q^G\simeq V^G/C_2(V^G)$, so it suffices to show that $Q$ is finite dimensional. \item \emph{The Bounds.} Choose a finite-dimensional graded $G$-stable subspace $U\subset V_{>0}$ that strongly generates $V$. Then $Q$ is spanned by classes of monomials \[ u^1_{-n_1}\cdots u^\ell_{-n_\ell}\ket{0}, \qquad u^i\in U,\quad n_i\ge1. \] We call $\ell$ the \emph{length} and $\max_i n_i$ the \emph{height} of a nonempty monomial. To prove that $Q$ is finite dimensional, it suffices to bound both quantities for a spanning family. The length bound follows from $C_2$-cofiniteness and standard finite-group invariant theory, using a filtration by the sum of the generating states' weights (Section~\ref{sec:length}). This part of the argument is comparatively straightforward. The main difficulty is the height bound, which controls how large the mode indices can be. Its proof occupies Sections~\ref{sec:fields} and~\ref{sec:cutoff}. \item \emph{The Height Bound.} We obtain a bound after adjoining auxiliary fields, then transfer it to the original $n$-point functions. \begin{enumerate} \renewcommand{\labelenumii}{\Alph{enumii}.} \item \emph{Differential Equations.} We derive differential equations that constrain the $n$-point functions after passing to $Q$. \item \emph{Auxiliary Fields.} We extend these functions by adjoining suitable auxiliary fields. Their properties, together with the differential equations and commutative algebra, give a weight bound for the extended $n$-point functions. \item \emph{Contractions.} Taking vacuum contractions removes the auxiliary fields and transfers the weight bound to the original $n$-point functions. This gives the required height bound. \end{enumerate} \end{enumerate} Together, the length and height bounds give a finite spanning set for $Q$, proving Theorem~\ref{thm:main}. The proof of the height bound does not use $C_2$-cofiniteness. Under the same simplicity and grading assumptions, this allows us to show that finite strong generation also passes to $V^G$. \paragraph{Proof Strategy for Theorem~\ref{thm:sfg}.} The argument has four steps. \begin{enumerate} \item \emph{The Associated Graded Algebra.} Choose a finite-dimensional graded $G$-stable strong generating space for $V$. A filtration lets us pass to a commutative associated graded algebra. We present this as a quotient of a symmetric algebra on infinitely many copies of the generating space, one for each derivative order. \item \emph{Noether's Bound.} Applying Noether's theorem to finitely many copies at a time gives a possibly infinite family of invariant generators of polynomial degree at most $|G|$. This bounds the number of factors in the corresponding lifts. \item \emph{Lifting and the Height Bound.} We lift these generators to invariant normally ordered products involving at most $|G|$ factors. Their classes generate the commutative algebra $V^G/C_2(V^G)$. The height bound makes all sufficiently high-weight lifts vanish in this quotient, leaving only finitely many generators. \item \emph{Strong Generation.} The images of this finite family then span $(V^G)_{>0}/C_1(V^G)$. Li's strong-generation criterion shows that the family strongly generates $V^G$, proving Theorem~\ref{thm:sfg}. \end{enumerate} \paragraph{Acknowledgements and Use of Generative AI.} Generative AI, including ChatGPT, was used extensively under the author's direction for proof exploration, calculations, checking arguments and references, and drafting and revising the exposition. This assistance contributed substantial portions of the initial arguments and text. The author takes responsibility for the mathematical claims and final text. \section{The Length Bound}\label{sec:length} Throughout, $V$ is a simple vertex algebra with a connected nonnegative integer grading, and $G$ is a finite group of grading-preserving automorphisms. We may replace $G$ by its image in $\operatorname{Aut}(V)$ without changing $V^G$, and hence assume the action is faithful. Write $r=|G|$, $B=V^G$, and $P_G=r^{-1}\sum_{g\in G}g$ for the Reynolds projection on any $G$-module under consideration. We use the mode convention $Y(a,z)=\sum_{m\in\mathbb Z}a_mz^{-m-1}$ and write $T$ for translation. For homogeneous $a\in V_h$, write $\wt a=h$. Normally ordered products are associated to the right. In this section, assume that $R_V$ is finite dimensional. \subsection{Strong Generators}\label{subsec:generators} A subset $S\subset V$ \emph{strongly generates} $V$ if $V$ is spanned by the monomials \begin{equation}\label{eq:words} W=u^1_{-n_1}\cdots u^{\ell}_{-n_\ell}\ket{0}, \qquad \ell\ge0,\quad u^i\in S,\quad n_i\ge1. \end{equation} The empty product is the vacuum. If $S$ can be chosen finite, $V$ is \emph{strongly finitely generated}. A subspace strongly generates $V$ if its elements do. For homogeneous positive-weight $u^i$, the monomial \eqref{eq:words} has length $\ell$ and weight \begin{equation}\label{eq:wordweight} \wt W=\sum_i\wt u^i+\sum_i(n_i-1). \end{equation} For a nonempty monomial, its height is $\max_i n_i$; in particular, $\operatorname{ht}(W)\le\wt W$. The identity $(T^{n-1}u)_{-1}=(n-1)!u_{-n}$ expresses these monomials as right-associated normally ordered products of derivatives, up to nonzero scalar factors \cite[Section 3]{LianLinshaw}. Write $R_{V,+}=\bigoplus_{w>0}(R_V)_w$. Choose a graded linear section $s:R_{V,+}\to V_{>0}$ of the quotient map. Set \[ s_G(x)=\frac1r\sum_{g\in G}g\,s(g^{-1}x),\qquad U=s_G(R_{V,+})\subset V_{>0}. \] The map $s_G$ is again a section and is $G$-equivariant, so $U$ is a finite-dimensional graded $G$-stable space of lifts. Let \[ C_1(V)=TV+\Span\{a_{-1}b:a,b\in V_{>0}\}. \] The identities $a_{-2}b=(Ta)_{-1}b$ and $T\ket{0}=0$ give $C_2(V)\subseteq C_1(V)$. Hence $V_{>0}=U+C_1(V)$, and induction on weight shows that $U$ strongly generates $V$; see \cite[Theorem 4.10]{Li}. In particular, each $V_w$ is finite dimensional: after choosing a finite homogeneous basis of $U$, both the length and the sum of the derivative orders of a monomial of weight $w$ are bounded. \subsection{The Relative Quotient} For a vertex algebra $W$ and a $W$-module $M$, write \[ R_W(M)=M/\Span\{w_{-2}m:w\in W,\ m\in M\}. \] Thus $R_W(W)=R_W$. Regard $V$ as a module over $B=V^G$ and abbreviate $R_B(V)$ to $Q$. Set \begin{equation}\label{eq:relative} D_B=\Span\{b_{-2}v:b\in B,\ v\in V\},\qquad Q=V/D_B,\qquad q:V\longrightarrow Q. \end{equation} The subspace $D_B$ is graded and $G$-stable, so $Q$ inherits a grading and a $G$-action, and $q$ is $G$-equivariant. Translation gives $(T^{m-2}b)_{-2}=(m-1)!b_{-m}$, so \begin{equation}\label{eq:negative} q(b_{-m}v)=0\qquad(b\in B,\ m\ge2). \end{equation} Applying $P_G$ to $\sum_jb^j_{-2}v_j\in D_B\cap B$ replaces each $v_j$ by $P_Gv_j\in B$. Hence \begin{equation}\label{eq:exact} D_B\cap B=C_2(B),\qquad Q^G\simeq R_B. \end{equation} For the second identification, an invariant class $q(v)$ has the invariant representative $P_Gv$, since $q(P_Gv)=P_Gq(v)=q(v)$. The kernel of $q|_B$ is $D_B\cap B=C_2(B)$. It is therefore enough to show that $Q$ is finite dimensional. We use $Q$ as a graded vector space; all field operations below are performed in $V$ before applying $q$. \subsection{The Length Estimate}\label{subsec:length} \begin{lemma}\label{lem:orbit} Let $A$ be a commutative $G$-algebra and $J$ a $G$-stable ideal. Then \[ J^r\subseteq A J^G, \] where $A J^G$ is the ideal generated by the invariant elements of $J$. \end{lemma} \begin{proof} For $x\in J$, the nonleading coefficients of $\prod_{g\in G}(t-gx)$ lie in $J^G$. Evaluating at $t=x$ therefore gives $x^r\in A J^G$. Apply the same argument in $A[t_1,\ldots,t_r]$, with $G$ fixing the variables, to $x=t_1x_1+\cdots+t_rx_r$, where $x_i\in J$. The coefficient of $t_1\cdots t_r$ is $r!x_1\cdots x_r$; it belongs to $A J^G$, since taking invariants commutes with adjoining the fixed variables. Since $r!\ne0$ over $\C$, the claim follows. \end{proof} The positive-weight ideal $R_{V,+}$ of the finite-dimensional, connected graded algebra $R_V$ is nilpotent. Choose $\nu\ge1$ with \begin{equation}\label{eq:nu} R_{V,+}^{\nu}=0. \end{equation} For $p\ge0$, define \begin{equation}\label{eq:goodfiltration} F_pV=\Span\left\{ u^1_{-n_1}\cdots u^{\ell}_{-n_\ell}\ket{0}: \ell\ge0,\ n_i\ge1,\ u^i\in U\text{ homogeneous},\quad \sum_i\wt u^i\le p\right\}. \end{equation} The empty monomial is $\ket{0}$; set $F_{-1}V=0$, so $F_0V=\C\ket{0}$. This filtration uses the sum of the weights of the generating states, independently of the mode indices. Since $[T,u_{-n}]=n u_{-n-1}$ and $T\ket{0}=0$, translation preserves the filtration while raising the weight by one. In particular, $T^ju\in V_{h+j}\cap F_hV$ for $u\in U\cap V_h$. Since $U\subset V_{>0}$ is a graded strong generating subspace and $V_0=\C\ket{0}$, Li's theorem \cite[Theorem 4.11]{Li} gives the good-filtration identities \[ (F_pV)_mF_qV\subseteq \begin{cases} F_{p+q}V,&m<0,\\ F_{p+q-1}V,&m\ge0. \end{cases} \] Thus $A=\gr_FV$ is a commutative differential algebra. For homogeneous $u\in U$, let $\bar u$ be its class in $F_{\wt u}V/F_{\wt u-1}V$. The elements $T^j\bar u$, $j\ge0$, generate $A$, where $T$ denotes the induced derivation. Each $F_pV$ is graded and $G$-stable, and \begin{equation}\label{eq:filtrationdrop} V_w\subseteq F_wV,\qquad C_2(V)\cap V_w\subseteq F_{w-1}V. \end{equation} The first inclusion follows from \eqref{eq:wordweight}. For $a\in V_h$ and $b\in V_k$, the filtration properties give $a_{-2}b=(Ta)_{-1}b\in F_{h+k}V$. This vector lies in $V_{h+k+1}$, proving the second inclusion. \begin{lemma}[Length Bound]\label{lem:length} Every class in $Q$ is represented by a linear combination of monomials \eqref{eq:words} in $U$ of length at most \[ L=r\nu-1. \] \end{lemma} \begin{proof} Let $\mathfrak a\subset A$ be the ideal generated by the symbols of elements of $U$. We first show \begin{equation}\label{eq:idealnil} \mathfrak a^{\nu}=0. \end{equation} Take $\nu$ homogeneous elements of $U$, of total weight $w$. Their normally ordered product lies in $C_2(V)$ by \eqref{eq:nu}, so its symbol in filtration degree $w$ vanishes by \eqref{eq:filtrationdrop}. That symbol is the product of the $\nu$ generating symbols. These products generate $\mathfrak a^\nu$ as an ideal, proving \eqref{eq:idealnil}. Consider the commutative algebra \[ \bar A=A/A\,T(A^G),\qquad J=(\bar A)_{>0}, \] where the subscript denotes positive filtration degree. The quotient is graded, with degree-zero part $\C\ket{0}$, and $J$ is generated by the images of the $T^j\bar u$. We claim \begin{equation}\label{eq:invnil} (J^G)^{\nu}=0. \end{equation} Taking invariants gives \[ \bar A^G\simeq A^G/\bigl(A^G T(A^G)\bigr), \] since applying $P_G$ to the coefficients in $\sum_j a_jT(f_j)$, $f_j\in A^G$, gives $(A\,T(A^G))^G=A^G T(A^G)$. Every positive-degree invariant is a linear combination of the elements $P_G(m)$, where $m$ is a monomial in the $T^j\bar u$. The identity \[ P_G((Ta)b)=T P_G(ab)-P_G(a(Tb)) \] gives, by iteration, \[ P_G((T^k\bar u)b)\equiv(-1)^kP_G(\bar u\,T^kb) \pmod{A\,T(A^G)}. \] Each positive-degree monomial has a factor $T^k\bar u$, so this identity applies to every $P_G(m)$. Moreover, $P_G(\bar u\,T^kb)=r^{-1}\sum_g(g\bar u)T^k(gb)$ belongs to $\mathfrak a\cap A^G$, because $U$ is $G$-stable. Thus every element of $J^G$ has a representative in $\mathfrak a\cap A^G$. Products of $\nu$ such representatives vanish by \eqref{eq:idealnil}, proving \eqref{eq:invnil}. Applying Lemma~\ref{lem:orbit} in $\bar A$ and taking $\nu$th powers gives \[ J^{r\nu}=(J^r)^\nu \subseteq(\bar A J^G)^\nu =\bar A(J^G)^\nu=0. \] Thus $\bar A$ is spanned by monomials in $T^j\bar u$ of length at most $r\nu-1$. Give $D_B$ and $Q$ the induced filtrations, so $F_pQ=(F_pV+D_B)/D_B$. Taking invariants in each filtration quotient gives $\gr_FB=A^G$: an invariant symbol has an invariant lift obtained by applying $P_G$ to any lift. If $b\in B\cap F_pV$, $v\in F_sV$, and $\sigma_j$ denotes the class in $F_jV/F_{j-1}V$, then \[ \sigma_s(v)\,T\sigma_p(b) =\sigma_{p+s}(b_{-2}v)\in\gr_FD_B. \] Consequently \begin{equation}\label{eq:gradedquotient} A\,T(A^G)\subseteq\gr_FD_B, \qquad \bar A\twoheadrightarrow\gr_FQ \end{equation} as graded vector spaces, since $\gr_FQ=A/\gr_FD_B$. For a class in $F_pQ$, express its degree-$p$ symbol using the spanning monomials of $\bar A$ and lift those monomials to normally ordered products in $V$. Each lift has length at most $L$, and subtracting their linear combination leaves a class in $F_{p-1}Q$. Induction terminates at $F_{-1}Q=0$. The filtration and all the maps used are compatible with weight, so the lifts can be chosen to preserve weight. \end{proof} It remains to bound the heights of these monomials. We do this by bounding the weights in $Q$ of coefficients in products of a fixed number of fields. \section{The Differential Equations}\label{sec:fields} We retain $B=V^G$, $D_B$, and $q:V\to Q$, but no longer assume $C_2$-cofiniteness. The arguments in this section use only the connected nonnegative grading and the finite group action. Lemmas~\ref{lem:regularity} and~\ref{lem:iterate} are standard consequences of the vertex algebra axioms and the grading. We defer their proofs to Appendix~\ref{app:voa}, where we give the details for completeness. \subsection{\texorpdfstring{$n$}{n}-Point Functions} Fix $n\ge1$ and finite-dimensional homogeneous $G$-stable subspaces $E_i\subset V_{h_i}$. Put $\mathbf{z}=(z_1,\ldots,z_n)$, $E=\bigotimes_{i=1}^nE_i$, and \begin{equation}\label{eq:Delta} \Delta(\mathbf{z})=\prod_{i0$, there is a homogeneous $w\in V_k$, $k>0$, such that \[ u_{h+k-1}w\ne0\quad\text{in }\C\ket{0}. \] \end{lemma} We can now construct the auxiliary state $\xi$. For each $g\ne1$, faithfulness gives a homogeneous $a_g$ with $ga_g\ne a_g$. Its weight is positive because $G$ fixes $V_0=\C\ket{0}$. Set $u_g=(1-g)a_g$. By Lemma~\ref{lem:detect}, we can choose and rescale a homogeneous $w_g$ so that \begin{equation}\label{eq:pairunit} (u_g)_{h_g+k_g-1}w_g=\ket{0}, \qquad h_g=\wt u_g,\quad k_g=\wt w_g. \end{equation} All higher modes vanish by the grading. Let \[ U_g=\Span\{ha_g:h\in G\},\qquad W_g=\Span\{hw_g:h\in G\}. \] The auxiliary spaces $U_g$ and $W_g$ are finite dimensional and homogeneous, and $a_g,u_g\in U_g$. Fix finite-dimensional homogeneous $G$-stable subspaces $E_i\subset V_{h_i}$, $1\le i\le k$, and put $E=\bigotimes_iE_i$. Fix an order of $G\setminus\{1\}$, and use it in the tensor products below. Adjoin $2r-1$ copies of each pair $(u_g,w_g)$ by setting \[ \widetilde E=E\otimes\bigotimes_{g\ne1}(U_g\otimes W_g)^{\otimes(2r-1)}, \qquad \xi=\bigotimes_{g\ne1}(u_g\otimes w_g)^{\otimes(2r-1)}. \] The tensor factors in this enlarged space retain the displayed order. We use $2r-1$ copies so that any set of at most $2r-2$ tensor positions leaves at least one prescribed copy of each $u_g$ outside it. This will be used in the proof of Lemma~\ref{lem:support}. Write the pairs in $\xi$ as $(u_t,w_t)$, $1\le t\le N$, where \begin{equation}\label{eq:N} N=(r-1)(2r-1). \end{equation} For $r=1$, these tensor products are empty: $N=0$, $\widetilde E=E$, and $\xi=1\in\C$. \subsection{The Universal Module} We now encode the differential equations of Lemma~\ref{lem:PDE} as relations over a polynomial ring. The enlarged space $\widetilde E$ has $n=k+2N$ factors; let $d_I$ be the orders of the corresponding differential equations and set \begin{equation}\label{eq:M} R=\C[p_1,\ldots,p_n],\qquad p_I=\sum_{i\in I}p_i,\qquad M=\frac{R\otimes\widetilde E} {\displaystyle\sum_{\varnothing\ne I\subseteq[n]} R\,p_I^{d_I}P_I\widetilde E}. \end{equation} The variable $p_i$ represents differentiation in the variable of the $i$th field. Let $\mathcal N$ be the $R$-submodule of $M$ generated by the classes of $e\otimes\xi$, $e\in E$. This submodule is finitely generated because $E$ is finite dimensional. We only need a bound for these prescribed tensors, not for all of $M$: we will show that every monomial of sufficiently large degree annihilates $\mathcal N$. \begin{lemma}[Support at the Origin]\label{lem:support} For some integer $q_0\ge1$, \begin{equation}\label{eq:origin} (p_1,\ldots,p_n)^{q_0}\mathcal N=0. \end{equation} \end{lemma} \begin{proof} For $\boldsymbol{\lambda}=(\lambda_1,\ldots,\lambda_n)\in\C^n$, write $\m_{\boldsymbol{\lambda}}=(p_1-\lambda_1,\ldots,p_n-\lambda_n)$. We first show that $\mathcal N_{\m_{\boldsymbol{\lambda}}}=0$ whenever $\boldsymbol{\lambda}\ne\mathbf{0}$, and then use finite generation to obtain the required power of $(p_1,\ldots,p_n)$. Fix such a $\boldsymbol{\lambda}$ and put $S=\{i:\lambda_i\ne0\}$. The relations in \eqref{eq:M} give $p_I^{d_I}P_Iv=0$ in $M$ for every $v\in\widetilde E$ and every nonempty $I\subseteq[n]$. If $p_I(\boldsymbol{\lambda})\ne0$, then $p_I$ is a unit in $R_{\m_{\boldsymbol{\lambda}}}$, so \[ P_Iv=0\quad\text{in }M_{\m_{\boldsymbol{\lambda}}} \qquad(v\in\widetilde E). \] We will apply this observation in two cases. Suppose first that $|S|\ge2r-1$. Choose a line through the origin in $\C$ containing none of the nonzero $\lambda_i$. One of its open half-planes contains at least $r$ of these numbers; choose a set $J\subseteq S$ of $r$ corresponding indices. There is a real-linear functional $\ell:\C\to\mathbb R$ with $\ell(\lambda_i)>0$ for all $i\in J$. Hence, for every nonempty $I\subseteq J$, \[ \ell\bigl(p_I(\boldsymbol{\lambda})\bigr) =\sum_{i\in I}\ell(\lambda_i)>0. \] Thus $p_I(\boldsymbol{\lambda})\ne0$, and the image of $P_I\widetilde E$ in $M_{\m_{\boldsymbol{\lambda}}}$ is zero. Applying \eqref{eq:invariantsubsets} to the $r$ factors indexed by $J$, and tensoring with the remaining factors, gives \[ \widetilde E =\sum_{\varnothing\ne I\subseteq J}P_I\widetilde E. \] Since these tensors generate $M$ over $R$, we obtain $M_{\m_{\boldsymbol{\lambda}}}=0$, and hence $\mathcal N_{\m_{\boldsymbol{\lambda}}}=0$. Now suppose $1\le|S|\le2r-2$, and choose $j\in S$. In our tensor order, the original factors occupy positions $1,\ldots,k$, and the $t$th auxiliary pair occupies positions $k+2t-1$ and $k+2t$, with $u_t$ first and $w_t$ second. For each $g\ne1$, let $A_g\subseteq[n]$ be the positions of the prescribed copies of $u_g$ in $\xi$. The sets $A_g$ are pairwise disjoint, each with $2r-1$ elements. Since $|S|\le2r-2$, we can choose \[ i_g\in A_g\setminus S\qquad(g\ne1). \] The chosen positions are distinct, and none equals $j$. It suffices by linearity to consider a pure tensor $e=e_1\otimes\cdots\otimes e_k\in E$. For $J\subseteq G\setminus\{1\}$, put \[ I_J=\{j\}\cup\{i_g:g\in J\}. \] Let $\theta_J\in\widetilde E$ be obtained from $e\otimes\xi$ by replacing the selected copy of $u_g$ at position $i_g$ by $a_g$ if $g\in J$ and by $ga_g$ otherwise, leaving all other factors unchanged. These replacements lie in $U_g$. The explicit identity in the proof of Lemma~\ref{lem:averaging}, with distinguished position $j$ and positions $i_g$ indexed by $g\ne1$, gives \[ e\otimes\xi = r\sum_{J\subseteq G\setminus\{1\}} (-1)^{r-1-|J|}P_{I_J}\theta_J. \] The factor at $j$ may be original or auxiliary, since the identity allows an arbitrary vector there. All tensor factors remain in their fixed order. The sum includes $J=\varnothing$, for which $I_J=\{j\}$. For every $J$, the selected positions $i_g$ lie outside $S$, so \[ p_{I_J}(\boldsymbol{\lambda}) =\lambda_j+\sum_{g\in J}\lambda_{i_g} =\lambda_j\ne0. \] Our initial observation therefore makes every $P_{I_J}\theta_J$ zero in $M_{\m_{\boldsymbol{\lambda}}}$. Thus every generator $e\otimes\xi$ of $\mathcal N$ vanishes after localization, proving $\mathcal N_{\m_{\boldsymbol{\lambda}}}=0$ in this case as well. If $\mathcal N=0$, take $q_0=1$. Otherwise, finite generation gives $\Supp\mathcal N=V(\Ann_R\mathcal N)$. We have excluded every maximal ideal except $\m_{\mathbf{0}}$ from this closed set. The Nullstellensatz therefore gives \[ (p_1,\ldots,p_n)\subseteq\sqrt{\Ann_R\mathcal N}. \] Choose integers $\nu_i\ge1$ such that $p_i^{\nu_i}\mathcal N=0$, and put \[ q_0=1+\sum_{i=1}^n(\nu_i-1). \] A monomial not divisible by any $p_i^{\nu_i}$ has total degree at most $\sum_i(\nu_i-1)$. Hence every monomial of total degree at least $q_0$ annihilates $\mathcal N$, proving \eqref{eq:origin}. \end{proof} \subsection{Proof of the Weight Bound} We recall the weight bound before proving it. \begin{restatedcutoff} Let $X\subset V$ be a finite-dimensional graded $G$-stable subspace and let $s\ge1$. There is an integer $H=H(X,s)\ge0$ such that, for every $1\le k\le s$ and $a_1,\ldots,a_k\in X$, all coefficients of \[ qY(a_1,z_1)\cdots Y(a_k,z_k)\ket{0} \] belong to $\bigoplus_{w=0}^H Q_w$. \end{restatedcutoff} \begin{proof}[Proof of Proposition~\ref{prop:cutoff}] By linearity, it suffices to take $a_i$ in homogeneous components $E_i=X\cap V_{h_i}$. We fix $k$ and $h_1,\ldots,h_k$ and prove a bound independent of the $a_i$ and the mode indices. Since $k\le s$ and $X$ has finitely many weights, the maximum of these bounds and $0$ gives $H(X,s)$. Form $\widetilde E$ as above. Give the original fields variables $\mathbf{z}=(z_1,\ldots,z_k)$ and the $t$th auxiliary pair variables $x_t,y_t$. Write $\mathbf{x}=(x_1,\ldots,x_N)$ and $\mathbf{y}=(y_1,\ldots,y_N)$, and put $H_t=\wt u_t$ and $K_t=\wt w_t$. For auxiliary states $b_t,c_t$ in their respective tensor factors, write, in tensor order, \[ \begin{aligned} (A_1,\ldots,A_n)&=(a_1,\ldots,a_k,b_1,c_1,\ldots,b_N,c_N),\\ \boldsymbol{\zeta}=(\zeta_1,\ldots,\zeta_n) &=(z_1,\ldots,z_k,x_1,y_1,\ldots,x_N,y_N). \end{aligned} \] Let $\eta_i$ be the weight of the $i$th tensor factor and $\widetilde e=A_1\otimes\cdots\otimes A_n$. Define \begin{equation}\label{eq:enlargedDelta} \widetilde\Delta(\mathbf{z},\mathbf{x},\mathbf{y})=\prod_{iH:=\max\{0,\widetilde h+q_0-1-\widetilde D\}. \end{equation} We now remove the auxiliary pairs one at a time. Let $H_V^{(j)}$ be the regularized product of the original fields and the auxiliary pairs $j+1,\ldots,N$, retaining their original order, and put $H_Q^{(j)}=qH_V^{(j)}$. Write $\Delta^{(j)}$ for its regularizing polynomial, which contains only the factors among these remaining fields. Thus \[ H_Q^{(0)}=F,\qquad H_Q^{(N)}=H_Q. \] We suppress the dependence on $e$. Suppose that the first $j$ pairs have been removed, and consider the next pair, indexed by $t=j+1$. Set $x_t=y_t+d$. The regularizing polynomial factors as \[ \left.\Delta^{(j)}\right|_{x_t=y_t+d} = d^{H_t+K_t}\,B_t^{(j)}(d)\,\Delta^{(j+1)}. \] Here $d^{H_t+K_t}$ is the factor within the $t$th pair, $B_t^{(j)}(d)$ contains the factors between this pair and the other remaining fields, and $\Delta^{(j+1)}$ contains the factors among those other fields. We retain the original orientation of every factor. By \eqref{eq:pairunit} and the grading, \[ d^{H_t+K_t}Y(u_t,d)w_t = \ket{0}+d\rho_t(d), \qquad \rho_t(d)\in V[[d]]. \] Applying Lemma~\ref{lem:iterate} to this pair, with common coordinate $y_t$, and then applying $q$, gives \[ \left.H_Q^{(j)}\right|_{x_t=y_t+d} = B_t^{(j)}(d)\, q\!\left( Y(\ket{0}+d\rho_t(d),y_t)\, H_V^{(j+1)} \right). \] The field action is interpreted coefficientwise in the remaining variables. Every factor on the right has only nonnegative powers of $d$. Taking its constant coefficient and using $Y(\ket{0},y_t)=\mathrm{id}$, we obtain \[ \left.H_Q^{(j)}\right|_{x_t=y_t} = B_t^{(j)}(0)\,H_Q^{(j+1)}. \] The specialization on the left is defined because $H_Q^{(j)}$ is a power series by Lemma~\ref{lem:regularity}. To keep track of the scalar factors accumulated during these removals, put \[ F^{(j)} = \left.F\right|_{x_1=y_1,\ldots,x_j=y_j}. \] We prove inductively that $F^{(j)}=C_jH_Q^{(j)}$, where $C_j$ is a scalar polynomial. The assertion holds for $j=0$, with $C_0=1$. Using the one-pair identity above, the induction step is \[ \begin{aligned} F^{(j+1)} &= \left.(C_jH_Q^{(j)})\right|_{x_t=y_t}\\ &= \left.C_j\right|_{x_t=y_t}\, B_t^{(j)}(0)\,H_Q^{(j+1)}\\ &= C_{j+1}H_Q^{(j+1)}, \end{aligned} \qquad C_{j+1} := \left.C_j\right|_{x_t=y_t}B_t^{(j)}(0). \] The displayed definition makes $C_{j+1}$ a scalar polynomial. This proves the induction. After removing all $N$ pairs, we therefore have \begin{equation}\label{eq:collision} F(\mathbf{z},\mathbf{y},\mathbf{y}) = C(\mathbf{z},\mathbf{y})\,H_Q(\mathbf{z}), \end{equation} where $C=C_N$, $H_Q$ is defined by $E_1,\ldots,E_k$, and \begin{equation}\label{eq:C} \begin{aligned} C(\mathbf{z},\mathbf{y})={}&\prod_{i=1}^k\prod_{t=1}^N (z_i-y_t)^{2h_i+H_t+K_t}\\ &\cdot\prod_{1\le tH$, project \eqref{eq:collision} to $Q_w$. Fix $e\in E$. Equation \eqref{eq:weightbound} gives \[ C(\mathbf{z},\mathbf{y})(H_Q)_w(\mathbf{z})(e)=0. \] By Lemma~\ref{lem:regularity}, $(H_Q)_w(\mathbf{z})(e)$ is a polynomial in $z_1,\ldots,z_k$ with coefficients in $Q_w$. Choose a basis $v_1,\ldots,v_m$ of the finite-dimensional subspace spanned by these coefficients, and write \[ (H_Q)_w(\mathbf{z})(e) =\sum_{i=1}^m f_i(\mathbf{z})v_i, \qquad f_i(\mathbf{z})\in\C[z_1,\ldots,z_k]. \] The identity $C(\mathbf{z},\mathbf{y})(H_Q)_w(\mathbf{z})(e)=0$ therefore becomes \[ \sum_{i=1}^m C(\mathbf{z},\mathbf{y})f_i(\mathbf{z})v_i=0. \] Comparing the coefficient of each monomial in $\mathbf{z},\mathbf{y}$ and using the linear independence of the $v_i$, we obtain the scalar polynomial equalities \[ C(\mathbf{z},\mathbf{y})f_i(\mathbf{z})=0 \qquad(1\le i\le m). \] Since $C\ne0$ and $\C[\mathbf{z},\mathbf{y}]$ is an integral domain, every $f_i$ is zero. Thus $(H_Q)_w(\mathbf{z})(e)=0$. As $e$ was arbitrary, $(H_Q)_w=0$. Finally, multiplication by $\Delta$ is invertible on the ordered Laurent space $Q((z_1))\cdots((z_k))$. Its inverse is multiplication by $\Delta^{-1}$, expanded according to the original order of the fields. This operation commutes with projection to $Q_w$. Thus the weight-$w$ component of the original product is zero as well. \end{proof} \section{Cofiniteness and Its Consequences}\label{sec:finish} We now combine the length and height bounds to prove $C_2$-cofiniteness of $V^G$. \begin{restatedmain} Let $V$ be a simple vertex algebra with a connected nonnegative integer grading, and let $G$ be a finite group of grading-preserving automorphisms. If $V$ is $C_2$-cofinite, then $V^G$ is $C_2$-cofinite. \end{restatedmain} \begin{proof}[Proof of Theorem~\ref{thm:main}] Let $U$ be the finite-dimensional graded $G$-stable strong generating space of Section~\ref{subsec:generators}, and choose $\nu$ with $R_{V,+}^{\nu}=0$. By Lemma~\ref{lem:length}, $Q$ is spanned by classes of monomials of length at most $L=r\nu-1$. Apply Proposition~\ref{prop:cutoff} with $X=U$ and $s=\max\{1,L\}$. With $H=H(U,s)$, we obtain $Q_w=0$ for $w>H$. Each $V_w$ is finite dimensional, and hence so is $Q=\bigoplus_{w\le H}Q_w$. Equation \eqref{eq:exact} now gives $\dim B/C_2(B)<\infty$. More explicitly, for a homogeneous basis $u^1,\ldots,u^d$ of $U$, \[ Q=\Span\left\{q(\ket{0}),\; q(u^{i_1}_{-n_1}\cdots u^{i_\ell}_{-n_\ell}\ket{0}): 1\le\ell\le L,\ 1\le i_j\le d,\ 1\le n_j\le H\right\}. \] The height bound follows from \eqref{eq:wordweight}. \end{proof} Recall that a vertex operator algebra is \emph{strongly rational} if it is simple, of CFT type, self-contragredient, rational, and $C_2$-cofinite. Here rationality means complete reducibility of admissible modules. \begin{samepage} \begin{corollary}\label{cor:rational} Let $V$ be a strongly rational vertex operator algebra and $G$ a finite group of automorphisms. Then $V^G$ is strongly rational. \end{corollary} \begin{proof} Theorem~\ref{thm:main} gives $C_2$-cofiniteness of $V^G$. McRae's theorem \cite[Corollary 4.23]{McRae} then gives strong rationality. \end{proof} \end{samepage} We next use the height bound to prove finite strong generation, without assuming $C_2$-cofiniteness. \begin{restatedsfg} Let $V$ be a simple, strongly finitely generated vertex algebra with a connected nonnegative integer grading. For every finite group $G$ of grading-preserving automorphisms, $V^G$ is strongly finitely generated. \end{restatedsfg} \begin{proof}[Proof of Theorem~\ref{thm:sfg}] Write $B=V^G$, $r=|G|$, and retain the relative quotient $q:V\to Q=V/D_B$ from \eqref{eq:relative}. We first explain the structure of the proof. Starting with a finite-dimensional graded $G$-stable strong generating space $X$ for $V$, we use a good filtration to pass to a commutative associated graded algebra. Following the standard approach through commutative invariant theory \cite[Sections~7--8]{LianLinshaw}, we present this algebra as a quotient of a symmetric algebra $S$ on copies of $X$, one for each derivative order. Applying Noether's bound to the subalgebras involving finitely many copies gives generators for $S^G$ of polynomial degree at most $|G|$. Their images in the associated graded algebra lift to invariant normally ordered products, giving a possibly infinite strong generating family $\mathcal X_G$ for $V^G$. The degree bound ensures that each defining word has at most $|G|$ factors, so Proposition~\ref{prop:cutoff} applies with $s=|G|$. It gives a weight cutoff above which these invariant lifts lie in $C_2(B)$. The remaining family $\mathcal X_G^{\mathrm{fin}}$ is finite, and its image generates $R_B=B/C_2(B)$. Its image consequently spans $B_{>0}/C_1(B)$, so Li's strong-generation criterion shows that $\mathcal X_G^{\mathrm{fin}}$ strongly generates $V^G$. \paragraph{A Finite Graded Generating Space and Its Filtration.} Choose a finite-dimensional graded $G$-stable subspace $X\subset V_{>0}$ that strongly generates $V$. Such a space is obtained by taking the $G$-span of a finite homogeneous strong generating set of positive weight. Each $V_w$ is then finite dimensional, since only finitely many normally ordered monomials in a homogeneous basis of $X$ and its derivatives have weight $w$. Define $F_pV$ by \eqref{eq:goodfiltration}, with $X$ in place of $U$. This definition and Li's good-filtration theorem \cite[Theorem~4.11]{Li} do not require $C_2$-cofiniteness. Set $A=\gr_FV$ and give $B$ the induced filtration $F_pB=B\cap F_pV$. The filtration is preserved by $G$ and by translation, and $A$ is a commutative differential algebra. In particular, \begin{equation}\label{eq:fixedgraded} \gr_FB\;=\;(\gr_FV)^G=A^G. \end{equation} To check the surjectivity implicit in \eqref{eq:fixedgraded}, let $\alpha\in(F_pV/F_{p-1}V)^G$ and choose a lift $v\in F_pV$. Then $P_Gv\in F_pB$ has symbol $P_G\alpha=\alpha$. This proves \eqref{eq:fixedgraded} in every filtration degree. \paragraph{Noether's Uniform Degree Bound.} Fix a finite homogeneous basis $\mathcal X$ of $X$. For each $j\ge0$, let $X^{(j)}$ be a copy of the $G$-module $X$, and write $x^{(j)}\in X^{(j)}$ for the copy of $x\in X$. Set \[ S=\Sym\left(\bigoplus_{j\ge0}X^{(j)}\right), \qquad S_{\le J}=\Sym\left(\bigoplus_{0\le j\le J}X^{(j)}\right). \] The map $x^{(j)}\mapsto T^j\bar x$ defines a surjective, $G$-equivariant homomorphism of commutative algebras \begin{equation}\label{eq:jetmap} \rho:S\twoheadrightarrow A, \qquad \rho(S^G)=A^G. \end{equation} For the second assertion, take a preimage of an invariant element of $A$ and apply $P_G$; its image is unchanged. For a homogeneous $x\in X\cap V_h$, give $x^{(j)}$ the two degrees \[ \deg_F(x^{(j)})=h, \qquad \wt(x^{(j)})=h+j. \] The map $\rho$ preserves both gradings, although a generator may map to zero. We also write $\deg_{\mathrm{pol}}$ for ordinary polynomial degree on $S$, defined by $\deg_{\mathrm{pol}}(x^{(j)})=1$. Thus \[ \deg_{\mathrm{pol}}\!\left(x_1^{(j_1)}\cdots x_d^{(j_d)}\right)=d. \] We use this degree in $S$ to bound the number of factors; we do not need it to descend to a grading of $A$. Every polynomial in $S$ belongs to some $S_{\le J}$, and $S_{\le J}$ is $G$-stable. Consequently \[ S^G=\bigcup_{J\ge0}S_{\le J}^G. \] For each $J$, the classical Noether bound \cite{Noether} states that $S_{\le J}^G$ is generated as a commutative algebra by invariants of \emph{ordinary polynomial degree at most $|G|$}. Since the invariant subspace of each homogeneous polynomial degree is the image of $P_G$, it is spanned by the elements $P_G(\mathsf{m})$, where $\mathsf{m}$ is a monomial of that degree in the variables $x^{(j)}$, with $0\le j\le J$ and $x$ ranging over our chosen homogeneous basis $\mathcal X$ of $X$. Thus the family \begin{equation}\label{eq:noetherfamily} \{P_G(\mathsf{m}):\mathsf{m}\text{ a monomial in the }x^{(j)},\quad 1\le\deg_{\mathrm{pol}}\mathsf{m}\le |G|\} \end{equation} generates $S^G$, and its image under $\rho$ generates $A^G$. This is only a \emph{bounded-degree} assertion: because $j$ is unbounded, \eqref{eq:noetherfamily} is not yet a finite set. \paragraph{Lifting Invariant Polynomials to $V^G$.} Recall that $B=V^G$ and that $x^{(j)}\in X^{(j)}$ denotes the copy of the basis vector $x\in\mathcal X$ in the $j$th copy of $X$. Fix an ordering of these symbols and write each monomial in that order. For such a monomial \[ \mathsf{m}=x_1^{(j_1)}\cdots x_d^{(j_d)}, \qquad 1\le d\le |G|,\quad x_i\in\mathcal X, \] write $h_i=\wt x_i$ and define the invariant state \begin{equation}\label{eq:invariantlift} \begin{aligned} \beta_{\mathsf{m}}&=P_G\bigl((T^{j_1}x_1)_{-1}\cdots (T^{j_d}x_d)_{-1}\ket{0}\bigr)\\ &=\left(\prod_{i=1}^d j_i!\right) P_G\bigl((x_1)_{-j_1-1}\cdots (x_d)_{-j_d-1}\ket{0}\bigr)\in B. \end{aligned} \end{equation} Set \[ \mathcal X_G=\{\beta_{\mathsf{m}}: \mathsf{m}\text{ is as in }\eqref{eq:noetherfamily}\}\subseteq B. \] The state $\beta_{\mathsf{m}}$ is homogeneous of weight \begin{equation}\label{eq:invariantliftweight} \wt\beta_{\mathsf{m}}=\sum_{i=1}^d(h_i+j_i) \quad\text{if }\beta_{\mathsf{m}}\ne0, \end{equation} and lies in $F_pV$, where $p=\sum_i h_i$. Writing $\sigma_p$ for the projection to $F_pV/F_{p-1}V$, the good-filtration product rule gives \[ \sigma_p(\beta_{\mathsf{m}})=\rho(P_G(\mathsf{m})). \] This equality also holds when $\beta_{\mathsf{m}}$ has lower filtration degree and both sides are zero. We claim that $\mathcal X_G$ strongly generates $B$. Indeed, let $b\in B\cap F_pV$ be homogeneous. By \eqref{eq:fixedgraded}, \eqref{eq:jetmap}, and \eqref{eq:noetherfamily}, its filtration-$p$ symbol is a polynomial in the symbols of the elements of $\mathcal X_G$. Both filtration degree and weight are preserved, so this polynomial may be chosen homogeneous of weight $\wt b$ and filtration degree $p$. Replace products in the polynomial by normally ordered products of the corresponding $\beta_{\mathsf{m}}$. The resulting state $b'\in B\cap F_pV$ has the same filtration-$p$ symbol as $b$. Hence \[ b-b'\in B\cap F_{p-1}V. \] Induction on $p$, starting with $F_0B=\C\ket{0}$, proves the claim. In particular, the image of $\mathcal X_G$ generates the commutative algebra $R_B=B/C_2(B)$: normally ordered products become products of classes, and derivatives vanish in $R_B$. \paragraph{The Weight Cutoff Makes the Family Finite.} Recall that, using Noether's bound, we constructed $\mathcal X_G$ from monomials of polynomial degree at most $|G|$. Thus the corresponding words in \eqref{eq:invariantlift} have at most $r=|G|$ factors. We therefore apply Proposition~\ref{prop:cutoff} to $X$ with $s=r$, and denote the resulting weight bound by $H_X=H(X,r)$. Set \[ \mathcal X_G^{\mathrm{fin}} =\{\beta\in\mathcal X_G\setminus\{0\}:\wt\beta\le H_X\}. \] We will show that this set is finite and that its image generates $R_B$. The word in the second line of \eqref{eq:invariantlift} is the coefficient \[ (x_1)_{-j_1-1}\cdots(x_d)_{-j_d-1}\ket{0} = [z_1^{j_1}\cdots z_d^{j_d}] Y(x_1,z_1)\cdots Y(x_d,z_d)\ket{0}. \] There are at most $r$ fields, regardless of the derivative orders $j_i$. Proposition~\ref{prop:cutoff} therefore bounds the weight of its class in $Q$ by $H_X$. Since the word is homogeneous of the weight in \eqref{eq:invariantliftweight}, this gives \begin{equation}\label{eq:highliftzero} \wt\beta_{\mathsf{m}}>H_X \quad\Longrightarrow\quad q(\beta_{\mathsf{m}})=0 \quad\Longrightarrow\quad \beta_{\mathsf{m}}\in D_B\cap B=C_2(B). \end{equation} The last implication uses \eqref{eq:exact}; the equality $qP_G=P_Gq$ follows from $G$-equivariance of $q$. Proposition~\ref{prop:cutoff} is used here only for the words of length at most $r$; no global weight bound on $Q$ is claimed. Only finitely many basis monomials $\mathsf{m}$ in \eqref{eq:noetherfamily} have $\sum_i(h_i+j_i)\le H_X$: there are at most $r$ factors, the $x_i$ belong to a finite basis of positive-weight states, and the nonnegative integers $j_i$ have bounded sum. Hence $\mathcal X_G^{\mathrm{fin}}$ is finite. Every omitted element of $\mathcal X_G$ has zero image in $R_B$ by \eqref{eq:highliftzero}, so the image of $\mathcal X_G^{\mathrm{fin}}$ generates $R_B$ as a commutative algebra. \paragraph{From $R_B$ to Strong Finite Generation.} Recall \[ C_1(B)=TB+\Span\{a_{-1}b:a,b\in B_{>0}\}, \qquad C_2(B)\subseteq C_1(B). \] Because the elements of $\mathcal X_G^{\mathrm{fin}}$ have positive weight and their images generate $R_B$, each homogeneous positive-weight element of $B$ is, modulo $C_2(B)$, a sum of linear terms in elements of $\mathcal X_G^{\mathrm{fin}}$ and normally ordered products of two or more such terms. Each such product lies in $C_1(B)$ by definition. Thus the images of $\mathcal X_G^{\mathrm{fin}}$ span the \emph{vector space} $B_{>0}/C_1(B)$; no algebra structure on this quotient is needed. Equivalently, \[ B_{>0}=\Span\mathcal X_G^{\mathrm{fin}}+C_1(B). \] The fixed-point algebra $B$ inherits the connected nonnegative integer grading. Li's strong-generation criterion \cite[Theorem~4.10]{Li} states that, in this setting, a homogeneous set strongly generates $B$ if and only if its image spans $B_{>0}/C_1(B)$. The criterion now implies that the finite set $\mathcal X_G^{\mathrm{fin}}$ strongly generates $V^G$. This proves Theorem~\ref{thm:sfg}. \end{proof} \begin{remark} In Theorems~\ref{thm:main} and~\ref{thm:sfg}, simplicity can be replaced by the following condition: for each $g\in G$ acting nontrivially on $V$, there are homogeneous $a_g,w_g$ such that $u_g=(1-g)a_g\ne0$ and \[ (u_g)_{\wt u_g+\wt w_g-1}w_g\in\C^\times\ket{0}. \] These pairs replace those supplied by Lemma~\ref{lem:detect}. \end{remark} \appendix \section{Technical VOA Appendix}\label{app:voa} We give the proofs deferred from Sections~\ref{sec:fields} and~\ref{sec:cutoff}, retaining their notation. \subsection{Regularity}\label{app:regularity} \begin{proof}[Proof of Lemma~\ref{lem:regularity}] For homogeneous $a_i,a_j$, the grading gives $(a_i)_\ell a_j=0$ for $\ell\ge h_i+h_j$. The commutator formula therefore makes $(z_i-z_j)^{h_i+h_j}$ a locality factor. After multiplication by $\Delta$, locality allows any one field $Y(a_i,z_i)$ to be moved to the rightmost position, without changing the scalar factors. Creation gives $Y(a_i,z_i)\ket{0}=e^{z_iT}a_i\in V[[z_i]]$. The other fields do not involve $z_i$, and multiplication by $\Delta$ introduces no negative powers. Thus the coefficient of any monomial with a negative $z_i$-power is zero. Since $i$ was arbitrary, $H_V(\mathbf z)(e)\in V[[z_1,\ldots,z_n]]$. The mode weight formula gives weight $\sum_i h_i+s$ for a coefficient of total degree $s$ before multiplication by $\Delta$. Since $\Delta$ is homogeneous, the coefficient of total degree $d$ after multiplication has the weight in \eqref{eq:degreeweight}. For fixed $w$, its total degree must therefore be $w-\sum_i h_i+\deg\Delta$. There are finitely many monomials of that degree; if it is negative, the projection is zero. No finite-dimensionality assumption on $V_w$ is needed here. \end{proof} \subsection{Isolating the Cross Factor}\label{app:iterate} \begin{proof}[Proof of Lemma~\ref{lem:iterate}] We first prove the identity on a fixed homogeneous vector. We then apply it to the complementary coefficients, one at a time. \smallskip\noindent \emph{The Identity on a Fixed Vector.} For this calculation only, relabel the indices of $I$ as $1,\ldots,m$, preserving their order, and put $x_m=0$. Write $\Delta_I$ for the corresponding internal polynomial and $C(\mathbf{x})$ for $C_I(\mathbf{x})$ in these labels. Fix $v\in V_d$ and put \[ F_v(\mathbf{z})=\Delta_I(\mathbf{z}) Y(a_1,z_1)\cdots Y(a_m,z_m)v, \qquad K_i=h_i+d,\qquad \Pi(\mathbf{z})=\prod_{i=1}^m z_i^{K_i}. \] Adjoin the field $Y(v,s)$ immediately before the vacuum: \[ \Psi(\mathbf{z};s)= \Delta_I(\mathbf{z})\prod_{i=1}^m(z_i-s)^{K_i} Y(a_1,z_1)\cdots Y(a_m,z_m)Y(v,s)\ket{0}. \] Lemma~\ref{lem:regularity} gives $\Psi\in V[[\mathbf{z},s]]$. Creation and constant-coefficient extraction in $s$ give \[ \Psi(\mathbf{z};0)=\Pi(\mathbf{z})F_v(\mathbf{z}). \] Thus all remaining coordinate poles of $F_v$ are bounded: \[ F_v\in\mathcal A:= V[[z_1,\ldots,z_m]][z_1^{-1},\ldots,z_m^{-1}]. \] There is a linear reexpansion map \[ \sigma:\mathcal A\longrightarrow V((t))[[x_1,\ldots,x_{m-1}]] \] defined by ordinary substitution $z_i=t+x_i$, $z_m=t$ on power-series numerators, and by \[ z_i^{-1}\longmapsto (t+x_i)^{-1}=\sum_{j\ge0}(-1)^j t^{-j-1}x_i^j \quad(i0$, since $Y(\ket{0},z)u=u$. Skew symmetry $Y(u,z)a=e^{zT}Y(a,-z)u$ gives \[ u_{h+k-1}a=(-1)^{h+k}a_{h+k-1}u. \] The terms involving positive powers of $T$ have zero vacuum component. 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