\documentclass[11pt]{article}
\usepackage[margin=1in]{geometry}
\usepackage[T1]{fontenc}
\usepackage{lmodern,amsmath,amssymb,amsthm,mathtools,booktabs,array}
\usepackage[hidelinks]{hyperref}
\usepackage{microtype,needspace}
\setlength{\emergencystretch}{2em}
\hypersetup{pdftitle={C2-cofiniteness of logarithmic W-algebras},pdfauthor={Andrew Riesen}}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{corollary}[theorem]{Corollary}
\theoremstyle{remark}\newtheorem{remark}[theorem]{Remark}
\newcommand{\C}{\mathbb C}\newcommand{\Z}{\mathbb Z}
\newcommand{\g}{\mathfrak g}\newcommand{\h}{\mathfrak h}
\newcommand{\n}{\mathfrak n}\newcommand{\V}{\mathcal V}
\newcommand{\W}{\mathcal W}\newcommand{\Res}{\operatorname{Res}}
\newcommand{\Span}{\operatorname{span}}\newcommand{\wt}{\operatorname{wt}}
\newcommand{\Ad}{\operatorname{Ad}}\newcommand{\gr}{\operatorname{gr}}
\newcommand{\DS}{\mathrm{DS}}\newcommand{\ch}{\operatorname{ch}}
\newcommand{\im}{\operatorname{im}}\newcommand{\id}{\operatorname{id}}
\numberwithin{equation}{section}

\title{$C_2$-cofiniteness of logarithmic $W$-algebras}
\author{Andrew Riesen}
\date{14 October 2026}
\begin{document}
\maketitle
\begin{abstract}
Let $\mathfrak{g}$ be a finite-dimensional simple complex Lie algebra with root lattice $Q$ and lacing number $t$, equipped with an invariant bilinear form normalized so that long roots have squared length two. We prove $C_2$-cofiniteness of the
Feigin--Tipunin logarithmic $W$-algebra $\W_p(Q)$ for
$p=tm\geq2$ and $m\geq h^\vee_{{}^L\g}-1$, where
$h^\vee_{{}^L\g}$ is the dual Coxeter number of the Langlands dual.
Drinfeld--Sokolov reduction of induced affine Weyl modules gives
graded generators for the multiplicity spaces of the conformal group
action. Their degree bounds show that the intrinsic $C_2$-algebra
is a finite module over the polynomial $C_2$-algebra of its principal
invariant subalgebra. Adjoint contractions produce a finite relation
ideal in this polynomial ring. Heisenberg normal ordering and a
constant-coefficient differential equation show that its common zero
is the origin. The argument gives explicit sufficient bounds and
strong finite generation in the stated parameter range.
\end{abstract}

\section{Introduction and statement of the result}
The triplet vertex algebra $\W(p)$ is a fundamental example of a nonrational, $C_2$-cofinite vertex operator algebra \cite{AM08}. Its higher-rank generalizations were proposed by Feigin and Tipunin through lattice-screening constructions \cite{FT}, and subsequently studied by Sugimoto, who established their geometric construction and simplicity properties \cite{S21,S23}. More recently, the shift-system framework of Sugimoto and Li has provided a uniform lattice construction for all finite simple Lie types, together with an identification of the principal invariant algebra and its multiplicity modules \cite{SL}.

These structural results describe the algebra as a sum of group
representations tensored with principal $W$-algebra modules.
To prove $C_2$-cofiniteness, one must control the intrinsic quotient
of the whole screening kernel. The invariant subalgebra has a
polynomial $C_2$-algebra, so both the number of surviving group types
and the polynomial action must be made finite. We prove these two
finiteness statements and combine them.

Let $\Phi$ be the root system of $\g$, with positive roots $\Phi_+$,
simple roots $\alpha_1,\ldots,\alpha_r$, root lattice $Q$, weight
lattice $P$, dominant cone $P_+$ and Weyl group $W$. Let $G$ be
the adjoint algebraic group and $\h$ the chosen Cartan subalgebra.
Write $Q_+=\sum_i\Z_{\geq0}\alpha_i$.
The invariant form identifies the Cartan space with its dual.
Use the positive-definite real root form,
extended complex bilinearly, normalized by
\[
 (\alpha,\alpha)=2\quad(\alpha\text{ long}),\qquad
 (\alpha,\alpha)=2/t\quad(\alpha\text{ short}),
\]
where $t\in\{1,2,3\}$ is the lacing number. Define
\[
 \alpha^\vee=\frac{2\alpha}{(\alpha,\alpha)},\qquad
 \rho=\frac12\sum_{\alpha>0}\alpha,\qquad
 \rho^\vee=\frac12\sum_{\alpha>0}\alpha^\vee.
\]
Thus $(\rho^\vee,\alpha_i)=1$ and
$(\rho,\alpha_i)=(\alpha_i,\alpha_i)/2$. Write $h,h^\vee$
for the Coxeter and dual Coxeter numbers of $\g$, and
$h_L^\vee=h^\vee_{{}^L\g}$ for the dual Coxeter number of its
Langlands dual.

Let $p=tm$, $m\in\Z_{>0}$. The lattice $\sqrt pQ$ is even and integral:
its simple-root Gram matrix has integral entries and even diagonal
entries. The divisibility is necessary for this ordinary lattice
construction: a scaled root of minimal length has squared length
$2p/t$, which must be an even integer.
Give its lattice vertex algebra the conformal vector
\begin{equation}\label{eq:background}
 a_p=\sqrt p\,\rho-\frac{\rho^\vee}{\sqrt p},\qquad
 \omega=\frac12\sum_j b^j(-1)b_j(-1)\mathbf1+a_p(-2)\mathbf1,
\end{equation}
where $(b^j),(b_j)$ are dual Cartan bases. The central charge and
the weight of a pure exponential are
\begin{equation}\label{eq:weight}
 c=r-12(a_p,a_p),\qquad
 \Delta(\beta)=\frac12(\beta,\beta)-(a_p,\beta).
\end{equation}
With $Y(u,z)=\sum_{j\in\Z}u_{(j)}z^{-j-1}$, define
\begin{equation}\label{eq:kernel}
 \V=\W_p(Q)=\bigcap_{i=1}^r
 \ker\Res_zY(e^{-\alpha_i^\vee/\sqrt p},z)
 \ \subset V_{\sqrt pQ}.
\end{equation}
These short screenings map to charged lattice modules. Their kernels
on the vacuum lattice space form the vertex algebra considered here.
This is $W_{\sqrt pQ}$ in \cite[Section 2.4.1]{SL}; in simply-laced
type it is also denoted $W(p)_Q$ in \cite{S21}.
Both $e^{\sqrt p\alpha_i}$ and $e^{-\alpha_i^\vee/\sqrt p}$
have conformal weight one under \eqref{eq:background}.
Write $T=L_{-1}=\omega_{(0)}$ for the translation operator.

For a vertex algebra $B$ and a $B$-module $M$, set
\begin{equation}\label{eq:C2}
 \begin{aligned}
 C_2(B)&=\Span\{u_{(-2)}v:u,v\in B\},\qquad R_B=B/C_2(B),\\
 C_2^B(M)&=\Span\{u_{(-2)}v:u\in B,v\in M\}.
 \end{aligned}
\end{equation}
The algebra $R_B$ is the commutative $C_2$ Poisson algebra of $B$.
Write $R_B(M)=M/C_2^B(M)$. Its $R_B$-action, like the product in
$R_B$, is induced by the $(-1)$ product.

\begin{theorem}\label{thm:main}
For every finite-dimensional simple complex Lie algebra and every
parameter
\begin{equation}\label{eq:range}
 p=tm\geq2,\qquad m\in\Z_{>0},\qquad m\geq h_L^\vee-1,
\end{equation}
the logarithmic $W$-algebra $\W_p(Q)$ is $C_2$-cofinite.
More precisely, for $A=\V^G$, $R_\V$ is a finite $R_A$-module.
The contraction ideal $I_{\Phi,p}\subset R_A$ defined in
Section~\ref{sec:ideal} has finite codimension and annihilates
$R_\V$. Strong generators can be chosen of weights at most
$\max\{h,pR_\Phi^2+(p+1)R_\Phi\}$, where the sufficient cutoff
$R_\Phi$ is defined in \eqref{eq:cutoff}.
\end{theorem}

The parameter range is
\begin{center}
\begin{tabular}{ccc}
\toprule
Type & Lacing number $t$ & Allowed $p$\\
\midrule
$A_1$ & 1 & $p\geq2$\\
$A_n$ ($n\geq2$) & 1 & $p\geq n$\\
$D_n$ ($n\geq4$) & 1 & $p\geq2n-3$\\
$E_6$ & 1 & $p\geq11$\\
$E_7$ & 1 & $p\geq17$\\
$E_8$ & 1 & $p\geq29$\\
$B_n$ ($n\geq2$) & 2 & even $p\geq2n$\\
$C_n$ ($n\geq2$) & 2 & even $p\geq4n-4$\\
$F_4$ & 2 & even $p\geq16$\\
$G_2$ & 3 & $3\mid p$, $p\geq9$\\
\bottomrule
\end{tabular}
\end{center}
The lower bound in \eqref{eq:range} is the vacuum strong condition
in the structural theorem of \cite{SL}. The theorem concerns the
ordinary lattice family \eqref{eq:kernel}. Smaller admissible parameters,
and the odd-parameter type-$B$ construction with a fermion factor,
are outside its statement.

The published module identification reduces each multiplicity space
to the principal reduction of an induced affine Weyl module.
Classical reduction with its finite-dimensional coefficient representation
determines the number and degrees of generators in the intrinsic
module $C_2$ quotient. The linear spread of these degrees is smaller
than the quadratic growth of lattice-exponential products in sufficiently
large group types, which then vanish in $R_\V$. Invariant contractions
of the adjoint component give relations in $R_A$. Their Cartan symbols
form exponential polynomials with nonzero constant terms at each root
exponent. A finite set of coefficients and a differential equation
force their common zero to be the origin, making the polynomial action
finite dimensional.

\subsection*{AI contribution}
OpenAI's ChatGPT was used extensively to develop proof strategies and
mathematical arguments, draft and revise the text, assist with literature
searches, and write code for exact symbolic computations. The author
directed this work and revised the manuscript.
\section{The group decomposition and the principal invariant algebra}\label{sec:structure}
Use the triangular decomposition
$\g=\n_-\oplus\h\oplus\n_+$ associated with $\Phi_+$.
Let $N\subset G$ be the positive maximal unipotent subgroup and
$\n=\operatorname{Lie}N=\n_+$. Fix a principal
$\mathfrak{sl}_2$-triple $(e,2\rho^\vee,f)$ with $f\in\n_-$.
The principal Kostant slice is $\mathcal S=f+\g^e$,
where $\g^e=\{y\in\g:[e,y]=0\}$.
We identify $\g$ with $\g^*$ by the invariant form; $\n^\perp$
is the annihilator of $\n$.

Distinguish the shifted level $\kappa$ from the affine level $k$.
The notation $\W^\ell(\g)$ denotes the universal principal algebra
at affine level $\ell$, and $\W_\ell(\g)$ its simple graded quotient.
For the algebra used in the reduction argument,
\begin{equation}\label{eq:levels}
 k=\frac1p-h^\vee,\quad \kappa=k+h^\vee=\frac1p,
 \qquad \check k=m-h^\vee_L,\quad
 \check\kappa=\check k+h^\vee_L=m.
\end{equation}
Thus $t\kappa\check\kappa=1$, the Feigin--Frenkel duality relation.
We perform untwisted principal reduction on $\widehat\g$ at $k$.

\begin{lemma}\label{lem:structure}
Under \eqref{eq:range}, $\V$ is simple, self-contragredient and of CFT
type, with a conformal algebraic $G$-action. It has a decomposition
\begin{equation}\label{eq:decomp}
 \V=\bigoplus_{\lambda\in P_+\cap Q}L(\lambda)\otimes M_\lambda,
 \qquad A=\V^G=\bigcap_i\ker(e^{\sqrt p\alpha_i})_{(0)}|_{\mathcal H},
\end{equation}
Here $L(\lambda)$ is the irreducible finite-dimensional $\g$-module
of highest weight $\lambda$, and $\mathcal H$ is the neutral Heisenberg algebra. The multiplicity
space $M_\lambda$ is a simple $A$-module, with a one-dimensional top
generated by $e^{-\sqrt p\lambda}$ in its highest-weight realization.
Its lowest conformal degree is
\begin{equation}\label{eq:top}
 H(\lambda)=(\rho^\vee,\lambda),\quad
 J(\lambda)=(\rho,\lambda),\quad
 h_\lambda=\frac p2(\lambda,\lambda)+pJ(\lambda)-H(\lambda).
\end{equation}
Through its specified Miura inclusion $A\subset\mathcal H$, one has
\begin{equation}\label{eq:actual-algebra}
 A\simeq\W^k(\g)\simeq\W_k(\g).
\end{equation}
Under this identification,
\begin{equation}\label{eq:moduleid}
 M_\lambda\simeq T^{1/p}_{\lambda,0}
 =H^0_{\DS,+}\bigl(\widehat V^k(L(\lambda))\bigr)
\end{equation}
as $A$-modules with their conformal grading.

\end{lemma}
\begin{proof}
The vacuum specialization of \cite[Theorem 3.2, Corollary 3.4,
equations (4.3)--(4.6), Theorem 4.3]{SL} gives the group action,
simplicity, the Miura identification \eqref{eq:actual-algebra}, and
\eqref{eq:moduleid}. Self-contragredience is the zero-coset case of
its Theorem 3.2(2). Let $\theta_s$ be the highest short root, taking
$\theta_s=\theta$ in simply-laced type. In the coroot-space convention
of that source, ${}^L\theta=t\theta_s$ and
$(\rho^\vee,\theta_s)=h_L^\vee-1$. Hence its vacuum strong condition is
\[
 (\rho^\vee,{}^L\theta)\leq p
 \quad\Longleftrightarrow\quad
 t(h_L^\vee-1)\leq tm.
\]
Thus the endpoint in \eqref{eq:range} is included. Its Lemma 4.1
identifies the one-dimensional top.

The algebra and module identifications at the rational shifted level
$\kappa=1/p$ are those of \cite[Theorem 4.3]{SL}. In its notation,
the coset parameter is zero and the dominant index is our $\lambda$,
so the module is $T^{1/p}_{\lambda,0}$. The algebra identification is
through its Miura inclusion in equation (4.3). Thus
\eqref{eq:moduleid} is a graded isomorphism for the specified
$A\subset\mathcal H$ action and transports the intrinsic subspace
$C_2^A$ itself. The convention for $T^{\kappa}_{\lambda,0}$ is that of
\cite[Sections 2.1--2.2]{AF}; the rational-level identification used
here is supplied by \cite{SL}.

The centre of the simply connected group acts on $L(\lambda)$ by the
class of $\lambda$ in $P/Q$; all indices here belong to $Q$. Hence the
action factors through the adjoint group. Formula \eqref{eq:weight}
at $\beta=-\sqrt p\lambda$ gives \eqref{eq:top}.
Dominant weights have nonnegative simple-root coordinates, because the
inverse finite Cartan matrix has positive entries. On a simple root,
\[
 (p\rho-\rho^\vee,\alpha_i)
     =\frac p2(\alpha_i,\alpha_i)-1>0
\]
in our parameter range. Consequently $h_\lambda>0$ for nonzero
$\lambda$. Relative Fock degrees are nonnegative integers, and the
vacuum multiplicity space has a one-dimensional degree-zero part.
This proves CFT type. All homogeneous pieces are finite dimensional.
\end{proof}

For later use, if $\mu-\lambda=\delta\in Q_+\setminus\{0\}$ and both
weights are dominant, then
\begin{equation}\label{eq:monotone}
 h_\mu-h_\lambda
 =\frac p2(\delta,\mu+\lambda)+(p\rho-\rho^\vee,\delta)>0.
\end{equation}
The first term is nonnegative by dominance; the second is strictly
positive. The short screening on $e^{-\sqrt p\lambda}$ starts with
$z^{(\alpha_i^\vee,\lambda)}$, so has zero residue. Its group Cartan
weight is $\lambda$, since the group Cartan operators on a lattice
charge $\beta$ are $-\alpha_i^\vee(0)/\sqrt p$. If $L(\mu)$ contains
that weight, then $\mu-\lambda\in Q_+$. Formula \eqref{eq:monotone}
therefore shows that this exponential is in precisely the
$\lambda$-isotypic component at its lowest degree. This component
contains a single copy of $L(\lambda)$ in degree $h_\lambda$.

\begin{lemma}\label{lem:invariants}
Heisenberg symbol projection induces an injection
\begin{equation}\label{eq:RA}
 R_A\hookrightarrow R_{\mathcal H}=\C[\h^*],\qquad
 \im R_A=\C[\h^*]^W=\C[u_1,\ldots,u_r].
\end{equation}
The degrees of $u_i$ are the basic invariant degrees. Every point of
$\operatorname{Spec}R_A$ lifts to a Cartan point.
\end{lemma}
\begin{proof}
The principal PBW and Miura symbol statements
\cite[Section 5, equations (23)--(25); proof of Lemma 6.1]{ACL}
apply to the universal algebra $\W^k(\g)$ at any noncritical level.
Here $A$ is this algebra by \eqref{eq:actual-algebra}, and
$k+h^\vee=1/p\neq0$. At Li degree zero the cited symbol theorem
gives the Chevalley restriction map
$\C[\mathcal S]\simeq\C[\h^*]^W\subset\C[\h^*]$, for the principal
Kostant slice $\mathcal S$. Under the physical lattice currents,
the level-$1/p$ Miura current is $b'_i=-\beta_i/p$,
$\beta_i=\sqrt p\alpha_i$, with Gram matrix
$(b'_i,b'_j)=(\alpha_i,\alpha_j)/p$. Its screening
$\exp(-\kappa^{-1}\int b'_i)$ is exactly $e^{\sqrt p\alpha_i}$.
This is the embedding in \cite[equation (4.3)]{SL}. Scaling the Cartan
variables by a common nonzero scalar preserves their invariant ring.
This proves \eqref{eq:RA} for the specified inclusion $A\subset\mathcal H$.
In particular the symbol map has zero kernel: if an element of $A$
has zero Heisenberg symbol, its class in $R_A$ is zero. This is the
degree-zero assertion of the cited Miura symbol theorem.

The Cartan coordinate ring is integral over its invariants: each
polynomial $a$ satisfies the monic invariant-coefficient equation
$\prod_{w\in W}(X-wa)=0$. Lying over, over $\C$, gives the final assertion.
\end{proof}

\section{Reduced Weyl modules and their generator degrees}\label{sec:modules}
For $E=L(\lambda)$ let
\[
 \widehat V^k(E)=U(\widehat\g)\otimes_{U(\g[t]\oplus\C K)}E,
 \qquad t\g[t]E=0,\quad K|_E=k,\quad
 N_\lambda=H^0_{\DS,+}(\widehat V^k(E)).
\]
Our convention is Arakawa's $H_f^{\infty/2+i}$ for a principal
nilpotent $f\in\n_-$ and character $\Psi(xt^j)=\delta_{j,-1}(f,x)$.
It is the zero-twist instance of \cite[Section 2.1]{AF}.
The principal good-grading element is $\rho^\vee$.

\begin{proposition}\label{prop:modulebound}
The intrinsic module quotient $R_A(N_\lambda)$ is a free $R_A$-module
of rank $\dim L(\lambda)$. A vector of weight $\nu$ in $L(\lambda)$
contributes a generator in degree
\begin{equation}\label{eq:generator}
 \Delta^{\mathrm{aff}}_\lambda-(\rho^\vee,\nu),\qquad
 \Delta^{\mathrm{aff}}_\lambda
 =\frac{(\lambda,\lambda+2\rho)}{2\kappa}
 =\frac p2(\lambda,\lambda+2\rho).
\end{equation}
Thus $R_A(M_\lambda)$ has at most $\dim L(\lambda)$ homogeneous
generators, all in $[h_\lambda,h_\lambda+2H(\lambda)]$.
Their homogeneous lifts strongly span $M_\lambda$ using negative $A$-modes.
\end{proposition}
\begin{proof}
Let $F^\bullet$ denote the intrinsic module Li filtration. For $E=L(\lambda)$, the affine PBW description is
\[
 \widehat V^k(E)=U(t^{-1}\g[t^{-1}])\otimes E.
\]
Giving $xt^{-j}$ degree $j$ makes its graded pieces finite-dimensional algebraic $\g$-modules. The affine Sugawara energies are $\Delta^{\mathrm{aff}}_\lambda+\Z_{\geq0}$, with finite-dimensional eigenspaces, and the $\g$-action is locally finite. The module is generated by $E$ and has a Weyl flag of length one. Thus it is a finitely generated object of $KL_k$ in the sense of \cite[Section~2.6]{Ara}, at the level in \eqref{eq:levels}.

The intrinsic affine Li-degree-zero quotient is
\begin{equation}\label{eq:affine-module-quotient}
 \widehat V^k(E)/C_2^{V^k(\g)}(\widehat V^k(E))
 \simeq \C[\g^*]\otimes E,
\end{equation}
with polynomial multiplication on the first factor and diagonal $\g$-action. Theorem~4.5.9(ii) and Corollary~4.5.10(ii) of \cite{Ara} identify the filtration induced by reduction with the intrinsic module filtration and give
\begin{equation}\label{eq:classical}
 R_A(N_\lambda)
 \simeq \bigl(\C[f+\n^\perp]\otimes E\bigr)^N.
\end{equation}
Here the principal grading is integral, so $\n=\n_+$ and there is no neutral-ghost factor. The moment constraints act on the coordinate factor in \eqref{eq:affine-module-quotient}. This identifies the intrinsic module quotient as an $R_A$-module.

For clarity, $F^1N_\lambda=C_2^A(N_\lambda)$. The intrinsic module Li filtration has this property by its definition; equivalently, modes of index at most $-2$ are translated $(-2)$-products and their span is stable under the remaining negative modes. In the present convention,
\[
 a_{(-j)}v=\frac{(T^{j-2}a)_{(-2)}v}{(j-1)!}\qquad(j\geq2).
\]

The principal Kostant isomorphism is
\[
 N\times \mathcal S\longrightarrow f+\n^\perp,
 \qquad (n,s)\longmapsto\Ad(n)s,
 \qquad \mathcal S=f+\g^e.
\]
View $\C[f+\n^\perp]\otimes E$ as polynomial $E$-valued functions. An invariant for the diagonal action satisfies
\[
 F(\Ad(n)x)=nF(x).
\]
Consequently restriction to $\mathcal S$ is an isomorphism
\begin{equation}\label{eq:kostant-module}
 \bigl(\C[f+\n^\perp]\otimes E\bigr)^N
 \longrightarrow \C[\mathcal S]\otimes E.
\end{equation}
Its inverse sends $q\otimes v$ to
\[
 F_{q,v}(\Ad(n)s)=q(s)nv.
\]
This is polynomial because the Kostant map is an isomorphism and the $N$-representation on $E$ is algebraic. It is $\C[\mathcal S]$-linear. Since $R_A=\C[\mathcal S]$, equations \eqref{eq:classical}--\eqref{eq:kostant-module} prove freeness of rank $\dim E$.

We next compute the generator degrees for use in Section~\ref{sec:finite}. Let
\[
 \Delta_\lambda^{\mathrm{aff}}
 =\frac{(\lambda,\lambda+2\rho)}{2(k+h^\vee)}
 =\frac p2(\lambda,\lambda+2\rho).
\]
The reduced Hamiltonian is the affine Hamiltonian minus $\rho^\vee(0)$, with the charged-ghost contribution. On the function realization its action is
\begin{equation}\label{eq:reduced-grading}
 (\mathcal G_zF)(x)
 =z^{\Delta_\lambda^{\mathrm{aff}}}z^{-\rho^\vee}
 F\bigl(z\Ad(z^{\rho^\vee})x\bigr).
\end{equation}
The coordinate action fixes the principal $f$ and preserves $\mathcal S$. Under the Kostant map it sends
\[
 (n,s)\longmapsto
 \left(z^{\rho^\vee}nz^{-\rho^\vee},\,
 z\Ad(z^{\rho^\vee})s\right).
\]
For $v\in E_\nu$, put $F_v=F_{1,v}$. Substitution into \eqref{eq:reduced-grading} gives
\begin{align*}
 (\mathcal G_zF_v)(\Ad(n)s)
 &=z^{\Delta_\lambda^{\mathrm{aff}}}z^{-\rho^\vee}
   (z^{\rho^\vee}nz^{-\rho^\vee})v\\
 &=z^{\Delta_\lambda^{\mathrm{aff}}-(\rho^\vee,\nu)}nv.
\end{align*}
Thus the generator associated to $v$ has degree
\begin{equation}\label{eq:generator-calculated}
 \Delta_\lambda^{\mathrm{aff}}-(\rho^\vee,\nu).
\end{equation}
For weights of $E$, the extreme values of $(\rho^\vee,\nu)$ are $H(\lambda)$ and $-H(\lambda)$, because $\rho^\vee$ is dominant and $w_0\rho^\vee=-\rho^\vee$. The lower endpoint is
\[
 \Delta_\lambda^{\mathrm{aff}}-H(\lambda)
 =\frac p2(\lambda,\lambda)+p(\rho,\lambda)-(\rho^\vee,\lambda)
 =h_\lambda.
\]
All generators therefore have degrees in $[h_\lambda,h_\lambda+2H(\lambda)]$. Transport them by the graded isomorphism \eqref{eq:moduleid}.

Finally choose homogeneous lifts of these module generators and of the generators of $R_A$. Modulo $C_2^A(M_\lambda)$ every vector is a polynomial in the latter acting on the former. Lift this expression using $(-1)$-products. The remainder is a sum of $a_{(-2)}v$. Terms with $a=\mathbf1$ vanish, and every other term has inner vector of strictly smaller conformal degree. Induction on the nonnegative integral relative degree strongly spans $M_\lambda$ by the chosen lifts.
\end{proof}

\section{Finitely many surviving group types}\label{sec:finite}
We write $V_\lambda=L(\lambda)\otimes M_\lambda$ and $P_\lambda$
for its isotypic projector. Each $P_\lambda$ commutes with all $A$-modes
and preserves $C_2(\V)$: the group action commutes with invariant fields,
and $C_2(\V)$ is group-stable on each finite semisimple homogeneous piece.

\begin{lemma}\label{lem:coefficient}
Let $U,W\subset\V$ be lower-bounded graded $A$-modules strongly spanned
by homogeneous finite sets $S_U,S_W$ using negative $A$-modes.
Let $\Pi\in\operatorname{End}(\V)$ commute with $A$-modes.
Every coefficient $\Pi(u_{(j)}v)$ is a
linear combination of negative $A$-modes applied to coefficients
$\Pi(s_{(l)}t)$, $s\in S_U,t\in S_W$. If $\Pi$ preserves $C_2(\V)$
and these generator coefficients vanish for $l\geq-1$, then all the
projected coefficients belong to $C_2(\V)$.
\end{lemma}
\begin{proof}
Set $I(u)_{(j)}v=\Pi(u_{(j)}v)$. For $a\in A_{>0}$ and $k\geq1$,
the iterate and commutator formulas give
\begin{align*}
 I(a_{(-k)}u)_{(j)}v
 &=\sum_{i\geq0}(-1)^i\binom{-k}{i}
   \bigl(a_{(-k-i)}I(u)_{(j+i)}v
          -(-1)^k I(u)_{(j-k-i)}a_{(i)}v\bigr),\\
 I(u)_{(j)}a_{(-k)}v
 &=a_{(-k)}I(u)_{(j)}v
       -\sum_{i\geq0}\binom{-k}{i}I(a_{(i)}u)_{(j-k-i)}v.
\end{align*}
The sums truncate on fixed inputs. Induct, for all $j$ together,
on the sum of the two relative input weights. Removing the outer
negative mode reduces this sum by $\wt(a)+k-1>0$. In an iterate
correction the pair $(a_{(-k)}u,v)$ is replaced by $(u,a_{(i)}v)$;
the total input weight decreases by $k+i>0$. The commutator
correction has the same decrease with $(a_{(i)}u,v)$. All external
output modes are negative.
This proves the first assertion. Modes of index at most $-2$ are
translated $(-2)$ products, hence lie in $C_2(\V)$ after projection.
For $j\geq2$, $b_{(-j)}v=(T^{j-2}b)_{(-2)}v/(j-1)!$.
Stability under $(-1)$ products follows from
\[
 a_{(-1)}b_{(-2)}v=b_{(-2)}a_{(-1)}v+
 \sum_{i\geq0}(-1)^i(a_{(i)}b)_{(-3-i)}v.
\]
All correction terms are again translated $(-2)$ products.
Thus that space is stable under every negative $A$-mode.
Together with the assumed vanishing at indices at least $-1$, this
proves the second assertion.
\end{proof}

\begin{proposition}\label{prop:eliminate}
If $\lambda=\mu+\nu$ for dominant root-lattice weights and
\begin{equation}\label{eq:gap}
 p(\mu,\nu)>2H(\mu)+2H(\nu),
\end{equation}
then $V_\lambda\subset C_2(\V)$.
\end{proposition}
\begin{proof}
By Proposition~\ref{prop:modulebound}, generator-weight sums for
$V_\mu,V_\nu$ are at most
$h_\mu+h_\nu+2H(\mu)+2H(\nu)<h_\lambda$, since
$h_{\mu+\nu}-h_\mu-h_\nu=p(\mu,\nu)$. Their projected products
of index at least $-1$ vanish by this degree bound.
Lemma~\ref{lem:coefficient} puts every projected coefficient in
$C_2(\V)$. Let $\mathcal C_\lambda$ be the span of all these projected
coefficients. It is $G$-stable by equivariance. For $a\in A$ and any
$q\in\Z$, the commutator identity gives
\[
 a_{(q)}P_\lambda(u_{(j)}v)
 =P_\lambda(u_{(j)}a_{(q)}v)
  +\sum_{i\geq0}\binom qi
       P_\lambda((a_{(i)}u)_{(q+j-i)}v).
\]
Since $V_\mu$ and $V_\nu$ are $A$-modules, every term belongs to
$\mathcal C_\lambda$. Hence this coefficient span is stable under
all $A$-modes. It contains
\[
 (e^{-\sqrt p\mu})_{(-1-p(\mu,\nu))}e^{-\sqrt p\nu}
     =\epsilon(-\sqrt p\mu,-\sqrt p\nu)e^{-\sqrt p\lambda}.
\]
The cocycle value is nonzero. The exponential is its own isotypic
projection by \eqref{eq:monotone}; its $G$-orbit spans
$L(\lambda)\otimes\C v_\lambda$ in the top degree, where
$v_\lambda$ is the multiplicity highest vector. The $A$-submodule
generated by $v_\lambda$ is all of the simple module $M_\lambda$.
Thus $\mathcal C_\lambda=V_\lambda$, proving the assertion.
\end{proof}

Here is an explicit uniform cutoff for each type. Use the Cartan
matrix convention $C_{ij}=(\alpha_i^\vee,\alpha_j)$, and fundamental
weights $\varpi_i$. Let
\begin{align}\label{eq:cutoff}
 \tau&=\det C,\quad G_{ij}=(\varpi_i,\varpi_j),\quad
 H_i=(\rho^\vee,\varpi_i),\quad
 c_0=\min_{i,j}\frac{\tau G_{ij}}{H_j},\nonumber\\
 p_*&=\max\{2,t(h_L^\vee-1)\},\quad
 n_0=\left\lceil\frac4{p_*c_0}\right\rceil,\quad L=n_0\tau,\quad
 B=L\max_iH_i,\nonumber\\
 R_\Phi&=\left\lceil\max\{2B,(L-1)\textstyle\sum_i H_i\}\right\rceil.
\end{align}
Write $D_{ii}=(\alpha_i,\alpha_i)/2$.
The simple-root Gram matrix is $DC$, and the fundamental weights
are the columns of $C^{-1}$ in the simple-root basis. Hence
\[
 G=(C^{-1})^{\mathsf T}D,\qquad H_i=\sum_j(C^{-1})_{ji}.
\]
These entries are positive. Write $C=2I-\mathcal A$.
The matrix $\mathcal A$ has nonnegative entries and its connected
graph is the Dynkin graph. Since $C$ is symmetrizable positive definite,
its eigenvalues are positive. Perron--Frobenius therefore gives
$\operatorname{spr}(\mathcal A)<2$, and
$C^{-1}=\frac12\sum_{j\geq0}(\mathcal A/2)^j$ has positive entries.
The rank-one case is immediate. The adjugate formula gives $\tau\varpi_i\in Q$.

\begin{lemma}\label{lem:cutoff}
For $H(\lambda)>R_\Phi$, $\lambda\in P_+\cap Q$, the decomposition
in Proposition~\ref{prop:eliminate} exists for every $p$ in
\eqref{eq:range}.
\end{lemma}
\begin{proof}
Write $\lambda=\sum_i a_i\varpi_i$, $a_i\in\Z_{\geq0}$.
If all $a_i\leq L-1$, then
$H(\lambda)\leq(L-1)\sum_iH_i$, a contradiction. Choose $a_i\geq L$
and put $\mu=L\varpi_i$, $\nu=\lambda-\mu$. Both are dominant and
belong to $Q$. We have $H(\mu)\leq B$ and
$H(\nu)>H(\mu)$. For every $j$,
\[
 p(\mu,\varpi_j)=pn_0\tau G_{ij}\geq p_*n_0c_0H_j\geq4H_j.
\]
Thus $p(\mu,\nu)\geq4H(\nu)>2H(\mu)+2H(\nu)$.
\end{proof}

It follows that $R_\V$ is a finite $R_A$-module, generated by at most
\begin{equation}\label{eq:K}
 K_\Phi=\sum_{\substack{\lambda\in P_+\cap Q\\H(\lambda)\leq R_\Phi}}
                   (\dim L(\lambda))^2
\end{equation}
elements. There are finitely many indices because all $H_i>0$.
Explicitly, the inclusions $C_2^A(M_\lambda)\subset C_2(\V)$ and
the eliminated components give the surjection
\begin{equation}\label{eq:finite-surjection}
 \bigoplus_{H(\lambda)\leq R_\Phi}L(\lambda)\otimes R_A(M_\lambda)
       \twoheadrightarrow R_\V.
\end{equation}

\section{Invariant adjoint contractions}\label{sec:contractions}
Let $\theta$ be the highest long root. Since
$(\rho,\theta)=h^\vee-1$ and
$(\rho^\vee,\theta)=h-1$, its extremal exponential has weight
\begin{equation}\label{eq:d}
 E=e^{-\sqrt p\theta},\qquad d=h_\theta=ph^\vee-h+1.
\end{equation}
Its group orbit spans $\mathcal E\simeq\g$ in $\V_d$.

\begin{lemma}\label{lem:adjoint-form}
The subspace $\mathcal E$ is the unique adjoint copy in $\V_d$ and
consists of Virasoro primary vectors. The normalized invariant bilinear
form of $\V$ restricts nondegenerately to $\mathcal E$ and pairs
opposite lattice charges. The nonzero charge spaces of $\mathcal E$
are one-dimensional and indexed by $\sqrt p\Phi$; its neutral space
has dimension $r$.
\end{lemma}
\begin{proof}
The decomposition in Lemma~\ref{lem:structure} and the one-dimensional
top of $M_\theta$ give the multiplicity assertion. For $n>0$,
\[
 L_n=\frac12\sum_{j,k}:b^j(k)b_j(n-k):-(n+1)a_p(n).
\]
Every summand annihilates the Fock highest vector $E$. Thus $E$ is
primary. The conformal $G$-action makes every vector of $\mathcal E$
primary as well.

Let $\langle\, ,\,\rangle$ be the invariant bilinear form normalized
by $\langle\mathbf1,\mathbf1\rangle=1$. Simplicity and
self-contragredience make it nondegenerate. Pullback by a conformal
automorphism gives another normalized invariant form, which agrees
with it by uniqueness. The form is therefore $G$-invariant and pairs
only equal conformal degrees. In the finite semisimple $G$-module
$\V_d$, an adjoint copy can pair only with an adjoint copy. Since
there is exactly one such copy, the form restricts nondegenerately
to $\mathcal E$.

Denote the group Cartan operators by $h_i$. They act on a vector of charge $\beta$ by
$-(\alpha_i^\vee,\beta)/\sqrt p$. Infinitesimal $G$-invariance gives
\[
 \langle h_i u,v\rangle+\langle u,h_i v\rangle=0.
\]
If vectors of charges $\beta,\gamma$ pair nontrivially, this identity
implies $(\alpha_i^\vee,\beta+\gamma)=0$ for every $i$, hence
$\gamma=-\beta$. The weights of the adjoint representation are the
roots with multiplicity one and zero with multiplicity $r$.
\end{proof}

Choose a basis $(E_a)$ of $\mathcal E$ with each $E_a$ in a single
lattice charge space, and
put $\mathcal B_{ab}=\langle E_a,E_b\rangle$ and let
$\mathcal B^{ab}$ be the entries of its inverse. Set
\begin{equation}\label{eq:contraction}
 \mathcal Q(z)=\sum_{a,b}\mathcal B^{ab}Y(E_a,z)E_b,\quad
 \mathcal Q_s=\sum_{a,b}\mathcal B^{ab}(E_a)_{(-s)}E_b,\quad f_s=[\mathcal Q_s]_{R_A}.
\end{equation}
Invariance of the tensor implies $\mathcal Q_s\in A$. Its degree is $2d+s-1$.
For $s\geq2$, it is in $C_2(\V)$. Let $\varphi:R_A\to R_\V$
be the algebra map induced by inclusion. Then $\varphi(f_s)=0$, so
$f_s\cdot[v]=\varphi(f_s)[v]=0$ for every $[v]\in R_\V$.

Let $\pi_x$ denote Heisenberg symbol projection followed by evaluation
at the Cartan point $x$: $a(-1)$ evaluates to $a(x)$, and
$a(-n)$ evaluates to zero for $n\geq2$. We identify the root space with its dual by
the normalized form and write $\alpha(x)$ for this evaluation.

\begin{lemma}\label{lem:exppoly}
For $\xi_\alpha=\sqrt p\,\alpha(x)$, the normalized contraction
$H_x(z)\in\C[[z]]$ has the form
\begin{equation}\label{eq:exppoly}
 H_x(z)=z^{2d}\pi_x\mathcal Q(z)
     =P_0(z)+\sum_{\alpha\in\Phi}e^{\xi_\alpha z}P_\alpha(z),
\end{equation}
where
\begin{equation}\label{eq:bounds}
 \deg P_0\leq2d,\qquad
 \deg P_\alpha\leq2d-p(\alpha,\alpha),\qquad
 P_\alpha(0)=(-1)^d.
\end{equation}
Moreover $P_0(0)=r(-1)^d$. The degree bound retains the squared
length of each root.
\end{lemma}
\begin{proof}
By Lemma~\ref{lem:adjoint-form}, every nonzero term has opposite input charges. Consider monomials
\[
 u=\prod_{i=1}^{a}a_i(-m_i)e^\beta,
 \qquad
 v=\prod_{j=1}^{b}b_j(-n_j)e^{-\beta},
 \qquad m_i,n_j\geq1.
\]
Introduce independent auxiliary variables $s_i,t_j$ and define the formal states
\[
 U(\mathbf s)=\exp\!\left(\sum_i s_i a_i(-m_i)\right)e^\beta,
 \quad
 V(\mathbf t)=\exp\!\left(\sum_j t_j b_j(-n_j)\right)e^{-\beta}.
\]
Their coefficients are ordinary Fock states. Set
\begin{align}
 L_i&=\delta_{m_i,1}a_i(x)+(-1)^{m_i}(a_i,\beta)z^{-m_i},\notag\\
 R_j&=\delta_{n_j,1}b_j(x)-(\beta,b_j)z^{-n_j},\label{eq:LRK}\\
 K_{ij}&=(-1)^{m_i-1}n_j
 \binom{m_i+n_j-1}{m_i-1}(a_i,b_j)z^{-m_i-n_j}.\notag
\end{align}
The coefficientwise generating identity is
\begin{align}\label{eq:wick-generating}
 \pi_xY(U(\mathbf s),z)V(\mathbf t)
 &=\epsilon(\beta,-\beta)z^{-(\beta,\beta)}e^{\beta(x)z}\notag\\
 &\quad\cdot\exp\!\left(\sum_i s_iL_i+\sum_j t_jR_j
 +\sum_{i,j}s_it_jK_{ij}\right).
\end{align}
Here the left and right auxiliary degrees, rather than powers of $z$, are used to extract individual states.

To verify \eqref{eq:wick-generating}, use
\[
 Y(a(-m)w,z)=:\frac{\partial^{m-1}a(z)}{(m-1)!}Y(w,z):.
\]
The left creation parts stand before the exponential field; the left annihilation parts stand after it. Among creation modes, and among nonnegative modes, Heisenberg commutators vanish. Thus no pairing of two left oscillators occurs. The surviving creation term is $a(-1)$ only when $m=1$. The left zero mode on charge $-\beta$ gives $(-1)^m(a,\beta)z^{-m}$. These are the two terms of $L_i$.

The annihilation exponential in $Y(e^\beta,z)$ shifts a right oscillator by
\[
 b(-n)\longmapsto b(-n)-(\beta,b)z^{-n},
\]
which gives $R_j$ after projection. Finally the scalar commutator of a positive left mode with a right oscillator is
\[
 \frac{\partial_z^{m-1}}{(m-1)!}
 \bigl(n(a,b)z^{-n-1}\bigr)
 =(-1)^{m-1}n\binom{m+n-1}{m-1}(a,b)z^{-m-n}.
\]
It gives $K_{ij}$. All these commutators are scalar, so their exponential is exactly the last exponential in \eqref{eq:wick-generating}. The creation exponential projects to $e^{\beta(x)z}$; the remaining lattice factor is $\epsilon(\beta,-\beta)z^{-(\beta,\beta)}$. This proves the identity.

Taking the coefficient of $s_1\cdots s_a t_1\cdots t_b$ gives
\begin{equation}\label{eq:wick-symbol}
 \pi_xY(u,z)v=
 \epsilon(\beta,-\beta)z^{-(\beta,\beta)}e^{\beta(x)z}
 \sum_M\prod_{(i,j)\in M}K_{ij}
 \prod_{i\ \mathrm{unpaired}}L_i
 \prod_{j\ \mathrm{unpaired}}R_j,
\end{equation}
where $M$ runs through partial matchings of \emph{labelled} left and right oscillators. Labelling preserves multiplicities for repeated oscillators.

Suppose $u,v$ both have conformal weight $d$. Their total oscillator energy is
\[
 N_\beta=2d-\Delta(\beta)-\Delta(-\beta)
 =2d-(\beta,\beta).
\]
Every monomial in the sum in \eqref{eq:wick-symbol} has $z$-degree between $-N_\beta$ and zero: each oscillator is either unpaired or used once in a cross-contraction. Consequently multiplication by $z^{N_\beta}$ turns that sum into a polynomial of degree at most $N_\beta$. Linearity gives the same bound for arbitrary homogeneous states of these charges. This proves the polynomial bounds in \eqref{eq:bounds}, both for $\beta=\sqrt p\alpha$ and for $\beta=0$.

For primary vectors $u,v$ of weight $d$, invariance of the bilinear form gives
\begin{equation}\label{eq:vacuum-pairing}
 u_{(2d-1)}v=(-1)^d \langle u,v\rangle\mathbf1.
\end{equation}
Indeed pair $Y(u,z)v$ with $\mathbf1$ and use
\[
 \langle Y(u,z)v,\mathbf1\rangle
 =(-1)^d z^{-2d}\langle v,Y(u,z^{-1})\mathbf1\rangle,
 \qquad Y(u,z^{-1})\mathbf1=e^{z^{-1}T}u.
\]
Only the term $u$ survives the pairing by conformal degree.

Fix a root charge $\beta$. Its line pairs nontrivially with the opposite line; write their pairing as $b_\beta\neq0$. Their form matrix and inverse are
\[
 \begin{pmatrix}0&b_\beta\\b_\beta&0\end{pmatrix},
 \qquad
 \begin{pmatrix}0&b_\beta^{-1}\\b_\beta^{-1}&0\end{pmatrix}.
\]
There is exactly one ordered summand with left charge $\beta$. By \eqref{eq:vacuum-pairing} its normalized constant term is $(-1)^d$. The opposite ordered summand has the opposite exponential. The same inverse-form contraction on the neutral block has constant term $r(-1)^d$. This establishes the claimed constants for every compatible choice of lattice cocycle.
\end{proof}

\begin{lemma}\label{lem:nocancel}
If $H_x(z)$ is a polynomial, then $x=0$.
\end{lemma}
\begin{proof}
Group the roots by equal scalar exponents $\xi=\sqrt p\alpha(x)$. For each nonzero $\xi$, the grouped coefficient polynomial has constant term
\[
 \#\{\alpha\in\Phi:\sqrt p\alpha(x)=\xi\}\,(-1)^d\neq0.
\]
This includes collisions between roots of different lengths. Distinct scalar exponentials are linearly independent over $\C[z]$: apply sufficiently large powers of $\partial_z-\eta$ to kill all other exponent groups, including the polynomial group at exponent zero. On the remaining coefficient at $\xi$, each resulting factor $\partial_z+\xi-\eta$ is invertible on the finite-dimensional space of polynomials of bounded degree, since its constant coefficient is nonzero. Hence no nonzero exponent group can occur in a polynomial $H_x$. Every root must evaluate to zero at $x$, and the roots span the Cartan space.
\end{proof}

Skew symmetry gives $\mathcal Q(z)=e^{zT}\mathcal Q(-z)$. All its coefficients are
neutral and symbol projection kills translation. Hence $f_s=0$ for
even $s$; the relation ideal may equivalently be generated by the odd
indices in its defining range.

\section{A finite relation ideal and proof of the theorem}\label{sec:ideal}
Define the length-dependent ODE multiplicities and relation cutoff
\begin{equation}\label{eq:S}
 m_\alpha=2d-p(\alpha,\alpha)+1,\qquad
 S=1+\sum_{\alpha\in\Phi}m_\alpha,\qquad
 I_{\Phi,p}=(f_s:2\leq s\leq S)\subset R_A.
\end{equation}
They are positive integers. At both opposite root charges there are
weight-$d$ vectors, so their total oscillator energy is
$2d-p(\alpha,\alpha)\geq0$. Equivalently, putting
$n_l=|\Phi_l|$, $n_s=|\Phi_s|$ in non-simply-laced type,
\begin{equation}\label{eq:Sexpanded}
 S=1+n_l(2d-2p+1)+n_s(2d-2p/t+1).
\end{equation}
In all types one may instead write
$S=1+|\Phi|(2d+1)-2prh^\vee$, using
$\sum_{\alpha\in\Phi}(\alpha,\alpha)=2rh^\vee$.

\begin{proposition}\label{prop:radical}
The radical of $I_{\Phi,p}$ is the homogeneous maximal ideal of
$R_A=\C[u_1,\ldots,u_r]$. In particular,
$D_{\Phi,p}:=\dim_\C R_A/I_{\Phi,p}<\infty$.
\end{proposition}
\begin{proof}
Let $c\in\operatorname{Spec} R_A$ be a common zero of the displayed generators and lift it to a Cartan point $x$ using \eqref{eq:RA}. Put $\xi_\alpha=\sqrt p\alpha(x)$. Lemma~\ref{lem:exppoly} implies that the formal series $H_x\in\C[[z]]$ is annihilated by the monic constant-coefficient differential operator
\begin{equation}\label{eq:ODE}
 \mathcal D_x=\partial_z^{2d+1}\prod_{\alpha\in\Phi}(\partial_z-\xi_\alpha)^{m_\alpha},
 \qquad q=\operatorname{ord}\mathcal D_x=2d+S.
\end{equation}
Coincident exponents merely repeat factors; zero exponents add factors of $\partial_z$. Thus the same operator annihilates the series at these points.

Let $J_x$ be the Taylor polynomial of $H_x$ through degree $2d$. Since the factors in $\mathcal D_x$ commute and $\partial_z^{2d+1}J_x=0$, both $H_x$ and $J_x$ are annihilated by $\mathcal D_x$.

The coefficient of $z^{2d+s-1}$ in $H_x$ is $f_s(c)$. The series $H_x-J_x$ therefore has zero coefficients in exactly the following consecutive initial range:
\begin{center}
\begin{tabular}{@{}ll@{}}
\toprule
Degrees & Reason for vanishing \\
\midrule
$0,\ldots,2d$ & Definition of $J_x$ \\
$2d+1,\ldots,2d+S-1=q-1$ & Relations $f_2(c)=\cdots=f_S(c)=0$ \\
\bottomrule
\end{tabular}
\end{center}
There are $q$ initial coefficients, as required for an operator of order $q$.

The uniqueness argument is formal. Write
\[
 \mathcal D_x=\partial_z^q+\sum_{j=0}^{q-1}c_j\partial_z^j,
 \qquad H_x-J_x=\sum_{n\geq0}a_nz^n.
\]
The coefficient of $z^n$ in $\mathcal D_x(H_x-J_x)=0$ is
\begin{equation}\label{eq:formal-recursion}
 \frac{(n+q)!}{n!}a_{n+q}
 +\sum_{j=0}^{q-1}c_j\frac{(n+j)!}{n!}a_{n+j}=0.
\end{equation}
The leading factor is nonzero over $\C$. Starting from $a_0=\cdots=a_{q-1}=0$, induction gives $a_n=0$ for every $n$. Thus $H_x=J_x$ is polynomial. Lemma~\ref{lem:nocancel} gives $x=0$, whence $c=0$.

Conversely each $f_s$ is homogeneous of positive conformal weight $2d+s-1$, and its Miura symbol vanishes at the origin. The common zero set is therefore exactly the origin. The Nullstellensatz gives the radical assertion. For each $i$, some power $u_i^{N_i}$ belongs to $I_{\Phi,p}$; the monomials with exponents below these powers span the quotient. This proves finite codimension.
\end{proof}

\begin{proof}[Proof of Theorem~\ref{thm:main}]
By \eqref{eq:contraction}, the algebra map $\varphi:R_A\to R_\V$
satisfies $\varphi(I_{\Phi,p})=0$. Hence $I_{\Phi,p}$ annihilates
the entire $R_A$-module $R_\V$. The finite-module surjection
\eqref{eq:finite-surjection} therefore factors through
\[
 \bigoplus_{H(\lambda)\leq R_\Phi}
 L(\lambda)\otimes
 \frac{R_A(M_\lambda)}{I_{\Phi,p}R_A(M_\lambda)}
       \twoheadrightarrow R_\V.
\]
Each summand is a quotient of at most $(\dim L(\lambda))^2$
copies of $R_A/I_{\Phi,p}$, by Proposition~\ref{prop:modulebound}.
Thus
\begin{equation}\label{eq:dimension}
 \dim_\C R_\V\leq K_\Phi D_{\Phi,p}<\infty.
\end{equation}
Homogeneous lifts of algebra generators of the finite-dimensional
intrinsic quotient strongly generate the CFT-type vertex algebra by
weight induction on the remaining $(-2)$ products.
\end{proof}

\begin{corollary}
Strong generators can be chosen of weights at most
$\max\{h,pR_\Phi^2+(p+1)R_\Phi\}$.
\end{corollary}
\begin{proof}
For a dominant root-lattice weight $\lambda=\sum b_i\alpha_i$,
$b_i\geq0$ and $H(\lambda)=\sum b_i$. Long roots have the maximum
length $\sqrt2$, so $\|\lambda\|\leq\sqrt2H(\lambda)$.
Also $J(\lambda)=\sum b_i(\alpha_i,\alpha_i)/2\leq H(\lambda)$.
Equation \eqref{eq:top} gives
$h_\lambda+2H(\lambda)\leq pH(\lambda)^2+(p+1)H(\lambda)$.
On the surviving components this is at most the claimed bound;
the invariant algebra has basic generator degrees at most $h$.
Choose lifts of these invariant generators and of the homogeneous
module generators in \eqref{eq:finite-surjection}, including bases of
their finite $L(\lambda)$ factors. Their classes algebra-generate
$R_\V$: polynomial combinations of the invariant classes multiply
the module generators to span every element of $R_\V$.
For a homogeneous vector of $\V$, lift such an expression by $(-1)$
products. The remainder lies in $C_2(\V)$ and is a sum of
$a_{(-2)}b$ with positive-weight $a$ and strictly smaller-weight
$a,b$. Induction on conformal weight gives strong generation by the
chosen lifts, with the stated weight bound.
\end{proof}

\begin{corollary}
Tensor products of these screening-kernel algebras are $C_2$-cofinite.
In particular the corresponding construction for a semisimple root
system, with allowed parameters on its simple factors, has the same property.
\end{corollary}
\begin{proof}
For vertex algebras $U,V$,
$R_{U\otimes V}\simeq R_U\otimes R_V$. This follows by expanding
the tensor-product fields and using translation to put modes of index
at most $-2$ in the tensor factors' $C_2$ spaces. Alternatively use
the tensor product of their Li filtrations at degree zero.
The factorwise screening kernels also tensor, by elementary linear
algebra over $\C$. Finiteness is preserved.
\end{proof}


\appendix
\section{Source conventions}
All internal numbers in citations refer to the arXiv versions specified
in the bibliography. We use \cite[Theorem 2.13(1), Theorem 3.2,
Corollary 3.4, Theorem 4.3]{SL} in the ordinary even-lattice case,
with vacuum coset parameter zero. Its strong condition is
$m\geq h_L^\vee-1$; together with $p=tm\geq2$ this is
\eqref{eq:range}. The module $T^{1/p}_{\lambda,0}$ is the untwisted
principal reduction of the induced affine Weyl module at
$k=1/p-h^\vee$. The Miura embedding is the one in
\cite[equation (4.3)]{SL}, transported by $b'_i=-\beta_i/p$.

For intrinsic quotients, \cite[Theorem 4.5.9(ii), Corollary 4.5.10(ii)]{Ara}
are applied to a finitely generated $KL_k$ module with a Weyl flag
of length one. Its PBW description and finite coefficient
representation are given in Section~\ref{sec:modules}.
For the invariant algebra, the noncritical PBW and Miura symbol statements
are \cite[Section 5, equations (23)--(25); proof of Lemma 6.1]{ACL}.
The shifted level is $\kappa=1/p\ne0$ throughout. The specified
Miura realization identifies $A$ with the universal principal
algebra, and the graded module isomorphism \eqref{eq:moduleid}
identifies the intrinsic module quotients used in
Proposition~\ref{prop:modulebound}.
\clearpage
\begin{thebibliography}{99}
\small\raggedright\setlength{\itemsep}{1pt}
\bibitem{SL} S. Sugimoto and H. Li,
\emph{A Lie algebraic pattern behind logarithmic CFTs},
Comm. Math. Phys. \textbf{407} (2026), article 202.
Numbered source: \href{https://arxiv.org/abs/2409.07381v4}{arXiv:2409.07381v4}.
\bibitem{Ara} T. Arakawa,
\emph{Associated varieties of modules over Kac--Moody algebras and
$C_2$-cofiniteness of $W$-algebras}, Int. Math. Res. Not. IMRN
\textbf{2015} (2015), no. 22, 11605--11666.
Numbered source: \href{https://arxiv.org/abs/1004.1554v5}{arXiv:1004.1554v5}.
\bibitem{AF} T. Arakawa and E. Frenkel,
\emph{Quantum Langlands duality of representations of $\mathcal W$-algebras},
Compos. Math. \textbf{155} (2019), no. 12, 2235--2262.
Numbered source: \href{https://arxiv.org/abs/1807.01536v3}{arXiv:1807.01536v3}.
\bibitem{ACL} T. Arakawa, T. Creutzig and A. R. Linshaw,
\emph{$W$-algebras as coset vertex algebras}, Invent. Math.
\textbf{218} (2019), no. 1, 145--195.
Numbered source: \href{https://arxiv.org/abs/1801.03822v4}{arXiv:1801.03822v4}.

\bibitem{AM08} D. Adamovi\'c and A. Milas,
\emph{On the triplet vertex algebra $\mathcal W(p)$}, Adv. Math.
\textbf{217} (2008), 2664--2699.
\href{https://arxiv.org/abs/0707.1857v2}{arXiv:0707.1857v2}.
\bibitem{FT} B. L. Feigin and I. Yu. Tipunin,
\emph{Logarithmic CFTs connected with simple Lie algebras}, preprint (2010).
\href{https://arxiv.org/abs/1002.5047}{arXiv:1002.5047}.
\bibitem{S21} S. Sugimoto, \emph{On the Feigin--Tipunin conjecture},
Selecta Math. (N.S.) \textbf{27} (2021), article 86.
Numbered source: \href{https://arxiv.org/abs/2004.05769v3}{arXiv:2004.05769v3}.
\bibitem{S23} S. Sugimoto, \emph{Simplicity of higher rank triplet
$W$-algebras}, Int. Math. Res. Not. IMRN \textbf{2023} (2023),
no. 8, 7169--7199.
Numbered source: \href{https://arxiv.org/abs/2105.00638v3}{arXiv:2105.00638v3}.
\end{thebibliography}
\end{document}