% Working manuscript, 23 October 2026. Standalone pdfLaTeX source. \documentclass[11pt,a4paper]{article} \usepackage[T1]{fontenc} \usepackage[utf8]{inputenc} \usepackage{lmodern,microtype} \usepackage[margin=25mm]{geometry} \usepackage{amsmath,amssymb,amsthm,mathtools,mathrsfs} \usepackage{booktabs,array,longtable,enumitem,listings} \usepackage[dvipsnames]{xcolor} \usepackage[colorlinks=true,linkcolor=MidnightBlue,citecolor=MidnightBlue,urlcolor=MidnightBlue]{hyperref} \usepackage{bookmark,xurl} \usepackage{fancyhdr} \usepackage{needspace} \pagestyle{fancy}\fancyhf{} \fancyhead[L]{\small Untwisted cyclic Haagerup--Izumi centers} \fancyhead[R]{} \fancyfoot[C]{\thepage} \setlength{\headheight}{14pt} \setlength{\emergencystretch}{3em} \setlength{\parskip}{2.5pt} \setlength{\parindent}{1.2em} \setlist{topsep=4pt,itemsep=3pt} \lstset{language=Python,basicstyle=\ttfamily\scriptsize, breaklines=true,columns=fullflexible,keepspaces=true, numbers=left,numberstyle=\tiny\color{gray},numbersep=6pt, frame=single,rulecolor=\color{gray},showstringspaces=false} \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{proposition}[theorem]{Proposition} \newtheorem{lemma}[theorem]{Lemma} \newtheorem{corollary}[theorem]{Corollary} \theoremstyle{definition}\newtheorem{definition}[theorem]{Definition} \theoremstyle{remark}\newtheorem{remark}[theorem]{Remark} \newcommand{\C}{\mathbb C}\newcommand{\Z}{\mathbb Z}\newcommand{\Q}{\mathbb Q} \newcommand{\one}{\mathbf1}\newcommand{\B}{\mathcal B}\newcommand{\W}{W_N} \newcommand{\FP}{\operatorname{FPdim}}\newcommand{\FPdim}{\operatorname{FPdim}} \newcommand{\Ind}{\operatorname{Ind}}\newcommand{\Irr}{\operatorname{Irr}} \newcommand{\Hom}{\operatorname{Hom}}\newcommand{\End}{\operatorname{End}} \newcommand{\Rep}{\operatorname{Rep}}\newcommand{\Ext}{\operatorname{Ext}} \newcommand{\rk}{\operatorname{rk}}\newcommand{\Tr}{\operatorname{Tr}} \newcommand{\wt}{\operatorname{wt}}\newcommand{\loc}{\mathrm{loc}} \newcommand{\ptr}{\operatorname{ptr}}\newcommand{\ch}{\operatorname{ch}} \newcommand{\diag}{\operatorname{diag}}\renewcommand{\Vec}{\operatorname{Vec}} \newcommand{\bo}{\boxtimes}\newcommand{\Cat}{\mathcal C} \newcommand{\Par}{\mathcal P}\newcommand{\Neu}{\mathcal D} \newcommand{\Mod}{\mathcal M}\newcommand{\HI}{\mathcal H} \newcommand{\Frame}{\mathcal B}\newcommand{\mm}{\mathfrak m} \newcommand{\eps}{\varepsilon}\newcommand{\NN}{\mathbb Z_{\geq0}} \newcommand{\ds}{\displaystyle} \newcommand{\parityrank}{p_N} \setcounter{tocdepth}{1} \hypersetup{pdftitle={Vertex operator algebra realizations of untwisted cyclic Haagerup--Izumi centers},pdfauthor={David E. Evans and Andrew Riesen},pdfsubject={Working manuscript, 23 October 2026},pdfkeywords={Haagerup--Izumi categories, vertex operator algebras}} \title{\textbf{Vertex operator algebra realizations of\\untwisted Haagerup--Izumi centers}} \author{David E. Evans\and Andrew Riesen} \date{23 October 2026} \begin{document} \maketitle \begin{abstract} For every odd integer $N\geq3$, we construct an untwisted Haagerup--Izumi fusion category $\mathcal H_N$ on the negative spherical branch and a simple, self-contragredient, regular, $C_2$-cofinite vertex operator algebra $E_N$ of central charge $-16N$, together with an equivalence of modular fusion categories: \[ \operatorname{Rep}(E_N)\simeq Z(\mathcal H_N). \] The VOA $E_N$ is rational, has exactly $N^2+3$ irreducible modules, and is lower bounded and integer graded, with least conformal degree $-(N-1)/2$. The proof begins with a CFT-type cyclic Virasoro extension of central charge $-5N$, whose regularity we establish. Two categorical condensations recover $\mathcal H_N$, and a finite extension by shifted lattice modules removes the auxiliary pointed factor. We identify the complete ribbon modular data of $Z(\mathcal H_N)$ with $\mathrm{MD}_{-}(\mathbb Z/N;\mathbb Z/(N^2+4),b\mapsto b^2/(N^2+4))$ in the notation of Gannon--Schopieray--Yadav, for every odd $N$. \end{abstract} \section{Introduction and main theorem} \label{sec:intro} The representation category of a rational vertex operator algebra, under the standard finiteness and duality hypotheses, records its fusion, braiding and conformal twists. The converse realization problem asks for a vertex operator algebra with prescribed categorical data. Haagerup--Izumi categories provide a family of test cases: their simple objects consist of a group of invertibles and a single orbit of noninvertibles, while their Drinfeld centers are modular tensor categories. We construct vertex operator algebra realizations for every odd cyclic order $N\geq3$, with the negative spherical dimension specified below. The first algebra, $W_N$, is a CFT-type extension of a tensor product of $N$ irrational Virasoro algebras. Its representation category contains a nondegenerate pointed factor $\mathcal P_N$ of rank $2^N$. We construct an untwisted cyclic HI category $\mathcal H_N$ and prove: \[ \operatorname{Rep}(W_N)\simeq \mathcal P_N\boxtimes Z(\mathcal H_N), \qquad c(W_N)=-5N. \] An explicit extension using shifted $E_7$ lattice modules cancels $\mathcal P_N$ and gives a second algebra $E_N$, with central charge $-16N$ and representation category $Z(\mathcal H_N)$. Both algebras are simple, self-contragredient, regular and $C_2$-cofinite. Their grading conventions and complete invariants are stated in Theorem~\ref{thm:main}. Two features of the proof are essential. First, the Virasoro factors are not rational, so regularity of their extension requires a separate argument. We prove $C_2$-cofiniteness and split extensions of the vacuum before using modular traces. Second, a single conformal-weight residue does not distinguish all exceptional labels when $N^2+4$ is composite. Keeping the partial conformal weights resolves these labels and makes the branching calculation sufficient to exhaust the simple modules. The center equivalence then follows from two specified condensation algebras. \paragraph{Origin of the construction.} Evans \cite[Section 6.2]{evanssfold} proposed the cyclic Virasoro sum $V_\varphi$ with parameters $\psi_i+\psi_{i-1}^{-1}=k_i$. For $(k_0,k_1,\ldots,k_{N-1})=(N+2,2,\ldots,2)$, that sum is exactly the underlying frame module of $W_N$ used here. The cited section presents a heuristic cyclic closure of the duality-wall gluing of Creutzig--Gaiotto \cite{cgduality}. The work below supplies the uniform VOA construction, regularity proof, module exhaustion and ribbon center identification for this cyclic family. The earlier physical proposals of Gang--Kim, Gang--Kim--Lee and Jeong--Sohn \cite{gangkim,gangkimlee,jeongsohn} provide the S-fold and character context; the hyperbolic-word description is developed in \cite{evanswords}. Flohr's screened one-variable algebras and theta-difference characters \cite{flohr1993,flohr1994} are an earlier precedent for these methods. Their character calculations do not supply the $N$-variable regularity argument proved below. \paragraph{Relation to categorical existence results.} The categorical background includes Izumi's endomorphism and tube algebras \cite{izumi}, the Evans--Gannon grafting of modular data \cite[Section 3.3]{evansgannon2011}, and their near-group and nonunitary constructions \cite{evansgannon2014,evansgannon2017}. Existence for every odd cyclic order is already established in \cite{huangcyclic} and \cite[Theorem A(1)]{gsy}; the latter includes the pseudo-unitary Galois conjugate. Its Theorem 5.1 identifies the untwisted center modular data. Thus the result here is a VOA and condensation realization, rather than a new categorical existence claim. We also identify the full modular data of our center with the negative cyclic class $(N^2+4)^-_1$ of Gannon--Schopieray--Yadav, including a bijection of simple labels. This is distinct from an equivalence with a prescribed category. At order fifteen, their examples have identical center modular data but are pairwise not categorically Morita equivalent \cite[Corollary 5.9]{gsy}. Our center equivalence retains the condensation algebras and their multiplication; the modular-data comparison is a subsequent theorem. The announced $c=8$ Haagerup VOA \cite{riesenc8} concerns a separate positive-central-charge realization. \subsection{Conventions and statement} Throughout, $N\geq3$ is odd. All categories are over $\mathbb C$. A fusion category is finite, semisimple, rigid, and has simple unit. A modular tensor category carries its specified ribbon structure; dimensions below are spherical dimensions unless marked $\operatorname{FPdim}$. For a fusion category $\mathcal H$, $Z(\mathcal H)$ denotes its Drinfeld center with the ribbon structure induced by the stated spherical structure. A VOA is lower bounded and integer graded, with finite-dimensional homogeneous spaces. It is of CFT type when its negative degrees vanish and its degree-zero space is $\mathbb C\mathbf1$. A VOA is \emph{regular} when every weak module is a direct sum of irreducible ordinary modules. Regularity implies rationality, meaning complete reducibility of admissible modules, for both $W_N$ and $E_N$. The regularity of $E_N$ is proved directly, without assuming CFT type. We use $C_2$-cofinite in the sense \[ C_2(V)=\operatorname{span}_{\mathbb C}\{a_{-2}b:a,b\in V\}, \qquad \dim V/C_2(V)<\infty. \] This condition depends on the vertex operators, not on the choice of conformal vector. Self-contragredience means an isomorphism $V\cong V^\vee$ with the graded contragredient module for the stated conformal vector. The notation $\operatorname{Rep}(V)$ denotes the category of grading-restricted generalized modules; for the regular VOAs considered here it consists of finite direct sums of ordinary simple modules. For a fusion category, its rank is the number of isomorphism classes of simple objects. We write $\boxtimes$ for Deligne products of categories and, with a VOA subscript when needed, for fusion products of modules. \begin{definition} An untwisted cyclic Haagerup--Izumi (HI) fusion category of odd order $N$ has simple objects $g$ and $g\rho$, $g\in G=\mathbb Z/N$, with: \[ gh=g+h,\qquad \rho g=(-g)\rho,\qquad \rho^2=\mathbf1\oplus\bigoplus_{g\in G}g\rho, \] and its pointed subcategory is tensor equivalent to $\operatorname{Vec}_{G}$ with trivial associator. This condition does not prescribe the remaining associator entries. \end{definition} Set: \[ \mu=N^2+4,\qquad t=\frac{\sqrt\mu-N}{2},\qquad \delta=t^{-1}=\frac{N+\sqrt\mu}{2},\qquad \parityrank=2^N. \] Thus $00). \] The frame is $\mathcal B=\bigotimes_iL(c_i,0)$, and $L_i(R,S)$ abbreviates the simple Virasoro module of central charge $c_i$ and highest weight $h_i(R,S)$. For positive integral $R,S$, it is the irreducible quotient at the degenerate Kac weight. The positive cycle form $K=K_+$ is defined without an ambiguous small-cycle matrix convention by: \[ x^{\mathsf t}K_+x=\sum_i(x_i-x_{i+1})^2+Nx_0^2. \] The antiperiodic form $K_-$ replaces the closing difference $(x_{N-1}-x_0)^2$ by $(x_{N-1}+x_0)^2$. Their determinants are $N^2$ and $\mu$. Use the even positive lattice $L=(\mathbb Z^N,2K_+)$ with basis $\beta_i$. Write $B=\sum_i\beta_i$, $\varepsilon=e^B$, and $D=L(-1)$. In character formulas, $q=e^{2\pi i\tau}$ and $q_i=e^{2\pi i\tau_i}$. Quantum-group parameters denoted by $q$ are introduced locally; the parity rank is always $\parityrank$. For $m\in\mathbb Z_{\geq0}^N$, write: \[ M_m=\bigotimes_iL_i(2m_i+1,2m_{i+1}+1),\qquad R_j=M_{e_j}. \] The state and module called $C$ in the vacuum calculation correspond to $m=\mathbf1$, whereas $\mathcal C=\operatorname{Rep}(W_N)$ is the full representation category. A proper interval containing site $0$ is called \emph{heavy}; an \emph{ordinary} proper interval avoids $0$. We also write $W=W_N$ when $N$ is fixed. Bold $\mathbf1$ denotes the vacuum or the all-ones coordinate vector according to its ambient space; the distinction is explicit whenever both occur together. The main categorical notation is collected here for reference. \begin{center}\small \begin{tabular}{@{}ll@{}} \toprule Symbol & Meaning\\ \midrule $\mathcal B$ & The tensor product of the $N$ Virasoro frame algebras\\ $A$ & The deformed lattice VOA, also its algebra object over $W_N$\\ $\mathcal C$ & $\operatorname{Rep}(W_N)$ after regularity is established\\ $P=(\mathbb Z/2)^N$, $\parityrank$ & The parity-label group and its order $2^N$\\ $\mathcal P=\mathcal P_N$ & The pointed ribbon subcategory on the $J_\epsilon$\\ $\mathcal D=\mathcal P'$ & The centralizer of $\mathcal P$ in $\mathcal C$\\ $\mathcal M=\mathcal D_A$ & Right $A$-modules in $\mathcal D$\\ $\mathcal D_A^{\mathrm{loc}}$ & The full subcategory of local $A$-modules\\ $\mathcal H_N$ & The fusion category obtained by the second condensation\\ \bottomrule \end{tabular} \end{center} \subsection{Structure of the proof} \label{sec:proof-structure} Regularity precedes the module classification and modular traces. \emph{Construction (Part~I).} Cyclic Virasoro gluing gives $W_N$ (Theorem~\ref{c:cyclic-gluing}). Theorem~\ref{c:cyclic-c2} proves $C_2$-cofiniteness, and the explicit commuting lattice frame gives a conformal embedding (Theorem~\ref{c:screening-frame} and Corollary~\ref{c:cyclic-embedding}). Theorem~\ref{c:currents} constructs the parity simple currents. \emph{Regularity (Part~II).} Proposition~\ref{neutral:prop:two-lowest-weights} reduces vacuum extensions to lowest weights zero and one. Theorems~\ref{r:thm:h0} and~\ref{r:thm:h1} exclude these cases, making the vacuum projective. Lemma~\ref{lem:projective-unit} then gives rigidity and semisimplicity, and Theorem~\ref{r:thm:regularity} passes to all weak modules. The parity factorization follows in Section~\ref{sec:parity-factor}. \emph{Modules and traces (Part~III).} We classify the periodic lattice sectors and prove the needed uniqueness results for rational frame sectors. Formal refined traces retain the joint partial weights and detect every exceptional label, including those not separated by a single conformal-weight residue when $N^2+4$ is composite. \emph{Center identification and realization (Part~IV).} Exact branching and the dimension identity in Section~\ref{r:sec:budget} exhaust the simple modules. Two condensations use the lattice algebra $A$ and the regular algebra $D_H$ of a cyclic Lagrangian subgroup: \[ \mathcal D=\mathcal P_N' \quad\longmapsto\quad \mathcal M=\mathcal D_A \quad\longmapsto\quad \mathcal H_N=\mathcal M_{D_H}. \] The second local category is trivial. Corollary~\ref{cor:our-center} identifies the ribbon center; Section~\ref{sec:gsy-modular} computes its full modular data. Theorem~\ref{thm:parity-cancellation-voa} cancels the parity factor by an explicit $E_7$ extension and proves its finiteness, duality, weak-module regularity and grading assertions. The appendices supply the interval and Zhu calculations and the symbolic frame certificate. Section~\ref{c:app:frame} explains the reduction to that finite certificate. No theorem from a companion twisted-case manuscript is assumed. \paragraph{AI disclosure.} Generative AI was used extensively in author-directed proof exploration, symbolic calculations, code development, and drafting, and produced substantial portions of the initial arguments and exposition. The authors formulated the research objective, guided the mathematical strategy, and revised the arguments, notation, and presentation. The authors take responsibility for the mathematical claims and final text. The accompanying calculations and code make the finite checks inspectable; their role in the proof is explained in the manuscript. Independent human verification of the complete argument remains pending. Corrections and feedback on the mathematics, exposition, attribution, and historical account are welcome. \paragraph{Acknowledgments.} The work of the first author was supported in part by a grant of access to OpenAI models through the ChatGPT for Academic Researchers programme. \clearpage \section*{Contents} \begingroup \small \setlength{\parskip}{0pt} \makeatletter \renewcommand{\l@section}{\@dottedtocline{1}{0pt}{2.5em}} \renewcommand{\l@part}[2]{% \addpenalty{-\@highpenalty}% \addvspace{0.8em}% {\bfseries\raggedright #1\par}\nobreak} \@starttoc{toc} \makeatother \endgroup \clearpage \part{Construction and lattice realization} \section{Cyclic gluing of generic Virasoro algebras} \label{sec:cyclic-gluing} Fix an odd integer $N\ge3$ and retain the parameters and Virasoro modules introduced in Section~\ref{sec:intro}. Thus $\psi_i=(t+i+1)/(t+i)$ and $t(t+N)=1$. We construct $W_N$ as an extension of the frame $\mathcal B$ by transporting a canonical diagonal algebra. \begin{lemma}[Signs of the quantum parameter on integer spins]\label{c:quantum-sign} Let $q\in\mathbb C^\times$ be not a root of unity. The type-I integer-spin ribbon categories for $q$ and $-q$ are ribbon equivalent. The corresponding category for $q^{-1}$ is braid-reversed equivalent to that for $q$. \end{lemma} \begin{proof} Use $\Delta E=E\otimes K+1\otimes E$ and $\Delta F=F\otimes1+K^{-1}\otimes F$. The map from $U_{-q}(\mathfrak{sl}_2)$ to $U_q(\mathfrak{sl}_2)$ given by $K'\mapsto K$, $E'\mapsto-E$, and $F'\mapsto F$ is a Hopf isomorphism. On even highest weights it preserves the type-I condition. In the standard universal $R$-matrix its $n$th nilpotent coefficient is \[ q^{n(n-1)/2}\frac{(q-q^{-1})^n}{[n]_q!}E^n\otimes F^n. \] Since $[n]_{-q}=(-1)^{n-1}[n]_q$, changing $q$ to $-q$ changes the scalar coefficient by $(-1)^n$, which is cancelled by $E'\mapsto-E$. The diagonal factor $q^{H\otimes H/2}$ is unchanged on even weights: the product of two even weights divided by two is even. The ribbon twist is unchanged as well, since its exponent on highest weight $2m$ is $(2m)(2m+2)/2=2m(m+1)$. For inversion, the Hopf isomorphism $K'\mapsto K$, $E'\mapsto KF$, $F'\mapsto EK^{-1}$ from $U_{q^{-1}}$ to $U_q$ carries the standard $R$-matrix to $R_{21}^{-1}$ and the ribbon twist to its inverse; this follows term by term from the same $R$-matrix formula. These identities give the stated ribbon equivalences. \end{proof} \begin{theorem}[Cyclic gluing]\label{c:cyclic-gluing} There is a simple VOA $W_N$ of CFT type and central charge $-5N$, with frame decomposition: \[ W_N=\bigoplus_{r_0,\ldots,r_{N-1}\in2\Z_{\ge0}+1} \bigotimes_{i=0}^{N-1}L_i(r_i,r_{i+1}). \] It is self-contragredient, $(W_N)_1=0$, and $\dim(W_N)_2=N(N+1)/2$. \end{theorem} \begin{proof} At an irrational Virasoro parameter, the category of finite-length $C_1$-cofinite modules is a semisimple rigid vertex tensor category \cite{virasoro}. Its odd first-row and first-column subcategories are generic integer-spin quantum $\mathfrak{sl}_2$ ribbon categories \cite[Theorem 1.1]{c:Koshida}. They centralize one another: the unique product is $L_i(r,s)$ and \[ h_i(r,s)-h_i(r,1)-h_i(1,s)=-\frac{(r-1)(s-1)}2, \] so balancing gives trivial monodromy for odd $r,s$. Consequently, the tensor-product functor is braided on these odd-label subcategories. It maps distinct simple pairs bijectively to the odd-label simples, by the fusion rules in \cite[Theorem 5.2.5]{virasoro}. The cyclic relations are \[ \psi_0+\psi_{N-1}^{-1}=N+2,\qquad \psi_i+\psi_{i-1}^{-1}=2\quad(1\le i0$. Both $u$ and $m-u$ are then nonzero. The coefficient $(v_u)_{(-\Delta-1)}v_{m-u}$ belongs to $C_2(W_N)$. Its projection to $M_m$ is a nonzero multiple of $v_m$: the canonical multiplication has a nonzero top fusion channel, and its highest-input, highest-output coefficient is nonzero. To see the latter assertion directly, pair its coefficients with all descendants of the highest output vector. Virasoro Ward identities determine these pairings recursively from the leading highest-weight coefficient. If that coefficient vanished, nondegeneracy of the irreducible target's contravariant form would make the entire intertwiner vanish. Every other summand in this coefficient has labels coordinatewise at most $m$ and strictly smaller total label. Thus $[v_m]$ reduces to lower-label summands. Their images are stable under all $D_i$ and $x_i$, so the entire image of $M_m$ reduces as well. Induction on $\sum_i m_i$ leaves only $m\in\mathcal L_N$. This is a finite set, since $m_i\le1+\lfloor N/2\rfloor$. In each surviving module put $r_i=2m_i+1$. The singular vector at site $i$ has level $d_i=r_i r_{i+1}$ and can be normalized to have leading term $(L^{(i)}_{-1})^{d_i}$; see \cite[Proposition 2.2.2]{virasoro}. Modulo $C_2$, it gives a monic relation for $D_i$ with coefficients in $\C[x_i]$. All remaining terms have smaller $D_i$-degree. Commutators with $L^{(i)}_{-2}$ vanish modulo $C_2$. Hence the images \[ \prod_i D_i^{a_i}v_m,\qquad m\in\mathcal L_N,\quad 0\le a_ib>0$ be irrational. If \[ F'=(C_{a,X}\otimes I+I\otimes C_{b,Y})F \] has a nonzero vector solution with Laurent-polynomial entries, then $aX=bY$. \end{lemma} \begin{proof} First suppose $X,Y\ne0$ and put $A=aX$, $B'=bY$. If $A\ne B'$, the constant coefficient matrix has a simple zero eigenvalue: its two constituent spectra are $\{0,\pm2\sqrt A\}$ and $\{0,\pm2\sqrt{B'}\}$. The right zero eigenvector is $e_0\otimes e_0$. Comparing the highest Laurent power and projecting the next equation to the left zero eigenspace shows that this power must be zero. The residue matrix annihilates the right zero eigenvector. The formal solution with prescribed constant coefficient is therefore unique: the recursion is invertible off the zero eigenspace, and its scalar equation at order $z^{-n-1}$ has coefficient $-n$ for $n\ge1$. The normalized scalar solution with zero exponential part is \[ f_{a,X}(z)=\sum_{n\ge0}u_nz^{-2n},\qquad u_n=\frac{(a)_n(\tfrac12-a)_n(1-2a)_n}{n!A^n}. \] Substitution into \eqref{c:scalar-ode} proves the recurrence; explicitly the coefficient factorization is \[ -2j(2j+1)(2j+2)+8aj(2j+1)+4ah_aj-4ah_a^2 =-4(a+j)(2a-2j-1)(2a-j-1). \] Consequently the first component of the unique tensor-product solution is $f_{a,X}f_{b,Y}$. None of its Pochhammer parameters is a nonpositive integer, and the gamma-function asymptotic gives \begin{equation}\label{c:factorial-asymptotic} u_n=\frac{(n!)^2 n^{-3/2-2a}}{\Gamma(a)\Gamma(\tfrac12-a)\Gamma(1-2a)A^n} (1+O(n^{-1})). \end{equation} Let $v_n$ be the analogous coefficients for $b,Y$. Their convolution cannot terminate. The following estimates apply also when $X$ and $Y$ are complex. If $|A|<|B'|$, then \[ \sum_{j=0}^n u_{n-j}v_j=u_n(1+o(1)); \] if $|A|>|B'|$, the analogous statement holds with $v_n$. For the first assertion, split the sum at $n/2$. After division by $|u_n|$, the terms with $1\le j\le n/2$ are bounded by a constant times $\binom nj^{-2}(|A|/|B'|)^j$, because both power exponents in \eqref{c:factorial-asymptotic} are negative. Their sum is $O(n^{-2})$. The other half has the exponentially small factor $(|A|/|B'|)^{n/2}$ times a fixed power of $n$. Initial finitely many coefficients are absorbed in the constants. If $|A|=|B'|$, split at $n/2$ and compare each half with its adjacent endpoint. For $1\le j\le n/2$, the first half is bounded by $C|v_n|\binom nj^{-2}$ after absorbing the fixed negative powers of $j$ and the bounded ratio $(n-j)/n$; the other half is bounded by $C|u_n|\binom nj^{-2}$ after reversing $j$. Since $\sum_{j=1}^{\lfloor n/2\rfloor}\binom nj^{-2}=O(n^{-2})$, we obtain \[ \sum_{j=0}^n u_jv_{n-j}=u_n+v_n+ O(n^{-2})(|u_n|+|v_n|). \] Since $a>b$, $|u_n/v_n|=O(n^{-2(a-b)})\to0$, so again the convolution is nonzero for all sufficiently large $n$. Thus $f_{a,X}f_{b,Y}$ is not a Laurent polynomial, a contradiction. If $X=0$ and $Y\ne0$, rescale the first-factor coordinates to $(f,zf',z^2f'')$. Its connection becomes $E_a/z$, where \[ E_a=\begin{pmatrix}0&1&0\\0&1&1\\4ah_a^2&2ah_a&2-4a\end{pmatrix},\qquad \operatorname{Spec}(E_a)=\{2a,1-2a,2-4a\}. \] The second constant matrix $C_0$ has a simple zero eigenspace. If the leading coefficient is $z^m v$, the highest-power equation places $v$ in $\C^3\otimes\ker C_0$. Projecting the next equation to this kernel gives $m v=(E_a\otimes1)v$: the projected residue of the second factor is zero. Thus the integer $m$ would be an eigenvalue of $E_a$, impossible because all three are irrational. The case $Y=0$, $X\ne0$ is identical with $b$. This proves the lemma. \end{proof} \subsection*{Null vectors transport the stress values around the cycle} Fix an algebra homomorphism $\ell:R\to\C$ and an index $i$. Let $v=v_{e_i}$, with $(3,1)$ label at site $i$, $(1,3)$ at site $i-1$, and vacuum labels elsewhere. Set \[ a=\psi_i>1,\quad b=\psi_{i-1}^{-1}\in(0,1),\quad X=\ell(T_i),\quad Y=\ell(T_{i-1}), \] and abbreviate the two translations by $D_A,D_B$. Define \[ F_{pq}(z)=\ell\big(Y(D_A^pD_B^qv,z)v\big),\qquad 0\le p,q\le2. \] Every entry is a Laurent polynomial with nonpositive powers: all positive-power coefficients belong to $C_2$. Moreover $F_{00}\ne0$, since its most singular coefficient is the nonzero vacuum pairing of $v$. The level-three null relation is \[ D_A^3v-4aL^A_{-2}D_Av+2ah_aL^A_{-3}v=0, \] and likewise for $b$. For any $D_B$-descendant $u$ of $v$, normal ordering and Lemma~\ref{c:partial-translations} give \begin{align*} \ell(Y(L^A_{-2}u,z)v) &=X\ell(Y(u,z)v)-z^{-1}\ell(Y(D_Au,z)v)+h_az^{-2}\ell(Y(u,z)v),\\ \ell(Y(L^A_{-3}u,z)v) &=z^{-2}\ell(Y(D_Au,z)v)-2h_az^{-3}\ell(Y(u,z)v). \end{align*} Indeed, on the left only $L^A_{-2}$ survives; on the right only $L^A_{-1}$ and $L^A_0$ act nontrivially on the highest vector, and the $L^A_{-1}$ action is moved to the inserted vector using $\ell D_A=0$. The null relation therefore gives \[ F_{3q}=4aXF_{1q}-4az^{-1}F_{2q}+2ah_az^{-2}F_{1q}+4ah_a^2z^{-3}F_{0q}, \] with the analogous relation for $F_{p3}$. Total translation gives $F'_{pq}=F_{p+1,q}+F_{p,q+1}$. Thus the nine entries satisfy precisely the tensor-sum system of Lemma~\ref{c:laurent-obstruction}. It follows that \begin{equation}\label{c:stress-transport} \psi_i\ell(T_i)=\psi_{i-1}^{-1}\ell(T_{i-1})\qquad\text{for every }i. \end{equation} Finally, \[ \prod_i\psi_i=\frac{t+N}{t}=t^{-2}. \] Iterating \eqref{c:stress-transport} gives $\ell(T_0)=t^4\ell(T_0)$. Since $00$. Writing $[\omega_0{}_z\varepsilon]=D\varepsilon$, sesquilinearity and skew-symmetry give \[ [\omega_0{}_z D^2\varepsilon]+[D^2\varepsilon{}_z\omega_0] =(D+z)^2D\varepsilon-z^2D\varepsilon =(D+2z)D^2\varepsilon. \] The self-bracket of $D^2\varepsilon$ also vanishes. Hence $\omega_\lambda$ satisfies the Virasoro relation with the same central charge. The mode formula $(D^2\varepsilon)_{(n)}=n(n-1)\varepsilon_{(n-2)}$ shows that the translation and grading operators are unchanged. The positive-definite lattice VOA is regular \cite[Theorem 3.16]{c:DLMRegular}. Weak modules depend only on the underlying vertex algebra, whose translation operator has not changed. Every weak module is therefore a direct sum of the standard lattice-coset modules. On each such module the new grading is the positive quadratic lattice energy minus the linear term $(B,k)/2$; it is bounded below, diagonalizable, and has finite-dimensional eigenspaces. The $D^2\varepsilon$ term does not change this grading. Thus these simple modules are ordinary for $\omega_\lambda$ and $A_N$ is regular. \end{proof} \section{The uniform screening frame and its certificate}\label{c:app:frame} We construct commuting Virasoro vectors in the ambient lattice algebra. Each vector has a correction linear in the coupling $u$; their sum first changes at order $u^N$. Put $r_i=t+i$, $p_i=(r_i+1)^{-1}$, and \[ P_- =\prod_{i=0}^{N-1}r_i,\qquad P_+=\prod_{i=0}^{N-1}(r_i+1),\qquad s=(-1)^N,\qquad C_N=\frac{s}{2}(P_-+P_+). \] Thus $P_+/P_-=(t+N)/t$. Pairing $r_jr_{N-j}=1+j(N-j)$ gives the useful closed formula \[ C_N=-\frac{\sqrt{N^2+4}}2 \prod_{j=1}^{(N-1)/2}\bigl(1+j(N-j)\bigr)\ne0. \] Use the orthogonal Heisenberg currents $A_i=\sqrt{2\psi_i}\,H_i$, with $(A_i,A_j)=2\psi_i\delta_{ij}$. For a charge $m=\sum m_i\beta_i$, set \[ p_i(m)=m_i-\frac{m_{i+1}}{\psi_i},\qquad h_i(m)=\psi_i p_i(m)^2-(\psi_i-1)p_i(m). \] In these coordinates \[ T_i=\frac{A_i(-1)^2\one}{4\psi_i} +\frac{\psi_i-1}{2\psi_i}A_i(-2)\one, \qquad h(m)=\sum_i h_i(m)=\sum_i(m_i-m_{i+1})^2+Nm_0(m_0-1). \] All lattice pairings are even, so the exponential fields below can use the trivial lattice cocycle. \begin{theorem}[Uniform Virasoro frame]\label{c:screening-frame} For every odd $N\ge3$, there are explicitly defined finite sums $\widetilde T_i(u)\in V_L[u]$ satisfying \[ [\widetilde T_i(u){}_{z}\widetilde T_j(u)] =\delta_{ij}\big((D+2z)\widetilde T_i(u)+c_i z^3\one/12\big), \qquad \sum_i\widetilde T_i(u)=\omega_0+C_Nu^N D^2\varepsilon. \] They have $\widetilde T_i(0)=T_i$, and their degree-one corrections are \[ u\big(e^{\beta_i}-e^{\beta_{i+1}}\big). \] Consequently, every nonzero $\lambda\in\C$ admits a conformal embedding \[ \bigotimes_i L(c_i,0)\ \hookrightarrow\ (V_L,\omega_0+\lambda D^2\varepsilon). \] This gives the Virasoro frame embedding. The extension of this map to $W_N$ is established in Corollary~\ref{c:cyclic-embedding}. \end{theorem} \subsection*{Finite formulas for the frame} A proper cyclic interval is an oriented consecutive set $I=[a,\ldots,b]$ of length $\elli)&-\dfrac{sP_-r_jp_j}{2r_i(r_i+1)}\\[3pt] a_ia_j\ (ji)&sP_-p_ip_j\\ a_ja_k\ (j\ne k,\ i\notin\{j,k\})&0 \end{array} \] The order $<$ in this table is the ordinary order $0<1<\cdots0$. The first linear bracket is $\sum_{k=0}^3 z^k L_i(k-1)(S_je^B)/k!$; the reversed one is obtained by skew-symmetry, replacing $z$ by $-z-D$ and negating. These formulas explain the mode action in the certificate. \subsection*{A nondegenerate invariant form on the ambient algebra} \begin{proposition}\label{c:screening-form} For $\lambda\ne0$, $(V_L,\omega_\lambda)$ is regular and has a unique normalized nondegenerate invariant bilinear form. Its weight-zero space is two-dimensional, so it is not of CFT type. \end{proposition} \begin{proof} The total grading is unchanged by the $D^2\varepsilon$ correction. It has \[ (V_L)_0=\C\one\oplus\C\varepsilon,\qquad (V_L)_1=\bigoplus_i\big(\C A_i(-1)\one\oplus\C A_i(-1)\varepsilon\big). \] The degree-one-to-zero action is \[ L_1^{(\lambda)}A_i(-1)\one =-2(\psi_i-1)(\one+2\lambda\varepsilon),\qquad L_1^{(\lambda)}A_i(-1)\varepsilon=0. \] Therefore $(V_L)_0/L_1(V_L)_1$ is one-dimensional, and the vacuum has nonzero image exactly when $\lambda\ne0$. The invariant-form theorem \cite{c:LiForm} gives a unique form with $\langle\one,\one\rangle=1$. Its radical is an ideal of the underlying simple lattice vertex algebra, hence is zero. On weight zero its matrix is \[ \begin{pmatrix}1&-1/(2\lambda)\\-1/(2\lambda)&0\end{pmatrix}. \] Regularity follows from Proposition~\ref{c:ambient-deformation}. \end{proof} \section{Fock modules and signed Kac labels}\label{c:fock-orientation} We fix the free-field conventions used below, since the distinction between a Verma module and its contragredient is essential. Let $[a_m,a_n]=m\delta_{m+n,0}$ and \[ Q=\frac{\psi-1}{\sqrt{2\psi}},\qquad T=\tfrac12 a(-1)^2\one+Q a(-2)\one, \qquad c=1-12Q^2, \] where $\psi>0$ is irrational. The Fock space $F_\xi$ has ground momentum $a_0=\xi$, and its highest weight is $h(\xi)=\xi(\xi-2Q)/2$. Put \[ \alpha_+=\sqrt{2\psi},\qquad \alpha_-=-\sqrt{2/\psi},\qquad \xi_{r,s}=\frac{1-r}{2}\alpha_++\frac{1-s}{2}\alpha_-. \] The classical free-field determinant formula \cite[Proposition 2.7]{c:TK} for the ground-vector map $M(c,h(\xi))\longrightarrow F_\xi$ is, at level $n$ and up to a nonzero basis constant, \begin{equation}\label{c:fock-determinant} \det\Phi_{\xi,n} =C_n\prod_{\substack{r,s\ge1\\rs\le n}} (\xi-\xi_{r,s})^{p(n-rs)},\qquad C_n\ne0. \end{equation} Here $p(n)$ is the partition number. To check conventions against \cite{c:TK}, set $w=\xi-Q$, $\lambda=-Q$, and $s_\pm=\alpha_\pm$. Their raising operator $L_n$ is our $L_{-n}$; their determinant factor $w+rs_+/2+ss_-/2$ is exactly $\xi-\xi_{r,s}$. At level one this gives $L_{-1}|\xi\rangle=\xi a_{-1}|\xi\rangle$. Fock contragredience is $F_\xi^\vee\simeq F_{2Q-\xi}$; the pairing is obtained by transposing nonzero oscillator modes as $a_n\mapsto-a_{-n}$ and the zero mode as $a_0\mapsto2Q-a_0$. \begin{lemma}[Signed Kac coordinates]\label{c:signed-fock} For nonzero integers $r,s$ and irrational $\psi$ the following hold. If $r,s>0$, then $F_{\xi_{r,s}}$ is the contragredient Verma module at $h_{r,s}$, with unique simple socle $L(r,s)$. If $r,s<0$, it is the Verma module at $h_{-r,-s}$; its simple socle is the irreducible partner Verma module and is not $C_1$-cofinite. If $rs<0$, it is an irreducible Verma module and is not $C_1$-cofinite. \end{lemma} \begin{proof} Irrationality implies that $\xi_{r,s}=\xi_{u,v}$ only when $(r,s)=(u,v)$. Thus \eqref{c:fock-determinant} identifies $F_{\xi_{r,s}}$ with its Verma module whenever at least one of $r,s$ is negative. Reflection sends $(r,s)$ to $(-r,-s)$, so contragredience proves the assertion when both are positive. The generic Verma-module classification gives one primitive singular vector at level $rs$ for a positive pair and an irreducible partner Verma module below it. A mixed-sign pair is not a positive Kac weight, even after reflection, so its Verma module is irreducible. The $C_1$-cofinite simple modules are exactly the positive Kac modules \cite[Theorems 2.2.4 and 3.1.4]{virasoro}, which proves the remaining assertions. \end{proof} At site $i$ of our lattice, the momentum of charge $k$ is $\xi=\alpha_+k_i+\alpha_-k_{i+1}$, hence its signed Kac coordinates are $(1-2k_i,1-2k_{i+1})$. In particular the charges $-m$ and $\mathbf1+m$ have, respectively, contragredient-Verma and Verma orientation for the positive pair $(2m_i+1,2m_{i+1}+1)$. \section{Simple generator modules for the deformed frame}\label{c:sec:uniform-socle} Retain the lattice $L$, the vector $B$, and the exponential $\varepsilon=e^B$ from Section~\ref{c:ambient}. Fix a nonzero coupling in the frame of Theorem~\ref{c:screening-frame}, so that: \[ \sum_i\widetilde T_i=\omega_0+\lambda D^2\varepsilon, \qquad \lambda\ne0. \] Each correction to $T_i$ has a strictly positive charge in the cone $\sum_i\mathbb Z_{\geq0}\beta_i$. The ambient normalized invariant form satisfies $\langle1,\varepsilon\rangle=-1/(2\lambda)$. We write $L_i(n)$ for the modes of $\widetilde T_i$. For $m\in\mathbb Z_{\geq0}^N$, set \[ r_i=2m_i+1,\quad \eta_i=h_i(r_i,r_{i+1}),\quad \ell_i=r_ir_{i+1},\quad H=\sum_i \eta_i=\sum_i(m_i-m_{i+1})^2+Nm_0(m_0+1). \] The two distinguished charges are $\mu_-=-\sum_i m_i\beta_i$ and $\mu_+=B+\sum_i m_i\beta_i$. Their undeformed Fock modules are, respectively, the tensor product of the contragredient Verma modules $M_i^\vee$ and the tensor product of the Verma modules $M_i$, by Lemma~\ref{c:signed-fock}. Here $M_i=M(c_i,\eta_i)$ has one primitive singular vector, at level $\ell_i$, and its simple quotient is $L_i(r_i,r_{i+1})$. \begin{proposition}[Simple generator socles]\label{c:generator-socles} For each $m=e_j$ with $0\leq j0$, all grades needed below are at most $H+3=5$. Equation \eqref{c:socle-common-C} gives $\kappa=k_0(k_0-1)\geq0$; if $\kappa>0$, then $h(k)\geq2N>5$. Hence $\kappa=0$. For the cyclic elementary generator $m=e_0$, the required grades are at most $H+3=2N+5<6N$. Here $\kappa=k_0(k_0-1)-2$. If $\kappa=-2$, every site $i>0$ would satisfy $k_i(k_i-1)=-2a_i<0$, which is impossible. If $\kappa>0$, then $k_0(k_0-1)\geq6$ contradicts the grade bound. Thus $\kappa=0$ for the closing generator as well. Thus throughout these finite ranges $k_i\in\{-m_i,m_i+1\}$. Write $k_i=(1+\epsilon_i r_i)/2$, $\epsilon_i\in\{\pm1\}$. Direct substitution gives \begin{equation}\label{c:socle-domain-wall} h_i(k)=\eta_i+ \begin{cases}0,&\epsilon_i=\epsilon_{i+1},\\ \ell_i,&\epsilon_i\ne\epsilon_{i+1}. \end{cases} \end{equation} A nonconstant cyclic sign sequence has at least two domain walls. Consequently, at the joint weight $\eta=(\eta_i)$, and at each joint weight $\eta+\ell_i e_i$, the only Fock charge quotients are $\mu_-$ and $\mu_+$. Below $\eta$ in any one coordinate there are no quotients. Oscillators add nonnegative integers separately in each coordinate, so they do not change these conclusions. \subsection*{Spectral projections, without a semisimplicity assumption} All arguments take place in finite-dimensional total-weight spaces. The commuting operators $L_i(0)$ are triangular for the positive-charge filtration; their associated graded operators are the undeformed Fock $L_i(0)$ operators. Their joint generalized spectral projections are polynomials in these commuting operators and preserve the filtration. One can equivalently choose a linear combination separating all the finitely many joint eigenvalue tuples and use its ordinary spectral projections. This construction does not require the individual $L_i(0)$ to be semisimple. The joint generalized eigenspace at $\eta$ has dimension two. Its intersection with the cone-filtered subspace supported at charges $\mu_++\sum_i\mathbb Z_{\geq0}\beta_i$ has dimension one. To justify the dimension count, restrict the polynomial spectral projector to this invariant filtered subspace. Its associated graded image is the sum of the Fock eigenspaces with tuple $\eta$ in that cone. Of the two possible charges, only $\mu_+$ belongs to the cone, and only its ground vector has that tuple. Thus the image has dimension one. The commuting nilpotent parts vanish on a one-dimensional space. It is therefore spanned by a genuine joint eigenvector $e$ with lowest charge $\mu_+$ and coefficient one at $e^{\mu_+}$. Projecting $e^{\mu_-}$ gives a vector $f$ with lowest charge $\mu_-$ and coefficient one there. Thus \[ L_i(0)e=\eta_i e,\qquad L_i(0)f=\eta_i f+b_i e. \] The absence of spectral targets below $\eta$ implies $L_i(n)e=L_i(n)f=0$ for all $n>0$. The submodule $E$ generated by $e$ is the tensor product of the Verma modules $M_i$: its map from that tensor product is injective, since its lowest-charge projection is exactly the isomorphism to the positive Fock module at $\mu_+$. Let $S_i$ be the primitive singular operator in $M_i$, with coefficients fixed at $\eta_i$. The leading $\mu_-$ part of $S_i f$ vanishes, since that Fock module is contragredient Verma. The spectral calculation at $\eta+\ell_i e_i$ therefore places $S_i f$ in $E$. In more detail, inside the cone-filtered subspace above $\mu_-$, the kernel of projection to the $\mu_-$ Fock quotient has generalized-block dimension $p(\ell_i)$, supplied entirely by the $\mu_+$ Fock quotient; the corresponding ordinary weight space of $E$ has that same dimension. They are equal. This remains valid before the coefficients $b_i$ have been shown to vanish. \subsection*{Eliminating the possible Jordan coupling} Each frame vector is quasi-primary for the total conformal vector, so the ambient invariant form satisfies $\langle L_i(n)x,y\rangle=\langle x,L_i(-n)y\rangle$. The module $E$ is isotropic. Indeed $e$ is a total primary of weight $H$, and every charge in it is at least $\mu_+\geq B$. Therefore the weight-zero product $e_{(2H-1)}e$ cannot have charge $0$ or $B$, the only weight-zero lattice charges. This proves $\langle e,e\rangle=0$, and contravariance proves $\langle E,E\rangle=0$. On the other hand, \begin{equation}\label{c:socle-cross-pair} \langle e,f\rangle=(-1)^H\langle1,e_{(2H-1)}f\rangle =-\frac1{2\lambda}\ne0. \end{equation} Here $H$ is even, the lowest-charge product is $e^{\mu_+}_{(2H-1)}e^{\mu_-}=\varepsilon$, and every correction has strictly larger charge and hence cannot contribute to weight zero. Let $P_i(z)$ be the Shapovalov norm polynomial of the fixed vector $S_i$: on a generalized highest vector, $S_i^\dagger S_i$ acts as $P_i(L_i(0))$. The Kac determinant has a simple zero at $\eta_i$ in level $\ell_i$, because $\psi_i$ is irrational and the only positive Kac pair with this highest weight is $(r_i,r_{i+1})$. Thus the level-$\ell_i$ Gram matrix has a one-dimensional kernel and \[ P_i(\eta_i)=0,\qquad P_i'(\eta_i)\ne0. \] The last assertion follows directly by differentiating the determinant: for a symmetric matrix with a one-dimensional kernel, its adjugate is a nonzero scalar times the outer product of a kernel vector with itself. A simple determinant zero is therefore equivalent to a nonzero derivative of that vector's norm. Since $S_i f\in E$, isotropy and contravariance now give \[ 0=\langle S_i f,S_i f\rangle =\langle f,P_i(L_i(0))f\rangle =b_iP_i'(\eta_i)\langle f,e\rangle. \] Equation \eqref{c:socle-cross-pair} proves $b_i=0$ for every site. \subsection*{The common singular-vector kernel} Now $f$ is a genuine joint highest vector. Thus $S_i f$ is a joint singular vector in $E$ at weight $\eta+\ell_i e_i$, and \[ S_i f=c_i S_i e \] for some scalar $c_i$. Operators belonging to different Virasoro factors commute, and $E$ is a tensor product of Verma modules. Hence \[ (c_i-c_j)S_iS_j e=0,\qquad S_iS_j e\ne0, \] so all $c_i$ have a common value $c$. The vector $v=f-ce$ is killed by every primitive singular operator. It therefore generates the simple tensor product $M_m$. Its leading negative exponential has coefficient one. No other such simple submodule can occur in this two-dimensional bottom space, since a kernel vector in $E$ would contradict its Verma structure. This proves the proposition. \section{Closure and classification of the frame socle} We first show that the sum of simple $C_1$-cofinite frame submodules is closed under vertex products. The category of lower-bounded $C_1$-cofinite modules for $\mathcal B=\bigotimes_iL(c_i,0)$ is the Deligne product of the generic Virasoro $C_1$ categories, and is therefore semisimple. This follows by applying McRae's tensor-product theorem inductively \cite[Theorems 3.7 and 4.13]{c:McRaeDeligne}. The required closure under contragredients holds in each generic Virasoro factor by \cite{virasoro}. For two simple $C_1$-cofinite frame submodules $M,N$ of the ambient algebra, let $P$ be the span of the coefficients of $Y(M,z)N$. The intertwining-operator Jacobi identity makes $P$ a frame submodule. It inherits a lower-bounded, finite-dimensional total-weight grading from the lattice. The intertwining operator with target $P$ is surjective by this definition, so Miyamoto's image theorem, in the form of \cite[Corollary 2.14]{cmy}, makes $P$ $C_1$-cofinite. The preceding tensor-product theorem then makes $P$ a finite direct sum of simple $C_1$ frame modules. Thus the sum of all such simple submodules is closed under vertex products. This argument does not assume that the full ambient lattice module is locally $C_1$-cofinite. \begin{lemma}[Aligned socle and multiplicity]\label{c:aligned-C1-socle} Every simple $C_1$-cofinite frame submodule of the ambient lattice is of type $M_m$, for a unique $m\in\mathbb Z_{\geq0}^N$, and its highest vector has a nonzero component at $e^{-\sum_i m_i\beta_i}$. Each such type occurs with multiplicity at most one. \end{lemma} \begin{proof} Let $v$ be the highest vector of a simple $C_1$ frame submodule $M$. Filter by the scalar charge height $d(k)=\sum_i k_i$. The vector $v$ has finite charge support; let $d_0$ be its minimum height. Every deformed frame mode preserves or strictly increases this height, since every correction charge lies in the positive cone. The same lower bound therefore holds on the module generated by $v$. For any charge $k$ at height $d_0$ with nonzero component in $v$, projection to that charge is a nonzero module homomorphism \[ M\longrightarrow F(k)=\bigotimes_iF_i(k). \] Indeed a correction could contribute at $k$ only from a charge of height strictly less than $d_0$, which is absent. Simplicity of $M$ makes this map injective. Here the two signed Kac coordinates of $F_i(k)$ are \[ R_i=1-2k_i,\qquad S_i=1-2k_{i+1}. \] They are nonzero odd integers. Lemma~\ref{c:signed-fock} gives the following three possibilities. If $R_i,S_i>0$, then $F_i(k)$ is a contragredient Verma module, whose unique simple socle is $L_i(R_i,S_i)$ and is $C_1$-cofinite. If $R_i,S_i<0$, it is the Verma module at the reflected positive Kac label, whose simple socle is the irreducible partner Verma module; that partner is not $C_1$-cofinite. If the signs are opposite, the Fock module itself is an irreducible Verma module and is not $C_1$-cofinite. Irrationality of $\psi_i$ excludes any additional positive Kac label in the last two cases. By the tensor-product description above, $M$ is an external tensor product of simple $C_1$ Virasoro modules. Its nonzero embedding in $F(k)$ forces such a simple socle in every factor. This can also be seen directly on its highest vector: the highest-weight and primitive-null equations act separately on the finite-dimensional site weight spaces, so their joint solution space is the tensor product of the corresponding site solution spaces. Consequently $R_i,S_i>0$ at every site, equivalently $k_i\leq0$ for every $i$. Writing $m=-k$, the embedded socle is exactly $M_m$. Its highest vector in $F(k)$ is the Fock ground exponential, so the coefficient at $e^{-m}$ is nonzero. The label uniquely determines $m$: equality of two positive Kac weights at irrational $\psi_i$ forces equality of their positive Kac pairs. Evaluation of the highest-vector coefficient at this one ground exponential is therefore an injective linear map \[ \operatorname{Hom}_{\mathcal B}(M_m,V_L)\longrightarrow\mathbb C. \] If a nonzero linear combination had coefficient zero, applying the preceding argument to its nonzero simple image would require that same coefficient to be nonzero. This contradiction proves the multiplicity assertion. \end{proof} \section{Nonzero self-couplings of all elementary generators} The elementary generators have an explicit nonzero self-channel. The calculation is uniform for all edges $j$, including $j=0$. Set \[ a=\psi_j,\quad b=\psi_{j-1}^{-1},\quad k=a+b, \qquad A=2a,\quad B'=2b,\quad H=2k-2. \] For an ordinary edge $k=2$; for the cyclic edge $k=N+2$. The symbol $B'$ in this section is a scalar and is not the lattice vector $B$. Use the two orthogonal currents $X=A_j$ and $Z=-bA_{j-1}$; their norms are $A$ and $B'$ and $\beta_j=X+Z$. The corresponding exponential $E=e^{\beta_j}$ has individual frame weights $(1,1)$. The charge-$\beta_j$ frame corrections are exactly $+uE$ and $-uE$. Define the Schur states by \[ \sum_{p\geq0}S_p(X)z^p =\exp\left(\sum_{n\geq1}\frac{X(-n)}n z^n\right), \] and similarly for $Z$. The normalized simple generator $v_j$ has lowest term $e^{-\beta_j}$. At charge zero its state is exactly \begin{equation}\label{c:edge-neutral-state} uP,\qquad P=\sum_{p=0}^H\frac{S_p(X)S_{H-p}(Z)}{A-1-p}. \end{equation} Indeed, $E_{(1)}e^{-\beta_j}$ is the Schur state of total degree $H=\beta_j^2-2$, and the original partial $L_j(0)$ acts by $p$ on its summand of $X$-degree $p$. The eigenvalue equation at charge zero therefore gives \eqref{c:edge-neutral-state}. Its denominators are nonzero because $a$ is irrational. No other charge in the global frame can enter this equation: both $v_j$ and the frame corrections are supported in the nonnegative cone above their respective lowest charges. Let $\sigma_j$ denote the coefficient of the normalized highest vector $v_j$ in $v_{j(H-1)}v_j$. The leading charge $-\beta_j$ can come only from the ordered charge pairs $(-\beta_j,0)$ and $(0,-\beta_j)$. Contracting $X(-n)$ against $e^{-\beta_j}$ replaces it by $(-1)^n A$, and similarly replaces $Z(-n)$ by $(-1)^nB'$. Thus the Schur states evaluate to $(-1)^p(A)_p/p!$ and $(-1)^{H-p}(B')_{H-p}/(H-p)!$. Since $H$ is even, their total sign is positive. Skew-symmetry gives the same coefficient for the reversed charge pair; derivative terms cannot change the leading highest vector at its minimum weight. Consequently \begin{equation}\label{c:edge-sigma-sum} \frac{\sigma_j}{u} =2\sum_{p=0}^H \frac{(A)_p(B')_{H-p}}{p!(H-p)!(A-1-p)}, \qquad A+B'=H+2. \end{equation} Vacuum descendants have only nonnegative charge and cannot contribute to this leading negative exponential. The next possible diagonal channel, with edge spin two, has weight $6k-4>H$, so the coefficient in \eqref{c:edge-sigma-sum} is indeed the self-channel scalar. \begin{lemma}[An elementary finite-sum identity] For every integer $H\geq0$, as an identity of rational functions of $A$, \begin{equation}\label{c:edge-sigma-product} 2\sum_{p=0}^H \frac{(A)_p(H+2-A)_{H-p}}{p!(H-p)!(A-1-p)} = 2^{H+1}\frac{\displaystyle\prod_{j=1}^H(2A-2j-1)} {\displaystyle\prod_{j=1}^{H+1}(A-j)}. \end{equation} \end{lemma} \begin{proof} The only possible finite poles on the left are simple poles at $A=p+1$, $0\leq p\leq H$. The residue there is \[ 2\binom{2p}{p}\binom{2H-2p}{H-p}. \] The right side has the same residue: evaluating its numerator at $A=p+1$ gives the product of the odd integers up to $2p-1$ and, with sign $(-1)^{H-p}$, the odd integers up to $2(H-p)-1$; the derivative of its denominator is $(-1)^{H-p}p!(H-p)!$. The double-factorial formula gives precisely the displayed central-binomial residue. It remains to check that the left side is proper. For $0\leq p\leq H$, the finite Newton expansion is \[ \frac1{A-1-p} =\sum_{r=0}^p \frac{p(p-1)\cdots(p-r+1)}{(A-1)(A-2)\cdots(A-r-1)}. \] Vandermonde convolution gives, for $A+B'=H+2$, \[ \sum_{p=0}^H p(p-1)\cdots(p-r+1) \frac{(A)_p(B')_{H-p}}{p!(H-p)!} =\frac{(A)_r(A+B'+r)_{H-r}}{(H-r)!}. \] After substitution, every term has numerator of degree $r$ in $A$ and denominator of degree $r+1$, since $A+B'$ is constant. The left side is therefore $O(A^{-1})$. The right side is also proper. Their difference is a polynomial vanishing at infinity, hence zero. \end{proof} All zeros and poles of \eqref{c:edge-sigma-product} occur at rational $A$. Therefore $\sigma_j\ne0$ for every edge in the cyclic frame and every nonzero common coupling $u$. \subsection{Exact ordinary-root coefficients} \begin{proposition}\label{c:ordinary-root-coefficients} Let $1\le j0$. Then the spin-one objects $X_{e_i}$ generate every integer-spin factor, hence their product category. This argument takes place in the support of the already simple quotient; it does not presume that those operator-product channels were nonzero in $S$. Finally the unital subalgebra generated by the $M_g$ satisfies the same hypotheses: it is again a multiplicity-free diagonal subobject, and it has the same nonzero generator vacuum channels. Applying the proof to it shows that it too has every diagonal summand, so its inclusion in $S$ is an isomorphism. \end{proof} \paragraph{Necessity of the two-point hypothesis.} Multiplicity-free diagonal support and associativity alone are not enough. For example, a Cartan-product degeneration can retain the same diagonal object while killing lower channels, including a nontrivial generator's vacuum channel. The proof above uses precisely the vacuum channels to rule out such degenerations before applying the mirror theorem. \subsection{A nonzero self-channel forces the vacuum channel} \begin{lemma}[The spin-one associativity test]\label{c:spin-one-vacuum} In the setup of Theorem~\ref{c:diagonal-recognition}, let $M=M_{e_i}$ occur in $S$. If the projection of $\mu_S:M\otimes M\to S$ to $M$ is nonzero, then its projection to the unit is nonzero. No simplicity assumption on $S$ and no prior occurrence of $M_{2e_i}$ are needed. \end{lemma} \begin{proof} Only one generic $SO(3)_q$ factor is involved. Write its spin-one object as $X$ and its three tensor-square summands as $X_0=\mathbf1,X_1=X,X_2$. Use the nonzero canonical diagonal multiplication to fix bases for every allowed diagonal multiplication channel. Express the multiplication of $S$ in these bases as scalars $a_{ij}^{\ell}$, assigning scalar zero if the output summand is absent. Unit normalization gives $a_{01}^1=a_{10}^1=1$. Because the reference multiplication is itself commutative, commutativity of $S$ gives $a_{1j}^1=a_{j1}^1$; this statement includes any braiding scalars and does not assume that a naked fusion basis has trivial braiding. Put $T=\operatorname{Hom}(X^{\otimes3},X)$, which has dimension three. Dual bases in the second factor identify the corresponding nine-dimensional space of diagonal-category morphisms with $\operatorname{End}(T)$. The left-associated multiplication of $M^{\otimes3}$ into $M$ is the diagonal endomorphism \[ D=\operatorname{diag}\bigl(a_{11}^0,\ (a_{11}^1)^2, a_{11}^2a_{21}^1\bigr) \] in the left fusion-tree basis. The right-associated multiplication is the same diagonal endomorphism in the right fusion-tree basis. If $F$ is the spin-one Racah change-of-basis matrix, the second factor changes by its inverse transpose: the evaluation pairing of the two fusion spaces must remain the identity. Explicitly, if $v'_a=\sum_bF_{ba}v_b$, then $(v'^a)=\sum_b(F^{-1})_{ab}v^b$. Consequently a diagonal channel tensor $\sum_a d_av'_a\otimes v'^a$ has matrix $FDF^{-1}$ in the unprimed bases. Associativity therefore says \[ D=FDF^{-1},\qquad FD=DF. \] The vacuum-to-spin-one entry $F_{01}$ is nonzero. For completeness, in the usual normalized pivotal fusion-tree bases the vacuum row is, up to nonzero signs, \[ F_{0j}=\frac{\sqrt{\dim X_j}}{\dim X},\qquad j=0,1,2. \] This follows by closing the vacuum cup-cap pair against the two trivalent vertices; the remaining closed diagram is the nonzero theta evaluation. All quantum dimensions in this row are nonzero for generic $q$. In particular $F_{01}=\pm1/\sqrt{\dim X}\ne0$. The $(0,1)$ entry of $FD=DF$ now yields $a_{11}^0=(a_{11}^1)^2$. The implication in the statement is independent of all the nonzero choices of channel bases. \end{proof} \subsection{Application to the cyclic frame} Let $\mathcal B=\bigotimes_i L(c_i,0)$ denote the Virasoro frame subalgebra in the deformed lattice construction, and let \[ \mathscr S(V_L)=\sum\{M\subset V_L: M\text{ is a simple }C_1\text{-cofinite }\mathcal B\text{-module}\}. \] Miyamoto's image theorem (in the form of Creutzig--McRae--Yang, \emph{Direct limit completions}, Corollary 2.14), together with generic Virasoro semisimplicity, makes this a vertex subalgebra. By Lemma~\ref{c:aligned-C1-socle}, its $\mathcal B$-module decomposition is a subobject of \[ \bigoplus_{m\geq0}\bigotimes_i L_i(2m_i+1,2m_{i+1}+1). \] On odd labels the row/column interchange has trivial double braiding, so the edgewise rearrangement identifies this with the diagonal categorical setup of Theorem~\ref{c:diagonal-recognition}. \begin{corollary}[Uniform cyclic embedding]\label{c:cyclic-embedding} For every odd $N\geq3$ and every nonzero coupling, the cyclic VOA $W_N$ embeds conformally in the deformed lattice VOA. More precisely: \[ \mathscr S(V_L)\simeq W_N \] as an extension of $\mathcal B$, with canonical diagonal multiplication. \end{corollary} \begin{proof} The uniform calculation \eqref{c:edge-sigma-product} gives a nonzero self-channel for every $M_{e_i}$. Lemma~\ref{c:spin-one-vacuum} therefore gives a nonzero vacuum channel for every elementary generator. These objects tensor-generate the product of integer-spin categories. Theorem~\ref{c:diagonal-recognition} applies and identifies the full socle, with its multiplication, with $W_N$. In particular the frame and the elementary highest vectors $v_{e_i}$ vertex-generate $W_N$, and all fusion-allowed diagonal channels are nonzero. \end{proof} \subsection{The orthogonal splitting and a second weight-zero simple module} \begin{proposition}\label{c:ambient-splitting} For $\lambda\ne0$ the conformal embedding just constructed gives a $W_N$-module decomposition \[ A_N=W_N\oplus X,\qquad X=W_N^\perp. \] The line $X_0$ is spanned by $\zeta=\one+2\lambda\varepsilon$. There is a simple $W_N$-module $T\not\simeq W_N$ of lowest weight zero, obtained as a quotient of the cyclic submodule generated by $\zeta$. If $C$ denotes the highest vector of the all-ones frame summand, normalized to have leading charge $e^{-B}$, then $o(C)\zeta=2\lambda\zeta$. \end{proposition} \begin{proof} The normalized ambient invariant form restricts to a nonzero invariant form on the simple CFT-type subalgebra $W_N$. Its restriction is therefore nondegenerate. Both algebras have finite-dimensional homogeneous spaces, so grade-by-grade orthogonal projection gives the asserted module splitting. The weight-zero Gram matrix in Proposition~\ref{c:screening-form} identifies $X_0$ as stated. The all-ones summand is present by the diagonal recognition theorem. Lemma~\ref{c:aligned-C1-socle} supplies its normalized lowest charge $-B$. Its total highest weight is $2N$. Consequently the charge-zero coefficient of $o(C)\varepsilon$ is one, while weight considerations give $o(C)\varepsilon=\one+b\varepsilon$ for some $b$. Since $C$ is primary of even weight and $o(C)\one=0$, invariance gives \[ 0=\langle o(C)\varepsilon,\one\rangle =1-\frac{b}{2\lambda}. \] Thus $b=2\lambda$ and $o(C)\zeta=2\lambda\zeta$. The cyclic submodule generated by $\zeta$ has finite length: it is grading restricted over the CFT-type $C_2$-cofinite algebra $W_N$, so the finite-category theorem applies \cite{huang}. Any simple quotient has nonzero image of its cyclic generator and therefore has lowest weight zero. Its $C$ zero mode acts there by $2\lambda\ne0$, whereas it acts by zero on the vacuum top. The simple quotient is consequently different from $W_N$. \end{proof} \section{Parity simple currents} \label{sec:parity-currents} We extend the odd-label gluing to all parity sectors. Continue to write $\mathcal B=\bigotimes_iL(c_i,0)$ and let $K=K_+$ be the cycle form from Section~\ref{sec:intro}. Its diagonal entries are $k_0=N+2$ and $k_i=2$ for $i\ne0$, and its cycle off-diagonal entries are $-1$. For $N=3$, each distinct off-diagonal entry is $-1$. The identities needed below are: \[ x^TKx=\sum_i(x_i-x_{i+1})^2+Nx_0^2, \qquad K\mathbf1=Ne_0,\qquad \det K=N^2. \] The canonical odd-label gluing is: \[ W=\bigoplus_{r_i\geq1,\ r_i\text{ odd}} \bigotimes_iL_i(r_i,r_{i+1}). \] Here $W=W_N$ is the extension constructed in Theorem~\ref{c:cyclic-gluing}. The ambient category used throughout is $\mathscr V_{\mathcal B}$, the category of $C_1$-cofinite grading-restricted generalized $\mathcal B$-modules. The generic Virasoro results \cite[Theorems 3.1.4,4.2.6,5.1.2,5.5.3]{virasoro}, followed by \cite[Theorem 1.2, equivalently Theorem 4.13]{c:McRaeDeligne}, show that $\mathscr V_{\mathcal B}$ is exactly the semisimple rigid Deligne product of the corresponding generic Virasoro categories; it is closed under contragredients. Thus \cite[Theorems 7.1,7.5,7.7]{cmy} applies to its direct-limit completion, to algebra extensions there, and to their local modules. These theorems do not require $W$ to be rational or $C_2$-cofinite. \subsection{Construction and unrestricted fusion} For $\epsilon\in P=(\mathbb Z/2\mathbb Z)^N$, define \[ J_\epsilon=\bigoplus_{\substack{r_i\geq1\\ r_i\equiv1+\epsilon_i\pmod2}} \bigotimes_iL_i(r_i,r_{i+1}). \] Represent the entries of $\epsilon$ by $0,1$ when evaluating integral quadratic expressions. \begin{theorem}\label{c:currents} Each $J_\epsilon$ is a simple local grading-restricted $W$-module, and \[ J_\epsilon\bo_W J_\eta\simeq J_{\epsilon+\eta}. \] These fusion products have the intertwining-operator universal property against every grading-restricted generalized $W$-module, not only against modules whose $\mathcal B$-restriction belongs to $\Ind(\mathscr V_{\mathcal B})$. The full finite pointed ribbon subcategory $\mathcal P$ on these modules has self-braiding quadratic form, conformal twist, and categorical dimension \begin{align*} Q(\epsilon)&=\exp\left(\frac{\pi i}{2}\epsilon^TK\epsilon\right),\\ \theta_\epsilon&=\exp\left(\frac{\pi i}{2} (\epsilon^TK\epsilon+2N\epsilon_0)\right),\\ d_\epsilon&=(-1)^{\epsilon_0}. \end{align*} Its monodromy is $(-1)^{\epsilon^TK\eta}$, hence is nondegenerate. \end{theorem} \begin{proof} We first work in $\Ind(\mathscr V_{\mathcal B})$. The generic first-row category is braided equivalent to type-I $U_q(\mathfrak{sl}_2)$ modules with $q=e^{\pi i\psi_i}$; see \cite[Theorem 1.1, Theorems 5.10--5.11]{c:Koshida}. The first-column category uses $e^{\pi i\psi_i^{-1}}$. At edge $i$, the parameters of the row at $i$ and column at $i-1$ sum to $k_i$. Moriwaki's integer parameter shift theorem for type A \cite[Theorem 2.16]{c:Moriwaki} and quantum-coordinate-ring functor \cite[Lemma 3.4, Proposition 3.5]{c:Moriwaki} therefore transport the $\mathbb Z/2$-graded pieces of $\mathcal O_q(SL_2)$ to a lax braided functor whose source quadratic form has value $i^{k_i}$ on the odd grade. Reordering the edge factors and multiplying the row and column factors at each site gives a strong monoidal equivalence onto $\mathscr V_{\mathcal B}$; the monoidal structure is the usual braiding interchange, and $L_i(r,1)\bo L_i(1,s)\simeq L_i(r,s)$ by \cite[Theorem 5.2.5]{virasoro}. This functor is not braided for the naive product braiding. Indeed, balancing gives \[ h_i(r,s)-h_i(r,1)-h_i(1,s) =-\frac{(r-1)(s-1)}2, \] so the row/column double braiding is $(-1)^{(r-1)(s-1)}$. One must multiply the source braiding by the bicharacter \[ \beta(\epsilon,\eta)=(-1)^{\sum_i\epsilon_{i+1}\eta_i}. \] The exponent is bilinear over $\mathbb F_2$, so $\beta(\epsilon+\eta,\zeta)=\beta(\epsilon,\zeta)\beta(\eta,\zeta)$ and likewise in the second argument. Multiplying a braiding by $\beta$ therefore preserves both hexagons with the same associator. Its polarization is $\beta(\epsilon,\eta)\beta(\eta,\epsilon) =(-1)^{\sum_i(\epsilon_{i+1}\eta_i+\eta_{i+1}\epsilon_i)}$, exactly the row--column monodromy introduced by the interchange. This is the correction in the braided compatibility diagram for the multiplication functor. Transposing the convention corresponds to reversing that interchange. The resulting source self-braiding is \[ \prod_i i^{k_i\epsilon_i}\, (-1)^{\sum_i\epsilon_i\epsilon_{i+1}} =\exp\left(\frac{\pi i}{2}\epsilon^TK\epsilon\right). \] The zero-grade image is the canonical odd-label algebra $W$, and the $\epsilon$-grade image is $J_\epsilon$. Normalized abelian cocycles have trivial components whenever one entry is zero. The lax braided functor therefore supplies ordinary local $W$-module structures on every $J_\epsilon$ and balanced multiplication maps between them. These balanced maps are isomorphisms, not merely nonzero maps. Before the transports, $\mathcal O_q(SL_2)$ is strongly $\mathbb Z/2$-graded. In its standard presentation $a,b,c,d$ are odd and $ad-qbc=1$, so $1\in\mathcal O_1\mathcal O_1$. Strong grading implies \[ \mathcal O_a\otimes_{\mathcal O_0}\mathcal O_b \xrightarrow{\sim}\mathcal O_{a+b}. \] For example a finite decomposition $1=\sum_jx_jy_j$ in odd elements supplies the inverse to multiplication. The map is equivariant, hence so is its inverse. Taking $N$ factors and transporting through the grading twists and strong monoidal equivalences preserves these relative tensor isomorphisms. Thus $J_\epsilon\bo_WJ_\eta\simeq J_{\epsilon+\eta}$ in the local direct-limit category. Since $W$ is simple, invertibility also implies simplicity of all these modules. The total top weight of a $\mathcal B$-summand is \[ \sum_i h_i(r_i,r_{i+1})=\frac{r^TKr-N}{4}. \] For $r=\mathbf1+v$, with $v_i\geq0$, this equals \[ \frac14\sum_i(v_i-v_{i+1})^2+\frac N4v_0(v_0+2). \] Positive definiteness gives a lower bound and finite-dimensional weight spaces for each parity sum. Reduction modulo integers gives the displayed conformal twist. Dividing the twist by the self-braiding quadratic form gives the displayed dimension character. The polarization of $Q$ is $(-1)^{\epsilon^TK\eta}$; $\det K$ is odd, so it is nondegenerate. It remains to verify that the calculated fusion products are universal outside the chosen ambient category. Let $\mathcal Y$ be any $W$-intertwining operator of type $\binom X{J_\epsilon\ J_\eta}$, where $X$ is a grading-restricted generalized $W$-module. Restrict to finite $\mathscr V_{\mathcal B}$-subobjects $U\subset J_\epsilon$ and $V\subset J_\eta$. By \cite[Corollary 2.14]{cmy}, the image of this $\mathcal B$-intertwining operator is $C_1$-cofinite and grading restricted. That corollary permits an arbitrary generalized target; it is the direct-limit version of Miyamoto's image theorem \cite{c:Miyamoto}. Consequently the full image $\operatorname{Im}\mathcal Y$ is a union of $\mathscr V_{\mathcal B}$-submodules and belongs to $\Ind(\mathscr V_{\mathcal B})$. The intertwining Jacobi identity makes it a $W$-submodule. Factor $\mathcal Y$ through that image and use the already-established local-category tensor universal property. Composing with the inclusion into $X$ proves the claimed unrestricted universal property. \end{proof} \begin{remark}[Comparison of braiding conventions] The braided identification above uses the first-row equivalence of \cite{c:Koshida}. With the Kac weights fixed here, balancing gives row--column monodromy $(-1)^{(r-1)(s-1)}$. In particular, it equals $-1$ when both highest-weight parities are odd. Thus the trivial row--column braiding asserted in \cite[Proposition 6.3]{c:Koshida} does not agree with these weights. The monoidal multiplication functor of its Proposition 6.2 is sufficient for the construction; the displayed bicharacter supplies the required braiding correction. \end{remark} \Needspace{12\baselineskip} \part{Vacuum extensions and regularity} \section{Reduction of vacuum extensions to lowest weights zero and one} \label{neutral:sec:neutral-reduction} We prove regularity by splitting every extension of the vacuum by a simple module. The first step reduces this problem to simple modules of lowest weight zero or one. Let $W=W_N$ be the vertex operator algebra of Theorem~\ref{c:cyclic-gluing}. We use its conformal frame $\mathcal B=\bigotimes_{i=0}^{N-1}L(c_i,0)$, its frame decomposition, and generation by the frame and the elementary summands $R_j=M_{e_j}$; see Corollary~\ref{c:cyclic-embedding}. Thus \[ h_i(r,s)=\frac{(r\psi_i-s)^2-(\psi_i-1)^2}{4\psi_i},\qquad \psi_i=\frac{t+i+1}{t+i},\qquad t(t+N)=1, \] and, with cyclic indices, \[ \psi_j+\psi_{j-1}^{-1}=k_j, \qquad k_0=N+2,\quad k_j=2\quad(1\leq j0. \] The pairs are disjoint because equality of two signed Kac weights at irrational $\psi$ forces equality of their signed labels up to simultaneous reversal. \begin{lemma}\label{neutral:lem:frame-blocks} The restriction of a grading-restricted generalized $W$-module $E$ to $\mathcal B$ is the direct sum of its joint Virasoro block components. The projections onto these components are canonical $\mathcal B$-module maps and are exact on short exact sequences. Every fixed joint block component has finite length. \end{lemma} \begin{proof} The subspace generated by finitely many vectors under the joint positive Borel algebra is finite-dimensional. Indeed, positive modes lower total conformal weight, partial zero modes preserve total weight, the total grading is bounded below, and each total generalized weight space is finite-dimensional. The image of this Borel algebra is solvable: the positive modes strictly lower the finite set of relevant total weights and the zero modes commute. Lie's theorem therefore gives a finite flag with one-dimensional quotients, on which positive modes vanish. PBW induction is exact. Consequently the frame module generated by the chosen vectors is a quotient of a module with a finite filtration by tensor products of ordinary Virasoro Verma modules. It has finite length. We justify block separation also for generalized partial zero modes. In a finite generalized highest-weight module for one Virasoro factor, the nilpotent part of $L(0)$ commutes with all modes. In an extension of two nonisomorphic ordinary simple modules, that nilpotent part therefore factors through a homomorphism from the quotient simple to the submodule simple, and is zero. Such an extension is ordinary. An ordinary extension between two simples in different displayed blocks splits. If their highest weights are incongruent modulo $\mathbb Z$, weight cosets split it. Otherwise, dualize if necessary so that the quotient has the smaller highest weight. Its highest vector lifts to a highest vector of the extension, since the submodule has no weights below its larger highest weight. The resulting Verma-module map has zero intersection with the submodule: the Verma structure theorem lists no constituent of that highest weight in a different block. Its image is thus a section. Restricted contragredience is exact, so this also proves the other orientation. Finite-length induction now gives the canonical block decomposition for finite generalized highest-weight modules. The restriction of all of $E$ to one factor can have infinite partial weight multiplicities. For each vector, however, its one-factor positive-Borel orbit is finite by the total-weight argument above. The induced module has finite partial weight spaces and finite Verma length. Thus this restriction is a union of finite generalized highest-weight submodules. Their canonical block decompositions agree under inclusions. They define canonical projections on the union. The remaining factors commute with the chosen one and preserve its block components. Iterating gives the joint projections. Their functoriality and the direct-sum decompositions prove exactness. A fixed joint block contains at most $2^N$ simple frame types. Let $D$ be their finite set of total lowest weights. Every finite submodule of this component has length at most \[ \sum_{d\in D}\dim E_{[d]}. \] This follows by additivity of generalized weight-space dimensions along a composition series: each constituent contributes at least one dimension at its own total lowest weight. The lengths of the finitely generated submodules are therefore bounded integers. Choose one of maximal length. Adjoining any further vector gives a finite-length submodule of the same length, so the inclusion is an equality. Thus the chosen submodule is the entire joint block component. \end{proof} \subsection{The aligned odd collection} Call a joint frame block \emph{aligned and odd} if it has a simple constituent \[ \bigotimes_iL_i(r_i,s_i),\qquad r_i,s_i\in2\mathbb Z+1,\qquad r_i^2=s_{i-1}^2. \] Linkage preserves this condition because it changes only signs of the Kac labels. Let $\mathscr P$ be this collection of blocks, and let $E_{\mathscr P}$ be the sum of their components in $E$. \begin{lemma}\label{neutral:lem:aligned-odd-stable} If the total generalized conformal weights of $E$ are integers, then $E_{\mathscr P}$ is a $W$-submodule and has no negative total weights. The restriction of $W$ to the frame belongs to $\mathscr P$. \end{lemma} \begin{proof} We first record the necessary local null constraint, including the case of signed Kac labels. For a nonzero intertwining operator with first input $L(3,1)$, second input $L(r,s)$, and a simple highest-weight output of weight $x$, one has \[ (x-y)\bigl[(x-y-2\psi+1)(x-y-1)-4\psi y\bigr]=0, \qquad y=h(r,s). \] Its three roots are $h(r-2,s),h(r,s),h(r+2,s)$. The reciprocal formula for first input $L(1,3)$ changes $s$ by $-2,0,2$. The polynomial is the image of the elementary level-three singular vector in the Virasoro Zhu bimodule \cite[Proposition 5.2.3]{virasoro}. Necessity for arbitrary simple highest-weight second and output modules follows from the corrected injectivity theorem \cite[Lemma 5.5 and Propositions 5.6--5.8]{huangyang}: both the second module and the output contragredient are generated by their lowest spaces, and the first Zhu bimodule is cyclic on its highest class. Thus a nonzero intertwiner cannot have zero highest coefficient. No sufficiency assertion or fusion decomposition for signed-label modules is being used. For the joint frame these same necessary relations hold separately at each factor. Project the action of an elementary summand to a fixed source and target frame block. Both blocks have finite length by Lemma~\ref{neutral:lem:frame-blocks}. If this projected intertwiner is nonzero, restrict it to the first nonzero source-filtration term; it factors through that term's simple quotient. The span of all coefficients is a frame submodule, by the intertwiner commutator formula. Choose the smallest target-filtration term containing this coefficient span and project to its simple quotient. Minimality makes the resulting intertwiner between simple subquotients nonzero. The preceding necessary constraints therefore determine the possible target blocks even when the original blocks are not semisimple. The summand $R_j$ changes only $r_j$ and $s_{j-1}$, each by $0,\pm2$. Let $\Delta_r$ and $\Delta_s$ be the respective changes of their squares. The possibly irrational part of the change of total highest weight is \[ \frac14\bigl(\psi_j\Delta_r+\psi_{j-1}^{-1}\Delta_s\bigr) =\frac14\bigl(\psi_j(\Delta_r-\Delta_s)+k_j\Delta_s\bigr). \] All other terms are rational. Both highest weights occur in the integer-graded module $E$, so irrationality of $\psi_j$ forces $\Delta_r=\Delta_s$. Alignment and odd parity are preserved. The frame and every $R_j$ therefore preserve $E_{\mathscr P}$; the vertex-generation theorem proves $W$-stability. For nonnegativity, reverse both labels at each site if necessary to write $r_i=\rho_i>0$ and $s_i=\epsilon_i\rho_{i+1}$, where every $\rho_i$ is odd and $\epsilon_i\in\{1,-1\}$. Direct substitution gives \begin{equation}\label{neutral:eq:aligned-positive-weight} 4\sum_i h_i(r_i,s_i) =\sum_i(\rho_i-\rho_{i+1})^2+N(\rho_0^2-1) +4\sum_{i:\epsilon_i=-1}\rho_i\rho_{i+1}\geq0. \end{equation} Frame descendants only increase total weight. Finally, the constructed summands $M_m$ of $W$ have $r_i=s_{i-1}=2m_i+1>0$, so belong to this collection. \end{proof} \begin{lemma}[Vacuum-like vectors]\label{neutral:lem:vacuum-like} If a vector $v$ of a weak $W$-module is killed by the total translation operator $D=L(-1)$, then $a_nv=0$ for every $a\in W$ and every $n\geq0$. The map $a\mapsto a_{-1}v$ is a $W$-module homomorphism from $W$ to the module containing $v$. \end{lemma} \begin{proof} Lower truncation gives $a_nv=0$ for all sufficiently large integers $n$. Descending induction using $[D,a_n]=-n a_{n-1}$ proves the vanishing for $n\geq0$. The module Jacobi identity then gives $b_m(a_{-1}v)=(b_ma)_{-1}v$, which is the asserted homomorphism. \end{proof} \begin{proposition}\label{neutral:prop:two-lowest-weights} Let $S$ be a simple grading-restricted $W$-module. Every nonsplit grading-restricted generalized extension \[ 0\longrightarrow S\longrightarrow E \longrightarrow W\longrightarrow0 \] has $E=E_{\mathscr P}$, has $S\not\cong W$, and has lowest weight $h(S)\in\{0,1\}$. On its lowest space, the possible joint generalized partial spectra are $\{0\}$ if $h(S)=0$, and $\{e_0,\ldots,e_{N-1}\}$ if $h(S)=1$. In the weight-zero case the joint generalized spectrum on all of $E_0$ is also zero and $L(0)=0$ there exactly. \end{proposition} \begin{proof} An irreducible lower-bounded generalized module is ordinary: the kernel of the nilpotent part of $L(0)$ is a nonzero submodule, hence the entire module. Its weights lie in one coset modulo $\mathbb Z$. If the coset differs from the vacuum coset, the two total weight cosets split the extension. In the remaining case $E$ is integer graded. The map $E\to W$ restricts to a surjection $E_{\mathscr P}\to W$, by exactness of the frame-block projections. Its kernel is $S\cap E_{\mathscr P}$, which is either zero or all of $S$ by simplicity and Lemma~\ref{neutral:lem:aligned-odd-stable}. If the kernel is zero, this restriction is an isomorphism onto $W$ and gives a section. If the kernel is $S$, then $E_{\mathscr P}$ contains $S$ and surjects onto $E/S$, so $E=E_{\mathscr P}$. Lemma~\ref{neutral:lem:aligned-odd-stable} then excludes negative weights. If $h(S)>1$, the unique vacuum lift $v\in E_0$ has $Dv=0$ because $E_1=0$. Lemma~\ref{neutral:lem:vacuum-like} splits the extension. The same argument handles $S\cong W$: here $E_1=0$ because $W_1=0$, and any lift of the vacuum is killed by $D$. Hence only the two stated lowest weights remain, with $S\not\cong W$. To determine the partial spectra, apply \eqref{neutral:eq:aligned-positive-weight}. If the positive odd sequence $(\rho_i)$ is nonconstant, its cyclic differences include at least two nonzero even integers, so its total lowest weight is at least two. If it is constant and greater than one, that weight is at least $2N$. At weights zero and one, therefore, all $\rho_i=1$. No negative sign gives the vacuum tuple $(0,\ldots,0)$; exactly one negative sign at site $j$ gives the tuple $e_j$. The simple vacuum frame module has no total weight-one descendants. Composition series of the finite frame blocks consequently give exactly the stated possible generalized spectra, with no assertion of partial semisimplicity. Finally, the nilpotent part of total $L(0)$ on $E$ vanishes. It is a module endomorphism, kills $S$, is zero on the quotient, and so factors through a homomorphism $W\to S$. That homomorphism is zero because $S\not\cong W$. In particular $L(0)=0$ on $E_0$ in the weight-zero case. \end{proof} \subsection{Exact identities for a weight-one vacuum defect} Suppose now $h(S)=1$, choose the unique vacuum lift $v\in E_0$, and set \[ H_i=L_i(0),\qquad w_i=L_i(-1)v,\qquad w=\sum_iw_i. \] Every positive-weight zero mode and every degree-lowering mode kills $v$. The $w_i$ are jointly highest and satisfy $H_iw_j=\delta_{ij}w_j$. A nonsplit extension has $w\ne0$ by Lemma~\ref{neutral:lem:vacuum-like}. For a positive-weight joint frame eigenvector $a\in W$, of total weight $p$ and partial weights $h_i$, put $r_a=a_{p-2}v$. The following identities hold without a semisimplicity assumption on the partial zero modes acting on $S_1$: \begin{align} o(a)w_i&=(h_i-H_i)r_a,& o(a)w&=(p-1)r_a,\label{neutral:eq:neutral-first-image}\\ (L_i(-n)a)_{p+n-2}v&=(-1)^n o(a)w_i &&(n\geq1).\label{neutral:eq:neutral-desc-first-image} \end{align} Indeed, the iterate formula for $(\omega_i{}_{1-n}a)_{p+n-2}v$ leaves only the term containing $\omega_i{}_0v=w_i$: all $a$-modes of degree at most zero on $v$ vanish and $\omega_i{}_kv=0$ for $k\geq1$. This gives \eqref{neutral:eq:neutral-desc-first-image}. The commutator with $H_i$ at mode $p-2$ gives the first identity in \eqref{neutral:eq:neutral-first-image}; summing it uses $\sum_iH_i=1$ on $S_1$. \begin{lemma}\label{neutral:lem:first-image-cubic} If the local factor of a joint primary $a$ is $(3,1)$ or $(1,3)$, with partial weight $h$, then its first image satisfies \[ H_i(H_i-1)(H_i-h)r_a=0. \] In particular, its components at partial eigenvalues zero and one are ordinary at that site. \end{lemma} \begin{proof} The local singular vector is \[ (L_i(-1)^3-2(h+1)L_i(-2)L_i(-1)+h(h+1)L_i(-3))a=0. \] Write $X=H_i$. Equations \eqref{neutral:eq:neutral-first-image}--\eqref{neutral:eq:neutral-desc-first-image} give, respectively, the following polynomials on $r_a$ for its three terms: \[ (X-h)(X-h-1)(X-h-2),\quad -(X-h)(X-h-1),\quad X-h. \] Their stated linear combination is $(X-h)X(X-1)$. Here $h$ is irrational, so these roots are distinct. The assertion follows on each finite generalized eigenspace. This is a direct first-image mode computation, rather than an ordinary Zhu calculation applied to a degree-raising mode. \end{proof} For homogeneous states $a,b$ of weights $p,q\geq2$, the usual mode commutator and \eqref{neutral:eq:neutral-first-image} give \begin{equation}\label{neutral:eq:neutral-defect-product} o(a)o(b)w=(q-1)\sum_{\ell=0}^{p-1} \binom{p-1}{\ell}\frac{o(a_\ell b)w}{p+q-\ell-2}. \end{equation} To see this, replace $o(b)w$ by $(q-1)b_{q-2}v$, commute $o(a)$ past $b_{q-2}$, and use $o(a)v=0$. The summand is the first image of $a_\ell b$, whose weight is $p+q-\ell-1$. For two weight-two states, skew symmetry and $W_1=0$ reduce this to \[ \{o(a),o(b)\}w=o(a_1b)w. \] Together with the ordinary-root identities proved below, this anticommutator relation shows that $U=\operatorname{span}\{w_i\}$ is invariant under every ordinary interval zero mode: one adjoins adjacent disjoint intervals inductively. We will use this invariant subspace without requiring a description of all of $S_1$. We also use the exact partial-conformal Zhu identity \begin{equation}\label{neutral:eq:neutral-partial-Zhu} [L_i(-n)a]=(-1)^n\bigl(-[\omega_i]*[a]+n[a]*[\omega_i]+h_i[a]\bigr), \qquad n\geq1. \end{equation} For $n=1$ this is the Zhu commutator formula; $n=2$ follows by expanding $[\omega_i]*[a]$. The higher circle relations give the recurrence $[L_i(-n-2)a]+2[L_i(-n-1)a]+[L_i(-n)a]=0$, proving all remaining cases. When it is iterated, $h_i$ is the weight of the descendant currently being reduced, not the weight of its original primary. \subsection{An algebraic criterion for second-order nilpotence} The following form of Nakayama's lemma controls the partial nilpotent operators on the first cyclic image. \begin{lemma}\label{neutral:lem:cut-Nakayama} Let $V$ be finite-dimensional, let commuting operators $H_i$ have joint generalized spectrum $e_j$ on a component, and suppose $\sum_i H_i=1$. Let $r$ be a vector in that component. Suppose polynomials $K_0,\ldots,K_{N-2}$ vanish at $e_j$, their gradients are independent modulo $\sum_i(H_i-\delta_{ij})=0$, and every $K_a(H)r$ is an ordinary simultaneous eigenvector. Then \[ (H_i-\delta_{ij})(H_k-\delta_{kj})r=0\qquad\text{for all }i,k. \] \end{lemma} \begin{proof} On the cyclic space generated by $r$, let $\mathcal A$ be the finite commutative algebra generated by $n_i=H_i-\delta_{ij}$. It is local with nilpotent maximal ideal $\mathfrak m=(n_i)$. Invertibility of the gradients says that the $K_a$ span $\mathfrak m/\mathfrak m^2$; Nakayama's lemma makes them generators of $\mathfrak m$. Ordinariness of their values gives $\mathfrak mK_a(H)r=0$, so $\mathfrak m^2r=0$. The maximal ideal is nilpotent because $\mathcal A$ is a finite local algebra. The argument therefore requires no additional bound from a $(3,3)$ singular vector. \end{proof} \subsection{The ordinary-root defect calculation} The ordinary roots determine the initial defect space without any classification of its ambient module. We give the calculation because it is used in every interval equation below. \begin{lemma}\label{neutral:lem:ordinary-root-defect} For an ordinary root $R=R_j$, $1\leq j3),\qquad \kappa\ne0. \] The other singular coefficients are level-one, level-two, and level-three descendants of \(C\). For a rank-one parameter \(p\), the \((3,1)\) followed by \((1,3)\) descendant coefficients are \begin{align*} v_1&=\frac p{p-1}L_{-1}C,\\ v_2&=-\frac{2p}{(p-2)(2p-1)}L_{-2}C +\frac{p^2(2p-3)}{2(p-2)(p-1)(2p-1)}L_{-1}^2C,\\ v_3&=-\frac{p+2}{3(p-2)}L_{-3}C -\frac{2p}{3(p-2)(p-1)}L_{-2}L_{-1}C +\frac{p^2}{6(p-2)(p-1)}L_{-1}^3C. \end{align*} Reversing the inputs gives \begin{align*} \widetilde v_1&=-\frac1{p-1}L_{-1}C,\\ \widetilde v_2&=-\frac{2p}{(p-2)(2p-1)}L_{-2}C +\frac{3p-2}{2(p-2)(p-1)(2p-1)}L_{-1}^2C,\\ \widetilde v_3&=\frac{2p+1}{3(2p-1)}L_{-3}C -\frac{2p}{3(p-1)(2p-1)}L_{-2}L_{-1}C +\frac1{6(p-1)(2p-1)}L_{-1}^3C. \end{align*} The Ward equations for \(L_1,L_2,L_3\) determine these coefficients uniquely, since the output has no singular vector below level nine. \subsection{The all-cut polynomial system} For each prefix cut \(a=0,\ldots,N-2\), let \(B_a=P_{[0,a]}\), \(A_a=P_{[a+1,N-1]}\), and normalize \[ (B_a)_3A_a=\kappa_a C,\qquad v_a=\kappa_a^{-1}o(B_a)o(A_a)w,\qquad \kappa_a\ne0. \] Put \[ r=t+a,\qquad Z=\sum_{i2K=4N+4\), so it cannot occur in the Zhu square. Every contribution of the self-channel annihilates \(U\). At a local degenerate weight \(h\), evaluation of the normalized vacuum channel on Virasoro highest weight one has exact series \[ V_h(z)={}_3F_2\!\left( \begin{matrix}-h-1,-h,2\\1-h/2,(1-3h)/2\end{matrix}; -\frac{z^2}{4(1+z)}\right). \] Appendix \ref{r:app:heavy-vacuum-obstruction} derives this formula from the level-three null relation and proves its uniqueness as the normalized vacuum solution. On \(w_0\), the active partial weights are \((1,0)\), so the other endpoint vacuum series evaluates to \(1\). The resulting Zhu coefficient is \begin{equation}\label{r:eq:finalnonzero} \begin{split} P_K(a)&=[z^{2K}](1+z)^K V_a(z)\\ &=\left(-\frac14\right)^K \frac{(-a-1)_K(-a)_K(2)_K} {(1-a/2)_K((1-3a)/2)_K\,K!}\ne0. \end{split} \end{equation} To see the extraction directly, write \(V_a(z)=\sum_nd_nz^{2n}(1+z)^{-n}\). For \(nK\), its lowest degree is \(2n>2K\). Only \(d_K\) remains. Irrationality of \(a\) excludes all zeros and poles of the finite Pochhammer products. Let \(\nu_0\ne0\) be the vacuum-channel coefficient. Its nonvanishing follows from the nondegenerate invariant form on the multiplicity-one self-contragredient frame summand \(M_{e_0}\). Applying \(R_0*R_0\) to \(w_0\) gives \[ 0=\nu_0P_K(a)w_0\ne0, \] a contradiction. This proves the theorem. \end{proof} Only the closing-root first-image vanishing is used in this proof; no description of repeated cyclic action on the full target top is required. \section{Vacuum projectivity and regularity} \label{r:sec:projectivity} The passage from vacuum projectivity to semisimplicity is categorical. We isolate it from the extension calculations, using the rigidity criterion of Etingof--Penneys \cite[Theorem 1.1]{ep}. \begin{lemma}[A projective-unit criterion]\label{lem:projective-unit} Let $\mathcal C$ be a finite $\mathbb C$-linear abelian braided monoidal category with simple unit $\one$ and right-exact tensor product. Suppose it has a contravariant duality $X\mapsto X'$ with natural isomorphisms \[ \operatorname{Hom}(Y\boxtimes X,\one) \cong\operatorname{Hom}(Y,X'). \] If $\one$ is projective, then $\mathcal C$ is rigid and semisimple. \end{lemma} \begin{proof} For every nonzero $X$, the identity of $X'$ corresponds to a nonzero map $e_X:X'\boxtimes X\to\one$. Since the unit is simple, $e_X$ is an epimorphism; projectivity supplies a section. Thus $\one$ is a direct summand of $X'\boxtimes X$. Every indecomposable $X$ is therefore non-negligible in the sense of Etingof--Penneys. This summand argument does not assume the snake identities. We verify moderate growth without using tensor exactness. Write $\ell$ for composition length, let $S_i$ run through the finitely many simple objects, and set $c_X=\max_i\ell(S_i\boxtimes X)$. Right exactness and induction on a composition series give: \[ \ell(Y\boxtimes X)\le c_X\ell(Y),\qquad \ell(X^{\boxtimes n})\le\ell(X)c_X^{n-1}. \] Every simple endomorphism algebra is $\mathbb C$. Induction on length in the two variables, using left exactness of Hom, gives $\dim\operatorname{Hom}(A,B)\le\ell(A)\ell(B)$. Consequently: \[ \dim\operatorname{End}(X^{\boxtimes n}) \le\ell(X)^2c_X^{2n-2}d(k)}$. Thus the deformed block is $F(k)$. The same minimum-height argument excludes all other blocks. \end{proof} \begin{proposition}[Downward charge transitions] \label{g:prop:downward-links} For every elementary generator $R_j$ and every $k$, the component of its vertex action from $E_k$ to $E_{k-\beta_j}$ is nonzero. \end{proposition} \begin{proof} The normalized highest vector $v_j\in R_j$ has lowest charge $-\beta_j$ and leading term $e^{-\beta_j}$, including for the closing generator $j=0$. All its other charges have strictly larger scalar height. The normalized $f_k$ has lowest charge $k$. Therefore the minimum-charge part of $Y(v_j,z)f_k$ is exactly $Y(e^{-\beta_j},z)e^k$, which is nonzero. Projecting to the canonical frame block of $F(k-\beta_j)$ retains this leading term. Hence the asserted block component is nonzero. \end{proof} A nonzero submodule could a priori be supported on a proper subset closed under downward charge transitions. To prove simplicity, we also need the upward transitions. They are obtained from the monodromy of the level-three BPZ equation in the next section. \section{BPZ monodromy and simplicity of the periodic sectors} \label{g:sec:bpz-periodic-simplicity} We prove that every elementary generator acts nontrivially in both charge directions. The argument combines the frame decomposition in Proposition~\ref{g:prop:periodic-frame} with Mimachi's irreducibility criterion for the Dotsenko--Fateev equation \cite[Theorem 11]{mimachi}. At this stage we use only the frame restrictions of the lattice modules; the corresponding statement for arbitrary modules is proved in Section~\ref{g:sec:generic-frame-semisimple}. \subsection{Frame labels and vacuum coefficients} Write $\mathcal B=\bigotimes_iL(c_i,0)$ and $M_\xi=V_{L+\xi}\downarrow W_N$, where the odd-order component of $\xi\in L^*/L$ is nonzero. The exact frame decomposition is \[ M_\xi=\bigoplus_{k\in\xi+\mathbb Z^N}E_k. \] Every $E_k$ is an irreducible tensor product of generic Virasoro Verma modules, occurs once, and has highest vector $f_k$ with rational Kac labels \[ R_i=1-2k_i,\qquad S_i=1-2k_{i+1}. \] All these labels have odd denominator. At every site at least one of $R_i,S_i$ is nonintegral. The summands $E_k$ are pairwise nonisomorphic. The elementary generator $v_j\in R_j$ has $(3,1)$ at site $j$, $(1,3)$ at site $j-1$, and the vacuum elsewhere. Put \[ a=\psi_j,\qquad b=\psi_{j-1}^{-1},\qquad \kappa_j=a+b=\begin{cases}2,&j\ne0,\\N+2,&j=0.\end{cases} \] Both $a,b$ are positive irrational numbers. At site $j-1$ use the equivalent parametrization $b=\psi_{j-1}^{-1}$, interchanging the two Kac labels. Thus both active factors of $v_j$ are of type $(3,1)$, at parameters $a$ and $b$ respectively. Their source momenta are \[ p=aR_j-S_j,\qquad q=bS_{j-1}-R_{j-1}. \] In particular \[ p+q=\kappa_j-2-2(K_+k)_j\in\mathbb Z. \] The generator has a nonzero vacuum two-point coefficient: \[ (v_j)_{2\operatorname{wt}(v_j)-1}v_j=\nu_j\mathbf1, \qquad \nu_j\ne0. \] This holds also for $j=0$. Indeed the cyclic construction is a multiplicity-free sum of pairwise distinct self-dual simple frame modules. Its nondegenerate invariant form restricts nondegenerately to each of them; distinct summands are orthogonal. Its restriction to the one-dimensional highest-weight line of $R_j$ is therefore nonzero. Equivalently, one may use directly the nonzero vacuum channel of the canonical diagonal algebra in the cyclic-gluing theorem. \subsection{Exact identification of the third-order local system} For an irrational parameter $a$, write \[ d=2a-1,\qquad H=\frac{p^2-(a-1)^2}{4a}. \] The level-three singular relation for the $(3,1)$ primary $X$ is \[ \bigl(L_{-1}^3-4aL_{-2}L_{-1}+2a(2a-1)L_{-3}\bigr)X=0. \] Consider the Virasoro four-point blocks $F(z)=\langle H|X(1)X(z)|H\rangle$. Ward identities give \[ F'''-4aU F''+(-4aV+2adT)F'+2adW F=0, \tag{BPZ}\label{g:eq:bpz} \] where \begin{align*} U&=\frac{1-2z}{z(z-1)},\\ V&=\frac{H}{z^2}+\frac{d}{(z-1)^2} -\frac{2d+1}{z(z-1)},\\ T&=\frac{3z^2-3z+1}{z^2(z-1)^2},\\ W&=-\frac{2H}{z^3}-\frac{2d}{(z-1)^3} -\frac{2d}{z^2}+\frac{2d}{(z-1)^2}. \end{align*} For example, translation and dilation give $\partial_0F=(z-1)F'+2dF$ and $\partial_1F=-zF'-2dF$. Applying these identities to the $L_{-1}X$ insertion gives the displayed $V$ coefficient. Thus the operator includes its accessory parameter, rather than being inferred only from its local exponents. To compare this equation with \cite[equation (1.1)]{mimachi}, denote the Dotsenko--Fateev parameters by $A,B,C,G$ and set: \[ A=a-1-p,\qquad B=C=-2a,\qquad G=2a. \] His equation (1.1) is \begin{align*} &z^2(z-1)^2I'''+(K_1z+K_2(z-1))z(z-1)I''\\ &\qquad +(L_1z^2+L_2(z-1)^2+L_3z(z-1))I' +(M_1z+M_2(z-1))I=0, \end{align*} with \begin{align*} K_1&=-G-3B-3C,\qquad K_2=-G-3A-3C,\\ L_1&=(B+C)(2B+2C+G+1),\\ L_2&=(A+C)(2A+2C+G+1),\\ L_3&=(B+C)(2A+2C+G+1)+(A+C)(2B+2C+G+1)\\ &\quad +(C-1)(A+B+C)+(3C+G)(A+B+C+G+1),\\ M_1&=-C(2B+2C+G+1)(2A+2B+2C+G+2),\\ M_2&=-C(2A+2C+G+1)(2A+2B+2C+G+2). \end{align*} The substitution \[ F(z)=z^{-A}(1-z)^{2a}I(z) \] turns this equation exactly into \eqref{g:eq:bpz}. One checks this by setting $w=-A/z+2a/(z-1)$ and replacing \begin{align*} I'&=z^A(1-z)^{-2a}(F'-wF),\\ I''&=z^A(1-z)^{-2a}(F''-2wF'+(w^2-w')F),\\ I'''&=z^A(1-z)^{-2a} (F'''-3wF''+(3w^2-3w')F'+(-w^3+3ww'-w'')F). \end{align*} Substitution recovers each of the three monic coefficients in \eqref{g:eq:bpz}. The two local systems therefore differ by a rank-one twist, which preserves irreducibility. \begin{lemma}[A sufficient generic irreducibility criterion] \label{g:lem:bpz-irreducible} The rank-three local system \eqref{g:eq:bpz} is irreducible if \[ p\notin(a+\mathbb Z)\cup(-a+\mathbb Z) \cup(2a+\mathbb Z)\cup(-2a+\mathbb Z) \cup(\tfrac12+\mathbb Z). \] If $p=aR-S$, with $R,S$ rational of odd denominator and not both integral, this condition holds. Under this additional rational-label hypothesis, its three exponents at zero \[ \lambda_+=1-a+p,\qquad \lambda_0=1-2a,\qquad \lambda_-=1-a-p \] are pairwise distinct modulo $\mathbb Z$. \end{lemma} \begin{proof} Mimachi's Theorem 11 requires the following eleven quantities to be nonintegral: \begin{gather*} A,\ A+G/2,\ B,\ B+G/2,\ C,\ C+G/2,\\ A+B+C+G/2,\ A+B+C+G,\ (G+1)/2,\\ A+C+(G+1)/2,\ B+C+(G+1)/2. \end{gather*} After substitution the nonautomatic conditions are exactly $a-1-p,2a-1-p,-2a-1-p,-a-1-p,-p-1/2\notin\mathbb Z$. All other conditions follow from irrationality of $a$. If $p=ma+n$ with $m,n\in\mathbb Z$, irrationality gives $R=m$ and $S=-n$, contrary to the hypothesis. If $2p\in\mathbb Z$, irrationality gives $R=0$ and $2S\in\mathbb Z$; the odd-denominator condition then makes $S$ integral, again a contradiction. These observations prove that the rational-label hypothesis implies the criterion and also gives nonresonance of the three exponents, whose differences are $a+p,a-p,2p$. Nonresonance is not asserted from the irreducibility criterion alone: that criterion need not exclude $2p\in\mathbb Z$. \end{proof} \subsection{Factorization and a finite-jet invariance lemma} Fix $k,j$ and let $f_k^*$ be the grade-zero functional on the frame summand $E_k$, normalized by $f_k^*(f_k)=1$ and extended by zero on the other summands. This is a homogeneous vector in the restricted dual of $M_\xi$, and is a highest vector for the contragredient frame action. Consider \[ \mathcal F(z)=f_k^*Y(v_j,1)Y(v_j,z)f_k. \] It is a rational function of $z$ by the ordinary VOA module Jacobi identity. In particular, it is single-valued about both zero and one. The exact frame decomposition justifies the following factorization without a rationality assumption on the frame or on $W_N$. Project the inner action onto an intermediate $E_l$. The singular relation for each active $(3,1)$ factor restricts its intermediate highest weight to the three values with momentum $p,p\pm2a$ at site $j$, and $q,q\pm2b$ at site $j-1$. This is also obtained directly from the indicial polynomial of \eqref{g:eq:bpz}. Every other site is unchanged. All these intermediate Verma modules are generic: changing a rational Kac label by an even integer does not make a nonintegral pair integral. Their nondegenerate Shapovalov forms and the Virasoro Ward identities uniquely determine each projected four-point block from its leading three-point constants. The PBW bases and the inverse Shapovalov forms are tensor products, so these projected blocks factor into the product of the two rank-three Virasoro blocks. This is a direct formal-series factorization; regular-singular ODE convergence makes it analytic near zero. Let $\boldsymbol f(z)=(f_+(z),f_0(z),f_-(z))^T$ and $\boldsymbol g(z)=(g_+(z),g_0(z),g_-(z))^T$ be the normalized Frobenius bases of the two equations. There is therefore a constant matrix $Q$ such that \[ \mathcal F(z)=\boldsymbol f(z)^TQ\boldsymbol g(z). \] More generally the two-variable expression \[ \mathcal H(z_1,z_2)=\boldsymbol f(z_1)^TQ\boldsymbol g(z_2) \] has the property \begin{equation} \left.\partial_{z_1}^r\partial_{z_2}^s \mathcal H(z_1,z_2)\right|_{z_1=z_2=z} =f_k^*Y(v_j,1) Y\bigl(L_j(-1)^rL_{j-1}(-1)^sv_j,z\bigr)f_k. \label{g:eq:nine-descendants} \end{equation} Here the componentwise $L(-1)$ derivative identity is applied to the factorized frame intertwiners. Its constants are the same $Q$: the descendants are obtained by differentiating the same moving insertion. Each right-hand side is again a rational VOA module matrix coefficient. \begin{lemma}[Nine rational derivatives force tensor invariance] \label{g:lem:nine-jets} The matrix $Q$ defines a monodromy-invariant tensor in the tensor product of the two rank-three local systems. \end{lemma} \begin{proof} After continuation along a loop about zero or one, write the new coefficient matrix, in the original local bases, as $Q'$. Rationality of \eqref{g:eq:nine-descendants} for $0\le r,s\le2$ implies \[ \boldsymbol f^{(r)}(z)^T(Q'-Q)\boldsymbol g^{(s)}(z)=0 \qquad(0\le r,s\le2). \] At any common ordinary point of the two differential equations, their two Wronskian matrices are invertible. Stacking these nine equations therefore gives $Q'=Q$. The two loops generate the fundamental group. The nine descendant coefficients are essential here. The scalar identity for $\mathcal F$ alone need not determine $Q$, because multiplication of the two solution spaces can have a kernel. \end{proof} \subsection{All elementary charge channels are nonzero} \begin{proposition} \label{g:prop:all-periodic-channels} For every $k\in\xi+\mathbb Z^N$ and every elementary index $j$, both components \[ R_j\times E_k\longrightarrow E_{k-\beta_j},\qquad R_j\times E_k\longrightarrow E_{k+\beta_j} \] of the $W_N$ vertex action are nonzero. This includes the closing index $j=0$. \end{proposition} \begin{proof} The two source Kac pairs satisfy Lemma~\ref{g:lem:bpz-irreducible}, so both rank-three monodromy representations are irreducible. The invariant tensor $Q$ is nonzero. Indeed the vacuum channel of $Y(v_j,1)Y(v_j,z)$ gives \[ \mathcal F(z)=\nu_j(1-z)^{-2\operatorname{wt}(v_j)} +\text{terms with smaller pole order}, \qquad \nu_j\ne0. \] The remaining $W_N$ states have positive conformal weight, and $f_k^*(f_k)=1$. Thus $\mathcal F$ is not zero. An invariant tensor between two irreducible three-dimensional representations is a homomorphism from the dual of one to the other. Since it is nonzero, it is an isomorphism. Hence $Q$ is invertible. The intermediate frame summands in the fixed lattice coset are limited to: \[ E_{k-\beta_j},\qquad E_k,\qquad E_{k+\beta_j}. \] Indeed, equality of rational Kac weights at an irrational parameter identifies the two Kac pairs up to simultaneous sign. A negative sign between two pairs whose charges are in the same coset would force both adjacent odd-discriminant coordinates to be zero, which is excluded. Thus at all inactive sites the labels agree literally, and at the two active sites their changes must be the same $0,2,$ or $-2$. In the two Frobenius bases, $Q$ is consequently diagonal, with entries belonging to these three intermediate summands. Equivalently, this diagonality follows from local monodromy. The exponents for the second factor are $1-b+q,1-2b,1-b-q$. Since $p+q\in\mathbb Z$, the matched pairs have integral sums. The unmatched corner pairs would require $2p\in\mathbb Z$; a pair involving just one middle exponent would require $p\in\pm a+\mathbb Z$. Both are impossible by the arithmetic in Lemma~\ref{g:lem:bpz-irreducible}. Invertibility makes each diagonal entry nonzero. The entry for $E_{k\pm\beta_j}$ is the product of the two corresponding three-point coefficients, for the passage from $E_k$ to that intermediate summand and back. Its nonvanishing implies that each of those two actions is nonzero, proving the claim. \end{proof} \begin{theorem}[Simplicity of every nonzero odd lattice restriction] \label{g:thm:periodic-simple} For every odd $N\ge3$ and every $\xi\in L^*/L$ whose odd-order component is nonzero, the restriction \[ V_{L+\xi}\downarrow W_N \] is simple. Its character is therefore the genuine simple-module character \[ \operatorname{ch}M_\xi(\tau) =\frac1{\eta(\tau)^N} \sum_{k\in\xi+\mathbb Z^N} q^{(k-\mathbf1/2)^TK_+(k-\mathbf1/2)}. \] Here $k$ is the coordinate column in the basis $(\beta_i)$, so the lattice vector $B/2$ has coordinate column $\mathbf1/2$. \end{theorem} \begin{proof} A nonzero submodule is a direct sum of some of the pairwise nonisomorphic simple frame summands $E_k$. Proposition \ref{g:prop:all-periodic-channels} makes its charge support closed under both $k\mapsto k+\beta_j$ and $k\mapsto k-\beta_j$, for every $j$. The lattice graph is connected, so its support is the entire coset. The submodule is all of $M_\xi$. The character is the ambient lattice character computed above. \end{proof} \section{Frame semisimplicity in generic rational sectors} \label{g:sec:generic-frame-semisimple} We next pass from the lattice modules to simple $W_N$-modules in nonzero rational sectors. The main issue is that semisimplicity of the total conformal operator does not imply semisimplicity of its commuting partial operators. The level-three singular vectors give polynomial relations for those partial operators separately; their simple roots eliminate the possible nilpotent parts. \begin{lemma}[Structure of a generic generalized frame block] \label{g:lem:generic-generalized-Verma} Let $\Lambda$ be a joint frame block whose unique simple object is the irreducible Verma tensor $F_\lambda$ of joint highest weight $\lambda$. Every finite-length module $Y$ in this block has the form \[ Y\cong U(\mathfrak n_-)\otimes T, \qquad T=Y_{[\lambda]}, \] where all positive modes annihilate $T$ and $L_i(0)|_T=\lambda_i+N_i$ for commuting nilpotent $N_i$. Its ordinary frame socle is \begin{equation} \operatorname{soc}_{\mathcal B}(Y) =U(\mathfrak n_-)\otimes\bigcap_i\ker N_i. \label{g:eq:generic-socle} \end{equation} Both $Y$ and its restricted contragredient are generated by their joint highest spaces. Moreover the Zhu annihilation space is exactly $\Omega(Y)=T$: there are no additional vectors annihilated by all positive Virasoro modes at a strictly higher joint degree. \end{lemma} \begin{proof} All simple constituents have weights in $\lambda+\mathbb Z_{\geq0}^N$, so the same is true of $Y$, and positive modes kill $T$. Induction from this highest space gives a generalized Verma tensor $\mathcal V(T)$ and a map $\mathcal V(T)\to Y$. The commuting nilpotents on $T$ admit a flag with one-dimensional trivial nilpotent quotients. Induction is exact by PBW. Hence $\mathcal V(T)$ has a finite flag of irreducible ordinary Verma tensors $F_\lambda$. The cokernel of the displayed map has zero joint highest space; since every nonzero object in the block has a constituent $F_\lambda$, exact highest-space extraction makes that cokernel zero. The kernel has zero joint highest space by construction and vanishes for the same reason. This proves the asserted description. A vector in $\bigcap_i\ker N_i$ generates an ordinary irreducible Verma tensor, and every simple submodule must start at joint weight $\lambda$. This proves \eqref{g:eq:generic-socle}. Restricted contragredience preserves finite length and the single simple type, so the same argument applies to $Y'$. For the last assertion, the Shapovalov matrix at every positive joint degree is invertible at the ordinary generic highest weight $\lambda$. Replacing its entries by their values at the commuting operators $\lambda_i+N_i$ changes this invertible matrix by a nilpotent matrix over the finite local algebra generated by the $N_i$. It remains invertible. Therefore a positive-degree vector killed by all positive Virasoro modes must be zero. The highest space is killed by every degree-lowering mode of the frame VOA, proving $\Omega(Y)=T$. \end{proof} \begin{lemma}[The elementary generalized top constraint] \label{g:lem:generic-top-constraint} Let $F$ be an ordinary irreducible Verma tensor with rational Kac pairs of odd denominator, with no Kac pair consisting of two integers. Let $Y$ be a finite-length generic joint block, ordinary for the \emph{total} conformal zero mode. Every frame intertwining operator obtained by restricting the $W_N$ vertex action, \[ \mathcal I:\quad R_j\times F\longrightarrow Y, \] has image contained in $\operatorname{soc}_{\mathcal B}(Y)$. \end{lemma} \begin{proof} Write $T=Y_{[\lambda]}$ and $L_i(0)|_T=\lambda_i+N_i$ as in Lemma~\ref{g:lem:generic-generalized-Verma}. Since total $L(0)$ is semisimple, $\sum_iN_i=0$, but individual $N_i$ need not vanish a priori. Let $f$ be the highest vector of $F$ with partial weights $h_i$. For a homogeneous $u\in R_j$, project $\mathcal I(u,z)f$ to $T$ and take its unique total-weight-compatible coefficient. Denote it by $t(u)\in T$. The standard necessary leading-coefficient Zhu map, obtained directly by the intertwining Jacobi identity, sends the left and right actions of the $i$th conformal vector to $\lambda_i+N_i$ and $h_i$, respectively. This assertion is valid for a nonsemisimple partial $L_i(0)$: the residue proof uses only the zero-mode operators and the annihilation of the source and target highest spaces by positive modes. In particular, with $x_i,y_i$ denoting the two Zhu actions, the identities for a state of partial weight $d_i$ are \begin{equation}\label{g:eq:partial-zhu} \begin{aligned} [L_i(-1)u]&=(x_i-y_i-d_i)[u],\\ [L_i(-2)u]&=(2y_i-x_i+d_i)[u],\\ [L_i(-3)u]&=(x_i-3y_i-d_i)[u]. \end{aligned} \end{equation} For example, the first follows already from \[ [L_i(0),\mathcal I(u,z)] =z\mathcal I(L_i(-1)u,z)+\mathcal I(L_i(0)u,z). \] The remaining formulas follow from the left/right Zhu products and the relation $[L_i(-3)u]+2[L_i(-2)u]+[L_i(-1)u]=0$. Successive applications of these relations give the PBW reduction of the Zhu bimodule of $R_j$: it is generated by its highest class under the commuting $x_i,y_i$. The argument uses this necessary leading-coefficient map, rather than a converse fusion-rule statement. The reduction is performed separately at each Virasoro factor. Indeed: \[ A(\mathcal B)=\mathbb C[x_0,\ldots,x_{N-1}],\qquad x_i=[e_i], \] and modes from distinct factors commute. Each factor of the root bimodule is generated by its highest class under its own left and right zero modes, so the tensor bimodule $A(R_j)$ is generated by the full root highest class. All partial variables remain independent. Lemma~\ref{g:lem:generic-generalized-Verma} identifies the target annihilation space $\Omega(Y)$ with $T$. At an active factor use the parameter $a$, so that its root highest weight is $d=2a-1$ and its singular vector is \[ (L_{-1}^3-4aL_{-2}L_{-1}+2a(2a-1)L_{-3})u=0. \] Set $s=x-y-d$. Its Zhu image, from \eqref{g:eq:partial-zhu}, is the polynomial \begin{equation} P_a(x,y)=s(s-1)(s-2)-4a(y-s+1)s +2a(2a-1)(s-2y). \label{g:eq:zhu-cubic} \end{equation} For $h_a(p)=(p^2-(a-1)^2)/(4a)$, direct factorization gives \begin{equation} P_a(x,h_a(p)) =\prod_{m=-1}^1\bigl(x-h_a(p+2ma)\bigr). \label{g:eq:zhu-factorization} \end{equation} If $p=aR-S$ with the stated rational labels, the differences of these three roots are $a+p,a-p,2p$, all nonzero; indeed each is nonintegral by irrationality and the odd-denominator condition. Apply \eqref{g:eq:zhu-factorization} to $t(u_j)$, where $u_j$ is the highest vector of $R_j$. If $\lambda_i$ is not one of the three roots, the polynomial in $\lambda_i+N_i$ is invertible and $t(u_j)=0$. If $\lambda_i$ is a root, it is simple, and \[ P_a(\lambda_i+N_i,h_i)=N_i U_i(N_i),\qquad U_i(0)\ne0. \] The nilpotent polynomial $U_i(N_i)$ is invertible, so $N_i t(u_j)=0$. At the other active factor apply the same calculation with reciprocal parameter, converting $(1,3)$ to $(3,1)$. At every inactive vacuum factor the Zhu relation is $x_i=y_i$; it likewise makes the top coefficient zero unless $\lambda_i=h_i$, and otherwise forces $N_it(u_j)=0$. Thus $t(u_j)\in T_0:=\bigcap_i\ker N_i$. Cyclicity of the root Zhu bimodule now puts \emph{every} $t(u)$ in $T_0$, because each partial zero mode preserves $T_0$. Pass to the quotient $\overline Y=Y/(U(\mathfrak n_-)T_0)$. All highest coefficients of the resulting intertwiner vanish. This implies that the intertwiner itself vanishes: $F$ is generated by its highest vector, and $\overline Y'$ is generated by its highest space by Lemma~\ref{g:lem:generic-generalized-Verma}. Move their negative Virasoro modes across the matrix coefficient by the intertwining Jacobi identity. Every matrix coefficient is thereby reduced to a highest-target/highest-source coefficient with some descendant in the first input, all of which vanish. Equivalently, the generic Shapovalov matrices remain invertible after the nilpotent highest weight substitution, giving the same induction at each degree. Thus the original image lies in $U(\mathfrak n_-)T_0$, the ordinary frame socle in \eqref{g:eq:generic-socle}. \end{proof} \begin{theorem}[Semisimple frame restriction in a nonzero rational sector] \label{g:thm:generic-sector-frame-semisimple} Let $X$ be a simple grading-restricted generalized $W_N$-module containing an ordinary simple frame module \[ F=\bigotimes_iL(c_i,h_i(R_i,S_i)), \] where every $R_i,S_i$ is rational with odd denominator and no pair $(R_i,S_i)$ consists of two integers. Then \[ X\big|_{\mathcal B} \] is a direct sum of ordinary irreducible Verma tensors, with finite multiplicity at each joint highest weight. In particular the conclusion holds for every simple module in any of the nonzero periodic or antiperiodic odd coarse sectors. \end{theorem} \begin{proof} The nilpotent part of total $L(0)$ is a natural $W_N$-endomorphism. On a simple grading-restricted generalized module it vanishes, so $X$ is ordinary for total $L(0)$. Let $X^{\mathrm{ord}}_{\mathrm{rat}}$ be the sum of the ordinary simple frame submodules whose Kac pairs satisfy the hypotheses. It contains $F$, and hence is nonzero. We show that every elementary generator preserves this sum. Take a simple frame summand $F_0$ in this sum and project the action of $R_j$ on $F_0$ to a joint frame block $Y$. By Lemma~\ref{neutral:lem:frame-blocks}, the block $Y$ has finite length. If the projected intertwiner is nonzero, the frame submodule spanned by its coefficients has a simple quotient $Q$, and the induced intertwiner to $Q$ is nonzero. Both $F_0$ and $Q'$ are generated by their highest spaces. The necessary leading-coefficient Zhu map is therefore injective, and the relations \eqref{g:eq:partial-zhu}--\eqref{g:eq:zhu-factorization} apply to its highest coefficient. At each active factor the target momentum is $p$, $p+2a$, or $p-2a$; at each inactive factor the highest weight is unchanged. Thus the Kac pairs of $Q$ are obtained from those of $F_0$ by even integral shifts and simultaneous sign changes. Each still has odd denominators and at least one nonintegral entry. Consequently $Q$ lies in a generic singleton block. Since $Q$ is a constituent of $Y$, the target block $Y$ is that same generic block. Lemma~\ref{g:lem:generic-top-constraint} now puts the entire projected image in the ordinary frame socle of $Y$, all of whose simple summands satisfy the required rationality condition. Each coefficient has only finitely many block components, so the action of $R_j$ preserves $X^{\mathrm{ord}}_{\mathrm{rat}}$. The frame and the elementary generators vertex-generate $W_N$. Thus $X^{\mathrm{ord}}_{\mathrm{rat}}$ is a nonzero $W_N$-submodule of $X$, and simplicity gives $X^{\mathrm{ord}}_{\mathrm{rat}}=X$. Finite-dimensional total conformal-weight spaces bound the multiplicity of each fixed joint highest weight. \end{proof} \begin{remark}[The leading-coefficient map] The Zhu-map input above is the necessary leading-coefficient map of Huang--Yang \cite[Lemma 5.5 and Propositions 5.6--5.8]{huangyang}, in their corrected formulation at level $0$. Its injectivity applies when the second input and the contragredient of the target are generated by their highest spaces, as verified above. This use does not require an unrestricted Frenkel--Zhu fusion-rule isomorphism. The direct residue calculation also keeps the partial conformal operators separate. Frame semisimplicity is the conclusion of this section; multiplicity one and uniqueness require the transport argument that follows. \end{remark} \section{Multiplicity one and uniqueness in rational sectors} \label{g:sec:generic-sector-classification} We now determine the multiplicities and the module structure on each regular rational grid. Besides frame semisimplicity and the BPZ calculation, we use the following property of the canonical diagonal algebra: multiplication is nonzero in every allowed diagonal frame fusion channel. In particular, with $R_j=M_{e_j}$ in its usual diagonal labeling, \begin{equation} R_j\cdot R_j\longrightarrow\mathbf1,\quad R_j,\quad M_{2e_j} \quad\hbox{are all nonzero}, \label{g:eq:root-self-products} \end{equation} and, for $i\ne j$, \begin{equation} R_i\boxtimes_{\mathcal B}R_j=M_{e_i+e_j},\qquad R_i\cdot R_j\longrightarrow M_{e_i+e_j}\ne0. \label{g:eq:mixed-root-product} \end{equation} The fusion in \eqref{g:eq:mixed-root-product} has just one simple summand. At a shared Virasoro site it is $(3,1)\boxtimes(1,3)=(3,3)$; at every other site one factor is vacuum. The canonical diagonal tensor-category description also proves this directly. Finally, the frame and the $R_j$ \emph{vertex-generate} $W_N$, by the construction above. This vertex-generation statement suffices for the submodule arguments below. \subsection{Regular rational label grids} Choose signs $\sigma_i\in\{1,-1\}$, and define \[ (K_\sigma r)_j=\kappa_jr_j -\sigma_{j-1}r_{j-1}-\sigma_jr_{j+1}, \qquad \kappa_0=N+2,\quad\kappa_j=2\ (j\ne0). \] After a diagonal sign change, this is $K_+$ when $\prod_i\sigma_i=1$, and $K_-$ when $\prod_i\sigma_i=-1$. Its determinant is respectively $N^2$ or $N^2+4$, so it is invertible. For $r\in\mathbb Q^N$ put \[ E(r)=\bigotimes_i L\bigl(c_i,h_i(r_i,\sigma_i r_{i+1})\bigr). \] Call $r$ regular here if its entries have odd denominators and no two cyclically adjacent entries are both integral. Every factor of $E(r)$ is then a generic irreducible Verma module in a singleton frame block. The term ``regular'' in this section refers only to this explicit arithmetic condition, not to regularity of a VOA. Fix such an $r^{(0)}$ with $K_\sigma r^{(0)}\in\mathbb Z^N$, and set \[ \mathcal O(r^{(0)})=r^{(0)}+2\mathbb Z^N. \] Every point of this grid has the same regularity property. The frame simples $E(r)$, for $r\in\mathcal O(r^{(0)})$, are pairwise distinct. Indeed equality of weights forces simultaneous sign equality of the two rational Kac labels at each irrational Virasoro parameter. A negative sign at a site, for two points in the same grid, would force both adjacent coordinates to be integral. This is excluded. The argument permits isolated integral or zero coordinates. The nonzero odd periodic and antiperiodic discriminant classes are regular in this sense. Periodic residue vectors are the nonzero affine progressions modulo $N$. The antiperiodic representatives, verified below, have coordinates $s(N-2i)/(N^2+4)$ modulo $\mathbb Z$, where $s\ne0$ in $\mathbb Z/(N^2+4)\mathbb Z$. Two adjacent periodic entries can vanish only when the progression is zero. In the antiperiodic case, adjacent numerators have difference $2$, except across the closing edge, where their sum is $2$. Since $N^2+4$ is odd, two vanishing adjacent entries would again force $s=0$. \begin{lemma}[The root action stays on one grid] \label{g:lem:generic-grid-closure} Suppose a simple $W_N$-module $X$ contains the frame simple $E(r^{(0)})$ with the above hypotheses. Then its frame restriction is semisimple, and every frame summand belongs to $\{E(r):r\in\mathcal O(r^{(0)})\}$. \end{lemma} \begin{proof} The generic frame-semisimplicity theorem applies to the nonintegral rational coarse tuple. Alternatively apply its top-polynomial argument successively from the given simple frame submodule: shifting a Kac label by $0,2,-2$ preserves its nonintegral residue, so every possible target block remains generic. The frame socle is stable under each root, hence under their vertex-generated VOA. At a point $r$ set \[ a=\psi_j,\quad b=\psi_{j-1}^{-1}=\kappa_j-a, \quad p=ar_j-\sigma_jr_{j+1}, \quad q=b\sigma_{j-1}r_j-r_{j-1}. \] Then $p+\sigma_{j-1}q=(K_\sigma r)_j\in\mathbb Z$. All nine elementary combinations of the two three-channel null polynomials are available as necessary frame fusion possibilities. An actual module mode changes the total weight by an integer. The mixed middle/corner channels would require $p\in\pm a+\mathbb Z$ or $q\in\pm b+\mathbb Z$; the wrongly paired corner channels would require $2p\in\mathbb Z$. All are excluded by regularity and irrationality, as in the BPZ irreducibility lemma. Thus the only possible targets are \[ E(r-2e_j),\quad E(r),\quad E(r+2e_j). \] The grid is closed under all root actions. Its sum of frame summands in $X$ is a nonzero $W_N$-submodule, so simplicity makes it all of $X$. \end{proof} Write the resulting frame decomposition as \[ X\big|_{\mathcal B} =\bigoplus_{r\in\mathcal O(r^{(0)})}E(r)\otimes T_r, \qquad\dim T_r<\infty, \] initially allowing $T_r=0$. Fix normalized highest vectors of all $E(r)$ and normalized frame intertwiners for every allowed channel. Each such intertwiner space has dimension at most one, by the highest-weight Ward recursion and the nondegenerate generic Shapovalov forms. Denote the three multiplicity maps of the root action by \[ A_j^\epsilon(r):T_r\longrightarrow T_{r+2\epsilon e_j}, \qquad\epsilon\in\{-1,0,1\}. \] \subsection{The two-root invariant tensor with multiplicity spaces} \begin{lemma}[Return maps and middle maps are scalar] \label{g:lem:scalar-root-maps} For every $r$ with $T_r\ne0$, all adjacent spaces $T_{r\pm2e_j}$ are nonzero and the maps $A_j^\pm(r)$ are isomorphisms. Their oppositely oriented compositions are nonzero scalar multiples of the identity. Moreover $A_j^0(r)$ is a scalar multiple of the identity. All these scalars depend only on the normalized $W_N$ multiplication and the frame labels, and not on $X$ or on the dimension of $T_r$. \end{lemma} \begin{proof} Use the operator-valued four-point coefficient \[ \Phi_j(z)=\operatorname{pr}_{E(r)^{\rm top}\otimes T_r} Y(v_j,1)Y(v_j,z)\big|_{E(r)^{\rm top}\otimes T_r}. \] By choosing a vector and a functional in $T_r$ one obtains ordinary rational matrix coefficients. The factorization and nine-derivative Wronskian proof apply to each matrix entry. Thus its coefficient tensor belongs to \[ (V_a\otimes V_b)^{\pi_1}\otimes\operatorname{End}(T_r), \] where $V_a,V_b$ are the two irreducible rank-three BPZ local systems. The existence of the nonzero vacuum coefficient shows these systems are dual; Schur's lemma gives $\dim(V_a\otimes V_b)^{\pi_1}=1$. Normalize a generator of this one-dimensional invariant space by its vacuum-channel coefficient. The vacuum term of $\Phi_j$ is the fixed nonzero number $\nu_j$ times $\operatorname{id}_{T_r}$. It follows that the entire invariant tensor equals this normalized scalar tensor multiplied by $\nu_j\operatorname{id}_{T_r}$. In the channel at $z=0$ the invariant tensor has the three allowed entries, one for each $E(r+2\epsilon e_j)$. It is invertible as a $3\times3$ tensor, so all three entries are nonzero. The two extreme entries are, with the fixed three-point normalizations, \[ A_j^{-\epsilon}(r+2\epsilon e_j)A_j^\epsilon(r) =\lambda_{j,r,\epsilon}\operatorname{id}_{T_r}, \qquad \lambda_{j,r,\epsilon}\ne0. \] Applying the same assertion at the neighboring point gives the reverse composition as a nonzero scalar identity. Hence these maps are isomorphisms and every neighboring multiplicity is present. For the middle map use the channel at $z=1$, where the two roots fuse first. By \eqref{g:eq:root-self-products} the three possible frame intermediates are $\mathbf1,R_j,M_{2e_j}$. The individual-factor local exponents are distinct modulo integers, making the three paired tensor channels independent: in one factor they are $2-4a,1-2a,2a$, whose differences are nonintegral because $a$ is irrational. The other factor has parameter $b=\kappa_j-a$. The tensor in this basis is again diagonal and invertible. Its $R_j$ entry is the nonzero multiplication constant in $R_j\cdot R_j\to R_j$ times $A_j^0(r)$. Since every tensor entry is a fixed scalar times $\operatorname{id}_{T_r}$, the middle map is scalar too. The scalar is nonzero. \end{proof} \subsection{Mixed generators and scalar square transport} \begin{lemma}[Every elementary square has scalar transport] \label{g:lem:scalar-squares} For $i\ne j$ there is a nonzero number $\gamma_{ij}(r)$, depending only on the normalized frame and algebra data, such that \begin{equation} A_i^+(r+2e_j)A_j^+(r) =\gamma_{ij}(r)A_j^+(r+2e_i)A_i^+(r). \label{g:eq:scalar-square} \end{equation} \end{lemma} \begin{proof} Project a two-root matrix coefficient from the highest space of $E(r)$ to that of $E(r+2e_i+2e_j)$. In the product ordering where $R_j$ acts first, there is just one possible intermediate frame simple, namely $E(r+2e_j)$. The other ordering similarly has only $E(r+2e_i)$. The three-channel grid closure and the injectivity of the label map give these unique intermediate summands. By \eqref{g:eq:mixed-root-product}, the product of the two root states belongs to the single simple frame module $P=M_{e_i+e_j}$, and its multiplication intertwiner is nonzero. The frame intertwiner space \[ \mathcal I_{\mathcal B}\binom{E(r+2e_i+2e_j)}{P\quad E(r)} \] has dimension at most one: all modules are highest-weight modules, the target Verma factors are generic, and Ward recursion determines all three-point coefficients from the highest coefficient. Let $D_{ij}(r):T_r\to T_{r+2e_i+2e_j}$ be the multiplicity map for this channel of the $P$ action, if present. Associativity expresses each ordered product through the same $P$ action and the same one-dimensional frame intertwiner space. Thus each ordered composition in \eqref{g:eq:scalar-square} is a scalar multiple of $D_{ij}(r)$. Those scalars are the ordinary frame change-of-channel constants together with the fixed multiplication in \eqref{g:eq:mixed-root-product}; they do not depend on the module. Both ordered compositions are invertible by the preceding lemma. Hence $D_{ij}(r)$ and both scalar constants are nonzero. Dividing gives \eqref{g:eq:scalar-square}, with $\gamma_{ij}(r)\ne0$. In particular the relevant frame intertwiner exists: otherwise associativity would make an invertible ordered product vanish. \end{proof} \begin{theorem}[One copy of every grid point] \label{g:thm:generic-grid-multiplicity-one} If a simple $W_N$-module contains $E(r^{(0)})$ as above, then \[ X\big|_{\mathcal B} =\bigoplus_{r\in r^{(0)}+2\mathbb Z^N}E(r) \] with multiplicity one at every point. \end{theorem} \begin{proof} The scalar-return lemma makes every grid point present, all multiplicity spaces have a common finite dimension, and transport along every edge is invertible. Every closed lattice walk is a product of elementary squares and backtracking pairs. By \eqref{g:eq:scalar-square} and the return identities, its transport is a scalar automorphism. These scalar holonomies need not equal one. Choose a nonzero subspace $U\subseteq T_{r^{(0)}}$, and transport it to $U_r\subseteq T_r$ along any grid path. The result is path-independent as a \emph{subspace}, because scalar automorphisms preserve every subspace. The sum \[ \bigoplus_r E(r)\otimes U_r \] is stable under the frame and under each root: the two extreme maps are the edge transports and their scalar inverses, while the middle map is scalar. Vertex generation makes it a $W_N$-submodule. Simplicity implies that $U$ cannot be a proper nonzero subspace. Hence $\dim T_{r^{(0)}}=1$ and all multiplicities equal one. \end{proof} \begin{theorem}[Uniqueness on a fixed regular grid] \label{g:thm:generic-grid-uniqueness} Any two simple $W_N$-modules containing the same regular frame simple $E(r^{(0)})$, with $K_\sigma r^{(0)}\in\mathbb Z^N$, are isomorphic. \end{theorem} \begin{proof} They have the same multiplicity-one frame decomposition by the preceding theorem. Fix the same normalized frame intertwiners for both. The scalar return identities, middle maps, and square constants are determined by the same frame local systems and the same fixed $W_N$ multiplication, so they agree in both modules. Choose a nonzero linear isomorphism between the multiplicity lines at one base point and extend it by the positive and negative edge maps. It is independent of the path: the ratio of the two transports around any square or backtracking pair is one, because the scalar holonomies agree. It intertwines the extreme root maps by construction and the middle maps because their scalars agree. It therefore intertwines every root and the frame, hence their vertex-generated VOA $W_N$. \end{proof} \subsection{Theta characters and the global sign} The simple module in a regular grid, whenever it exists, has character \begin{equation} \boxed{\qquad \operatorname{ch}X_{[r^{(0)},\sigma]}(\tau) =\frac1{\eta(\tau)^N} \sum_{m\in\mathbb Z^N} q^{\frac14(r^{(0)}+2m)^TK_\sigma(r^{(0)}+2m)}. \qquad} \label{g:eq:grid-character} \end{equation} Indeed the oscillator-normalized exponent of $E(r)$ is \[ \sum_i\frac{(\psi_i r_i-\sigma_i r_{i+1})^2}{4\psi_i} =\frac14r^TK_\sigma r. \] Consequently, once the modular-support argument establishes the existence of a nonzero antiperiodic grid module, its entire character is given by \eqref{g:eq:grid-character}. The global sign gives the same set of frame simples: $E(r)=E(-r)$ and negation bijects the two grids. The uniqueness theorem consequently identifies their simple modules. For the periodic lattice restriction $r=\mathbf1-2k$, this global sign is \[ k\longmapsto\mathbf1-k. \] On lattice vectors this is the reflection in $B/2$. It sends the lattice coset $\xi$ to $-\xi$. Thus, for nonzero odd component, \[ M_\xi\cong M_{-\xi} \qquad\hbox{and hence}\qquad M_\xi'\cong M_\xi. \] Different regular grids can agree as sets of frame simples only under the sign/gauge identifications described next. A coarse label alone does not identify different parity lifts modulo $2\mathbb Z^N$. \subsection{Alignment and canonical sign gauges} The next proposition derives rational alignment from local monodromy. It applies to generic blocks with the stated nonresonance conditions; integral and resonant blocks are treated separately later. \begin{proposition}[Alignment from the local monodromy spectra] \label{g:prop:monodromy-alignment} Suppose a generic singleton-block frame simple in a semisimple frame restriction has momenta $p_i$, so that the preceding highest-weight factorization applies, and at each elementary edge both rank-three BPZ local systems are irreducible with nonresonant local exponents. Suppose in addition that their middle eigenvalues cannot match a corner eigenvalue; it suffices that \[ p_j\notin\pm\psi_j+\mathbb Z,\qquad -p_{j-1}/\psi_{j-1} \notin\pm\psi_{j-1}^{-1}+\mathbb Z. \] Then signs $\sigma_i$ and integers $n_i$ exist such that \[ p_j-\sigma_{j-1}p_{j-1}/\psi_{j-1}=n_j. \] There is a rational vector $r$ with \[ p_i=\psi_i r_i-\sigma_i r_{i+1},\qquad K_\sigma r=n. \] In particular its denominators are odd, and its sign holonomy is periodic or antiperiodic. If this $r$ has no adjacent integral pair, the preceding classification applies. \end{proposition} \begin{proof} The same rational-derivative argument gives a nonzero invariant tensor between the two irreducible local systems at each edge. Their local eigenvalues at zero therefore match inversely. The middle eigenvalues match each other because $\psi_j+\psi_{j-1}^{-1}=\kappa_j\in\mathbb Z$. The stated exclusions rule out matching middle with corner. Matching the two remaining eigenvalues gives $p_j\pm q_j\in\mathbb Z$, with $q_j=-p_{j-1}/\psi_{j-1}$, proving the first assertion. Define linear maps \[ (Dr)_i=\psi_i r_i-\sigma_i r_{i+1},\qquad (Lp)_j=p_j-\sigma_{j-1}p_{j-1}/\psi_{j-1}. \] The cyclic parameter identity gives $LD=K_\sigma$. Since $K_\sigma$ is invertible, both square matrices $L,D$ are invertible. Setting $r=K_\sigma^{-1}n$ gives $Dr=p$. The matrix $K_\sigma$ is integral with odd determinant, so $r\in\mathbb Q^N$ has odd denominators. \end{proof} \begin{corollary}[Exhaustion of the regular rational sectors] \label{g:cor:regular-rational-exhaustion} Let $X$ be a simple $W_N$-module containing a frame simple with rational Kac pairs of odd denominator, at least one entry of each pair being nonintegral. Then $X$ is one of the regular periodic or antiperiodic grid modules just classified. \end{corollary} \begin{proof} The generic frame-semisimplicity theorem applies. For both the row and reciprocal-column parametrizations, the rational-label irreducibility and nonresonance criterion holds at every site. Proposition~\ref{g:prop:monodromy-alignment} therefore gives a canonical periodic or antiperiodic rational vector $r\in K_\pm^{-1}\mathbb Z^N$. Irrationality identifies its Kac pairs with the original pairs up to simultaneous sign, so no adjacent coordinates are both integral. Its odd-discriminant component is consequently nonzero. The periodic Smith form gives one of the nonzero affine-progressions modulo $N$; the antiperiodic Smith form gives $s(N-2i)/(N^2+4)$ modulo $\mathbb Z$ with $s\ne0$. The grid classification and uniqueness theorems apply. \end{proof} For the regular rational labels used in this section, $2p_i$ and $2p_i/\psi_i$ are nonintegral. Hence precisely one of the two possible edge-sign matching conditions can hold. Changing the independently chosen momentum signs $p_i\mapsto\varepsilon_ip_i$ consequently changes the unique edge signs by \[ \sigma_i\mapsto \varepsilon_i\varepsilon_{i+1}\sigma_i. \] Their product is invariant, including at points where a coordinate $r_i$ is zero or integral. Once $\sigma$ is fixed to the standard periodic or antiperiodic representative, the only remaining gauge is global sign. This establishes the sign identification using local monodromy and integrality; it does not assume sign propagation through nonzero coordinates. \section{Refined modular traces after regularity} \label{sec:ordinary-jets} Regularity gives finitely many simple $W_N$-modules and ordinary trace functions on each of them. We refine these traces by recording the commuting partial conformal weights. To avoid a separate convergence problem in several variables, we work with formal Taylor germs along the diagonal. The Ward identities determine all their coefficients from convergent one-variable traces. Let $V$ be a regular, $C_2$-cofinite CFT-type VOA with a commuting conformal frame $\omega=\sum_{i=0}^{N-1}e_i$, and write $L_i(n)$ for its modes and $c_i$ for its central charges. For a simple $V$-module $M$, put \[ F_M(v;\boldsymbol\tau)= \operatorname{Tr}_M\left(o(v) \prod_i e^{2\pi i\tau_i(L_i(0)-c_i/24)}\right). \] Initially this expression is interpreted coefficientwise in the total $q$-grading and as a formal Taylor germ at $\boldsymbol\tau=(\tau,\ldots,\tau)$. The Ward recursion below proves that each Taylor coefficient is holomorphic for $\tau$ in the upper half-plane. All the partial zero modes commute and preserve each finite-dimensional total-weight space. For a vector that is not ordinary homogeneous, $o(v)$ is extended linearly from its homogeneous components. The analytic theorem in \cite[Theorem 1]{km} assumes convergence on the full product of upper half-planes. We use its Ward recursion at the formal diagonal germ, and prove that version below; no such global convergence hypothesis is silently applied to the unknown modules. For ordinary homogeneous $v$, use the square-bracket coordinate \[ Y[v,z]=Y(v,e^{2\pi iz}-1)e^{2\pi iz\operatorname{wt}(v)}, \qquad Y[v,z]=\sum_{m\in\mathbb Z}v[m]z^{-m-1}, \] extended linearly, and set \[ \widetilde e_i=(2\pi i)^2(e_i-c_i\one/24),\qquad L_i[m]=\widetilde e_i[m+1]. \] The Eisenstein normalization is \[ G_{2k}(\tau)=2\zeta(2k)+ \frac{2(2\pi i)^{2k}}{(2k-1)!} \sum_{n\geq1}\sigma_{2k-1}(n)e^{2\pi in\tau}. \] For a bracket-frame homogeneous vacuum descendant $v$ of partial weights $w_i$, the trace Ward identities are \begin{align} F_M(L_i[-1]v)&=0,\notag\\ F_M(L_i[-2]v)&=2\pi i\partial_iF_M(v) +w_iG_2(\tau_i)F_M(v) +\sum_{k\ge2}G_{2k}(\tau_i)F_M(L_i[2k-2]v). \label{eq:ordinary-ward} \end{align} Only finitely many terms occur for each $v$. The residue proof uses mode commutators and cyclicity of trace on finite total-weight spaces; it is valid as an identity of formal germs. For a Taylor coefficient of a fixed order, solve this identity for $\partial_iF_M(v)$ and iterate. Only finitely many frame descendants are introduced at that order, and the diagonal values are convergent ordinary one-point functions. Derivatives of the Eisenstein series are holomorphic as well. This proves the claimed convergence of individual Taylor coefficients, without convergence of the full multivariable series. The identities are \cite[Lemmas 4--5 and equation (2.5)]{km}. \begin{lemma}[Formal covariance with the ordinary modular matrix] \label{lem:ordinary-jets} Let $v$ be a bracket-frame homogeneous vector in the vacuum subalgebra generated by the commuting $e_i$, with partial weights $w_i$. Let $S_{M,X}^{\mathrm{an}}$ be the coefficients in the full insertion-dependent one-point $S$-transformation of $V$. Then, as formal germs along the diagonal, \[ F_M(v;-1/\tau_0,\ldots,-1/\tau_{N-1}) =\prod_i\tau_i^{w_i} \sum_{X\in\Irr(V)} S_{M,X}^{\mathrm{an}} F_X(v;\boldsymbol\tau). \] \end{lemma} \begin{proof} We prove the analogous identity for each $\gamma=\left(\begin{smallmatrix}a&b\\c&d\end{smallmatrix}\right) \in\operatorname{SL}_2(\mathbb Z)$. Put $j_i=c\tau_i+d$ and subtract the proposed right-hand side from $\prod_i j_i^{-w_i}F_M(v;\gamma\boldsymbol\tau)$, calling the result $H_v$. Full one-point modularity makes $H_v$ vanish on the diagonal for every frame vacuum descendant $v$. In this normalization, \[ G_2(\gamma\tau)=(c\tau+d)^2G_2(\tau)-2\pi ic(c\tau+d). \] The anomaly cancels the derivative of $j_i^{-w_i}$ in the covariant operator $2\pi i\partial_i+w_iG_2$. Higher Eisenstein series have their usual modular weights. Thus \eqref{eq:ordinary-ward} implies \[ 2\pi i\partial_iH_v =H_{L_i[-2]v}-w_iG_2(\tau_i)H_v -\sum_{k\ge2}G_{2k}(\tau_i)H_{L_i[2k-2]v}. \] Induct simultaneously for every frame descendant $v$ on the total order of a multi-index of partial derivatives. Order zero is the ordinary modular identity on the diagonal. Apply a derivative of order $m$ to the displayed recursion. Every term on its right is an Eisenstein coefficient times a derivative of some $H_u$ of order at most $m$, and hence vanishes on the diagonal by the induction hypothesis. Thus every derivative of order $m+1$ vanishes there. All transverse Taylor coefficients therefore vanish, proving the formal identity. This is the formal Ward-recursion argument underlying \cite{km}; it requires convergence of the individual one-variable coefficients only. \end{proof} \begin{lemma}[Ordinary trace support in a frame block] \label{lem:ordinary-support} For a grading-restricted $W_N$-module, multiplication of its refined ordinary character by $\eta_\Pi=\prod_i\eta(\tau_i)$ leaves only the joint endpoints of the irrational Virasoro linkage blocks in its frame restriction. In a singleton block with irreducible Verma tensor of joint highest weight $\lambda$, its contribution is \[ m_\lambda\prod_iq_i^{\lambda_i-(c_i-1)/24}, \qquad m_\lambda\in\mathbb Z_{\ge0}. \] For a finite linear combination of refined characters of simple $W_N$-modules, a nonzero coefficient at such a \emph{singleton-block} endpoint implies that at least one simple module with a nonzero coefficient in that linear combination contains the corresponding frame simple. This assertion is not made for a subtracted endpoint of a two-element linkage block. \end{lemma} \begin{proof} The frame-block lemma gives a finite-length restriction in each fixed joint block, with finitely many simple types. Ordinary trace is additive on a composition series, including when partial zero modes are not semisimple: in a simultaneous upper triangular basis the exponential has the prescribed diagonal eigenvalues, and the strictly upper triangular terms have zero trace. Its block trace is therefore the sum of the ordinary frame-simple characters with their composition multiplicities. At an irrational Virasoro parameter, an ordinary simple is either an irreducible Verma module or the quotient of a Verma module by its single irreducible singular submodule. Its character is accordingly one Verma character or the difference of two. Multiplication by eta cancels the oscillator factor, leaving just the one or two linkage endpoints. Tensoring gives the stated support description. A singleton block with positive multiplicity is a nonzero finite-length module and has a simple socle of that unique type. Only the same linkage block can contribute that endpoint. Hence, even in a linear combination with complex coefficients, a nonzero singleton endpoint requires a nonzero block in one of its summands, proving the stated submodule conclusion. The possibly infinite sum of blocks causes no ambiguity. At each fixed total $q$-degree only finitely many joint weight spaces occur. To see that the formal germ distinguishes them, fix a total exponent and let $\lambda^{(1)},\ldots,\lambda^{(r)}$ be the distinct joint exponent tuples occurring there. They have the same coordinate sum. Choose a transverse direction $u$ with $\sum_i u_i=0$ such that $u\cdot\lambda^{(1)},\ldots,u\cdot\lambda^{(r)}$ are distinct; this avoids finitely many proper hyperplanes. The directional derivatives of orders $0,\ldots,r-1$ of \[ \sum_{a=1}^r c_a\exp(2\pi i z\,u\cdot\lambda^{(a)}) \] at $z=0$ form a Vandermonde system in the $c_a$, so its vanishing forces every $c_a=0$. The same argument applies after multiplication by eta, which contributes only finitely many terms at each bounded degree. Coefficients can therefore be compared degree by degree in the formal germs of Lemma~\ref{lem:ordinary-jets}. \end{proof} The theta transforms used below have real, positive-definite quadratic exponents. Comparing all transverse derivatives separates joint weight tuples even when their total conformal weights coincide. This distinction is necessary at composite $N^2+4$: a nonzero discriminant label can have square zero modulo $N^2+4$, so its conformal residue alone need not separate it from the zero label. \subsection{The trace normalization and categorical dimensions} \label{sec:analytic-normalization} We next relate the analytic modular matrix to categorical dimensions, fixing compatible conventions for the Hopf link. The argument applies to a simple, self-contragredient, regular, $C_2$-cofinite, nonnegatively graded VOA $V$; it does not require $V_0=\mathbb C\mathbf1$. Its finite module category is modular by \cite[Corollary 5.19]{mcrae}. Let $S^{\rm an}$ be the matrix acting on the full insertion-dependent ordinary traces. Its square is the contragredient permutation $C$. Here this is an identity of full trace functions, not an inference from scalar characters. Indeed the defining contragredient vertex operator uses the coordinate inversion $x\mapsto x^{-1}$. At the insertion $x=1$, the coordinate $f(z)=e^{2\pi iz}-1$ satisfies \[ f(-z)=-\frac{f(z)}{1+f(z)}. \] Thus inversion in the exponential coordinate becomes $z\mapsto-z$. For a bracket-homogeneous insertion of weight $w$, the contragredient trace is consequently $(-1)^w$ times the original trace. This is exactly the slash action of $-I\in\operatorname{SL}_2(\mathbb Z)$. Linear independence with all insertions identifies its matrix with $C$, proving $(S^{\rm an})^2=C$. Write $d_X=\dim X$ and $\Delta=\sum_Xd_X^2$. Choose the orientation of the categorical Hopf-link matrix $\widetilde S$ compatible with the analytic generator. For clarity, McRae's geometrically modified convention uses $(-1/\tau)^{L(0)}$, whereas the ordinary bracket convention uses $\tau^{-w}$. These represent the two inverse lifts of $\tau\mapsto-1/\tau$; their matrices differ by $C$. Reversing the Hopf-link orientation likewise dualizes one index. Both choices of $\widetilde S$ are symmetric and satisfy \[ \widetilde S_{\mathbf1,X}=d_X,\qquad \widetilde S^2=\Delta C. \] These are the usual ribbon-category identities \cite[Section 4]{ko}, with no positivity condition on the $d_X$. In the matching convention, apply the open-Hopf-link formula \cite[Theorem 5.2, equation (5.5)]{mcrae} directly to $V$. All simple $V$-modules are rigid, and regularity makes the modular transform of the vacuum trace a sum of ordinary simple-module traces. On a simple $Y$, the open Hopf link is the scalar $\widetilde S_{X,Y}/d_Y$. Linear independence of the full insertion-dependent traces therefore gives \[ S^{\rm an}_{X,Y} =S^{\rm an}_{\mathbf1,Y}\frac{\widetilde S_{X,Y}}{d_Y}. \] This argument applies separately to $V=W_N$ and to the deformed lattice VOA $V=A$: both are simple, self-contragredient, regular, $C_2$-cofinite and nonnegatively graded, as proved above. The theorem requires no one-dimensional degree-zero space. In particular this application to $A$ does not use the orbifold-specific extension setting of \cite[Remark 5.21]{mcrae}. Set \[ D=\operatorname{diag}\left( S^{\rm an}_{\mathbf1,Y}/d_Y\right)_Y. \] Spherical dimensions of simple objects are nonzero, so $D$ is well-defined. The preceding identity reads $S^{\rm an}=\widetilde SD$. Using the two square identities gives \[ D\widetilde SD=\Delta^{-1}\widetilde S. \] Its $(\mathbf1,\mathbf1)$ entry gives $D_{\mathbf1}^2=\Delta^{-1}$; its $(\mathbf1,Y)$ entries then give $D_Y=D_{\mathbf1}$ for every $Y$. Consequently \begin{equation}\label{eq:analytic-vacuum-row} S^{\rm an}_{\mathbf1,X} =d_X S^{\rm an}_{\mathbf1,\mathbf1},\qquad S^{\rm an}_{\mathbf1,\mathbf1}\ne0. \end{equation} In particular the analytic matrix is symmetric, in either compatible orientation. If $J$ centralizes $X$, its closed Hopf link is $d_Jd_X$ in both orientations, and hence \begin{equation}\label{eq:analytic-centralizer-row} \frac{S^{\rm an}_{J,X}}{S^{\rm an}_{\mathbf1,X}}=d_J. \end{equation} These two identities are independent of the compatible orientation chosen. They provide the lattice-index calculation and the categorical dimension ratios extracted from the refined parity rows, without requiring a unique module of minimal conformal weight. \section{The multivariable theta identity and its Fourier transform} \label{sec:multivariable-theta} \label{g:sec:theta} The difference of two theta quotients is the multivariable analogue of Flohr's one-variable character formula \cite[equations (4.2)--(4.5) and Table 4.2]{flohr1993}. The periodic and antiperiodic metric sectors are the two sectors in the Evans--Gannon grafting \cite[Section 3.3]{evansgannon2011}, proved for untwisted HI centers in \cite[Theorem 5.1]{gsy}. Here the separate frame variables are retained to resolve labels that the single total conformal weight does not separate. Let $\sigma=(\sigma_i)$ have all signs positive, or just its closing sign negative. Write these two choices as $+$ and $-$, and define \[ (B_\sigma r)_i=\sqrt{\psi_i}\,r_i -\frac{\sigma_i r_{i+1}}{\sqrt{\psi_i}}. \] Direct multiplication, using the cyclic parameter relations, gives \[ B_\sigma^TB_\sigma=K_\sigma,\qquad \det B_+=t^{-1}-t=N,\qquad \det B_-=t^{-1}+t=\sqrt\mu. \] For $v\in\frac12\Z^N$, put \[ \Theta_{\sigma,v}(\boldsymbol\tau) =\sum_{x\in\Z^N+v} \exp\left(2\pi i\sum_i\tau_i(B_\sigma x)_i^2\right), \qquad \eta_\Pi(\boldsymbol\tau)=\prod_i\eta(\tau_i). \] \begin{lemma}[All-parity refined identity]\label{g:lem:theta} For $v_\epsilon=(\one+\epsilon)/2$ modulo $\Z^N$, \[ \ch J_\epsilon(\boldsymbol\tau) =\frac{\Theta_{+,v_\epsilon}(\boldsymbol\tau) -\Theta_{-,v_\epsilon}(\boldsymbol\tau)} {2\eta_\Pi(\boldsymbol\tau)}. \] \end{lemma} \begin{proof} Expand the product of generic Virasoro character differences in the known frame decomposition of $J_\epsilon$. Its positive labels have parity $1+\epsilon_i$. Each term is indexed by edge signs. Independent changes of signs of the labels gauge all edge signs except their product; they preserve \emph{each} partial exponent, so separate $\tau_i$ are permitted. Overall label reversal gives the factor two. Completing positive labels to full signed theta sums introduces zero coordinates for even labels. They cancel. If $j$ is the least zero coordinate, reverse the labels at $0,\ldots,j-1$. This toggles the closing sign and the sign on an edge incident to that zero, preserving each partial square. If $j=0$ the closing cross term already vanishes. This is an involutive, parity-preserving cancellation of the extra terms. \end{proof} Define \[ \mathcal F_a(\boldsymbol\tau)= 2^{-N}\sum_{\epsilon\in(\Z/2)^N} (-1)^{\epsilon\cdot a} \ch J_\epsilon(-1/\tau_0,\ldots,-1/\tau_{N-1}). \] Poisson summation with the matrix $B_\sigma^T\diag(\tau_i)B_\sigma$, followed by finite Fourier orthogonality, gives \begin{equation}\label{g:eq:poisson} \begin{split} \eta_\Pi\mathcal F_a ={}&(-1)^{|a|}2^{-N/2-1} \left[ \frac1N\sum_{\substack{z\in\Z^N\\z\equiv a\ (2)}} e^{\,2\pi i\sum_i\tau_i(B_+^{-T}z)_i^2/4}\right.\\[-2pt] &\hspace{34mm}\left. -\frac1{\sqrt\mu} \sum_{\substack{z\in\Z^N\\z\equiv a\ (2)}} e^{\,2\pi i\sum_i\tau_i(B_-^{-T}z)_i^2/4} \right]. \end{split} \end{equation} The determinant factors cancel the eta automorphy factors. The shift phase is $(-1)^{z\cdot(\one+\epsilon)}$, so the Fourier sum keeps exactly $z\equiv a\pmod2$. All coefficients within either bracketed branch have the same sign. The theta expressions converge on $\mathbb H^N$; for comparison to ordinary traces, only their formal diagonal germs are needed. \section{Exceptional modules for all nonzero discriminant labels} \label{g:sec:exceptional} Set \[ v_i=\frac{N-2i}{\mu},\qquad v_N=-v_0=-\frac N\mu. \] Direct multiplication gives \[ K_-v=e_0. \] The interior equations are second differences, the final equation is $2(N-2N+2)-(N-2N+4)+N=0$, and the zeroth equation is $(N+2)N-(N-2)+(2-N)=\mu$. The class of $v$ has order $\mu$, since $v_{(N-1)/2}=1/\mu$. It generates the odd discriminant group. For $s\not\equiv0\pmod\mu$ define the joint highest weights \begin{equation}\label{g:eq:exceptionaltuple} \lambda_i(s)=h_i(sv_i,sv_{i+1}),\qquad 0\leq i0}. \end{equation} There is no local summand, and a tensor functor between fusion categories cannot annihilate a nonzero object. \subsection{A sum-of-squares dimension identity} The global dimension identity is \[ N^2a^2 =1+(a-1)^2+\frac{N^2-1}{2}a^2+r^2\sum_Zn_Z^2. \] Substituting \(a=2+kr\) and \(r^2=N^2(1+kr)\) gives \begin{equation}\label{r:eq:budget} \boxed{\sum_Zn_Z^2 =2+\frac{N^2-1}{2}k^2.} \end{equation} The sum runs over all neutral simples other than $\mathbf1$, $T$, and the periodic family. In particular, it includes both the exceptional modules constructed in Section~\ref{g:sec:exceptional} and any additional simples. Thus the identity is available before exhaustion has been proved. \subsection{Extracting the modular coefficient ratio} For a regular module \(Z\), let \(a_Z=z\bmod2\) be its parity Fourier label in the refined dual-lattice formula. The refined Poisson identity has coefficient \[ 2^{-N/2-1}N^{-1}(-1)^{(\one+\epsilon)\cdot a_Z} \quad\hbox{or}\quad -2^{-N/2-1}\mu^{-1/2}(-1)^{(\one+\epsilon)\cdot a_Z}. \] More explicitly, for a regular label choose $z_0$ in the appropriate discriminant class and let $\Gamma=z_0+2K_\pm\Z^N$ and $-\Gamma=-z_0+2K_\pm\Z^N$. These grids are distinct: equality would make the nonzero odd part of the class two-torsion. Their parity labels agree because $-z_0\equiv z_0\pmod2$, and $z\mapsto-z$ preserves every squared partial exponent. Consequently their contributions combine as \[ \frac{c_\pm}{2} \left(\sum_{z\in\Gamma}q^{\mathbf h(z)} +\sum_{z\in-\Gamma}q^{\mathbf h(z)}\right) =c_\pm\sum_{z\in\Gamma}q^{\mathbf h(z)}, \] where $q^{\mathbf h(z)}$ denotes the multivariable exponential in \eqref{g:eq:poisson}. Here $c_+=2^{-N/2}N^{-1}(-1)^{(\one+\epsilon)\cdot a_Z}$ and $c_-=-2^{-N/2}\mu^{-1/2}(-1)^{(\one+\epsilon)\cdot a_Z}$. This accounts for the factor two before a module coefficient is read off. The exact nonintegral frame support uniquely isolates its regular simple and excludes every unknown integral module. Hence \begin{align} S^W_{J_\epsilon,Y} &=2^{-N/2}N^{-1}(-1)^{(\one+\epsilon)\cdot a_Y},\label{r:eq:SY}\\ S^W_{J_\epsilon,X} &=-2^{-N/2}\mu^{-1/2}(-1)^{(\one+\epsilon)\cdot a_X}. \label{r:eq:SX} \end{align} Rationality ensures that the full one-point modular action here is the ordinary modular action; the formal frame-germ covariance uses the same coefficients. This argument does not depend on linear independence of scalar characters. For a representative \(Z\in\Neu\), trivial monodromy with \(\Par\) implies \[ \frac{S^W_{J_\epsilon,Z}}{S^W_{\one,Z}} =\dim J_\epsilon. \] Equations \eqref{r:eq:SY}--\eqref{r:eq:SX} therefore force the same \(a_Z\) for every neutral representative in both regular families. Removing the common parity vacuum entry gives the exact ratio \begin{equation}\label{r:eq:Sratio} \boxed{\frac{S^\Neu_{\one,X_s}} {S^\Neu_{\one,Y_{[e]}}} =-\frac N{\sqrt{N^2+4}}.} \end{equation} \subsection{Determining the integer multiplicities} Let \(x=\dim_{\Mod}R\). The local invertibles have spherical dimension one: in a spherical category their dimensions are \(\pm1\), and their group has odd order. Dimension-preserving induction gives \[ x^2=N^2+N^2kx,\qquad \dim Y_{[e]}=2+kx,\qquad \dim Z=n_Zx. \] The number \(x\) may be negative. The quadratic relation implies \[ \left(\frac{x}{2+kx}\right)^2 =\frac{N^2}{N^2k^2+4}. \] For each of the \(M=(N^2+3)/2\) known exceptional simples, equation \eqref{r:eq:Sratio} yields \begin{equation}\label{r:eq:nconstraint} n_s^2(N^2+4)=N^2k^2+4. \end{equation} All \(n_s\) are positive integers, hence have the same value \(n\). Subtract their contribution from \eqref{r:eq:budget}: \begin{equation}\label{r:eq:unused} \boxed{\sum_{\text{additional }Z}n_Z^2 =2+\frac{N^2-1}{2}k^2-\frac{N^2+3}{2}n^2 =\frac{2(1-k^2)}{N^2+4}.} \end{equation} Every term on the left is the square of a positive integer. Nonnegativity therefore gives $k\leq1$. If $k=0$, \eqref{r:eq:nconstraint} gives \(n^2=4/(N^2+4)<1\), impossible. Hence $k=n=1$, and the sum over additional simples vanishes. No additional simple can occur, since it would contribute a positive integer square. This proves both the asserted multiplicities and exhaustion. The negative sign of \eqref{r:eq:Sratio} selects \[ x=-Nt,\qquad \dim T=t^2,\qquad \dim Y=1+t^2,\qquad \dim X=-Nt. \] The positive FP root is \(r=N\delta\), giving \[ \FPdim T=\delta^2,\qquad \FPdim Y=1+\delta^2,\qquad \FPdim X=N\delta. \] \begin{center} \begin{tabular}{llll} \toprule Neutral simple & Number & Induction to \(\Neu_A\) & Spherical dimension\\ \midrule \(\one\) & \(1\) & \(\one\) & \(1\)\\ \(T\) & \(1\) & \(\one\oplus R\) & \(t^2\)\\ \(Y_{[e]}\) & \((N^2-1)/2\) & \(e\oplus(-e)\oplus R\) & \(1+t^2\)\\ \(X_s\) & \((N^2+3)/2\) & \(R\) & \(-Nt\)\\ \bottomrule \end{tabular} \end{center} \section{Reconstruction of the untwisted ribbon center} \label{sec:actual-center} \subsection{Cyclic de-equivariantization gives the HI rules} \label{sec:cyclic-reconstruction} The near-group setting originates in Izumi's work \cite{izumi} and the constructions and doubles of Evans--Gannon \cite{evansgannon2014,evansgannon2017}. The equivariantization construction in \cite[Section 6]{gsy} relates cyclic HI categories to near-group categories. For the converse used here, the near-group fusion rules must be accompanied by a specified central Tannakian algebra. The next theorem records these hypotheses and reconstructs the fusion rules after de-equivariantization. \begin{theorem}\label{thm:cyclic-converse} Let $N\geq3$ be odd. Let $\mathcal B$ be a complex fusion category whose simple objects are the group $E=(\mathbb Z/N)^2$ of invertibles and one object $R$, with \[ R^2=\bigoplus_{e\in E}e\oplus N^2R. \] Suppose a central Tannakian functor $\operatorname{Rep}(\Gamma)\to Z(\mathcal B)$, with $\Gamma\cong\mathbb Z/N$, has an underlying fully faithful functor identifying its simple objects with a subgroup $H\subset E$ of order $N$. Let $D_H$ be the image of its regular algebra and put $\mathcal H=\mathcal B_{D_H}$, using its specified central structure. Then $\mathcal H$ has precisely the simple objects $g$ and $g\rho$, $g\in G=E/H\cong\mathbb Z/N$, and \[ gh=g+h,\qquad \rho g=(-g)\rho,\qquad \rho^2=\mathbf1\oplus\bigoplus_{g\in G}g\rho. \] In particular, $\mathcal H$ is a fusion category with the cyclic Haagerup--Izumi fusion rules. \end{theorem} \begin{proof} De-equivariantization by the stated central Tannakian algebra is a fusion category, with a tensor action of $\Gamma$ whose equivariantization is $\mathcal B$; the underlying full faithfulness is precisely what ensures a simple unit \cite[Theorem 4.18, Proposition 4.19, Remark 4.21]{dgno}. Let $F:\mathcal B\to\mathcal H$ be free induction. The subgroup $H$ is cyclic of order $N$, because its simple objects are the characters of the cyclic group $\Gamma$. A generator of $H\subset(\mathbb Z/N)^2$ is a primitive vector modulo $N$ and extends to a basis; hence $E/H\cong\mathbb Z/N$. For invertibles $e,f$, \[ \operatorname{Hom}_{\mathcal H}(F(e),F(f)) \cong\bigoplus_{h\in H}\operatorname{Hom}_{\mathcal B}(e,fh). \] Thus the $F(e)$ give $N$ distinct simple invertibles, indexed by $G$. Moreover, \[ \operatorname{End}_{\mathcal H}F(R) \cong\bigoplus_{h\in H}\operatorname{Hom}_{\mathcal B}(R,Rh). \] Every summand is one-dimensional. Choosing an isomorphism $R\to Rh$ in each degree, composition multiplies such basis vectors by nonzero scalars. The endomorphism algebra is therefore a twisted group algebra of $H$. A cyclic group has trivial $H^2(H,\mathbb C^\times)$: after choosing a lift of its generator, rescale that lift so that its $N$th power is $1$. Hence this algebra is $\mathbb C[H]\cong\mathbb C^N$. It follows that $F(R)=Q_1\oplus\cdots\oplus Q_N$, with all $Q_i$ simple, distinct and of multiplicity one. Adjunction excludes the $F(e)$ among the $Q_i$. Dominance of $F$ shows that this list exhausts the simples of $\mathcal H$. Put $\delta=(N+\sqrt{N^2+4})/2$. Then $\operatorname{FPdim}R=N\delta$. Tensor functors preserve Frobenius--Perron dimensions, and de-equivariantization divides the global dimension by $N$. Writing $d_i=\operatorname{FPdim}Q_i$ gives \[ \sum_i d_i=N\delta,\qquad \sum_i d_i^2 =\frac{N^2+N^2\delta^2}{N}-N=N\delta^2. \] Equality in Cauchy--Schwarz forces every $d_i=\delta>1$. For any $Q_i$, write the total invertible and noninvertible multiplicities of $Q_iQ_i^*$ as $a,b$. Since $\delta^2=1+N\delta$ is irrational, $a+b\delta=1+N\delta$ gives $a=1$, $b=N$. Frobenius reciprocity identifies $a$ with the order of the left $G$-stabilizer. Thus the left action is free, and hence transitive; the same argument with $Q_i^*Q_i$ handles the right action. Choose $\rho=Q_1$ and write $\rho g=\varphi(g)\rho$, where $\varphi\in\operatorname{Aut}(G)$. The deck action fixes the isomorphism class of each $F(e)$: twisting the regular-algebra action is undone by rescaling its $H$-graded free factor. It is transitive on the $Q_i$. Indeed each deck orbit of noninvertible simples supplies a noninvertible equivariant simple, whereas $\mathcal B$ has just one. Since there are $N$ deck elements and $N$ such simples, the action is free and transitive. As it fixes every invertible, it commutes with left translation by $G$. Thus the deck action on the noninvertible labels consists of all translations $g\rho\mapsto(g+a)\rho$. Write $\rho^2=g_0\oplus\bigoplus_g m_g\,g\rho$. The same dimension calculation shows there is exactly one invertible term and $\sum_gm_g=N$. Applying the deck translation $a$, and comparing unique invertible terms with $(a\rho)^2=(a+\varphi(a))\rho^2$, gives $a+\varphi(a)=0$ for every $a$. Hence $\varphi=-1$. Now $(a\rho)^2=\rho^2$, so the same comparison of noninvertible terms gives $m_{g+a}=m_g$; consequently all $m_g=1$. Finally associativity of $\rho^3$ gives $g_0\rho=(-g_0)\rho$ after cancelling the common sums. The free $G$-action yields $2g_0=0$. Oddness of $N$ gives $g_0=0$, as required. \end{proof} \subsection{The complementary subgroup fixes the pointed associator} \begin{corollary}\label{cor:untwisted} In Theorem \ref{thm:cyclic-converse}, suppose the specified central Tannakian category is one Lagrangian factor of a braided embedding \[ \mathcal E^{\mathrm{rev}}\longrightarrow Z(\mathcal B),\qquad \mathcal E\simeq\mathcal C(G\times\widehat G,q),\quad q(g,\chi)=\chi(g), \] whose underlying functor identifies all $N^2$ invertibles of $\mathcal B$. Then the pointed subcategory of $\mathcal H$ is tensor equivalent to $\operatorname{Vec}_G$ with trivial associator. \end{corollary} \begin{proof} The hyperbolic category has a skeletal realization with associator $1$ and braiding $c_{(g,\chi),(h,\psi)}=\psi(g)$; replacing this braiding by its inverse-transpose realizes the reverse category. Both factors $H=\{0\}\times\widehat G$ and $K=G\times\{1\}$ are therefore tensor subcategories with trivial associator. Use the regular algebra on $H$. The underlying central embedding and free induction restrict to a tensor functor $\operatorname{Vec}_K\to\mathcal H$. The adjunction calculation in the theorem shows that its $N$ simple images are distinct and exhaust the invertibles of $\mathcal H$; it is fully faithful and hence an equivalence onto $\mathcal H_{\mathrm{pt}}$. This proves the associator assertion. It does not require $K$ to remain a central subcategory after induction. \end{proof} \subsection{Preservation of the ribbon structure} \begin{theorem}\label{thm:ribbon-center} Let $\mathcal D$ be a modular tensor category, and let $A$ be a connected \'etale algebra with $\theta_A=1$ and $\dim_{\mathcal D}A\ne0$. Suppose $\mathcal B=\mathcal D_A$ has the near-group rules in Theorem \ref{thm:cyclic-converse} and its local category is the standard hyperbolic ribbon category of order $N^2$, with all its objects invertible in $\mathcal B$. Then there is a connected \'etale algebra $A'$ over $A$ such that $\mathcal H=\mathcal D_{A'}$ has the untwisted HI rules and \[ \mathcal D\simeq Z(\mathcal H) \] as ribbon categories. If $\dim_{\mathcal B}R=-Nt$, then $\dim_{\mathcal H}\rho=-t$. \end{theorem} \begin{proof} The central embedding $(\mathcal D_A^0)^{\mathrm{rev}}\to Z(\mathcal D_A)$ is part of the center equivalence \cite[Corollary 3.30]{dmno}. It is fully faithful after forgetting to $\mathcal D_A$ on the local subcategory. Choose either cyclic Lagrangian factor $H$ and its regular algebra. By \cite[Section 3.6 and Proposition 3.16]{dmno}, this algebra in $\mathcal D_A^0$ gives a connected \'etale algebra $A'$ over $A$ in $\mathcal D$, and algebra-module formation in stages identifies $\mathcal D_{A'}$ with the corresponding central de-equivariantization. Our inverse-braiding right-module convention agrees with the reverse local factor in the displayed center embedding. Theorem \ref{thm:cyclic-converse} and Corollary \ref{cor:untwisted} apply to it. The local modules after the second condensation have group $H^\perp/H=0$, so $\mathcal D_{A'}^0\simeq\operatorname{Vec}$. Equivalently $\operatorname{FPdim}(A')^2=\operatorname{FPdim}\mathcal D$. The central lift of free induction is therefore a braided equivalence $\mathcal D\to Z(\mathcal D_{A'})$ \cite[Corollary 4.1]{dmno}. For completeness, this equivalence preserves the required ribbon structures, not only the braiding. Local restriction preserves twists, so $\theta_{A'}=1$. Each local invertible has dimension $1$, and the module-dimension formula gives $\dim_{\mathcal D}A'=N\dim_{\mathcal D}A\ne0$. A connected \'etale algebra has nondegenerate multiplication pairing against its normalized projection to the unit: the radical is a right ideal, while its regular module is simple since its endomorphism algebra is $\operatorname{Hom}(\mathbf1,A')=\mathbb C$ in a semisimple module category; the pairing is nonzero on the unit. Thus $A'$ is a rigid algebra in the sense of \cite[Definition 1.11]{ko}. Theorems 1.17--1.18 and Remark 1.19 of that paper give a spherical structure on $\mathcal D_{A'}$ with \[ \dim_{\mathcal D_{A'}}F_{A'}(X)=\dim_{\mathcal D}X. \] The induced spherical structure on its center has the dimension of the underlying object. Hence the braided center equivalence preserves the dimension of every simple. Two pivotal structures on a fusion category differ by a tensor automorphism of the identity, scalar on each simple; that scalar multiplies its nonzero pivotal dimension. Equality of all simple dimensions therefore makes the two pivotal structures equal. In a braided category the pivotal structure and braiding determine the twist, proving the ribbon assertion. The same dimension formulas in two stages give $\dim F_H(R)=\dim R$. Its $N$ noninvertible summands form one orbit under the odd-order invertibles, all of dimension $1$. Their common dimension is consequently $\dim R/N=-t$. \end{proof} \paragraph{Scope of the identification.} The theorem constructs the particular fusion category $\mathcal H=\mathcal D_{A'}$ and identifies its full ribbon center. Here ``untwisted'' means that its pointed associator is trivial. It does not identify $\mathcal H$ with every category having these fusion rules, or with a separately prescribed HI solution. Such uniqueness cannot be inferred from modular data; distinct order-fifteen HI categories with identical center modular data are exhibited in \cite[Corollary 5.9]{gsy}. \begin{corollary}\label{cor:our-center} For the algebra $A=\mathbf1\oplus T$ constructed in $\mathcal D=\Par'$ above, the category $\mathcal D$ is ribbon equivalent to the modular tensor category $Z(\mathcal H_N)$, where $\mathcal H_N$ has the untwisted cyclic HI fusion rules and $\dim\rho=-t$. \end{corollary} \begin{proof} The branching and multiplicity budget prove that $\mathcal D_A$ has exactly the invertibles $E=(\mathbb Z/N)^2$ and $R$, with $R^2=\sum_E e+N^2R$ and $\dim R=-Nt$. Its local ribbon category is the standard hyperbolic category, and $\dim A=1+t^2\ne0$. Theorem~\ref{thm:ribbon-center} applies. In particular, $\mathcal H_N$ is a spherical fusion category over $\mathbb C$. Its Drinfeld center is therefore a modular tensor category with the induced ribbon structure by the general center theorem \cite[Section 2.9]{dgno}. \end{proof} \paragraph{Galois consequence.} Choose a number-field form of the fusion category and its spherical structure, and extend the automorphism $\sqrt\mu\mapsto-\sqrt\mu$ to its algebraic closure. Applied to the associativity and duality maps, it preserves the fusion rules and the trivial pointed associator, and sends $-t$ to $\delta$. All simple spherical dimensions of the conjugate are therefore its Frobenius--Perron dimensions. Hence the conjugate is pseudo-unitary. This is the same consequence already included in \cite[Theorem A(1) and Section 1.3]{gsy}; pseudo-unitarity here means equality of global and Frobenius--Perron dimensions. It does not assert a positive-definite unitary structure or a Galois-conjugate VOA construction. \section{Modular data of the center} \label{sec:gsy-modular} The center equivalence in Corollary~\ref{cor:our-center} determines a ribbon category. We now identify its complete modular data in the notation of Gannon--Schopieray--Yadav \cite[Section 5 and Appendix F.1]{gsy}. The comparison is uniform in odd $N$, including when $N^2+4$ is composite or has repeated prime factors. In their notation the answer is \[ \mathcal{MD}_{-}\bigl(\Z/N;\Z/(N^2+4),\ b\longmapsto b^2/(N^2+4)\bigr). \] The subscript $-$ specifies the spherical dimension $\dim\rho=-t$. The cyclic group and its quadratic form are additional information; the spherical sign alone does not determine them in general. \subsection{Quadratic forms and the exceptional labels} Write $e(x)=\exp(2\pi ix)$ and $\mu=N^2+4$. In this section quadratic forms are written additively in $\Q/\Z$, with their $e$-values giving the twists. The local odd lattice category has metric group \[ E=(\Z/N)^2,\qquad q_E(x,y)=\frac{xy}{N}\pmod\Z, \qquad B_E((x,y),(x',y'))=\frac{xy'+x'y}{N}\pmod\Z. \] Here the coordinates are obtained from the difference-of-squares coordinates in Part~IV by $x=a+b$, $y=a-b$. The letters $a,b$ below will instead label the exceptional cyclic group. Recall $v=K_-^{-1}e_0$, with $v_i=(N-2i)/\mu$ and $v_N=-v_0$. For each nonzero $a\in\Z/\mu$, choose the unique odd residue $s$ modulo $2\mu$ satisfying $s\equiv2a\pmod\mu$. Denote by $X^{\mathrm{neu}}_{[a]}$ the exceptional simple with frame grid \[ r\in sv+2\Z^N. \] The subscript $[a]$ means the orbit $\{a,-a\}$. Its dual-theta label has parity $e_0$, which is exactly the neutral label forced by \eqref{r:eq:SY}--\eqref{r:eq:SX} and $\dim J_\epsilon=(-1)^{\epsilon_0}$. Changing $s$ by $2\mu$ preserves the grid, while replacing $s$ by $-s$ gives the isomorphic opposite-grid module. This notation fixes the neutral representative as well as its odd discriminant label. The total conformal weight at $r=sv$ is \[ h(s)=\frac N4\left(\frac{s^2}{\mu}-1\right). \] Writing $s=2a+\mu k$, with $k$ odd, gives \[ h(s)-\frac{Na^2}{\mu} =Nak+\frac N4(\mu k^2-1)\in\Z, \] because $\mu\equiv1\pmod4$. Thus the exceptional twists are governed by \begin{equation}\label{eq:gsy-natural-form} H_N^{\mathrm{met}}=\Z/\mu,\qquad q_N(a)=\frac{Na^2}{\mu}\pmod\Z,\qquad B_N(a,a')=\frac{2Naa'}{\mu}\pmod\Z. \end{equation} Nondegeneracy follows from $\gcd(2N,\mu)=1$. \begin{lemma}[Uniform metric isometry]\label{lem:gsy-isometry} For every odd $N\geq3$, put \[ u_N=\frac{N^2-N+2}{2}. \] Then multiplication by $u_N$ identifies $(\Z/\mu,q_N)$ with $(\Z/\mu,q_0)$, where $q_0(b)=b^2/\mu\pmod\Z$. In particular, \begin{equation}\label{eq:gsy-gauss} \sum_{a\bmod\mu}e\bigl(q_N(a)\bigr)=\sqrt\mu. \end{equation} \end{lemma} \begin{proof} There is the integer identity \[ u_N^2-N=\mu\left(\frac{N-1}{2}\right)^2. \] Hence $u_N^2\equiv N\pmod\mu$. Since $N$ is a unit modulo $\mu$, so is $u_N$, and $q_0(u_Na)=q_N(a)$ for every $a$. The ordinary quadratic Gauss sum for the positive odd integer $\mu\equiv1\pmod4$ is $\sum_{b\bmod\mu}e(b^2/\mu)=\sqrt\mu$; the formula holds without a square-free hypothesis. Applying the isometry proves \eqref{eq:gsy-gauss}. This is the sign required in \cite[Theorem 5.1]{gsy}, where $r_-=-\sqrt\mu$. \end{proof} \subsection{The complete modular-data theorem} We use the normalized categorical Hopf-link matrix with \begin{equation}\label{eq:gsy-normalization} S_{\one,\one}=\frac1{\dim\mathcal H_N} =\frac1{N(1+t^2)}>0. \end{equation} Write $T^{\mathrm{rib}}=\diag(\theta_Z)$ for the ribbon twist matrix. This is the $T$ used in the center formulas of \cite{gsy}. The object called $T$ earlier in the manuscript remains the simple summand of $A=\one\oplus T$; the superscript distinguishes the matrix. \begin{theorem}[Negative-branch GSY modular data]\label{thm:gsy-modular} The ribbon categories $\mathcal D=\mathcal P_N'$, $Z(\mathcal H_N)$, and $\Rep(E_N)$ have modular data \begin{equation}\label{eq:gsy-identification} (S,T^{\mathrm{rib}}) =\mathcal{MD}_{-}\bigl(\Z/N;\Z/\mu,q_0\bigr), \qquad q_0(b)=\frac{b^2}{\mu}\pmod\Z, \end{equation} under the following bijection of simple labels: \begin{center} \begin{tabular}{@{}lll@{}} \toprule Simple of $\mathcal D$ & GSY label & Ribbon twist\\ \midrule $\one$ & $0_+$ & $1$\\ $T$ & $0_-$ & $1$\\ $Y_{[(x,y)]}$ & $\mathfrak a_{[x,y]}$ & $e(xy/N)$\\ $X^{\mathrm{neu}}_{[a]}$ & $\mathfrak d_{[u_Na]}$ & $e(Na^2/\mu)$\\ \bottomrule \end{tabular} \end{center} The periodic labels range over $((\Z/N)^2\setminus\{0\})/\{\pm1\}$ and the exceptional labels over $(\Z/\mu\setminus\{0\})/\{\pm1\}$. To display every entry, set \[ \alpha=\frac12\left(\frac1N+\frac1{\sqrt\mu}\right), \qquad \beta=\frac12\left(\frac1N-\frac1{\sqrt\mu}\right), \qquad r_G=\frac{N^2-1}{2},\quad r_H=\frac{\mu-1}{2}. \] In the ordered GSY basis $(0_+,0_-),(\mathfrak a),(\mathfrak d)$, \begin{equation}\label{eq:gsy-full-S} S=\begin{pmatrix} \alpha&\beta&N^{-1}\mathbf1_{1\times r_G}&-\mu^{-1/2}\mathbf1_{1\times r_H}\\ \beta&\alpha&N^{-1}\mathbf1_{1\times r_G}&\mu^{-1/2}\mathbf1_{1\times r_H}\\ N^{-1}\mathbf1_{r_G\times1}&N^{-1}\mathbf1_{r_G\times1}&C_E&0\\ -\mu^{-1/2}\mathbf1_{r_H\times1}&\mu^{-1/2}\mathbf1_{r_H\times1}&0&C_H \end{pmatrix}, \end{equation} where \begin{align} (C_E)_{[x,y],[x',y']} &=\frac2N\cos\left(\frac{2\pi(xy'+x'y)}N\right), \label{eq:gsy-CE}\\ (C_H)_{[b],[b']} &=\frac2{\sqrt\mu}\cos\left(\frac{4\pi bb'}\mu\right). \label{eq:gsy-CH} \end{align} The diagonal entries of $T^{\mathrm{rib}}$ in this basis are $1,1,e(xy/N),e(b^2/\mu)$. Thus the comparison fixes the tensor unit, the full $S$-matrix, and the twists, rather than only their spectra or the fusion rules. \end{theorem} \begin{proof} It suffices to work in $\mathcal D$: the two ribbon equivalences are Corollary~\ref{cor:our-center} and Theorem~\ref{thm:parity-cancellation-voa}. We first keep the natural exceptional coordinate $a$ of \eqref{eq:gsy-natural-form}. \emph{Normalization and distinguished rows.} The dimensions computed in Section~\ref{r:sec:budget} are $1,t^2,1+t^2,-Nt$ for $\one,T,Y,X$, respectively. Since $1+t^2=t\sqrt\mu$, the vacuum row in normalization \eqref{eq:gsy-normalization} is \[ S_{\one,\one}=\alpha,\qquad S_{\one,T}=\beta,\qquad S_{\one,Y}=\frac1N,\qquad S_{\one,X}=-\frac1{\sqrt\mu}. \] The remaining distinguished entries also follow directly from the branching matrix. Write $\mathcal E=\mathcal D_A^{\loc}$ and let $b^0$ be restriction from its odd local simples to $\mathcal D$. Its row at the local unit is $\one+T$, and its row at a nonzero $e\in E$ is $Y_{[e]}$. In the categorical normalizations $S^{\mathcal E}_{0,0}=1/N$ and $S^{\mathcal D}_{\one,\one}=1/[N\dim A]$, one has \[ S^{\mathcal E}b^0=b^0S^{\mathcal D}. \] Indeed, factor the analytic branching identity \eqref{r:eq:analytic} by the common parity matrix and use \eqref{r:eq:index}. If $S_W^{\rm an}=c_W\widetilde S_W$ and $S_L^{\rm an}=c_L\widetilde S_L$, that index identity is $c_L=c_W\dim A$; the unnormalized neutral and local Hopf-link matrices are $N\dim A\,S^{\mathcal D}$ and $N S^{\mathcal E}$, respectively. The scalar factors therefore cancel. The local unit row now gives \[ S_{\one,T}+S_{T,T}=\frac1N,\qquad S_{\one,Y}+S_{T,Y}=\frac2N,\qquad S_{\one,X}+S_{T,X}=0. \] Together with the vacuum row these imply \[ S_{T,T}=\alpha,\qquad S_{T,Y}=\frac1N,\qquad S_{T,X}=\frac1{\sqrt\mu}. \] These are precisely the tractable entries of \cite[Section 5.1]{gsy}. The two distinguished labels are fixed by their dimensions and by $A=\one\oplus T$; the periodic and exceptional families have dimensions $1+t^2$ and $-Nt$. The twists of $\one$ and $T$ are $1$ because $A$ is the conformal extension algebra. The local lattice twists give $e(xy/N)$ for $Y$, and the calculation preceding Lemma~\ref{lem:gsy-isometry} gives $e(Na^2/\mu)$ for $X^{\mathrm{neu}}_{[a]}$. \emph{Comparing the regular Fourier blocks.} For a rational vector $w$, write \[ f_{\sigma,w}(\boldsymbol\tau) =\frac1{\eta_\Pi(\boldsymbol\tau)} \sum_{m\in\Z^N+w} e\left(\sum_i\tau_i(B_\sigma m)_i^2\right). \] Multiplicity one and the grid classification identify the refined exceptional character with $f_{-,sv/2}$. Poisson summation gives the finite transform \begin{equation}\label{eq:gsy-grid-fourier} f_{\sigma,w}(-1/\boldsymbol\tau) =\frac1{\sqrt{\det(2K_\sigma)}} \sum_{z\in\Z^N/(2K_\sigma\Z^N)} e(-w^{\mathsf t}z)\, f_{\sigma,K_\sigma^{-1}z/2}(\boldsymbol\tau). \end{equation} Here $2K_\sigma w$ is integral for the grids being transformed. The formula follows by Poisson summation for the quadratic matrix $B_\sigma^{\mathsf t}\diag(\tau_i)B_\sigma$; its inverse sends $z$ to the displayed dual grid. The determinant factor is $2^{-N/2}/N$ in the periodic case and $2^{-N/2}/\sqrt\mu$ in the antiperiodic case. It holds on the convergent theta functions, hence on their formal diagonal germs. To pass to the normalization of $\mathcal D$, there is one common scalar $\kappa$ such that $S^{W,\mathrm{an}}_{Z,Z'}=\kappa S^{\mathcal D}_{Z,Z'}$ for neutral simples $Z,Z'$. This follows from the analytic/Hopf-link comparison in Section~\ref{sec:analytic-normalization} and the ribbon factorization by $\mathcal P_N$. For neutral $Y$, its Fourier label is $e_0$, so \eqref{r:eq:SY} and $S^{\mathcal D}_{\one,Y}=1/N$ give \begin{equation}\label{eq:gsy-parity-normalization} \kappa=-2^{-N/2}. \end{equation} In particular the analytic sign is determined, not replaced by a positivity assumption. For two neutral exceptional labels choose odd $s,s'$ with $s\equiv2a$ and $s'\equiv2a'$ modulo $\mu$. The two opposite target grids in \eqref{eq:gsy-grid-fourier} contribute \[ \frac{2^{1-N/2}}{\sqrt\mu} \cos\left(\frac{\pi Nss'}\mu\right) =-\frac{2^{1-N/2}}{\sqrt\mu} \cos\left(\frac{4\pi Naa'}\mu\right). \] For the equality, write $s=2a+\mu k$, $s'=2a'+\mu k'$ with $k,k'$ odd: the additional phase in $Nss'/(2\mu)$ is $N\mu kk'/2$ modulo integers, a half-integer. Division by \eqref{eq:gsy-parity-normalization} therefore gives \begin{equation}\label{eq:gsy-natural-XX} S^{\mathcal D}_{X^{\mathrm{neu}}_{[a]},X^{\mathrm{neu}}_{[a']}} =\frac2{\sqrt\mu}\cos\left(\frac{4\pi Naa'}\mu\right). \end{equation} For the periodic block, let $\xi_e$ be the odd lattice class indexed by $e\in E$. The neutral character uses the theta coset $\xi_e-\mathbf1/2$. Pairing an odd-order class with the two-primary class $\mathbf1/2$ is integral, and \[ 2(\mathbf1/2)^{\mathsf t}K_+(\mathbf1/2)=N/2. \] Thus the Fourier phase between two neutral periodic cosets is $-e(-B_E(e,e'))$. Pairing the opposite grids and dividing by $\kappa$ gives \eqref{eq:gsy-CE}. A periodic theta transform has no antiperiodic regular support, and conversely, so $S_{Y,X}=0$. These coefficient comparisons use the disjoint joint frame supports and uniqueness of regular grid modules from Sections~\ref{g:sec:generic-sector-classification} and~\ref{g:sec:exceptional}, together with Lemma~\ref{lem:ordinary-jets}. They do not assume linear independence of scalar characters, nor do they distinguish labels merely by their total conformal weight. They therefore also apply at composite and nonsquare-free $\mu$. Finally put $b=u_Na$. Lemma~\ref{lem:gsy-isometry} turns \eqref{eq:gsy-natural-XX} into \eqref{eq:gsy-CH} and turns the exceptional twist into $e(b^2/\mu)$. In Appendix~F.1 of \cite{gsy}, substituting $\sigma=-1$, $y=N/\sqrt\mu$, and $B_H(b,b')=2bb'/\mu$ gives exactly \eqref{eq:gsy-full-S}--\eqref{eq:gsy-CH}. Equation \eqref{eq:gsy-gauss} supplies their required Gauss-sum normalization. Every block and every twist agrees under the displayed bijection. \end{proof} \subsection{Ribbon twists and the VOA character normalization} The theorem compares ribbon modular data. For an ordinary VOA trace, the translation matrix includes the central-charge scalar: \[ T^{\mathrm{an}}=e(-c/24)T^{\mathrm{rib}}. \] Consequently, for the parity-cancelled realization $E_N$, \[ T^{E_N,\mathrm{an}}=e(2N/3)T^{\mathrm{rib}}. \] One must remove this scalar when comparing character transformations with the center matrices in \cite{gsy}. It has cube $1$, consistently with $c(E_N)=-16N\equiv0\pmod8$. For $W_N$, the parity factor must also be retained: \[ T^{W_N,\mathrm{an}} =e(5N/24)\bigl(T^{\mathcal P_N,\mathrm{rib}} \otimes T^{\mathrm{rib}}\bigr). \] The full character representation of $W_N$ is therefore not the center representation by itself. The vacuum row of \eqref{eq:gsy-full-S} recovers precisely $1,t^2,1+t^2,-Nt$. In particular, the negative branch is fixed with the vacuum at $0_+$; no interchange of $0_+$ and $0_-$ is being used. All the labels are self-dual, so the categorical normalization obeys $S^2=I$ and $(ST^{\mathrm{rib}})^3=I$. \subsection{Matching the tabulated categories and the scope of the result} \label{sec:small-comparison} The compact GSY modular label for Theorem~\ref{thm:gsy-modular} is $\mu^-_1$. The explicit isometry in Lemma~\ref{lem:gsy-isometry} removes the need for separate square-root searches at small orders. For completeness, the following are all rows of \cite[Appendix F, Table 2]{gsy} with this modular label at each listed $N$. A range of superscripts includes every row in that range. The table is a comparison of exact modular labels, not a calculation from the rounded numerical HI matrices printed there. \begin{center} \small \begin{tabular}{@{}rrll@{}} \toprule $N$ & $\mu$ & Modular label & Matching GSY rows\\ \midrule 3 & 13 & $13^-_1$ & $\alpha(3)$\\ 5 & 29 & $29^-_1$ & $\alpha(5)$\\ 7 & 53 & $53^-_1$ & $\alpha(7)$\\ 9 & 85 & $85^-_1$ & $\alpha^{(1)}(9)$\\ 11 & 125 & $125^-_1$ & $\alpha_A^{(1)}(11)$\\ 13 & 173 & $173^-_1$ & $\alpha(13)$\\ 15 & 229 & $229^-_1$ & $\alpha^{(1)}(15),\alpha^{(2)}(15),\alpha^{(3)}(15)$\\ 17 & 293 & $293^-_1$ & $\alpha(17)$\\ 19 & 365 & $365^-_1$ & $\alpha^{(1)}(19)$\\ 21 & 445 & $445^-_1$ & $\alpha^{(1)}(21),\alpha^{(3)}(21)$\\ 23 & 533 & $533^-_1$ & $\alpha^{(1)}(23)$\\ 25 & 629 & $629^-_1$ & $\alpha^{(1)}(25)$\\ 27 & 733 & $733^-_1$ & $\alpha^{(1)}(27),\alpha^{(2)}(27),\alpha^{(3)}(27)$\\ 29 & 845 & $845^-_1$ & $\alpha_A^{(1)}(29)$\\ 31 & 965 & $965^-_1$ & $\alpha^{(1)}(31)$\\ 33 & 1093 & $1093^-_1$ & $\alpha^{(1)}(33),\ldots,\alpha^{(5)}(33)$\\ 35 & 1229 & $1229^-_1$ & $\alpha^{(1)}(35),\alpha^{(2)}(35),\alpha^{(3)}(35)$\\ 37 & 1373 & $1373^-_1$ & $\alpha^{(1)}(37),\alpha^{(2)}(37),\alpha^{(3)}(37)$\\ 39 & 1525 & $1525^-_1$ & $\alpha_A^{(2)}(39),\alpha_A^{(4)}(39)$\\ 41 & 1685 & $1685^-_1$ & $\alpha^{(1)}(41)$\\ 43 & 1853 & $1853^-_1$ & $\alpha^{(1)}(43)$\\ \bottomrule \end{tabular} \end{center} Thus the realization has the modular data of the listed negative spherical GSY centers. In particular it selects $85^-_1$, rather than $85^-_3$, at $N=9$, and the cyclic $125^-_1$, rather than the noncyclic $25^-_1\times5^-_1$, at $N=11$. For arbitrary odd $N$, the uniform assertion is the exact datum \eqref{eq:gsy-identification} in the GSY classification of center modular data. Their general theorem permits other metric groups and forms. It does not by itself assign $\mu^-_1$ to every category in their uniform double-sine construction, and no such additional assignment is needed or asserted here. Nor does this theorem identify the associator of $\mathcal H_N$ with that of a separately prescribed GSY category. Already at $N=15$, the three matching rows have the same modular data but are pairwise not categorically Morita equivalent \cite[Corollary 5.9]{gsy}. An equivalence with a named category would require comparison of associativity data, or an equivalence of centers retaining the canonical Lagrangian algebra. At $N=3$ the present theorem gives the negative-branch Haagerup modular data; the classification results in \cite{grossmansnyder,grossmanizumi} are not being used to promote that equality to a named-category equivalence. The ribbon equivalence with the center of the category constructed here remains Corollary~\ref{cor:our-center}. \section{An explicit VOA realization of the parity centralizer} \label{sec:parity-cancellation} We use ``vertex operator algebra'' in the lower-bounded, integer-graded sense: negative conformal degrees are allowed, but there are only finitely many of them and every homogeneous space is finite-dimensional. The extension below is simple, self-contragredient, regular (hence rational), and $C_2$-cofinite. Its regularity means that every weak module is a direct sum of irreducible ordinary modules; its $C_2$-cofiniteness is proved separately from this assertion. \subsection{The parity coordinates} Let $W=W_N$ and retain the parity subcategory $\mathcal P\subset\operatorname{Rep}(W_N)$ from Theorem~\ref{c:currents}. Its parity modules are \[ J_\epsilon= \bigoplus_{\substack{r_i\geq1\cr r_i\equiv1+\epsilon_i\ ({\rm mod}\ 2)}} \bigotimes_{i=0}^{N-1}L_i(r_i,r_{i+1}), \qquad \epsilon\in P=(\mathbb Z/2)^N. \] The parity-current construction gives $J_\epsilon\boxtimes_WJ_\eta\cong J_{\epsilon+\eta}$. With the cycle matrix $K$ of diagonal $(N+2,2,\ldots,2)$ and off-diagonal cycle entries $-1$, its self-braiding quadratic form, twist, and dimension are \begin{equation}\label{eq:parity-data} Q(\epsilon)=i^{\epsilon^{\mathsf t}K\epsilon},\qquad \theta_\epsilon=i^{\epsilon^{\mathsf t}K\epsilon+2N\epsilon_0}, \qquad d_\epsilon=(-1)^{\epsilon_0}. \end{equation} For $N=3$ each off-diagonal entry is $-1$, so that $x^{\mathsf t}Kx=\sum_i(x_i-x_{i+1})^2+Nx_0^2$ still holds. Here is an explicit change of parity coordinates, needed to specify the cancellation algebra. Set \[ (S\epsilon)_i=\epsilon_i+\epsilon_{i+1}+\epsilon_0 \quad\text{in }\mathbb F_2. \] If $S\epsilon=0$, all successive differences equal $\epsilon_0$; summing them and using oddness of $N$ gives $\epsilon_0=0$, and then every coordinate is zero. Thus $S$ is invertible. Let $T(\epsilon)$ be the number of cyclic transitions of the binary word $\epsilon$; it is even. Then \[ \epsilon^{\mathsf t}K\epsilon=T(\epsilon)+N\epsilon_0, \qquad |S\epsilon|= \begin{cases} T(\epsilon),&\epsilon_0=0,\\ N-T(\epsilon),&\epsilon_0=1. \end{cases} \] Substitution in \eqref{eq:parity-data} gives \begin{equation}\label{eq:semion-coordinates} Q(\epsilon)=i^{|S\epsilon|},\qquad \theta_\epsilon=(-i)^{|S\epsilon|},\qquad d_\epsilon=(-1)^{|S\epsilon|}. \end{equation} The classification of pointed braided categories by their quadratic forms, with the additional specified ribbon twist, therefore identifies the parity ribbon category as $\mathcal P\simeq(\mathcal S^-)^{\boxtimes N}$. The nontrivial object of $\mathcal S^-$ has self-braiding $i$, twist $-i$, and dimension $-1$. \subsection{A minuscule conformal shift of the \texorpdfstring{$E_7$}{E7} lattice} Choose the standard even unimodular $E_8$ lattice in $\mathbb R^8$, put $\alpha=e_7+e_8$, and let \[ K_7=E_8\cap\alpha^\perp\cong E_7,\qquad u=e_6+\tfrac12e_7-\tfrac12e_8. \] Writing $\beta=e_6+e_7\in E_8$, we have $u=\beta-\alpha/2$. Hence \[ u\in K_7^*\setminus K_7,\quad 2u\in K_7, \quad u^2=\tfrac32, \quad K_7^*/K_7\cong\mathbb Z/2. \] Define \[ U=(V_{K_7},\omega_{K_7}+u(-2)\mathbf1). \] The fields, vacuum and translation are those of the lattice vertex algebra. Its Virasoro modes and momentum weights are \begin{align*} L_u(n)&=L_{K_7}(n)-(n+1)u(n),& c(U)&=7-12u^2=-11,\\ \operatorname{wt}_u(e^p) &=\frac{p^2}{2}-(u,p) =\frac{(p-u)^2}{2}-\frac34. \end{align*} The shifted VOA is integer graded since $u\in K_7^*$. It is nonnegatively graded: for $p\in K_7$, the vector $\beta-p\in E_8$ has inner product $1$ with $\alpha$, and therefore has squared norm at least $2$. Its orthogonal projection $u-p$ has squared norm at least $2-1/2=3/2$. The positive definite square-completion formula also gives finite weight spaces. The two simple weak modules are the shifted versions of $V_{K_7}$ and $V_{K_7+u}$. Their vertex operators are unchanged. The ordinary lattice VOA is regular, and each lattice module remains ordinary for $L_u(0)$ by the same square-completion formula. Thus $U$ is regular. It is $C_2$-cofinite because the state-field map and hence its $C_2$-space are unchanged. Self-contragredience follows from $2u\in K_7$ \cite[Theorems 3.1, 4.1, 4.4]{dongmason}. Write $M=V_{K_7+u}$ with the shifted conformal structure. For $p=u+a$, $a\in K_7$, its weights are $a^2/2-3/4$ plus oscillator degrees. Thus it has a one-dimensional lowest space of weight $-3/4$ and twist $i$. The change of conformal vector preserves the braided category: apply \cite[Proposition 2.17]{mcrae} to $v=u(-1)\mathbf1$, for which $v_0\omega_{K_7}=0$; both conformal gradings are nonnegative. The lattice fusion and self-braiding are consequently \[ M\boxtimes_U M\cong U, \qquad c_{M,M}=e^{\pi i u^2}=-i. \] The order-two pointed ribbon category has $d(M)=\theta_M/c_{M,M}=-1$. Its associator on the triple of nontrivial labels is $-1$ in the usual semion gauge. These data give the ribbon equivalence \begin{equation}\label{eq:inverse-E7} \operatorname{Rep}(U)\simeq(\mathcal S^-)^{\mathrm{rev}}. \end{equation} In particular no theorem about the representation category of a twisted HI VOA is needed to construct this auxiliary factor. \subsection{The diagonal extension and all its weak modules} \begin{theorem}\label{thm:parity-cancellation-voa} Let $W_N$ be the VOA constructed above, with its established parity ribbon factor, and put $\mathcal D=\mathcal P'\subset\operatorname{Rep}(W_N)$. Set $M^0=U$, $M^1=M$. Then \[ E_N=\bigoplus_{\epsilon\in P} J_\epsilon\otimes\bigotimes_{i=0}^{N-1}M^{(S\epsilon)_i} \] has a simple, self-contragredient, regular, $C_2$-cofinite VOA extension structure with \[ c(E_N)=-16N,\qquad \operatorname{Rep}(E_N)\simeq\mathcal D \] as ribbon categories. In particular, $E_N$ is rational and has exactly $N^2+3$ irreducible ordinary modules. Its least conformal degree is $-(N-1)/2$, and its homogeneous spaces are finite-dimensional. Thus it is a lower-bounded integer-graded VOA, and is not of CFT type. \end{theorem} \begin{proof} Let $V=W_N\otimes U^{\otimes N}$. Every tensor factor is nonnegatively graded and $C_2$-cofinite. Their finite tensor product has these properties too. The Deligne product theorem \cite[Theorem 5.1]{c:McRaeDeligne} gives \[ \operatorname{Rep}(V) \simeq\mathcal P\boxtimes\mathcal D\boxtimes\mathcal P^{\mathrm{rev}}. \] The displayed object $E_N$ is the diagonal regular algebra in the first and third factors, tensored with the unit of $\mathcal D$. Explicitly, all its self-braidings and twists cancel by \eqref{eq:semion-coordinates}--\eqref{eq:inverse-E7}. The diagonal pointed subcategory is Tannakian, and its regular algebra is connected, commutative and separable. Its multiplication is nonzero between every pair of its simple-current components. It has dimension $|P|=2^N$, since each paired component has dimension $1$. The weights in this finite direct sum are integral and lower bounded, with finite-dimensional homogeneous spaces, as verified below. The extension correspondence \cite[Theorem 7.5]{cmy} therefore supplies the VOA structure. Its unit map embeds $V$ since $V$ is simple. Separability and connectedness make its regular algebra module simple, which is exactly simplicity of the extension VOA. \emph{$C_2$-cofiniteness.} Equip the same underlying vertex algebra as $V$ with the unshifted lattice conformal vectors and write \[ V^{\mathrm{un}}=W_N\otimes V_{K_7}^{\otimes N}. \] This is a CFT-type, $C_2$-cofinite VOA. As a weak $V^{\mathrm{un}}$-module, $E_N$ is the finite direct sum of the irreducible modules \[ J_\epsilon\otimes \bigotimes_{i=0}^{N-1}V_{K_7+(S\epsilon)_i u}, \qquad \epsilon\in P. \] Irreducibility follows from the tensor-product decomposition, and the conformal shift has not changed the module vertex operators. Each summand is $C_2$-cofinite as a $V^{\mathrm{un}}$-module by \cite[Proposition 5.2]{abd}. Consequently the subspace \[ C_2^{V^{\mathrm{un}}}(E_N) =\operatorname{span}_{\mathbb C} \{v_{-2}e:v\in V^{\mathrm{un}},\ e\in E_N\} \] has finite codimension in $E_N$. The underlying vertex algebra $V^{\mathrm{un}}$ embeds in $E_N$, so \[ C_2^{V^{\mathrm{un}}}(E_N)\subseteq C_2(E_N), \qquad \dim E_N/C_2(E_N) \leq\dim E_N/C_2^{V^{\mathrm{un}}}(E_N)<\infty. \] Here $V^{\mathrm{un}}$ is used only to apply the CFT-type module theorem: its conformal vector need not agree with that of $E_N$. The vertex operators, and hence the displayed $C_2$-spaces, are unchanged. This proves $C_2$-cofiniteness with the stated conformal structure on $E_N$. \emph{The module category and regularity.} The local-module category of the diagonal algebra in $\mathcal P\boxtimes\mathcal P^{\mathrm{rev}}$ is $\operatorname{Vec}$; this follows either from its Lagrangian dimension or directly from the diagonal subgroup's orthogonal complement. Hence the local algebra modules in $\operatorname{Rep}(V)$ form the ribbon category $\mathcal D$. The VOA/local-algebra correspondence is a braided tensor equivalence \cite[Theorem 3.65]{ckm}; twists agree because both are $e^{2\pi i L(0)}$ on the same underlying module. This identifies the full ordinary category, and also proves regularity rather than only semisimplicity within a chosen finite category. Indeed the displayed Deligne product is semisimple. By \cite[Proposition 2.11]{mcrae}, every cyclic weak $V$-module is grading-restricted generalized, since $V$ is nonnegatively graded and $C_2$-cofinite. It is therefore a finite direct sum of ordinary simples. Every weak $V$-module is the sum of its cyclic submodules, and hence a sum of simple submodules. Such a module is semisimple; this proves regularity of $V$ in the weak-module sense. Every weak $E_N$-module restricts to a direct sum of ordinary simple $V$-modules, hence belongs to $\operatorname{Ind}(\operatorname{Rep}(V))$. The extension correspondence on direct limits \cite[Theorems 7.5 and 7.7]{cmy} identifies it with a local module for the same finite separable algebra there. Here the passage to arbitrary weak modules can be checked on finite pieces. Let $Q$ be such a local algebra module and let $X\subset Q$ be any finite $V$-subobject. The action map \[ E_N\boxtimes_V X\longrightarrow Q \] has finite image, since its source lies in the finite category $\operatorname{Rep}(V)$. Associativity makes that image an $E_N$-submodule. Separability makes every finite algebra module semisimple. These finite images cover $Q$, because $Q$ is a union of finite $V$-subobjects and the algebra action is unital. Consequently $Q$ is a sum of simple algebra submodules, hence a direct sum of them. Each simple summand is finite as a $V$-object and is ordinary for the common conformal vector: its total $L(0)$ is diagonalizable, lower bounded, and has finite-dimensional eigenspaces. Locality passes to subobjects and direct summands. Thus every weak $E_N$-module is a direct sum of irreducible ordinary $E_N$-modules. In particular every admissible module is completely reducible, so $E_N$ is rational. The ribbon equivalence and the previously established rank of $\mathcal D$ give its $N^2+3$ simple modules. \emph{Self-contragredience.} The conformal subalgebra $V=W_N\otimes U^{\otimes N}$ is self-contragredient, since all its tensor factors are. The finite $V$-module decomposition of $E_N$ contains $V$ as a direct summand. Taking graded contragredients therefore shows that the ordinary $E_N$-module $E_N^\vee$, restricted to $V$, contains $V^\vee\cong V$. Induction--restriction adjunction \cite[Lemma 2.61 and Theorem 3.65]{ckm} gives \[ \operatorname{Hom}_{E_N}(E_N,E_N^\vee) \cong\operatorname{Hom}_{V}(V,E_N^\vee)\ne0, \] because induction sends $V$ to $E_N$. The adjoint $E_N$-module is irreducible by simplicity, and its graded contragredient is also irreducible. A nonzero morphism between them is therefore an isomorphism. Thus $E_N\cong E_N^\vee$ for its stated conformal vector; this argument does not change that conformal vector or assume CFT type. \emph{Conformal degrees and central charge.} Write $r=\mathbf1+v$ in a frame summand of $J_\epsilon$. Its lowest total weight is \[ \frac14\sum_i(v_i-v_{i+1})^2+\frac N4v_0(v_0+2). \] The constraints $v_i\geq0$ and $v_i\equiv\epsilon_i\pmod2$ imply a squared difference of at least $1$ at every parity transition and $v_0\geq\epsilon_0$; $v=\epsilon$ attains all these bounds. Consequently \[ h_{\min}(J_\epsilon) =\frac{T(\epsilon)+3N\epsilon_0}{4}. \] Adding the auxiliary minimum $-3|S\epsilon|/4$ gives the integral minimum of the corresponding extension component: \[ h_{\min}(E_{N,\epsilon})= \begin{cases} -T(\epsilon)/2,&\epsilon_0=0,\\ T(\epsilon),&\epsilon_0=1. \end{cases} \] All higher weights differ from these by integers. The maximum possible even transition number on an odd cycle is $N-1$, achieved by a word with $\epsilon_0=0$. This proves the stated least degree. Central charges add to $-5N+N(-11)=-16N$. \end{proof} \begin{proof}[Completion of the proof of Theorem~\ref{thm:main}] The cyclic construction, invariant form, and $C_2$ argument establish the stated properties of $W_N$; Section~\ref{r:sec:projectivity} proves its regularity. The parity factorization is Section~\ref{sec:parity-factor}. The lattice branching and the vanishing remainder in the dimension identity give $|\operatorname{Irr}\mathcal D|=N^2+3$. The parity factorization then gives $|\operatorname{Irr}(W_N)|=2^N(N^2+3)$. Corollary~\ref{cor:our-center} constructs $\mathcal H_N$ and gives the ribbon-center identification with $\dim\rho=-t$. Theorem~\ref{thm:parity-cancellation-voa} supplies the explicit VOA $E_N$, its simplicity, self-contragredience, $C_2$-cofiniteness, regularity (and hence rationality), its central charge, its least degree, and the required ribbon equivalence. Theorem~\ref{thm:gsy-modular}, with the metric isometry of Lemma~\ref{lem:gsy-isometry}, proves the modular-data identification in part (4). These are all the assertions of the theorem. \end{proof} \Needspace{12\baselineskip} \appendix \part*{Technical appendices} \addcontentsline{toc}{part}{Technical appendices} \section{Ordinary first images of proper cyclic intervals} \label{app:ordinary-interval-images} \label{neutral:app:proper-intervals} Keep the hypothetical weight-one vacuum extension and put $H_*=2N+2$. For a proper connected binary root support $J$ containing zero, let $B_J$ be its primary and $r_J=(B_J)_{H_*-2}v$. \begin{lemma}\label{neutral:lem:proper-first-images} Every $r_J$ is a sum of ordinary simultaneous partial eigenvectors. Its support consists only of coordinate spectra at the active frame sites of $B_J$. \end{lemma} \begin{proof} At a vacuum frame site, $L_i(-1)B_J=0$ and $h_i^{J}=0$. Equations \eqref{neutral:eq:neutral-first-image}-- \eqref{neutral:eq:neutral-desc-first-image} therefore give $H_ir_J=0$ exactly. At its two endpoints, Lemma~\ref{neutral:lem:first-image-cubic} makes the partial operator semisimple on $r_J$. Only internal $(3,3)$ factors can initially have partial nilpotents. For $J=\{0\}$ there are no such internal factors, proving the base case. First let $J=[0,k]$, with $0\leq k\leq N-2$, and write $B_k=B_J$. For $0\leq ak}n_\ell\right). \] This is the repeated partial Zhu identity \eqref{neutral:eq:neutral-partial-Zhu}, with the partial weight updated after each inner descendant. Let $V_d^+(p)$ and $V_d^-(p)$ be the local level-$d$ coefficients for ordered inputs $(3,1),(1,3)$ and $(1,3),(3,1)$, respectively. Through level three they are the coefficients displayed in Section~\ref{r:sec:cyclic}, with $V_0^+=V_0^-=1$. Write $Q_d^\pm(x,y;p)$ for the polynomial obtained by substituting $Z_{\mathbf n}$ into $V_d^\pm$. For three disjoint groups of sites, let $X,Y,Z$ denote the sums of their left partial conformal operators, let $U,V,T$ denote independent right variables, and let $h_X,h_Y,h_Z$ be the corresponding partial weights of $C$. For a polynomial $P$, write $P_{abc}=P(U=a,V=b,T=c)$ and put \begin{align*} \mathcal F(P)={}&(H-1)P_{000} +(h_X-X)(P_{100}-P_{000})\\ &+(h_Y-Y)(P_{010}-P_{000}) +(h_Z-Z)(P_{001}-P_{000}),\qquad H=2N. \end{align*} The right partial operators on $U=\operatorname{span}\{w_i\}$ are orthogonal coordinate idempotents. Since $o(C)w=(H-1)r_C$ and $o(C)w_i=(h_i-H_i)r_C$, the action of the descendant encoded by $P$ on $w$ is exactly $\mathcal F(P)r_C$. There is no division by $r_C$ in this definition. \subsection{The prefix cuts} Fix $0\leq a\leq N-2$ and put $r=t+a$, \[ X=H_a,\qquad Y=H_{N-1},\qquad Z=\sum_{i1\) and \[ B(r)-A= \frac{r((2t-3)r^2+(8t-9)r+2t)}{6t(r-1)(r+2)}<0. \] Thus \(B(r)-1\) follows from \(4t^3+16t^2+t-6<0\) on \(00\). Every subsequent prefix row has \(r>1\), multiplier \[ \frac{r(r^2+4r+1)}{(r+1)(r^2+4r-2)}\in(0,1), \] and common fixed point \[ v_*=-\frac{(t-1)(2t^2+4t-1)} {t(t+1)(2t-1)}>0. \] Each step is a convex combination of its predecessor and \(v_*\). Thus \(u_{N-1}>0\), contradicting the terminal value \(-1\). \subsection{The two remaining rank-three cases} Put \(t=(\sqrt{13}-3)/2\). At \(j=0\), exact reduction modulo \(t^2+3t-1\) gives \[ u_1+1=\frac{53-175t}{2(16t-5)}<0,\qquad v_*+1=\frac{7t-2}{16t-5}<0. \] The next step is a convex combination of two values below \(-1\), so cannot reach \(-1\). The signs follow from \(2/70, \] using \(2/70\). The sum relation then gives \(n_0=z(t-1/(t+1))\). Substitution in any remaining interval with \(a=1\) leaves \[ -\frac{(3t+1)(t^2+1)}{6(t-1)}z=0. \] The scalar substitution behind this residual is \begin{align*} &g_{\rm left}(t,s)\cdot \left(t-\frac1{t+1},\frac1{s(s+1)},\frac1{t+1}-\frac1s\right)\\ &\qquad-\frac{(t+1)(2t+s^2+s-4)}{2(t-1)(s-1)(s+2)} =-\frac{(3t+1)(t^2+1)}{6(t-1)}. \end{align*} The coefficient is nonzero, so \(u=0\) and every \(n_i=0\) on this component. On component \(e_{N-1}\), put \(u=r_{R_0,N-1}\). The transport formulas are \[ r_{B,N-1}=\begin{cases} \dfrac{r(r+1)}{t(t+1)}u,&b=N-1,\\[4pt] \dfrac{2r(r+1)(\sigma+1)(\sigma+2)}{t(t+1)D(s)}u,&b\le N-2. \end{cases} \] For completeness, the corresponding scalar recurrences are \[ \frac{T_{N-1}(a+1,b)}{T_{N-1}(a,b)}=\frac{r+2}{r},\qquad \frac{T_{N-1}(a,b-1)}{T_{N-1}(a,b)}= \begin{cases} \dfrac{2(\sigma+2)}{(\sigma-2)(\sigma-1)\sigma},&b=N-1,\\[3pt] \dfrac{s+2}{s-2},&b\le N-2. \end{cases} \] They use the same operator at $e_{N-1}$ and the same initial value. Set \(z=2(\sigma+1)(\sigma+2)u/[t(t+1)]\). The off-component equations are again \eqref{r:eq:triangle}, now for \(0\le i0\). The sum fixes \(n_{N-1}\). A remaining interval ending at \(N-1\) gives \[ -\frac{(3\sigma+2)(\sigma^2+2\sigma+2)} {6(\sigma+2)}z=0. \] In this case the corresponding scalar identity, with $s=\sigma$, is \begin{align*} &g_{\rm right}(r,s)\cdot \left(\frac1{r(r+1)},\frac1s-s-1,\frac1{r+1}-\frac1s\right)\\ &\qquad-\frac{s(r^2+r-2s-6)}{2(r-1)(r+2)(s+2)} =-\frac{(3s+2)(s^2+2s+2)}{6(s+2)}. \end{align*} Again \(u=0\) and every \(n_i=0\). The two endpoints are the only allowed components of \(r_{R_0}\), so \(r_{R_0}=0\). \subsection{Exact rank-three endpoint calculation} For \(N=3\), use the four unknown vectors \((n_0,n_1,n_2,u)\), with \(u=r_{R_0,j}\). The row order is the sum relation followed by the interval equations \([1,1],[1,2],[2,2]\). Substituting the same gradients and transport formulas, and reducing modulo \(t^2+3t-1\), gives the matrices below. Explicitly, for $(a,b)=(1,1),(1,2),(2,2)$, choose $g=(g_X,g_Y,g_Z)$ to be the left, right, or off row according as $j=a-1$, $j=b$, or neither. With $\mathbf e_i$ the coordinate row vectors, the corresponding matrix row is \[ \left(g_X\mathbf e_{a-1}+g_Y\mathbf e_b+ g_Z\sum_{a\le i0\). Thus \(r_{R_0}=0\) also for \(N=3\). The displayed matrices follow by substituting the three intervals into \eqref{neutral:eq:explicit-all-interval-K} and the transport relations, then reducing modulo \(t^2+3t-1\). \subsection{Propagation and the final Jacobian step} Suppose a proper heavy interval \(B\) satisfies \(r_B=0\), so \(o(B)U=0\). Adjoin an ordinary interval \(A\) to form another proper heavy interval \(B'\). Its unique order-two OPE has \[ A_1B=\kappa B',\qquad A_0B=\kappa\sum_i c_iL_i(-1)B',\qquad \sum_i c_i h_i^{B'}=1. \] For right growth, \(c_i\) is \(-(t+a)\) at the shared site, one at each new site, and zero elsewhere. For left growth, its shared-site value is \(t+b+1\), again with new-site values one. Put \(Q=\sum_i c_iH_i\). On every top, the partial Zhu formula and the usual mode commutator give \[ [o(A),o(B)]=o(A_0B)+o(A_1B)=\kappa[Q,o(B')]. \] The scalar terms cancel because their partial-weight pairing is one. The left side kills \(U\), as \(o(A)U\subset U\). The first image \(r_{B'}\) is ordinary. Project the identity, applied to \(w_i\), to a target component \(e_j\): \[ (c_j-c_i)(h_i^{B'}-\delta_{ij})r_{B',j}=0. \] On the active support, shared and new sites have different coefficients. For any target \(j\), choose an active \(i\) with \(c_i\ne c_j\); then \(i\ne j\). Every active partial weight is nonzero: internally it is \(2/[(t+i)(t+i+1)]\), and at an endpoint it is an irrational \((3,1)\) or \((1,3)\) weight. Hence \(r_{B',j}=0\). Starting at \(R_0\) and adjoining roots on either side reaches every proper heavy cyclic interval, proving \(r_B=0\) for all of them. Every right side in \eqref{r:eq:K} is now zero. Appendix \ref{r:app:jac}'s invertible Jacobian therefore gives \(\mm r_{C,j}=0\). This proves Theorem~\ref{r:thm:first} for every odd \(N\ge3\). \section{The weight-zero top has vacuum open-chain action} \label{r:app:h0-open-chain} We use the frame decomposition of \(W\), its nonzero diagonal multiplication, the elementary cubic null relations, and Appendix~\ref{neutral:app:filtered-spanning}. Let $M$ be a finite-dimensional representation of $A(W)$ whose commuting partial conformal operators $H_i$ have joint generalized eigenvalue zero and satisfy \[ \sum_{i=0}^{N-1}H_i=0. \] These hypotheses hold on the relevant neutral aligned simple tops of lowest weight zero. They also hold on a vacuum extension by a distinct such simple: the total conformal nilpotent factors through a homomorphism from the vacuum to the distinct simple and hence vanishes. We do not assert the result for arbitrary logarithmic tops with a nonzero total conformal nilpotent. \subsection{Eliminating the partial nilpotents} For the ordinary elementary root $R_j$, $1\leq j0. \] Its two active partial weights are $a,b$ at sites $j,j-1$. The local cubic relation is \[ (x-y)\bigl((x-y)^2-2\psi(x+y)+2\psi-1\bigr)=0. \] The second factor is invertible on left and right multiplication by a nilpotent $H_i$, since $2\psi-1\ne0$. Thus $[H_i,R_j]=0$ for every $i$. Let $\mathcal R\subseteq\End(M)$ be the commutative algebra generated by the $H_i$, and let $\mathfrak m=(H_0,\ldots,H_{N-1})$ be its nilpotent maximal ideal. The operators $R_j$ commute with $\mathcal R$ by the preceding null relation. The partial Zhu formula \begin{equation} [L_i(-n)v]=(-1)^n\bigl(-H_i*[v]+n[v]*H_i+h_i[v]\bigr),\qquad n\geq1, \label{r:eq:h0-partial-zhu} \end{equation} gives $[D_aR_j]=-aR_j$. In particular $[R_j,D_aR_j]=0$. Only the vacuum and root self-channel occur in $R_j*R_j$ and in \[ [R_j,D_aR_j]=[(R_j)_0D_aR_j]+[(R_j)_1D_aR_j]. \] All states used have weight at most four, whereas the next aligned channel has lowest weight eight. Write $\nu_j,\sigma_j\ne0$ for the vacuum and self-channel coefficients. Set \begin{align*} u(h)&=\frac{h(h+1)}{3(h-4)(3h-1)}, &d(a)&=(a-4)(a+2)(2a-3)(2a-1),\\ \kappa(a)&=-\frac{a(a-2)(5a^2-10a+2)}{4d(a)}, &\eta(a)&=-\frac{7a^2(a-2)^2(a-1)}{8d(a)}. \end{align*} The commutator has the form $P(H)+Q(H)R_j=0$, where $P(0)=0$ and $Q(0)=2\sigma_j\eta(a)\ne0$. Since $Q(H)$ is invertible in $\mathcal R$, this expresses $R_j$ as $-Q(H)^{-1}P(H)\in\mathfrak m$. Modulo $\mathfrak m^2$, the product and half the commutator give \begin{align*} 0&\equiv\nu_j\bigl(u(a)H_j+u(b)H_{j-1}\bigr) +\sigma_j\kappa(a)R_j,\\ 0&\equiv\nu_j\bigl(-b\,u(a)H_j+a\,u(b)H_{j-1}\bigr) +\sigma_j\eta(a)R_j. \end{align*} Eliminating $R_j$ uses the exact identity \[ \bigl((\eta+b\kappa)u(a),(\eta-a\kappa)u(b)\bigr) =\frac{a^2(a-2)^2}{24d(a)}(a+1,a-3). \] All denominators, as well as $\sigma_j$, $\eta(a)$ and the common factor $a^2(a-2)^2/(24d(a))$, are nonzero: each possible zero would force $a$ to be rational. No division by $\kappa(a)$ is needed. We obtain \begin{equation} (t+j+1)H_j-(t+j-1)H_{j-1}\in\mathfrak m^2. \label{r:eq:h0-recurrence} \end{equation} Iteration gives \[ H_j\equiv\frac{t(t+1)}{(t+j)(t+j+1)}H_0\pmod{\mathfrak m^2}. \] All coefficients are positive. Their sum, together with $\sum_iH_i=0$, gives $H_0\in\mathfrak m^2$, then $\mathfrak m=\mathfrak m^2$. Nilpotence forces $\mathfrak m=0$. Therefore \begin{equation} H_i=0\quad(0\leq i0$ (already $4k-1>0$ when $j=2$). It never vanishes. Hence every ordinary interval acts by zero. The full ordinary open-chain Zhu image therefore acts by its vacuum character on the entire top $M$. \section{Completion of the weight-zero Zhu calculation} \label{r:app:h0-completion} Retain the hypotheses and conclusion of Appendix~\ref{r:app:h0-open-chain}. We now show that the proper heavy intervals also act trivially. Only the full-cycle class $C$ remains; its quadratic relation will determine the quotient governing lowest weight zero. \subsection{The known ground character and maximal heavy intervals} Lemma~\ref{neutral:lem:second-ground-character} constructs a simple ground character $T_0$ on which every proper interval vanishes and $C$ acts by $2\lambda\ne0$. The argument uses only the quotient $A/W$, charge support and the invariant pairing; semisimplicity of the lattice restriction is not required. Fix $1\leq j0. \] Therefore $B=0$. The scalar requires only level two. At site $j$ the two inputs have weight $1+2/q$ and the output is the vacuum, whose evaluated series is $1$. At site $j-1$ the first, second, and output weights are \[ p=\frac{q-2}{q},\qquad x=\frac2{q(q-1)},\qquad w=\frac{q+1}{q-1}. \] The evaluated local series is \[ 1-z+\frac{3q^2+10q+4}{(q+3)(3q+2)}z^2+O(z^3). \] Indeed, writing the second descendant as $\alpha L(-2)v+\beta L(-1)^2v$, the Ward equations are \begin{align*} 3\alpha+2(2w+1)\beta &=\frac{(w+p-x+1)(w+p-x)}{2w},\\ (4w+c/2)\alpha+6w\beta&=w+2p-x, \qquad c=13-6\left(\frac q{q-1}+\frac{q-1}q\right). \end{align*} Its Zhu evaluation is $w\alpha+w(w+1)\beta$. Multiplication of the series by $(1+z)^2$ and extraction of its degree-two coefficient gives $\kappa_-$. All other first inputs are the tensor unit. This proves that every proper heavy interval vanishes. \subsection{The universal cyclic quadratic and its consequences} The full Zhu algebra is generated by the partial frame classes, all proper interval highest classes, and $C$. The preceding results leave only $C$. The exact lowest-weight formula \[ h(m)=\sum_i(m_i-m_{i+1})^2+Nm_0(m_0+1) \] shows that a frame summand with $m_0\geq2$ starts at weight at least $6N$. Every state in $C*C$ has weight at most $4N<6N$, so $C*C$ lies in the root-zero filtration piece $F_1W$. The filtered maximum-plateau Zhu spanning lemma gives \[ [F_1W]\subseteq A_0+ \sum_{P\text{ proper heavy or }P=C}A_0[P]A_0, \] where $A_0$ is the ordinary open-chain image. Its vacuum action and the heavy annihilation show that $C^2=\gamma\mathbf1+\delta C$ with fixed scalars. The vacuum character gives $\gamma=0$, and $T_0$ gives $\delta=2\lambda$. Consequently \[ C(C-2\lambda)=0, \qquad A(W_N)/\langle H_0,\ldots,H_{N-1}\rangle \cong\frac{\mathbb C[X]}{X(X-2\lambda)} \cong\mathbb C\oplus\mathbb C. \] The vacuum and $T_0$ supply both distinct characters. Thus the surjection from $\mathbb C[X]/(X(X-2\lambda))$ to the quotient is an isomorphism. Every relevant finite top with generalized partial spectrum zero and exact total conformal operator zero first reduces to this quotient by Appendix~\ref{r:app:h0-open-chain}. It is therefore a direct sum of the known vacuum and $T$ characters. The Zhu correspondence excludes any additional neutral aligned simple of lowest weight zero. In a vacuum extension by a distinct such simple, the top extension splits over this semisimple quotient; the previously proved injection of vacuum extensions into Zhu-top extensions then splits the full module extension. Vacuum self-extensions split by Proposition~\ref{neutral:prop:two-lowest-weights}. Thus every vacuum extension by a simple target of lowest weight zero splits. \section{The heavy-root vacuum coefficient excludes a weight-one defect} \label{r:app:heavy-vacuum-obstruction} This calculation uses the closing-root first-image annihilation from Theorem~\ref{r:thm:first}. Suppose a nonsplit vacuum extension has a simple target of lowest weight one. Lemma~\ref{neutral:lem:ordinary-root-defect} gives nonzero vectors $w_j$ with \[ U=\operatorname{span}\{w_0,\ldots,w_{N-1}\},\qquad H_iw_j=\delta_{ij}w_j. \] Theorem~\ref{r:thm:first} gives $o(R_0)U=0$ for the closing root in every odd rank, including $N=3$. Its total and active local weights are \[ H=2N+2,\qquad a=1+\frac2t,\qquad b=1-2t, \qquad a+b=H,\quad t^2+Nt-1=0. \] Every partial Virasoro descendant of $R_0$ also annihilates $U$: iterate \[ [L_i(-n)v]=(-1)^n\bigl(-H_i*[v]+n[v]*H_i+h_i[v]\bigr) \] and use $H_iU\subset U$. The aligned channels of $R_0\times R_0$ are the vacuum, $R_0$, and $2e_0$. The last has lowest weight $6N+8$, above the maximum weight $2H=4N+4$ in $R_0*R_0$. Thus its self-channel annihilates $U$, and it remains to evaluate the vacuum channel on $w_0$. \subsection{The local evaluated vacuum block} Let $h$ be an irrational local $(3,1)$ or $(1,3)$ weight, with \[ c=\frac{-3h^2+7h-2}{h+1}. \] Its highest vector has the null relation \begin{equation} \bigl(L(-1)^3-2(h+1)L(-2)L(-1)+h(h+1)L(-3)\bigr)v_h=0. \label{r:eq:heavy-null} \end{equation} Normalize the vacuum intertwiner as \[ I(v_h,z)v_h=z^{-2h}\sum_{n\geq0}v_nz^n,\qquad v_0=\mathbf1. \] Evaluate these vacuum descendants in the highest-weight-one Virasoro module, with normalized highest vector and dual $\xi,\xi'$, and put \[ V_h(z)=\sum_{n\geq0}\langle \xi',o(v_n)\xi\rangle z^n, \qquad F(x)=(x-1)^{-2h}V_h(x-1). \] This is the formal four-point block with weights $1,h,h,1$ at $\infty,x,1,0$, in the vacuum channel between the two $h$ insertions. It is defined by the formal vacuum intertwiner and the ordinary Virasoro module action, and has normalization $V_h(0)=1$. Set \begin{align*} A(x)&=\frac{x-1}{x}-\frac{x}{x-1},\\ B(x)&=\frac{1+2h}{x}-\frac{1+2h}{x-1} +\frac h{(x-1)^2}+\frac1{x^2},\\ C(x)&=-\frac{x-1}{x^2}+\frac{x}{(x-1)^2},\\ D(x)&=-\frac{2h}{x^2}+\frac{2h}{(x-1)^2} -\frac{2h}{(x-1)^3}-\frac2{x^3}. \end{align*} The null relation gives \begin{equation} F'''-2(h+1)AF''+ \bigl(-2(h+1)B+h(h+1)C\bigr)F'+h(h+1)DF=0. \label{r:eq:heavy-bpz} \end{equation} For a direct Ward verification, allow the other insertions to vary at $y,s$. At $y=1,s=0$, translation and scaling give \[ \partial_yF=-xF'-2hF,\qquad \partial_sF=(x-1)F'+2hF. \] The $L(-2)$ insertion is $AF'+B_0F$, with \[ B_0=\frac1{x^2}+\frac h{(x-1)^2}+\frac{2h}{x}-\frac{2h}{x-1}. \] Differentiating the global identities for the first descendant gives $AF''+BF'$ for $L(-2)L(-1)$. Since $L(-3)=[L(-1),L(-2)]$, its insertion is $CF'+DF$, with $C=A'+B_0-B$ and $D=B_0'$. This proves \eqref{r:eq:heavy-bpz}, including the outer-weight-one terms. \subsection{Uniform solution of the vacuum channel} Put $u=(x-1)^2/x$, $\theta=u\,d/du$, and substitute $F=(x-1)^{-2h}G(u)$. After removing $(x-1)^{-2h}$, the left side of \eqref{r:eq:heavy-bpz} equals exactly \begin{equation} \frac{4(x+1)}{x^2(x-1)^3} \left[ \theta\left(\theta-\frac h2\right) \left(\theta-\frac{1+3h}{2}\right) +\frac u4(\theta-h-1)(\theta-h)(\theta+2) \right]G(u). \label{r:eq:heavy-transformed} \end{equation} One can verify the identity directly by expanding the bracket as an operator in $d/du$. Its coefficients on $G''',G'',G',G$, respectively, are \[ u^3(1+u/4),\qquad \frac{u^2}{2}(5-4h+(2-h)u),\qquad \frac u4(3h^2-7h+2+h(h-5)u),\qquad \frac u2h(h+1). \] The chain rule with $u'=(x^2-1)/x^2$ gives exactly the four coefficients of the left-hand side after its common factor has been removed. Construct the ordinary power-series solution $G(0)=1$. Its recurrence has no zero denominator because $h$ is irrational, and gives \begin{equation} G(u)=\sum_{n\geq0}d_nu^n,\qquad d_n=\left(-\frac14\right)^n \frac{(-h-1)_n(-h)_n(2)_n} {(1-h/2)_n((1-3h)/2)_n\,n!}, \label{r:eq:heavy-dn} \end{equation} where $(x)_n=x(x+1)\cdots(x+n-1)$. The exponents of \eqref{r:eq:heavy-bpz} at $x=1$ are $-2h,-h,h+1$, with nonintegral pairwise differences because $h$ is irrational. The normalized Frobenius solution with exponent $-2h$ is therefore unique. Thus the series just constructed is the vacuum block, and its dependence on $u$ follows from the equation and normalization. Consequently \[ V_h(z)={}_3F_2\!\left( \begin{matrix}-h-1,-h,2\\1-h/2,(1-3h)/2\end{matrix}; -\frac{z^2}{4(1+z)}\right). \] For comparison with the low-level Ward equations, $d_1=2h/c$ and \[ d_2=\frac{h^2(h+1)}{(h-4)(h-2)(3h-1)} \] agree with the level-two and level-four vacuum Ward equations. \subsection{The coefficient cannot vanish} On $w_0$ the two active partial Zhu weights are $1,0$. At weight zero, every positive vacuum descendant evaluates to zero; the second local series is exactly $1$. Therefore, after dividing out the nonzero root vacuum coefficient $\nu_0$, the vacuum contribution to $R_0*R_0$ is \[ P_H(h)=[z^{2H}](1+z)^HV_h(z),\qquad h=a=1+2/t. \] Using \eqref{r:eq:heavy-dn}, \[ (1+z)^HV_h(z)=\sum_{n\geq0}d_nz^{2n}(1+z)^{H-n}. \] For $nH$ its lowest degree is $2n>2H$. Hence only $n=H$ contributes: \begin{equation} \boxed{\quad P_H(h)=\left(-\frac14\right)^H \frac{(-h-1)_H(-h)_H(2)_H} {(1-h/2)_H((1-3h)/2)_H\,H!}\ne0. \quad} \label{r:eq:heavy-nonzero} \end{equation} Every possible zero or pole of these finite Pochhammer products would make $h$ rational, whereas $1+2/t$ is irrational. The scalar $\nu_0$ is nonzero by the nondegenerate invariant form on the multiplicity-one self-contragredient summand \(M_{e_0}\). Finally apply $R_0*R_0$ to $w_0\ne0$. Its left side is zero because $R_0U=0$. Its self-channel descendants annihilate $U$, and the higher channel is excluded by weight. Its vacuum channel equals $\nu_0P_H(a)w_0\ne0$, a contradiction. Thus no nonsplit vacuum extension with a lowest-weight-one simple target exists. Together with the weight-zero analysis and Proposition~\ref{neutral:prop:two-lowest-weights}, this proves that every extension of the vacuum by a simple module splits. \section{The uniform symbolic frame certificate} \label{app:certificate} The screening-frame construction has one computer-assisted algebraic step. The charge bound in Section~\ref{c:app:frame} reduces the Virasoro relations to proper intervals, charge $B$, and finitely many tails $B+I$. The proper-interval and tail equations are proved there. The program reproduced below checks the remaining charge-$B$ identities with $N$, $t$, and the ordered site positions left indeterminate. A coefficient involves at most two frame indices and three oscillator indices. The program enumerates their equality and order patterns; its interval-count formula includes the end sites. After the denominators prescribed by the frame formulas are cleared, every residual numerator expands to zero. The frame proof separately establishes that those denominators are nonzero at the parameters used in the theorem. Together with the mathematical reduction to these patterns, this makes the calculation uniform in $N$. The program checks the following numbers of coefficient identities: \begin{center} \begin{tabular}{@{}lrrrrr@{}} \toprule Distinct site indices & 1 & 2 & 3 & 4 & 5\\ Coefficient identities & 7 & 40 & 69 & 46 & 10\\ \bottomrule \end{tabular} \end{center} All 172 residual numerators vanish identically. The calculation uses exact symbolic arithmetic, with no numerical substitution for the rank or site positions. It checks these identities, not the surrounding representation-theoretic arguments. To reproduce the calculation, save the listing as \nolinkurl{certify_uniform_epsilon.py} and run it with Python~3 and SymPy. It prints a JSON report and writes the same report beside the script. The program requires no external mathematical data and makes no network requests. The listing is self-contained and requires no supplementary files. \subsection{Complete program} \begin{lstlisting} """Uniform symbolic verification of the charge-B Virasoro-frame equations. N, t, and all distinct site positions are indeterminates. This enumerates order/equality types of at most five site indices, NOT numerical ranks. Interval sums use the exact counting formula in the frame construction. """ from collections import Counter from functools import lru_cache from itertools import combinations_with_replacement, product from pathlib import Path import json import time import sympy as s # Polynomial operations for oscillator monomials. def poly_add(*args): out = {} for p in args: for k,v in p.items(): out[k] = out.get(k,0)+v return {k:v for k,v in out.items() if v != 0} def scale(p,c): return {k:c*v for k,v in p.items() if c*v != 0} def mul(p,q): out={} for a,x in p.items(): for b,y in q.items(): k=tuple(u+v for u,v in zip(a,b)) out[k]=out.get(k,0)+x*y return {k:v for k,v in out.items() if v != 0} N,t=s.symbols('N t') def ba(*args): ans={} for p in args: for key,c in p.items():ans[key]=ans.get(key,0)+c return {key:c for key,c in ans.items() if c != 0} def bm(p,q): ans={} for (M,H),a in p.items(): for (M2,H2),b in q.items(): key=(M|M2,H|H2);ans[key]=ans.get(key,0)+a*b return {key:c for key,c in ans.items() if c != 0} def bs(p,c):return {key:v*c for key,v in p.items() if v*c != 0} BC={(frozenset(),frozenset()):s.S.One} def setup(m): positions=s.symbols(f'j0:{m}') r=[t+j for j in positions];p=[1/(x+1) for x in r] psi=[(x+1)/x for x in r];R=(t+N)/t def bv(kind,i,offset=0): point=(i,offset) return {(frozenset([point]) if kind=='M' else frozenset(), frozenset([point]) if kind=='H' else frozenset()):s.S.One} mom=[];u=[];v=[] for i in range(m): mi,mnext=bv('M',i),bv('M',i,1) hi,hnext=bv('H',i),bv('H',i,1) mom.append(ba(mi,bs(mnext,-r[i]/(r[i]+1)))) u.append(ba(bs(mi,1/r[i]),bs(mnext,-1/(r[i]+1)))) gi=ba(BC,bs(mi,-1),bs(hi,R-1)) gn=ba(BC,bs(mnext,-1),bs(hnext,R-1)) v.append(ba(bs(gi,1/r[i]),bs(gn,-1/(r[i]+1)))) comp=[ba(bs(BC,p[i]),bs(mom[i],-1)) for i in range(m)] def val(point):return positions[point[0]]+point[1] def interval_sum(pol): terms=[] for (M,H),c in pol.items(): upper=min(H) if H else (m,0) upperval=val(upper) if H else N if M: lo,hi=min(M),max(M) count=0 if upper<=hi else val(lo)*(upperval-val(hi)) else:count=upperval*(upperval-1)/2 if count!=0:terms.append(c*count) return s.Add(*terms) @lru_cache(None) def source(i,j,indices): left=bm(u[i],v[j]);right=bm(v[i],u[j]) for k in indices: left=bm(left,mom[k]);right=bm(right,comp[k]) return interval_sum(ba(left,right)) variables=3*m one={(0,)*variables:s.S.One} def atom(i,k): key=[0]*variables;key[3*i+k-1]=1;return {tuple(key):s.S.One} atoms={(i,k):atom(i,k) for i in range(m) for k in (1,2,3)} def D(pol,i): ans=mul(scale(atoms[i,1],p[i]),pol) for key,c in pol.items(): for k in (1,2): power=key[3*i+k-1] if power: kk=list(key);kk[3*i+k-1]-=1;kk[3*i+k]+=1 ans=poly_add(ans,{tuple(kk):k*power*c}) return ans def Dt(pol):return poly_add(*(D(pol,i) for i in range(m))) def action(pol,i,mode): if mode==-1:return D(pol,i) if mode==0:return {key:c*sum(k*key[3*i+k-1] for k in (1,2,3)) for key,c in pol.items()} ans={} for key,c in pol.items(): if mode==1 and key[3*i+1]: kk=list(key);kk[3*i+1]-=1;kk[3*i]+=1 ans=poly_add(ans,{tuple(kk):2*c*key[3*i+1]}) if mode==2: if key[3*i]==2: kk=list(key);kk[3*i]-=2 ans=poly_add(ans,{tuple(kk):2*psi[i]*c}) if key[3*i+1]: kk=list(key);kk[3*i+1]-=1 ans=poly_add(ans,{tuple(kk):-2*psi[i]*p[i]*c}) return ans corrections=[] for i in range(m): pol=scale(atoms[i,2],(1+R)*p[i]/2) for j in range(m): diag=(R*p[i]-r[i]*p[i]**2)/2 if j==i else -(R if j