\documentclass[11pt]{article} \usepackage[T1]{fontenc} \usepackage{lmodern,microtype,amsmath,amssymb,amsthm,mathtools,booktabs,xurl} \usepackage[margin=25mm]{geometry} \usepackage[colorlinks=true,allcolors=blue]{hyperref} \setlength{\emergencystretch}{2em} \newtheorem{theorem}{Theorem}[section] \newtheorem{proposition}[theorem]{Proposition} \newtheorem{lemma}[theorem]{Lemma} \newtheorem{corollary}[theorem]{Corollary} \newcommand{\C}{\mathbb C} \newcommand{\Q}{\mathbb Q} \newcommand{\Z}{\mathbb Z} \newcommand{\one}{\mathbf1} \DeclareMathOperator{\Rep}{Rep} \DeclareMathOperator{\FPdim}{FPdim} \DeclareMathOperator{\Dim}{Dim} \DeclareMathOperator{\Hom}{Hom} \DeclareMathOperator{\End}{End} \DeclareMathOperator{\Tr}{Tr} \renewcommand{\Vec}{\operatorname{Vec}} \title{An order-five Haagerup--Izumi vertex operator algebra\\ at central charge eight} \author{Tzu-Chen Huang} \date{12 October 2026} \begin{document} \maketitle \begin{abstract} We construct an explicit unitary strongly rational vertex operator algebra $U$ of central charge eight inside $V_{A_4\oplus A_4}$ and identify $\Rep(U)$ with the Drinfeld center of the positive order-five Haagerup--Izumi category. The construction adjoins one algebraic weight-two field to a diagonal affine $\mathfrak{sl}_2$ at level forty and the ambient conformal vector. Exact finite lattice identities imply strong generation and $C_2$-cofiniteness. A finite integral order in the actual Zhu algebra descends semisimplicity from a $1590$-dimensional upper presentation over $\mathbb F_{10069}$. To determine the modules, we combine the good-prime block bound with lattice character growth and a Wronskian inequality. The resulting full branching has an irreducible congruence complement of dimension at most fifteen; its exact classification forces all fourteen denominator-$29$ sectors. Finally, the actual holomorphic $E_8$ extension determines the boundary fusion category and its associator. The proof is computer-assisted, with exact arithmetic certificates separated from exploratory numerical evidence. \end{abstract} \section{Statement and construction} The Haagerup--Izumi fusion category $\mathcal H_5$ has simple objects $\alpha_g,\alpha_g\rho$, $g\in\Z/5$, with \begin{equation}\label{eq:HI} \alpha_g\alpha_h=\alpha_{g+h},\qquad \rho\alpha_g=\alpha_{-g}\rho,\qquad \rho^2=\one\oplus\bigoplus_g\alpha_g\rho. \end{equation} Its positive dimension is $d=(5+\sqrt{29})/2$. The chiral realization problem concerns the braided center $Z(\mathcal H_5)$, rather than $\mathcal H_5$ itself. Evans--Gannon proposed VOA realizations of this family and described their modular targets~\cite{EG10}. The recent order-three lattice constructions~\cite{Riesen,HuangHI3} motivate the present lattice approach. \begin{theorem}\label{thm:main} The explicitly specified subalgebra \[ U=\langle E,H_c,F,\omega_B,u\rangle\subset B=V_{A_4\oplus A_4} \] is simple, unitary and strongly rational, with $c=8$, $U_1\cong\mathfrak{sl}_2$ at level $40$, and a ribbon equivalence \[ \Rep(U)\simeq Z(\mathcal H_5). \] It has $28$ simple modules and \[ \chi_U=q^{-1/3}(1+3q+38q^2+250q^3+1575q^4+7545q^5+O(q^6)). \] The lattice restriction is $B\downarrow_U=U\oplus\pi$, with $h_\pi=1$, $\dim\pi_1=45$, and $\operatorname{qdim}\pi=d^2$. \end{theorem} Here is a complete specification of the generators. Use simple-root coordinates on $A_4\oplus A_4$, with Cartan matrix $G$, and cocycle \[ \epsilon(a,b)=(-1)^{\sum_i a_i b_i+\sum_{i>j}a_iG_{ij}b_j}. \] Set $c=(4,6,6,4,4,6,6,4)$ and \[ E=\sum_{i=1}^8 e^{\alpha_i},\qquad F=-\sum_{i=1}^8 c_i e^{-\alpha_i},\qquad H_c=\sum_{i=1}^8c_i\alpha_i(-1)\one. \] Then $E_{(0)}F=H_c$, $H_{c(0)}E=2E$, $E_{(1)}F=40\one$ and $H_{c(1)}H_c=80\one$. The vector $\omega_B$ is the usual rank-eight lattice conformal vector. In each $A_4$ write $\gamma_{ij}=e_i-e_j$, $\theta=\gamma_{15}$ and $p=\theta+\gamma_{24}$. The field $u$ is the sum of twelve norm-four exponentials with the following coefficients and momenta, in corresponding order: \[ \begin{array}{c|l} s,t&(p,0),(0,p)\\ a,b,c_0&(\gamma_{13},\theta),(\gamma_{24},\theta),(\gamma_{35},\theta)\\ d_0,e_0,f_0,g_0&(\gamma_{14},\gamma_{14}),(\gamma_{14},\gamma_{25}), (\gamma_{25},\gamma_{14}),(\gamma_{25},\gamma_{25})\\ h,i_0,j_0&(\theta,\gamma_{13}),(\theta,\gamma_{24}),(\theta,\gamma_{35}). \end{array} \] The dependent coefficients are $b=a+d_0-e_0$, $c_0=b+f_0-g_0$, $i_0=h+d_0-f_0$, $j_0=i_0+e_0-g_0$. Put $R_0=\sqrt{145}$, \[ z=\frac{3\sqrt{29}-5R_0}{58},\qquad D=\frac{9125+125R_0}{20736} +\frac{365\sqrt5+25\sqrt{29}}{6912},\qquad \tau=i\sqrt D. \] All square roots in these formulas are positive before multiplication by $i$. The independent parameter column is $A+\tau B_0$, where \begin{equation}\label{eq:seed} A=\begin{pmatrix} 1\\-(73+R_0)/72\\(31\sqrt5+11\sqrt{29})/144\\ (-3\sqrt5+\sqrt{29})/16\\ -(1+R_0)/32-(\sqrt5+5\sqrt{29})/288\\ -(1+R_0)/32+(\sqrt5+5\sqrt{29})/288\\ -(1+R_0)/32+(\sqrt5+5\sqrt{29})/288\\ -(1+R_0)/32-(\sqrt5+5\sqrt{29})/288 \end{pmatrix},\quad B_0=\begin{pmatrix}0\\0\\1+z\\1-z\\-1\\z\\-z\\1\end{pmatrix}. \end{equation} The order is $(s,t,a,h,d_0,e_0,f_0,g_0)$. All computations take place in the degree-eight field $K=\Q(\sqrt5,\sqrt{29},\tau)$. The vector $u$ is an affine highest vector of horizontal weight $12$. In this Chevalley gauge, compact PCT conjugation is \[ \Theta\bigl(\alpha_{i_1}(-m_1)\cdots\alpha_{i_r}(-m_r)e^a\bigr) =(-1)^r\Bigl(\prod_i c_i^{a_i}\Bigr) \alpha_{i_1}(-m_1)\cdots\alpha_{i_r}(-m_r)e^{-a}, \] with antilinear coefficients. Exact calculation gives $\Theta(u)=F_{(0)}^{12}u/12!$ and $F_{(0)}^{13}u=0$. Thus the generating set is PCT stable. The positive lattice form restricts to $U$; the invariant-form argument with $U_0=\C\one$ makes $U$ simple and self-contragredient. \section{Finite closure and \texorpdfstring{$C_2$}{C2}-cofiniteness} Let $J=\langle E,H_c,F\rangle$ and let $A$ be the exact weight-two space obtained by closing the current descendants, $\omega_B$ and the current multiplet of $u$ under weight-preserving products. With $T=L(-1)$ define \begin{align*} N_3&=J_{(-1)}A+TA,&S_3&=N_3+A_{(0)}A,\\ N_4&=J_{(-1)}S_3+TS_3+A_{(-1)}A,&W_4&=N_4+A_{(0)}S_3,\\ N_5&=J_{(-1)}W_4+TW_4+A_{(-1)}S_3. \end{align*} These definitions initially give finite subspaces, without asserting that they exhaust the graded pieces of $U$. \begin{lemma}[Recursive closure]\label{lem:closure} Let $G_1,\ldots,G_h$ be finite positive-weight spaces, and let $\mathcal N$ be the span of all parenthesized differential normal words in them. Suppose all singular products of $G_1+G_2$ with the $G_i$ belong to $\mathcal N$, and for $i\ge3$ suppose \[ G_i\subset (G_2)_{(0)}G_{i-1} +\mathcal N_i(G_1+\cdots+G_{i-1}). \] Then $\mathcal N$ is a vertex subalgebra. \end{lemma} \begin{proof} Zero modes are derivations, so the assumed $G_2$ zero products first prove $(G_2)_{(0)}\mathcal N\subset\mathcal N$. Wick induction handles any pair having a derivative or composite input from generator brackets of smaller total weight. For the remaining generator pairs, induct lexicographically on total weight and left weight. A recursive term $s=a_{(0)}b$ satisfies \[ s_{(n)}t=a_{(0)}(b_{(n)}t)-b_{(n)}(a_{(0)}t). \] The first term has smaller total input weight. In the second, a composite or derivative in the known normal expression of $a_{(0)}t$ is handled by Wick induction; a single generator has the same total weight and smaller left weight. All generator brackets close, and Wick induction finishes. No PBW independence is assumed. \end{proof} \begin{proposition}[Exact lattice certificate]\label{prop:finite} For the seed \eqref{eq:seed}, \[ \dim A=38,\quad \dim S_3=250,\quad \dim N_4=1286,\quad\dim W_4=1575,\quad\dim N_5=7545. \] The hypotheses of Lemma~\ref{lem:closure} hold with $(G_1,G_2,G_3,G_4)=(J,A,S_3,W_4)$. \end{proposition} \begin{proof}[Exact computational proof] Lattice fields are expanded by their cocycle and the finite truncations of the Heisenberg exponential formula. Current highest vectors reduce the computations to rational Clebsch--Gordan projections over $K$. Each asserted dependency is checked by reconstruction in the full ambient lattice basis, not only on a selected minor. Independent specialization ranks certify the selected vectors' independence. The primitive complements have dimensions $\dim(S_3/N_3)=112$ and $\dim(W_4/N_4)=289$, in $14$ and $35$ current multiplets. The exact positive-product certificate checks all $5550$ projected components on these complements. Current products follow from their recursive provenance. The zero-mode derivation and skew-symmetry identities give $A_{(0)}N_4\subset N_5$. The remaining $1074$ projected components of $A_{(0)}(W_4/N_4)$ all belong to the same $7545$-dimensional exact normal space. The coverage audit enumerates every required component and checks the producer and witness hashes; see Section~\ref{sec:certificates}. Lemma~\ref{lem:closure} now applies in characteristic zero. \end{proof} It follows that $U_1=J$ and $U\ne B$. The graded dimensions through weight five are now dimensions of the actual VOA. In particular, finite vertex generation has not been substituted for strong generation. \begin{proposition}\label{prop:c2} $U$ is $C_2$-cofinite. \end{proposition} \begin{proof} The positive Heisenberg bound is $L_0\ge H_0^2/160$. If a field has weight $g$ and nonzero charge $\ell$, its $N$th normal power vanishes whenever $N>160g/\ell^2$. Thus charged classes in $U/C_2(U)$ are nilpotent. In characteristic zero the nilradical is stable under derivations, so all components of nontrivial current multiplets are nilpotent as well. The strong spaces have only two essential neutral classes: $\omega_C=\omega_B-\omega_{40}$ and a fixed second weight-two singlet $x$. The affine Sugawara class is a polynomial in currents; the weight-three singlets are translates and the weight-four singlets are normal expressions in lower fields. In the reduced $C_2$ quotient, two exact singlet projections of $A_{(0)}S_3$ give homogeneous quadratics $q_1(\omega_C,x)=q_2(\omega_C,x)=0$. Their coefficients come from brackets of nontrivial current multiplets, which vanish in this reduced Poisson quotient. The four linear multiples $\omega_Cq_1,xq_1,\omega_Cq_2,xq_2$ span all neutral cubics; the exact $4\times4$ coefficient determinant is nonzero. Reducedness forces both neutral classes to vanish. Strong generation makes the $C_2$ algebra finitely generated, and each positive-degree generator is nilpotent. Bounded monomials span it. \end{proof} \section{An integral order and rationality} An affine extension makes the Zhu computation finite and small. Let $a$ be the unique current-highest-weight-eight vector in $A$ and $\lambda=H_c/2=(2,3,3,2;2,3,3,2)$. The extremal weight-ten, charge-forty lattice space is $\C e^\lambda$. Exact calculation gives \[ a_{(-1)}^5\one=\kappa e^\lambda,\qquad\kappa\ne0. \] One rational coordinate of $\kappa$ is $-5918853670082299150421875/5210532790841206823544$. It can be computed by five applications of $a_{(3)}$ to $e^{-\lambda}$, using the invariant pairing. PCT and affine fusion therefore embed \[ \mathcal D=L_{40}(\mathfrak{sl}_2)\oplus L_{40}(40)\subset U. \] Its internal grading agrees with the ambient grading on $\mathcal D$, so $A(\mathcal D)\to A(U)$ is defined although its Sugawara conformal vector differs from $\omega_B$. Its simple tops have current types $0,2,\ldots,20$, with two blocks at $20$. Let $R$ be the DVR in $K$ above $p=10069$ with residues $\sqrt5=2054$, $\sqrt{29}=531$, $\tau=515$. All finite closure and recursive-provenance coefficients are integral at this prime. An audit of $10\,571\,523$ rational coordinates finds no denominator divisible by $p$, and the neutral cubic determinant has residue $7638$. Unit-minor checks for the exact highest-vector bases and normal targets prove saturation of the required integral spans; remaining low weight-five memberships have explicit integral witnesses. Let $U_R$ be the resulting generated vertex algebra. \begin{lemma}\label{lem:order} The $R$-span $C_R$ of the images of $U_R$ in $A(U_K)$ is a finite free order. Its current highest-line idempotent $e$ is full over $R$ and in the special fiber. \end{lemma} \begin{proof} Use negative-mode words for integral strong generation. The proof of Lemma~\ref{lem:closure} uses integral binomial identities and divided translations, so requires no arbitrary factorial inversions. Set \[ Q_R=U_R\Big/\sum_{n\ge2,\,b,c\in U_R}R\,b_{(-n)}c. \] Retaining all higher negative modes is essential in positive characteristic. The integral Borcherds identities make $Q_R$ a commutative graded Poisson algebra, generated by the finite strong fields, with finite graded pieces. Its special fiber is finite-dimensional. For a highest field of weight $g\le4$ and current highest weight $\ell>0$, put $N=\lfloor160g/\ell^2\rfloor+1$. Its class satisfies $a^N=0$. Applying the finite lowering exponential and a Vandermonde matrix with unit determinant shows every component is nilpotent of exponent $(\ell+1)(N-1)+1\le4810$, so the spherical dimensions are positive. Let $r'\le28$ be the dimension of the ordinary-character span. Choose a basis of actual characters containing the desired character; their lowest weights are all nonnegative. The nonzero modular Wronskian has weight $r'(r'-1)$ and finite-order multiplier. Its order at infinity is at least $\sum_i(h_i-1/3)$, even when leading exponents coincide. The valence formula gives \[ h\le\sum_i h_i\le\frac{r'}3+\frac{r'(r'-1)}{12}\le\frac{217}{3}. \] This argument does not require independence of all simple characters. \end{proof} \begin{proposition}\label{prop:branch} The full lattice restrictions are $B_{0,0}=U\oplus\pi$ and $B_{x,y}=G_{[x,y]}$ for nonzero $(x,y)$. \end{proposition} \begin{proof} An additional simple of weight $j/5$ would have $0\le j\le361$. Its residue requires $j\equiv5k/29\pmod {10069}$, whereas the smallest nonnegative representative among these residues is $697$. Thus the lattice restrictions contain only known simples. Their ground spaces exhaust the lowest spaces, and their actual weights permit no further summand at a higher integer grade. In the vacuum coset, $U_0$ and the dimensions $3+45=48$ at weight one exhaust the possibilities. Rationality ensures these restrictions are finite semisimple sums without arbitrary shifts in conformal weight. \end{proof} The positive-dimensional lattice extension is a connected separable commutative algebra, so its branching matrix satisfies $bS_U=S_Bb$, $bT_U=T_Bb$~\cite{HKL,DMNO}. Its row space has dimension thirteen. Let $W$ be its invariant orthogonal complement, of dimension at most fifteen. It has basis $v=(e_0-e_\pi)/\sqrt2$ and the additional labels. Remove the central-charge character from $T$, leaving the twists. \begin{lemma}\label{lem:Weil} $W$ is the nonsquare even Weil representation of level $29$. In particular $\dim W=15$ and $S_{vv}=-1/\sqrt{29}$. \end{lemma} \begin{proof} All simples are self-dual and have positive dimensions, so $S$ is real orthogonal. The twist-one line in $W$ is uniquely $\C v$. The other twists are distinct: if $h_i-h_j=n\in\Z$, reduction gives $29n-(k_i-k_j)\equiv0\pmod {10069}$. Its absolute value is below $10069$ by Lemma~\ref{lem:bounds}, so it is zero, contradicting the distinct residues of \eqref{eq:K} modulo $29$. Branching gives $S_{\pi j}=-S_{0j}$ for an additional label, so $S_{vj}=\sqrt2S_{0j}>0$. With simple $T$ spectrum, these connections to $v$ force $W$ to be irreducible. The congruence theorem~\cite{DLN} applies to $W$. We use the complete Nobs--Wolfart congruence classification as implemented in \texttt{SL2Reps}~\cite{NWW}. Among all $6268$ irreducibles of dimensions $1$ through $15$, precisely $113$ have $S^2=1$, simple $T$ spectrum and exactly one twist one. For each other phase $a/N$, the exact test \[ h=a/N+n\le217/3,\qquad n\ge0,\qquad h\equiv k/29\pmod {10069} \] leaves only the trivial representation and the nonsquare even Weil representation of level $29$. There is at most one integer $n$ for each residue. An independent exact-fraction replay verifies all $113$ spectral filters. The trivial possibility is excluded without guessing any index. For $u_0=(e_0+e_\pi)/\sqrt2$, branching gives $S_{u_0u_0}=1/5$. Positivity of $S_{00}$ and $S_{0\pi}$ implies $|S_{vv}|<1/5$, whereas the trivial representation has $S_{vv}=1$. In the nonsquare Weil representation, the Gauss sum gives $S_{vv}=-1/\sqrt{29}$; an independent exact Fourier/character-table calculation checks this identification. \end{proof} Thus all fourteen additional simples occur. Their phases together with their preserved residues and the bound determine the actual lowest weights $k/29$, not merely their classes modulo one. Their top dimensions are \[ \begin{array}{c|rrrrrrr} 29h&8&11&12&14&15&17&18\\\hline \dim M_h&3&1&3&7&5&13&8\\\hline 29h&19&21&26&27&31&32&39\\\hline \dim M_h&17&21&31&41&62&58&134. \end{array} \] Consequently the actual Zhu algebra has dimension $62388$, and its current highest-line corner has dimension $1588$. The two excluded scalar blocks account for the difference from $\dim P=1590$. Finally, \begin{equation}\label{eq:S00} S_{00}=\frac12\left(\frac15-\frac1{\sqrt{29}}\right). \end{equation} \section{The extension identifies the category} The isotropic discriminant line $L=\{(x,0)\}$ extends $B$ to $H=V_{E_8}$. Proposition~\ref{prop:branch} identifies its actual algebra object in $\mathcal C=\Rep(U)$ as \[ \mathcal E=\one\oplus\pi\oplus2G_{10}\oplus2G_{20}. \] It is Lagrangian: its local modules are $\Vec$ because $H$ is holomorphic. The center theorem~\cite{DMNO} gives $\mathcal C\simeq Z(\mathcal T)$ for the fusion category $\mathcal T=\mathcal C_{\mathcal E}$ of right modules. Use the left action induced by the inverse braiding to fix the center orientation. Positive spherical structures are preserved. \begin{proposition}\label{prop:fusion} $\mathcal T$ has the fusion rules \eqref{eq:HI}, positive dimension $d=(5+\sqrt{29})/2$, and untwisted pointed subcategory $\Vec_{\Z/5}$. \end{proposition} \begin{proof} Let $F:Z(\mathcal T)\to\mathcal T$ be forgetful and $I$ its adjoint. Since $\mathcal E=I(\one)$, \[ \operatorname{rank}\mathcal T=\dim\End(\mathcal E)=1+1+4+4=10. \] Successive induction from the intermediate algebra $\one\oplus\pi$ embeds five distinct invertibles $\alpha_g$. The complementary isotropic line $\{(0,y)\}$ gives coherent representatives: the pointed metric category has trivial associator and braiding $c((x,y),(x',y'))=\zeta_5^{xy'}$, so its restriction to that line is trivial. This proves the pointed associator assertion. For $g\ne0$, the underlying central object $I(\alpha_g)$ is the sum of the five lattice restrictions in $L+(0,g)$, none occurring in $\mathcal E$. Thus $\Hom(\mathcal E,I(\alpha_g))=0$. Using $FI(\one)=\bigoplus_Y Y\otimes Y^*$, adjunction rules out $\alpha_gY\simeq Y$ for any simple $Y$. The ten simples form two free orbits of five. Duality gives free right action and a self-dual representative $\rho$ in the second orbit. Equation~\eqref{eq:S00} gives \[ \Dim\mathcal T=S_{00}^{-1} =\frac{145+25\sqrt{29}}2=5+5d^2, \] so that orbit is noninvertible. The commuting left and right actions give $\rho\alpha_g=\alpha_{\sigma(g)}\rho$. The only invertible constituent of $\rho^2$ is $\one$, once, and comparison with its right translates gives $\sigma^2=1$. The multiplicities $1,1,2,2$ in $I(\one)$ are the irreducible degrees of the complexified Grothendieck algebra~\cite[Theorem 2.13]{Ostrik}. This algebra is noncommutative; hence $\sigma=-1$. Conjugation by $\alpha_1$ fixes $\rho^2$ and permutes its five possible noninvertible constituents transitively. Their multiplicities are equal, say $m$, and $d^2=1+5md$ forces $m=1$. \end{proof} For completeness, we specify the associator input in the final step. It cannot be replaced by agreement of modular data. \begin{lemma}[Associator normalization]\label{lem:associator} A complex fusion category with \eqref{eq:HI} and untwisted pointed subcategory is tensor equivalent to an Evans--Gannon category $\mathcal C(\Z/5;\pm,\omega,M)$. \end{lemma} \begin{proof}[Normalization argument] Choose coherent pointed multiplication and the maps moving an invertible through $\rho$. The remaining crossing discrepancy is a scalar $2$-cocycle, removable because $H^2(\Z/5,\C^\times)=0$. Tensor words can then be strictified with object monoid $(\Z/5)\rtimes_{-1}\mathbb N$; this does not trivialize the simple objects' associator. The categorical splitting $\rho^2=\one\oplus\bigoplus_g\alpha_g\rho$ gives inclusions $s,t_g$ and inverse projections $s',t'_g$ obeying the Leavitt relations. They embed faithfully in the padded algebra \[ \mathcal B=\bigoplus_{\gamma\in(\Z/5)\rtimes_{-1}\Z} \varinjlim_n\Hom(\rho^n,\gamma\rho^n). \] Transition maps tensor on the right by $\rho$ and are injective because $\one$ splits from $\rho^2$. Multiplication uses common right multiples of tensor words; left tensoring gives endomorphisms. The induced map from the simple Leavitt algebra $L(1,6)$ is unital and therefore injective~\cite{Leavitt}. The Evans--Gannon deconstruction~\cite[Section 4]{EG15} now takes place in actual categorical Hom spaces. The four relevant spaces for the images of $s,s',t_g,t'_g$ have dimensions $6,6,27,27$, as obtained directly from \eqref{eq:HI}. Their splitting bases give all coefficient comparisons used there; no assertion about every intertwiner of an ambient endomorphism algebra is required. The scalar equations normalize to $\delta^2=5\delta+1$, $\omega^3=1$ and the matrix equations of \cite[Theorem 2]{EG15}. The resulting endomorphisms preserve $L(1,6)$. Repeated splitting of the final $\rho^2$ reduces every tensor word to the ten simple types; the corresponding inclusions and projections express every morphism in that Leavitt algebra. Thus reconstruction is full and faithful as well as essentially surjective. Expanded scalar equations and the padding construction are supplied in the companion categorical note. \end{proof} The positive global dimension in Proposition~\ref{prop:fusion} selects the positive branch of Lemma~\ref{lem:associator}. Evans--Gannon's complete order-five classification has two positive tensor classes~\cite[Theorem 3(iii)]{EG15}, which are Morita equivalent~\cite[Theorem 6.4 and Section 6.6]{GS}. One is the standard Izumi category; their positive centers are ribbon equivalent. This proves the categorical assertion of Theorem~\ref{thm:main}. In particular, the argument uses the actual extension multiplication and the recovered associator. \section{Certificates and scope}\label{sec:certificates} The proof uses exact rational arithmetic in a fixed number field, exact prime-field linear algebra, and exact cyclotomic calculations. The principal machine records in the accompanying research bundle are listed below; the companion notes give the detailed mathematical reductions and replay commands. \begin{center}\small \begin{tabular}{@{}p{.29\linewidth}p{.66\linewidth}@{}} \toprule Conclusion&Certificate filename (common prefix \texttt{n5\_})\\\midrule Complete finite closure&\path{c8_closure_certificate.json}\\ Neutral $C_2$ relations&\path{c8_c2_bracket_direct.json}, \path{c8_c2_quadratic_exact.json}\\ Integral inputs&\path{c8_integrality_audit.json}, \path{c8_integral_spanning_mod.json}\\ Finite Zhu upper algebra&\path{c8_zhu_finite_quartic5_diagonal_zero_mod.json}\\ Nondegenerate regular trace&\path{c8_zhu_trace_quartic5_diagonal_zero_mod.json}\\ Center and simple profiles&\path{c8_zhu_center_quartic5_diagonal_zero_mod.json}, \path{c8_zhu_modules_quartic5_diagonal_zero_mod.json}\\ Weight and rank filter&\path{rank_bound_arithmetic.json}, \path{sl2_rank_bound_verified.json}\\ Independent Weil check&\path{psl29_weil_verified.json}\\ Provenance&\path{verification_manifest.json}\\\bottomrule \end{tabular} \end{center} The closure audit verifies exhaustive coverage and the recorded full-state witnesses; it is distinct from replaying each producer. The Zhu quotient checks distinguish a spanning set from a basis. The congruence filter is independently replayed with exact fractions, the degree enumeration is checked against the level enumeration and independent Chinese-remainder counts, and the external representation constructors are checked against GAP character tables. Source revisions and historical guard versions are retained rather than silently exempted from hash validation. The result is not a Lean-formalized proof. \section*{AI Disclosure} AI systems were used in the exploration and development of the note. \begin{thebibliography}{99} \bibitem{EG10} D.~E.~Evans and T.~Gannon, \emph{The exoticness and realisability of twisted Haagerup--Izumi modular data}, \href{https://arxiv.org/abs/1006.1326}{arXiv:1006.1326}. \bibitem{Riesen} A.~Riesen, \emph{The Haagerup VOA at $c=8$}, \href{https://arxiv.org/abs/2610.13933}{arXiv:2610.13933}, announcement. \bibitem{HuangHI3} T.-C.~Huang, \emph{Haagerup conformal field theory at central charge eight}, to appear on arXiv. \bibitem{McR} R.~McRae, \emph{On rationality for $C_2$-cofinite vertex operator algebras}, \href{https://arxiv.org/abs/2108.01898v3}{arXiv:2108.01898v3}. \bibitem{DLN} C.~Dong, X.~Lin and S.-H.~Ng, \emph{Congruence property in conformal field theory}, \href{https://arxiv.org/abs/1201.6644}{arXiv:1201.6644}. \bibitem{NWW} S.-H.~Ng, Y.~Wang and S.~Wilson, \emph{On symmetric representations of $\mathrm{SL}_2(\Z)$}, \href{https://arxiv.org/abs/2203.15701}{arXiv:2203.15701}; \href{https://snw-0.github.io/sl2-reps/}{SL2Reps 1.1}. \bibitem{HKL} Y.-Z.~Huang, A.~Kirillov Jr. and J.~Lepowsky, \emph{Braided tensor categories and extensions of vertex operator algebras}, \href{https://arxiv.org/abs/1406.3420}{arXiv:1406.3420}. \bibitem{DMNO} A.~Davydov, M.~M\"uger, D.~Nikshych and V.~Ostrik, \emph{The Witt group of non-degenerate braided fusion categories}, \href{https://arxiv.org/abs/1009.2117}{arXiv:1009.2117}. \bibitem{Ostrik} V.~Ostrik, \emph{Pivotal fusion categories of rank 3}, \href{https://arxiv.org/abs/1309.4822}{arXiv:1309.4822}. \bibitem{EG15} D.~E.~Evans and T.~Gannon, \emph{Non-unitary fusion categories and their doubles via endomorphisms}, \href{https://arxiv.org/abs/1506.03546}{arXiv:1506.03546}. \bibitem{Leavitt} G.~Abrams and G.~Aranda Pino, \emph{The Leavitt path algebra of a graph}, \href{https://arxiv.org/abs/math/0509494}{arXiv:math/0509494}. \bibitem{GS} P.~Grossman and N.~Snyder, \emph{Quantum subgroups of the Haagerup fusion categories}, \href{https://arxiv.org/abs/1102.2631}{arXiv:1102.2631}, Section 6.6. \end{thebibliography} \end{document}