\documentclass[11pt,letterpaper]{article} \usepackage[T1]{fontenc} \usepackage[utf8]{inputenc} \usepackage{microtype} \usepackage[margin=1.05in]{geometry} \usepackage{amsmath,amssymb,amsthm,mathtools,mathrsfs} \usepackage{booktabs,enumitem} \usepackage{aliascnt} \usepackage[numbers,sort&compress]{natbib} \usepackage{xcolor} \definecolor{paperlink}{RGB}{0,0,114} \usepackage[colorlinks=true,linktoc=all,linkcolor=paperlink,citecolor=paperlink,urlcolor=paperlink]{hyperref} \usepackage[nameinlink,capitalize,noabbrev]{cleveref} \pagestyle{plain} \setlength{\parindent}{1.25em} \setlength{\parskip}{0pt} \setcounter{tocdepth}{1} \newtheorem{theorem}{Theorem}[section] \newaliascnt{proposition}{theorem} \newtheorem{proposition}[proposition]{Proposition} \aliascntresetthe{proposition} \newaliascnt{lemma}{theorem} \newtheorem{lemma}[lemma]{Lemma} \aliascntresetthe{lemma} \newaliascnt{corollary}{theorem} \newtheorem{corollary}[corollary]{Corollary} \aliascntresetthe{corollary} \theoremstyle{definition} \newaliascnt{definition}{theorem} \newtheorem{definition}[definition]{Definition} \aliascntresetthe{definition} \newaliascnt{remark}{theorem} \newtheorem{remark}[remark]{Remark} \aliascntresetthe{remark} \newaliascnt{example}{theorem} \newtheorem{example}[example]{Example} \aliascntresetthe{example} \numberwithin{equation}{section} \newcommand{\C}{\mathbb C} \newcommand{\A}{\mathscr A} \newcommand{\TL}{\mathrm{TL}} \newcommand{\EPTL}{\mathrm{EPTL}} \newcommand{\Hom}{\operatorname{Hom}} \newcommand{\End}{\operatorname{End}} \newcommand{\Ext}{\operatorname{Ext}} \newcommand{\rad}{\operatorname{rad}} \newcommand{\soc}{\operatorname{soc}} \newcommand{\im}{\operatorname{im}} \newcommand{\rank}{\operatorname{rank}} \newcommand{\Ann}{\operatorname{Ann}} \newcommand{\supp}{\operatorname{Supp}} \newcommand{\id}{\mathrm{id}} \newcommand{\Cat}{\mathrm{Cat}} \newcommand{\Fbar}{\overline F} \newcommand{\Dcat}{\mathscr D_\beta} \setlist[enumerate]{leftmargin=2.5em,label=\textup{(\roman*)}} \setlength{\emergencystretch}{2em} \title{Marked-point modules and central nilpotents at roots of unity} \hypersetup{pdftitle={Marked-point modules and central nilpotents at roots of unity},pdfauthor={Zhenyu Sun},pdfkeywords={periodic Temperley-Lieb algebra, marked-point modules, roots of unity, extension groups, central nilpotents}} \author{Zhenyu Sun} \date{} \begin{document} \maketitle \begin{abstract} We study extensions in the two-marked-point representation of the periodic Temperley--Lieb algebra at $q=e^{i\pi/p}$, $p\in\mathbb Z_{\geq3}$. For every positive even size $N$, we construct a submodule $Z_N$ extending $T_N=W_{N;2,\varepsilon}$ by $W_N=W_{N;0,\varepsilon q}$, where $\varepsilon=\pm1$. The dual trees of noncrossing matchings give an explicit injection $\Theta_N:T_N\to W_N$. Both central braid transfer elements have nilpotent part $g\Theta_N\pi_N$ on $Z_N$, where $\pi_N$ is the quotient map and $g=x+x^{-1}+\varepsilon(y+y^{-1})$ for the marked-point parameters $x,y\in\C^\times$. The extension splits exactly when $g=0$, and the formula determines the central Jordan forms at every even size. In the window $2\leq N<2p-2$, $Z_N$ is a joint central summand and, when $g\ne0$, has simple layers $T_N/S_N/T_N$. An explicit intertwiner and annular localization identify its two-layer quotient with a periodic XXZ sector. The associated basic image algebras have dimensions five and four; the quotient preserves first extensions between simples and kills a nonzero Yoneda product in degree two. An annular operator detects the extension with coefficient $2\varepsilon g$. Beyond the isolated window, we determine a five-simple image algebra and compute a $43$-dimensional joint central summand at $p=3,N=6$, on which the two central nilpotents generate a four-dimensional algebra. Exact matrix calculations and verification code accompany these finite computations. \end{abstract} \tableofcontents \clearpage \section{Introduction and main results} \label{sec:intro} Periodic Temperley--Lieb representations describe connectivity sectors, twisted spin chains, and bulk operator insertions. At roots of unity, their structure is not determined by their simple composition factors: the extensions between those factors and the nilpotent action of central elements must also be specified. Two-marked-point link states provide a setting in which these data can be calculated together. Graham and Lehrer developed the standard modules and their successor homomorphisms; Erdmann and Green studied finite quotients and extensions \cite{GrahamLehrer1998,ErdmannGreen1999}. Pinet and Saint-Aubin determined the Loewy structures of periodic XXZ sectors, including the root-of-unity cases \cite{PinetSaintAubin2023}. Together with the standard-module structure, their result supplies the two opposite extensions used in our isolated-channel comparison. Ikhlef and Morin-Duchesne introduced two-marked connectivity modules and analyzed periodic fusion \cite{IkhlefMorinDuchesne2022,IkhlefMorinDuchesne2024}. Their non-generic analysis for $q$ away from roots of unity produces a three-layer module and a parameter-dependent central Jordan action. At a root of unity, additional standard constituents can enter the same central spectrum, and a generic spectral decomposition need not specialize term by term. We give a direct construction of the first marked channel at the root of unity itself. Its maps are described by two-node diagrams and the incidence matrices of finite trees. Fix \[ q=e^{i\pi/p},\qquad p\in\mathbb Z_{\geq3},\qquad \beta=-q-q^{-1},\qquad \varepsilon\in\{1,-1\}, \qquad w=\varepsilon q, \] and let $x,y\in\C^\times$. Set \[ f=\varepsilon(q+q^{-1}),\qquad g=x+x^{-1}+\varepsilon(y+y^{-1}). \] For $N=2n$, put $C_n=(n+1)^{-1}\binom{2n}{n}$ and abbreviate \[ W_N=W_{N;0,w},\qquad T_N=W_{N;2,\varepsilon},\qquad Y_N=Y_{0,0,x,y,[z,w]}(N). \] The last module is independent of $z$. \begin{theorem}[The first marked channel] \label{thm:main-marked} For every even $N\geq2$, there is an explicitly based submodule $Z_N\subset Y_N$ and a natural injective standard homomorphism $\Theta_N:T_N\to W_N$ such that \[ 0\longrightarrow W_N\longrightarrow Z_N \xrightarrow{\pi_N}T_N\longrightarrow0, \qquad (F-f)|_{Z_N}=(\Fbar-f)|_{Z_N}=g\Theta_N\pi_N. \] This sequence splits exactly when $g=0$. If $g\ne0$, the Jordan form of each central element is \[ J_2(f)^{\oplus nC_n}\oplus J_1(f)^{\oplus C_n}. \] If $2\leq N<2p-2$, then $Z_N$ is the joint generalized $(f,f)$-eigenspace in $Y_N$, $T_N$ is simple, and $W_N$ has simple head $S_N$ and radical $\Theta_NT_N$. For $g\ne0$, the module $Z_N$ is uniserial with layers $T_N/S_N/T_N$. \end{theorem} Here and below, layers in a displayed quotient notation are read from head to socle. The first part of the theorem holds at arbitrarily large even sizes with $p$ fixed. The size restriction is used to identify simple layers and to separate the channel as a summand of the full marked module. The proof is given in \Cref{sec:marked-proved}. The three-layer module changes the image algebra in a precise way. Write $P_N=Z_N$ in the nonsplit isolated case and $E_N=P_N/\soc P_N$. \begin{theorem}[Finite images and the lifting obstruction] \label{thm:main-image} Under the isolated nonsplit assumptions of \Cref{thm:main-marked}, one has \[ E_N\cong X^+_{N;0,\varepsilon q}. \] Let $B_N$ be the image algebra on $P_N$ and $R_N$ its quotient acting on $W_N\oplus E_N$. Their basic algebras are \[ B_0=\C Q/(ba),\qquad R_0=\C Q/(ab,ba), \qquad Q:\ S\xrightarrow{a}T\xrightarrow{b}S, \] where products are compositions, with the rightmost arrow traversed first. The basic kernel $\C ab$ is central and has square zero. Inflation from $R_N$ to $B_N$ is an isomorphism on first extensions between simple modules. If $\alpha\in\Ext^1(S_N,T_N)$ and $\xi\in\Ext^1(T_N,S_N)$ are the two nonzero classes, then \[ \alpha\circ\xi\ne0\ \text{over }R_N, \qquad \alpha\circ\xi=0\ \text{over }B_N. \] The latter equality is realized by $0\to W_N\to P_N\to T_N\to0$. \end{theorem} The comparison with the XXZ module is proved through a full annular corner and an explicit two-node intertwiner. If $s=\dim S_N=C_n$ and $t=\dim T_N=nC_n$, the physical image algebras have dimensions \[ \dim B_N=s^2+2st+2t^2, \qquad \dim R_N=(s+t)^2. \] Their quotient kernel has dimension $t^2$ and is generated by a central nilpotent. For $t>1$, only its one-dimensional scalar subspace is central; in the basic algebra the entire kernel is central. We also compute a normalized annular extension datum. Cup and cap maps from two nodes to $N$ transport $e_1-e_0$ to an operator annihilating both endpoints of $0\to S_N\to E_N\to T_N\to0$. Its action on a specified lift is $2\varepsilon g$ times a specified nonzero lower vector. For $p=5,N=6,x=1,y=2,\varepsilon=1$, the central coefficient is $9/2$ and this annular coefficient is $9$. For a specified standard--XXZ generating list at $p=3,N=6$, we determine a five-simple image algebra of dimension $328$, including its quiver, quadratic relations, projective covers and second extensions. Adding the inverse-twist XXZ sector changes the image dimension to $397$ and changes its Morita type. The non-isolated marked module exhibits a further change. For $p=3,N=6,x=1,y=2,w=q$, exact diagrammatic matrices give a $43$-dimensional joint central summand. On it, $F$ has Jordan form \[ J_3(1)^{\oplus2}\oplus J_2(1)^{\oplus17}\oplus J_1(1)^{\oplus3}. \] The restrictions of the two central elements generate \[ \C[u,v]/(u^3,uv,v^2), \qquad u=F-1,\quad v=F-\Fbar. \] The $35$-dimensional depth-one submodule of \Cref{thm:main-marked} persists inside this larger summand. The matrices, exact rank calculations, and an independent verification program are provided with the paper. Ikhlef and Morin-Duchesne construct a distributive, commutative, associative fusion bifunctor on families of periodic modules \cite{IkhlefMorinDuchesne2026Fusion}. Our cup and cap maps act on families in the same diagrammatic category. The uncoiled quotients of Langlois-R\'emillard and Morin-Duchesne \cite{LangloisRemillardMorinDuchesne2026} give another class of finite algebras; our image-algebra arguments use faithfulness and module structure, without a cellularity assumption. The ADE construction of \cite{IkhlefMorinDuchesne2026} associates local operators to irreducible constituents. Here the annular coefficient is defined on a specified extension with fixed endpoint vectors; comparison with a continuum logarithmic coupling requires additional normalization and limiting data. Section~\ref{sec:annular-preliminaries} fixes the diagrammatic and twist conventions. Section~\ref{sec:marked-proved} constructs the marked extension and computes its central action. Section~\ref{sec:image-algebras} identifies its finite image algebras and their extension groups. Sections~\ref{sec:five-simple-image} and~\ref{sec:exact-marked} treat the five-simple image and the exact marked-module calculations. Section~\ref{sec:outlook} discusses higher channels and module families beyond the isolated window. \section{Annular diagrams, twists and standard modules} \label{sec:annular-preliminaries} Let $\Dcat$ be the complex linear annular Temperley--Lieb category on nonnegative even objects. A morphism $M\to N$ is a noncrossing connectivity diagram in an annulus with $M$ nodes on its inner boundary and $N$ nodes on its outer boundary. Contractible closed curves are evaluated as $\beta$; essential closed curves are retained as diagrams. Composition is concatenation. Write $\mathsf A_N=\End_{\Dcat}(N)$. This diagrammatic convention includes $N=0,2$ without importing relations valid only at larger sizes. For $N\geq4$, the local cup--cap diagrams $e_i$ and the rotation $\Omega$ obey \[ e_i^2=\beta e_i,\qquad e_ie_{i\pm1}e_i=e_i,\qquad e_ie_j=e_je_i\quad(i-j\not\equiv\pm1\pmod N), \qquad\Omega e_i\Omega^{-1}=e_{i-1}. \] Indices belong to $\mathbb Z/N\mathbb Z$. We use the enlarged periodic diagram algebra, also denoted $\EPTL_N(\beta)$ in the marked-module literature. An annular module family is a collection of spaces $M_N$ equipped with the action of every morphism in $\Dcat$. If $I:M\to N$ adds $k=(N-M)/2$ adjacent cups and $C:N\to M$ is its reflected cap diagram, then \begin{equation} CI=\beta^k1_M. \label{eq:basic-cupcap} \end{equation} Thus $I$ is injective on every such family when $\beta\ne0$. This elementary contraction will be used both for the marked modules and for the spin-chain family. \subsection{Standard labels and actual twists} The standard module $W_{N;d,z}$ has $d$ defects, $0\leq d\leq N$ even, and twist $z\in\C^\times$. Its dimension is \[ \dim W_{N;d,z}=\binom{N}{(N-d)/2}. \] For $d=0$ the noncontractible-loop value is $z+z^{-1}$ and $W_{N;0,z}\cong W_{N;0,z^{-1}}$. The standard and simple label set therefore identifies $(0,z)$ with $(0,z^{-1})$. A spin-chain label $X^+_{N;d,z}$ retains the actual complex number $z$. We impose no inversion identification on this latter label set. The successor homomorphisms are controlled by $t=d+2r>d$ and \begin{equation} \begin{array}{lll} \mathrm{(A)}&z^2=q^t,&u=zq^{-r},\\ \mathrm{(B)}&z^2=q^{-t},&u=zq^r. \end{array} \label{eq:successors} \end{equation} At the parameters used here, the standard modules have simple heads and multiplicity-one composition factors described by their successor closures. The maps $W_{N;t,u}\to W_{N;d,z}$ associated with successors are injective. We use these standard facts in the form of \cite[Theorem~2.2 and Proposition~3.2]{PinetSaintAubin2023}. Actual Loewy arrows must be read from the corresponding module structure; the two successor conditions alone do not prescribe XXZ arrow directions. For any finite list of actual seed twists, take the direct sum of the specified standard and XXZ modules and form its finite image algebra. Its accessible simple labels are taken in the standard label set, while its generators retain their actual twists. \Cref{cor:unconditional-resonance-blocks} proves the associated block statement. This convention also allows us to specify individual finite images by an explicit list of modules in the non-isolated example. \subsection{The two central braid transfer elements} Set $b_i=q^{1/2}1+q^{-1/2}e_i$, so that $b_i^{-1}=q^{-1/2}1+q^{1/2}e_i$. Let $I_{\rm aux}:N\to N+2$ append an empty adjacent cup, and let $C_{\rm seam}:N+2\to N$ cap the last and first nodes and retain the intervening nodes in their cyclic order. With indices $0,\ldots,N+1$ on the intermediate boundary, define \begin{equation} F_N=C_{\rm seam}b_0b_1\cdots b_{N-1}I_{\rm aux}, \qquad \Fbar_N=C_{\rm seam}b_0^{-1}b_1^{-1}\cdots b_{N-1}^{-1}I_{\rm aux}. \label{eq:hoop-definition} \end{equation} The rightmost factor acts first. Since $N$ is even, the total powers of $q$ in the skein expansion are integral. The braid and cup--cap identities allow this closed auxiliary strand to pass across every annular diagram. Consequently these two elements are central and form natural endomorphisms of every diagrammatic module family. These are the central braid transfer elements used in \cite[Section~2.3]{IkhlefMorinDuchesne2024}. Their action on standards is recorded in \eqref{eq:marked-central-characters}. Formula~\eqref{eq:hoop-definition} is also the definition used in the accompanying computation. It specifies the actual diagram operator before any spectral decomposition is performed. \section{The first marked channel at every even size} \label{sec:marked-proved} Put $q=\exp(i\pi/p)$ with $p\in\mathbb Z_{\geq3}$, and write \[ c=q+q^{-1},\qquad \beta=-c,\qquad w=\varepsilon q,\qquad f=\varepsilon c, \qquad \varepsilon\in\{1,-1\}. \] Thus $c\neq0$ and $\beta\neq0$. Fix $x,y\in\mathbb C^\times$ and set \begin{equation} A=x+x^{-1},\qquad B=y+y^{-1},\qquad g=A+\varepsilon B =\frac{(y+\varepsilon x)(\varepsilon xy+1)}{xy}. \label{eq:marked-g-explicit} \end{equation} We use the diagram category and the endomorphism algebras $\mathsf A_N$ of \Cref{sec:annular-preliminaries}. Cup and cap diagrams act between the different sizes in each module family. For even $N=2n\geq2$, abbreviate \[ Y_N=Y_{0,0,x,y,[z,\varepsilon q]}(N),\qquad W_N=W_{N;0,\varepsilon q},\qquad T_N=W_{N;2,\varepsilon}. \] The parameter $z$ does not occur in $ Y_N$. The defining loop weights are $\beta,A,B,f$ according as a removed loop surrounds neither marked point, only the first, only the second, or both. This is the two-marked module of \cite[Sections~3.2--3.3]{IkhlefMorinDuchesne2024}. \subsection{A basis described by faces} Let $\mathcal M_n$ be the noncrossing perfect matchings of the $2n$ boundary nodes of a disc. For $\pi\in\mathcal M_n$, write $\mathcal F(\pi)$ for its faces and $D_\pi$ for its dual graph. The graph $D_\pi$ is a tree with $n+1$ vertices and $n$ edges. A basis of $ Y_N$ is \[ v_{\pi;a,b},\qquad \pi\in\mathcal M_n,\quad a,b\in\mathcal F(\pi). \] The two marked points lie in the ordered pair of faces $(a,b)$. The depth of this state is the distance between $a$ and $b$ in $D_\pi$. Let $ Y_N^{\leq r}$ be the span of the states of depth at most $r$. Concatenation cannot increase depth, so these spaces are submodules. The diagonal states $v_{\pi;a,a}$ identify $ Y_N^{\leq0}$ with $ W_N$. The descriptions by faces and by link states are identical: after forgetting the two marked points, the remaining curves give $\pi$, and their isotopy class with the marked points retained is determined by the two faces. In particular, \[ \dim Y_N=(n+1)^2C_n,\qquad \dim W_N=(n+1)C_n,\qquad C_n=\frac1{n+1}\binom{2n}{n}. \] \begin{lemma}[The first depth quotient] \label{lem:marked-depth-one} For every value of $\beta$, there is a decomposition \begin{equation} Y_N^{\leq1}/ W_N \cong W_{N;2,1}\oplus W_{N;2,-1}. \label{eq:marked-depth-one-decomp} \end{equation} Modulo $ W_N$, the interchange $v_{\pi;a,b}\mapsto v_{\pi;b,a}$ commutes with the annular action. Its $\varepsilon$-eigenspace is the summand $W_{N;2,\varepsilon}$. \end{lemma} \begin{proof} At depth one the two faces share an edge of $D_\pi$. Cut the corresponding arc of $\pi$ to obtain two defects. The two possible orders of the faces specify the two orders of these defects at the puncture. Changing this order is the inner rotation of the two defects. In the quotient by depth zero, any contraction involving a loop surrounding one marked point vanishes, because it lowers the depth. The surviving concatenations use only the contractible-loop weight $\beta$ and the inner rotation. Thus this quotient is the two-defect link space with the regular representation of the two-element rotation group at its inner boundary. The two Fourier projections $(1\pm R)/2$ give the two standard modules in \eqref{eq:marked-depth-one-decomp}. This argument uses a change of basis over $\mathbb C$ and no rational function of $q$. \end{proof} Choose one orientation $(a,b)$ for each edge of each $D_\pi$ and put \begin{equation} t_{\pi;a,b}=v_{\pi;a,b}+\varepsilon v_{\pi;b,a}, \qquad r_{\pi;a,b}=v_{\pi;a,a}+\varepsilon v_{\pi;b,b}. \label{eq:marked-tree-vectors} \end{equation} Reversing the chosen orientation multiplies both vectors by $\varepsilon$. Define $ Z_N$ to be the inverse image of the $\varepsilon$-summand in \eqref{eq:marked-depth-one-decomp}. It is a submodule with the explicit basis \begin{equation} \{v_{\pi;a,a}\}_{\pi,a}\ \cup\ \{t_{\pi;a,b}\}_{\pi,\{a,b\}\in E(D_\pi)}. \label{eq:marked-Z-basis} \end{equation} In particular, \begin{equation} 0\longrightarrow W_N\longrightarrow Z_N \xrightarrow{\,\pi_N\,} T_N\longrightarrow0, \qquad \dim Z_N=(2n+1)C_n. \label{eq:marked-global-extension} \end{equation} We normalize the standard basis vector associated with the cut edge $(\pi;a,b)$ to be $\pi_N(t_{\pi;a,b})$. At object zero we set $T_0=0$ and $W_0=Z_0=Y_0$, the one-dimensional empty-matching space. These conventions include caps from two nodes to zero in the module families. \subsection{The two-node calculation} At $N=2$, write the basis as $(a,b,u,v)$, with $a,b$ the two diagonal-face states and $u,v$ the two ordered pairs of distinct faces. Choose the labelling so that $\Omega a=b$, $\Omega u=v$. Direct concatenation gives \begin{equation} e_1= \begin{pmatrix} \beta&f&B&A\\0&0&0&0\\0&0&0&0\\0&0&0&0 \end{pmatrix},\qquad e_0= \begin{pmatrix} 0&0&0&0\\f&\beta&A&B\\0&0&0&0\\0&0&0&0 \end{pmatrix},\qquad \Omega= \begin{pmatrix} 0&1&0&0\\1&0&0&0\\0&0&0&1\\0&0&1&0 \end{pmatrix}. \label{eq:marked-two-node-generators} \end{equation} For the two central braid transfer elements of \cite[Section~2.3]{IkhlefMorinDuchesne2024}, the same concatenation gives \begin{equation} F_2=\overline F_2= \begin{pmatrix} f&0&A&B\\0&f&B&A\\0&0&0&c\\0&0&c&0 \end{pmatrix}. \label{eq:marked-two-node-F} \end{equation} These matrices are obtained by evaluating the four loop weights directly at the specified root of unity. Set $r=a+\varepsilon b$ and $t=u+\varepsilon v$. Then \begin{equation} \begin{gathered} W_2=\mathbb Ca+\mathbb Cb,\qquad Z_2= W_2+\mathbb Ct,\\ e_0r=e_1r=0,\quad\Omega r=\varepsilon r,\quad e_1t=\varepsilon g a,\quad e_0t=g b,\quad\Omega t=\varepsilon t,\\ (F_2-f)t=(\overline F_2-f)t=gr, \qquad (F_2-f) W_2=(\overline F_2-f) W_2=0. \end{gathered} \label{eq:marked-two-node-relations} \end{equation} The radical of $ W_2$ is $\mathbb Cr$. Its quotient is the one-dimensional module with $e_0=e_1=\beta$ and $\Omega=-\varepsilon$. \subsection{Insertion and the central nilpotent} The annular category acts on the families $ Y_N$ and on the standard families. The braid push-through identities give, for every annular diagram $D:M\to N$, \begin{equation} F_ND=DF_M,\qquad \overline F_ND=D\overline F_M. \label{eq:marked-natural-F} \end{equation} Every depth-zero or depth-one link state is obtained from a two-node state by such a diagram. One way to see this is to remove cups bounding an unmarked leaf of the dual tree until two boundary nodes remain. For a state of depth at most one, every removed leaf can be chosen away from the path between the marked faces. Reinserting the cups recovers the original state. Equivalently, an edge of $D_\pi$ can be cut to a two-defect diagram $D:2\to N$; with suitable ordering, \[ Da=v_{\pi;a,a},\quad Db=v_{\pi;b,b},\quad Du=v_{\pi;a,b},\quad Dv=v_{\pi;b,a}. \] For a diagonal state, choose any edge incident to its marked face; such an edge exists for $N\geq2$. The same insertion realizes the state as $Da$ or $Db$. This also proves generation at depth zero. \begin{lemma}[The first standard inclusion] \label{lem:marked-Theta} There is a natural injective homomorphism \begin{equation} \Theta_N: T_N\longrightarrow W_N, \qquad \Theta_N\bigl(\pi_N(t_{\pi;a,b})\bigr)=r_{\pi;a,b}. \label{eq:marked-Theta} \end{equation} It is the Graham--Lehrer successor homomorphism, normalized at two nodes by $\Theta_2(\pi_2(t))=r$. In compositions with $\pi_N$ we view its image inside $W_N\subset Z_N$ and suppress this inclusion. \end{lemma} \begin{proof} At two nodes the vector $r$ is annihilated by both caps and has rotation eigenvalue $\varepsilon$, by \eqref{eq:marked-two-node-relations}. The cap assertion follows also from $e_j=c_j^\dagger c_j$ and the injectivity of $c_j^\dagger$, since $c_jc_j^\dagger=\beta\mathbf1$ and $\beta\neq0$. The defining presentation of the two-defect standard module therefore allows one to replace its two defects by $r$. Concatenation gives the indicated natural homomorphism. This is also the specialization of the successor inclusion in \cite[Proposition~3.2]{PinetSaintAubin2023}; here the insertion has two terms and no quantum-integer denominator. For a direct proof of injectivity, fix $\pi$. In the diagonal-face basis of $ W_N$, the columns of this map are $v_a+\varepsilon v_b$ for the edges of the tree $D_\pi$. They are linearly independent: a leaf occurs in only one column, so its coefficient first vanishes in any linear dependence; remove that edge and continue by induction. The subspaces for distinct matchings are disjoint. Consequently all columns of $\Theta_N$ are independent. \end{proof} \begin{theorem}[The first marked extension at every even size] \label{thm:marked-all-size} For every even $N=2n\geq2$ and every integer $p\geq3$, the submodule $ Z_N\subset Y_N$ defined in \eqref{eq:marked-Z-basis} satisfies \eqref{eq:marked-global-extension} and \begin{equation} (F_N-f)|_{ Z_N} = (\overline F_N-f)|_{ Z_N} =g\Theta_N\pi_N. \label{eq:marked-global-nilpotent} \end{equation} The sequence \eqref{eq:marked-global-extension} splits if and only if \begin{equation} g=0 \quad\Longleftrightarrow\quad y=-\varepsilon x\ \text{ or }\ y=-\varepsilon x^{-1}. \label{eq:marked-global-split-locus} \end{equation} If $g\neq0$, the Jordan form of either central element on $ Z_N$ is \begin{equation} J_2(f)^{\oplus\binom{N}{N/2-1}} \ \oplus\ J_1(f)^{\oplus C_{N/2}}. \label{eq:marked-global-Jordan} \end{equation} \end{theorem} \begin{proof} The short exact sequence was established in Lemma~\ref{lem:marked-depth-one}. The two sides of \eqref{eq:marked-global-nilpotent} are natural for annular diagrams, and they agree on the two-node generators by \eqref{eq:marked-two-node-relations}. Inserting the two-node states into each matching proves the identity on the basis \eqref{eq:marked-Z-basis}. More explicitly, \[ (F_N-f)v_{\pi;a,a}=0,\qquad (F_N-f)t_{\pi;a,b}=g\,r_{\pi;a,b}, \] and the same equations hold for $\overline F_N$. Since $\pi_N\Theta_N=0$, the nilpotent square is zero. When $g\neq0$, its rank is $\dim T_N$ by Lemma~\ref{lem:marked-Theta}. A square-zero endomorphism of rank $r$ on a $d$-dimensional space has $r$ Jordan blocks of size two and $d-2r$ of size one. This proves \eqref{eq:marked-global-Jordan}. Both $F_N$ and $\overline F_N$ act by $f$ on each standard module $ W_N$ and $ T_N$. A splitting of \eqref{eq:marked-global-extension} would therefore force their action on $ Z_N$ to be scalar. Equation~\eqref{eq:marked-global-nilpotent} excludes this when $g\neq0$. If $g=0$, the vector $t$ at two nodes satisfies $e_0t=e_1t=0$ and $\Omega t=\varepsilon t$. Let $\mathcal C_N$ be the span of the edge vectors $t_{\pi;a,b}$. An empty cup or a rotation sends such a vector to another edge vector, up to the orientation factor $\varepsilon$. Under a cap, the two ordered states either remain at depth one with the same contractible-loop coefficient, or both become the same diagonal state. To obtain the latter case, the cap must merge the two marked faces. Reconnecting two different arcs merges faces at dual-tree distance two and therefore cannot merge this adjacent pair. The cap must instead close their separating arc. The two oriented terms then contribute a combined coefficient $A+\varepsilon B=g$ or $B+\varepsilon A=\varepsilon g$, which vanishes. Thus $\mathcal C_N$ is preserved by cups, caps and rotation, including caps crossing the seam. These diagrams generate the annular category. Since $\pi_N|_{\mathcal C_N}$ is an isomorphism of vector spaces, its inverse is a natural splitting $s_N:T_N\to Z_N$. In the chosen basis it sends $\pi_N(t_{\pi;a,b})$ to $t_{\pi;a,b}$. The factorization in \eqref{eq:marked-g-explicit} proves the stated parameter locus. \end{proof} \begin{remark}[An explicit projector on the first depth space] \label{rem:marked-projector} On $ Y_N^{\leq1}$ one has \[ (F_N-f)^2(F_N+f)=0. \] Thus \[ Q_N=1-\frac{(F_N-f)^2}{4f^2} \] is an idempotent with image $ Z_N$. This formula has only the nonzero denominator $4f^2$. It is compatible with cup and cap maps at every size. \end{remark} \subsection{The isolated window and the full two-marked module} For completeness, the same cutting argument used for depth one gives \begin{equation} Y_N^{\leq r}/ Y_N^{\leq r-1} \cong\bigoplus_{\omega^{2r}=1}W_{N;2r,\omega}, \qquad 1\leq r\leq N/2. \label{eq:marked-all-depth-filtration} \end{equation} Indeed, cutting the $r$ arcs along the path between the marked faces yields $2r$ defects and a cyclic choice of their origin; the regular action of this cyclic group is diagonalized by the ordinary Fourier transform. Contractions lowering the depth vanish in the quotient. Consequently \eqref{eq:marked-all-depth-filtration} holds at the present root of unity, even when one of its standard summands is reducible. On $W_{N;2r,\omega}$ the two central eigenvalues are \begin{equation} f_{r,\omega}=\omega q^r+\omega^{-1}q^{-r},\qquad \bar f_{r,\omega}=\omega q^{-r}+\omega^{-1}q^r. \label{eq:marked-central-characters} \end{equation} \begin{lemma}[Joint central isolation] \label{lem:marked-joint-isolation} Assume $2\leq N<2p-2$. Among the positive-depth standard summands in \eqref{eq:marked-all-depth-filtration}, the only one with joint central character $(f,f)$ is $W_{N;2,\varepsilon}$. \end{lemma} \begin{proof} If both expressions in \eqref{eq:marked-central-characters} equal $f$, then \[ (\omega-\omega^{-1})(q^r-q^{-r})=0. \] Here $1\leq r\leq N/21$ the kernel is not contained in the centre. At the basic-algebra level the quotient is $B_0\twoheadrightarrow R_0=B_0/(ab)$ and its kernel is the central one-dimensional space $\C ab$. \end{enumerate} \end{theorem} \begin{proof} Pass first to a basic corner, and denote the resulting faithful module by $V$. Its space at $S$ is one-dimensional, and its space at $T$ is two-dimensional. Choose a vector $u$ lifting the head, a vector $v$ in the middle layer, and a vector $w$ spanning the socle. Elements of the basic algebra preserve the radical series. On $\C u\oplus\C w$ their two diagonal entries agree, since the head and socle are the same simple module. Thus the algebra embeds into the five-dimensional space spanned by the two vertex projections and the maps \[ b:u\longmapsto v,\qquad a:v\longmapsto w, \qquad ab:u\longmapsto w. \] The first two radical corners are nonzero: otherwise one of the successive extensions in $P$ would split or the radical series would have length at most two. Each corner is one-dimensional, so we may rescale its generator to obtain the displayed maps. Their product $ab$ is nonzero, whereas $ba=0$. The five maps are linearly independent and exhaust the image. This proves the description of $B_0$. Its projectives are $B_0e_S=S/T$ and $B_0e_T=T/S/T$. In the basic module, a commuting endomorphism is a common scalar on the three layers, together with an arbitrary multiple of $u\mapsto w$. This proves the endomorphism assertion; Morita equivalence preserves this endomorphism algebra. The same matrix calculation gives the stated centre of $B_0$. For completeness, the full matrix description can be obtained directly. Choose a semisimple complement to $\operatorname{rad}B$ and decompose the three layers as modules over that complement. The semisimple algebra acts by $\operatorname{Mat}_t(\C)$ on both copies of $T$, with the same matrix on those copies, and by $\operatorname{Mat}_s(\C)$ on $S$. The nonzero radical corners from $T$ to $S$ and from $S$ to $T$ are bimodules for these full matrix algebras. A nonzero such bimodule contains every matrix in the corresponding rectangular block. Products of the two rectangular blocks span every matrix in the lower-left $t\times t$ block. No other blocks are allowed by the radical filtration. This proves \eqref{eq:full-three-layer-image}. Commuting an arbitrary matrix with the map $\eta$ that identifies the first and third summands gives exactly the same block conditions, proving the centralizer assertion. Finally, restriction to $W$ and passage to $E$ annihilate precisely the lower-left block $C$. The resulting quotient has the two opposite length-two extensions as a faithful module and is the algebra of Theorem~\ref{thm:opposite-extension-image}. Multiplying the central map $\eta$ by the repeated block $U$ produces every lower-left matrix $C$. The description of the kernel, its dimension and its square-zero property follow. Its elements commute with all matrices $U$ only when $C$ is scalar, which proves the final qualification. \end{proof} \begin{remark} \label{rem:image-size-example} For the isolated channel with $N=6$, $\dim S=5$ and $\dim T=15$. The three-layer module has dimension $35$, its faithful image algebra has dimension $625$, and the image on its two length-two subquotients has dimension $400$. The quotient kernel has dimension $225$, while its intersection with the centre is one-dimensional. \end{remark} \subsection{First extensions, second extensions and lifting} Write $\alpha\in\Ext^1(S,T)$ for the class of $W=S/T$ and $\xi\in\Ext^1(T,S)$ for the class of $E=T/S$. In products such as $\xi\alpha$, the rightmost extension is traversed first; thus $\xi\alpha\in\Ext^2(S,S)$ and $\alpha\xi\in\Ext^2(T,T)$. \begin{theorem}[The change in Yoneda extensions] \label{thm:yoneda-obstruction} Let $B$ and $R$ be the image algebras of Theorem~\ref{thm:three-layer-image}. Inflation along $B\twoheadrightarrow R$ induces isomorphisms \[ \Ext^1_R(U,V)\xrightarrow{\sim}\Ext^1_B(U,V) \qquad(U,V\in\{S,T\}). \] For both algebras the two cross-extension spaces are one-dimensional and the two self-extension spaces vanish. The higher extensions satisfy \begin{align*} \dim\Ext^n_R(S,S)=\dim\Ext^n_R(T,T)&= \begin{cases}1,&n\text{ even},\\0,&n\text{ odd},\end{cases}\\ \dim\Ext^n_R(S,T)=\dim\Ext^n_R(T,S)&= \begin{cases}0,&n\text{ even},\\1,&n\text{ odd},\end{cases} \qquad(n\ge0), \end{align*} whereas \[ \Ext^2_B(S,S)=\C,\qquad \Ext^2_B(S,T)=\Ext^2_B(T,S)=\Ext^2_B(T,T)=0, \qquad\operatorname{gldim}B=2. \] All alternating Yoneda products in degree one are nonzero over $R$. Over $B$ one has \[ \xi\alpha\ne0,\qquad \alpha\xi=0. \] Consequently inflation is an isomorphism on $\Ext^2(S,S)$ and annihilates $\Ext^2_R(T,T)$; it annihilates all groups in degrees at least three. \end{theorem} \begin{proof} It suffices to work with the basic algebras. For $R_0$, the projectives are $W=S/T$ and $E=T/S$. The syzygies of the simple modules alternate: \[ \Omega_R S=T,\qquad\Omega_R T=S. \] The minimal projective resolutions give the displayed dimensions. Splicing the two short exact sequences gives the successive minimal syzygy sequences, so every composable alternating product is nonzero. For $B_0$, the projectives are $W$ and $P$. There are minimal resolutions \[ 0\longrightarrow W\longrightarrow P\longrightarrow T\longrightarrow0 \] and \[ 0\longrightarrow W\longrightarrow P\longrightarrow W \longrightarrow S\longrightarrow0. \] In the second resolution, the map $P\to W$ is the quotient onto $T$ followed by the inclusion of the socle of $W$. Applying $\Hom(-,S)$ and $\Hom(-,T)$ gives all the asserted dimensions. The projective dimensions of $T$ and $S$ are respectively one and two, so the global dimension is two. The two inflated first extensions remain nonsplit, because inflation is fully faithful and does not change whether a section exists. Since both cross-extension spaces have dimension one, the first-extension maps are isomorphisms. Apply $\Hom_B(-,S)$ to $0\to T\to W\to S\to0$. Projectivity of $W$ shows that the connecting map $\Ext^1_B(T,S)\to\Ext^2_B(S,S)$ is an isomorphism. This connecting map is multiplication by $\alpha$, and hence $\xi\alpha\ne0$. The other product is zero because $T$ has projective dimension one. Naturality of Yoneda products proves the assertions about inflation in degree two; the higher-degree assertion follows from global dimension. \end{proof} \begin{corollary}[The three-layer module removes a lifting obstruction] \label{cor:marked-lifting-obstruction} Over $R$, the extension $\xi$ has no lift to an extension of $T$ by $W$. Over $B$, each class in $\Ext^1_B(T,S)$ has a unique lift in $\Ext^1_B(T,W)$, and a nonzero lift is represented by \[ 0\longrightarrow W\longrightarrow P\longrightarrow T\longrightarrow0, \qquad\Ext^1_B(T,W)\simeq\C. \] Thus adjoining $P$ does not create a new first extension between $S$ and $T$. It makes an existing first extension liftable through $W$. \end{corollary} \begin{proof} In the long exact sequence obtained by applying $\Hom(T,-)$ to $0\to T\to W\to S\to0$, the obstruction to lifting $\xi\in\Ext^1(T,S)$ lies in $\Ext^2(T,T)$ and is $\alpha\xi$. Over $R$ this class is nonzero by Theorem~\ref{thm:yoneda-obstruction}; indeed the connecting map is an isomorphism and $\Ext^1_R(T,W)=0$. Over $B$ the obstruction vanishes. The same exact sequence, using $\Hom(T,S)=0$ and $\Ext^1_B(T,T)=0$, shows that $\Ext^1_B(T,W)\to\Ext^1_B(T,S)$ is an isomorphism. The displayed sequence is the asserted lift. \end{proof} \subsection{Compatibility with a separately constructed XXZ module} The preceding results concern $P$ and its own subquotients. An additional length-two module $X=T/S$ over the original algebra need not be isomorphic to $E$ merely because it has the same Loewy diagram. \begin{proposition}[A compatibility criterion] \label{prop:xxz-compatibility} Let $P=T/S/T$ and let $X=T/S$ be a nonsplit $\mathscr A$-module. The following conditions are equivalent: \begin{enumerate} \item $X\simeq P/\operatorname{soc}P$ as $\mathscr A$-modules; \item $\Hom_{\mathscr A}(P,X)\ne0$; \item $\operatorname{Ann}_{\mathscr A}P\subseteq \operatorname{Ann}_{\mathscr A}X$; \item after identifying the simple endpoints, the extension classes of $X$ and $P/\operatorname{soc}P$ span the same line in $\Ext^1_{\mathscr A}(T,S)$. \end{enumerate} If these conditions hold and $W=\operatorname{rad}P$, then \[ A(P\oplus W\oplus X)=A(P),\qquad A(W\oplus X)=A(P)/(\eta). \] \end{proposition} \begin{proof} The first condition implies the second and third. A nonzero map $P\to X$ cannot have image contained in the simple socle $S$, since $\Hom_{\mathscr A}(P,S)=0$. It is therefore onto. Its kernel has length one and is the unique simple submodule $\operatorname{soc}P$, proving the converse to the second condition. Under the third condition, $X$ is a module for $B=A(P)$. Theorem \ref{thm:yoneda-obstruction} shows that $\Ext^1_B(T,S)$ is one-dimensional. Hence its nonsplit length-two modules are all isomorphic to $E$. The equivalence with the fourth condition is the usual action of $\operatorname{Aut}(S)\times\operatorname{Aut}(T)=(\C^\times)^2$ on extensions, which rescales their classes. The two image-algebra identities now follow from the subquotient descriptions and Theorem~\ref{thm:three-layer-image}. \end{proof} \subsection{Full corners and annular localisation} \begin{lemma}[Fullness on a specified set of simple modules] \label{lem:full-corner-support} Let $A$ be a finite-dimensional algebra and $e$ an idempotent. If $eL\ne0$ for every simple $A$-module $L$, then $AeA=A$. In this case $e(-)$ is a Morita equivalence from $A$-mod to $eAe$-mod and reflects isomorphisms, exact sequences, radical and socle filtrations, projectivity and Yoneda extension classes. \end{lemma} \begin{proof} If $A/AeA$ were nonzero, it would have a simple quotient annihilated by $e$, contradicting the hypothesis. Thus $e$ is full. The equivalence has quasi-inverse $Ae\otimes_{eAe}(-)$; the stated preservation and reflection properties follow from exactness of an equivalence. \end{proof} \begin{lemma}[The annular two-site corner] \label{lem:annular-two-site-corner} Let $\mathsf A_N$ be the endomorphism algebra of $N$ in the annular Temperley--Lieb diagram category, with contractible loop value $\beta\ne0$, and let $N=2+2k$. Choose a cup diagram $I:2\to N$ adding $k$ adjacent cups and its reflected cap diagram $C:N\to2$, so that $CI=\beta^k1_2$. Then \[ e=\beta^{-k}IC\in\mathsf A_N \] is an idempotent and \[ j:\mathsf A_2\xrightarrow{\sim}e\mathsf A_Ne, \qquad j(a)=\beta^{-k}IaC \] is an algebra isomorphism, with inverse $x\mapsto\beta^{-k}CxI$. If $A$ is the finite image of $\mathsf A_N$ on a module $M$, then $eAe$ is exactly the image of $\mathsf A_2$ on $eM$ under this identification. \end{lemma} \begin{proof} The equation $CI=\beta^k1_2$ gives $e^2=e$ and $j(a)j(b)=j(ab)$. It also verifies the stated inverse. For any $x\in\mathsf A_N$, \[ exe=\beta^{-2k}ICxIC=j(\beta^{-k}CxI), \] which proves surjectivity onto the entire corner. Passing to a representation image gives the final assertion. \end{proof} \begin{lemma}[Localisation of the periodic XXZ sector] \label{lem:xxz-two-site-localisation} Use the XXZ convention in which the local generator on the ordered basis $|++\rangle,|+-\rangle,|-+\rangle,|--\rangle$ has the sole nonzero block \[ \begin{pmatrix}-q&1\\1&-q^{-1}\end{pmatrix} \] on the middle two basis vectors, and the rotation is \[ \Omega_N|s_1s_2\cdots s_N\rangle =z^{-s_1}|s_2\cdots s_Ns_1\rangle, \qquad s_i\in\{1,-1\}. \] Then the idempotent of Lemma~\ref{lem:annular-two-site-corner} satisfies \[ eX^+_{N;0,z}\simeq X^+_{2;0,z} \] as $\mathsf A_2$-modules. The analogous statement holds for the charge $2$ sector. \end{lemma} \begin{proof} Take the cup vector and cap functional \[ c=-q|+-\rangle+|-+\rangle, \qquad \ell(|+-\rangle)=1,\quad \ell(|-+\rangle)=-q^{-1},\quad \ell(|++\rangle)=\ell(|--\rangle)=0. \] They satisfy $c\ell=e_i$ on the selected adjacent pair and $\ell c=\beta=-q-q^{-1}$. Their two snake contractions are identity maps. The seam matrix $D_z=\operatorname{diag}(z^{-1},z)$ satisfies $(D_z\otimes D_z)c=c$ and $\ell(D_z\otimes D_z)=\ell$. These identities give the annular diagram action with the stated local generator and rotation. Place the added cups away from the seam. Their spin-space maps are \[ I_{\mathrm{spin}}(u)=u\otimes c^{\otimes k},\qquad C_{\mathrm{spin}}=1\otimes\ell^{\otimes k}. \] The image of $e$ is precisely the image of $I_{\mathrm{spin}}$. Every cup has charge zero, so its charge-zero part is identified with the two-dimensional charge-zero space on the active two sites. Cup--cap contraction in $j(a)$ cancels the $k$ factors $\beta$ and leaves the two-site action of $a$. In particular the remaining rotation matrix is \[ \begin{pmatrix}0&z\\z^{-1}&0\end{pmatrix}. \] This proves the assertion. In charge $2$, the active two-site vector is $|++\rangle$, and the same contractions apply. \end{proof} \begin{proposition}[Transport from an isolated two-site channel] \label{prop:isolated-channel-transport} Let $M_N=P_N\oplus W_N\oplus X_N$ be a finite-dimensional $\mathsf A_N$-module. Assume independently that all its simple composition factors belong to two classes $S_N,T_N$, and that under the annular corner isomorphism one has $eS_N\simeq S_2$ and $eT_N\simeq T_2$, where $S_2,T_2$ are nonisomorphic simple modules. Suppose in addition that \[ eP_N\simeq P_2=T_2/S_2/T_2,\qquad eW_N\simeq\operatorname{rad}P_2,\qquad eX_N\simeq P_2/\operatorname{soc}P_2. \] Then $P_N=T_N/S_N/T_N$, $\operatorname{rad}P_N\simeq W_N$, and $P_N/\operatorname{soc}P_N\simeq X_N$. All conclusions of Theorems~\ref{thm:three-layer-image} and \ref{thm:yoneda-obstruction} hold for the joint image on $M_N$. \end{proposition} \begin{proof} Let $A=A(M_N)$. The support assumption and the nonvanishing of $e$ on the two simple classes imply that $e$ is full by Lemma~\ref{lem:full-corner-support}. Its corner algebra is the image of $\mathsf A_2$ on the three displayed modules. The last two are subquotients of $P_2$, so this corner algebra is $A(P_2)$. The Morita equivalence transports the radical series of $P_2$, the identifications of its subquotients, and all homological conclusions to $A$. In particular $W_N$ and $X_N$ are subquotients of $P_N$, so the joint image is $A(P_N)$. \end{proof} \begin{remark} The support assumption in Proposition~\ref{prop:isolated-channel-transport} must be established before invoking fullness. The equality $eP_N\simeq P_2$ alone does not exclude additional composition factors annihilated by $e$. \end{remark} \subsection{The isolated marked image and its XXZ quotient} \label{subsec:isolated-images} We now apply the image-algebra results to the modules constructed in \Cref{sec:marked-proved}. Fix $2\leq N<2p-2$ and $g\ne0$, and write $P_N=Z_N$ and $E_N=P_N/\Theta_NT_N$. \begin{theorem}[The marked and XXZ comparison] \label{thm:isolated-xxz-images} There is an isomorphism of annular modules \[ E_N\cong X^+_{N;0,\varepsilon q}. \] Consequently, if \[ B_N=A(P_N),\qquad R_N=A\bigl(W_N\oplus X^+_{N;0,\varepsilon q}\bigr), \] then $B_N\twoheadrightarrow R_N$ is the quotient of \Cref{thm:three-layer-image}, and all conclusions of \Cref{thm:yoneda-obstruction,cor:marked-lifting-obstruction} apply. In particular, $W_N$ is the projective cover of $S_N$ in both image categories. The projective cover of $T_N$ is $P_N$ over $B_N$ and $X^+_{N;0,\varepsilon q}$ over $R_N$. \end{theorem} \begin{proof} Form the finite image on $P_N\oplus W_N\oplus X^+_{N;0,\varepsilon q}$. By \Cref{thm:marked-isolated-proved} and the XXZ composition-factor theorem \cite[Theorem~2.4]{PinetSaintAubin2023}, its simple support is exactly $\{S_N,T_N\}$. For the annular idempotent $e=\beta^{-(N-2)/2}IC$, the standard localization and exactness give \[ eT_N\cong T_2,\qquad eW_N\cong W_2,\qquad eS_N\cong S_2. \] The two simple corner modules are distinct: their rotation eigenvalues are $\varepsilon$ and $-\varepsilon$. Thus $e$ is full by \Cref{lem:full-corner-support}. The marked localization gives $eP_N\cong P_2$ and $eE_N\cong E_2$. The explicit spin cup--cap calculation in \Cref{lem:xxz-two-site-localisation} identifies $eX^+_{N;0,\varepsilon q}$ with $X^+_{2;0,\varepsilon q}$. The intertwiner of \Cref{lem:marked-two-node-xxz} identifies the last module with $E_2$. The Morita equivalence of the full corner therefore identifies $E_N$ and the stated XXZ module. Both $W_N$ and this XXZ module are subquotients of $P_N$, so the joint image is $A(P_N)$. \Cref{thm:three-layer-image} now gives the quotient and the projective-cover statements. \end{proof} In the normalization of the two-node vectors $a,t,r$, the central action is $g\nu_N$, where $\nu_N=\Theta_N\pi_N$ identifies the head and socle of $P_N$. Its image algebra has centre \[ Z(B_N)=\C 1\oplus\C\nu_N, \] whereas $Z(R_N)=\C$. The quotient kernel is the ideal generated by $\nu_N$, of dimension $(nC_n)^2$. \begin{corollary}[A complete detector in the isolated image] \label{cor:complete-annular-detector} Fix the endpoint identifications obtained by localization at two nodes. Evaluation of the annular operator $U_N$ of \Cref{prop:marked-annular-datum} on the specified lifted vector defines a nonzero linear functional on the one-dimensional space $\Ext^1_{B_N}(T_N,S_N)$, and hence identifies that space with $\C$. On the displayed extension class $[E_N]$, its value is $2\varepsilon g$. \end{corollary} \begin{proof} The operator annihilates the two endpoints, so changing the lift does not change its value. Formation of the upper-right action block of an extension is linear under Baer sum, and evaluation at the fixed corner vectors is linear. \Cref{prop:marked-annular-datum} gives the nonzero value $2\varepsilon g$ on $E_N$. The dimension assertion follows from \Cref{thm:yoneda-obstruction}. \end{proof} This detector is attached to a specified annular operator and specified cup--cap normalization. It supplies an algebraic extension parameter without choosing an ADE field or a continuum normalization. \section{A five-simple image algebra beyond the isolated window} \label{sec:five-simple-image} The image algebra depends on the actual twists of the chosen XXZ modules. We therefore specify the generating module explicitly. Throughout this section let $q=e^{i\pi/3}$, $N=6$, and \begin{equation} M=W_{6;0,q}\oplus X^+_{6;0,q}\oplus X^+_{6;2,1}, \qquad A=\operatorname{im}\bigl(\mathsf A_6\to\End_{\C}M\bigr). \label{eq:c3-selected-image} \end{equation} The Graham--Lehrer injection $W_{6;2,1}\hookrightarrow W_{6;0,q}$ shows that adjoining $W_{6;2,1}$ does not change this image. We do not identify the twists $q$ and $q^{-1}$ in \eqref{eq:c3-selected-image}. Write the five simple classes as \[ a=L_{6;0,q},\quad b=L_{6;2,1},\quad c=L_{6;4,-1}, \quad d=L_{6;6,-q},\quad e=L_{6;6,q^2}. \] Their dimensions are respectively $1,13,4,1,1$. The standard and XXZ structure theorems give the following radical layers: \begin{center} \begin{tabular}{c|c|c|c} Module & head & first radical layer & second radical layer\\ \hline $W_{6;0,q}$ & $a$ & $b\oplus c$ & $d\oplus e$\\ $X^+_{6;0,q}$ & $b$ & $a\oplus d\oplus e$ & $c$\\ $X^+_{6;2,1}$ & $d$ & $b$ & $e$ \end{tabular} \end{center} In the first row every middle constituent is joined to both bottom constituents. In the second row the head is joined to all three middle constituents, each of which is joined to the bottom. The third row is a chain. These are the orientations in \cite[Theorems~2.2 and~2.4]{PinetSaintAubin2023}. \subsection{Exact reconstruction of the image} \begin{proposition}[The image dimension] \label{prop:c3-exact-dimension} The algebra $A$ in \eqref{eq:c3-selected-image} has dimension $328$. \end{proposition} \begin{proof} The assertion is established by an exact finite matrix calculation over $K=\mathbb Q[q]/(q^2-q+1)$. We describe the matrices and the finite closure check so that both bounds on the dimension are reproducible. The basis of $W_{6;0,q}$ is indexed by the three-element subsets of $\{0,1,\ldots,5\}$ giving the initial endpoints of the annular cups. To reconstruct the cups, read the corresponding balanced parenthesis word cyclically, starting at a minimum of its partial sums, and match parentheses by a stack. Rotation decreases every endpoint by one modulo six. The generator $e_1$ joins endpoints $0,1$ and reconnects their former partners. A contractible closed component contributes $\beta=-1$; a noncontractible closed component contributes $q+q^{-1}=1$. This gives two $20\times20$ matrices with integer entries. For the two XXZ summands use the spin words of charge zero and charge two, respectively. The local $e_1$ matrix is the usual block $\left(\begin{smallmatrix}-q&1\\1&-q^{-1}\end{smallmatrix}\right)$ on $|+-\rangle,|-+\rangle$, and rotation is \[ |s_1\cdots s_6\rangle\longmapsto z^{-s_1}|s_2\cdots s_6s_1\rangle \] with $z=q$ and $z=1$, respectively. The two sector dimensions are $20$ and $15$. Let $E,O$ be the simultaneous matrices for $e_1,\Omega$ on the three summands. On each summand $O^6=1$, as checked directly; hence $O^{-1}=O^5$, and $E,O$ generate the entire periodic image. Starting with $1$, append a vector whenever exact row reduction shows that right multiplication of a previously retained vector by $E$ or $O$ leaves the current span. Coordinates are the concatenated matrix entries of the three summands, a vector space of dimension $20^2+20^2+15^2=1025$. Arithmetic represents $u+vq\in K$ by the pair $(u,v)\in\mathbb Q^2$ and uses \[ (u+vq)(r+sq)=(ur-vs)+(us+vr+vs)q. \] The resulting basis has $328$ vectors with distinct leading coordinates. All $656$ right products of those vectors by $E,O$ lie in their span. Thus their span contains the unit, is invariant under both generators, and is exactly the image algebra. Linear independence gives the lower bound $328$, while the verified closure gives the upper bound. The calculation also checks all affine Temperley--Lieb defining relations exactly before computing the span. The complete construction, exact arithmetic, closure computation and serialized basis are supplied in the supporting files listed below. \end{proof} \subsection{Quiver, relations and projective covers} For vertices $i,j$, denote an arrow from $i$ to $j$ by $u_{ij}$, and read products with the rightmost arrow first. \begin{theorem}[The five-simple algebra] \label{thm:c3-quiver} The algebra $A$ is Morita equivalent to the $19$-dimensional algebra $B=\C Q/I$ with arrows \begin{equation} a\to b,c;\qquad b\to a,d,e;\qquad c\to d,e; \qquad d\to b,c;\qquad e\to c. \label{eq:c3-arrows} \end{equation} The nonzero paths of length two are exactly the paths occurring in \begin{equation} \begin{aligned} u_{bd}u_{ab}&=u_{cd}u_{ac},\\ u_{be}u_{ab}&=u_{ce}u_{ac},\\ u_{ac}u_{ba}&=u_{dc}u_{bd}=u_{ec}u_{be},\\ &u_{be}u_{db}. \label{eq:c3-nonzero-paths} \end{aligned} \end{equation} All other paths of length two are zero. These quadratic relations imply that every path of length three is zero. In particular, \[ \dim B/\operatorname{rad}B=5,\qquad \dim\operatorname{rad}B/\operatorname{rad}^2B=10,\qquad \dim\operatorname{rad}^2B=4. \] For the full image algebra, \[ \dim A/\operatorname{rad}A=188,\quad \dim\operatorname{rad}A/\operatorname{rad}^2A=85,\quad \dim\operatorname{rad}^2A=55. \] Both centres are $\C$, and both algebras have infinite global dimension. \end{theorem} \begin{proof} Write $J=\operatorname{rad}A$. All three summands of the faithful module have radical length at most three, so $J^3M=0$ and hence $J^3=0$. Pass to a basic corner. Every nonzero vertex space in each of the three modules over the basic corner has dimension one. The action of $J^2$ can only map $a$ to $d,e$ in the first module, $b$ to $c$ in the second, and $d$ to $e$ in the third. Each of these four corners is nonzero by the stated radical layers. Faithfulness bounds each corner by one dimension, so the basic $J^2$ has exactly these four one-dimensional corners. In the full algebra, a one-dimensional basic corner from $i$ to $j$ corresponds to a matrix bimodule of dimension $(\dim i)(\dim j)$. The four dimensions are therefore $1,1,52,1$, whose sum is $55$. The adjacent radical layers of the three modules exhibit every arrow in \eqref{eq:c3-arrows}. Thus the basic first radical layer contains at least these ten independent corners. Weighted by the dimensions $1,13,4,1,1$ of the simple modules, they contribute $85$ to the full first radical layer. The semisimple quotient has dimension $1^2+13^2+4^2+1^2+1^2=188$. Proposition \ref{prop:c3-exact-dimension} gives \[ 328=188+85+55, \] so there can be no further arrows and no extra arrow multiplicities. The ten arrow corners are disjoint from the four second-radical corners, so choosing arrow representatives introduces no quadratic correction terms. Each composition in the first two lines of \eqref{eq:c3-nonzero-paths} acts nontrivially on the first module. Each composition in the third line acts nontrivially on the second, and the final path acts nontrivially on the third. The four resulting corners of $J^2$ are one-dimensional. Rescaling the four arrows $u_{cd},u_{ce},u_{dc},u_{ec}$ makes the proportionality constants in the first three lines equal to one. Every other length-two path acts as zero on all three modules. For example, $u_{ce}u_{dc}$ is zero although the corner from $d$ to $e$ is nonzero: its two arrows act on different generating summands. These relations kill every cubic path. For instance, extending the path $u_{bd}u_{ab}$ by $u_{dc}$ gives $u_{ac}u_{ba}u_{ab}=0$; extending $u_{ac}u_{ba}$ by $u_{cd}$ gives $u_{bd}u_{ab}u_{ba}=0$; and extending $u_{be}u_{db}$ by $u_{ec}$ gives $u_{ac}u_{ba}u_{db}=0$. The other extensions contain a zero quadratic subpath or reduce to one of these. The quotient therefore has a basis of five idempotents, ten arrows and four length-two paths. Its surjection to the basic algebra is an isomorphism by the dimension count. The centre is $\C$ by Theorem~\ref{thm:multiplicity-free-centre}, since the support graph is connected. The Cartan matrix displayed below is singular. A finite-dimensional algebra of finite global dimension has an invertible Cartan matrix over $\mathbb Z$: finite projective resolutions express every simple class as an integral combination of projective classes. Hence the present algebra has infinite global dimension. \end{proof} \begin{corollary}[Projectives in the five-simple image] \label{cor:c3-projectives} With rows indexed by simple modules and columns by their projective covers in the order $a,b,c,d,e$, the Cartan matrix is \[ C_B=\begin{pmatrix} 1&1&0&0&0\\ 1&1&0&1&0\\ 1&1&1&1&1\\ 1&1&1&1&0\\ 1&1&1&1&1 \end{pmatrix}. \] The Loewy layers and physical dimensions of the projectives are \begin{center} \begin{tabular}{c|c|c|c|c} Projective & head & first radical layer & second radical layer & dimension\\ \hline $P_a$ & $a$ & $b\oplus c$ & $d\oplus e$ & $20$\\ $P_b$ & $b$ & $a\oplus d\oplus e$ & $c$ & $20$\\ $P_c$ & $c$ & $d\oplus e$ & $0$ & $6$\\ $P_d$ & $d$ & $b\oplus c$ & $e$ & $19$\\ $P_e$ & $e$ & $c$ & $0$ & $5$ \end{tabular} \end{center} In particular, \[ P_a\simeq W_{6;0,q},\qquad P_b\simeq X^+_{6;0,q},\qquad P_c\simeq W_{6;4,-1}. \] The standard module $W_{6;2,1}$ is not projective in this image algebra. \end{corollary} \begin{proof} The columns are the spaces of paths starting at the corresponding vertices, using Theorem~\ref{thm:c3-quiver}. This gives the Cartan matrix and all layers. Weighting the composition multiplicities by $1,13,4,1,1$ gives the physical dimensions. Each of the three displayed module isomorphisms follows by mapping the appropriate projective cover onto the known module with that simple head and comparing dimensions. The standard module $W_{6;4,-1}$ is an $A$-module because it embeds into $W_{6;0,q}$ by a Graham--Lehrer map. Finally $W_{6;2,1}$ has dimension $15$ and head $b$, whereas its projective cover $P_b$ has dimension $20$. \end{proof} \begin{corollary}[Second extensions] \label{cor:c3-second-extensions} The dimensions of $\Ext^2_B(i,j)$ are \[ \begin{array}{c|ccccc} &a&b&c&d&e\\ \hline a&1&0&0&1&1\\ b&0&2&2&0&0\\ c&0&1&2&0&0\\ d&1&0&0&2&1\\ e&0&0&0&1&1 \end{array}. \] The same dimensions hold for the full image algebra $A$. \end{corollary} \begin{proof} The defining ideal is generated by the sixteen independent quadratic relations described in Theorem~\ref{thm:c3-quiver}. Their components from $i$ to $j$ are the minimal relation spaces, which are dual to $\Ext^2_B(i,j)$. Counting the zero paths and the displayed linear relations between parallel paths gives the table. Morita equivalence preserves extension groups. \end{proof} \subsection{Supporting calculation} The file \texttt{certify\_c3\_image.py} performs the characteristic-zero calculation in Proposition~\ref{prop:c3-exact-dimension}; it uses only Python rational arithmetic and NumPy for integer matrix construction. The companion \texttt{compute\_c3\_image.py} contains the cyclic link-state basis construction and also provides an optional finite-field exploration. The dimension certificate uses the rational calculation. The exact retained basis is recorded in \texttt{c3\_exact\_basis.json}, with each coefficient represented as two rational numbers in the basis $1,q$. The small file \texttt{c3\_exact\_result.json} records the dimension and closure count. Running \[ \texttt{python anc/code/certify\_c3\_image.py} \] from the archive root reconstructs the matrices, verifies all defining relations, recomputes the independent basis and checks closure under both generators. \subsection{The effect of the inverse twist} \begin{proposition}[Adjoining the inverse-twist sector] \label{prop:c3-inverse-twist} Let \[ M'=W_{6;0,q}\oplus X^+_{6;0,q}\oplus X^+_{6;0,q^{-1}}\oplus X^+_{6;2,1}, \qquad A'=A(M'). \] Then \[ \dim A'=397,\qquad \dim A'_{\mathrm{basic}}=22. \] Relative to the quiver of Theorem~\ref{thm:c3-quiver}, the basic algebra has the two additional arrows $c\to a$ and $e\to b$, each of multiplicity one, and the additional one-dimensional second-radical corner from $c$ to $b$. Consequently \[ \dim\operatorname{rad}A'/\operatorname{rad}^2A'=102,\qquad \dim\operatorname{rad}^2A'=107, \] and $\operatorname{rad}^3A'=0$, $Z(A')=\C$. In particular the two twist lists give non-Morita-equivalent image algebras despite having the same simple labels and resonance component. \end{proposition} \begin{proof} The additional XXZ module has radical layers $c$, $a\oplus d\oplus e$, $b$, with all the indicated adjacent arrows nonzero. This follows from the spin-flip comparison and the XXZ structure theorem in \cite[Proposition~2.3 and Theorem~2.4]{PinetSaintAubin2023}. It adds the two stated first-radical directions and the second-radical direction $c\to b$. The basic second radical therefore has dimension five; its full dimension is $55+4\cdot13=107$. The two new arrow corners contribute $4\cdot1+1\cdot13=17$, in addition to the previous contribution $85$. The same exact rational span computation as in Proposition~\ref{prop:c3-exact-dimension}, now with the four matrices of dimensions $20,20,20,15$, gives $397$ independent vectors and verifies closure in all $794$ generator products. Thus \[ 397=188+(85+17)+(55+52), \] which rules out any further directions or multiplicities. The basic dimension is $5+12+5=22$. Radical cube zero follows from the radical lengths of the four generating modules, and the centre follows from Theorem~\ref{thm:multiplicity-free-centre}. Morita equivalence preserves the basic algebra and the first-extension quiver, so the change in arrow count proves the final assertion. \end{proof} This calculation is reproduced by \[ \texttt{python anc/code/certify\_c3\_image.py {-}{-}include-inverse}. \] Its exact basis and closure summary are recorded in \texttt{c3\_inverse\_exact\_basis.json} and \texttt{c3\_inverse\_exact\_result.json}. All generated certificate files are written to the archive's \path{anc/data/} directory by default. The separate script \path{anc/code/verify_quiver.py} enumerates the path spaces of the presentation. It verifies the sixteen-dimensional quadratic relation space, the vanishing of all thirty-eight cubic paths, and the Cartan matrix, dimensions of projective covers and second-extension table using exact rational arithmetic. \section{Exact computations of the marked central action} \label{sec:exact-marked} The calculations in this section start from the face basis of \Cref{sec:marked-proved} and the hoop diagrams in \eqref{eq:hoop-definition}. They provide finite checks of the all-size theorem and determine the larger central component in a non-isolated example. All rank decisions are made in a cyclotomic number field. \subsection{Matrix construction and exact ranks} Vertices and boundary gaps are numbered $0,\ldots,N-1$ cyclically. For a matching $\pi$, the complementary faces are the cycles of $i\mapsto\pi(i+1\bmod N)$; a face is stored as its sorted set of gaps. The generator $e_i$ joins $i,i+1$ and reconnects their former partners. The marked faces are carried through this surgery, and each erased loop receives one of the weights $\beta,A,B,f$ according to the marked points that it encloses. Rotation decreases every node and gap label by one. Each generator column is obtained by this concatenation rule. For the hoop matrices, the program inserts the auxiliary cup, expands the $N$ crossings, and applies the seam cap. A term with $k$ local cup--cap smoothings in $F$ has the scalar factor $q^{n-k}$. The exchanged skein coefficients give $\Fbar$. Thus the construction uses integral powers of $q$, without choosing a square root of $q$. All entries are evaluated in $K=\mathbb Q(e^{i\pi/p})$. \Cref{tab:exact-marked} records exact ranks on the full joint primary component $\mathcal H$ containing $Z_N$. Here $x=1$, $\varepsilon=1$, $w=q$, and $H=(F-q-q^{-1})|_{\mathcal H}$. \begin{table}[htbp] \centering \begin{tabular}{cccccc} \toprule $p$&$N$&$y$&$\dim Y_N$&$\dim\mathcal H$& $(\rank H,\rank H^2,\rank H^3)$\\ \midrule 5&2&2&4&3&$(1,0,0)$\\ 5&4&2&18&10&$(4,0,0)$\\ 5&6&2&80&35&$(15,0,0)$\\ 5&6&$-1$&80&35&$(0,0,0)$\\ 3&6&2&80&43&$(21,2,0)$\\ \bottomrule \end{tabular} \caption{Exact ranks from the two-marked link-state action.} \label{tab:exact-marked} \end{table} In the $p=5,N=6,y=2$ example, $\mathcal H=Z_6$, $F=\Fbar$ on this component, and \[ F|_{\mathcal H}\sim J_2(\varphi)^{\oplus15} \oplus J_1(\varphi)^{\oplus5},\qquad \varphi=2\cos(\pi/5). \] On the full $80$-dimensional space the ranks of the first four powers of $F-\varphi$ are $60,45,45,45$. For $y=-1$, both hoops restrict to $\varphi1_{\mathcal H}$, in agreement with the exact splitting locus. The algebra generated by the hoops on $\mathcal H$ is therefore the algebra of dual numbers in the former case and $\C$ in the latter. Additional exported examples check both signs of $\varepsilon$ at six sites for $p=3$ and $p=5$. \subsection{The non-isolated six-site component} \begin{proposition} \label{prop:nonisolated-marked-central} Let $q=e^{i\pi/3}$ and $Y=Y_{0,0,1,2,[z,q]}(6)$. Then \[ \mathcal H=\ker(F-1)^3=\ker(\Fbar-1)^3 \] is the joint generalized $(1,1)$-eigenspace and has dimension $43$. On $\mathcal H$, put $U=F-1$, $V=\Fbar-1$, and $D=U-V$. One has \[ U^2=UV=V^2,\qquad U^3=V^3=0,\qquad UD=D^2=0, \] and \[ \rank U=\rank V=21,\qquad \rank U^2=2, \qquad \rank D=2. \] The two hoops generate the algebra \[ \C[F|_{\mathcal H},\Fbar|_{\mathcal H}] \cong \C[u,d]/(u^3,ud,d^2), \qquad u\longmapsto U,\quad d\longmapsto D. \] It has dimension $4$, radical dimension $3$, and radical-square dimension $1$. The Jordan form of $F|_{\mathcal H}$ is \[ J_3(1)^{\oplus2}\oplus J_2(1)^{\oplus17} \oplus J_1(1)^{\oplus3}. \] \end{proposition} \begin{proof} The depth filtration and the joint characters in \eqref{eq:marked-central-characters} give the standard factors \[ W_{6;0,q},\quad W_{6;2,1},\quad W_{6;4,-1},\quad W_{6;6,q^2},\quad W_{6;6,q^{-2}} \] in the joint $(1,1)$-component. Their dimensions are $20,15,6,1,1$, whose sum is $43$. The following exact calculation determines the central action and identifies this component as the stated kernels. The full $80\times80$ matrix $F-1$ has successive ranks $58,39,37,37$. The same ranks hold for $\Fbar-1$ and for the vertically stacked pairs of corresponding powers. The latter equalities identify the two kernels, and stabilization at the third power identifies the primary component. The supplied rank certificates consist of a nonzero minor and a full-column-rank right kernel for each of the first three powers of $F-1$. The verification program checks these certificates and recomputes all ranks over $K=\mathbb Q[q]/(q^2-q+1)$. They therefore establish the stated ranks over $\C$. The exported invariant kernel basis gives the restricted matrices, whose products verify the displayed relations. For an explicit independence witness, use the supplied ordered $43$-dimensional kernel basis. Index the rows by $1,U,D,U^2$ and the columns by the matrix positions $(1,1),(1,2),(1,29),(2,29)$, with one-based indices. The resulting evaluation matrix is \[ \begin{pmatrix} 1&0&0&0\\ 0&9/2&-3/2&6q-27/2\\ 0&0&0&12q-6\\ 0&0&-81/4&0 \end{pmatrix}. \] Its determinant is $\frac{2187}{4}(2q-1)\ne0$. The stated quotient algebra has basis $1,u,d,u^2$, so its surjection onto the hoop algebra is an isomorphism. The Jordan multiplicities follow from the ranks $43,21,2,0$ of $1,U,U^2,U^3$. \end{proof} The submodule $Z_6$ has dimension $35$ inside this $43$-dimensional component. Both hoops coincide on $Z_6$ and have square-zero shifted action of rank $15$ there. The larger component thus contains central nilpotence beyond the first-depth construction. This calculation determines the algebra generated by the two hoops on $\mathcal H$; the indecomposable decomposition of $\mathcal H$ as a module for the full periodic algebra is a separate question. \subsection{Reproduction and data certificates} The accompanying archive contains supplementary programs and data in \path{anc/}, including the construction program \path{anc/code/compute_marked.py}, the separate verification program \path{anc/code/verify_marked_data.py}, and seven exact matrix files in \path{anc/data/}. From the archive root, the non-isolated example is reproduced by \begin{verbatim} python -m pip install numpy==2.3.5 sympy==1.14.0 \ python-flint==0.9.0 python anc/code/compute_marked.py --n 6 --p 3 --exact \ --out anc/data/exact_p3_n6_nonsplit.json python anc/code/verify_marked_data.py anc/data/exact_p3_n6_nonsplit.json \end{verbatim} The verification program reads the exported matrices independently of the construction algorithm. It checks the local and affine Temperley--Lieb relations, the rotation relations, central commutation, the scalar depth-zero action, the primary component, and the invariance and central-action identities of the explicit tree basis. It also checks the field metadata and the scalars $f$ and $g$ against the recorded parameters. All seven examples can be verified with \begin{verbatim} python anc/code/verify_marked_data.py anc/data/exact_*.json \end{verbatim} Each JSON file records the field polynomial, a coefficient dictionary, the ordered link-state basis, all generator and hoop matrices, an ordered primary kernel basis, the restricted matrices, and the explicit tree basis. A coefficient is a list of rational strings in ascending powers $1,q,\ldots,q^{[K:\mathbb Q]-1}$. A sparse entry is \texttt{[row,column,coefficient\_id]}, with zero-based indices. If $K_0$ is the exported primary basis matrix, the restricted matrices satisfy $GK_0=K_0G_{\mathcal H}$ for every generator and both hoops. For each rank certificate the verifier checks the nonzero minor, the matrix--kernel product, and an identity submatrix of the kernel basis. These supply both bounds for the asserted rank. \section{Module families and further questions} \label{sec:outlook} The depth-one construction is compatible with every annular diagram. At fixed $p$, it therefore gives a sequence of actual modules $ Z_N$ for all even sizes, with natural inclusions $\Theta_N$ and identical central-action formulas. The first singular threshold $N=2p-2$ changes the composition series of the standard factors. It does not interrupt the construction of the extension or its square-zero central action. The six-site computation at $p=3$ illustrates how this family embeds in a larger joint central component with cubic nilpotence. Several further questions arise from these results. For higher marked channels $w=\varepsilon q^m$, $m>1$, joint central characters again separate the proposed pair of standard factors whenever $2m\leq N<2p-2m$. Indeed the argument of \Cref{lem:marked-joint-isolation} forces $\omega=\pm1$ and then $r=m$ or $r=p-m$. Constructing the corresponding higher-depth analogue of the tree-incidence map requires additional relations between several cut arcs. A useful objective is an explicit map with controlled rank at the root of unity, together with its parameter-dependent splitting equation. The five-simple image of \Cref{sec:five-simple-image} and the $43$-dimensional marked component record different central behavior at the same root of unity and system size. Determining the full representation image of that marked component, its indecomposable summands, and its comparison with the five-simple quotients would extend the two-simple analysis of \Cref{sec:image-algebras}. The four-dimensional algebra generated by the two hoops supplies a specific constraint on such a determination. The annular module families can also be compared with the fusion bifunctor of \cite{IkhlefMorinDuchesne2026Fusion}. Natural questions are whether the computed extension classes are preserved under fusion, which finite-image quotients can be imposed after fusion, and how the second-extension obstruction changes under these operations. A scaling-limit problem can now be formulated on the fixed-parameter family $ Z_N$. Its additional inputs are a topology on selected states, a normalization of the inclusion and quotient maps, and a specified physical evolution or dilation operator. The annular coefficient $2\varepsilon g$ can then be compared with matrix elements of that operator. Establishing this comparison would connect the finite-dimensional extension calculation with bulk logarithmic correlation functions. \clearpage \bibliographystyle{abbrvnat} \bibliography{reference} \end{document}