\section{Introduction}
Let $k$ be a field and consider the path algebra $kQ$ of the quiver
$Q$. A pair of indecomposable $kQ$-modules $(Y,X)$ is called an \emph{orthogonal
	exceptional pair} if the modules are exceptional (i.e. without self extension) and $\operatorname{Hom}(X,Y)=\operatorname{Hom}(Y,X)=\operatorname{Ext}^{1}(X,Y)=0$.
Denote by $\mathcal{F}(X,Y)$ the full subcategory of objects having a
filtration with factors $X$ and $Y$. Observe that $\mathcal{F}(X,Y)$
is an exact, hereditary, abelian subcategory equivalent to the category
of finite dimensional $k$-representations of the quiver having two
vertices and $d=\dim_k\operatorname{Ext}^{1}(Y,X)$ identically oriented arrows.

We know due to Hubery (see \cite{Hubery}) that in the tame case we
have $\dim_k\operatorname{Ext}^{1}(Y,X)\leq2$; moreover, if equality holds then
${\underline\dim}(X\oplus Y)=\delta$ and $\partial Y=1$. Thus in
the case when $\dim_k\operatorname{Ext}^{1}(Y,X)=2$ we have $X=P$ indecomposable
preprojective of defect $-1$ and $Y=I$ indecomposable preinjective
having dimension vector $\delta-{\underline\dim}P$ and defect $1$. This pair
$(I,P)$ is then called a \emph{Kronecker pair} since the category $\mathcal{F}(P,I)$
is equivalent to the category of finite dimensional $k$-representations
of the Kronecker quiver $K$.

The following theorem by Schofield (see \cite{Ringel2,Schofield})
makes it possible to construct exceptional modules as extensions of smaller
exceptional ones. This procedure is generally called \emph{Schofield
	induction} and it is a very useful tool in representation theory of algebras.
\begin{thm}
	\label{thm:Schofield}(Schofield, \cite{Hubery,Ringel2,Schofield})
	If $Z$ is exceptional but not simple, then $Z\in\mathcal{F}(X,Y)$
	for some orthogonal exceptional pair $(Y,X)$, and $Z$ is not a simple
	object in $\mathcal{F}(X,Y)$. In fact, there are precisely $s(Z)-1$
	such pairs, where $s(Z)$ is the support of $Z$ (i.e.  $s(Z)=\sharp \{i\in Q_0\ | \ Z_i\neq 0\}$).
\end{thm}

Note that in the theorem above the condition requiring $Z$ not to
be a simple object in $\mathcal{F}(Y,X)$ is equivalent to the existence
of an exact sequence of the form ~$\xymatrix{0\ar[r] & uX\ar[r] & Z\ar[r] & vY\ar[r] & 0}
$ with $u,v$ positive and uniquely determined by $X,Y,Z$.

\begin{defn}
Consider an orthogonal exceptional pair $(Y,X)$. A short exact sequence of the form $\xymatrix{0\ar[r] & uX\ar[r] & Z\ar[r] & vY\ar[r] & 0}$ with $u,v$ positive integers and $Z$ exceptional is called a \emph{Schofield sequence} and the pair $(Y,X)$ a \emph{Schofield pair} associated to $Z$.
\end{defn}

As Ringel states in \cite{Ringel3} whereas it is easy to construct
$Z$ given $X$ and $Y$, there is no convenient procedure yet to
determine the possible modules $X$ (called \emph{Schofield submodules}
of $Z$) and then $Y$ (called \emph{Schofield factors of $Z$}), when $Z$ is given. In the Dynkin case Bo
Chen's Theorem provides a method to find at least some of these modules
$X$, namely the Gabriel-Roiter submodules of $Z$ (see \cite{Chen, Ringel3}).

In the present paper we describe all the tame Schofield sequences
using some numerical criteria for the characterization of these sequences.
As a result we give in the tame cases an explicit procedure to obtain
all the possible Schofield submodules and factor modules for a given exceptional module.
We will notice that all the tame Schofield sequences are field independent,
thus of purely combinatorial nature, depending only on the oriented
quiver.

\section{Tame quivers and roots}

We begin with a short survey of the problem context. For further details
we refer to \cite{Skow1,Aus,Kac,Skow2}.

Let $Q=(Q_{0},Q_{1})$ be a simply-laced tame quiver without oriented
cycles. This means $Q$ is a directed acyclic graph with vertex set
$Q_{0}$ and arrow set $Q_{1}$ and with its underlying undirected
graph a so-called Euclidean (affine) diagram of type $\widetilde{\mathbb{A}}_{m}$, $\widetilde{\mathbb{D}}_{m}$, $\widetilde{\mathbb{E}}_{6}$, $\widetilde{\mathbb{E}}_{7}$, $\widetilde{\mathbb{E}}_{8}$.
Suppose that the vertex set $Q_{0}$ has $n$ elements and for an
arrow $\alpha\in Q_{1}$ we denote by $t(\alpha),h(\alpha)\in Q_{0}$
the tail, respectively the head of $\alpha$.

The \emph{Euler form} of $Q$ is a bilinear form on $\mathbb{Z}Q_{0}\cong\mathbb{Z}^{n}$
given by $\langle x,y\rangle=\sum_{i\in Q_{0}}x_{i}y_{i}-\sum_{\alpha\in Q_{1}}x_{t(\alpha)}y_{h(\alpha)}$.
Its quadratic form $q_{Q}$ (called \emph{Tits form}) is independent
of the orientation of $Q$ and in the tame case it is positive semi-definite
with radical $\{a\in\mathbb{Z}Q_{0}|q_{Q}(a)=0\}=\mathbb{Z}\delta$,
where $\delta$ is called \emph{minimal positive imaginary root} of
the corresponding Kac-Moody root system (see \cite{Kac}). The \emph{defect}
of $x\in\mathbb{Z}Q_{0}$ is then $\partial x=\langle\delta,x\rangle$,
the \emph{absolute defect} being the absolute value $|\partial x|$.
The vectors $a\in\mathbb{N}Q_{0}$ with $q_{Q}(a)=1$ are called \emph{positive
real roots}.

Below is the list of all the Euclidean diagrams marking each vertex
$i\in Q_{0}$ with the component $\delta_{i}$ (the number of vertices
being $n$).

Type $\widetilde{\mathbb{A}}_m$, with $m\geq1$ (the case $m=0$ having only
cyclic orientation) and $n=m+1$:

\[
\xymatrix{ & 1\ar@{-}[r] & \dots\ar@{-}[r] & 1\ar@{-}[dr]\\
1\ar@{-}[ur]\ar@{-}[dr] &  &  &  & 1\\
 & 1\ar@{-}[r] & \dots\ar@{-}[r] & 1\ar@{-}[ur]
}
\]

The \emph{Kronecker quiver} is the particular quiver of type $\widetilde{\mathbb{A}}_{1}$
with non cyclic orientation
\[
\xymatrix{{\bullet} & {\bullet}\ar@<1ex>[l]\ar@<-1ex>[l]}
.
\]

Type $\widetilde{\mathbb{D}}_{m}$, with $m\geq4$ and $n=m+1$:

\[
\xymatrix{1\ar@{-}[dr] &  &  &  &  & 1\\
 & 2\ar@{-}[r] & 2\ar@{-}[r] & \dots\ar@{-}[r] & 2\ar@{-}[ur]\\
1\ar@{-}[ur] &  &  &  &  & 1\ar@{-}[ul]
}
\]

Type $\widetilde{\mathbb{E}}_{6}$, with $n=7$:
\[
\xymatrix{ &  & 1\ar@{-}[d]\\
 &  & 2\ar@{-}[d]\\
1\ar@{-}[r] & 2\ar@{-}[r] & 3\ar@{-}[r] & 2\ar@{-}[r] & 1
}
\]

Type $\widetilde{\mathbb{E}}_{7}$, with $n=8$:
\[
\xymatrix{ &  &  & 2\ar@{-}[d]\\
1\ar@{-}[r] & 2\ar@{-}[r] & 3\ar@{-}[r] & 4\ar@{-}[r] & 3\ar@{-}[r] & 2\ar@{-}[r] & 1
}
\]

Type $\widetilde{\mathbb{E}}_{8}$, with $n=9$:
\[
\xymatrix{ &  & 3\ar@{-}[d]\\
2\ar@{-}[r] & 4\ar@{-}[r] & 6\ar@{-}[r] & 5\ar@{-}[r] & 4\ar@{-}[r] & 3\ar@{-}[r] & 2\ar@{-}[r] & 1
}
\]

\section{Tame hereditary algebras. Representations of tame quivers}

Let $k$ be a field and consider the path algebra $kQ$ of the quiver
$Q$. The algebra $kQ$ is a finite dimensional
\emph{tame hereditary algebra} and in fact all connected elementary
tame hereditary algebras have this form.

The category of finite dimensional right modules (which is abelian
and Krull--Schmidt) is denoted by $\text{\rm mod-} kQ$. For a module $M\in\text{\rm mod-} kQ$
, $[M]$ denotes its isomorphism class, $|M|$ its length and $uM=M\oplus\dots\oplus M$ ($u$-times).

The category $\text{\rm mod-} kQ$ can and will be identified with the category
$\text{{\rm rep-}}kQ$ of  finite dimensional $k$-representations
of the quiver $Q$. Recall that a $k$-representation of $Q$ is defined
as a set of finite dimensional $k$-spaces $\{V_{i}|i\in Q_{0}\}$
corresponding to the vertices, together with $k$-linear maps $V_{\alpha}:V_{t(\alpha)}\to V_{h(\alpha)}$
corresponding to the arrows $\alpha\in Q_{1}$. The dimension vector of a
module $M=(V_{i},V_{\alpha})\in\text{\rm mod-} kQ=\text{{\rm rep-}}kQ$ is then
${\underline\dim}M=(\dim_kV_{i})_{i\in Q_{0}}\in\mathbb{Z}Q_{0}$. For two (dimension) vectors $d,d'\in \mathbb{Z}Q_0$ we say that $d\leq d'$ if $d_i \leq d'_i$ for all $i\in Q_0$.

Let $S(i)$, $P(i)$ and $I(i)$ be the indecomposable simple, projective
and injective modules corresponding to the vertex $i$ and consider
the \emph{Cartan matrix of the algebra $kQ$} $C_{Q}$ with the $j$-th column being ${\underline\dim}P(j)$  (see Chapter III of \cite{Skow1}).
The \emph{Coxeter matrix} is defined as $\Phi_{Q}=-C_{Q}^{t}C_{Q}^{-1}$.
Then $\Phi_{Q}\delta=\delta$ and the Euler form satisfies $\langle a,b\rangle=aC_{Q}^{-t}b^{t}=-\langle b,\Phi_{Q}a\rangle$,
where $a,b\in\mathbb{Z}Q_{0}$. Moreover, (because our algebra is hereditary)
for two modules $M,N\in\text{\rm mod-} kQ$ we get
\[
\langle{\underline\dim}M,{\underline\dim}N\rangle=\dim_k\operatorname{Hom}(M,N)-\dim_k\operatorname{Ext}^{1}(M,N).
\]
Let $\partial M=\partial(\underline\dim M)=\langle\delta,{\underline\dim}M\rangle=-\langle{\underline\dim}M,\delta\rangle$
be the \emph{defect} of the module $M$.

Consider the Auslander--Reiten translates $\tau=D\operatorname{Ext}^{1}(-,kQ)$ and
$\tau^{-1}=\operatorname{Ext}^{1}(D(kQ),-)$, where $D=\operatorname{Hom}_k(-,k)$.
An indecomposable module $M$ is \emph{preprojective} (\emph{preinjective})
if there exists a positive integer $m$ such that $\tau^{m}(M)=0$
($\tau^{-m}(M)=0$). Otherwise $M$ is said to be \emph{regular}.
Note that an indecomposable module $M$ is preprojective (preinjective,
regular) if and only if $\partial M<0$ ($\partial M>0$, $\partial M=0$).
A module is called preprojective (preinjective, regular) if all its
indecomposable components are preprojective (preinjective, regular).
We will use the notation $P$, $I$ and $R$ in these cases.

The category $\text{\rm mod-} kQ$ is well known, its Auslander--Reiten quiver
being completely described. The indecomposable modules are either
preprojective, preinjective or regular. Thus up to isomorphism the
indecomposable preprojective modules are $P(m,i)=\tau^{-m}P(i)$ and the indecomposable preinjectives are  $I(m,i)=\tau^{m}I(i)$ where $m\in\mathbb{N}$ and $i\in Q_{0}$. Note
that ${\underline\dim}P(m,i)=\Phi_{Q}^{-m}{\underline\dim}P(i)$
and $\partial P(m,i)=\partial P(i)=-\delta_{i}$, respectively  ${\underline\dim}I(m,i)=\Phi_{Q}^{m}{\underline\dim}I(i)$
and $\partial I(m,i)=\partial I(i)=\delta_{i}$. Thus the possible
defects of indecomposable preprojectives are $-1$ in the $\widetilde{\mathbb{A}}_{m}$
case, $-1$ or $-2$ in the $\widetilde{\mathbb{D}}_{m}$ case, $-1$, $-2$ or
$-3$ in the $\widetilde{\mathbb{E}}_{6}$ case, ranging from $-4$ to $-1$ in
the $\widetilde{\mathbb{E}}_{7}$ case and ranging from $-6$ to $-1$ in the $\widetilde{\mathbb{E}}_{8}$
case. Indecomposable preprojectives and preinjectives are \emph{exceptional}
modules (i.e. they are indecomposable modules without self extension
and with one dimensional endomorphism space), are \emph{directing}
(see \cite{Skow1} pages 357, 358 for details) and also uniquely determined
up to isomorphism by their dimension vectors which are positive real
roots, so they can be considered independently from the field $k$.

The category of regular modules is an abelian, exact subcategory which
decomposes into a direct sum of serial categories with Auslander--Reiten
quiver of the form $\mathbb ZA_{\infty}/m$, called \emph{tubes of rank} $m$. These tubes are indexed by the scheme theoretic, closed points of the projective line $\mathbb{P}_{k}^{1}$,
the degree of such a point being $d_{a}=[\kappa(a): k]$ (where $\kappa(a)$ is the residue field at the point $a$).
A tube of rank 1 is called \emph{homogeneous}, otherwise it is called
\emph{non-homogeneous} or \emph{exceptional}. We have at most 3 non-homogeneous
tubes indexed by points $a$ of degree $d_{a}=1$. All the other tubes
are homogeneous. We assume that the non-homogeneous tubes are labeled
by the elements of some subset $E$ of $\{0,1,\infty\}$, whereas the homogeneous
tubes are labeled by the points of the scheme $\mathbb{H}_{k}=\mathbb{H}_{\mathbb{Z}}\otimes k$
for some open subscheme $\mathbb{H}_{\mathbb{Z}}\subset\mathbb{P}_{\mathbb{Z}}^{1}$. For $x\in\mathbb{H}_{k}$ the indecomposables on a homogeneous
tube $\mathcal{T}_{x}$ are denoted by $R_{x}(1)\subset R_{x}(2)\subset\dots$.
Note that these modules depend on the field $k$ (sometimes this being
emphasized by the notation $R_{x}^{k}(t)$) and ${\underline\dim}R_{x}(t)=td_{x}\delta$.
The module $R_{x}(1)$ is called \emph{homogeneous regular simple}
and $t$ is called the \emph{regular length} of $R_{x}(t)$. The homogeneous
regular simple $R_{x}(1)$ is the \emph{regular socle} and also the \emph{regular top}
of $R_{x}(t)$. For a partition $\lambda=(\lambda_{1},\dots,\lambda_{n})$
let $R_{x}(\lambda)=R_{x}(\lambda_{1})\oplus\dots\oplus R_{x}(\lambda_{n})$.
The indecomposable modules on the mouth of the non-homogeneous tube
$\mathcal{T}_{e}$ of rank $m$ (where $e\in E$) are labeled $R_{e}^{l}(1)$
($l\in\{1,\dots,m\}$) and are called \emph{non-homogeneous regular
simples}. We have $\tau R_{e}^{l}(1)=R_{e}^{l-1}(1)$ for $l\geq2$,
$\tau R_{e}^{1}(1)=R_{e}^{m}(1)$ and $\sum_{l=1}^{m}{\underline\dim}R_{e}^{l}(1)=\delta$.
We denote by $R_{e}^{l}(t)$ the non-homogeneous indecomposable regular
with \emph{regular socle} $R_{e}^{l}(1)$ and \emph{regular length} $t$
(and \emph{regular top} $R_{e}^{l}(t)/R_{e}^{l}(t-1)=R_{e}^{l'}(1)$).
Note that $R_{e}^{l}(t)$ is uniquely determined by the values $e$,
$l$ and $t$ thus can be treated field independently; moreover, ${\underline\dim}R_{e}^{l}(t)$
is a positive real root (uniquely corresponding to the module) unless
$t$ is a multiple of $m$. Finally note that ${\underline\dim}R_{e}^{l}(sm)=s\delta$.

We should remark here that for any positive real root $a$ we have
a unique indecomposable module of dimension vector $a$ (see \cite{Kac,Dlab1}).

We list below some known facts on morphisms and
extensions (see \cite{Szanto} for all the details).
\begin{lem}
\label{lem:basic}
\begin{enumerate}
\item[i)] For $P$ preprojective, $I$ preinjective, $R$ regular module we
have $\operatorname{Hom}({R},{P})=\operatorname{Hom}({I},{P})=\operatorname{Hom}({I},{R})=\operatorname{Ext}^{1}({P},{R})=\operatorname{Ext}^{1}({P},{I})=\operatorname{Ext}^{1}({R},{I})=0.$
\item[ii)] If $x\neq x'$ and $R_{x}$ (respectively $R_{x'}$) is a regular
with components from the tube $\mathcal{T}_{x}$ (respectively $\mathcal{T}_{x'}$),
then $\operatorname{Hom}(R_{x},R_{x'})=\operatorname{Ext}^{1}(R_{x},R_{x'})=0.$
\item[iii)]  For $\mathcal{T}_{x}$ homogeneous and $R_{x}(t)$, $R_{x}(t')$
indecomposables from $\mathcal{T}_{x}$ we have $\dim_k\operatorname{Hom}(R_{x}(t),R_{x}(t'))=\dim_k\operatorname{Ext}^{1}(R_{x}(t),R_{x}(t'))=\min(t,t')d_{x}$.
\item[iv)]  For $\mathcal{T}_{e}$ non-homogeneous of rank $m$ and $R_{e}^{l}(t)$
an indecomposable from $\mathcal{T}_{e}$ we have $\dim_k\operatorname{End}(R_{e}^{l}(t))=s+1$
for $sm<t\leq(s+1)m$ and $\dim_k\operatorname{Ext}^{1}(R_{e}^{l}(t),R_{e}^{l}(t))=s$
for $sm\leq t<(s+1)m$, thus $R_{e}^{l}(t)$ is exceptional if and
only if $0<t<m$. Moreover, in this case $R_{e}^{l}(t)$ is uniquely
determined by its unique regular composition series, with distinct
regular composition factors.
\item[v)]  For $P$ indecomposable preprojective and $I$ indecomposable preinjective
modules we have $\dim_k\operatorname{End}(P)=\dim_k\operatorname{End}(I)=1$ and $\operatorname{Ext}^{1}(P,P)=\operatorname{Ext}^{1}(I,I)=0$,
so they are exceptional modules. Moreover, they are also directing.
\end{enumerate}
\end{lem}
\begin{lem} \label{lem:preproj} (\cite{Szanto}) Let $P$ be a preprojective indecomposable with defect
	$\partial P=-1$, $P'$ a preprojective module and $R$ a regular
	indecomposable. Then we have:
\begin{enumerate}
    \item[i)] Every nonzero morphism $f:P\to P'$ is a monomorphism.
    \item[ii)] For every nonzero morphism $f:P\to R$, $f$ is either a
	monomorphism or $\operatorname{Im} f$ is regular. In particular, if $R$ is
	regular simple and $\operatorname{Im} f$ is regular then $f$ is an epimorphism.
	\item[iii)]Suppose that $\underline\dim P>\delta$. Then $P$ projects to a unique	regular simple $R^P_e(1)$ from the mouth of each non-homogeneous tube
	$\mathcal{T}_e$; moreover, $\dim_k\operatorname{Hom}(P,R^P_e(1))=\langle \underline\dim P, \underline\dim R^P_e(1)\rangle=1$.
	\item[iv)] Suppose that $\underline\dim P<\delta$. Then, depending on its dimension vector, $P$ embeds in or projects onto
	a unique regular simple $R^P_e(1)$ from each non-homogeneous
	tube $\mathcal{T}_e$; moreover, $\dim_k\operatorname{Hom}(P,R^P_e(1))=\langle \underline\dim  P, \underline\dim R^P_e(1)\rangle=1$.
\end{enumerate}
\end{lem}
\begin{lem}\label{lem:preinjextension}(\cite{Szanto}) Let $I$ be an indecomposable preinjective of
		defect 1, $P$ an indecomposable preprojective of defect $-1$ and
		$X\ncong P\oplus I$. There exists a short exact sequence $0\to P\to X\to I\to 0$ iff $X$ satisfies the
		following conditions:
\begin{enumerate}
    \item[i)] $X$ is a regular module with $\underline\dim
		X=\underline\dim I+\underline\dim P$;
	\item[ii)] if $X$ has an indecomposable component from a non-homogeneous tube
		$\mathcal{T}_e$, then the regular top of this component is $R^{P}_e(1)$;
	\item[iii)] the indecomposable components of $X$ are taken from
		pairwise different tubes.
\end{enumerate}
\end{lem}

We end this section with some facts on reflections.

Let $i$ be a sink in the quiver $Q$. Denote by $s_i$ the reflection induced by the vertex $i$ and by $\sigma_i Q$ the quiver obtained by reversing all arrows involving $i$. Let $\text{\rm mod-} kQ\langle i\rangle$ be the full subcategory of modules not containing the simple module corresponding to the vertex $i$ as a direct summand.

We consider the reflection functors $S^+_i:\text{\rm mod-} kQ\to\text{\rm mod-} k\sigma_iQ$ and  $S^-_i:\text{\rm mod-} k\sigma_i Q\to\text{\rm mod-} kQ$. For all details concerning reflection functors we refer the reader to \cite{bernstein,Skow1}. It is well known that for an indecomposable $M$ we have $S^+_iM\neq 0$ iff $M\ncong S(i)$, where $S(i)$ is the projective simple corresponding to the sink $i$. Moreover, in this case $S^+_iM$ is indecomposable and $\underline\dim S^+_iM=s_i(\underline\dim M)$. Also the functors $S^+_i$, $S^-_i$ induce quasi-inverse equivalences between $\text{\rm mod-} kQ\langle i\rangle$ and $\text{\rm mod-} k\sigma Q\langle i\rangle$. It is easy to see that for $M\in\text{\rm mod-} kQ\langle i\rangle$ indecomposable we have that $\partial S^+_iM=\partial M$ and if $R$ is a simple homogeneous (respectively non-homogeneous) regular, then so is $S^+_iR$.

The following lemma is taken from \cite{Skow1}.
\begin{lem}\label{lem:reflections} Let $Q$, $Q'$ be trees having the same underlying graph. There exists a sequence $i_1,...,i_t$ of vertices of $Q$ such that for each $s\in\{1,\dots,t\}$ the vertex $i_s$ is a sink in $\sigma_{i_{s-1}}\dots\sigma_{i_1}Q$ and $\sigma_{i_{t}}\dots\sigma_{i_1}Q=Q'$.
\end{lem}

All the information above can be dualized for sources instead of sinks.

\section{Tame Schofield sequences}

We begin with a numerical criterion characterizing
tame Schofield sequences.
\begin{prop}
\label{prop:SchofieldCharaterization}Suppose $X$, $Y$, $Z$ are
exceptional indecomposables such that $u\underline\dim X+v\underline\dim Y=\underline\dim Z$.
Then we have a Schofield sequence
\[
\xymatrix{0\ar[r] & uX\ar[r] & Z\ar[r] & vY\ar[r] & 0}
\]
if and only if $\left\langle \underline\dim X,\underline\dim Y\right\rangle =0$. Moreover,
in this case either $u=v=1$ or $\left|u-v\right|=1$ with $\partial X=-1$,
$\partial Y=1$, $\underline\dim X+\underline\dim Y=\delta$ and $\partial Z=\pm1$.
In the latter case we will speak about a \emph{special Schofield sequence}. Finally $u$, $v$ and thus $Y$ are uniquely determined by $X$, $Z$ and also $u$, $v$ and thus $X$ are uniquely determined by $Z$, $Y$ .
\end{prop}

\begin{proof}
The necessity is trivial. For the converse, note that we cannot
have simultaneously $\operatorname{Hom}(X,Y)\neq0$ and $\operatorname{Ext}^{1}(X,Y)\neq0$. By
Lemma \ref{lem:basic} this is trivial if $X$ and $Y$ are not of
the same type (preprojective, regular or preinjective). If both $X$
and $Y$ are preprojective (respectively preinjective), they are directing,
hence the assertion is true. If both $X$ and $Y$ are regular (non-homogeneous
exceptional) then we must have $u=v=1$ and also $\operatorname{Hom}(X,Y)=0$. This
holds because $Z$ being a regular exceptional, by Lemma \ref{lem:basic}
iii) it has a unique regular composition series with distinct regular
composition factors. Hence this is true also for $X$ and $Y$; moreover,
there is no common regular composition factor for $X$ and $Y$. In
case of a nonzero morphism $f:X\to Y$, $X$ would project onto the
regular module $f(X)\subseteq Y$, so the regular composition factors of $f(X)$ would appear in both $X$ and $Y$, which is impossible.

Since $0=\left\langle \underline\dim X,\underline\dim Y\right\rangle =\dim_{k}\operatorname{Hom}(X,Y)-\dim_{k}\operatorname{Ext}^{1}(X,Y)$,
it follows that $\operatorname{Hom}(X,Y)=\operatorname{Ext}^{1}(X,Y)=0$ using the previous observation.

Since $\underline\dim Z$, $\underline\dim X$ and $\underline\dim Y$ are all positive roots
and $u\underline\dim X+v\underline\dim Y=\underline\dim Z$ it follows that
\begin{align*}
1 & =\left\langle \underline\dim Z,\underline\dim Z\right\rangle \\
 & =u^{2}\left\langle \underline\dim X,\underline\dim X\right\rangle +uv\left\langle \underline\dim X,\underline\dim Y\right\rangle +uv\left\langle \underline\dim Y,\underline\dim X\right\rangle +v^{2}\left\langle \underline\dim Y,\underline\dim Y\right\rangle \\
 & =u^{2}+v^{2}+uv\left\langle \underline\dim X,\underline\dim Y\right\rangle +uv\left\langle \underline\dim Y,\underline\dim X\right\rangle \\
 & =u^{2}+v^{2}+uv\left\langle \underline\dim Y,\underline\dim X\right\rangle .
\end{align*}
So $\left\langle \underline\dim Y,\underline\dim X\right\rangle =\frac{1-u^{2}-v^{2}}{uv}<0$
(here $u,v\ge1$), which implies that $\operatorname{Hom}(Y,X)=0$ (using the fact proven above that $\operatorname{Hom}(Y,X)$ and $\operatorname{Ext}^1(Y,X)$ can't both be nonzero) and  $\dim_k\operatorname{Ext}^{1}(Y,X)=\frac{-1+u^{2}+v^{2}}{uv}$.
So we already have that $X$ and $Y$ form an orthogonal exceptional
pair; therefore, by using Lemma 2 from \cite{Hubery1}
we get that the value of $\dim_{k}\operatorname{Ext}^{1}(Y,X)$ is either $1$ or
$2$. Moreover, if the value is 2 we must have $\partial X=-1$ and $\underline\dim X+\underline\dim Y=\delta$.

In the case when $\dim_k\operatorname{Ext}^{1}(Y,X)=1=\frac{-1+u^{2}+v^{2}}{uv}$,
we must have $u=v=1$, so $\underline\dim Z=\underline\dim X+\underline\dim Y$ and we know (by \cite{Hubery1}) that $\mathcal{F}(Y,X)$ is an exact, hereditary, abelian subcategory equivalent to the category of modules over the finite dimensional path algebra of the Dynkin quiver of type $A_1$ ($\xymatrix{{\bullet} & {\bullet}\ar[l]}$) over $k$ ($X,Y$ corresponding to the simple Dynkin modules). Knowing the Auslander-Reiten quiver of this Dynkin algebra and the fact that $\underline\dim Z=\underline\dim X+\underline\dim Y$ it follows that there is an Auslander-Reiten sequence $0\to X\to Z\to Y\to 0$, which will be our Schofield sequence.

In case $\dim_k\operatorname{Ext}^{1}(Y,X)=2$, we have $(u-v)^{2}=1$, so $\left|u-v\right|=1$,
$\partial X=-1$, $\underline\dim X+\underline\dim Y=\delta$. We also know (by \cite{Hubery1}) that in this case $\mathcal{F}(Y,X)$ is an exact, hereditary, abelian subcategory equivalent to the category of modules over the Kronecker algebra $kK$, where $K$ is the Kronecker quiver ($X,Y$ corresponding to the simple Kronecker modules). Knowing the Auslander-Reiten quiver of the Kronecker algebra and the fact that $\underline\dim Z=u\underline\dim X+v\underline\dim Y$ it follows that there is a short exact sequence $0\to uX\to Z\to vY\to 0$, which will be our Schofield sequence.

Note that $u$, $v$ and thus $Y$ are uniquely determined by $X$, $Z$. Indeed using the results above it is enough to consider the case $\partial X=-1$ and $\partial Z=\pm 1$. If we have two Schofield sequences, a special one $0\to (u\mp 1)X\to Z\to uY'\to 0$ with $\underline\dim Y'=\delta-\underline\dim X$ and a non-special one $0\to X\to Z\to Y\to 0$, then $\partial Y=2$ or 0 and
\begin{align*}
1 & =\left\langle \underline\dim Z,\underline\dim Z\right\rangle \\
&=\left\langle\underline\dim Z,(u\mp 1)\underline\dim X+u\underline\dim Y'\right\rangle\\
&=\left\langle\underline\dim X+\underline\dim Y,u\delta\mp\underline\dim X\right\rangle\\
& =u\left\langle \underline\dim X,\delta\right\rangle\mp\left\langle \underline\dim X,\underline\dim X\right\rangle +u\left\langle \underline\dim Y,\delta\right\rangle \mp\left\langle \underline\dim Y,\underline\dim X\right\rangle \\
& =u(1-\partial Y).
\end{align*}
It follows that either $u=-1$, a contradiction or $u=1$ which is again impossible (since then $\underline\dim Y=\underline\dim Z-\underline\dim X=\delta$).

Similarly follows that $u,v$ and thus $X$ are uniquely determined by $Z$ and $Y$.
\end{proof}

We can deduce from the proof above that:

\begin{cor} \label{cor:Schofield conditions}Suppose $X$, $Y$, $Z$ are
	exceptional indecomposables such that $\underline\dim X+\underline\dim Y=\underline\dim Z$. Then the following are equivalent:
	\begin{enumerate}

    \item[i)] we have a non-special Schofield sequence
	\[
	\xymatrix{0\ar[r] & X\ar[r] & Z\ar[r] & Y\ar[r] & 0;}
	\]
     \item[ii)] $\dim_k\operatorname{Ext}^1(Y,X)=1$;

	 \item[iii)] in case $X$ and $Z$ are not both regular $\dim_k\operatorname{Hom}(X,Z)=1$ and in case $X$ and $Z$ are both regular they share the same regular socle;

	 \item[iv)] in case $Y$ and $Z$ are not both regular $\dim_k\operatorname{Hom}(Z,Y)=1$ and in case $Y$ and $Z$ are both regular they share the same regular top.
\end{enumerate}
\end{cor}
\begin{proof} $i)\Rightarrow ii)$  Use the proof of the previous proposition.

$ii)\Rightarrow i)$ Note that $\dim_k\operatorname{Ext}^1(Y,X)=1$ implies $\left\langle \underline\dim Y,\underline\dim X\right\rangle =-1$, so because $1=\left\langle \underline\dim Z,\underline\dim Z\right\rangle =\left\langle \underline\dim X,\underline\dim X\right\rangle +\left\langle \underline\dim X,\underline\dim Y\right\rangle +\left\langle \underline\dim Y,\underline\dim X\right\rangle +\left\langle \underline\dim Y,\underline\dim Y\right\rangle=1+\left\langle \underline\dim X,\underline\dim Y\right\rangle$, we obtain $\left\langle \underline\dim X,\underline\dim Y\right\rangle =0$.

$i)\Rightarrow iii)$ In case $X$,$Z$ are not both regular, from $0=\left\langle \underline\dim X,\underline\dim Y\right\rangle=\left\langle \underline\dim X,\underline\dim Z-\underline\dim X\right\rangle$ it follows that $\left\langle \underline\dim X,\underline\dim Z\right\rangle=\left\langle \underline\dim X,\underline\dim X\right\rangle=1$. Also by Lemma \ref{lem:basic} $\operatorname{Hom}(X,Z)$ and $\operatorname{Ext}^1(X,Z)$ can't be both nonzero, so the assertion follows.

In case $X$, $Z$ are both regular the assertion follows, since we have a monomorphism $X\to Z$.

$iii)\Rightarrow i)$ If $\dim_k\operatorname{Hom}(X,Z)=1$ and $X$, $Z$ are not both regular exceptional modules,  then by Lemma \ref{lem:basic} $\operatorname{Ext}^1(X,Z)=0$, which means that $1=\langle\underline\dim X,\underline\dim Z\rangle=\langle \underline\dim X,\underline\dim X+\underline\dim Y\rangle=1+\langle \underline\dim X,\underline\dim Y\rangle$, that is $\langle \underline\dim X,\underline\dim Y\rangle=0$.

If both $X$ and $Z$ are regular sharing the same regular socle then we will have a unique regular short exact sequence (due to uniseriality)
$0\to X\to Z\to Y\to 0$, thus we have the long Hom sequences
$$0\to\operatorname{Hom}(Z,X)\to\operatorname{Hom}(Z,Z)\to\operatorname{Hom}(Z,Y)\to\operatorname{Ext}^1(Z,X)\to\operatorname{Ext}^1(Z,Z)\to\operatorname{Ext}^1(Z,Y)\to 0,$$
$$0\to\operatorname{Hom}(Y,Y)\to\operatorname{Hom}(Z,Y)\to\operatorname{Hom}(X,Y)\to\operatorname{Ext}^1(Y,Y)\to\operatorname{Ext}^1(Z,Y)\to\operatorname{Ext}^1(X,Y)\to 0.$$
Knowing that the modules are exceptional we obtain that $\operatorname{Ext}^1(Z,Y)=\operatorname{Ext}^1(X,Y)=0$. Using the previous proof we have $\operatorname{Hom}(X,Y)=0$, so we get $\langle\underline\dim X,\underline\dim Y\rangle=0$.

Dually we obtain that $i)\Leftrightarrow iv)$.
\end{proof}

Concerning the special Schofield sequences we can prove the following:
\begin{prop}
\label{prop:SchofieldSpecial}Let $Z$ be an exceptional indecomposable
of defect $-1$ (i.e. preprojective) with $\underline\dim Z>\delta$. Then
it has a unique special Schofield sequence which is of the form
\[
\xymatrix{0\ar[r] & (u+1)X\ar[r] & Z\ar[r] & uY\ar[r] & 0}
\]
 with $\partial X=-1$, $\underline\dim X+\underline\dim Y=\delta$ and thus $\underline\dim Z=u\delta +\underline\dim X$.

Dually, let $Z$ be an exceptional indecomposable of defect $1$ (i.e.
preinjective) with $\underline\dim Z>\delta$. Then it has a unique special
Schofield sequence which is of the form
\[
\xymatrix{0\ar[r] & vX\ar[r] & Z\ar[r] & (v+1)Y\ar[r] & 0}
\]
 with $\partial X=-1$, $\underline\dim X+\underline\dim Y=\delta$ and thus $\underline\dim Z=v\delta +\underline\dim Y$.
\end{prop}

\begin{proof}
For the existence, write $\underline\dim Z$ as $\underline\dim Z=u\delta+x$, where
$x<\delta$. Since $\left\langle x,x\right\rangle =\left\langle \underline\dim Z-u\delta,\underline\dim Z-u\delta\right\rangle =1$,
$x$ is a positive real root, so there exists a unique indecomposable $X$
such that $\underline\dim X=x$; moreover, since $\partial x=\left\langle \delta,x\right\rangle =\left\langle \delta,\underline\dim Z-u\delta\right\rangle =\partial Z=-1$
it follows that $X$ is an exceptional preprojective (of defect $-1$).
Since $\delta-x$ is also a positive real root, let $Y$ be the unique
indecomposable with $\underline\dim Y=\delta-x$. We have that $\partial Y=1$,
hence $Y$ is a preinjective indecomposable, thus exceptional. Of
course we have $\underline\dim X+\underline\dim Y=\delta$; moreover, $\left\langle x,\delta-x\right\rangle =-\partial x-1=1-1=0$.
Therefore, $\left\langle \underline\dim X,\underline\dim Y\right\rangle =0$ and $\underline\dim Z=(u+1)\cdot\underline\dim X+u\cdot\underline\dim Y$
and by the previous theorem we have the required short exact sequence.

For the uniqueness, if there is another sequence $0\to(u'+1)X'\to Z\to u'Y'\to0$
fulfilling the requirements, then $\underline\dim Z=u'\delta+\underline\dim X'$ with
$\underline\dim X'<\delta$, $\partial X'=-1$, so $u'=u$ and $\underline\dim X'=x=\underline\dim X$.
\end{proof}

We will describe now some non-special Schofield sequences associated to $Z$ of defect $\pm 1$ or 0.
\begin{prop} If $Z$ is a preprojective indecomposable of defect $-1$, $R$ is an exceptional regular module having regular top $R^Z_e(1)$ and $\underline\dim Z>\underline\dim R$, then $R$ is a non-special Schofield factor of $Z$. A dual statement can be formulated for the case when $Z$ is preinjective of defect 1.
\end{prop}
\begin{proof}
By Lemma \ref{lem:preproj} we obtain that there is a projection $Z\to R^Z_e(1)$ and $\dim_k\operatorname{Hom} (Z,R^Z_e(1))=\dim_k\operatorname{Hom} (Z,R)=1$. If $f:Z\to R$ is a nonzero morphism then by Lemma \ref{lem:preproj} $\operatorname{Im} f$ is regular exceptional. In case $\operatorname{Im} f\neq R$, then the regular composition series of $\operatorname{Im} f$ does not contain  $R^Z_e(1)$, so $\dim_k\operatorname{Hom}(Z,\operatorname{Im} f)=0$, a contradiction. Therefore, $f$ must be an epimorphism, but then $\partial\operatorname{Ker} f=-1$, so $\operatorname{Ker} f$ is a preprojective indecomposable of defect $-1$. Applying Corollary \ref{cor:Schofield conditions} iv) we are done.
\end{proof}
\begin{cor}\label{cor: defect 1}
\begin{enumerate}
    \item[i)] In case $\partial Z=-1$ and $\underline\dim Z>\delta$, all its non-special Schofield factors are the regular exceptionals having regular top $R^Z_e(1)$ for $e\in E$. A dual statement is true for $\partial Z=1$.
    \item[ii)]If Q is of $\widetilde{\mathbb{A}}_m$ type then the statement formulated in a) is valid for any indecomposable $Z$ of defect $\partial Z=\pm 1$.
\end{enumerate}
\end{cor}
\begin{proof} i) We know that we have $s(Z)-2=|Q_0|-2$ different non-special Schofield factors, since $\underline\dim Z>\delta$. But the number of the specified regular exceptionals matches this number (just look at the rank of the non-homogeneous tubes in all the cases, the exceptional regulars being all the non-homogeneous regular indecomposables of dimension vector less than $\delta$).

ii) Just note that in the $\widetilde{\mathbb{A}}_m$ case the absolute defect is at most 1, so a Schofield factor in case of $\partial Z=-1$ must be regular exceptional with top $R^Z_e(1)$.
\end{proof}

\begin{prop} \label{prop:regular} Suppose $Z$ is an exceptional regular from the non-homogeneous tube $\mathcal{T}_e$ and $P$ is an indecomposable preprojective of defect $-1$. Then $P$ is a non-special Schofield submodule in $Z$ iff the regular top of $Z$ is $R^P_e(1)$ and $\underline\dim P<\underline\dim Z$.
\end{prop}

\begin{proof} Using Lemma \ref{lem:preinjextension} the necessity follows. For the sufficiency suppose the regular top of $Z$ is $R^P_e(1)$ and $\underline\dim P<\underline\dim Z$. Then $\dim_k\operatorname{Hom} (P,Z)=1$ and if $Z'\subset Z$ is the regular radical of $Z$ (i.e. $Z/Z'=R^P_e(1)$) we have that $\operatorname{Hom}(P,Z')=0$. By Lemma \ref{lem:preproj} this means that nonzero morphisms from $\operatorname{Hom} (P,Z)$ are monomorphisms. So consider $f:P\to Z$ such a monomorphism. Then $Z/P$ has defect 1 and we show that we can't have regular components in it. Indeed the projection from $Z$ to such a component would have a proper regular exceptional kernel containing $\operatorname{Im} f=P$, which is impossible due to $\operatorname{Hom}(P,Z')=0$. So $Z/P$ is preinjective indecomposable and we are done.
\end{proof}
\begin{cor} \label{cor:regular} In the $\widetilde{\mathbb{A}}_m$ case consider the regular exceptional $R_e(t)$ with regular top $R_0$. Then its	 non-special Schofield submodules  are the regulars $R_e(t')$ with $t'<t$ and the indecomposable preprojectives $P$ of defect $-1$, having dimension vector less than $\underline\dim R_e(t)$ and satisfying $R^P_e(1)=R_0$ (i.e. $\dim_k\operatorname{Hom}(P,R_0)=1$).
\end{cor}

By Corollaries \ref{cor: defect 1} and \ref{cor:regular} we already know all the Schofield pairs associated to exceptional modules in the $\widetilde{\mathbb{A}}_m$ case.

In the other tame cases (when the quiver is a tree) we will use reflections and AR-translations in order to obtain all the Schofield pairs over any orientation.

The following proposition is the result of the general compatibility of exact sequences with reflections.
\begin{prop}
\label{prop:Reflection} Consider the sink $i$, its reflection functor $S^+_i$ and suppose $X$ is an indecomposable module which is not the projective simple corresponding to the sink $i$. If $0\to uX\to Z\to vY\to 0$ is a Schofield sequence, then so is $0\to uS^+_iX\to S^+_iZ\to vS^+_iY\to 0$. The assertion remains valid also for the functor $S^-_i$, in the case $i$ is a source and $Y$ is not the injective simple corresponding to $i$. Moreover, we obtain the same result for the AR-translations $\tau$ (in the case $X$ is not an indecomposable projective) and $\tau^{-1}$ (in the case $Y$ is not an indecomposable injective)
\end{prop}

As we can see from the proposition above a Schofield sequence $0\to uX\to Z\to vY\to 0$ vanishes under the reflection $S^+_i$ in the case $X$ is the projective simple corresponding to the sink $i$. This might suggest that the set of all the Schofield pairs associated to $Z$ is not compatible with reflections. However, for non-special Schofield pairs the contrary is true.
\begin{prop}
\label{prop:ReflectionSupport} Consider the sink $i$ and its reflection functor $S^+_i$. If $\mathcal{M}=\{(Y_j,X_j)|j\in J\}$ is the set of all non-special Schofield pairs associated to $Z$, then $S^+_i\mathcal{M}=\{(S^+_iY_j,S^+_iX_j)|j\in J, S^+_iX_j\neq 0\}$ is the set of all non-special Schofield pairs associated to $S^+_iZ$. Dually if $i$ is a source and $\mathcal{M}=\{(Y_j,X_j)|j\in J\}$ is the set of all non-special Schofield pairs associated to $Z$, then $S^-_i\mathcal{M}=\{(S^-_iY_j,S^-_iX_j)|j\in J, S^-_iY_j\neq 0\}$ is the set of all non-special Schofield pairs associated to $S^-_iZ$.
\end{prop}
\begin{proof} Suppose $S^+_iX_j=0$ for some $j$. It follows that $X_j=S(i)$ is the projective simple corresponding to the sink $i$ and, moreover, due to Proposition \ref{prop:SchofieldCharaterization} we have a single Schofield pair with this second term.
Suppose this pair $(Y_j,X_j)$ with $X_j=S(i)$ is not special. Since $\langle\underline\dim Y_j,\underline\dim S(i)\rangle=-1$, we have $(\underline\dim Y_j)_i-\sum_{t}(\underline\dim Y_j)_t=-1$ (where $t$ are the neighbours of $i$ in the quiver), so $(\underline\dim S^+_iZ)_i=-(\underline\dim Z)_i+\sum_{t}(\underline\dim Z)_t=0<(\underline\dim Z)_i$ and $(\underline\dim S^+_iZ)_l=(\underline\dim Z)_l$ (for $l\neq i$). This means that $s(S^+_iZ)=s(Z)-1$, so the number of Schofield pairs associated to $S^+_iZ$ (which equals $s(S^+_iZ)-1$ by Theorem \ref{thm:Schofield}) is one less than the number of Schofield pairs associated to $Z$ (which equals $s(Z)-1$).
\end{proof}

\begin{thm}
\label{thm:SchofieldGeneralCanonical}In case of a canonically oriented
tame tree quiver, for a given exceptional module $Z$ all its Schofield pairs are listed in the Appendix (Section \ref{sec:Appendix}).
\end{thm}

\begin{proof}
Let $Q$ be a canonically oriented Euclidean quiver (the canonical
orientations correspond to those given in \cite{Skow2} and are also
shown in this document). We determine the dimension vectors of all
exceptional modules, falling below (at least) $3\delta$. Say $M$
is such an exceptional module, which is not simple. Then, we perform
an ``exhaustive search'' for non-special Schofield pairs $(Y,X)$
using the criterion $\left\langle \underline\dim X,\underline\dim Y\right\rangle =0$
from Proposition \ref{prop:SchofieldCharaterization}, finding in this way
exactly $s(Z)-1$ pairs (if $|\partial Z|>1$
or $|\partial Z|=1$ and $\underline\dim Z<\delta$), respectively $s(Z)-2$
pairs (if $|\partial Z|=1$ and $\dim Z>\delta$).
But this is exactly the number of different Schofield pairs possible
under Theorem \ref{thm:Schofield}, with the additional information
on special Schofield pairs provided by Proposition \ref{prop:SchofieldSpecial}. The cases $\underline\dim Z>3\delta$ will easily follow using AR-translations and Proposition \ref{prop:Reflection}.
\end{proof}
The computations required in the previous proof were carried out using
the computer algebra system GAP (see \cite{GAP}) but any other computer
algebra system, standalone implementation in any programming language,
or considerable amounts of paper, spare time and patience may be used
to verify the results, since only vector and matrix operations are
needed, over small integers.

\medskip

We are ready now to describe the procedure of obtaining all the Schofield pairs of any exceptional module $Z$ over any tame quiver not of type $\widetilde{\mathbb{A}}_m$ (this type being already discussed).

Consider the tame tree quiver $Q$ and denote by $Q'$ the canonically oriented quiver having the same underlying graph.

{\bf Case 1} If $Z$ is preinjective or regular.

We know by Lemma \ref{lem:reflections} that there exists a sequence $i_1,\dots,i_t$ of vertices in $Q$ such that for each $s\in\{1,\dots,t\}$ the vertex $i_s$ is a sink in $\sigma_{i_{s-1}}\dots\sigma_{i_1}Q$ and $\sigma_{i_{t}}\dots\sigma_{i_1}Q=Q'$. We  perform the following steps:
\begin{description}
\item[{Step 1.}]  We fix the special Schofield pair associated to $Z$ (if $\underline\dim Z>\delta$) using Proposition \ref{prop:SchofieldSpecial}.
\item[{Step 2.}] To determine the non-special Schofield pairs associated to $Z$ consider the reflected exceptional module
$S^+_{i_t}\dots S^+_{i_1} Z$ in the canonically oriented quiver $Q'=\sigma_{i_{t}}\dots\sigma_{i_1}Q$. Note that since $Z$ is not preprojective $S^+_{i_t}\dots S^+_{i_1} Z\neq 0$.
\item[{Step 3.}] Using Theorem \ref{thm:SchofieldGeneralCanonical} and the Appendix we determine the non-special Schofield pairs of $S^+_{i_t}\dots S^+_{i_1} Z$ in $Q'$.
\item[{Step 4.}] Reflect back these Schofield pairs using the functor $S^-_{i_1}\dots S^-_{i_t}$. By Proposition \ref{prop:ReflectionSupport} these pairs will be all the non-special Schofield pairs of $Z$.
\end{description}

{\bf Case 2} If $Z$ is preprojective.

We know by Lemma \ref{lem:reflections} that there exists a sequence $i_1,\dots,i_t$ of vertices in $Q$ such that for each $s\in\{1,\dots,t\}$ the vertex $i_s$ is a source in $\sigma_{i_{s-1}}\dots\sigma_{i_1}Q$ and $\sigma_{i_{t}}\dots\sigma_{i_1}Q=Q'$. We  perform the following steps:
\begin{description}
\item[{Step 1.}]  We fix the special Schofield pair associated to $Z$ (if $\underline\dim Z>\delta$) using Proposition \ref{prop:SchofieldSpecial}.
\item[{Step 2.}] To determine the non-special Schofield pairs associated to $Z$ consider the reflected exceptional module
$S^-_{i_t}\dots S^-_{i_1} Z$ in the canonically oriented quiver $Q'=\sigma_{i_{t}}\dots\sigma_{i_1}Q$. Note that since $Z$ is not preinjective $S^-_{i_t}\dots S^-_{i_1} Z\neq 0$.
\item[{Step 3.}] Using Theorem \ref{thm:SchofieldGeneralCanonical} and the Appendix we determine the non-special Schofield pairs of $S^-_{i_t}\dots S^-_{i_1} Z$ in $Q'$.
\item[{Step 4.}] Reflect back these Schofield pairs using the functor $S^+_{i_1}\dots S^+_{i_t}$. By Proposition \ref{prop:ReflectionSupport} these pairs will be all the non-special Schofield pairs of $Z$.
\end{description}

\subsection*{Acknowledgment}

We are extremely grateful for the thorough and professional report
of the anonymous referee and for the advices which made the presentation
more precise.

\subsection[Schofield pairs for the quiver $\Delta(\widetilde{\mathbb{A}}_{1,2})$]{Schofield pairs for the quiver $\Delta(\widetilde{\mathbb{A}}_{1,2})$ -- {\usebox{\boxAlAA}}}
\[
\vcenter{\hbox{\xymatrix{1 &   & 3\ar[ll]\ar[dl]\\
  & 2\ar[ul] &  }}}\qquad C_{\Delta(\widetilde{\mathbb{A}}_{1,2})} = \begin{bmatrix}1 & 1 & 2\\
0 & 1 & 1\\
0 & 0 & 1\end{bmatrix}\quad\Phi_{\Delta(\widetilde{\mathbb{A}}_{1,2})} = \begin{bmatrix}-1 & 1 & 1\\
-1 & 0 & 2\\
-2 & 1 & 2\end{bmatrix}
\]
\subsubsection{Schofield pairs associated to preprojective exceptional modules}

\begin{figure}[ht]

\begin{center}
\begin{scaletikzpicturetowidth}{\textwidth}
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	\draw[gray,dashed,->] (P_1_1__Rationals) -- (P_2_1__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
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	\draw[blue,->] (P_1_3__Rationals) -- (P_2_1__Rationals);
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	\draw[blue,->] (P_2_1__Rationals) to [out=135,in=-135] (P_2_2__Rationals);
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	\draw[blue,->] (P_2_2__Rationals) -- (P_2_3__Rationals);
	\draw[gray,dashed,->] (P_2_1__Rationals) -- (P_3_1__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[blue,->] (P_2_2__Rationals) -- (P_3_1__Rationals);
	\draw[blue,->] (P_2_3__Rationals) -- (P_3_1__Rationals);
	\draw[gray,dashed,->] (P_2_2__Rationals) -- (P_3_2__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[blue,->] (P_2_3__Rationals) -- (P_3_2__Rationals);
	\draw[blue,->] (P_3_1__Rationals) to [out=135,in=-135] (P_3_2__Rationals);
	\draw[gray,dashed,->] (P_2_3__Rationals) -- (P_3_3__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[blue,->] (P_3_1__Rationals) -- (P_3_3__Rationals);
	\draw[blue,->] (P_3_2__Rationals) -- (P_3_3__Rationals);
\end{tikzpicture}
\end{scaletikzpicturetowidth}
\end{center}

\end{figure}
\medskip{}
\subsubsection*{Modules of the form $P(n,1)$}

Defect: $\partial P(n,1) = -1$, for $n\ge 0$.

\begin{fleqn}
\begin{align*}
P(0,1):\ & - \\
P(1,1):\ & \bigl(R_{0}^{1}(1),P(0,3)\bigr),\ \bigl(I(0,3),2P(0,2)\bigr)\\
P(n,1):\ & \bigl(R_{0}^{(n-1)\bmod 2+1}(1),P(n-1,3)\bigr),\ \bigl(uI,(u+1)P\bigr),\ n>1\\
\end{align*}
\end{fleqn}
\subsubsection*{Modules of the form $P(n,2)$}

Defect: $\partial P(n,2) = -1$, for $n\ge 0$.

\begin{fleqn}
\begin{align*}
P(0,2):\ & \bigl(R_{0}^{1}(1),P(0,1)\bigr)\\
P(1,2):\ & \bigl(R_{0}^{2}(1),P(1,1)\bigr),\ \bigl(2I(0,2),3P(0,1)\bigr)\\
P(n,2):\ & \bigl(R_{0}^{n\bmod 2+1}(1),P(n,1)\bigr),\ \bigl(uI,(u+1)P\bigr),\ n>1\\
\end{align*}
\end{fleqn}
\subsubsection*{Modules of the form $P(n,3)$}

Defect: $\partial P(n,3) = -1$, for $n\ge 0$.

\begin{fleqn}
\begin{align*}
P(0,3):\ & \bigl(R_{0}^{2}(1),P(0,2)\bigr),\ \bigl(I(0,2),2P(0,1)\bigr)\\
P(n,3):\ & \bigl(R_{0}^{(n+1)\bmod 2+1}(1),P(n,2)\bigr),\ \bigl(uI,(u+1)P\bigr),\ n>0\\
\end{align*}
\end{fleqn}
\subsubsection{Schofield pairs associated to preinjective exceptional modules}

\begin{figure}[ht]

\begin{center}
\begin{scaletikzpicturetowidth}{\textwidth}
\begin{tikzpicture}[scale=\tikzscale]
	\node[seloneAlAA, label={[shift={(0.4,-0.2)}, brown]\tiny $I(0,1)$}] (I_0_1__Rationals) at (0,0) {$\begin{smallmatrix}1&&2\\&1\end{smallmatrix}$};
	\node[noselAlAA, label={[shift={(0.4,-0.2)}, brown]\tiny $I(0,2)$}] (I_0_2__Rationals) at (0,-2) {$\begin{smallmatrix}0&&1\\&1\end{smallmatrix}$};
	\node[noselAlAA, label={[shift={(0.4,-0.2)}, brown]\tiny $I(0,3)$}] (I_0_3__Rationals) at (0,-1) {$\begin{smallmatrix}0&&1\\&0\end{smallmatrix}$};
	\node[noselAlAA, label={[shift={(0.4,-0.2)}, brown]\tiny $I(1,1)$}] (I_1_1__Rationals) at (-4,0) {$\begin{smallmatrix}2&&3\\&3\end{smallmatrix}$};
	\node[seloneAlAA, label={[shift={(0.4,-0.2)}, brown]\tiny $I(1,2)$}] (I_1_2__Rationals) at (-4,-2) {$\begin{smallmatrix}2&&3\\&2\end{smallmatrix}$};
	\node[seloneAlAA, label={[shift={(0.4,-0.2)}, brown]\tiny $I(1,3)$}] (I_1_3__Rationals) at (-4,-1) {$\begin{smallmatrix}1&&2\\&2\end{smallmatrix}$};
	\node[noselAlAA, label={[shift={(0.4,-0.2)}, brown]\tiny $I(2,1)$}] (I_2_1__Rationals) at (-8,0) {$\begin{smallmatrix}4&&5\\&4\end{smallmatrix}$};
	\node[noselAlAA, label={[shift={(0.4,-0.2)}, brown]\tiny $I(2,2)$}] (I_2_2__Rationals) at (-8,-2) {$\begin{smallmatrix}3&&4\\&4\end{smallmatrix}$};
	\node[noselAlAA, label={[shift={(0.4,-0.2)}, brown]\tiny $I(2,3)$}] (I_2_3__Rationals) at (-8,-1) {$\begin{smallmatrix}3&&4\\&3\end{smallmatrix}$};
	\node[noselAlAA, label={[shift={(0.4,-0.2)}, brown]\tiny $I(3,1)$}] (I_3_1__Rationals) at (-12,0) {$\begin{smallmatrix}5&&6\\&6\end{smallmatrix}$};
	\node[noselAlAA, label={[shift={(0.4,-0.2)}, brown]\tiny $I(3,2)$}] (I_3_2__Rationals) at (-12,-2) {$\begin{smallmatrix}5&&6\\&5\end{smallmatrix}$};
	\node[noselAlAA, label={[shift={(0.4,-0.2)}, brown]\tiny $I(3,3)$}] (I_3_3__Rationals) at (-12,-1) {$\begin{smallmatrix}4&&5\\&5\end{smallmatrix}$};
	\node at (-13,-1.) {$\dots$};
	\draw[red,->] (I_1_2__Rationals) -- (I_0_1__Rationals);
	\draw[red,->] (I_1_3__Rationals) -- (I_0_1__Rationals);
	\draw[red,->] (I_0_1__Rationals) to [out=135,in=-135] (I_0_2__Rationals);
	\draw[red,->] (I_1_3__Rationals) -- (I_0_2__Rationals);
	\draw[red,->] (I_0_1__Rationals) -- (I_0_3__Rationals);
	\draw[red,->] (I_0_2__Rationals) -- (I_0_3__Rationals);
	\draw[gray,dashed,->] (I_0_1__Rationals) -- (I_1_1__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_2_2__Rationals) -- (I_1_1__Rationals);
	\draw[red,->] (I_2_3__Rationals) -- (I_1_1__Rationals);
	\draw[gray,dashed,->] (I_0_2__Rationals) -- (I_1_2__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_1_1__Rationals) to [out=135,in=-135] (I_1_2__Rationals);
	\draw[red,->] (I_2_3__Rationals) -- (I_1_2__Rationals);
	\draw[gray,dashed,->] (I_0_3__Rationals) -- (I_1_3__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_1_1__Rationals) -- (I_1_3__Rationals);
	\draw[red,->] (I_1_2__Rationals) -- (I_1_3__Rationals);
	\draw[gray,dashed,->] (I_1_1__Rationals) -- (I_2_1__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_3_2__Rationals) -- (I_2_1__Rationals);
	\draw[red,->] (I_3_3__Rationals) -- (I_2_1__Rationals);
	\draw[gray,dashed,->] (I_1_2__Rationals) -- (I_2_2__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_2_1__Rationals) to [out=135,in=-135] (I_2_2__Rationals);
	\draw[red,->] (I_3_3__Rationals) -- (I_2_2__Rationals);
	\draw[gray,dashed,->] (I_1_3__Rationals) -- (I_2_3__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_2_1__Rationals) -- (I_2_3__Rationals);
	\draw[red,->] (I_2_2__Rationals) -- (I_2_3__Rationals);
	\draw[gray,dashed,->] (I_2_1__Rationals) -- (I_3_1__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[gray,dashed,->] (I_2_2__Rationals) -- (I_3_2__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_3_1__Rationals) to [out=135,in=-135] (I_3_2__Rationals);
	\draw[gray,dashed,->] (I_2_3__Rationals) -- (I_3_3__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_3_1__Rationals) -- (I_3_3__Rationals);
	\draw[red,->] (I_3_2__Rationals) -- (I_3_3__Rationals);
\end{tikzpicture}
\end{scaletikzpicturetowidth}
\end{center}

\end{figure}
\medskip{}
\subsubsection*{Modules of the form $I(n,1)$}

Defect: $\partial I(n,1) = 1$, for $n\ge 0$.

\begin{fleqn}
\begin{align*}
I(0,1):\ & \bigl(I(0,2),R_{0}^{2}(1)\bigr),\ \bigl(2I(0,3),P(0,2)\bigr)\\
I(n,1):\ & \bigl(I(n,2),R_{0}^{(-n+1)\bmod 2+1}(1)\bigr),\ \bigl((v+1)I,vP\bigr),\ n>0\\
\end{align*}
\end{fleqn}
\subsubsection*{Modules of the form $I(n,2)$}

Defect: $\partial I(n,2) = 1$, for $n\ge 0$.

\begin{fleqn}
\begin{align*}
I(0,2):\ & \bigl(I(0,3),R_{0}^{1}(1)\bigr)\\
I(1,2):\ & \bigl(I(1,3),R_{0}^{2}(1)\bigr),\ \bigl(3I(0,3),2P(0,2)\bigr)\\
I(n,2):\ & \bigl(I(n,3),R_{0}^{(-n+2)\bmod 2+1}(1)\bigr),\ \bigl((v+1)I,vP\bigr),\ n>1\\
\end{align*}
\end{fleqn}
\subsubsection*{Modules of the form $I(n,3)$}

Defect: $\partial I(n,3) = 1$, for $n\ge 0$.

\begin{fleqn}
\begin{align*}
I(0,3):\ & - \\
I(1,3):\ & \bigl(I(0,1),R_{0}^{1}(1)\bigr),\ \bigl(2I(0,2),P(0,1)\bigr)\\
I(n,3):\ & \bigl(I(n-1,1),R_{0}^{(-n+1)\bmod 2+1}(1)\bigr),\ \bigl((v+1)I,vP\bigr),\ n>1\\
\end{align*}
\end{fleqn}
\subsubsection{Schofield pairs associated to regular exceptional modules}

\subsubsection*{The non-homogeneous tube $\mathcal{T}_{0}^{\Delta(\widetilde{\mathbb{A}}_{1,2})}$}

\begin{figure}[ht]

\begin{center}
\captionof{figure}{\vspace*{-10pt}$\mathcal{T}_{0}^{\Delta(\widetilde{\mathbb{A}}_{1,2})}$}
\begin{scaletikzpicturetowidth}{0.40000000000000002\textwidth}
\begin{tikzpicture}[scale=\tikzscale]
	\node[seltwoAlAA, label={[shift={(0.0,-0.45)}, brown]\tiny $R_{0}^{1}(1)$}] (RN_1_1_1__Rationals) at (4,0) {$\begin{smallmatrix}0&&0\\&1\end{smallmatrix}$};
	\node[seltwoAlAA, label={[shift={(0.0,-0.45)}, brown]\tiny $R_{0}^{2}(1)$}] (RN_1_2_1__Rationals) at (8,0) {$\begin{smallmatrix}1&&1\\&0\end{smallmatrix}$};
	\node[seltwoAlAA, label={[shift={(0.0,-0.45)}, brown]\tiny $R_{0}^{1}(1)$}] (RN_1_1_1__Rationals_repeated) at (12,0) {$\begin{smallmatrix}0&&0\\&1\end{smallmatrix}$};
	\node[noselAlAA, label={[shift={(0.0,-0.45)}, brown]\tiny $R_{0}^{1}(2)$}] (RN_1_1_2__Rationals) at (6,2) {$\begin{smallmatrix}1&&1\\&1\end{smallmatrix}$};
	\node[noselAlAA, label={[shift={(0.0,-0.45)}, brown]\tiny $R_{0}^{2}(2)$}] (RN_1_2_2__Rationals) at (10,2) {$\begin{smallmatrix}1&&1\\&1\end{smallmatrix}$};
	\node[noselAlAA, label={[shift={(0.0,-0.45)}, brown]\tiny $R_{0}^{2}(3)$}] (RN_1_2_3__Rationals) at (4,4) {$\begin{smallmatrix}2&&2\\&1\end{smallmatrix}$};
	\node[noselAlAA, label={[shift={(0.0,-0.45)}, brown]\tiny $R_{0}^{1}(3)$}] (RN_1_1_3__Rationals) at (8,4) {$\begin{smallmatrix}1&&1\\&2\end{smallmatrix}$};
	\node[noselAlAA, label={[shift={(0.0,-0.45)}, brown]\tiny $R_{0}^{2}(3)$}] (RN_1_2_3__Rationals_repeated) at (12,4) {$\begin{smallmatrix}2&&2\\&1\end{smallmatrix}$};
	\draw[blue,->] (RN_1_1_1__Rationals) -- (RN_1_1_2__Rationals);
	\draw[gray,dashed,->] (RN_1_1_1__Rationals) -- (RN_1_2_1__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[red,->] (RN_1_1_2__Rationals) -- (RN_1_2_1__Rationals);
	\draw[gray,dashed,->] (RN_1_1_2__Rationals) -- (RN_1_2_2__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[blue,->] (RN_1_1_2__Rationals) -- (RN_1_1_3__Rationals);
	\draw[gray,dashed,->] (RN_1_2_1__Rationals) -- (RN_1_1_1__Rationals_repeated) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[blue,->] (RN_1_2_1__Rationals) -- (RN_1_2_2__Rationals);
	\draw[red,->] (RN_1_2_2__Rationals) -- (RN_1_1_1__Rationals_repeated);
	\draw[gray,dashed,->,out=219,in=-39] (RN_1_2_2__Rationals) to node[midway,sloped,above]{{\tiny $\tau^{-1}$}} (RN_1_1_2__Rationals);
	\draw[blue,->] (RN_1_2_2__Rationals) -- (RN_1_2_3__Rationals_repeated);
	\draw[red,->] (RN_1_2_3__Rationals) -- (RN_1_1_2__Rationals);
	\draw[gray,dashed,->] (RN_1_2_3__Rationals) -- (RN_1_1_3__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[red,->] (RN_1_1_3__Rationals) -- (RN_1_2_2__Rationals);
	\draw[gray,dashed,->] (RN_1_1_3__Rationals) -- (RN_1_2_3__Rationals_repeated) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[dashed,-] (RN_1_1_1__Rationals) -- (RN_1_2_3__Rationals);
	\draw[dashed,-] (RN_1_1_1__Rationals_repeated) -- (RN_1_2_3__Rationals_repeated);
\end{tikzpicture}
\end{scaletikzpicturetowidth}
\end{center}

\end{figure}
\begin{fleqn}
\begin{align*}
R_{0}^{1}(1):\ & - \\
R_{0}^{2}(1):\ & \bigl(I(0,3),P(0,1)\bigr)\\
\end{align*}
\end{fleqn}
\clearpage

\subsection[Schofield pairs for the quiver $\Delta(\widetilde{\mathbb{A}}_{1,3})$]{Schofield pairs for the quiver $\Delta(\widetilde{\mathbb{A}}_{1,3})$ -- {\usebox{\boxAlAAA}}}
\[
\vcenter{\hbox{\xymatrix{1 &   &   & 4\ar[lll]\ar[dl]\\
  & 2\ar[ul] & 3\ar[l] &  }}}\qquad C_{\Delta(\widetilde{\mathbb{A}}_{1,3})} = \begin{bmatrix}1 & 1 & 1 & 2\\
0 & 1 & 1 & 1\\
0 & 0 & 1 & 1\\
0 & 0 & 0 & 1\end{bmatrix}\quad\Phi_{\Delta(\widetilde{\mathbb{A}}_{1,3})} = \begin{bmatrix}-1 & 1 & 0 & 1\\
-1 & 0 & 1 & 1\\
-1 & 0 & 0 & 2\\
-2 & 1 & 0 & 2\end{bmatrix}
\]
\subsubsection{Schofield pairs associated to preprojective exceptional modules}

\begin{figure}[ht]

\begin{center}
\begin{scaletikzpicturetowidth}{\textwidth}
\begin{tikzpicture}[scale=\tikzscale]
	\node[noselAlAAA, label={[shift={(0.4,-0.2)}, brown]\tiny $P(0,1)$}] (P_0_1__Rationals) at (0,0) {$\begin{smallmatrix}1&&&0\\&0&0\end{smallmatrix}$};
	\node[noselAlAAA, label={[shift={(0.4,-0.2)}, brown]\tiny $P(0,2)$}] (P_0_2__Rationals) at (0,-3) {$\begin{smallmatrix}1&&&0\\&1&0\end{smallmatrix}$};
	\node[noselAlAAA, label={[shift={(0.4,-0.2)}, brown]\tiny $P(0,3)$}] (P_0_3__Rationals) at (0,-2) {$\begin{smallmatrix}1&&&0\\&1&1\end{smallmatrix}$};
	\node[seloneAlAAA, label={[shift={(0.4,-0.2)}, brown]\tiny $P(0,4)$}] (P_0_4__Rationals) at (0,-1) {$\begin{smallmatrix}2&&&1\\&1&1\end{smallmatrix}$};
	\node[seloneAlAAA, label={[shift={(0.4,-0.2)}, brown]\tiny $P(1,1)$}] (P_1_1__Rationals) at (4,0) {$\begin{smallmatrix}2&&&1\\&2&1\end{smallmatrix}$};
	\node[seloneAlAAA, label={[shift={(0.4,-0.2)}, brown]\tiny $P(1,2)$}] (P_1_2__Rationals) at (4,-3) {$\begin{smallmatrix}2&&&1\\&2&2\end{smallmatrix}$};
	\node[seloneAlAAA, label={[shift={(0.4,-0.2)}, brown]\tiny $P(1,3)$}] (P_1_3__Rationals) at (4,-2) {$\begin{smallmatrix}3&&&2\\&2&2\end{smallmatrix}$};
	\node[noselAlAAA, label={[shift={(0.4,-0.2)}, brown]\tiny $P(1,4)$}] (P_1_4__Rationals) at (4,-1) {$\begin{smallmatrix}3&&&2\\&3&2\end{smallmatrix}$};
	\node[noselAlAAA, label={[shift={(0.4,-0.2)}, brown]\tiny $P(2,1)$}] (P_2_1__Rationals) at (8,0) {$\begin{smallmatrix}3&&&2\\&3&3\end{smallmatrix}$};
	\node[noselAlAAA, label={[shift={(0.4,-0.2)}, brown]\tiny $P(2,2)$}] (P_2_2__Rationals) at (8,-3) {$\begin{smallmatrix}4&&&3\\&3&3\end{smallmatrix}$};
	\node[noselAlAAA, label={[shift={(0.4,-0.2)}, brown]\tiny $P(2,3)$}] (P_2_3__Rationals) at (8,-2) {$\begin{smallmatrix}4&&&3\\&4&3\end{smallmatrix}$};
	\node[noselAlAAA, label={[shift={(0.4,-0.2)}, brown]\tiny $P(2,4)$}] (P_2_4__Rationals) at (8,-1) {$\begin{smallmatrix}4&&&3\\&4&4\end{smallmatrix}$};
	\node[noselAlAAA, label={[shift={(0.4,-0.2)}, brown]\tiny $P(3,1)$}] (P_3_1__Rationals) at (12,0) {$\begin{smallmatrix}5&&&4\\&4&4\end{smallmatrix}$};
	\node[noselAlAAA, label={[shift={(0.4,-0.2)}, brown]\tiny $P(3,2)$}] (P_3_2__Rationals) at (12,-3) {$\begin{smallmatrix}5&&&4\\&5&4\end{smallmatrix}$};
	\node[noselAlAAA, label={[shift={(0.4,-0.2)}, brown]\tiny $P(3,3)$}] (P_3_3__Rationals) at (12,-2) {$\begin{smallmatrix}5&&&4\\&5&5\end{smallmatrix}$};
	\node[noselAlAAA, label={[shift={(0.4,-0.2)}, brown]\tiny $P(3,4)$}] (P_3_4__Rationals) at (12,-1) {$\begin{smallmatrix}6&&&5\\&5&5\end{smallmatrix}$};
	\node at (13,-1.5) {$\dots$};
	\draw[blue,->] (P_0_1__Rationals) to [out=135,in=-135] (P_0_2__Rationals);
	\draw[blue,->] (P_0_2__Rationals) -- (P_0_3__Rationals);
	\draw[blue,->] (P_0_1__Rationals) -- (P_0_4__Rationals);
	\draw[blue,->] (P_0_3__Rationals) -- (P_0_4__Rationals);
	\draw[gray,dashed,->] (P_0_1__Rationals) -- (P_1_1__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[blue,->] (P_0_2__Rationals) -- (P_1_1__Rationals);
	\draw[blue,->] (P_0_4__Rationals) -- (P_1_1__Rationals);
	\draw[gray,dashed,->] (P_0_2__Rationals) -- (P_1_2__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[blue,->] (P_0_3__Rationals) -- (P_1_2__Rationals);
	\draw[blue,->] (P_1_1__Rationals) to [out=135,in=-135] (P_1_2__Rationals);
	\draw[gray,dashed,->] (P_0_3__Rationals) -- (P_1_3__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[blue,->] (P_0_4__Rationals) -- (P_1_3__Rationals);
	\draw[blue,->] (P_1_2__Rationals) -- (P_1_3__Rationals);
	\draw[gray,dashed,->] (P_0_4__Rationals) -- (P_1_4__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[blue,->] (P_1_1__Rationals) -- (P_1_4__Rationals);
	\draw[blue,->] (P_1_3__Rationals) -- (P_1_4__Rationals);
	\draw[gray,dashed,->] (P_1_1__Rationals) -- (P_2_1__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[blue,->] (P_1_2__Rationals) -- (P_2_1__Rationals);
	\draw[blue,->] (P_1_4__Rationals) -- (P_2_1__Rationals);
	\draw[gray,dashed,->] (P_1_2__Rationals) -- (P_2_2__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[blue,->] (P_1_3__Rationals) -- (P_2_2__Rationals);
	\draw[blue,->] (P_2_1__Rationals) to [out=135,in=-135] (P_2_2__Rationals);
	\draw[gray,dashed,->] (P_1_3__Rationals) -- (P_2_3__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[blue,->] (P_1_4__Rationals) -- (P_2_3__Rationals);
	\draw[blue,->] (P_2_2__Rationals) -- (P_2_3__Rationals);
	\draw[gray,dashed,->] (P_1_4__Rationals) -- (P_2_4__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[blue,->] (P_2_1__Rationals) -- (P_2_4__Rationals);
	\draw[blue,->] (P_2_3__Rationals) -- (P_2_4__Rationals);
	\draw[gray,dashed,->] (P_2_1__Rationals) -- (P_3_1__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[blue,->] (P_2_2__Rationals) -- (P_3_1__Rationals);
	\draw[blue,->] (P_2_4__Rationals) -- (P_3_1__Rationals);
	\draw[gray,dashed,->] (P_2_2__Rationals) -- (P_3_2__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[blue,->] (P_2_3__Rationals) -- (P_3_2__Rationals);
	\draw[blue,->] (P_3_1__Rationals) to [out=135,in=-135] (P_3_2__Rationals);
	\draw[gray,dashed,->] (P_2_3__Rationals) -- (P_3_3__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[blue,->] (P_2_4__Rationals) -- (P_3_3__Rationals);
	\draw[blue,->] (P_3_2__Rationals) -- (P_3_3__Rationals);
	\draw[gray,dashed,->] (P_2_4__Rationals) -- (P_3_4__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[blue,->] (P_3_1__Rationals) -- (P_3_4__Rationals);
	\draw[blue,->] (P_3_3__Rationals) -- (P_3_4__Rationals);
\end{tikzpicture}
\end{scaletikzpicturetowidth}
\end{center}

\end{figure}
\medskip{}
\subsubsection*{Modules of the form $P(n,1)$}

Defect: $\partial P(n,1) = -1$, for $n\ge 0$.

\begin{fleqn}
\begin{align*}
P(0,1):\ & - \\
P(1,1):\ & \bigl(R_{0}^{2}(2),P(0,3)\bigr),\ \bigl(R_{0}^{3}(1),P(0,4)\bigr),\ \bigl(I(0,3),2P(0,2)\bigr)\\
P(n,1):\ & \bigl(R_{0}^{n\bmod 3+1}(2),P(n-1,3)\bigr),\ \bigl(R_{0}^{(n+1)\bmod 3+1}(1),P(n-1,4)\bigr),\ \bigl(uI,(u+1)P\bigr),\ n>1\\
\end{align*}
\end{fleqn}
\subsubsection*{Modules of the form $P(n,2)$}

Defect: $\partial P(n,2) = -1$, for $n\ge 0$.

\begin{fleqn}
\begin{align*}
P(0,2):\ & \bigl(R_{0}^{3}(1),P(0,1)\bigr)\\
P(1,2):\ & \bigl(R_{0}^{3}(2),P(0,4)\bigr),\ \bigl(R_{0}^{1}(1),P(1,1)\bigr),\ \bigl(I(0,4),2P(0,3)\bigr)\\
P(n,2):\ & \bigl(R_{0}^{(n+1)\bmod 3+1}(2),P(n-1,4)\bigr),\ \bigl(R_{0}^{(n-1)\bmod 3+1}(1),P(n,1)\bigr),\ \bigl(uI,(u+1)P\bigr),\ n>1\\
\end{align*}
\end{fleqn}
\subsubsection*{Modules of the form $P(n,3)$}

Defect: $\partial P(n,3) = -1$, for $n\ge 0$.

\begin{fleqn}
\begin{align*}
P(0,3):\ & \bigl(R_{0}^{3}(2),P(0,1)\bigr),\ \bigl(R_{0}^{1}(1),P(0,2)\bigr)\\
P(1,3):\ & \bigl(R_{0}^{1}(2),P(1,1)\bigr),\ \bigl(R_{0}^{2}(1),P(1,2)\bigr),\ \bigl(2I(0,2),3P(0,1)\bigr)\\
P(n,3):\ & \bigl(R_{0}^{(n-1)\bmod 3+1}(2),P(n,1)\bigr),\ \bigl(R_{0}^{n\bmod 3+1}(1),P(n,2)\bigr),\ \bigl(uI,(u+1)P\bigr),\ n>1\\
\end{align*}
\end{fleqn}
\subsubsection*{Modules of the form $P(n,4)$}

Defect: $\partial P(n,4) = -1$, for $n\ge 0$.

\begin{fleqn}
\begin{align*}
P(0,4):\ & \bigl(R_{0}^{1}(2),P(0,2)\bigr),\ \bigl(R_{0}^{2}(1),P(0,3)\bigr),\ \bigl(I(0,2),2P(0,1)\bigr)\\
P(n,4):\ & \bigl(R_{0}^{n\bmod 3+1}(2),P(n,2)\bigr),\ \bigl(R_{0}^{(n+1)\bmod 3+1}(1),P(n,3)\bigr),\ \bigl(uI,(u+1)P\bigr),\ n>0\\
\end{align*}
\end{fleqn}
\subsubsection{Schofield pairs associated to preinjective exceptional modules}

\begin{figure}[ht]

\begin{center}
\begin{scaletikzpicturetowidth}{\textwidth}
\begin{tikzpicture}[scale=\tikzscale]
	\node[seloneAlAAA, label={[shift={(0.4,-0.2)}, brown]\tiny $I(0,1)$}] (I_0_1__Rationals) at (0,0) {$\begin{smallmatrix}1&&&2\\&1&1\end{smallmatrix}$};
	\node[noselAlAAA, label={[shift={(0.4,-0.2)}, brown]\tiny $I(0,2)$}] (I_0_2__Rationals) at (0,-3) {$\begin{smallmatrix}0&&&1\\&1&1\end{smallmatrix}$};
	\node[noselAlAAA, label={[shift={(0.4,-0.2)}, brown]\tiny $I(0,3)$}] (I_0_3__Rationals) at (0,-2) {$\begin{smallmatrix}0&&&1\\&0&1\end{smallmatrix}$};
	\node[noselAlAAA, label={[shift={(0.4,-0.2)}, brown]\tiny $I(0,4)$}] (I_0_4__Rationals) at (0,-1) {$\begin{smallmatrix}0&&&1\\&0&0\end{smallmatrix}$};
	\node[noselAlAAA, label={[shift={(0.4,-0.2)}, brown]\tiny $I(1,1)$}] (I_1_1__Rationals) at (-4,0) {$\begin{smallmatrix}2&&&3\\&2&3\end{smallmatrix}$};
	\node[seloneAlAAA, label={[shift={(0.4,-0.2)}, brown]\tiny $I(1,2)$}] (I_1_2__Rationals) at (-4,-3) {$\begin{smallmatrix}2&&&3\\&2&2\end{smallmatrix}$};
	\node[seloneAlAAA, label={[shift={(0.4,-0.2)}, brown]\tiny $I(1,3)$}] (I_1_3__Rationals) at (-4,-2) {$\begin{smallmatrix}1&&&2\\&2&2\end{smallmatrix}$};
	\node[seloneAlAAA, label={[shift={(0.4,-0.2)}, brown]\tiny $I(1,4)$}] (I_1_4__Rationals) at (-4,-1) {$\begin{smallmatrix}1&&&2\\&1&2\end{smallmatrix}$};
	\node[noselAlAAA, label={[shift={(0.4,-0.2)}, brown]\tiny $I(2,1)$}] (I_2_1__Rationals) at (-8,0) {$\begin{smallmatrix}3&&&4\\&4&4\end{smallmatrix}$};
	\node[noselAlAAA, label={[shift={(0.4,-0.2)}, brown]\tiny $I(2,2)$}] (I_2_2__Rationals) at (-8,-3) {$\begin{smallmatrix}3&&&4\\&3&4\end{smallmatrix}$};
	\node[noselAlAAA, label={[shift={(0.4,-0.2)}, brown]\tiny $I(2,3)$}] (I_2_3__Rationals) at (-8,-2) {$\begin{smallmatrix}3&&&4\\&3&3\end{smallmatrix}$};
	\node[noselAlAAA, label={[shift={(0.4,-0.2)}, brown]\tiny $I(2,4)$}] (I_2_4__Rationals) at (-8,-1) {$\begin{smallmatrix}2&&&3\\&3&3\end{smallmatrix}$};
	\node[noselAlAAA, label={[shift={(0.4,-0.2)}, brown]\tiny $I(3,1)$}] (I_3_1__Rationals) at (-12,0) {$\begin{smallmatrix}5&&&6\\&5&5\end{smallmatrix}$};
	\node[noselAlAAA, label={[shift={(0.4,-0.2)}, brown]\tiny $I(3,2)$}] (I_3_2__Rationals) at (-12,-3) {$\begin{smallmatrix}4&&&5\\&5&5\end{smallmatrix}$};
	\node[noselAlAAA, label={[shift={(0.4,-0.2)}, brown]\tiny $I(3,3)$}] (I_3_3__Rationals) at (-12,-2) {$\begin{smallmatrix}4&&&5\\&4&5\end{smallmatrix}$};
	\node[noselAlAAA, label={[shift={(0.4,-0.2)}, brown]\tiny $I(3,4)$}] (I_3_4__Rationals) at (-12,-1) {$\begin{smallmatrix}4&&&5\\&4&4\end{smallmatrix}$};
	\node at (-13,-1.5) {$\dots$};
	\draw[red,->] (I_1_2__Rationals) -- (I_0_1__Rationals);
	\draw[red,->] (I_1_4__Rationals) -- (I_0_1__Rationals);
	\draw[red,->] (I_0_1__Rationals) to [out=135,in=-135] (I_0_2__Rationals);
	\draw[red,->] (I_1_3__Rationals) -- (I_0_2__Rationals);
	\draw[red,->] (I_0_2__Rationals) -- (I_0_3__Rationals);
	\draw[red,->] (I_1_4__Rationals) -- (I_0_3__Rationals);
	\draw[red,->] (I_0_1__Rationals) -- (I_0_4__Rationals);
	\draw[red,->] (I_0_3__Rationals) -- (I_0_4__Rationals);
	\draw[gray,dashed,->] (I_0_1__Rationals) -- (I_1_1__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_2_2__Rationals) -- (I_1_1__Rationals);
	\draw[red,->] (I_2_4__Rationals) -- (I_1_1__Rationals);
	\draw[gray,dashed,->] (I_0_2__Rationals) -- (I_1_2__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_1_1__Rationals) to [out=135,in=-135] (I_1_2__Rationals);
	\draw[red,->] (I_2_3__Rationals) -- (I_1_2__Rationals);
	\draw[gray,dashed,->] (I_0_3__Rationals) -- (I_1_3__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_1_2__Rationals) -- (I_1_3__Rationals);
	\draw[red,->] (I_2_4__Rationals) -- (I_1_3__Rationals);
	\draw[gray,dashed,->] (I_0_4__Rationals) -- (I_1_4__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_1_1__Rationals) -- (I_1_4__Rationals);
	\draw[red,->] (I_1_3__Rationals) -- (I_1_4__Rationals);
	\draw[gray,dashed,->] (I_1_1__Rationals) -- (I_2_1__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_3_2__Rationals) -- (I_2_1__Rationals);
	\draw[red,->] (I_3_4__Rationals) -- (I_2_1__Rationals);
	\draw[gray,dashed,->] (I_1_2__Rationals) -- (I_2_2__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_2_1__Rationals) to [out=135,in=-135] (I_2_2__Rationals);
	\draw[red,->] (I_3_3__Rationals) -- (I_2_2__Rationals);
	\draw[gray,dashed,->] (I_1_3__Rationals) -- (I_2_3__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_2_2__Rationals) -- (I_2_3__Rationals);
	\draw[red,->] (I_3_4__Rationals) -- (I_2_3__Rationals);
	\draw[gray,dashed,->] (I_1_4__Rationals) -- (I_2_4__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_2_1__Rationals) -- (I_2_4__Rationals);
	\draw[red,->] (I_2_3__Rationals) -- (I_2_4__Rationals);
	\draw[gray,dashed,->] (I_2_1__Rationals) -- (I_3_1__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[gray,dashed,->] (I_2_2__Rationals) -- (I_3_2__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_3_1__Rationals) to [out=135,in=-135] (I_3_2__Rationals);
	\draw[gray,dashed,->] (I_2_3__Rationals) -- (I_3_3__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_3_2__Rationals) -- (I_3_3__Rationals);
	\draw[gray,dashed,->] (I_2_4__Rationals) -- (I_3_4__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_3_1__Rationals) -- (I_3_4__Rationals);
	\draw[red,->] (I_3_3__Rationals) -- (I_3_4__Rationals);
\end{tikzpicture}
\end{scaletikzpicturetowidth}
\end{center}

\end{figure}
\medskip{}
\subsubsection*{Modules of the form $I(n,1)$}

Defect: $\partial I(n,1) = 1$, for $n\ge 0$.

\begin{fleqn}
\begin{align*}
I(0,1):\ & \bigl(I(0,2),R_{0}^{2}(1)\bigr),\ \bigl(I(0,3),R_{0}^{2}(2)\bigr),\ \bigl(2I(0,4),P(0,3)\bigr)\\
I(n,1):\ & \bigl(I(n,2),R_{0}^{(-n+1)\bmod 3+1}(1)\bigr),\ \bigl(I(n,3),R_{0}^{(-n+1)\bmod 3+1}(2)\bigr),\ \bigl((v+1)I,vP\bigr),\ n>0\\
\end{align*}
\end{fleqn}
\subsubsection*{Modules of the form $I(n,2)$}

Defect: $\partial I(n,2) = 1$, for $n\ge 0$.

\begin{fleqn}
\begin{align*}
I(0,2):\ & \bigl(I(0,3),R_{0}^{3}(1)\bigr),\ \bigl(I(0,4),R_{0}^{3}(2)\bigr)\\
I(1,2):\ & \bigl(I(1,3),R_{0}^{2}(1)\bigr),\ \bigl(I(1,4),R_{0}^{2}(2)\bigr),\ \bigl(3I(0,4),2P(0,3)\bigr)\\
I(n,2):\ & \bigl(I(n,3),R_{0}^{(-n+2)\bmod 3+1}(1)\bigr),\ \bigl(I(n,4),R_{0}^{(-n+2)\bmod 3+1}(2)\bigr),\ \bigl((v+1)I,vP\bigr),\ n>1\\
\end{align*}
\end{fleqn}
\subsubsection*{Modules of the form $I(n,3)$}

Defect: $\partial I(n,3) = 1$, for $n\ge 0$.

\begin{fleqn}
\begin{align*}
I(0,3):\ & \bigl(I(0,4),R_{0}^{1}(1)\bigr)\\
I(1,3):\ & \bigl(I(0,1),R_{0}^{3}(2)\bigr),\ \bigl(I(1,4),R_{0}^{3}(1)\bigr),\ \bigl(2I(0,2),P(0,1)\bigr)\\
I(n,3):\ & \bigl(I(n-1,1),R_{0}^{(-n+3)\bmod 3+1}(2)\bigr),\ \bigl(I(n,4),R_{0}^{(-n+3)\bmod 3+1}(1)\bigr),\ \bigl((v+1)I,vP\bigr),\ n>1\\
\end{align*}
\end{fleqn}
\subsubsection*{Modules of the form $I(n,4)$}

Defect: $\partial I(n,4) = 1$, for $n\ge 0$.

\begin{fleqn}
\begin{align*}
I(0,4):\ & - \\
I(1,4):\ & \bigl(I(0,1),R_{0}^{1}(1)\bigr),\ \bigl(I(0,2),R_{0}^{1}(2)\bigr),\ \bigl(2I(0,3),P(0,2)\bigr)\\
I(n,4):\ & \bigl(I(n-1,1),R_{0}^{(-n+1)\bmod 3+1}(1)\bigr),\ \bigl(I(n-1,2),R_{0}^{(-n+1)\bmod 3+1}(2)\bigr),\ \bigl((v+1)I,vP\bigr),\ n>1\\
\end{align*}
\end{fleqn}
\subsubsection{Schofield pairs associated to regular exceptional modules}

\subsubsection*{The non-homogeneous tube $\mathcal{T}_{0}^{\Delta(\widetilde{\mathbb{A}}_{1,3})}$}

\begin{figure}[ht]

\begin{center}
\captionof{figure}{\vspace*{-10pt}$\mathcal{T}_{0}^{\Delta(\widetilde{\mathbb{A}}_{1,3})}$}
\begin{scaletikzpicturetowidth}{0.69999999999999996\textwidth}
\begin{tikzpicture}[scale=\tikzscale]
	\node[seltwoAlAAA, label={[shift={(0.0,-0.45)}, brown]\tiny $R_{0}^{1}(1)$}] (RN_1_1_1__Rationals) at (4,0) {$\begin{smallmatrix}0&&&0\\&0&1\end{smallmatrix}$};
	\node[seltwoAlAAA, label={[shift={(0.0,-0.45)}, brown]\tiny $R_{0}^{2}(1)$}] (RN_1_2_1__Rationals) at (8,0) {$\begin{smallmatrix}1&&&1\\&0&0\end{smallmatrix}$};
	\node[seltwoAlAAA, label={[shift={(0.0,-0.45)}, brown]\tiny $R_{0}^{3}(1)$}] (RN_1_3_1__Rationals) at (12,0) {$\begin{smallmatrix}0&&&0\\&1&0\end{smallmatrix}$};
	\node[seltwoAlAAA, label={[shift={(0.0,-0.45)}, brown]\tiny $R_{0}^{1}(1)$}] (RN_1_1_1__Rationals_repeated) at (16,0) {$\begin{smallmatrix}0&&&0\\&0&1\end{smallmatrix}$};
	\node[seltwoAlAAA, label={[shift={(0.0,-0.45)}, brown]\tiny $R_{0}^{1}(2)$}] (RN_1_1_2__Rationals) at (6,2) {$\begin{smallmatrix}1&&&1\\&0&1\end{smallmatrix}$};
	\node[seltwoAlAAA, label={[shift={(0.0,-0.45)}, brown]\tiny $R_{0}^{2}(2)$}] (RN_1_2_2__Rationals) at (10,2) {$\begin{smallmatrix}1&&&1\\&1&0\end{smallmatrix}$};
	\node[seltwoAlAAA, label={[shift={(0.0,-0.45)}, brown]\tiny $R_{0}^{3}(2)$}] (RN_1_3_2__Rationals) at (14,2) {$\begin{smallmatrix}0&&&0\\&1&1\end{smallmatrix}$};
	\node[noselAlAAA, label={[shift={(0.0,-0.45)}, brown]\tiny $R_{0}^{3}(3)$}] (RN_1_3_3__Rationals) at (4,4) {$\begin{smallmatrix}1&&&1\\&1&1\end{smallmatrix}$};
	\node[noselAlAAA, label={[shift={(0.0,-0.45)}, brown]\tiny $R_{0}^{1}(3)$}] (RN_1_1_3__Rationals) at (8,4) {$\begin{smallmatrix}1&&&1\\&1&1\end{smallmatrix}$};
	\node[noselAlAAA, label={[shift={(0.0,-0.45)}, brown]\tiny $R_{0}^{2}(3)$}] (RN_1_2_3__Rationals) at (12,4) {$\begin{smallmatrix}1&&&1\\&1&1\end{smallmatrix}$};
	\node[noselAlAAA, label={[shift={(0.0,-0.45)}, brown]\tiny $R_{0}^{3}(3)$}] (RN_1_3_3__Rationals_repeated) at (16,4) {$\begin{smallmatrix}1&&&1\\&1&1\end{smallmatrix}$};
	\node[noselAlAAA, label={[shift={(0.0,-0.45)}, brown]\tiny $R_{0}^{3}(4)$}] (RN_1_3_4__Rationals) at (6,6) {$\begin{smallmatrix}1&&&1\\&2&1\end{smallmatrix}$};
	\node[noselAlAAA, label={[shift={(0.0,-0.45)}, brown]\tiny $R_{0}^{1}(4)$}] (RN_1_1_4__Rationals) at (10,6) {$\begin{smallmatrix}1&&&1\\&1&2\end{smallmatrix}$};
	\node[noselAlAAA, label={[shift={(0.0,-0.45)}, brown]\tiny $R_{0}^{2}(4)$}] (RN_1_2_4__Rationals) at (14,6) {$\begin{smallmatrix}2&&&2\\&1&1\end{smallmatrix}$};
	\draw[blue,->] (RN_1_1_1__Rationals) -- (RN_1_1_2__Rationals);
	\draw[gray,dashed,->] (RN_1_1_1__Rationals) -- (RN_1_2_1__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[red,->] (RN_1_1_2__Rationals) -- (RN_1_2_1__Rationals);
	\draw[gray,dashed,->] (RN_1_1_2__Rationals) -- (RN_1_2_2__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[blue,->] (RN_1_1_2__Rationals) -- (RN_1_1_3__Rationals);
	\draw[blue,->] (RN_1_2_1__Rationals) -- (RN_1_2_2__Rationals);
	\draw[gray,dashed,->] (RN_1_2_1__Rationals) -- (RN_1_3_1__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[red,->] (RN_1_2_2__Rationals) -- (RN_1_3_1__Rationals);
	\draw[gray,dashed,->] (RN_1_2_2__Rationals) -- (RN_1_3_2__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[blue,->] (RN_1_2_2__Rationals) -- (RN_1_2_3__Rationals);
	\draw[gray,dashed,->] (RN_1_3_1__Rationals) -- (RN_1_1_1__Rationals_repeated) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[blue,->] (RN_1_3_1__Rationals) -- (RN_1_3_2__Rationals);
	\draw[red,->] (RN_1_3_2__Rationals) -- (RN_1_1_1__Rationals_repeated);
	\draw[gray,dashed,->,out=200,in=-20] (RN_1_3_2__Rationals) to node[midway,sloped,above]{{\tiny $\tau^{-1}$}} (RN_1_1_2__Rationals);
	\draw[blue,->] (RN_1_3_2__Rationals) -- (RN_1_3_3__Rationals_repeated);
	\draw[red,->] (RN_1_3_3__Rationals) -- (RN_1_1_2__Rationals);
	\draw[blue,->] (RN_1_3_3__Rationals) -- (RN_1_3_4__Rationals);
	\draw[gray,dashed,->] (RN_1_3_3__Rationals) -- (RN_1_1_3__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[red,->] (RN_1_3_4__Rationals) -- (RN_1_1_3__Rationals);
	\draw[gray,dashed,->] (RN_1_3_4__Rationals) -- (RN_1_1_4__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[red,->] (RN_1_1_3__Rationals) -- (RN_1_2_2__Rationals);
	\draw[blue,->] (RN_1_1_3__Rationals) -- (RN_1_1_4__Rationals);
	\draw[gray,dashed,->] (RN_1_1_3__Rationals) -- (RN_1_2_3__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[red,->] (RN_1_1_4__Rationals) -- (RN_1_2_3__Rationals);
	\draw[gray,dashed,->] (RN_1_1_4__Rationals) -- (RN_1_2_4__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[red,->] (RN_1_2_3__Rationals) -- (RN_1_3_2__Rationals);
	\draw[gray,dashed,->] (RN_1_2_3__Rationals) -- (RN_1_3_3__Rationals_repeated) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[blue,->] (RN_1_2_3__Rationals) -- (RN_1_2_4__Rationals);
	\draw[red,->] (RN_1_2_4__Rationals) -- (RN_1_3_3__Rationals_repeated);
	\draw[gray,dashed,->,out=200,in=-20] (RN_1_2_4__Rationals) to node[midway,sloped,above]{{\tiny $\tau^{-1}$}} (RN_1_3_4__Rationals);
	\draw[dashed,-] (RN_1_1_1__Rationals) -- (RN_1_3_3__Rationals);
	\draw[dashed,-] (RN_1_1_1__Rationals_repeated) -- (RN_1_3_3__Rationals_repeated);
	\node (LeftTopPhantom_1) at (4,6) {};
	\draw[dashed,-] (RN_1_3_3__Rationals) -- (LeftTopPhantom_1);
	\node (RightTopPhantom_1) at (16,6) {};
	\draw[dashed,-] (RN_1_3_3__Rationals_repeated) -- (RightTopPhantom_1);
\end{tikzpicture}
\end{scaletikzpicturetowidth}
\end{center}

\end{figure}
\begin{fleqn}
\begin{align*}
R_{0}^{1}(1):\ & - \\
R_{0}^{1}(2):\ & \bigl(R_{0}^{2}(1),R_{0}^{1}(1)\bigr),\ \bigl(I(0,3),P(0,1)\bigr)\\
R_{0}^{2}(1):\ & \bigl(I(0,4),P(0,1)\bigr)\\
R_{0}^{2}(2):\ & \bigl(R_{0}^{3}(1),R_{0}^{2}(1)\bigr),\ \bigl(I(0,4),P(0,2)\bigr)\\
R_{0}^{3}(1):\ & - \\
R_{0}^{3}(2):\ & \bigl(R_{0}^{1}(1),R_{0}^{3}(1)\bigr)\\
\end{align*}
\end{fleqn}
\clearpage

\subsection[Schofield pairs for the quiver $\Delta(\widetilde{\mathbb{A}}_{1,4})$]{Schofield pairs for the quiver $\Delta(\widetilde{\mathbb{A}}_{1,4})$ -- {\usebox{\boxAlAAAA}}}
\[
\vcenter{\hbox{\xymatrix{1 &   &   &   & 5\ar[llll]\ar[dl]\\
  & 2\ar[ul] & 3\ar[l] & 4\ar[l] &  }}}\qquad C_{\Delta(\widetilde{\mathbb{A}}_{1,4})} = \begin{bmatrix}1 & 1 & 1 & 1 & 2\\
0 & 1 & 1 & 1 & 1\\
0 & 0 & 1 & 1 & 1\\
0 & 0 & 0 & 1 & 1\\
0 & 0 & 0 & 0 & 1\end{bmatrix}\quad\Phi_{\Delta(\widetilde{\mathbb{A}}_{1,4})} = \begin{bmatrix}-1 & 1 & 0 & 0 & 1\\
-1 & 0 & 1 & 0 & 1\\
-1 & 0 & 0 & 1 & 1\\
-1 & 0 & 0 & 0 & 2\\
-2 & 1 & 0 & 0 & 2\end{bmatrix}
\]
\subsubsection{Schofield pairs associated to preprojective exceptional modules}

\begin{figure}[ht]

\begin{center}
\begin{scaletikzpicturetowidth}{\textwidth}
\begin{tikzpicture}[scale=\tikzscale]
	\node[noselAlAAAA, label={[shift={(0.4,-0.2)}, brown]\tiny $P(0,1)$}] (P_0_1__Rationals) at (0,0) {$\begin{smallmatrix}1&&&&0\\&0&0&0\end{smallmatrix}$};
	\node[noselAlAAAA, label={[shift={(0.4,-0.2)}, brown]\tiny $P(0,2)$}] (P_0_2__Rationals) at (0,-4) {$\begin{smallmatrix}1&&&&0\\&1&0&0\end{smallmatrix}$};
	\node[noselAlAAAA, label={[shift={(0.4,-0.2)}, brown]\tiny $P(0,3)$}] (P_0_3__Rationals) at (0,-3) {$\begin{smallmatrix}1&&&&0\\&1&1&0\end{smallmatrix}$};
	\node[noselAlAAAA, label={[shift={(0.4,-0.2)}, brown]\tiny $P(0,4)$}] (P_0_4__Rationals) at (0,-2) {$\begin{smallmatrix}1&&&&0\\&1&1&1\end{smallmatrix}$};
	\node[seloneAlAAAA, label={[shift={(0.4,-0.2)}, brown]\tiny $P(0,5)$}] (P_0_5__Rationals) at (0,-1) {$\begin{smallmatrix}2&&&&1\\&1&1&1\end{smallmatrix}$};
	\node[seloneAlAAAA, label={[shift={(0.4,-0.2)}, brown]\tiny $P(1,1)$}] (P_1_1__Rationals) at (4,0) {$\begin{smallmatrix}2&&&&1\\&2&1&1\end{smallmatrix}$};
	\node[seloneAlAAAA, label={[shift={(0.4,-0.2)}, brown]\tiny $P(1,2)$}] (P_1_2__Rationals) at (4,-4) {$\begin{smallmatrix}2&&&&1\\&2&2&1\end{smallmatrix}$};
	\node[seloneAlAAAA, label={[shift={(0.4,-0.2)}, brown]\tiny $P(1,3)$}] (P_1_3__Rationals) at (4,-3) {$\begin{smallmatrix}2&&&&1\\&2&2&2\end{smallmatrix}$};
	\node[seloneAlAAAA, label={[shift={(0.4,-0.2)}, brown]\tiny $P(1,4)$}] (P_1_4__Rationals) at (4,-2) {$\begin{smallmatrix}3&&&&2\\&2&2&2\end{smallmatrix}$};
	\node[noselAlAAAA, label={[shift={(0.4,-0.2)}, brown]\tiny $P(1,5)$}] (P_1_5__Rationals) at (4,-1) {$\begin{smallmatrix}3&&&&2\\&3&2&2\end{smallmatrix}$};
	\node[noselAlAAAA, label={[shift={(0.4,-0.2)}, brown]\tiny $P(2,1)$}] (P_2_1__Rationals) at (8,0) {$\begin{smallmatrix}3&&&&2\\&3&3&2\end{smallmatrix}$};
	\node[noselAlAAAA, label={[shift={(0.4,-0.2)}, brown]\tiny $P(2,2)$}] (P_2_2__Rationals) at (8,-4) {$\begin{smallmatrix}3&&&&2\\&3&3&3\end{smallmatrix}$};
	\node[noselAlAAAA, label={[shift={(0.4,-0.2)}, brown]\tiny $P(2,3)$}] (P_2_3__Rationals) at (8,-3) {$\begin{smallmatrix}4&&&&3\\&3&3&3\end{smallmatrix}$};
	\node[noselAlAAAA, label={[shift={(0.4,-0.2)}, brown]\tiny $P(2,4)$}] (P_2_4__Rationals) at (8,-2) {$\begin{smallmatrix}4&&&&3\\&4&3&3\end{smallmatrix}$};
	\node[noselAlAAAA, label={[shift={(0.4,-0.2)}, brown]\tiny $P(2,5)$}] (P_2_5__Rationals) at (8,-1) {$\begin{smallmatrix}4&&&&3\\&4&4&3\end{smallmatrix}$};
	\node[noselAlAAAA, label={[shift={(0.4,-0.2)}, brown]\tiny $P(3,1)$}] (P_3_1__Rationals) at (12,0) {$\begin{smallmatrix}4&&&&3\\&4&4&4\end{smallmatrix}$};
	\node[noselAlAAAA, label={[shift={(0.4,-0.2)}, brown]\tiny $P(3,2)$}] (P_3_2__Rationals) at (12,-4) {$\begin{smallmatrix}5&&&&4\\&4&4&4\end{smallmatrix}$};
	\node[noselAlAAAA, label={[shift={(0.4,-0.2)}, brown]\tiny $P(3,3)$}] (P_3_3__Rationals) at (12,-3) {$\begin{smallmatrix}5&&&&4\\&5&4&4\end{smallmatrix}$};
	\node[noselAlAAAA, label={[shift={(0.4,-0.2)}, brown]\tiny $P(3,4)$}] (P_3_4__Rationals) at (12,-2) {$\begin{smallmatrix}5&&&&4\\&5&5&4\end{smallmatrix}$};
	\node[noselAlAAAA, label={[shift={(0.4,-0.2)}, brown]\tiny $P(3,5)$}] (P_3_5__Rationals) at (12,-1) {$\begin{smallmatrix}5&&&&4\\&5&5&5\end{smallmatrix}$};
	\node at (13,-2.) {$\dots$};
	\draw[blue,->] (P_0_1__Rationals) to [out=135,in=-135] (P_0_2__Rationals);
	\draw[blue,->] (P_0_2__Rationals) -- (P_0_3__Rationals);
	\draw[blue,->] (P_0_3__Rationals) -- (P_0_4__Rationals);
	\draw[blue,->] (P_0_1__Rationals) -- (P_0_5__Rationals);
	\draw[blue,->] (P_0_4__Rationals) -- (P_0_5__Rationals);
	\draw[gray,dashed,->] (P_0_1__Rationals) -- (P_1_1__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[blue,->] (P_0_2__Rationals) -- (P_1_1__Rationals);
	\draw[blue,->] (P_0_5__Rationals) -- (P_1_1__Rationals);
	\draw[gray,dashed,->] (P_0_2__Rationals) -- (P_1_2__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[blue,->] (P_0_3__Rationals) -- (P_1_2__Rationals);
	\draw[blue,->] (P_1_1__Rationals) to [out=135,in=-135] (P_1_2__Rationals);
	\draw[gray,dashed,->] (P_0_3__Rationals) -- (P_1_3__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[blue,->] (P_0_4__Rationals) -- (P_1_3__Rationals);
	\draw[blue,->] (P_1_2__Rationals) -- (P_1_3__Rationals);
	\draw[gray,dashed,->] (P_0_4__Rationals) -- (P_1_4__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[blue,->] (P_0_5__Rationals) -- (P_1_4__Rationals);
	\draw[blue,->] (P_1_3__Rationals) -- (P_1_4__Rationals);
	\draw[gray,dashed,->] (P_0_5__Rationals) -- (P_1_5__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[blue,->] (P_1_1__Rationals) -- (P_1_5__Rationals);
	\draw[blue,->] (P_1_4__Rationals) -- (P_1_5__Rationals);
	\draw[gray,dashed,->] (P_1_1__Rationals) -- (P_2_1__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[blue,->] (P_1_2__Rationals) -- (P_2_1__Rationals);
	\draw[blue,->] (P_1_5__Rationals) -- (P_2_1__Rationals);
	\draw[gray,dashed,->] (P_1_2__Rationals) -- (P_2_2__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[blue,->] (P_1_3__Rationals) -- (P_2_2__Rationals);
	\draw[blue,->] (P_2_1__Rationals) to [out=135,in=-135] (P_2_2__Rationals);
	\draw[gray,dashed,->] (P_1_3__Rationals) -- (P_2_3__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[blue,->] (P_1_4__Rationals) -- (P_2_3__Rationals);
	\draw[blue,->] (P_2_2__Rationals) -- (P_2_3__Rationals);
	\draw[gray,dashed,->] (P_1_4__Rationals) -- (P_2_4__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[blue,->] (P_1_5__Rationals) -- (P_2_4__Rationals);
	\draw[blue,->] (P_2_3__Rationals) -- (P_2_4__Rationals);
	\draw[gray,dashed,->] (P_1_5__Rationals) -- (P_2_5__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[blue,->] (P_2_1__Rationals) -- (P_2_5__Rationals);
	\draw[blue,->] (P_2_4__Rationals) -- (P_2_5__Rationals);
	\draw[gray,dashed,->] (P_2_1__Rationals) -- (P_3_1__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[blue,->] (P_2_2__Rationals) -- (P_3_1__Rationals);
	\draw[blue,->] (P_2_5__Rationals) -- (P_3_1__Rationals);
	\draw[gray,dashed,->] (P_2_2__Rationals) -- (P_3_2__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[blue,->] (P_2_3__Rationals) -- (P_3_2__Rationals);
	\draw[blue,->] (P_3_1__Rationals) to [out=135,in=-135] (P_3_2__Rationals);
	\draw[gray,dashed,->] (P_2_3__Rationals) -- (P_3_3__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[blue,->] (P_2_4__Rationals) -- (P_3_3__Rationals);
	\draw[blue,->] (P_3_2__Rationals) -- (P_3_3__Rationals);
	\draw[gray,dashed,->] (P_2_4__Rationals) -- (P_3_4__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[blue,->] (P_2_5__Rationals) -- (P_3_4__Rationals);
	\draw[blue,->] (P_3_3__Rationals) -- (P_3_4__Rationals);
	\draw[gray,dashed,->] (P_2_5__Rationals) -- (P_3_5__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[blue,->] (P_3_1__Rationals) -- (P_3_5__Rationals);
	\draw[blue,->] (P_3_4__Rationals) -- (P_3_5__Rationals);
\end{tikzpicture}
\end{scaletikzpicturetowidth}
\end{center}

\end{figure}
\medskip{}
\subsubsection*{Modules of the form $P(n,1)$}

Defect: $\partial P(n,1) = -1$, for $n\ge 0$.

\begin{fleqn}
\begin{align*}
P(0,1):\ & - \\
P(1,1):\ & \bigl(R_{0}^{1}(3),P(0,3)\bigr),\ \bigl(R_{0}^{2}(2),P(0,4)\bigr),\ \bigl(R_{0}^{3}(1),P(0,5)\bigr),\ \bigl(I(0,3),2P(0,2)\bigr)\\
P(n,1):\ & \bigl(R_{0}^{(n-1)\bmod 4+1}(3),P(n-1,3)\bigr),\ \bigl(R_{0}^{n\bmod 4+1}(2),P(n-1,4)\bigr),\ \bigl(R_{0}^{(n+1)\bmod 4+1}(1),P(n-1,5)\bigr)\\
 & \bigl(uI,(u+1)P\bigr),\ n>1\\
\end{align*}
\end{fleqn}
\subsubsection*{Modules of the form $P(n,2)$}

Defect: $\partial P(n,2) = -1$, for $n\ge 0$.

\begin{fleqn}
\begin{align*}
P(0,2):\ & \bigl(R_{0}^{3}(1),P(0,1)\bigr)\\
P(1,2):\ & \bigl(R_{0}^{2}(3),P(0,4)\bigr),\ \bigl(R_{0}^{3}(2),P(0,5)\bigr),\ \bigl(R_{0}^{4}(1),P(1,1)\bigr),\ \bigl(I(0,4),2P(0,3)\bigr)\\
P(n,2):\ & \bigl(R_{0}^{n\bmod 4+1}(3),P(n-1,4)\bigr),\ \bigl(R_{0}^{(n+1)\bmod 4+1}(2),P(n-1,5)\bigr),\ \bigl(R_{0}^{(n+2)\bmod 4+1}(1),P(n,1)\bigr)\\
 & \bigl(uI,(u+1)P\bigr),\ n>1\\
\end{align*}
\end{fleqn}
\subsubsection*{Modules of the form $P(n,3)$}

Defect: $\partial P(n,3) = -1$, for $n\ge 0$.

\begin{fleqn}
\begin{align*}
P(0,3):\ & \bigl(R_{0}^{3}(2),P(0,1)\bigr),\ \bigl(R_{0}^{4}(1),P(0,2)\bigr)\\
P(1,3):\ & \bigl(R_{0}^{3}(3),P(0,5)\bigr),\ \bigl(R_{0}^{4}(2),P(1,1)\bigr),\ \bigl(R_{0}^{1}(1),P(1,2)\bigr),\ \bigl(I(0,5),2P(0,4)\bigr)\\
P(n,3):\ & \bigl(R_{0}^{(n+1)\bmod 4+1}(3),P(n-1,5)\bigr),\ \bigl(R_{0}^{(n+2)\bmod 4+1}(2),P(n,1)\bigr),\ \bigl(R_{0}^{(n-1)\bmod 4+1}(1),P(n,2)\bigr)\\
 & \bigl(uI,(u+1)P\bigr),\ n>1\\
\end{align*}
\end{fleqn}
\subsubsection*{Modules of the form $P(n,4)$}

Defect: $\partial P(n,4) = -1$, for $n\ge 0$.

\begin{fleqn}
\begin{align*}
P(0,4):\ & \bigl(R_{0}^{3}(3),P(0,1)\bigr),\ \bigl(R_{0}^{4}(2),P(0,2)\bigr),\ \bigl(R_{0}^{1}(1),P(0,3)\bigr)\\
P(1,4):\ & \bigl(R_{0}^{4}(3),P(1,1)\bigr),\ \bigl(R_{0}^{1}(2),P(1,2)\bigr),\ \bigl(R_{0}^{2}(1),P(1,3)\bigr),\ \bigl(2I(0,2),3P(0,1)\bigr)\\
P(n,4):\ & \bigl(R_{0}^{(n+2)\bmod 4+1}(3),P(n,1)\bigr),\ \bigl(R_{0}^{(n-1)\bmod 4+1}(2),P(n,2)\bigr),\ \bigl(R_{0}^{n\bmod 4+1}(1),P(n,3)\bigr)\\
 & \bigl(uI,(u+1)P\bigr),\ n>1\\
\end{align*}
\end{fleqn}
\subsubsection*{Modules of the form $P(n,5)$}

Defect: $\partial P(n,5) = -1$, for $n\ge 0$.

\begin{fleqn}
\begin{align*}
P(0,5):\ & \bigl(R_{0}^{4}(3),P(0,2)\bigr),\ \bigl(R_{0}^{1}(2),P(0,3)\bigr),\ \bigl(R_{0}^{2}(1),P(0,4)\bigr),\ \bigl(I(0,2),2P(0,1)\bigr)\\
P(n,5):\ & \bigl(R_{0}^{(n+3)\bmod 4+1}(3),P(n,2)\bigr),\ \bigl(R_{0}^{n\bmod 4+1}(2),P(n,3)\bigr),\ \bigl(R_{0}^{(n+1)\bmod 4+1}(1),P(n,4)\bigr)\\
 & \bigl(uI,(u+1)P\bigr),\ n>0\\
\end{align*}
\end{fleqn}
\subsubsection{Schofield pairs associated to preinjective exceptional modules}

\begin{figure}[ht]

\begin{center}
\begin{scaletikzpicturetowidth}{\textwidth}
\begin{tikzpicture}[scale=\tikzscale]
	\node[seloneAlAAAA, label={[shift={(0.4,-0.2)}, brown]\tiny $I(0,1)$}] (I_0_1__Rationals) at (0,0) {$\begin{smallmatrix}1&&&&2\\&1&1&1\end{smallmatrix}$};
	\node[noselAlAAAA, label={[shift={(0.4,-0.2)}, brown]\tiny $I(0,2)$}] (I_0_2__Rationals) at (0,-4) {$\begin{smallmatrix}0&&&&1\\&1&1&1\end{smallmatrix}$};
	\node[noselAlAAAA, label={[shift={(0.4,-0.2)}, brown]\tiny $I(0,3)$}] (I_0_3__Rationals) at (0,-3) {$\begin{smallmatrix}0&&&&1\\&0&1&1\end{smallmatrix}$};
	\node[noselAlAAAA, label={[shift={(0.4,-0.2)}, brown]\tiny $I(0,4)$}] (I_0_4__Rationals) at (0,-2) {$\begin{smallmatrix}0&&&&1\\&0&0&1\end{smallmatrix}$};
	\node[noselAlAAAA, label={[shift={(0.4,-0.2)}, brown]\tiny $I(0,5)$}] (I_0_5__Rationals) at (0,-1) {$\begin{smallmatrix}0&&&&1\\&0&0&0\end{smallmatrix}$};
	\node[noselAlAAAA, label={[shift={(0.4,-0.2)}, brown]\tiny $I(1,1)$}] (I_1_1__Rationals) at (-4,0) {$\begin{smallmatrix}2&&&&3\\&2&2&3\end{smallmatrix}$};
	\node[seloneAlAAAA, label={[shift={(0.4,-0.2)}, brown]\tiny $I(1,2)$}] (I_1_2__Rationals) at (-4,-4) {$\begin{smallmatrix}2&&&&3\\&2&2&2\end{smallmatrix}$};
	\node[seloneAlAAAA, label={[shift={(0.4,-0.2)}, brown]\tiny $I(1,3)$}] (I_1_3__Rationals) at (-4,-3) {$\begin{smallmatrix}1&&&&2\\&2&2&2\end{smallmatrix}$};
	\node[seloneAlAAAA, label={[shift={(0.4,-0.2)}, brown]\tiny $I(1,4)$}] (I_1_4__Rationals) at (-4,-2) {$\begin{smallmatrix}1&&&&2\\&1&2&2\end{smallmatrix}$};
	\node[seloneAlAAAA, label={[shift={(0.4,-0.2)}, brown]\tiny $I(1,5)$}] (I_1_5__Rationals) at (-4,-1) {$\begin{smallmatrix}1&&&&2\\&1&1&2\end{smallmatrix}$};
	\node[noselAlAAAA, label={[shift={(0.4,-0.2)}, brown]\tiny $I(2,1)$}] (I_2_1__Rationals) at (-8,0) {$\begin{smallmatrix}3&&&&4\\&3&4&4\end{smallmatrix}$};
	\node[noselAlAAAA, label={[shift={(0.4,-0.2)}, brown]\tiny $I(2,2)$}] (I_2_2__Rationals) at (-8,-4) {$\begin{smallmatrix}3&&&&4\\&3&3&4\end{smallmatrix}$};
	\node[noselAlAAAA, label={[shift={(0.4,-0.2)}, brown]\tiny $I(2,3)$}] (I_2_3__Rationals) at (-8,-3) {$\begin{smallmatrix}3&&&&4\\&3&3&3\end{smallmatrix}$};
	\node[noselAlAAAA, label={[shift={(0.4,-0.2)}, brown]\tiny $I(2,4)$}] (I_2_4__Rationals) at (-8,-2) {$\begin{smallmatrix}2&&&&3\\&3&3&3\end{smallmatrix}$};
	\node[noselAlAAAA, label={[shift={(0.4,-0.2)}, brown]\tiny $I(2,5)$}] (I_2_5__Rationals) at (-8,-1) {$\begin{smallmatrix}2&&&&3\\&2&3&3\end{smallmatrix}$};
	\node[noselAlAAAA, label={[shift={(0.4,-0.2)}, brown]\tiny $I(3,1)$}] (I_3_1__Rationals) at (-12,0) {$\begin{smallmatrix}4&&&&5\\&5&5&5\end{smallmatrix}$};
	\node[noselAlAAAA, label={[shift={(0.4,-0.2)}, brown]\tiny $I(3,2)$}] (I_3_2__Rationals) at (-12,-4) {$\begin{smallmatrix}4&&&&5\\&4&5&5\end{smallmatrix}$};
	\node[noselAlAAAA, label={[shift={(0.4,-0.2)}, brown]\tiny $I(3,3)$}] (I_3_3__Rationals) at (-12,-3) {$\begin{smallmatrix}4&&&&5\\&4&4&5\end{smallmatrix}$};
	\node[noselAlAAAA, label={[shift={(0.4,-0.2)}, brown]\tiny $I(3,4)$}] (I_3_4__Rationals) at (-12,-2) {$\begin{smallmatrix}4&&&&5\\&4&4&4\end{smallmatrix}$};
	\node[noselAlAAAA, label={[shift={(0.4,-0.2)}, brown]\tiny $I(3,5)$}] (I_3_5__Rationals) at (-12,-1) {$\begin{smallmatrix}3&&&&4\\&4&4&4\end{smallmatrix}$};
	\node at (-13,-2.) {$\dots$};
	\draw[red,->] (I_1_2__Rationals) -- (I_0_1__Rationals);
	\draw[red,->] (I_1_5__Rationals) -- (I_0_1__Rationals);
	\draw[red,->] (I_0_1__Rationals) to [out=135,in=-135] (I_0_2__Rationals);
	\draw[red,->] (I_1_3__Rationals) -- (I_0_2__Rationals);
	\draw[red,->] (I_0_2__Rationals) -- (I_0_3__Rationals);
	\draw[red,->] (I_1_4__Rationals) -- (I_0_3__Rationals);
	\draw[red,->] (I_0_3__Rationals) -- (I_0_4__Rationals);
	\draw[red,->] (I_1_5__Rationals) -- (I_0_4__Rationals);
	\draw[red,->] (I_0_1__Rationals) -- (I_0_5__Rationals);
	\draw[red,->] (I_0_4__Rationals) -- (I_0_5__Rationals);
	\draw[gray,dashed,->] (I_0_1__Rationals) -- (I_1_1__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_2_2__Rationals) -- (I_1_1__Rationals);
	\draw[red,->] (I_2_5__Rationals) -- (I_1_1__Rationals);
	\draw[gray,dashed,->] (I_0_2__Rationals) -- (I_1_2__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_1_1__Rationals) to [out=135,in=-135] (I_1_2__Rationals);
	\draw[red,->] (I_2_3__Rationals) -- (I_1_2__Rationals);
	\draw[gray,dashed,->] (I_0_3__Rationals) -- (I_1_3__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_1_2__Rationals) -- (I_1_3__Rationals);
	\draw[red,->] (I_2_4__Rationals) -- (I_1_3__Rationals);
	\draw[gray,dashed,->] (I_0_4__Rationals) -- (I_1_4__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_1_3__Rationals) -- (I_1_4__Rationals);
	\draw[red,->] (I_2_5__Rationals) -- (I_1_4__Rationals);
	\draw[gray,dashed,->] (I_0_5__Rationals) -- (I_1_5__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_1_1__Rationals) -- (I_1_5__Rationals);
	\draw[red,->] (I_1_4__Rationals) -- (I_1_5__Rationals);
	\draw[gray,dashed,->] (I_1_1__Rationals) -- (I_2_1__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_3_2__Rationals) -- (I_2_1__Rationals);
	\draw[red,->] (I_3_5__Rationals) -- (I_2_1__Rationals);
	\draw[gray,dashed,->] (I_1_2__Rationals) -- (I_2_2__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_2_1__Rationals) to [out=135,in=-135] (I_2_2__Rationals);
	\draw[red,->] (I_3_3__Rationals) -- (I_2_2__Rationals);
	\draw[gray,dashed,->] (I_1_3__Rationals) -- (I_2_3__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_2_2__Rationals) -- (I_2_3__Rationals);
	\draw[red,->] (I_3_4__Rationals) -- (I_2_3__Rationals);
	\draw[gray,dashed,->] (I_1_4__Rationals) -- (I_2_4__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_2_3__Rationals) -- (I_2_4__Rationals);
	\draw[red,->] (I_3_5__Rationals) -- (I_2_4__Rationals);
	\draw[gray,dashed,->] (I_1_5__Rationals) -- (I_2_5__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_2_1__Rationals) -- (I_2_5__Rationals);
	\draw[red,->] (I_2_4__Rationals) -- (I_2_5__Rationals);
	\draw[gray,dashed,->] (I_2_1__Rationals) -- (I_3_1__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[gray,dashed,->] (I_2_2__Rationals) -- (I_3_2__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_3_1__Rationals) to [out=135,in=-135] (I_3_2__Rationals);
	\draw[gray,dashed,->] (I_2_3__Rationals) -- (I_3_3__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_3_2__Rationals) -- (I_3_3__Rationals);
	\draw[gray,dashed,->] (I_2_4__Rationals) -- (I_3_4__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_3_3__Rationals) -- (I_3_4__Rationals);
	\draw[gray,dashed,->] (I_2_5__Rationals) -- (I_3_5__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_3_1__Rationals) -- (I_3_5__Rationals);
	\draw[red,->] (I_3_4__Rationals) -- (I_3_5__Rationals);
\end{tikzpicture}
\end{scaletikzpicturetowidth}
\end{center}

\end{figure}
\medskip{}
\subsubsection*{Modules of the form $I(n,1)$}

Defect: $\partial I(n,1) = 1$, for $n\ge 0$.

\begin{fleqn}
\begin{align*}
I(0,1):\ & \bigl(I(0,2),R_{0}^{2}(1)\bigr),\ \bigl(I(0,3),R_{0}^{2}(2)\bigr),\ \bigl(I(0,4),R_{0}^{2}(3)\bigr),\ \bigl(2I(0,5),P(0,4)\bigr)\\
I(n,1):\ & \bigl(I(n,2),R_{0}^{(-n+1)\bmod 4+1}(1)\bigr),\ \bigl(I(n,3),R_{0}^{(-n+1)\bmod 4+1}(2)\bigr),\ \bigl(I(n,4),R_{0}^{(-n+1)\bmod 4+1}(3)\bigr)\\
 & \bigl((v+1)I,vP\bigr),\ n>0\\
\end{align*}
\end{fleqn}
\subsubsection*{Modules of the form $I(n,2)$}

Defect: $\partial I(n,2) = 1$, for $n\ge 0$.

\begin{fleqn}
\begin{align*}
I(0,2):\ & \bigl(I(0,3),R_{0}^{3}(1)\bigr),\ \bigl(I(0,4),R_{0}^{3}(2)\bigr),\ \bigl(I(0,5),R_{0}^{3}(3)\bigr)\\
I(1,2):\ & \bigl(I(1,3),R_{0}^{2}(1)\bigr),\ \bigl(I(1,4),R_{0}^{2}(2)\bigr),\ \bigl(I(1,5),R_{0}^{2}(3)\bigr),\ \bigl(3I(0,5),2P(0,4)\bigr)\\
I(n,2):\ & \bigl(I(n,3),R_{0}^{(-n+2)\bmod 4+1}(1)\bigr),\ \bigl(I(n,4),R_{0}^{(-n+2)\bmod 4+1}(2)\bigr),\ \bigl(I(n,5),R_{0}^{(-n+2)\bmod 4+1}(3)\bigr)\\
 & \bigl((v+1)I,vP\bigr),\ n>1\\
\end{align*}
\end{fleqn}
\subsubsection*{Modules of the form $I(n,3)$}

Defect: $\partial I(n,3) = 1$, for $n\ge 0$.

\begin{fleqn}
\begin{align*}
I(0,3):\ & \bigl(I(0,4),R_{0}^{4}(1)\bigr),\ \bigl(I(0,5),R_{0}^{4}(2)\bigr)\\
I(1,3):\ & \bigl(I(0,1),R_{0}^{3}(3)\bigr),\ \bigl(I(1,4),R_{0}^{3}(1)\bigr),\ \bigl(I(1,5),R_{0}^{3}(2)\bigr),\ \bigl(2I(0,2),P(0,1)\bigr)\\
I(n,3):\ & \bigl(I(n-1,1),R_{0}^{(-n+3)\bmod 4+1}(3)\bigr),\ \bigl(I(n,4),R_{0}^{(-n+3)\bmod 4+1}(1)\bigr),\ \bigl(I(n,5),R_{0}^{(-n+3)\bmod 4+1}(2)\bigr)\\
 & \bigl((v+1)I,vP\bigr),\ n>1\\
\end{align*}
\end{fleqn}
\subsubsection*{Modules of the form $I(n,4)$}

Defect: $\partial I(n,4) = 1$, for $n\ge 0$.

\begin{fleqn}
\begin{align*}
I(0,4):\ & \bigl(I(0,5),R_{0}^{1}(1)\bigr)\\
I(1,4):\ & \bigl(I(0,1),R_{0}^{4}(2)\bigr),\ \bigl(I(0,2),R_{0}^{4}(3)\bigr),\ \bigl(I(1,5),R_{0}^{4}(1)\bigr),\ \bigl(2I(0,3),P(0,2)\bigr)\\
I(n,4):\ & \bigl(I(n-1,1),R_{0}^{(-n+4)\bmod 4+1}(2)\bigr),\ \bigl(I(n-1,2),R_{0}^{(-n+4)\bmod 4+1}(3)\bigr),\ \bigl(I(n,5),R_{0}^{(-n+4)\bmod 4+1}(1)\bigr)\\
 & \bigl((v+1)I,vP\bigr),\ n>1\\
\end{align*}
\end{fleqn}
\subsubsection*{Modules of the form $I(n,5)$}

Defect: $\partial I(n,5) = 1$, for $n\ge 0$.

\begin{fleqn}
\begin{align*}
I(0,5):\ & - \\
I(1,5):\ & \bigl(I(0,1),R_{0}^{1}(1)\bigr),\ \bigl(I(0,2),R_{0}^{1}(2)\bigr),\ \bigl(I(0,3),R_{0}^{1}(3)\bigr),\ \bigl(2I(0,4),P(0,3)\bigr)\\
I(n,5):\ & \bigl(I(n-1,1),R_{0}^{(-n+1)\bmod 4+1}(1)\bigr),\ \bigl(I(n-1,2),R_{0}^{(-n+1)\bmod 4+1}(2)\bigr),\ \bigl(I(n-1,3),R_{0}^{(-n+1)\bmod 4+1}(3)\bigr)\\
 & \bigl((v+1)I,vP\bigr),\ n>1\\
\end{align*}
\end{fleqn}
\subsubsection{Schofield pairs associated to regular exceptional modules}

\subsubsection*{The non-homogeneous tube $\mathcal{T}_{0}^{\Delta(\widetilde{\mathbb{A}}_{1,4})}$}

\begin{figure}[ht]

\begin{center}
\captionof{figure}{\vspace*{-10pt}$\mathcal{T}_{0}^{\Delta(\widetilde{\mathbb{A}}_{1,4})}$}
\begin{scaletikzpicturetowidth}{1.\textwidth}
\begin{tikzpicture}[scale=\tikzscale]
	\node[seltwoAlAAAA, label={[shift={(0.0,-0.45)}, brown]\tiny $R_{0}^{1}(1)$}] (RN_1_1_1__Rationals) at (4,0) {$\begin{smallmatrix}0&&&&0\\&0&0&1\end{smallmatrix}$};
	\node[seltwoAlAAAA, label={[shift={(0.0,-0.45)}, brown]\tiny $R_{0}^{2}(1)$}] (RN_1_2_1__Rationals) at (8,0) {$\begin{smallmatrix}1&&&&1\\&0&0&0\end{smallmatrix}$};
	\node[seltwoAlAAAA, label={[shift={(0.0,-0.45)}, brown]\tiny $R_{0}^{3}(1)$}] (RN_1_3_1__Rationals) at (12,0) {$\begin{smallmatrix}0&&&&0\\&1&0&0\end{smallmatrix}$};
	\node[seltwoAlAAAA, label={[shift={(0.0,-0.45)}, brown]\tiny $R_{0}^{4}(1)$}] (RN_1_4_1__Rationals) at (16,0) {$\begin{smallmatrix}0&&&&0\\&0&1&0\end{smallmatrix}$};
	\node[seltwoAlAAAA, label={[shift={(0.0,-0.45)}, brown]\tiny $R_{0}^{1}(1)$}] (RN_1_1_1__Rationals_repeated) at (20,0) {$\begin{smallmatrix}0&&&&0\\&0&0&1\end{smallmatrix}$};
	\node[seltwoAlAAAA, label={[shift={(0.0,-0.45)}, brown]\tiny $R_{0}^{1}(2)$}] (RN_1_1_2__Rationals) at (6,2) {$\begin{smallmatrix}1&&&&1\\&0&0&1\end{smallmatrix}$};
	\node[seltwoAlAAAA, label={[shift={(0.0,-0.45)}, brown]\tiny $R_{0}^{2}(2)$}] (RN_1_2_2__Rationals) at (10,2) {$\begin{smallmatrix}1&&&&1\\&1&0&0\end{smallmatrix}$};
	\node[seltwoAlAAAA, label={[shift={(0.0,-0.45)}, brown]\tiny $R_{0}^{3}(2)$}] (RN_1_3_2__Rationals) at (14,2) {$\begin{smallmatrix}0&&&&0\\&1&1&0\end{smallmatrix}$};
	\node[seltwoAlAAAA, label={[shift={(0.0,-0.45)}, brown]\tiny $R_{0}^{4}(2)$}] (RN_1_4_2__Rationals) at (18,2) {$\begin{smallmatrix}0&&&&0\\&0&1&1\end{smallmatrix}$};
	\node[seltwoAlAAAA, label={[shift={(0.0,-0.45)}, brown]\tiny $R_{0}^{4}(3)$}] (RN_1_4_3__Rationals) at (4,4) {$\begin{smallmatrix}1&&&&1\\&0&1&1\end{smallmatrix}$};
	\node[seltwoAlAAAA, label={[shift={(0.0,-0.45)}, brown]\tiny $R_{0}^{1}(3)$}] (RN_1_1_3__Rationals) at (8,4) {$\begin{smallmatrix}1&&&&1\\&1&0&1\end{smallmatrix}$};
	\node[seltwoAlAAAA, label={[shift={(0.0,-0.45)}, brown]\tiny $R_{0}^{2}(3)$}] (RN_1_2_3__Rationals) at (12,4) {$\begin{smallmatrix}1&&&&1\\&1&1&0\end{smallmatrix}$};
	\node[seltwoAlAAAA, label={[shift={(0.0,-0.45)}, brown]\tiny $R_{0}^{3}(3)$}] (RN_1_3_3__Rationals) at (16,4) {$\begin{smallmatrix}0&&&&0\\&1&1&1\end{smallmatrix}$};
	\node[seltwoAlAAAA, label={[shift={(0.0,-0.45)}, brown]\tiny $R_{0}^{4}(3)$}] (RN_1_4_3__Rationals_repeated) at (20,4) {$\begin{smallmatrix}1&&&&1\\&0&1&1\end{smallmatrix}$};
	\node[noselAlAAAA, label={[shift={(0.0,-0.45)}, brown]\tiny $R_{0}^{4}(4)$}] (RN_1_4_4__Rationals) at (6,6) {$\begin{smallmatrix}1&&&&1\\&1&1&1\end{smallmatrix}$};
	\node[noselAlAAAA, label={[shift={(0.0,-0.45)}, brown]\tiny $R_{0}^{1}(4)$}] (RN_1_1_4__Rationals) at (10,6) {$\begin{smallmatrix}1&&&&1\\&1&1&1\end{smallmatrix}$};
	\node[noselAlAAAA, label={[shift={(0.0,-0.45)}, brown]\tiny $R_{0}^{2}(4)$}] (RN_1_2_4__Rationals) at (14,6) {$\begin{smallmatrix}1&&&&1\\&1&1&1\end{smallmatrix}$};
	\node[noselAlAAAA, label={[shift={(0.0,-0.45)}, brown]\tiny $R_{0}^{3}(4)$}] (RN_1_3_4__Rationals) at (18,6) {$\begin{smallmatrix}1&&&&1\\&1&1&1\end{smallmatrix}$};
	\node[noselAlAAAA, label={[shift={(0.0,-0.45)}, brown]\tiny $R_{0}^{3}(5)$}] (RN_1_3_5__Rationals) at (4,8) {$\begin{smallmatrix}1&&&&1\\&2&1&1\end{smallmatrix}$};
	\node[noselAlAAAA, label={[shift={(0.0,-0.45)}, brown]\tiny $R_{0}^{4}(5)$}] (RN_1_4_5__Rationals) at (8,8) {$\begin{smallmatrix}1&&&&1\\&1&2&1\end{smallmatrix}$};
	\node[noselAlAAAA, label={[shift={(0.0,-0.45)}, brown]\tiny $R_{0}^{1}(5)$}] (RN_1_1_5__Rationals) at (12,8) {$\begin{smallmatrix}1&&&&1\\&1&1&2\end{smallmatrix}$};
	\node[noselAlAAAA, label={[shift={(0.0,-0.45)}, brown]\tiny $R_{0}^{2}(5)$}] (RN_1_2_5__Rationals) at (16,8) {$\begin{smallmatrix}2&&&&2\\&1&1&1\end{smallmatrix}$};
	\node[noselAlAAAA, label={[shift={(0.0,-0.45)}, brown]\tiny $R_{0}^{3}(5)$}] (RN_1_3_5__Rationals_repeated) at (20,8) {$\begin{smallmatrix}1&&&&1\\&2&1&1\end{smallmatrix}$};
	\draw[blue,->] (RN_1_1_1__Rationals) -- (RN_1_1_2__Rationals);
	\draw[gray,dashed,->] (RN_1_1_1__Rationals) -- (RN_1_2_1__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[red,->] (RN_1_1_2__Rationals) -- (RN_1_2_1__Rationals);
	\draw[gray,dashed,->] (RN_1_1_2__Rationals) -- (RN_1_2_2__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[blue,->] (RN_1_1_2__Rationals) -- (RN_1_1_3__Rationals);
	\draw[blue,->] (RN_1_2_1__Rationals) -- (RN_1_2_2__Rationals);
	\draw[gray,dashed,->] (RN_1_2_1__Rationals) -- (RN_1_3_1__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[red,->] (RN_1_2_2__Rationals) -- (RN_1_3_1__Rationals);
	\draw[gray,dashed,->] (RN_1_2_2__Rationals) -- (RN_1_3_2__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[blue,->] (RN_1_2_2__Rationals) -- (RN_1_2_3__Rationals);
	\draw[blue,->] (RN_1_3_1__Rationals) -- (RN_1_3_2__Rationals);
	\draw[gray,dashed,->] (RN_1_3_1__Rationals) -- (RN_1_4_1__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[red,->] (RN_1_3_2__Rationals) -- (RN_1_4_1__Rationals);
	\draw[gray,dashed,->] (RN_1_3_2__Rationals) -- (RN_1_4_2__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[blue,->] (RN_1_3_2__Rationals) -- (RN_1_3_3__Rationals);
	\draw[gray,dashed,->] (RN_1_4_1__Rationals) -- (RN_1_1_1__Rationals_repeated) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[blue,->] (RN_1_4_1__Rationals) -- (RN_1_4_2__Rationals);
	\draw[red,->] (RN_1_4_2__Rationals) -- (RN_1_1_1__Rationals_repeated);
	\draw[gray,dashed,->,out=193,in=-13] (RN_1_4_2__Rationals) to node[midway,sloped,above]{{\tiny $\tau^{-1}$}} (RN_1_1_2__Rationals);
	\draw[blue,->] (RN_1_4_2__Rationals) -- (RN_1_4_3__Rationals_repeated);
	\draw[red,->] (RN_1_4_3__Rationals) -- (RN_1_1_2__Rationals);
	\draw[blue,->] (RN_1_4_3__Rationals) -- (RN_1_4_4__Rationals);
	\draw[gray,dashed,->] (RN_1_4_3__Rationals) -- (RN_1_1_3__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[red,->] (RN_1_4_4__Rationals) -- (RN_1_1_3__Rationals);
	\draw[gray,dashed,->] (RN_1_4_4__Rationals) -- (RN_1_1_4__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[blue,->] (RN_1_4_4__Rationals) -- (RN_1_4_5__Rationals);
	\draw[red,->] (RN_1_1_3__Rationals) -- (RN_1_2_2__Rationals);
	\draw[blue,->] (RN_1_1_3__Rationals) -- (RN_1_1_4__Rationals);
	\draw[gray,dashed,->] (RN_1_1_3__Rationals) -- (RN_1_2_3__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[red,->] (RN_1_1_4__Rationals) -- (RN_1_2_3__Rationals);
	\draw[gray,dashed,->] (RN_1_1_4__Rationals) -- (RN_1_2_4__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[blue,->] (RN_1_1_4__Rationals) -- (RN_1_1_5__Rationals);
	\draw[red,->] (RN_1_2_3__Rationals) -- (RN_1_3_2__Rationals);
	\draw[blue,->] (RN_1_2_3__Rationals) -- (RN_1_2_4__Rationals);
	\draw[gray,dashed,->] (RN_1_2_3__Rationals) -- (RN_1_3_3__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[red,->] (RN_1_2_4__Rationals) -- (RN_1_3_3__Rationals);
	\draw[gray,dashed,->] (RN_1_2_4__Rationals) -- (RN_1_3_4__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[blue,->] (RN_1_2_4__Rationals) -- (RN_1_2_5__Rationals);
	\draw[red,->] (RN_1_3_3__Rationals) -- (RN_1_4_2__Rationals);
	\draw[gray,dashed,->] (RN_1_3_3__Rationals) -- (RN_1_4_3__Rationals_repeated) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[blue,->] (RN_1_3_3__Rationals) -- (RN_1_3_4__Rationals);
	\draw[red,->] (RN_1_3_4__Rationals) -- (RN_1_4_3__Rationals_repeated);
	\draw[gray,dashed,->,out=193,in=-13] (RN_1_3_4__Rationals) to node[midway,sloped,above]{{\tiny $\tau^{-1}$}} (RN_1_4_4__Rationals);
	\draw[blue,->] (RN_1_3_4__Rationals) -- (RN_1_3_5__Rationals_repeated);
	\draw[red,->] (RN_1_3_5__Rationals) -- (RN_1_4_4__Rationals);
	\draw[gray,dashed,->] (RN_1_3_5__Rationals) -- (RN_1_4_5__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[red,->] (RN_1_4_5__Rationals) -- (RN_1_1_4__Rationals);
	\draw[gray,dashed,->] (RN_1_4_5__Rationals) -- (RN_1_1_5__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[red,->] (RN_1_1_5__Rationals) -- (RN_1_2_4__Rationals);
	\draw[gray,dashed,->] (RN_1_1_5__Rationals) -- (RN_1_2_5__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[red,->] (RN_1_2_5__Rationals) -- (RN_1_3_4__Rationals);
	\draw[gray,dashed,->] (RN_1_2_5__Rationals) -- (RN_1_3_5__Rationals_repeated) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[dashed,-] (RN_1_1_1__Rationals) -- (RN_1_4_3__Rationals);
	\draw[dashed,-] (RN_1_1_1__Rationals_repeated) -- (RN_1_4_3__Rationals_repeated);
	\draw[dashed,-] (RN_1_4_3__Rationals) -- (RN_1_3_5__Rationals);
	\draw[dashed,-] (RN_1_4_3__Rationals_repeated) -- (RN_1_3_5__Rationals_repeated);
\end{tikzpicture}
\end{scaletikzpicturetowidth}
\end{center}

\end{figure}
\begin{fleqn}
\begin{align*}
R_{0}^{1}(1):\ & - \\
R_{0}^{1}(2):\ & \bigl(R_{0}^{2}(1),R_{0}^{1}(1)\bigr),\ \bigl(I(0,4),P(0,1)\bigr)\\
R_{0}^{2}(1):\ & \bigl(I(0,5),P(0,1)\bigr)\\
R_{0}^{2}(2):\ & \bigl(R_{0}^{3}(1),R_{0}^{2}(1)\bigr),\ \bigl(I(0,5),P(0,2)\bigr)\\
R_{0}^{3}(1):\ & - \\
R_{0}^{3}(2):\ & \bigl(R_{0}^{4}(1),R_{0}^{3}(1)\bigr)\\
R_{0}^{4}(1):\ & - \\
R_{0}^{4}(2):\ & \bigl(R_{0}^{1}(1),R_{0}^{4}(1)\bigr)\\
R_{0}^{4}(3):\ & \bigl(R_{0}^{1}(2),R_{0}^{4}(1)\bigr),\ \bigl(R_{0}^{2}(1),R_{0}^{4}(2)\bigr),\ \bigl(I(0,3),P(0,1)\bigr)\\
R_{0}^{1}(3):\ & \bigl(R_{0}^{2}(2),R_{0}^{1}(1)\bigr),\ \bigl(R_{0}^{3}(1),R_{0}^{1}(2)\bigr),\ \bigl(I(0,4),P(0,2)\bigr)\\
R_{0}^{2}(3):\ & \bigl(R_{0}^{3}(2),R_{0}^{2}(1)\bigr),\ \bigl(R_{0}^{4}(1),R_{0}^{2}(2)\bigr),\ \bigl(I(0,5),P(0,3)\bigr)\\
R_{0}^{3}(3):\ & \bigl(R_{0}^{4}(2),R_{0}^{3}(1)\bigr),\ \bigl(R_{0}^{1}(1),R_{0}^{3}(2)\bigr)\\
\end{align*}
\end{fleqn}
\clearpage

\subsection[Schofield pairs for the quiver $\Delta(\widetilde{\mathbb{A}}_{2,2})$]{Schofield pairs for the quiver $\Delta(\widetilde{\mathbb{A}}_{2,2})$ -- {\usebox{\boxAAlAA}}}
\[
\vcenter{\hbox{\xymatrix{  & 2\ar[dl] &   &  \\
1 &   & 4\ar[ul]\ar[dl] &  \\
  & 3\ar[ul] &   &  }}}\qquad C_{\Delta(\widetilde{\mathbb{A}}_{2,2})} = \begin{bmatrix}1 & 1 & 1 & 2\\
0 & 1 & 0 & 1\\
0 & 0 & 1 & 1\\
0 & 0 & 0 & 1\end{bmatrix}\quad\Phi_{\Delta(\widetilde{\mathbb{A}}_{2,2})} = \begin{bmatrix}-1 & 1 & 1 & 0\\
-1 & 0 & 1 & 1\\
-1 & 1 & 0 & 1\\
-2 & 1 & 1 & 1\end{bmatrix}
\]
\subsubsection{Schofield pairs associated to preprojective exceptional modules}

\begin{figure}[ht]

\begin{center}
\begin{scaletikzpicturetowidth}{\textwidth}
\begin{tikzpicture}[scale=\tikzscale]
	\node[noselAAlAA, label={[shift={(0.4,-0.2)}, brown]\tiny $P(0,1)$}] (P_0_1__Rationals) at (0,0) {$\begin{smallmatrix}&0\\1&&0\\&0\end{smallmatrix}$};
	\node[noselAAlAA, label={[shift={(0.4,-0.2)}, brown]\tiny $P(0,2)$}] (P_0_2__Rationals) at (0,-1) {$\begin{smallmatrix}&1\\1&&0\\&0\end{smallmatrix}$};
	\node[noselAAlAA, label={[shift={(0.4,-0.2)}, brown]\tiny $P(0,3)$}] (P_0_3__Rationals) at (0,-3) {$\begin{smallmatrix}&0\\1&&0\\&1\end{smallmatrix}$};
	\node[seloneAAlAA, label={[shift={(0.4,-0.2)}, brown]\tiny $P(0,4)$}] (P_0_4__Rationals) at (0,-2) {$\begin{smallmatrix}&1\\2&&1\\&1\end{smallmatrix}$};
	\node[noselAAlAA, label={[shift={(0.4,-0.2)}, brown]\tiny $P(1,1)$}] (P_1_1__Rationals) at (4,0) {$\begin{smallmatrix}&1\\1&&0\\&1\end{smallmatrix}$};
	\node[seloneAAlAA, label={[shift={(0.4,-0.2)}, brown]\tiny $P(1,2)$}] (P_1_2__Rationals) at (4,-1) {$\begin{smallmatrix}&1\\2&&1\\&2\end{smallmatrix}$};
	\node[seloneAAlAA, label={[shift={(0.4,-0.2)}, brown]\tiny $P(1,3)$}] (P_1_3__Rationals) at (4,-3) {$\begin{smallmatrix}&2\\2&&1\\&1\end{smallmatrix}$};
	\node[noselAAlAA, label={[shift={(0.4,-0.2)}, brown]\tiny $P(1,4)$}] (P_1_4__Rationals) at (4,-2) {$\begin{smallmatrix}&2\\2&&1\\&2\end{smallmatrix}$};
	\node[seloneAAlAA, label={[shift={(0.4,-0.2)}, brown]\tiny $P(2,1)$}] (P_2_1__Rationals) at (8,0) {$\begin{smallmatrix}&2\\3&&2\\&2\end{smallmatrix}$};
	\node[noselAAlAA, label={[shift={(0.4,-0.2)}, brown]\tiny $P(2,2)$}] (P_2_2__Rationals) at (8,-1) {$\begin{smallmatrix}&3\\3&&2\\&2\end{smallmatrix}$};
	\node[noselAAlAA, label={[shift={(0.4,-0.2)}, brown]\tiny $P(2,3)$}] (P_2_3__Rationals) at (8,-3) {$\begin{smallmatrix}&2\\3&&2\\&3\end{smallmatrix}$};
	\node[noselAAlAA, label={[shift={(0.4,-0.2)}, brown]\tiny $P(2,4)$}] (P_2_4__Rationals) at (8,-2) {$\begin{smallmatrix}&3\\4&&3\\&3\end{smallmatrix}$};
	\node[noselAAlAA, label={[shift={(0.4,-0.2)}, brown]\tiny $P(3,1)$}] (P_3_1__Rationals) at (12,0) {$\begin{smallmatrix}&3\\3&&2\\&3\end{smallmatrix}$};
	\node[noselAAlAA, label={[shift={(0.4,-0.2)}, brown]\tiny $P(3,2)$}] (P_3_2__Rationals) at (12,-1) {$\begin{smallmatrix}&3\\4&&3\\&4\end{smallmatrix}$};
	\node[noselAAlAA, label={[shift={(0.4,-0.2)}, brown]\tiny $P(3,3)$}] (P_3_3__Rationals) at (12,-3) {$\begin{smallmatrix}&4\\4&&3\\&3\end{smallmatrix}$};
	\node[noselAAlAA, label={[shift={(0.4,-0.2)}, brown]\tiny $P(3,4)$}] (P_3_4__Rationals) at (12,-2) {$\begin{smallmatrix}&4\\4&&3\\&4\end{smallmatrix}$};
	\node at (13,-1.5) {$\dots$};
	\draw[blue,->] (P_0_1__Rationals) -- (P_0_2__Rationals);
	\draw[blue,->] (P_0_1__Rationals) to [out=135,in=-135] (P_0_3__Rationals);
	\draw[blue,->] (P_0_2__Rationals) -- (P_0_4__Rationals);
	\draw[blue,->] (P_0_3__Rationals) -- (P_0_4__Rationals);
	\draw[gray,dashed,->] (P_0_1__Rationals) -- (P_1_1__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[blue,->] (P_0_2__Rationals) -- (P_1_1__Rationals);
	\draw[blue,->] (P_0_3__Rationals) -- (P_1_1__Rationals);
	\draw[gray,dashed,->] (P_0_2__Rationals) -- (P_1_2__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[blue,->] (P_0_4__Rationals) -- (P_1_2__Rationals);
	\draw[blue,->] (P_1_1__Rationals) -- (P_1_2__Rationals);
	\draw[gray,dashed,->] (P_0_3__Rationals) -- (P_1_3__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[blue,->] (P_0_4__Rationals) -- (P_1_3__Rationals);
	\draw[blue,->] (P_1_1__Rationals) to [out=135,in=-135] (P_1_3__Rationals);
	\draw[gray,dashed,->] (P_0_4__Rationals) -- (P_1_4__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[blue,->] (P_1_2__Rationals) -- (P_1_4__Rationals);
	\draw[blue,->] (P_1_3__Rationals) -- (P_1_4__Rationals);
	\draw[gray,dashed,->] (P_1_1__Rationals) -- (P_2_1__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[blue,->] (P_1_2__Rationals) -- (P_2_1__Rationals);
	\draw[blue,->] (P_1_3__Rationals) -- (P_2_1__Rationals);
	\draw[gray,dashed,->] (P_1_2__Rationals) -- (P_2_2__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[blue,->] (P_1_4__Rationals) -- (P_2_2__Rationals);
	\draw[blue,->] (P_2_1__Rationals) -- (P_2_2__Rationals);
	\draw[gray,dashed,->] (P_1_3__Rationals) -- (P_2_3__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[blue,->] (P_1_4__Rationals) -- (P_2_3__Rationals);
	\draw[blue,->] (P_2_1__Rationals) to [out=135,in=-135] (P_2_3__Rationals);
	\draw[gray,dashed,->] (P_1_4__Rationals) -- (P_2_4__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[blue,->] (P_2_2__Rationals) -- (P_2_4__Rationals);
	\draw[blue,->] (P_2_3__Rationals) -- (P_2_4__Rationals);
	\draw[gray,dashed,->] (P_2_1__Rationals) -- (P_3_1__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[blue,->] (P_2_2__Rationals) -- (P_3_1__Rationals);
	\draw[blue,->] (P_2_3__Rationals) -- (P_3_1__Rationals);
	\draw[gray,dashed,->] (P_2_2__Rationals) -- (P_3_2__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[blue,->] (P_2_4__Rationals) -- (P_3_2__Rationals);
	\draw[blue,->] (P_3_1__Rationals) -- (P_3_2__Rationals);
	\draw[gray,dashed,->] (P_2_3__Rationals) -- (P_3_3__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[blue,->] (P_2_4__Rationals) -- (P_3_3__Rationals);
	\draw[blue,->] (P_3_1__Rationals) to [out=135,in=-135] (P_3_3__Rationals);
	\draw[gray,dashed,->] (P_2_4__Rationals) -- (P_3_4__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[blue,->] (P_3_2__Rationals) -- (P_3_4__Rationals);
	\draw[blue,->] (P_3_3__Rationals) -- (P_3_4__Rationals);
\end{tikzpicture}
\end{scaletikzpicturetowidth}
\end{center}

\end{figure}
\medskip{}
\subsubsection*{Modules of the form $P(n,1)$}

Defect: $\partial P(n,1) = -1$, for $n\ge 0$.

\begin{fleqn}
\begin{align*}
P(0,1):\ & - \\
P(1,1):\ & \bigl(R_{0}^{1}(1),P(0,2)\bigr),\ \bigl(R_{\infty}^{1}(1),P(0,3)\bigr)\\
P(2,1):\ & \bigl(R_{0}^{2}(1),P(1,2)\bigr),\ \bigl(R_{\infty}^{2}(1),P(1,3)\bigr),\ \bigl(2I(1,4),3P(0,1)\bigr)\\
P(n,1):\ & \bigl(R_{0}^{(n-1)\bmod 2+1}(1),P(n-1,2)\bigr),\ \bigl(R_{\infty}^{(n-1)\bmod 2+1}(1),P(n-1,3)\bigr),\ \bigl(uI,(u+1)P\bigr),\ n>2\\
\end{align*}
\end{fleqn}
\subsubsection*{Modules of the form $P(n,2)$}

Defect: $\partial P(n,2) = -1$, for $n\ge 0$.

\begin{fleqn}
\begin{align*}
P(0,2):\ & \bigl(R_{\infty}^{1}(1),P(0,1)\bigr)\\
P(1,2):\ & \bigl(R_{0}^{1}(1),P(0,4)\bigr),\ \bigl(R_{\infty}^{2}(1),P(1,1)\bigr),\ \bigl(I(0,2),2P(0,3)\bigr)\\
P(n,2):\ & \bigl(R_{0}^{(n-1)\bmod 2+1}(1),P(n-1,4)\bigr),\ \bigl(R_{\infty}^{n\bmod 2+1}(1),P(n,1)\bigr),\ \bigl(uI,(u+1)P\bigr),\ n>1\\
\end{align*}
\end{fleqn}
\subsubsection*{Modules of the form $P(n,3)$}

Defect: $\partial P(n,3) = -1$, for $n\ge 0$.

\begin{fleqn}
\begin{align*}
P(0,3):\ & \bigl(R_{0}^{1}(1),P(0,1)\bigr)\\
P(1,3):\ & \bigl(R_{\infty}^{1}(1),P(0,4)\bigr),\ \bigl(R_{0}^{2}(1),P(1,1)\bigr),\ \bigl(I(0,3),2P(0,2)\bigr)\\
P(n,3):\ & \bigl(R_{\infty}^{(n-1)\bmod 2+1}(1),P(n-1,4)\bigr),\ \bigl(R_{0}^{n\bmod 2+1}(1),P(n,1)\bigr),\ \bigl(uI,(u+1)P\bigr),\ n>1\\
\end{align*}
\end{fleqn}
\subsubsection*{Modules of the form $P(n,4)$}

Defect: $\partial P(n,4) = -1$, for $n\ge 0$.

\begin{fleqn}
\begin{align*}
P(0,4):\ & \bigl(R_{\infty}^{2}(1),P(0,2)\bigr),\ \bigl(R_{0}^{2}(1),P(0,3)\bigr),\ \bigl(I(1,4),2P(0,1)\bigr)\\
P(n,4):\ & \bigl(R_{\infty}^{(n+1)\bmod 2+1}(1),P(n,2)\bigr),\ \bigl(R_{0}^{(n+1)\bmod 2+1}(1),P(n,3)\bigr),\ \bigl(uI,(u+1)P\bigr),\ n>0\\
\end{align*}
\end{fleqn}
\subsubsection{Schofield pairs associated to preinjective exceptional modules}

\begin{figure}[ht]

\begin{center}
\begin{scaletikzpicturetowidth}{\textwidth}
\begin{tikzpicture}[scale=\tikzscale]
	\node[seloneAAlAA, label={[shift={(0.4,-0.2)}, brown]\tiny $I(0,1)$}] (I_0_1__Rationals) at (0,0) {$\begin{smallmatrix}&1\\1&&2\\&1\end{smallmatrix}$};
	\node[noselAAlAA, label={[shift={(0.4,-0.2)}, brown]\tiny $I(0,2)$}] (I_0_2__Rationals) at (0,-1) {$\begin{smallmatrix}&1\\0&&1\\&0\end{smallmatrix}$};
	\node[noselAAlAA, label={[shift={(0.4,-0.2)}, brown]\tiny $I(0,3)$}] (I_0_3__Rationals) at (0,-3) {$\begin{smallmatrix}&0\\0&&1\\&1\end{smallmatrix}$};
	\node[noselAAlAA, label={[shift={(0.4,-0.2)}, brown]\tiny $I(0,4)$}] (I_0_4__Rationals) at (0,-2) {$\begin{smallmatrix}&0\\0&&1\\&0\end{smallmatrix}$};
	\node[noselAAlAA, label={[shift={(0.4,-0.2)}, brown]\tiny $I(1,1)$}] (I_1_1__Rationals) at (-4,0) {$\begin{smallmatrix}&2\\1&&2\\&2\end{smallmatrix}$};
	\node[seloneAAlAA, label={[shift={(0.4,-0.2)}, brown]\tiny $I(1,2)$}] (I_1_2__Rationals) at (-4,-1) {$\begin{smallmatrix}&1\\1&&2\\&2\end{smallmatrix}$};
	\node[seloneAAlAA, label={[shift={(0.4,-0.2)}, brown]\tiny $I(1,3)$}] (I_1_3__Rationals) at (-4,-3) {$\begin{smallmatrix}&2\\1&&2\\&1\end{smallmatrix}$};
	\node[noselAAlAA, label={[shift={(0.4,-0.2)}, brown]\tiny $I(1,4)$}] (I_1_4__Rationals) at (-4,-2) {$\begin{smallmatrix}&1\\0&&1\\&1\end{smallmatrix}$};
	\node[noselAAlAA, label={[shift={(0.4,-0.2)}, brown]\tiny $I(2,1)$}] (I_2_1__Rationals) at (-8,0) {$\begin{smallmatrix}&3\\3&&4\\&3\end{smallmatrix}$};
	\node[noselAAlAA, label={[shift={(0.4,-0.2)}, brown]\tiny $I(2,2)$}] (I_2_2__Rationals) at (-8,-1) {$\begin{smallmatrix}&3\\2&&3\\&2\end{smallmatrix}$};
	\node[noselAAlAA, label={[shift={(0.4,-0.2)}, brown]\tiny $I(2,3)$}] (I_2_3__Rationals) at (-8,-3) {$\begin{smallmatrix}&2\\2&&3\\&3\end{smallmatrix}$};
	\node[seloneAAlAA, label={[shift={(0.4,-0.2)}, brown]\tiny $I(2,4)$}] (I_2_4__Rationals) at (-8,-2) {$\begin{smallmatrix}&2\\2&&3\\&2\end{smallmatrix}$};
	\node[noselAAlAA, label={[shift={(0.4,-0.2)}, brown]\tiny $I(3,1)$}] (I_3_1__Rationals) at (-12,0) {$\begin{smallmatrix}&4\\3&&4\\&4\end{smallmatrix}$};
	\node[noselAAlAA, label={[shift={(0.4,-0.2)}, brown]\tiny $I(3,2)$}] (I_3_2__Rationals) at (-12,-1) {$\begin{smallmatrix}&3\\3&&4\\&4\end{smallmatrix}$};
	\node[noselAAlAA, label={[shift={(0.4,-0.2)}, brown]\tiny $I(3,3)$}] (I_3_3__Rationals) at (-12,-3) {$\begin{smallmatrix}&4\\3&&4\\&3\end{smallmatrix}$};
	\node[noselAAlAA, label={[shift={(0.4,-0.2)}, brown]\tiny $I(3,4)$}] (I_3_4__Rationals) at (-12,-2) {$\begin{smallmatrix}&3\\2&&3\\&3\end{smallmatrix}$};
	\node at (-13,-1.5) {$\dots$};
	\draw[red,->] (I_1_2__Rationals) -- (I_0_1__Rationals);
	\draw[red,->] (I_1_3__Rationals) -- (I_0_1__Rationals);
	\draw[red,->] (I_0_1__Rationals) -- (I_0_2__Rationals);
	\draw[red,->] (I_1_4__Rationals) -- (I_0_2__Rationals);
	\draw[red,->] (I_0_1__Rationals) to [out=135,in=-135] (I_0_3__Rationals);
	\draw[red,->] (I_1_4__Rationals) -- (I_0_3__Rationals);
	\draw[red,->] (I_0_2__Rationals) -- (I_0_4__Rationals);
	\draw[red,->] (I_0_3__Rationals) -- (I_0_4__Rationals);
	\draw[gray,dashed,->] (I_0_1__Rationals) -- (I_1_1__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_2_2__Rationals) -- (I_1_1__Rationals);
	\draw[red,->] (I_2_3__Rationals) -- (I_1_1__Rationals);
	\draw[gray,dashed,->] (I_0_2__Rationals) -- (I_1_2__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_1_1__Rationals) -- (I_1_2__Rationals);
	\draw[red,->] (I_2_4__Rationals) -- (I_1_2__Rationals);
	\draw[gray,dashed,->] (I_0_3__Rationals) -- (I_1_3__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_1_1__Rationals) to [out=135,in=-135] (I_1_3__Rationals);
	\draw[red,->] (I_2_4__Rationals) -- (I_1_3__Rationals);
	\draw[gray,dashed,->] (I_0_4__Rationals) -- (I_1_4__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_1_2__Rationals) -- (I_1_4__Rationals);
	\draw[red,->] (I_1_3__Rationals) -- (I_1_4__Rationals);
	\draw[gray,dashed,->] (I_1_1__Rationals) -- (I_2_1__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_3_2__Rationals) -- (I_2_1__Rationals);
	\draw[red,->] (I_3_3__Rationals) -- (I_2_1__Rationals);
	\draw[gray,dashed,->] (I_1_2__Rationals) -- (I_2_2__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_2_1__Rationals) -- (I_2_2__Rationals);
	\draw[red,->] (I_3_4__Rationals) -- (I_2_2__Rationals);
	\draw[gray,dashed,->] (I_1_3__Rationals) -- (I_2_3__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_2_1__Rationals) to [out=135,in=-135] (I_2_3__Rationals);
	\draw[red,->] (I_3_4__Rationals) -- (I_2_3__Rationals);
	\draw[gray,dashed,->] (I_1_4__Rationals) -- (I_2_4__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_2_2__Rationals) -- (I_2_4__Rationals);
	\draw[red,->] (I_2_3__Rationals) -- (I_2_4__Rationals);
	\draw[gray,dashed,->] (I_2_1__Rationals) -- (I_3_1__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[gray,dashed,->] (I_2_2__Rationals) -- (I_3_2__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_3_1__Rationals) -- (I_3_2__Rationals);
	\draw[gray,dashed,->] (I_2_3__Rationals) -- (I_3_3__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_3_1__Rationals) to [out=135,in=-135] (I_3_3__Rationals);
	\draw[gray,dashed,->] (I_2_4__Rationals) -- (I_3_4__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_3_2__Rationals) -- (I_3_4__Rationals);
	\draw[red,->] (I_3_3__Rationals) -- (I_3_4__Rationals);
\end{tikzpicture}
\end{scaletikzpicturetowidth}
\end{center}

\end{figure}
\medskip{}
\subsubsection*{Modules of the form $I(n,1)$}

Defect: $\partial I(n,1) = 1$, for $n\ge 0$.

\begin{fleqn}
\begin{align*}
I(0,1):\ & \bigl(I(0,2),R_{\infty}^{2}(1)\bigr),\ \bigl(I(0,3),R_{0}^{2}(1)\bigr),\ \bigl(2I(0,4),P(1,1)\bigr)\\
I(n,1):\ & \bigl(I(n,2),R_{\infty}^{(-n+1)\bmod 2+1}(1)\bigr),\ \bigl(I(n,3),R_{0}^{(-n+1)\bmod 2+1}(1)\bigr),\ \bigl((v+1)I,vP\bigr),\ n>0\\
\end{align*}
\end{fleqn}
\subsubsection*{Modules of the form $I(n,2)$}

Defect: $\partial I(n,2) = 1$, for $n\ge 0$.

\begin{fleqn}
\begin{align*}
I(0,2):\ & \bigl(I(0,4),R_{\infty}^{1}(1)\bigr)\\
I(1,2):\ & \bigl(I(0,1),R_{0}^{1}(1)\bigr),\ \bigl(I(1,4),R_{\infty}^{2}(1)\bigr),\ \bigl(2I(0,3),P(0,2)\bigr)\\
I(n,2):\ & \bigl(I(n-1,1),R_{0}^{(-n+1)\bmod 2+1}(1)\bigr),\ \bigl(I(n,4),R_{\infty}^{(-n+2)\bmod 2+1}(1)\bigr),\ \bigl((v+1)I,vP\bigr),\ n>1\\
\end{align*}
\end{fleqn}
\subsubsection*{Modules of the form $I(n,3)$}

Defect: $\partial I(n,3) = 1$, for $n\ge 0$.

\begin{fleqn}
\begin{align*}
I(0,3):\ & \bigl(I(0,4),R_{0}^{1}(1)\bigr)\\
I(1,3):\ & \bigl(I(0,1),R_{\infty}^{1}(1)\bigr),\ \bigl(I(1,4),R_{0}^{2}(1)\bigr),\ \bigl(2I(0,2),P(0,3)\bigr)\\
I(n,3):\ & \bigl(I(n-1,1),R_{\infty}^{(-n+1)\bmod 2+1}(1)\bigr),\ \bigl(I(n,4),R_{0}^{(-n+2)\bmod 2+1}(1)\bigr),\ \bigl((v+1)I,vP\bigr),\ n>1\\
\end{align*}
\end{fleqn}
\subsubsection*{Modules of the form $I(n,4)$}

Defect: $\partial I(n,4) = 1$, for $n\ge 0$.

\begin{fleqn}
\begin{align*}
I(0,4):\ & - \\
I(1,4):\ & \bigl(I(0,2),R_{0}^{1}(1)\bigr),\ \bigl(I(0,3),R_{\infty}^{1}(1)\bigr)\\
I(2,4):\ & \bigl(I(1,2),R_{0}^{2}(1)\bigr),\ \bigl(I(1,3),R_{\infty}^{2}(1)\bigr),\ \bigl(3I(0,4),2P(1,1)\bigr)\\
I(n,4):\ & \bigl(I(n-1,2),R_{0}^{(-n+3)\bmod 2+1}(1)\bigr),\ \bigl(I(n-1,3),R_{\infty}^{(-n+3)\bmod 2+1}(1)\bigr),\ \bigl((v+1)I,vP\bigr),\ n>2\\
\end{align*}
\end{fleqn}
\subsubsection{Schofield pairs associated to regular exceptional modules}

\subsubsection*{The non-homogeneous tube $\mathcal{T}_{0}^{\Delta(\widetilde{\mathbb{A}}_{2,2})}$}

\begin{figure}[ht]

\begin{center}
\captionof{figure}{\vspace*{-10pt}$\mathcal{T}_{0}^{\Delta(\widetilde{\mathbb{A}}_{2,2})}$}
\begin{scaletikzpicturetowidth}{0.40000000000000002\textwidth}
\begin{tikzpicture}[scale=\tikzscale]
	\node[seltwoAAlAA, label={[shift={(0.0,-0.45)}, brown]\tiny $R_{0}^{1}(1)$}] (RN_1_1_1__Rationals) at (4,0) {$\begin{smallmatrix}&0\\0&&0\\&1\end{smallmatrix}$};
	\node[seltwoAAlAA, label={[shift={(0.0,-0.45)}, brown]\tiny $R_{0}^{2}(1)$}] (RN_1_2_1__Rationals) at (8,0) {$\begin{smallmatrix}&1\\1&&1\\&0\end{smallmatrix}$};
	\node[seltwoAAlAA, label={[shift={(0.0,-0.45)}, brown]\tiny $R_{0}^{1}(1)$}] (RN_1_1_1__Rationals_repeated) at (12,0) {$\begin{smallmatrix}&0\\0&&0\\&1\end{smallmatrix}$};
	\node[noselAAlAA, label={[shift={(0.0,-0.45)}, brown]\tiny $R_{0}^{1}(2)$}] (RN_1_1_2__Rationals) at (6,2) {$\begin{smallmatrix}&1\\1&&1\\&1\end{smallmatrix}$};
	\node[noselAAlAA, label={[shift={(0.0,-0.45)}, brown]\tiny $R_{0}^{2}(2)$}] (RN_1_2_2__Rationals) at (10,2) {$\begin{smallmatrix}&1\\1&&1\\&1\end{smallmatrix}$};
	\node[noselAAlAA, label={[shift={(0.0,-0.45)}, brown]\tiny $R_{0}^{2}(3)$}] (RN_1_2_3__Rationals) at (4,4) {$\begin{smallmatrix}&2\\2&&2\\&1\end{smallmatrix}$};
	\node[noselAAlAA, label={[shift={(0.0,-0.45)}, brown]\tiny $R_{0}^{1}(3)$}] (RN_1_1_3__Rationals) at (8,4) {$\begin{smallmatrix}&1\\1&&1\\&2\end{smallmatrix}$};
	\node[noselAAlAA, label={[shift={(0.0,-0.45)}, brown]\tiny $R_{0}^{2}(3)$}] (RN_1_2_3__Rationals_repeated) at (12,4) {$\begin{smallmatrix}&2\\2&&2\\&1\end{smallmatrix}$};
	\draw[blue,->] (RN_1_1_1__Rationals) -- (RN_1_1_2__Rationals);
	\draw[gray,dashed,->] (RN_1_1_1__Rationals) -- (RN_1_2_1__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[red,->] (RN_1_1_2__Rationals) -- (RN_1_2_1__Rationals);
	\draw[gray,dashed,->] (RN_1_1_2__Rationals) -- (RN_1_2_2__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[blue,->] (RN_1_1_2__Rationals) -- (RN_1_1_3__Rationals);
	\draw[gray,dashed,->] (RN_1_2_1__Rationals) -- (RN_1_1_1__Rationals_repeated) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[blue,->] (RN_1_2_1__Rationals) -- (RN_1_2_2__Rationals);
	\draw[red,->] (RN_1_2_2__Rationals) -- (RN_1_1_1__Rationals_repeated);
	\draw[gray,dashed,->,out=219,in=-39] (RN_1_2_2__Rationals) to node[midway,sloped,above]{{\tiny $\tau^{-1}$}} (RN_1_1_2__Rationals);
	\draw[blue,->] (RN_1_2_2__Rationals) -- (RN_1_2_3__Rationals_repeated);
	\draw[red,->] (RN_1_2_3__Rationals) -- (RN_1_1_2__Rationals);
	\draw[gray,dashed,->] (RN_1_2_3__Rationals) -- (RN_1_1_3__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[red,->] (RN_1_1_3__Rationals) -- (RN_1_2_2__Rationals);
	\draw[gray,dashed,->] (RN_1_1_3__Rationals) -- (RN_1_2_3__Rationals_repeated) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[dashed,-] (RN_1_1_1__Rationals) -- (RN_1_2_3__Rationals);
	\draw[dashed,-] (RN_1_1_1__Rationals_repeated) -- (RN_1_2_3__Rationals_repeated);
\end{tikzpicture}
\end{scaletikzpicturetowidth}
\end{center}

\end{figure}
\begin{fleqn}
\begin{align*}
R_{0}^{1}(1):\ & - \\
R_{0}^{2}(1):\ & \bigl(I(0,2),P(0,1)\bigr),\ \bigl(I(0,4),P(0,2)\bigr)\\
\end{align*}
\end{fleqn}
\subsubsection*{The non-homogeneous tube $\mathcal{T}_{\infty}^{\Delta(\widetilde{\mathbb{A}}_{2,2})}$}

\begin{figure}[ht]

\begin{center}
\captionof{figure}{\vspace*{-10pt}$\mathcal{T}_{\infty}^{\Delta(\widetilde{\mathbb{A}}_{2,2})}$}
\begin{scaletikzpicturetowidth}{0.40000000000000002\textwidth}
\begin{tikzpicture}[scale=\tikzscale]
	\node[seltwoAAlAA, label={[shift={(0.0,-0.45)}, brown]\tiny $R_{\infty}^{1}(1)$}] (RN_2_1_1__Rationals) at (4,0) {$\begin{smallmatrix}&1\\0&&0\\&0\end{smallmatrix}$};
	\node[seltwoAAlAA, label={[shift={(0.0,-0.45)}, brown]\tiny $R_{\infty}^{2}(1)$}] (RN_2_2_1__Rationals) at (8,0) {$\begin{smallmatrix}&0\\1&&1\\&1\end{smallmatrix}$};
	\node[seltwoAAlAA, label={[shift={(0.0,-0.45)}, brown]\tiny $R_{\infty}^{1}(1)$}] (RN_2_1_1__Rationals_repeated) at (12,0) {$\begin{smallmatrix}&1\\0&&0\\&0\end{smallmatrix}$};
	\node[noselAAlAA, label={[shift={(0.0,-0.45)}, brown]\tiny $R_{\infty}^{1}(2)$}] (RN_2_1_2__Rationals) at (6,2) {$\begin{smallmatrix}&1\\1&&1\\&1\end{smallmatrix}$};
	\node[noselAAlAA, label={[shift={(0.0,-0.45)}, brown]\tiny $R_{\infty}^{2}(2)$}] (RN_2_2_2__Rationals) at (10,2) {$\begin{smallmatrix}&1\\1&&1\\&1\end{smallmatrix}$};
	\node[noselAAlAA, label={[shift={(0.0,-0.45)}, brown]\tiny $R_{\infty}^{2}(3)$}] (RN_2_2_3__Rationals) at (4,4) {$\begin{smallmatrix}&1\\2&&2\\&2\end{smallmatrix}$};
	\node[noselAAlAA, label={[shift={(0.0,-0.45)}, brown]\tiny $R_{\infty}^{1}(3)$}] (RN_2_1_3__Rationals) at (8,4) {$\begin{smallmatrix}&2\\1&&1\\&1\end{smallmatrix}$};
	\node[noselAAlAA, label={[shift={(0.0,-0.45)}, brown]\tiny $R_{\infty}^{2}(3)$}] (RN_2_2_3__Rationals_repeated) at (12,4) {$\begin{smallmatrix}&1\\2&&2\\&2\end{smallmatrix}$};
	\draw[blue,->] (RN_2_1_1__Rationals) -- (RN_2_1_2__Rationals);
	\draw[gray,dashed,->] (RN_2_1_1__Rationals) -- (RN_2_2_1__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[red,->] (RN_2_1_2__Rationals) -- (RN_2_2_1__Rationals);
	\draw[gray,dashed,->] (RN_2_1_2__Rationals) -- (RN_2_2_2__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[blue,->] (RN_2_1_2__Rationals) -- (RN_2_1_3__Rationals);
	\draw[gray,dashed,->] (RN_2_2_1__Rationals) -- (RN_2_1_1__Rationals_repeated) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[blue,->] (RN_2_2_1__Rationals) -- (RN_2_2_2__Rationals);
	\draw[red,->] (RN_2_2_2__Rationals) -- (RN_2_1_1__Rationals_repeated);
	\draw[gray,dashed,->,out=219,in=-39] (RN_2_2_2__Rationals) to node[midway,sloped,above]{{\tiny $\tau^{-1}$}} (RN_2_1_2__Rationals);
	\draw[blue,->] (RN_2_2_2__Rationals) -- (RN_2_2_3__Rationals_repeated);
	\draw[red,->] (RN_2_2_3__Rationals) -- (RN_2_1_2__Rationals);
	\draw[gray,dashed,->] (RN_2_2_3__Rationals) -- (RN_2_1_3__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[red,->] (RN_2_1_3__Rationals) -- (RN_2_2_2__Rationals);
	\draw[gray,dashed,->] (RN_2_1_3__Rationals) -- (RN_2_2_3__Rationals_repeated) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[dashed,-] (RN_2_1_1__Rationals) -- (RN_2_2_3__Rationals);
	\draw[dashed,-] (RN_2_1_1__Rationals_repeated) -- (RN_2_2_3__Rationals_repeated);
\end{tikzpicture}
\end{scaletikzpicturetowidth}
\end{center}

\end{figure}
\begin{fleqn}
\begin{align*}
R_{\infty}^{1}(1):\ & - \\
R_{\infty}^{2}(1):\ & \bigl(I(0,3),P(0,1)\bigr),\ \bigl(I(0,4),P(0,3)\bigr)\\
\end{align*}
\end{fleqn}
\clearpage

\subsection[Schofield pairs for the quiver $\Delta(\widetilde{\mathbb{A}}_{2,3})$]{Schofield pairs for the quiver $\Delta(\widetilde{\mathbb{A}}_{2,3})$ -- {\usebox{\boxAAlAAA}}}
\[
\vcenter{\hbox{\xymatrix{  & 2\ar[dl] &   &   &  \\
1 &   &   & 5\ar[ull]\ar[dl] &  \\
  & 3\ar[ul] & 4\ar[l] &   &  }}}\qquad C_{\Delta(\widetilde{\mathbb{A}}_{2,3})} = \begin{bmatrix}1 & 1 & 1 & 1 & 2\\
0 & 1 & 0 & 0 & 1\\
0 & 0 & 1 & 1 & 1\\
0 & 0 & 0 & 1 & 1\\
0 & 0 & 0 & 0 & 1\end{bmatrix}\quad\Phi_{\Delta(\widetilde{\mathbb{A}}_{2,3})} = \begin{bmatrix}-1 & 1 & 1 & 0 & 0\\
-1 & 0 & 1 & 0 & 1\\
-1 & 1 & 0 & 1 & 0\\
-1 & 1 & 0 & 0 & 1\\
-2 & 1 & 1 & 0 & 1\end{bmatrix}
\]
\subsubsection{Schofield pairs associated to preprojective exceptional modules}

\begin{figure}[ht]

\begin{center}
\begin{scaletikzpicturetowidth}{\textwidth}
\begin{tikzpicture}[scale=\tikzscale]
	\node[noselAAlAAA, label={[shift={(0.4,-0.2)}, brown]\tiny $P(0,1)$}] (P_0_1__Rationals) at (0,0) {$\begin{smallmatrix}&0\\1&&&0\\&0&0\end{smallmatrix}$};
	\node[noselAAlAAA, label={[shift={(0.4,-0.2)}, brown]\tiny $P(0,2)$}] (P_0_2__Rationals) at (0,-1) {$\begin{smallmatrix}&1\\1&&&0\\&0&0\end{smallmatrix}$};
	\node[noselAAlAAA, label={[shift={(0.4,-0.2)}, brown]\tiny $P(0,3)$}] (P_0_3__Rationals) at (0,-4) {$\begin{smallmatrix}&0\\1&&&0\\&1&0\end{smallmatrix}$};
	\node[noselAAlAAA, label={[shift={(0.4,-0.2)}, brown]\tiny $P(0,4)$}] (P_0_4__Rationals) at (0,-3) {$\begin{smallmatrix}&0\\1&&&0\\&1&1\end{smallmatrix}$};
	\node[seloneAAlAAA, label={[shift={(0.4,-0.2)}, brown]\tiny $P(0,5)$}] (P_0_5__Rationals) at (0,-2) {$\begin{smallmatrix}&1\\2&&&1\\&1&1\end{smallmatrix}$};
	\node[noselAAlAAA, label={[shift={(0.4,-0.2)}, brown]\tiny $P(1,1)$}] (P_1_1__Rationals) at (4,0) {$\begin{smallmatrix}&1\\1&&&0\\&1&0\end{smallmatrix}$};
	\node[seloneAAlAAA, label={[shift={(0.4,-0.2)}, brown]\tiny $P(1,2)$}] (P_1_2__Rationals) at (4,-1) {$\begin{smallmatrix}&1\\2&&&1\\&2&1\end{smallmatrix}$};
	\node[noselAAlAAA, label={[shift={(0.4,-0.2)}, brown]\tiny $P(1,3)$}] (P_1_3__Rationals) at (4,-4) {$\begin{smallmatrix}&1\\1&&&0\\&1&1\end{smallmatrix}$};
	\node[seloneAAlAAA, label={[shift={(0.4,-0.2)}, brown]\tiny $P(1,4)$}] (P_1_4__Rationals) at (4,-3) {$\begin{smallmatrix}&2\\2&&&1\\&1&1\end{smallmatrix}$};
	\node[noselAAlAAA, label={[shift={(0.4,-0.2)}, brown]\tiny $P(1,5)$}] (P_1_5__Rationals) at (4,-2) {$\begin{smallmatrix}&2\\2&&&1\\&2&1\end{smallmatrix}$};
	\node[seloneAAlAAA, label={[shift={(0.4,-0.2)}, brown]\tiny $P(2,1)$}] (P_2_1__Rationals) at (8,0) {$\begin{smallmatrix}&1\\2&&&1\\&2&2\end{smallmatrix}$};
	\node[noselAAlAAA, label={[shift={(0.4,-0.2)}, brown]\tiny $P(2,2)$}] (P_2_2__Rationals) at (8,-1) {$\begin{smallmatrix}&2\\2&&&1\\&2&2\end{smallmatrix}$};
	\node[seloneAAlAAA, label={[shift={(0.4,-0.2)}, brown]\tiny $P(2,3)$}] (P_2_3__Rationals) at (8,-4) {$\begin{smallmatrix}&2\\3&&&2\\&2&2\end{smallmatrix}$};
	\node[noselAAlAAA, label={[shift={(0.4,-0.2)}, brown]\tiny $P(2,4)$}] (P_2_4__Rationals) at (8,-3) {$\begin{smallmatrix}&2\\3&&&2\\&3&2\end{smallmatrix}$};
	\node[noselAAlAAA, label={[shift={(0.4,-0.2)}, brown]\tiny $P(2,5)$}] (P_2_5__Rationals) at (8,-2) {$\begin{smallmatrix}&2\\3&&&2\\&3&3\end{smallmatrix}$};
	\node[noselAAlAAA, label={[shift={(0.4,-0.2)}, brown]\tiny $P(3,1)$}] (P_3_1__Rationals) at (12,0) {$\begin{smallmatrix}&3\\3&&&2\\&2&2\end{smallmatrix}$};
	\node[noselAAlAAA, label={[shift={(0.4,-0.2)}, brown]\tiny $P(3,2)$}] (P_3_2__Rationals) at (12,-1) {$\begin{smallmatrix}&3\\4&&&3\\&3&3\end{smallmatrix}$};
	\node[noselAAlAAA, label={[shift={(0.4,-0.2)}, brown]\tiny $P(3,3)$}] (P_3_3__Rationals) at (12,-4) {$\begin{smallmatrix}&3\\3&&&2\\&3&2\end{smallmatrix}$};
	\node[noselAAlAAA, label={[shift={(0.4,-0.2)}, brown]\tiny $P(3,4)$}] (P_3_4__Rationals) at (12,-3) {$\begin{smallmatrix}&3\\3&&&2\\&3&3\end{smallmatrix}$};
	\node[noselAAlAAA, label={[shift={(0.4,-0.2)}, brown]\tiny $P(3,5)$}] (P_3_5__Rationals) at (12,-2) {$\begin{smallmatrix}&4\\4&&&3\\&3&3\end{smallmatrix}$};
	\node at (13,-2.) {$\dots$};
	\draw[blue,->] (P_0_1__Rationals) -- (P_0_2__Rationals);
	\draw[blue,->] (P_0_1__Rationals) to [out=135,in=-135] (P_0_3__Rationals);
	\draw[blue,->] (P_0_3__Rationals) -- (P_0_4__Rationals);
	\draw[blue,->] (P_0_2__Rationals) -- (P_0_5__Rationals);
	\draw[blue,->] (P_0_4__Rationals) -- (P_0_5__Rationals);
	\draw[gray,dashed,->] (P_0_1__Rationals) -- (P_1_1__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[blue,->] (P_0_2__Rationals) -- (P_1_1__Rationals);
	\draw[blue,->] (P_0_3__Rationals) -- (P_1_1__Rationals);
	\draw[gray,dashed,->] (P_0_2__Rationals) -- (P_1_2__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[blue,->] (P_0_5__Rationals) -- (P_1_2__Rationals);
	\draw[blue,->] (P_1_1__Rationals) -- (P_1_2__Rationals);
	\draw[gray,dashed,->] (P_0_3__Rationals) -- (P_1_3__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[blue,->] (P_0_4__Rationals) -- (P_1_3__Rationals);
	\draw[blue,->] (P_1_1__Rationals) to [out=135,in=-135] (P_1_3__Rationals);
	\draw[gray,dashed,->] (P_0_4__Rationals) -- (P_1_4__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[blue,->] (P_0_5__Rationals) -- (P_1_4__Rationals);
	\draw[blue,->] (P_1_3__Rationals) -- (P_1_4__Rationals);
	\draw[gray,dashed,->] (P_0_5__Rationals) -- (P_1_5__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[blue,->] (P_1_2__Rationals) -- (P_1_5__Rationals);
	\draw[blue,->] (P_1_4__Rationals) -- (P_1_5__Rationals);
	\draw[gray,dashed,->] (P_1_1__Rationals) -- (P_2_1__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[blue,->] (P_1_2__Rationals) -- (P_2_1__Rationals);
	\draw[blue,->] (P_1_3__Rationals) -- (P_2_1__Rationals);
	\draw[gray,dashed,->] (P_1_2__Rationals) -- (P_2_2__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[blue,->] (P_1_5__Rationals) -- (P_2_2__Rationals);
	\draw[blue,->] (P_2_1__Rationals) -- (P_2_2__Rationals);
	\draw[gray,dashed,->] (P_1_3__Rationals) -- (P_2_3__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[blue,->] (P_1_4__Rationals) -- (P_2_3__Rationals);
	\draw[blue,->] (P_2_1__Rationals) to [out=135,in=-135] (P_2_3__Rationals);
	\draw[gray,dashed,->] (P_1_4__Rationals) -- (P_2_4__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[blue,->] (P_1_5__Rationals) -- (P_2_4__Rationals);
	\draw[blue,->] (P_2_3__Rationals) -- (P_2_4__Rationals);
	\draw[gray,dashed,->] (P_1_5__Rationals) -- (P_2_5__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[blue,->] (P_2_2__Rationals) -- (P_2_5__Rationals);
	\draw[blue,->] (P_2_4__Rationals) -- (P_2_5__Rationals);
	\draw[gray,dashed,->] (P_2_1__Rationals) -- (P_3_1__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[blue,->] (P_2_2__Rationals) -- (P_3_1__Rationals);
	\draw[blue,->] (P_2_3__Rationals) -- (P_3_1__Rationals);
	\draw[gray,dashed,->] (P_2_2__Rationals) -- (P_3_2__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[blue,->] (P_2_5__Rationals) -- (P_3_2__Rationals);
	\draw[blue,->] (P_3_1__Rationals) -- (P_3_2__Rationals);
	\draw[gray,dashed,->] (P_2_3__Rationals) -- (P_3_3__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[blue,->] (P_2_4__Rationals) -- (P_3_3__Rationals);
	\draw[blue,->] (P_3_1__Rationals) to [out=135,in=-135] (P_3_3__Rationals);
	\draw[gray,dashed,->] (P_2_4__Rationals) -- (P_3_4__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[blue,->] (P_2_5__Rationals) -- (P_3_4__Rationals);
	\draw[blue,->] (P_3_3__Rationals) -- (P_3_4__Rationals);
	\draw[gray,dashed,->] (P_2_5__Rationals) -- (P_3_5__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[blue,->] (P_3_2__Rationals) -- (P_3_5__Rationals);
	\draw[blue,->] (P_3_4__Rationals) -- (P_3_5__Rationals);
\end{tikzpicture}
\end{scaletikzpicturetowidth}
\end{center}

\end{figure}
\medskip{}
\subsubsection*{Modules of the form $P(n,1)$}

Defect: $\partial P(n,1) = -1$, for $n\ge 0$.

\begin{fleqn}
\begin{align*}
P(0,1):\ & - \\
P(1,1):\ & \bigl(R_{0}^{3}(1),P(0,2)\bigr),\ \bigl(R_{\infty}^{1}(1),P(0,3)\bigr)\\
P(2,1):\ & \bigl(R_{0}^{3}(2),P(0,5)\bigr),\ \bigl(R_{0}^{1}(1),P(1,2)\bigr),\ \bigl(R_{\infty}^{2}(1),P(1,3)\bigr),\ \bigl(I(0,2),2P(0,4)\bigr)\\
P(n,1):\ & \bigl(R_{0}^{n\bmod 3+1}(2),P(n-2,5)\bigr),\ \bigl(R_{0}^{(n-2)\bmod 3+1}(1),P(n-1,2)\bigr),\ \bigl(R_{\infty}^{(n-1)\bmod 2+1}(1),P(n-1,3)\bigr)\\
 & \bigl(uI,(u+1)P\bigr),\ n>2\\
\end{align*}
\end{fleqn}
\subsubsection*{Modules of the form $P(n,2)$}

Defect: $\partial P(n,2) = -1$, for $n\ge 0$.

\begin{fleqn}
\begin{align*}
P(0,2):\ & \bigl(R_{\infty}^{1}(1),P(0,1)\bigr)\\
P(1,2):\ & \bigl(R_{0}^{2}(2),P(0,4)\bigr),\ \bigl(R_{0}^{3}(1),P(0,5)\bigr),\ \bigl(R_{\infty}^{2}(1),P(1,1)\bigr),\ \bigl(I(1,5),2P(0,3)\bigr)\\
P(n,2):\ & \bigl(R_{0}^{n\bmod 3+1}(2),P(n-1,4)\bigr),\ \bigl(R_{0}^{(n+1)\bmod 3+1}(1),P(n-1,5)\bigr),\ \bigl(R_{\infty}^{n\bmod 2+1}(1),P(n,1)\bigr)\\
 & \bigl(uI,(u+1)P\bigr),\ n>1\\
\end{align*}
\end{fleqn}
\subsubsection*{Modules of the form $P(n,3)$}

Defect: $\partial P(n,3) = -1$, for $n\ge 0$.

\begin{fleqn}
\begin{align*}
P(0,3):\ & \bigl(R_{0}^{3}(1),P(0,1)\bigr)\\
P(1,3):\ & \bigl(R_{0}^{3}(2),P(0,2)\bigr),\ \bigl(R_{\infty}^{1}(1),P(0,4)\bigr),\ \bigl(R_{0}^{1}(1),P(1,1)\bigr)\\
P(2,3):\ & \bigl(R_{0}^{1}(2),P(1,2)\bigr),\ \bigl(R_{\infty}^{2}(1),P(1,4)\bigr),\ \bigl(R_{0}^{2}(1),P(2,1)\bigr),\ \bigl(2I(1,4),3P(0,1)\bigr)\\
P(n,3):\ & \bigl(R_{0}^{(n-2)\bmod 3+1}(2),P(n-1,2)\bigr),\ \bigl(R_{\infty}^{(n-1)\bmod 2+1}(1),P(n-1,4)\bigr),\ \bigl(R_{0}^{(n-1)\bmod 3+1}(1),P(n,1)\bigr)\\
 & \bigl(uI,(u+1)P\bigr),\ n>2\\
\end{align*}
\end{fleqn}
\subsubsection*{Modules of the form $P(n,4)$}

Defect: $\partial P(n,4) = -1$, for $n\ge 0$.

\begin{fleqn}
\begin{align*}
P(0,4):\ & \bigl(R_{0}^{3}(2),P(0,1)\bigr),\ \bigl(R_{0}^{1}(1),P(0,3)\bigr)\\
P(1,4):\ & \bigl(R_{\infty}^{1}(1),P(0,5)\bigr),\ \bigl(R_{0}^{1}(2),P(1,1)\bigr),\ \bigl(R_{0}^{2}(1),P(1,3)\bigr),\ \bigl(I(0,3),2P(0,2)\bigr)\\
P(n,4):\ & \bigl(R_{\infty}^{(n-1)\bmod 2+1}(1),P(n-1,5)\bigr),\ \bigl(R_{0}^{(n-1)\bmod 3+1}(2),P(n,1)\bigr),\ \bigl(R_{0}^{n\bmod 3+1}(1),P(n,3)\bigr)\\
 & \bigl(uI,(u+1)P\bigr),\ n>1\\
\end{align*}
\end{fleqn}
\subsubsection*{Modules of the form $P(n,5)$}

Defect: $\partial P(n,5) = -1$, for $n\ge 0$.

\begin{fleqn}
\begin{align*}
P(0,5):\ & \bigl(R_{\infty}^{2}(1),P(0,2)\bigr),\ \bigl(R_{0}^{1}(2),P(0,3)\bigr),\ \bigl(R_{0}^{2}(1),P(0,4)\bigr),\ \bigl(I(1,4),2P(0,1)\bigr)\\
P(n,5):\ & \bigl(R_{\infty}^{(n+1)\bmod 2+1}(1),P(n,2)\bigr),\ \bigl(R_{0}^{n\bmod 3+1}(2),P(n,3)\bigr),\ \bigl(R_{0}^{(n+1)\bmod 3+1}(1),P(n,4)\bigr)\\
 & \bigl(uI,(u+1)P\bigr),\ n>0\\
\end{align*}
\end{fleqn}
\subsubsection{Schofield pairs associated to preinjective exceptional modules}

\begin{figure}[ht]

\begin{center}
\begin{scaletikzpicturetowidth}{\textwidth}
\begin{tikzpicture}[scale=\tikzscale]
	\node[seloneAAlAAA, label={[shift={(0.4,-0.2)}, brown]\tiny $I(0,1)$}] (I_0_1__Rationals) at (0,0) {$\begin{smallmatrix}&1\\1&&&2\\&1&1\end{smallmatrix}$};
	\node[noselAAlAAA, label={[shift={(0.4,-0.2)}, brown]\tiny $I(0,2)$}] (I_0_2__Rationals) at (0,-1) {$\begin{smallmatrix}&1\\0&&&1\\&0&0\end{smallmatrix}$};
	\node[noselAAlAAA, label={[shift={(0.4,-0.2)}, brown]\tiny $I(0,3)$}] (I_0_3__Rationals) at (0,-4) {$\begin{smallmatrix}&0\\0&&&1\\&1&1\end{smallmatrix}$};
	\node[noselAAlAAA, label={[shift={(0.4,-0.2)}, brown]\tiny $I(0,4)$}] (I_0_4__Rationals) at (0,-3) {$\begin{smallmatrix}&0\\0&&&1\\&0&1\end{smallmatrix}$};
	\node[noselAAlAAA, label={[shift={(0.4,-0.2)}, brown]\tiny $I(0,5)$}] (I_0_5__Rationals) at (0,-2) {$\begin{smallmatrix}&0\\0&&&1\\&0&0\end{smallmatrix}$};
	\node[noselAAlAAA, label={[shift={(0.4,-0.2)}, brown]\tiny $I(1,1)$}] (I_1_1__Rationals) at (-4,0) {$\begin{smallmatrix}&2\\1&&&2\\&1&2\end{smallmatrix}$};
	\node[seloneAAlAAA, label={[shift={(0.4,-0.2)}, brown]\tiny $I(1,2)$}] (I_1_2__Rationals) at (-4,-1) {$\begin{smallmatrix}&1\\1&&&2\\&1&2\end{smallmatrix}$};
	\node[seloneAAlAAA, label={[shift={(0.4,-0.2)}, brown]\tiny $I(1,3)$}] (I_1_3__Rationals) at (-4,-4) {$\begin{smallmatrix}&2\\1&&&2\\&1&1\end{smallmatrix}$};
	\node[noselAAlAAA, label={[shift={(0.4,-0.2)}, brown]\tiny $I(1,4)$}] (I_1_4__Rationals) at (-4,-3) {$\begin{smallmatrix}&1\\0&&&1\\&1&1\end{smallmatrix}$};
	\node[noselAAlAAA, label={[shift={(0.4,-0.2)}, brown]\tiny $I(1,5)$}] (I_1_5__Rationals) at (-4,-2) {$\begin{smallmatrix}&1\\0&&&1\\&0&1\end{smallmatrix}$};
	\node[noselAAlAAA, label={[shift={(0.4,-0.2)}, brown]\tiny $I(2,1)$}] (I_2_1__Rationals) at (-8,0) {$\begin{smallmatrix}&2\\2&&&3\\&3&3\end{smallmatrix}$};
	\node[noselAAlAAA, label={[shift={(0.4,-0.2)}, brown]\tiny $I(2,2)$}] (I_2_2__Rationals) at (-8,-1) {$\begin{smallmatrix}&2\\1&&&2\\&2&2\end{smallmatrix}$};
	\node[noselAAlAAA, label={[shift={(0.4,-0.2)}, brown]\tiny $I(2,3)$}] (I_2_3__Rationals) at (-8,-4) {$\begin{smallmatrix}&2\\2&&&3\\&2&3\end{smallmatrix}$};
	\node[seloneAAlAAA, label={[shift={(0.4,-0.2)}, brown]\tiny $I(2,4)$}] (I_2_4__Rationals) at (-8,-3) {$\begin{smallmatrix}&2\\2&&&3\\&2&2\end{smallmatrix}$};
	\node[seloneAAlAAA, label={[shift={(0.4,-0.2)}, brown]\tiny $I(2,5)$}] (I_2_5__Rationals) at (-8,-2) {$\begin{smallmatrix}&1\\1&&&2\\&2&2\end{smallmatrix}$};
	\node[noselAAlAAA, label={[shift={(0.4,-0.2)}, brown]\tiny $I(3,1)$}] (I_3_1__Rationals) at (-12,0) {$\begin{smallmatrix}&4\\3&&&4\\&3&3\end{smallmatrix}$};
	\node[noselAAlAAA, label={[shift={(0.4,-0.2)}, brown]\tiny $I(3,2)$}] (I_3_2__Rationals) at (-12,-1) {$\begin{smallmatrix}&3\\3&&&4\\&3&3\end{smallmatrix}$};
	\node[noselAAlAAA, label={[shift={(0.4,-0.2)}, brown]\tiny $I(3,3)$}] (I_3_3__Rationals) at (-12,-4) {$\begin{smallmatrix}&3\\2&&&3\\&3&3\end{smallmatrix}$};
	\node[noselAAlAAA, label={[shift={(0.4,-0.2)}, brown]\tiny $I(3,4)$}] (I_3_4__Rationals) at (-12,-3) {$\begin{smallmatrix}&3\\2&&&3\\&2&3\end{smallmatrix}$};
	\node[noselAAlAAA, label={[shift={(0.4,-0.2)}, brown]\tiny $I(3,5)$}] (I_3_5__Rationals) at (-12,-2) {$\begin{smallmatrix}&3\\2&&&3\\&2&2\end{smallmatrix}$};
	\node at (-13,-2.) {$\dots$};
	\draw[red,->] (I_1_2__Rationals) -- (I_0_1__Rationals);
	\draw[red,->] (I_1_3__Rationals) -- (I_0_1__Rationals);
	\draw[red,->] (I_0_1__Rationals) -- (I_0_2__Rationals);
	\draw[red,->] (I_1_5__Rationals) -- (I_0_2__Rationals);
	\draw[red,->] (I_0_1__Rationals) to [out=135,in=-135] (I_0_3__Rationals);
	\draw[red,->] (I_1_4__Rationals) -- (I_0_3__Rationals);
	\draw[red,->] (I_0_3__Rationals) -- (I_0_4__Rationals);
	\draw[red,->] (I_1_5__Rationals) -- (I_0_4__Rationals);
	\draw[red,->] (I_0_2__Rationals) -- (I_0_5__Rationals);
	\draw[red,->] (I_0_4__Rationals) -- (I_0_5__Rationals);
	\draw[gray,dashed,->] (I_0_1__Rationals) -- (I_1_1__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_2_2__Rationals) -- (I_1_1__Rationals);
	\draw[red,->] (I_2_3__Rationals) -- (I_1_1__Rationals);
	\draw[gray,dashed,->] (I_0_2__Rationals) -- (I_1_2__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_1_1__Rationals) -- (I_1_2__Rationals);
	\draw[red,->] (I_2_5__Rationals) -- (I_1_2__Rationals);
	\draw[gray,dashed,->] (I_0_3__Rationals) -- (I_1_3__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_1_1__Rationals) to [out=135,in=-135] (I_1_3__Rationals);
	\draw[red,->] (I_2_4__Rationals) -- (I_1_3__Rationals);
	\draw[gray,dashed,->] (I_0_4__Rationals) -- (I_1_4__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_1_3__Rationals) -- (I_1_4__Rationals);
	\draw[red,->] (I_2_5__Rationals) -- (I_1_4__Rationals);
	\draw[gray,dashed,->] (I_0_5__Rationals) -- (I_1_5__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_1_2__Rationals) -- (I_1_5__Rationals);
	\draw[red,->] (I_1_4__Rationals) -- (I_1_5__Rationals);
	\draw[gray,dashed,->] (I_1_1__Rationals) -- (I_2_1__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_3_2__Rationals) -- (I_2_1__Rationals);
	\draw[red,->] (I_3_3__Rationals) -- (I_2_1__Rationals);
	\draw[gray,dashed,->] (I_1_2__Rationals) -- (I_2_2__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_2_1__Rationals) -- (I_2_2__Rationals);
	\draw[red,->] (I_3_5__Rationals) -- (I_2_2__Rationals);
	\draw[gray,dashed,->] (I_1_3__Rationals) -- (I_2_3__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_2_1__Rationals) to [out=135,in=-135] (I_2_3__Rationals);
	\draw[red,->] (I_3_4__Rationals) -- (I_2_3__Rationals);
	\draw[gray,dashed,->] (I_1_4__Rationals) -- (I_2_4__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_2_3__Rationals) -- (I_2_4__Rationals);
	\draw[red,->] (I_3_5__Rationals) -- (I_2_4__Rationals);
	\draw[gray,dashed,->] (I_1_5__Rationals) -- (I_2_5__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_2_2__Rationals) -- (I_2_5__Rationals);
	\draw[red,->] (I_2_4__Rationals) -- (I_2_5__Rationals);
	\draw[gray,dashed,->] (I_2_1__Rationals) -- (I_3_1__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[gray,dashed,->] (I_2_2__Rationals) -- (I_3_2__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_3_1__Rationals) -- (I_3_2__Rationals);
	\draw[gray,dashed,->] (I_2_3__Rationals) -- (I_3_3__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_3_1__Rationals) to [out=135,in=-135] (I_3_3__Rationals);
	\draw[gray,dashed,->] (I_2_4__Rationals) -- (I_3_4__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_3_3__Rationals) -- (I_3_4__Rationals);
	\draw[gray,dashed,->] (I_2_5__Rationals) -- (I_3_5__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_3_2__Rationals) -- (I_3_5__Rationals);
	\draw[red,->] (I_3_4__Rationals) -- (I_3_5__Rationals);
\end{tikzpicture}
\end{scaletikzpicturetowidth}
\end{center}

\end{figure}
\medskip{}
\subsubsection*{Modules of the form $I(n,1)$}

Defect: $\partial I(n,1) = 1$, for $n\ge 0$.

\begin{fleqn}
\begin{align*}
I(0,1):\ & \bigl(I(0,2),R_{\infty}^{2}(1)\bigr),\ \bigl(I(0,3),R_{0}^{2}(1)\bigr),\ \bigl(I(0,4),R_{0}^{2}(2)\bigr),\ \bigl(2I(0,5),P(1,3)\bigr)\\
I(n,1):\ & \bigl(I(n,2),R_{\infty}^{(-n+1)\bmod 2+1}(1)\bigr),\ \bigl(I(n,3),R_{0}^{(-n+1)\bmod 3+1}(1)\bigr),\ \bigl(I(n,4),R_{0}^{(-n+1)\bmod 3+1}(2)\bigr)\\
 & \bigl((v+1)I,vP\bigr),\ n>0\\
\end{align*}
\end{fleqn}
\subsubsection*{Modules of the form $I(n,2)$}

Defect: $\partial I(n,2) = 1$, for $n\ge 0$.

\begin{fleqn}
\begin{align*}
I(0,2):\ & \bigl(I(0,5),R_{\infty}^{1}(1)\bigr)\\
I(1,2):\ & \bigl(I(0,1),R_{0}^{1}(1)\bigr),\ \bigl(I(0,3),R_{0}^{1}(2)\bigr),\ \bigl(I(1,5),R_{\infty}^{2}(1)\bigr),\ \bigl(2I(0,4),P(1,1)\bigr)\\
I(n,2):\ & \bigl(I(n-1,1),R_{0}^{(-n+1)\bmod 3+1}(1)\bigr),\ \bigl(I(n-1,3),R_{0}^{(-n+1)\bmod 3+1}(2)\bigr),\ \bigl(I(n,5),R_{\infty}^{(-n+2)\bmod 2+1}(1)\bigr)\\
 & \bigl((v+1)I,vP\bigr),\ n>1\\
\end{align*}
\end{fleqn}
\subsubsection*{Modules of the form $I(n,3)$}

Defect: $\partial I(n,3) = 1$, for $n\ge 0$.

\begin{fleqn}
\begin{align*}
I(0,3):\ & \bigl(I(0,4),R_{0}^{3}(1)\bigr),\ \bigl(I(0,5),R_{0}^{3}(2)\bigr)\\
I(1,3):\ & \bigl(I(0,1),R_{\infty}^{1}(1)\bigr),\ \bigl(I(1,4),R_{0}^{2}(1)\bigr),\ \bigl(I(1,5),R_{0}^{2}(2)\bigr),\ \bigl(2I(0,2),P(0,4)\bigr)\\
I(n,3):\ & \bigl(I(n-1,1),R_{\infty}^{(-n+1)\bmod 2+1}(1)\bigr),\ \bigl(I(n,4),R_{0}^{(-n+2)\bmod 3+1}(1)\bigr),\ \bigl(I(n,5),R_{0}^{(-n+2)\bmod 3+1}(2)\bigr)\\
 & \bigl((v+1)I,vP\bigr),\ n>1\\
\end{align*}
\end{fleqn}
\subsubsection*{Modules of the form $I(n,4)$}

Defect: $\partial I(n,4) = 1$, for $n\ge 0$.

\begin{fleqn}
\begin{align*}
I(0,4):\ & \bigl(I(0,5),R_{0}^{1}(1)\bigr)\\
I(1,4):\ & \bigl(I(0,2),R_{0}^{3}(2)\bigr),\ \bigl(I(0,3),R_{\infty}^{1}(1)\bigr),\ \bigl(I(1,5),R_{0}^{3}(1)\bigr)\\
I(2,4):\ & \bigl(I(1,2),R_{0}^{2}(2)\bigr),\ \bigl(I(1,3),R_{\infty}^{2}(1)\bigr),\ \bigl(I(2,5),R_{0}^{2}(1)\bigr),\ \bigl(3I(0,5),2P(1,3)\bigr)\\
I(n,4):\ & \bigl(I(n-1,2),R_{0}^{(-n+3)\bmod 3+1}(2)\bigr),\ \bigl(I(n-1,3),R_{\infty}^{(-n+3)\bmod 2+1}(1)\bigr),\ \bigl(I(n,5),R_{0}^{(-n+3)\bmod 3+1}(1)\bigr)\\
 & \bigl((v+1)I,vP\bigr),\ n>2\\
\end{align*}
\end{fleqn}
\subsubsection*{Modules of the form $I(n,5)$}

Defect: $\partial I(n,5) = 1$, for $n\ge 0$.

\begin{fleqn}
\begin{align*}
I(0,5):\ & - \\
I(1,5):\ & \bigl(I(0,2),R_{0}^{1}(1)\bigr),\ \bigl(I(0,4),R_{\infty}^{1}(1)\bigr)\\
I(2,5):\ & \bigl(I(0,1),R_{0}^{3}(2)\bigr),\ \bigl(I(1,2),R_{0}^{3}(1)\bigr),\ \bigl(I(1,4),R_{\infty}^{2}(1)\bigr),\ \bigl(2I(0,3),P(0,2)\bigr)\\
I(n,5):\ & \bigl(I(n-2,1),R_{0}^{(-n+4)\bmod 3+1}(2)\bigr),\ \bigl(I(n-1,2),R_{0}^{(-n+4)\bmod 3+1}(1)\bigr),\ \bigl(I(n-1,4),R_{\infty}^{(-n+3)\bmod 2+1}(1)\bigr)\\
 & \bigl((v+1)I,vP\bigr),\ n>2\\
\end{align*}
\end{fleqn}
\subsubsection{Schofield pairs associated to regular exceptional modules}

\subsubsection*{The non-homogeneous tube $\mathcal{T}_{0}^{\Delta(\widetilde{\mathbb{A}}_{2,3})}$}

\begin{figure}[ht]

\begin{center}
\captionof{figure}{\vspace*{-10pt}$\mathcal{T}_{0}^{\Delta(\widetilde{\mathbb{A}}_{2,3})}$}
\begin{scaletikzpicturetowidth}{0.69999999999999996\textwidth}
\begin{tikzpicture}[scale=\tikzscale]
	\node[seltwoAAlAAA, label={[shift={(0.0,-0.45)}, brown]\tiny $R_{0}^{1}(1)$}] (RN_1_1_1__Rationals) at (4,0) {$\begin{smallmatrix}&0\\0&&&0\\&0&1\end{smallmatrix}$};
	\node[seltwoAAlAAA, label={[shift={(0.0,-0.45)}, brown]\tiny $R_{0}^{2}(1)$}] (RN_1_2_1__Rationals) at (8,0) {$\begin{smallmatrix}&1\\1&&&1\\&0&0\end{smallmatrix}$};
	\node[seltwoAAlAAA, label={[shift={(0.0,-0.45)}, brown]\tiny $R_{0}^{3}(1)$}] (RN_1_3_1__Rationals) at (12,0) {$\begin{smallmatrix}&0\\0&&&0\\&1&0\end{smallmatrix}$};
	\node[seltwoAAlAAA, label={[shift={(0.0,-0.45)}, brown]\tiny $R_{0}^{1}(1)$}] (RN_1_1_1__Rationals_repeated) at (16,0) {$\begin{smallmatrix}&0\\0&&&0\\&0&1\end{smallmatrix}$};
	\node[seltwoAAlAAA, label={[shift={(0.0,-0.45)}, brown]\tiny $R_{0}^{1}(2)$}] (RN_1_1_2__Rationals) at (6,2) {$\begin{smallmatrix}&1\\1&&&1\\&0&1\end{smallmatrix}$};
	\node[seltwoAAlAAA, label={[shift={(0.0,-0.45)}, brown]\tiny $R_{0}^{2}(2)$}] (RN_1_2_2__Rationals) at (10,2) {$\begin{smallmatrix}&1\\1&&&1\\&1&0\end{smallmatrix}$};
	\node[seltwoAAlAAA, label={[shift={(0.0,-0.45)}, brown]\tiny $R_{0}^{3}(2)$}] (RN_1_3_2__Rationals) at (14,2) {$\begin{smallmatrix}&0\\0&&&0\\&1&1\end{smallmatrix}$};
	\node[noselAAlAAA, label={[shift={(0.0,-0.45)}, brown]\tiny $R_{0}^{3}(3)$}] (RN_1_3_3__Rationals) at (4,4) {$\begin{smallmatrix}&1\\1&&&1\\&1&1\end{smallmatrix}$};
	\node[noselAAlAAA, label={[shift={(0.0,-0.45)}, brown]\tiny $R_{0}^{1}(3)$}] (RN_1_1_3__Rationals) at (8,4) {$\begin{smallmatrix}&1\\1&&&1\\&1&1\end{smallmatrix}$};
	\node[noselAAlAAA, label={[shift={(0.0,-0.45)}, brown]\tiny $R_{0}^{2}(3)$}] (RN_1_2_3__Rationals) at (12,4) {$\begin{smallmatrix}&1\\1&&&1\\&1&1\end{smallmatrix}$};
	\node[noselAAlAAA, label={[shift={(0.0,-0.45)}, brown]\tiny $R_{0}^{3}(3)$}] (RN_1_3_3__Rationals_repeated) at (16,4) {$\begin{smallmatrix}&1\\1&&&1\\&1&1\end{smallmatrix}$};
	\node[noselAAlAAA, label={[shift={(0.0,-0.45)}, brown]\tiny $R_{0}^{3}(4)$}] (RN_1_3_4__Rationals) at (6,6) {$\begin{smallmatrix}&1\\1&&&1\\&2&1\end{smallmatrix}$};
	\node[noselAAlAAA, label={[shift={(0.0,-0.45)}, brown]\tiny $R_{0}^{1}(4)$}] (RN_1_1_4__Rationals) at (10,6) {$\begin{smallmatrix}&1\\1&&&1\\&1&2\end{smallmatrix}$};
	\node[noselAAlAAA, label={[shift={(0.0,-0.45)}, brown]\tiny $R_{0}^{2}(4)$}] (RN_1_2_4__Rationals) at (14,6) {$\begin{smallmatrix}&2\\2&&&2\\&1&1\end{smallmatrix}$};
	\draw[blue,->] (RN_1_1_1__Rationals) -- (RN_1_1_2__Rationals);
	\draw[gray,dashed,->] (RN_1_1_1__Rationals) -- (RN_1_2_1__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[red,->] (RN_1_1_2__Rationals) -- (RN_1_2_1__Rationals);
	\draw[gray,dashed,->] (RN_1_1_2__Rationals) -- (RN_1_2_2__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[blue,->] (RN_1_1_2__Rationals) -- (RN_1_1_3__Rationals);
	\draw[blue,->] (RN_1_2_1__Rationals) -- (RN_1_2_2__Rationals);
	\draw[gray,dashed,->] (RN_1_2_1__Rationals) -- (RN_1_3_1__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[red,->] (RN_1_2_2__Rationals) -- (RN_1_3_1__Rationals);
	\draw[gray,dashed,->] (RN_1_2_2__Rationals) -- (RN_1_3_2__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[blue,->] (RN_1_2_2__Rationals) -- (RN_1_2_3__Rationals);
	\draw[gray,dashed,->] (RN_1_3_1__Rationals) -- (RN_1_1_1__Rationals_repeated) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[blue,->] (RN_1_3_1__Rationals) -- (RN_1_3_2__Rationals);
	\draw[red,->] (RN_1_3_2__Rationals) -- (RN_1_1_1__Rationals_repeated);
	\draw[gray,dashed,->,out=200,in=-20] (RN_1_3_2__Rationals) to node[midway,sloped,above]{{\tiny $\tau^{-1}$}} (RN_1_1_2__Rationals);
	\draw[blue,->] (RN_1_3_2__Rationals) -- (RN_1_3_3__Rationals_repeated);
	\draw[red,->] (RN_1_3_3__Rationals) -- (RN_1_1_2__Rationals);
	\draw[blue,->] (RN_1_3_3__Rationals) -- (RN_1_3_4__Rationals);
	\draw[gray,dashed,->] (RN_1_3_3__Rationals) -- (RN_1_1_3__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[red,->] (RN_1_3_4__Rationals) -- (RN_1_1_3__Rationals);
	\draw[gray,dashed,->] (RN_1_3_4__Rationals) -- (RN_1_1_4__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[red,->] (RN_1_1_3__Rationals) -- (RN_1_2_2__Rationals);
	\draw[blue,->] (RN_1_1_3__Rationals) -- (RN_1_1_4__Rationals);
	\draw[gray,dashed,->] (RN_1_1_3__Rationals) -- (RN_1_2_3__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[red,->] (RN_1_1_4__Rationals) -- (RN_1_2_3__Rationals);
	\draw[gray,dashed,->] (RN_1_1_4__Rationals) -- (RN_1_2_4__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[red,->] (RN_1_2_3__Rationals) -- (RN_1_3_2__Rationals);
	\draw[gray,dashed,->] (RN_1_2_3__Rationals) -- (RN_1_3_3__Rationals_repeated) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[blue,->] (RN_1_2_3__Rationals) -- (RN_1_2_4__Rationals);
	\draw[red,->] (RN_1_2_4__Rationals) -- (RN_1_3_3__Rationals_repeated);
	\draw[gray,dashed,->,out=200,in=-20] (RN_1_2_4__Rationals) to node[midway,sloped,above]{{\tiny $\tau^{-1}$}} (RN_1_3_4__Rationals);
	\draw[dashed,-] (RN_1_1_1__Rationals) -- (RN_1_3_3__Rationals);
	\draw[dashed,-] (RN_1_1_1__Rationals_repeated) -- (RN_1_3_3__Rationals_repeated);
	\node (LeftTopPhantom_1) at (4,6) {};
	\draw[dashed,-] (RN_1_3_3__Rationals) -- (LeftTopPhantom_1);
	\node (RightTopPhantom_1) at (16,6) {};
	\draw[dashed,-] (RN_1_3_3__Rationals_repeated) -- (RightTopPhantom_1);
\end{tikzpicture}
\end{scaletikzpicturetowidth}
\end{center}

\end{figure}
\begin{fleqn}
\begin{align*}
R_{0}^{1}(1):\ & - \\
R_{0}^{1}(2):\ & \bigl(R_{0}^{2}(1),R_{0}^{1}(1)\bigr),\ \bigl(I(1,5),P(0,1)\bigr),\ \bigl(I(0,4),P(0,2)\bigr)\\
R_{0}^{2}(1):\ & \bigl(I(0,2),P(0,1)\bigr),\ \bigl(I(0,5),P(0,2)\bigr)\\
R_{0}^{2}(2):\ & \bigl(R_{0}^{3}(1),R_{0}^{2}(1)\bigr),\ \bigl(I(0,2),P(0,3)\bigr),\ \bigl(I(0,5),P(1,1)\bigr)\\
R_{0}^{3}(1):\ & - \\
R_{0}^{3}(2):\ & \bigl(R_{0}^{1}(1),R_{0}^{3}(1)\bigr)\\
\end{align*}
\end{fleqn}
\subsubsection*{The non-homogeneous tube $\mathcal{T}_{\infty}^{\Delta(\widetilde{\mathbb{A}}_{2,3})}$}

\begin{figure}[ht]

\begin{center}
\captionof{figure}{\vspace*{-10pt}$\mathcal{T}_{\infty}^{\Delta(\widetilde{\mathbb{A}}_{2,3})}$}
\begin{scaletikzpicturetowidth}{0.5\textwidth}
\begin{tikzpicture}[scale=\tikzscale]
	\node[seltwoAAlAAA, label={[shift={(0.0,-0.45)}, brown]\tiny $R_{\infty}^{1}(1)$}] (RN_2_1_1__Rationals) at (4,0) {$\begin{smallmatrix}&1\\0&&&0\\&0&0\end{smallmatrix}$};
	\node[seltwoAAlAAA, label={[shift={(0.0,-0.45)}, brown]\tiny $R_{\infty}^{2}(1)$}] (RN_2_2_1__Rationals) at (8,0) {$\begin{smallmatrix}&0\\1&&&1\\&1&1\end{smallmatrix}$};
	\node[seltwoAAlAAA, label={[shift={(0.0,-0.45)}, brown]\tiny $R_{\infty}^{1}(1)$}] (RN_2_1_1__Rationals_repeated) at (12,0) {$\begin{smallmatrix}&1\\0&&&0\\&0&0\end{smallmatrix}$};
	\node[noselAAlAAA, label={[shift={(0.0,-0.45)}, brown]\tiny $R_{\infty}^{1}(2)$}] (RN_2_1_2__Rationals) at (6,2) {$\begin{smallmatrix}&1\\1&&&1\\&1&1\end{smallmatrix}$};
	\node[noselAAlAAA, label={[shift={(0.0,-0.45)}, brown]\tiny $R_{\infty}^{2}(2)$}] (RN_2_2_2__Rationals) at (10,2) {$\begin{smallmatrix}&1\\1&&&1\\&1&1\end{smallmatrix}$};
	\node[noselAAlAAA, label={[shift={(0.0,-0.45)}, brown]\tiny $R_{\infty}^{2}(3)$}] (RN_2_2_3__Rationals) at (4,4) {$\begin{smallmatrix}&1\\2&&&2\\&2&2\end{smallmatrix}$};
	\node[noselAAlAAA, label={[shift={(0.0,-0.45)}, brown]\tiny $R_{\infty}^{1}(3)$}] (RN_2_1_3__Rationals) at (8,4) {$\begin{smallmatrix}&2\\1&&&1\\&1&1\end{smallmatrix}$};
	\node[noselAAlAAA, label={[shift={(0.0,-0.45)}, brown]\tiny $R_{\infty}^{2}(3)$}] (RN_2_2_3__Rationals_repeated) at (12,4) {$\begin{smallmatrix}&1\\2&&&2\\&2&2\end{smallmatrix}$};
	\draw[blue,->] (RN_2_1_1__Rationals) -- (RN_2_1_2__Rationals);
	\draw[gray,dashed,->] (RN_2_1_1__Rationals) -- (RN_2_2_1__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[red,->] (RN_2_1_2__Rationals) -- (RN_2_2_1__Rationals);
	\draw[gray,dashed,->] (RN_2_1_2__Rationals) -- (RN_2_2_2__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[blue,->] (RN_2_1_2__Rationals) -- (RN_2_1_3__Rationals);
	\draw[gray,dashed,->] (RN_2_2_1__Rationals) -- (RN_2_1_1__Rationals_repeated) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[blue,->] (RN_2_2_1__Rationals) -- (RN_2_2_2__Rationals);
	\draw[red,->] (RN_2_2_2__Rationals) -- (RN_2_1_1__Rationals_repeated);
	\draw[gray,dashed,->,out=219,in=-39] (RN_2_2_2__Rationals) to node[midway,sloped,above]{{\tiny $\tau^{-1}$}} (RN_2_1_2__Rationals);
	\draw[blue,->] (RN_2_2_2__Rationals) -- (RN_2_2_3__Rationals_repeated);
	\draw[red,->] (RN_2_2_3__Rationals) -- (RN_2_1_2__Rationals);
	\draw[gray,dashed,->] (RN_2_2_3__Rationals) -- (RN_2_1_3__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[red,->] (RN_2_1_3__Rationals) -- (RN_2_2_2__Rationals);
	\draw[gray,dashed,->] (RN_2_1_3__Rationals) -- (RN_2_2_3__Rationals_repeated) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[dashed,-] (RN_2_1_1__Rationals) -- (RN_2_2_3__Rationals);
	\draw[dashed,-] (RN_2_1_1__Rationals_repeated) -- (RN_2_2_3__Rationals_repeated);
\end{tikzpicture}
\end{scaletikzpicturetowidth}
\end{center}

\end{figure}
\begin{fleqn}
\begin{align*}
R_{\infty}^{1}(1):\ & - \\
R_{\infty}^{2}(1):\ & \bigl(I(0,3),P(0,1)\bigr),\ \bigl(I(0,4),P(0,3)\bigr),\ \bigl(I(0,5),P(0,4)\bigr)\\
\end{align*}
\end{fleqn}
\clearpage

\subsection[Schofield pairs for the quiver $\Delta(\widetilde{\mathbb{A}}_{2,4})$]{Schofield pairs for the quiver $\Delta(\widetilde{\mathbb{A}}_{2,4})$ -- {\usebox{\boxAAlAAAA}}}
\[
\vcenter{\hbox{\xymatrix{  & 2\ar[dl] &   &   &   &  \\
1 &   &   &   & 6\ar[ulll]\ar[dl] &  \\
  & 3\ar[ul] & 4\ar[l] & 5\ar[l] &   &  }}}\qquad C_{\Delta(\widetilde{\mathbb{A}}_{2,4})} = \begin{bmatrix}1 & 1 & 1 & 1 & 1 & 2\\
0 & 1 & 0 & 0 & 0 & 1\\
0 & 0 & 1 & 1 & 1 & 1\\
0 & 0 & 0 & 1 & 1 & 1\\
0 & 0 & 0 & 0 & 1 & 1\\
0 & 0 & 0 & 0 & 0 & 1\end{bmatrix}\quad\Phi_{\Delta(\widetilde{\mathbb{A}}_{2,4})} = \begin{bmatrix}-1 & 1 & 1 & 0 & 0 & 0\\
-1 & 0 & 1 & 0 & 0 & 1\\
-1 & 1 & 0 & 1 & 0 & 0\\
-1 & 1 & 0 & 0 & 1 & 0\\
-1 & 1 & 0 & 0 & 0 & 1\\
-2 & 1 & 1 & 0 & 0 & 1\end{bmatrix}
\]
\subsubsection{Schofield pairs associated to preprojective exceptional modules}

\begin{figure}[ht]

\begin{center}
\begin{scaletikzpicturetowidth}{\textwidth}
\begin{tikzpicture}[scale=\tikzscale]
	\node[noselAAlAAAA, label={[shift={(0.4,-0.2)}, brown]\tiny $P(0,1)$}] (P_0_1__Rationals) at (0,0) {$\begin{smallmatrix}&0\\1&&&&0\\&0&0&0\end{smallmatrix}$};
	\node[noselAAlAAAA, label={[shift={(0.4,-0.2)}, brown]\tiny $P(0,2)$}] (P_0_2__Rationals) at (0,-1) {$\begin{smallmatrix}&1\\1&&&&0\\&0&0&0\end{smallmatrix}$};
	\node[noselAAlAAAA, label={[shift={(0.4,-0.2)}, brown]\tiny $P(0,3)$}] (P_0_3__Rationals) at (0,-5) {$\begin{smallmatrix}&0\\1&&&&0\\&1&0&0\end{smallmatrix}$};
	\node[noselAAlAAAA, label={[shift={(0.4,-0.2)}, brown]\tiny $P(0,4)$}] (P_0_4__Rationals) at (0,-4) {$\begin{smallmatrix}&0\\1&&&&0\\&1&1&0\end{smallmatrix}$};
	\node[noselAAlAAAA, label={[shift={(0.4,-0.2)}, brown]\tiny $P(0,5)$}] (P_0_5__Rationals) at (0,-3) {$\begin{smallmatrix}&0\\1&&&&0\\&1&1&1\end{smallmatrix}$};
	\node[seloneAAlAAAA, label={[shift={(0.4,-0.2)}, brown]\tiny $P(0,6)$}] (P_0_6__Rationals) at (0,-2) {$\begin{smallmatrix}&1\\2&&&&1\\&1&1&1\end{smallmatrix}$};
	\node[noselAAlAAAA, label={[shift={(0.4,-0.2)}, brown]\tiny $P(1,1)$}] (P_1_1__Rationals) at (4,0) {$\begin{smallmatrix}&1\\1&&&&0\\&1&0&0\end{smallmatrix}$};
	\node[seloneAAlAAAA, label={[shift={(0.4,-0.2)}, brown]\tiny $P(1,2)$}] (P_1_2__Rationals) at (4,-1) {$\begin{smallmatrix}&1\\2&&&&1\\&2&1&1\end{smallmatrix}$};
	\node[noselAAlAAAA, label={[shift={(0.4,-0.2)}, brown]\tiny $P(1,3)$}] (P_1_3__Rationals) at (4,-5) {$\begin{smallmatrix}&1\\1&&&&0\\&1&1&0\end{smallmatrix}$};
	\node[noselAAlAAAA, label={[shift={(0.4,-0.2)}, brown]\tiny $P(1,4)$}] (P_1_4__Rationals) at (4,-4) {$\begin{smallmatrix}&1\\1&&&&0\\&1&1&1\end{smallmatrix}$};
	\node[seloneAAlAAAA, label={[shift={(0.4,-0.2)}, brown]\tiny $P(1,5)$}] (P_1_5__Rationals) at (4,-3) {$\begin{smallmatrix}&2\\2&&&&1\\&1&1&1\end{smallmatrix}$};
	\node[noselAAlAAAA, label={[shift={(0.4,-0.2)}, brown]\tiny $P(1,6)$}] (P_1_6__Rationals) at (4,-2) {$\begin{smallmatrix}&2\\2&&&&1\\&2&1&1\end{smallmatrix}$};
	\node[seloneAAlAAAA, label={[shift={(0.4,-0.2)}, brown]\tiny $P(2,1)$}] (P_2_1__Rationals) at (8,0) {$\begin{smallmatrix}&1\\2&&&&1\\&2&2&1\end{smallmatrix}$};
	\node[noselAAlAAAA, label={[shift={(0.4,-0.2)}, brown]\tiny $P(2,2)$}] (P_2_2__Rationals) at (8,-1) {$\begin{smallmatrix}&2\\2&&&&1\\&2&2&1\end{smallmatrix}$};
	\node[seloneAAlAAAA, label={[shift={(0.4,-0.2)}, brown]\tiny $P(2,3)$}] (P_2_3__Rationals) at (8,-5) {$\begin{smallmatrix}&1\\2&&&&1\\&2&2&2\end{smallmatrix}$};
	\node[seloneAAlAAAA, label={[shift={(0.4,-0.2)}, brown]\tiny $P(2,4)$}] (P_2_4__Rationals) at (8,-4) {$\begin{smallmatrix}&2\\3&&&&2\\&2&2&2\end{smallmatrix}$};
	\node[noselAAlAAAA, label={[shift={(0.4,-0.2)}, brown]\tiny $P(2,5)$}] (P_2_5__Rationals) at (8,-3) {$\begin{smallmatrix}&2\\3&&&&2\\&3&2&2\end{smallmatrix}$};
	\node[noselAAlAAAA, label={[shift={(0.4,-0.2)}, brown]\tiny $P(2,6)$}] (P_2_6__Rationals) at (8,-2) {$\begin{smallmatrix}&2\\3&&&&2\\&3&3&2\end{smallmatrix}$};
	\node[noselAAlAAAA, label={[shift={(0.4,-0.2)}, brown]\tiny $P(3,1)$}] (P_3_1__Rationals) at (12,0) {$\begin{smallmatrix}&2\\2&&&&1\\&2&2&2\end{smallmatrix}$};
	\node[noselAAlAAAA, label={[shift={(0.4,-0.2)}, brown]\tiny $P(3,2)$}] (P_3_2__Rationals) at (12,-1) {$\begin{smallmatrix}&2\\3&&&&2\\&3&3&3\end{smallmatrix}$};
	\node[noselAAlAAAA, label={[shift={(0.4,-0.2)}, brown]\tiny $P(3,3)$}] (P_3_3__Rationals) at (12,-5) {$\begin{smallmatrix}&3\\3&&&&2\\&2&2&2\end{smallmatrix}$};
	\node[noselAAlAAAA, label={[shift={(0.4,-0.2)}, brown]\tiny $P(3,4)$}] (P_3_4__Rationals) at (12,-4) {$\begin{smallmatrix}&3\\3&&&&2\\&3&2&2\end{smallmatrix}$};
	\node[noselAAlAAAA, label={[shift={(0.4,-0.2)}, brown]\tiny $P(3,5)$}] (P_3_5__Rationals) at (12,-3) {$\begin{smallmatrix}&3\\3&&&&2\\&3&3&2\end{smallmatrix}$};
	\node[noselAAlAAAA, label={[shift={(0.4,-0.2)}, brown]\tiny $P(3,6)$}] (P_3_6__Rationals) at (12,-2) {$\begin{smallmatrix}&3\\3&&&&2\\&3&3&3\end{smallmatrix}$};
	\node at (13,-2.5) {$\dots$};
	\draw[blue,->] (P_0_1__Rationals) -- (P_0_2__Rationals);
	\draw[blue,->] (P_0_1__Rationals) to [out=135,in=-135] (P_0_3__Rationals);
	\draw[blue,->] (P_0_3__Rationals) -- (P_0_4__Rationals);
	\draw[blue,->] (P_0_4__Rationals) -- (P_0_5__Rationals);
	\draw[blue,->] (P_0_2__Rationals) -- (P_0_6__Rationals);
	\draw[blue,->] (P_0_5__Rationals) -- (P_0_6__Rationals);
	\draw[gray,dashed,->] (P_0_1__Rationals) -- (P_1_1__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[blue,->] (P_0_2__Rationals) -- (P_1_1__Rationals);
	\draw[blue,->] (P_0_3__Rationals) -- (P_1_1__Rationals);
	\draw[gray,dashed,->] (P_0_2__Rationals) -- (P_1_2__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[blue,->] (P_0_6__Rationals) -- (P_1_2__Rationals);
	\draw[blue,->] (P_1_1__Rationals) -- (P_1_2__Rationals);
	\draw[gray,dashed,->] (P_0_3__Rationals) -- (P_1_3__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[blue,->] (P_0_4__Rationals) -- (P_1_3__Rationals);
	\draw[blue,->] (P_1_1__Rationals) to [out=135,in=-135] (P_1_3__Rationals);
	\draw[gray,dashed,->] (P_0_4__Rationals) -- (P_1_4__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[blue,->] (P_0_5__Rationals) -- (P_1_4__Rationals);
	\draw[blue,->] (P_1_3__Rationals) -- (P_1_4__Rationals);
	\draw[gray,dashed,->] (P_0_5__Rationals) -- (P_1_5__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[blue,->] (P_0_6__Rationals) -- (P_1_5__Rationals);
	\draw[blue,->] (P_1_4__Rationals) -- (P_1_5__Rationals);
	\draw[gray,dashed,->] (P_0_6__Rationals) -- (P_1_6__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[blue,->] (P_1_2__Rationals) -- (P_1_6__Rationals);
	\draw[blue,->] (P_1_5__Rationals) -- (P_1_6__Rationals);
	\draw[gray,dashed,->] (P_1_1__Rationals) -- (P_2_1__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[blue,->] (P_1_2__Rationals) -- (P_2_1__Rationals);
	\draw[blue,->] (P_1_3__Rationals) -- (P_2_1__Rationals);
	\draw[gray,dashed,->] (P_1_2__Rationals) -- (P_2_2__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[blue,->] (P_1_6__Rationals) -- (P_2_2__Rationals);
	\draw[blue,->] (P_2_1__Rationals) -- (P_2_2__Rationals);
	\draw[gray,dashed,->] (P_1_3__Rationals) -- (P_2_3__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[blue,->] (P_1_4__Rationals) -- (P_2_3__Rationals);
	\draw[blue,->] (P_2_1__Rationals) to [out=135,in=-135] (P_2_3__Rationals);
	\draw[gray,dashed,->] (P_1_4__Rationals) -- (P_2_4__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[blue,->] (P_1_5__Rationals) -- (P_2_4__Rationals);
	\draw[blue,->] (P_2_3__Rationals) -- (P_2_4__Rationals);
	\draw[gray,dashed,->] (P_1_5__Rationals) -- (P_2_5__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[blue,->] (P_1_6__Rationals) -- (P_2_5__Rationals);
	\draw[blue,->] (P_2_4__Rationals) -- (P_2_5__Rationals);
	\draw[gray,dashed,->] (P_1_6__Rationals) -- (P_2_6__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[blue,->] (P_2_2__Rationals) -- (P_2_6__Rationals);
	\draw[blue,->] (P_2_5__Rationals) -- (P_2_6__Rationals);
	\draw[gray,dashed,->] (P_2_1__Rationals) -- (P_3_1__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[blue,->] (P_2_2__Rationals) -- (P_3_1__Rationals);
	\draw[blue,->] (P_2_3__Rationals) -- (P_3_1__Rationals);
	\draw[gray,dashed,->] (P_2_2__Rationals) -- (P_3_2__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[blue,->] (P_2_6__Rationals) -- (P_3_2__Rationals);
	\draw[blue,->] (P_3_1__Rationals) -- (P_3_2__Rationals);
	\draw[gray,dashed,->] (P_2_3__Rationals) -- (P_3_3__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[blue,->] (P_2_4__Rationals) -- (P_3_3__Rationals);
	\draw[blue,->] (P_3_1__Rationals) to [out=135,in=-135] (P_3_3__Rationals);
	\draw[gray,dashed,->] (P_2_4__Rationals) -- (P_3_4__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[blue,->] (P_2_5__Rationals) -- (P_3_4__Rationals);
	\draw[blue,->] (P_3_3__Rationals) -- (P_3_4__Rationals);
	\draw[gray,dashed,->] (P_2_5__Rationals) -- (P_3_5__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[blue,->] (P_2_6__Rationals) -- (P_3_5__Rationals);
	\draw[blue,->] (P_3_4__Rationals) -- (P_3_5__Rationals);
	\draw[gray,dashed,->] (P_2_6__Rationals) -- (P_3_6__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[blue,->] (P_3_2__Rationals) -- (P_3_6__Rationals);
	\draw[blue,->] (P_3_5__Rationals) -- (P_3_6__Rationals);
\end{tikzpicture}
\end{scaletikzpicturetowidth}
\end{center}

\end{figure}
\medskip{}
\subsubsection*{Modules of the form $P(n,1)$}

Defect: $\partial P(n,1) = -1$, for $n\ge 0$.

\begin{fleqn}
\begin{align*}
P(0,1):\ & - \\
P(1,1):\ & \bigl(R_{0}^{3}(1),P(0,2)\bigr),\ \bigl(R_{\infty}^{1}(1),P(0,3)\bigr)\\
P(2,1):\ & \bigl(R_{0}^{2}(3),P(0,5)\bigr),\ \bigl(R_{0}^{3}(2),P(0,6)\bigr),\ \bigl(R_{0}^{4}(1),P(1,2)\bigr),\ \bigl(R_{\infty}^{2}(1),P(1,3)\bigr),\ \bigl(I(1,6),2P(0,4)\bigr)\\
P(n,1):\ & \bigl(R_{0}^{(n-1)\bmod 4+1}(3),P(n-2,5)\bigr),\ \bigl(R_{0}^{n\bmod 4+1}(2),P(n-2,6)\bigr),\ \bigl(R_{0}^{(n+1)\bmod 4+1}(1),P(n-1,2)\bigr)\\
 & \bigl(R_{\infty}^{(n-1)\bmod 2+1}(1),P(n-1,3)\bigr),\ \bigl(uI,(u+1)P\bigr),\ n>2\\
\end{align*}
\end{fleqn}
\subsubsection*{Modules of the form $P(n,2)$}

Defect: $\partial P(n,2) = -1$, for $n\ge 0$.

\begin{fleqn}
\begin{align*}
P(0,2):\ & \bigl(R_{\infty}^{1}(1),P(0,1)\bigr)\\
P(1,2):\ & \bigl(R_{0}^{1}(3),P(0,4)\bigr),\ \bigl(R_{0}^{2}(2),P(0,5)\bigr),\ \bigl(R_{0}^{3}(1),P(0,6)\bigr),\ \bigl(R_{\infty}^{2}(1),P(1,1)\bigr),\ \bigl(I(1,5),2P(0,3)\bigr)\\
P(n,2):\ & \bigl(R_{0}^{(n-1)\bmod 4+1}(3),P(n-1,4)\bigr),\ \bigl(R_{0}^{n\bmod 4+1}(2),P(n-1,5)\bigr),\ \bigl(R_{0}^{(n+1)\bmod 4+1}(1),P(n-1,6)\bigr)\\
 & \bigl(R_{\infty}^{n\bmod 2+1}(1),P(n,1)\bigr),\ \bigl(uI,(u+1)P\bigr),\ n>1\\
\end{align*}
\end{fleqn}
\subsubsection*{Modules of the form $P(n,3)$}

Defect: $\partial P(n,3) = -1$, for $n\ge 0$.

\begin{fleqn}
\begin{align*}
P(0,3):\ & \bigl(R_{0}^{3}(1),P(0,1)\bigr)\\
P(1,3):\ & \bigl(R_{0}^{3}(2),P(0,2)\bigr),\ \bigl(R_{\infty}^{1}(1),P(0,4)\bigr),\ \bigl(R_{0}^{4}(1),P(1,1)\bigr)\\
P(2,3):\ & \bigl(R_{0}^{3}(3),P(0,6)\bigr),\ \bigl(R_{0}^{4}(2),P(1,2)\bigr),\ \bigl(R_{\infty}^{2}(1),P(1,4)\bigr),\ \bigl(R_{0}^{1}(1),P(2,1)\bigr),\ \bigl(I(0,2),2P(0,5)\bigr)\\
P(n,3):\ & \bigl(R_{0}^{n\bmod 4+1}(3),P(n-2,6)\bigr),\ \bigl(R_{0}^{(n+1)\bmod 4+1}(2),P(n-1,2)\bigr),\ \bigl(R_{\infty}^{(n-1)\bmod 2+1}(1),P(n-1,4)\bigr)\\
 & \bigl(R_{0}^{(n-2)\bmod 4+1}(1),P(n,1)\bigr),\ \bigl(uI,(u+1)P\bigr),\ n>2\\
\end{align*}
\end{fleqn}
\subsubsection*{Modules of the form $P(n,4)$}

Defect: $\partial P(n,4) = -1$, for $n\ge 0$.

\begin{fleqn}
\begin{align*}
P(0,4):\ & \bigl(R_{0}^{3}(2),P(0,1)\bigr),\ \bigl(R_{0}^{4}(1),P(0,3)\bigr)\\
P(1,4):\ & \bigl(R_{0}^{3}(3),P(0,2)\bigr),\ \bigl(R_{\infty}^{1}(1),P(0,5)\bigr),\ \bigl(R_{0}^{4}(2),P(1,1)\bigr),\ \bigl(R_{0}^{1}(1),P(1,3)\bigr)\\
P(2,4):\ & \bigl(R_{0}^{4}(3),P(1,2)\bigr),\ \bigl(R_{\infty}^{2}(1),P(1,5)\bigr),\ \bigl(R_{0}^{1}(2),P(2,1)\bigr),\ \bigl(R_{0}^{2}(1),P(2,3)\bigr),\ \bigl(2I(1,4),3P(0,1)\bigr)\\
P(n,4):\ & \bigl(R_{0}^{(n+1)\bmod 4+1}(3),P(n-1,2)\bigr),\ \bigl(R_{\infty}^{(n-1)\bmod 2+1}(1),P(n-1,5)\bigr),\ \bigl(R_{0}^{(n-2)\bmod 4+1}(2),P(n,1)\bigr)\\
 & \bigl(R_{0}^{(n-1)\bmod 4+1}(1),P(n,3)\bigr),\ \bigl(uI,(u+1)P\bigr),\ n>2\\
\end{align*}
\end{fleqn}
\subsubsection*{Modules of the form $P(n,5)$}

Defect: $\partial P(n,5) = -1$, for $n\ge 0$.

\begin{fleqn}
\begin{align*}
P(0,5):\ & \bigl(R_{0}^{3}(3),P(0,1)\bigr),\ \bigl(R_{0}^{4}(2),P(0,3)\bigr),\ \bigl(R_{0}^{1}(1),P(0,4)\bigr)\\
P(1,5):\ & \bigl(R_{\infty}^{1}(1),P(0,6)\bigr),\ \bigl(R_{0}^{4}(3),P(1,1)\bigr),\ \bigl(R_{0}^{1}(2),P(1,3)\bigr),\ \bigl(R_{0}^{2}(1),P(1,4)\bigr),\ \bigl(I(0,3),2P(0,2)\bigr)\\
P(n,5):\ & \bigl(R_{\infty}^{(n-1)\bmod 2+1}(1),P(n-1,6)\bigr),\ \bigl(R_{0}^{(n+2)\bmod 4+1}(3),P(n,1)\bigr),\ \bigl(R_{0}^{(n-1)\bmod 4+1}(2),P(n,3)\bigr)\\
 & \bigl(R_{0}^{n\bmod 4+1}(1),P(n,4)\bigr),\ \bigl(uI,(u+1)P\bigr),\ n>1\\
\end{align*}
\end{fleqn}
\subsubsection*{Modules of the form $P(n,6)$}

Defect: $\partial P(n,6) = -1$, for $n\ge 0$.

\begin{fleqn}
\begin{align*}
P(0,6):\ & \bigl(R_{\infty}^{2}(1),P(0,2)\bigr),\ \bigl(R_{0}^{4}(3),P(0,3)\bigr),\ \bigl(R_{0}^{1}(2),P(0,4)\bigr),\ \bigl(R_{0}^{2}(1),P(0,5)\bigr),\ \bigl(I(1,4),2P(0,1)\bigr)\\
P(n,6):\ & \bigl(R_{\infty}^{(n+1)\bmod 2+1}(1),P(n,2)\bigr),\ \bigl(R_{0}^{(n+3)\bmod 4+1}(3),P(n,3)\bigr),\ \bigl(R_{0}^{n\bmod 4+1}(2),P(n,4)\bigr)\\
 & \bigl(R_{0}^{(n+1)\bmod 4+1}(1),P(n,5)\bigr),\ \bigl(uI,(u+1)P\bigr),\ n>0\\
\end{align*}
\end{fleqn}
\subsubsection{Schofield pairs associated to preinjective exceptional modules}

\begin{figure}[ht]

\begin{center}
\begin{scaletikzpicturetowidth}{\textwidth}
\begin{tikzpicture}[scale=\tikzscale]
	\node[seloneAAlAAAA, label={[shift={(0.4,-0.2)}, brown]\tiny $I(0,1)$}] (I_0_1__Rationals) at (0,0) {$\begin{smallmatrix}&1\\1&&&&2\\&1&1&1\end{smallmatrix}$};
	\node[noselAAlAAAA, label={[shift={(0.4,-0.2)}, brown]\tiny $I(0,2)$}] (I_0_2__Rationals) at (0,-1) {$\begin{smallmatrix}&1\\0&&&&1\\&0&0&0\end{smallmatrix}$};
	\node[noselAAlAAAA, label={[shift={(0.4,-0.2)}, brown]\tiny $I(0,3)$}] (I_0_3__Rationals) at (0,-5) {$\begin{smallmatrix}&0\\0&&&&1\\&1&1&1\end{smallmatrix}$};
	\node[noselAAlAAAA, label={[shift={(0.4,-0.2)}, brown]\tiny $I(0,4)$}] (I_0_4__Rationals) at (0,-4) {$\begin{smallmatrix}&0\\0&&&&1\\&0&1&1\end{smallmatrix}$};
	\node[noselAAlAAAA, label={[shift={(0.4,-0.2)}, brown]\tiny $I(0,5)$}] (I_0_5__Rationals) at (0,-3) {$\begin{smallmatrix}&0\\0&&&&1\\&0&0&1\end{smallmatrix}$};
	\node[noselAAlAAAA, label={[shift={(0.4,-0.2)}, brown]\tiny $I(0,6)$}] (I_0_6__Rationals) at (0,-2) {$\begin{smallmatrix}&0\\0&&&&1\\&0&0&0\end{smallmatrix}$};
	\node[noselAAlAAAA, label={[shift={(0.4,-0.2)}, brown]\tiny $I(1,1)$}] (I_1_1__Rationals) at (-4,0) {$\begin{smallmatrix}&2\\1&&&&2\\&1&1&2\end{smallmatrix}$};
	\node[seloneAAlAAAA, label={[shift={(0.4,-0.2)}, brown]\tiny $I(1,2)$}] (I_1_2__Rationals) at (-4,-1) {$\begin{smallmatrix}&1\\1&&&&2\\&1&1&2\end{smallmatrix}$};
	\node[seloneAAlAAAA, label={[shift={(0.4,-0.2)}, brown]\tiny $I(1,3)$}] (I_1_3__Rationals) at (-4,-5) {$\begin{smallmatrix}&2\\1&&&&2\\&1&1&1\end{smallmatrix}$};
	\node[noselAAlAAAA, label={[shift={(0.4,-0.2)}, brown]\tiny $I(1,4)$}] (I_1_4__Rationals) at (-4,-4) {$\begin{smallmatrix}&1\\0&&&&1\\&1&1&1\end{smallmatrix}$};
	\node[noselAAlAAAA, label={[shift={(0.4,-0.2)}, brown]\tiny $I(1,5)$}] (I_1_5__Rationals) at (-4,-3) {$\begin{smallmatrix}&1\\0&&&&1\\&0&1&1\end{smallmatrix}$};
	\node[noselAAlAAAA, label={[shift={(0.4,-0.2)}, brown]\tiny $I(1,6)$}] (I_1_6__Rationals) at (-4,-2) {$\begin{smallmatrix}&1\\0&&&&1\\&0&0&1\end{smallmatrix}$};
	\node[noselAAlAAAA, label={[shift={(0.4,-0.2)}, brown]\tiny $I(2,1)$}] (I_2_1__Rationals) at (-8,0) {$\begin{smallmatrix}&2\\2&&&&3\\&2&3&3\end{smallmatrix}$};
	\node[noselAAlAAAA, label={[shift={(0.4,-0.2)}, brown]\tiny $I(2,2)$}] (I_2_2__Rationals) at (-8,-1) {$\begin{smallmatrix}&2\\1&&&&2\\&1&2&2\end{smallmatrix}$};
	\node[noselAAlAAAA, label={[shift={(0.4,-0.2)}, brown]\tiny $I(2,3)$}] (I_2_3__Rationals) at (-8,-5) {$\begin{smallmatrix}&2\\2&&&&3\\&2&2&3\end{smallmatrix}$};
	\node[seloneAAlAAAA, label={[shift={(0.4,-0.2)}, brown]\tiny $I(2,4)$}] (I_2_4__Rationals) at (-8,-4) {$\begin{smallmatrix}&2\\2&&&&3\\&2&2&2\end{smallmatrix}$};
	\node[seloneAAlAAAA, label={[shift={(0.4,-0.2)}, brown]\tiny $I(2,5)$}] (I_2_5__Rationals) at (-8,-3) {$\begin{smallmatrix}&1\\1&&&&2\\&2&2&2\end{smallmatrix}$};
	\node[seloneAAlAAAA, label={[shift={(0.4,-0.2)}, brown]\tiny $I(2,6)$}] (I_2_6__Rationals) at (-8,-2) {$\begin{smallmatrix}&1\\1&&&&2\\&1&2&2\end{smallmatrix}$};
	\node[noselAAlAAAA, label={[shift={(0.4,-0.2)}, brown]\tiny $I(3,1)$}] (I_3_1__Rationals) at (-12,0) {$\begin{smallmatrix}&3\\2&&&&3\\&3&3&3\end{smallmatrix}$};
	\node[noselAAlAAAA, label={[shift={(0.4,-0.2)}, brown]\tiny $I(3,2)$}] (I_3_2__Rationals) at (-12,-1) {$\begin{smallmatrix}&2\\2&&&&3\\&3&3&3\end{smallmatrix}$};
	\node[noselAAlAAAA, label={[shift={(0.4,-0.2)}, brown]\tiny $I(3,3)$}] (I_3_3__Rationals) at (-12,-5) {$\begin{smallmatrix}&3\\2&&&&3\\&2&3&3\end{smallmatrix}$};
	\node[noselAAlAAAA, label={[shift={(0.4,-0.2)}, brown]\tiny $I(3,4)$}] (I_3_4__Rationals) at (-12,-4) {$\begin{smallmatrix}&3\\2&&&&3\\&2&2&3\end{smallmatrix}$};
	\node[noselAAlAAAA, label={[shift={(0.4,-0.2)}, brown]\tiny $I(3,5)$}] (I_3_5__Rationals) at (-12,-3) {$\begin{smallmatrix}&3\\2&&&&3\\&2&2&2\end{smallmatrix}$};
	\node[noselAAlAAAA, label={[shift={(0.4,-0.2)}, brown]\tiny $I(3,6)$}] (I_3_6__Rationals) at (-12,-2) {$\begin{smallmatrix}&2\\1&&&&2\\&2&2&2\end{smallmatrix}$};
	\node at (-13,-2.5) {$\dots$};
	\draw[red,->] (I_1_2__Rationals) -- (I_0_1__Rationals);
	\draw[red,->] (I_1_3__Rationals) -- (I_0_1__Rationals);
	\draw[red,->] (I_0_1__Rationals) -- (I_0_2__Rationals);
	\draw[red,->] (I_1_6__Rationals) -- (I_0_2__Rationals);
	\draw[red,->] (I_0_1__Rationals) to [out=135,in=-135] (I_0_3__Rationals);
	\draw[red,->] (I_1_4__Rationals) -- (I_0_3__Rationals);
	\draw[red,->] (I_0_3__Rationals) -- (I_0_4__Rationals);
	\draw[red,->] (I_1_5__Rationals) -- (I_0_4__Rationals);
	\draw[red,->] (I_0_4__Rationals) -- (I_0_5__Rationals);
	\draw[red,->] (I_1_6__Rationals) -- (I_0_5__Rationals);
	\draw[red,->] (I_0_2__Rationals) -- (I_0_6__Rationals);
	\draw[red,->] (I_0_5__Rationals) -- (I_0_6__Rationals);
	\draw[gray,dashed,->] (I_0_1__Rationals) -- (I_1_1__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_2_2__Rationals) -- (I_1_1__Rationals);
	\draw[red,->] (I_2_3__Rationals) -- (I_1_1__Rationals);
	\draw[gray,dashed,->] (I_0_2__Rationals) -- (I_1_2__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_1_1__Rationals) -- (I_1_2__Rationals);
	\draw[red,->] (I_2_6__Rationals) -- (I_1_2__Rationals);
	\draw[gray,dashed,->] (I_0_3__Rationals) -- (I_1_3__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_1_1__Rationals) to [out=135,in=-135] (I_1_3__Rationals);
	\draw[red,->] (I_2_4__Rationals) -- (I_1_3__Rationals);
	\draw[gray,dashed,->] (I_0_4__Rationals) -- (I_1_4__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_1_3__Rationals) -- (I_1_4__Rationals);
	\draw[red,->] (I_2_5__Rationals) -- (I_1_4__Rationals);
	\draw[gray,dashed,->] (I_0_5__Rationals) -- (I_1_5__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_1_4__Rationals) -- (I_1_5__Rationals);
	\draw[red,->] (I_2_6__Rationals) -- (I_1_5__Rationals);
	\draw[gray,dashed,->] (I_0_6__Rationals) -- (I_1_6__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_1_2__Rationals) -- (I_1_6__Rationals);
	\draw[red,->] (I_1_5__Rationals) -- (I_1_6__Rationals);
	\draw[gray,dashed,->] (I_1_1__Rationals) -- (I_2_1__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_3_2__Rationals) -- (I_2_1__Rationals);
	\draw[red,->] (I_3_3__Rationals) -- (I_2_1__Rationals);
	\draw[gray,dashed,->] (I_1_2__Rationals) -- (I_2_2__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_2_1__Rationals) -- (I_2_2__Rationals);
	\draw[red,->] (I_3_6__Rationals) -- (I_2_2__Rationals);
	\draw[gray,dashed,->] (I_1_3__Rationals) -- (I_2_3__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_2_1__Rationals) to [out=135,in=-135] (I_2_3__Rationals);
	\draw[red,->] (I_3_4__Rationals) -- (I_2_3__Rationals);
	\draw[gray,dashed,->] (I_1_4__Rationals) -- (I_2_4__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_2_3__Rationals) -- (I_2_4__Rationals);
	\draw[red,->] (I_3_5__Rationals) -- (I_2_4__Rationals);
	\draw[gray,dashed,->] (I_1_5__Rationals) -- (I_2_5__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_2_4__Rationals) -- (I_2_5__Rationals);
	\draw[red,->] (I_3_6__Rationals) -- (I_2_5__Rationals);
	\draw[gray,dashed,->] (I_1_6__Rationals) -- (I_2_6__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_2_2__Rationals) -- (I_2_6__Rationals);
	\draw[red,->] (I_2_5__Rationals) -- (I_2_6__Rationals);
	\draw[gray,dashed,->] (I_2_1__Rationals) -- (I_3_1__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[gray,dashed,->] (I_2_2__Rationals) -- (I_3_2__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_3_1__Rationals) -- (I_3_2__Rationals);
	\draw[gray,dashed,->] (I_2_3__Rationals) -- (I_3_3__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_3_1__Rationals) to [out=135,in=-135] (I_3_3__Rationals);
	\draw[gray,dashed,->] (I_2_4__Rationals) -- (I_3_4__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_3_3__Rationals) -- (I_3_4__Rationals);
	\draw[gray,dashed,->] (I_2_5__Rationals) -- (I_3_5__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_3_4__Rationals) -- (I_3_5__Rationals);
	\draw[gray,dashed,->] (I_2_6__Rationals) -- (I_3_6__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_3_2__Rationals) -- (I_3_6__Rationals);
	\draw[red,->] (I_3_5__Rationals) -- (I_3_6__Rationals);
\end{tikzpicture}
\end{scaletikzpicturetowidth}
\end{center}

\end{figure}
\medskip{}
\subsubsection*{Modules of the form $I(n,1)$}

Defect: $\partial I(n,1) = 1$, for $n\ge 0$.

\begin{fleqn}
\begin{align*}
I(0,1):\ & \bigl(I(0,2),R_{\infty}^{2}(1)\bigr),\ \bigl(I(0,3),R_{0}^{2}(1)\bigr),\ \bigl(I(0,4),R_{0}^{2}(2)\bigr),\ \bigl(I(0,5),R_{0}^{2}(3)\bigr),\ \bigl(2I(0,6),P(1,4)\bigr)\\
I(n,1):\ & \bigl(I(n,2),R_{\infty}^{(-n+1)\bmod 2+1}(1)\bigr),\ \bigl(I(n,3),R_{0}^{(-n+1)\bmod 4+1}(1)\bigr),\ \bigl(I(n,4),R_{0}^{(-n+1)\bmod 4+1}(2)\bigr)\\
 & \bigl(I(n,5),R_{0}^{(-n+1)\bmod 4+1}(3)\bigr),\ \bigl((v+1)I,vP\bigr),\ n>0\\
\end{align*}
\end{fleqn}
\subsubsection*{Modules of the form $I(n,2)$}

Defect: $\partial I(n,2) = 1$, for $n\ge 0$.

\begin{fleqn}
\begin{align*}
I(0,2):\ & \bigl(I(0,6),R_{\infty}^{1}(1)\bigr)\\
I(1,2):\ & \bigl(I(0,1),R_{0}^{1}(1)\bigr),\ \bigl(I(0,3),R_{0}^{1}(2)\bigr),\ \bigl(I(0,4),R_{0}^{1}(3)\bigr),\ \bigl(I(1,6),R_{\infty}^{2}(1)\bigr),\ \bigl(2I(0,5),P(1,3)\bigr)\\
I(n,2):\ & \bigl(I(n-1,1),R_{0}^{(-n+1)\bmod 4+1}(1)\bigr),\ \bigl(I(n-1,3),R_{0}^{(-n+1)\bmod 4+1}(2)\bigr),\ \bigl(I(n-1,4),R_{0}^{(-n+1)\bmod 4+1}(3)\bigr)\\
 & \bigl(I(n,6),R_{\infty}^{(-n+2)\bmod 2+1}(1)\bigr),\ \bigl((v+1)I,vP\bigr),\ n>1\\
\end{align*}
\end{fleqn}
\subsubsection*{Modules of the form $I(n,3)$}

Defect: $\partial I(n,3) = 1$, for $n\ge 0$.

\begin{fleqn}
\begin{align*}
I(0,3):\ & \bigl(I(0,4),R_{0}^{3}(1)\bigr),\ \bigl(I(0,5),R_{0}^{3}(2)\bigr),\ \bigl(I(0,6),R_{0}^{3}(3)\bigr)\\
I(1,3):\ & \bigl(I(0,1),R_{\infty}^{1}(1)\bigr),\ \bigl(I(1,4),R_{0}^{2}(1)\bigr),\ \bigl(I(1,5),R_{0}^{2}(2)\bigr),\ \bigl(I(1,6),R_{0}^{2}(3)\bigr),\ \bigl(2I(0,2),P(0,5)\bigr)\\
I(n,3):\ & \bigl(I(n-1,1),R_{\infty}^{(-n+1)\bmod 2+1}(1)\bigr),\ \bigl(I(n,4),R_{0}^{(-n+2)\bmod 4+1}(1)\bigr),\ \bigl(I(n,5),R_{0}^{(-n+2)\bmod 4+1}(2)\bigr)\\
 & \bigl(I(n,6),R_{0}^{(-n+2)\bmod 4+1}(3)\bigr),\ \bigl((v+1)I,vP\bigr),\ n>1\\
\end{align*}
\end{fleqn}
\subsubsection*{Modules of the form $I(n,4)$}

Defect: $\partial I(n,4) = 1$, for $n\ge 0$.

\begin{fleqn}
\begin{align*}
I(0,4):\ & \bigl(I(0,5),R_{0}^{4}(1)\bigr),\ \bigl(I(0,6),R_{0}^{4}(2)\bigr)\\
I(1,4):\ & \bigl(I(0,2),R_{0}^{3}(3)\bigr),\ \bigl(I(0,3),R_{\infty}^{1}(1)\bigr),\ \bigl(I(1,5),R_{0}^{3}(1)\bigr),\ \bigl(I(1,6),R_{0}^{3}(2)\bigr)\\
I(2,4):\ & \bigl(I(1,2),R_{0}^{2}(3)\bigr),\ \bigl(I(1,3),R_{\infty}^{2}(1)\bigr),\ \bigl(I(2,5),R_{0}^{2}(1)\bigr),\ \bigl(I(2,6),R_{0}^{2}(2)\bigr),\ \bigl(3I(0,6),2P(1,4)\bigr)\\
I(n,4):\ & \bigl(I(n-1,2),R_{0}^{(-n+3)\bmod 4+1}(3)\bigr),\ \bigl(I(n-1,3),R_{\infty}^{(-n+3)\bmod 2+1}(1)\bigr),\ \bigl(I(n,5),R_{0}^{(-n+3)\bmod 4+1}(1)\bigr)\\
 & \bigl(I(n,6),R_{0}^{(-n+3)\bmod 4+1}(2)\bigr),\ \bigl((v+1)I,vP\bigr),\ n>2\\
\end{align*}
\end{fleqn}
\subsubsection*{Modules of the form $I(n,5)$}

Defect: $\partial I(n,5) = 1$, for $n\ge 0$.

\begin{fleqn}
\begin{align*}
I(0,5):\ & \bigl(I(0,6),R_{0}^{1}(1)\bigr)\\
I(1,5):\ & \bigl(I(0,2),R_{0}^{4}(2)\bigr),\ \bigl(I(0,4),R_{\infty}^{1}(1)\bigr),\ \bigl(I(1,6),R_{0}^{4}(1)\bigr)\\
I(2,5):\ & \bigl(I(0,1),R_{0}^{3}(3)\bigr),\ \bigl(I(1,2),R_{0}^{3}(2)\bigr),\ \bigl(I(1,4),R_{\infty}^{2}(1)\bigr),\ \bigl(I(2,6),R_{0}^{3}(1)\bigr),\ \bigl(2I(0,3),P(0,2)\bigr)\\
I(n,5):\ & \bigl(I(n-2,1),R_{0}^{(-n+4)\bmod 4+1}(3)\bigr),\ \bigl(I(n-1,2),R_{0}^{(-n+4)\bmod 4+1}(2)\bigr),\ \bigl(I(n-1,4),R_{\infty}^{(-n+3)\bmod 2+1}(1)\bigr)\\
 & \bigl(I(n,6),R_{0}^{(-n+4)\bmod 4+1}(1)\bigr),\ \bigl((v+1)I,vP\bigr),\ n>2\\
\end{align*}
\end{fleqn}
\subsubsection*{Modules of the form $I(n,6)$}

Defect: $\partial I(n,6) = 1$, for $n\ge 0$.

\begin{fleqn}
\begin{align*}
I(0,6):\ & - \\
I(1,6):\ & \bigl(I(0,2),R_{0}^{1}(1)\bigr),\ \bigl(I(0,5),R_{\infty}^{1}(1)\bigr)\\
I(2,6):\ & \bigl(I(0,1),R_{0}^{4}(2)\bigr),\ \bigl(I(0,3),R_{0}^{4}(3)\bigr),\ \bigl(I(1,2),R_{0}^{4}(1)\bigr),\ \bigl(I(1,5),R_{\infty}^{2}(1)\bigr),\ \bigl(2I(0,4),P(1,1)\bigr)\\
I(n,6):\ & \bigl(I(n-2,1),R_{0}^{(-n+5)\bmod 4+1}(2)\bigr),\ \bigl(I(n-2,3),R_{0}^{(-n+5)\bmod 4+1}(3)\bigr),\ \bigl(I(n-1,2),R_{0}^{(-n+5)\bmod 4+1}(1)\bigr)\\
 & \bigl(I(n-1,5),R_{\infty}^{(-n+3)\bmod 2+1}(1)\bigr),\ \bigl((v+1)I,vP\bigr),\ n>2\\
\end{align*}
\end{fleqn}
\subsubsection{Schofield pairs associated to regular exceptional modules}

\subsubsection*{The non-homogeneous tube $\mathcal{T}_{0}^{\Delta(\widetilde{\mathbb{A}}_{2,4})}$}

\begin{figure}[ht]

\begin{center}
\captionof{figure}{\vspace*{-10pt}$\mathcal{T}_{0}^{\Delta(\widetilde{\mathbb{A}}_{2,4})}$}
\begin{scaletikzpicturetowidth}{1.\textwidth}
\begin{tikzpicture}[scale=\tikzscale]
	\node[seltwoAAlAAAA, label={[shift={(0.0,-0.45)}, brown]\tiny $R_{0}^{1}(1)$}] (RN_1_1_1__Rationals) at (4,0) {$\begin{smallmatrix}&0\\0&&&&0\\&0&0&1\end{smallmatrix}$};
	\node[seltwoAAlAAAA, label={[shift={(0.0,-0.45)}, brown]\tiny $R_{0}^{2}(1)$}] (RN_1_2_1__Rationals) at (8,0) {$\begin{smallmatrix}&1\\1&&&&1\\&0&0&0\end{smallmatrix}$};
	\node[seltwoAAlAAAA, label={[shift={(0.0,-0.45)}, brown]\tiny $R_{0}^{3}(1)$}] (RN_1_3_1__Rationals) at (12,0) {$\begin{smallmatrix}&0\\0&&&&0\\&1&0&0\end{smallmatrix}$};
	\node[seltwoAAlAAAA, label={[shift={(0.0,-0.45)}, brown]\tiny $R_{0}^{4}(1)$}] (RN_1_4_1__Rationals) at (16,0) {$\begin{smallmatrix}&0\\0&&&&0\\&0&1&0\end{smallmatrix}$};
	\node[seltwoAAlAAAA, label={[shift={(0.0,-0.45)}, brown]\tiny $R_{0}^{1}(1)$}] (RN_1_1_1__Rationals_repeated) at (20,0) {$\begin{smallmatrix}&0\\0&&&&0\\&0&0&1\end{smallmatrix}$};
	\node[seltwoAAlAAAA, label={[shift={(0.0,-0.45)}, brown]\tiny $R_{0}^{1}(2)$}] (RN_1_1_2__Rationals) at (6,2) {$\begin{smallmatrix}&1\\1&&&&1\\&0&0&1\end{smallmatrix}$};
	\node[seltwoAAlAAAA, label={[shift={(0.0,-0.45)}, brown]\tiny $R_{0}^{2}(2)$}] (RN_1_2_2__Rationals) at (10,2) {$\begin{smallmatrix}&1\\1&&&&1\\&1&0&0\end{smallmatrix}$};
	\node[seltwoAAlAAAA, label={[shift={(0.0,-0.45)}, brown]\tiny $R_{0}^{3}(2)$}] (RN_1_3_2__Rationals) at (14,2) {$\begin{smallmatrix}&0\\0&&&&0\\&1&1&0\end{smallmatrix}$};
	\node[seltwoAAlAAAA, label={[shift={(0.0,-0.45)}, brown]\tiny $R_{0}^{4}(2)$}] (RN_1_4_2__Rationals) at (18,2) {$\begin{smallmatrix}&0\\0&&&&0\\&0&1&1\end{smallmatrix}$};
	\node[seltwoAAlAAAA, label={[shift={(0.0,-0.45)}, brown]\tiny $R_{0}^{4}(3)$}] (RN_1_4_3__Rationals) at (4,4) {$\begin{smallmatrix}&1\\1&&&&1\\&0&1&1\end{smallmatrix}$};
	\node[seltwoAAlAAAA, label={[shift={(0.0,-0.45)}, brown]\tiny $R_{0}^{1}(3)$}] (RN_1_1_3__Rationals) at (8,4) {$\begin{smallmatrix}&1\\1&&&&1\\&1&0&1\end{smallmatrix}$};
	\node[seltwoAAlAAAA, label={[shift={(0.0,-0.45)}, brown]\tiny $R_{0}^{2}(3)$}] (RN_1_2_3__Rationals) at (12,4) {$\begin{smallmatrix}&1\\1&&&&1\\&1&1&0\end{smallmatrix}$};
	\node[seltwoAAlAAAA, label={[shift={(0.0,-0.45)}, brown]\tiny $R_{0}^{3}(3)$}] (RN_1_3_3__Rationals) at (16,4) {$\begin{smallmatrix}&0\\0&&&&0\\&1&1&1\end{smallmatrix}$};
	\node[seltwoAAlAAAA, label={[shift={(0.0,-0.45)}, brown]\tiny $R_{0}^{4}(3)$}] (RN_1_4_3__Rationals_repeated) at (20,4) {$\begin{smallmatrix}&1\\1&&&&1\\&0&1&1\end{smallmatrix}$};
	\node[noselAAlAAAA, label={[shift={(0.0,-0.45)}, brown]\tiny $R_{0}^{4}(4)$}] (RN_1_4_4__Rationals) at (6,6) {$\begin{smallmatrix}&1\\1&&&&1\\&1&1&1\end{smallmatrix}$};
	\node[noselAAlAAAA, label={[shift={(0.0,-0.45)}, brown]\tiny $R_{0}^{1}(4)$}] (RN_1_1_4__Rationals) at (10,6) {$\begin{smallmatrix}&1\\1&&&&1\\&1&1&1\end{smallmatrix}$};
	\node[noselAAlAAAA, label={[shift={(0.0,-0.45)}, brown]\tiny $R_{0}^{2}(4)$}] (RN_1_2_4__Rationals) at (14,6) {$\begin{smallmatrix}&1\\1&&&&1\\&1&1&1\end{smallmatrix}$};
	\node[noselAAlAAAA, label={[shift={(0.0,-0.45)}, brown]\tiny $R_{0}^{3}(4)$}] (RN_1_3_4__Rationals) at (18,6) {$\begin{smallmatrix}&1\\1&&&&1\\&1&1&1\end{smallmatrix}$};
	\node[noselAAlAAAA, label={[shift={(0.0,-0.45)}, brown]\tiny $R_{0}^{3}(5)$}] (RN_1_3_5__Rationals) at (4,8) {$\begin{smallmatrix}&1\\1&&&&1\\&2&1&1\end{smallmatrix}$};
	\node[noselAAlAAAA, label={[shift={(0.0,-0.45)}, brown]\tiny $R_{0}^{4}(5)$}] (RN_1_4_5__Rationals) at (8,8) {$\begin{smallmatrix}&1\\1&&&&1\\&1&2&1\end{smallmatrix}$};
	\node[noselAAlAAAA, label={[shift={(0.0,-0.45)}, brown]\tiny $R_{0}^{1}(5)$}] (RN_1_1_5__Rationals) at (12,8) {$\begin{smallmatrix}&1\\1&&&&1\\&1&1&2\end{smallmatrix}$};
	\node[noselAAlAAAA, label={[shift={(0.0,-0.45)}, brown]\tiny $R_{0}^{2}(5)$}] (RN_1_2_5__Rationals) at (16,8) {$\begin{smallmatrix}&2\\2&&&&2\\&1&1&1\end{smallmatrix}$};
	\node[noselAAlAAAA, label={[shift={(0.0,-0.45)}, brown]\tiny $R_{0}^{3}(5)$}] (RN_1_3_5__Rationals_repeated) at (20,8) {$\begin{smallmatrix}&1\\1&&&&1\\&2&1&1\end{smallmatrix}$};
	\draw[blue,->] (RN_1_1_1__Rationals) -- (RN_1_1_2__Rationals);
	\draw[gray,dashed,->] (RN_1_1_1__Rationals) -- (RN_1_2_1__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[red,->] (RN_1_1_2__Rationals) -- (RN_1_2_1__Rationals);
	\draw[gray,dashed,->] (RN_1_1_2__Rationals) -- (RN_1_2_2__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[blue,->] (RN_1_1_2__Rationals) -- (RN_1_1_3__Rationals);
	\draw[blue,->] (RN_1_2_1__Rationals) -- (RN_1_2_2__Rationals);
	\draw[gray,dashed,->] (RN_1_2_1__Rationals) -- (RN_1_3_1__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[red,->] (RN_1_2_2__Rationals) -- (RN_1_3_1__Rationals);
	\draw[gray,dashed,->] (RN_1_2_2__Rationals) -- (RN_1_3_2__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[blue,->] (RN_1_2_2__Rationals) -- (RN_1_2_3__Rationals);
	\draw[blue,->] (RN_1_3_1__Rationals) -- (RN_1_3_2__Rationals);
	\draw[gray,dashed,->] (RN_1_3_1__Rationals) -- (RN_1_4_1__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[red,->] (RN_1_3_2__Rationals) -- (RN_1_4_1__Rationals);
	\draw[gray,dashed,->] (RN_1_3_2__Rationals) -- (RN_1_4_2__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[blue,->] (RN_1_3_2__Rationals) -- (RN_1_3_3__Rationals);
	\draw[gray,dashed,->] (RN_1_4_1__Rationals) -- (RN_1_1_1__Rationals_repeated) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[blue,->] (RN_1_4_1__Rationals) -- (RN_1_4_2__Rationals);
	\draw[red,->] (RN_1_4_2__Rationals) -- (RN_1_1_1__Rationals_repeated);
	\draw[gray,dashed,->,out=193,in=-13] (RN_1_4_2__Rationals) to node[midway,sloped,above]{{\tiny $\tau^{-1}$}} (RN_1_1_2__Rationals);
	\draw[blue,->] (RN_1_4_2__Rationals) -- (RN_1_4_3__Rationals_repeated);
	\draw[red,->] (RN_1_4_3__Rationals) -- (RN_1_1_2__Rationals);
	\draw[blue,->] (RN_1_4_3__Rationals) -- (RN_1_4_4__Rationals);
	\draw[gray,dashed,->] (RN_1_4_3__Rationals) -- (RN_1_1_3__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[red,->] (RN_1_4_4__Rationals) -- (RN_1_1_3__Rationals);
	\draw[gray,dashed,->] (RN_1_4_4__Rationals) -- (RN_1_1_4__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[blue,->] (RN_1_4_4__Rationals) -- (RN_1_4_5__Rationals);
	\draw[red,->] (RN_1_1_3__Rationals) -- (RN_1_2_2__Rationals);
	\draw[blue,->] (RN_1_1_3__Rationals) -- (RN_1_1_4__Rationals);
	\draw[gray,dashed,->] (RN_1_1_3__Rationals) -- (RN_1_2_3__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[red,->] (RN_1_1_4__Rationals) -- (RN_1_2_3__Rationals);
	\draw[gray,dashed,->] (RN_1_1_4__Rationals) -- (RN_1_2_4__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[blue,->] (RN_1_1_4__Rationals) -- (RN_1_1_5__Rationals);
	\draw[red,->] (RN_1_2_3__Rationals) -- (RN_1_3_2__Rationals);
	\draw[blue,->] (RN_1_2_3__Rationals) -- (RN_1_2_4__Rationals);
	\draw[gray,dashed,->] (RN_1_2_3__Rationals) -- (RN_1_3_3__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[red,->] (RN_1_2_4__Rationals) -- (RN_1_3_3__Rationals);
	\draw[gray,dashed,->] (RN_1_2_4__Rationals) -- (RN_1_3_4__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[blue,->] (RN_1_2_4__Rationals) -- (RN_1_2_5__Rationals);
	\draw[red,->] (RN_1_3_3__Rationals) -- (RN_1_4_2__Rationals);
	\draw[gray,dashed,->] (RN_1_3_3__Rationals) -- (RN_1_4_3__Rationals_repeated) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[blue,->] (RN_1_3_3__Rationals) -- (RN_1_3_4__Rationals);
	\draw[red,->] (RN_1_3_4__Rationals) -- (RN_1_4_3__Rationals_repeated);
	\draw[gray,dashed,->,out=193,in=-13] (RN_1_3_4__Rationals) to node[midway,sloped,above]{{\tiny $\tau^{-1}$}} (RN_1_4_4__Rationals);
	\draw[blue,->] (RN_1_3_4__Rationals) -- (RN_1_3_5__Rationals_repeated);
	\draw[red,->] (RN_1_3_5__Rationals) -- (RN_1_4_4__Rationals);
	\draw[gray,dashed,->] (RN_1_3_5__Rationals) -- (RN_1_4_5__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[red,->] (RN_1_4_5__Rationals) -- (RN_1_1_4__Rationals);
	\draw[gray,dashed,->] (RN_1_4_5__Rationals) -- (RN_1_1_5__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[red,->] (RN_1_1_5__Rationals) -- (RN_1_2_4__Rationals);
	\draw[gray,dashed,->] (RN_1_1_5__Rationals) -- (RN_1_2_5__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[red,->] (RN_1_2_5__Rationals) -- (RN_1_3_4__Rationals);
	\draw[gray,dashed,->] (RN_1_2_5__Rationals) -- (RN_1_3_5__Rationals_repeated) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[dashed,-] (RN_1_1_1__Rationals) -- (RN_1_4_3__Rationals);
	\draw[dashed,-] (RN_1_1_1__Rationals_repeated) -- (RN_1_4_3__Rationals_repeated);
	\draw[dashed,-] (RN_1_4_3__Rationals) -- (RN_1_3_5__Rationals);
	\draw[dashed,-] (RN_1_4_3__Rationals_repeated) -- (RN_1_3_5__Rationals_repeated);
\end{tikzpicture}
\end{scaletikzpicturetowidth}
\end{center}

\end{figure}
\begin{fleqn}
\begin{align*}
R_{0}^{1}(1):\ & - \\
R_{0}^{1}(2):\ & \bigl(R_{0}^{2}(1),R_{0}^{1}(1)\bigr),\ \bigl(I(1,6),P(0,1)\bigr),\ \bigl(I(0,5),P(0,2)\bigr)\\
R_{0}^{2}(1):\ & \bigl(I(0,2),P(0,1)\bigr),\ \bigl(I(0,6),P(0,2)\bigr)\\
R_{0}^{2}(2):\ & \bigl(R_{0}^{3}(1),R_{0}^{2}(1)\bigr),\ \bigl(I(0,2),P(0,3)\bigr),\ \bigl(I(0,6),P(1,1)\bigr)\\
R_{0}^{3}(1):\ & - \\
R_{0}^{3}(2):\ & \bigl(R_{0}^{4}(1),R_{0}^{3}(1)\bigr)\\
R_{0}^{4}(1):\ & - \\
R_{0}^{4}(2):\ & \bigl(R_{0}^{1}(1),R_{0}^{4}(1)\bigr)\\
R_{0}^{4}(3):\ & \bigl(R_{0}^{1}(2),R_{0}^{4}(1)\bigr),\ \bigl(R_{0}^{2}(1),R_{0}^{4}(2)\bigr),\ \bigl(I(1,5),P(0,1)\bigr),\ \bigl(I(0,4),P(0,2)\bigr)\\
R_{0}^{1}(3):\ & \bigl(R_{0}^{2}(2),R_{0}^{1}(1)\bigr),\ \bigl(R_{0}^{3}(1),R_{0}^{1}(2)\bigr),\ \bigl(I(1,6),P(0,3)\bigr),\ \bigl(I(0,5),P(1,1)\bigr)\\
R_{0}^{2}(3):\ & \bigl(R_{0}^{3}(2),R_{0}^{2}(1)\bigr),\ \bigl(R_{0}^{4}(1),R_{0}^{2}(2)\bigr),\ \bigl(I(0,2),P(0,4)\bigr),\ \bigl(I(0,6),P(1,3)\bigr)\\
R_{0}^{3}(3):\ & \bigl(R_{0}^{4}(2),R_{0}^{3}(1)\bigr),\ \bigl(R_{0}^{1}(1),R_{0}^{3}(2)\bigr)\\
\end{align*}
\end{fleqn}
\subsubsection*{The non-homogeneous tube $\mathcal{T}_{\infty}^{\Delta(\widetilde{\mathbb{A}}_{2,4})}$}

\begin{figure}[ht]

\begin{center}
\captionof{figure}{\vspace*{-10pt}$\mathcal{T}_{\infty}^{\Delta(\widetilde{\mathbb{A}}_{2,4})}$}
\begin{scaletikzpicturetowidth}{0.5\textwidth}
\begin{tikzpicture}[scale=\tikzscale]
	\node[seltwoAAlAAAA, label={[shift={(0.0,-0.45)}, brown]\tiny $R_{\infty}^{1}(1)$}] (RN_2_1_1__Rationals) at (4,0) {$\begin{smallmatrix}&1\\0&&&&0\\&0&0&0\end{smallmatrix}$};
	\node[seltwoAAlAAAA, label={[shift={(0.0,-0.45)}, brown]\tiny $R_{\infty}^{2}(1)$}] (RN_2_2_1__Rationals) at (8,0) {$\begin{smallmatrix}&0\\1&&&&1\\&1&1&1\end{smallmatrix}$};
	\node[seltwoAAlAAAA, label={[shift={(0.0,-0.45)}, brown]\tiny $R_{\infty}^{1}(1)$}] (RN_2_1_1__Rationals_repeated) at (12,0) {$\begin{smallmatrix}&1\\0&&&&0\\&0&0&0\end{smallmatrix}$};
	\node[noselAAlAAAA, label={[shift={(0.0,-0.45)}, brown]\tiny $R_{\infty}^{1}(2)$}] (RN_2_1_2__Rationals) at (6,2) {$\begin{smallmatrix}&1\\1&&&&1\\&1&1&1\end{smallmatrix}$};
	\node[noselAAlAAAA, label={[shift={(0.0,-0.45)}, brown]\tiny $R_{\infty}^{2}(2)$}] (RN_2_2_2__Rationals) at (10,2) {$\begin{smallmatrix}&1\\1&&&&1\\&1&1&1\end{smallmatrix}$};
	\node[noselAAlAAAA, label={[shift={(0.0,-0.45)}, brown]\tiny $R_{\infty}^{2}(3)$}] (RN_2_2_3__Rationals) at (4,4) {$\begin{smallmatrix}&1\\2&&&&2\\&2&2&2\end{smallmatrix}$};
	\node[noselAAlAAAA, label={[shift={(0.0,-0.45)}, brown]\tiny $R_{\infty}^{1}(3)$}] (RN_2_1_3__Rationals) at (8,4) {$\begin{smallmatrix}&2\\1&&&&1\\&1&1&1\end{smallmatrix}$};
	\node[noselAAlAAAA, label={[shift={(0.0,-0.45)}, brown]\tiny $R_{\infty}^{2}(3)$}] (RN_2_2_3__Rationals_repeated) at (12,4) {$\begin{smallmatrix}&1\\2&&&&2\\&2&2&2\end{smallmatrix}$};
	\draw[blue,->] (RN_2_1_1__Rationals) -- (RN_2_1_2__Rationals);
	\draw[gray,dashed,->] (RN_2_1_1__Rationals) -- (RN_2_2_1__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[red,->] (RN_2_1_2__Rationals) -- (RN_2_2_1__Rationals);
	\draw[gray,dashed,->] (RN_2_1_2__Rationals) -- (RN_2_2_2__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[blue,->] (RN_2_1_2__Rationals) -- (RN_2_1_3__Rationals);
	\draw[gray,dashed,->] (RN_2_2_1__Rationals) -- (RN_2_1_1__Rationals_repeated) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[blue,->] (RN_2_2_1__Rationals) -- (RN_2_2_2__Rationals);
	\draw[red,->] (RN_2_2_2__Rationals) -- (RN_2_1_1__Rationals_repeated);
	\draw[gray,dashed,->,out=219,in=-39] (RN_2_2_2__Rationals) to node[midway,sloped,above]{{\tiny $\tau^{-1}$}} (RN_2_1_2__Rationals);
	\draw[blue,->] (RN_2_2_2__Rationals) -- (RN_2_2_3__Rationals_repeated);
	\draw[red,->] (RN_2_2_3__Rationals) -- (RN_2_1_2__Rationals);
	\draw[gray,dashed,->] (RN_2_2_3__Rationals) -- (RN_2_1_3__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[red,->] (RN_2_1_3__Rationals) -- (RN_2_2_2__Rationals);
	\draw[gray,dashed,->] (RN_2_1_3__Rationals) -- (RN_2_2_3__Rationals_repeated) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[dashed,-] (RN_2_1_1__Rationals) -- (RN_2_2_3__Rationals);
	\draw[dashed,-] (RN_2_1_1__Rationals_repeated) -- (RN_2_2_3__Rationals_repeated);
\end{tikzpicture}
\end{scaletikzpicturetowidth}
\end{center}

\end{figure}
\begin{fleqn}
\begin{align*}
R_{\infty}^{1}(1):\ & - \\
R_{\infty}^{2}(1):\ & \bigl(I(0,3),P(0,1)\bigr),\ \bigl(I(0,4),P(0,3)\bigr),\ \bigl(I(0,5),P(0,4)\bigr),\ \bigl(I(0,6),P(0,5)\bigr)\\
\end{align*}
\end{fleqn}
\clearpage

\subsection[Schofield pairs for the quiver $\Delta(\widetilde{\mathbb{A}}_{3,3})$]{Schofield pairs for the quiver $\Delta(\widetilde{\mathbb{A}}_{3,3})$ -- {\usebox{\boxAAAlAAA}}}
\[
\vcenter{\hbox{\xymatrix{  & 2\ar[dl] & 3\ar[l] &   &   &  \\
1 &   &   & 6\ar[ul]\ar[dl] &   &  \\
  & 4\ar[ul] & 5\ar[l] &   &   &  }}}\qquad C_{\Delta(\widetilde{\mathbb{A}}_{3,3})} = \begin{bmatrix}1 & 1 & 1 & 1 & 1 & 2\\
0 & 1 & 1 & 0 & 0 & 1\\
0 & 0 & 1 & 0 & 0 & 1\\
0 & 0 & 0 & 1 & 1 & 1\\
0 & 0 & 0 & 0 & 1 & 1\\
0 & 0 & 0 & 0 & 0 & 1\end{bmatrix}\quad\Phi_{\Delta(\widetilde{\mathbb{A}}_{3,3})} = \begin{bmatrix}-1 & 1 & 0 & 1 & 0 & 0\\
-1 & 0 & 1 & 1 & 0 & 0\\
-1 & 0 & 0 & 1 & 0 & 1\\
-1 & 1 & 0 & 0 & 1 & 0\\
-1 & 1 & 0 & 0 & 0 & 1\\
-2 & 1 & 0 & 1 & 0 & 1\end{bmatrix}
\]
\subsubsection{Schofield pairs associated to preprojective exceptional modules}

\begin{figure}[ht]

\begin{center}
\begin{scaletikzpicturetowidth}{\textwidth}
\begin{tikzpicture}[scale=\tikzscale]
	\node[noselAAAlAAA, label={[shift={(0.4,-0.2)}, brown]\tiny $P(0,1)$}] (P_0_1__Rationals) at (0,0) {$\begin{smallmatrix}&0&0\\1&&&0\\&0&0\end{smallmatrix}$};
	\node[noselAAAlAAA, label={[shift={(0.4,-0.2)}, brown]\tiny $P(0,2)$}] (P_0_2__Rationals) at (0,-1) {$\begin{smallmatrix}&1&0\\1&&&0\\&0&0\end{smallmatrix}$};
	\node[noselAAAlAAA, label={[shift={(0.4,-0.2)}, brown]\tiny $P(0,3)$}] (P_0_3__Rationals) at (0,-2) {$\begin{smallmatrix}&1&1\\1&&&0\\&0&0\end{smallmatrix}$};
	\node[noselAAAlAAA, label={[shift={(0.4,-0.2)}, brown]\tiny $P(0,4)$}] (P_0_4__Rationals) at (0,-5) {$\begin{smallmatrix}&0&0\\1&&&0\\&1&0\end{smallmatrix}$};
	\node[noselAAAlAAA, label={[shift={(0.4,-0.2)}, brown]\tiny $P(0,5)$}] (P_0_5__Rationals) at (0,-4) {$\begin{smallmatrix}&0&0\\1&&&0\\&1&1\end{smallmatrix}$};
	\node[seloneAAAlAAA, label={[shift={(0.4,-0.2)}, brown]\tiny $P(0,6)$}] (P_0_6__Rationals) at (0,-3) {$\begin{smallmatrix}&1&1\\2&&&1\\&1&1\end{smallmatrix}$};
	\node[noselAAAlAAA, label={[shift={(0.4,-0.2)}, brown]\tiny $P(1,1)$}] (P_1_1__Rationals) at (4,0) {$\begin{smallmatrix}&1&0\\1&&&0\\&1&0\end{smallmatrix}$};
	\node[noselAAAlAAA, label={[shift={(0.4,-0.2)}, brown]\tiny $P(1,2)$}] (P_1_2__Rationals) at (4,-1) {$\begin{smallmatrix}&1&1\\1&&&0\\&1&0\end{smallmatrix}$};
	\node[seloneAAAlAAA, label={[shift={(0.4,-0.2)}, brown]\tiny $P(1,3)$}] (P_1_3__Rationals) at (4,-2) {$\begin{smallmatrix}&1&1\\2&&&1\\&2&1\end{smallmatrix}$};
	\node[noselAAAlAAA, label={[shift={(0.4,-0.2)}, brown]\tiny $P(1,4)$}] (P_1_4__Rationals) at (4,-5) {$\begin{smallmatrix}&1&0\\1&&&0\\&1&1\end{smallmatrix}$};
	\node[seloneAAAlAAA, label={[shift={(0.4,-0.2)}, brown]\tiny $P(1,5)$}] (P_1_5__Rationals) at (4,-4) {$\begin{smallmatrix}&2&1\\2&&&1\\&1&1\end{smallmatrix}$};
	\node[noselAAAlAAA, label={[shift={(0.4,-0.2)}, brown]\tiny $P(1,6)$}] (P_1_6__Rationals) at (4,-3) {$\begin{smallmatrix}&2&1\\2&&&1\\&2&1\end{smallmatrix}$};
	\node[noselAAAlAAA, label={[shift={(0.4,-0.2)}, brown]\tiny $P(2,1)$}] (P_2_1__Rationals) at (8,0) {$\begin{smallmatrix}&1&1\\1&&&0\\&1&1\end{smallmatrix}$};
	\node[seloneAAAlAAA, label={[shift={(0.4,-0.2)}, brown]\tiny $P(2,2)$}] (P_2_2__Rationals) at (8,-1) {$\begin{smallmatrix}&1&1\\2&&&1\\&2&2\end{smallmatrix}$};
	\node[noselAAAlAAA, label={[shift={(0.4,-0.2)}, brown]\tiny $P(2,3)$}] (P_2_3__Rationals) at (8,-2) {$\begin{smallmatrix}&2&1\\2&&&1\\&2&2\end{smallmatrix}$};
	\node[seloneAAAlAAA, label={[shift={(0.4,-0.2)}, brown]\tiny $P(2,4)$}] (P_2_4__Rationals) at (8,-5) {$\begin{smallmatrix}&2&2\\2&&&1\\&1&1\end{smallmatrix}$};
	\node[noselAAAlAAA, label={[shift={(0.4,-0.2)}, brown]\tiny $P(2,5)$}] (P_2_5__Rationals) at (8,-4) {$\begin{smallmatrix}&2&2\\2&&&1\\&2&1\end{smallmatrix}$};
	\node[noselAAAlAAA, label={[shift={(0.4,-0.2)}, brown]\tiny $P(2,6)$}] (P_2_6__Rationals) at (8,-3) {$\begin{smallmatrix}&2&2\\2&&&1\\&2&2\end{smallmatrix}$};
	\node[seloneAAAlAAA, label={[shift={(0.4,-0.2)}, brown]\tiny $P(3,1)$}] (P_3_1__Rationals) at (12,0) {$\begin{smallmatrix}&2&2\\3&&&2\\&2&2\end{smallmatrix}$};
	\node[noselAAAlAAA, label={[shift={(0.4,-0.2)}, brown]\tiny $P(3,2)$}] (P_3_2__Rationals) at (12,-1) {$\begin{smallmatrix}&3&2\\3&&&2\\&2&2\end{smallmatrix}$};
	\node[noselAAAlAAA, label={[shift={(0.4,-0.2)}, brown]\tiny $P(3,3)$}] (P_3_3__Rationals) at (12,-2) {$\begin{smallmatrix}&3&3\\3&&&2\\&2&2\end{smallmatrix}$};
	\node[noselAAAlAAA, label={[shift={(0.4,-0.2)}, brown]\tiny $P(3,4)$}] (P_3_4__Rationals) at (12,-5) {$\begin{smallmatrix}&2&2\\3&&&2\\&3&2\end{smallmatrix}$};
	\node[noselAAAlAAA, label={[shift={(0.4,-0.2)}, brown]\tiny $P(3,5)$}] (P_3_5__Rationals) at (12,-4) {$\begin{smallmatrix}&2&2\\3&&&2\\&3&3\end{smallmatrix}$};
	\node[noselAAAlAAA, label={[shift={(0.4,-0.2)}, brown]\tiny $P(3,6)$}] (P_3_6__Rationals) at (12,-3) {$\begin{smallmatrix}&3&3\\4&&&3\\&3&3\end{smallmatrix}$};
	\node at (13,-2.5) {$\dots$};
	\draw[blue,->] (P_0_1__Rationals) -- (P_0_2__Rationals);
	\draw[blue,->] (P_0_2__Rationals) -- (P_0_3__Rationals);
	\draw[blue,->] (P_0_1__Rationals) to [out=135,in=-135] (P_0_4__Rationals);
	\draw[blue,->] (P_0_4__Rationals) -- (P_0_5__Rationals);
	\draw[blue,->] (P_0_3__Rationals) -- (P_0_6__Rationals);
	\draw[blue,->] (P_0_5__Rationals) -- (P_0_6__Rationals);
	\draw[gray,dashed,->] (P_0_1__Rationals) -- (P_1_1__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[blue,->] (P_0_2__Rationals) -- (P_1_1__Rationals);
	\draw[blue,->] (P_0_4__Rationals) -- (P_1_1__Rationals);
	\draw[gray,dashed,->] (P_0_2__Rationals) -- (P_1_2__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[blue,->] (P_0_3__Rationals) -- (P_1_2__Rationals);
	\draw[blue,->] (P_1_1__Rationals) -- (P_1_2__Rationals);
	\draw[gray,dashed,->] (P_0_3__Rationals) -- (P_1_3__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[blue,->] (P_0_6__Rationals) -- (P_1_3__Rationals);
	\draw[blue,->] (P_1_2__Rationals) -- (P_1_3__Rationals);
	\draw[gray,dashed,->] (P_0_4__Rationals) -- (P_1_4__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[blue,->] (P_0_5__Rationals) -- (P_1_4__Rationals);
	\draw[blue,->] (P_1_1__Rationals) to [out=135,in=-135] (P_1_4__Rationals);
	\draw[gray,dashed,->] (P_0_5__Rationals) -- (P_1_5__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[blue,->] (P_0_6__Rationals) -- (P_1_5__Rationals);
	\draw[blue,->] (P_1_4__Rationals) -- (P_1_5__Rationals);
	\draw[gray,dashed,->] (P_0_6__Rationals) -- (P_1_6__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[blue,->] (P_1_3__Rationals) -- (P_1_6__Rationals);
	\draw[blue,->] (P_1_5__Rationals) -- (P_1_6__Rationals);
	\draw[gray,dashed,->] (P_1_1__Rationals) -- (P_2_1__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[blue,->] (P_1_2__Rationals) -- (P_2_1__Rationals);
	\draw[blue,->] (P_1_4__Rationals) -- (P_2_1__Rationals);
	\draw[gray,dashed,->] (P_1_2__Rationals) -- (P_2_2__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[blue,->] (P_1_3__Rationals) -- (P_2_2__Rationals);
	\draw[blue,->] (P_2_1__Rationals) -- (P_2_2__Rationals);
	\draw[gray,dashed,->] (P_1_3__Rationals) -- (P_2_3__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[blue,->] (P_1_6__Rationals) -- (P_2_3__Rationals);
	\draw[blue,->] (P_2_2__Rationals) -- (P_2_3__Rationals);
	\draw[gray,dashed,->] (P_1_4__Rationals) -- (P_2_4__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[blue,->] (P_1_5__Rationals) -- (P_2_4__Rationals);
	\draw[blue,->] (P_2_1__Rationals) to [out=135,in=-135] (P_2_4__Rationals);
	\draw[gray,dashed,->] (P_1_5__Rationals) -- (P_2_5__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[blue,->] (P_1_6__Rationals) -- (P_2_5__Rationals);
	\draw[blue,->] (P_2_4__Rationals) -- (P_2_5__Rationals);
	\draw[gray,dashed,->] (P_1_6__Rationals) -- (P_2_6__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[blue,->] (P_2_3__Rationals) -- (P_2_6__Rationals);
	\draw[blue,->] (P_2_5__Rationals) -- (P_2_6__Rationals);
	\draw[gray,dashed,->] (P_2_1__Rationals) -- (P_3_1__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[blue,->] (P_2_2__Rationals) -- (P_3_1__Rationals);
	\draw[blue,->] (P_2_4__Rationals) -- (P_3_1__Rationals);
	\draw[gray,dashed,->] (P_2_2__Rationals) -- (P_3_2__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[blue,->] (P_2_3__Rationals) -- (P_3_2__Rationals);
	\draw[blue,->] (P_3_1__Rationals) -- (P_3_2__Rationals);
	\draw[gray,dashed,->] (P_2_3__Rationals) -- (P_3_3__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[blue,->] (P_2_6__Rationals) -- (P_3_3__Rationals);
	\draw[blue,->] (P_3_2__Rationals) -- (P_3_3__Rationals);
	\draw[gray,dashed,->] (P_2_4__Rationals) -- (P_3_4__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[blue,->] (P_2_5__Rationals) -- (P_3_4__Rationals);
	\draw[blue,->] (P_3_1__Rationals) to [out=135,in=-135] (P_3_4__Rationals);
	\draw[gray,dashed,->] (P_2_5__Rationals) -- (P_3_5__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[blue,->] (P_2_6__Rationals) -- (P_3_5__Rationals);
	\draw[blue,->] (P_3_4__Rationals) -- (P_3_5__Rationals);
	\draw[gray,dashed,->] (P_2_6__Rationals) -- (P_3_6__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[blue,->] (P_3_3__Rationals) -- (P_3_6__Rationals);
	\draw[blue,->] (P_3_5__Rationals) -- (P_3_6__Rationals);
\end{tikzpicture}
\end{scaletikzpicturetowidth}
\end{center}

\end{figure}
\medskip{}
\subsubsection*{Modules of the form $P(n,1)$}

Defect: $\partial P(n,1) = -1$, for $n\ge 0$.

\begin{fleqn}
\begin{align*}
P(0,1):\ & - \\
P(1,1):\ & \bigl(R_{0}^{3}(1),P(0,2)\bigr),\ \bigl(R_{\infty}^{3}(1),P(0,4)\bigr)\\
P(2,1):\ & \bigl(R_{0}^{3}(2),P(0,3)\bigr),\ \bigl(R_{\infty}^{3}(2),P(0,5)\bigr),\ \bigl(R_{0}^{1}(1),P(1,2)\bigr),\ \bigl(R_{\infty}^{1}(1),P(1,4)\bigr)\\
P(3,1):\ & \bigl(R_{0}^{1}(2),P(1,3)\bigr),\ \bigl(R_{\infty}^{1}(2),P(1,5)\bigr),\ \bigl(R_{0}^{2}(1),P(2,2)\bigr),\ \bigl(R_{\infty}^{2}(1),P(2,4)\bigr),\ \bigl(2I(2,6),3P(0,1)\bigr)\\
P(n,1):\ & \bigl(R_{0}^{(n-3)\bmod 3+1}(2),P(n-2,3)\bigr),\ \bigl(R_{\infty}^{(n-3)\bmod 3+1}(2),P(n-2,5)\bigr),\ \bigl(R_{0}^{(n-2)\bmod 3+1}(1),P(n-1,2)\bigr)\\
 & \bigl(R_{\infty}^{(n-2)\bmod 3+1}(1),P(n-1,4)\bigr),\ \bigl(uI,(u+1)P\bigr),\ n>3\\
\end{align*}
\end{fleqn}
\subsubsection*{Modules of the form $P(n,2)$}

Defect: $\partial P(n,2) = -1$, for $n\ge 0$.

\begin{fleqn}
\begin{align*}
P(0,2):\ & \bigl(R_{\infty}^{3}(1),P(0,1)\bigr)\\
P(1,2):\ & \bigl(R_{0}^{3}(1),P(0,3)\bigr),\ \bigl(R_{\infty}^{3}(2),P(0,4)\bigr),\ \bigl(R_{\infty}^{1}(1),P(1,1)\bigr)\\
P(2,2):\ & \bigl(R_{0}^{3}(2),P(0,6)\bigr),\ \bigl(R_{0}^{1}(1),P(1,3)\bigr),\ \bigl(R_{\infty}^{1}(2),P(1,4)\bigr),\ \bigl(R_{\infty}^{2}(1),P(2,1)\bigr),\ \bigl(I(0,2),2P(0,5)\bigr)\\
P(n,2):\ & \bigl(R_{0}^{n\bmod 3+1}(2),P(n-2,6)\bigr),\ \bigl(R_{0}^{(n-2)\bmod 3+1}(1),P(n-1,3)\bigr),\ \bigl(R_{\infty}^{(n-2)\bmod 3+1}(2),P(n-1,4)\bigr)\\
 & \bigl(R_{\infty}^{(n-1)\bmod 3+1}(1),P(n,1)\bigr),\ \bigl(uI,(u+1)P\bigr),\ n>2\\
\end{align*}
\end{fleqn}
\subsubsection*{Modules of the form $P(n,3)$}

Defect: $\partial P(n,3) = -1$, for $n\ge 0$.

\begin{fleqn}
\begin{align*}
P(0,3):\ & \bigl(R_{\infty}^{3}(2),P(0,1)\bigr),\ \bigl(R_{\infty}^{1}(1),P(0,2)\bigr)\\
P(1,3):\ & \bigl(R_{0}^{2}(2),P(0,5)\bigr),\ \bigl(R_{0}^{3}(1),P(0,6)\bigr),\ \bigl(R_{\infty}^{1}(2),P(1,1)\bigr),\ \bigl(R_{\infty}^{2}(1),P(1,2)\bigr),\ \bigl(I(1,3),2P(0,4)\bigr)\\
P(n,3):\ & \bigl(R_{0}^{n\bmod 3+1}(2),P(n-1,5)\bigr),\ \bigl(R_{0}^{(n+1)\bmod 3+1}(1),P(n-1,6)\bigr),\ \bigl(R_{\infty}^{(n-1)\bmod 3+1}(2),P(n,1)\bigr)\\
 & \bigl(R_{\infty}^{n\bmod 3+1}(1),P(n,2)\bigr),\ \bigl(uI,(u+1)P\bigr),\ n>1\\
\end{align*}
\end{fleqn}
\subsubsection*{Modules of the form $P(n,4)$}

Defect: $\partial P(n,4) = -1$, for $n\ge 0$.

\begin{fleqn}
\begin{align*}
P(0,4):\ & \bigl(R_{0}^{3}(1),P(0,1)\bigr)\\
P(1,4):\ & \bigl(R_{0}^{3}(2),P(0,2)\bigr),\ \bigl(R_{\infty}^{3}(1),P(0,5)\bigr),\ \bigl(R_{0}^{1}(1),P(1,1)\bigr)\\
P(2,4):\ & \bigl(R_{\infty}^{3}(2),P(0,6)\bigr),\ \bigl(R_{0}^{1}(2),P(1,2)\bigr),\ \bigl(R_{\infty}^{1}(1),P(1,5)\bigr),\ \bigl(R_{0}^{2}(1),P(2,1)\bigr),\ \bigl(I(0,4),2P(0,3)\bigr)\\
P(n,4):\ & \bigl(R_{\infty}^{n\bmod 3+1}(2),P(n-2,6)\bigr),\ \bigl(R_{0}^{(n-2)\bmod 3+1}(2),P(n-1,2)\bigr),\ \bigl(R_{\infty}^{(n-2)\bmod 3+1}(1),P(n-1,5)\bigr)\\
 & \bigl(R_{0}^{(n-1)\bmod 3+1}(1),P(n,1)\bigr),\ \bigl(uI,(u+1)P\bigr),\ n>2\\
\end{align*}
\end{fleqn}
\subsubsection*{Modules of the form $P(n,5)$}

Defect: $\partial P(n,5) = -1$, for $n\ge 0$.

\begin{fleqn}
\begin{align*}
P(0,5):\ & \bigl(R_{0}^{3}(2),P(0,1)\bigr),\ \bigl(R_{0}^{1}(1),P(0,4)\bigr)\\
P(1,5):\ & \bigl(R_{\infty}^{2}(2),P(0,3)\bigr),\ \bigl(R_{\infty}^{3}(1),P(0,6)\bigr),\ \bigl(R_{0}^{1}(2),P(1,1)\bigr),\ \bigl(R_{0}^{2}(1),P(1,4)\bigr),\ \bigl(I(1,5),2P(0,2)\bigr)\\
P(n,5):\ & \bigl(R_{\infty}^{n\bmod 3+1}(2),P(n-1,3)\bigr),\ \bigl(R_{\infty}^{(n+1)\bmod 3+1}(1),P(n-1,6)\bigr),\ \bigl(R_{0}^{(n-1)\bmod 3+1}(2),P(n,1)\bigr)\\
 & \bigl(R_{0}^{n\bmod 3+1}(1),P(n,4)\bigr),\ \bigl(uI,(u+1)P\bigr),\ n>1\\
\end{align*}
\end{fleqn}
\subsubsection*{Modules of the form $P(n,6)$}

Defect: $\partial P(n,6) = -1$, for $n\ge 0$.

\begin{fleqn}
\begin{align*}
P(0,6):\ & \bigl(R_{\infty}^{1}(2),P(0,2)\bigr),\ \bigl(R_{\infty}^{2}(1),P(0,3)\bigr),\ \bigl(R_{0}^{1}(2),P(0,4)\bigr),\ \bigl(R_{0}^{2}(1),P(0,5)\bigr),\ \bigl(I(2,6),2P(0,1)\bigr)\\
P(n,6):\ & \bigl(R_{\infty}^{n\bmod 3+1}(2),P(n,2)\bigr),\ \bigl(R_{\infty}^{(n+1)\bmod 3+1}(1),P(n,3)\bigr),\ \bigl(R_{0}^{n\bmod 3+1}(2),P(n,4)\bigr)\\
 & \bigl(R_{0}^{(n+1)\bmod 3+1}(1),P(n,5)\bigr),\ \bigl(uI,(u+1)P\bigr),\ n>0\\
\end{align*}
\end{fleqn}
\subsubsection{Schofield pairs associated to preinjective exceptional modules}

\begin{figure}[ht]

\begin{center}
\begin{scaletikzpicturetowidth}{\textwidth}
\begin{tikzpicture}[scale=\tikzscale]
	\node[seloneAAAlAAA, label={[shift={(0.4,-0.2)}, brown]\tiny $I(0,1)$}] (I_0_1__Rationals) at (0,0) {$\begin{smallmatrix}&1&1\\1&&&2\\&1&1\end{smallmatrix}$};
	\node[noselAAAlAAA, label={[shift={(0.4,-0.2)}, brown]\tiny $I(0,2)$}] (I_0_2__Rationals) at (0,-1) {$\begin{smallmatrix}&1&1\\0&&&1\\&0&0\end{smallmatrix}$};
	\node[noselAAAlAAA, label={[shift={(0.4,-0.2)}, brown]\tiny $I(0,3)$}] (I_0_3__Rationals) at (0,-2) {$\begin{smallmatrix}&0&1\\0&&&1\\&0&0\end{smallmatrix}$};
	\node[noselAAAlAAA, label={[shift={(0.4,-0.2)}, brown]\tiny $I(0,4)$}] (I_0_4__Rationals) at (0,-5) {$\begin{smallmatrix}&0&0\\0&&&1\\&1&1\end{smallmatrix}$};
	\node[noselAAAlAAA, label={[shift={(0.4,-0.2)}, brown]\tiny $I(0,5)$}] (I_0_5__Rationals) at (0,-4) {$\begin{smallmatrix}&0&0\\0&&&1\\&0&1\end{smallmatrix}$};
	\node[noselAAAlAAA, label={[shift={(0.4,-0.2)}, brown]\tiny $I(0,6)$}] (I_0_6__Rationals) at (0,-3) {$\begin{smallmatrix}&0&0\\0&&&1\\&0&0\end{smallmatrix}$};
	\node[noselAAAlAAA, label={[shift={(0.4,-0.2)}, brown]\tiny $I(1,1)$}] (I_1_1__Rationals) at (-4,0) {$\begin{smallmatrix}&1&2\\1&&&2\\&1&2\end{smallmatrix}$};
	\node[seloneAAAlAAA, label={[shift={(0.4,-0.2)}, brown]\tiny $I(1,2)$}] (I_1_2__Rationals) at (-4,-1) {$\begin{smallmatrix}&1&1\\1&&&2\\&1&2\end{smallmatrix}$};
	\node[noselAAAlAAA, label={[shift={(0.4,-0.2)}, brown]\tiny $I(1,3)$}] (I_1_3__Rationals) at (-4,-2) {$\begin{smallmatrix}&1&1\\0&&&1\\&0&1\end{smallmatrix}$};
	\node[seloneAAAlAAA, label={[shift={(0.4,-0.2)}, brown]\tiny $I(1,4)$}] (I_1_4__Rationals) at (-4,-5) {$\begin{smallmatrix}&1&2\\1&&&2\\&1&1\end{smallmatrix}$};
	\node[noselAAAlAAA, label={[shift={(0.4,-0.2)}, brown]\tiny $I(1,5)$}] (I_1_5__Rationals) at (-4,-4) {$\begin{smallmatrix}&0&1\\0&&&1\\&1&1\end{smallmatrix}$};
	\node[noselAAAlAAA, label={[shift={(0.4,-0.2)}, brown]\tiny $I(1,6)$}] (I_1_6__Rationals) at (-4,-3) {$\begin{smallmatrix}&0&1\\0&&&1\\&0&1\end{smallmatrix}$};
	\node[noselAAAlAAA, label={[shift={(0.4,-0.2)}, brown]\tiny $I(2,1)$}] (I_2_1__Rationals) at (-8,0) {$\begin{smallmatrix}&2&2\\1&&&2\\&2&2\end{smallmatrix}$};
	\node[noselAAAlAAA, label={[shift={(0.4,-0.2)}, brown]\tiny $I(2,2)$}] (I_2_2__Rationals) at (-8,-1) {$\begin{smallmatrix}&1&2\\1&&&2\\&2&2\end{smallmatrix}$};
	\node[seloneAAAlAAA, label={[shift={(0.4,-0.2)}, brown]\tiny $I(2,3)$}] (I_2_3__Rationals) at (-8,-2) {$\begin{smallmatrix}&1&1\\1&&&2\\&2&2\end{smallmatrix}$};
	\node[noselAAAlAAA, label={[shift={(0.4,-0.2)}, brown]\tiny $I(2,4)$}] (I_2_4__Rationals) at (-8,-5) {$\begin{smallmatrix}&2&2\\1&&&2\\&1&2\end{smallmatrix}$};
	\node[seloneAAAlAAA, label={[shift={(0.4,-0.2)}, brown]\tiny $I(2,5)$}] (I_2_5__Rationals) at (-8,-4) {$\begin{smallmatrix}&2&2\\1&&&2\\&1&1\end{smallmatrix}$};
	\node[noselAAAlAAA, label={[shift={(0.4,-0.2)}, brown]\tiny $I(2,6)$}] (I_2_6__Rationals) at (-8,-3) {$\begin{smallmatrix}&1&1\\0&&&1\\&1&1\end{smallmatrix}$};
	\node[noselAAAlAAA, label={[shift={(0.4,-0.2)}, brown]\tiny $I(3,1)$}] (I_3_1__Rationals) at (-12,0) {$\begin{smallmatrix}&3&3\\3&&&4\\&3&3\end{smallmatrix}$};
	\node[noselAAAlAAA, label={[shift={(0.4,-0.2)}, brown]\tiny $I(3,2)$}] (I_3_2__Rationals) at (-12,-1) {$\begin{smallmatrix}&3&3\\2&&&3\\&2&2\end{smallmatrix}$};
	\node[noselAAAlAAA, label={[shift={(0.4,-0.2)}, brown]\tiny $I(3,3)$}] (I_3_3__Rationals) at (-12,-2) {$\begin{smallmatrix}&2&3\\2&&&3\\&2&2\end{smallmatrix}$};
	\node[noselAAAlAAA, label={[shift={(0.4,-0.2)}, brown]\tiny $I(3,4)$}] (I_3_4__Rationals) at (-12,-5) {$\begin{smallmatrix}&2&2\\2&&&3\\&3&3\end{smallmatrix}$};
	\node[noselAAAlAAA, label={[shift={(0.4,-0.2)}, brown]\tiny $I(3,5)$}] (I_3_5__Rationals) at (-12,-4) {$\begin{smallmatrix}&2&2\\2&&&3\\&2&3\end{smallmatrix}$};
	\node[seloneAAAlAAA, label={[shift={(0.4,-0.2)}, brown]\tiny $I(3,6)$}] (I_3_6__Rationals) at (-12,-3) {$\begin{smallmatrix}&2&2\\2&&&3\\&2&2\end{smallmatrix}$};
	\node at (-13,-2.5) {$\dots$};
	\draw[red,->] (I_1_2__Rationals) -- (I_0_1__Rationals);
	\draw[red,->] (I_1_4__Rationals) -- (I_0_1__Rationals);
	\draw[red,->] (I_0_1__Rationals) -- (I_0_2__Rationals);
	\draw[red,->] (I_1_3__Rationals) -- (I_0_2__Rationals);
	\draw[red,->] (I_0_2__Rationals) -- (I_0_3__Rationals);
	\draw[red,->] (I_1_6__Rationals) -- (I_0_3__Rationals);
	\draw[red,->] (I_0_1__Rationals) to [out=135,in=-135] (I_0_4__Rationals);
	\draw[red,->] (I_1_5__Rationals) -- (I_0_4__Rationals);
	\draw[red,->] (I_0_4__Rationals) -- (I_0_5__Rationals);
	\draw[red,->] (I_1_6__Rationals) -- (I_0_5__Rationals);
	\draw[red,->] (I_0_3__Rationals) -- (I_0_6__Rationals);
	\draw[red,->] (I_0_5__Rationals) -- (I_0_6__Rationals);
	\draw[gray,dashed,->] (I_0_1__Rationals) -- (I_1_1__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_2_2__Rationals) -- (I_1_1__Rationals);
	\draw[red,->] (I_2_4__Rationals) -- (I_1_1__Rationals);
	\draw[gray,dashed,->] (I_0_2__Rationals) -- (I_1_2__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_1_1__Rationals) -- (I_1_2__Rationals);
	\draw[red,->] (I_2_3__Rationals) -- (I_1_2__Rationals);
	\draw[gray,dashed,->] (I_0_3__Rationals) -- (I_1_3__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_1_2__Rationals) -- (I_1_3__Rationals);
	\draw[red,->] (I_2_6__Rationals) -- (I_1_3__Rationals);
	\draw[gray,dashed,->] (I_0_4__Rationals) -- (I_1_4__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_1_1__Rationals) to [out=135,in=-135] (I_1_4__Rationals);
	\draw[red,->] (I_2_5__Rationals) -- (I_1_4__Rationals);
	\draw[gray,dashed,->] (I_0_5__Rationals) -- (I_1_5__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_1_4__Rationals) -- (I_1_5__Rationals);
	\draw[red,->] (I_2_6__Rationals) -- (I_1_5__Rationals);
	\draw[gray,dashed,->] (I_0_6__Rationals) -- (I_1_6__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_1_3__Rationals) -- (I_1_6__Rationals);
	\draw[red,->] (I_1_5__Rationals) -- (I_1_6__Rationals);
	\draw[gray,dashed,->] (I_1_1__Rationals) -- (I_2_1__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_3_2__Rationals) -- (I_2_1__Rationals);
	\draw[red,->] (I_3_4__Rationals) -- (I_2_1__Rationals);
	\draw[gray,dashed,->] (I_1_2__Rationals) -- (I_2_2__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_2_1__Rationals) -- (I_2_2__Rationals);
	\draw[red,->] (I_3_3__Rationals) -- (I_2_2__Rationals);
	\draw[gray,dashed,->] (I_1_3__Rationals) -- (I_2_3__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_2_2__Rationals) -- (I_2_3__Rationals);
	\draw[red,->] (I_3_6__Rationals) -- (I_2_3__Rationals);
	\draw[gray,dashed,->] (I_1_4__Rationals) -- (I_2_4__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_2_1__Rationals) to [out=135,in=-135] (I_2_4__Rationals);
	\draw[red,->] (I_3_5__Rationals) -- (I_2_4__Rationals);
	\draw[gray,dashed,->] (I_1_5__Rationals) -- (I_2_5__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_2_4__Rationals) -- (I_2_5__Rationals);
	\draw[red,->] (I_3_6__Rationals) -- (I_2_5__Rationals);
	\draw[gray,dashed,->] (I_1_6__Rationals) -- (I_2_6__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_2_3__Rationals) -- (I_2_6__Rationals);
	\draw[red,->] (I_2_5__Rationals) -- (I_2_6__Rationals);
	\draw[gray,dashed,->] (I_2_1__Rationals) -- (I_3_1__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[gray,dashed,->] (I_2_2__Rationals) -- (I_3_2__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_3_1__Rationals) -- (I_3_2__Rationals);
	\draw[gray,dashed,->] (I_2_3__Rationals) -- (I_3_3__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_3_2__Rationals) -- (I_3_3__Rationals);
	\draw[gray,dashed,->] (I_2_4__Rationals) -- (I_3_4__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_3_1__Rationals) to [out=135,in=-135] (I_3_4__Rationals);
	\draw[gray,dashed,->] (I_2_5__Rationals) -- (I_3_5__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_3_4__Rationals) -- (I_3_5__Rationals);
	\draw[gray,dashed,->] (I_2_6__Rationals) -- (I_3_6__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_3_3__Rationals) -- (I_3_6__Rationals);
	\draw[red,->] (I_3_5__Rationals) -- (I_3_6__Rationals);
\end{tikzpicture}
\end{scaletikzpicturetowidth}
\end{center}

\end{figure}
\medskip{}
\subsubsection*{Modules of the form $I(n,1)$}

Defect: $\partial I(n,1) = 1$, for $n\ge 0$.

\begin{fleqn}
\begin{align*}
I(0,1):\ & \bigl(I(0,2),R_{\infty}^{2}(1)\bigr),\ \bigl(I(0,3),R_{\infty}^{2}(2)\bigr),\ \bigl(I(0,4),R_{0}^{2}(1)\bigr),\ \bigl(I(0,5),R_{0}^{2}(2)\bigr),\ \bigl(2I(0,6),P(2,1)\bigr)\\
I(n,1):\ & \bigl(I(n,2),R_{\infty}^{(-n+1)\bmod 3+1}(1)\bigr),\ \bigl(I(n,3),R_{\infty}^{(-n+1)\bmod 3+1}(2)\bigr),\ \bigl(I(n,4),R_{0}^{(-n+1)\bmod 3+1}(1)\bigr)\\
 & \bigl(I(n,5),R_{0}^{(-n+1)\bmod 3+1}(2)\bigr),\ \bigl((v+1)I,vP\bigr),\ n>0\\
\end{align*}
\end{fleqn}
\subsubsection*{Modules of the form $I(n,2)$}

Defect: $\partial I(n,2) = 1$, for $n\ge 0$.

\begin{fleqn}
\begin{align*}
I(0,2):\ & \bigl(I(0,3),R_{\infty}^{3}(1)\bigr),\ \bigl(I(0,6),R_{\infty}^{3}(2)\bigr)\\
I(1,2):\ & \bigl(I(0,1),R_{0}^{1}(1)\bigr),\ \bigl(I(0,4),R_{0}^{1}(2)\bigr),\ \bigl(I(1,3),R_{\infty}^{2}(1)\bigr),\ \bigl(I(1,6),R_{\infty}^{2}(2)\bigr),\ \bigl(2I(0,5),P(1,2)\bigr)\\
I(n,2):\ & \bigl(I(n-1,1),R_{0}^{(-n+1)\bmod 3+1}(1)\bigr),\ \bigl(I(n-1,4),R_{0}^{(-n+1)\bmod 3+1}(2)\bigr),\ \bigl(I(n,3),R_{\infty}^{(-n+2)\bmod 3+1}(1)\bigr)\\
 & \bigl(I(n,6),R_{\infty}^{(-n+2)\bmod 3+1}(2)\bigr),\ \bigl((v+1)I,vP\bigr),\ n>1\\
\end{align*}
\end{fleqn}
\subsubsection*{Modules of the form $I(n,3)$}

Defect: $\partial I(n,3) = 1$, for $n\ge 0$.

\begin{fleqn}
\begin{align*}
I(0,3):\ & \bigl(I(0,6),R_{\infty}^{1}(1)\bigr)\\
I(1,3):\ & \bigl(I(0,2),R_{0}^{1}(1)\bigr),\ \bigl(I(0,5),R_{\infty}^{3}(2)\bigr),\ \bigl(I(1,6),R_{\infty}^{3}(1)\bigr)\\
I(2,3):\ & \bigl(I(0,1),R_{0}^{3}(2)\bigr),\ \bigl(I(1,2),R_{0}^{3}(1)\bigr),\ \bigl(I(1,5),R_{\infty}^{2}(2)\bigr),\ \bigl(I(2,6),R_{\infty}^{2}(1)\bigr),\ \bigl(2I(0,4),P(0,3)\bigr)\\
I(n,3):\ & \bigl(I(n-2,1),R_{0}^{(-n+4)\bmod 3+1}(2)\bigr),\ \bigl(I(n-1,2),R_{0}^{(-n+4)\bmod 3+1}(1)\bigr),\ \bigl(I(n-1,5),R_{\infty}^{(-n+3)\bmod 3+1}(2)\bigr)\\
 & \bigl(I(n,6),R_{\infty}^{(-n+3)\bmod 3+1}(1)\bigr),\ \bigl((v+1)I,vP\bigr),\ n>2\\
\end{align*}
\end{fleqn}
\subsubsection*{Modules of the form $I(n,4)$}

Defect: $\partial I(n,4) = 1$, for $n\ge 0$.

\begin{fleqn}
\begin{align*}
I(0,4):\ & \bigl(I(0,5),R_{0}^{3}(1)\bigr),\ \bigl(I(0,6),R_{0}^{3}(2)\bigr)\\
I(1,4):\ & \bigl(I(0,1),R_{\infty}^{1}(1)\bigr),\ \bigl(I(0,2),R_{\infty}^{1}(2)\bigr),\ \bigl(I(1,5),R_{0}^{2}(1)\bigr),\ \bigl(I(1,6),R_{0}^{2}(2)\bigr),\ \bigl(2I(0,3),P(1,4)\bigr)\\
I(n,4):\ & \bigl(I(n-1,1),R_{\infty}^{(-n+1)\bmod 3+1}(1)\bigr),\ \bigl(I(n-1,2),R_{\infty}^{(-n+1)\bmod 3+1}(2)\bigr),\ \bigl(I(n,5),R_{0}^{(-n+2)\bmod 3+1}(1)\bigr)\\
 & \bigl(I(n,6),R_{0}^{(-n+2)\bmod 3+1}(2)\bigr),\ \bigl((v+1)I,vP\bigr),\ n>1\\
\end{align*}
\end{fleqn}
\subsubsection*{Modules of the form $I(n,5)$}

Defect: $\partial I(n,5) = 1$, for $n\ge 0$.

\begin{fleqn}
\begin{align*}
I(0,5):\ & \bigl(I(0,6),R_{0}^{1}(1)\bigr)\\
I(1,5):\ & \bigl(I(0,3),R_{0}^{3}(2)\bigr),\ \bigl(I(0,4),R_{\infty}^{1}(1)\bigr),\ \bigl(I(1,6),R_{0}^{3}(1)\bigr)\\
I(2,5):\ & \bigl(I(0,1),R_{\infty}^{3}(2)\bigr),\ \bigl(I(1,3),R_{0}^{2}(2)\bigr),\ \bigl(I(1,4),R_{\infty}^{3}(1)\bigr),\ \bigl(I(2,6),R_{0}^{2}(1)\bigr),\ \bigl(2I(0,2),P(0,5)\bigr)\\
I(n,5):\ & \bigl(I(n-2,1),R_{\infty}^{(-n+4)\bmod 3+1}(2)\bigr),\ \bigl(I(n-1,3),R_{0}^{(-n+3)\bmod 3+1}(2)\bigr),\ \bigl(I(n-1,4),R_{\infty}^{(-n+4)\bmod 3+1}(1)\bigr)\\
 & \bigl(I(n,6),R_{0}^{(-n+3)\bmod 3+1}(1)\bigr),\ \bigl((v+1)I,vP\bigr),\ n>2\\
\end{align*}
\end{fleqn}
\subsubsection*{Modules of the form $I(n,6)$}

Defect: $\partial I(n,6) = 1$, for $n\ge 0$.

\begin{fleqn}
\begin{align*}
I(0,6):\ & - \\
I(1,6):\ & \bigl(I(0,3),R_{0}^{1}(1)\bigr),\ \bigl(I(0,5),R_{\infty}^{1}(1)\bigr)\\
I(2,6):\ & \bigl(I(0,2),R_{0}^{3}(2)\bigr),\ \bigl(I(0,4),R_{\infty}^{3}(2)\bigr),\ \bigl(I(1,3),R_{0}^{3}(1)\bigr),\ \bigl(I(1,5),R_{\infty}^{3}(1)\bigr)\\
I(3,6):\ & \bigl(I(1,2),R_{0}^{2}(2)\bigr),\ \bigl(I(1,4),R_{\infty}^{2}(2)\bigr),\ \bigl(I(2,3),R_{0}^{2}(1)\bigr),\ \bigl(I(2,5),R_{\infty}^{2}(1)\bigr),\ \bigl(3I(0,6),2P(2,1)\bigr)\\
I(n,6):\ & \bigl(I(n-2,2),R_{0}^{(-n+4)\bmod 3+1}(2)\bigr),\ \bigl(I(n-2,4),R_{\infty}^{(-n+4)\bmod 3+1}(2)\bigr),\ \bigl(I(n-1,3),R_{0}^{(-n+4)\bmod 3+1}(1)\bigr)\\
 & \bigl(I(n-1,5),R_{\infty}^{(-n+4)\bmod 3+1}(1)\bigr),\ \bigl((v+1)I,vP\bigr),\ n>3\\
\end{align*}
\end{fleqn}
\subsubsection{Schofield pairs associated to regular exceptional modules}

\subsubsection*{The non-homogeneous tube $\mathcal{T}_{0}^{\Delta(\widetilde{\mathbb{A}}_{3,3})}$}

\begin{figure}[ht]

\begin{center}
\captionof{figure}{\vspace*{-10pt}$\mathcal{T}_{0}^{\Delta(\widetilde{\mathbb{A}}_{3,3})}$}
\begin{scaletikzpicturetowidth}{0.80000000000000004\textwidth}
\begin{tikzpicture}[scale=\tikzscale]
	\node[seltwoAAAlAAA, label={[shift={(0.0,-0.45)}, brown]\tiny $R_{0}^{1}(1)$}] (RN_1_1_1__Rationals) at (4,0) {$\begin{smallmatrix}&0&0\\0&&&0\\&0&1\end{smallmatrix}$};
	\node[seltwoAAAlAAA, label={[shift={(0.0,-0.45)}, brown]\tiny $R_{0}^{2}(1)$}] (RN_1_2_1__Rationals) at (8,0) {$\begin{smallmatrix}&1&1\\1&&&1\\&0&0\end{smallmatrix}$};
	\node[seltwoAAAlAAA, label={[shift={(0.0,-0.45)}, brown]\tiny $R_{0}^{3}(1)$}] (RN_1_3_1__Rationals) at (12,0) {$\begin{smallmatrix}&0&0\\0&&&0\\&1&0\end{smallmatrix}$};
	\node[seltwoAAAlAAA, label={[shift={(0.0,-0.45)}, brown]\tiny $R_{0}^{1}(1)$}] (RN_1_1_1__Rationals_repeated) at (16,0) {$\begin{smallmatrix}&0&0\\0&&&0\\&0&1\end{smallmatrix}$};
	\node[seltwoAAAlAAA, label={[shift={(0.0,-0.45)}, brown]\tiny $R_{0}^{1}(2)$}] (RN_1_1_2__Rationals) at (6,2) {$\begin{smallmatrix}&1&1\\1&&&1\\&0&1\end{smallmatrix}$};
	\node[seltwoAAAlAAA, label={[shift={(0.0,-0.45)}, brown]\tiny $R_{0}^{2}(2)$}] (RN_1_2_2__Rationals) at (10,2) {$\begin{smallmatrix}&1&1\\1&&&1\\&1&0\end{smallmatrix}$};
	\node[seltwoAAAlAAA, label={[shift={(0.0,-0.45)}, brown]\tiny $R_{0}^{3}(2)$}] (RN_1_3_2__Rationals) at (14,2) {$\begin{smallmatrix}&0&0\\0&&&0\\&1&1\end{smallmatrix}$};
	\node[noselAAAlAAA, label={[shift={(0.0,-0.45)}, brown]\tiny $R_{0}^{3}(3)$}] (RN_1_3_3__Rationals) at (4,4) {$\begin{smallmatrix}&1&1\\1&&&1\\&1&1\end{smallmatrix}$};
	\node[noselAAAlAAA, label={[shift={(0.0,-0.45)}, brown]\tiny $R_{0}^{1}(3)$}] (RN_1_1_3__Rationals) at (8,4) {$\begin{smallmatrix}&1&1\\1&&&1\\&1&1\end{smallmatrix}$};
	\node[noselAAAlAAA, label={[shift={(0.0,-0.45)}, brown]\tiny $R_{0}^{2}(3)$}] (RN_1_2_3__Rationals) at (12,4) {$\begin{smallmatrix}&1&1\\1&&&1\\&1&1\end{smallmatrix}$};
	\node[noselAAAlAAA, label={[shift={(0.0,-0.45)}, brown]\tiny $R_{0}^{3}(3)$}] (RN_1_3_3__Rationals_repeated) at (16,4) {$\begin{smallmatrix}&1&1\\1&&&1\\&1&1\end{smallmatrix}$};
	\node[noselAAAlAAA, label={[shift={(0.0,-0.45)}, brown]\tiny $R_{0}^{3}(4)$}] (RN_1_3_4__Rationals) at (6,6) {$\begin{smallmatrix}&1&1\\1&&&1\\&2&1\end{smallmatrix}$};
	\node[noselAAAlAAA, label={[shift={(0.0,-0.45)}, brown]\tiny $R_{0}^{1}(4)$}] (RN_1_1_4__Rationals) at (10,6) {$\begin{smallmatrix}&1&1\\1&&&1\\&1&2\end{smallmatrix}$};
	\node[noselAAAlAAA, label={[shift={(0.0,-0.45)}, brown]\tiny $R_{0}^{2}(4)$}] (RN_1_2_4__Rationals) at (14,6) {$\begin{smallmatrix}&2&2\\2&&&2\\&1&1\end{smallmatrix}$};
	\draw[blue,->] (RN_1_1_1__Rationals) -- (RN_1_1_2__Rationals);
	\draw[gray,dashed,->] (RN_1_1_1__Rationals) -- (RN_1_2_1__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[red,->] (RN_1_1_2__Rationals) -- (RN_1_2_1__Rationals);
	\draw[gray,dashed,->] (RN_1_1_2__Rationals) -- (RN_1_2_2__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[blue,->] (RN_1_1_2__Rationals) -- (RN_1_1_3__Rationals);
	\draw[blue,->] (RN_1_2_1__Rationals) -- (RN_1_2_2__Rationals);
	\draw[gray,dashed,->] (RN_1_2_1__Rationals) -- (RN_1_3_1__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[red,->] (RN_1_2_2__Rationals) -- (RN_1_3_1__Rationals);
	\draw[gray,dashed,->] (RN_1_2_2__Rationals) -- (RN_1_3_2__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[blue,->] (RN_1_2_2__Rationals) -- (RN_1_2_3__Rationals);
	\draw[gray,dashed,->] (RN_1_3_1__Rationals) -- (RN_1_1_1__Rationals_repeated) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[blue,->] (RN_1_3_1__Rationals) -- (RN_1_3_2__Rationals);
	\draw[red,->] (RN_1_3_2__Rationals) -- (RN_1_1_1__Rationals_repeated);
	\draw[gray,dashed,->,out=200,in=-20] (RN_1_3_2__Rationals) to node[midway,sloped,above]{{\tiny $\tau^{-1}$}} (RN_1_1_2__Rationals);
	\draw[blue,->] (RN_1_3_2__Rationals) -- (RN_1_3_3__Rationals_repeated);
	\draw[red,->] (RN_1_3_3__Rationals) -- (RN_1_1_2__Rationals);
	\draw[blue,->] (RN_1_3_3__Rationals) -- (RN_1_3_4__Rationals);
	\draw[gray,dashed,->] (RN_1_3_3__Rationals) -- (RN_1_1_3__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[red,->] (RN_1_3_4__Rationals) -- (RN_1_1_3__Rationals);
	\draw[gray,dashed,->] (RN_1_3_4__Rationals) -- (RN_1_1_4__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[red,->] (RN_1_1_3__Rationals) -- (RN_1_2_2__Rationals);
	\draw[blue,->] (RN_1_1_3__Rationals) -- (RN_1_1_4__Rationals);
	\draw[gray,dashed,->] (RN_1_1_3__Rationals) -- (RN_1_2_3__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[red,->] (RN_1_1_4__Rationals) -- (RN_1_2_3__Rationals);
	\draw[gray,dashed,->] (RN_1_1_4__Rationals) -- (RN_1_2_4__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[red,->] (RN_1_2_3__Rationals) -- (RN_1_3_2__Rationals);
	\draw[gray,dashed,->] (RN_1_2_3__Rationals) -- (RN_1_3_3__Rationals_repeated) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[blue,->] (RN_1_2_3__Rationals) -- (RN_1_2_4__Rationals);
	\draw[red,->] (RN_1_2_4__Rationals) -- (RN_1_3_3__Rationals_repeated);
	\draw[gray,dashed,->,out=200,in=-20] (RN_1_2_4__Rationals) to node[midway,sloped,above]{{\tiny $\tau^{-1}$}} (RN_1_3_4__Rationals);
	\draw[dashed,-] (RN_1_1_1__Rationals) -- (RN_1_3_3__Rationals);
	\draw[dashed,-] (RN_1_1_1__Rationals_repeated) -- (RN_1_3_3__Rationals_repeated);
	\node (LeftTopPhantom_1) at (4,6) {};
	\draw[dashed,-] (RN_1_3_3__Rationals) -- (LeftTopPhantom_1);
	\node (RightTopPhantom_1) at (16,6) {};
	\draw[dashed,-] (RN_1_3_3__Rationals_repeated) -- (RightTopPhantom_1);
\end{tikzpicture}
\end{scaletikzpicturetowidth}
\end{center}

\end{figure}
\begin{fleqn}
\begin{align*}
R_{0}^{1}(1):\ & - \\
R_{0}^{1}(2):\ & \bigl(R_{0}^{2}(1),R_{0}^{1}(1)\bigr),\ \bigl(I(1,3),P(0,1)\bigr),\ \bigl(I(1,6),P(0,2)\bigr),\ \bigl(I(0,5),P(0,3)\bigr)\\
R_{0}^{2}(1):\ & \bigl(I(0,2),P(0,1)\bigr),\ \bigl(I(0,3),P(0,2)\bigr),\ \bigl(I(0,6),P(0,3)\bigr)\\
R_{0}^{2}(2):\ & \bigl(R_{0}^{3}(1),R_{0}^{2}(1)\bigr),\ \bigl(I(0,2),P(0,4)\bigr),\ \bigl(I(0,3),P(1,1)\bigr),\ \bigl(I(0,6),P(1,2)\bigr)\\
R_{0}^{3}(1):\ & - \\
R_{0}^{3}(2):\ & \bigl(R_{0}^{1}(1),R_{0}^{3}(1)\bigr)\\
\end{align*}
\end{fleqn}
\subsubsection*{The non-homogeneous tube $\mathcal{T}_{\infty}^{\Delta(\widetilde{\mathbb{A}}_{3,3})}$}

\begin{figure}[ht]

\begin{center}
\captionof{figure}{\vspace*{-10pt}$\mathcal{T}_{\infty}^{\Delta(\widetilde{\mathbb{A}}_{3,3})}$}
\begin{scaletikzpicturetowidth}{0.80000000000000004\textwidth}
\begin{tikzpicture}[scale=\tikzscale]
	\node[seltwoAAAlAAA, label={[shift={(0.0,-0.45)}, brown]\tiny $R_{\infty}^{1}(1)$}] (RN_2_1_1__Rationals) at (4,0) {$\begin{smallmatrix}&0&1\\0&&&0\\&0&0\end{smallmatrix}$};
	\node[seltwoAAAlAAA, label={[shift={(0.0,-0.45)}, brown]\tiny $R_{\infty}^{2}(1)$}] (RN_2_2_1__Rationals) at (8,0) {$\begin{smallmatrix}&0&0\\1&&&1\\&1&1\end{smallmatrix}$};
	\node[seltwoAAAlAAA, label={[shift={(0.0,-0.45)}, brown]\tiny $R_{\infty}^{3}(1)$}] (RN_2_3_1__Rationals) at (12,0) {$\begin{smallmatrix}&1&0\\0&&&0\\&0&0\end{smallmatrix}$};
	\node[seltwoAAAlAAA, label={[shift={(0.0,-0.45)}, brown]\tiny $R_{\infty}^{1}(1)$}] (RN_2_1_1__Rationals_repeated) at (16,0) {$\begin{smallmatrix}&0&1\\0&&&0\\&0&0\end{smallmatrix}$};
	\node[seltwoAAAlAAA, label={[shift={(0.0,-0.45)}, brown]\tiny $R_{\infty}^{1}(2)$}] (RN_2_1_2__Rationals) at (6,2) {$\begin{smallmatrix}&0&1\\1&&&1\\&1&1\end{smallmatrix}$};
	\node[seltwoAAAlAAA, label={[shift={(0.0,-0.45)}, brown]\tiny $R_{\infty}^{2}(2)$}] (RN_2_2_2__Rationals) at (10,2) {$\begin{smallmatrix}&1&0\\1&&&1\\&1&1\end{smallmatrix}$};
	\node[seltwoAAAlAAA, label={[shift={(0.0,-0.45)}, brown]\tiny $R_{\infty}^{3}(2)$}] (RN_2_3_2__Rationals) at (14,2) {$\begin{smallmatrix}&1&1\\0&&&0\\&0&0\end{smallmatrix}$};
	\node[noselAAAlAAA, label={[shift={(0.0,-0.45)}, brown]\tiny $R_{\infty}^{3}(3)$}] (RN_2_3_3__Rationals) at (4,4) {$\begin{smallmatrix}&1&1\\1&&&1\\&1&1\end{smallmatrix}$};
	\node[noselAAAlAAA, label={[shift={(0.0,-0.45)}, brown]\tiny $R_{\infty}^{1}(3)$}] (RN_2_1_3__Rationals) at (8,4) {$\begin{smallmatrix}&1&1\\1&&&1\\&1&1\end{smallmatrix}$};
	\node[noselAAAlAAA, label={[shift={(0.0,-0.45)}, brown]\tiny $R_{\infty}^{2}(3)$}] (RN_2_2_3__Rationals) at (12,4) {$\begin{smallmatrix}&1&1\\1&&&1\\&1&1\end{smallmatrix}$};
	\node[noselAAAlAAA, label={[shift={(0.0,-0.45)}, brown]\tiny $R_{\infty}^{3}(3)$}] (RN_2_3_3__Rationals_repeated) at (16,4) {$\begin{smallmatrix}&1&1\\1&&&1\\&1&1\end{smallmatrix}$};
	\node[noselAAAlAAA, label={[shift={(0.0,-0.45)}, brown]\tiny $R_{\infty}^{3}(4)$}] (RN_2_3_4__Rationals) at (6,6) {$\begin{smallmatrix}&2&1\\1&&&1\\&1&1\end{smallmatrix}$};
	\node[noselAAAlAAA, label={[shift={(0.0,-0.45)}, brown]\tiny $R_{\infty}^{1}(4)$}] (RN_2_1_4__Rationals) at (10,6) {$\begin{smallmatrix}&1&2\\1&&&1\\&1&1\end{smallmatrix}$};
	\node[noselAAAlAAA, label={[shift={(0.0,-0.45)}, brown]\tiny $R_{\infty}^{2}(4)$}] (RN_2_2_4__Rationals) at (14,6) {$\begin{smallmatrix}&1&1\\2&&&2\\&2&2\end{smallmatrix}$};
	\draw[blue,->] (RN_2_1_1__Rationals) -- (RN_2_1_2__Rationals);
	\draw[gray,dashed,->] (RN_2_1_1__Rationals) -- (RN_2_2_1__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[red,->] (RN_2_1_2__Rationals) -- (RN_2_2_1__Rationals);
	\draw[gray,dashed,->] (RN_2_1_2__Rationals) -- (RN_2_2_2__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[blue,->] (RN_2_1_2__Rationals) -- (RN_2_1_3__Rationals);
	\draw[blue,->] (RN_2_2_1__Rationals) -- (RN_2_2_2__Rationals);
	\draw[gray,dashed,->] (RN_2_2_1__Rationals) -- (RN_2_3_1__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[red,->] (RN_2_2_2__Rationals) -- (RN_2_3_1__Rationals);
	\draw[gray,dashed,->] (RN_2_2_2__Rationals) -- (RN_2_3_2__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[blue,->] (RN_2_2_2__Rationals) -- (RN_2_2_3__Rationals);
	\draw[gray,dashed,->] (RN_2_3_1__Rationals) -- (RN_2_1_1__Rationals_repeated) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[blue,->] (RN_2_3_1__Rationals) -- (RN_2_3_2__Rationals);
	\draw[red,->] (RN_2_3_2__Rationals) -- (RN_2_1_1__Rationals_repeated);
	\draw[gray,dashed,->,out=200,in=-20] (RN_2_3_2__Rationals) to node[midway,sloped,above]{{\tiny $\tau^{-1}$}} (RN_2_1_2__Rationals);
	\draw[blue,->] (RN_2_3_2__Rationals) -- (RN_2_3_3__Rationals_repeated);
	\draw[red,->] (RN_2_3_3__Rationals) -- (RN_2_1_2__Rationals);
	\draw[blue,->] (RN_2_3_3__Rationals) -- (RN_2_3_4__Rationals);
	\draw[gray,dashed,->] (RN_2_3_3__Rationals) -- (RN_2_1_3__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[red,->] (RN_2_3_4__Rationals) -- (RN_2_1_3__Rationals);
	\draw[gray,dashed,->] (RN_2_3_4__Rationals) -- (RN_2_1_4__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[red,->] (RN_2_1_3__Rationals) -- (RN_2_2_2__Rationals);
	\draw[blue,->] (RN_2_1_3__Rationals) -- (RN_2_1_4__Rationals);
	\draw[gray,dashed,->] (RN_2_1_3__Rationals) -- (RN_2_2_3__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[red,->] (RN_2_1_4__Rationals) -- (RN_2_2_3__Rationals);
	\draw[gray,dashed,->] (RN_2_1_4__Rationals) -- (RN_2_2_4__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[red,->] (RN_2_2_3__Rationals) -- (RN_2_3_2__Rationals);
	\draw[gray,dashed,->] (RN_2_2_3__Rationals) -- (RN_2_3_3__Rationals_repeated) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[blue,->] (RN_2_2_3__Rationals) -- (RN_2_2_4__Rationals);
	\draw[red,->] (RN_2_2_4__Rationals) -- (RN_2_3_3__Rationals_repeated);
	\draw[gray,dashed,->,out=200,in=-20] (RN_2_2_4__Rationals) to node[midway,sloped,above]{{\tiny $\tau^{-1}$}} (RN_2_3_4__Rationals);
	\draw[dashed,-] (RN_2_1_1__Rationals) -- (RN_2_3_3__Rationals);
	\draw[dashed,-] (RN_2_1_1__Rationals_repeated) -- (RN_2_3_3__Rationals_repeated);
	\node (LeftTopPhantom_2) at (4,6) {};
	\draw[dashed,-] (RN_2_3_3__Rationals) -- (LeftTopPhantom_2);
	\node (RightTopPhantom_2) at (16,6) {};
	\draw[dashed,-] (RN_2_3_3__Rationals_repeated) -- (RightTopPhantom_2);
\end{tikzpicture}
\end{scaletikzpicturetowidth}
\end{center}

\end{figure}
\begin{fleqn}
\begin{align*}
R_{\infty}^{1}(1):\ & - \\
R_{\infty}^{1}(2):\ & \bigl(R_{\infty}^{2}(1),R_{\infty}^{1}(1)\bigr),\ \bigl(I(1,5),P(0,1)\bigr),\ \bigl(I(1,6),P(0,4)\bigr),\ \bigl(I(0,3),P(0,5)\bigr)\\
R_{\infty}^{2}(1):\ & \bigl(I(0,4),P(0,1)\bigr),\ \bigl(I(0,5),P(0,4)\bigr),\ \bigl(I(0,6),P(0,5)\bigr)\\
R_{\infty}^{2}(2):\ & \bigl(R_{\infty}^{3}(1),R_{\infty}^{2}(1)\bigr),\ \bigl(I(0,4),P(0,2)\bigr),\ \bigl(I(0,5),P(1,1)\bigr),\ \bigl(I(0,6),P(1,4)\bigr)\\
R_{\infty}^{3}(1):\ & - \\
R_{\infty}^{3}(2):\ & \bigl(R_{\infty}^{1}(1),R_{\infty}^{3}(1)\bigr)\\
\end{align*}
\end{fleqn}
\clearpage

\subsection[Schofield pairs for the quiver $\Delta(\widetilde{\mathbb{A}}_{3,4})$]{Schofield pairs for the quiver $\Delta(\widetilde{\mathbb{A}}_{3,4})$ -- {\usebox{\boxAAAlAAAA}}}
\[
\xymatrix{  & 2\ar[dl] & 3\ar[l] &   &   &   &  \\
1 &   &   &   & 7\ar[ull]\ar[dl] &   &  \\
  & 4\ar[ul] & 5\ar[l] & 6\ar[l] &   &   &  }
\]
\medskip{}
\[
 C_{\Delta(\widetilde{\mathbb{A}}_{3,4})} = \begin{bmatrix}1 & 1 & 1 & 1 & 1 & 1 & 2\\
0 & 1 & 1 & 0 & 0 & 0 & 1\\
0 & 0 & 1 & 0 & 0 & 0 & 1\\
0 & 0 & 0 & 1 & 1 & 1 & 1\\
0 & 0 & 0 & 0 & 1 & 1 & 1\\
0 & 0 & 0 & 0 & 0 & 1 & 1\\
0 & 0 & 0 & 0 & 0 & 0 & 1\end{bmatrix}\qquad\Phi_{\Delta(\widetilde{\mathbb{A}}_{3,4})} = \begin{bmatrix}-1 & 1 & 0 & 1 & 0 & 0 & 0\\
-1 & 0 & 1 & 1 & 0 & 0 & 0\\
-1 & 0 & 0 & 1 & 0 & 0 & 1\\
-1 & 1 & 0 & 0 & 1 & 0 & 0\\
-1 & 1 & 0 & 0 & 0 & 1 & 0\\
-1 & 1 & 0 & 0 & 0 & 0 & 1\\
-2 & 1 & 0 & 1 & 0 & 0 & 1\end{bmatrix}
\]

\medskip{}\subsubsection{Schofield pairs associated to preprojective exceptional modules}

\begin{figure}[ht]

\begin{center}
\begin{scaletikzpicturetowidth}{\textwidth}
\begin{tikzpicture}[scale=\tikzscale]
	\node[noselAAAlAAAA, label={[shift={(0.4,-0.2)}, brown]\tiny $P(0,1)$}] (P_0_1__Rationals) at (0,0) {$\begin{smallmatrix}&0&0\\1&&&&0\\&0&0&0\end{smallmatrix}$};
	\node[noselAAAlAAAA, label={[shift={(0.4,-0.2)}, brown]\tiny $P(0,2)$}] (P_0_2__Rationals) at (0,-1) {$\begin{smallmatrix}&1&0\\1&&&&0\\&0&0&0\end{smallmatrix}$};
	\node[noselAAAlAAAA, label={[shift={(0.4,-0.2)}, brown]\tiny $P(0,3)$}] (P_0_3__Rationals) at (0,-2) {$\begin{smallmatrix}&1&1\\1&&&&0\\&0&0&0\end{smallmatrix}$};
	\node[noselAAAlAAAA, label={[shift={(0.4,-0.2)}, brown]\tiny $P(0,4)$}] (P_0_4__Rationals) at (0,-6) {$\begin{smallmatrix}&0&0\\1&&&&0\\&1&0&0\end{smallmatrix}$};
	\node[noselAAAlAAAA, label={[shift={(0.4,-0.2)}, brown]\tiny $P(0,5)$}] (P_0_5__Rationals) at (0,-5) {$\begin{smallmatrix}&0&0\\1&&&&0\\&1&1&0\end{smallmatrix}$};
	\node[noselAAAlAAAA, label={[shift={(0.4,-0.2)}, brown]\tiny $P(0,6)$}] (P_0_6__Rationals) at (0,-4) {$\begin{smallmatrix}&0&0\\1&&&&0\\&1&1&1\end{smallmatrix}$};
	\node[seloneAAAlAAAA, label={[shift={(0.4,-0.2)}, brown]\tiny $P(0,7)$}] (P_0_7__Rationals) at (0,-3) {$\begin{smallmatrix}&1&1\\2&&&&1\\&1&1&1\end{smallmatrix}$};
	\node[noselAAAlAAAA, label={[shift={(0.4,-0.2)}, brown]\tiny $P(1,1)$}] (P_1_1__Rationals) at (4,0) {$\begin{smallmatrix}&1&0\\1&&&&0\\&1&0&0\end{smallmatrix}$};
	\node[noselAAAlAAAA, label={[shift={(0.4,-0.2)}, brown]\tiny $P(1,2)$}] (P_1_2__Rationals) at (4,-1) {$\begin{smallmatrix}&1&1\\1&&&&0\\&1&0&0\end{smallmatrix}$};
	\node[seloneAAAlAAAA, label={[shift={(0.4,-0.2)}, brown]\tiny $P(1,3)$}] (P_1_3__Rationals) at (4,-2) {$\begin{smallmatrix}&1&1\\2&&&&1\\&2&1&1\end{smallmatrix}$};
	\node[noselAAAlAAAA, label={[shift={(0.4,-0.2)}, brown]\tiny $P(1,4)$}] (P_1_4__Rationals) at (4,-6) {$\begin{smallmatrix}&1&0\\1&&&&0\\&1&1&0\end{smallmatrix}$};
	\node[noselAAAlAAAA, label={[shift={(0.4,-0.2)}, brown]\tiny $P(1,5)$}] (P_1_5__Rationals) at (4,-5) {$\begin{smallmatrix}&1&0\\1&&&&0\\&1&1&1\end{smallmatrix}$};
	\node[seloneAAAlAAAA, label={[shift={(0.4,-0.2)}, brown]\tiny $P(1,6)$}] (P_1_6__Rationals) at (4,-4) {$\begin{smallmatrix}&2&1\\2&&&&1\\&1&1&1\end{smallmatrix}$};
	\node[noselAAAlAAAA, label={[shift={(0.4,-0.2)}, brown]\tiny $P(1,7)$}] (P_1_7__Rationals) at (4,-3) {$\begin{smallmatrix}&2&1\\2&&&&1\\&2&1&1\end{smallmatrix}$};
	\node[noselAAAlAAAA, label={[shift={(0.4,-0.2)}, brown]\tiny $P(2,1)$}] (P_2_1__Rationals) at (8,0) {$\begin{smallmatrix}&1&1\\1&&&&0\\&1&1&0\end{smallmatrix}$};
	\node[seloneAAAlAAAA, label={[shift={(0.4,-0.2)}, brown]\tiny $P(2,2)$}] (P_2_2__Rationals) at (8,-1) {$\begin{smallmatrix}&1&1\\2&&&&1\\&2&2&1\end{smallmatrix}$};
	\node[noselAAAlAAAA, label={[shift={(0.4,-0.2)}, brown]\tiny $P(2,3)$}] (P_2_3__Rationals) at (8,-2) {$\begin{smallmatrix}&2&1\\2&&&&1\\&2&2&1\end{smallmatrix}$};
	\node[noselAAAlAAAA, label={[shift={(0.4,-0.2)}, brown]\tiny $P(2,4)$}] (P_2_4__Rationals) at (8,-6) {$\begin{smallmatrix}&1&1\\1&&&&0\\&1&1&1\end{smallmatrix}$};
	\node[seloneAAAlAAAA, label={[shift={(0.4,-0.2)}, brown]\tiny $P(2,5)$}] (P_2_5__Rationals) at (8,-5) {$\begin{smallmatrix}&2&2\\2&&&&1\\&1&1&1\end{smallmatrix}$};
	\node[noselAAAlAAAA, label={[shift={(0.4,-0.2)}, brown]\tiny $P(2,6)$}] (P_2_6__Rationals) at (8,-4) {$\begin{smallmatrix}&2&2\\2&&&&1\\&2&1&1\end{smallmatrix}$};
	\node[noselAAAlAAAA, label={[shift={(0.4,-0.2)}, brown]\tiny $P(2,7)$}] (P_2_7__Rationals) at (8,-3) {$\begin{smallmatrix}&2&2\\2&&&&1\\&2&2&1\end{smallmatrix}$};
	\node[seloneAAAlAAAA, label={[shift={(0.4,-0.2)}, brown]\tiny $P(3,1)$}] (P_3_1__Rationals) at (12,0) {$\begin{smallmatrix}&1&1\\2&&&&1\\&2&2&2\end{smallmatrix}$};
	\node[noselAAAlAAAA, label={[shift={(0.4,-0.2)}, brown]\tiny $P(3,2)$}] (P_3_2__Rationals) at (12,-1) {$\begin{smallmatrix}&2&1\\2&&&&1\\&2&2&2\end{smallmatrix}$};
	\node[noselAAAlAAAA, label={[shift={(0.4,-0.2)}, brown]\tiny $P(3,3)$}] (P_3_3__Rationals) at (12,-2) {$\begin{smallmatrix}&2&2\\2&&&&1\\&2&2&2\end{smallmatrix}$};
	\node[seloneAAAlAAAA, label={[shift={(0.4,-0.2)}, brown]\tiny $P(3,4)$}] (P_3_4__Rationals) at (12,-6) {$\begin{smallmatrix}&2&2\\3&&&&2\\&2&2&2\end{smallmatrix}$};
	\node[noselAAAlAAAA, label={[shift={(0.4,-0.2)}, brown]\tiny $P(3,5)$}] (P_3_5__Rationals) at (12,-5) {$\begin{smallmatrix}&2&2\\3&&&&2\\&3&2&2\end{smallmatrix}$};
	\node[noselAAAlAAAA, label={[shift={(0.4,-0.2)}, brown]\tiny $P(3,6)$}] (P_3_6__Rationals) at (12,-4) {$\begin{smallmatrix}&2&2\\3&&&&2\\&3&3&2\end{smallmatrix}$};
	\node[noselAAAlAAAA, label={[shift={(0.4,-0.2)}, brown]\tiny $P(3,7)$}] (P_3_7__Rationals) at (12,-3) {$\begin{smallmatrix}&2&2\\3&&&&2\\&3&3&3\end{smallmatrix}$};
	\node at (13,-3.) {$\dots$};
	\draw[blue,->] (P_0_1__Rationals) -- (P_0_2__Rationals);
	\draw[blue,->] (P_0_2__Rationals) -- (P_0_3__Rationals);
	\draw[blue,->] (P_0_1__Rationals) to [out=135,in=-135] (P_0_4__Rationals);
	\draw[blue,->] (P_0_4__Rationals) -- (P_0_5__Rationals);
	\draw[blue,->] (P_0_5__Rationals) -- (P_0_6__Rationals);
	\draw[blue,->] (P_0_3__Rationals) -- (P_0_7__Rationals);
	\draw[blue,->] (P_0_6__Rationals) -- (P_0_7__Rationals);
	\draw[gray,dashed,->] (P_0_1__Rationals) -- (P_1_1__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[blue,->] (P_0_2__Rationals) -- (P_1_1__Rationals);
	\draw[blue,->] (P_0_4__Rationals) -- (P_1_1__Rationals);
	\draw[gray,dashed,->] (P_0_2__Rationals) -- (P_1_2__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[blue,->] (P_0_3__Rationals) -- (P_1_2__Rationals);
	\draw[blue,->] (P_1_1__Rationals) -- (P_1_2__Rationals);
	\draw[gray,dashed,->] (P_0_3__Rationals) -- (P_1_3__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[blue,->] (P_0_7__Rationals) -- (P_1_3__Rationals);
	\draw[blue,->] (P_1_2__Rationals) -- (P_1_3__Rationals);
	\draw[gray,dashed,->] (P_0_4__Rationals) -- (P_1_4__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[blue,->] (P_0_5__Rationals) -- (P_1_4__Rationals);
	\draw[blue,->] (P_1_1__Rationals) to [out=135,in=-135] (P_1_4__Rationals);
	\draw[gray,dashed,->] (P_0_5__Rationals) -- (P_1_5__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[blue,->] (P_0_6__Rationals) -- (P_1_5__Rationals);
	\draw[blue,->] (P_1_4__Rationals) -- (P_1_5__Rationals);
	\draw[gray,dashed,->] (P_0_6__Rationals) -- (P_1_6__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[blue,->] (P_0_7__Rationals) -- (P_1_6__Rationals);
	\draw[blue,->] (P_1_5__Rationals) -- (P_1_6__Rationals);
	\draw[gray,dashed,->] (P_0_7__Rationals) -- (P_1_7__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[blue,->] (P_1_3__Rationals) -- (P_1_7__Rationals);
	\draw[blue,->] (P_1_6__Rationals) -- (P_1_7__Rationals);
	\draw[gray,dashed,->] (P_1_1__Rationals) -- (P_2_1__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[blue,->] (P_1_2__Rationals) -- (P_2_1__Rationals);
	\draw[blue,->] (P_1_4__Rationals) -- (P_2_1__Rationals);
	\draw[gray,dashed,->] (P_1_2__Rationals) -- (P_2_2__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[blue,->] (P_1_3__Rationals) -- (P_2_2__Rationals);
	\draw[blue,->] (P_2_1__Rationals) -- (P_2_2__Rationals);
	\draw[gray,dashed,->] (P_1_3__Rationals) -- (P_2_3__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[blue,->] (P_1_7__Rationals) -- (P_2_3__Rationals);
	\draw[blue,->] (P_2_2__Rationals) -- (P_2_3__Rationals);
	\draw[gray,dashed,->] (P_1_4__Rationals) -- (P_2_4__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[blue,->] (P_1_5__Rationals) -- (P_2_4__Rationals);
	\draw[blue,->] (P_2_1__Rationals) to [out=135,in=-135] (P_2_4__Rationals);
	\draw[gray,dashed,->] (P_1_5__Rationals) -- (P_2_5__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[blue,->] (P_1_6__Rationals) -- (P_2_5__Rationals);
	\draw[blue,->] (P_2_4__Rationals) -- (P_2_5__Rationals);
	\draw[gray,dashed,->] (P_1_6__Rationals) -- (P_2_6__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[blue,->] (P_1_7__Rationals) -- (P_2_6__Rationals);
	\draw[blue,->] (P_2_5__Rationals) -- (P_2_6__Rationals);
	\draw[gray,dashed,->] (P_1_7__Rationals) -- (P_2_7__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[blue,->] (P_2_3__Rationals) -- (P_2_7__Rationals);
	\draw[blue,->] (P_2_6__Rationals) -- (P_2_7__Rationals);
	\draw[gray,dashed,->] (P_2_1__Rationals) -- (P_3_1__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[blue,->] (P_2_2__Rationals) -- (P_3_1__Rationals);
	\draw[blue,->] (P_2_4__Rationals) -- (P_3_1__Rationals);
	\draw[gray,dashed,->] (P_2_2__Rationals) -- (P_3_2__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[blue,->] (P_2_3__Rationals) -- (P_3_2__Rationals);
	\draw[blue,->] (P_3_1__Rationals) -- (P_3_2__Rationals);
	\draw[gray,dashed,->] (P_2_3__Rationals) -- (P_3_3__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[blue,->] (P_2_7__Rationals) -- (P_3_3__Rationals);
	\draw[blue,->] (P_3_2__Rationals) -- (P_3_3__Rationals);
	\draw[gray,dashed,->] (P_2_4__Rationals) -- (P_3_4__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[blue,->] (P_2_5__Rationals) -- (P_3_4__Rationals);
	\draw[blue,->] (P_3_1__Rationals) to [out=135,in=-135] (P_3_4__Rationals);
	\draw[gray,dashed,->] (P_2_5__Rationals) -- (P_3_5__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[blue,->] (P_2_6__Rationals) -- (P_3_5__Rationals);
	\draw[blue,->] (P_3_4__Rationals) -- (P_3_5__Rationals);
	\draw[gray,dashed,->] (P_2_6__Rationals) -- (P_3_6__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[blue,->] (P_2_7__Rationals) -- (P_3_6__Rationals);
	\draw[blue,->] (P_3_5__Rationals) -- (P_3_6__Rationals);
	\draw[gray,dashed,->] (P_2_7__Rationals) -- (P_3_7__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[blue,->] (P_3_3__Rationals) -- (P_3_7__Rationals);
	\draw[blue,->] (P_3_6__Rationals) -- (P_3_7__Rationals);
\end{tikzpicture}
\end{scaletikzpicturetowidth}
\end{center}

\end{figure}
\medskip{}
\subsubsection*{Modules of the form $P(n,1)$}

Defect: $\partial P(n,1) = -1$, for $n\ge 0$.

\begin{fleqn}
\begin{align*}
P(0,1):\ & - \\
P(1,1):\ & \bigl(R_{0}^{3}(1),P(0,2)\bigr),\ \bigl(R_{\infty}^{3}(1),P(0,4)\bigr)\\
P(2,1):\ & \bigl(R_{0}^{3}(2),P(0,3)\bigr),\ \bigl(R_{\infty}^{3}(2),P(0,5)\bigr),\ \bigl(R_{0}^{4}(1),P(1,2)\bigr),\ \bigl(R_{\infty}^{1}(1),P(1,4)\bigr)\\
P(3,1):\ & \bigl(R_{0}^{3}(3),P(0,7)\bigr),\ \bigl(R_{0}^{4}(2),P(1,3)\bigr),\ \bigl(R_{\infty}^{1}(2),P(1,5)\bigr),\ \bigl(R_{0}^{1}(1),P(2,2)\bigr),\ \bigl(R_{\infty}^{2}(1),P(2,4)\bigr)\\
 & \bigl(I(0,2),2P(0,6)\bigr)\\
P(n,1):\ & \bigl(R_{0}^{(n-1)\bmod 4+1}(3),P(n-3,7)\bigr),\ \bigl(R_{0}^{n\bmod 4+1}(2),P(n-2,3)\bigr),\ \bigl(R_{\infty}^{(n-3)\bmod 3+1}(2),P(n-2,5)\bigr)\\
 & \bigl(R_{0}^{(n-3)\bmod 4+1}(1),P(n-1,2)\bigr),\ \bigl(R_{\infty}^{(n-2)\bmod 3+1}(1),P(n-1,4)\bigr),\ \bigl(uI,(u+1)P\bigr),\ n>3\\
\end{align*}
\end{fleqn}
\subsubsection*{Modules of the form $P(n,2)$}

Defect: $\partial P(n,2) = -1$, for $n\ge 0$.

\begin{fleqn}
\begin{align*}
P(0,2):\ & \bigl(R_{\infty}^{3}(1),P(0,1)\bigr)\\
P(1,2):\ & \bigl(R_{0}^{3}(1),P(0,3)\bigr),\ \bigl(R_{\infty}^{3}(2),P(0,4)\bigr),\ \bigl(R_{\infty}^{1}(1),P(1,1)\bigr)\\
P(2,2):\ & \bigl(R_{0}^{2}(3),P(0,6)\bigr),\ \bigl(R_{0}^{3}(2),P(0,7)\bigr),\ \bigl(R_{0}^{4}(1),P(1,3)\bigr),\ \bigl(R_{\infty}^{1}(2),P(1,4)\bigr),\ \bigl(R_{\infty}^{2}(1),P(2,1)\bigr)\\
 & \bigl(I(1,3),2P(0,5)\bigr)\\
P(n,2):\ & \bigl(R_{0}^{(n-1)\bmod 4+1}(3),P(n-2,6)\bigr),\ \bigl(R_{0}^{n\bmod 4+1}(2),P(n-2,7)\bigr),\ \bigl(R_{0}^{(n+1)\bmod 4+1}(1),P(n-1,3)\bigr)\\
 & \bigl(R_{\infty}^{(n-2)\bmod 3+1}(2),P(n-1,4)\bigr),\ \bigl(R_{\infty}^{(n-1)\bmod 3+1}(1),P(n,1)\bigr),\ \bigl(uI,(u+1)P\bigr),\ n>2\\
\end{align*}
\end{fleqn}
\subsubsection*{Modules of the form $P(n,3)$}

Defect: $\partial P(n,3) = -1$, for $n\ge 0$.

\begin{fleqn}
\begin{align*}
P(0,3):\ & \bigl(R_{\infty}^{3}(2),P(0,1)\bigr),\ \bigl(R_{\infty}^{1}(1),P(0,2)\bigr)\\
P(1,3):\ & \bigl(R_{0}^{1}(3),P(0,5)\bigr),\ \bigl(R_{0}^{2}(2),P(0,6)\bigr),\ \bigl(R_{0}^{3}(1),P(0,7)\bigr),\ \bigl(R_{\infty}^{1}(2),P(1,1)\bigr),\ \bigl(R_{\infty}^{2}(1),P(1,2)\bigr)\\
 & \bigl(I(2,7),2P(0,4)\bigr)\\
P(n,3):\ & \bigl(R_{0}^{(n-1)\bmod 4+1}(3),P(n-1,5)\bigr),\ \bigl(R_{0}^{n\bmod 4+1}(2),P(n-1,6)\bigr),\ \bigl(R_{0}^{(n+1)\bmod 4+1}(1),P(n-1,7)\bigr)\\
 & \bigl(R_{\infty}^{(n-1)\bmod 3+1}(2),P(n,1)\bigr),\ \bigl(R_{\infty}^{n\bmod 3+1}(1),P(n,2)\bigr),\ \bigl(uI,(u+1)P\bigr),\ n>1\\
\end{align*}
\end{fleqn}
\subsubsection*{Modules of the form $P(n,4)$}

Defect: $\partial P(n,4) = -1$, for $n\ge 0$.

\begin{fleqn}
\begin{align*}
P(0,4):\ & \bigl(R_{0}^{3}(1),P(0,1)\bigr)\\
P(1,4):\ & \bigl(R_{0}^{3}(2),P(0,2)\bigr),\ \bigl(R_{\infty}^{3}(1),P(0,5)\bigr),\ \bigl(R_{0}^{4}(1),P(1,1)\bigr)\\
P(2,4):\ & \bigl(R_{0}^{3}(3),P(0,3)\bigr),\ \bigl(R_{\infty}^{3}(2),P(0,6)\bigr),\ \bigl(R_{0}^{4}(2),P(1,2)\bigr),\ \bigl(R_{\infty}^{1}(1),P(1,5)\bigr),\ \bigl(R_{0}^{1}(1),P(2,1)\bigr)\\
P(3,4):\ & \bigl(R_{0}^{4}(3),P(1,3)\bigr),\ \bigl(R_{\infty}^{1}(2),P(1,6)\bigr),\ \bigl(R_{0}^{1}(2),P(2,2)\bigr),\ \bigl(R_{\infty}^{2}(1),P(2,5)\bigr),\ \bigl(R_{0}^{2}(1),P(3,1)\bigr)\\
 & \bigl(2I(2,6),3P(0,1)\bigr)\\
P(n,4):\ & \bigl(R_{0}^{n\bmod 4+1}(3),P(n-2,3)\bigr),\ \bigl(R_{\infty}^{(n-3)\bmod 3+1}(2),P(n-2,6)\bigr),\ \bigl(R_{0}^{(n-3)\bmod 4+1}(2),P(n-1,2)\bigr)\\
 & \bigl(R_{\infty}^{(n-2)\bmod 3+1}(1),P(n-1,5)\bigr),\ \bigl(R_{0}^{(n-2)\bmod 4+1}(1),P(n,1)\bigr),\ \bigl(uI,(u+1)P\bigr),\ n>3\\
\end{align*}
\end{fleqn}
\subsubsection*{Modules of the form $P(n,5)$}

Defect: $\partial P(n,5) = -1$, for $n\ge 0$.

\begin{fleqn}
\begin{align*}
P(0,5):\ & \bigl(R_{0}^{3}(2),P(0,1)\bigr),\ \bigl(R_{0}^{4}(1),P(0,4)\bigr)\\
P(1,5):\ & \bigl(R_{0}^{3}(3),P(0,2)\bigr),\ \bigl(R_{\infty}^{3}(1),P(0,6)\bigr),\ \bigl(R_{0}^{4}(2),P(1,1)\bigr),\ \bigl(R_{0}^{1}(1),P(1,4)\bigr)\\
P(2,5):\ & \bigl(R_{\infty}^{3}(2),P(0,7)\bigr),\ \bigl(R_{0}^{4}(3),P(1,2)\bigr),\ \bigl(R_{\infty}^{1}(1),P(1,6)\bigr),\ \bigl(R_{0}^{1}(2),P(2,1)\bigr),\ \bigl(R_{0}^{2}(1),P(2,4)\bigr)\\
 & \bigl(I(0,4),2P(0,3)\bigr)\\
P(n,5):\ & \bigl(R_{\infty}^{n\bmod 3+1}(2),P(n-2,7)\bigr),\ \bigl(R_{0}^{(n+1)\bmod 4+1}(3),P(n-1,2)\bigr),\ \bigl(R_{\infty}^{(n-2)\bmod 3+1}(1),P(n-1,6)\bigr)\\
 & \bigl(R_{0}^{(n-2)\bmod 4+1}(2),P(n,1)\bigr),\ \bigl(R_{0}^{(n-1)\bmod 4+1}(1),P(n,4)\bigr),\ \bigl(uI,(u+1)P\bigr),\ n>2\\
\end{align*}
\end{fleqn}
\subsubsection*{Modules of the form $P(n,6)$}

Defect: $\partial P(n,6) = -1$, for $n\ge 0$.

\begin{fleqn}
\begin{align*}
P(0,6):\ & \bigl(R_{0}^{3}(3),P(0,1)\bigr),\ \bigl(R_{0}^{4}(2),P(0,4)\bigr),\ \bigl(R_{0}^{1}(1),P(0,5)\bigr)\\
P(1,6):\ & \bigl(R_{\infty}^{2}(2),P(0,3)\bigr),\ \bigl(R_{\infty}^{3}(1),P(0,7)\bigr),\ \bigl(R_{0}^{4}(3),P(1,1)\bigr),\ \bigl(R_{0}^{1}(2),P(1,4)\bigr),\ \bigl(R_{0}^{2}(1),P(1,5)\bigr)\\
 & \bigl(I(1,5),2P(0,2)\bigr)\\
P(n,6):\ & \bigl(R_{\infty}^{n\bmod 3+1}(2),P(n-1,3)\bigr),\ \bigl(R_{\infty}^{(n+1)\bmod 3+1}(1),P(n-1,7)\bigr),\ \bigl(R_{0}^{(n+2)\bmod 4+1}(3),P(n,1)\bigr)\\
 & \bigl(R_{0}^{(n-1)\bmod 4+1}(2),P(n,4)\bigr),\ \bigl(R_{0}^{n\bmod 4+1}(1),P(n,5)\bigr),\ \bigl(uI,(u+1)P\bigr),\ n>1\\
\end{align*}
\end{fleqn}
\subsubsection*{Modules of the form $P(n,7)$}

Defect: $\partial P(n,7) = -1$, for $n\ge 0$.

\begin{fleqn}
\begin{align*}
P(0,7):\ & \bigl(R_{\infty}^{1}(2),P(0,2)\bigr),\ \bigl(R_{\infty}^{2}(1),P(0,3)\bigr),\ \bigl(R_{0}^{4}(3),P(0,4)\bigr),\ \bigl(R_{0}^{1}(2),P(0,5)\bigr),\ \bigl(R_{0}^{2}(1),P(0,6)\bigr)\\
 & \bigl(I(2,6),2P(0,1)\bigr)\\
P(n,7):\ & \bigl(R_{\infty}^{n\bmod 3+1}(2),P(n,2)\bigr),\ \bigl(R_{\infty}^{(n+1)\bmod 3+1}(1),P(n,3)\bigr),\ \bigl(R_{0}^{(n+3)\bmod 4+1}(3),P(n,4)\bigr)\\
 & \bigl(R_{0}^{n\bmod 4+1}(2),P(n,5)\bigr),\ \bigl(R_{0}^{(n+1)\bmod 4+1}(1),P(n,6)\bigr),\ \bigl(uI,(u+1)P\bigr),\ n>0\\
\end{align*}
\end{fleqn}
\subsubsection{Schofield pairs associated to preinjective exceptional modules}

\begin{figure}[ht]

\begin{center}
\begin{scaletikzpicturetowidth}{\textwidth}
\begin{tikzpicture}[scale=\tikzscale]
	\node[seloneAAAlAAAA, label={[shift={(0.4,-0.2)}, brown]\tiny $I(0,1)$}] (I_0_1__Rationals) at (0,0) {$\begin{smallmatrix}&1&1\\1&&&&2\\&1&1&1\end{smallmatrix}$};
	\node[noselAAAlAAAA, label={[shift={(0.4,-0.2)}, brown]\tiny $I(0,2)$}] (I_0_2__Rationals) at (0,-1) {$\begin{smallmatrix}&1&1\\0&&&&1\\&0&0&0\end{smallmatrix}$};
	\node[noselAAAlAAAA, label={[shift={(0.4,-0.2)}, brown]\tiny $I(0,3)$}] (I_0_3__Rationals) at (0,-2) {$\begin{smallmatrix}&0&1\\0&&&&1\\&0&0&0\end{smallmatrix}$};
	\node[noselAAAlAAAA, label={[shift={(0.4,-0.2)}, brown]\tiny $I(0,4)$}] (I_0_4__Rationals) at (0,-6) {$\begin{smallmatrix}&0&0\\0&&&&1\\&1&1&1\end{smallmatrix}$};
	\node[noselAAAlAAAA, label={[shift={(0.4,-0.2)}, brown]\tiny $I(0,5)$}] (I_0_5__Rationals) at (0,-5) {$\begin{smallmatrix}&0&0\\0&&&&1\\&0&1&1\end{smallmatrix}$};
	\node[noselAAAlAAAA, label={[shift={(0.4,-0.2)}, brown]\tiny $I(0,6)$}] (I_0_6__Rationals) at (0,-4) {$\begin{smallmatrix}&0&0\\0&&&&1\\&0&0&1\end{smallmatrix}$};
	\node[noselAAAlAAAA, label={[shift={(0.4,-0.2)}, brown]\tiny $I(0,7)$}] (I_0_7__Rationals) at (0,-3) {$\begin{smallmatrix}&0&0\\0&&&&1\\&0&0&0\end{smallmatrix}$};
	\node[noselAAAlAAAA, label={[shift={(0.4,-0.2)}, brown]\tiny $I(1,1)$}] (I_1_1__Rationals) at (-4,0) {$\begin{smallmatrix}&1&2\\1&&&&2\\&1&1&2\end{smallmatrix}$};
	\node[seloneAAAlAAAA, label={[shift={(0.4,-0.2)}, brown]\tiny $I(1,2)$}] (I_1_2__Rationals) at (-4,-1) {$\begin{smallmatrix}&1&1\\1&&&&2\\&1&1&2\end{smallmatrix}$};
	\node[noselAAAlAAAA, label={[shift={(0.4,-0.2)}, brown]\tiny $I(1,3)$}] (I_1_3__Rationals) at (-4,-2) {$\begin{smallmatrix}&1&1\\0&&&&1\\&0&0&1\end{smallmatrix}$};
	\node[seloneAAAlAAAA, label={[shift={(0.4,-0.2)}, brown]\tiny $I(1,4)$}] (I_1_4__Rationals) at (-4,-6) {$\begin{smallmatrix}&1&2\\1&&&&2\\&1&1&1\end{smallmatrix}$};
	\node[noselAAAlAAAA, label={[shift={(0.4,-0.2)}, brown]\tiny $I(1,5)$}] (I_1_5__Rationals) at (-4,-5) {$\begin{smallmatrix}&0&1\\0&&&&1\\&1&1&1\end{smallmatrix}$};
	\node[noselAAAlAAAA, label={[shift={(0.4,-0.2)}, brown]\tiny $I(1,6)$}] (I_1_6__Rationals) at (-4,-4) {$\begin{smallmatrix}&0&1\\0&&&&1\\&0&1&1\end{smallmatrix}$};
	\node[noselAAAlAAAA, label={[shift={(0.4,-0.2)}, brown]\tiny $I(1,7)$}] (I_1_7__Rationals) at (-4,-3) {$\begin{smallmatrix}&0&1\\0&&&&1\\&0&0&1\end{smallmatrix}$};
	\node[noselAAAlAAAA, label={[shift={(0.4,-0.2)}, brown]\tiny $I(2,1)$}] (I_2_1__Rationals) at (-8,0) {$\begin{smallmatrix}&2&2\\1&&&&2\\&1&2&2\end{smallmatrix}$};
	\node[noselAAAlAAAA, label={[shift={(0.4,-0.2)}, brown]\tiny $I(2,2)$}] (I_2_2__Rationals) at (-8,-1) {$\begin{smallmatrix}&1&2\\1&&&&2\\&1&2&2\end{smallmatrix}$};
	\node[seloneAAAlAAAA, label={[shift={(0.4,-0.2)}, brown]\tiny $I(2,3)$}] (I_2_3__Rationals) at (-8,-2) {$\begin{smallmatrix}&1&1\\1&&&&2\\&1&2&2\end{smallmatrix}$};
	\node[noselAAAlAAAA, label={[shift={(0.4,-0.2)}, brown]\tiny $I(2,4)$}] (I_2_4__Rationals) at (-8,-6) {$\begin{smallmatrix}&2&2\\1&&&&2\\&1&1&2\end{smallmatrix}$};
	\node[seloneAAAlAAAA, label={[shift={(0.4,-0.2)}, brown]\tiny $I(2,5)$}] (I_2_5__Rationals) at (-8,-5) {$\begin{smallmatrix}&2&2\\1&&&&2\\&1&1&1\end{smallmatrix}$};
	\node[noselAAAlAAAA, label={[shift={(0.4,-0.2)}, brown]\tiny $I(2,6)$}] (I_2_6__Rationals) at (-8,-4) {$\begin{smallmatrix}&1&1\\0&&&&1\\&1&1&1\end{smallmatrix}$};
	\node[noselAAAlAAAA, label={[shift={(0.4,-0.2)}, brown]\tiny $I(2,7)$}] (I_2_7__Rationals) at (-8,-3) {$\begin{smallmatrix}&1&1\\0&&&&1\\&0&1&1\end{smallmatrix}$};
	\node[noselAAAlAAAA, label={[shift={(0.4,-0.2)}, brown]\tiny $I(3,1)$}] (I_3_1__Rationals) at (-12,0) {$\begin{smallmatrix}&2&2\\2&&&&3\\&3&3&3\end{smallmatrix}$};
	\node[noselAAAlAAAA, label={[shift={(0.4,-0.2)}, brown]\tiny $I(3,2)$}] (I_3_2__Rationals) at (-12,-1) {$\begin{smallmatrix}&2&2\\1&&&&2\\&2&2&2\end{smallmatrix}$};
	\node[noselAAAlAAAA, label={[shift={(0.4,-0.2)}, brown]\tiny $I(3,3)$}] (I_3_3__Rationals) at (-12,-2) {$\begin{smallmatrix}&1&2\\1&&&&2\\&2&2&2\end{smallmatrix}$};
	\node[noselAAAlAAAA, label={[shift={(0.4,-0.2)}, brown]\tiny $I(3,4)$}] (I_3_4__Rationals) at (-12,-6) {$\begin{smallmatrix}&2&2\\2&&&&3\\&2&3&3\end{smallmatrix}$};
	\node[noselAAAlAAAA, label={[shift={(0.4,-0.2)}, brown]\tiny $I(3,5)$}] (I_3_5__Rationals) at (-12,-5) {$\begin{smallmatrix}&2&2\\2&&&&3\\&2&2&3\end{smallmatrix}$};
	\node[seloneAAAlAAAA, label={[shift={(0.4,-0.2)}, brown]\tiny $I(3,6)$}] (I_3_6__Rationals) at (-12,-4) {$\begin{smallmatrix}&2&2\\2&&&&3\\&2&2&2\end{smallmatrix}$};
	\node[seloneAAAlAAAA, label={[shift={(0.4,-0.2)}, brown]\tiny $I(3,7)$}] (I_3_7__Rationals) at (-12,-3) {$\begin{smallmatrix}&1&1\\1&&&&2\\&2&2&2\end{smallmatrix}$};
	\node at (-13,-3.) {$\dots$};
	\draw[red,->] (I_1_2__Rationals) -- (I_0_1__Rationals);
	\draw[red,->] (I_1_4__Rationals) -- (I_0_1__Rationals);
	\draw[red,->] (I_0_1__Rationals) -- (I_0_2__Rationals);
	\draw[red,->] (I_1_3__Rationals) -- (I_0_2__Rationals);
	\draw[red,->] (I_0_2__Rationals) -- (I_0_3__Rationals);
	\draw[red,->] (I_1_7__Rationals) -- (I_0_3__Rationals);
	\draw[red,->] (I_0_1__Rationals) to [out=135,in=-135] (I_0_4__Rationals);
	\draw[red,->] (I_1_5__Rationals) -- (I_0_4__Rationals);
	\draw[red,->] (I_0_4__Rationals) -- (I_0_5__Rationals);
	\draw[red,->] (I_1_6__Rationals) -- (I_0_5__Rationals);
	\draw[red,->] (I_0_5__Rationals) -- (I_0_6__Rationals);
	\draw[red,->] (I_1_7__Rationals) -- (I_0_6__Rationals);
	\draw[red,->] (I_0_3__Rationals) -- (I_0_7__Rationals);
	\draw[red,->] (I_0_6__Rationals) -- (I_0_7__Rationals);
	\draw[gray,dashed,->] (I_0_1__Rationals) -- (I_1_1__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_2_2__Rationals) -- (I_1_1__Rationals);
	\draw[red,->] (I_2_4__Rationals) -- (I_1_1__Rationals);
	\draw[gray,dashed,->] (I_0_2__Rationals) -- (I_1_2__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_1_1__Rationals) -- (I_1_2__Rationals);
	\draw[red,->] (I_2_3__Rationals) -- (I_1_2__Rationals);
	\draw[gray,dashed,->] (I_0_3__Rationals) -- (I_1_3__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_1_2__Rationals) -- (I_1_3__Rationals);
	\draw[red,->] (I_2_7__Rationals) -- (I_1_3__Rationals);
	\draw[gray,dashed,->] (I_0_4__Rationals) -- (I_1_4__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_1_1__Rationals) to [out=135,in=-135] (I_1_4__Rationals);
	\draw[red,->] (I_2_5__Rationals) -- (I_1_4__Rationals);
	\draw[gray,dashed,->] (I_0_5__Rationals) -- (I_1_5__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_1_4__Rationals) -- (I_1_5__Rationals);
	\draw[red,->] (I_2_6__Rationals) -- (I_1_5__Rationals);
	\draw[gray,dashed,->] (I_0_6__Rationals) -- (I_1_6__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_1_5__Rationals) -- (I_1_6__Rationals);
	\draw[red,->] (I_2_7__Rationals) -- (I_1_6__Rationals);
	\draw[gray,dashed,->] (I_0_7__Rationals) -- (I_1_7__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_1_3__Rationals) -- (I_1_7__Rationals);
	\draw[red,->] (I_1_6__Rationals) -- (I_1_7__Rationals);
	\draw[gray,dashed,->] (I_1_1__Rationals) -- (I_2_1__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_3_2__Rationals) -- (I_2_1__Rationals);
	\draw[red,->] (I_3_4__Rationals) -- (I_2_1__Rationals);
	\draw[gray,dashed,->] (I_1_2__Rationals) -- (I_2_2__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_2_1__Rationals) -- (I_2_2__Rationals);
	\draw[red,->] (I_3_3__Rationals) -- (I_2_2__Rationals);
	\draw[gray,dashed,->] (I_1_3__Rationals) -- (I_2_3__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_2_2__Rationals) -- (I_2_3__Rationals);
	\draw[red,->] (I_3_7__Rationals) -- (I_2_3__Rationals);
	\draw[gray,dashed,->] (I_1_4__Rationals) -- (I_2_4__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_2_1__Rationals) to [out=135,in=-135] (I_2_4__Rationals);
	\draw[red,->] (I_3_5__Rationals) -- (I_2_4__Rationals);
	\draw[gray,dashed,->] (I_1_5__Rationals) -- (I_2_5__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_2_4__Rationals) -- (I_2_5__Rationals);
	\draw[red,->] (I_3_6__Rationals) -- (I_2_5__Rationals);
	\draw[gray,dashed,->] (I_1_6__Rationals) -- (I_2_6__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_2_5__Rationals) -- (I_2_6__Rationals);
	\draw[red,->] (I_3_7__Rationals) -- (I_2_6__Rationals);
	\draw[gray,dashed,->] (I_1_7__Rationals) -- (I_2_7__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_2_3__Rationals) -- (I_2_7__Rationals);
	\draw[red,->] (I_2_6__Rationals) -- (I_2_7__Rationals);
	\draw[gray,dashed,->] (I_2_1__Rationals) -- (I_3_1__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[gray,dashed,->] (I_2_2__Rationals) -- (I_3_2__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_3_1__Rationals) -- (I_3_2__Rationals);
	\draw[gray,dashed,->] (I_2_3__Rationals) -- (I_3_3__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_3_2__Rationals) -- (I_3_3__Rationals);
	\draw[gray,dashed,->] (I_2_4__Rationals) -- (I_3_4__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_3_1__Rationals) to [out=135,in=-135] (I_3_4__Rationals);
	\draw[gray,dashed,->] (I_2_5__Rationals) -- (I_3_5__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_3_4__Rationals) -- (I_3_5__Rationals);
	\draw[gray,dashed,->] (I_2_6__Rationals) -- (I_3_6__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_3_5__Rationals) -- (I_3_6__Rationals);
	\draw[gray,dashed,->] (I_2_7__Rationals) -- (I_3_7__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_3_3__Rationals) -- (I_3_7__Rationals);
	\draw[red,->] (I_3_6__Rationals) -- (I_3_7__Rationals);
\end{tikzpicture}
\end{scaletikzpicturetowidth}
\end{center}

\end{figure}
\medskip{}
\subsubsection*{Modules of the form $I(n,1)$}

Defect: $\partial I(n,1) = 1$, for $n\ge 0$.

\begin{fleqn}
\begin{align*}
I(0,1):\ & \bigl(I(0,2),R_{\infty}^{2}(1)\bigr),\ \bigl(I(0,3),R_{\infty}^{2}(2)\bigr),\ \bigl(I(0,4),R_{0}^{2}(1)\bigr),\ \bigl(I(0,5),R_{0}^{2}(2)\bigr),\ \bigl(I(0,6),R_{0}^{2}(3)\bigr)\\
 & \bigl(2I(0,7),P(2,4)\bigr)\\
I(n,1):\ & \bigl(I(n,2),R_{\infty}^{(-n+1)\bmod 3+1}(1)\bigr),\ \bigl(I(n,3),R_{\infty}^{(-n+1)\bmod 3+1}(2)\bigr),\ \bigl(I(n,4),R_{0}^{(-n+1)\bmod 4+1}(1)\bigr)\\
 & \bigl(I(n,5),R_{0}^{(-n+1)\bmod 4+1}(2)\bigr),\ \bigl(I(n,6),R_{0}^{(-n+1)\bmod 4+1}(3)\bigr),\ \bigl((v+1)I,vP\bigr),\ n>0\\
\end{align*}
\end{fleqn}
\subsubsection*{Modules of the form $I(n,2)$}

Defect: $\partial I(n,2) = 1$, for $n\ge 0$.

\begin{fleqn}
\begin{align*}
I(0,2):\ & \bigl(I(0,3),R_{\infty}^{3}(1)\bigr),\ \bigl(I(0,7),R_{\infty}^{3}(2)\bigr)\\
I(1,2):\ & \bigl(I(0,1),R_{0}^{1}(1)\bigr),\ \bigl(I(0,4),R_{0}^{1}(2)\bigr),\ \bigl(I(0,5),R_{0}^{1}(3)\bigr),\ \bigl(I(1,3),R_{\infty}^{2}(1)\bigr),\ \bigl(I(1,7),R_{\infty}^{2}(2)\bigr)\\
 & \bigl(2I(0,6),P(2,1)\bigr)\\
I(n,2):\ & \bigl(I(n-1,1),R_{0}^{(-n+1)\bmod 4+1}(1)\bigr),\ \bigl(I(n-1,4),R_{0}^{(-n+1)\bmod 4+1}(2)\bigr),\ \bigl(I(n-1,5),R_{0}^{(-n+1)\bmod 4+1}(3)\bigr)\\
 & \bigl(I(n,3),R_{\infty}^{(-n+2)\bmod 3+1}(1)\bigr),\ \bigl(I(n,7),R_{\infty}^{(-n+2)\bmod 3+1}(2)\bigr),\ \bigl((v+1)I,vP\bigr),\ n>1\\
\end{align*}
\end{fleqn}
\subsubsection*{Modules of the form $I(n,3)$}

Defect: $\partial I(n,3) = 1$, for $n\ge 0$.

\begin{fleqn}
\begin{align*}
I(0,3):\ & \bigl(I(0,7),R_{\infty}^{1}(1)\bigr)\\
I(1,3):\ & \bigl(I(0,2),R_{0}^{1}(1)\bigr),\ \bigl(I(0,6),R_{\infty}^{3}(2)\bigr),\ \bigl(I(1,7),R_{\infty}^{3}(1)\bigr)\\
I(2,3):\ & \bigl(I(0,1),R_{0}^{4}(2)\bigr),\ \bigl(I(0,4),R_{0}^{4}(3)\bigr),\ \bigl(I(1,2),R_{0}^{4}(1)\bigr),\ \bigl(I(1,6),R_{\infty}^{2}(2)\bigr),\ \bigl(I(2,7),R_{\infty}^{2}(1)\bigr)\\
 & \bigl(2I(0,5),P(1,2)\bigr)\\
I(n,3):\ & \bigl(I(n-2,1),R_{0}^{(-n+5)\bmod 4+1}(2)\bigr),\ \bigl(I(n-2,4),R_{0}^{(-n+5)\bmod 4+1}(3)\bigr),\ \bigl(I(n-1,2),R_{0}^{(-n+5)\bmod 4+1}(1)\bigr)\\
 & \bigl(I(n-1,6),R_{\infty}^{(-n+3)\bmod 3+1}(2)\bigr),\ \bigl(I(n,7),R_{\infty}^{(-n+3)\bmod 3+1}(1)\bigr),\ \bigl((v+1)I,vP\bigr),\ n>2\\
\end{align*}
\end{fleqn}
\subsubsection*{Modules of the form $I(n,4)$}

Defect: $\partial I(n,4) = 1$, for $n\ge 0$.

\begin{fleqn}
\begin{align*}
I(0,4):\ & \bigl(I(0,5),R_{0}^{3}(1)\bigr),\ \bigl(I(0,6),R_{0}^{3}(2)\bigr),\ \bigl(I(0,7),R_{0}^{3}(3)\bigr)\\
I(1,4):\ & \bigl(I(0,1),R_{\infty}^{1}(1)\bigr),\ \bigl(I(0,2),R_{\infty}^{1}(2)\bigr),\ \bigl(I(1,5),R_{0}^{2}(1)\bigr),\ \bigl(I(1,6),R_{0}^{2}(2)\bigr),\ \bigl(I(1,7),R_{0}^{2}(3)\bigr)\\
 & \bigl(2I(0,3),P(1,5)\bigr)\\
I(n,4):\ & \bigl(I(n-1,1),R_{\infty}^{(-n+1)\bmod 3+1}(1)\bigr),\ \bigl(I(n-1,2),R_{\infty}^{(-n+1)\bmod 3+1}(2)\bigr),\ \bigl(I(n,5),R_{0}^{(-n+2)\bmod 4+1}(1)\bigr)\\
 & \bigl(I(n,6),R_{0}^{(-n+2)\bmod 4+1}(2)\bigr),\ \bigl(I(n,7),R_{0}^{(-n+2)\bmod 4+1}(3)\bigr),\ \bigl((v+1)I,vP\bigr),\ n>1\\
\end{align*}
\end{fleqn}
\subsubsection*{Modules of the form $I(n,5)$}

Defect: $\partial I(n,5) = 1$, for $n\ge 0$.

\begin{fleqn}
\begin{align*}
I(0,5):\ & \bigl(I(0,6),R_{0}^{4}(1)\bigr),\ \bigl(I(0,7),R_{0}^{4}(2)\bigr)\\
I(1,5):\ & \bigl(I(0,3),R_{0}^{3}(3)\bigr),\ \bigl(I(0,4),R_{\infty}^{1}(1)\bigr),\ \bigl(I(1,6),R_{0}^{3}(1)\bigr),\ \bigl(I(1,7),R_{0}^{3}(2)\bigr)\\
I(2,5):\ & \bigl(I(0,1),R_{\infty}^{3}(2)\bigr),\ \bigl(I(1,3),R_{0}^{2}(3)\bigr),\ \bigl(I(1,4),R_{\infty}^{3}(1)\bigr),\ \bigl(I(2,6),R_{0}^{2}(1)\bigr),\ \bigl(I(2,7),R_{0}^{2}(2)\bigr)\\
 & \bigl(2I(0,2),P(0,6)\bigr)\\
I(n,5):\ & \bigl(I(n-2,1),R_{\infty}^{(-n+4)\bmod 3+1}(2)\bigr),\ \bigl(I(n-1,3),R_{0}^{(-n+3)\bmod 4+1}(3)\bigr),\ \bigl(I(n-1,4),R_{\infty}^{(-n+4)\bmod 3+1}(1)\bigr)\\
 & \bigl(I(n,6),R_{0}^{(-n+3)\bmod 4+1}(1)\bigr),\ \bigl(I(n,7),R_{0}^{(-n+3)\bmod 4+1}(2)\bigr),\ \bigl((v+1)I,vP\bigr),\ n>2\\
\end{align*}
\end{fleqn}
\subsubsection*{Modules of the form $I(n,6)$}

Defect: $\partial I(n,6) = 1$, for $n\ge 0$.

\begin{fleqn}
\begin{align*}
I(0,6):\ & \bigl(I(0,7),R_{0}^{1}(1)\bigr)\\
I(1,6):\ & \bigl(I(0,3),R_{0}^{4}(2)\bigr),\ \bigl(I(0,5),R_{\infty}^{1}(1)\bigr),\ \bigl(I(1,7),R_{0}^{4}(1)\bigr)\\
I(2,6):\ & \bigl(I(0,2),R_{0}^{3}(3)\bigr),\ \bigl(I(0,4),R_{\infty}^{3}(2)\bigr),\ \bigl(I(1,3),R_{0}^{3}(2)\bigr),\ \bigl(I(1,5),R_{\infty}^{3}(1)\bigr),\ \bigl(I(2,7),R_{0}^{3}(1)\bigr)\\
I(3,6):\ & \bigl(I(1,2),R_{0}^{2}(3)\bigr),\ \bigl(I(1,4),R_{\infty}^{2}(2)\bigr),\ \bigl(I(2,3),R_{0}^{2}(2)\bigr),\ \bigl(I(2,5),R_{\infty}^{2}(1)\bigr),\ \bigl(I(3,7),R_{0}^{2}(1)\bigr)\\
 & \bigl(3I(0,7),2P(2,4)\bigr)\\
I(n,6):\ & \bigl(I(n-2,2),R_{0}^{(-n+4)\bmod 4+1}(3)\bigr),\ \bigl(I(n-2,4),R_{\infty}^{(-n+4)\bmod 3+1}(2)\bigr),\ \bigl(I(n-1,3),R_{0}^{(-n+4)\bmod 4+1}(2)\bigr)\\
 & \bigl(I(n-1,5),R_{\infty}^{(-n+4)\bmod 3+1}(1)\bigr),\ \bigl(I(n,7),R_{0}^{(-n+4)\bmod 4+1}(1)\bigr),\ \bigl((v+1)I,vP\bigr),\ n>3\\
\end{align*}
\end{fleqn}
\subsubsection*{Modules of the form $I(n,7)$}

Defect: $\partial I(n,7) = 1$, for $n\ge 0$.

\begin{fleqn}
\begin{align*}
I(0,7):\ & - \\
I(1,7):\ & \bigl(I(0,3),R_{0}^{1}(1)\bigr),\ \bigl(I(0,6),R_{\infty}^{1}(1)\bigr)\\
I(2,7):\ & \bigl(I(0,2),R_{0}^{4}(2)\bigr),\ \bigl(I(0,5),R_{\infty}^{3}(2)\bigr),\ \bigl(I(1,3),R_{0}^{4}(1)\bigr),\ \bigl(I(1,6),R_{\infty}^{3}(1)\bigr)\\
I(3,7):\ & \bigl(I(0,1),R_{0}^{3}(3)\bigr),\ \bigl(I(1,2),R_{0}^{3}(2)\bigr),\ \bigl(I(1,5),R_{\infty}^{2}(2)\bigr),\ \bigl(I(2,3),R_{0}^{3}(1)\bigr),\ \bigl(I(2,6),R_{\infty}^{2}(1)\bigr)\\
 & \bigl(2I(0,4),P(0,3)\bigr)\\
I(n,7):\ & \bigl(I(n-3,1),R_{0}^{(-n+5)\bmod 4+1}(3)\bigr),\ \bigl(I(n-2,2),R_{0}^{(-n+5)\bmod 4+1}(2)\bigr),\ \bigl(I(n-2,5),R_{\infty}^{(-n+4)\bmod 3+1}(2)\bigr)\\
 & \bigl(I(n-1,3),R_{0}^{(-n+5)\bmod 4+1}(1)\bigr),\ \bigl(I(n-1,6),R_{\infty}^{(-n+4)\bmod 3+1}(1)\bigr),\ \bigl((v+1)I,vP\bigr),\ n>3\\
\end{align*}
\end{fleqn}
\subsubsection{Schofield pairs associated to regular exceptional modules}

\subsubsection*{The non-homogeneous tube $\mathcal{T}_{0}^{\Delta(\widetilde{\mathbb{A}}_{3,4})}$}

\begin{figure}[ht]

\begin{center}
\captionof{figure}{\vspace*{-10pt}$\mathcal{T}_{0}^{\Delta(\widetilde{\mathbb{A}}_{3,4})}$}
\begin{scaletikzpicturetowidth}{1.\textwidth}
\begin{tikzpicture}[scale=\tikzscale]
	\node[seltwoAAAlAAAA, label={[shift={(0.0,-0.45)}, brown]\tiny $R_{0}^{1}(1)$}] (RN_1_1_1__Rationals) at (4,0) {$\begin{smallmatrix}&0&0\\0&&&&0\\&0&0&1\end{smallmatrix}$};
	\node[seltwoAAAlAAAA, label={[shift={(0.0,-0.45)}, brown]\tiny $R_{0}^{2}(1)$}] (RN_1_2_1__Rationals) at (8,0) {$\begin{smallmatrix}&1&1\\1&&&&1\\&0&0&0\end{smallmatrix}$};
	\node[seltwoAAAlAAAA, label={[shift={(0.0,-0.45)}, brown]\tiny $R_{0}^{3}(1)$}] (RN_1_3_1__Rationals) at (12,0) {$\begin{smallmatrix}&0&0\\0&&&&0\\&1&0&0\end{smallmatrix}$};
	\node[seltwoAAAlAAAA, label={[shift={(0.0,-0.45)}, brown]\tiny $R_{0}^{4}(1)$}] (RN_1_4_1__Rationals) at (16,0) {$\begin{smallmatrix}&0&0\\0&&&&0\\&0&1&0\end{smallmatrix}$};
	\node[seltwoAAAlAAAA, label={[shift={(0.0,-0.45)}, brown]\tiny $R_{0}^{1}(1)$}] (RN_1_1_1__Rationals_repeated) at (20,0) {$\begin{smallmatrix}&0&0\\0&&&&0\\&0&0&1\end{smallmatrix}$};
	\node[seltwoAAAlAAAA, label={[shift={(0.0,-0.45)}, brown]\tiny $R_{0}^{1}(2)$}] (RN_1_1_2__Rationals) at (6,2) {$\begin{smallmatrix}&1&1\\1&&&&1\\&0&0&1\end{smallmatrix}$};
	\node[seltwoAAAlAAAA, label={[shift={(0.0,-0.45)}, brown]\tiny $R_{0}^{2}(2)$}] (RN_1_2_2__Rationals) at (10,2) {$\begin{smallmatrix}&1&1\\1&&&&1\\&1&0&0\end{smallmatrix}$};
	\node[seltwoAAAlAAAA, label={[shift={(0.0,-0.45)}, brown]\tiny $R_{0}^{3}(2)$}] (RN_1_3_2__Rationals) at (14,2) {$\begin{smallmatrix}&0&0\\0&&&&0\\&1&1&0\end{smallmatrix}$};
	\node[seltwoAAAlAAAA, label={[shift={(0.0,-0.45)}, brown]\tiny $R_{0}^{4}(2)$}] (RN_1_4_2__Rationals) at (18,2) {$\begin{smallmatrix}&0&0\\0&&&&0\\&0&1&1\end{smallmatrix}$};
	\node[seltwoAAAlAAAA, label={[shift={(0.0,-0.45)}, brown]\tiny $R_{0}^{4}(3)$}] (RN_1_4_3__Rationals) at (4,4) {$\begin{smallmatrix}&1&1\\1&&&&1\\&0&1&1\end{smallmatrix}$};
	\node[seltwoAAAlAAAA, label={[shift={(0.0,-0.45)}, brown]\tiny $R_{0}^{1}(3)$}] (RN_1_1_3__Rationals) at (8,4) {$\begin{smallmatrix}&1&1\\1&&&&1\\&1&0&1\end{smallmatrix}$};
	\node[seltwoAAAlAAAA, label={[shift={(0.0,-0.45)}, brown]\tiny $R_{0}^{2}(3)$}] (RN_1_2_3__Rationals) at (12,4) {$\begin{smallmatrix}&1&1\\1&&&&1\\&1&1&0\end{smallmatrix}$};
	\node[seltwoAAAlAAAA, label={[shift={(0.0,-0.45)}, brown]\tiny $R_{0}^{3}(3)$}] (RN_1_3_3__Rationals) at (16,4) {$\begin{smallmatrix}&0&0\\0&&&&0\\&1&1&1\end{smallmatrix}$};
	\node[seltwoAAAlAAAA, label={[shift={(0.0,-0.45)}, brown]\tiny $R_{0}^{4}(3)$}] (RN_1_4_3__Rationals_repeated) at (20,4) {$\begin{smallmatrix}&1&1\\1&&&&1\\&0&1&1\end{smallmatrix}$};
	\node[noselAAAlAAAA, label={[shift={(0.0,-0.45)}, brown]\tiny $R_{0}^{4}(4)$}] (RN_1_4_4__Rationals) at (6,6) {$\begin{smallmatrix}&1&1\\1&&&&1\\&1&1&1\end{smallmatrix}$};
	\node[noselAAAlAAAA, label={[shift={(0.0,-0.45)}, brown]\tiny $R_{0}^{1}(4)$}] (RN_1_1_4__Rationals) at (10,6) {$\begin{smallmatrix}&1&1\\1&&&&1\\&1&1&1\end{smallmatrix}$};
	\node[noselAAAlAAAA, label={[shift={(0.0,-0.45)}, brown]\tiny $R_{0}^{2}(4)$}] (RN_1_2_4__Rationals) at (14,6) {$\begin{smallmatrix}&1&1\\1&&&&1\\&1&1&1\end{smallmatrix}$};
	\node[noselAAAlAAAA, label={[shift={(0.0,-0.45)}, brown]\tiny $R_{0}^{3}(4)$}] (RN_1_3_4__Rationals) at (18,6) {$\begin{smallmatrix}&1&1\\1&&&&1\\&1&1&1\end{smallmatrix}$};
	\node[noselAAAlAAAA, label={[shift={(0.0,-0.45)}, brown]\tiny $R_{0}^{3}(5)$}] (RN_1_3_5__Rationals) at (4,8) {$\begin{smallmatrix}&1&1\\1&&&&1\\&2&1&1\end{smallmatrix}$};
	\node[noselAAAlAAAA, label={[shift={(0.0,-0.45)}, brown]\tiny $R_{0}^{4}(5)$}] (RN_1_4_5__Rationals) at (8,8) {$\begin{smallmatrix}&1&1\\1&&&&1\\&1&2&1\end{smallmatrix}$};
	\node[noselAAAlAAAA, label={[shift={(0.0,-0.45)}, brown]\tiny $R_{0}^{1}(5)$}] (RN_1_1_5__Rationals) at (12,8) {$\begin{smallmatrix}&1&1\\1&&&&1\\&1&1&2\end{smallmatrix}$};
	\node[noselAAAlAAAA, label={[shift={(0.0,-0.45)}, brown]\tiny $R_{0}^{2}(5)$}] (RN_1_2_5__Rationals) at (16,8) {$\begin{smallmatrix}&2&2\\2&&&&2\\&1&1&1\end{smallmatrix}$};
	\node[noselAAAlAAAA, label={[shift={(0.0,-0.45)}, brown]\tiny $R_{0}^{3}(5)$}] (RN_1_3_5__Rationals_repeated) at (20,8) {$\begin{smallmatrix}&1&1\\1&&&&1\\&2&1&1\end{smallmatrix}$};
	\draw[blue,->] (RN_1_1_1__Rationals) -- (RN_1_1_2__Rationals);
	\draw[gray,dashed,->] (RN_1_1_1__Rationals) -- (RN_1_2_1__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[red,->] (RN_1_1_2__Rationals) -- (RN_1_2_1__Rationals);
	\draw[gray,dashed,->] (RN_1_1_2__Rationals) -- (RN_1_2_2__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[blue,->] (RN_1_1_2__Rationals) -- (RN_1_1_3__Rationals);
	\draw[blue,->] (RN_1_2_1__Rationals) -- (RN_1_2_2__Rationals);
	\draw[gray,dashed,->] (RN_1_2_1__Rationals) -- (RN_1_3_1__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[red,->] (RN_1_2_2__Rationals) -- (RN_1_3_1__Rationals);
	\draw[gray,dashed,->] (RN_1_2_2__Rationals) -- (RN_1_3_2__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[blue,->] (RN_1_2_2__Rationals) -- (RN_1_2_3__Rationals);
	\draw[blue,->] (RN_1_3_1__Rationals) -- (RN_1_3_2__Rationals);
	\draw[gray,dashed,->] (RN_1_3_1__Rationals) -- (RN_1_4_1__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[red,->] (RN_1_3_2__Rationals) -- (RN_1_4_1__Rationals);
	\draw[gray,dashed,->] (RN_1_3_2__Rationals) -- (RN_1_4_2__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[blue,->] (RN_1_3_2__Rationals) -- (RN_1_3_3__Rationals);
	\draw[gray,dashed,->] (RN_1_4_1__Rationals) -- (RN_1_1_1__Rationals_repeated) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[blue,->] (RN_1_4_1__Rationals) -- (RN_1_4_2__Rationals);
	\draw[red,->] (RN_1_4_2__Rationals) -- (RN_1_1_1__Rationals_repeated);
	\draw[gray,dashed,->,out=193,in=-13] (RN_1_4_2__Rationals) to node[midway,sloped,above]{{\tiny $\tau^{-1}$}} (RN_1_1_2__Rationals);
	\draw[blue,->] (RN_1_4_2__Rationals) -- (RN_1_4_3__Rationals_repeated);
	\draw[red,->] (RN_1_4_3__Rationals) -- (RN_1_1_2__Rationals);
	\draw[blue,->] (RN_1_4_3__Rationals) -- (RN_1_4_4__Rationals);
	\draw[gray,dashed,->] (RN_1_4_3__Rationals) -- (RN_1_1_3__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[red,->] (RN_1_4_4__Rationals) -- (RN_1_1_3__Rationals);
	\draw[gray,dashed,->] (RN_1_4_4__Rationals) -- (RN_1_1_4__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[blue,->] (RN_1_4_4__Rationals) -- (RN_1_4_5__Rationals);
	\draw[red,->] (RN_1_1_3__Rationals) -- (RN_1_2_2__Rationals);
	\draw[blue,->] (RN_1_1_3__Rationals) -- (RN_1_1_4__Rationals);
	\draw[gray,dashed,->] (RN_1_1_3__Rationals) -- (RN_1_2_3__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[red,->] (RN_1_1_4__Rationals) -- (RN_1_2_3__Rationals);
	\draw[gray,dashed,->] (RN_1_1_4__Rationals) -- (RN_1_2_4__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[blue,->] (RN_1_1_4__Rationals) -- (RN_1_1_5__Rationals);
	\draw[red,->] (RN_1_2_3__Rationals) -- (RN_1_3_2__Rationals);
	\draw[blue,->] (RN_1_2_3__Rationals) -- (RN_1_2_4__Rationals);
	\draw[gray,dashed,->] (RN_1_2_3__Rationals) -- (RN_1_3_3__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[red,->] (RN_1_2_4__Rationals) -- (RN_1_3_3__Rationals);
	\draw[gray,dashed,->] (RN_1_2_4__Rationals) -- (RN_1_3_4__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[blue,->] (RN_1_2_4__Rationals) -- (RN_1_2_5__Rationals);
	\draw[red,->] (RN_1_3_3__Rationals) -- (RN_1_4_2__Rationals);
	\draw[gray,dashed,->] (RN_1_3_3__Rationals) -- (RN_1_4_3__Rationals_repeated) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[blue,->] (RN_1_3_3__Rationals) -- (RN_1_3_4__Rationals);
	\draw[red,->] (RN_1_3_4__Rationals) -- (RN_1_4_3__Rationals_repeated);
	\draw[gray,dashed,->,out=193,in=-13] (RN_1_3_4__Rationals) to node[midway,sloped,above]{{\tiny $\tau^{-1}$}} (RN_1_4_4__Rationals);
	\draw[blue,->] (RN_1_3_4__Rationals) -- (RN_1_3_5__Rationals_repeated);
	\draw[red,->] (RN_1_3_5__Rationals) -- (RN_1_4_4__Rationals);
	\draw[gray,dashed,->] (RN_1_3_5__Rationals) -- (RN_1_4_5__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[red,->] (RN_1_4_5__Rationals) -- (RN_1_1_4__Rationals);
	\draw[gray,dashed,->] (RN_1_4_5__Rationals) -- (RN_1_1_5__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[red,->] (RN_1_1_5__Rationals) -- (RN_1_2_4__Rationals);
	\draw[gray,dashed,->] (RN_1_1_5__Rationals) -- (RN_1_2_5__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[red,->] (RN_1_2_5__Rationals) -- (RN_1_3_4__Rationals);
	\draw[gray,dashed,->] (RN_1_2_5__Rationals) -- (RN_1_3_5__Rationals_repeated) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[dashed,-] (RN_1_1_1__Rationals) -- (RN_1_4_3__Rationals);
	\draw[dashed,-] (RN_1_1_1__Rationals_repeated) -- (RN_1_4_3__Rationals_repeated);
	\draw[dashed,-] (RN_1_4_3__Rationals) -- (RN_1_3_5__Rationals);
	\draw[dashed,-] (RN_1_4_3__Rationals_repeated) -- (RN_1_3_5__Rationals_repeated);
\end{tikzpicture}
\end{scaletikzpicturetowidth}
\end{center}

\end{figure}
\begin{fleqn}
\begin{align*}
R_{0}^{1}(1):\ & - \\
R_{0}^{1}(2):\ & \bigl(R_{0}^{2}(1),R_{0}^{1}(1)\bigr),\ \bigl(I(1,3),P(0,1)\bigr),\ \bigl(I(1,7),P(0,2)\bigr),\ \bigl(I(0,6),P(0,3)\bigr)\\
R_{0}^{2}(1):\ & \bigl(I(0,2),P(0,1)\bigr),\ \bigl(I(0,3),P(0,2)\bigr),\ \bigl(I(0,7),P(0,3)\bigr)\\
R_{0}^{2}(2):\ & \bigl(R_{0}^{3}(1),R_{0}^{2}(1)\bigr),\ \bigl(I(0,2),P(0,4)\bigr),\ \bigl(I(0,3),P(1,1)\bigr),\ \bigl(I(0,7),P(1,2)\bigr)\\
R_{0}^{3}(1):\ & - \\
R_{0}^{3}(2):\ & \bigl(R_{0}^{4}(1),R_{0}^{3}(1)\bigr)\\
R_{0}^{4}(1):\ & - \\
R_{0}^{4}(2):\ & \bigl(R_{0}^{1}(1),R_{0}^{4}(1)\bigr)\\
R_{0}^{4}(3):\ & \bigl(R_{0}^{1}(2),R_{0}^{4}(1)\bigr),\ \bigl(R_{0}^{2}(1),R_{0}^{4}(2)\bigr),\ \bigl(I(2,7),P(0,1)\bigr),\ \bigl(I(1,6),P(0,2)\bigr),\ \bigl(I(0,5),P(0,3)\bigr)\\
R_{0}^{1}(3):\ & \bigl(R_{0}^{2}(2),R_{0}^{1}(1)\bigr),\ \bigl(R_{0}^{3}(1),R_{0}^{1}(2)\bigr),\ \bigl(I(1,3),P(0,4)\bigr),\ \bigl(I(1,7),P(1,1)\bigr),\ \bigl(I(0,6),P(1,2)\bigr)\\
R_{0}^{2}(3):\ & \bigl(R_{0}^{3}(2),R_{0}^{2}(1)\bigr),\ \bigl(R_{0}^{4}(1),R_{0}^{2}(2)\bigr),\ \bigl(I(0,2),P(0,5)\bigr),\ \bigl(I(0,3),P(1,4)\bigr),\ \bigl(I(0,7),P(2,1)\bigr)\\
R_{0}^{3}(3):\ & \bigl(R_{0}^{4}(2),R_{0}^{3}(1)\bigr),\ \bigl(R_{0}^{1}(1),R_{0}^{3}(2)\bigr)\\
\end{align*}
\end{fleqn}
\subsubsection*{The non-homogeneous tube $\mathcal{T}_{\infty}^{\Delta(\widetilde{\mathbb{A}}_{3,4})}$}

\begin{figure}[ht]

\begin{center}
\captionof{figure}{\vspace*{-10pt}$\mathcal{T}_{\infty}^{\Delta(\widetilde{\mathbb{A}}_{3,4})}$}
\begin{scaletikzpicturetowidth}{0.80000000000000004\textwidth}
\begin{tikzpicture}[scale=\tikzscale]
	\node[seltwoAAAlAAAA, label={[shift={(0.0,-0.45)}, brown]\tiny $R_{\infty}^{1}(1)$}] (RN_2_1_1__Rationals) at (4,0) {$\begin{smallmatrix}&0&1\\0&&&&0\\&0&0&0\end{smallmatrix}$};
	\node[seltwoAAAlAAAA, label={[shift={(0.0,-0.45)}, brown]\tiny $R_{\infty}^{2}(1)$}] (RN_2_2_1__Rationals) at (8,0) {$\begin{smallmatrix}&0&0\\1&&&&1\\&1&1&1\end{smallmatrix}$};
	\node[seltwoAAAlAAAA, label={[shift={(0.0,-0.45)}, brown]\tiny $R_{\infty}^{3}(1)$}] (RN_2_3_1__Rationals) at (12,0) {$\begin{smallmatrix}&1&0\\0&&&&0\\&0&0&0\end{smallmatrix}$};
	\node[seltwoAAAlAAAA, label={[shift={(0.0,-0.45)}, brown]\tiny $R_{\infty}^{1}(1)$}] (RN_2_1_1__Rationals_repeated) at (16,0) {$\begin{smallmatrix}&0&1\\0&&&&0\\&0&0&0\end{smallmatrix}$};
	\node[seltwoAAAlAAAA, label={[shift={(0.0,-0.45)}, brown]\tiny $R_{\infty}^{1}(2)$}] (RN_2_1_2__Rationals) at (6,2) {$\begin{smallmatrix}&0&1\\1&&&&1\\&1&1&1\end{smallmatrix}$};
	\node[seltwoAAAlAAAA, label={[shift={(0.0,-0.45)}, brown]\tiny $R_{\infty}^{2}(2)$}] (RN_2_2_2__Rationals) at (10,2) {$\begin{smallmatrix}&1&0\\1&&&&1\\&1&1&1\end{smallmatrix}$};
	\node[seltwoAAAlAAAA, label={[shift={(0.0,-0.45)}, brown]\tiny $R_{\infty}^{3}(2)$}] (RN_2_3_2__Rationals) at (14,2) {$\begin{smallmatrix}&1&1\\0&&&&0\\&0&0&0\end{smallmatrix}$};
	\node[noselAAAlAAAA, label={[shift={(0.0,-0.45)}, brown]\tiny $R_{\infty}^{3}(3)$}] (RN_2_3_3__Rationals) at (4,4) {$\begin{smallmatrix}&1&1\\1&&&&1\\&1&1&1\end{smallmatrix}$};
	\node[noselAAAlAAAA, label={[shift={(0.0,-0.45)}, brown]\tiny $R_{\infty}^{1}(3)$}] (RN_2_1_3__Rationals) at (8,4) {$\begin{smallmatrix}&1&1\\1&&&&1\\&1&1&1\end{smallmatrix}$};
	\node[noselAAAlAAAA, label={[shift={(0.0,-0.45)}, brown]\tiny $R_{\infty}^{2}(3)$}] (RN_2_2_3__Rationals) at (12,4) {$\begin{smallmatrix}&1&1\\1&&&&1\\&1&1&1\end{smallmatrix}$};
	\node[noselAAAlAAAA, label={[shift={(0.0,-0.45)}, brown]\tiny $R_{\infty}^{3}(3)$}] (RN_2_3_3__Rationals_repeated) at (16,4) {$\begin{smallmatrix}&1&1\\1&&&&1\\&1&1&1\end{smallmatrix}$};
	\node[noselAAAlAAAA, label={[shift={(0.0,-0.45)}, brown]\tiny $R_{\infty}^{3}(4)$}] (RN_2_3_4__Rationals) at (6,6) {$\begin{smallmatrix}&2&1\\1&&&&1\\&1&1&1\end{smallmatrix}$};
	\node[noselAAAlAAAA, label={[shift={(0.0,-0.45)}, brown]\tiny $R_{\infty}^{1}(4)$}] (RN_2_1_4__Rationals) at (10,6) {$\begin{smallmatrix}&1&2\\1&&&&1\\&1&1&1\end{smallmatrix}$};
	\node[noselAAAlAAAA, label={[shift={(0.0,-0.45)}, brown]\tiny $R_{\infty}^{2}(4)$}] (RN_2_2_4__Rationals) at (14,6) {$\begin{smallmatrix}&1&1\\2&&&&2\\&2&2&2\end{smallmatrix}$};
	\draw[blue,->] (RN_2_1_1__Rationals) -- (RN_2_1_2__Rationals);
	\draw[gray,dashed,->] (RN_2_1_1__Rationals) -- (RN_2_2_1__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[red,->] (RN_2_1_2__Rationals) -- (RN_2_2_1__Rationals);
	\draw[gray,dashed,->] (RN_2_1_2__Rationals) -- (RN_2_2_2__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[blue,->] (RN_2_1_2__Rationals) -- (RN_2_1_3__Rationals);
	\draw[blue,->] (RN_2_2_1__Rationals) -- (RN_2_2_2__Rationals);
	\draw[gray,dashed,->] (RN_2_2_1__Rationals) -- (RN_2_3_1__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[red,->] (RN_2_2_2__Rationals) -- (RN_2_3_1__Rationals);
	\draw[gray,dashed,->] (RN_2_2_2__Rationals) -- (RN_2_3_2__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[blue,->] (RN_2_2_2__Rationals) -- (RN_2_2_3__Rationals);
	\draw[gray,dashed,->] (RN_2_3_1__Rationals) -- (RN_2_1_1__Rationals_repeated) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[blue,->] (RN_2_3_1__Rationals) -- (RN_2_3_2__Rationals);
	\draw[red,->] (RN_2_3_2__Rationals) -- (RN_2_1_1__Rationals_repeated);
	\draw[gray,dashed,->,out=200,in=-20] (RN_2_3_2__Rationals) to node[midway,sloped,above]{{\tiny $\tau^{-1}$}} (RN_2_1_2__Rationals);
	\draw[blue,->] (RN_2_3_2__Rationals) -- (RN_2_3_3__Rationals_repeated);
	\draw[red,->] (RN_2_3_3__Rationals) -- (RN_2_1_2__Rationals);
	\draw[blue,->] (RN_2_3_3__Rationals) -- (RN_2_3_4__Rationals);
	\draw[gray,dashed,->] (RN_2_3_3__Rationals) -- (RN_2_1_3__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[red,->] (RN_2_3_4__Rationals) -- (RN_2_1_3__Rationals);
	\draw[gray,dashed,->] (RN_2_3_4__Rationals) -- (RN_2_1_4__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[red,->] (RN_2_1_3__Rationals) -- (RN_2_2_2__Rationals);
	\draw[blue,->] (RN_2_1_3__Rationals) -- (RN_2_1_4__Rationals);
	\draw[gray,dashed,->] (RN_2_1_3__Rationals) -- (RN_2_2_3__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[red,->] (RN_2_1_4__Rationals) -- (RN_2_2_3__Rationals);
	\draw[gray,dashed,->] (RN_2_1_4__Rationals) -- (RN_2_2_4__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[red,->] (RN_2_2_3__Rationals) -- (RN_2_3_2__Rationals);
	\draw[gray,dashed,->] (RN_2_2_3__Rationals) -- (RN_2_3_3__Rationals_repeated) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[blue,->] (RN_2_2_3__Rationals) -- (RN_2_2_4__Rationals);
	\draw[red,->] (RN_2_2_4__Rationals) -- (RN_2_3_3__Rationals_repeated);
	\draw[gray,dashed,->,out=200,in=-20] (RN_2_2_4__Rationals) to node[midway,sloped,above]{{\tiny $\tau^{-1}$}} (RN_2_3_4__Rationals);
	\draw[dashed,-] (RN_2_1_1__Rationals) -- (RN_2_3_3__Rationals);
	\draw[dashed,-] (RN_2_1_1__Rationals_repeated) -- (RN_2_3_3__Rationals_repeated);
	\node (LeftTopPhantom_2) at (4,6) {};
	\draw[dashed,-] (RN_2_3_3__Rationals) -- (LeftTopPhantom_2);
	\node (RightTopPhantom_2) at (16,6) {};
	\draw[dashed,-] (RN_2_3_3__Rationals_repeated) -- (RightTopPhantom_2);
\end{tikzpicture}
\end{scaletikzpicturetowidth}
\end{center}

\end{figure}
\begin{fleqn}
\begin{align*}
R_{\infty}^{1}(1):\ & - \\
R_{\infty}^{1}(2):\ & \bigl(R_{\infty}^{2}(1),R_{\infty}^{1}(1)\bigr),\ \bigl(I(1,5),P(0,1)\bigr),\ \bigl(I(1,6),P(0,4)\bigr),\ \bigl(I(1,7),P(0,5)\bigr),\ \bigl(I(0,3),P(0,6)\bigr)\\
R_{\infty}^{2}(1):\ & \bigl(I(0,4),P(0,1)\bigr),\ \bigl(I(0,5),P(0,4)\bigr),\ \bigl(I(0,6),P(0,5)\bigr),\ \bigl(I(0,7),P(0,6)\bigr)\\
R_{\infty}^{2}(2):\ & \bigl(R_{\infty}^{3}(1),R_{\infty}^{2}(1)\bigr),\ \bigl(I(0,4),P(0,2)\bigr),\ \bigl(I(0,5),P(1,1)\bigr),\ \bigl(I(0,6),P(1,4)\bigr),\ \bigl(I(0,7),P(1,5)\bigr)\\
R_{\infty}^{3}(1):\ & - \\
R_{\infty}^{3}(2):\ & \bigl(R_{\infty}^{1}(1),R_{\infty}^{3}(1)\bigr)\\
\end{align*}
\end{fleqn}
\clearpage

\subsection[Schofield pairs for the quiver $\Delta(\widetilde{\mathbb{A}}_{3,5})$]{Schofield pairs for the quiver $\Delta(\widetilde{\mathbb{A}}_{3,5})$ -- {\usebox{\boxAAAlAAAAA}}}
\[
\xymatrix{  & 2\ar[dl] & 3\ar[l] &   &   &   &   &  \\
1 &   &   &   &   & 8\ar[ulll]\ar[dl] &   &  \\
  & 4\ar[ul] & 5\ar[l] & 6\ar[l] & 7\ar[l] &   &   &  }
\]
\medskip{}
\[
 C_{\Delta(\widetilde{\mathbb{A}}_{3,5})} = \begin{bmatrix}1 & 1 & 1 & 1 & 1 & 1 & 1 & 2\\
0 & 1 & 1 & 0 & 0 & 0 & 0 & 1\\
0 & 0 & 1 & 0 & 0 & 0 & 0 & 1\\
0 & 0 & 0 & 1 & 1 & 1 & 1 & 1\\
0 & 0 & 0 & 0 & 1 & 1 & 1 & 1\\
0 & 0 & 0 & 0 & 0 & 1 & 1 & 1\\
0 & 0 & 0 & 0 & 0 & 0 & 1 & 1\\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 1\end{bmatrix}\qquad\Phi_{\Delta(\widetilde{\mathbb{A}}_{3,5})} = \begin{bmatrix}-1 & 1 & 0 & 1 & 0 & 0 & 0 & 0\\
-1 & 0 & 1 & 1 & 0 & 0 & 0 & 0\\
-1 & 0 & 0 & 1 & 0 & 0 & 0 & 1\\
-1 & 1 & 0 & 0 & 1 & 0 & 0 & 0\\
-1 & 1 & 0 & 0 & 0 & 1 & 0 & 0\\
-1 & 1 & 0 & 0 & 0 & 0 & 1 & 0\\
-1 & 1 & 0 & 0 & 0 & 0 & 0 & 1\\
-2 & 1 & 0 & 1 & 0 & 0 & 0 & 1\end{bmatrix}
\]

\medskip{}\subsubsection{Schofield pairs associated to preprojective exceptional modules}

\begin{figure}[ht]

\begin{center}
\begin{scaletikzpicturetowidth}{\textwidth}
\begin{tikzpicture}[scale=\tikzscale]
	\node[noselAAAlAAAAA, label={[shift={(0.4,-0.2)}, brown]\tiny $P(0,1)$}] (P_0_1__Rationals) at (0,0) {$\begin{smallmatrix}&0&0\\1&&&&&0\\&0&0&0&0\end{smallmatrix}$};
	\node[noselAAAlAAAAA, label={[shift={(0.4,-0.2)}, brown]\tiny $P(0,2)$}] (P_0_2__Rationals) at (0,-1) {$\begin{smallmatrix}&1&0\\1&&&&&0\\&0&0&0&0\end{smallmatrix}$};
	\node[noselAAAlAAAAA, label={[shift={(0.4,-0.2)}, brown]\tiny $P(0,3)$}] (P_0_3__Rationals) at (0,-2) {$\begin{smallmatrix}&1&1\\1&&&&&0\\&0&0&0&0\end{smallmatrix}$};
	\node[noselAAAlAAAAA, label={[shift={(0.4,-0.2)}, brown]\tiny $P(0,4)$}] (P_0_4__Rationals) at (0,-7) {$\begin{smallmatrix}&0&0\\1&&&&&0\\&1&0&0&0\end{smallmatrix}$};
	\node[noselAAAlAAAAA, label={[shift={(0.4,-0.2)}, brown]\tiny $P(0,5)$}] (P_0_5__Rationals) at (0,-6) {$\begin{smallmatrix}&0&0\\1&&&&&0\\&1&1&0&0\end{smallmatrix}$};
	\node[noselAAAlAAAAA, label={[shift={(0.4,-0.2)}, brown]\tiny $P(0,6)$}] (P_0_6__Rationals) at (0,-5) {$\begin{smallmatrix}&0&0\\1&&&&&0\\&1&1&1&0\end{smallmatrix}$};
	\node[noselAAAlAAAAA, label={[shift={(0.4,-0.2)}, brown]\tiny $P(0,7)$}] (P_0_7__Rationals) at (0,-4) {$\begin{smallmatrix}&0&0\\1&&&&&0\\&1&1&1&1\end{smallmatrix}$};
	\node[seloneAAAlAAAAA, label={[shift={(0.4,-0.2)}, brown]\tiny $P(0,8)$}] (P_0_8__Rationals) at (0,-3) {$\begin{smallmatrix}&1&1\\2&&&&&1\\&1&1&1&1\end{smallmatrix}$};
	\node[noselAAAlAAAAA, label={[shift={(0.4,-0.2)}, brown]\tiny $P(1,1)$}] (P_1_1__Rationals) at (4,0) {$\begin{smallmatrix}&1&0\\1&&&&&0\\&1&0&0&0\end{smallmatrix}$};
	\node[noselAAAlAAAAA, label={[shift={(0.4,-0.2)}, brown]\tiny $P(1,2)$}] (P_1_2__Rationals) at (4,-1) {$\begin{smallmatrix}&1&1\\1&&&&&0\\&1&0&0&0\end{smallmatrix}$};
	\node[seloneAAAlAAAAA, label={[shift={(0.4,-0.2)}, brown]\tiny $P(1,3)$}] (P_1_3__Rationals) at (4,-2) {$\begin{smallmatrix}&1&1\\2&&&&&1\\&2&1&1&1\end{smallmatrix}$};
	\node[noselAAAlAAAAA, label={[shift={(0.4,-0.2)}, brown]\tiny $P(1,4)$}] (P_1_4__Rationals) at (4,-7) {$\begin{smallmatrix}&1&0\\1&&&&&0\\&1&1&0&0\end{smallmatrix}$};
	\node[noselAAAlAAAAA, label={[shift={(0.4,-0.2)}, brown]\tiny $P(1,5)$}] (P_1_5__Rationals) at (4,-6) {$\begin{smallmatrix}&1&0\\1&&&&&0\\&1&1&1&0\end{smallmatrix}$};
	\node[noselAAAlAAAAA, label={[shift={(0.4,-0.2)}, brown]\tiny $P(1,6)$}] (P_1_6__Rationals) at (4,-5) {$\begin{smallmatrix}&1&0\\1&&&&&0\\&1&1&1&1\end{smallmatrix}$};
	\node[seloneAAAlAAAAA, label={[shift={(0.4,-0.2)}, brown]\tiny $P(1,7)$}] (P_1_7__Rationals) at (4,-4) {$\begin{smallmatrix}&2&1\\2&&&&&1\\&1&1&1&1\end{smallmatrix}$};
	\node[noselAAAlAAAAA, label={[shift={(0.4,-0.2)}, brown]\tiny $P(1,8)$}] (P_1_8__Rationals) at (4,-3) {$\begin{smallmatrix}&2&1\\2&&&&&1\\&2&1&1&1\end{smallmatrix}$};
	\node[noselAAAlAAAAA, label={[shift={(0.4,-0.2)}, brown]\tiny $P(2,1)$}] (P_2_1__Rationals) at (8,0) {$\begin{smallmatrix}&1&1\\1&&&&&0\\&1&1&0&0\end{smallmatrix}$};
	\node[seloneAAAlAAAAA, label={[shift={(0.4,-0.2)}, brown]\tiny $P(2,2)$}] (P_2_2__Rationals) at (8,-1) {$\begin{smallmatrix}&1&1\\2&&&&&1\\&2&2&1&1\end{smallmatrix}$};
	\node[noselAAAlAAAAA, label={[shift={(0.4,-0.2)}, brown]\tiny $P(2,3)$}] (P_2_3__Rationals) at (8,-2) {$\begin{smallmatrix}&2&1\\2&&&&&1\\&2&2&1&1\end{smallmatrix}$};
	\node[noselAAAlAAAAA, label={[shift={(0.4,-0.2)}, brown]\tiny $P(2,4)$}] (P_2_4__Rationals) at (8,-7) {$\begin{smallmatrix}&1&1\\1&&&&&0\\&1&1&1&0\end{smallmatrix}$};
	\node[noselAAAlAAAAA, label={[shift={(0.4,-0.2)}, brown]\tiny $P(2,5)$}] (P_2_5__Rationals) at (8,-6) {$\begin{smallmatrix}&1&1\\1&&&&&0\\&1&1&1&1\end{smallmatrix}$};
	\node[seloneAAAlAAAAA, label={[shift={(0.4,-0.2)}, brown]\tiny $P(2,6)$}] (P_2_6__Rationals) at (8,-5) {$\begin{smallmatrix}&2&2\\2&&&&&1\\&1&1&1&1\end{smallmatrix}$};
	\node[noselAAAlAAAAA, label={[shift={(0.4,-0.2)}, brown]\tiny $P(2,7)$}] (P_2_7__Rationals) at (8,-4) {$\begin{smallmatrix}&2&2\\2&&&&&1\\&2&1&1&1\end{smallmatrix}$};
	\node[noselAAAlAAAAA, label={[shift={(0.4,-0.2)}, brown]\tiny $P(2,8)$}] (P_2_8__Rationals) at (8,-3) {$\begin{smallmatrix}&2&2\\2&&&&&1\\&2&2&1&1\end{smallmatrix}$};
	\node[seloneAAAlAAAAA, label={[shift={(0.4,-0.2)}, brown]\tiny $P(3,1)$}] (P_3_1__Rationals) at (12,0) {$\begin{smallmatrix}&1&1\\2&&&&&1\\&2&2&2&1\end{smallmatrix}$};
	\node[noselAAAlAAAAA, label={[shift={(0.4,-0.2)}, brown]\tiny $P(3,2)$}] (P_3_2__Rationals) at (12,-1) {$\begin{smallmatrix}&2&1\\2&&&&&1\\&2&2&2&1\end{smallmatrix}$};
	\node[noselAAAlAAAAA, label={[shift={(0.4,-0.2)}, brown]\tiny $P(3,3)$}] (P_3_3__Rationals) at (12,-2) {$\begin{smallmatrix}&2&2\\2&&&&&1\\&2&2&2&1\end{smallmatrix}$};
	\node[seloneAAAlAAAAA, label={[shift={(0.4,-0.2)}, brown]\tiny $P(3,4)$}] (P_3_4__Rationals) at (12,-7) {$\begin{smallmatrix}&1&1\\2&&&&&1\\&2&2&2&2\end{smallmatrix}$};
	\node[seloneAAAlAAAAA, label={[shift={(0.4,-0.2)}, brown]\tiny $P(3,5)$}] (P_3_5__Rationals) at (12,-6) {$\begin{smallmatrix}&2&2\\3&&&&&2\\&2&2&2&2\end{smallmatrix}$};
	\node[noselAAAlAAAAA, label={[shift={(0.4,-0.2)}, brown]\tiny $P(3,6)$}] (P_3_6__Rationals) at (12,-5) {$\begin{smallmatrix}&2&2\\3&&&&&2\\&3&2&2&2\end{smallmatrix}$};
	\node[noselAAAlAAAAA, label={[shift={(0.4,-0.2)}, brown]\tiny $P(3,7)$}] (P_3_7__Rationals) at (12,-4) {$\begin{smallmatrix}&2&2\\3&&&&&2\\&3&3&2&2\end{smallmatrix}$};
	\node[noselAAAlAAAAA, label={[shift={(0.4,-0.2)}, brown]\tiny $P(3,8)$}] (P_3_8__Rationals) at (12,-3) {$\begin{smallmatrix}&2&2\\3&&&&&2\\&3&3&3&2\end{smallmatrix}$};
	\node at (13,-3.5) {$\dots$};
	\draw[blue,->] (P_0_1__Rationals) -- (P_0_2__Rationals);
	\draw[blue,->] (P_0_2__Rationals) -- (P_0_3__Rationals);
	\draw[blue,->] (P_0_1__Rationals) to [out=135,in=-135] (P_0_4__Rationals);
	\draw[blue,->] (P_0_4__Rationals) -- (P_0_5__Rationals);
	\draw[blue,->] (P_0_5__Rationals) -- (P_0_6__Rationals);
	\draw[blue,->] (P_0_6__Rationals) -- (P_0_7__Rationals);
	\draw[blue,->] (P_0_3__Rationals) -- (P_0_8__Rationals);
	\draw[blue,->] (P_0_7__Rationals) -- (P_0_8__Rationals);
	\draw[gray,dashed,->] (P_0_1__Rationals) -- (P_1_1__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[blue,->] (P_0_2__Rationals) -- (P_1_1__Rationals);
	\draw[blue,->] (P_0_4__Rationals) -- (P_1_1__Rationals);
	\draw[gray,dashed,->] (P_0_2__Rationals) -- (P_1_2__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[blue,->] (P_0_3__Rationals) -- (P_1_2__Rationals);
	\draw[blue,->] (P_1_1__Rationals) -- (P_1_2__Rationals);
	\draw[gray,dashed,->] (P_0_3__Rationals) -- (P_1_3__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[blue,->] (P_0_8__Rationals) -- (P_1_3__Rationals);
	\draw[blue,->] (P_1_2__Rationals) -- (P_1_3__Rationals);
	\draw[gray,dashed,->] (P_0_4__Rationals) -- (P_1_4__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[blue,->] (P_0_5__Rationals) -- (P_1_4__Rationals);
	\draw[blue,->] (P_1_1__Rationals) to [out=135,in=-135] (P_1_4__Rationals);
	\draw[gray,dashed,->] (P_0_5__Rationals) -- (P_1_5__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[blue,->] (P_0_6__Rationals) -- (P_1_5__Rationals);
	\draw[blue,->] (P_1_4__Rationals) -- (P_1_5__Rationals);
	\draw[gray,dashed,->] (P_0_6__Rationals) -- (P_1_6__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[blue,->] (P_0_7__Rationals) -- (P_1_6__Rationals);
	\draw[blue,->] (P_1_5__Rationals) -- (P_1_6__Rationals);
	\draw[gray,dashed,->] (P_0_7__Rationals) -- (P_1_7__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[blue,->] (P_0_8__Rationals) -- (P_1_7__Rationals);
	\draw[blue,->] (P_1_6__Rationals) -- (P_1_7__Rationals);
	\draw[gray,dashed,->] (P_0_8__Rationals) -- (P_1_8__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[blue,->] (P_1_3__Rationals) -- (P_1_8__Rationals);
	\draw[blue,->] (P_1_7__Rationals) -- (P_1_8__Rationals);
	\draw[gray,dashed,->] (P_1_1__Rationals) -- (P_2_1__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[blue,->] (P_1_2__Rationals) -- (P_2_1__Rationals);
	\draw[blue,->] (P_1_4__Rationals) -- (P_2_1__Rationals);
	\draw[gray,dashed,->] (P_1_2__Rationals) -- (P_2_2__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[blue,->] (P_1_3__Rationals) -- (P_2_2__Rationals);
	\draw[blue,->] (P_2_1__Rationals) -- (P_2_2__Rationals);
	\draw[gray,dashed,->] (P_1_3__Rationals) -- (P_2_3__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[blue,->] (P_1_8__Rationals) -- (P_2_3__Rationals);
	\draw[blue,->] (P_2_2__Rationals) -- (P_2_3__Rationals);
	\draw[gray,dashed,->] (P_1_4__Rationals) -- (P_2_4__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[blue,->] (P_1_5__Rationals) -- (P_2_4__Rationals);
	\draw[blue,->] (P_2_1__Rationals) to [out=135,in=-135] (P_2_4__Rationals);
	\draw[gray,dashed,->] (P_1_5__Rationals) -- (P_2_5__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[blue,->] (P_1_6__Rationals) -- (P_2_5__Rationals);
	\draw[blue,->] (P_2_4__Rationals) -- (P_2_5__Rationals);
	\draw[gray,dashed,->] (P_1_6__Rationals) -- (P_2_6__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[blue,->] (P_1_7__Rationals) -- (P_2_6__Rationals);
	\draw[blue,->] (P_2_5__Rationals) -- (P_2_6__Rationals);
	\draw[gray,dashed,->] (P_1_7__Rationals) -- (P_2_7__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[blue,->] (P_1_8__Rationals) -- (P_2_7__Rationals);
	\draw[blue,->] (P_2_6__Rationals) -- (P_2_7__Rationals);
	\draw[gray,dashed,->] (P_1_8__Rationals) -- (P_2_8__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[blue,->] (P_2_3__Rationals) -- (P_2_8__Rationals);
	\draw[blue,->] (P_2_7__Rationals) -- (P_2_8__Rationals);
	\draw[gray,dashed,->] (P_2_1__Rationals) -- (P_3_1__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[blue,->] (P_2_2__Rationals) -- (P_3_1__Rationals);
	\draw[blue,->] (P_2_4__Rationals) -- (P_3_1__Rationals);
	\draw[gray,dashed,->] (P_2_2__Rationals) -- (P_3_2__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[blue,->] (P_2_3__Rationals) -- (P_3_2__Rationals);
	\draw[blue,->] (P_3_1__Rationals) -- (P_3_2__Rationals);
	\draw[gray,dashed,->] (P_2_3__Rationals) -- (P_3_3__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[blue,->] (P_2_8__Rationals) -- (P_3_3__Rationals);
	\draw[blue,->] (P_3_2__Rationals) -- (P_3_3__Rationals);
	\draw[gray,dashed,->] (P_2_4__Rationals) -- (P_3_4__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[blue,->] (P_2_5__Rationals) -- (P_3_4__Rationals);
	\draw[blue,->] (P_3_1__Rationals) to [out=135,in=-135] (P_3_4__Rationals);
	\draw[gray,dashed,->] (P_2_5__Rationals) -- (P_3_5__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[blue,->] (P_2_6__Rationals) -- (P_3_5__Rationals);
	\draw[blue,->] (P_3_4__Rationals) -- (P_3_5__Rationals);
	\draw[gray,dashed,->] (P_2_6__Rationals) -- (P_3_6__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[blue,->] (P_2_7__Rationals) -- (P_3_6__Rationals);
	\draw[blue,->] (P_3_5__Rationals) -- (P_3_6__Rationals);
	\draw[gray,dashed,->] (P_2_7__Rationals) -- (P_3_7__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[blue,->] (P_2_8__Rationals) -- (P_3_7__Rationals);
	\draw[blue,->] (P_3_6__Rationals) -- (P_3_7__Rationals);
	\draw[gray,dashed,->] (P_2_8__Rationals) -- (P_3_8__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[blue,->] (P_3_3__Rationals) -- (P_3_8__Rationals);
	\draw[blue,->] (P_3_7__Rationals) -- (P_3_8__Rationals);
\end{tikzpicture}
\end{scaletikzpicturetowidth}
\end{center}

\end{figure}
\medskip{}
\subsubsection*{Modules of the form $P(n,1)$}

Defect: $\partial P(n,1) = -1$, for $n\ge 0$.

\begin{fleqn}
\begin{align*}
P(0,1):\ & - \\
P(1,1):\ & \bigl(R_{0}^{3}(1),P(0,2)\bigr),\ \bigl(R_{\infty}^{3}(1),P(0,4)\bigr)\\
P(2,1):\ & \bigl(R_{0}^{3}(2),P(0,3)\bigr),\ \bigl(R_{\infty}^{3}(2),P(0,5)\bigr),\ \bigl(R_{0}^{4}(1),P(1,2)\bigr),\ \bigl(R_{\infty}^{1}(1),P(1,4)\bigr)\\
P(3,1):\ & \bigl(R_{0}^{2}(4),P(0,7)\bigr),\ \bigl(R_{0}^{3}(3),P(0,8)\bigr),\ \bigl(R_{0}^{4}(2),P(1,3)\bigr),\ \bigl(R_{\infty}^{1}(2),P(1,5)\bigr),\ \bigl(R_{0}^{5}(1),P(2,2)\bigr)\\
 & \bigl(R_{\infty}^{2}(1),P(2,4)\bigr),\ \bigl(I(1,3),2P(0,6)\bigr)\\
P(n,1):\ & \bigl(R_{0}^{(n-2)\bmod 5+1}(4),P(n-3,7)\bigr),\ \bigl(R_{0}^{(n-1)\bmod 5+1}(3),P(n-3,8)\bigr),\ \bigl(R_{0}^{n\bmod 5+1}(2),P(n-2,3)\bigr)\\
 & \bigl(R_{\infty}^{(n-3)\bmod 3+1}(2),P(n-2,5)\bigr),\ \bigl(R_{0}^{(n+1)\bmod 5+1}(1),P(n-1,2)\bigr),\ \bigl(R_{\infty}^{(n-2)\bmod 3+1}(1),P(n-1,4)\bigr)\\
 & \bigl(uI,(u+1)P\bigr),\ n>3\\
\end{align*}
\end{fleqn}
\subsubsection*{Modules of the form $P(n,2)$}

Defect: $\partial P(n,2) = -1$, for $n\ge 0$.

\begin{fleqn}
\begin{align*}
P(0,2):\ & \bigl(R_{\infty}^{3}(1),P(0,1)\bigr)\\
P(1,2):\ & \bigl(R_{0}^{3}(1),P(0,3)\bigr),\ \bigl(R_{\infty}^{3}(2),P(0,4)\bigr),\ \bigl(R_{\infty}^{1}(1),P(1,1)\bigr)\\
P(2,2):\ & \bigl(R_{0}^{1}(4),P(0,6)\bigr),\ \bigl(R_{0}^{2}(3),P(0,7)\bigr),\ \bigl(R_{0}^{3}(2),P(0,8)\bigr),\ \bigl(R_{0}^{4}(1),P(1,3)\bigr),\ \bigl(R_{\infty}^{1}(2),P(1,4)\bigr)\\
 & \bigl(R_{\infty}^{2}(1),P(2,1)\bigr),\ \bigl(I(2,8),2P(0,5)\bigr)\\
P(n,2):\ & \bigl(R_{0}^{(n-2)\bmod 5+1}(4),P(n-2,6)\bigr),\ \bigl(R_{0}^{(n-1)\bmod 5+1}(3),P(n-2,7)\bigr),\ \bigl(R_{0}^{n\bmod 5+1}(2),P(n-2,8)\bigr)\\
 & \bigl(R_{0}^{(n+1)\bmod 5+1}(1),P(n-1,3)\bigr),\ \bigl(R_{\infty}^{(n-2)\bmod 3+1}(2),P(n-1,4)\bigr),\ \bigl(R_{\infty}^{(n-1)\bmod 3+1}(1),P(n,1)\bigr)\\
 & \bigl(uI,(u+1)P\bigr),\ n>2\\
\end{align*}
\end{fleqn}
\subsubsection*{Modules of the form $P(n,3)$}

Defect: $\partial P(n,3) = -1$, for $n\ge 0$.

\begin{fleqn}
\begin{align*}
P(0,3):\ & \bigl(R_{\infty}^{3}(2),P(0,1)\bigr),\ \bigl(R_{\infty}^{1}(1),P(0,2)\bigr)\\
P(1,3):\ & \bigl(R_{0}^{5}(4),P(0,5)\bigr),\ \bigl(R_{0}^{1}(3),P(0,6)\bigr),\ \bigl(R_{0}^{2}(2),P(0,7)\bigr),\ \bigl(R_{0}^{3}(1),P(0,8)\bigr),\ \bigl(R_{\infty}^{1}(2),P(1,1)\bigr)\\
 & \bigl(R_{\infty}^{2}(1),P(1,2)\bigr),\ \bigl(I(2,7),2P(0,4)\bigr)\\
P(n,3):\ & \bigl(R_{0}^{(n+3)\bmod 5+1}(4),P(n-1,5)\bigr),\ \bigl(R_{0}^{(n-1)\bmod 5+1}(3),P(n-1,6)\bigr),\ \bigl(R_{0}^{n\bmod 5+1}(2),P(n-1,7)\bigr)\\
 & \bigl(R_{0}^{(n+1)\bmod 5+1}(1),P(n-1,8)\bigr),\ \bigl(R_{\infty}^{(n-1)\bmod 3+1}(2),P(n,1)\bigr),\ \bigl(R_{\infty}^{n\bmod 3+1}(1),P(n,2)\bigr)\\
 & \bigl(uI,(u+1)P\bigr),\ n>1\\
\end{align*}
\end{fleqn}
\subsubsection*{Modules of the form $P(n,4)$}

Defect: $\partial P(n,4) = -1$, for $n\ge 0$.

\begin{fleqn}
\begin{align*}
P(0,4):\ & \bigl(R_{0}^{3}(1),P(0,1)\bigr)\\
P(1,4):\ & \bigl(R_{0}^{3}(2),P(0,2)\bigr),\ \bigl(R_{\infty}^{3}(1),P(0,5)\bigr),\ \bigl(R_{0}^{4}(1),P(1,1)\bigr)\\
P(2,4):\ & \bigl(R_{0}^{3}(3),P(0,3)\bigr),\ \bigl(R_{\infty}^{3}(2),P(0,6)\bigr),\ \bigl(R_{0}^{4}(2),P(1,2)\bigr),\ \bigl(R_{\infty}^{1}(1),P(1,5)\bigr),\ \bigl(R_{0}^{5}(1),P(2,1)\bigr)\\
P(3,4):\ & \bigl(R_{0}^{3}(4),P(0,8)\bigr),\ \bigl(R_{0}^{4}(3),P(1,3)\bigr),\ \bigl(R_{\infty}^{1}(2),P(1,6)\bigr),\ \bigl(R_{0}^{5}(2),P(2,2)\bigr),\ \bigl(R_{\infty}^{2}(1),P(2,5)\bigr)\\
 & \bigl(R_{0}^{1}(1),P(3,1)\bigr),\ \bigl(I(0,2),2P(0,7)\bigr)\\
P(n,4):\ & \bigl(R_{0}^{(n-1)\bmod 5+1}(4),P(n-3,8)\bigr),\ \bigl(R_{0}^{n\bmod 5+1}(3),P(n-2,3)\bigr),\ \bigl(R_{\infty}^{(n-3)\bmod 3+1}(2),P(n-2,6)\bigr)\\
 & \bigl(R_{0}^{(n+1)\bmod 5+1}(2),P(n-1,2)\bigr),\ \bigl(R_{\infty}^{(n-2)\bmod 3+1}(1),P(n-1,5)\bigr),\ \bigl(R_{0}^{(n-3)\bmod 5+1}(1),P(n,1)\bigr)\\
 & \bigl(uI,(u+1)P\bigr),\ n>3\\
\end{align*}
\end{fleqn}
\subsubsection*{Modules of the form $P(n,5)$}

Defect: $\partial P(n,5) = -1$, for $n\ge 0$.

\begin{fleqn}
\begin{align*}
P(0,5):\ & \bigl(R_{0}^{3}(2),P(0,1)\bigr),\ \bigl(R_{0}^{4}(1),P(0,4)\bigr)\\
P(1,5):\ & \bigl(R_{0}^{3}(3),P(0,2)\bigr),\ \bigl(R_{\infty}^{3}(1),P(0,6)\bigr),\ \bigl(R_{0}^{4}(2),P(1,1)\bigr),\ \bigl(R_{0}^{5}(1),P(1,4)\bigr)\\
P(2,5):\ & \bigl(R_{0}^{3}(4),P(0,3)\bigr),\ \bigl(R_{\infty}^{3}(2),P(0,7)\bigr),\ \bigl(R_{0}^{4}(3),P(1,2)\bigr),\ \bigl(R_{\infty}^{1}(1),P(1,6)\bigr),\ \bigl(R_{0}^{5}(2),P(2,1)\bigr)\\
 & \bigl(R_{0}^{1}(1),P(2,4)\bigr)\\
P(3,5):\ & \bigl(R_{0}^{4}(4),P(1,3)\bigr),\ \bigl(R_{\infty}^{1}(2),P(1,7)\bigr),\ \bigl(R_{0}^{5}(3),P(2,2)\bigr),\ \bigl(R_{\infty}^{2}(1),P(2,6)\bigr),\ \bigl(R_{0}^{1}(2),P(3,1)\bigr)\\
 & \bigl(R_{0}^{2}(1),P(3,4)\bigr),\ \bigl(2I(2,6),3P(0,1)\bigr)\\
P(n,5):\ & \bigl(R_{0}^{n\bmod 5+1}(4),P(n-2,3)\bigr),\ \bigl(R_{\infty}^{(n-3)\bmod 3+1}(2),P(n-2,7)\bigr),\ \bigl(R_{0}^{(n+1)\bmod 5+1}(3),P(n-1,2)\bigr)\\
 & \bigl(R_{\infty}^{(n-2)\bmod 3+1}(1),P(n-1,6)\bigr),\ \bigl(R_{0}^{(n-3)\bmod 5+1}(2),P(n,1)\bigr),\ \bigl(R_{0}^{(n-2)\bmod 5+1}(1),P(n,4)\bigr)\\
 & \bigl(uI,(u+1)P\bigr),\ n>3\\
\end{align*}
\end{fleqn}
\subsubsection*{Modules of the form $P(n,6)$}

Defect: $\partial P(n,6) = -1$, for $n\ge 0$.

\begin{fleqn}
\begin{align*}
P(0,6):\ & \bigl(R_{0}^{3}(3),P(0,1)\bigr),\ \bigl(R_{0}^{4}(2),P(0,4)\bigr),\ \bigl(R_{0}^{5}(1),P(0,5)\bigr)\\
P(1,6):\ & \bigl(R_{0}^{3}(4),P(0,2)\bigr),\ \bigl(R_{\infty}^{3}(1),P(0,7)\bigr),\ \bigl(R_{0}^{4}(3),P(1,1)\bigr),\ \bigl(R_{0}^{5}(2),P(1,4)\bigr),\ \bigl(R_{0}^{1}(1),P(1,5)\bigr)\\
P(2,6):\ & \bigl(R_{\infty}^{3}(2),P(0,8)\bigr),\ \bigl(R_{0}^{4}(4),P(1,2)\bigr),\ \bigl(R_{\infty}^{1}(1),P(1,7)\bigr),\ \bigl(R_{0}^{5}(3),P(2,1)\bigr),\ \bigl(R_{0}^{1}(2),P(2,4)\bigr)\\
 & \bigl(R_{0}^{2}(1),P(2,5)\bigr),\ \bigl(I(0,4),2P(0,3)\bigr)\\
P(n,6):\ & \bigl(R_{\infty}^{n\bmod 3+1}(2),P(n-2,8)\bigr),\ \bigl(R_{0}^{(n+1)\bmod 5+1}(4),P(n-1,2)\bigr),\ \bigl(R_{\infty}^{(n-2)\bmod 3+1}(1),P(n-1,7)\bigr)\\
 & \bigl(R_{0}^{(n+2)\bmod 5+1}(3),P(n,1)\bigr),\ \bigl(R_{0}^{(n-2)\bmod 5+1}(2),P(n,4)\bigr),\ \bigl(R_{0}^{(n-1)\bmod 5+1}(1),P(n,5)\bigr)\\
 & \bigl(uI,(u+1)P\bigr),\ n>2\\
\end{align*}
\end{fleqn}
\subsubsection*{Modules of the form $P(n,7)$}

Defect: $\partial P(n,7) = -1$, for $n\ge 0$.

\begin{fleqn}
\begin{align*}
P(0,7):\ & \bigl(R_{0}^{3}(4),P(0,1)\bigr),\ \bigl(R_{0}^{4}(3),P(0,4)\bigr),\ \bigl(R_{0}^{5}(2),P(0,5)\bigr),\ \bigl(R_{0}^{1}(1),P(0,6)\bigr)\\
P(1,7):\ & \bigl(R_{\infty}^{2}(2),P(0,3)\bigr),\ \bigl(R_{\infty}^{3}(1),P(0,8)\bigr),\ \bigl(R_{0}^{4}(4),P(1,1)\bigr),\ \bigl(R_{0}^{5}(3),P(1,4)\bigr),\ \bigl(R_{0}^{1}(2),P(1,5)\bigr)\\
 & \bigl(R_{0}^{2}(1),P(1,6)\bigr),\ \bigl(I(1,5),2P(0,2)\bigr)\\
P(n,7):\ & \bigl(R_{\infty}^{n\bmod 3+1}(2),P(n-1,3)\bigr),\ \bigl(R_{\infty}^{(n+1)\bmod 3+1}(1),P(n-1,8)\bigr),\ \bigl(R_{0}^{(n+2)\bmod 5+1}(4),P(n,1)\bigr)\\
 & \bigl(R_{0}^{(n+3)\bmod 5+1}(3),P(n,4)\bigr),\ \bigl(R_{0}^{(n-1)\bmod 5+1}(2),P(n,5)\bigr),\ \bigl(R_{0}^{n\bmod 5+1}(1),P(n,6)\bigr)\\
 & \bigl(uI,(u+1)P\bigr),\ n>1\\
\end{align*}
\end{fleqn}
\subsubsection*{Modules of the form $P(n,8)$}

Defect: $\partial P(n,8) = -1$, for $n\ge 0$.

\begin{fleqn}
\begin{align*}
P(0,8):\ & \bigl(R_{\infty}^{1}(2),P(0,2)\bigr),\ \bigl(R_{\infty}^{2}(1),P(0,3)\bigr),\ \bigl(R_{0}^{4}(4),P(0,4)\bigr),\ \bigl(R_{0}^{5}(3),P(0,5)\bigr),\ \bigl(R_{0}^{1}(2),P(0,6)\bigr)\\
 & \bigl(R_{0}^{2}(1),P(0,7)\bigr),\ \bigl(I(2,6),2P(0,1)\bigr)\\
P(n,8):\ & \bigl(R_{\infty}^{n\bmod 3+1}(2),P(n,2)\bigr),\ \bigl(R_{\infty}^{(n+1)\bmod 3+1}(1),P(n,3)\bigr),\ \bigl(R_{0}^{(n+3)\bmod 5+1}(4),P(n,4)\bigr)\\
 & \bigl(R_{0}^{(n+4)\bmod 5+1}(3),P(n,5)\bigr),\ \bigl(R_{0}^{n\bmod 5+1}(2),P(n,6)\bigr),\ \bigl(R_{0}^{(n+1)\bmod 5+1}(1),P(n,7)\bigr)\\
 & \bigl(uI,(u+1)P\bigr),\ n>0\\
\end{align*}
\end{fleqn}
\subsubsection{Schofield pairs associated to preinjective exceptional modules}

\begin{figure}[ht]

\begin{center}
\begin{scaletikzpicturetowidth}{\textwidth}
\begin{tikzpicture}[scale=\tikzscale]
	\node[seloneAAAlAAAAA, label={[shift={(0.4,-0.2)}, brown]\tiny $I(0,1)$}] (I_0_1__Rationals) at (0,0) {$\begin{smallmatrix}&1&1\\1&&&&&2\\&1&1&1&1\end{smallmatrix}$};
	\node[noselAAAlAAAAA, label={[shift={(0.4,-0.2)}, brown]\tiny $I(0,2)$}] (I_0_2__Rationals) at (0,-1) {$\begin{smallmatrix}&1&1\\0&&&&&1\\&0&0&0&0\end{smallmatrix}$};
	\node[noselAAAlAAAAA, label={[shift={(0.4,-0.2)}, brown]\tiny $I(0,3)$}] (I_0_3__Rationals) at (0,-2) {$\begin{smallmatrix}&0&1\\0&&&&&1\\&0&0&0&0\end{smallmatrix}$};
	\node[noselAAAlAAAAA, label={[shift={(0.4,-0.2)}, brown]\tiny $I(0,4)$}] (I_0_4__Rationals) at (0,-7) {$\begin{smallmatrix}&0&0\\0&&&&&1\\&1&1&1&1\end{smallmatrix}$};
	\node[noselAAAlAAAAA, label={[shift={(0.4,-0.2)}, brown]\tiny $I(0,5)$}] (I_0_5__Rationals) at (0,-6) {$\begin{smallmatrix}&0&0\\0&&&&&1\\&0&1&1&1\end{smallmatrix}$};
	\node[noselAAAlAAAAA, label={[shift={(0.4,-0.2)}, brown]\tiny $I(0,6)$}] (I_0_6__Rationals) at (0,-5) {$\begin{smallmatrix}&0&0\\0&&&&&1\\&0&0&1&1\end{smallmatrix}$};
	\node[noselAAAlAAAAA, label={[shift={(0.4,-0.2)}, brown]\tiny $I(0,7)$}] (I_0_7__Rationals) at (0,-4) {$\begin{smallmatrix}&0&0\\0&&&&&1\\&0&0&0&1\end{smallmatrix}$};
	\node[noselAAAlAAAAA, label={[shift={(0.4,-0.2)}, brown]\tiny $I(0,8)$}] (I_0_8__Rationals) at (0,-3) {$\begin{smallmatrix}&0&0\\0&&&&&1\\&0&0&0&0\end{smallmatrix}$};
	\node[noselAAAlAAAAA, label={[shift={(0.4,-0.2)}, brown]\tiny $I(1,1)$}] (I_1_1__Rationals) at (-4,0) {$\begin{smallmatrix}&1&2\\1&&&&&2\\&1&1&1&2\end{smallmatrix}$};
	\node[seloneAAAlAAAAA, label={[shift={(0.4,-0.2)}, brown]\tiny $I(1,2)$}] (I_1_2__Rationals) at (-4,-1) {$\begin{smallmatrix}&1&1\\1&&&&&2\\&1&1&1&2\end{smallmatrix}$};
	\node[noselAAAlAAAAA, label={[shift={(0.4,-0.2)}, brown]\tiny $I(1,3)$}] (I_1_3__Rationals) at (-4,-2) {$\begin{smallmatrix}&1&1\\0&&&&&1\\&0&0&0&1\end{smallmatrix}$};
	\node[seloneAAAlAAAAA, label={[shift={(0.4,-0.2)}, brown]\tiny $I(1,4)$}] (I_1_4__Rationals) at (-4,-7) {$\begin{smallmatrix}&1&2\\1&&&&&2\\&1&1&1&1\end{smallmatrix}$};
	\node[noselAAAlAAAAA, label={[shift={(0.4,-0.2)}, brown]\tiny $I(1,5)$}] (I_1_5__Rationals) at (-4,-6) {$\begin{smallmatrix}&0&1\\0&&&&&1\\&1&1&1&1\end{smallmatrix}$};
	\node[noselAAAlAAAAA, label={[shift={(0.4,-0.2)}, brown]\tiny $I(1,6)$}] (I_1_6__Rationals) at (-4,-5) {$\begin{smallmatrix}&0&1\\0&&&&&1\\&0&1&1&1\end{smallmatrix}$};
	\node[noselAAAlAAAAA, label={[shift={(0.4,-0.2)}, brown]\tiny $I(1,7)$}] (I_1_7__Rationals) at (-4,-4) {$\begin{smallmatrix}&0&1\\0&&&&&1\\&0&0&1&1\end{smallmatrix}$};
	\node[noselAAAlAAAAA, label={[shift={(0.4,-0.2)}, brown]\tiny $I(1,8)$}] (I_1_8__Rationals) at (-4,-3) {$\begin{smallmatrix}&0&1\\0&&&&&1\\&0&0&0&1\end{smallmatrix}$};
	\node[noselAAAlAAAAA, label={[shift={(0.4,-0.2)}, brown]\tiny $I(2,1)$}] (I_2_1__Rationals) at (-8,0) {$\begin{smallmatrix}&2&2\\1&&&&&2\\&1&1&2&2\end{smallmatrix}$};
	\node[noselAAAlAAAAA, label={[shift={(0.4,-0.2)}, brown]\tiny $I(2,2)$}] (I_2_2__Rationals) at (-8,-1) {$\begin{smallmatrix}&1&2\\1&&&&&2\\&1&1&2&2\end{smallmatrix}$};
	\node[seloneAAAlAAAAA, label={[shift={(0.4,-0.2)}, brown]\tiny $I(2,3)$}] (I_2_3__Rationals) at (-8,-2) {$\begin{smallmatrix}&1&1\\1&&&&&2\\&1&1&2&2\end{smallmatrix}$};
	\node[noselAAAlAAAAA, label={[shift={(0.4,-0.2)}, brown]\tiny $I(2,4)$}] (I_2_4__Rationals) at (-8,-7) {$\begin{smallmatrix}&2&2\\1&&&&&2\\&1&1&1&2\end{smallmatrix}$};
	\node[seloneAAAlAAAAA, label={[shift={(0.4,-0.2)}, brown]\tiny $I(2,5)$}] (I_2_5__Rationals) at (-8,-6) {$\begin{smallmatrix}&2&2\\1&&&&&2\\&1&1&1&1\end{smallmatrix}$};
	\node[noselAAAlAAAAA, label={[shift={(0.4,-0.2)}, brown]\tiny $I(2,6)$}] (I_2_6__Rationals) at (-8,-5) {$\begin{smallmatrix}&1&1\\0&&&&&1\\&1&1&1&1\end{smallmatrix}$};
	\node[noselAAAlAAAAA, label={[shift={(0.4,-0.2)}, brown]\tiny $I(2,7)$}] (I_2_7__Rationals) at (-8,-4) {$\begin{smallmatrix}&1&1\\0&&&&&1\\&0&1&1&1\end{smallmatrix}$};
	\node[noselAAAlAAAAA, label={[shift={(0.4,-0.2)}, brown]\tiny $I(2,8)$}] (I_2_8__Rationals) at (-8,-3) {$\begin{smallmatrix}&1&1\\0&&&&&1\\&0&0&1&1\end{smallmatrix}$};
	\node[noselAAAlAAAAA, label={[shift={(0.4,-0.2)}, brown]\tiny $I(3,1)$}] (I_3_1__Rationals) at (-12,0) {$\begin{smallmatrix}&2&2\\2&&&&&3\\&2&3&3&3\end{smallmatrix}$};
	\node[noselAAAlAAAAA, label={[shift={(0.4,-0.2)}, brown]\tiny $I(3,2)$}] (I_3_2__Rationals) at (-12,-1) {$\begin{smallmatrix}&2&2\\1&&&&&2\\&1&2&2&2\end{smallmatrix}$};
	\node[noselAAAlAAAAA, label={[shift={(0.4,-0.2)}, brown]\tiny $I(3,3)$}] (I_3_3__Rationals) at (-12,-2) {$\begin{smallmatrix}&1&2\\1&&&&&2\\&1&2&2&2\end{smallmatrix}$};
	\node[noselAAAlAAAAA, label={[shift={(0.4,-0.2)}, brown]\tiny $I(3,4)$}] (I_3_4__Rationals) at (-12,-7) {$\begin{smallmatrix}&2&2\\2&&&&&3\\&2&2&3&3\end{smallmatrix}$};
	\node[noselAAAlAAAAA, label={[shift={(0.4,-0.2)}, brown]\tiny $I(3,5)$}] (I_3_5__Rationals) at (-12,-6) {$\begin{smallmatrix}&2&2\\2&&&&&3\\&2&2&2&3\end{smallmatrix}$};
	\node[seloneAAAlAAAAA, label={[shift={(0.4,-0.2)}, brown]\tiny $I(3,6)$}] (I_3_6__Rationals) at (-12,-5) {$\begin{smallmatrix}&2&2\\2&&&&&3\\&2&2&2&2\end{smallmatrix}$};
	\node[seloneAAAlAAAAA, label={[shift={(0.4,-0.2)}, brown]\tiny $I(3,7)$}] (I_3_7__Rationals) at (-12,-4) {$\begin{smallmatrix}&1&1\\1&&&&&2\\&2&2&2&2\end{smallmatrix}$};
	\node[seloneAAAlAAAAA, label={[shift={(0.4,-0.2)}, brown]\tiny $I(3,8)$}] (I_3_8__Rationals) at (-12,-3) {$\begin{smallmatrix}&1&1\\1&&&&&2\\&1&2&2&2\end{smallmatrix}$};
	\node at (-13,-3.5) {$\dots$};
	\draw[red,->] (I_1_2__Rationals) -- (I_0_1__Rationals);
	\draw[red,->] (I_1_4__Rationals) -- (I_0_1__Rationals);
	\draw[red,->] (I_0_1__Rationals) -- (I_0_2__Rationals);
	\draw[red,->] (I_1_3__Rationals) -- (I_0_2__Rationals);
	\draw[red,->] (I_0_2__Rationals) -- (I_0_3__Rationals);
	\draw[red,->] (I_1_8__Rationals) -- (I_0_3__Rationals);
	\draw[red,->] (I_0_1__Rationals) to [out=135,in=-135] (I_0_4__Rationals);
	\draw[red,->] (I_1_5__Rationals) -- (I_0_4__Rationals);
	\draw[red,->] (I_0_4__Rationals) -- (I_0_5__Rationals);
	\draw[red,->] (I_1_6__Rationals) -- (I_0_5__Rationals);
	\draw[red,->] (I_0_5__Rationals) -- (I_0_6__Rationals);
	\draw[red,->] (I_1_7__Rationals) -- (I_0_6__Rationals);
	\draw[red,->] (I_0_6__Rationals) -- (I_0_7__Rationals);
	\draw[red,->] (I_1_8__Rationals) -- (I_0_7__Rationals);
	\draw[red,->] (I_0_3__Rationals) -- (I_0_8__Rationals);
	\draw[red,->] (I_0_7__Rationals) -- (I_0_8__Rationals);
	\draw[gray,dashed,->] (I_0_1__Rationals) -- (I_1_1__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_2_2__Rationals) -- (I_1_1__Rationals);
	\draw[red,->] (I_2_4__Rationals) -- (I_1_1__Rationals);
	\draw[gray,dashed,->] (I_0_2__Rationals) -- (I_1_2__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_1_1__Rationals) -- (I_1_2__Rationals);
	\draw[red,->] (I_2_3__Rationals) -- (I_1_2__Rationals);
	\draw[gray,dashed,->] (I_0_3__Rationals) -- (I_1_3__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_1_2__Rationals) -- (I_1_3__Rationals);
	\draw[red,->] (I_2_8__Rationals) -- (I_1_3__Rationals);
	\draw[gray,dashed,->] (I_0_4__Rationals) -- (I_1_4__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_1_1__Rationals) to [out=135,in=-135] (I_1_4__Rationals);
	\draw[red,->] (I_2_5__Rationals) -- (I_1_4__Rationals);
	\draw[gray,dashed,->] (I_0_5__Rationals) -- (I_1_5__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_1_4__Rationals) -- (I_1_5__Rationals);
	\draw[red,->] (I_2_6__Rationals) -- (I_1_5__Rationals);
	\draw[gray,dashed,->] (I_0_6__Rationals) -- (I_1_6__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_1_5__Rationals) -- (I_1_6__Rationals);
	\draw[red,->] (I_2_7__Rationals) -- (I_1_6__Rationals);
	\draw[gray,dashed,->] (I_0_7__Rationals) -- (I_1_7__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_1_6__Rationals) -- (I_1_7__Rationals);
	\draw[red,->] (I_2_8__Rationals) -- (I_1_7__Rationals);
	\draw[gray,dashed,->] (I_0_8__Rationals) -- (I_1_8__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_1_3__Rationals) -- (I_1_8__Rationals);
	\draw[red,->] (I_1_7__Rationals) -- (I_1_8__Rationals);
	\draw[gray,dashed,->] (I_1_1__Rationals) -- (I_2_1__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_3_2__Rationals) -- (I_2_1__Rationals);
	\draw[red,->] (I_3_4__Rationals) -- (I_2_1__Rationals);
	\draw[gray,dashed,->] (I_1_2__Rationals) -- (I_2_2__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_2_1__Rationals) -- (I_2_2__Rationals);
	\draw[red,->] (I_3_3__Rationals) -- (I_2_2__Rationals);
	\draw[gray,dashed,->] (I_1_3__Rationals) -- (I_2_3__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_2_2__Rationals) -- (I_2_3__Rationals);
	\draw[red,->] (I_3_8__Rationals) -- (I_2_3__Rationals);
	\draw[gray,dashed,->] (I_1_4__Rationals) -- (I_2_4__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_2_1__Rationals) to [out=135,in=-135] (I_2_4__Rationals);
	\draw[red,->] (I_3_5__Rationals) -- (I_2_4__Rationals);
	\draw[gray,dashed,->] (I_1_5__Rationals) -- (I_2_5__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_2_4__Rationals) -- (I_2_5__Rationals);
	\draw[red,->] (I_3_6__Rationals) -- (I_2_5__Rationals);
	\draw[gray,dashed,->] (I_1_6__Rationals) -- (I_2_6__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_2_5__Rationals) -- (I_2_6__Rationals);
	\draw[red,->] (I_3_7__Rationals) -- (I_2_6__Rationals);
	\draw[gray,dashed,->] (I_1_7__Rationals) -- (I_2_7__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_2_6__Rationals) -- (I_2_7__Rationals);
	\draw[red,->] (I_3_8__Rationals) -- (I_2_7__Rationals);
	\draw[gray,dashed,->] (I_1_8__Rationals) -- (I_2_8__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_2_3__Rationals) -- (I_2_8__Rationals);
	\draw[red,->] (I_2_7__Rationals) -- (I_2_8__Rationals);
	\draw[gray,dashed,->] (I_2_1__Rationals) -- (I_3_1__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[gray,dashed,->] (I_2_2__Rationals) -- (I_3_2__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_3_1__Rationals) -- (I_3_2__Rationals);
	\draw[gray,dashed,->] (I_2_3__Rationals) -- (I_3_3__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_3_2__Rationals) -- (I_3_3__Rationals);
	\draw[gray,dashed,->] (I_2_4__Rationals) -- (I_3_4__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_3_1__Rationals) to [out=135,in=-135] (I_3_4__Rationals);
	\draw[gray,dashed,->] (I_2_5__Rationals) -- (I_3_5__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_3_4__Rationals) -- (I_3_5__Rationals);
	\draw[gray,dashed,->] (I_2_6__Rationals) -- (I_3_6__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_3_5__Rationals) -- (I_3_6__Rationals);
	\draw[gray,dashed,->] (I_2_7__Rationals) -- (I_3_7__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_3_6__Rationals) -- (I_3_7__Rationals);
	\draw[gray,dashed,->] (I_2_8__Rationals) -- (I_3_8__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_3_3__Rationals) -- (I_3_8__Rationals);
	\draw[red,->] (I_3_7__Rationals) -- (I_3_8__Rationals);
\end{tikzpicture}
\end{scaletikzpicturetowidth}
\end{center}

\end{figure}
\medskip{}
\subsubsection*{Modules of the form $I(n,1)$}

Defect: $\partial I(n,1) = 1$, for $n\ge 0$.

\begin{fleqn}
\begin{align*}
I(0,1):\ & \bigl(I(0,2),R_{\infty}^{2}(1)\bigr),\ \bigl(I(0,3),R_{\infty}^{2}(2)\bigr),\ \bigl(I(0,4),R_{0}^{2}(1)\bigr),\ \bigl(I(0,5),R_{0}^{2}(2)\bigr),\ \bigl(I(0,6),R_{0}^{2}(3)\bigr)\\
 & \bigl(I(0,7),R_{0}^{2}(4)\bigr),\ \bigl(2I(0,8),P(2,5)\bigr)\\
I(n,1):\ & \bigl(I(n,2),R_{\infty}^{(-n+1)\bmod 3+1}(1)\bigr),\ \bigl(I(n,3),R_{\infty}^{(-n+1)\bmod 3+1}(2)\bigr),\ \bigl(I(n,4),R_{0}^{(-n+1)\bmod 5+1}(1)\bigr)\\
 & \bigl(I(n,5),R_{0}^{(-n+1)\bmod 5+1}(2)\bigr),\ \bigl(I(n,6),R_{0}^{(-n+1)\bmod 5+1}(3)\bigr),\ \bigl(I(n,7),R_{0}^{(-n+1)\bmod 5+1}(4)\bigr)\\
 & \bigl((v+1)I,vP\bigr),\ n>0\\
\end{align*}
\end{fleqn}
\subsubsection*{Modules of the form $I(n,2)$}

Defect: $\partial I(n,2) = 1$, for $n\ge 0$.

\begin{fleqn}
\begin{align*}
I(0,2):\ & \bigl(I(0,3),R_{\infty}^{3}(1)\bigr),\ \bigl(I(0,8),R_{\infty}^{3}(2)\bigr)\\
I(1,2):\ & \bigl(I(0,1),R_{0}^{1}(1)\bigr),\ \bigl(I(0,4),R_{0}^{1}(2)\bigr),\ \bigl(I(0,5),R_{0}^{1}(3)\bigr),\ \bigl(I(0,6),R_{0}^{1}(4)\bigr),\ \bigl(I(1,3),R_{\infty}^{2}(1)\bigr)\\
 & \bigl(I(1,8),R_{\infty}^{2}(2)\bigr),\ \bigl(2I(0,7),P(2,4)\bigr)\\
I(n,2):\ & \bigl(I(n-1,1),R_{0}^{(-n+1)\bmod 5+1}(1)\bigr),\ \bigl(I(n-1,4),R_{0}^{(-n+1)\bmod 5+1}(2)\bigr),\ \bigl(I(n-1,5),R_{0}^{(-n+1)\bmod 5+1}(3)\bigr)\\
 & \bigl(I(n-1,6),R_{0}^{(-n+1)\bmod 5+1}(4)\bigr),\ \bigl(I(n,3),R_{\infty}^{(-n+2)\bmod 3+1}(1)\bigr),\ \bigl(I(n,8),R_{\infty}^{(-n+2)\bmod 3+1}(2)\bigr)\\
 & \bigl((v+1)I,vP\bigr),\ n>1\\
\end{align*}
\end{fleqn}
\subsubsection*{Modules of the form $I(n,3)$}

Defect: $\partial I(n,3) = 1$, for $n\ge 0$.

\begin{fleqn}
\begin{align*}
I(0,3):\ & \bigl(I(0,8),R_{\infty}^{1}(1)\bigr)\\
I(1,3):\ & \bigl(I(0,2),R_{0}^{1}(1)\bigr),\ \bigl(I(0,7),R_{\infty}^{3}(2)\bigr),\ \bigl(I(1,8),R_{\infty}^{3}(1)\bigr)\\
I(2,3):\ & \bigl(I(0,1),R_{0}^{5}(2)\bigr),\ \bigl(I(0,4),R_{0}^{5}(3)\bigr),\ \bigl(I(0,5),R_{0}^{5}(4)\bigr),\ \bigl(I(1,2),R_{0}^{5}(1)\bigr),\ \bigl(I(1,7),R_{\infty}^{2}(2)\bigr)\\
 & \bigl(I(2,8),R_{\infty}^{2}(1)\bigr),\ \bigl(2I(0,6),P(2,1)\bigr)\\
I(n,3):\ & \bigl(I(n-2,1),R_{0}^{(-n+6)\bmod 5+1}(2)\bigr),\ \bigl(I(n-2,4),R_{0}^{(-n+6)\bmod 5+1}(3)\bigr),\ \bigl(I(n-2,5),R_{0}^{(-n+6)\bmod 5+1}(4)\bigr)\\
 & \bigl(I(n-1,2),R_{0}^{(-n+6)\bmod 5+1}(1)\bigr),\ \bigl(I(n-1,7),R_{\infty}^{(-n+3)\bmod 3+1}(2)\bigr),\ \bigl(I(n,8),R_{\infty}^{(-n+3)\bmod 3+1}(1)\bigr)\\
 & \bigl((v+1)I,vP\bigr),\ n>2\\
\end{align*}
\end{fleqn}
\subsubsection*{Modules of the form $I(n,4)$}

Defect: $\partial I(n,4) = 1$, for $n\ge 0$.

\begin{fleqn}
\begin{align*}
I(0,4):\ & \bigl(I(0,5),R_{0}^{3}(1)\bigr),\ \bigl(I(0,6),R_{0}^{3}(2)\bigr),\ \bigl(I(0,7),R_{0}^{3}(3)\bigr),\ \bigl(I(0,8),R_{0}^{3}(4)\bigr)\\
I(1,4):\ & \bigl(I(0,1),R_{\infty}^{1}(1)\bigr),\ \bigl(I(0,2),R_{\infty}^{1}(2)\bigr),\ \bigl(I(1,5),R_{0}^{2}(1)\bigr),\ \bigl(I(1,6),R_{0}^{2}(2)\bigr),\ \bigl(I(1,7),R_{0}^{2}(3)\bigr)\\
 & \bigl(I(1,8),R_{0}^{2}(4)\bigr),\ \bigl(2I(0,3),P(1,6)\bigr)\\
I(n,4):\ & \bigl(I(n-1,1),R_{\infty}^{(-n+1)\bmod 3+1}(1)\bigr),\ \bigl(I(n-1,2),R_{\infty}^{(-n+1)\bmod 3+1}(2)\bigr),\ \bigl(I(n,5),R_{0}^{(-n+2)\bmod 5+1}(1)\bigr)\\
 & \bigl(I(n,6),R_{0}^{(-n+2)\bmod 5+1}(2)\bigr),\ \bigl(I(n,7),R_{0}^{(-n+2)\bmod 5+1}(3)\bigr),\ \bigl(I(n,8),R_{0}^{(-n+2)\bmod 5+1}(4)\bigr)\\
 & \bigl((v+1)I,vP\bigr),\ n>1\\
\end{align*}
\end{fleqn}
\subsubsection*{Modules of the form $I(n,5)$}

Defect: $\partial I(n,5) = 1$, for $n\ge 0$.

\begin{fleqn}
\begin{align*}
I(0,5):\ & \bigl(I(0,6),R_{0}^{4}(1)\bigr),\ \bigl(I(0,7),R_{0}^{4}(2)\bigr),\ \bigl(I(0,8),R_{0}^{4}(3)\bigr)\\
I(1,5):\ & \bigl(I(0,3),R_{0}^{3}(4)\bigr),\ \bigl(I(0,4),R_{\infty}^{1}(1)\bigr),\ \bigl(I(1,6),R_{0}^{3}(1)\bigr),\ \bigl(I(1,7),R_{0}^{3}(2)\bigr),\ \bigl(I(1,8),R_{0}^{3}(3)\bigr)\\
I(2,5):\ & \bigl(I(0,1),R_{\infty}^{3}(2)\bigr),\ \bigl(I(1,3),R_{0}^{2}(4)\bigr),\ \bigl(I(1,4),R_{\infty}^{3}(1)\bigr),\ \bigl(I(2,6),R_{0}^{2}(1)\bigr),\ \bigl(I(2,7),R_{0}^{2}(2)\bigr)\\
 & \bigl(I(2,8),R_{0}^{2}(3)\bigr),\ \bigl(2I(0,2),P(0,7)\bigr)\\
I(n,5):\ & \bigl(I(n-2,1),R_{\infty}^{(-n+4)\bmod 3+1}(2)\bigr),\ \bigl(I(n-1,3),R_{0}^{(-n+3)\bmod 5+1}(4)\bigr),\ \bigl(I(n-1,4),R_{\infty}^{(-n+4)\bmod 3+1}(1)\bigr)\\
 & \bigl(I(n,6),R_{0}^{(-n+3)\bmod 5+1}(1)\bigr),\ \bigl(I(n,7),R_{0}^{(-n+3)\bmod 5+1}(2)\bigr),\ \bigl(I(n,8),R_{0}^{(-n+3)\bmod 5+1}(3)\bigr)\\
 & \bigl((v+1)I,vP\bigr),\ n>2\\
\end{align*}
\end{fleqn}
\subsubsection*{Modules of the form $I(n,6)$}

Defect: $\partial I(n,6) = 1$, for $n\ge 0$.

\begin{fleqn}
\begin{align*}
I(0,6):\ & \bigl(I(0,7),R_{0}^{5}(1)\bigr),\ \bigl(I(0,8),R_{0}^{5}(2)\bigr)\\
I(1,6):\ & \bigl(I(0,3),R_{0}^{4}(3)\bigr),\ \bigl(I(0,5),R_{\infty}^{1}(1)\bigr),\ \bigl(I(1,7),R_{0}^{4}(1)\bigr),\ \bigl(I(1,8),R_{0}^{4}(2)\bigr)\\
I(2,6):\ & \bigl(I(0,2),R_{0}^{3}(4)\bigr),\ \bigl(I(0,4),R_{\infty}^{3}(2)\bigr),\ \bigl(I(1,3),R_{0}^{3}(3)\bigr),\ \bigl(I(1,5),R_{\infty}^{3}(1)\bigr),\ \bigl(I(2,7),R_{0}^{3}(1)\bigr)\\
 & \bigl(I(2,8),R_{0}^{3}(2)\bigr)\\
I(3,6):\ & \bigl(I(1,2),R_{0}^{2}(4)\bigr),\ \bigl(I(1,4),R_{\infty}^{2}(2)\bigr),\ \bigl(I(2,3),R_{0}^{2}(3)\bigr),\ \bigl(I(2,5),R_{\infty}^{2}(1)\bigr),\ \bigl(I(3,7),R_{0}^{2}(1)\bigr)\\
 & \bigl(I(3,8),R_{0}^{2}(2)\bigr),\ \bigl(3I(0,8),2P(2,5)\bigr)\\
I(n,6):\ & \bigl(I(n-2,2),R_{0}^{(-n+4)\bmod 5+1}(4)\bigr),\ \bigl(I(n-2,4),R_{\infty}^{(-n+4)\bmod 3+1}(2)\bigr),\ \bigl(I(n-1,3),R_{0}^{(-n+4)\bmod 5+1}(3)\bigr)\\
 & \bigl(I(n-1,5),R_{\infty}^{(-n+4)\bmod 3+1}(1)\bigr),\ \bigl(I(n,7),R_{0}^{(-n+4)\bmod 5+1}(1)\bigr),\ \bigl(I(n,8),R_{0}^{(-n+4)\bmod 5+1}(2)\bigr)\\
 & \bigl((v+1)I,vP\bigr),\ n>3\\
\end{align*}
\end{fleqn}
\subsubsection*{Modules of the form $I(n,7)$}

Defect: $\partial I(n,7) = 1$, for $n\ge 0$.

\begin{fleqn}
\begin{align*}
I(0,7):\ & \bigl(I(0,8),R_{0}^{1}(1)\bigr)\\
I(1,7):\ & \bigl(I(0,3),R_{0}^{5}(2)\bigr),\ \bigl(I(0,6),R_{\infty}^{1}(1)\bigr),\ \bigl(I(1,8),R_{0}^{5}(1)\bigr)\\
I(2,7):\ & \bigl(I(0,2),R_{0}^{4}(3)\bigr),\ \bigl(I(0,5),R_{\infty}^{3}(2)\bigr),\ \bigl(I(1,3),R_{0}^{4}(2)\bigr),\ \bigl(I(1,6),R_{\infty}^{3}(1)\bigr),\ \bigl(I(2,8),R_{0}^{4}(1)\bigr)\\
I(3,7):\ & \bigl(I(0,1),R_{0}^{3}(4)\bigr),\ \bigl(I(1,2),R_{0}^{3}(3)\bigr),\ \bigl(I(1,5),R_{\infty}^{2}(2)\bigr),\ \bigl(I(2,3),R_{0}^{3}(2)\bigr),\ \bigl(I(2,6),R_{\infty}^{2}(1)\bigr)\\
 & \bigl(I(3,8),R_{0}^{3}(1)\bigr),\ \bigl(2I(0,4),P(0,3)\bigr)\\
I(n,7):\ & \bigl(I(n-3,1),R_{0}^{(-n+5)\bmod 5+1}(4)\bigr),\ \bigl(I(n-2,2),R_{0}^{(-n+5)\bmod 5+1}(3)\bigr),\ \bigl(I(n-2,5),R_{\infty}^{(-n+4)\bmod 3+1}(2)\bigr)\\
 & \bigl(I(n-1,3),R_{0}^{(-n+5)\bmod 5+1}(2)\bigr),\ \bigl(I(n-1,6),R_{\infty}^{(-n+4)\bmod 3+1}(1)\bigr),\ \bigl(I(n,8),R_{0}^{(-n+5)\bmod 5+1}(1)\bigr)\\
 & \bigl((v+1)I,vP\bigr),\ n>3\\
\end{align*}
\end{fleqn}
\subsubsection*{Modules of the form $I(n,8)$}

Defect: $\partial I(n,8) = 1$, for $n\ge 0$.

\begin{fleqn}
\begin{align*}
I(0,8):\ & - \\
I(1,8):\ & \bigl(I(0,3),R_{0}^{1}(1)\bigr),\ \bigl(I(0,7),R_{\infty}^{1}(1)\bigr)\\
I(2,8):\ & \bigl(I(0,2),R_{0}^{5}(2)\bigr),\ \bigl(I(0,6),R_{\infty}^{3}(2)\bigr),\ \bigl(I(1,3),R_{0}^{5}(1)\bigr),\ \bigl(I(1,7),R_{\infty}^{3}(1)\bigr)\\
I(3,8):\ & \bigl(I(0,1),R_{0}^{4}(3)\bigr),\ \bigl(I(0,4),R_{0}^{4}(4)\bigr),\ \bigl(I(1,2),R_{0}^{4}(2)\bigr),\ \bigl(I(1,6),R_{\infty}^{2}(2)\bigr),\ \bigl(I(2,3),R_{0}^{4}(1)\bigr)\\
 & \bigl(I(2,7),R_{\infty}^{2}(1)\bigr),\ \bigl(2I(0,5),P(1,2)\bigr)\\
I(n,8):\ & \bigl(I(n-3,1),R_{0}^{(-n+6)\bmod 5+1}(3)\bigr),\ \bigl(I(n-3,4),R_{0}^{(-n+6)\bmod 5+1}(4)\bigr),\ \bigl(I(n-2,2),R_{0}^{(-n+6)\bmod 5+1}(2)\bigr)\\
 & \bigl(I(n-2,6),R_{\infty}^{(-n+4)\bmod 3+1}(2)\bigr),\ \bigl(I(n-1,3),R_{0}^{(-n+6)\bmod 5+1}(1)\bigr),\ \bigl(I(n-1,7),R_{\infty}^{(-n+4)\bmod 3+1}(1)\bigr)\\
 & \bigl((v+1)I,vP\bigr),\ n>3\\
\end{align*}
\end{fleqn}
\subsubsection{Schofield pairs associated to regular exceptional modules}

\subsubsection*{The non-homogeneous tube $\mathcal{T}_{0}^{\Delta(\widetilde{\mathbb{A}}_{3,5})}$}

\begin{figure}[ht]

\begin{center}
\captionof{figure}{\vspace*{-10pt}$\mathcal{T}_{0}^{\Delta(\widetilde{\mathbb{A}}_{3,5})}$}
\begin{scaletikzpicturetowidth}{1.\textwidth}
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	\draw[blue,->] (RN_1_4_5__Rationals) -- (RN_1_4_6__Rationals);
	\draw[gray,dashed,->] (RN_1_4_5__Rationals) -- (RN_1_5_5__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[red,->] (RN_1_4_6__Rationals) -- (RN_1_5_5__Rationals);
	\draw[gray,dashed,->] (RN_1_4_6__Rationals) -- (RN_1_5_6__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[red,->] (RN_1_5_5__Rationals) -- (RN_1_1_4__Rationals);
	\draw[blue,->] (RN_1_5_5__Rationals) -- (RN_1_5_6__Rationals);
	\draw[gray,dashed,->] (RN_1_5_5__Rationals) -- (RN_1_1_5__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[red,->] (RN_1_5_6__Rationals) -- (RN_1_1_5__Rationals);
	\draw[gray,dashed,->] (RN_1_5_6__Rationals) -- (RN_1_1_6__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[red,->] (RN_1_1_5__Rationals) -- (RN_1_2_4__Rationals);
	\draw[blue,->] (RN_1_1_5__Rationals) -- (RN_1_1_6__Rationals);
	\draw[gray,dashed,->] (RN_1_1_5__Rationals) -- (RN_1_2_5__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[red,->] (RN_1_1_6__Rationals) -- (RN_1_2_5__Rationals);
	\draw[gray,dashed,->] (RN_1_1_6__Rationals) -- (RN_1_2_6__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[red,->] (RN_1_2_5__Rationals) -- (RN_1_3_4__Rationals);
	\draw[blue,->] (RN_1_2_5__Rationals) -- (RN_1_2_6__Rationals);
	\draw[gray,dashed,->] (RN_1_2_5__Rationals) -- (RN_1_3_5__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[red,->] (RN_1_2_6__Rationals) -- (RN_1_3_5__Rationals);
	\draw[gray,dashed,->] (RN_1_2_6__Rationals) -- (RN_1_3_6__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[red,->] (RN_1_3_5__Rationals) -- (RN_1_4_4__Rationals);
	\draw[gray,dashed,->] (RN_1_3_5__Rationals) -- (RN_1_4_5__Rationals_repeated) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[blue,->] (RN_1_3_5__Rationals) -- (RN_1_3_6__Rationals);
	\draw[red,->] (RN_1_3_6__Rationals) -- (RN_1_4_5__Rationals_repeated);
	\draw[gray,dashed,->,out=190,in=-10] (RN_1_3_6__Rationals) to node[midway,sloped,above]{{\tiny $\tau^{-1}$}} (RN_1_4_6__Rationals);
	\draw[dashed,-] (RN_1_1_1__Rationals) -- (RN_1_5_3__Rationals);
	\draw[dashed,-] (RN_1_1_1__Rationals_repeated) -- (RN_1_5_3__Rationals_repeated);
	\draw[dashed,-] (RN_1_5_3__Rationals) -- (RN_1_4_5__Rationals);
	\draw[dashed,-] (RN_1_5_3__Rationals_repeated) -- (RN_1_4_5__Rationals_repeated);
	\node (LeftTopPhantom_1) at (4,10) {};
	\draw[dashed,-] (RN_1_4_5__Rationals) -- (LeftTopPhantom_1);
	\node (RightTopPhantom_1) at (24,10) {};
	\draw[dashed,-] (RN_1_4_5__Rationals_repeated) -- (RightTopPhantom_1);
\end{tikzpicture}
\end{scaletikzpicturetowidth}
\end{center}

\end{figure}
\begin{fleqn}
\begin{align*}
R_{0}^{1}(1):\ & - \\
R_{0}^{1}(2):\ & \bigl(R_{0}^{2}(1),R_{0}^{1}(1)\bigr),\ \bigl(I(1,3),P(0,1)\bigr),\ \bigl(I(1,8),P(0,2)\bigr),\ \bigl(I(0,7),P(0,3)\bigr)\\
R_{0}^{2}(1):\ & \bigl(I(0,2),P(0,1)\bigr),\ \bigl(I(0,3),P(0,2)\bigr),\ \bigl(I(0,8),P(0,3)\bigr)\\
R_{0}^{2}(2):\ & \bigl(R_{0}^{3}(1),R_{0}^{2}(1)\bigr),\ \bigl(I(0,2),P(0,4)\bigr),\ \bigl(I(0,3),P(1,1)\bigr),\ \bigl(I(0,8),P(1,2)\bigr)\\
R_{0}^{3}(1):\ & - \\
R_{0}^{3}(2):\ & \bigl(R_{0}^{4}(1),R_{0}^{3}(1)\bigr)\\
R_{0}^{4}(1):\ & - \\
R_{0}^{4}(2):\ & \bigl(R_{0}^{5}(1),R_{0}^{4}(1)\bigr)\\
R_{0}^{5}(1):\ & - \\
R_{0}^{5}(2):\ & \bigl(R_{0}^{1}(1),R_{0}^{5}(1)\bigr)\\
R_{0}^{5}(3):\ & \bigl(R_{0}^{1}(2),R_{0}^{5}(1)\bigr),\ \bigl(R_{0}^{2}(1),R_{0}^{5}(2)\bigr),\ \bigl(I(2,8),P(0,1)\bigr),\ \bigl(I(1,7),P(0,2)\bigr),\ \bigl(I(0,6),P(0,3)\bigr)\\
R_{0}^{5}(4):\ & \bigl(R_{0}^{1}(3),R_{0}^{5}(1)\bigr),\ \bigl(R_{0}^{2}(2),R_{0}^{5}(2)\bigr),\ \bigl(R_{0}^{3}(1),R_{0}^{5}(3)\bigr),\ \bigl(I(2,8),P(0,4)\bigr),\ \bigl(I(1,7),P(1,1)\bigr)\\
 & \bigl(I(0,6),P(1,2)\bigr)\\
R_{0}^{1}(3):\ & \bigl(R_{0}^{2}(2),R_{0}^{1}(1)\bigr),\ \bigl(R_{0}^{3}(1),R_{0}^{1}(2)\bigr),\ \bigl(I(1,3),P(0,4)\bigr),\ \bigl(I(1,8),P(1,1)\bigr),\ \bigl(I(0,7),P(1,2)\bigr)\\
R_{0}^{1}(4):\ & \bigl(R_{0}^{2}(3),R_{0}^{1}(1)\bigr),\ \bigl(R_{0}^{3}(2),R_{0}^{1}(2)\bigr),\ \bigl(R_{0}^{4}(1),R_{0}^{1}(3)\bigr),\ \bigl(I(1,3),P(0,5)\bigr),\ \bigl(I(1,8),P(1,4)\bigr)\\
 & \bigl(I(0,7),P(2,1)\bigr)\\
R_{0}^{2}(3):\ & \bigl(R_{0}^{3}(2),R_{0}^{2}(1)\bigr),\ \bigl(R_{0}^{4}(1),R_{0}^{2}(2)\bigr),\ \bigl(I(0,2),P(0,5)\bigr),\ \bigl(I(0,3),P(1,4)\bigr),\ \bigl(I(0,8),P(2,1)\bigr)\\
R_{0}^{2}(4):\ & \bigl(R_{0}^{3}(3),R_{0}^{2}(1)\bigr),\ \bigl(R_{0}^{4}(2),R_{0}^{2}(2)\bigr),\ \bigl(R_{0}^{5}(1),R_{0}^{2}(3)\bigr),\ \bigl(I(0,2),P(0,6)\bigr),\ \bigl(I(0,3),P(1,5)\bigr)\\
 & \bigl(I(0,8),P(2,4)\bigr)\\
R_{0}^{3}(3):\ & \bigl(R_{0}^{4}(2),R_{0}^{3}(1)\bigr),\ \bigl(R_{0}^{5}(1),R_{0}^{3}(2)\bigr)\\
R_{0}^{3}(4):\ & \bigl(R_{0}^{4}(3),R_{0}^{3}(1)\bigr),\ \bigl(R_{0}^{5}(2),R_{0}^{3}(2)\bigr),\ \bigl(R_{0}^{1}(1),R_{0}^{3}(3)\bigr)\\
R_{0}^{4}(3):\ & \bigl(R_{0}^{5}(2),R_{0}^{4}(1)\bigr),\ \bigl(R_{0}^{1}(1),R_{0}^{4}(2)\bigr)\\
R_{0}^{4}(4):\ & \bigl(R_{0}^{5}(3),R_{0}^{4}(1)\bigr),\ \bigl(R_{0}^{1}(2),R_{0}^{4}(2)\bigr),\ \bigl(R_{0}^{2}(1),R_{0}^{4}(3)\bigr),\ \bigl(I(2,7),P(0,1)\bigr),\ \bigl(I(1,6),P(0,2)\bigr)\\
 & \bigl(I(0,5),P(0,3)\bigr)\\
\end{align*}
\end{fleqn}
\subsubsection*{The non-homogeneous tube $\mathcal{T}_{\infty}^{\Delta(\widetilde{\mathbb{A}}_{3,5})}$}

\begin{figure}[ht]

\begin{center}
\captionof{figure}{\vspace*{-10pt}$\mathcal{T}_{\infty}^{\Delta(\widetilde{\mathbb{A}}_{3,5})}$}
\begin{scaletikzpicturetowidth}{0.80000000000000004\textwidth}
\begin{tikzpicture}[scale=\tikzscale]
	\node[seltwoAAAlAAAAA, label={[shift={(0.0,-0.45)}, brown]\tiny $R_{\infty}^{1}(1)$}] (RN_2_1_1__Rationals) at (4,0) {$\begin{smallmatrix}&0&1\\0&&&&&0\\&0&0&0&0\end{smallmatrix}$};
	\node[seltwoAAAlAAAAA, label={[shift={(0.0,-0.45)}, brown]\tiny $R_{\infty}^{2}(1)$}] (RN_2_2_1__Rationals) at (8,0) {$\begin{smallmatrix}&0&0\\1&&&&&1\\&1&1&1&1\end{smallmatrix}$};
	\node[seltwoAAAlAAAAA, label={[shift={(0.0,-0.45)}, brown]\tiny $R_{\infty}^{3}(1)$}] (RN_2_3_1__Rationals) at (12,0) {$\begin{smallmatrix}&1&0\\0&&&&&0\\&0&0&0&0\end{smallmatrix}$};
	\node[seltwoAAAlAAAAA, label={[shift={(0.0,-0.45)}, brown]\tiny $R_{\infty}^{1}(1)$}] (RN_2_1_1__Rationals_repeated) at (16,0) {$\begin{smallmatrix}&0&1\\0&&&&&0\\&0&0&0&0\end{smallmatrix}$};
	\node[seltwoAAAlAAAAA, label={[shift={(0.0,-0.45)}, brown]\tiny $R_{\infty}^{1}(2)$}] (RN_2_1_2__Rationals) at (6,2) {$\begin{smallmatrix}&0&1\\1&&&&&1\\&1&1&1&1\end{smallmatrix}$};
	\node[seltwoAAAlAAAAA, label={[shift={(0.0,-0.45)}, brown]\tiny $R_{\infty}^{2}(2)$}] (RN_2_2_2__Rationals) at (10,2) {$\begin{smallmatrix}&1&0\\1&&&&&1\\&1&1&1&1\end{smallmatrix}$};
	\node[seltwoAAAlAAAAA, label={[shift={(0.0,-0.45)}, brown]\tiny $R_{\infty}^{3}(2)$}] (RN_2_3_2__Rationals) at (14,2) {$\begin{smallmatrix}&1&1\\0&&&&&0\\&0&0&0&0\end{smallmatrix}$};
	\node[noselAAAlAAAAA, label={[shift={(0.0,-0.45)}, brown]\tiny $R_{\infty}^{3}(3)$}] (RN_2_3_3__Rationals) at (4,4) {$\begin{smallmatrix}&1&1\\1&&&&&1\\&1&1&1&1\end{smallmatrix}$};
	\node[noselAAAlAAAAA, label={[shift={(0.0,-0.45)}, brown]\tiny $R_{\infty}^{1}(3)$}] (RN_2_1_3__Rationals) at (8,4) {$\begin{smallmatrix}&1&1\\1&&&&&1\\&1&1&1&1\end{smallmatrix}$};
	\node[noselAAAlAAAAA, label={[shift={(0.0,-0.45)}, brown]\tiny $R_{\infty}^{2}(3)$}] (RN_2_2_3__Rationals) at (12,4) {$\begin{smallmatrix}&1&1\\1&&&&&1\\&1&1&1&1\end{smallmatrix}$};
	\node[noselAAAlAAAAA, label={[shift={(0.0,-0.45)}, brown]\tiny $R_{\infty}^{3}(3)$}] (RN_2_3_3__Rationals_repeated) at (16,4) {$\begin{smallmatrix}&1&1\\1&&&&&1\\&1&1&1&1\end{smallmatrix}$};
	\node[noselAAAlAAAAA, label={[shift={(0.0,-0.45)}, brown]\tiny $R_{\infty}^{3}(4)$}] (RN_2_3_4__Rationals) at (6,6) {$\begin{smallmatrix}&2&1\\1&&&&&1\\&1&1&1&1\end{smallmatrix}$};
	\node[noselAAAlAAAAA, label={[shift={(0.0,-0.45)}, brown]\tiny $R_{\infty}^{1}(4)$}] (RN_2_1_4__Rationals) at (10,6) {$\begin{smallmatrix}&1&2\\1&&&&&1\\&1&1&1&1\end{smallmatrix}$};
	\node[noselAAAlAAAAA, label={[shift={(0.0,-0.45)}, brown]\tiny $R_{\infty}^{2}(4)$}] (RN_2_2_4__Rationals) at (14,6) {$\begin{smallmatrix}&1&1\\2&&&&&2\\&2&2&2&2\end{smallmatrix}$};
	\draw[blue,->] (RN_2_1_1__Rationals) -- (RN_2_1_2__Rationals);
	\draw[gray,dashed,->] (RN_2_1_1__Rationals) -- (RN_2_2_1__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[red,->] (RN_2_1_2__Rationals) -- (RN_2_2_1__Rationals);
	\draw[gray,dashed,->] (RN_2_1_2__Rationals) -- (RN_2_2_2__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[blue,->] (RN_2_1_2__Rationals) -- (RN_2_1_3__Rationals);
	\draw[blue,->] (RN_2_2_1__Rationals) -- (RN_2_2_2__Rationals);
	\draw[gray,dashed,->] (RN_2_2_1__Rationals) -- (RN_2_3_1__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[red,->] (RN_2_2_2__Rationals) -- (RN_2_3_1__Rationals);
	\draw[gray,dashed,->] (RN_2_2_2__Rationals) -- (RN_2_3_2__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[blue,->] (RN_2_2_2__Rationals) -- (RN_2_2_3__Rationals);
	\draw[gray,dashed,->] (RN_2_3_1__Rationals) -- (RN_2_1_1__Rationals_repeated) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[blue,->] (RN_2_3_1__Rationals) -- (RN_2_3_2__Rationals);
	\draw[red,->] (RN_2_3_2__Rationals) -- (RN_2_1_1__Rationals_repeated);
	\draw[gray,dashed,->,out=200,in=-20] (RN_2_3_2__Rationals) to node[midway,sloped,above]{{\tiny $\tau^{-1}$}} (RN_2_1_2__Rationals);
	\draw[blue,->] (RN_2_3_2__Rationals) -- (RN_2_3_3__Rationals_repeated);
	\draw[red,->] (RN_2_3_3__Rationals) -- (RN_2_1_2__Rationals);
	\draw[blue,->] (RN_2_3_3__Rationals) -- (RN_2_3_4__Rationals);
	\draw[gray,dashed,->] (RN_2_3_3__Rationals) -- (RN_2_1_3__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[red,->] (RN_2_3_4__Rationals) -- (RN_2_1_3__Rationals);
	\draw[gray,dashed,->] (RN_2_3_4__Rationals) -- (RN_2_1_4__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[red,->] (RN_2_1_3__Rationals) -- (RN_2_2_2__Rationals);
	\draw[blue,->] (RN_2_1_3__Rationals) -- (RN_2_1_4__Rationals);
	\draw[gray,dashed,->] (RN_2_1_3__Rationals) -- (RN_2_2_3__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[red,->] (RN_2_1_4__Rationals) -- (RN_2_2_3__Rationals);
	\draw[gray,dashed,->] (RN_2_1_4__Rationals) -- (RN_2_2_4__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[red,->] (RN_2_2_3__Rationals) -- (RN_2_3_2__Rationals);
	\draw[gray,dashed,->] (RN_2_2_3__Rationals) -- (RN_2_3_3__Rationals_repeated) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[blue,->] (RN_2_2_3__Rationals) -- (RN_2_2_4__Rationals);
	\draw[red,->] (RN_2_2_4__Rationals) -- (RN_2_3_3__Rationals_repeated);
	\draw[gray,dashed,->,out=200,in=-20] (RN_2_2_4__Rationals) to node[midway,sloped,above]{{\tiny $\tau^{-1}$}} (RN_2_3_4__Rationals);
	\draw[dashed,-] (RN_2_1_1__Rationals) -- (RN_2_3_3__Rationals);
	\draw[dashed,-] (RN_2_1_1__Rationals_repeated) -- (RN_2_3_3__Rationals_repeated);
	\node (LeftTopPhantom_2) at (4,6) {};
	\draw[dashed,-] (RN_2_3_3__Rationals) -- (LeftTopPhantom_2);
	\node (RightTopPhantom_2) at (16,6) {};
	\draw[dashed,-] (RN_2_3_3__Rationals_repeated) -- (RightTopPhantom_2);
\end{tikzpicture}
\end{scaletikzpicturetowidth}
\end{center}

\end{figure}
\begin{fleqn}
\begin{align*}
R_{\infty}^{1}(1):\ & - \\
R_{\infty}^{1}(2):\ & \bigl(R_{\infty}^{2}(1),R_{\infty}^{1}(1)\bigr),\ \bigl(I(1,5),P(0,1)\bigr),\ \bigl(I(1,6),P(0,4)\bigr),\ \bigl(I(1,7),P(0,5)\bigr),\ \bigl(I(1,8),P(0,6)\bigr)\\
 & \bigl(I(0,3),P(0,7)\bigr)\\
R_{\infty}^{2}(1):\ & \bigl(I(0,4),P(0,1)\bigr),\ \bigl(I(0,5),P(0,4)\bigr),\ \bigl(I(0,6),P(0,5)\bigr),\ \bigl(I(0,7),P(0,6)\bigr),\ \bigl(I(0,8),P(0,7)\bigr)\\
R_{\infty}^{2}(2):\ & \bigl(R_{\infty}^{3}(1),R_{\infty}^{2}(1)\bigr),\ \bigl(I(0,4),P(0,2)\bigr),\ \bigl(I(0,5),P(1,1)\bigr),\ \bigl(I(0,6),P(1,4)\bigr),\ \bigl(I(0,7),P(1,5)\bigr)\\
 & \bigl(I(0,8),P(1,6)\bigr)\\
R_{\infty}^{3}(1):\ & - \\
R_{\infty}^{3}(2):\ & \bigl(R_{\infty}^{1}(1),R_{\infty}^{3}(1)\bigr)\\
\end{align*}
\end{fleqn}
\clearpage

\subsection[Schofield pairs for the quiver $\Delta(\widetilde{\mathbb{D}}_4)$]{Schofield pairs for the quiver $\Delta(\widetilde{\mathbb{D}}_4)$ -- {\usebox{\boxDDDDD}}}
\[
\vcenter{\hbox{\xymatrix{2 &   & 3\ar[dl]\\
  & 5\ar[dl]\ar[ul] &  \\
1 &   & 4\ar[ul]}}}\qquad C_{\Delta(\widetilde{\mathbb{D}}_{4})} = \begin{bmatrix}1 & 0 & 1 & 1 & 1\\
0 & 1 & 1 & 1 & 1\\
0 & 0 & 1 & 0 & 0\\
0 & 0 & 0 & 1 & 0\\
0 & 0 & 1 & 1 & 1\end{bmatrix}\quad\Phi_{\Delta(\widetilde{\mathbb{D}}_{4})} = \begin{bmatrix}-1 & 0 & 0 & 0 & 1\\
0 & -1 & 0 & 0 & 1\\
-1 & -1 & 0 & 1 & 1\\
-1 & -1 & 1 & 0 & 1\\
-1 & -1 & 1 & 1 & 1\end{bmatrix}
\]
\subsubsection{Schofield pairs associated to preprojective exceptional modules}

\begin{figure}[ht]

\begin{center}
\begin{scaletikzpicturetowidth}{\textwidth}
\begin{tikzpicture}[scale=\tikzscale]
	\node[noselDDDDD, label={[shift={(0.0,-0.45)}, brown]\tiny $P(0,1)$}] (P_0_1__Rationals) at (0,0) {$\begin{smallmatrix}0&&0\\&0\\1&&0\end{smallmatrix}$};
	\node[noselDDDDD, label={[shift={(0.0,-0.45)}, brown]\tiny $P(0,2)$}] (P_0_2__Rationals) at (0,1) {$\begin{smallmatrix}1&&0\\&0\\0&&0\end{smallmatrix}$};
	\node[noselDDDDD, label={[shift={(0.0,-0.45)}, brown]\tiny $P(0,3)$}] (P_0_3__Rationals) at (4,4) {$\begin{smallmatrix}1&&1\\&1\\1&&0\end{smallmatrix}$};
	\node[noselDDDDD, label={[shift={(0.0,-0.45)}, brown]\tiny $P(0,4)$}] (P_0_4__Rationals) at (4,3) {$\begin{smallmatrix}1&&0\\&1\\1&&1\end{smallmatrix}$};
	\node[noselDDDDD, label={[shift={(0.0,-0.45)}, brown]\tiny $P(0,5)$}] (P_0_5__Rationals) at (2,2) {$\begin{smallmatrix}1&&0\\&1\\1&&0\end{smallmatrix}$};
	\node[noselDDDDD, label={[shift={(0.0,-0.45)}, brown]\tiny $P(1,1)$}] (P_1_1__Rationals) at (4,0) {$\begin{smallmatrix}1&&0\\&1\\0&&0\end{smallmatrix}$};
	\node[noselDDDDD, label={[shift={(0.0,-0.45)}, brown]\tiny $P(1,2)$}] (P_1_2__Rationals) at (4,1) {$\begin{smallmatrix}0&&0\\&1\\1&&0\end{smallmatrix}$};
	\node[noselDDDDD, label={[shift={(0.0,-0.45)}, brown]\tiny $P(1,3)$}] (P_1_3__Rationals) at (8,4) {$\begin{smallmatrix}1&&0\\&2\\1&&1\end{smallmatrix}$};
	\node[noselDDDDD, label={[shift={(0.0,-0.45)}, brown]\tiny $P(1,4)$}] (P_1_4__Rationals) at (8,3) {$\begin{smallmatrix}1&&1\\&2\\1&&0\end{smallmatrix}$};
	\node[seloneDDDDD, label={[shift={(0.0,-0.45)}, brown]\tiny $P(1,5)$}] (P_1_5__Rationals) at (6,2) {$\begin{smallmatrix}2&&1\\&3\\2&&1\end{smallmatrix}$};
	\node[seloneDDDDD, label={[shift={(0.0,-0.45)}, brown]\tiny $P(2,1)$}] (P_2_1__Rationals) at (8,0) {$\begin{smallmatrix}1&&1\\&2\\2&&1\end{smallmatrix}$};
	\node[seloneDDDDD, label={[shift={(0.0,-0.45)}, brown]\tiny $P(2,2)$}] (P_2_2__Rationals) at (8,1) {$\begin{smallmatrix}2&&1\\&2\\1&&1\end{smallmatrix}$};
	\node[seloneDDDDD, label={[shift={(0.0,-0.45)}, brown]\tiny $P(2,3)$}] (P_2_3__Rationals) at (12,4) {$\begin{smallmatrix}2&&2\\&3\\2&&1\end{smallmatrix}$};
	\node[seloneDDDDD, label={[shift={(0.0,-0.45)}, brown]\tiny $P(2,4)$}] (P_2_4__Rationals) at (12,3) {$\begin{smallmatrix}2&&1\\&3\\2&&2\end{smallmatrix}$};
	\node[noselDDDDD, label={[shift={(0.0,-0.45)}, brown]\tiny $P(2,5)$}] (P_2_5__Rationals) at (10,2) {$\begin{smallmatrix}3&&2\\&5\\3&&2\end{smallmatrix}$};
	\node[noselDDDDD, label={[shift={(0.0,-0.45)}, brown]\tiny $P(3,1)$}] (P_3_1__Rationals) at (12,0) {$\begin{smallmatrix}2&&1\\&3\\1&&1\end{smallmatrix}$};
	\node[noselDDDDD, label={[shift={(0.0,-0.45)}, brown]\tiny $P(3,2)$}] (P_3_2__Rationals) at (12,1) {$\begin{smallmatrix}1&&1\\&3\\2&&1\end{smallmatrix}$};
	\node[noselDDDDD, label={[shift={(0.0,-0.45)}, brown]\tiny $P(3,3)$}] (P_3_3__Rationals) at (16,4) {$\begin{smallmatrix}2&&1\\&4\\2&&2\end{smallmatrix}$};
	\node[noselDDDDD, label={[shift={(0.0,-0.45)}, brown]\tiny $P(3,4)$}] (P_3_4__Rationals) at (16,3) {$\begin{smallmatrix}2&&2\\&4\\2&&1\end{smallmatrix}$};
	\node[noselDDDDD, label={[shift={(0.0,-0.45)}, brown]\tiny $P(3,5)$}] (P_3_5__Rationals) at (14,2) {$\begin{smallmatrix}4&&3\\&7\\4&&3\end{smallmatrix}$};
	\node at (16,2) {$\dots$};
	\draw[blue,->] (P_0_5__Rationals) -- (P_0_3__Rationals);
	\draw[blue,->] (P_0_5__Rationals) -- (P_0_4__Rationals);
	\draw[blue,->] (P_0_1__Rationals) -- (P_0_5__Rationals);
	\draw[blue,->] (P_0_2__Rationals) -- (P_0_5__Rationals);
	\draw[gray,dashed,->] (P_0_1__Rationals) -- (P_1_1__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[red,->] (P_0_5__Rationals) -- (P_1_1__Rationals);
	\draw[gray,dashed,->] (P_0_2__Rationals) -- (P_1_2__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[red,->] (P_0_5__Rationals) -- (P_1_2__Rationals);
	\draw[gray,dashed,->] (P_0_3__Rationals) -- (P_1_3__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[red,->] (P_1_5__Rationals) -- (P_1_3__Rationals);
	\draw[gray,dashed,->] (P_0_4__Rationals) -- (P_1_4__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[red,->] (P_1_5__Rationals) -- (P_1_4__Rationals);
	\draw[blue,->] (P_0_3__Rationals) -- (P_1_5__Rationals);
	\draw[blue,->] (P_0_4__Rationals) -- (P_1_5__Rationals);
	\draw[gray,dashed,->] (P_0_5__Rationals) -- (P_1_5__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[blue,->] (P_1_1__Rationals) -- (P_1_5__Rationals);
	\draw[blue,->] (P_1_2__Rationals) -- (P_1_5__Rationals);
	\draw[gray,dashed,->] (P_1_1__Rationals) -- (P_2_1__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[red,->] (P_1_5__Rationals) -- (P_2_1__Rationals);
	\draw[gray,dashed,->] (P_1_2__Rationals) -- (P_2_2__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[red,->] (P_1_5__Rationals) -- (P_2_2__Rationals);
	\draw[gray,dashed,->] (P_1_3__Rationals) -- (P_2_3__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[red,->] (P_2_5__Rationals) -- (P_2_3__Rationals);
	\draw[gray,dashed,->] (P_1_4__Rationals) -- (P_2_4__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[red,->] (P_2_5__Rationals) -- (P_2_4__Rationals);
	\draw[blue,->] (P_1_3__Rationals) -- (P_2_5__Rationals);
	\draw[blue,->] (P_1_4__Rationals) -- (P_2_5__Rationals);
	\draw[gray,dashed,->] (P_1_5__Rationals) -- (P_2_5__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[blue,->] (P_2_1__Rationals) -- (P_2_5__Rationals);
	\draw[blue,->] (P_2_2__Rationals) -- (P_2_5__Rationals);
	\draw[gray,dashed,->] (P_2_1__Rationals) -- (P_3_1__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[red,->] (P_2_5__Rationals) -- (P_3_1__Rationals);
	\draw[gray,dashed,->] (P_2_2__Rationals) -- (P_3_2__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[red,->] (P_2_5__Rationals) -- (P_3_2__Rationals);
	\draw[gray,dashed,->] (P_2_3__Rationals) -- (P_3_3__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[red,->] (P_3_5__Rationals) -- (P_3_3__Rationals);
	\draw[gray,dashed,->] (P_2_4__Rationals) -- (P_3_4__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[red,->] (P_3_5__Rationals) -- (P_3_4__Rationals);
	\draw[blue,->] (P_2_3__Rationals) -- (P_3_5__Rationals);
	\draw[blue,->] (P_2_4__Rationals) -- (P_3_5__Rationals);
	\draw[gray,dashed,->] (P_2_5__Rationals) -- (P_3_5__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[blue,->] (P_3_1__Rationals) -- (P_3_5__Rationals);
	\draw[blue,->] (P_3_2__Rationals) -- (P_3_5__Rationals);
\end{tikzpicture}
\end{scaletikzpicturetowidth}
\end{center}

\end{figure}
\medskip{}
\subsubsection*{Modules of the form $P(n,1)$}

Defect: $\partial P(n,1) = -1$, for $n\ge 0$.

\begin{fleqn}
\begin{align*}
P(0,1):\ & - \\
P(1,1):\ & \bigl(R_{1}^{1}(1),P(0,2)\bigr)\\
P(2,1):\ & \bigl(R_{0}^{2}(1),P(0,3)\bigr),\ \bigl(R_{\infty}^{2}(1),P(0,4)\bigr),\ \bigl(R_{1}^{2}(1),P(1,2)\bigr),\ \bigl(I(1,1),2P(0,1)\bigr)\\
P(n,1):\ & \bigl(R_{0}^{(n-1)\bmod 2+1}(1),P(n-2,3)\bigr),\ \bigl(R_{\infty}^{(n-1)\bmod 2+1}(1),P(n-2,4)\bigr),\ \bigl(R_{1}^{(n-1)\bmod 2+1}(1),P(n-1,2)\bigr)\\
 & \bigl(uI,(u+1)P\bigr),\ n>2\\
\end{align*}
\end{fleqn}
\subsubsection*{Modules of the form $P(n,2)$}

Defect: $\partial P(n,2) = -1$, for $n\ge 0$.

\begin{fleqn}
\begin{align*}
P(0,2):\ & - \\
P(1,2):\ & \bigl(R_{1}^{1}(1),P(0,1)\bigr)\\
P(2,2):\ & \bigl(R_{\infty}^{1}(1),P(0,3)\bigr),\ \bigl(R_{0}^{1}(1),P(0,4)\bigr),\ \bigl(R_{1}^{2}(1),P(1,1)\bigr),\ \bigl(I(1,2),2P(0,2)\bigr)\\
P(n,2):\ & \bigl(R_{\infty}^{(n-2)\bmod 2+1}(1),P(n-2,3)\bigr),\ \bigl(R_{0}^{(n-2)\bmod 2+1}(1),P(n-2,4)\bigr),\ \bigl(R_{1}^{(n-1)\bmod 2+1}(1),P(n-1,1)\bigr)\\
 & \bigl(uI,(u+1)P\bigr),\ n>2\\
\end{align*}
\end{fleqn}
\subsubsection*{Modules of the form $P(n,3)$}

Defect: $\partial P(n,3) = -1$, for $n\ge 0$.

\begin{fleqn}
\begin{align*}
P(0,3):\ & \bigl(R_{0}^{1}(1),P(0,1)\bigr),\ \bigl(R_{\infty}^{2}(1),P(0,2)\bigr),\ \bigl(I(0,3),P(0,5)\bigr)\\
P(1,3):\ & \bigl(R_{1}^{1}(1),P(0,4)\bigr),\ \bigl(R_{0}^{2}(1),P(1,1)\bigr),\ \bigl(R_{\infty}^{1}(1),P(1,2)\bigr)\\
P(2,3):\ & \bigl(R_{1}^{2}(1),P(1,4)\bigr),\ \bigl(R_{0}^{1}(1),P(2,1)\bigr),\ \bigl(R_{\infty}^{2}(1),P(2,2)\bigr),\ \bigl(I(1,3),2P(0,3)\bigr)\\
P(n,3):\ & \bigl(R_{1}^{(n-1)\bmod 2+1}(1),P(n-1,4)\bigr),\ \bigl(R_{0}^{(n-2)\bmod 2+1}(1),P(n,1)\bigr),\ \bigl(R_{\infty}^{(n-1)\bmod 2+1}(1),P(n,2)\bigr)\\
 & \bigl(uI,(u+1)P\bigr),\ n>2\\
\end{align*}
\end{fleqn}
\subsubsection*{Modules of the form $P(n,4)$}

Defect: $\partial P(n,4) = -1$, for $n\ge 0$.

\begin{fleqn}
\begin{align*}
P(0,4):\ & \bigl(R_{\infty}^{1}(1),P(0,1)\bigr),\ \bigl(R_{0}^{2}(1),P(0,2)\bigr),\ \bigl(I(0,4),P(0,5)\bigr)\\
P(1,4):\ & \bigl(R_{1}^{1}(1),P(0,3)\bigr),\ \bigl(R_{\infty}^{2}(1),P(1,1)\bigr),\ \bigl(R_{0}^{1}(1),P(1,2)\bigr)\\
P(2,4):\ & \bigl(R_{1}^{2}(1),P(1,3)\bigr),\ \bigl(R_{\infty}^{1}(1),P(2,1)\bigr),\ \bigl(R_{0}^{2}(1),P(2,2)\bigr),\ \bigl(I(1,4),2P(0,4)\bigr)\\
P(n,4):\ & \bigl(R_{1}^{(n-1)\bmod 2+1}(1),P(n-1,3)\bigr),\ \bigl(R_{\infty}^{(n-2)\bmod 2+1}(1),P(n,1)\bigr),\ \bigl(R_{0}^{(n-1)\bmod 2+1}(1),P(n,2)\bigr)\\
 & \bigl(uI,(u+1)P\bigr),\ n>2\\
\end{align*}
\end{fleqn}
\subsubsection*{Modules of the form $P(n,5)$}

Defect: $\partial P(n,5) = -2$, for $n\ge 0$.

\begin{fleqn}
\begin{align*}
P(0,5):\ & \bigl(P(1,1),P(0,1)\bigr),\ \bigl(P(1,2),P(0,2)\bigr)\\
P(1,5):\ & \bigl(P(1,3),P(0,3)\bigr),\ \bigl(P(1,4),P(0,4)\bigr),\ \bigl(P(2,1),P(1,1)\bigr),\ \bigl(P(2,2),P(1,2)\bigr)\\
P(n,5):\ & \bigl(P(n,3),P(n-1,3)\bigr),\ \bigl(P(n,4),P(n-1,4)\bigr),\ \bigl(P(n+1,1),P(n,1)\bigr)\\
 & \bigl(P(n+1,2),P(n,2)\bigr),\ n>1\\
\end{align*}
\end{fleqn}
\subsubsection{Schofield pairs associated to preinjective exceptional modules}

\begin{figure}[ht]

\begin{center}
\begin{scaletikzpicturetowidth}{\textwidth}
\begin{tikzpicture}[scale=\tikzscale]
	\node[noselDDDDD, label={[shift={(0.0,-0.45)}, brown]\tiny $I(0,1)$}] (I_0_1__Rationals) at (0,0) {$\begin{smallmatrix}0&&1\\&1\\1&&1\end{smallmatrix}$};
	\node[noselDDDDD, label={[shift={(0.0,-0.45)}, brown]\tiny $I(0,2)$}] (I_0_2__Rationals) at (0,1) {$\begin{smallmatrix}1&&1\\&1\\0&&1\end{smallmatrix}$};
	\node[noselDDDDD, label={[shift={(0.0,-0.45)}, brown]\tiny $I(0,3)$}] (I_0_3__Rationals) at (4,4) {$\begin{smallmatrix}0&&1\\&0\\0&&0\end{smallmatrix}$};
	\node[noselDDDDD, label={[shift={(0.0,-0.45)}, brown]\tiny $I(0,4)$}] (I_0_4__Rationals) at (4,3) {$\begin{smallmatrix}0&&0\\&0\\0&&1\end{smallmatrix}$};
	\node[noselDDDDD, label={[shift={(0.0,-0.45)}, brown]\tiny $I(0,5)$}] (I_0_5__Rationals) at (2,2) {$\begin{smallmatrix}0&&1\\&1\\0&&1\end{smallmatrix}$};
	\node[noselDDDDD, label={[shift={(0.0,-0.45)}, brown]\tiny $I(1,1)$}] (I_1_1__Rationals) at (-4,0) {$\begin{smallmatrix}1&&1\\&2\\0&&1\end{smallmatrix}$};
	\node[noselDDDDD, label={[shift={(0.0,-0.45)}, brown]\tiny $I(1,2)$}] (I_1_2__Rationals) at (-4,1) {$\begin{smallmatrix}0&&1\\&2\\1&&1\end{smallmatrix}$};
	\node[noselDDDDD, label={[shift={(0.0,-0.45)}, brown]\tiny $I(1,3)$}] (I_1_3__Rationals) at (0,4) {$\begin{smallmatrix}0&&0\\&1\\0&&1\end{smallmatrix}$};
	\node[noselDDDDD, label={[shift={(0.0,-0.45)}, brown]\tiny $I(1,4)$}] (I_1_4__Rationals) at (0,3) {$\begin{smallmatrix}0&&1\\&1\\0&&0\end{smallmatrix}$};
	\node[seloneDDDDD, label={[shift={(0.0,-0.45)}, brown]\tiny $I(1,5)$}] (I_1_5__Rationals) at (-2,2) {$\begin{smallmatrix}1&&2\\&3\\1&&2\end{smallmatrix}$};
	\node[seloneDDDDD, label={[shift={(0.0,-0.45)}, brown]\tiny $I(2,1)$}] (I_2_1__Rationals) at (-8,0) {$\begin{smallmatrix}1&&2\\&3\\2&&2\end{smallmatrix}$};
	\node[seloneDDDDD, label={[shift={(0.0,-0.45)}, brown]\tiny $I(2,2)$}] (I_2_2__Rationals) at (-8,1) {$\begin{smallmatrix}2&&2\\&3\\1&&2\end{smallmatrix}$};
	\node[seloneDDDDD, label={[shift={(0.0,-0.45)}, brown]\tiny $I(2,3)$}] (I_2_3__Rationals) at (-4,4) {$\begin{smallmatrix}1&&2\\&2\\1&&1\end{smallmatrix}$};
	\node[seloneDDDDD, label={[shift={(0.0,-0.45)}, brown]\tiny $I(2,4)$}] (I_2_4__Rationals) at (-4,3) {$\begin{smallmatrix}1&&1\\&2\\1&&2\end{smallmatrix}$};
	\node[noselDDDDD, label={[shift={(0.0,-0.45)}, brown]\tiny $I(2,5)$}] (I_2_5__Rationals) at (-6,2) {$\begin{smallmatrix}2&&3\\&5\\2&&3\end{smallmatrix}$};
	\node[noselDDDDD, label={[shift={(0.0,-0.45)}, brown]\tiny $I(3,1)$}] (I_3_1__Rationals) at (-12,0) {$\begin{smallmatrix}2&&2\\&4\\1&&2\end{smallmatrix}$};
	\node[noselDDDDD, label={[shift={(0.0,-0.45)}, brown]\tiny $I(3,2)$}] (I_3_2__Rationals) at (-12,1) {$\begin{smallmatrix}1&&2\\&4\\2&&2\end{smallmatrix}$};
	\node[noselDDDDD, label={[shift={(0.0,-0.45)}, brown]\tiny $I(3,3)$}] (I_3_3__Rationals) at (-8,4) {$\begin{smallmatrix}1&&1\\&3\\1&&2\end{smallmatrix}$};
	\node[noselDDDDD, label={[shift={(0.0,-0.45)}, brown]\tiny $I(3,4)$}] (I_3_4__Rationals) at (-8,3) {$\begin{smallmatrix}1&&2\\&3\\1&&1\end{smallmatrix}$};
	\node[noselDDDDD, label={[shift={(0.0,-0.45)}, brown]\tiny $I(3,5)$}] (I_3_5__Rationals) at (-10,2) {$\begin{smallmatrix}3&&4\\&7\\3&&4\end{smallmatrix}$};
	\node at (-12,2) {$\dots$};
	\draw[red,->] (I_1_5__Rationals) -- (I_0_1__Rationals);
	\draw[red,->] (I_1_5__Rationals) -- (I_0_2__Rationals);
	\draw[red,->] (I_0_5__Rationals) -- (I_0_3__Rationals);
	\draw[red,->] (I_0_5__Rationals) -- (I_0_4__Rationals);
	\draw[red,->] (I_0_1__Rationals) -- (I_0_5__Rationals);
	\draw[red,->] (I_0_2__Rationals) -- (I_0_5__Rationals);
	\draw[blue,->] (I_1_3__Rationals) -- (I_0_5__Rationals);
	\draw[blue,->] (I_1_4__Rationals) -- (I_0_5__Rationals);
	\draw[gray,dashed,->] (I_0_1__Rationals) -- (I_1_1__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_2_5__Rationals) -- (I_1_1__Rationals);
	\draw[gray,dashed,->] (I_0_2__Rationals) -- (I_1_2__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_2_5__Rationals) -- (I_1_2__Rationals);
	\draw[gray,dashed,->] (I_0_3__Rationals) -- (I_1_3__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_1_5__Rationals) -- (I_1_3__Rationals);
	\draw[gray,dashed,->] (I_0_4__Rationals) -- (I_1_4__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_1_5__Rationals) -- (I_1_4__Rationals);
	\draw[gray,dashed,->] (I_0_5__Rationals) -- (I_1_5__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[blue,->] (I_1_1__Rationals) -- (I_1_5__Rationals);
	\draw[blue,->] (I_1_2__Rationals) -- (I_1_5__Rationals);
	\draw[blue,->] (I_2_3__Rationals) -- (I_1_5__Rationals);
	\draw[blue,->] (I_2_4__Rationals) -- (I_1_5__Rationals);
	\draw[gray,dashed,->] (I_1_1__Rationals) -- (I_2_1__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_3_5__Rationals) -- (I_2_1__Rationals);
	\draw[gray,dashed,->] (I_1_2__Rationals) -- (I_2_2__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_3_5__Rationals) -- (I_2_2__Rationals);
	\draw[gray,dashed,->] (I_1_3__Rationals) -- (I_2_3__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_2_5__Rationals) -- (I_2_3__Rationals);
	\draw[gray,dashed,->] (I_1_4__Rationals) -- (I_2_4__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_2_5__Rationals) -- (I_2_4__Rationals);
	\draw[gray,dashed,->] (I_1_5__Rationals) -- (I_2_5__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[blue,->] (I_2_1__Rationals) -- (I_2_5__Rationals);
	\draw[blue,->] (I_2_2__Rationals) -- (I_2_5__Rationals);
	\draw[blue,->] (I_3_3__Rationals) -- (I_2_5__Rationals);
	\draw[blue,->] (I_3_4__Rationals) -- (I_2_5__Rationals);
	\draw[gray,dashed,->] (I_2_1__Rationals) -- (I_3_1__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[gray,dashed,->] (I_2_2__Rationals) -- (I_3_2__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[gray,dashed,->] (I_2_3__Rationals) -- (I_3_3__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_3_5__Rationals) -- (I_3_3__Rationals);
	\draw[gray,dashed,->] (I_2_4__Rationals) -- (I_3_4__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_3_5__Rationals) -- (I_3_4__Rationals);
	\draw[gray,dashed,->] (I_2_5__Rationals) -- (I_3_5__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[blue,->] (I_3_1__Rationals) -- (I_3_5__Rationals);
	\draw[blue,->] (I_3_2__Rationals) -- (I_3_5__Rationals);
\end{tikzpicture}
\end{scaletikzpicturetowidth}
\end{center}

\end{figure}
\medskip{}
\subsubsection*{Modules of the form $I(n,1)$}

Defect: $\partial I(n,1) = 1$, for $n\ge 0$.

\begin{fleqn}
\begin{align*}
I(0,1):\ & \bigl(I(0,3),R_{0}^{2}(1)\bigr),\ \bigl(I(0,4),R_{\infty}^{2}(1)\bigr),\ \bigl(I(0,5),P(0,1)\bigr)\\
I(1,1):\ & \bigl(I(0,2),R_{1}^{1}(1)\bigr),\ \bigl(I(1,3),R_{0}^{1}(1)\bigr),\ \bigl(I(1,4),R_{\infty}^{1}(1)\bigr)\\
I(2,1):\ & \bigl(I(1,2),R_{1}^{2}(1)\bigr),\ \bigl(I(2,3),R_{0}^{2}(1)\bigr),\ \bigl(I(2,4),R_{\infty}^{2}(1)\bigr),\ \bigl(2I(0,1),P(1,1)\bigr)\\
I(n,1):\ & \bigl(I(n-1,2),R_{1}^{(-n+3)\bmod 2+1}(1)\bigr),\ \bigl(I(n,3),R_{0}^{(-n+3)\bmod 2+1}(1)\bigr),\ \bigl(I(n,4),R_{\infty}^{(-n+3)\bmod 2+1}(1)\bigr)\\
 & \bigl((v+1)I,vP\bigr),\ n>2\\
\end{align*}
\end{fleqn}
\subsubsection*{Modules of the form $I(n,2)$}

Defect: $\partial I(n,2) = 1$, for $n\ge 0$.

\begin{fleqn}
\begin{align*}
I(0,2):\ & \bigl(I(0,3),R_{\infty}^{1}(1)\bigr),\ \bigl(I(0,4),R_{0}^{1}(1)\bigr),\ \bigl(I(0,5),P(0,2)\bigr)\\
I(1,2):\ & \bigl(I(0,1),R_{1}^{1}(1)\bigr),\ \bigl(I(1,3),R_{\infty}^{2}(1)\bigr),\ \bigl(I(1,4),R_{0}^{2}(1)\bigr)\\
I(2,2):\ & \bigl(I(1,1),R_{1}^{2}(1)\bigr),\ \bigl(I(2,3),R_{\infty}^{1}(1)\bigr),\ \bigl(I(2,4),R_{0}^{1}(1)\bigr),\ \bigl(2I(0,2),P(1,2)\bigr)\\
I(n,2):\ & \bigl(I(n-1,1),R_{1}^{(-n+3)\bmod 2+1}(1)\bigr),\ \bigl(I(n,3),R_{\infty}^{(-n+2)\bmod 2+1}(1)\bigr),\ \bigl(I(n,4),R_{0}^{(-n+2)\bmod 2+1}(1)\bigr)\\
 & \bigl((v+1)I,vP\bigr),\ n>2\\
\end{align*}
\end{fleqn}
\subsubsection*{Modules of the form $I(n,3)$}

Defect: $\partial I(n,3) = 1$, for $n\ge 0$.

\begin{fleqn}
\begin{align*}
I(0,3):\ & - \\
I(1,3):\ & \bigl(I(0,4),R_{1}^{1}(1)\bigr)\\
I(2,3):\ & \bigl(I(0,1),R_{0}^{1}(1)\bigr),\ \bigl(I(0,2),R_{\infty}^{2}(1)\bigr),\ \bigl(I(1,4),R_{1}^{2}(1)\bigr),\ \bigl(2I(0,3),P(1,3)\bigr)\\
I(n,3):\ & \bigl(I(n-2,1),R_{0}^{(-n+2)\bmod 2+1}(1)\bigr),\ \bigl(I(n-2,2),R_{\infty}^{(-n+3)\bmod 2+1}(1)\bigr),\ \bigl(I(n-1,4),R_{1}^{(-n+3)\bmod 2+1}(1)\bigr)\\
 & \bigl((v+1)I,vP\bigr),\ n>2\\
\end{align*}
\end{fleqn}
\subsubsection*{Modules of the form $I(n,4)$}

Defect: $\partial I(n,4) = 1$, for $n\ge 0$.

\begin{fleqn}
\begin{align*}
I(0,4):\ & - \\
I(1,4):\ & \bigl(I(0,3),R_{1}^{1}(1)\bigr)\\
I(2,4):\ & \bigl(I(0,1),R_{\infty}^{1}(1)\bigr),\ \bigl(I(0,2),R_{0}^{2}(1)\bigr),\ \bigl(I(1,3),R_{1}^{2}(1)\bigr),\ \bigl(2I(0,4),P(1,4)\bigr)\\
I(n,4):\ & \bigl(I(n-2,1),R_{\infty}^{(-n+2)\bmod 2+1}(1)\bigr),\ \bigl(I(n-2,2),R_{0}^{(-n+3)\bmod 2+1}(1)\bigr),\ \bigl(I(n-1,3),R_{1}^{(-n+3)\bmod 2+1}(1)\bigr)\\
 & \bigl((v+1)I,vP\bigr),\ n>2\\
\end{align*}
\end{fleqn}
\subsubsection*{Modules of the form $I(n,5)$}

Defect: $\partial I(n,5) = 2$, for $n\ge 0$.

\begin{fleqn}
\begin{align*}
I(0,5):\ & \bigl(I(0,3),I(1,3)\bigr),\ \bigl(I(0,4),I(1,4)\bigr)\\
I(1,5):\ & \bigl(I(0,1),I(1,1)\bigr),\ \bigl(I(0,2),I(1,2)\bigr),\ \bigl(I(1,3),I(2,3)\bigr),\ \bigl(I(1,4),I(2,4)\bigr)\\
I(n,5):\ & \bigl(I(n-1,1),I(n,1)\bigr),\ \bigl(I(n-1,2),I(n,2)\bigr),\ \bigl(I(n,3),I(n+1,3)\bigr)\\
 & \bigl(I(n,4),I(n+1,4)\bigr),\ n>1\\
\end{align*}
\end{fleqn}
\subsubsection{Schofield pairs associated to regular exceptional modules}

\subsubsection*{The non-homogeneous tube $\mathcal{T}_{1}^{\Delta(\widetilde{\mathbb{D}}_{4})}$}

\begin{figure}[ht]

\begin{center}
\captionof{figure}{\vspace*{-10pt}$\mathcal{T}_{1}^{\Delta(\widetilde{\mathbb{D}}_{4})}$}
\begin{scaletikzpicturetowidth}{0.40000000000000002\textwidth}
\begin{tikzpicture}[scale=\tikzscale]
	\node[seltwoDDDDD, label={[shift={(0.0,-0.45)}, brown]\tiny $R_{1}^{1}(1)$}] (RN_1_1_1__Rationals) at (4,0) {$\begin{smallmatrix}0&&0\\&1\\0&&0\end{smallmatrix}$};
	\node[seltwoDDDDD, label={[shift={(0.0,-0.45)}, brown]\tiny $R_{1}^{2}(1)$}] (RN_1_2_1__Rationals) at (8,0) {$\begin{smallmatrix}1&&1\\&1\\1&&1\end{smallmatrix}$};
	\node[seltwoDDDDD, label={[shift={(0.0,-0.45)}, brown]\tiny $R_{1}^{1}(1)$}] (RN_1_1_1__Rationals_repeated) at (12,0) {$\begin{smallmatrix}0&&0\\&1\\0&&0\end{smallmatrix}$};
	\node[noselDDDDD, label={[shift={(0.0,-0.45)}, brown]\tiny $R_{1}^{1}(2)$}] (RN_1_1_2__Rationals) at (6,2) {$\begin{smallmatrix}1&&1\\&2\\1&&1\end{smallmatrix}$};
	\node[noselDDDDD, label={[shift={(0.0,-0.45)}, brown]\tiny $R_{1}^{2}(2)$}] (RN_1_2_2__Rationals) at (10,2) {$\begin{smallmatrix}1&&1\\&2\\1&&1\end{smallmatrix}$};
	\node[noselDDDDD, label={[shift={(0.0,-0.45)}, brown]\tiny $R_{1}^{2}(3)$}] (RN_1_2_3__Rationals) at (4,4) {$\begin{smallmatrix}2&&2\\&3\\2&&2\end{smallmatrix}$};
	\node[noselDDDDD, label={[shift={(0.0,-0.45)}, brown]\tiny $R_{1}^{1}(3)$}] (RN_1_1_3__Rationals) at (8,4) {$\begin{smallmatrix}1&&1\\&3\\1&&1\end{smallmatrix}$};
	\node[noselDDDDD, label={[shift={(0.0,-0.45)}, brown]\tiny $R_{1}^{2}(3)$}] (RN_1_2_3__Rationals_repeated) at (12,4) {$\begin{smallmatrix}2&&2\\&3\\2&&2\end{smallmatrix}$};
	\draw[blue,->] (RN_1_1_1__Rationals) -- (RN_1_1_2__Rationals);
	\draw[gray,dashed,->] (RN_1_1_1__Rationals) -- (RN_1_2_1__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[red,->] (RN_1_1_2__Rationals) -- (RN_1_2_1__Rationals);
	\draw[gray,dashed,->] (RN_1_1_2__Rationals) -- (RN_1_2_2__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[blue,->] (RN_1_1_2__Rationals) -- (RN_1_1_3__Rationals);
	\draw[gray,dashed,->] (RN_1_2_1__Rationals) -- (RN_1_1_1__Rationals_repeated) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[blue,->] (RN_1_2_1__Rationals) -- (RN_1_2_2__Rationals);
	\draw[red,->] (RN_1_2_2__Rationals) -- (RN_1_1_1__Rationals_repeated);
	\draw[gray,dashed,->,out=219,in=-39] (RN_1_2_2__Rationals) to node[midway,sloped,above]{{\tiny $\tau^{-1}$}} (RN_1_1_2__Rationals);
	\draw[blue,->] (RN_1_2_2__Rationals) -- (RN_1_2_3__Rationals_repeated);
	\draw[red,->] (RN_1_2_3__Rationals) -- (RN_1_1_2__Rationals);
	\draw[gray,dashed,->] (RN_1_2_3__Rationals) -- (RN_1_1_3__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[red,->] (RN_1_1_3__Rationals) -- (RN_1_2_2__Rationals);
	\draw[gray,dashed,->] (RN_1_1_3__Rationals) -- (RN_1_2_3__Rationals_repeated) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[dashed,-] (RN_1_1_1__Rationals) -- (RN_1_2_3__Rationals);
	\draw[dashed,-] (RN_1_1_1__Rationals_repeated) -- (RN_1_2_3__Rationals_repeated);
\end{tikzpicture}
\end{scaletikzpicturetowidth}
\end{center}

\end{figure}
\begin{fleqn}
\begin{align*}
R_{1}^{1}(1):\ & - \\
R_{1}^{2}(1):\ & \bigl(I(0,2),P(0,1)\bigr),\ \bigl(I(0,1),P(0,2)\bigr),\ \bigl(I(0,4),P(0,3)\bigr),\ \bigl(I(0,3),P(0,4)\bigr)\\
\end{align*}
\end{fleqn}
\subsubsection*{The non-homogeneous tube $\mathcal{T}_{\infty}^{\Delta(\widetilde{\mathbb{D}}_{4})}$}

\begin{figure}[ht]

\begin{center}
\captionof{figure}{\vspace*{-10pt}$\mathcal{T}_{\infty}^{\Delta(\widetilde{\mathbb{D}}_{4})}$}
\begin{scaletikzpicturetowidth}{0.40000000000000002\textwidth}
\begin{tikzpicture}[scale=\tikzscale]
	\node[seltwoDDDDD, label={[shift={(0.0,-0.45)}, brown]\tiny $R_{\infty}^{1}(1)$}] (RN_2_1_1__Rationals) at (4,0) {$\begin{smallmatrix}1&&0\\&1\\0&&1\end{smallmatrix}$};
	\node[seltwoDDDDD, label={[shift={(0.0,-0.45)}, brown]\tiny $R_{\infty}^{2}(1)$}] (RN_2_2_1__Rationals) at (8,0) {$\begin{smallmatrix}0&&1\\&1\\1&&0\end{smallmatrix}$};
	\node[seltwoDDDDD, label={[shift={(0.0,-0.45)}, brown]\tiny $R_{\infty}^{1}(1)$}] (RN_2_1_1__Rationals_repeated) at (12,0) {$\begin{smallmatrix}1&&0\\&1\\0&&1\end{smallmatrix}$};
	\node[noselDDDDD, label={[shift={(0.0,-0.45)}, brown]\tiny $R_{\infty}^{1}(2)$}] (RN_2_1_2__Rationals) at (6,2) {$\begin{smallmatrix}1&&1\\&2\\1&&1\end{smallmatrix}$};
	\node[noselDDDDD, label={[shift={(0.0,-0.45)}, brown]\tiny $R_{\infty}^{2}(2)$}] (RN_2_2_2__Rationals) at (10,2) {$\begin{smallmatrix}1&&1\\&2\\1&&1\end{smallmatrix}$};
	\node[noselDDDDD, label={[shift={(0.0,-0.45)}, brown]\tiny $R_{\infty}^{2}(3)$}] (RN_2_2_3__Rationals) at (4,4) {$\begin{smallmatrix}1&&2\\&3\\2&&1\end{smallmatrix}$};
	\node[noselDDDDD, label={[shift={(0.0,-0.45)}, brown]\tiny $R_{\infty}^{1}(3)$}] (RN_2_1_3__Rationals) at (8,4) {$\begin{smallmatrix}2&&1\\&3\\1&&2\end{smallmatrix}$};
	\node[noselDDDDD, label={[shift={(0.0,-0.45)}, brown]\tiny $R_{\infty}^{2}(3)$}] (RN_2_2_3__Rationals_repeated) at (12,4) {$\begin{smallmatrix}1&&2\\&3\\2&&1\end{smallmatrix}$};
	\draw[blue,->] (RN_2_1_1__Rationals) -- (RN_2_1_2__Rationals);
	\draw[gray,dashed,->] (RN_2_1_1__Rationals) -- (RN_2_2_1__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[red,->] (RN_2_1_2__Rationals) -- (RN_2_2_1__Rationals);
	\draw[gray,dashed,->] (RN_2_1_2__Rationals) -- (RN_2_2_2__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[blue,->] (RN_2_1_2__Rationals) -- (RN_2_1_3__Rationals);
	\draw[gray,dashed,->] (RN_2_2_1__Rationals) -- (RN_2_1_1__Rationals_repeated) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[blue,->] (RN_2_2_1__Rationals) -- (RN_2_2_2__Rationals);
	\draw[red,->] (RN_2_2_2__Rationals) -- (RN_2_1_1__Rationals_repeated);
	\draw[gray,dashed,->,out=219,in=-39] (RN_2_2_2__Rationals) to node[midway,sloped,above]{{\tiny $\tau^{-1}$}} (RN_2_1_2__Rationals);
	\draw[blue,->] (RN_2_2_2__Rationals) -- (RN_2_2_3__Rationals_repeated);
	\draw[red,->] (RN_2_2_3__Rationals) -- (RN_2_1_2__Rationals);
	\draw[gray,dashed,->] (RN_2_2_3__Rationals) -- (RN_2_1_3__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[red,->] (RN_2_1_3__Rationals) -- (RN_2_2_2__Rationals);
	\draw[gray,dashed,->] (RN_2_1_3__Rationals) -- (RN_2_2_3__Rationals_repeated) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[dashed,-] (RN_2_1_1__Rationals) -- (RN_2_2_3__Rationals);
	\draw[dashed,-] (RN_2_1_1__Rationals_repeated) -- (RN_2_2_3__Rationals_repeated);
\end{tikzpicture}
\end{scaletikzpicturetowidth}
\end{center}

\end{figure}
\begin{fleqn}
\begin{align*}
R_{\infty}^{1}(1):\ & \bigl(I(1,3),P(0,2)\bigr),\ \bigl(I(0,4),P(1,1)\bigr)\\
R_{\infty}^{2}(1):\ & \bigl(I(1,4),P(0,1)\bigr),\ \bigl(I(0,3),P(1,2)\bigr)\\
\end{align*}
\end{fleqn}
\subsubsection*{The non-homogeneous tube $\mathcal{T}_{0}^{\Delta(\widetilde{\mathbb{D}}_{4})}$}

\begin{figure}[ht]

\begin{center}
\captionof{figure}{\vspace*{-10pt}$\mathcal{T}_{0}^{\Delta(\widetilde{\mathbb{D}}_{4})}$}
\begin{scaletikzpicturetowidth}{0.40000000000000002\textwidth}
\begin{tikzpicture}[scale=\tikzscale]
	\node[seltwoDDDDD, label={[shift={(0.0,-0.45)}, brown]\tiny $R_{0}^{1}(1)$}] (RN_3_1_1__Rationals) at (4,0) {$\begin{smallmatrix}1&&1\\&1\\0&&0\end{smallmatrix}$};
	\node[seltwoDDDDD, label={[shift={(0.0,-0.45)}, brown]\tiny $R_{0}^{2}(1)$}] (RN_3_2_1__Rationals) at (8,0) {$\begin{smallmatrix}0&&0\\&1\\1&&1\end{smallmatrix}$};
	\node[seltwoDDDDD, label={[shift={(0.0,-0.45)}, brown]\tiny $R_{0}^{1}(1)$}] (RN_3_1_1__Rationals_repeated) at (12,0) {$\begin{smallmatrix}1&&1\\&1\\0&&0\end{smallmatrix}$};
	\node[noselDDDDD, label={[shift={(0.0,-0.45)}, brown]\tiny $R_{0}^{1}(2)$}] (RN_3_1_2__Rationals) at (6,2) {$\begin{smallmatrix}1&&1\\&2\\1&&1\end{smallmatrix}$};
	\node[noselDDDDD, label={[shift={(0.0,-0.45)}, brown]\tiny $R_{0}^{2}(2)$}] (RN_3_2_2__Rationals) at (10,2) {$\begin{smallmatrix}1&&1\\&2\\1&&1\end{smallmatrix}$};
	\node[noselDDDDD, label={[shift={(0.0,-0.45)}, brown]\tiny $R_{0}^{2}(3)$}] (RN_3_2_3__Rationals) at (4,4) {$\begin{smallmatrix}1&&1\\&3\\2&&2\end{smallmatrix}$};
	\node[noselDDDDD, label={[shift={(0.0,-0.45)}, brown]\tiny $R_{0}^{1}(3)$}] (RN_3_1_3__Rationals) at (8,4) {$\begin{smallmatrix}2&&2\\&3\\1&&1\end{smallmatrix}$};
	\node[noselDDDDD, label={[shift={(0.0,-0.45)}, brown]\tiny $R_{0}^{2}(3)$}] (RN_3_2_3__Rationals_repeated) at (12,4) {$\begin{smallmatrix}1&&1\\&3\\2&&2\end{smallmatrix}$};
	\draw[blue,->] (RN_3_1_1__Rationals) -- (RN_3_1_2__Rationals);
	\draw[gray,dashed,->] (RN_3_1_1__Rationals) -- (RN_3_2_1__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[red,->] (RN_3_1_2__Rationals) -- (RN_3_2_1__Rationals);
	\draw[gray,dashed,->] (RN_3_1_2__Rationals) -- (RN_3_2_2__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[blue,->] (RN_3_1_2__Rationals) -- (RN_3_1_3__Rationals);
	\draw[gray,dashed,->] (RN_3_2_1__Rationals) -- (RN_3_1_1__Rationals_repeated) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[blue,->] (RN_3_2_1__Rationals) -- (RN_3_2_2__Rationals);
	\draw[red,->] (RN_3_2_2__Rationals) -- (RN_3_1_1__Rationals_repeated);
	\draw[gray,dashed,->,out=219,in=-39] (RN_3_2_2__Rationals) to node[midway,sloped,above]{{\tiny $\tau^{-1}$}} (RN_3_1_2__Rationals);
	\draw[blue,->] (RN_3_2_2__Rationals) -- (RN_3_2_3__Rationals_repeated);
	\draw[red,->] (RN_3_2_3__Rationals) -- (RN_3_1_2__Rationals);
	\draw[gray,dashed,->] (RN_3_2_3__Rationals) -- (RN_3_1_3__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[red,->] (RN_3_1_3__Rationals) -- (RN_3_2_2__Rationals);
	\draw[gray,dashed,->] (RN_3_1_3__Rationals) -- (RN_3_2_3__Rationals_repeated) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[dashed,-] (RN_3_1_1__Rationals) -- (RN_3_2_3__Rationals);
	\draw[dashed,-] (RN_3_1_1__Rationals_repeated) -- (RN_3_2_3__Rationals_repeated);
\end{tikzpicture}
\end{scaletikzpicturetowidth}
\end{center}

\end{figure}
\begin{fleqn}
\begin{align*}
R_{0}^{1}(1):\ & \bigl(I(1,4),P(0,2)\bigr),\ \bigl(I(0,3),P(1,1)\bigr)\\
R_{0}^{2}(1):\ & \bigl(I(1,3),P(0,1)\bigr),\ \bigl(I(0,4),P(1,2)\bigr)\\
\end{align*}
\end{fleqn}
\clearpage

\subsection[Schofield pairs for the quiver $\Delta(\widetilde{\mathbb{D}}_5)$]{Schofield pairs for the quiver $\Delta(\widetilde{\mathbb{D}}_5)$ -- {\usebox{\boxDDDDDD}}}
\[
\vcenter{\hbox{\xymatrix{2 &   &   & 3\ar[dl]\\
  & 5\ar[dl]\ar[ul] & 6\ar[l] &  \\
1 &   &   & 4\ar[ul]}}}\qquad C_{\Delta(\widetilde{\mathbb{D}}_{5})} = \begin{bmatrix}1 & 0 & 1 & 1 & 1 & 1\\
0 & 1 & 1 & 1 & 1 & 1\\
0 & 0 & 1 & 0 & 0 & 0\\
0 & 0 & 0 & 1 & 0 & 0\\
0 & 0 & 1 & 1 & 1 & 1\\
0 & 0 & 1 & 1 & 0 & 1\end{bmatrix}\quad\Phi_{\Delta(\widetilde{\mathbb{D}}_{5})} = \begin{bmatrix}-1 & 0 & 0 & 0 & 1 & 0\\
0 & -1 & 0 & 0 & 1 & 0\\
-1 & -1 & 0 & 1 & 1 & 0\\
-1 & -1 & 1 & 0 & 1 & 0\\
-1 & -1 & 0 & 0 & 1 & 1\\
-1 & -1 & 1 & 1 & 1 & 0\end{bmatrix}
\]
\subsubsection{Schofield pairs associated to preprojective exceptional modules}

\begin{figure}[ht]

\begin{center}
\begin{scaletikzpicturetowidth}{\textwidth}
\begin{tikzpicture}[scale=\tikzscale]
	\node[noselDDDDDD, label={[shift={(0.0,-0.45)}, brown]\tiny $P(0,1)$}] (P_0_1__Rationals) at (0,0) {$\begin{smallmatrix}0&&&0\\&0&0\\1&&&0\end{smallmatrix}$};
	\node[noselDDDDDD, label={[shift={(0.0,-0.45)}, brown]\tiny $P(0,2)$}] (P_0_2__Rationals) at (0,1) {$\begin{smallmatrix}1&&&0\\&0&0\\0&&&0\end{smallmatrix}$};
	\node[noselDDDDDD, label={[shift={(0.0,-0.45)}, brown]\tiny $P(0,3)$}] (P_0_3__Rationals) at (6,6) {$\begin{smallmatrix}1&&&1\\&1&1\\1&&&0\end{smallmatrix}$};
	\node[noselDDDDDD, label={[shift={(0.0,-0.45)}, brown]\tiny $P(0,4)$}] (P_0_4__Rationals) at (6,5) {$\begin{smallmatrix}1&&&0\\&1&1\\1&&&1\end{smallmatrix}$};
	\node[noselDDDDDD, label={[shift={(0.0,-0.45)}, brown]\tiny $P(0,5)$}] (P_0_5__Rationals) at (2,2) {$\begin{smallmatrix}1&&&0\\&1&0\\1&&&0\end{smallmatrix}$};
	\node[noselDDDDDD, label={[shift={(0.0,-0.45)}, brown]\tiny $P(0,6)$}] (P_0_6__Rationals) at (4,4) {$\begin{smallmatrix}1&&&0\\&1&1\\1&&&0\end{smallmatrix}$};
	\node[noselDDDDDD, label={[shift={(0.0,-0.45)}, brown]\tiny $P(1,1)$}] (P_1_1__Rationals) at (4,0) {$\begin{smallmatrix}1&&&0\\&1&0\\0&&&0\end{smallmatrix}$};
	\node[noselDDDDDD, label={[shift={(0.0,-0.45)}, brown]\tiny $P(1,2)$}] (P_1_2__Rationals) at (4,1) {$\begin{smallmatrix}0&&&0\\&1&0\\1&&&0\end{smallmatrix}$};
	\node[noselDDDDDD, label={[shift={(0.0,-0.45)}, brown]\tiny $P(1,3)$}] (P_1_3__Rationals) at (10,6) {$\begin{smallmatrix}1&&&0\\&2&1\\1&&&1\end{smallmatrix}$};
	\node[noselDDDDDD, label={[shift={(0.0,-0.45)}, brown]\tiny $P(1,4)$}] (P_1_4__Rationals) at (10,5) {$\begin{smallmatrix}1&&&1\\&2&1\\1&&&0\end{smallmatrix}$};
	\node[noselDDDDDD, label={[shift={(0.0,-0.45)}, brown]\tiny $P(1,5)$}] (P_1_5__Rationals) at (6,2) {$\begin{smallmatrix}1&&&0\\&2&1\\1&&&0\end{smallmatrix}$};
	\node[seloneDDDDDD, label={[shift={(0.0,-0.45)}, brown]\tiny $P(1,6)$}] (P_1_6__Rationals) at (8,4) {$\begin{smallmatrix}2&&&1\\&3&2\\2&&&1\end{smallmatrix}$};
	\node[noselDDDDDD, label={[shift={(0.0,-0.45)}, brown]\tiny $P(2,1)$}] (P_2_1__Rationals) at (8,0) {$\begin{smallmatrix}0&&&0\\&1&1\\1&&&0\end{smallmatrix}$};
	\node[noselDDDDDD, label={[shift={(0.0,-0.45)}, brown]\tiny $P(2,2)$}] (P_2_2__Rationals) at (8,1) {$\begin{smallmatrix}1&&&0\\&1&1\\0&&&0\end{smallmatrix}$};
	\node[noselDDDDDD, label={[shift={(0.0,-0.45)}, brown]\tiny $P(2,3)$}] (P_2_3__Rationals) at (14,6) {$\begin{smallmatrix}1&&&1\\&2&2\\1&&&0\end{smallmatrix}$};
	\node[noselDDDDDD, label={[shift={(0.0,-0.45)}, brown]\tiny $P(2,4)$}] (P_2_4__Rationals) at (14,5) {$\begin{smallmatrix}1&&&0\\&2&2\\1&&&1\end{smallmatrix}$};
	\node[seloneDDDDDD, label={[shift={(0.0,-0.45)}, brown]\tiny $P(2,5)$}] (P_2_5__Rationals) at (10,2) {$\begin{smallmatrix}2&&&1\\&3&3\\2&&&1\end{smallmatrix}$};
	\node[noselDDDDDD, label={[shift={(0.0,-0.45)}, brown]\tiny $P(2,6)$}] (P_2_6__Rationals) at (12,4) {$\begin{smallmatrix}2&&&1\\&4&3\\2&&&1\end{smallmatrix}$};
	\node[seloneDDDDDD, label={[shift={(0.0,-0.45)}, brown]\tiny $P(3,1)$}] (P_3_1__Rationals) at (12,0) {$\begin{smallmatrix}2&&&1\\&2&2\\1&&&1\end{smallmatrix}$};
	\node[seloneDDDDDD, label={[shift={(0.0,-0.45)}, brown]\tiny $P(3,2)$}] (P_3_2__Rationals) at (12,1) {$\begin{smallmatrix}1&&&1\\&2&2\\2&&&1\end{smallmatrix}$};
	\node[seloneDDDDDD, label={[shift={(0.0,-0.45)}, brown]\tiny $P(3,3)$}] (P_3_3__Rationals) at (18,6) {$\begin{smallmatrix}2&&&1\\&3&3\\2&&&2\end{smallmatrix}$};
	\node[seloneDDDDDD, label={[shift={(0.0,-0.45)}, brown]\tiny $P(3,4)$}] (P_3_4__Rationals) at (18,5) {$\begin{smallmatrix}2&&&2\\&3&3\\2&&&1\end{smallmatrix}$};
	\node[noselDDDDDD, label={[shift={(0.0,-0.45)}, brown]\tiny $P(3,5)$}] (P_3_5__Rationals) at (14,2) {$\begin{smallmatrix}3&&&2\\&5&4\\3&&&2\end{smallmatrix}$};
	\node[noselDDDDDD, label={[shift={(0.0,-0.45)}, brown]\tiny $P(3,6)$}] (P_3_6__Rationals) at (16,4) {$\begin{smallmatrix}3&&&2\\&5&5\\3&&&2\end{smallmatrix}$};
	\node at (18,4) {$\dots$};
	\draw[blue,->] (P_0_6__Rationals) -- (P_0_3__Rationals);
	\draw[blue,->] (P_0_6__Rationals) -- (P_0_4__Rationals);
	\draw[blue,->] (P_0_1__Rationals) -- (P_0_5__Rationals);
	\draw[blue,->] (P_0_2__Rationals) -- (P_0_5__Rationals);
	\draw[blue,->] (P_0_5__Rationals) -- (P_0_6__Rationals);
	\draw[gray,dashed,->] (P_0_1__Rationals) -- (P_1_1__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[red,->] (P_0_5__Rationals) -- (P_1_1__Rationals);
	\draw[gray,dashed,->] (P_0_2__Rationals) -- (P_1_2__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[red,->] (P_0_5__Rationals) -- (P_1_2__Rationals);
	\draw[gray,dashed,->] (P_0_3__Rationals) -- (P_1_3__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[red,->] (P_1_6__Rationals) -- (P_1_3__Rationals);
	\draw[gray,dashed,->] (P_0_4__Rationals) -- (P_1_4__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[red,->] (P_1_6__Rationals) -- (P_1_4__Rationals);
	\draw[gray,dashed,->] (P_0_5__Rationals) -- (P_1_5__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[blue,->] (P_0_6__Rationals) -- (P_1_5__Rationals);
	\draw[blue,->] (P_1_1__Rationals) -- (P_1_5__Rationals);
	\draw[blue,->] (P_1_2__Rationals) -- (P_1_5__Rationals);
	\draw[blue,->] (P_0_3__Rationals) -- (P_1_6__Rationals);
	\draw[blue,->] (P_0_4__Rationals) -- (P_1_6__Rationals);
	\draw[gray,dashed,->] (P_0_6__Rationals) -- (P_1_6__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[blue,->] (P_1_5__Rationals) -- (P_1_6__Rationals);
	\draw[gray,dashed,->] (P_1_1__Rationals) -- (P_2_1__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[red,->] (P_1_5__Rationals) -- (P_2_1__Rationals);
	\draw[gray,dashed,->] (P_1_2__Rationals) -- (P_2_2__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[red,->] (P_1_5__Rationals) -- (P_2_2__Rationals);
	\draw[gray,dashed,->] (P_1_3__Rationals) -- (P_2_3__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[red,->] (P_2_6__Rationals) -- (P_2_3__Rationals);
	\draw[gray,dashed,->] (P_1_4__Rationals) -- (P_2_4__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[red,->] (P_2_6__Rationals) -- (P_2_4__Rationals);
	\draw[gray,dashed,->] (P_1_5__Rationals) -- (P_2_5__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[blue,->] (P_1_6__Rationals) -- (P_2_5__Rationals);
	\draw[blue,->] (P_2_1__Rationals) -- (P_2_5__Rationals);
	\draw[blue,->] (P_2_2__Rationals) -- (P_2_5__Rationals);
	\draw[blue,->] (P_1_3__Rationals) -- (P_2_6__Rationals);
	\draw[blue,->] (P_1_4__Rationals) -- (P_2_6__Rationals);
	\draw[gray,dashed,->] (P_1_6__Rationals) -- (P_2_6__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[blue,->] (P_2_5__Rationals) -- (P_2_6__Rationals);
	\draw[gray,dashed,->] (P_2_1__Rationals) -- (P_3_1__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[red,->] (P_2_5__Rationals) -- (P_3_1__Rationals);
	\draw[gray,dashed,->] (P_2_2__Rationals) -- (P_3_2__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[red,->] (P_2_5__Rationals) -- (P_3_2__Rationals);
	\draw[gray,dashed,->] (P_2_3__Rationals) -- (P_3_3__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[red,->] (P_3_6__Rationals) -- (P_3_3__Rationals);
	\draw[gray,dashed,->] (P_2_4__Rationals) -- (P_3_4__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[red,->] (P_3_6__Rationals) -- (P_3_4__Rationals);
	\draw[gray,dashed,->] (P_2_5__Rationals) -- (P_3_5__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[blue,->] (P_2_6__Rationals) -- (P_3_5__Rationals);
	\draw[blue,->] (P_3_1__Rationals) -- (P_3_5__Rationals);
	\draw[blue,->] (P_3_2__Rationals) -- (P_3_5__Rationals);
	\draw[blue,->] (P_2_3__Rationals) -- (P_3_6__Rationals);
	\draw[blue,->] (P_2_4__Rationals) -- (P_3_6__Rationals);
	\draw[gray,dashed,->] (P_2_6__Rationals) -- (P_3_6__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[blue,->] (P_3_5__Rationals) -- (P_3_6__Rationals);
\end{tikzpicture}
\end{scaletikzpicturetowidth}
\end{center}

\end{figure}
\medskip{}
\subsubsection*{Modules of the form $P(n,1)$}

Defect: $\partial P(n,1) = -1$, for $n\ge 0$.

\begin{fleqn}
\begin{align*}
P(0,1):\ & - \\
P(1,1):\ & \bigl(R_{1}^{3}(1),P(0,2)\bigr)\\
P(2,1):\ & \bigl(R_{1}^{3}(2),P(0,1)\bigr),\ \bigl(R_{1}^{1}(1),P(1,2)\bigr)\\
P(3,1):\ & \bigl(R_{\infty}^{1}(1),P(0,3)\bigr),\ \bigl(R_{0}^{1}(1),P(0,4)\bigr),\ \bigl(R_{1}^{1}(2),P(1,1)\bigr),\ \bigl(R_{1}^{2}(1),P(2,2)\bigr),\ \bigl(I(2,1),2P(0,2)\bigr)\\
P(n,1):\ & \bigl(R_{\infty}^{(n-3)\bmod 2+1}(1),P(n-3,3)\bigr),\ \bigl(R_{0}^{(n-3)\bmod 2+1}(1),P(n-3,4)\bigr),\ \bigl(R_{1}^{(n-3)\bmod 3+1}(2),P(n-2,1)\bigr)\\
 & \bigl(R_{1}^{(n-2)\bmod 3+1}(1),P(n-1,2)\bigr),\ \bigl(uI,(u+1)P\bigr),\ n>3\\
\end{align*}
\end{fleqn}
\subsubsection*{Modules of the form $P(n,2)$}

Defect: $\partial P(n,2) = -1$, for $n\ge 0$.

\begin{fleqn}
\begin{align*}
P(0,2):\ & - \\
P(1,2):\ & \bigl(R_{1}^{3}(1),P(0,1)\bigr)\\
P(2,2):\ & \bigl(R_{1}^{3}(2),P(0,2)\bigr),\ \bigl(R_{1}^{1}(1),P(1,1)\bigr)\\
P(3,2):\ & \bigl(R_{0}^{2}(1),P(0,3)\bigr),\ \bigl(R_{\infty}^{2}(1),P(0,4)\bigr),\ \bigl(R_{1}^{1}(2),P(1,2)\bigr),\ \bigl(R_{1}^{2}(1),P(2,1)\bigr),\ \bigl(I(2,2),2P(0,1)\bigr)\\
P(n,2):\ & \bigl(R_{0}^{(n-2)\bmod 2+1}(1),P(n-3,3)\bigr),\ \bigl(R_{\infty}^{(n-2)\bmod 2+1}(1),P(n-3,4)\bigr),\ \bigl(R_{1}^{(n-3)\bmod 3+1}(2),P(n-2,2)\bigr)\\
 & \bigl(R_{1}^{(n-2)\bmod 3+1}(1),P(n-1,1)\bigr),\ \bigl(uI,(u+1)P\bigr),\ n>3\\
\end{align*}
\end{fleqn}
\subsubsection*{Modules of the form $P(n,3)$}

Defect: $\partial P(n,3) = -1$, for $n\ge 0$.

\begin{fleqn}
\begin{align*}
P(0,3):\ & \bigl(R_{0}^{1}(1),P(0,1)\bigr),\ \bigl(R_{\infty}^{2}(1),P(0,2)\bigr),\ \bigl(I(1,4),P(0,5)\bigr),\ \bigl(I(0,3),P(0,6)\bigr)\\
P(1,3):\ & \bigl(R_{1}^{3}(1),P(0,4)\bigr),\ \bigl(R_{0}^{2}(1),P(1,1)\bigr),\ \bigl(R_{\infty}^{1}(1),P(1,2)\bigr),\ \bigl(I(0,4),P(1,5)\bigr)\\
P(2,3):\ & \bigl(R_{1}^{3}(2),P(0,3)\bigr),\ \bigl(R_{1}^{1}(1),P(1,4)\bigr),\ \bigl(R_{0}^{1}(1),P(2,1)\bigr),\ \bigl(R_{\infty}^{2}(1),P(2,2)\bigr)\\
P(3,3):\ & \bigl(R_{1}^{1}(2),P(1,3)\bigr),\ \bigl(R_{1}^{2}(1),P(2,4)\bigr),\ \bigl(R_{0}^{2}(1),P(3,1)\bigr),\ \bigl(R_{\infty}^{1}(1),P(3,2)\bigr),\ \bigl(I(2,3),2P(0,4)\bigr)\\
P(n,3):\ & \bigl(R_{1}^{(n-3)\bmod 3+1}(2),P(n-2,3)\bigr),\ \bigl(R_{1}^{(n-2)\bmod 3+1}(1),P(n-1,4)\bigr),\ \bigl(R_{0}^{(n-2)\bmod 2+1}(1),P(n,1)\bigr)\\
 & \bigl(R_{\infty}^{(n-3)\bmod 2+1}(1),P(n,2)\bigr),\ \bigl(uI,(u+1)P\bigr),\ n>3\\
\end{align*}
\end{fleqn}
\subsubsection*{Modules of the form $P(n,4)$}

Defect: $\partial P(n,4) = -1$, for $n\ge 0$.

\begin{fleqn}
\begin{align*}
P(0,4):\ & \bigl(R_{\infty}^{1}(1),P(0,1)\bigr),\ \bigl(R_{0}^{2}(1),P(0,2)\bigr),\ \bigl(I(1,3),P(0,5)\bigr),\ \bigl(I(0,4),P(0,6)\bigr)\\
P(1,4):\ & \bigl(R_{1}^{3}(1),P(0,3)\bigr),\ \bigl(R_{\infty}^{2}(1),P(1,1)\bigr),\ \bigl(R_{0}^{1}(1),P(1,2)\bigr),\ \bigl(I(0,3),P(1,5)\bigr)\\
P(2,4):\ & \bigl(R_{1}^{3}(2),P(0,4)\bigr),\ \bigl(R_{1}^{1}(1),P(1,3)\bigr),\ \bigl(R_{\infty}^{1}(1),P(2,1)\bigr),\ \bigl(R_{0}^{2}(1),P(2,2)\bigr)\\
P(3,4):\ & \bigl(R_{1}^{1}(2),P(1,4)\bigr),\ \bigl(R_{1}^{2}(1),P(2,3)\bigr),\ \bigl(R_{\infty}^{2}(1),P(3,1)\bigr),\ \bigl(R_{0}^{1}(1),P(3,2)\bigr),\ \bigl(I(2,4),2P(0,3)\bigr)\\
P(n,4):\ & \bigl(R_{1}^{(n-3)\bmod 3+1}(2),P(n-2,4)\bigr),\ \bigl(R_{1}^{(n-2)\bmod 3+1}(1),P(n-1,3)\bigr),\ \bigl(R_{\infty}^{(n-2)\bmod 2+1}(1),P(n,1)\bigr)\\
 & \bigl(R_{0}^{(n-3)\bmod 2+1}(1),P(n,2)\bigr),\ \bigl(uI,(u+1)P\bigr),\ n>3\\
\end{align*}
\end{fleqn}
\subsubsection*{Modules of the form $P(n,5)$}

Defect: $\partial P(n,5) = -2$, for $n\ge 0$.

\begin{fleqn}
\begin{align*}
P(0,5):\ & \bigl(P(1,1),P(0,1)\bigr),\ \bigl(P(1,2),P(0,2)\bigr)\\
P(1,5):\ & \bigl(R_{1}^{3}(1),P(0,6)\bigr),\ \bigl(P(2,1),P(1,1)\bigr),\ \bigl(P(2,2),P(1,2)\bigr)\\
P(2,5):\ & \bigl(P(2,4),P(0,3)\bigr),\ \bigl(P(2,3),P(0,4)\bigr),\ \bigl(R_{1}^{1}(1),P(1,6)\bigr),\ \bigl(P(3,1),P(2,1)\bigr),\ \bigl(P(3,2),P(2,2)\bigr)\\
P(n,5):\ & \bigl(P(n,4),P(n-2,3)\bigr),\ \bigl(P(n,3),P(n-2,4)\bigr),\ \bigl(R_{1}^{(n-2)\bmod 3+1}(1),P(n-1,6)\bigr)\\
 & \bigl(P(n+1,1),P(n,1)\bigr),\ \bigl(P(n+1,2),P(n,2)\bigr),\ n>2\\
\end{align*}
\end{fleqn}
\subsubsection*{Modules of the form $P(n,6)$}

Defect: $\partial P(n,6) = -2$, for $n\ge 0$.

\begin{fleqn}
\begin{align*}
P(0,6):\ & \bigl(P(2,2),P(0,1)\bigr),\ \bigl(P(2,1),P(0,2)\bigr),\ \bigl(R_{1}^{1}(1),P(0,5)\bigr)\\
P(1,6):\ & \bigl(P(1,3),P(0,3)\bigr),\ \bigl(P(1,4),P(0,4)\bigr),\ \bigl(P(3,2),P(1,1)\bigr),\ \bigl(P(3,1),P(1,2)\bigr),\ \bigl(R_{1}^{2}(1),P(1,5)\bigr)\\
P(n,6):\ & \bigl(P(n,3),P(n-1,3)\bigr),\ \bigl(P(n,4),P(n-1,4)\bigr),\ \bigl(P(n+2,2),P(n,1)\bigr)\\
 & \bigl(P(n+2,1),P(n,2)\bigr),\ \bigl(R_{1}^{n\bmod 3+1}(1),P(n,5)\bigr),\ n>1\\
\end{align*}
\end{fleqn}
\subsubsection{Schofield pairs associated to preinjective exceptional modules}

\begin{figure}[ht]

\begin{center}
\begin{scaletikzpicturetowidth}{\textwidth}
\begin{tikzpicture}[scale=\tikzscale]
	\node[noselDDDDDD, label={[shift={(0.0,-0.45)}, brown]\tiny $I(0,1)$}] (I_0_1__Rationals) at (0,0) {$\begin{smallmatrix}0&&&1\\&1&1\\1&&&1\end{smallmatrix}$};
	\node[noselDDDDDD, label={[shift={(0.0,-0.45)}, brown]\tiny $I(0,2)$}] (I_0_2__Rationals) at (0,1) {$\begin{smallmatrix}1&&&1\\&1&1\\0&&&1\end{smallmatrix}$};
	\node[noselDDDDDD, label={[shift={(0.0,-0.45)}, brown]\tiny $I(0,3)$}] (I_0_3__Rationals) at (6,6) {$\begin{smallmatrix}0&&&1\\&0&0\\0&&&0\end{smallmatrix}$};
	\node[noselDDDDDD, label={[shift={(0.0,-0.45)}, brown]\tiny $I(0,4)$}] (I_0_4__Rationals) at (6,5) {$\begin{smallmatrix}0&&&0\\&0&0\\0&&&1\end{smallmatrix}$};
	\node[noselDDDDDD, label={[shift={(0.0,-0.45)}, brown]\tiny $I(0,5)$}] (I_0_5__Rationals) at (2,2) {$\begin{smallmatrix}0&&&1\\&1&1\\0&&&1\end{smallmatrix}$};
	\node[noselDDDDDD, label={[shift={(0.0,-0.45)}, brown]\tiny $I(0,6)$}] (I_0_6__Rationals) at (4,4) {$\begin{smallmatrix}0&&&1\\&0&1\\0&&&1\end{smallmatrix}$};
	\node[noselDDDDDD, label={[shift={(0.0,-0.45)}, brown]\tiny $I(1,1)$}] (I_1_1__Rationals) at (-4,0) {$\begin{smallmatrix}1&&&1\\&1&2\\0&&&1\end{smallmatrix}$};
	\node[noselDDDDDD, label={[shift={(0.0,-0.45)}, brown]\tiny $I(1,2)$}] (I_1_2__Rationals) at (-4,1) {$\begin{smallmatrix}0&&&1\\&1&2\\1&&&1\end{smallmatrix}$};
	\node[noselDDDDDD, label={[shift={(0.0,-0.45)}, brown]\tiny $I(1,3)$}] (I_1_3__Rationals) at (2,6) {$\begin{smallmatrix}0&&&0\\&0&1\\0&&&1\end{smallmatrix}$};
	\node[noselDDDDDD, label={[shift={(0.0,-0.45)}, brown]\tiny $I(1,4)$}] (I_1_4__Rationals) at (2,5) {$\begin{smallmatrix}0&&&1\\&0&1\\0&&&0\end{smallmatrix}$};
	\node[seloneDDDDDD, label={[shift={(0.0,-0.45)}, brown]\tiny $I(1,5)$}] (I_1_5__Rationals) at (-2,2) {$\begin{smallmatrix}1&&&2\\&2&3\\1&&&2\end{smallmatrix}$};
	\node[noselDDDDDD, label={[shift={(0.0,-0.45)}, brown]\tiny $I(1,6)$}] (I_1_6__Rationals) at (0,4) {$\begin{smallmatrix}0&&&1\\&1&2\\0&&&1\end{smallmatrix}$};
	\node[noselDDDDDD, label={[shift={(0.0,-0.45)}, brown]\tiny $I(2,1)$}] (I_2_1__Rationals) at (-8,0) {$\begin{smallmatrix}0&&&1\\&2&2\\1&&&1\end{smallmatrix}$};
	\node[noselDDDDDD, label={[shift={(0.0,-0.45)}, brown]\tiny $I(2,2)$}] (I_2_2__Rationals) at (-8,1) {$\begin{smallmatrix}1&&&1\\&2&2\\0&&&1\end{smallmatrix}$};
	\node[noselDDDDDD, label={[shift={(0.0,-0.45)}, brown]\tiny $I(2,3)$}] (I_2_3__Rationals) at (-2,6) {$\begin{smallmatrix}0&&&1\\&1&1\\0&&&0\end{smallmatrix}$};
	\node[noselDDDDDD, label={[shift={(0.0,-0.45)}, brown]\tiny $I(2,4)$}] (I_2_4__Rationals) at (-2,5) {$\begin{smallmatrix}0&&&0\\&1&1\\0&&&1\end{smallmatrix}$};
	\node[noselDDDDDD, label={[shift={(0.0,-0.45)}, brown]\tiny $I(2,5)$}] (I_2_5__Rationals) at (-6,2) {$\begin{smallmatrix}1&&&2\\&3&4\\1&&&2\end{smallmatrix}$};
	\node[seloneDDDDDD, label={[shift={(0.0,-0.45)}, brown]\tiny $I(2,6)$}] (I_2_6__Rationals) at (-4,4) {$\begin{smallmatrix}1&&&2\\&3&3\\1&&&2\end{smallmatrix}$};
	\node[seloneDDDDDD, label={[shift={(0.0,-0.45)}, brown]\tiny $I(3,1)$}] (I_3_1__Rationals) at (-12,0) {$\begin{smallmatrix}2&&&2\\&3&3\\1&&&2\end{smallmatrix}$};
	\node[seloneDDDDDD, label={[shift={(0.0,-0.45)}, brown]\tiny $I(3,2)$}] (I_3_2__Rationals) at (-12,1) {$\begin{smallmatrix}1&&&2\\&3&3\\2&&&2\end{smallmatrix}$};
	\node[seloneDDDDDD, label={[shift={(0.0,-0.45)}, brown]\tiny $I(3,3)$}] (I_3_3__Rationals) at (-6,6) {$\begin{smallmatrix}1&&&1\\&2&2\\1&&&2\end{smallmatrix}$};
	\node[seloneDDDDDD, label={[shift={(0.0,-0.45)}, brown]\tiny $I(3,4)$}] (I_3_4__Rationals) at (-6,5) {$\begin{smallmatrix}1&&&2\\&2&2\\1&&&1\end{smallmatrix}$};
	\node[noselDDDDDD, label={[shift={(0.0,-0.45)}, brown]\tiny $I(3,5)$}] (I_3_5__Rationals) at (-10,2) {$\begin{smallmatrix}2&&&3\\&5&5\\2&&&3\end{smallmatrix}$};
	\node[noselDDDDDD, label={[shift={(0.0,-0.45)}, brown]\tiny $I(3,6)$}] (I_3_6__Rationals) at (-8,4) {$\begin{smallmatrix}2&&&3\\&4&5\\2&&&3\end{smallmatrix}$};
	\node at (-12,2) {$\dots$};
	\draw[red,->] (I_1_5__Rationals) -- (I_0_1__Rationals);
	\draw[red,->] (I_1_5__Rationals) -- (I_0_2__Rationals);
	\draw[red,->] (I_0_6__Rationals) -- (I_0_3__Rationals);
	\draw[red,->] (I_0_6__Rationals) -- (I_0_4__Rationals);
	\draw[red,->] (I_0_1__Rationals) -- (I_0_5__Rationals);
	\draw[red,->] (I_0_2__Rationals) -- (I_0_5__Rationals);
	\draw[red,->] (I_1_6__Rationals) -- (I_0_5__Rationals);
	\draw[red,->] (I_0_5__Rationals) -- (I_0_6__Rationals);
	\draw[blue,->] (I_1_3__Rationals) -- (I_0_6__Rationals);
	\draw[blue,->] (I_1_4__Rationals) -- (I_0_6__Rationals);
	\draw[gray,dashed,->] (I_0_1__Rationals) -- (I_1_1__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_2_5__Rationals) -- (I_1_1__Rationals);
	\draw[gray,dashed,->] (I_0_2__Rationals) -- (I_1_2__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_2_5__Rationals) -- (I_1_2__Rationals);
	\draw[gray,dashed,->] (I_0_3__Rationals) -- (I_1_3__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_1_6__Rationals) -- (I_1_3__Rationals);
	\draw[gray,dashed,->] (I_0_4__Rationals) -- (I_1_4__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_1_6__Rationals) -- (I_1_4__Rationals);
	\draw[gray,dashed,->] (I_0_5__Rationals) -- (I_1_5__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[blue,->] (I_1_1__Rationals) -- (I_1_5__Rationals);
	\draw[blue,->] (I_1_2__Rationals) -- (I_1_5__Rationals);
	\draw[red,->] (I_2_6__Rationals) -- (I_1_5__Rationals);
	\draw[gray,dashed,->] (I_0_6__Rationals) -- (I_1_6__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_1_5__Rationals) -- (I_1_6__Rationals);
	\draw[blue,->] (I_2_3__Rationals) -- (I_1_6__Rationals);
	\draw[blue,->] (I_2_4__Rationals) -- (I_1_6__Rationals);
	\draw[gray,dashed,->] (I_1_1__Rationals) -- (I_2_1__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_3_5__Rationals) -- (I_2_1__Rationals);
	\draw[gray,dashed,->] (I_1_2__Rationals) -- (I_2_2__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_3_5__Rationals) -- (I_2_2__Rationals);
	\draw[gray,dashed,->] (I_1_3__Rationals) -- (I_2_3__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_2_6__Rationals) -- (I_2_3__Rationals);
	\draw[gray,dashed,->] (I_1_4__Rationals) -- (I_2_4__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_2_6__Rationals) -- (I_2_4__Rationals);
	\draw[gray,dashed,->] (I_1_5__Rationals) -- (I_2_5__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[blue,->] (I_2_1__Rationals) -- (I_2_5__Rationals);
	\draw[blue,->] (I_2_2__Rationals) -- (I_2_5__Rationals);
	\draw[red,->] (I_3_6__Rationals) -- (I_2_5__Rationals);
	\draw[gray,dashed,->] (I_1_6__Rationals) -- (I_2_6__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_2_5__Rationals) -- (I_2_6__Rationals);
	\draw[blue,->] (I_3_3__Rationals) -- (I_2_6__Rationals);
	\draw[blue,->] (I_3_4__Rationals) -- (I_2_6__Rationals);
	\draw[gray,dashed,->] (I_2_1__Rationals) -- (I_3_1__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[gray,dashed,->] (I_2_2__Rationals) -- (I_3_2__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[gray,dashed,->] (I_2_3__Rationals) -- (I_3_3__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_3_6__Rationals) -- (I_3_3__Rationals);
	\draw[gray,dashed,->] (I_2_4__Rationals) -- (I_3_4__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_3_6__Rationals) -- (I_3_4__Rationals);
	\draw[gray,dashed,->] (I_2_5__Rationals) -- (I_3_5__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[blue,->] (I_3_1__Rationals) -- (I_3_5__Rationals);
	\draw[blue,->] (I_3_2__Rationals) -- (I_3_5__Rationals);
	\draw[gray,dashed,->] (I_2_6__Rationals) -- (I_3_6__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_3_5__Rationals) -- (I_3_6__Rationals);
\end{tikzpicture}
\end{scaletikzpicturetowidth}
\end{center}

\end{figure}
\medskip{}
\subsubsection*{Modules of the form $I(n,1)$}

Defect: $\partial I(n,1) = 1$, for $n\ge 0$.

\begin{fleqn}
\begin{align*}
I(0,1):\ & \bigl(I(0,3),R_{0}^{2}(1)\bigr),\ \bigl(I(0,4),R_{\infty}^{2}(1)\bigr),\ \bigl(I(0,5),P(0,1)\bigr),\ \bigl(I(0,6),P(1,2)\bigr)\\
I(1,1):\ & \bigl(I(0,2),R_{1}^{1}(1)\bigr),\ \bigl(I(1,3),R_{0}^{1}(1)\bigr),\ \bigl(I(1,4),R_{\infty}^{1}(1)\bigr),\ \bigl(I(1,6),P(0,2)\bigr)\\
I(2,1):\ & \bigl(I(0,1),R_{1}^{3}(2)\bigr),\ \bigl(I(1,2),R_{1}^{3}(1)\bigr),\ \bigl(I(2,3),R_{0}^{2}(1)\bigr),\ \bigl(I(2,4),R_{\infty}^{2}(1)\bigr)\\
I(3,1):\ & \bigl(I(1,1),R_{1}^{2}(2)\bigr),\ \bigl(I(2,2),R_{1}^{2}(1)\bigr),\ \bigl(I(3,3),R_{0}^{1}(1)\bigr),\ \bigl(I(3,4),R_{\infty}^{1}(1)\bigr),\ \bigl(2I(0,2),P(2,1)\bigr)\\
I(n,1):\ & \bigl(I(n-2,1),R_{1}^{(-n+4)\bmod 3+1}(2)\bigr),\ \bigl(I(n-1,2),R_{1}^{(-n+4)\bmod 3+1}(1)\bigr),\ \bigl(I(n,3),R_{0}^{(-n+3)\bmod 2+1}(1)\bigr)\\
 & \bigl(I(n,4),R_{\infty}^{(-n+3)\bmod 2+1}(1)\bigr),\ \bigl((v+1)I,vP\bigr),\ n>3\\
\end{align*}
\end{fleqn}
\subsubsection*{Modules of the form $I(n,2)$}

Defect: $\partial I(n,2) = 1$, for $n\ge 0$.

\begin{fleqn}
\begin{align*}
I(0,2):\ & \bigl(I(0,3),R_{\infty}^{1}(1)\bigr),\ \bigl(I(0,4),R_{0}^{1}(1)\bigr),\ \bigl(I(0,5),P(0,2)\bigr),\ \bigl(I(0,6),P(1,1)\bigr)\\
I(1,2):\ & \bigl(I(0,1),R_{1}^{1}(1)\bigr),\ \bigl(I(1,3),R_{\infty}^{2}(1)\bigr),\ \bigl(I(1,4),R_{0}^{2}(1)\bigr),\ \bigl(I(1,6),P(0,1)\bigr)\\
I(2,2):\ & \bigl(I(0,2),R_{1}^{3}(2)\bigr),\ \bigl(I(1,1),R_{1}^{3}(1)\bigr),\ \bigl(I(2,3),R_{\infty}^{1}(1)\bigr),\ \bigl(I(2,4),R_{0}^{1}(1)\bigr)\\
I(3,2):\ & \bigl(I(1,2),R_{1}^{2}(2)\bigr),\ \bigl(I(2,1),R_{1}^{2}(1)\bigr),\ \bigl(I(3,3),R_{\infty}^{2}(1)\bigr),\ \bigl(I(3,4),R_{0}^{2}(1)\bigr),\ \bigl(2I(0,1),P(2,2)\bigr)\\
I(n,2):\ & \bigl(I(n-2,2),R_{1}^{(-n+4)\bmod 3+1}(2)\bigr),\ \bigl(I(n-1,1),R_{1}^{(-n+4)\bmod 3+1}(1)\bigr),\ \bigl(I(n,3),R_{\infty}^{(-n+4)\bmod 2+1}(1)\bigr)\\
 & \bigl(I(n,4),R_{0}^{(-n+4)\bmod 2+1}(1)\bigr),\ \bigl((v+1)I,vP\bigr),\ n>3\\
\end{align*}
\end{fleqn}
\subsubsection*{Modules of the form $I(n,3)$}

Defect: $\partial I(n,3) = 1$, for $n\ge 0$.

\begin{fleqn}
\begin{align*}
I(0,3):\ & - \\
I(1,3):\ & \bigl(I(0,4),R_{1}^{1}(1)\bigr)\\
I(2,3):\ & \bigl(I(0,3),R_{1}^{3}(2)\bigr),\ \bigl(I(1,4),R_{1}^{3}(1)\bigr)\\
I(3,3):\ & \bigl(I(0,1),R_{\infty}^{1}(1)\bigr),\ \bigl(I(0,2),R_{0}^{2}(1)\bigr),\ \bigl(I(1,3),R_{1}^{2}(2)\bigr),\ \bigl(I(2,4),R_{1}^{2}(1)\bigr),\ \bigl(2I(0,4),P(2,3)\bigr)\\
I(n,3):\ & \bigl(I(n-3,1),R_{\infty}^{(-n+3)\bmod 2+1}(1)\bigr),\ \bigl(I(n-3,2),R_{0}^{(-n+4)\bmod 2+1}(1)\bigr),\ \bigl(I(n-2,3),R_{1}^{(-n+4)\bmod 3+1}(2)\bigr)\\
 & \bigl(I(n-1,4),R_{1}^{(-n+4)\bmod 3+1}(1)\bigr),\ \bigl((v+1)I,vP\bigr),\ n>3\\
\end{align*}
\end{fleqn}
\subsubsection*{Modules of the form $I(n,4)$}

Defect: $\partial I(n,4) = 1$, for $n\ge 0$.

\begin{fleqn}
\begin{align*}
I(0,4):\ & - \\
I(1,4):\ & \bigl(I(0,3),R_{1}^{1}(1)\bigr)\\
I(2,4):\ & \bigl(I(0,4),R_{1}^{3}(2)\bigr),\ \bigl(I(1,3),R_{1}^{3}(1)\bigr)\\
I(3,4):\ & \bigl(I(0,1),R_{0}^{1}(1)\bigr),\ \bigl(I(0,2),R_{\infty}^{2}(1)\bigr),\ \bigl(I(1,4),R_{1}^{2}(2)\bigr),\ \bigl(I(2,3),R_{1}^{2}(1)\bigr),\ \bigl(2I(0,3),P(2,4)\bigr)\\
I(n,4):\ & \bigl(I(n-3,1),R_{0}^{(-n+3)\bmod 2+1}(1)\bigr),\ \bigl(I(n-3,2),R_{\infty}^{(-n+4)\bmod 2+1}(1)\bigr),\ \bigl(I(n-2,4),R_{1}^{(-n+4)\bmod 3+1}(2)\bigr)\\
 & \bigl(I(n-1,3),R_{1}^{(-n+4)\bmod 3+1}(1)\bigr),\ \bigl((v+1)I,vP\bigr),\ n>3\\
\end{align*}
\end{fleqn}
\subsubsection*{Modules of the form $I(n,5)$}

Defect: $\partial I(n,5) = 2$, for $n\ge 0$.

\begin{fleqn}
\begin{align*}
I(0,5):\ & \bigl(I(0,3),I(2,4)\bigr),\ \bigl(I(0,4),I(2,3)\bigr),\ \bigl(I(0,6),R_{1}^{3}(1)\bigr)\\
I(1,5):\ & \bigl(I(0,1),I(1,1)\bigr),\ \bigl(I(0,2),I(1,2)\bigr),\ \bigl(I(1,3),I(3,4)\bigr),\ \bigl(I(1,4),I(3,3)\bigr),\ \bigl(I(1,6),R_{1}^{2}(1)\bigr)\\
I(n,5):\ & \bigl(I(n-1,1),I(n,1)\bigr),\ \bigl(I(n-1,2),I(n,2)\bigr),\ \bigl(I(n,3),I(n+2,4)\bigr)\\
 & \bigl(I(n,4),I(n+2,3)\bigr),\ \bigl(I(n,6),R_{1}^{(-n+2)\bmod 3+1}(1)\bigr),\ n>1\\
\end{align*}
\end{fleqn}
\subsubsection*{Modules of the form $I(n,6)$}

Defect: $\partial I(n,6) = 2$, for $n\ge 0$.

\begin{fleqn}
\begin{align*}
I(0,6):\ & \bigl(I(0,3),I(1,3)\bigr),\ \bigl(I(0,4),I(1,4)\bigr)\\
I(1,6):\ & \bigl(I(0,5),R_{1}^{1}(1)\bigr),\ \bigl(I(1,3),I(2,3)\bigr),\ \bigl(I(1,4),I(2,4)\bigr)\\
I(2,6):\ & \bigl(I(0,1),I(2,2)\bigr),\ \bigl(I(0,2),I(2,1)\bigr),\ \bigl(I(1,5),R_{1}^{3}(1)\bigr),\ \bigl(I(2,3),I(3,3)\bigr),\ \bigl(I(2,4),I(3,4)\bigr)\\
I(n,6):\ & \bigl(I(n-2,1),I(n,2)\bigr),\ \bigl(I(n-2,2),I(n,1)\bigr),\ \bigl(I(n-1,5),R_{1}^{(-n+4)\bmod 3+1}(1)\bigr)\\
 & \bigl(I(n,3),I(n+1,3)\bigr),\ \bigl(I(n,4),I(n+1,4)\bigr),\ n>2\\
\end{align*}
\end{fleqn}
\subsubsection{Schofield pairs associated to regular exceptional modules}

\subsubsection*{The non-homogeneous tube $\mathcal{T}_{1}^{\Delta(\widetilde{\mathbb{D}}_{5})}$}

\begin{figure}[ht]

\begin{center}
\captionof{figure}{\vspace*{-10pt}$\mathcal{T}_{1}^{\Delta(\widetilde{\mathbb{D}}_{5})}$}
\begin{scaletikzpicturetowidth}{0.59999999999999998\textwidth}
\begin{tikzpicture}[scale=\tikzscale]
	\node[seltwoDDDDDD, label={[shift={(0.0,-0.45)}, brown]\tiny $R_{1}^{1}(1)$}] (RN_1_1_1__Rationals) at (4,0) {$\begin{smallmatrix}0&&&0\\&0&1\\0&&&0\end{smallmatrix}$};
	\node[seltwoDDDDDD, label={[shift={(0.0,-0.45)}, brown]\tiny $R_{1}^{2}(1)$}] (RN_1_2_1__Rationals) at (8,0) {$\begin{smallmatrix}1&&&1\\&1&1\\1&&&1\end{smallmatrix}$};
	\node[seltwoDDDDDD, label={[shift={(0.0,-0.45)}, brown]\tiny $R_{1}^{3}(1)$}] (RN_1_3_1__Rationals) at (12,0) {$\begin{smallmatrix}0&&&0\\&1&0\\0&&&0\end{smallmatrix}$};
	\node[seltwoDDDDDD, label={[shift={(0.0,-0.45)}, brown]\tiny $R_{1}^{1}(1)$}] (RN_1_1_1__Rationals_repeated) at (16,0) {$\begin{smallmatrix}0&&&0\\&0&1\\0&&&0\end{smallmatrix}$};
	\node[seltwoDDDDDD, label={[shift={(0.0,-0.45)}, brown]\tiny $R_{1}^{1}(2)$}] (RN_1_1_2__Rationals) at (6,2) {$\begin{smallmatrix}1&&&1\\&1&2\\1&&&1\end{smallmatrix}$};
	\node[seltwoDDDDDD, label={[shift={(0.0,-0.45)}, brown]\tiny $R_{1}^{2}(2)$}] (RN_1_2_2__Rationals) at (10,2) {$\begin{smallmatrix}1&&&1\\&2&1\\1&&&1\end{smallmatrix}$};
	\node[seltwoDDDDDD, label={[shift={(0.0,-0.45)}, brown]\tiny $R_{1}^{3}(2)$}] (RN_1_3_2__Rationals) at (14,2) {$\begin{smallmatrix}0&&&0\\&1&1\\0&&&0\end{smallmatrix}$};
	\node[noselDDDDDD, label={[shift={(0.0,-0.45)}, brown]\tiny $R_{1}^{3}(3)$}] (RN_1_3_3__Rationals) at (4,4) {$\begin{smallmatrix}1&&&1\\&2&2\\1&&&1\end{smallmatrix}$};
	\node[noselDDDDDD, label={[shift={(0.0,-0.45)}, brown]\tiny $R_{1}^{1}(3)$}] (RN_1_1_3__Rationals) at (8,4) {$\begin{smallmatrix}1&&&1\\&2&2\\1&&&1\end{smallmatrix}$};
	\node[noselDDDDDD, label={[shift={(0.0,-0.45)}, brown]\tiny $R_{1}^{2}(3)$}] (RN_1_2_3__Rationals) at (12,4) {$\begin{smallmatrix}1&&&1\\&2&2\\1&&&1\end{smallmatrix}$};
	\node[noselDDDDDD, label={[shift={(0.0,-0.45)}, brown]\tiny $R_{1}^{3}(3)$}] (RN_1_3_3__Rationals_repeated) at (16,4) {$\begin{smallmatrix}1&&&1\\&2&2\\1&&&1\end{smallmatrix}$};
	\node[noselDDDDDD, label={[shift={(0.0,-0.45)}, brown]\tiny $R_{1}^{3}(4)$}] (RN_1_3_4__Rationals) at (6,6) {$\begin{smallmatrix}1&&&1\\&3&2\\1&&&1\end{smallmatrix}$};
	\node[noselDDDDDD, label={[shift={(0.0,-0.45)}, brown]\tiny $R_{1}^{1}(4)$}] (RN_1_1_4__Rationals) at (10,6) {$\begin{smallmatrix}1&&&1\\&2&3\\1&&&1\end{smallmatrix}$};
	\node[noselDDDDDD, label={[shift={(0.0,-0.45)}, brown]\tiny $R_{1}^{2}(4)$}] (RN_1_2_4__Rationals) at (14,6) {$\begin{smallmatrix}2&&&2\\&3&3\\2&&&2\end{smallmatrix}$};
	\draw[blue,->] (RN_1_1_1__Rationals) -- (RN_1_1_2__Rationals);
	\draw[gray,dashed,->] (RN_1_1_1__Rationals) -- (RN_1_2_1__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[red,->] (RN_1_1_2__Rationals) -- (RN_1_2_1__Rationals);
	\draw[gray,dashed,->] (RN_1_1_2__Rationals) -- (RN_1_2_2__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[blue,->] (RN_1_1_2__Rationals) -- (RN_1_1_3__Rationals);
	\draw[blue,->] (RN_1_2_1__Rationals) -- (RN_1_2_2__Rationals);
	\draw[gray,dashed,->] (RN_1_2_1__Rationals) -- (RN_1_3_1__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[red,->] (RN_1_2_2__Rationals) -- (RN_1_3_1__Rationals);
	\draw[gray,dashed,->] (RN_1_2_2__Rationals) -- (RN_1_3_2__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[blue,->] (RN_1_2_2__Rationals) -- (RN_1_2_3__Rationals);
	\draw[gray,dashed,->] (RN_1_3_1__Rationals) -- (RN_1_1_1__Rationals_repeated) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[blue,->] (RN_1_3_1__Rationals) -- (RN_1_3_2__Rationals);
	\draw[red,->] (RN_1_3_2__Rationals) -- (RN_1_1_1__Rationals_repeated);
	\draw[gray,dashed,->,out=200,in=-20] (RN_1_3_2__Rationals) to node[midway,sloped,above]{{\tiny $\tau^{-1}$}} (RN_1_1_2__Rationals);
	\draw[blue,->] (RN_1_3_2__Rationals) -- (RN_1_3_3__Rationals_repeated);
	\draw[red,->] (RN_1_3_3__Rationals) -- (RN_1_1_2__Rationals);
	\draw[blue,->] (RN_1_3_3__Rationals) -- (RN_1_3_4__Rationals);
	\draw[gray,dashed,->] (RN_1_3_3__Rationals) -- (RN_1_1_3__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[red,->] (RN_1_3_4__Rationals) -- (RN_1_1_3__Rationals);
	\draw[gray,dashed,->] (RN_1_3_4__Rationals) -- (RN_1_1_4__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[red,->] (RN_1_1_3__Rationals) -- (RN_1_2_2__Rationals);
	\draw[blue,->] (RN_1_1_3__Rationals) -- (RN_1_1_4__Rationals);
	\draw[gray,dashed,->] (RN_1_1_3__Rationals) -- (RN_1_2_3__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[red,->] (RN_1_1_4__Rationals) -- (RN_1_2_3__Rationals);
	\draw[gray,dashed,->] (RN_1_1_4__Rationals) -- (RN_1_2_4__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[red,->] (RN_1_2_3__Rationals) -- (RN_1_3_2__Rationals);
	\draw[gray,dashed,->] (RN_1_2_3__Rationals) -- (RN_1_3_3__Rationals_repeated) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[blue,->] (RN_1_2_3__Rationals) -- (RN_1_2_4__Rationals);
	\draw[red,->] (RN_1_2_4__Rationals) -- (RN_1_3_3__Rationals_repeated);
	\draw[gray,dashed,->,out=200,in=-20] (RN_1_2_4__Rationals) to node[midway,sloped,above]{{\tiny $\tau^{-1}$}} (RN_1_3_4__Rationals);
	\draw[dashed,-] (RN_1_1_1__Rationals) -- (RN_1_3_3__Rationals);
	\draw[dashed,-] (RN_1_1_1__Rationals_repeated) -- (RN_1_3_3__Rationals_repeated);
	\node (LeftTopPhantom_1) at (4,6) {};
	\draw[dashed,-] (RN_1_3_3__Rationals) -- (LeftTopPhantom_1);
	\node (RightTopPhantom_1) at (16,6) {};
	\draw[dashed,-] (RN_1_3_3__Rationals_repeated) -- (RightTopPhantom_1);
\end{tikzpicture}
\end{scaletikzpicturetowidth}
\end{center}

\end{figure}
\begin{fleqn}
\begin{align*}
R_{1}^{1}(1):\ & - \\
R_{1}^{1}(2):\ & \bigl(R_{1}^{2}(1),R_{1}^{1}(1)\bigr),\ \bigl(I(1,1),P(0,1)\bigr),\ \bigl(I(1,2),P(0,2)\bigr),\ \bigl(I(1,3),P(0,3)\bigr),\ \bigl(I(1,4),P(0,4)\bigr)\\
R_{1}^{2}(1):\ & \bigl(I(0,2),P(0,1)\bigr),\ \bigl(I(0,1),P(0,2)\bigr),\ \bigl(I(0,4),P(0,3)\bigr),\ \bigl(I(0,3),P(0,4)\bigr),\ \bigl(I(0,6),P(0,5)\bigr)\\
R_{1}^{2}(2):\ & \bigl(R_{1}^{3}(1),R_{1}^{2}(1)\bigr),\ \bigl(I(0,1),P(1,1)\bigr),\ \bigl(I(0,2),P(1,2)\bigr),\ \bigl(I(0,3),P(1,3)\bigr),\ \bigl(I(0,4),P(1,4)\bigr)\\
R_{1}^{3}(1):\ & - \\
R_{1}^{3}(2):\ & \bigl(R_{1}^{1}(1),R_{1}^{3}(1)\bigr)\\
\end{align*}
\end{fleqn}
\subsubsection*{The non-homogeneous tube $\mathcal{T}_{\infty}^{\Delta(\widetilde{\mathbb{D}}_{5})}$}

\begin{figure}[ht]

\begin{center}
\captionof{figure}{\vspace*{-10pt}$\mathcal{T}_{\infty}^{\Delta(\widetilde{\mathbb{D}}_{5})}$}
\begin{scaletikzpicturetowidth}{0.5\textwidth}
\begin{tikzpicture}[scale=\tikzscale]
	\node[seltwoDDDDDD, label={[shift={(0.0,-0.45)}, brown]\tiny $R_{\infty}^{1}(1)$}] (RN_2_1_1__Rationals) at (4,0) {$\begin{smallmatrix}1&&&0\\&1&1\\0&&&1\end{smallmatrix}$};
	\node[seltwoDDDDDD, label={[shift={(0.0,-0.45)}, brown]\tiny $R_{\infty}^{2}(1)$}] (RN_2_2_1__Rationals) at (8,0) {$\begin{smallmatrix}0&&&1\\&1&1\\1&&&0\end{smallmatrix}$};
	\node[seltwoDDDDDD, label={[shift={(0.0,-0.45)}, brown]\tiny $R_{\infty}^{1}(1)$}] (RN_2_1_1__Rationals_repeated) at (12,0) {$\begin{smallmatrix}1&&&0\\&1&1\\0&&&1\end{smallmatrix}$};
	\node[noselDDDDDD, label={[shift={(0.0,-0.45)}, brown]\tiny $R_{\infty}^{1}(2)$}] (RN_2_1_2__Rationals) at (6,2) {$\begin{smallmatrix}1&&&1\\&2&2\\1&&&1\end{smallmatrix}$};
	\node[noselDDDDDD, label={[shift={(0.0,-0.45)}, brown]\tiny $R_{\infty}^{2}(2)$}] (RN_2_2_2__Rationals) at (10,2) {$\begin{smallmatrix}1&&&1\\&2&2\\1&&&1\end{smallmatrix}$};
	\node[noselDDDDDD, label={[shift={(0.0,-0.45)}, brown]\tiny $R_{\infty}^{2}(3)$}] (RN_2_2_3__Rationals) at (4,4) {$\begin{smallmatrix}1&&&2\\&3&3\\2&&&1\end{smallmatrix}$};
	\node[noselDDDDDD, label={[shift={(0.0,-0.45)}, brown]\tiny $R_{\infty}^{1}(3)$}] (RN_2_1_3__Rationals) at (8,4) {$\begin{smallmatrix}2&&&1\\&3&3\\1&&&2\end{smallmatrix}$};
	\node[noselDDDDDD, label={[shift={(0.0,-0.45)}, brown]\tiny $R_{\infty}^{2}(3)$}] (RN_2_2_3__Rationals_repeated) at (12,4) {$\begin{smallmatrix}1&&&2\\&3&3\\2&&&1\end{smallmatrix}$};
	\draw[blue,->] (RN_2_1_1__Rationals) -- (RN_2_1_2__Rationals);
	\draw[gray,dashed,->] (RN_2_1_1__Rationals) -- (RN_2_2_1__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[red,->] (RN_2_1_2__Rationals) -- (RN_2_2_1__Rationals);
	\draw[gray,dashed,->] (RN_2_1_2__Rationals) -- (RN_2_2_2__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[blue,->] (RN_2_1_2__Rationals) -- (RN_2_1_3__Rationals);
	\draw[gray,dashed,->] (RN_2_2_1__Rationals) -- (RN_2_1_1__Rationals_repeated) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[blue,->] (RN_2_2_1__Rationals) -- (RN_2_2_2__Rationals);
	\draw[red,->] (RN_2_2_2__Rationals) -- (RN_2_1_1__Rationals_repeated);
	\draw[gray,dashed,->,out=219,in=-39] (RN_2_2_2__Rationals) to node[midway,sloped,above]{{\tiny $\tau^{-1}$}} (RN_2_1_2__Rationals);
	\draw[blue,->] (RN_2_2_2__Rationals) -- (RN_2_2_3__Rationals_repeated);
	\draw[red,->] (RN_2_2_3__Rationals) -- (RN_2_1_2__Rationals);
	\draw[gray,dashed,->] (RN_2_2_3__Rationals) -- (RN_2_1_3__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[red,->] (RN_2_1_3__Rationals) -- (RN_2_2_2__Rationals);
	\draw[gray,dashed,->] (RN_2_1_3__Rationals) -- (RN_2_2_3__Rationals_repeated) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[dashed,-] (RN_2_1_1__Rationals) -- (RN_2_2_3__Rationals);
	\draw[dashed,-] (RN_2_1_1__Rationals_repeated) -- (RN_2_2_3__Rationals_repeated);
\end{tikzpicture}
\end{scaletikzpicturetowidth}
\end{center}

\end{figure}
\begin{fleqn}
\begin{align*}
R_{\infty}^{1}(1):\ & \bigl(I(2,4),P(0,2)\bigr),\ \bigl(I(1,3),P(1,1)\bigr),\ \bigl(I(0,4),P(2,2)\bigr)\\
R_{\infty}^{2}(1):\ & \bigl(I(2,3),P(0,1)\bigr),\ \bigl(I(1,4),P(1,2)\bigr),\ \bigl(I(0,3),P(2,1)\bigr)\\
\end{align*}
\end{fleqn}
\subsubsection*{The non-homogeneous tube $\mathcal{T}_{0}^{\Delta(\widetilde{\mathbb{D}}_{5})}$}

\begin{figure}[ht]

\begin{center}
\captionof{figure}{\vspace*{-10pt}$\mathcal{T}_{0}^{\Delta(\widetilde{\mathbb{D}}_{5})}$}
\begin{scaletikzpicturetowidth}{0.5\textwidth}
\begin{tikzpicture}[scale=\tikzscale]
	\node[seltwoDDDDDD, label={[shift={(0.0,-0.45)}, brown]\tiny $R_{0}^{1}(1)$}] (RN_3_1_1__Rationals) at (4,0) {$\begin{smallmatrix}1&&&1\\&1&1\\0&&&0\end{smallmatrix}$};
	\node[seltwoDDDDDD, label={[shift={(0.0,-0.45)}, brown]\tiny $R_{0}^{2}(1)$}] (RN_3_2_1__Rationals) at (8,0) {$\begin{smallmatrix}0&&&0\\&1&1\\1&&&1\end{smallmatrix}$};
	\node[seltwoDDDDDD, label={[shift={(0.0,-0.45)}, brown]\tiny $R_{0}^{1}(1)$}] (RN_3_1_1__Rationals_repeated) at (12,0) {$\begin{smallmatrix}1&&&1\\&1&1\\0&&&0\end{smallmatrix}$};
	\node[noselDDDDDD, label={[shift={(0.0,-0.45)}, brown]\tiny $R_{0}^{1}(2)$}] (RN_3_1_2__Rationals) at (6,2) {$\begin{smallmatrix}1&&&1\\&2&2\\1&&&1\end{smallmatrix}$};
	\node[noselDDDDDD, label={[shift={(0.0,-0.45)}, brown]\tiny $R_{0}^{2}(2)$}] (RN_3_2_2__Rationals) at (10,2) {$\begin{smallmatrix}1&&&1\\&2&2\\1&&&1\end{smallmatrix}$};
	\node[noselDDDDDD, label={[shift={(0.0,-0.45)}, brown]\tiny $R_{0}^{2}(3)$}] (RN_3_2_3__Rationals) at (4,4) {$\begin{smallmatrix}1&&&1\\&3&3\\2&&&2\end{smallmatrix}$};
	\node[noselDDDDDD, label={[shift={(0.0,-0.45)}, brown]\tiny $R_{0}^{1}(3)$}] (RN_3_1_3__Rationals) at (8,4) {$\begin{smallmatrix}2&&&2\\&3&3\\1&&&1\end{smallmatrix}$};
	\node[noselDDDDDD, label={[shift={(0.0,-0.45)}, brown]\tiny $R_{0}^{2}(3)$}] (RN_3_2_3__Rationals_repeated) at (12,4) {$\begin{smallmatrix}1&&&1\\&3&3\\2&&&2\end{smallmatrix}$};
	\draw[blue,->] (RN_3_1_1__Rationals) -- (RN_3_1_2__Rationals);
	\draw[gray,dashed,->] (RN_3_1_1__Rationals) -- (RN_3_2_1__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[red,->] (RN_3_1_2__Rationals) -- (RN_3_2_1__Rationals);
	\draw[gray,dashed,->] (RN_3_1_2__Rationals) -- (RN_3_2_2__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[blue,->] (RN_3_1_2__Rationals) -- (RN_3_1_3__Rationals);
	\draw[gray,dashed,->] (RN_3_2_1__Rationals) -- (RN_3_1_1__Rationals_repeated) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[blue,->] (RN_3_2_1__Rationals) -- (RN_3_2_2__Rationals);
	\draw[red,->] (RN_3_2_2__Rationals) -- (RN_3_1_1__Rationals_repeated);
	\draw[gray,dashed,->,out=219,in=-39] (RN_3_2_2__Rationals) to node[midway,sloped,above]{{\tiny $\tau^{-1}$}} (RN_3_1_2__Rationals);
	\draw[blue,->] (RN_3_2_2__Rationals) -- (RN_3_2_3__Rationals_repeated);
	\draw[red,->] (RN_3_2_3__Rationals) -- (RN_3_1_2__Rationals);
	\draw[gray,dashed,->] (RN_3_2_3__Rationals) -- (RN_3_1_3__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[red,->] (RN_3_1_3__Rationals) -- (RN_3_2_2__Rationals);
	\draw[gray,dashed,->] (RN_3_1_3__Rationals) -- (RN_3_2_3__Rationals_repeated) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[dashed,-] (RN_3_1_1__Rationals) -- (RN_3_2_3__Rationals);
	\draw[dashed,-] (RN_3_1_1__Rationals_repeated) -- (RN_3_2_3__Rationals_repeated);
\end{tikzpicture}
\end{scaletikzpicturetowidth}
\end{center}

\end{figure}
\begin{fleqn}
\begin{align*}
R_{0}^{1}(1):\ & \bigl(I(2,3),P(0,2)\bigr),\ \bigl(I(1,4),P(1,1)\bigr),\ \bigl(I(0,3),P(2,2)\bigr)\\
R_{0}^{2}(1):\ & \bigl(I(2,4),P(0,1)\bigr),\ \bigl(I(1,3),P(1,2)\bigr),\ \bigl(I(0,4),P(2,1)\bigr)\\
\end{align*}
\end{fleqn}
\clearpage

\subsection[Schofield pairs for the quiver $\Delta(\widetilde{\mathbb{D}}_6)$]{Schofield pairs for the quiver $\Delta(\widetilde{\mathbb{D}}_6)$ -- {\usebox{\boxDDDDDDD}}}
\[
\vcenter{\hbox{\xymatrix{2 &   &   &   & 3\ar[dl]\\
  & 5\ar[dl]\ar[ul] & 6\ar[l] & 7\ar[l] &  \\
1 &   &   &   & 4\ar[ul]}}}\qquad C_{\Delta(\widetilde{\mathbb{D}}_{6})} = \begin{bmatrix}1 & 0 & 1 & 1 & 1 & 1 & 1\\
0 & 1 & 1 & 1 & 1 & 1 & 1\\
0 & 0 & 1 & 0 & 0 & 0 & 0\\
0 & 0 & 0 & 1 & 0 & 0 & 0\\
0 & 0 & 1 & 1 & 1 & 1 & 1\\
0 & 0 & 1 & 1 & 0 & 1 & 1\\
0 & 0 & 1 & 1 & 0 & 0 & 1\end{bmatrix}\quad\Phi_{\Delta(\widetilde{\mathbb{D}}_{6})} = \begin{bmatrix}-1 & 0 & 0 & 0 & 1 & 0 & 0\\
0 & -1 & 0 & 0 & 1 & 0 & 0\\
-1 & -1 & 0 & 1 & 1 & 0 & 0\\
-1 & -1 & 1 & 0 & 1 & 0 & 0\\
-1 & -1 & 0 & 0 & 1 & 1 & 0\\
-1 & -1 & 0 & 0 & 1 & 0 & 1\\
-1 & -1 & 1 & 1 & 1 & 0 & 0\end{bmatrix}
\]
\subsubsection{Schofield pairs associated to preprojective exceptional modules}

\begin{figure}[ht]

\begin{center}
\begin{scaletikzpicturetowidth}{\textwidth}
\begin{tikzpicture}[scale=\tikzscale]
	\node[noselDDDDDDD, label={[shift={(0.0,-0.45)}, brown]\tiny $P(0,1)$}] (P_0_1__Rationals) at (0,0) {$\begin{smallmatrix}0&&&&0\\&0&0&0\\1&&&&0\end{smallmatrix}$};
	\node[noselDDDDDDD, label={[shift={(0.0,-0.45)}, brown]\tiny $P(0,2)$}] (P_0_2__Rationals) at (0,1) {$\begin{smallmatrix}1&&&&0\\&0&0&0\\0&&&&0\end{smallmatrix}$};
	\node[noselDDDDDDD, label={[shift={(0.0,-0.45)}, brown]\tiny $P(0,3)$}] (P_0_3__Rationals) at (8,8) {$\begin{smallmatrix}1&&&&1\\&1&1&1\\1&&&&0\end{smallmatrix}$};
	\node[noselDDDDDDD, label={[shift={(0.0,-0.45)}, brown]\tiny $P(0,4)$}] (P_0_4__Rationals) at (8,7) {$\begin{smallmatrix}1&&&&0\\&1&1&1\\1&&&&1\end{smallmatrix}$};
	\node[noselDDDDDDD, label={[shift={(0.0,-0.45)}, brown]\tiny $P(0,5)$}] (P_0_5__Rationals) at (2,2) {$\begin{smallmatrix}1&&&&0\\&1&0&0\\1&&&&0\end{smallmatrix}$};
	\node[noselDDDDDDD, label={[shift={(0.0,-0.45)}, brown]\tiny $P(0,6)$}] (P_0_6__Rationals) at (4,4) {$\begin{smallmatrix}1&&&&0\\&1&1&0\\1&&&&0\end{smallmatrix}$};
	\node[noselDDDDDDD, label={[shift={(0.0,-0.45)}, brown]\tiny $P(0,7)$}] (P_0_7__Rationals) at (6,6) {$\begin{smallmatrix}1&&&&0\\&1&1&1\\1&&&&0\end{smallmatrix}$};
	\node[noselDDDDDDD, label={[shift={(0.0,-0.45)}, brown]\tiny $P(1,1)$}] (P_1_1__Rationals) at (4,0) {$\begin{smallmatrix}1&&&&0\\&1&0&0\\0&&&&0\end{smallmatrix}$};
	\node[noselDDDDDDD, label={[shift={(0.0,-0.45)}, brown]\tiny $P(1,2)$}] (P_1_2__Rationals) at (4,1) {$\begin{smallmatrix}0&&&&0\\&1&0&0\\1&&&&0\end{smallmatrix}$};
	\node[noselDDDDDDD, label={[shift={(0.0,-0.45)}, brown]\tiny $P(1,3)$}] (P_1_3__Rationals) at (12,8) {$\begin{smallmatrix}1&&&&0\\&2&1&1\\1&&&&1\end{smallmatrix}$};
	\node[noselDDDDDDD, label={[shift={(0.0,-0.45)}, brown]\tiny $P(1,4)$}] (P_1_4__Rationals) at (12,7) {$\begin{smallmatrix}1&&&&1\\&2&1&1\\1&&&&0\end{smallmatrix}$};
	\node[noselDDDDDDD, label={[shift={(0.0,-0.45)}, brown]\tiny $P(1,5)$}] (P_1_5__Rationals) at (6,2) {$\begin{smallmatrix}1&&&&0\\&2&1&0\\1&&&&0\end{smallmatrix}$};
	\node[noselDDDDDDD, label={[shift={(0.0,-0.45)}, brown]\tiny $P(1,6)$}] (P_1_6__Rationals) at (8,4) {$\begin{smallmatrix}1&&&&0\\&2&1&1\\1&&&&0\end{smallmatrix}$};
	\node[seloneDDDDDDD, label={[shift={(0.0,-0.45)}, brown]\tiny $P(1,7)$}] (P_1_7__Rationals) at (10,6) {$\begin{smallmatrix}2&&&&1\\&3&2&2\\2&&&&1\end{smallmatrix}$};
	\node[noselDDDDDDD, label={[shift={(0.0,-0.45)}, brown]\tiny $P(2,1)$}] (P_2_1__Rationals) at (8,0) {$\begin{smallmatrix}0&&&&0\\&1&1&0\\1&&&&0\end{smallmatrix}$};
	\node[noselDDDDDDD, label={[shift={(0.0,-0.45)}, brown]\tiny $P(2,2)$}] (P_2_2__Rationals) at (8,1) {$\begin{smallmatrix}1&&&&0\\&1&1&0\\0&&&&0\end{smallmatrix}$};
	\node[noselDDDDDDD, label={[shift={(0.0,-0.45)}, brown]\tiny $P(2,3)$}] (P_2_3__Rationals) at (16,8) {$\begin{smallmatrix}1&&&&1\\&2&2&1\\1&&&&0\end{smallmatrix}$};
	\node[noselDDDDDDD, label={[shift={(0.0,-0.45)}, brown]\tiny $P(2,4)$}] (P_2_4__Rationals) at (16,7) {$\begin{smallmatrix}1&&&&0\\&2&2&1\\1&&&&1\end{smallmatrix}$};
	\node[noselDDDDDDD, label={[shift={(0.0,-0.45)}, brown]\tiny $P(2,5)$}] (P_2_5__Rationals) at (10,2) {$\begin{smallmatrix}1&&&&0\\&2&2&1\\1&&&&0\end{smallmatrix}$};
	\node[seloneDDDDDDD, label={[shift={(0.0,-0.45)}, brown]\tiny $P(2,6)$}] (P_2_6__Rationals) at (12,4) {$\begin{smallmatrix}2&&&&1\\&3&3&2\\2&&&&1\end{smallmatrix}$};
	\node[noselDDDDDDD, label={[shift={(0.0,-0.45)}, brown]\tiny $P(2,7)$}] (P_2_7__Rationals) at (14,6) {$\begin{smallmatrix}2&&&&1\\&4&3&2\\2&&&&1\end{smallmatrix}$};
	\node[noselDDDDDDD, label={[shift={(0.0,-0.45)}, brown]\tiny $P(3,1)$}] (P_3_1__Rationals) at (12,0) {$\begin{smallmatrix}1&&&&0\\&1&1&1\\0&&&&0\end{smallmatrix}$};
	\node[noselDDDDDDD, label={[shift={(0.0,-0.45)}, brown]\tiny $P(3,2)$}] (P_3_2__Rationals) at (12,1) {$\begin{smallmatrix}0&&&&0\\&1&1&1\\1&&&&0\end{smallmatrix}$};
	\node[noselDDDDDDD, label={[shift={(0.0,-0.45)}, brown]\tiny $P(3,3)$}] (P_3_3__Rationals) at (20,8) {$\begin{smallmatrix}1&&&&0\\&2&2&2\\1&&&&1\end{smallmatrix}$};
	\node[noselDDDDDDD, label={[shift={(0.0,-0.45)}, brown]\tiny $P(3,4)$}] (P_3_4__Rationals) at (20,7) {$\begin{smallmatrix}1&&&&1\\&2&2&2\\1&&&&0\end{smallmatrix}$};
	\node[seloneDDDDDDD, label={[shift={(0.0,-0.45)}, brown]\tiny $P(3,5)$}] (P_3_5__Rationals) at (14,2) {$\begin{smallmatrix}2&&&&1\\&3&3&3\\2&&&&1\end{smallmatrix}$};
	\node[noselDDDDDDD, label={[shift={(0.0,-0.45)}, brown]\tiny $P(3,6)$}] (P_3_6__Rationals) at (16,4) {$\begin{smallmatrix}2&&&&1\\&4&3&3\\2&&&&1\end{smallmatrix}$};
	\node[noselDDDDDDD, label={[shift={(0.0,-0.45)}, brown]\tiny $P(3,7)$}] (P_3_7__Rationals) at (18,6) {$\begin{smallmatrix}2&&&&1\\&4&4&3\\2&&&&1\end{smallmatrix}$};
	\node at (20,6) {$\dots$};
	\draw[blue,->] (P_0_7__Rationals) -- (P_0_3__Rationals);
	\draw[blue,->] (P_0_7__Rationals) -- (P_0_4__Rationals);
	\draw[blue,->] (P_0_1__Rationals) -- (P_0_5__Rationals);
	\draw[blue,->] (P_0_2__Rationals) -- (P_0_5__Rationals);
	\draw[blue,->] (P_0_5__Rationals) -- (P_0_6__Rationals);
	\draw[blue,->] (P_0_6__Rationals) -- (P_0_7__Rationals);
	\draw[gray,dashed,->] (P_0_1__Rationals) -- (P_1_1__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[red,->] (P_0_5__Rationals) -- (P_1_1__Rationals);
	\draw[gray,dashed,->] (P_0_2__Rationals) -- (P_1_2__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[red,->] (P_0_5__Rationals) -- (P_1_2__Rationals);
	\draw[gray,dashed,->] (P_0_3__Rationals) -- (P_1_3__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[red,->] (P_1_7__Rationals) -- (P_1_3__Rationals);
	\draw[gray,dashed,->] (P_0_4__Rationals) -- (P_1_4__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[red,->] (P_1_7__Rationals) -- (P_1_4__Rationals);
	\draw[gray,dashed,->] (P_0_5__Rationals) -- (P_1_5__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[blue,->] (P_0_6__Rationals) -- (P_1_5__Rationals);
	\draw[blue,->] (P_1_1__Rationals) -- (P_1_5__Rationals);
	\draw[blue,->] (P_1_2__Rationals) -- (P_1_5__Rationals);
	\draw[gray,dashed,->] (P_0_6__Rationals) -- (P_1_6__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[blue,->] (P_0_7__Rationals) -- (P_1_6__Rationals);
	\draw[blue,->] (P_1_5__Rationals) -- (P_1_6__Rationals);
	\draw[blue,->] (P_0_3__Rationals) -- (P_1_7__Rationals);
	\draw[blue,->] (P_0_4__Rationals) -- (P_1_7__Rationals);
	\draw[gray,dashed,->] (P_0_7__Rationals) -- (P_1_7__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[blue,->] (P_1_6__Rationals) -- (P_1_7__Rationals);
	\draw[gray,dashed,->] (P_1_1__Rationals) -- (P_2_1__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[red,->] (P_1_5__Rationals) -- (P_2_1__Rationals);
	\draw[gray,dashed,->] (P_1_2__Rationals) -- (P_2_2__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[red,->] (P_1_5__Rationals) -- (P_2_2__Rationals);
	\draw[gray,dashed,->] (P_1_3__Rationals) -- (P_2_3__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[red,->] (P_2_7__Rationals) -- (P_2_3__Rationals);
	\draw[gray,dashed,->] (P_1_4__Rationals) -- (P_2_4__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[red,->] (P_2_7__Rationals) -- (P_2_4__Rationals);
	\draw[gray,dashed,->] (P_1_5__Rationals) -- (P_2_5__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[blue,->] (P_1_6__Rationals) -- (P_2_5__Rationals);
	\draw[blue,->] (P_2_1__Rationals) -- (P_2_5__Rationals);
	\draw[blue,->] (P_2_2__Rationals) -- (P_2_5__Rationals);
	\draw[gray,dashed,->] (P_1_6__Rationals) -- (P_2_6__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[blue,->] (P_1_7__Rationals) -- (P_2_6__Rationals);
	\draw[blue,->] (P_2_5__Rationals) -- (P_2_6__Rationals);
	\draw[blue,->] (P_1_3__Rationals) -- (P_2_7__Rationals);
	\draw[blue,->] (P_1_4__Rationals) -- (P_2_7__Rationals);
	\draw[gray,dashed,->] (P_1_7__Rationals) -- (P_2_7__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[blue,->] (P_2_6__Rationals) -- (P_2_7__Rationals);
	\draw[gray,dashed,->] (P_2_1__Rationals) -- (P_3_1__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[red,->] (P_2_5__Rationals) -- (P_3_1__Rationals);
	\draw[gray,dashed,->] (P_2_2__Rationals) -- (P_3_2__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[red,->] (P_2_5__Rationals) -- (P_3_2__Rationals);
	\draw[gray,dashed,->] (P_2_3__Rationals) -- (P_3_3__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[red,->] (P_3_7__Rationals) -- (P_3_3__Rationals);
	\draw[gray,dashed,->] (P_2_4__Rationals) -- (P_3_4__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[red,->] (P_3_7__Rationals) -- (P_3_4__Rationals);
	\draw[gray,dashed,->] (P_2_5__Rationals) -- (P_3_5__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[blue,->] (P_2_6__Rationals) -- (P_3_5__Rationals);
	\draw[blue,->] (P_3_1__Rationals) -- (P_3_5__Rationals);
	\draw[blue,->] (P_3_2__Rationals) -- (P_3_5__Rationals);
	\draw[gray,dashed,->] (P_2_6__Rationals) -- (P_3_6__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[blue,->] (P_2_7__Rationals) -- (P_3_6__Rationals);
	\draw[blue,->] (P_3_5__Rationals) -- (P_3_6__Rationals);
	\draw[blue,->] (P_2_3__Rationals) -- (P_3_7__Rationals);
	\draw[blue,->] (P_2_4__Rationals) -- (P_3_7__Rationals);
	\draw[gray,dashed,->] (P_2_7__Rationals) -- (P_3_7__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[blue,->] (P_3_6__Rationals) -- (P_3_7__Rationals);
\end{tikzpicture}
\end{scaletikzpicturetowidth}
\end{center}

\end{figure}
\begin{figure}[ht]

\begin{center}
\begin{scaletikzpicturetowidth}{\textwidth}
\begin{tikzpicture}[scale=\tikzscale]
	\node at (0,2) {$\dots$};
	\node[seloneDDDDDDD, label={[shift={(0.0,-0.45)}, brown]\tiny $P(4,1)$}] (P_4_1__Rationals) at (0,0) {$\begin{smallmatrix}1&&&&1\\&2&2&2\\2&&&&1\end{smallmatrix}$};
	\node[seloneDDDDDDD, label={[shift={(0.0,-0.45)}, brown]\tiny $P(4,2)$}] (P_4_2__Rationals) at (0,1) {$\begin{smallmatrix}2&&&&1\\&2&2&2\\1&&&&1\end{smallmatrix}$};
	\node[seloneDDDDDDD, label={[shift={(0.0,-0.45)}, brown]\tiny $P(4,3)$}] (P_4_3__Rationals) at (8,8) {$\begin{smallmatrix}2&&&&2\\&3&3&3\\2&&&&1\end{smallmatrix}$};
	\node[seloneDDDDDDD, label={[shift={(0.0,-0.45)}, brown]\tiny $P(4,4)$}] (P_4_4__Rationals) at (8,7) {$\begin{smallmatrix}2&&&&1\\&3&3&3\\2&&&&2\end{smallmatrix}$};
	\node[noselDDDDDDD, label={[shift={(0.0,-0.45)}, brown]\tiny $P(4,5)$}] (P_4_5__Rationals) at (2,2) {$\begin{smallmatrix}3&&&&2\\&5&4&4\\3&&&&2\end{smallmatrix}$};
	\node[noselDDDDDDD, label={[shift={(0.0,-0.45)}, brown]\tiny $P(4,6)$}] (P_4_6__Rationals) at (4,4) {$\begin{smallmatrix}3&&&&2\\&5&5&4\\3&&&&2\end{smallmatrix}$};
	\node[noselDDDDDDD, label={[shift={(0.0,-0.45)}, brown]\tiny $P(4,7)$}] (P_4_7__Rationals) at (6,6) {$\begin{smallmatrix}3&&&&2\\&5&5&5\\3&&&&2\end{smallmatrix}$};
	\node[noselDDDDDDD, label={[shift={(0.0,-0.45)}, brown]\tiny $P(5,1)$}] (P_5_1__Rationals) at (4,0) {$\begin{smallmatrix}2&&&&1\\&3&2&2\\1&&&&1\end{smallmatrix}$};
	\node[noselDDDDDDD, label={[shift={(0.0,-0.45)}, brown]\tiny $P(5,2)$}] (P_5_2__Rationals) at (4,1) {$\begin{smallmatrix}1&&&&1\\&3&2&2\\2&&&&1\end{smallmatrix}$};
	\node[noselDDDDDDD, label={[shift={(0.0,-0.45)}, brown]\tiny $P(5,3)$}] (P_5_3__Rationals) at (12,8) {$\begin{smallmatrix}2&&&&1\\&4&3&3\\2&&&&2\end{smallmatrix}$};
	\node[noselDDDDDDD, label={[shift={(0.0,-0.45)}, brown]\tiny $P(5,4)$}] (P_5_4__Rationals) at (12,7) {$\begin{smallmatrix}2&&&&2\\&4&3&3\\2&&&&1\end{smallmatrix}$};
	\node[noselDDDDDDD, label={[shift={(0.0,-0.45)}, brown]\tiny $P(5,5)$}] (P_5_5__Rationals) at (6,2) {$\begin{smallmatrix}3&&&&2\\&6&5&4\\3&&&&2\end{smallmatrix}$};
	\node[noselDDDDDDD, label={[shift={(0.0,-0.45)}, brown]\tiny $P(5,6)$}] (P_5_6__Rationals) at (8,4) {$\begin{smallmatrix}3&&&&2\\&6&5&5\\3&&&&2\end{smallmatrix}$};
	\node[noselDDDDDDD, label={[shift={(0.0,-0.45)}, brown]\tiny $P(5,7)$}] (P_5_7__Rationals) at (10,6) {$\begin{smallmatrix}4&&&&3\\&7&6&6\\4&&&&3\end{smallmatrix}$};
	\node[noselDDDDDDD, label={[shift={(0.0,-0.45)}, brown]\tiny $P(6,1)$}] (P_6_1__Rationals) at (8,0) {$\begin{smallmatrix}1&&&&1\\&3&3&2\\2&&&&1\end{smallmatrix}$};
	\node[noselDDDDDDD, label={[shift={(0.0,-0.45)}, brown]\tiny $P(6,2)$}] (P_6_2__Rationals) at (8,1) {$\begin{smallmatrix}2&&&&1\\&3&3&2\\1&&&&1\end{smallmatrix}$};
	\node[noselDDDDDDD, label={[shift={(0.0,-0.45)}, brown]\tiny $P(6,3)$}] (P_6_3__Rationals) at (16,8) {$\begin{smallmatrix}2&&&&2\\&4&4&3\\2&&&&1\end{smallmatrix}$};
	\node[noselDDDDDDD, label={[shift={(0.0,-0.45)}, brown]\tiny $P(6,4)$}] (P_6_4__Rationals) at (16,7) {$\begin{smallmatrix}2&&&&1\\&4&4&3\\2&&&&2\end{smallmatrix}$};
	\node[noselDDDDDDD, label={[shift={(0.0,-0.45)}, brown]\tiny $P(6,5)$}] (P_6_5__Rationals) at (10,2) {$\begin{smallmatrix}3&&&&2\\&6&6&5\\3&&&&2\end{smallmatrix}$};
	\node[noselDDDDDDD, label={[shift={(0.0,-0.45)}, brown]\tiny $P(6,6)$}] (P_6_6__Rationals) at (12,4) {$\begin{smallmatrix}4&&&&3\\&7&7&6\\4&&&&3\end{smallmatrix}$};
	\node[noselDDDDDDD, label={[shift={(0.0,-0.45)}, brown]\tiny $P(6,7)$}] (P_6_7__Rationals) at (14,6) {$\begin{smallmatrix}4&&&&3\\&8&7&6\\4&&&&3\end{smallmatrix}$};
	\node[noselDDDDDDD, label={[shift={(0.0,-0.45)}, brown]\tiny $P(7,1)$}] (P_7_1__Rationals) at (12,0) {$\begin{smallmatrix}2&&&&1\\&3&3&3\\1&&&&1\end{smallmatrix}$};
	\node[noselDDDDDDD, label={[shift={(0.0,-0.45)}, brown]\tiny $P(7,2)$}] (P_7_2__Rationals) at (12,1) {$\begin{smallmatrix}1&&&&1\\&3&3&3\\2&&&&1\end{smallmatrix}$};
	\node[noselDDDDDDD, label={[shift={(0.0,-0.45)}, brown]\tiny $P(7,3)$}] (P_7_3__Rationals) at (20,8) {$\begin{smallmatrix}2&&&&1\\&4&4&4\\2&&&&2\end{smallmatrix}$};
	\node[noselDDDDDDD, label={[shift={(0.0,-0.45)}, brown]\tiny $P(7,4)$}] (P_7_4__Rationals) at (20,7) {$\begin{smallmatrix}2&&&&2\\&4&4&4\\2&&&&1\end{smallmatrix}$};
	\node[noselDDDDDDD, label={[shift={(0.0,-0.45)}, brown]\tiny $P(7,5)$}] (P_7_5__Rationals) at (14,2) {$\begin{smallmatrix}4&&&&3\\&7&7&7\\4&&&&3\end{smallmatrix}$};
	\node[noselDDDDDDD, label={[shift={(0.0,-0.45)}, brown]\tiny $P(7,6)$}] (P_7_6__Rationals) at (16,4) {$\begin{smallmatrix}4&&&&3\\&8&7&7\\4&&&&3\end{smallmatrix}$};
	\node[noselDDDDDDD, label={[shift={(0.0,-0.45)}, brown]\tiny $P(7,7)$}] (P_7_7__Rationals) at (18,6) {$\begin{smallmatrix}4&&&&3\\&8&8&7\\4&&&&3\end{smallmatrix}$};
	\node at (20,6) {$\dots$};
	\draw[red,->] (P_4_7__Rationals) -- (P_4_3__Rationals);
	\draw[red,->] (P_4_7__Rationals) -- (P_4_4__Rationals);
	\draw[blue,->] (P_4_1__Rationals) -- (P_4_5__Rationals);
	\draw[blue,->] (P_4_2__Rationals) -- (P_4_5__Rationals);
	\draw[blue,->] (P_4_5__Rationals) -- (P_4_6__Rationals);
	\draw[blue,->] (P_4_6__Rationals) -- (P_4_7__Rationals);
	\draw[gray,dashed,->] (P_4_1__Rationals) -- (P_5_1__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[red,->] (P_4_5__Rationals) -- (P_5_1__Rationals);
	\draw[gray,dashed,->] (P_4_2__Rationals) -- (P_5_2__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[red,->] (P_4_5__Rationals) -- (P_5_2__Rationals);
	\draw[gray,dashed,->] (P_4_3__Rationals) -- (P_5_3__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[red,->] (P_5_7__Rationals) -- (P_5_3__Rationals);
	\draw[gray,dashed,->] (P_4_4__Rationals) -- (P_5_4__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[red,->] (P_5_7__Rationals) -- (P_5_4__Rationals);
	\draw[gray,dashed,->] (P_4_5__Rationals) -- (P_5_5__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[blue,->] (P_4_6__Rationals) -- (P_5_5__Rationals);
	\draw[blue,->] (P_5_1__Rationals) -- (P_5_5__Rationals);
	\draw[blue,->] (P_5_2__Rationals) -- (P_5_5__Rationals);
	\draw[gray,dashed,->] (P_4_6__Rationals) -- (P_5_6__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[blue,->] (P_4_7__Rationals) -- (P_5_6__Rationals);
	\draw[blue,->] (P_5_5__Rationals) -- (P_5_6__Rationals);
	\draw[blue,->] (P_4_3__Rationals) -- (P_5_7__Rationals);
	\draw[blue,->] (P_4_4__Rationals) -- (P_5_7__Rationals);
	\draw[gray,dashed,->] (P_4_7__Rationals) -- (P_5_7__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[blue,->] (P_5_6__Rationals) -- (P_5_7__Rationals);
	\draw[gray,dashed,->] (P_5_1__Rationals) -- (P_6_1__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[red,->] (P_5_5__Rationals) -- (P_6_1__Rationals);
	\draw[gray,dashed,->] (P_5_2__Rationals) -- (P_6_2__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[red,->] (P_5_5__Rationals) -- (P_6_2__Rationals);
	\draw[gray,dashed,->] (P_5_3__Rationals) -- (P_6_3__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[red,->] (P_6_7__Rationals) -- (P_6_3__Rationals);
	\draw[gray,dashed,->] (P_5_4__Rationals) -- (P_6_4__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[red,->] (P_6_7__Rationals) -- (P_6_4__Rationals);
	\draw[gray,dashed,->] (P_5_5__Rationals) -- (P_6_5__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[blue,->] (P_5_6__Rationals) -- (P_6_5__Rationals);
	\draw[blue,->] (P_6_1__Rationals) -- (P_6_5__Rationals);
	\draw[blue,->] (P_6_2__Rationals) -- (P_6_5__Rationals);
	\draw[gray,dashed,->] (P_5_6__Rationals) -- (P_6_6__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[blue,->] (P_5_7__Rationals) -- (P_6_6__Rationals);
	\draw[blue,->] (P_6_5__Rationals) -- (P_6_6__Rationals);
	\draw[blue,->] (P_5_3__Rationals) -- (P_6_7__Rationals);
	\draw[blue,->] (P_5_4__Rationals) -- (P_6_7__Rationals);
	\draw[gray,dashed,->] (P_5_7__Rationals) -- (P_6_7__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[blue,->] (P_6_6__Rationals) -- (P_6_7__Rationals);
	\draw[gray,dashed,->] (P_6_1__Rationals) -- (P_7_1__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[red,->] (P_6_5__Rationals) -- (P_7_1__Rationals);
	\draw[gray,dashed,->] (P_6_2__Rationals) -- (P_7_2__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[red,->] (P_6_5__Rationals) -- (P_7_2__Rationals);
	\draw[gray,dashed,->] (P_6_3__Rationals) -- (P_7_3__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[red,->] (P_7_7__Rationals) -- (P_7_3__Rationals);
	\draw[gray,dashed,->] (P_6_4__Rationals) -- (P_7_4__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[red,->] (P_7_7__Rationals) -- (P_7_4__Rationals);
	\draw[gray,dashed,->] (P_6_5__Rationals) -- (P_7_5__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[blue,->] (P_6_6__Rationals) -- (P_7_5__Rationals);
	\draw[blue,->] (P_7_1__Rationals) -- (P_7_5__Rationals);
	\draw[blue,->] (P_7_2__Rationals) -- (P_7_5__Rationals);
	\draw[gray,dashed,->] (P_6_6__Rationals) -- (P_7_6__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[blue,->] (P_6_7__Rationals) -- (P_7_6__Rationals);
	\draw[blue,->] (P_7_5__Rationals) -- (P_7_6__Rationals);
	\draw[blue,->] (P_6_3__Rationals) -- (P_7_7__Rationals);
	\draw[blue,->] (P_6_4__Rationals) -- (P_7_7__Rationals);
	\draw[gray,dashed,->] (P_6_7__Rationals) -- (P_7_7__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[blue,->] (P_7_6__Rationals) -- (P_7_7__Rationals);
\end{tikzpicture}
\end{scaletikzpicturetowidth}
\end{center}

\end{figure}
\medskip{}
\subsubsection*{Modules of the form $P(n,1)$}

Defect: $\partial P(n,1) = -1$, for $n\ge 0$.

\begin{fleqn}
\begin{align*}
P(0,1):\ & - \\
P(1,1):\ & \bigl(R_{1}^{3}(1),P(0,2)\bigr)\\
P(2,1):\ & \bigl(R_{1}^{3}(2),P(0,1)\bigr),\ \bigl(R_{1}^{4}(1),P(1,2)\bigr)\\
P(3,1):\ & \bigl(R_{1}^{3}(3),P(0,2)\bigr),\ \bigl(R_{1}^{4}(2),P(1,1)\bigr),\ \bigl(R_{1}^{1}(1),P(2,2)\bigr)\\
P(4,1):\ & \bigl(R_{0}^{2}(1),P(0,3)\bigr),\ \bigl(R_{\infty}^{2}(1),P(0,4)\bigr),\ \bigl(R_{1}^{4}(3),P(1,2)\bigr),\ \bigl(R_{1}^{1}(2),P(2,1)\bigr),\ \bigl(R_{1}^{2}(1),P(3,2)\bigr)\\
 & \bigl(I(3,1),2P(0,1)\bigr)\\
P(n,1):\ & \bigl(R_{0}^{(n-3)\bmod 2+1}(1),P(n-4,3)\bigr),\ \bigl(R_{\infty}^{(n-3)\bmod 2+1}(1),P(n-4,4)\bigr),\ \bigl(R_{1}^{(n-1)\bmod 4+1}(3),P(n-3,2)\bigr)\\
 & \bigl(R_{1}^{(n-4)\bmod 4+1}(2),P(n-2,1)\bigr),\ \bigl(R_{1}^{(n-3)\bmod 4+1}(1),P(n-1,2)\bigr),\ \bigl(uI,(u+1)P\bigr),\ n>4\\
\end{align*}
\end{fleqn}
\subsubsection*{Modules of the form $P(n,2)$}

Defect: $\partial P(n,2) = -1$, for $n\ge 0$.

\begin{fleqn}
\begin{align*}
P(0,2):\ & - \\
P(1,2):\ & \bigl(R_{1}^{3}(1),P(0,1)\bigr)\\
P(2,2):\ & \bigl(R_{1}^{3}(2),P(0,2)\bigr),\ \bigl(R_{1}^{4}(1),P(1,1)\bigr)\\
P(3,2):\ & \bigl(R_{1}^{3}(3),P(0,1)\bigr),\ \bigl(R_{1}^{4}(2),P(1,2)\bigr),\ \bigl(R_{1}^{1}(1),P(2,1)\bigr)\\
P(4,2):\ & \bigl(R_{\infty}^{1}(1),P(0,3)\bigr),\ \bigl(R_{0}^{1}(1),P(0,4)\bigr),\ \bigl(R_{1}^{4}(3),P(1,1)\bigr),\ \bigl(R_{1}^{1}(2),P(2,2)\bigr),\ \bigl(R_{1}^{2}(1),P(3,1)\bigr)\\
 & \bigl(I(3,2),2P(0,2)\bigr)\\
P(n,2):\ & \bigl(R_{\infty}^{(n-4)\bmod 2+1}(1),P(n-4,3)\bigr),\ \bigl(R_{0}^{(n-4)\bmod 2+1}(1),P(n-4,4)\bigr),\ \bigl(R_{1}^{(n-1)\bmod 4+1}(3),P(n-3,1)\bigr)\\
 & \bigl(R_{1}^{(n-4)\bmod 4+1}(2),P(n-2,2)\bigr),\ \bigl(R_{1}^{(n-3)\bmod 4+1}(1),P(n-1,1)\bigr),\ \bigl(uI,(u+1)P\bigr),\ n>4\\
\end{align*}
\end{fleqn}
\subsubsection*{Modules of the form $P(n,3)$}

Defect: $\partial P(n,3) = -1$, for $n\ge 0$.

\begin{fleqn}
\begin{align*}
P(0,3):\ & \bigl(R_{0}^{1}(1),P(0,1)\bigr),\ \bigl(R_{\infty}^{2}(1),P(0,2)\bigr),\ \bigl(I(2,3),P(0,5)\bigr),\ \bigl(I(1,4),P(0,6)\bigr),\ \bigl(I(0,3),P(0,7)\bigr)\\
P(1,3):\ & \bigl(R_{1}^{3}(1),P(0,4)\bigr),\ \bigl(R_{0}^{2}(1),P(1,1)\bigr),\ \bigl(R_{\infty}^{1}(1),P(1,2)\bigr),\ \bigl(I(1,3),P(1,5)\bigr),\ \bigl(I(0,4),P(1,6)\bigr)\\
P(2,3):\ & \bigl(R_{1}^{3}(2),P(0,3)\bigr),\ \bigl(R_{1}^{4}(1),P(1,4)\bigr),\ \bigl(R_{0}^{1}(1),P(2,1)\bigr),\ \bigl(R_{\infty}^{2}(1),P(2,2)\bigr),\ \bigl(I(0,3),P(2,5)\bigr)\\
P(3,3):\ & \bigl(R_{1}^{3}(3),P(0,4)\bigr),\ \bigl(R_{1}^{4}(2),P(1,3)\bigr),\ \bigl(R_{1}^{1}(1),P(2,4)\bigr),\ \bigl(R_{0}^{2}(1),P(3,1)\bigr),\ \bigl(R_{\infty}^{1}(1),P(3,2)\bigr)\\
P(4,3):\ & \bigl(R_{1}^{4}(3),P(1,4)\bigr),\ \bigl(R_{1}^{1}(2),P(2,3)\bigr),\ \bigl(R_{1}^{2}(1),P(3,4)\bigr),\ \bigl(R_{0}^{1}(1),P(4,1)\bigr),\ \bigl(R_{\infty}^{2}(1),P(4,2)\bigr)\\
 & \bigl(I(3,3),2P(0,3)\bigr)\\
P(n,3):\ & \bigl(R_{1}^{(n-1)\bmod 4+1}(3),P(n-3,4)\bigr),\ \bigl(R_{1}^{(n-4)\bmod 4+1}(2),P(n-2,3)\bigr),\ \bigl(R_{1}^{(n-3)\bmod 4+1}(1),P(n-1,4)\bigr)\\
 & \bigl(R_{0}^{(n-4)\bmod 2+1}(1),P(n,1)\bigr),\ \bigl(R_{\infty}^{(n-3)\bmod 2+1}(1),P(n,2)\bigr),\ \bigl(uI,(u+1)P\bigr),\ n>4\\
\end{align*}
\end{fleqn}
\subsubsection*{Modules of the form $P(n,4)$}

Defect: $\partial P(n,4) = -1$, for $n\ge 0$.

\begin{fleqn}
\begin{align*}
P(0,4):\ & \bigl(R_{\infty}^{1}(1),P(0,1)\bigr),\ \bigl(R_{0}^{2}(1),P(0,2)\bigr),\ \bigl(I(2,4),P(0,5)\bigr),\ \bigl(I(1,3),P(0,6)\bigr),\ \bigl(I(0,4),P(0,7)\bigr)\\
P(1,4):\ & \bigl(R_{1}^{3}(1),P(0,3)\bigr),\ \bigl(R_{\infty}^{2}(1),P(1,1)\bigr),\ \bigl(R_{0}^{1}(1),P(1,2)\bigr),\ \bigl(I(1,4),P(1,5)\bigr),\ \bigl(I(0,3),P(1,6)\bigr)\\
P(2,4):\ & \bigl(R_{1}^{3}(2),P(0,4)\bigr),\ \bigl(R_{1}^{4}(1),P(1,3)\bigr),\ \bigl(R_{\infty}^{1}(1),P(2,1)\bigr),\ \bigl(R_{0}^{2}(1),P(2,2)\bigr),\ \bigl(I(0,4),P(2,5)\bigr)\\
P(3,4):\ & \bigl(R_{1}^{3}(3),P(0,3)\bigr),\ \bigl(R_{1}^{4}(2),P(1,4)\bigr),\ \bigl(R_{1}^{1}(1),P(2,3)\bigr),\ \bigl(R_{\infty}^{2}(1),P(3,1)\bigr),\ \bigl(R_{0}^{1}(1),P(3,2)\bigr)\\
P(4,4):\ & \bigl(R_{1}^{4}(3),P(1,3)\bigr),\ \bigl(R_{1}^{1}(2),P(2,4)\bigr),\ \bigl(R_{1}^{2}(1),P(3,3)\bigr),\ \bigl(R_{\infty}^{1}(1),P(4,1)\bigr),\ \bigl(R_{0}^{2}(1),P(4,2)\bigr)\\
 & \bigl(I(3,4),2P(0,4)\bigr)\\
P(n,4):\ & \bigl(R_{1}^{(n-1)\bmod 4+1}(3),P(n-3,3)\bigr),\ \bigl(R_{1}^{(n-4)\bmod 4+1}(2),P(n-2,4)\bigr),\ \bigl(R_{1}^{(n-3)\bmod 4+1}(1),P(n-1,3)\bigr)\\
 & \bigl(R_{\infty}^{(n-4)\bmod 2+1}(1),P(n,1)\bigr),\ \bigl(R_{0}^{(n-3)\bmod 2+1}(1),P(n,2)\bigr),\ \bigl(uI,(u+1)P\bigr),\ n>4\\
\end{align*}
\end{fleqn}
\subsubsection*{Modules of the form $P(n,5)$}

Defect: $\partial P(n,5) = -2$, for $n\ge 0$.

\begin{fleqn}
\begin{align*}
P(0,5):\ & \bigl(P(1,1),P(0,1)\bigr),\ \bigl(P(1,2),P(0,2)\bigr)\\
P(1,5):\ & \bigl(R_{1}^{3}(1),P(0,6)\bigr),\ \bigl(P(2,1),P(1,1)\bigr),\ \bigl(P(2,2),P(1,2)\bigr)\\
P(2,5):\ & \bigl(R_{1}^{3}(2),P(0,7)\bigr),\ \bigl(R_{1}^{4}(1),P(1,6)\bigr),\ \bigl(P(3,1),P(2,1)\bigr),\ \bigl(P(3,2),P(2,2)\bigr)\\
P(3,5):\ & \bigl(P(3,3),P(0,3)\bigr),\ \bigl(P(3,4),P(0,4)\bigr),\ \bigl(R_{1}^{4}(2),P(1,7)\bigr),\ \bigl(R_{1}^{1}(1),P(2,6)\bigr),\ \bigl(P(4,1),P(3,1)\bigr)\\
 & \bigl(P(4,2),P(3,2)\bigr)\\
P(n,5):\ & \bigl(P(n,3),P(n-3,3)\bigr),\ \bigl(P(n,4),P(n-3,4)\bigr),\ \bigl(R_{1}^{n\bmod 4+1}(2),P(n-2,7)\bigr)\\
 & \bigl(R_{1}^{(n-3)\bmod 4+1}(1),P(n-1,6)\bigr),\ \bigl(P(n+1,1),P(n,1)\bigr),\ \bigl(P(n+1,2),P(n,2)\bigr),\ n>3\\
\end{align*}
\end{fleqn}
\subsubsection*{Modules of the form $P(n,6)$}

Defect: $\partial P(n,6) = -2$, for $n\ge 0$.

\begin{fleqn}
\begin{align*}
P(0,6):\ & \bigl(P(2,2),P(0,1)\bigr),\ \bigl(P(2,1),P(0,2)\bigr),\ \bigl(R_{1}^{4}(1),P(0,5)\bigr)\\
P(1,6):\ & \bigl(R_{1}^{3}(1),P(0,7)\bigr),\ \bigl(P(3,2),P(1,1)\bigr),\ \bigl(P(3,1),P(1,2)\bigr),\ \bigl(R_{1}^{1}(1),P(1,5)\bigr)\\
P(2,6):\ & \bigl(P(2,4),P(0,3)\bigr),\ \bigl(P(2,3),P(0,4)\bigr),\ \bigl(R_{1}^{4}(1),P(1,7)\bigr),\ \bigl(P(4,2),P(2,1)\bigr),\ \bigl(P(4,1),P(2,2)\bigr)\\
 & \bigl(R_{1}^{2}(1),P(2,5)\bigr)\\
P(n,6):\ & \bigl(P(n,4),P(n-2,3)\bigr),\ \bigl(P(n,3),P(n-2,4)\bigr),\ \bigl(R_{1}^{(n+1)\bmod 4+1}(1),P(n-1,7)\bigr)\\
 & \bigl(P(n+2,2),P(n,1)\bigr),\ \bigl(P(n+2,1),P(n,2)\bigr),\ \bigl(R_{1}^{(n-1)\bmod 4+1}(1),P(n,5)\bigr),\ n>2\\
\end{align*}
\end{fleqn}
\subsubsection*{Modules of the form $P(n,7)$}

Defect: $\partial P(n,7) = -2$, for $n\ge 0$.

\begin{fleqn}
\begin{align*}
P(0,7):\ & \bigl(P(3,1),P(0,1)\bigr),\ \bigl(P(3,2),P(0,2)\bigr),\ \bigl(R_{1}^{4}(2),P(0,5)\bigr),\ \bigl(R_{1}^{1}(1),P(0,6)\bigr)\\
P(1,7):\ & \bigl(P(1,3),P(0,3)\bigr),\ \bigl(P(1,4),P(0,4)\bigr),\ \bigl(P(4,1),P(1,1)\bigr),\ \bigl(P(4,2),P(1,2)\bigr),\ \bigl(R_{1}^{1}(2),P(1,5)\bigr)\\
 & \bigl(R_{1}^{2}(1),P(1,6)\bigr)\\
P(n,7):\ & \bigl(P(n,3),P(n-1,3)\bigr),\ \bigl(P(n,4),P(n-1,4)\bigr),\ \bigl(P(n+3,1),P(n,1)\bigr)\\
 & \bigl(P(n+3,2),P(n,2)\bigr),\ \bigl(R_{1}^{(n-1)\bmod 4+1}(2),P(n,5)\bigr),\ \bigl(R_{1}^{n\bmod 4+1}(1),P(n,6)\bigr),\ n>1\\
\end{align*}
\end{fleqn}
\subsubsection{Schofield pairs associated to preinjective exceptional modules}

\begin{figure}[ht]

\begin{center}
\begin{scaletikzpicturetowidth}{\textwidth}
\begin{tikzpicture}[scale=\tikzscale]
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	\node[noselDDDDDDD, label={[shift={(0.0,-0.45)}, brown]\tiny $I(0,2)$}] (I_0_2__Rationals) at (0,1) {$\begin{smallmatrix}1&&&&1\\&1&1&1\\0&&&&1\end{smallmatrix}$};
	\node[noselDDDDDDD, label={[shift={(0.0,-0.45)}, brown]\tiny $I(0,3)$}] (I_0_3__Rationals) at (8,8) {$\begin{smallmatrix}0&&&&1\\&0&0&0\\0&&&&0\end{smallmatrix}$};
	\node[noselDDDDDDD, label={[shift={(0.0,-0.45)}, brown]\tiny $I(0,4)$}] (I_0_4__Rationals) at (8,7) {$\begin{smallmatrix}0&&&&0\\&0&0&0\\0&&&&1\end{smallmatrix}$};
	\node[noselDDDDDDD, label={[shift={(0.0,-0.45)}, brown]\tiny $I(0,5)$}] (I_0_5__Rationals) at (2,2) {$\begin{smallmatrix}0&&&&1\\&1&1&1\\0&&&&1\end{smallmatrix}$};
	\node[noselDDDDDDD, label={[shift={(0.0,-0.45)}, brown]\tiny $I(0,6)$}] (I_0_6__Rationals) at (4,4) {$\begin{smallmatrix}0&&&&1\\&0&1&1\\0&&&&1\end{smallmatrix}$};
	\node[noselDDDDDDD, label={[shift={(0.0,-0.45)}, brown]\tiny $I(0,7)$}] (I_0_7__Rationals) at (6,6) {$\begin{smallmatrix}0&&&&1\\&0&0&1\\0&&&&1\end{smallmatrix}$};
	\node[noselDDDDDDD, label={[shift={(0.0,-0.45)}, brown]\tiny $I(1,1)$}] (I_1_1__Rationals) at (-4,0) {$\begin{smallmatrix}1&&&&1\\&1&1&2\\0&&&&1\end{smallmatrix}$};
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	\node[noselDDDDDDD, label={[shift={(0.0,-0.45)}, brown]\tiny $I(2,7)$}] (I_2_7__Rationals) at (-2,6) {$\begin{smallmatrix}0&&&&1\\&1&2&2\\0&&&&1\end{smallmatrix}$};
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	\node[noselDDDDDDD, label={[shift={(0.0,-0.45)}, brown]\tiny $I(3,2)$}] (I_3_2__Rationals) at (-12,1) {$\begin{smallmatrix}0&&&&1\\&2&2&2\\1&&&&1\end{smallmatrix}$};
	\node[noselDDDDDDD, label={[shift={(0.0,-0.45)}, brown]\tiny $I(3,3)$}] (I_3_3__Rationals) at (-4,8) {$\begin{smallmatrix}0&&&&0\\&1&1&1\\0&&&&1\end{smallmatrix}$};
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	\node[noselDDDDDDD, label={[shift={(0.0,-0.45)}, brown]\tiny $I(3,5)$}] (I_3_5__Rationals) at (-10,2) {$\begin{smallmatrix}1&&&&2\\&3&4&4\\1&&&&2\end{smallmatrix}$};
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	\node[seloneDDDDDDD, label={[shift={(0.0,-0.45)}, brown]\tiny $I(3,7)$}] (I_3_7__Rationals) at (-6,6) {$\begin{smallmatrix}1&&&&2\\&3&3&3\\1&&&&2\end{smallmatrix}$};
	\node at (-12,2) {$\dots$};
	\draw[red,->] (I_1_5__Rationals) -- (I_0_1__Rationals);
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	\draw[red,->] (I_0_7__Rationals) -- (I_0_3__Rationals);
	\draw[red,->] (I_0_7__Rationals) -- (I_0_4__Rationals);
	\draw[red,->] (I_0_1__Rationals) -- (I_0_5__Rationals);
	\draw[red,->] (I_0_2__Rationals) -- (I_0_5__Rationals);
	\draw[red,->] (I_1_6__Rationals) -- (I_0_5__Rationals);
	\draw[red,->] (I_0_5__Rationals) -- (I_0_6__Rationals);
	\draw[red,->] (I_1_7__Rationals) -- (I_0_6__Rationals);
	\draw[red,->] (I_0_6__Rationals) -- (I_0_7__Rationals);
	\draw[blue,->] (I_1_3__Rationals) -- (I_0_7__Rationals);
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	\draw[red,->] (I_2_5__Rationals) -- (I_1_1__Rationals);
	\draw[gray,dashed,->] (I_0_2__Rationals) -- (I_1_2__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_2_5__Rationals) -- (I_1_2__Rationals);
	\draw[gray,dashed,->] (I_0_3__Rationals) -- (I_1_3__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_1_7__Rationals) -- (I_1_3__Rationals);
	\draw[gray,dashed,->] (I_0_4__Rationals) -- (I_1_4__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_1_7__Rationals) -- (I_1_4__Rationals);
	\draw[gray,dashed,->] (I_0_5__Rationals) -- (I_1_5__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[blue,->] (I_1_1__Rationals) -- (I_1_5__Rationals);
	\draw[blue,->] (I_1_2__Rationals) -- (I_1_5__Rationals);
	\draw[red,->] (I_2_6__Rationals) -- (I_1_5__Rationals);
	\draw[gray,dashed,->] (I_0_6__Rationals) -- (I_1_6__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_1_5__Rationals) -- (I_1_6__Rationals);
	\draw[red,->] (I_2_7__Rationals) -- (I_1_6__Rationals);
	\draw[gray,dashed,->] (I_0_7__Rationals) -- (I_1_7__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_1_6__Rationals) -- (I_1_7__Rationals);
	\draw[blue,->] (I_2_3__Rationals) -- (I_1_7__Rationals);
	\draw[blue,->] (I_2_4__Rationals) -- (I_1_7__Rationals);
	\draw[gray,dashed,->] (I_1_1__Rationals) -- (I_2_1__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_3_5__Rationals) -- (I_2_1__Rationals);
	\draw[gray,dashed,->] (I_1_2__Rationals) -- (I_2_2__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_3_5__Rationals) -- (I_2_2__Rationals);
	\draw[gray,dashed,->] (I_1_3__Rationals) -- (I_2_3__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_2_7__Rationals) -- (I_2_3__Rationals);
	\draw[gray,dashed,->] (I_1_4__Rationals) -- (I_2_4__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_2_7__Rationals) -- (I_2_4__Rationals);
	\draw[gray,dashed,->] (I_1_5__Rationals) -- (I_2_5__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[blue,->] (I_2_1__Rationals) -- (I_2_5__Rationals);
	\draw[blue,->] (I_2_2__Rationals) -- (I_2_5__Rationals);
	\draw[red,->] (I_3_6__Rationals) -- (I_2_5__Rationals);
	\draw[gray,dashed,->] (I_1_6__Rationals) -- (I_2_6__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_2_5__Rationals) -- (I_2_6__Rationals);
	\draw[red,->] (I_3_7__Rationals) -- (I_2_6__Rationals);
	\draw[gray,dashed,->] (I_1_7__Rationals) -- (I_2_7__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_2_6__Rationals) -- (I_2_7__Rationals);
	\draw[blue,->] (I_3_3__Rationals) -- (I_2_7__Rationals);
	\draw[blue,->] (I_3_4__Rationals) -- (I_2_7__Rationals);
	\draw[gray,dashed,->] (I_2_1__Rationals) -- (I_3_1__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[gray,dashed,->] (I_2_2__Rationals) -- (I_3_2__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[gray,dashed,->] (I_2_3__Rationals) -- (I_3_3__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_3_7__Rationals) -- (I_3_3__Rationals);
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	\draw[red,->] (I_3_7__Rationals) -- (I_3_4__Rationals);
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	\draw[red,->] (I_3_6__Rationals) -- (I_3_7__Rationals);
\end{tikzpicture}
\end{scaletikzpicturetowidth}
\end{center}

\end{figure}
\begin{figure}[ht]

\begin{center}
\begin{scaletikzpicturetowidth}{\textwidth}
\begin{tikzpicture}[scale=\tikzscale]
	\node at (8,6) {$\dots$};
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	\draw[blue,->] (I_4_2__Rationals) -- (I_4_5__Rationals);
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	\draw[blue,->] (I_5_3__Rationals) -- (I_4_7__Rationals);
	\draw[blue,->] (I_5_4__Rationals) -- (I_4_7__Rationals);
	\draw[gray,dashed,->] (I_4_1__Rationals) -- (I_5_1__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_6_5__Rationals) -- (I_5_1__Rationals);
	\draw[gray,dashed,->] (I_4_2__Rationals) -- (I_5_2__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_6_5__Rationals) -- (I_5_2__Rationals);
	\draw[gray,dashed,->] (I_4_3__Rationals) -- (I_5_3__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_5_7__Rationals) -- (I_5_3__Rationals);
	\draw[gray,dashed,->] (I_4_4__Rationals) -- (I_5_4__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_5_7__Rationals) -- (I_5_4__Rationals);
	\draw[gray,dashed,->] (I_4_5__Rationals) -- (I_5_5__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[blue,->] (I_5_1__Rationals) -- (I_5_5__Rationals);
	\draw[blue,->] (I_5_2__Rationals) -- (I_5_5__Rationals);
	\draw[red,->] (I_6_6__Rationals) -- (I_5_5__Rationals);
	\draw[gray,dashed,->] (I_4_6__Rationals) -- (I_5_6__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_5_5__Rationals) -- (I_5_6__Rationals);
	\draw[red,->] (I_6_7__Rationals) -- (I_5_6__Rationals);
	\draw[gray,dashed,->] (I_4_7__Rationals) -- (I_5_7__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_5_6__Rationals) -- (I_5_7__Rationals);
	\draw[blue,->] (I_6_3__Rationals) -- (I_5_7__Rationals);
	\draw[blue,->] (I_6_4__Rationals) -- (I_5_7__Rationals);
	\draw[gray,dashed,->] (I_5_1__Rationals) -- (I_6_1__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_7_5__Rationals) -- (I_6_1__Rationals);
	\draw[gray,dashed,->] (I_5_2__Rationals) -- (I_6_2__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_7_5__Rationals) -- (I_6_2__Rationals);
	\draw[gray,dashed,->] (I_5_3__Rationals) -- (I_6_3__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_6_7__Rationals) -- (I_6_3__Rationals);
	\draw[gray,dashed,->] (I_5_4__Rationals) -- (I_6_4__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_6_7__Rationals) -- (I_6_4__Rationals);
	\draw[gray,dashed,->] (I_5_5__Rationals) -- (I_6_5__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[blue,->] (I_6_1__Rationals) -- (I_6_5__Rationals);
	\draw[blue,->] (I_6_2__Rationals) -- (I_6_5__Rationals);
	\draw[red,->] (I_7_6__Rationals) -- (I_6_5__Rationals);
	\draw[gray,dashed,->] (I_5_6__Rationals) -- (I_6_6__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_6_5__Rationals) -- (I_6_6__Rationals);
	\draw[red,->] (I_7_7__Rationals) -- (I_6_6__Rationals);
	\draw[gray,dashed,->] (I_5_7__Rationals) -- (I_6_7__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_6_6__Rationals) -- (I_6_7__Rationals);
	\draw[blue,->] (I_7_3__Rationals) -- (I_6_7__Rationals);
	\draw[blue,->] (I_7_4__Rationals) -- (I_6_7__Rationals);
	\draw[gray,dashed,->] (I_6_1__Rationals) -- (I_7_1__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[gray,dashed,->] (I_6_2__Rationals) -- (I_7_2__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[gray,dashed,->] (I_6_3__Rationals) -- (I_7_3__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_7_7__Rationals) -- (I_7_3__Rationals);
	\draw[gray,dashed,->] (I_6_4__Rationals) -- (I_7_4__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_7_7__Rationals) -- (I_7_4__Rationals);
	\draw[gray,dashed,->] (I_6_5__Rationals) -- (I_7_5__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[blue,->] (I_7_1__Rationals) -- (I_7_5__Rationals);
	\draw[blue,->] (I_7_2__Rationals) -- (I_7_5__Rationals);
	\draw[gray,dashed,->] (I_6_6__Rationals) -- (I_7_6__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_7_5__Rationals) -- (I_7_6__Rationals);
	\draw[gray,dashed,->] (I_6_7__Rationals) -- (I_7_7__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_7_6__Rationals) -- (I_7_7__Rationals);
\end{tikzpicture}
\end{scaletikzpicturetowidth}
\end{center}

\end{figure}
\medskip{}
\subsubsection*{Modules of the form $I(n,1)$}

Defect: $\partial I(n,1) = 1$, for $n\ge 0$.

\begin{fleqn}
\begin{align*}
I(0,1):\ & \bigl(I(0,3),R_{0}^{2}(1)\bigr),\ \bigl(I(0,4),R_{\infty}^{2}(1)\bigr),\ \bigl(I(0,5),P(0,1)\bigr),\ \bigl(I(0,6),P(1,2)\bigr),\ \bigl(I(0,7),P(2,1)\bigr)\\
I(1,1):\ & \bigl(I(0,2),R_{1}^{1}(1)\bigr),\ \bigl(I(1,3),R_{0}^{1}(1)\bigr),\ \bigl(I(1,4),R_{\infty}^{1}(1)\bigr),\ \bigl(I(1,6),P(0,2)\bigr),\ \bigl(I(1,7),P(1,1)\bigr)\\
I(2,1):\ & \bigl(I(0,1),R_{1}^{4}(2)\bigr),\ \bigl(I(1,2),R_{1}^{4}(1)\bigr),\ \bigl(I(2,3),R_{0}^{2}(1)\bigr),\ \bigl(I(2,4),R_{\infty}^{2}(1)\bigr),\ \bigl(I(2,7),P(0,1)\bigr)\\
I(3,1):\ & \bigl(I(0,2),R_{1}^{3}(3)\bigr),\ \bigl(I(1,1),R_{1}^{3}(2)\bigr),\ \bigl(I(2,2),R_{1}^{3}(1)\bigr),\ \bigl(I(3,3),R_{0}^{1}(1)\bigr),\ \bigl(I(3,4),R_{\infty}^{1}(1)\bigr)\\
I(4,1):\ & \bigl(I(1,2),R_{1}^{2}(3)\bigr),\ \bigl(I(2,1),R_{1}^{2}(2)\bigr),\ \bigl(I(3,2),R_{1}^{2}(1)\bigr),\ \bigl(I(4,3),R_{0}^{2}(1)\bigr),\ \bigl(I(4,4),R_{\infty}^{2}(1)\bigr)\\
 & \bigl(2I(0,1),P(3,1)\bigr)\\
I(n,1):\ & \bigl(I(n-3,2),R_{1}^{(-n+5)\bmod 4+1}(3)\bigr),\ \bigl(I(n-2,1),R_{1}^{(-n+5)\bmod 4+1}(2)\bigr),\ \bigl(I(n-1,2),R_{1}^{(-n+5)\bmod 4+1}(1)\bigr)\\
 & \bigl(I(n,3),R_{0}^{(-n+5)\bmod 2+1}(1)\bigr),\ \bigl(I(n,4),R_{\infty}^{(-n+5)\bmod 2+1}(1)\bigr),\ \bigl((v+1)I,vP\bigr),\ n>4\\
\end{align*}
\end{fleqn}
\subsubsection*{Modules of the form $I(n,2)$}

Defect: $\partial I(n,2) = 1$, for $n\ge 0$.

\begin{fleqn}
\begin{align*}
I(0,2):\ & \bigl(I(0,3),R_{\infty}^{1}(1)\bigr),\ \bigl(I(0,4),R_{0}^{1}(1)\bigr),\ \bigl(I(0,5),P(0,2)\bigr),\ \bigl(I(0,6),P(1,1)\bigr),\ \bigl(I(0,7),P(2,2)\bigr)\\
I(1,2):\ & \bigl(I(0,1),R_{1}^{1}(1)\bigr),\ \bigl(I(1,3),R_{\infty}^{2}(1)\bigr),\ \bigl(I(1,4),R_{0}^{2}(1)\bigr),\ \bigl(I(1,6),P(0,1)\bigr),\ \bigl(I(1,7),P(1,2)\bigr)\\
I(2,2):\ & \bigl(I(0,2),R_{1}^{4}(2)\bigr),\ \bigl(I(1,1),R_{1}^{4}(1)\bigr),\ \bigl(I(2,3),R_{\infty}^{1}(1)\bigr),\ \bigl(I(2,4),R_{0}^{1}(1)\bigr),\ \bigl(I(2,7),P(0,2)\bigr)\\
I(3,2):\ & \bigl(I(0,1),R_{1}^{3}(3)\bigr),\ \bigl(I(1,2),R_{1}^{3}(2)\bigr),\ \bigl(I(2,1),R_{1}^{3}(1)\bigr),\ \bigl(I(3,3),R_{\infty}^{2}(1)\bigr),\ \bigl(I(3,4),R_{0}^{2}(1)\bigr)\\
I(4,2):\ & \bigl(I(1,1),R_{1}^{2}(3)\bigr),\ \bigl(I(2,2),R_{1}^{2}(2)\bigr),\ \bigl(I(3,1),R_{1}^{2}(1)\bigr),\ \bigl(I(4,3),R_{\infty}^{1}(1)\bigr),\ \bigl(I(4,4),R_{0}^{1}(1)\bigr)\\
 & \bigl(2I(0,2),P(3,2)\bigr)\\
I(n,2):\ & \bigl(I(n-3,1),R_{1}^{(-n+5)\bmod 4+1}(3)\bigr),\ \bigl(I(n-2,2),R_{1}^{(-n+5)\bmod 4+1}(2)\bigr),\ \bigl(I(n-1,1),R_{1}^{(-n+5)\bmod 4+1}(1)\bigr)\\
 & \bigl(I(n,3),R_{\infty}^{(-n+4)\bmod 2+1}(1)\bigr),\ \bigl(I(n,4),R_{0}^{(-n+4)\bmod 2+1}(1)\bigr),\ \bigl((v+1)I,vP\bigr),\ n>4\\
\end{align*}
\end{fleqn}
\subsubsection*{Modules of the form $I(n,3)$}

Defect: $\partial I(n,3) = 1$, for $n\ge 0$.

\begin{fleqn}
\begin{align*}
I(0,3):\ & - \\
I(1,3):\ & \bigl(I(0,4),R_{1}^{1}(1)\bigr)\\
I(2,3):\ & \bigl(I(0,3),R_{1}^{4}(2)\bigr),\ \bigl(I(1,4),R_{1}^{4}(1)\bigr)\\
I(3,3):\ & \bigl(I(0,4),R_{1}^{3}(3)\bigr),\ \bigl(I(1,3),R_{1}^{3}(2)\bigr),\ \bigl(I(2,4),R_{1}^{3}(1)\bigr)\\
I(4,3):\ & \bigl(I(0,1),R_{0}^{1}(1)\bigr),\ \bigl(I(0,2),R_{\infty}^{2}(1)\bigr),\ \bigl(I(1,4),R_{1}^{2}(3)\bigr),\ \bigl(I(2,3),R_{1}^{2}(2)\bigr),\ \bigl(I(3,4),R_{1}^{2}(1)\bigr)\\
 & \bigl(2I(0,3),P(3,3)\bigr)\\
I(n,3):\ & \bigl(I(n-4,1),R_{0}^{(-n+4)\bmod 2+1}(1)\bigr),\ \bigl(I(n-4,2),R_{\infty}^{(-n+5)\bmod 2+1}(1)\bigr),\ \bigl(I(n-3,4),R_{1}^{(-n+5)\bmod 4+1}(3)\bigr)\\
 & \bigl(I(n-2,3),R_{1}^{(-n+5)\bmod 4+1}(2)\bigr),\ \bigl(I(n-1,4),R_{1}^{(-n+5)\bmod 4+1}(1)\bigr),\ \bigl((v+1)I,vP\bigr),\ n>4\\
\end{align*}
\end{fleqn}
\subsubsection*{Modules of the form $I(n,4)$}

Defect: $\partial I(n,4) = 1$, for $n\ge 0$.

\begin{fleqn}
\begin{align*}
I(0,4):\ & - \\
I(1,4):\ & \bigl(I(0,3),R_{1}^{1}(1)\bigr)\\
I(2,4):\ & \bigl(I(0,4),R_{1}^{4}(2)\bigr),\ \bigl(I(1,3),R_{1}^{4}(1)\bigr)\\
I(3,4):\ & \bigl(I(0,3),R_{1}^{3}(3)\bigr),\ \bigl(I(1,4),R_{1}^{3}(2)\bigr),\ \bigl(I(2,3),R_{1}^{3}(1)\bigr)\\
I(4,4):\ & \bigl(I(0,1),R_{\infty}^{1}(1)\bigr),\ \bigl(I(0,2),R_{0}^{2}(1)\bigr),\ \bigl(I(1,3),R_{1}^{2}(3)\bigr),\ \bigl(I(2,4),R_{1}^{2}(2)\bigr),\ \bigl(I(3,3),R_{1}^{2}(1)\bigr)\\
 & \bigl(2I(0,4),P(3,4)\bigr)\\
I(n,4):\ & \bigl(I(n-4,1),R_{\infty}^{(-n+4)\bmod 2+1}(1)\bigr),\ \bigl(I(n-4,2),R_{0}^{(-n+5)\bmod 2+1}(1)\bigr),\ \bigl(I(n-3,3),R_{1}^{(-n+5)\bmod 4+1}(3)\bigr)\\
 & \bigl(I(n-2,4),R_{1}^{(-n+5)\bmod 4+1}(2)\bigr),\ \bigl(I(n-1,3),R_{1}^{(-n+5)\bmod 4+1}(1)\bigr),\ \bigl((v+1)I,vP\bigr),\ n>4\\
\end{align*}
\end{fleqn}
\subsubsection*{Modules of the form $I(n,5)$}

Defect: $\partial I(n,5) = 2$, for $n\ge 0$.

\begin{fleqn}
\begin{align*}
I(0,5):\ & \bigl(I(0,3),I(3,3)\bigr),\ \bigl(I(0,4),I(3,4)\bigr),\ \bigl(I(0,6),R_{1}^{3}(1)\bigr),\ \bigl(I(0,7),R_{1}^{3}(2)\bigr)\\
I(1,5):\ & \bigl(I(0,1),I(1,1)\bigr),\ \bigl(I(0,2),I(1,2)\bigr),\ \bigl(I(1,3),I(4,3)\bigr),\ \bigl(I(1,4),I(4,4)\bigr),\ \bigl(I(1,6),R_{1}^{2}(1)\bigr)\\
 & \bigl(I(1,7),R_{1}^{2}(2)\bigr)\\
I(n,5):\ & \bigl(I(n-1,1),I(n,1)\bigr),\ \bigl(I(n-1,2),I(n,2)\bigr),\ \bigl(I(n,3),I(n+3,3)\bigr)\\
 & \bigl(I(n,4),I(n+3,4)\bigr),\ \bigl(I(n,6),R_{1}^{(-n+2)\bmod 4+1}(1)\bigr),\ \bigl(I(n,7),R_{1}^{(-n+2)\bmod 4+1}(2)\bigr),\ n>1\\
\end{align*}
\end{fleqn}
\subsubsection*{Modules of the form $I(n,6)$}

Defect: $\partial I(n,6) = 2$, for $n\ge 0$.

\begin{fleqn}
\begin{align*}
I(0,6):\ & \bigl(I(0,3),I(2,4)\bigr),\ \bigl(I(0,4),I(2,3)\bigr),\ \bigl(I(0,7),R_{1}^{4}(1)\bigr)\\
I(1,6):\ & \bigl(I(0,5),R_{1}^{1}(1)\bigr),\ \bigl(I(1,3),I(3,4)\bigr),\ \bigl(I(1,4),I(3,3)\bigr),\ \bigl(I(1,7),R_{1}^{3}(1)\bigr)\\
I(2,6):\ & \bigl(I(0,1),I(2,2)\bigr),\ \bigl(I(0,2),I(2,1)\bigr),\ \bigl(I(1,5),R_{1}^{4}(1)\bigr),\ \bigl(I(2,3),I(4,4)\bigr),\ \bigl(I(2,4),I(4,3)\bigr)\\
 & \bigl(I(2,7),R_{1}^{2}(1)\bigr)\\
I(n,6):\ & \bigl(I(n-2,1),I(n,2)\bigr),\ \bigl(I(n-2,2),I(n,1)\bigr),\ \bigl(I(n-1,5),R_{1}^{(-n+5)\bmod 4+1}(1)\bigr)\\
 & \bigl(I(n,3),I(n+2,4)\bigr),\ \bigl(I(n,4),I(n+2,3)\bigr),\ \bigl(I(n,7),R_{1}^{(-n+3)\bmod 4+1}(1)\bigr),\ n>2\\
\end{align*}
\end{fleqn}
\subsubsection*{Modules of the form $I(n,7)$}

Defect: $\partial I(n,7) = 2$, for $n\ge 0$.

\begin{fleqn}
\begin{align*}
I(0,7):\ & \bigl(I(0,3),I(1,3)\bigr),\ \bigl(I(0,4),I(1,4)\bigr)\\
I(1,7):\ & \bigl(I(0,6),R_{1}^{1}(1)\bigr),\ \bigl(I(1,3),I(2,3)\bigr),\ \bigl(I(1,4),I(2,4)\bigr)\\
I(2,7):\ & \bigl(I(0,5),R_{1}^{4}(2)\bigr),\ \bigl(I(1,6),R_{1}^{4}(1)\bigr),\ \bigl(I(2,3),I(3,3)\bigr),\ \bigl(I(2,4),I(3,4)\bigr)\\
I(3,7):\ & \bigl(I(0,1),I(3,1)\bigr),\ \bigl(I(0,2),I(3,2)\bigr),\ \bigl(I(1,5),R_{1}^{3}(2)\bigr),\ \bigl(I(2,6),R_{1}^{3}(1)\bigr),\ \bigl(I(3,3),I(4,3)\bigr)\\
 & \bigl(I(3,4),I(4,4)\bigr)\\
I(n,7):\ & \bigl(I(n-3,1),I(n,1)\bigr),\ \bigl(I(n-3,2),I(n,2)\bigr),\ \bigl(I(n-2,5),R_{1}^{(-n+5)\bmod 4+1}(2)\bigr)\\
 & \bigl(I(n-1,6),R_{1}^{(-n+5)\bmod 4+1}(1)\bigr),\ \bigl(I(n,3),I(n+1,3)\bigr),\ \bigl(I(n,4),I(n+1,4)\bigr),\ n>3\\
\end{align*}
\end{fleqn}
\subsubsection{Schofield pairs associated to regular exceptional modules}

\subsubsection*{The non-homogeneous tube $\mathcal{T}_{1}^{\Delta(\widetilde{\mathbb{D}}_{6})}$}

\begin{figure}[ht]

\begin{center}
\captionof{figure}{\vspace*{-10pt}$\mathcal{T}_{1}^{\Delta(\widetilde{\mathbb{D}}_{6})}$}
\begin{scaletikzpicturetowidth}{0.75\textwidth}
\begin{tikzpicture}[scale=\tikzscale]
	\node[seltwoDDDDDDD, label={[shift={(0.0,-0.45)}, brown]\tiny $R_{1}^{1}(1)$}] (RN_1_1_1__Rationals) at (4,0) {$\begin{smallmatrix}0&&&&0\\&0&0&1\\0&&&&0\end{smallmatrix}$};
	\node[seltwoDDDDDDD, label={[shift={(0.0,-0.45)}, brown]\tiny $R_{1}^{2}(1)$}] (RN_1_2_1__Rationals) at (8,0) {$\begin{smallmatrix}1&&&&1\\&1&1&1\\1&&&&1\end{smallmatrix}$};
	\node[seltwoDDDDDDD, label={[shift={(0.0,-0.45)}, brown]\tiny $R_{1}^{3}(1)$}] (RN_1_3_1__Rationals) at (12,0) {$\begin{smallmatrix}0&&&&0\\&1&0&0\\0&&&&0\end{smallmatrix}$};
	\node[seltwoDDDDDDD, label={[shift={(0.0,-0.45)}, brown]\tiny $R_{1}^{4}(1)$}] (RN_1_4_1__Rationals) at (16,0) {$\begin{smallmatrix}0&&&&0\\&0&1&0\\0&&&&0\end{smallmatrix}$};
	\node[seltwoDDDDDDD, label={[shift={(0.0,-0.45)}, brown]\tiny $R_{1}^{1}(1)$}] (RN_1_1_1__Rationals_repeated) at (20,0) {$\begin{smallmatrix}0&&&&0\\&0&0&1\\0&&&&0\end{smallmatrix}$};
	\node[seltwoDDDDDDD, label={[shift={(0.0,-0.45)}, brown]\tiny $R_{1}^{1}(2)$}] (RN_1_1_2__Rationals) at (6,2) {$\begin{smallmatrix}1&&&&1\\&1&1&2\\1&&&&1\end{smallmatrix}$};
	\node[seltwoDDDDDDD, label={[shift={(0.0,-0.45)}, brown]\tiny $R_{1}^{2}(2)$}] (RN_1_2_2__Rationals) at (10,2) {$\begin{smallmatrix}1&&&&1\\&2&1&1\\1&&&&1\end{smallmatrix}$};
	\node[seltwoDDDDDDD, label={[shift={(0.0,-0.45)}, brown]\tiny $R_{1}^{3}(2)$}] (RN_1_3_2__Rationals) at (14,2) {$\begin{smallmatrix}0&&&&0\\&1&1&0\\0&&&&0\end{smallmatrix}$};
	\node[seltwoDDDDDDD, label={[shift={(0.0,-0.45)}, brown]\tiny $R_{1}^{4}(2)$}] (RN_1_4_2__Rationals) at (18,2) {$\begin{smallmatrix}0&&&&0\\&0&1&1\\0&&&&0\end{smallmatrix}$};
	\node[seltwoDDDDDDD, label={[shift={(0.0,-0.45)}, brown]\tiny $R_{1}^{4}(3)$}] (RN_1_4_3__Rationals) at (4,4) {$\begin{smallmatrix}1&&&&1\\&1&2&2\\1&&&&1\end{smallmatrix}$};
	\node[seltwoDDDDDDD, label={[shift={(0.0,-0.45)}, brown]\tiny $R_{1}^{1}(3)$}] (RN_1_1_3__Rationals) at (8,4) {$\begin{smallmatrix}1&&&&1\\&2&1&2\\1&&&&1\end{smallmatrix}$};
	\node[seltwoDDDDDDD, label={[shift={(0.0,-0.45)}, brown]\tiny $R_{1}^{2}(3)$}] (RN_1_2_3__Rationals) at (12,4) {$\begin{smallmatrix}1&&&&1\\&2&2&1\\1&&&&1\end{smallmatrix}$};
	\node[seltwoDDDDDDD, label={[shift={(0.0,-0.45)}, brown]\tiny $R_{1}^{3}(3)$}] (RN_1_3_3__Rationals) at (16,4) {$\begin{smallmatrix}0&&&&0\\&1&1&1\\0&&&&0\end{smallmatrix}$};
	\node[seltwoDDDDDDD, label={[shift={(0.0,-0.45)}, brown]\tiny $R_{1}^{4}(3)$}] (RN_1_4_3__Rationals_repeated) at (20,4) {$\begin{smallmatrix}1&&&&1\\&1&2&2\\1&&&&1\end{smallmatrix}$};
	\node[noselDDDDDDD, label={[shift={(0.0,-0.45)}, brown]\tiny $R_{1}^{4}(4)$}] (RN_1_4_4__Rationals) at (6,6) {$\begin{smallmatrix}1&&&&1\\&2&2&2\\1&&&&1\end{smallmatrix}$};
	\node[noselDDDDDDD, label={[shift={(0.0,-0.45)}, brown]\tiny $R_{1}^{1}(4)$}] (RN_1_1_4__Rationals) at (10,6) {$\begin{smallmatrix}1&&&&1\\&2&2&2\\1&&&&1\end{smallmatrix}$};
	\node[noselDDDDDDD, label={[shift={(0.0,-0.45)}, brown]\tiny $R_{1}^{2}(4)$}] (RN_1_2_4__Rationals) at (14,6) {$\begin{smallmatrix}1&&&&1\\&2&2&2\\1&&&&1\end{smallmatrix}$};
	\node[noselDDDDDDD, label={[shift={(0.0,-0.45)}, brown]\tiny $R_{1}^{3}(4)$}] (RN_1_3_4__Rationals) at (18,6) {$\begin{smallmatrix}1&&&&1\\&2&2&2\\1&&&&1\end{smallmatrix}$};
	\node[noselDDDDDDD, label={[shift={(0.0,-0.45)}, brown]\tiny $R_{1}^{3}(5)$}] (RN_1_3_5__Rationals) at (4,8) {$\begin{smallmatrix}1&&&&1\\&3&2&2\\1&&&&1\end{smallmatrix}$};
	\node[noselDDDDDDD, label={[shift={(0.0,-0.45)}, brown]\tiny $R_{1}^{4}(5)$}] (RN_1_4_5__Rationals) at (8,8) {$\begin{smallmatrix}1&&&&1\\&2&3&2\\1&&&&1\end{smallmatrix}$};
	\node[noselDDDDDDD, label={[shift={(0.0,-0.45)}, brown]\tiny $R_{1}^{1}(5)$}] (RN_1_1_5__Rationals) at (12,8) {$\begin{smallmatrix}1&&&&1\\&2&2&3\\1&&&&1\end{smallmatrix}$};
	\node[noselDDDDDDD, label={[shift={(0.0,-0.45)}, brown]\tiny $R_{1}^{2}(5)$}] (RN_1_2_5__Rationals) at (16,8) {$\begin{smallmatrix}2&&&&2\\&3&3&3\\2&&&&2\end{smallmatrix}$};
	\node[noselDDDDDDD, label={[shift={(0.0,-0.45)}, brown]\tiny $R_{1}^{3}(5)$}] (RN_1_3_5__Rationals_repeated) at (20,8) {$\begin{smallmatrix}1&&&&1\\&3&2&2\\1&&&&1\end{smallmatrix}$};
	\draw[blue,->] (RN_1_1_1__Rationals) -- (RN_1_1_2__Rationals);
	\draw[gray,dashed,->] (RN_1_1_1__Rationals) -- (RN_1_2_1__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[red,->] (RN_1_1_2__Rationals) -- (RN_1_2_1__Rationals);
	\draw[gray,dashed,->] (RN_1_1_2__Rationals) -- (RN_1_2_2__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[blue,->] (RN_1_1_2__Rationals) -- (RN_1_1_3__Rationals);
	\draw[blue,->] (RN_1_2_1__Rationals) -- (RN_1_2_2__Rationals);
	\draw[gray,dashed,->] (RN_1_2_1__Rationals) -- (RN_1_3_1__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[red,->] (RN_1_2_2__Rationals) -- (RN_1_3_1__Rationals);
	\draw[gray,dashed,->] (RN_1_2_2__Rationals) -- (RN_1_3_2__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[blue,->] (RN_1_2_2__Rationals) -- (RN_1_2_3__Rationals);
	\draw[blue,->] (RN_1_3_1__Rationals) -- (RN_1_3_2__Rationals);
	\draw[gray,dashed,->] (RN_1_3_1__Rationals) -- (RN_1_4_1__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[red,->] (RN_1_3_2__Rationals) -- (RN_1_4_1__Rationals);
	\draw[gray,dashed,->] (RN_1_3_2__Rationals) -- (RN_1_4_2__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[blue,->] (RN_1_3_2__Rationals) -- (RN_1_3_3__Rationals);
	\draw[gray,dashed,->] (RN_1_4_1__Rationals) -- (RN_1_1_1__Rationals_repeated) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[blue,->] (RN_1_4_1__Rationals) -- (RN_1_4_2__Rationals);
	\draw[red,->] (RN_1_4_2__Rationals) -- (RN_1_1_1__Rationals_repeated);
	\draw[gray,dashed,->,out=193,in=-13] (RN_1_4_2__Rationals) to node[midway,sloped,above]{{\tiny $\tau^{-1}$}} (RN_1_1_2__Rationals);
	\draw[blue,->] (RN_1_4_2__Rationals) -- (RN_1_4_3__Rationals_repeated);
	\draw[red,->] (RN_1_4_3__Rationals) -- (RN_1_1_2__Rationals);
	\draw[blue,->] (RN_1_4_3__Rationals) -- (RN_1_4_4__Rationals);
	\draw[gray,dashed,->] (RN_1_4_3__Rationals) -- (RN_1_1_3__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[red,->] (RN_1_4_4__Rationals) -- (RN_1_1_3__Rationals);
	\draw[gray,dashed,->] (RN_1_4_4__Rationals) -- (RN_1_1_4__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[blue,->] (RN_1_4_4__Rationals) -- (RN_1_4_5__Rationals);
	\draw[red,->] (RN_1_1_3__Rationals) -- (RN_1_2_2__Rationals);
	\draw[blue,->] (RN_1_1_3__Rationals) -- (RN_1_1_4__Rationals);
	\draw[gray,dashed,->] (RN_1_1_3__Rationals) -- (RN_1_2_3__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[red,->] (RN_1_1_4__Rationals) -- (RN_1_2_3__Rationals);
	\draw[gray,dashed,->] (RN_1_1_4__Rationals) -- (RN_1_2_4__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[blue,->] (RN_1_1_4__Rationals) -- (RN_1_1_5__Rationals);
	\draw[red,->] (RN_1_2_3__Rationals) -- (RN_1_3_2__Rationals);
	\draw[blue,->] (RN_1_2_3__Rationals) -- (RN_1_2_4__Rationals);
	\draw[gray,dashed,->] (RN_1_2_3__Rationals) -- (RN_1_3_3__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[red,->] (RN_1_2_4__Rationals) -- (RN_1_3_3__Rationals);
	\draw[gray,dashed,->] (RN_1_2_4__Rationals) -- (RN_1_3_4__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[blue,->] (RN_1_2_4__Rationals) -- (RN_1_2_5__Rationals);
	\draw[red,->] (RN_1_3_3__Rationals) -- (RN_1_4_2__Rationals);
	\draw[gray,dashed,->] (RN_1_3_3__Rationals) -- (RN_1_4_3__Rationals_repeated) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[blue,->] (RN_1_3_3__Rationals) -- (RN_1_3_4__Rationals);
	\draw[red,->] (RN_1_3_4__Rationals) -- (RN_1_4_3__Rationals_repeated);
	\draw[gray,dashed,->,out=193,in=-13] (RN_1_3_4__Rationals) to node[midway,sloped,above]{{\tiny $\tau^{-1}$}} (RN_1_4_4__Rationals);
	\draw[blue,->] (RN_1_3_4__Rationals) -- (RN_1_3_5__Rationals_repeated);
	\draw[red,->] (RN_1_3_5__Rationals) -- (RN_1_4_4__Rationals);
	\draw[gray,dashed,->] (RN_1_3_5__Rationals) -- (RN_1_4_5__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[red,->] (RN_1_4_5__Rationals) -- (RN_1_1_4__Rationals);
	\draw[gray,dashed,->] (RN_1_4_5__Rationals) -- (RN_1_1_5__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[red,->] (RN_1_1_5__Rationals) -- (RN_1_2_4__Rationals);
	\draw[gray,dashed,->] (RN_1_1_5__Rationals) -- (RN_1_2_5__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[red,->] (RN_1_2_5__Rationals) -- (RN_1_3_4__Rationals);
	\draw[gray,dashed,->] (RN_1_2_5__Rationals) -- (RN_1_3_5__Rationals_repeated) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[dashed,-] (RN_1_1_1__Rationals) -- (RN_1_4_3__Rationals);
	\draw[dashed,-] (RN_1_1_1__Rationals_repeated) -- (RN_1_4_3__Rationals_repeated);
	\draw[dashed,-] (RN_1_4_3__Rationals) -- (RN_1_3_5__Rationals);
	\draw[dashed,-] (RN_1_4_3__Rationals_repeated) -- (RN_1_3_5__Rationals_repeated);
\end{tikzpicture}
\end{scaletikzpicturetowidth}
\end{center}

\end{figure}
\begin{fleqn}
\begin{align*}
R_{1}^{1}(1):\ & - \\
R_{1}^{1}(2):\ & \bigl(R_{1}^{2}(1),R_{1}^{1}(1)\bigr),\ \bigl(I(1,1),P(0,1)\bigr),\ \bigl(I(1,2),P(0,2)\bigr),\ \bigl(I(1,3),P(0,3)\bigr),\ \bigl(I(1,4),P(0,4)\bigr)\\
 & \bigl(I(1,7),P(0,5)\bigr)\\
R_{1}^{2}(1):\ & \bigl(I(0,2),P(0,1)\bigr),\ \bigl(I(0,1),P(0,2)\bigr),\ \bigl(I(0,4),P(0,3)\bigr),\ \bigl(I(0,3),P(0,4)\bigr),\ \bigl(I(0,6),P(0,5)\bigr)\\
 & \bigl(I(0,7),P(0,6)\bigr)\\
R_{1}^{2}(2):\ & \bigl(R_{1}^{3}(1),R_{1}^{2}(1)\bigr),\ \bigl(I(0,1),P(1,1)\bigr),\ \bigl(I(0,2),P(1,2)\bigr),\ \bigl(I(0,3),P(1,3)\bigr),\ \bigl(I(0,4),P(1,4)\bigr)\\
 & \bigl(I(0,7),P(1,5)\bigr)\\
R_{1}^{3}(1):\ & - \\
R_{1}^{3}(2):\ & \bigl(R_{1}^{4}(1),R_{1}^{3}(1)\bigr)\\
R_{1}^{4}(1):\ & - \\
R_{1}^{4}(2):\ & \bigl(R_{1}^{1}(1),R_{1}^{4}(1)\bigr)\\
R_{1}^{4}(3):\ & \bigl(R_{1}^{1}(2),R_{1}^{4}(1)\bigr),\ \bigl(R_{1}^{2}(1),R_{1}^{4}(2)\bigr),\ \bigl(I(2,2),P(0,1)\bigr),\ \bigl(I(2,1),P(0,2)\bigr),\ \bigl(I(2,4),P(0,3)\bigr)\\
 & \bigl(I(2,3),P(0,4)\bigr)\\
R_{1}^{1}(3):\ & \bigl(R_{1}^{2}(2),R_{1}^{1}(1)\bigr),\ \bigl(R_{1}^{3}(1),R_{1}^{1}(2)\bigr),\ \bigl(I(1,2),P(1,1)\bigr),\ \bigl(I(1,1),P(1,2)\bigr),\ \bigl(I(1,4),P(1,3)\bigr)\\
 & \bigl(I(1,3),P(1,4)\bigr)\\
R_{1}^{2}(3):\ & \bigl(R_{1}^{3}(2),R_{1}^{2}(1)\bigr),\ \bigl(R_{1}^{4}(1),R_{1}^{2}(2)\bigr),\ \bigl(I(0,2),P(2,1)\bigr),\ \bigl(I(0,1),P(2,2)\bigr),\ \bigl(I(0,4),P(2,3)\bigr)\\
 & \bigl(I(0,3),P(2,4)\bigr)\\
R_{1}^{3}(3):\ & \bigl(R_{1}^{4}(2),R_{1}^{3}(1)\bigr),\ \bigl(R_{1}^{1}(1),R_{1}^{3}(2)\bigr)\\
\end{align*}
\end{fleqn}
\subsubsection*{The non-homogeneous tube $\mathcal{T}_{\infty}^{\Delta(\widetilde{\mathbb{D}}_{6})}$}

\begin{figure}[ht]

\begin{center}
\captionof{figure}{\vspace*{-10pt}$\mathcal{T}_{\infty}^{\Delta(\widetilde{\mathbb{D}}_{6})}$}
\begin{scaletikzpicturetowidth}{0.40000000000000002\textwidth}
\begin{tikzpicture}[scale=\tikzscale]
	\node[seltwoDDDDDDD, label={[shift={(0.0,-0.45)}, brown]\tiny $R_{\infty}^{1}(1)$}] (RN_2_1_1__Rationals) at (4,0) {$\begin{smallmatrix}1&&&&0\\&1&1&1\\0&&&&1\end{smallmatrix}$};
	\node[seltwoDDDDDDD, label={[shift={(0.0,-0.45)}, brown]\tiny $R_{\infty}^{2}(1)$}] (RN_2_2_1__Rationals) at (8,0) {$\begin{smallmatrix}0&&&&1\\&1&1&1\\1&&&&0\end{smallmatrix}$};
	\node[seltwoDDDDDDD, label={[shift={(0.0,-0.45)}, brown]\tiny $R_{\infty}^{1}(1)$}] (RN_2_1_1__Rationals_repeated) at (12,0) {$\begin{smallmatrix}1&&&&0\\&1&1&1\\0&&&&1\end{smallmatrix}$};
	\node[noselDDDDDDD, label={[shift={(0.0,-0.45)}, brown]\tiny $R_{\infty}^{1}(2)$}] (RN_2_1_2__Rationals) at (6,2) {$\begin{smallmatrix}1&&&&1\\&2&2&2\\1&&&&1\end{smallmatrix}$};
	\node[noselDDDDDDD, label={[shift={(0.0,-0.45)}, brown]\tiny $R_{\infty}^{2}(2)$}] (RN_2_2_2__Rationals) at (10,2) {$\begin{smallmatrix}1&&&&1\\&2&2&2\\1&&&&1\end{smallmatrix}$};
	\node[noselDDDDDDD, label={[shift={(0.0,-0.45)}, brown]\tiny $R_{\infty}^{2}(3)$}] (RN_2_2_3__Rationals) at (4,4) {$\begin{smallmatrix}1&&&&2\\&3&3&3\\2&&&&1\end{smallmatrix}$};
	\node[noselDDDDDDD, label={[shift={(0.0,-0.45)}, brown]\tiny $R_{\infty}^{1}(3)$}] (RN_2_1_3__Rationals) at (8,4) {$\begin{smallmatrix}2&&&&1\\&3&3&3\\1&&&&2\end{smallmatrix}$};
	\node[noselDDDDDDD, label={[shift={(0.0,-0.45)}, brown]\tiny $R_{\infty}^{2}(3)$}] (RN_2_2_3__Rationals_repeated) at (12,4) {$\begin{smallmatrix}1&&&&2\\&3&3&3\\2&&&&1\end{smallmatrix}$};
	\draw[blue,->] (RN_2_1_1__Rationals) -- (RN_2_1_2__Rationals);
	\draw[gray,dashed,->] (RN_2_1_1__Rationals) -- (RN_2_2_1__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[red,->] (RN_2_1_2__Rationals) -- (RN_2_2_1__Rationals);
	\draw[gray,dashed,->] (RN_2_1_2__Rationals) -- (RN_2_2_2__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[blue,->] (RN_2_1_2__Rationals) -- (RN_2_1_3__Rationals);
	\draw[gray,dashed,->] (RN_2_2_1__Rationals) -- (RN_2_1_1__Rationals_repeated) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[blue,->] (RN_2_2_1__Rationals) -- (RN_2_2_2__Rationals);
	\draw[red,->] (RN_2_2_2__Rationals) -- (RN_2_1_1__Rationals_repeated);
	\draw[gray,dashed,->,out=219,in=-39] (RN_2_2_2__Rationals) to node[midway,sloped,above]{{\tiny $\tau^{-1}$}} (RN_2_1_2__Rationals);
	\draw[blue,->] (RN_2_2_2__Rationals) -- (RN_2_2_3__Rationals_repeated);
	\draw[red,->] (RN_2_2_3__Rationals) -- (RN_2_1_2__Rationals);
	\draw[gray,dashed,->] (RN_2_2_3__Rationals) -- (RN_2_1_3__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[red,->] (RN_2_1_3__Rationals) -- (RN_2_2_2__Rationals);
	\draw[gray,dashed,->] (RN_2_1_3__Rationals) -- (RN_2_2_3__Rationals_repeated) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[dashed,-] (RN_2_1_1__Rationals) -- (RN_2_2_3__Rationals);
	\draw[dashed,-] (RN_2_1_1__Rationals_repeated) -- (RN_2_2_3__Rationals_repeated);
\end{tikzpicture}
\end{scaletikzpicturetowidth}
\end{center}

\end{figure}
\begin{fleqn}
\begin{align*}
R_{\infty}^{1}(1):\ & \bigl(I(3,3),P(0,2)\bigr),\ \bigl(I(2,4),P(1,1)\bigr),\ \bigl(I(1,3),P(2,2)\bigr),\ \bigl(I(0,4),P(3,1)\bigr)\\
R_{\infty}^{2}(1):\ & \bigl(I(3,4),P(0,1)\bigr),\ \bigl(I(2,3),P(1,2)\bigr),\ \bigl(I(1,4),P(2,1)\bigr),\ \bigl(I(0,3),P(3,2)\bigr)\\
\end{align*}
\end{fleqn}
\subsubsection*{The non-homogeneous tube $\mathcal{T}_{0}^{\Delta(\widetilde{\mathbb{D}}_{6})}$}

\begin{figure}[ht]

\begin{center}
\captionof{figure}{\vspace*{-10pt}$\mathcal{T}_{0}^{\Delta(\widetilde{\mathbb{D}}_{6})}$}
\begin{scaletikzpicturetowidth}{0.40000000000000002\textwidth}
\begin{tikzpicture}[scale=\tikzscale]
	\node[seltwoDDDDDDD, label={[shift={(0.0,-0.45)}, brown]\tiny $R_{0}^{1}(1)$}] (RN_3_1_1__Rationals) at (4,0) {$\begin{smallmatrix}1&&&&1\\&1&1&1\\0&&&&0\end{smallmatrix}$};
	\node[seltwoDDDDDDD, label={[shift={(0.0,-0.45)}, brown]\tiny $R_{0}^{2}(1)$}] (RN_3_2_1__Rationals) at (8,0) {$\begin{smallmatrix}0&&&&0\\&1&1&1\\1&&&&1\end{smallmatrix}$};
	\node[seltwoDDDDDDD, label={[shift={(0.0,-0.45)}, brown]\tiny $R_{0}^{1}(1)$}] (RN_3_1_1__Rationals_repeated) at (12,0) {$\begin{smallmatrix}1&&&&1\\&1&1&1\\0&&&&0\end{smallmatrix}$};
	\node[noselDDDDDDD, label={[shift={(0.0,-0.45)}, brown]\tiny $R_{0}^{1}(2)$}] (RN_3_1_2__Rationals) at (6,2) {$\begin{smallmatrix}1&&&&1\\&2&2&2\\1&&&&1\end{smallmatrix}$};
	\node[noselDDDDDDD, label={[shift={(0.0,-0.45)}, brown]\tiny $R_{0}^{2}(2)$}] (RN_3_2_2__Rationals) at (10,2) {$\begin{smallmatrix}1&&&&1\\&2&2&2\\1&&&&1\end{smallmatrix}$};
	\node[noselDDDDDDD, label={[shift={(0.0,-0.45)}, brown]\tiny $R_{0}^{2}(3)$}] (RN_3_2_3__Rationals) at (4,4) {$\begin{smallmatrix}1&&&&1\\&3&3&3\\2&&&&2\end{smallmatrix}$};
	\node[noselDDDDDDD, label={[shift={(0.0,-0.45)}, brown]\tiny $R_{0}^{1}(3)$}] (RN_3_1_3__Rationals) at (8,4) {$\begin{smallmatrix}2&&&&2\\&3&3&3\\1&&&&1\end{smallmatrix}$};
	\node[noselDDDDDDD, label={[shift={(0.0,-0.45)}, brown]\tiny $R_{0}^{2}(3)$}] (RN_3_2_3__Rationals_repeated) at (12,4) {$\begin{smallmatrix}1&&&&1\\&3&3&3\\2&&&&2\end{smallmatrix}$};
	\draw[blue,->] (RN_3_1_1__Rationals) -- (RN_3_1_2__Rationals);
	\draw[gray,dashed,->] (RN_3_1_1__Rationals) -- (RN_3_2_1__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[red,->] (RN_3_1_2__Rationals) -- (RN_3_2_1__Rationals);
	\draw[gray,dashed,->] (RN_3_1_2__Rationals) -- (RN_3_2_2__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[blue,->] (RN_3_1_2__Rationals) -- (RN_3_1_3__Rationals);
	\draw[gray,dashed,->] (RN_3_2_1__Rationals) -- (RN_3_1_1__Rationals_repeated) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[blue,->] (RN_3_2_1__Rationals) -- (RN_3_2_2__Rationals);
	\draw[red,->] (RN_3_2_2__Rationals) -- (RN_3_1_1__Rationals_repeated);
	\draw[gray,dashed,->,out=219,in=-39] (RN_3_2_2__Rationals) to node[midway,sloped,above]{{\tiny $\tau^{-1}$}} (RN_3_1_2__Rationals);
	\draw[blue,->] (RN_3_2_2__Rationals) -- (RN_3_2_3__Rationals_repeated);
	\draw[red,->] (RN_3_2_3__Rationals) -- (RN_3_1_2__Rationals);
	\draw[gray,dashed,->] (RN_3_2_3__Rationals) -- (RN_3_1_3__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[red,->] (RN_3_1_3__Rationals) -- (RN_3_2_2__Rationals);
	\draw[gray,dashed,->] (RN_3_1_3__Rationals) -- (RN_3_2_3__Rationals_repeated) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[dashed,-] (RN_3_1_1__Rationals) -- (RN_3_2_3__Rationals);
	\draw[dashed,-] (RN_3_1_1__Rationals_repeated) -- (RN_3_2_3__Rationals_repeated);
\end{tikzpicture}
\end{scaletikzpicturetowidth}
\end{center}

\end{figure}
\begin{fleqn}
\begin{align*}
R_{0}^{1}(1):\ & \bigl(I(3,4),P(0,2)\bigr),\ \bigl(I(2,3),P(1,1)\bigr),\ \bigl(I(1,4),P(2,2)\bigr),\ \bigl(I(0,3),P(3,1)\bigr)\\
R_{0}^{2}(1):\ & \bigl(I(3,3),P(0,1)\bigr),\ \bigl(I(2,4),P(1,2)\bigr),\ \bigl(I(1,3),P(2,1)\bigr),\ \bigl(I(0,4),P(3,2)\bigr)\\
\end{align*}
\end{fleqn}
\clearpage

\subsection[Schofield pairs for the quiver $\Delta(\widetilde{\mathbb{D}}_m)$]{Schofield pairs for the quiver $\Delta(\widetilde{\mathbb{D}}_m)$ -- {\usebox{\boxDm}}, for $m\ge 4$}
\[
\xymatrix{2 &  &  &  &  & 3\ar[dl]\\
 & 5\ar[dl]\ar[ul] & 6\ar[l] & \ar[l]\cdots & \ar[l]m+1\\
1 &  &  &  &  & 4\ar[ul]
}
\]

\[
C_{\Delta(\widetilde{\mathbb{D}}_{m})}=\begin{bmatrix}1 & 0 & 1 & 1 & 1 & 1 & 1 & \cdots & 1 & 1\\
0 & 1 & 1 & 1 & 1 & 1 & 1 & \cdots & 1 & 1\\
0 & 0 & 1 & 0 & 0 & 0 & 0 & \cdots & 0 & 0\\
0 & 0 & 0 & 1 & 0 & 0 & 0 & \cdots & 0 & 0\\
0 & 0 & 1 & 1 & 1 & 1 & 1 & \cdots & 1 & 1\\
0 & 0 & 1 & 1 & 0 & 1 & 1 & \cdots & 1 & 1\\
0 & 0 & 1 & 1 & 0 & 0 & 1 & \cdots & 1 & 1\\
\vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \ddots & \ddots & \vdots & \vdots\\
0 & 0 & 1 & 1 & 0 & 0 & 0 & \cdots & 1 & 1\\
0 & 0 & 1 & 1 & 0 & 0 & 0 & \cdots & 0 & 1
\end{bmatrix}
\quad
\Phi_{\Delta(\widetilde{\mathbb{D}}_{m})}=\begin{bmatrix}-1 & 0 & 0 & 0 & 1 & 0 & 0 & \cdots & 0\\
0 & -1 & 0 & 0 & 1 & 0 & 0 & \cdots & 0\\
-1 & -1 & 0 & 1 & 1 & 0 & 0 & \cdots & 0\\
-1 & -1 & 1 & 0 & 1 & 0 & 0 & \cdots & 0\\
-1 & -1 & 0 & 0 & 1 & 1 & 0 & \cdots & 0\\
-1 & -1 & 0 & 0 & 1 & 0 & 1 & \cdots & 0\\
\vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \ddots & \ddots & \vdots\\
-1 & -1 & 0 & 0 & 1 & 0 & 0 & \cdots & 1\\
-1 & -1 & 1 & 1 & 1 & 0 & 0 & \cdots & 0
\end{bmatrix}
\]

\subsubsection{Schofield pairs associated to preprojective exceptional modules}

$\underline\dim P(0,j)=j^{th}\text{ column of }C_{\Delta(\widetilde{\mathbb{D}}_{m})}$,\quad{}$\underline\dim I(0,i)=i^{th}\text{ row of }C_{\Delta(\widetilde{\mathbb{D}}_{m})}$\medskip{}

Throughout the lists below, in Schofield pairs of the form $(R,P)$
associated to the preprojective indecomposable $P'$, the preprojective
indecomposable $P$ is given by an explicit formula, while $R$ always
denotes the non-homogeneous regular with dimension vector $\underline\dim R=\underline\dim P'-\underline\dim P$
(for each such pair separately).

\subsubsection*{Modules of the form $P(n,1)$}

Defect: $\partial P(n,1)=-1$, for $n\ge0$. In the formulas below
$0\leq n_{1}<m-2$ and $n_{2}>m-2$:

\begin{fleqn}

\begin{align*}
P(n_{1},1):\  & \bigl(R,P(i,(n_{1}+i)\bmod2+1)\bigr),\ 0\leq i\leq n_{1}-1\\
P(m-2,1):\  & \bigl(R,P(i,(m+i)\bmod2+1)\bigr),\ 1\leq i\leq m-3\\
 & \bigl(R,P(0,3)\bigr),\ \bigl(R,P(0,4)\bigr),\ \bigl(I(m-3,1),2P(0,m\bmod2+1)\bigr)\\
P(n_{2},1):\  & \bigl(R,P(n_{2}-m+i+2,(m+i)\bmod2+1)\bigr),\ 1\leq i\leq m-3\\
 & \bigl(R,P(n_{2}-m+2,3)\bigr),\ \bigl(R,P(n_{2}-m+2,4)\bigr),\ \bigl(uI,(u+1)P\bigr)
\end{align*}

\end{fleqn}

\subsubsection*{Modules of the form $P(n,2)$}

Defect: $\partial P(n,2)=-1$, for $n\ge0$. In the formulas below
$0\leq n_{1}<m-2$ and $n_{2}>m-2$:

\begin{fleqn}

\begin{align*}
P(n_{1},2):\  & \bigl(R,P(i,(n_{1}+i+1)\bmod2+1)\bigr),\ 0\leq i\leq n_{1}-1\\
P(m-2,2):\  & \bigl(R,P(i,(m+i+1)\bmod2+1)\bigr),\ 1\leq i\leq m-3\\
 & \bigl(R,P(0,3)\bigr),\ \bigl(R,P(0,4)\bigr),\ \bigl(I(m-3,2),2P(0,(m+1)\bmod2+1)\bigr)\\
P(n_{2},2):\  & \bigl(R,P(n_{2}-m+i+2,(m+i+1)\bmod2+1)\bigr),\ 1\leq i\leq m-3\\
 & \bigl(R,P(n_{2}-m+2,3)\bigr),\ \bigl(R,P(n_{2}-m+2,4)\bigr),\ \bigl(uI,(u+1)P\bigr)
\end{align*}

\end{fleqn}

\subsubsection*{Modules of the form $P(n,3)$}

Defect: $\partial P(n,3)=-1$, for $n\ge0$. In the formulas below
$0\leq n_{1}<m-2$ and $n_{2}>m-2$:

\begin{fleqn}
\begin{align*}
P(n_{1},3):\  & \bigl(R,P(i,(n_{1}+i)\bmod2+3)\bigr),\ 0\leq i\leq n_{1}-1\\
 & \bigl(I(m-n_{1}-i+1,(m+i+1)\bmod2+3),P(n_{1},i)\bigr),\ 5\leq i\leq m-n_{1}+1\\
 & \bigl(R,P(n_{1},1)\bigr),\ \bigl(R,P(n_{1},2)\bigr)\\
P(m-2,3):\  & \bigl(R,P(i,(m+i)\bmod2+3)\bigr),\ 1\leq i\leq m-3\\
 & \bigl(R,P(m-2,1)\bigr),\ \bigl(R,P(m-2,2)\bigr),\ \bigl(I(m-3,3),2P(0,m\bmod2+3)\bigr)\\
P(n_{2},3):\  & \bigl(R,P(n_{2}-m+i+2,(m+i)\bmod2+3)\bigr),\ 1\leq i\leq m-3\\
 & \bigl(R,P(n_{2},1)\bigr),\ \bigl(R,P(n_{2},2)\bigr),\ \bigl(uI,(u+1)P\bigr)
\end{align*}

\end{fleqn}

\subsubsection*{Modules of the form $P(n,4)$}

Defect: $\partial P(n,4)=-1$, for $n\ge0$. In the formulas below
$0\leq n_{1}<m-2$ and $n_{2}>m-2$:

\begin{fleqn}
\begin{align*}
P(n_{1},4):\  & \bigl(R,P(i,(n_{1}+i+1)\bmod2+3)\bigr),\ 0\leq i\leq n_{1}-1\\
 & \bigl(I(m-n_{1}-i+1,(m+i)\bmod2+3),P(n_{1},i)\bigr),\ 5\leq i\leq m-n_{1}+1\\
 & \bigl(R,P(n_{1},1)\bigr),\ \bigl(R,P(n_{1},2)\bigr)\\
P(m-2,4):\  & \bigl(R,P(i,(m+i+1)\bmod2+3)\bigr),\ 1\leq i\leq m-3\\
 & \bigl(R,P(m-2,1)\bigr),\ \bigl(R,P(m-2,2)\bigr),\ \bigl(I(m-3,4),2P(0,(m+1)\bmod2+3)\bigr)\\
P(n_{2},4):\  & \bigl(R,P(n_{2}-m+i+2,(m+i+1)\bmod2+3)\bigr),\ 1\leq i\leq m-3\\
 & \bigl(R,P(n_{2},1)\bigr),\ \bigl(R,P(n_{2},2)\bigr),\ \bigl(uI,(u+1)P\bigr)
\end{align*}

\end{fleqn}

\subsubsection*{Modules of the form $P(n,j)$, for $5\protect\leq j\protect\leq m+1$}

Defect: $\partial P(n,j)=-2$, for $n\ge0$. In the formulas below
$0\leq n_{1}<m-j+1$ and $n_{2}>m-j+1$:

\begin{fleqn}

\begin{align*}
P(n_{1},j):\  & \bigl(R,P(i,j+n_{1}-i)\bigr),\ 0\leq i\leq n_{1}-1\\
 & \bigl(R,P(n_{1},i)\bigr),\ 5\leq i\leq j-1\\
 & \bigl(P(j-4+n_{1},(j+1)\bmod2+1),P(n_{1},1)\bigr),\ \bigl(P(j-4+n_{1},j\bmod2+1),P(n_{1},2)\bigr)\\
P(m-j+2,j):\  & \bigl(R,P(i+1,m-i+1)\bigr),\ 0\leq i\leq m-j\\
 & \bigl(R,P(m-j+2,i)\bigr),\ 5\leq i\leq j-1\\
 & \bigl(P(m-j+2,(m+j+1)\mod2+3),P(0,3)\bigr),\ \bigl(P(m-j+2,4-(m+j+1)\bmod2),P(0,4)\bigr),\\
 & \bigl(P(m-2,(j+1)\bmod2+1),P(m-j+2,1)\bigr),\ \bigl(P(m-2,j\bmod2+1),P(m-j+2,2)\bigr)\\
P(n_{2},j):\  & \bigl(R,P(n_{2}-m+j+i,m-i+1)\bigr),\ 0\leq i\leq m-j\\
 & \bigl(R,P(n_{2}+1,i)\bigr),\ 5\leq i\leq j-1\\
 & \bigl(P(n_{2}+1,(m+j+1)\mod2+3),P(n_{2}-m+j-1,3)\bigr),\\
 & \bigl(P(n_{2}+1,4-(m+j+1)\bmod2),P(n_{2}-m+j-1,4)\bigr),\\
 & \bigl(P(n_{2}+j-3,(j+1)\bmod2+1),P(n_{2}+1,1)\bigr),\\
 & \bigl(P(n_{2}+j-3,j\bmod2+1),P(n_{2}+1,2)\bigr)
\end{align*}

\end{fleqn}

\subsubsection{Schofield pairs associated to preinjective exceptional modules}

$\underline\dim P(0,j)=j^{th}\text{ column of }C_{\Delta(\widetilde{\mathbb{D}}_{m})}$,\quad{}$\underline\dim I(0,i)=i^{th}\text{ row of }C_{\Delta(\widetilde{\mathbb{D}}_{m})}$\medskip{}

Throughout the lists below, in Schofield pairs of the form $(I,R)$
associated to the preinjective indecomposable $I'$, the preinjective
indecomposable $I$ is given by an explicit formula, while $R$ always
denotes the non-homogeneous regular with dimension vector $\underline\dim R=\underline\dim I'-\underline\dim I$
(for each such pair separately).

\subsubsection*{Modules of the form $I(n,1)$}

Defect: $\partial I(n,1)=1$, for $n\ge0$. In the formulas below
$0\leq n_{1}<m-2$ and $n_{2}>m-2$:

\begin{fleqn}

\begin{align*}
I(n_{1},1):\  & \bigl(I(i,(n_{1}+i)\bmod2+1),R\bigr),\ 0\leq i\leq n_{1}-1\\
 & \bigl(I(n_{1},i),P(i-n_{1}-5,(i+1)\bmod2+1)\bigr),\ 5+n_{1}\leq i\leq m+1\\
 & \bigl(I(n_{1},3),R\bigr),\ \bigl(I(n_{1},4),R\bigr)\\
I(m-2,1):\  & \bigl(I(i,(m+i)\bmod2+1),R\bigr),\ 1\leq i\leq m-3\\
 & \bigl(I(m-2,3),R\bigr),\ \bigl(I(m-2,4),R\bigr),\ \bigl(2I(0,m\bmod2+1),P(m-3,1)\bigr)\\
I(n_{2},1):\  & \bigl(I(n_{2}-m+i+2,(m+i)\bmod2+1),R\bigr),\ 1\leq i\leq m-3\\
 & \bigl(I(n_{2},3),R\bigr),\ \bigl(I(n_{2},4),R\bigr),\ \bigl((v+1)I,vP\bigr)
\end{align*}

\end{fleqn}

\subsubsection*{Modules of the form $I(n,2)$}

Defect: $\partial I(n,2)=1$, for $n\ge0$. In the formulas below
$0\leq n_{1}<m-2$ and $n_{2}>m-2$:

\begin{fleqn}

\begin{align*}
I(n_{1},2):\  & \bigl(I(i,(n_{1}+i+1)\bmod2+1),R\bigr),\ 0\leq i\leq n_{1}-1\\
 & \bigl(I(n_{1},i),P(i-n_{1}-5,i\bmod2+1)\bigr),\ 5+n_{1}\leq i\leq m+1\\
 & \bigl(I(n_{1},3),R\bigr),\ \bigl(I(n_{1},4)\bigr)\\
I(m-2,2):\  & \bigl(I(i,(m+i+1)\bmod2+1),R\bigr),\ 1\leq i\leq m-3\\
 & \bigl(I(m-2,3),R\bigr),\ \bigl(I(m-2,4),R\bigr),\ \bigl(2I(0,(m+1)\bmod2+1),P(m-3,2)\bigr)\\
I(n_{2},2):\  & \bigl(I(n_{2}-m+i+2,(m+i+1)\bmod2+1),R\bigr),\ 1\leq i\leq m-3\\
 & \bigl(I(n_{2},3),R\bigr),\ \bigl(I(n_{2},4),R\bigr),\ \bigl((v+1)I,vP\bigr)
\end{align*}

\end{fleqn}

\subsubsection*{Modules of the form $I(n,3)$}

Defect: $\partial I(n,3)=1$, for $n\ge0$. In the formulas below
$0\leq n_{1}<m-2$ and $n_{2}>m-2$:

\begin{fleqn}

\begin{align*}
I(n_{1},3):\  & \bigl(I(i,(n_{1}+i)\bmod2+3),R\bigr),\ 0\leq i\leq n_{1}-1\\
I(m-2,3):\  & \bigl(I(i,(m+i)\bmod2+3),R\bigr),\ 1\leq i\leq m-3\\
 & \bigl(I(0,1),R\bigr),\ \bigl(I(0,2),R\bigr),\ \bigl(2I(0,m\bmod2+3),P(m-3,3)\bigr)\\
I(n_{2},3):\  & \bigl(I(n_{2}-m+i+2,(m+i)\bmod2+3),R\bigr),\ 1\leq i\leq m-3\\
 & \bigl(I(n_{2}-m+2,1),R\bigr),\ \bigl(I(n_{2}-m+2,2),R\bigr),\ \bigl((v+1)I,vP\bigr)
\end{align*}

\end{fleqn}

\subsubsection*{Modules of the form $I(n,4)$}

Defect: $\partial I(n,4)=1$, for $n\ge0$. In the formulas below
$0\leq n_{1}<m-2$ and $n_{2}>m-2$:

\begin{fleqn}

\begin{align*}
I(n_{1},4):\  & \bigl(I(i,(n_{1}+i+1)\bmod2+3),R\bigr),\ 0\leq i\leq n_{1}-1\\
I(m-2,4):\  & \bigl(I(i,(m+i+1)\bmod2+3),R\bigr),\ 1\leq i\leq m-3\\
 & \bigl(I(0,1),R\bigr),\ \bigl(I(0,2),R\bigr),\ \bigl(2I(0,(m+1)\bmod2+3),P(m-3,4)\bigr)\\
I(n_{2},4):\  & \bigl(I(n_{2}-m+i+2,(m+i+1)\bmod2+3),R\bigr),\ 1\leq i\leq m-3\\
 & \bigl(I(n_{2}-m+2,1),R\bigr),\ \bigl(I(n_{2}-m+2,2),R\bigr),\ \bigl((v+1)I,vP\bigr)
\end{align*}

\end{fleqn}

\subsubsection*{Modules of the form $I(n,j)$, for $5\protect\leq j\protect\leq m+1$}

Defect: $\partial I(n,j)=2$, for $n\ge0$. In the formulas below
$0\leq n_{1}<j-4$ and $n_{2}>j-4$:

\begin{fleqn}

\begin{align*}
I(n_{1},j):\  & \bigl(I(n_{1},i),R\bigr),\ j+1\leq i\leq m+1\\
 & \bigl(I(i-j+n_{1},i),R\bigr),\ j-n_{1}\leq i\leq j-1\\
 & \bigl(I(n_{1},3),I(m-j+n_{1}+2,(m+j+1)\bmod2+3)\bigr),\ \bigl(I(n_{1},4),I(m-j+n_{1}+2,(m+j)\bmod2+3)\bigr)\\
I(j-4,j):\  & \bigl(I(j-4,i),R\bigr),\ j+1\leq i\leq m+1\\
 & \bigl(I(i-4,i),R\bigr),\ 5\leq i\leq j-1\\
 & \bigl(I(j-4,3),I(m-2,(m+j+1)\bmod2+3)\bigr),\ \bigl(I(j-4,4),I(m-2,(m+j)\bmod2+3)\bigr),\\
 & \bigl(I(0,1),I(j-4,(j+1)\bmod2+1)\bigr),\ \bigl(I(0,2),I(j-4,j\bmod2+1)\bigr)\\
I(n_{2},j):\  & \bigl(I(n_{2},i),R\bigr),\ j+1\leq i\leq m+1\\
 & \bigl(I(n_{2}-j+i,i),R\bigr),\ 5\leq i\leq j-1\\
 & \bigl(I(n_{2},3),I(m-2,(m+j+1)\bmod2+3)\bigr),\ \bigl(I(n_{2},4),I(m-2,(m+j)\bmod2+3)\bigr),\\
 & \bigl(I(n_{2}-j+4,1),I(j-4,(j+1)\bmod2+1)\bigr),\ \bigl(I(n_{2}-j+4,2),I(j-4,j\bmod2+1)\bigr)
\end{align*}

\end{fleqn}

\subsubsection{Schofield pairs associated to regular exceptional modules}

\subsubsection*{The non-homogeneous tube $\mathcal{T}_{1}^{\Delta(\widetilde{\mathbb{D}}_{m})}$}

$\underline\dim R_{1}^{1}(1)=\begin{smallmatrix}0 &  &  &  &  &  & 0\\
 & 0 & 0 & \cdots & 0 & 1\\
0 &  &  &  &  &  & 0
\end{smallmatrix}$,$\quad\underline\dim R_{1}^{2}(1)=\begin{smallmatrix}1 &  &  &  &  &  & 1\\
 & 1 & 1 & 1 & \cdots & 1\\
1 &  &  &  &  &  & 1
\end{smallmatrix}$,$\quad\underline\dim R_{1}^{3}(1)=\begin{smallmatrix}0 &  &  &  &  &  & 0\\
 & 1 & 0 & 0 & \cdots & 0\\
0 &  &  &  &  &  & 0
\end{smallmatrix}$,\newline\newline$\underline\dim R_{1}^{4}(1)=\begin{smallmatrix}0 &  &  &  &  &  & 0\\
 & 0 & 1 & 0 & \cdots & 0\\
0 &  &  &  &  &  & 0
\end{smallmatrix}$,$\quad\dots$,$\quad\underline\dim R_{1}^{m-2}(1)=\begin{smallmatrix}0 &  &  &  &  &  & 0\\
 & 0 & \cdots & 0 & 1 & 0\\
0 &  &  &  &  &  & 0
\end{smallmatrix}$\newline

For all $1\leq t\leq m-3$ and $1\leq l\leq m-2$ we have the following
Schofield pairs for non-homogeneous regulars of the form $R_{1}^{l}(t)$,
where $t'=(l-3)\bmod(m-2)+1$:

\begin{fleqn}
\begin{align*}
R_{1}^{l}(t):\  & \bigl(R_{1}^{(l-1+i)\bmod(m-2)+1}(t-i),R_{1}^{l}(i)\bigr),\ 1\leq i\leq t-1\\
 & \bigl(I,P(t'-m+t+1,j)\bigr),\ 1\leq j\leq m-t+1,\ \text{if}\ t'-m+t+1\ge0,\\
 & \text{where }I\text{ is a preinjective exceptional with }\underline\dim I=\underline\dim R_{1}^{l}(t)-\underline\dim P(t'-m+t+1,j)
\end{align*}
\end{fleqn}

\subsubsection*{The non-homogeneous tube $\mathcal{T}_{\infty}^{\Delta(\widetilde{\mathbb{D}}_{m})}$}

$\underline\dim R_{\infty}^{1}(1)=\begin{smallmatrix}1 &  &  &  &  & 0\\
 & 1 & 1 & \cdots & 1\\
0 &  &  &  &  & 1
\end{smallmatrix}$,$\quad\underline\dim R_{\infty}^{2}(1)=\begin{smallmatrix}0 &  &  &  &  & 1\\
 & 1 & 1 & \cdots & 1\\
1 &  &  &  &  & 0
\end{smallmatrix}$

\begin{fleqn}
\begin{align*}
R_{\infty}^{1}(1):\  & \bigl(I(m-i-3,(m+i)\bmod2+3),P(i,2-(i\bmod2))\bigr),\ 0\leq i\leq m-3\\
R_{\infty}^{2}(1):\  & \bigl(I(m-i-3,(m+i+1)\bmod2+3),P(i,2-((i+1)\bmod2))\bigr),\ 0\leq i\leq m-3
\end{align*}
\end{fleqn}

\subsubsection*{The non-homogeneous tube $\mathcal{T}_{0}^{\Delta(\widetilde{\mathbb{D}}_{m})}$}

$\underline\dim R_{0}^{1}(1)=\begin{smallmatrix}1 &  &  &  &  & 1\\
 & 1 & 1 & \cdots & 1\\
0 &  &  &  &  & 0
\end{smallmatrix}$,$\quad\underline\dim R_{0}^{2}(0)=\begin{smallmatrix}0 &  &  &  &  & 0\\
 & 1 & 1 & \cdots & 1\\
1 &  &  &  &  & 1
\end{smallmatrix}$

\begin{fleqn}
\begin{align*}
R_{0}^{1}(1):\  & \bigl(I(m-i-3,(m+i+1)\bmod2+3),P(i,2-(i\bmod2))\bigr),\ 0\leq i\leq m-3\\
R_{0}^{2}(1):\  & \bigl(I(m-i-3,(m+i)\bmod2+3),P(i,2-((i+1)\bmod2))\bigr),\ 0\leq i\leq m-3
\end{align*}
\end{fleqn}
\subsection[Schofield pairs for the quiver $\Delta(\widetilde{\mathbb{E}}_6)$]{Schofield pairs for the quiver $\Delta(\widetilde{\mathbb{E}}_6)$ -- {\usebox{\boxEEEEEEE}}}
\[
\vcenter{\hbox{\xymatrix{  &   & 7\ar[d] &   &  \\
  &   & 6\ar[d] &   &  \\
1\ar[r] & 2\ar[r] & 3 & 4\ar[l] & 5\ar[l]}}}\qquad C_{\Delta(\widetilde{\mathbb{E}}_{6})} = \begin{bmatrix}1 & 0 & 0 & 0 & 0 & 0 & 0\\
1 & 1 & 0 & 0 & 0 & 0 & 0\\
1 & 1 & 1 & 1 & 1 & 1 & 1\\
0 & 0 & 0 & 1 & 1 & 0 & 0\\
0 & 0 & 0 & 0 & 1 & 0 & 0\\
0 & 0 & 0 & 0 & 0 & 1 & 1\\
0 & 0 & 0 & 0 & 0 & 0 & 1\end{bmatrix}\quad\Phi_{\Delta(\widetilde{\mathbb{E}}_{6})} = \begin{bmatrix}0 & 0 & -1 & 1 & 0 & 1 & 0\\
1 & 0 & -1 & 1 & 0 & 1 & 0\\
0 & 1 & -1 & 1 & 0 & 1 & 0\\
0 & 1 & -1 & 0 & 1 & 1 & 0\\
0 & 1 & -1 & 0 & 0 & 1 & 0\\
0 & 1 & -1 & 1 & 0 & 0 & 1\\
0 & 1 & -1 & 1 & 0 & 0 & 0\end{bmatrix}
\]
\subsubsection{Schofield pairs associated to preprojective exceptional modules}

\begin{figure}[ht]

\begin{center}
\begin{scaletikzpicturetowidth}{\textwidth}
\begin{tikzpicture}[scale=\tikzscale]
	\node[noselEEEEEEE, label={[shift={(0.4,-0.7)}, brown]\tiny $P(0,1)$}] (P_0_1__Rationals) at (4,6) {$\begin{smallmatrix}&&0\\&&0\\1&1&1&0&0\end{smallmatrix}$};
	\node[noselEEEEEEE, label={[shift={(0.4,-0.7)}, brown]\tiny $P(0,2)$}] (P_0_2__Rationals) at (2,3) {$\begin{smallmatrix}&&0\\&&0\\0&1&1&0&0\end{smallmatrix}$};
	\node[noselEEEEEEE, label={[shift={(0.4,-0.7)}, brown]\tiny $P(0,3)$}] (P_0_3__Rationals) at (0,0) {$\begin{smallmatrix}&&0\\&&0\\0&0&1&0&0\end{smallmatrix}$};
	\node[noselEEEEEEE, label={[shift={(0.4,-0.7)}, brown]\tiny $P(0,4)$}] (P_0_4__Rationals) at (2,-3) {$\begin{smallmatrix}&&0\\&&0\\0&0&1&1&0\end{smallmatrix}$};
	\node[noselEEEEEEE, label={[shift={(0.4,-0.7)}, brown]\tiny $P(0,5)$}] (P_0_5__Rationals) at (4,-6) {$\begin{smallmatrix}&&0\\&&0\\0&0&1&1&1\end{smallmatrix}$};
	\node[noselEEEEEEE, label={[shift={(0.4,-0.7)}, brown]\tiny $P(0,6)$}] (P_0_6__Rationals) at (2,1) {$\begin{smallmatrix}&&0\\&&1\\0&0&1&0&0\end{smallmatrix}$};
	\node[noselEEEEEEE, label={[shift={(0.4,-0.7)}, brown]\tiny $P(0,7)$}] (P_0_7__Rationals) at (4,2) {$\begin{smallmatrix}&&1\\&&1\\0&0&1&0&0\end{smallmatrix}$};
	\node[noselEEEEEEE, label={[shift={(0.4,-0.7)}, brown]\tiny $P(1,1)$}] (P_1_1__Rationals) at (8,6) {$\begin{smallmatrix}&&0\\&&1\\0&0&1&1&0\end{smallmatrix}$};
	\node[noselEEEEEEE, label={[shift={(0.4,-0.7)}, brown]\tiny $P(1,2)$}] (P_1_2__Rationals) at (6,3) {$\begin{smallmatrix}&&0\\&&1\\1&1&2&1&0\end{smallmatrix}$};
	\node[noselEEEEEEE, label={[shift={(0.4,-0.7)}, brown]\tiny $P(1,3)$}] (P_1_3__Rationals) at (4,0) {$\begin{smallmatrix}&&0\\&&1\\0&1&2&1&0\end{smallmatrix}$};
	\node[noselEEEEEEE, label={[shift={(0.4,-0.7)}, brown]\tiny $P(1,4)$}] (P_1_4__Rationals) at (6,-3) {$\begin{smallmatrix}&&0\\&&1\\0&1&2&1&1\end{smallmatrix}$};
	\node[noselEEEEEEE, label={[shift={(0.4,-0.7)}, brown]\tiny $P(1,5)$}] (P_1_5__Rationals) at (8,-6) {$\begin{smallmatrix}&&0\\&&1\\0&1&1&0&0\end{smallmatrix}$};
	\node[noselEEEEEEE, label={[shift={(0.4,-0.7)}, brown]\tiny $P(1,6)$}] (P_1_6__Rationals) at (6,1) {$\begin{smallmatrix}&&1\\&&1\\0&1&2&1&0\end{smallmatrix}$};
	\node[noselEEEEEEE, label={[shift={(0.4,-0.7)}, brown]\tiny $P(1,7)$}] (P_1_7__Rationals) at (8,2) {$\begin{smallmatrix}&&0\\&&0\\0&1&1&1&0\end{smallmatrix}$};
	\node[noselEEEEEEE, label={[shift={(0.4,-0.7)}, brown]\tiny $P(2,1)$}] (P_2_1__Rationals) at (12,6) {$\begin{smallmatrix}&&1\\&&1\\0&1&2&1&1\end{smallmatrix}$};
	\node[noselEEEEEEE, label={[shift={(0.4,-0.7)}, brown]\tiny $P(2,2)$}] (P_2_2__Rationals) at (10,3) {$\begin{smallmatrix}&&1\\&&2\\0&1&3&2&1\end{smallmatrix}$};
	\node[seloneEEEEEEE, label={[shift={(0.4,-0.7)}, brown]\tiny $P(2,3)$}] (P_2_3__Rationals) at (8,0) {$\begin{smallmatrix}&&1\\&&2\\1&2&4&2&1\end{smallmatrix}$};
	\node[noselEEEEEEE, label={[shift={(0.4,-0.7)}, brown]\tiny $P(2,4)$}] (P_2_4__Rationals) at (10,-3) {$\begin{smallmatrix}&&1\\&&2\\1&2&3&1&0\end{smallmatrix}$};
	\node[noselEEEEEEE, label={[shift={(0.4,-0.7)}, brown]\tiny $P(2,5)$}] (P_2_5__Rationals) at (12,-6) {$\begin{smallmatrix}&&1\\&&1\\1&1&2&1&0\end{smallmatrix}$};
	\node[noselEEEEEEE, label={[shift={(0.4,-0.7)}, brown]\tiny $P(2,6)$}] (P_2_6__Rationals) at (10,1) {$\begin{smallmatrix}&&0\\&&1\\1&2&3&2&1\end{smallmatrix}$};
	\node[noselEEEEEEE, label={[shift={(0.4,-0.7)}, brown]\tiny $P(2,7)$}] (P_2_7__Rationals) at (12,2) {$\begin{smallmatrix}&&0\\&&1\\1&1&2&1&1\end{smallmatrix}$};
	\node[noselEEEEEEE, label={[shift={(0.4,-0.7)}, brown]\tiny $P(3,1)$}] (P_3_1__Rationals) at (16,6) {$\begin{smallmatrix}&&0\\&&1\\1&2&2&1&0\end{smallmatrix}$};
	\node[seloneEEEEEEE, label={[shift={(0.4,-0.7)}, brown]\tiny $P(3,2)$}] (P_3_2__Rationals) at (14,3) {$\begin{smallmatrix}&&1\\&&2\\1&3&4&2&1\end{smallmatrix}$};
	\node[noselEEEEEEE, label={[shift={(0.4,-0.7)}, brown]\tiny $P(3,3)$}] (P_3_3__Rationals) at (12,0) {$\begin{smallmatrix}&&1\\&&3\\1&3&5&3&1\end{smallmatrix}$};
	\node[seloneEEEEEEE, label={[shift={(0.4,-0.7)}, brown]\tiny $P(3,4)$}] (P_3_4__Rationals) at (14,-3) {$\begin{smallmatrix}&&1\\&&2\\1&2&4&3&1\end{smallmatrix}$};
	\node[noselEEEEEEE, label={[shift={(0.4,-0.7)}, brown]\tiny $P(3,5)$}] (P_3_5__Rationals) at (16,-6) {$\begin{smallmatrix}&&0\\&&1\\0&1&2&2&1\end{smallmatrix}$};
	\node[seloneEEEEEEE, label={[shift={(0.4,-0.7)}, brown]\tiny $P(3,6)$}] (P_3_6__Rationals) at (14,1) {$\begin{smallmatrix}&&1\\&&3\\1&2&4&2&1\end{smallmatrix}$};
	\node[noselEEEEEEE, label={[shift={(0.4,-0.7)}, brown]\tiny $P(3,7)$}] (P_3_7__Rationals) at (16,2) {$\begin{smallmatrix}&&1\\&&2\\0&1&2&1&0\end{smallmatrix}$};
	\node[seloneEEEEEEE, label={[shift={(0.4,-0.7)}, brown]\tiny $P(4,1)$}] (P_4_1__Rationals) at (20,6) {$\begin{smallmatrix}&&1\\&&2\\1&1&3&2&1\end{smallmatrix}$};
	\node[noselEEEEEEE, label={[shift={(0.4,-0.7)}, brown]\tiny $P(4,2)$}] (P_4_2__Rationals) at (18,3) {$\begin{smallmatrix}&&1\\&&3\\2&3&5&3&1\end{smallmatrix}$};
	\node[noselEEEEEEE, label={[shift={(0.4,-0.7)}, brown]\tiny $P(4,3)$}] (P_4_3__Rationals) at (16,0) {$\begin{smallmatrix}&&2\\&&4\\2&4&7&4&2\end{smallmatrix}$};
	\node[noselEEEEEEE, label={[shift={(0.4,-0.7)}, brown]\tiny $P(4,4)$}] (P_4_4__Rationals) at (18,-3) {$\begin{smallmatrix}&&1\\&&3\\1&3&5&3&2\end{smallmatrix}$};
	\node[seloneEEEEEEE, label={[shift={(0.4,-0.7)}, brown]\tiny $P(4,5)$}] (P_4_5__Rationals) at (20,-6) {$\begin{smallmatrix}&&1\\&&2\\1&2&3&1&1\end{smallmatrix}$};
	\node[noselEEEEEEE, label={[shift={(0.4,-0.7)}, brown]\tiny $P(4,6)$}] (P_4_6__Rationals) at (18,1) {$\begin{smallmatrix}&&2\\&&3\\1&3&5&3&1\end{smallmatrix}$};
	\node[seloneEEEEEEE, label={[shift={(0.4,-0.7)}, brown]\tiny $P(4,7)$}] (P_4_7__Rationals) at (20,2) {$\begin{smallmatrix}&&1\\&&1\\1&2&3&2&1\end{smallmatrix}$};
	\node at (20,0) {$\dots$};
	\draw[blue,->] (P_0_2__Rationals) -- (P_0_1__Rationals);
	\draw[blue,->] (P_0_3__Rationals) -- (P_0_2__Rationals);
	\draw[blue,->] (P_0_3__Rationals) -- (P_0_4__Rationals);
	\draw[blue,->] (P_0_4__Rationals) -- (P_0_5__Rationals);
	\draw[blue,->] (P_0_3__Rationals) -- (P_0_6__Rationals);
	\draw[blue,->] (P_0_6__Rationals) -- (P_0_7__Rationals);
	\draw[gray,dashed,->] (P_0_1__Rationals) -- (P_1_1__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[red,->] (P_1_2__Rationals) -- (P_1_1__Rationals);
	\draw[blue,->] (P_0_1__Rationals) -- (P_1_2__Rationals);
	\draw[gray,dashed,->] (P_0_2__Rationals) -- (P_1_2__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[blue,->] (P_1_3__Rationals) -- (P_1_2__Rationals);
	\draw[blue,->] (P_0_2__Rationals) -- (P_1_3__Rationals);
	\draw[gray,dashed,->] (P_0_3__Rationals) -- (P_1_3__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[blue,->] (P_0_4__Rationals) -- (P_1_3__Rationals);
	\draw[blue,->] (P_0_6__Rationals) -- (P_1_3__Rationals);
	\draw[gray,dashed,->] (P_0_4__Rationals) -- (P_1_4__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[blue,->] (P_0_5__Rationals) -- (P_1_4__Rationals);
	\draw[blue,->] (P_1_3__Rationals) -- (P_1_4__Rationals);
	\draw[gray,dashed,->] (P_0_5__Rationals) -- (P_1_5__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[red,->] (P_1_4__Rationals) -- (P_1_5__Rationals);
	\draw[gray,dashed,->] (P_0_6__Rationals) -- (P_1_6__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[blue,->] (P_0_7__Rationals) -- (P_1_6__Rationals);
	\draw[blue,->] (P_1_3__Rationals) -- (P_1_6__Rationals);
	\draw[gray,dashed,->] (P_0_7__Rationals) -- (P_1_7__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[red,->] (P_1_6__Rationals) -- (P_1_7__Rationals);
	\draw[gray,dashed,->] (P_1_1__Rationals) -- (P_2_1__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[red,->] (P_2_2__Rationals) -- (P_2_1__Rationals);
	\draw[blue,->] (P_1_1__Rationals) -- (P_2_2__Rationals);
	\draw[gray,dashed,->] (P_1_2__Rationals) -- (P_2_2__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[red,->] (P_2_3__Rationals) -- (P_2_2__Rationals);
	\draw[blue,->] (P_1_2__Rationals) -- (P_2_3__Rationals);
	\draw[gray,dashed,->] (P_1_3__Rationals) -- (P_2_3__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[blue,->] (P_1_4__Rationals) -- (P_2_3__Rationals);
	\draw[blue,->] (P_1_6__Rationals) -- (P_2_3__Rationals);
	\draw[gray,dashed,->] (P_1_4__Rationals) -- (P_2_4__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[blue,->] (P_1_5__Rationals) -- (P_2_4__Rationals);
	\draw[red,->] (P_2_3__Rationals) -- (P_2_4__Rationals);
	\draw[gray,dashed,->] (P_1_5__Rationals) -- (P_2_5__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[red,->] (P_2_4__Rationals) -- (P_2_5__Rationals);
	\draw[gray,dashed,->] (P_1_6__Rationals) -- (P_2_6__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[blue,->] (P_1_7__Rationals) -- (P_2_6__Rationals);
	\draw[red,->] (P_2_3__Rationals) -- (P_2_6__Rationals);
	\draw[gray,dashed,->] (P_1_7__Rationals) -- (P_2_7__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[red,->] (P_2_6__Rationals) -- (P_2_7__Rationals);
	\draw[gray,dashed,->] (P_2_1__Rationals) -- (P_3_1__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[red,->] (P_3_2__Rationals) -- (P_3_1__Rationals);
	\draw[blue,->] (P_2_1__Rationals) -- (P_3_2__Rationals);
	\draw[gray,dashed,->] (P_2_2__Rationals) -- (P_3_2__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[red,->] (P_3_3__Rationals) -- (P_3_2__Rationals);
	\draw[blue,->] (P_2_2__Rationals) -- (P_3_3__Rationals);
	\draw[gray,dashed,->] (P_2_3__Rationals) -- (P_3_3__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[blue,->] (P_2_4__Rationals) -- (P_3_3__Rationals);
	\draw[blue,->] (P_2_6__Rationals) -- (P_3_3__Rationals);
	\draw[gray,dashed,->] (P_2_4__Rationals) -- (P_3_4__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[blue,->] (P_2_5__Rationals) -- (P_3_4__Rationals);
	\draw[red,->] (P_3_3__Rationals) -- (P_3_4__Rationals);
	\draw[gray,dashed,->] (P_2_5__Rationals) -- (P_3_5__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[red,->] (P_3_4__Rationals) -- (P_3_5__Rationals);
	\draw[gray,dashed,->] (P_2_6__Rationals) -- (P_3_6__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[blue,->] (P_2_7__Rationals) -- (P_3_6__Rationals);
	\draw[red,->] (P_3_3__Rationals) -- (P_3_6__Rationals);
	\draw[gray,dashed,->] (P_2_7__Rationals) -- (P_3_7__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[red,->] (P_3_6__Rationals) -- (P_3_7__Rationals);
	\draw[gray,dashed,->] (P_3_1__Rationals) -- (P_4_1__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[red,->] (P_4_2__Rationals) -- (P_4_1__Rationals);
	\draw[blue,->] (P_3_1__Rationals) -- (P_4_2__Rationals);
	\draw[gray,dashed,->] (P_3_2__Rationals) -- (P_4_2__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[red,->] (P_4_3__Rationals) -- (P_4_2__Rationals);
	\draw[blue,->] (P_3_2__Rationals) -- (P_4_3__Rationals);
	\draw[gray,dashed,->] (P_3_3__Rationals) -- (P_4_3__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[blue,->] (P_3_4__Rationals) -- (P_4_3__Rationals);
	\draw[blue,->] (P_3_6__Rationals) -- (P_4_3__Rationals);
	\draw[gray,dashed,->] (P_3_4__Rationals) -- (P_4_4__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[blue,->] (P_3_5__Rationals) -- (P_4_4__Rationals);
	\draw[red,->] (P_4_3__Rationals) -- (P_4_4__Rationals);
	\draw[gray,dashed,->] (P_3_5__Rationals) -- (P_4_5__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[red,->] (P_4_4__Rationals) -- (P_4_5__Rationals);
	\draw[gray,dashed,->] (P_3_6__Rationals) -- (P_4_6__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[blue,->] (P_3_7__Rationals) -- (P_4_6__Rationals);
	\draw[red,->] (P_4_3__Rationals) -- (P_4_6__Rationals);
	\draw[gray,dashed,->] (P_3_7__Rationals) -- (P_4_7__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[red,->] (P_4_6__Rationals) -- (P_4_7__Rationals);
\end{tikzpicture}
\end{scaletikzpicturetowidth}
\end{center}

\end{figure}
\begin{figure}[ht]

\begin{center}
\begin{scaletikzpicturetowidth}{\textwidth}
\begin{tikzpicture}[scale=\tikzscale]
	\node at (0,3) {$\dots$};
	\node at (0,-3) {$\dots$};
	\node[noselEEEEEEE, label={[shift={(0.4,-0.7)}, brown]\tiny $P(5,1)$}] (P_5_1__Rationals) at (4,6) {$\begin{smallmatrix}&&1\\&&2\\0&2&3&2&1\end{smallmatrix}$};
	\node[noselEEEEEEE, label={[shift={(0.4,-0.7)}, brown]\tiny $P(5,2)$}] (P_5_2__Rationals) at (2,3) {$\begin{smallmatrix}&&2\\&&4\\1&3&6&4&2\end{smallmatrix}$};
	\node[noselEEEEEEE, label={[shift={(0.4,-0.7)}, brown]\tiny $P(5,3)$}] (P_5_3__Rationals) at (0,0) {$\begin{smallmatrix}&&2\\&&5\\2&5&8&5&2\end{smallmatrix}$};
	\node[noselEEEEEEE, label={[shift={(0.4,-0.7)}, brown]\tiny $P(5,4)$}] (P_5_4__Rationals) at (2,-3) {$\begin{smallmatrix}&&2\\&&4\\2&4&6&3&1\end{smallmatrix}$};
	\node[noselEEEEEEE, label={[shift={(0.4,-0.7)}, brown]\tiny $P(5,5)$}] (P_5_5__Rationals) at (4,-6) {$\begin{smallmatrix}&&1\\&&2\\1&2&3&2&0\end{smallmatrix}$};
	\node[noselEEEEEEE, label={[shift={(0.4,-0.7)}, brown]\tiny $P(5,6)$}] (P_5_6__Rationals) at (2,1) {$\begin{smallmatrix}&&1\\&&3\\2&4&6&4&2\end{smallmatrix}$};
	\node[noselEEEEEEE, label={[shift={(0.4,-0.7)}, brown]\tiny $P(5,7)$}] (P_5_7__Rationals) at (4,2) {$\begin{smallmatrix}&&0\\&&2\\1&2&3&2&1\end{smallmatrix}$};
	\node[noselEEEEEEE, label={[shift={(0.4,-0.7)}, brown]\tiny $P(6,1)$}] (P_6_1__Rationals) at (8,6) {$\begin{smallmatrix}&&1\\&&2\\2&3&4&2&1\end{smallmatrix}$};
	\node[noselEEEEEEE, label={[shift={(0.4,-0.7)}, brown]\tiny $P(6,2)$}] (P_6_2__Rationals) at (6,3) {$\begin{smallmatrix}&&2\\&&4\\2&5&7&4&2\end{smallmatrix}$};
	\node[noselEEEEEEE, label={[shift={(0.4,-0.7)}, brown]\tiny $P(6,3)$}] (P_6_3__Rationals) at (4,0) {$\begin{smallmatrix}&&3\\&&6\\3&6&10&6&3\end{smallmatrix}$};
	\node[noselEEEEEEE, label={[shift={(0.4,-0.7)}, brown]\tiny $P(6,4)$}] (P_6_4__Rationals) at (6,-3) {$\begin{smallmatrix}&&2\\&&4\\2&4&7&5&2\end{smallmatrix}$};
	\node[noselEEEEEEE, label={[shift={(0.4,-0.7)}, brown]\tiny $P(6,5)$}] (P_6_5__Rationals) at (8,-6) {$\begin{smallmatrix}&&1\\&&2\\1&2&4&3&2\end{smallmatrix}$};
	\node[noselEEEEEEE, label={[shift={(0.4,-0.7)}, brown]\tiny $P(6,6)$}] (P_6_6__Rationals) at (6,1) {$\begin{smallmatrix}&&2\\&&5\\2&4&7&4&2\end{smallmatrix}$};
	\node[noselEEEEEEE, label={[shift={(0.4,-0.7)}, brown]\tiny $P(6,7)$}] (P_6_7__Rationals) at (8,2) {$\begin{smallmatrix}&&2\\&&3\\1&2&4&2&1\end{smallmatrix}$};
	\node[noselEEEEEEE, label={[shift={(0.4,-0.7)}, brown]\tiny $P(7,1)$}] (P_7_1__Rationals) at (12,6) {$\begin{smallmatrix}&&1\\&&3\\1&2&4&3&1\end{smallmatrix}$};
	\node[noselEEEEEEE, label={[shift={(0.4,-0.7)}, brown]\tiny $P(7,2)$}] (P_7_2__Rationals) at (10,3) {$\begin{smallmatrix}&&2\\&&5\\3&5&8&5&2\end{smallmatrix}$};
	\node[noselEEEEEEE, label={[shift={(0.4,-0.7)}, brown]\tiny $P(7,3)$}] (P_7_3__Rationals) at (8,0) {$\begin{smallmatrix}&&3\\&&7\\3&7&11&7&3\end{smallmatrix}$};
	\node[noselEEEEEEE, label={[shift={(0.4,-0.7)}, brown]\tiny $P(7,4)$}] (P_7_4__Rationals) at (10,-3) {$\begin{smallmatrix}&&2\\&&5\\2&5&8&5&3\end{smallmatrix}$};
	\node[noselEEEEEEE, label={[shift={(0.4,-0.7)}, brown]\tiny $P(7,5)$}] (P_7_5__Rationals) at (12,-6) {$\begin{smallmatrix}&&1\\&&3\\1&3&4&2&1\end{smallmatrix}$};
	\node[noselEEEEEEE, label={[shift={(0.4,-0.7)}, brown]\tiny $P(7,6)$}] (P_7_6__Rationals) at (10,1) {$\begin{smallmatrix}&&3\\&&5\\2&5&8&5&2\end{smallmatrix}$};
	\node[noselEEEEEEE, label={[shift={(0.4,-0.7)}, brown]\tiny $P(7,7)$}] (P_7_7__Rationals) at (12,2) {$\begin{smallmatrix}&&1\\&&2\\1&3&4&3&1\end{smallmatrix}$};
	\node[noselEEEEEEE, label={[shift={(0.4,-0.7)}, brown]\tiny $P(8,1)$}] (P_8_1__Rationals) at (16,6) {$\begin{smallmatrix}&&2\\&&3\\1&3&5&3&2\end{smallmatrix}$};
	\node[noselEEEEEEE, label={[shift={(0.4,-0.7)}, brown]\tiny $P(8,2)$}] (P_8_2__Rationals) at (14,3) {$\begin{smallmatrix}&&3\\&&6\\2&5&9&6&3\end{smallmatrix}$};
	\node[noselEEEEEEE, label={[shift={(0.4,-0.7)}, brown]\tiny $P(8,3)$}] (P_8_3__Rationals) at (12,0) {$\begin{smallmatrix}&&4\\&&8\\4&8&13&8&4\end{smallmatrix}$};
	\node[noselEEEEEEE, label={[shift={(0.4,-0.7)}, brown]\tiny $P(8,4)$}] (P_8_4__Rationals) at (14,-3) {$\begin{smallmatrix}&&3\\&&6\\3&6&9&5&2\end{smallmatrix}$};
	\node[noselEEEEEEE, label={[shift={(0.4,-0.7)}, brown]\tiny $P(8,5)$}] (P_8_5__Rationals) at (16,-6) {$\begin{smallmatrix}&&2\\&&3\\2&3&5&3&1\end{smallmatrix}$};
	\node[noselEEEEEEE, label={[shift={(0.4,-0.7)}, brown]\tiny $P(8,6)$}] (P_8_6__Rationals) at (14,1) {$\begin{smallmatrix}&&2\\&&5\\3&6&9&6&3\end{smallmatrix}$};
	\node[noselEEEEEEE, label={[shift={(0.4,-0.7)}, brown]\tiny $P(8,7)$}] (P_8_7__Rationals) at (16,2) {$\begin{smallmatrix}&&1\\&&3\\2&3&5&3&2\end{smallmatrix}$};
	\node[noselEEEEEEE, label={[shift={(0.4,-0.7)}, brown]\tiny $P(9,1)$}] (P_9_1__Rationals) at (20,6) {$\begin{smallmatrix}&&1\\&&3\\2&4&5&3&1\end{smallmatrix}$};
	\node[noselEEEEEEE, label={[shift={(0.4,-0.7)}, brown]\tiny $P(9,2)$}] (P_9_2__Rationals) at (18,3) {$\begin{smallmatrix}&&3\\&&6\\3&7&10&6&3\end{smallmatrix}$};
	\node[noselEEEEEEE, label={[shift={(0.4,-0.7)}, brown]\tiny $P(9,3)$}] (P_9_3__Rationals) at (16,0) {$\begin{smallmatrix}&&4\\&&9\\4&9&14&9&4\end{smallmatrix}$};
	\node[noselEEEEEEE, label={[shift={(0.4,-0.7)}, brown]\tiny $P(9,4)$}] (P_9_4__Rationals) at (18,-3) {$\begin{smallmatrix}&&3\\&&6\\3&6&10&7&3\end{smallmatrix}$};
	\node[noselEEEEEEE, label={[shift={(0.4,-0.7)}, brown]\tiny $P(9,5)$}] (P_9_5__Rationals) at (20,-6) {$\begin{smallmatrix}&&1\\&&3\\1&3&5&4&2\end{smallmatrix}$};
	\node[noselEEEEEEE, label={[shift={(0.4,-0.7)}, brown]\tiny $P(9,6)$}] (P_9_6__Rationals) at (18,1) {$\begin{smallmatrix}&&3\\&&7\\3&6&10&6&3\end{smallmatrix}$};
	\node[noselEEEEEEE, label={[shift={(0.4,-0.7)}, brown]\tiny $P(9,7)$}] (P_9_7__Rationals) at (20,2) {$\begin{smallmatrix}&&2\\&&4\\1&3&5&3&1\end{smallmatrix}$};
	\node at (20,0) {$\dots$};
	\draw[red,->] (P_5_2__Rationals) -- (P_5_1__Rationals);
	\draw[red,->] (P_5_3__Rationals) -- (P_5_2__Rationals);
	\draw[red,->] (P_5_3__Rationals) -- (P_5_4__Rationals);
	\draw[red,->] (P_5_4__Rationals) -- (P_5_5__Rationals);
	\draw[red,->] (P_5_3__Rationals) -- (P_5_6__Rationals);
	\draw[red,->] (P_5_6__Rationals) -- (P_5_7__Rationals);
	\draw[gray,dashed,->] (P_5_1__Rationals) -- (P_6_1__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[red,->] (P_6_2__Rationals) -- (P_6_1__Rationals);
	\draw[blue,->] (P_5_1__Rationals) -- (P_6_2__Rationals);
	\draw[gray,dashed,->] (P_5_2__Rationals) -- (P_6_2__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[red,->] (P_6_3__Rationals) -- (P_6_2__Rationals);
	\draw[blue,->] (P_5_2__Rationals) -- (P_6_3__Rationals);
	\draw[gray,dashed,->] (P_5_3__Rationals) -- (P_6_3__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[blue,->] (P_5_4__Rationals) -- (P_6_3__Rationals);
	\draw[blue,->] (P_5_6__Rationals) -- (P_6_3__Rationals);
	\draw[gray,dashed,->] (P_5_4__Rationals) -- (P_6_4__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[blue,->] (P_5_5__Rationals) -- (P_6_4__Rationals);
	\draw[red,->] (P_6_3__Rationals) -- (P_6_4__Rationals);
	\draw[gray,dashed,->] (P_5_5__Rationals) -- (P_6_5__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[red,->] (P_6_4__Rationals) -- (P_6_5__Rationals);
	\draw[gray,dashed,->] (P_5_6__Rationals) -- (P_6_6__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[blue,->] (P_5_7__Rationals) -- (P_6_6__Rationals);
	\draw[red,->] (P_6_3__Rationals) -- (P_6_6__Rationals);
	\draw[gray,dashed,->] (P_5_7__Rationals) -- (P_6_7__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[red,->] (P_6_6__Rationals) -- (P_6_7__Rationals);
	\draw[gray,dashed,->] (P_6_1__Rationals) -- (P_7_1__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[red,->] (P_7_2__Rationals) -- (P_7_1__Rationals);
	\draw[blue,->] (P_6_1__Rationals) -- (P_7_2__Rationals);
	\draw[gray,dashed,->] (P_6_2__Rationals) -- (P_7_2__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[red,->] (P_7_3__Rationals) -- (P_7_2__Rationals);
	\draw[blue,->] (P_6_2__Rationals) -- (P_7_3__Rationals);
	\draw[gray,dashed,->] (P_6_3__Rationals) -- (P_7_3__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[blue,->] (P_6_4__Rationals) -- (P_7_3__Rationals);
	\draw[blue,->] (P_6_6__Rationals) -- (P_7_3__Rationals);
	\draw[gray,dashed,->] (P_6_4__Rationals) -- (P_7_4__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[blue,->] (P_6_5__Rationals) -- (P_7_4__Rationals);
	\draw[red,->] (P_7_3__Rationals) -- (P_7_4__Rationals);
	\draw[gray,dashed,->] (P_6_5__Rationals) -- (P_7_5__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[red,->] (P_7_4__Rationals) -- (P_7_5__Rationals);
	\draw[gray,dashed,->] (P_6_6__Rationals) -- (P_7_6__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[blue,->] (P_6_7__Rationals) -- (P_7_6__Rationals);
	\draw[red,->] (P_7_3__Rationals) -- (P_7_6__Rationals);
	\draw[gray,dashed,->] (P_6_7__Rationals) -- (P_7_7__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[red,->] (P_7_6__Rationals) -- (P_7_7__Rationals);
	\draw[gray,dashed,->] (P_7_1__Rationals) -- (P_8_1__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[red,->] (P_8_2__Rationals) -- (P_8_1__Rationals);
	\draw[blue,->] (P_7_1__Rationals) -- (P_8_2__Rationals);
	\draw[gray,dashed,->] (P_7_2__Rationals) -- (P_8_2__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[red,->] (P_8_3__Rationals) -- (P_8_2__Rationals);
	\draw[blue,->] (P_7_2__Rationals) -- (P_8_3__Rationals);
	\draw[gray,dashed,->] (P_7_3__Rationals) -- (P_8_3__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[blue,->] (P_7_4__Rationals) -- (P_8_3__Rationals);
	\draw[blue,->] (P_7_6__Rationals) -- (P_8_3__Rationals);
	\draw[gray,dashed,->] (P_7_4__Rationals) -- (P_8_4__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[blue,->] (P_7_5__Rationals) -- (P_8_4__Rationals);
	\draw[red,->] (P_8_3__Rationals) -- (P_8_4__Rationals);
	\draw[gray,dashed,->] (P_7_5__Rationals) -- (P_8_5__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[red,->] (P_8_4__Rationals) -- (P_8_5__Rationals);
	\draw[gray,dashed,->] (P_7_6__Rationals) -- (P_8_6__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[blue,->] (P_7_7__Rationals) -- (P_8_6__Rationals);
	\draw[red,->] (P_8_3__Rationals) -- (P_8_6__Rationals);
	\draw[gray,dashed,->] (P_7_7__Rationals) -- (P_8_7__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[red,->] (P_8_6__Rationals) -- (P_8_7__Rationals);
	\draw[gray,dashed,->] (P_8_1__Rationals) -- (P_9_1__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[red,->] (P_9_2__Rationals) -- (P_9_1__Rationals);
	\draw[blue,->] (P_8_1__Rationals) -- (P_9_2__Rationals);
	\draw[gray,dashed,->] (P_8_2__Rationals) -- (P_9_2__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[red,->] (P_9_3__Rationals) -- (P_9_2__Rationals);
	\draw[blue,->] (P_8_2__Rationals) -- (P_9_3__Rationals);
	\draw[gray,dashed,->] (P_8_3__Rationals) -- (P_9_3__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[blue,->] (P_8_4__Rationals) -- (P_9_3__Rationals);
	\draw[blue,->] (P_8_6__Rationals) -- (P_9_3__Rationals);
	\draw[gray,dashed,->] (P_8_4__Rationals) -- (P_9_4__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[blue,->] (P_8_5__Rationals) -- (P_9_4__Rationals);
	\draw[red,->] (P_9_3__Rationals) -- (P_9_4__Rationals);
	\draw[gray,dashed,->] (P_8_5__Rationals) -- (P_9_5__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[red,->] (P_9_4__Rationals) -- (P_9_5__Rationals);
	\draw[gray,dashed,->] (P_8_6__Rationals) -- (P_9_6__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[blue,->] (P_8_7__Rationals) -- (P_9_6__Rationals);
	\draw[red,->] (P_9_3__Rationals) -- (P_9_6__Rationals);
	\draw[gray,dashed,->] (P_8_7__Rationals) -- (P_9_7__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[red,->] (P_9_6__Rationals) -- (P_9_7__Rationals);
\end{tikzpicture}
\end{scaletikzpicturetowidth}
\end{center}

\end{figure}
\medskip{}
\subsubsection*{Modules of the form $P(n,1)$}

Defect: $\partial P(n,1) = -1$, for $n\ge 0$.

\begin{fleqn}
\begin{align*}
P(0,1):\ & \bigl(I(0,1),P(0,2)\bigr),\ \bigl(I(0,2),P(0,3)\bigr)\\
P(1,1):\ & \bigl(I(1,7),P(0,4)\bigr),\ \bigl(I(1,5),P(0,6)\bigr)\\
P(2,1):\ & \bigl(I(2,1),P(0,2)\bigr),\ \bigl(R_{0}^{2}(1),P(0,5)\bigr),\ \bigl(R_{1}^{2}(1),P(0,7)\bigr),\ \bigl(I(0,7),P(1,4)\bigr),\ \bigl(I(0,5),P(1,6)\bigr)\\
P(3,1):\ & \bigl(R_{\infty}^{1}(1),P(0,1)\bigr),\ \bigl(I(1,1),P(1,2)\bigr),\ \bigl(R_{0}^{3}(1),P(1,5)\bigr),\ \bigl(R_{1}^{3}(1),P(1,7)\bigr)\\
P(4,1):\ & \bigl(R_{1}^{3}(2),P(0,5)\bigr),\ \bigl(R_{0}^{3}(2),P(0,7)\bigr),\ \bigl(R_{\infty}^{2}(1),P(1,1)\bigr),\ \bigl(I(0,1),P(2,2)\bigr),\ \bigl(R_{0}^{1}(1),P(2,5)\bigr)\\
 & \bigl(R_{1}^{1}(1),P(2,7)\bigr)\\
P(n,1):\ & \bigl(R_{1}^{(n-2)\bmod 3+1}(2),P(n-4,5)\bigr),\ \bigl(R_{0}^{(n-2)\bmod 3+1}(2),P(n-4,7)\bigr),\ \bigl(R_{\infty}^{(n-3)\bmod 2+1}(1),P(n-3,1)\bigr)\\
 & \bigl(I(-n+4,1),P(n-2,2)\bigr),\ \bigl(R_{0}^{(n-4)\bmod 3+1}(1),P(n-2,5)\bigr),\ \bigl(R_{1}^{(n-4)\bmod 3+1}(1),P(n-2,7)\bigr),\ n>4\\
\end{align*}
\end{fleqn}
\subsubsection*{Modules of the form $P(n,2)$}

Defect: $\partial P(n,2) = -2$, for $n\ge 0$.

\begin{fleqn}
\begin{align*}
P(0,2):\ & \bigl(I(1,1),P(0,3)\bigr)\\
P(1,2):\ & \bigl(P(1,1),P(0,1)\bigr),\ \bigl(R_{1}^{3}(1),P(0,4)\bigr),\ \bigl(R_{0}^{3}(1),P(0,6)\bigr),\ \bigl(I(0,1),P(1,3)\bigr)\\
P(2,2):\ & \bigl(P(3,7),P(0,5)\bigr),\ \bigl(P(3,5),P(0,7)\bigr),\ \bigl(P(2,1),P(1,1)\bigr),\ \bigl(R_{1}^{1}(1),P(1,4)\bigr),\ \bigl(R_{0}^{1}(1),P(1,6)\bigr)\\
P(3,2):\ & \bigl(P(5,1),P(0,1)\bigr),\ \bigl(P(4,7),P(1,5)\bigr),\ \bigl(P(4,5),P(1,7)\bigr),\ \bigl(P(3,1),P(2,1)\bigr),\ \bigl(R_{1}^{2}(1),P(2,4)\bigr)\\
 & \bigl(R_{0}^{2}(1),P(2,6)\bigr)\\
P(n,2):\ & \bigl(P(n+2,1),P(n-3,1)\bigr),\ \bigl(P(n+1,7),P(n-2,5)\bigr),\ \bigl(P(n+1,5),P(n-2,7)\bigr)\\
 & \bigl(P(n,1),P(n-1,1)\bigr),\ \bigl(R_{1}^{(n-2)\bmod 3+1}(1),P(n-1,4)\bigr),\ \bigl(R_{0}^{(n-2)\bmod 3+1}(1),P(n-1,6)\bigr),\ n>3\\
\end{align*}
\end{fleqn}
\subsubsection*{Modules of the form $P(n,3)$}

Defect: $\partial P(n,3) = -3$, for $n\ge 0$.

\begin{fleqn}
\begin{align*}
P(0,3):\ & - \\
P(1,3):\ & \bigl(P(1,1),P(0,2)\bigr),\ \bigl(P(1,5),P(0,4)\bigr),\ \bigl(P(1,7),P(0,6)\bigr)\\
P(2,3):\ & \bigl(P(2,2),P(0,1)\bigr),\ \bigl(P(2,4),P(0,5)\bigr),\ \bigl(P(2,6),P(0,7)\bigr),\ \bigl(P(2,1),P(1,2)\bigr),\ \bigl(P(2,5),P(1,4)\bigr)\\
 & \bigl(P(2,7),P(1,6)\bigr)\\
P(n,3):\ & \bigl(P(n,2),P(n-2,1)\bigr),\ \bigl(P(n,4),P(n-2,5)\bigr),\ \bigl(P(n,6),P(n-2,7)\bigr)\\
 & \bigl(P(n,1),P(n-1,2)\bigr),\ \bigl(P(n,5),P(n-1,4)\bigr),\ \bigl(P(n,7),P(n-1,6)\bigr),\ n>2\\
\end{align*}
\end{fleqn}
\subsubsection*{Modules of the form $P(n,4)$}

Defect: $\partial P(n,4) = -2$, for $n\ge 0$.

\begin{fleqn}
\begin{align*}
P(0,4):\ & \bigl(I(1,5),P(0,3)\bigr)\\
P(1,4):\ & \bigl(R_{0}^{1}(1),P(0,2)\bigr),\ \bigl(P(1,5),P(0,5)\bigr),\ \bigl(R_{1}^{2}(1),P(0,6)\bigr),\ \bigl(I(0,5),P(1,3)\bigr)\\
P(2,4):\ & \bigl(P(3,7),P(0,1)\bigr),\ \bigl(P(3,1),P(0,7)\bigr),\ \bigl(R_{0}^{2}(1),P(1,2)\bigr),\ \bigl(P(2,5),P(1,5)\bigr),\ \bigl(R_{1}^{3}(1),P(1,6)\bigr)\\
P(3,4):\ & \bigl(P(5,5),P(0,5)\bigr),\ \bigl(P(4,7),P(1,1)\bigr),\ \bigl(P(4,1),P(1,7)\bigr),\ \bigl(R_{0}^{3}(1),P(2,2)\bigr),\ \bigl(P(3,5),P(2,5)\bigr)\\
 & \bigl(R_{1}^{1}(1),P(2,6)\bigr)\\
P(n,4):\ & \bigl(P(n+2,5),P(n-3,5)\bigr),\ \bigl(P(n+1,7),P(n-2,1)\bigr),\ \bigl(P(n+1,1),P(n-2,7)\bigr)\\
 & \bigl(R_{0}^{(n-1)\bmod 3+1}(1),P(n-1,2)\bigr),\ \bigl(P(n,5),P(n-1,5)\bigr),\ \bigl(R_{1}^{(n-3)\bmod 3+1}(1),P(n-1,6)\bigr),\ n>3\\
\end{align*}
\end{fleqn}
\subsubsection*{Modules of the form $P(n,5)$}

Defect: $\partial P(n,5) = -1$, for $n\ge 0$.

\begin{fleqn}
\begin{align*}
P(0,5):\ & \bigl(I(0,4),P(0,3)\bigr),\ \bigl(I(0,5),P(0,4)\bigr)\\
P(1,5):\ & \bigl(I(1,7),P(0,2)\bigr),\ \bigl(I(1,1),P(0,6)\bigr)\\
P(2,5):\ & \bigl(R_{1}^{1}(1),P(0,1)\bigr),\ \bigl(I(2,5),P(0,4)\bigr),\ \bigl(R_{0}^{3}(1),P(0,7)\bigr),\ \bigl(I(0,7),P(1,2)\bigr),\ \bigl(I(0,1),P(1,6)\bigr)\\
P(3,5):\ & \bigl(R_{\infty}^{1}(1),P(0,5)\bigr),\ \bigl(R_{1}^{2}(1),P(1,1)\bigr),\ \bigl(I(1,5),P(1,4)\bigr),\ \bigl(R_{0}^{1}(1),P(1,7)\bigr)\\
P(4,5):\ & \bigl(R_{0}^{1}(2),P(0,1)\bigr),\ \bigl(R_{1}^{2}(2),P(0,7)\bigr),\ \bigl(R_{\infty}^{2}(1),P(1,5)\bigr),\ \bigl(R_{1}^{3}(1),P(2,1)\bigr),\ \bigl(I(0,5),P(2,4)\bigr)\\
 & \bigl(R_{0}^{2}(1),P(2,7)\bigr)\\
P(n,5):\ & \bigl(R_{0}^{(n-4)\bmod 3+1}(2),P(n-4,1)\bigr),\ \bigl(R_{1}^{(n-3)\bmod 3+1}(2),P(n-4,7)\bigr),\ \bigl(R_{\infty}^{(n-3)\bmod 2+1}(1),P(n-3,5)\bigr)\\
 & \bigl(R_{1}^{(n-2)\bmod 3+1}(1),P(n-2,1)\bigr),\ \bigl(I(-n+4,5),P(n-2,4)\bigr),\ \bigl(R_{0}^{(n-3)\bmod 3+1}(1),P(n-2,7)\bigr),\ n>4\\
\end{align*}
\end{fleqn}
\subsubsection*{Modules of the form $P(n,6)$}

Defect: $\partial P(n,6) = -2$, for $n\ge 0$.

\begin{fleqn}
\begin{align*}
P(0,6):\ & \bigl(I(1,7),P(0,3)\bigr)\\
P(1,6):\ & \bigl(R_{1}^{1}(1),P(0,2)\bigr),\ \bigl(R_{0}^{2}(1),P(0,4)\bigr),\ \bigl(P(1,7),P(0,7)\bigr),\ \bigl(I(0,7),P(1,3)\bigr)\\
P(2,6):\ & \bigl(P(3,5),P(0,1)\bigr),\ \bigl(P(3,1),P(0,5)\bigr),\ \bigl(R_{1}^{2}(1),P(1,2)\bigr),\ \bigl(R_{0}^{3}(1),P(1,4)\bigr),\ \bigl(P(2,7),P(1,7)\bigr)\\
P(3,6):\ & \bigl(P(5,7),P(0,7)\bigr),\ \bigl(P(4,5),P(1,1)\bigr),\ \bigl(P(4,1),P(1,5)\bigr),\ \bigl(R_{1}^{3}(1),P(2,2)\bigr),\ \bigl(R_{0}^{1}(1),P(2,4)\bigr)\\
 & \bigl(P(3,7),P(2,7)\bigr)\\
P(n,6):\ & \bigl(P(n+2,7),P(n-3,7)\bigr),\ \bigl(P(n+1,5),P(n-2,1)\bigr),\ \bigl(P(n+1,1),P(n-2,5)\bigr)\\
 & \bigl(R_{1}^{(n-1)\bmod 3+1}(1),P(n-1,2)\bigr),\ \bigl(R_{0}^{(n-3)\bmod 3+1}(1),P(n-1,4)\bigr),\ \bigl(P(n,7),P(n-1,7)\bigr),\ n>3\\
\end{align*}
\end{fleqn}
\subsubsection*{Modules of the form $P(n,7)$}

Defect: $\partial P(n,7) = -1$, for $n\ge 0$.

\begin{fleqn}
\begin{align*}
P(0,7):\ & \bigl(I(0,6),P(0,3)\bigr),\ \bigl(I(0,7),P(0,6)\bigr)\\
P(1,7):\ & \bigl(I(1,5),P(0,2)\bigr),\ \bigl(I(1,1),P(0,4)\bigr)\\
P(2,7):\ & \bigl(R_{0}^{1}(1),P(0,1)\bigr),\ \bigl(R_{1}^{3}(1),P(0,5)\bigr),\ \bigl(I(2,7),P(0,6)\bigr),\ \bigl(I(0,5),P(1,2)\bigr),\ \bigl(I(0,1),P(1,4)\bigr)\\
P(3,7):\ & \bigl(R_{\infty}^{1}(1),P(0,7)\bigr),\ \bigl(R_{0}^{2}(1),P(1,1)\bigr),\ \bigl(R_{1}^{1}(1),P(1,5)\bigr),\ \bigl(I(1,7),P(1,6)\bigr)\\
P(4,7):\ & \bigl(R_{1}^{1}(2),P(0,1)\bigr),\ \bigl(R_{0}^{2}(2),P(0,5)\bigr),\ \bigl(R_{\infty}^{2}(1),P(1,7)\bigr),\ \bigl(R_{0}^{3}(1),P(2,1)\bigr),\ \bigl(R_{1}^{2}(1),P(2,5)\bigr)\\
 & \bigl(I(0,7),P(2,6)\bigr)\\
P(n,7):\ & \bigl(R_{1}^{(n-4)\bmod 3+1}(2),P(n-4,1)\bigr),\ \bigl(R_{0}^{(n-3)\bmod 3+1}(2),P(n-4,5)\bigr),\ \bigl(R_{\infty}^{(n-3)\bmod 2+1}(1),P(n-3,7)\bigr)\\
 & \bigl(R_{0}^{(n-2)\bmod 3+1}(1),P(n-2,1)\bigr),\ \bigl(R_{1}^{(n-3)\bmod 3+1}(1),P(n-2,5)\bigr),\ \bigl(I(-n+4,7),P(n-2,6)\bigr),\ n>4\\
\end{align*}
\end{fleqn}
\subsubsection{Schofield pairs associated to preinjective exceptional modules}

\begin{figure}[ht]

\begin{center}
\begin{scaletikzpicturetowidth}{\textwidth}
\begin{tikzpicture}[scale=\tikzscale]
	\node[noselEEEEEEE, label={[shift={(0.4,-0.7)}, brown]\tiny $I(0,1)$}] (I_0_1__Rationals) at (4,6) {$\begin{smallmatrix}&&0\\&&0\\1&0&0&0&0\end{smallmatrix}$};
	\node[noselEEEEEEE, label={[shift={(0.4,-0.7)}, brown]\tiny $I(0,2)$}] (I_0_2__Rationals) at (2,3) {$\begin{smallmatrix}&&0\\&&0\\1&1&0&0&0\end{smallmatrix}$};
	\node[seloneEEEEEEE, label={[shift={(0.4,-0.7)}, brown]\tiny $I(0,3)$}] (I_0_3__Rationals) at (0,0) {$\begin{smallmatrix}&&1\\&&1\\1&1&1&1&1\end{smallmatrix}$};
	\node[noselEEEEEEE, label={[shift={(0.4,-0.7)}, brown]\tiny $I(0,4)$}] (I_0_4__Rationals) at (2,-3) {$\begin{smallmatrix}&&0\\&&0\\0&0&0&1&1\end{smallmatrix}$};
	\node[noselEEEEEEE, label={[shift={(0.4,-0.7)}, brown]\tiny $I(0,5)$}] (I_0_5__Rationals) at (4,-6) {$\begin{smallmatrix}&&0\\&&0\\0&0&0&0&1\end{smallmatrix}$};
	\node[noselEEEEEEE, label={[shift={(0.4,-0.7)}, brown]\tiny $I(0,6)$}] (I_0_6__Rationals) at (2,1) {$\begin{smallmatrix}&&1\\&&1\\0&0&0&0&0\end{smallmatrix}$};
	\node[noselEEEEEEE, label={[shift={(0.4,-0.7)}, brown]\tiny $I(0,7)$}] (I_0_7__Rationals) at (4,2) {$\begin{smallmatrix}&&1\\&&0\\0&0&0&0&0\end{smallmatrix}$};
	\node[noselEEEEEEE, label={[shift={(0.4,-0.7)}, brown]\tiny $I(1,1)$}] (I_1_1__Rationals) at (0,6) {$\begin{smallmatrix}&&0\\&&0\\0&1&0&0&0\end{smallmatrix}$};
	\node[noselEEEEEEE, label={[shift={(0.4,-0.7)}, brown]\tiny $I(1,2)$}] (I_1_2__Rationals) at (-2,3) {$\begin{smallmatrix}&&1\\&&1\\0&1&1&1&1\end{smallmatrix}$};
	\node[noselEEEEEEE, label={[shift={(0.4,-0.7)}, brown]\tiny $I(1,3)$}] (I_1_3__Rationals) at (-4,0) {$\begin{smallmatrix}&&1\\&&2\\1&2&2&2&1\end{smallmatrix}$};
	\node[noselEEEEEEE, label={[shift={(0.4,-0.7)}, brown]\tiny $I(1,4)$}] (I_1_4__Rationals) at (-2,-3) {$\begin{smallmatrix}&&1\\&&1\\1&1&1&1&0\end{smallmatrix}$};
	\node[noselEEEEEEE, label={[shift={(0.4,-0.7)}, brown]\tiny $I(1,5)$}] (I_1_5__Rationals) at (0,-6) {$\begin{smallmatrix}&&0\\&&0\\0&0&0&1&0\end{smallmatrix}$};
	\node[noselEEEEEEE, label={[shift={(0.4,-0.7)}, brown]\tiny $I(1,6)$}] (I_1_6__Rationals) at (-2,1) {$\begin{smallmatrix}&&0\\&&1\\1&1&1&1&1\end{smallmatrix}$};
	\node[noselEEEEEEE, label={[shift={(0.4,-0.7)}, brown]\tiny $I(1,7)$}] (I_1_7__Rationals) at (0,2) {$\begin{smallmatrix}&&0\\&&1\\0&0&0&0&0\end{smallmatrix}$};
	\node[noselEEEEEEE, label={[shift={(0.4,-0.7)}, brown]\tiny $I(2,1)$}] (I_2_1__Rationals) at (-4,6) {$\begin{smallmatrix}&&1\\&&1\\0&0&1&1&1\end{smallmatrix}$};
	\node[seloneEEEEEEE, label={[shift={(0.4,-0.7)}, brown]\tiny $I(2,2)$}] (I_2_2__Rationals) at (-6,3) {$\begin{smallmatrix}&&1\\&&2\\1&1&2&2&1\end{smallmatrix}$};
	\node[noselEEEEEEE, label={[shift={(0.4,-0.7)}, brown]\tiny $I(2,3)$}] (I_2_3__Rationals) at (-8,0) {$\begin{smallmatrix}&&2\\&&3\\2&3&4&3&2\end{smallmatrix}$};
	\node[seloneEEEEEEE, label={[shift={(0.4,-0.7)}, brown]\tiny $I(2,4)$}] (I_2_4__Rationals) at (-6,-3) {$\begin{smallmatrix}&&1\\&&2\\1&2&2&1&1\end{smallmatrix}$};
	\node[noselEEEEEEE, label={[shift={(0.4,-0.7)}, brown]\tiny $I(2,5)$}] (I_2_5__Rationals) at (-4,-6) {$\begin{smallmatrix}&&1\\&&1\\1&1&1&0&0\end{smallmatrix}$};
	\node[seloneEEEEEEE, label={[shift={(0.4,-0.7)}, brown]\tiny $I(2,6)$}] (I_2_6__Rationals) at (-6,1) {$\begin{smallmatrix}&&1\\&&1\\1&2&2&2&1\end{smallmatrix}$};
	\node[noselEEEEEEE, label={[shift={(0.4,-0.7)}, brown]\tiny $I(2,7)$}] (I_2_7__Rationals) at (-4,2) {$\begin{smallmatrix}&&0\\&&0\\1&1&1&1&1\end{smallmatrix}$};
	\node[noselEEEEEEE, label={[shift={(0.4,-0.7)}, brown]\tiny $I(3,1)$}] (I_3_1__Rationals) at (-8,6) {$\begin{smallmatrix}&&0\\&&1\\1&1&1&1&0\end{smallmatrix}$};
	\node[noselEEEEEEE, label={[shift={(0.4,-0.7)}, brown]\tiny $I(3,2)$}] (I_3_2__Rationals) at (-10,3) {$\begin{smallmatrix}&&1\\&&2\\2&3&3&2&1\end{smallmatrix}$};
	\node[noselEEEEEEE, label={[shift={(0.4,-0.7)}, brown]\tiny $I(3,3)$}] (I_3_3__Rationals) at (-12,0) {$\begin{smallmatrix}&&2\\&&4\\2&4&5&4&2\end{smallmatrix}$};
	\node[noselEEEEEEE, label={[shift={(0.4,-0.7)}, brown]\tiny $I(3,4)$}] (I_3_4__Rationals) at (-10,-3) {$\begin{smallmatrix}&&1\\&&2\\1&2&3&3&2\end{smallmatrix}$};
	\node[noselEEEEEEE, label={[shift={(0.4,-0.7)}, brown]\tiny $I(3,5)$}] (I_3_5__Rationals) at (-8,-6) {$\begin{smallmatrix}&&0\\&&1\\0&1&1&1&1\end{smallmatrix}$};
	\node[noselEEEEEEE, label={[shift={(0.4,-0.7)}, brown]\tiny $I(3,6)$}] (I_3_6__Rationals) at (-10,1) {$\begin{smallmatrix}&&2\\&&3\\1&2&3&2&1\end{smallmatrix}$};
	\node[noselEEEEEEE, label={[shift={(0.4,-0.7)}, brown]\tiny $I(3,7)$}] (I_3_7__Rationals) at (-8,2) {$\begin{smallmatrix}&&1\\&&1\\0&1&1&1&0\end{smallmatrix}$};
	\node[seloneEEEEEEE, label={[shift={(0.4,-0.7)}, brown]\tiny $I(4,1)$}] (I_4_1__Rationals) at (-12,6) {$\begin{smallmatrix}&&1\\&&1\\1&2&2&1&1\end{smallmatrix}$};
	\node[noselEEEEEEE, label={[shift={(0.4,-0.7)}, brown]\tiny $I(4,2)$}] (I_4_2__Rationals) at (-14,3) {$\begin{smallmatrix}&&2\\&&3\\1&3&4&3&2\end{smallmatrix}$};
	\node[noselEEEEEEE, label={[shift={(0.4,-0.7)}, brown]\tiny $I(4,3)$}] (I_4_3__Rationals) at (-16,0) {$\begin{smallmatrix}&&3\\&&5\\3&5&7&5&3\end{smallmatrix}$};
	\node[noselEEEEEEE, label={[shift={(0.4,-0.7)}, brown]\tiny $I(4,4)$}] (I_4_4__Rationals) at (-14,-3) {$\begin{smallmatrix}&&2\\&&3\\2&3&4&3&1\end{smallmatrix}$};
	\node[seloneEEEEEEE, label={[shift={(0.4,-0.7)}, brown]\tiny $I(4,5)$}] (I_4_5__Rationals) at (-12,-6) {$\begin{smallmatrix}&&1\\&&1\\1&1&2&2&1\end{smallmatrix}$};
	\node[noselEEEEEEE, label={[shift={(0.4,-0.7)}, brown]\tiny $I(4,6)$}] (I_4_6__Rationals) at (-14,1) {$\begin{smallmatrix}&&1\\&&3\\2&3&4&3&2\end{smallmatrix}$};
	\node[seloneEEEEEEE, label={[shift={(0.4,-0.7)}, brown]\tiny $I(4,7)$}] (I_4_7__Rationals) at (-12,2) {$\begin{smallmatrix}&&1\\&&2\\1&1&2&1&1\end{smallmatrix}$};
	\node at (-16,3) {$\dots$};
	\node at (-16,-3) {$\dots$};
	\draw[red,->] (I_0_2__Rationals) -- (I_0_1__Rationals);
	\draw[red,->] (I_0_3__Rationals) -- (I_0_2__Rationals);
	\draw[blue,->] (I_1_1__Rationals) -- (I_0_2__Rationals);
	\draw[blue,->] (I_1_2__Rationals) -- (I_0_3__Rationals);
	\draw[blue,->] (I_1_4__Rationals) -- (I_0_3__Rationals);
	\draw[blue,->] (I_1_6__Rationals) -- (I_0_3__Rationals);
	\draw[red,->] (I_0_3__Rationals) -- (I_0_4__Rationals);
	\draw[blue,->] (I_1_5__Rationals) -- (I_0_4__Rationals);
	\draw[red,->] (I_0_4__Rationals) -- (I_0_5__Rationals);
	\draw[red,->] (I_0_3__Rationals) -- (I_0_6__Rationals);
	\draw[blue,->] (I_1_7__Rationals) -- (I_0_6__Rationals);
	\draw[red,->] (I_0_6__Rationals) -- (I_0_7__Rationals);
	\draw[gray,dashed,->] (I_0_1__Rationals) -- (I_1_1__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_1_2__Rationals) -- (I_1_1__Rationals);
	\draw[gray,dashed,->] (I_0_2__Rationals) -- (I_1_2__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_1_3__Rationals) -- (I_1_2__Rationals);
	\draw[blue,->] (I_2_1__Rationals) -- (I_1_2__Rationals);
	\draw[gray,dashed,->] (I_0_3__Rationals) -- (I_1_3__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[blue,->] (I_2_2__Rationals) -- (I_1_3__Rationals);
	\draw[blue,->] (I_2_4__Rationals) -- (I_1_3__Rationals);
	\draw[blue,->] (I_2_6__Rationals) -- (I_1_3__Rationals);
	\draw[gray,dashed,->] (I_0_4__Rationals) -- (I_1_4__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_1_3__Rationals) -- (I_1_4__Rationals);
	\draw[blue,->] (I_2_5__Rationals) -- (I_1_4__Rationals);
	\draw[gray,dashed,->] (I_0_5__Rationals) -- (I_1_5__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_1_4__Rationals) -- (I_1_5__Rationals);
	\draw[gray,dashed,->] (I_0_6__Rationals) -- (I_1_6__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_1_3__Rationals) -- (I_1_6__Rationals);
	\draw[blue,->] (I_2_7__Rationals) -- (I_1_6__Rationals);
	\draw[gray,dashed,->] (I_0_7__Rationals) -- (I_1_7__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_1_6__Rationals) -- (I_1_7__Rationals);
	\draw[gray,dashed,->] (I_1_1__Rationals) -- (I_2_1__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_2_2__Rationals) -- (I_2_1__Rationals);
	\draw[gray,dashed,->] (I_1_2__Rationals) -- (I_2_2__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_2_3__Rationals) -- (I_2_2__Rationals);
	\draw[blue,->] (I_3_1__Rationals) -- (I_2_2__Rationals);
	\draw[gray,dashed,->] (I_1_3__Rationals) -- (I_2_3__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[blue,->] (I_3_2__Rationals) -- (I_2_3__Rationals);
	\draw[blue,->] (I_3_4__Rationals) -- (I_2_3__Rationals);
	\draw[blue,->] (I_3_6__Rationals) -- (I_2_3__Rationals);
	\draw[gray,dashed,->] (I_1_4__Rationals) -- (I_2_4__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_2_3__Rationals) -- (I_2_4__Rationals);
	\draw[blue,->] (I_3_5__Rationals) -- (I_2_4__Rationals);
	\draw[gray,dashed,->] (I_1_5__Rationals) -- (I_2_5__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_2_4__Rationals) -- (I_2_5__Rationals);
	\draw[gray,dashed,->] (I_1_6__Rationals) -- (I_2_6__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_2_3__Rationals) -- (I_2_6__Rationals);
	\draw[blue,->] (I_3_7__Rationals) -- (I_2_6__Rationals);
	\draw[gray,dashed,->] (I_1_7__Rationals) -- (I_2_7__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_2_6__Rationals) -- (I_2_7__Rationals);
	\draw[gray,dashed,->] (I_2_1__Rationals) -- (I_3_1__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_3_2__Rationals) -- (I_3_1__Rationals);
	\draw[gray,dashed,->] (I_2_2__Rationals) -- (I_3_2__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_3_3__Rationals) -- (I_3_2__Rationals);
	\draw[blue,->] (I_4_1__Rationals) -- (I_3_2__Rationals);
	\draw[gray,dashed,->] (I_2_3__Rationals) -- (I_3_3__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[blue,->] (I_4_2__Rationals) -- (I_3_3__Rationals);
	\draw[blue,->] (I_4_4__Rationals) -- (I_3_3__Rationals);
	\draw[blue,->] (I_4_6__Rationals) -- (I_3_3__Rationals);
	\draw[gray,dashed,->] (I_2_4__Rationals) -- (I_3_4__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_3_3__Rationals) -- (I_3_4__Rationals);
	\draw[blue,->] (I_4_5__Rationals) -- (I_3_4__Rationals);
	\draw[gray,dashed,->] (I_2_5__Rationals) -- (I_3_5__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_3_4__Rationals) -- (I_3_5__Rationals);
	\draw[gray,dashed,->] (I_2_6__Rationals) -- (I_3_6__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_3_3__Rationals) -- (I_3_6__Rationals);
	\draw[blue,->] (I_4_7__Rationals) -- (I_3_6__Rationals);
	\draw[gray,dashed,->] (I_2_7__Rationals) -- (I_3_7__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_3_6__Rationals) -- (I_3_7__Rationals);
	\draw[gray,dashed,->] (I_3_1__Rationals) -- (I_4_1__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_4_2__Rationals) -- (I_4_1__Rationals);
	\draw[gray,dashed,->] (I_3_2__Rationals) -- (I_4_2__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_4_3__Rationals) -- (I_4_2__Rationals);
	\draw[gray,dashed,->] (I_3_3__Rationals) -- (I_4_3__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[gray,dashed,->] (I_3_4__Rationals) -- (I_4_4__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_4_3__Rationals) -- (I_4_4__Rationals);
	\draw[gray,dashed,->] (I_3_5__Rationals) -- (I_4_5__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_4_4__Rationals) -- (I_4_5__Rationals);
	\draw[gray,dashed,->] (I_3_6__Rationals) -- (I_4_6__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_4_3__Rationals) -- (I_4_6__Rationals);
	\draw[gray,dashed,->] (I_3_7__Rationals) -- (I_4_7__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_4_6__Rationals) -- (I_4_7__Rationals);
\end{tikzpicture}
\end{scaletikzpicturetowidth}
\end{center}

\end{figure}
\begin{figure}[ht]

\begin{center}
\begin{scaletikzpicturetowidth}{\textwidth}
\begin{tikzpicture}[scale=\tikzscale]
	\node at (4,0) {$\dots$};
	\node[noselEEEEEEE, label={[shift={(0.4,-0.7)}, brown]\tiny $I(5,1)$}] (I_5_1__Rationals) at (4,6) {$\begin{smallmatrix}&&1\\&&2\\0&1&2&2&1\end{smallmatrix}$};
	\node[noselEEEEEEE, label={[shift={(0.4,-0.7)}, brown]\tiny $I(5,2)$}] (I_5_2__Rationals) at (2,3) {$\begin{smallmatrix}&&2\\&&4\\2&3&5&4&2\end{smallmatrix}$};
	\node[noselEEEEEEE, label={[shift={(0.4,-0.7)}, brown]\tiny $I(5,3)$}] (I_5_3__Rationals) at (0,0) {$\begin{smallmatrix}&&3\\&&6\\3&6&8&6&3\end{smallmatrix}$};
	\node[noselEEEEEEE, label={[shift={(0.4,-0.7)}, brown]\tiny $I(5,4)$}] (I_5_4__Rationals) at (2,-3) {$\begin{smallmatrix}&&2\\&&4\\2&4&5&3&2\end{smallmatrix}$};
	\node[noselEEEEEEE, label={[shift={(0.4,-0.7)}, brown]\tiny $I(5,5)$}] (I_5_5__Rationals) at (4,-6) {$\begin{smallmatrix}&&1\\&&2\\1&2&2&1&0\end{smallmatrix}$};
	\node[noselEEEEEEE, label={[shift={(0.4,-0.7)}, brown]\tiny $I(5,6)$}] (I_5_6__Rationals) at (2,1) {$\begin{smallmatrix}&&2\\&&3\\2&4&5&4&2\end{smallmatrix}$};
	\node[noselEEEEEEE, label={[shift={(0.4,-0.7)}, brown]\tiny $I(5,7)$}] (I_5_7__Rationals) at (4,2) {$\begin{smallmatrix}&&0\\&&1\\1&2&2&2&1\end{smallmatrix}$};
	\node[noselEEEEEEE, label={[shift={(0.4,-0.7)}, brown]\tiny $I(6,1)$}] (I_6_1__Rationals) at (0,6) {$\begin{smallmatrix}&&1\\&&2\\2&2&3&2&1\end{smallmatrix}$};
	\node[noselEEEEEEE, label={[shift={(0.4,-0.7)}, brown]\tiny $I(6,2)$}] (I_6_2__Rationals) at (-2,3) {$\begin{smallmatrix}&&2\\&&4\\3&5&6&4&2\end{smallmatrix}$};
	\node[noselEEEEEEE, label={[shift={(0.4,-0.7)}, brown]\tiny $I(6,3)$}] (I_6_3__Rationals) at (-4,0) {$\begin{smallmatrix}&&4\\&&7\\4&7&10&7&4\end{smallmatrix}$};
	\node[noselEEEEEEE, label={[shift={(0.4,-0.7)}, brown]\tiny $I(6,4)$}] (I_6_4__Rationals) at (-2,-3) {$\begin{smallmatrix}&&2\\&&4\\2&4&6&5&3\end{smallmatrix}$};
	\node[noselEEEEEEE, label={[shift={(0.4,-0.7)}, brown]\tiny $I(6,5)$}] (I_6_5__Rationals) at (0,-6) {$\begin{smallmatrix}&&1\\&&2\\1&2&3&2&2\end{smallmatrix}$};
	\node[noselEEEEEEE, label={[shift={(0.4,-0.7)}, brown]\tiny $I(6,6)$}] (I_6_6__Rationals) at (-2,1) {$\begin{smallmatrix}&&3\\&&5\\2&4&6&4&2\end{smallmatrix}$};
	\node[noselEEEEEEE, label={[shift={(0.4,-0.7)}, brown]\tiny $I(6,7)$}] (I_6_7__Rationals) at (0,2) {$\begin{smallmatrix}&&2\\&&2\\1&2&3&2&1\end{smallmatrix}$};
	\node[noselEEEEEEE, label={[shift={(0.4,-0.7)}, brown]\tiny $I(7,1)$}] (I_7_1__Rationals) at (-4,6) {$\begin{smallmatrix}&&1\\&&2\\1&3&3&2&1\end{smallmatrix}$};
	\node[noselEEEEEEE, label={[shift={(0.4,-0.7)}, brown]\tiny $I(7,2)$}] (I_7_2__Rationals) at (-6,3) {$\begin{smallmatrix}&&3\\&&5\\2&5&7&5&3\end{smallmatrix}$};
	\node[noselEEEEEEE, label={[shift={(0.4,-0.7)}, brown]\tiny $I(7,3)$}] (I_7_3__Rationals) at (-8,0) {$\begin{smallmatrix}&&4\\&&8\\4&8&11&8&4\end{smallmatrix}$};
	\node[noselEEEEEEE, label={[shift={(0.4,-0.7)}, brown]\tiny $I(7,4)$}] (I_7_4__Rationals) at (-6,-3) {$\begin{smallmatrix}&&3\\&&5\\3&5&7&5&2\end{smallmatrix}$};
	\node[noselEEEEEEE, label={[shift={(0.4,-0.7)}, brown]\tiny $I(7,5)$}] (I_7_5__Rationals) at (-4,-6) {$\begin{smallmatrix}&&1\\&&2\\1&2&3&3&1\end{smallmatrix}$};
	\node[noselEEEEEEE, label={[shift={(0.4,-0.7)}, brown]\tiny $I(7,6)$}] (I_7_6__Rationals) at (-6,1) {$\begin{smallmatrix}&&2\\&&5\\3&5&7&5&3\end{smallmatrix}$};
	\node[noselEEEEEEE, label={[shift={(0.4,-0.7)}, brown]\tiny $I(7,7)$}] (I_7_7__Rationals) at (-4,2) {$\begin{smallmatrix}&&1\\&&3\\1&2&3&2&1\end{smallmatrix}$};
	\node[noselEEEEEEE, label={[shift={(0.4,-0.7)}, brown]\tiny $I(8,1)$}] (I_8_1__Rationals) at (-8,6) {$\begin{smallmatrix}&&2\\&&3\\1&2&4&3&2\end{smallmatrix}$};
	\node[noselEEEEEEE, label={[shift={(0.4,-0.7)}, brown]\tiny $I(8,2)$}] (I_8_2__Rationals) at (-10,3) {$\begin{smallmatrix}&&3\\&&6\\3&5&8&6&3\end{smallmatrix}$};
	\node[noselEEEEEEE, label={[shift={(0.4,-0.7)}, brown]\tiny $I(8,3)$}] (I_8_3__Rationals) at (-12,0) {$\begin{smallmatrix}&&5\\&&9\\5&9&13&9&5\end{smallmatrix}$};
	\node[noselEEEEEEE, label={[shift={(0.4,-0.7)}, brown]\tiny $I(8,4)$}] (I_8_4__Rationals) at (-10,-3) {$\begin{smallmatrix}&&3\\&&6\\3&6&8&5&3\end{smallmatrix}$};
	\node[noselEEEEEEE, label={[shift={(0.4,-0.7)}, brown]\tiny $I(8,5)$}] (I_8_5__Rationals) at (-8,-6) {$\begin{smallmatrix}&&2\\&&3\\2&3&4&2&1\end{smallmatrix}$};
	\node[noselEEEEEEE, label={[shift={(0.4,-0.7)}, brown]\tiny $I(8,6)$}] (I_8_6__Rationals) at (-10,1) {$\begin{smallmatrix}&&3\\&&5\\3&6&8&6&3\end{smallmatrix}$};
	\node[noselEEEEEEE, label={[shift={(0.4,-0.7)}, brown]\tiny $I(8,7)$}] (I_8_7__Rationals) at (-8,2) {$\begin{smallmatrix}&&1\\&&2\\2&3&4&3&2\end{smallmatrix}$};
	\node[noselEEEEEEE, label={[shift={(0.4,-0.7)}, brown]\tiny $I(9,1)$}] (I_9_1__Rationals) at (-12,6) {$\begin{smallmatrix}&&1\\&&3\\2&3&4&3&1\end{smallmatrix}$};
	\node[noselEEEEEEE, label={[shift={(0.4,-0.7)}, brown]\tiny $I(9,2)$}] (I_9_2__Rationals) at (-14,3) {$\begin{smallmatrix}&&3\\&&6\\4&7&9&6&3\end{smallmatrix}$};
	\node[noselEEEEEEE, label={[shift={(0.4,-0.7)}, brown]\tiny $I(9,3)$}] (I_9_3__Rationals) at (-16,0) {$\begin{smallmatrix}&&5\\&&10\\5&10&14&10&5\end{smallmatrix}$};
	\node[noselEEEEEEE, label={[shift={(0.4,-0.7)}, brown]\tiny $I(9,4)$}] (I_9_4__Rationals) at (-14,-3) {$\begin{smallmatrix}&&3\\&&6\\3&6&9&7&4\end{smallmatrix}$};
	\node[noselEEEEEEE, label={[shift={(0.4,-0.7)}, brown]\tiny $I(9,5)$}] (I_9_5__Rationals) at (-12,-6) {$\begin{smallmatrix}&&1\\&&3\\1&3&4&3&2\end{smallmatrix}$};
	\node[noselEEEEEEE, label={[shift={(0.4,-0.7)}, brown]\tiny $I(9,6)$}] (I_9_6__Rationals) at (-14,1) {$\begin{smallmatrix}&&4\\&&7\\3&6&9&6&3\end{smallmatrix}$};
	\node[noselEEEEEEE, label={[shift={(0.4,-0.7)}, brown]\tiny $I(9,7)$}] (I_9_7__Rationals) at (-12,2) {$\begin{smallmatrix}&&2\\&&3\\1&3&4&3&1\end{smallmatrix}$};
	\node at (-16,3) {$\dots$};
	\node at (-16,-3) {$\dots$};
	\draw[red,->] (I_5_2__Rationals) -- (I_5_1__Rationals);
	\draw[red,->] (I_5_3__Rationals) -- (I_5_2__Rationals);
	\draw[blue,->] (I_6_1__Rationals) -- (I_5_2__Rationals);
	\draw[blue,->] (I_6_2__Rationals) -- (I_5_3__Rationals);
	\draw[blue,->] (I_6_4__Rationals) -- (I_5_3__Rationals);
	\draw[blue,->] (I_6_6__Rationals) -- (I_5_3__Rationals);
	\draw[red,->] (I_5_3__Rationals) -- (I_5_4__Rationals);
	\draw[blue,->] (I_6_5__Rationals) -- (I_5_4__Rationals);
	\draw[red,->] (I_5_4__Rationals) -- (I_5_5__Rationals);
	\draw[red,->] (I_5_3__Rationals) -- (I_5_6__Rationals);
	\draw[blue,->] (I_6_7__Rationals) -- (I_5_6__Rationals);
	\draw[red,->] (I_5_6__Rationals) -- (I_5_7__Rationals);
	\draw[gray,dashed,->] (I_5_1__Rationals) -- (I_6_1__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_6_2__Rationals) -- (I_6_1__Rationals);
	\draw[gray,dashed,->] (I_5_2__Rationals) -- (I_6_2__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_6_3__Rationals) -- (I_6_2__Rationals);
	\draw[blue,->] (I_7_1__Rationals) -- (I_6_2__Rationals);
	\draw[gray,dashed,->] (I_5_3__Rationals) -- (I_6_3__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[blue,->] (I_7_2__Rationals) -- (I_6_3__Rationals);
	\draw[blue,->] (I_7_4__Rationals) -- (I_6_3__Rationals);
	\draw[blue,->] (I_7_6__Rationals) -- (I_6_3__Rationals);
	\draw[gray,dashed,->] (I_5_4__Rationals) -- (I_6_4__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_6_3__Rationals) -- (I_6_4__Rationals);
	\draw[blue,->] (I_7_5__Rationals) -- (I_6_4__Rationals);
	\draw[gray,dashed,->] (I_5_5__Rationals) -- (I_6_5__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_6_4__Rationals) -- (I_6_5__Rationals);
	\draw[gray,dashed,->] (I_5_6__Rationals) -- (I_6_6__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_6_3__Rationals) -- (I_6_6__Rationals);
	\draw[blue,->] (I_7_7__Rationals) -- (I_6_6__Rationals);
	\draw[gray,dashed,->] (I_5_7__Rationals) -- (I_6_7__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_6_6__Rationals) -- (I_6_7__Rationals);
	\draw[gray,dashed,->] (I_6_1__Rationals) -- (I_7_1__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_7_2__Rationals) -- (I_7_1__Rationals);
	\draw[gray,dashed,->] (I_6_2__Rationals) -- (I_7_2__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_7_3__Rationals) -- (I_7_2__Rationals);
	\draw[blue,->] (I_8_1__Rationals) -- (I_7_2__Rationals);
	\draw[gray,dashed,->] (I_6_3__Rationals) -- (I_7_3__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[blue,->] (I_8_2__Rationals) -- (I_7_3__Rationals);
	\draw[blue,->] (I_8_4__Rationals) -- (I_7_3__Rationals);
	\draw[blue,->] (I_8_6__Rationals) -- (I_7_3__Rationals);
	\draw[gray,dashed,->] (I_6_4__Rationals) -- (I_7_4__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_7_3__Rationals) -- (I_7_4__Rationals);
	\draw[blue,->] (I_8_5__Rationals) -- (I_7_4__Rationals);
	\draw[gray,dashed,->] (I_6_5__Rationals) -- (I_7_5__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_7_4__Rationals) -- (I_7_5__Rationals);
	\draw[gray,dashed,->] (I_6_6__Rationals) -- (I_7_6__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_7_3__Rationals) -- (I_7_6__Rationals);
	\draw[blue,->] (I_8_7__Rationals) -- (I_7_6__Rationals);
	\draw[gray,dashed,->] (I_6_7__Rationals) -- (I_7_7__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_7_6__Rationals) -- (I_7_7__Rationals);
	\draw[gray,dashed,->] (I_7_1__Rationals) -- (I_8_1__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_8_2__Rationals) -- (I_8_1__Rationals);
	\draw[gray,dashed,->] (I_7_2__Rationals) -- (I_8_2__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_8_3__Rationals) -- (I_8_2__Rationals);
	\draw[blue,->] (I_9_1__Rationals) -- (I_8_2__Rationals);
	\draw[gray,dashed,->] (I_7_3__Rationals) -- (I_8_3__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[blue,->] (I_9_2__Rationals) -- (I_8_3__Rationals);
	\draw[blue,->] (I_9_4__Rationals) -- (I_8_3__Rationals);
	\draw[blue,->] (I_9_6__Rationals) -- (I_8_3__Rationals);
	\draw[gray,dashed,->] (I_7_4__Rationals) -- (I_8_4__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_8_3__Rationals) -- (I_8_4__Rationals);
	\draw[blue,->] (I_9_5__Rationals) -- (I_8_4__Rationals);
	\draw[gray,dashed,->] (I_7_5__Rationals) -- (I_8_5__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_8_4__Rationals) -- (I_8_5__Rationals);
	\draw[gray,dashed,->] (I_7_6__Rationals) -- (I_8_6__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_8_3__Rationals) -- (I_8_6__Rationals);
	\draw[blue,->] (I_9_7__Rationals) -- (I_8_6__Rationals);
	\draw[gray,dashed,->] (I_7_7__Rationals) -- (I_8_7__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_8_6__Rationals) -- (I_8_7__Rationals);
	\draw[gray,dashed,->] (I_8_1__Rationals) -- (I_9_1__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_9_2__Rationals) -- (I_9_1__Rationals);
	\draw[gray,dashed,->] (I_8_2__Rationals) -- (I_9_2__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_9_3__Rationals) -- (I_9_2__Rationals);
	\draw[gray,dashed,->] (I_8_3__Rationals) -- (I_9_3__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[gray,dashed,->] (I_8_4__Rationals) -- (I_9_4__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_9_3__Rationals) -- (I_9_4__Rationals);
	\draw[gray,dashed,->] (I_8_5__Rationals) -- (I_9_5__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_9_4__Rationals) -- (I_9_5__Rationals);
	\draw[gray,dashed,->] (I_8_6__Rationals) -- (I_9_6__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_9_3__Rationals) -- (I_9_6__Rationals);
	\draw[gray,dashed,->] (I_8_7__Rationals) -- (I_9_7__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_9_6__Rationals) -- (I_9_7__Rationals);
\end{tikzpicture}
\end{scaletikzpicturetowidth}
\end{center}

\end{figure}
\medskip{}
\subsubsection*{Modules of the form $I(n,1)$}

Defect: $\partial I(n,1) = 1$, for $n\ge 0$.

\begin{fleqn}
\begin{align*}
I(0,1):\ & - \\
I(1,1):\ & - \\
I(2,1):\ & \bigl(I(0,4),P(0,7)\bigr),\ \bigl(I(0,5),R_{1}^{1}(1)\bigr),\ \bigl(I(0,6),P(0,5)\bigr),\ \bigl(I(0,7),R_{0}^{1}(1)\bigr)\\
I(3,1):\ & \bigl(I(0,1),R_{\infty}^{1}(1)\bigr),\ \bigl(I(0,2),P(1,1)\bigr),\ \bigl(I(1,5),R_{1}^{3}(1)\bigr),\ \bigl(I(1,7),R_{0}^{3}(1)\bigr)\\
I(4,1):\ & \bigl(I(0,5),R_{0}^{2}(2)\bigr),\ \bigl(I(0,7),R_{1}^{2}(2)\bigr),\ \bigl(I(1,1),R_{\infty}^{2}(1)\bigr),\ \bigl(I(1,2),P(0,1)\bigr),\ \bigl(I(2,5),R_{1}^{2}(1)\bigr)\\
 & \bigl(I(2,7),R_{0}^{2}(1)\bigr)\\
I(n,1):\ & \bigl(I(n-4,5),R_{0}^{(-n+5)\bmod 3+1}(2)\bigr),\ \bigl(I(n-4,7),R_{1}^{(-n+5)\bmod 3+1}(2)\bigr),\ \bigl(I(n-3,1),R_{\infty}^{(-n+5)\bmod 2+1}(1)\bigr)\\
 & \bigl(I(n-3,2),P(-n+4,1)\bigr),\ \bigl(I(n-2,5),R_{1}^{(-n+5)\bmod 3+1}(1)\bigr),\ \bigl(I(n-2,7),R_{0}^{(-n+5)\bmod 3+1}(1)\bigr),\ n>4\\
\end{align*}
\end{fleqn}
\subsubsection*{Modules of the form $I(n,2)$}

Defect: $\partial I(n,2) = 2$, for $n\ge 0$.

\begin{fleqn}
\begin{align*}
I(0,2):\ & \bigl(I(0,1),I(1,1)\bigr)\\
I(1,2):\ & \bigl(I(0,4),R_{0}^{2}(1)\bigr),\ \bigl(I(0,5),I(3,7)\bigr),\ \bigl(I(0,6),R_{1}^{2}(1)\bigr),\ \bigl(I(0,7),I(3,5)\bigr),\ \bigl(I(1,1),I(2,1)\bigr)\\
I(2,2):\ & \bigl(I(0,1),I(5,1)\bigr),\ \bigl(I(1,4),R_{0}^{1}(1)\bigr),\ \bigl(I(1,5),I(4,7)\bigr),\ \bigl(I(1,6),R_{1}^{1}(1)\bigr),\ \bigl(I(1,7),I(4,5)\bigr)\\
 & \bigl(I(2,1),I(3,1)\bigr)\\
I(n,2):\ & \bigl(I(n-2,1),I(n+3,1)\bigr),\ \bigl(I(n-1,4),R_{0}^{(-n+2)\bmod 3+1}(1)\bigr),\ \bigl(I(n-1,5),I(n+2,7)\bigr)\\
 & \bigl(I(n-1,6),R_{1}^{(-n+2)\bmod 3+1}(1)\bigr),\ \bigl(I(n-1,7),I(n+2,5)\bigr),\ \bigl(I(n,1),I(n+1,1)\bigr),\ n>2\\
\end{align*}
\end{fleqn}
\subsubsection*{Modules of the form $I(n,3)$}

Defect: $\partial I(n,3) = 3$, for $n\ge 0$.

\begin{fleqn}
\begin{align*}
I(0,3):\ & \bigl(I(0,1),I(1,2)\bigr),\ \bigl(I(0,2),I(2,1)\bigr),\ \bigl(I(0,4),I(2,5)\bigr),\ \bigl(I(0,5),I(1,4)\bigr),\ \bigl(I(0,6),I(2,7)\bigr)\\
 & \bigl(I(0,7),I(1,6)\bigr)\\
I(n,3):\ & \bigl(I(n,1),I(n+1,2)\bigr),\ \bigl(I(n,2),I(n+2,1)\bigr),\ \bigl(I(n,4),I(n+2,5)\bigr)\\
 & \bigl(I(n,5),I(n+1,4)\bigr),\ \bigl(I(n,6),I(n+2,7)\bigr),\ \bigl(I(n,7),I(n+1,6)\bigr),\ n>0\\
\end{align*}
\end{fleqn}
\subsubsection*{Modules of the form $I(n,4)$}

Defect: $\partial I(n,4) = 2$, for $n\ge 0$.

\begin{fleqn}
\begin{align*}
I(0,4):\ & \bigl(I(0,5),I(1,5)\bigr)\\
I(1,4):\ & \bigl(I(0,1),I(3,7)\bigr),\ \bigl(I(0,2),R_{1}^{1}(1)\bigr),\ \bigl(I(0,6),R_{0}^{3}(1)\bigr),\ \bigl(I(0,7),I(3,1)\bigr),\ \bigl(I(1,5),I(2,5)\bigr)\\
I(2,4):\ & \bigl(I(0,5),I(5,5)\bigr),\ \bigl(I(1,1),I(4,7)\bigr),\ \bigl(I(1,2),R_{1}^{3}(1)\bigr),\ \bigl(I(1,6),R_{0}^{2}(1)\bigr),\ \bigl(I(1,7),I(4,1)\bigr)\\
 & \bigl(I(2,5),I(3,5)\bigr)\\
I(n,4):\ & \bigl(I(n-2,5),I(n+3,5)\bigr),\ \bigl(I(n-1,1),I(n+2,7)\bigr),\ \bigl(I(n-1,2),R_{1}^{(-n+4)\bmod 3+1}(1)\bigr)\\
 & \bigl(I(n-1,6),R_{0}^{(-n+3)\bmod 3+1}(1)\bigr),\ \bigl(I(n-1,7),I(n+2,1)\bigr),\ \bigl(I(n,5),I(n+1,5)\bigr),\ n>2\\
\end{align*}
\end{fleqn}
\subsubsection*{Modules of the form $I(n,5)$}

Defect: $\partial I(n,5) = 1$, for $n\ge 0$.

\begin{fleqn}
\begin{align*}
I(0,5):\ & - \\
I(1,5):\ & - \\
I(2,5):\ & \bigl(I(0,1),R_{0}^{2}(1)\bigr),\ \bigl(I(0,2),P(0,7)\bigr),\ \bigl(I(0,6),P(0,1)\bigr),\ \bigl(I(0,7),R_{1}^{3}(1)\bigr)\\
I(3,5):\ & \bigl(I(0,4),P(1,5)\bigr),\ \bigl(I(0,5),R_{\infty}^{1}(1)\bigr),\ \bigl(I(1,1),R_{0}^{1}(1)\bigr),\ \bigl(I(1,7),R_{1}^{2}(1)\bigr)\\
I(4,5):\ & \bigl(I(0,1),R_{1}^{1}(2)\bigr),\ \bigl(I(0,7),R_{0}^{3}(2)\bigr),\ \bigl(I(1,4),P(0,5)\bigr),\ \bigl(I(1,5),R_{\infty}^{2}(1)\bigr),\ \bigl(I(2,1),R_{0}^{3}(1)\bigr)\\
 & \bigl(I(2,7),R_{1}^{1}(1)\bigr)\\
I(n,5):\ & \bigl(I(n-4,1),R_{1}^{(-n+4)\bmod 3+1}(2)\bigr),\ \bigl(I(n-4,7),R_{0}^{(-n+6)\bmod 3+1}(2)\bigr),\ \bigl(I(n-3,4),P(-n+4,5)\bigr)\\
 & \bigl(I(n-3,5),R_{\infty}^{(-n+5)\bmod 2+1}(1)\bigr),\ \bigl(I(n-2,1),R_{0}^{(-n+6)\bmod 3+1}(1)\bigr),\ \bigl(I(n-2,7),R_{1}^{(-n+4)\bmod 3+1}(1)\bigr),\ n>4\\
\end{align*}
\end{fleqn}
\subsubsection*{Modules of the form $I(n,6)$}

Defect: $\partial I(n,6) = 2$, for $n\ge 0$.

\begin{fleqn}
\begin{align*}
I(0,6):\ & \bigl(I(0,7),I(1,7)\bigr)\\
I(1,6):\ & \bigl(I(0,1),I(3,5)\bigr),\ \bigl(I(0,2),R_{0}^{1}(1)\bigr),\ \bigl(I(0,4),R_{1}^{3}(1)\bigr),\ \bigl(I(0,5),I(3,1)\bigr),\ \bigl(I(1,7),I(2,7)\bigr)\\
I(2,6):\ & \bigl(I(0,7),I(5,7)\bigr),\ \bigl(I(1,1),I(4,5)\bigr),\ \bigl(I(1,2),R_{0}^{3}(1)\bigr),\ \bigl(I(1,4),R_{1}^{2}(1)\bigr),\ \bigl(I(1,5),I(4,1)\bigr)\\
 & \bigl(I(2,7),I(3,7)\bigr)\\
I(n,6):\ & \bigl(I(n-2,7),I(n+3,7)\bigr),\ \bigl(I(n-1,1),I(n+2,5)\bigr),\ \bigl(I(n-1,2),R_{0}^{(-n+4)\bmod 3+1}(1)\bigr)\\
 & \bigl(I(n-1,4),R_{1}^{(-n+3)\bmod 3+1}(1)\bigr),\ \bigl(I(n-1,5),I(n+2,1)\bigr),\ \bigl(I(n,7),I(n+1,7)\bigr),\ n>2\\
\end{align*}
\end{fleqn}
\subsubsection*{Modules of the form $I(n,7)$}

Defect: $\partial I(n,7) = 1$, for $n\ge 0$.

\begin{fleqn}
\begin{align*}
I(0,7):\ & - \\
I(1,7):\ & - \\
I(2,7):\ & \bigl(I(0,1),R_{1}^{2}(1)\bigr),\ \bigl(I(0,2),P(0,5)\bigr),\ \bigl(I(0,4),P(0,1)\bigr),\ \bigl(I(0,5),R_{0}^{3}(1)\bigr)\\
I(3,7):\ & \bigl(I(0,6),P(1,7)\bigr),\ \bigl(I(0,7),R_{\infty}^{1}(1)\bigr),\ \bigl(I(1,1),R_{1}^{1}(1)\bigr),\ \bigl(I(1,5),R_{0}^{2}(1)\bigr)\\
I(4,7):\ & \bigl(I(0,1),R_{0}^{1}(2)\bigr),\ \bigl(I(0,5),R_{1}^{3}(2)\bigr),\ \bigl(I(1,6),P(0,7)\bigr),\ \bigl(I(1,7),R_{\infty}^{2}(1)\bigr),\ \bigl(I(2,1),R_{1}^{3}(1)\bigr)\\
 & \bigl(I(2,5),R_{0}^{1}(1)\bigr)\\
I(n,7):\ & \bigl(I(n-4,1),R_{0}^{(-n+4)\bmod 3+1}(2)\bigr),\ \bigl(I(n-4,5),R_{1}^{(-n+6)\bmod 3+1}(2)\bigr),\ \bigl(I(n-3,6),P(-n+4,7)\bigr)\\
 & \bigl(I(n-3,7),R_{\infty}^{(-n+5)\bmod 2+1}(1)\bigr),\ \bigl(I(n-2,1),R_{1}^{(-n+6)\bmod 3+1}(1)\bigr),\ \bigl(I(n-2,5),R_{0}^{(-n+4)\bmod 3+1}(1)\bigr),\ n>4\\
\end{align*}
\end{fleqn}
\subsubsection{Schofield pairs associated to regular exceptional modules}

\subsubsection*{The non-homogeneous tube $\mathcal{T}_{1}^{\Delta(\widetilde{\mathbb{E}}_{6})}$}

\begin{figure}[ht]

\begin{center}
\captionof{figure}{\vspace*{-10pt}$\mathcal{T}_{1}^{\Delta(\widetilde{\mathbb{E}}_{6})}$}
\begin{scaletikzpicturetowidth}{0.69999999999999996\textwidth}
\begin{tikzpicture}[scale=\tikzscale]
	\node[seltwoEEEEEEE, label={[shift={(0.5,-0.65)}, brown]\tiny $R_{1}^{1}(1)$}] (RN_1_1_1__Rationals) at (4,0) {$\begin{smallmatrix}&&1\\&&1\\0&0&1&1&0\end{smallmatrix}$};
	\node[seltwoEEEEEEE, label={[shift={(0.5,-0.65)}, brown]\tiny $R_{1}^{2}(1)$}] (RN_1_2_1__Rationals) at (8,0) {$\begin{smallmatrix}&&0\\&&0\\0&1&1&1&1\end{smallmatrix}$};
	\node[seltwoEEEEEEE, label={[shift={(0.5,-0.65)}, brown]\tiny $R_{1}^{3}(1)$}] (RN_1_3_1__Rationals) at (12,0) {$\begin{smallmatrix}&&0\\&&1\\1&1&1&0&0\end{smallmatrix}$};
	\node[seltwoEEEEEEE, label={[shift={(0.5,-0.65)}, brown]\tiny $R_{1}^{1}(1)$}] (RN_1_1_1__Rationals_repeated) at (16,0) {$\begin{smallmatrix}&&1\\&&1\\0&0&1&1&0\end{smallmatrix}$};
	\node[seltwoEEEEEEE, label={[shift={(0.5,-0.65)}, brown]\tiny $R_{1}^{1}(2)$}] (RN_1_1_2__Rationals) at (6,2) {$\begin{smallmatrix}&&1\\&&1\\0&1&2&2&1\end{smallmatrix}$};
	\node[seltwoEEEEEEE, label={[shift={(0.5,-0.65)}, brown]\tiny $R_{1}^{2}(2)$}] (RN_1_2_2__Rationals) at (10,2) {$\begin{smallmatrix}&&0\\&&1\\1&2&2&1&1\end{smallmatrix}$};
	\node[seltwoEEEEEEE, label={[shift={(0.5,-0.65)}, brown]\tiny $R_{1}^{3}(2)$}] (RN_1_3_2__Rationals) at (14,2) {$\begin{smallmatrix}&&1\\&&2\\1&1&2&1&0\end{smallmatrix}$};
	\node[noselEEEEEEE, label={[shift={(0.5,-0.65)}, brown]\tiny $R_{1}^{3}(3)$}] (RN_1_3_3__Rationals) at (4,4) {$\begin{smallmatrix}&&1\\&&2\\1&2&3&2&1\end{smallmatrix}$};
	\node[noselEEEEEEE, label={[shift={(0.5,-0.65)}, brown]\tiny $R_{1}^{1}(3)$}] (RN_1_1_3__Rationals) at (8,4) {$\begin{smallmatrix}&&1\\&&2\\1&2&3&2&1\end{smallmatrix}$};
	\node[noselEEEEEEE, label={[shift={(0.5,-0.65)}, brown]\tiny $R_{1}^{2}(3)$}] (RN_1_2_3__Rationals) at (12,4) {$\begin{smallmatrix}&&1\\&&2\\1&2&3&2&1\end{smallmatrix}$};
	\node[noselEEEEEEE, label={[shift={(0.5,-0.65)}, brown]\tiny $R_{1}^{3}(3)$}] (RN_1_3_3__Rationals_repeated) at (16,4) {$\begin{smallmatrix}&&1\\&&2\\1&2&3&2&1\end{smallmatrix}$};
	\node[noselEEEEEEE, label={[shift={(0.5,-0.65)}, brown]\tiny $R_{1}^{3}(4)$}] (RN_1_3_4__Rationals) at (6,6) {$\begin{smallmatrix}&&1\\&&3\\2&3&4&2&1\end{smallmatrix}$};
	\node[noselEEEEEEE, label={[shift={(0.5,-0.65)}, brown]\tiny $R_{1}^{1}(4)$}] (RN_1_1_4__Rationals) at (10,6) {$\begin{smallmatrix}&&2\\&&3\\1&2&4&3&1\end{smallmatrix}$};
	\node[noselEEEEEEE, label={[shift={(0.5,-0.65)}, brown]\tiny $R_{1}^{2}(4)$}] (RN_1_2_4__Rationals) at (14,6) {$\begin{smallmatrix}&&1\\&&2\\1&3&4&3&2\end{smallmatrix}$};
	\draw[blue,->] (RN_1_1_1__Rationals) -- (RN_1_1_2__Rationals);
	\draw[gray,dashed,->] (RN_1_1_1__Rationals) -- (RN_1_2_1__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[red,->] (RN_1_1_2__Rationals) -- (RN_1_2_1__Rationals);
	\draw[gray,dashed,->] (RN_1_1_2__Rationals) -- (RN_1_2_2__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[blue,->] (RN_1_1_2__Rationals) -- (RN_1_1_3__Rationals);
	\draw[blue,->] (RN_1_2_1__Rationals) -- (RN_1_2_2__Rationals);
	\draw[gray,dashed,->] (RN_1_2_1__Rationals) -- (RN_1_3_1__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[red,->] (RN_1_2_2__Rationals) -- (RN_1_3_1__Rationals);
	\draw[gray,dashed,->] (RN_1_2_2__Rationals) -- (RN_1_3_2__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[blue,->] (RN_1_2_2__Rationals) -- (RN_1_2_3__Rationals);
	\draw[gray,dashed,->] (RN_1_3_1__Rationals) -- (RN_1_1_1__Rationals_repeated) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[blue,->] (RN_1_3_1__Rationals) -- (RN_1_3_2__Rationals);
	\draw[red,->] (RN_1_3_2__Rationals) -- (RN_1_1_1__Rationals_repeated);
	\draw[gray,dashed,->,out=200,in=-20] (RN_1_3_2__Rationals) to node[midway,sloped,above]{{\tiny $\tau^{-1}$}} (RN_1_1_2__Rationals);
	\draw[blue,->] (RN_1_3_2__Rationals) -- (RN_1_3_3__Rationals_repeated);
	\draw[red,->] (RN_1_3_3__Rationals) -- (RN_1_1_2__Rationals);
	\draw[blue,->] (RN_1_3_3__Rationals) -- (RN_1_3_4__Rationals);
	\draw[gray,dashed,->] (RN_1_3_3__Rationals) -- (RN_1_1_3__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[red,->] (RN_1_3_4__Rationals) -- (RN_1_1_3__Rationals);
	\draw[gray,dashed,->] (RN_1_3_4__Rationals) -- (RN_1_1_4__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[red,->] (RN_1_1_3__Rationals) -- (RN_1_2_2__Rationals);
	\draw[blue,->] (RN_1_1_3__Rationals) -- (RN_1_1_4__Rationals);
	\draw[gray,dashed,->] (RN_1_1_3__Rationals) -- (RN_1_2_3__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[red,->] (RN_1_1_4__Rationals) -- (RN_1_2_3__Rationals);
	\draw[gray,dashed,->] (RN_1_1_4__Rationals) -- (RN_1_2_4__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[red,->] (RN_1_2_3__Rationals) -- (RN_1_3_2__Rationals);
	\draw[gray,dashed,->] (RN_1_2_3__Rationals) -- (RN_1_3_3__Rationals_repeated) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[blue,->] (RN_1_2_3__Rationals) -- (RN_1_2_4__Rationals);
	\draw[red,->] (RN_1_2_4__Rationals) -- (RN_1_3_3__Rationals_repeated);
	\draw[gray,dashed,->,out=200,in=-20] (RN_1_2_4__Rationals) to node[midway,sloped,above]{{\tiny $\tau^{-1}$}} (RN_1_3_4__Rationals);
	\draw[dashed,-] (RN_1_1_1__Rationals) -- (RN_1_3_3__Rationals);
	\draw[dashed,-] (RN_1_1_1__Rationals_repeated) -- (RN_1_3_3__Rationals_repeated);
	\node (LeftTopPhantom_1) at (4,6) {};
	\draw[dashed,-] (RN_1_3_3__Rationals) -- (LeftTopPhantom_1);
	\node (RightTopPhantom_1) at (16,6) {};
	\draw[dashed,-] (RN_1_3_3__Rationals_repeated) -- (RightTopPhantom_1);
\end{tikzpicture}
\end{scaletikzpicturetowidth}
\end{center}

\end{figure}
\begin{fleqn}
\begin{align*}
R_{1}^{1}(1):\ & \bigl(I(0,6),P(0,4)\bigr),\ \bigl(I(1,5),P(0,7)\bigr),\ \bigl(I(0,7),P(1,1)\bigr)\\
R_{1}^{1}(2):\ & \bigl(R_{1}^{2}(1),R_{1}^{1}(1)\bigr),\ \bigl(I(3,7),P(0,5)\bigr),\ \bigl(I(2,1),P(1,7)\bigr),\ \bigl(I(1,5),P(2,1)\bigr),\ \bigl(I(0,7),P(3,5)\bigr)\\
R_{1}^{2}(1):\ & \bigl(I(0,4),P(0,2)\bigr),\ \bigl(I(1,1),P(0,5)\bigr),\ \bigl(I(0,5),P(1,7)\bigr)\\
R_{1}^{2}(2):\ & \bigl(R_{1}^{3}(1),R_{1}^{2}(1)\bigr),\ \bigl(I(3,5),P(0,1)\bigr),\ \bigl(I(2,7),P(1,5)\bigr),\ \bigl(I(1,1),P(2,7)\bigr),\ \bigl(I(0,5),P(3,1)\bigr)\\
R_{1}^{3}(1):\ & \bigl(I(1,7),P(0,1)\bigr),\ \bigl(I(0,2),P(0,6)\bigr),\ \bigl(I(0,1),P(1,5)\bigr)\\
R_{1}^{3}(2):\ & \bigl(R_{1}^{1}(1),R_{1}^{3}(1)\bigr),\ \bigl(I(3,1),P(0,7)\bigr),\ \bigl(I(2,5),P(1,1)\bigr),\ \bigl(I(1,7),P(2,5)\bigr),\ \bigl(I(0,1),P(3,7)\bigr)\\
\end{align*}
\end{fleqn}
\subsubsection*{The non-homogeneous tube $\mathcal{T}_{0}^{\Delta(\widetilde{\mathbb{E}}_{6})}$}

\begin{figure}[ht]

\begin{center}
\captionof{figure}{\vspace*{-10pt}$\mathcal{T}_{0}^{\Delta(\widetilde{\mathbb{E}}_{6})}$}
\begin{scaletikzpicturetowidth}{0.69999999999999996\textwidth}
\begin{tikzpicture}[scale=\tikzscale]
	\node[seltwoEEEEEEE, label={[shift={(0.5,-0.65)}, brown]\tiny $R_{0}^{1}(1)$}] (RN_2_1_1__Rationals) at (4,0) {$\begin{smallmatrix}&&0\\&&1\\0&0&1&1&1\end{smallmatrix}$};
	\node[seltwoEEEEEEE, label={[shift={(0.5,-0.65)}, brown]\tiny $R_{0}^{2}(1)$}] (RN_2_2_1__Rationals) at (8,0) {$\begin{smallmatrix}&&1\\&&1\\0&1&1&0&0\end{smallmatrix}$};
	\node[seltwoEEEEEEE, label={[shift={(0.5,-0.65)}, brown]\tiny $R_{0}^{3}(1)$}] (RN_2_3_1__Rationals) at (12,0) {$\begin{smallmatrix}&&0\\&&0\\1&1&1&1&0\end{smallmatrix}$};
	\node[seltwoEEEEEEE, label={[shift={(0.5,-0.65)}, brown]\tiny $R_{0}^{1}(1)$}] (RN_2_1_1__Rationals_repeated) at (16,0) {$\begin{smallmatrix}&&0\\&&1\\0&0&1&1&1\end{smallmatrix}$};
	\node[seltwoEEEEEEE, label={[shift={(0.5,-0.65)}, brown]\tiny $R_{0}^{1}(2)$}] (RN_2_1_2__Rationals) at (6,2) {$\begin{smallmatrix}&&1\\&&2\\0&1&2&1&1\end{smallmatrix}$};
	\node[seltwoEEEEEEE, label={[shift={(0.5,-0.65)}, brown]\tiny $R_{0}^{2}(2)$}] (RN_2_2_2__Rationals) at (10,2) {$\begin{smallmatrix}&&1\\&&1\\1&2&2&1&0\end{smallmatrix}$};
	\node[seltwoEEEEEEE, label={[shift={(0.5,-0.65)}, brown]\tiny $R_{0}^{3}(2)$}] (RN_2_3_2__Rationals) at (14,2) {$\begin{smallmatrix}&&0\\&&1\\1&1&2&2&1\end{smallmatrix}$};
	\node[noselEEEEEEE, label={[shift={(0.5,-0.65)}, brown]\tiny $R_{0}^{3}(3)$}] (RN_2_3_3__Rationals) at (4,4) {$\begin{smallmatrix}&&1\\&&2\\1&2&3&2&1\end{smallmatrix}$};
	\node[noselEEEEEEE, label={[shift={(0.5,-0.65)}, brown]\tiny $R_{0}^{1}(3)$}] (RN_2_1_3__Rationals) at (8,4) {$\begin{smallmatrix}&&1\\&&2\\1&2&3&2&1\end{smallmatrix}$};
	\node[noselEEEEEEE, label={[shift={(0.5,-0.65)}, brown]\tiny $R_{0}^{2}(3)$}] (RN_2_2_3__Rationals) at (12,4) {$\begin{smallmatrix}&&1\\&&2\\1&2&3&2&1\end{smallmatrix}$};
	\node[noselEEEEEEE, label={[shift={(0.5,-0.65)}, brown]\tiny $R_{0}^{3}(3)$}] (RN_2_3_3__Rationals_repeated) at (16,4) {$\begin{smallmatrix}&&1\\&&2\\1&2&3&2&1\end{smallmatrix}$};
	\node[noselEEEEEEE, label={[shift={(0.5,-0.65)}, brown]\tiny $R_{0}^{3}(4)$}] (RN_2_3_4__Rationals) at (6,6) {$\begin{smallmatrix}&&1\\&&2\\2&3&4&3&1\end{smallmatrix}$};
	\node[noselEEEEEEE, label={[shift={(0.5,-0.65)}, brown]\tiny $R_{0}^{1}(4)$}] (RN_2_1_4__Rationals) at (10,6) {$\begin{smallmatrix}&&1\\&&3\\1&2&4&3&2\end{smallmatrix}$};
	\node[noselEEEEEEE, label={[shift={(0.5,-0.65)}, brown]\tiny $R_{0}^{2}(4)$}] (RN_2_2_4__Rationals) at (14,6) {$\begin{smallmatrix}&&2\\&&3\\1&3&4&2&1\end{smallmatrix}$};
	\draw[blue,->] (RN_2_1_1__Rationals) -- (RN_2_1_2__Rationals);
	\draw[gray,dashed,->] (RN_2_1_1__Rationals) -- (RN_2_2_1__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[red,->] (RN_2_1_2__Rationals) -- (RN_2_2_1__Rationals);
	\draw[gray,dashed,->] (RN_2_1_2__Rationals) -- (RN_2_2_2__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[blue,->] (RN_2_1_2__Rationals) -- (RN_2_1_3__Rationals);
	\draw[blue,->] (RN_2_2_1__Rationals) -- (RN_2_2_2__Rationals);
	\draw[gray,dashed,->] (RN_2_2_1__Rationals) -- (RN_2_3_1__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[red,->] (RN_2_2_2__Rationals) -- (RN_2_3_1__Rationals);
	\draw[gray,dashed,->] (RN_2_2_2__Rationals) -- (RN_2_3_2__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[blue,->] (RN_2_2_2__Rationals) -- (RN_2_2_3__Rationals);
	\draw[gray,dashed,->] (RN_2_3_1__Rationals) -- (RN_2_1_1__Rationals_repeated) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[blue,->] (RN_2_3_1__Rationals) -- (RN_2_3_2__Rationals);
	\draw[red,->] (RN_2_3_2__Rationals) -- (RN_2_1_1__Rationals_repeated);
	\draw[gray,dashed,->,out=200,in=-20] (RN_2_3_2__Rationals) to node[midway,sloped,above]{{\tiny $\tau^{-1}$}} (RN_2_1_2__Rationals);
	\draw[blue,->] (RN_2_3_2__Rationals) -- (RN_2_3_3__Rationals_repeated);
	\draw[red,->] (RN_2_3_3__Rationals) -- (RN_2_1_2__Rationals);
	\draw[blue,->] (RN_2_3_3__Rationals) -- (RN_2_3_4__Rationals);
	\draw[gray,dashed,->] (RN_2_3_3__Rationals) -- (RN_2_1_3__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[red,->] (RN_2_3_4__Rationals) -- (RN_2_1_3__Rationals);
	\draw[gray,dashed,->] (RN_2_3_4__Rationals) -- (RN_2_1_4__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[red,->] (RN_2_1_3__Rationals) -- (RN_2_2_2__Rationals);
	\draw[blue,->] (RN_2_1_3__Rationals) -- (RN_2_1_4__Rationals);
	\draw[gray,dashed,->] (RN_2_1_3__Rationals) -- (RN_2_2_3__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[red,->] (RN_2_1_4__Rationals) -- (RN_2_2_3__Rationals);
	\draw[gray,dashed,->] (RN_2_1_4__Rationals) -- (RN_2_2_4__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[red,->] (RN_2_2_3__Rationals) -- (RN_2_3_2__Rationals);
	\draw[gray,dashed,->] (RN_2_2_3__Rationals) -- (RN_2_3_3__Rationals_repeated) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[blue,->] (RN_2_2_3__Rationals) -- (RN_2_2_4__Rationals);
	\draw[red,->] (RN_2_2_4__Rationals) -- (RN_2_3_3__Rationals_repeated);
	\draw[gray,dashed,->,out=200,in=-20] (RN_2_2_4__Rationals) to node[midway,sloped,above]{{\tiny $\tau^{-1}$}} (RN_2_3_4__Rationals);
	\draw[dashed,-] (RN_2_1_1__Rationals) -- (RN_2_3_3__Rationals);
	\draw[dashed,-] (RN_2_1_1__Rationals_repeated) -- (RN_2_3_3__Rationals_repeated);
	\node (LeftTopPhantom_2) at (4,6) {};
	\draw[dashed,-] (RN_2_3_3__Rationals) -- (LeftTopPhantom_2);
	\node (RightTopPhantom_2) at (16,6) {};
	\draw[dashed,-] (RN_2_3_3__Rationals_repeated) -- (RightTopPhantom_2);
\end{tikzpicture}
\end{scaletikzpicturetowidth}
\end{center}

\end{figure}
\begin{fleqn}
\begin{align*}
R_{0}^{1}(1):\ & \bigl(I(1,7),P(0,5)\bigr),\ \bigl(I(0,4),P(0,6)\bigr),\ \bigl(I(0,5),P(1,1)\bigr)\\
R_{0}^{1}(2):\ & \bigl(R_{0}^{2}(1),R_{0}^{1}(1)\bigr),\ \bigl(I(3,5),P(0,7)\bigr),\ \bigl(I(2,1),P(1,5)\bigr),\ \bigl(I(1,7),P(2,1)\bigr),\ \bigl(I(0,5),P(3,7)\bigr)\\
R_{0}^{2}(1):\ & \bigl(I(0,6),P(0,2)\bigr),\ \bigl(I(1,1),P(0,7)\bigr),\ \bigl(I(0,7),P(1,5)\bigr)\\
R_{0}^{2}(2):\ & \bigl(R_{0}^{3}(1),R_{0}^{2}(1)\bigr),\ \bigl(I(3,7),P(0,1)\bigr),\ \bigl(I(2,5),P(1,7)\bigr),\ \bigl(I(1,1),P(2,5)\bigr),\ \bigl(I(0,7),P(3,1)\bigr)\\
R_{0}^{3}(1):\ & \bigl(I(1,5),P(0,1)\bigr),\ \bigl(I(0,2),P(0,4)\bigr),\ \bigl(I(0,1),P(1,7)\bigr)\\
R_{0}^{3}(2):\ & \bigl(R_{0}^{1}(1),R_{0}^{3}(1)\bigr),\ \bigl(I(3,1),P(0,5)\bigr),\ \bigl(I(2,7),P(1,1)\bigr),\ \bigl(I(1,5),P(2,7)\bigr),\ \bigl(I(0,1),P(3,5)\bigr)\\
\end{align*}
\end{fleqn}
\subsubsection*{The non-homogeneous tube $\mathcal{T}_{\infty}^{\Delta(\widetilde{\mathbb{E}}_{6})}$}

\begin{figure}[ht]

\begin{center}
\captionof{figure}{\vspace*{-10pt}$\mathcal{T}_{\infty}^{\Delta(\widetilde{\mathbb{E}}_{6})}$}
\begin{scaletikzpicturetowidth}{0.59999999999999998\textwidth}
\begin{tikzpicture}[scale=\tikzscale]
	\node[seltwoEEEEEEE, label={[shift={(0.5,-0.65)}, brown]\tiny $R_{\infty}^{1}(1)$}] (RN_3_1_1__Rationals) at (4,0) {$\begin{smallmatrix}&&0\\&&1\\0&1&1&1&0\end{smallmatrix}$};
	\node[seltwoEEEEEEE, label={[shift={(0.5,-0.65)}, brown]\tiny $R_{\infty}^{2}(1)$}] (RN_3_2_1__Rationals) at (8,0) {$\begin{smallmatrix}&&1\\&&1\\1&1&2&1&1\end{smallmatrix}$};
	\node[seltwoEEEEEEE, label={[shift={(0.5,-0.65)}, brown]\tiny $R_{\infty}^{1}(1)$}] (RN_3_1_1__Rationals_repeated) at (12,0) {$\begin{smallmatrix}&&0\\&&1\\0&1&1&1&0\end{smallmatrix}$};
	\node[noselEEEEEEE, label={[shift={(0.5,-0.65)}, brown]\tiny $R_{\infty}^{1}(2)$}] (RN_3_1_2__Rationals) at (6,2) {$\begin{smallmatrix}&&1\\&&2\\1&2&3&2&1\end{smallmatrix}$};
	\node[noselEEEEEEE, label={[shift={(0.5,-0.65)}, brown]\tiny $R_{\infty}^{2}(2)$}] (RN_3_2_2__Rationals) at (10,2) {$\begin{smallmatrix}&&1\\&&2\\1&2&3&2&1\end{smallmatrix}$};
	\node[noselEEEEEEE, label={[shift={(0.5,-0.65)}, brown]\tiny $R_{\infty}^{2}(3)$}] (RN_3_2_3__Rationals) at (4,4) {$\begin{smallmatrix}&&2\\&&3\\2&3&5&3&2\end{smallmatrix}$};
	\node[noselEEEEEEE, label={[shift={(0.5,-0.65)}, brown]\tiny $R_{\infty}^{1}(3)$}] (RN_3_1_3__Rationals) at (8,4) {$\begin{smallmatrix}&&1\\&&3\\1&3&4&3&1\end{smallmatrix}$};
	\node[noselEEEEEEE, label={[shift={(0.5,-0.65)}, brown]\tiny $R_{\infty}^{2}(3)$}] (RN_3_2_3__Rationals_repeated) at (12,4) {$\begin{smallmatrix}&&2\\&&3\\2&3&5&3&2\end{smallmatrix}$};
	\draw[blue,->] (RN_3_1_1__Rationals) -- (RN_3_1_2__Rationals);
	\draw[gray,dashed,->] (RN_3_1_1__Rationals) -- (RN_3_2_1__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[red,->] (RN_3_1_2__Rationals) -- (RN_3_2_1__Rationals);
	\draw[gray,dashed,->] (RN_3_1_2__Rationals) -- (RN_3_2_2__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[blue,->] (RN_3_1_2__Rationals) -- (RN_3_1_3__Rationals);
	\draw[gray,dashed,->] (RN_3_2_1__Rationals) -- (RN_3_1_1__Rationals_repeated) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[blue,->] (RN_3_2_1__Rationals) -- (RN_3_2_2__Rationals);
	\draw[red,->] (RN_3_2_2__Rationals) -- (RN_3_1_1__Rationals_repeated);
	\draw[gray,dashed,->,out=219,in=-39] (RN_3_2_2__Rationals) to node[midway,sloped,above]{{\tiny $\tau^{-1}$}} (RN_3_1_2__Rationals);
	\draw[blue,->] (RN_3_2_2__Rationals) -- (RN_3_2_3__Rationals_repeated);
	\draw[red,->] (RN_3_2_3__Rationals) -- (RN_3_1_2__Rationals);
	\draw[gray,dashed,->] (RN_3_2_3__Rationals) -- (RN_3_1_3__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[red,->] (RN_3_1_3__Rationals) -- (RN_3_2_2__Rationals);
	\draw[gray,dashed,->] (RN_3_1_3__Rationals) -- (RN_3_2_3__Rationals_repeated) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[dashed,-] (RN_3_1_1__Rationals) -- (RN_3_2_3__Rationals);
	\draw[dashed,-] (RN_3_1_1__Rationals_repeated) -- (RN_3_2_3__Rationals_repeated);
\end{tikzpicture}
\end{scaletikzpicturetowidth}
\end{center}

\end{figure}
\begin{fleqn}
\begin{align*}
R_{\infty}^{1}(1):\ & \bigl(I(1,1),P(1,1)\bigr),\ \bigl(I(1,5),P(1,5)\bigr),\ \bigl(I(1,7),P(1,7)\bigr)\\
R_{\infty}^{2}(1):\ & \bigl(I(2,1),P(0,1)\bigr),\ \bigl(I(2,5),P(0,5)\bigr),\ \bigl(I(2,7),P(0,7)\bigr),\ \bigl(I(0,1),P(2,1)\bigr),\ \bigl(I(0,5),P(2,5)\bigr)\\
 & \bigl(I(0,7),P(2,7)\bigr)\\
\end{align*}
\end{fleqn}
\clearpage

\subsection[Schofield pairs for the quiver $\Delta(\widetilde{\mathbb{E}}_7)$]{Schofield pairs for the quiver $\Delta(\widetilde{\mathbb{E}}_7)$ -- {\usebox{\boxEEEEEEEE}}}
\[
\xymatrix{  &   &   & 8\ar[d] &   &   &  \\
1\ar[r] & 2\ar[r] & 3\ar[r] & 4 & 5\ar[l] & 6\ar[l] & 7\ar[l]}
\]
\medskip{}
\[
 C_{\Delta(\widetilde{\mathbb{E}}_{7})} = \begin{bmatrix}1 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\
1 & 1 & 0 & 0 & 0 & 0 & 0 & 0\\
1 & 1 & 1 & 0 & 0 & 0 & 0 & 0\\
1 & 1 & 1 & 1 & 1 & 1 & 1 & 1\\
0 & 0 & 0 & 0 & 1 & 1 & 1 & 0\\
0 & 0 & 0 & 0 & 0 & 1 & 1 & 0\\
0 & 0 & 0 & 0 & 0 & 0 & 1 & 0\\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 1\end{bmatrix}\qquad\Phi_{\Delta(\widetilde{\mathbb{E}}_{7})} = \begin{bmatrix}0 & 0 & 0 & -1 & 1 & 0 & 0 & 1\\
1 & 0 & 0 & -1 & 1 & 0 & 0 & 1\\
0 & 1 & 0 & -1 & 1 & 0 & 0 & 1\\
0 & 0 & 1 & -1 & 1 & 0 & 0 & 1\\
0 & 0 & 1 & -1 & 0 & 1 & 0 & 1\\
0 & 0 & 1 & -1 & 0 & 0 & 1 & 1\\
0 & 0 & 1 & -1 & 0 & 0 & 0 & 1\\
0 & 0 & 1 & -1 & 1 & 0 & 0 & 0\end{bmatrix}
\]

\medskip{}\subsubsection{Schofield pairs associated to preprojective exceptional modules}

\begin{figure}[ht]

\begin{center}
\begin{scaletikzpicturetowidth}{\textwidth}
\begin{tikzpicture}[scale=\tikzscale]
	\node[noselEEEEEEEE, label={[shift={(0.45,-0.55)}, brown]\tiny $P(0,1)$}] (P_0_1__Rationals) at (6,9) {$\begin{smallmatrix}&&&0\\1&1&1&1&0&0&0\end{smallmatrix}$};
	\node[noselEEEEEEEE, label={[shift={(0.45,-0.55)}, brown]\tiny $P(0,2)$}] (P_0_2__Rationals) at (4,6) {$\begin{smallmatrix}&&&0\\0&1&1&1&0&0&0\end{smallmatrix}$};
	\node[noselEEEEEEEE, label={[shift={(0.45,-0.55)}, brown]\tiny $P(0,3)$}] (P_0_3__Rationals) at (2,3) {$\begin{smallmatrix}&&&0\\0&0&1&1&0&0&0\end{smallmatrix}$};
	\node[noselEEEEEEEE, label={[shift={(0.45,-0.55)}, brown]\tiny $P(0,4)$}] (P_0_4__Rationals) at (0,0) {$\begin{smallmatrix}&&&0\\0&0&0&1&0&0&0\end{smallmatrix}$};
	\node[noselEEEEEEEE, label={[shift={(0.45,-0.55)}, brown]\tiny $P(0,5)$}] (P_0_5__Rationals) at (2,-3) {$\begin{smallmatrix}&&&0\\0&0&0&1&1&0&0\end{smallmatrix}$};
	\node[noselEEEEEEEE, label={[shift={(0.45,-0.55)}, brown]\tiny $P(0,6)$}] (P_0_6__Rationals) at (4,-6) {$\begin{smallmatrix}&&&0\\0&0&0&1&1&1&0\end{smallmatrix}$};
	\node[noselEEEEEEEE, label={[shift={(0.45,-0.55)}, brown]\tiny $P(0,7)$}] (P_0_7__Rationals) at (6,-9) {$\begin{smallmatrix}&&&0\\0&0&0&1&1&1&1\end{smallmatrix}$};
	\node[noselEEEEEEEE, label={[shift={(0.45,-0.55)}, brown]\tiny $P(0,8)$}] (P_0_8__Rationals) at (2,1) {$\begin{smallmatrix}&&&1\\0&0&0&1&0&0&0\end{smallmatrix}$};
	\node[noselEEEEEEEE, label={[shift={(0.45,-0.55)}, brown]\tiny $P(1,1)$}] (P_1_1__Rationals) at (10,9) {$\begin{smallmatrix}&&&1\\0&0&0&1&1&0&0\end{smallmatrix}$};
	\node[noselEEEEEEEE, label={[shift={(0.45,-0.55)}, brown]\tiny $P(1,2)$}] (P_1_2__Rationals) at (8,6) {$\begin{smallmatrix}&&&1\\1&1&1&2&1&0&0\end{smallmatrix}$};
	\node[noselEEEEEEEE, label={[shift={(0.45,-0.55)}, brown]\tiny $P(1,3)$}] (P_1_3__Rationals) at (6,3) {$\begin{smallmatrix}&&&1\\0&1&1&2&1&0&0\end{smallmatrix}$};
	\node[noselEEEEEEEE, label={[shift={(0.45,-0.55)}, brown]\tiny $P(1,4)$}] (P_1_4__Rationals) at (4,0) {$\begin{smallmatrix}&&&1\\0&0&1&2&1&0&0\end{smallmatrix}$};
	\node[noselEEEEEEEE, label={[shift={(0.45,-0.55)}, brown]\tiny $P(1,5)$}] (P_1_5__Rationals) at (6,-3) {$\begin{smallmatrix}&&&1\\0&0&1&2&1&1&0\end{smallmatrix}$};
	\node[noselEEEEEEEE, label={[shift={(0.45,-0.55)}, brown]\tiny $P(1,6)$}] (P_1_6__Rationals) at (8,-6) {$\begin{smallmatrix}&&&1\\0&0&1&2&1&1&1\end{smallmatrix}$};
	\node[noselEEEEEEEE, label={[shift={(0.45,-0.55)}, brown]\tiny $P(1,7)$}] (P_1_7__Rationals) at (10,-9) {$\begin{smallmatrix}&&&1\\0&0&1&1&0&0&0\end{smallmatrix}$};
	\node[noselEEEEEEEE, label={[shift={(0.45,-0.55)}, brown]\tiny $P(1,8)$}] (P_1_8__Rationals) at (6,1) {$\begin{smallmatrix}&&&0\\0&0&1&1&1&0&0\end{smallmatrix}$};
	\node[noselEEEEEEEE, label={[shift={(0.45,-0.55)}, brown]\tiny $P(2,1)$}] (P_2_1__Rationals) at (14,9) {$\begin{smallmatrix}&&&0\\0&0&1&1&1&1&0\end{smallmatrix}$};
	\node[noselEEEEEEEE, label={[shift={(0.45,-0.55)}, brown]\tiny $P(2,2)$}] (P_2_2__Rationals) at (12,6) {$\begin{smallmatrix}&&&1\\0&0&1&2&2&1&0\end{smallmatrix}$};
	\node[noselEEEEEEEE, label={[shift={(0.45,-0.55)}, brown]\tiny $P(2,3)$}] (P_2_3__Rationals) at (10,3) {$\begin{smallmatrix}&&&1\\1&1&2&3&2&1&0\end{smallmatrix}$};
	\node[noselEEEEEEEE, label={[shift={(0.45,-0.55)}, brown]\tiny $P(2,4)$}] (P_2_4__Rationals) at (8,0) {$\begin{smallmatrix}&&&1\\0&1&2&3&2&1&0\end{smallmatrix}$};
	\node[noselEEEEEEEE, label={[shift={(0.45,-0.55)}, brown]\tiny $P(2,5)$}] (P_2_5__Rationals) at (10,-3) {$\begin{smallmatrix}&&&1\\0&1&2&3&2&1&1\end{smallmatrix}$};
	\node[noselEEEEEEEE, label={[shift={(0.45,-0.55)}, brown]\tiny $P(2,6)$}] (P_2_6__Rationals) at (12,-6) {$\begin{smallmatrix}&&&1\\0&1&2&2&1&0&0\end{smallmatrix}$};
	\node[noselEEEEEEEE, label={[shift={(0.45,-0.55)}, brown]\tiny $P(2,7)$}] (P_2_7__Rationals) at (14,-9) {$\begin{smallmatrix}&&&0\\0&1&1&1&1&0&0\end{smallmatrix}$};
	\node[noselEEEEEEEE, label={[shift={(0.45,-0.55)}, brown]\tiny $P(2,8)$}] (P_2_8__Rationals) at (10,1) {$\begin{smallmatrix}&&&1\\0&1&1&2&1&1&0\end{smallmatrix}$};
	\node[noselEEEEEEEE, label={[shift={(0.45,-0.55)}, brown]\tiny $P(3,1)$}] (P_3_1__Rationals) at (18,9) {$\begin{smallmatrix}&&&1\\0&1&1&2&1&1&1\end{smallmatrix}$};
	\node[noselEEEEEEEE, label={[shift={(0.45,-0.55)}, brown]\tiny $P(3,2)$}] (P_3_2__Rationals) at (16,6) {$\begin{smallmatrix}&&&1\\0&1&2&3&2&2&1\end{smallmatrix}$};
	\node[noselEEEEEEEE, label={[shift={(0.45,-0.55)}, brown]\tiny $P(3,3)$}] (P_3_3__Rationals) at (14,3) {$\begin{smallmatrix}&&&2\\0&1&2&4&3&2&1\end{smallmatrix}$};
	\node[seloneEEEEEEEE, label={[shift={(0.45,-0.55)}, brown]\tiny $P(3,4)$}] (P_3_4__Rationals) at (12,0) {$\begin{smallmatrix}&&&2\\1&2&3&5&3&2&1\end{smallmatrix}$};
	\node[noselEEEEEEEE, label={[shift={(0.45,-0.55)}, brown]\tiny $P(3,5)$}] (P_3_5__Rationals) at (14,-3) {$\begin{smallmatrix}&&&2\\1&2&3&4&2&1&0\end{smallmatrix}$};
	\node[noselEEEEEEEE, label={[shift={(0.45,-0.55)}, brown]\tiny $P(3,6)$}] (P_3_6__Rationals) at (16,-6) {$\begin{smallmatrix}&&&1\\1&2&2&3&2&1&0\end{smallmatrix}$};
	\node[noselEEEEEEEE, label={[shift={(0.45,-0.55)}, brown]\tiny $P(3,7)$}] (P_3_7__Rationals) at (18,-9) {$\begin{smallmatrix}&&&1\\1&1&1&2&1&1&0\end{smallmatrix}$};
	\node[noselEEEEEEEE, label={[shift={(0.45,-0.55)}, brown]\tiny $P(3,8)$}] (P_3_8__Rationals) at (14,1) {$\begin{smallmatrix}&&&1\\1&1&2&3&2&1&1\end{smallmatrix}$};
	\node[noselEEEEEEEE, label={[shift={(0.45,-0.55)}, brown]\tiny $P(4,1)$}] (P_4_1__Rationals) at (22,9) {$\begin{smallmatrix}&&&1\\1&1&2&2&1&0&0\end{smallmatrix}$};
	\node[noselEEEEEEEE, label={[shift={(0.45,-0.55)}, brown]\tiny $P(4,2)$}] (P_4_2__Rationals) at (20,6) {$\begin{smallmatrix}&&&2\\1&2&3&4&2&1&1\end{smallmatrix}$};
	\node[seloneEEEEEEEE, label={[shift={(0.45,-0.55)}, brown]\tiny $P(4,3)$}] (P_4_3__Rationals) at (18,3) {$\begin{smallmatrix}&&&2\\1&2&4&5&3&2&1\end{smallmatrix}$};
	\node[noselEEEEEEEE, label={[shift={(0.45,-0.55)}, brown]\tiny $P(4,4)$}] (P_4_4__Rationals) at (16,0) {$\begin{smallmatrix}&&&3\\1&2&4&6&4&2&1\end{smallmatrix}$};
	\node[seloneEEEEEEEE, label={[shift={(0.45,-0.55)}, brown]\tiny $P(4,5)$}] (P_4_5__Rationals) at (18,-3) {$\begin{smallmatrix}&&&2\\1&2&3&5&4&2&1\end{smallmatrix}$};
	\node[noselEEEEEEEE, label={[shift={(0.45,-0.55)}, brown]\tiny $P(4,6)$}] (P_4_6__Rationals) at (20,-6) {$\begin{smallmatrix}&&&2\\1&1&2&4&3&2&1\end{smallmatrix}$};
	\node[noselEEEEEEEE, label={[shift={(0.45,-0.55)}, brown]\tiny $P(4,7)$}] (P_4_7__Rationals) at (22,-9) {$\begin{smallmatrix}&&&1\\0&0&1&2&2&1&1\end{smallmatrix}$};
	\node[noselEEEEEEEE, label={[shift={(0.45,-0.55)}, brown]\tiny $P(4,8)$}] (P_4_8__Rationals) at (18,1) {$\begin{smallmatrix}&&&2\\0&1&2&3&2&1&0\end{smallmatrix}$};
	\node at (20,3) {$\dots$};
	\node at (20,-3) {$\dots$};
	\draw[blue,->] (P_0_2__Rationals) -- (P_0_1__Rationals);
	\draw[blue,->] (P_0_3__Rationals) -- (P_0_2__Rationals);
	\draw[blue,->] (P_0_4__Rationals) -- (P_0_3__Rationals);
	\draw[blue,->] (P_0_4__Rationals) -- (P_0_5__Rationals);
	\draw[blue,->] (P_0_5__Rationals) -- (P_0_6__Rationals);
	\draw[blue,->] (P_0_6__Rationals) -- (P_0_7__Rationals);
	\draw[blue,->] (P_0_4__Rationals) -- (P_0_8__Rationals);
	\draw[gray,dashed,->] (P_0_1__Rationals) -- (P_1_1__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[red,->] (P_1_2__Rationals) -- (P_1_1__Rationals);
	\draw[blue,->] (P_0_1__Rationals) -- (P_1_2__Rationals);
	\draw[gray,dashed,->] (P_0_2__Rationals) -- (P_1_2__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[blue,->] (P_1_3__Rationals) -- (P_1_2__Rationals);
	\draw[blue,->] (P_0_2__Rationals) -- (P_1_3__Rationals);
	\draw[gray,dashed,->] (P_0_3__Rationals) -- (P_1_3__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[blue,->] (P_1_4__Rationals) -- (P_1_3__Rationals);
	\draw[blue,->] (P_0_3__Rationals) -- (P_1_4__Rationals);
	\draw[gray,dashed,->] (P_0_4__Rationals) -- (P_1_4__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[blue,->] (P_0_5__Rationals) -- (P_1_4__Rationals);
	\draw[blue,->] (P_0_8__Rationals) -- (P_1_4__Rationals);
	\draw[gray,dashed,->] (P_0_5__Rationals) -- (P_1_5__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[blue,->] (P_0_6__Rationals) -- (P_1_5__Rationals);
	\draw[blue,->] (P_1_4__Rationals) -- (P_1_5__Rationals);
	\draw[gray,dashed,->] (P_0_6__Rationals) -- (P_1_6__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[blue,->] (P_0_7__Rationals) -- (P_1_6__Rationals);
	\draw[blue,->] (P_1_5__Rationals) -- (P_1_6__Rationals);
	\draw[gray,dashed,->] (P_0_7__Rationals) -- (P_1_7__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[red,->] (P_1_6__Rationals) -- (P_1_7__Rationals);
	\draw[gray,dashed,->] (P_0_8__Rationals) -- (P_1_8__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[red,->] (P_1_4__Rationals) -- (P_1_8__Rationals);
	\draw[gray,dashed,->] (P_1_1__Rationals) -- (P_2_1__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[red,->] (P_2_2__Rationals) -- (P_2_1__Rationals);
	\draw[blue,->] (P_1_1__Rationals) -- (P_2_2__Rationals);
	\draw[gray,dashed,->] (P_1_2__Rationals) -- (P_2_2__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[red,->] (P_2_3__Rationals) -- (P_2_2__Rationals);
	\draw[blue,->] (P_1_2__Rationals) -- (P_2_3__Rationals);
	\draw[gray,dashed,->] (P_1_3__Rationals) -- (P_2_3__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[blue,->] (P_2_4__Rationals) -- (P_2_3__Rationals);
	\draw[blue,->] (P_1_3__Rationals) -- (P_2_4__Rationals);
	\draw[gray,dashed,->] (P_1_4__Rationals) -- (P_2_4__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[blue,->] (P_1_5__Rationals) -- (P_2_4__Rationals);
	\draw[blue,->] (P_1_8__Rationals) -- (P_2_4__Rationals);
	\draw[gray,dashed,->] (P_1_5__Rationals) -- (P_2_5__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[blue,->] (P_1_6__Rationals) -- (P_2_5__Rationals);
	\draw[blue,->] (P_2_4__Rationals) -- (P_2_5__Rationals);
	\draw[gray,dashed,->] (P_1_6__Rationals) -- (P_2_6__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[blue,->] (P_1_7__Rationals) -- (P_2_6__Rationals);
	\draw[red,->] (P_2_5__Rationals) -- (P_2_6__Rationals);
	\draw[gray,dashed,->] (P_1_7__Rationals) -- (P_2_7__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[red,->] (P_2_6__Rationals) -- (P_2_7__Rationals);
	\draw[gray,dashed,->] (P_1_8__Rationals) -- (P_2_8__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[red,->] (P_2_4__Rationals) -- (P_2_8__Rationals);
	\draw[gray,dashed,->] (P_2_1__Rationals) -- (P_3_1__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[red,->] (P_3_2__Rationals) -- (P_3_1__Rationals);
	\draw[blue,->] (P_2_1__Rationals) -- (P_3_2__Rationals);
	\draw[gray,dashed,->] (P_2_2__Rationals) -- (P_3_2__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[red,->] (P_3_3__Rationals) -- (P_3_2__Rationals);
	\draw[blue,->] (P_2_2__Rationals) -- (P_3_3__Rationals);
	\draw[gray,dashed,->] (P_2_3__Rationals) -- (P_3_3__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[red,->] (P_3_4__Rationals) -- (P_3_3__Rationals);
	\draw[blue,->] (P_2_3__Rationals) -- (P_3_4__Rationals);
	\draw[gray,dashed,->] (P_2_4__Rationals) -- (P_3_4__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[blue,->] (P_2_5__Rationals) -- (P_3_4__Rationals);
	\draw[blue,->] (P_2_8__Rationals) -- (P_3_4__Rationals);
	\draw[gray,dashed,->] (P_2_5__Rationals) -- (P_3_5__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[blue,->] (P_2_6__Rationals) -- (P_3_5__Rationals);
	\draw[red,->] (P_3_4__Rationals) -- (P_3_5__Rationals);
	\draw[gray,dashed,->] (P_2_6__Rationals) -- (P_3_6__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[blue,->] (P_2_7__Rationals) -- (P_3_6__Rationals);
	\draw[red,->] (P_3_5__Rationals) -- (P_3_6__Rationals);
	\draw[gray,dashed,->] (P_2_7__Rationals) -- (P_3_7__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[red,->] (P_3_6__Rationals) -- (P_3_7__Rationals);
	\draw[gray,dashed,->] (P_2_8__Rationals) -- (P_3_8__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[red,->] (P_3_4__Rationals) -- (P_3_8__Rationals);
	\draw[gray,dashed,->] (P_3_1__Rationals) -- (P_4_1__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[red,->] (P_4_2__Rationals) -- (P_4_1__Rationals);
	\draw[blue,->] (P_3_1__Rationals) -- (P_4_2__Rationals);
	\draw[gray,dashed,->] (P_3_2__Rationals) -- (P_4_2__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[red,->] (P_4_3__Rationals) -- (P_4_2__Rationals);
	\draw[blue,->] (P_3_2__Rationals) -- (P_4_3__Rationals);
	\draw[gray,dashed,->] (P_3_3__Rationals) -- (P_4_3__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[red,->] (P_4_4__Rationals) -- (P_4_3__Rationals);
	\draw[blue,->] (P_3_3__Rationals) -- (P_4_4__Rationals);
	\draw[gray,dashed,->] (P_3_4__Rationals) -- (P_4_4__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[blue,->] (P_3_5__Rationals) -- (P_4_4__Rationals);
	\draw[blue,->] (P_3_8__Rationals) -- (P_4_4__Rationals);
	\draw[gray,dashed,->] (P_3_5__Rationals) -- (P_4_5__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[blue,->] (P_3_6__Rationals) -- (P_4_5__Rationals);
	\draw[red,->] (P_4_4__Rationals) -- (P_4_5__Rationals);
	\draw[gray,dashed,->] (P_3_6__Rationals) -- (P_4_6__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[blue,->] (P_3_7__Rationals) -- (P_4_6__Rationals);
	\draw[red,->] (P_4_5__Rationals) -- (P_4_6__Rationals);
	\draw[gray,dashed,->] (P_3_7__Rationals) -- (P_4_7__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[red,->] (P_4_6__Rationals) -- (P_4_7__Rationals);
	\draw[gray,dashed,->] (P_3_8__Rationals) -- (P_4_8__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[red,->] (P_4_4__Rationals) -- (P_4_8__Rationals);
\end{tikzpicture}
\end{scaletikzpicturetowidth}
\end{center}

\end{figure}
\begin{figure}[ht]

\begin{center}
\begin{scaletikzpicturetowidth}{\textwidth}
\begin{tikzpicture}[scale=\tikzscale]
	\node at (0,3) {$\dots$};
	\node at (0,-3) {$\dots$};
	\node[noselEEEEEEEE, label={[shift={(0.45,-0.55)}, brown]\tiny $P(5,1)$}] (P_5_1__Rationals) at (6,9) {$\begin{smallmatrix}&&&1\\0&1&1&2&2&1&0\end{smallmatrix}$};
	\node[noselEEEEEEEE, label={[shift={(0.45,-0.55)}, brown]\tiny $P(5,2)$}] (P_5_2__Rationals) at (4,6) {$\begin{smallmatrix}&&&2\\1&2&3&4&3&1&0\end{smallmatrix}$};
	\node[noselEEEEEEEE, label={[shift={(0.45,-0.55)}, brown]\tiny $P(5,3)$}] (P_5_3__Rationals) at (2,3) {$\begin{smallmatrix}&&&3\\1&3&4&6&4&2&1\end{smallmatrix}$};
	\node[noselEEEEEEEE, label={[shift={(0.45,-0.55)}, brown]\tiny $P(5,4)$}] (P_5_4__Rationals) at (0,0) {$\begin{smallmatrix}&&&3\\1&3&5&7&5&3&1\end{smallmatrix}$};
	\node[noselEEEEEEEE, label={[shift={(0.45,-0.55)}, brown]\tiny $P(5,5)$}] (P_5_5__Rationals) at (2,-3) {$\begin{smallmatrix}&&&3\\1&2&4&6&4&3&1\end{smallmatrix}$};
	\node[noselEEEEEEEE, label={[shift={(0.45,-0.55)}, brown]\tiny $P(5,6)$}] (P_5_6__Rationals) at (4,-6) {$\begin{smallmatrix}&&&2\\0&1&3&4&3&2&1\end{smallmatrix}$};
	\node[noselEEEEEEEE, label={[shift={(0.45,-0.55)}, brown]\tiny $P(5,7)$}] (P_5_7__Rationals) at (6,-9) {$\begin{smallmatrix}&&&1\\0&1&2&2&1&1&0\end{smallmatrix}$};
	\node[seloneEEEEEEEE, label={[shift={(0.45,-0.55)}, brown]\tiny $P(5,8)$}] (P_5_8__Rationals) at (2,1) {$\begin{smallmatrix}&&&1\\1&2&3&4&3&2&1\end{smallmatrix}$};
	\node[noselEEEEEEEE, label={[shift={(0.45,-0.55)}, brown]\tiny $P(6,1)$}] (P_6_1__Rationals) at (10,9) {$\begin{smallmatrix}&&&1\\1&1&2&3&2&2&1\end{smallmatrix}$};
	\node[seloneEEEEEEEE, label={[shift={(0.45,-0.55)}, brown]\tiny $P(6,2)$}] (P_6_2__Rationals) at (8,6) {$\begin{smallmatrix}&&&2\\1&2&3&5&4&3&1\end{smallmatrix}$};
	\node[noselEEEEEEEE, label={[shift={(0.45,-0.55)}, brown]\tiny $P(6,3)$}] (P_6_3__Rationals) at (6,3) {$\begin{smallmatrix}&&&3\\2&3&5&7&5&3&1\end{smallmatrix}$};
	\node[noselEEEEEEEE, label={[shift={(0.45,-0.55)}, brown]\tiny $P(6,4)$}] (P_6_4__Rationals) at (4,0) {$\begin{smallmatrix}&&&4\\2&4&6&9&6&4&2\end{smallmatrix}$};
	\node[noselEEEEEEEE, label={[shift={(0.45,-0.55)}, brown]\tiny $P(6,5)$}] (P_6_5__Rationals) at (6,-3) {$\begin{smallmatrix}&&&3\\1&3&5&7&5&3&2\end{smallmatrix}$};
	\node[seloneEEEEEEEE, label={[shift={(0.45,-0.55)}, brown]\tiny $P(6,6)$}] (P_6_6__Rationals) at (8,-6) {$\begin{smallmatrix}&&&2\\1&3&4&5&3&2&1\end{smallmatrix}$};
	\node[noselEEEEEEEE, label={[shift={(0.45,-0.55)}, brown]\tiny $P(6,7)$}] (P_6_7__Rationals) at (10,-9) {$\begin{smallmatrix}&&&1\\1&2&2&3&2&1&1\end{smallmatrix}$};
	\node[noselEEEEEEEE, label={[shift={(0.45,-0.55)}, brown]\tiny $P(6,8)$}] (P_6_8__Rationals) at (6,1) {$\begin{smallmatrix}&&&3\\1&2&3&5&3&2&1\end{smallmatrix}$};
	\node[noselEEEEEEEE, label={[shift={(0.45,-0.55)}, brown]\tiny $P(7,1)$}] (P_7_1__Rationals) at (14,9) {$\begin{smallmatrix}&&&2\\0&1&2&3&2&1&1\end{smallmatrix}$};
	\node[noselEEEEEEEE, label={[shift={(0.45,-0.55)}, brown]\tiny $P(7,2)$}] (P_7_2__Rationals) at (12,6) {$\begin{smallmatrix}&&&3\\1&2&4&6&4&3&2\end{smallmatrix}$};
	\node[noselEEEEEEEE, label={[shift={(0.45,-0.55)}, brown]\tiny $P(7,3)$}] (P_7_3__Rationals) at (10,3) {$\begin{smallmatrix}&&&4\\1&3&5&8&6&4&2\end{smallmatrix}$};
	\node[noselEEEEEEEE, label={[shift={(0.45,-0.55)}, brown]\tiny $P(7,4)$}] (P_7_4__Rationals) at (8,0) {$\begin{smallmatrix}&&&5\\2&4&7&10&7&4&2\end{smallmatrix}$};
	\node[noselEEEEEEEE, label={[shift={(0.45,-0.55)}, brown]\tiny $P(7,5)$}] (P_7_5__Rationals) at (10,-3) {$\begin{smallmatrix}&&&4\\2&4&6&8&5&3&1\end{smallmatrix}$};
	\node[noselEEEEEEEE, label={[shift={(0.45,-0.55)}, brown]\tiny $P(7,6)$}] (P_7_6__Rationals) at (12,-6) {$\begin{smallmatrix}&&&3\\2&3&4&6&4&2&1\end{smallmatrix}$};
	\node[noselEEEEEEEE, label={[shift={(0.45,-0.55)}, brown]\tiny $P(7,7)$}] (P_7_7__Rationals) at (14,-9) {$\begin{smallmatrix}&&&2\\1&1&2&3&2&1&0\end{smallmatrix}$};
	\node[noselEEEEEEEE, label={[shift={(0.45,-0.55)}, brown]\tiny $P(7,8)$}] (P_7_8__Rationals) at (10,1) {$\begin{smallmatrix}&&&2\\1&2&4&5&4&2&1\end{smallmatrix}$};
	\node[noselEEEEEEEE, label={[shift={(0.45,-0.55)}, brown]\tiny $P(8,1)$}] (P_8_1__Rationals) at (18,9) {$\begin{smallmatrix}&&&1\\1&2&3&3&2&1&0\end{smallmatrix}$};
	\node[noselEEEEEEEE, label={[shift={(0.45,-0.55)}, brown]\tiny $P(8,2)$}] (P_8_2__Rationals) at (16,6) {$\begin{smallmatrix}&&&3\\1&3&5&6&4&2&1\end{smallmatrix}$};
	\node[noselEEEEEEEE, label={[shift={(0.45,-0.55)}, brown]\tiny $P(8,3)$}] (P_8_3__Rationals) at (14,3) {$\begin{smallmatrix}&&&4\\2&4&7&9&6&4&2\end{smallmatrix}$};
	\node[noselEEEEEEEE, label={[shift={(0.45,-0.55)}, brown]\tiny $P(8,4)$}] (P_8_4__Rationals) at (12,0) {$\begin{smallmatrix}&&&5\\2&5&8&11&8&5&2\end{smallmatrix}$};
	\node[noselEEEEEEEE, label={[shift={(0.45,-0.55)}, brown]\tiny $P(8,5)$}] (P_8_5__Rationals) at (14,-3) {$\begin{smallmatrix}&&&4\\2&4&6&9&7&4&2\end{smallmatrix}$};
	\node[noselEEEEEEEE, label={[shift={(0.45,-0.55)}, brown]\tiny $P(8,6)$}] (P_8_6__Rationals) at (16,-6) {$\begin{smallmatrix}&&&3\\1&2&4&6&5&3&1\end{smallmatrix}$};
	\node[noselEEEEEEEE, label={[shift={(0.45,-0.55)}, brown]\tiny $P(8,7)$}] (P_8_7__Rationals) at (18,-9) {$\begin{smallmatrix}&&&1\\0&1&2&3&3&2&1\end{smallmatrix}$};
	\node[noselEEEEEEEE, label={[shift={(0.45,-0.55)}, brown]\tiny $P(8,8)$}] (P_8_8__Rationals) at (14,1) {$\begin{smallmatrix}&&&3\\1&3&4&6&4&3&1\end{smallmatrix}$};
	\node[seloneEEEEEEEE, label={[shift={(0.45,-0.55)}, brown]\tiny $P(9,1)$}] (P_9_1__Rationals) at (22,9) {$\begin{smallmatrix}&&&2\\1&2&2&4&3&2&1\end{smallmatrix}$};
	\node[noselEEEEEEEE, label={[shift={(0.45,-0.55)}, brown]\tiny $P(9,2)$}] (P_9_2__Rationals) at (20,6) {$\begin{smallmatrix}&&&3\\2&4&5&7&5&3&1\end{smallmatrix}$};
	\node[noselEEEEEEEE, label={[shift={(0.45,-0.55)}, brown]\tiny $P(9,3)$}] (P_9_3__Rationals) at (18,3) {$\begin{smallmatrix}&&&5\\2&5&7&10&7&4&2\end{smallmatrix}$};
	\node[noselEEEEEEEE, label={[shift={(0.45,-0.55)}, brown]\tiny $P(9,4)$}] (P_9_4__Rationals) at (16,0) {$\begin{smallmatrix}&&&6\\3&6&9&13&9&6&3\end{smallmatrix}$};
	\node[noselEEEEEEEE, label={[shift={(0.45,-0.55)}, brown]\tiny $P(9,5)$}] (P_9_5__Rationals) at (18,-3) {$\begin{smallmatrix}&&&5\\2&4&7&10&7&5&2\end{smallmatrix}$};
	\node[noselEEEEEEEE, label={[shift={(0.45,-0.55)}, brown]\tiny $P(9,6)$}] (P_9_6__Rationals) at (20,-6) {$\begin{smallmatrix}&&&3\\1&3&5&7&5&4&2\end{smallmatrix}$};
	\node[seloneEEEEEEEE, label={[shift={(0.45,-0.55)}, brown]\tiny $P(9,7)$}] (P_9_7__Rationals) at (22,-9) {$\begin{smallmatrix}&&&2\\1&2&3&4&2&2&1\end{smallmatrix}$};
	\node[noselEEEEEEEE, label={[shift={(0.45,-0.55)}, brown]\tiny $P(9,8)$}] (P_9_8__Rationals) at (18,1) {$\begin{smallmatrix}&&&3\\2&3&5&7&5&3&2\end{smallmatrix}$};
	\node at (20,3) {$\dots$};
	\node at (20,-3) {$\dots$};
	\draw[red,->] (P_5_2__Rationals) -- (P_5_1__Rationals);
	\draw[red,->] (P_5_3__Rationals) -- (P_5_2__Rationals);
	\draw[red,->] (P_5_4__Rationals) -- (P_5_3__Rationals);
	\draw[red,->] (P_5_4__Rationals) -- (P_5_5__Rationals);
	\draw[red,->] (P_5_5__Rationals) -- (P_5_6__Rationals);
	\draw[red,->] (P_5_6__Rationals) -- (P_5_7__Rationals);
	\draw[red,->] (P_5_4__Rationals) -- (P_5_8__Rationals);
	\draw[gray,dashed,->] (P_5_1__Rationals) -- (P_6_1__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[red,->] (P_6_2__Rationals) -- (P_6_1__Rationals);
	\draw[blue,->] (P_5_1__Rationals) -- (P_6_2__Rationals);
	\draw[gray,dashed,->] (P_5_2__Rationals) -- (P_6_2__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[red,->] (P_6_3__Rationals) -- (P_6_2__Rationals);
	\draw[blue,->] (P_5_2__Rationals) -- (P_6_3__Rationals);
	\draw[gray,dashed,->] (P_5_3__Rationals) -- (P_6_3__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[red,->] (P_6_4__Rationals) -- (P_6_3__Rationals);
	\draw[blue,->] (P_5_3__Rationals) -- (P_6_4__Rationals);
	\draw[gray,dashed,->] (P_5_4__Rationals) -- (P_6_4__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[blue,->] (P_5_5__Rationals) -- (P_6_4__Rationals);
	\draw[blue,->] (P_5_8__Rationals) -- (P_6_4__Rationals);
	\draw[gray,dashed,->] (P_5_5__Rationals) -- (P_6_5__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[blue,->] (P_5_6__Rationals) -- (P_6_5__Rationals);
	\draw[red,->] (P_6_4__Rationals) -- (P_6_5__Rationals);
	\draw[gray,dashed,->] (P_5_6__Rationals) -- (P_6_6__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[blue,->] (P_5_7__Rationals) -- (P_6_6__Rationals);
	\draw[red,->] (P_6_5__Rationals) -- (P_6_6__Rationals);
	\draw[gray,dashed,->] (P_5_7__Rationals) -- (P_6_7__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[red,->] (P_6_6__Rationals) -- (P_6_7__Rationals);
	\draw[gray,dashed,->] (P_5_8__Rationals) -- (P_6_8__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[red,->] (P_6_4__Rationals) -- (P_6_8__Rationals);
	\draw[gray,dashed,->] (P_6_1__Rationals) -- (P_7_1__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[red,->] (P_7_2__Rationals) -- (P_7_1__Rationals);
	\draw[blue,->] (P_6_1__Rationals) -- (P_7_2__Rationals);
	\draw[gray,dashed,->] (P_6_2__Rationals) -- (P_7_2__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[red,->] (P_7_3__Rationals) -- (P_7_2__Rationals);
	\draw[blue,->] (P_6_2__Rationals) -- (P_7_3__Rationals);
	\draw[gray,dashed,->] (P_6_3__Rationals) -- (P_7_3__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[red,->] (P_7_4__Rationals) -- (P_7_3__Rationals);
	\draw[blue,->] (P_6_3__Rationals) -- (P_7_4__Rationals);
	\draw[gray,dashed,->] (P_6_4__Rationals) -- (P_7_4__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[blue,->] (P_6_5__Rationals) -- (P_7_4__Rationals);
	\draw[blue,->] (P_6_8__Rationals) -- (P_7_4__Rationals);
	\draw[gray,dashed,->] (P_6_5__Rationals) -- (P_7_5__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[blue,->] (P_6_6__Rationals) -- (P_7_5__Rationals);
	\draw[red,->] (P_7_4__Rationals) -- (P_7_5__Rationals);
	\draw[gray,dashed,->] (P_6_6__Rationals) -- (P_7_6__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[blue,->] (P_6_7__Rationals) -- (P_7_6__Rationals);
	\draw[red,->] (P_7_5__Rationals) -- (P_7_6__Rationals);
	\draw[gray,dashed,->] (P_6_7__Rationals) -- (P_7_7__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[red,->] (P_7_6__Rationals) -- (P_7_7__Rationals);
	\draw[gray,dashed,->] (P_6_8__Rationals) -- (P_7_8__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[red,->] (P_7_4__Rationals) -- (P_7_8__Rationals);
	\draw[gray,dashed,->] (P_7_1__Rationals) -- (P_8_1__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[red,->] (P_8_2__Rationals) -- (P_8_1__Rationals);
	\draw[blue,->] (P_7_1__Rationals) -- (P_8_2__Rationals);
	\draw[gray,dashed,->] (P_7_2__Rationals) -- (P_8_2__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[red,->] (P_8_3__Rationals) -- (P_8_2__Rationals);
	\draw[blue,->] (P_7_2__Rationals) -- (P_8_3__Rationals);
	\draw[gray,dashed,->] (P_7_3__Rationals) -- (P_8_3__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[red,->] (P_8_4__Rationals) -- (P_8_3__Rationals);
	\draw[blue,->] (P_7_3__Rationals) -- (P_8_4__Rationals);
	\draw[gray,dashed,->] (P_7_4__Rationals) -- (P_8_4__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[blue,->] (P_7_5__Rationals) -- (P_8_4__Rationals);
	\draw[blue,->] (P_7_8__Rationals) -- (P_8_4__Rationals);
	\draw[gray,dashed,->] (P_7_5__Rationals) -- (P_8_5__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[blue,->] (P_7_6__Rationals) -- (P_8_5__Rationals);
	\draw[red,->] (P_8_4__Rationals) -- (P_8_5__Rationals);
	\draw[gray,dashed,->] (P_7_6__Rationals) -- (P_8_6__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[blue,->] (P_7_7__Rationals) -- (P_8_6__Rationals);
	\draw[red,->] (P_8_5__Rationals) -- (P_8_6__Rationals);
	\draw[gray,dashed,->] (P_7_7__Rationals) -- (P_8_7__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[red,->] (P_8_6__Rationals) -- (P_8_7__Rationals);
	\draw[gray,dashed,->] (P_7_8__Rationals) -- (P_8_8__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[red,->] (P_8_4__Rationals) -- (P_8_8__Rationals);
	\draw[gray,dashed,->] (P_8_1__Rationals) -- (P_9_1__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[red,->] (P_9_2__Rationals) -- (P_9_1__Rationals);
	\draw[blue,->] (P_8_1__Rationals) -- (P_9_2__Rationals);
	\draw[gray,dashed,->] (P_8_2__Rationals) -- (P_9_2__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[red,->] (P_9_3__Rationals) -- (P_9_2__Rationals);
	\draw[blue,->] (P_8_2__Rationals) -- (P_9_3__Rationals);
	\draw[gray,dashed,->] (P_8_3__Rationals) -- (P_9_3__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[red,->] (P_9_4__Rationals) -- (P_9_3__Rationals);
	\draw[blue,->] (P_8_3__Rationals) -- (P_9_4__Rationals);
	\draw[gray,dashed,->] (P_8_4__Rationals) -- (P_9_4__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[blue,->] (P_8_5__Rationals) -- (P_9_4__Rationals);
	\draw[blue,->] (P_8_8__Rationals) -- (P_9_4__Rationals);
	\draw[gray,dashed,->] (P_8_5__Rationals) -- (P_9_5__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[blue,->] (P_8_6__Rationals) -- (P_9_5__Rationals);
	\draw[red,->] (P_9_4__Rationals) -- (P_9_5__Rationals);
	\draw[gray,dashed,->] (P_8_6__Rationals) -- (P_9_6__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[blue,->] (P_8_7__Rationals) -- (P_9_6__Rationals);
	\draw[red,->] (P_9_5__Rationals) -- (P_9_6__Rationals);
	\draw[gray,dashed,->] (P_8_7__Rationals) -- (P_9_7__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[red,->] (P_9_6__Rationals) -- (P_9_7__Rationals);
	\draw[gray,dashed,->] (P_8_8__Rationals) -- (P_9_8__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[red,->] (P_9_4__Rationals) -- (P_9_8__Rationals);
\end{tikzpicture}
\end{scaletikzpicturetowidth}
\end{center}

\end{figure}
\medskip{}
\subsubsection*{Modules of the form $P(n,1)$}

Defect: $\partial P(n,1) = -1$, for $n\ge 0$.

\begin{fleqn}
\begin{align*}
P(0,1):\ & \bigl(I(0,1),P(0,2)\bigr),\ \bigl(I(0,2),P(0,3)\bigr),\ \bigl(I(0,3),P(0,4)\bigr)\\
P(1,1):\ & \bigl(I(0,8),P(0,5)\bigr),\ \bigl(I(2,7),P(0,8)\bigr)\\
P(2,1):\ & \bigl(I(1,6),P(0,3)\bigr),\ \bigl(I(2,1),P(0,6)\bigr),\ \bigl(I(1,7),P(1,8)\bigr)\\
P(3,1):\ & \bigl(I(3,1),P(0,2)\bigr),\ \bigl(R_{1}^{3}(1),P(0,7)\bigr),\ \bigl(I(4,7),P(0,8)\bigr),\ \bigl(I(0,6),P(1,3)\bigr),\ \bigl(I(1,1),P(1,6)\bigr)\\
 & \bigl(I(0,7),P(2,8)\bigr)\\
P(4,1):\ & \bigl(R_{0}^{1}(1),P(0,1)\bigr),\ \bigl(I(2,1),P(1,2)\bigr),\ \bigl(R_{1}^{4}(1),P(1,7)\bigr),\ \bigl(I(3,7),P(1,8)\bigr),\ \bigl(I(0,1),P(2,6)\bigr)\\
P(5,1):\ & \bigl(I(5,1),P(0,6)\bigr),\ \bigl(R_{0}^{2}(1),P(1,1)\bigr),\ \bigl(I(1,1),P(2,2)\bigr),\ \bigl(R_{1}^{1}(1),P(2,7)\bigr),\ \bigl(I(2,7),P(2,8)\bigr)\\
P(6,1):\ & \bigl(R_{1}^{1}(2),P(0,1)\bigr),\ \bigl(R_{\infty}^{2}(1),P(0,7)\bigr),\ \bigl(I(4,1),P(1,6)\bigr),\ \bigl(R_{0}^{3}(1),P(2,1)\bigr),\ \bigl(I(0,1),P(3,2)\bigr)\\
 & \bigl(R_{1}^{2}(1),P(3,7)\bigr),\ \bigl(I(1,7),P(3,8)\bigr)\\
P(7,1):\ & \bigl(R_{1}^{2}(2),P(1,1)\bigr),\ \bigl(R_{\infty}^{1}(1),P(1,7)\bigr),\ \bigl(I(3,1),P(2,6)\bigr),\ \bigl(R_{0}^{1}(1),P(3,1)\bigr),\ \bigl(R_{1}^{3}(1),P(4,7)\bigr)\\
 & \bigl(I(0,7),P(4,8)\bigr)\\
P(8,1):\ & \bigl(R_{0}^{1}(2),P(0,1)\bigr),\ \bigl(R_{1}^{3}(2),P(2,1)\bigr),\ \bigl(R_{\infty}^{2}(1),P(2,7)\bigr),\ \bigl(I(2,1),P(3,6)\bigr),\ \bigl(R_{0}^{2}(1),P(4,1)\bigr)\\
 & \bigl(R_{1}^{4}(1),P(5,7)\bigr)\\
P(9,1):\ & \bigl(R_{1}^{3}(3),P(0,7)\bigr),\ \bigl(R_{0}^{2}(2),P(1,1)\bigr),\ \bigl(R_{1}^{4}(2),P(3,1)\bigr),\ \bigl(R_{\infty}^{1}(1),P(3,7)\bigr),\ \bigl(I(1,1),P(4,6)\bigr)\\
 & \bigl(R_{0}^{3}(1),P(5,1)\bigr),\ \bigl(R_{1}^{1}(1),P(6,7)\bigr)\\
P(n,1):\ & \bigl(R_{1}^{(n-7)\bmod 4+1}(3),P(n-9,7)\bigr),\ \bigl(R_{0}^{(n-8)\bmod 3+1}(2),P(n-8,1)\bigr),\ \bigl(R_{1}^{(n-6)\bmod 4+1}(2),P(n-6,1)\bigr)\\
 & \bigl(R_{\infty}^{(n-9)\bmod 2+1}(1),P(n-6,7)\bigr),\ \bigl(I(-n+10,1),P(n-5,6)\bigr),\ \bigl(R_{0}^{(n-7)\bmod 3+1}(1),P(n-4,1)\bigr)\\
 & \bigl(R_{1}^{(n-9)\bmod 4+1}(1),P(n-3,7)\bigr),\ n>9\\
\end{align*}
\end{fleqn}
\subsubsection*{Modules of the form $P(n,2)$}

Defect: $\partial P(n,2) = -2$, for $n\ge 0$.

\begin{fleqn}
\begin{align*}
P(0,2):\ & \bigl(I(1,1),P(0,3)\bigr),\ \bigl(I(1,2),P(0,4)\bigr)\\
P(1,2):\ & \bigl(P(1,1),P(0,1)\bigr),\ \bigl(I(3,7),P(0,5)\bigr),\ \bigl(R_{1}^{4}(1),P(0,8)\bigr),\ \bigl(I(0,1),P(1,3)\bigr),\ \bigl(I(0,2),P(1,4)\bigr)\\
P(2,2):\ & \bigl(R_{0}^{1}(1),P(0,6)\bigr),\ \bigl(P(2,1),P(1,1)\bigr),\ \bigl(I(2,7),P(1,5)\bigr),\ \bigl(R_{1}^{1}(1),P(1,8)\bigr)\\
P(3,2):\ & \bigl(R_{1}^{1}(2),P(0,2)\bigr),\ \bigl(P(5,7),P(0,7)\bigr),\ \bigl(R_{0}^{2}(1),P(1,6)\bigr),\ \bigl(P(3,1),P(2,1)\bigr),\ \bigl(I(1,7),P(2,5)\bigr)\\
 & \bigl(R_{1}^{2}(1),P(2,8)\bigr)\\
P(4,2):\ & \bigl(P(7,1),P(0,1)\bigr),\ \bigl(R_{1}^{2}(2),P(1,2)\bigr),\ \bigl(P(6,7),P(1,7)\bigr),\ \bigl(R_{0}^{3}(1),P(2,6)\bigr),\ \bigl(P(4,1),P(3,1)\bigr)\\
 & \bigl(I(0,7),P(3,5)\bigr),\ \bigl(R_{1}^{3}(1),P(3,8)\bigr)\\
P(5,2):\ & \bigl(P(8,1),P(1,1)\bigr),\ \bigl(R_{1}^{3}(2),P(2,2)\bigr),\ \bigl(P(7,7),P(2,7)\bigr),\ \bigl(R_{0}^{1}(1),P(3,6)\bigr),\ \bigl(P(5,1),P(4,1)\bigr)\\
 & \bigl(R_{1}^{4}(1),P(4,8)\bigr)\\
P(6,2):\ & \bigl(P(11,7),P(0,7)\bigr),\ \bigl(P(9,1),P(2,1)\bigr),\ \bigl(R_{1}^{4}(2),P(3,2)\bigr),\ \bigl(P(8,7),P(3,7)\bigr),\ \bigl(R_{0}^{2}(1),P(4,6)\bigr)\\
 & \bigl(P(6,1),P(5,1)\bigr),\ \bigl(R_{1}^{1}(1),P(5,8)\bigr)\\
P(n,2):\ & \bigl(P(n+5,7),P(n-6,7)\bigr),\ \bigl(P(n+3,1),P(n-4,1)\bigr),\ \bigl(R_{1}^{(n-3)\bmod 4+1}(2),P(n-3,2)\bigr)\\
 & \bigl(P(n+2,7),P(n-3,7)\bigr),\ \bigl(R_{0}^{(n-5)\bmod 3+1}(1),P(n-2,6)\bigr),\ \bigl(P(n,1),P(n-1,1)\bigr)\\
 & \bigl(R_{1}^{(n-6)\bmod 4+1}(1),P(n-1,8)\bigr),\ n>6\\
\end{align*}
\end{fleqn}
\subsubsection*{Modules of the form $P(n,3)$}

Defect: $\partial P(n,3) = -3$, for $n\ge 0$.

\begin{fleqn}
\begin{align*}
P(0,3):\ & \bigl(I(2,1),P(0,4)\bigr)\\
P(1,3):\ & \bigl(P(1,1),P(0,2)\bigr),\ \bigl(R_{1}^{3}(1),P(0,5)\bigr),\ \bigl(P(2,7),P(0,8)\bigr),\ \bigl(I(1,1),P(1,4)\bigr)\\
P(2,3):\ & \bigl(P(2,2),P(0,1)\bigr),\ \bigl(P(4,1),P(0,6)\bigr),\ \bigl(P(2,1),P(1,2)\bigr),\ \bigl(R_{1}^{4}(1),P(1,5)\bigr),\ \bigl(P(3,7),P(1,8)\bigr)\\
 & \bigl(I(0,1),P(2,4)\bigr)\\
P(3,3):\ & \bigl(P(4,8),P(0,7)\bigr),\ \bigl(P(3,2),P(1,1)\bigr),\ \bigl(P(5,1),P(1,6)\bigr),\ \bigl(P(3,1),P(2,2)\bigr),\ \bigl(R_{1}^{1}(1),P(2,5)\bigr)\\
 & \bigl(P(4,7),P(2,8)\bigr)\\
P(4,3):\ & \bigl(P(5,6),P(0,1)\bigr),\ \bigl(P(5,8),P(1,7)\bigr),\ \bigl(P(4,2),P(2,1)\bigr),\ \bigl(P(6,1),P(2,6)\bigr),\ \bigl(P(4,1),P(3,2)\bigr)\\
 & \bigl(R_{1}^{2}(1),P(3,5)\bigr),\ \bigl(P(5,7),P(3,8)\bigr)\\
P(n,3):\ & \bigl(P(n+1,6),P(n-4,1)\bigr),\ \bigl(P(n+1,8),P(n-3,7)\bigr),\ \bigl(P(n,2),P(n-2,1)\bigr)\\
 & \bigl(P(n+2,1),P(n-2,6)\bigr),\ \bigl(P(n,1),P(n-1,2)\bigr),\ \bigl(R_{1}^{(n-3)\bmod 4+1}(1),P(n-1,5)\bigr)\\
 & \bigl(P(n+1,7),P(n-1,8)\bigr),\ n>4\\
\end{align*}
\end{fleqn}
\subsubsection*{Modules of the form $P(n,4)$}

Defect: $\partial P(n,4) = -4$, for $n\ge 0$.

\begin{fleqn}
\begin{align*}
P(0,4):\ & - \\
P(1,4):\ & \bigl(P(1,1),P(0,3)\bigr),\ \bigl(P(1,7),P(0,5)\bigr),\ \bigl(P(1,8),P(0,8)\bigr)\\
P(2,4):\ & \bigl(P(2,2),P(0,2)\bigr),\ \bigl(P(2,6),P(0,6)\bigr),\ \bigl(P(2,1),P(1,3)\bigr),\ \bigl(P(2,7),P(1,5)\bigr),\ \bigl(P(2,8),P(1,8)\bigr)\\
P(3,4):\ & \bigl(P(3,3),P(0,1)\bigr),\ \bigl(P(3,5),P(0,7)\bigr),\ \bigl(P(3,2),P(1,2)\bigr),\ \bigl(P(3,6),P(1,6)\bigr),\ \bigl(P(3,1),P(2,3)\bigr)\\
 & \bigl(P(3,7),P(2,5)\bigr),\ \bigl(P(3,8),P(2,8)\bigr)\\
P(n,4):\ & \bigl(P(n,3),P(n-3,1)\bigr),\ \bigl(P(n,5),P(n-3,7)\bigr),\ \bigl(P(n,2),P(n-2,2)\bigr)\\
 & \bigl(P(n,6),P(n-2,6)\bigr),\ \bigl(P(n,1),P(n-1,3)\bigr),\ \bigl(P(n,7),P(n-1,5)\bigr)\\
 & \bigl(P(n,8),P(n-1,8)\bigr),\ n>3\\
\end{align*}
\end{fleqn}
\subsubsection*{Modules of the form $P(n,5)$}

Defect: $\partial P(n,5) = -3$, for $n\ge 0$.

\begin{fleqn}
\begin{align*}
P(0,5):\ & \bigl(I(2,7),P(0,4)\bigr)\\
P(1,5):\ & \bigl(R_{1}^{1}(1),P(0,3)\bigr),\ \bigl(P(1,7),P(0,6)\bigr),\ \bigl(P(2,1),P(0,8)\bigr),\ \bigl(I(1,7),P(1,4)\bigr)\\
P(2,5):\ & \bigl(P(4,7),P(0,2)\bigr),\ \bigl(P(2,6),P(0,7)\bigr),\ \bigl(R_{1}^{2}(1),P(1,3)\bigr),\ \bigl(P(2,7),P(1,6)\bigr),\ \bigl(P(3,1),P(1,8)\bigr)\\
 & \bigl(I(0,7),P(2,4)\bigr)\\
P(3,5):\ & \bigl(P(4,8),P(0,1)\bigr),\ \bigl(P(5,7),P(1,2)\bigr),\ \bigl(P(3,6),P(1,7)\bigr),\ \bigl(R_{1}^{3}(1),P(2,3)\bigr),\ \bigl(P(3,7),P(2,6)\bigr)\\
 & \bigl(P(4,1),P(2,8)\bigr)\\
P(4,5):\ & \bigl(P(5,2),P(0,7)\bigr),\ \bigl(P(5,8),P(1,1)\bigr),\ \bigl(P(6,7),P(2,2)\bigr),\ \bigl(P(4,6),P(2,7)\bigr),\ \bigl(R_{1}^{4}(1),P(3,3)\bigr)\\
 & \bigl(P(4,7),P(3,6)\bigr),\ \bigl(P(5,1),P(3,8)\bigr)\\
P(n,5):\ & \bigl(P(n+1,2),P(n-4,7)\bigr),\ \bigl(P(n+1,8),P(n-3,1)\bigr),\ \bigl(P(n+2,7),P(n-2,2)\bigr)\\
 & \bigl(P(n,6),P(n-2,7)\bigr),\ \bigl(R_{1}^{(n-1)\bmod 4+1}(1),P(n-1,3)\bigr),\ \bigl(P(n,7),P(n-1,6)\bigr)\\
 & \bigl(P(n+1,1),P(n-1,8)\bigr),\ n>4\\
\end{align*}
\end{fleqn}
\subsubsection*{Modules of the form $P(n,6)$}

Defect: $\partial P(n,6) = -2$, for $n\ge 0$.

\begin{fleqn}
\begin{align*}
P(0,6):\ & \bigl(I(1,6),P(0,4)\bigr),\ \bigl(I(1,7),P(0,5)\bigr)\\
P(1,6):\ & \bigl(I(3,1),P(0,3)\bigr),\ \bigl(P(1,7),P(0,7)\bigr),\ \bigl(R_{1}^{2}(1),P(0,8)\bigr),\ \bigl(I(0,6),P(1,4)\bigr),\ \bigl(I(0,7),P(1,5)\bigr)\\
P(2,6):\ & \bigl(R_{0}^{1}(1),P(0,2)\bigr),\ \bigl(I(2,1),P(1,3)\bigr),\ \bigl(P(2,7),P(1,7)\bigr),\ \bigl(R_{1}^{3}(1),P(1,8)\bigr)\\
P(3,6):\ & \bigl(P(5,1),P(0,1)\bigr),\ \bigl(R_{1}^{3}(2),P(0,6)\bigr),\ \bigl(R_{0}^{2}(1),P(1,2)\bigr),\ \bigl(I(1,1),P(2,3)\bigr),\ \bigl(P(3,7),P(2,7)\bigr)\\
 & \bigl(R_{1}^{4}(1),P(2,8)\bigr)\\
P(4,6):\ & \bigl(P(7,7),P(0,7)\bigr),\ \bigl(P(6,1),P(1,1)\bigr),\ \bigl(R_{1}^{4}(2),P(1,6)\bigr),\ \bigl(R_{0}^{3}(1),P(2,2)\bigr),\ \bigl(I(0,1),P(3,3)\bigr)\\
 & \bigl(P(4,7),P(3,7)\bigr),\ \bigl(R_{1}^{1}(1),P(3,8)\bigr)\\
P(5,6):\ & \bigl(P(8,7),P(1,7)\bigr),\ \bigl(P(7,1),P(2,1)\bigr),\ \bigl(R_{1}^{1}(2),P(2,6)\bigr),\ \bigl(R_{0}^{1}(1),P(3,2)\bigr),\ \bigl(P(5,7),P(4,7)\bigr)\\
 & \bigl(R_{1}^{2}(1),P(4,8)\bigr)\\
P(6,6):\ & \bigl(P(11,1),P(0,1)\bigr),\ \bigl(P(9,7),P(2,7)\bigr),\ \bigl(P(8,1),P(3,1)\bigr),\ \bigl(R_{1}^{2}(2),P(3,6)\bigr),\ \bigl(R_{0}^{2}(1),P(4,2)\bigr)\\
 & \bigl(P(6,7),P(5,7)\bigr),\ \bigl(R_{1}^{3}(1),P(5,8)\bigr)\\
P(n,6):\ & \bigl(P(n+5,1),P(n-6,1)\bigr),\ \bigl(P(n+3,7),P(n-4,7)\bigr),\ \bigl(P(n+2,1),P(n-3,1)\bigr)\\
 & \bigl(R_{1}^{(n-5)\bmod 4+1}(2),P(n-3,6)\bigr),\ \bigl(R_{0}^{(n-5)\bmod 3+1}(1),P(n-2,2)\bigr),\ \bigl(P(n,7),P(n-1,7)\bigr)\\
 & \bigl(R_{1}^{(n-4)\bmod 4+1}(1),P(n-1,8)\bigr),\ n>6\\
\end{align*}
\end{fleqn}
\subsubsection*{Modules of the form $P(n,7)$}

Defect: $\partial P(n,7) = -1$, for $n\ge 0$.

\begin{fleqn}
\begin{align*}
P(0,7):\ & \bigl(I(0,5),P(0,4)\bigr),\ \bigl(I(0,6),P(0,5)\bigr),\ \bigl(I(0,7),P(0,6)\bigr)\\
P(1,7):\ & \bigl(I(0,8),P(0,3)\bigr),\ \bigl(I(2,1),P(0,8)\bigr)\\
P(2,7):\ & \bigl(I(2,7),P(0,2)\bigr),\ \bigl(I(1,2),P(0,5)\bigr),\ \bigl(I(1,1),P(1,8)\bigr)\\
P(3,7):\ & \bigl(R_{1}^{1}(1),P(0,1)\bigr),\ \bigl(I(3,7),P(0,6)\bigr),\ \bigl(I(4,1),P(0,8)\bigr),\ \bigl(I(1,7),P(1,2)\bigr),\ \bigl(I(0,2),P(1,5)\bigr)\\
 & \bigl(I(0,1),P(2,8)\bigr)\\
P(4,7):\ & \bigl(R_{0}^{1}(1),P(0,7)\bigr),\ \bigl(R_{1}^{2}(1),P(1,1)\bigr),\ \bigl(I(2,7),P(1,6)\bigr),\ \bigl(I(3,1),P(1,8)\bigr),\ \bigl(I(0,7),P(2,2)\bigr)\\
P(5,7):\ & \bigl(I(5,7),P(0,2)\bigr),\ \bigl(R_{0}^{2}(1),P(1,7)\bigr),\ \bigl(R_{1}^{3}(1),P(2,1)\bigr),\ \bigl(I(1,7),P(2,6)\bigr),\ \bigl(I(2,1),P(2,8)\bigr)\\
P(6,7):\ & \bigl(R_{\infty}^{1}(1),P(0,1)\bigr),\ \bigl(R_{1}^{3}(2),P(0,7)\bigr),\ \bigl(I(4,7),P(1,2)\bigr),\ \bigl(R_{0}^{3}(1),P(2,7)\bigr),\ \bigl(R_{1}^{4}(1),P(3,1)\bigr)\\
 & \bigl(I(0,7),P(3,6)\bigr),\ \bigl(I(1,1),P(3,8)\bigr)\\
P(7,7):\ & \bigl(R_{\infty}^{2}(1),P(1,1)\bigr),\ \bigl(R_{1}^{4}(2),P(1,7)\bigr),\ \bigl(I(3,7),P(2,2)\bigr),\ \bigl(R_{0}^{1}(1),P(3,7)\bigr),\ \bigl(R_{1}^{1}(1),P(4,1)\bigr)\\
 & \bigl(I(0,1),P(4,8)\bigr)\\
P(8,7):\ & \bigl(R_{0}^{1}(2),P(0,7)\bigr),\ \bigl(R_{\infty}^{1}(1),P(2,1)\bigr),\ \bigl(R_{1}^{1}(2),P(2,7)\bigr),\ \bigl(I(2,7),P(3,2)\bigr),\ \bigl(R_{0}^{2}(1),P(4,7)\bigr)\\
 & \bigl(R_{1}^{2}(1),P(5,1)\bigr)\\
P(9,7):\ & \bigl(R_{1}^{1}(3),P(0,1)\bigr),\ \bigl(R_{0}^{2}(2),P(1,7)\bigr),\ \bigl(R_{\infty}^{2}(1),P(3,1)\bigr),\ \bigl(R_{1}^{2}(2),P(3,7)\bigr),\ \bigl(I(1,7),P(4,2)\bigr)\\
 & \bigl(R_{0}^{3}(1),P(5,7)\bigr),\ \bigl(R_{1}^{3}(1),P(6,1)\bigr)\\
P(n,7):\ & \bigl(R_{1}^{(n-9)\bmod 4+1}(3),P(n-9,1)\bigr),\ \bigl(R_{0}^{(n-8)\bmod 3+1}(2),P(n-8,7)\bigr),\ \bigl(R_{\infty}^{(n-8)\bmod 2+1}(1),P(n-6,1)\bigr)\\
 & \bigl(R_{1}^{(n-8)\bmod 4+1}(2),P(n-6,7)\bigr),\ \bigl(I(-n+10,7),P(n-5,2)\bigr),\ \bigl(R_{0}^{(n-7)\bmod 3+1}(1),P(n-4,7)\bigr)\\
 & \bigl(R_{1}^{(n-7)\bmod 4+1}(1),P(n-3,1)\bigr),\ n>9\\
\end{align*}
\end{fleqn}
\subsubsection*{Modules of the form $P(n,8)$}

Defect: $\partial P(n,8) = -2$, for $n\ge 0$.

\begin{fleqn}
\begin{align*}
P(0,8):\ & \bigl(I(0,8),P(0,4)\bigr)\\
P(1,8):\ & \bigl(I(2,7),P(0,3)\bigr),\ \bigl(I(2,1),P(0,5)\bigr)\\
P(2,8):\ & \bigl(R_{1}^{1}(1),P(0,2)\bigr),\ \bigl(R_{1}^{3}(1),P(0,6)\bigr),\ \bigl(R_{0}^{2}(1),P(0,8)\bigr),\ \bigl(I(1,7),P(1,3)\bigr),\ \bigl(I(1,1),P(1,5)\bigr)\\
P(3,8):\ & \bigl(P(4,7),P(0,1)\bigr),\ \bigl(P(4,1),P(0,7)\bigr),\ \bigl(R_{1}^{2}(1),P(1,2)\bigr),\ \bigl(R_{1}^{4}(1),P(1,6)\bigr),\ \bigl(R_{0}^{3}(1),P(1,8)\bigr)\\
 & \bigl(I(0,7),P(2,3)\bigr),\ \bigl(I(0,1),P(2,5)\bigr)\\
P(4,8):\ & \bigl(P(5,7),P(1,1)\bigr),\ \bigl(P(5,1),P(1,7)\bigr),\ \bigl(R_{1}^{3}(1),P(2,2)\bigr),\ \bigl(R_{1}^{1}(1),P(2,6)\bigr),\ \bigl(R_{0}^{1}(1),P(2,8)\bigr)\\
P(5,8):\ & \bigl(P(8,7),P(0,1)\bigr),\ \bigl(P(8,1),P(0,7)\bigr),\ \bigl(P(6,7),P(2,1)\bigr),\ \bigl(P(6,1),P(2,7)\bigr),\ \bigl(R_{1}^{4}(1),P(3,2)\bigr)\\
 & \bigl(R_{1}^{2}(1),P(3,6)\bigr),\ \bigl(R_{0}^{2}(1),P(3,8)\bigr)\\
P(n,8):\ & \bigl(P(n+3,7),P(n-5,1)\bigr),\ \bigl(P(n+3,1),P(n-5,7)\bigr),\ \bigl(P(n+1,7),P(n-3,1)\bigr)\\
 & \bigl(P(n+1,1),P(n-3,7)\bigr),\ \bigl(R_{1}^{(n-2)\bmod 4+1}(1),P(n-2,2)\bigr),\ \bigl(R_{1}^{(n-4)\bmod 4+1}(1),P(n-2,6)\bigr)\\
 & \bigl(R_{0}^{(n-4)\bmod 3+1}(1),P(n-2,8)\bigr),\ n>5\\
\end{align*}
\end{fleqn}
\subsubsection{Schofield pairs associated to preinjective exceptional modules}

\begin{figure}[ht]

\begin{center}
\begin{scaletikzpicturetowidth}{\textwidth}
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	\node[noselEEEEEEEE, label={[shift={(0.45,-0.55)}, brown]\tiny $I(3,4)$}] (I_3_4__Rationals) at (-12,0) {$\begin{smallmatrix}&&&3\\2&3&4&5&4&3&2\end{smallmatrix}$};
	\node[noselEEEEEEEE, label={[shift={(0.45,-0.55)}, brown]\tiny $I(3,5)$}] (I_3_5__Rationals) at (-10,-3) {$\begin{smallmatrix}&&&2\\1&2&3&3&2&2&1\end{smallmatrix}$};
	\node[noselEEEEEEEE, label={[shift={(0.45,-0.55)}, brown]\tiny $I(3,6)$}] (I_3_6__Rationals) at (-8,-6) {$\begin{smallmatrix}&&&1\\1&2&2&2&1&1&1\end{smallmatrix}$};
	\node[noselEEEEEEEE, label={[shift={(0.45,-0.55)}, brown]\tiny $I(3,7)$}] (I_3_7__Rationals) at (-6,-9) {$\begin{smallmatrix}&&&1\\1&1&1&1&0&0&0\end{smallmatrix}$};
	\node[seloneEEEEEEEE, label={[shift={(0.45,-0.55)}, brown]\tiny $I(3,8)$}] (I_3_8__Rationals) at (-10,1) {$\begin{smallmatrix}&&&1\\1&1&2&2&2&1&1\end{smallmatrix}$};
	\node[noselEEEEEEEE, label={[shift={(0.45,-0.55)}, brown]\tiny $I(4,1)$}] (I_4_1__Rationals) at (-10,9) {$\begin{smallmatrix}&&&0\\1&1&1&1&1&1&0\end{smallmatrix}$};
	\node[noselEEEEEEEE, label={[shift={(0.45,-0.55)}, brown]\tiny $I(4,2)$}] (I_4_2__Rationals) at (-12,6) {$\begin{smallmatrix}&&&1\\1&2&2&2&2&1&0\end{smallmatrix}$};
	\node[noselEEEEEEEE, label={[shift={(0.45,-0.55)}, brown]\tiny $I(4,3)$}] (I_4_3__Rationals) at (-14,3) {$\begin{smallmatrix}&&&2\\2&3&4&4&3&2&1\end{smallmatrix}$};
	\node[noselEEEEEEEE, label={[shift={(0.45,-0.55)}, brown]\tiny $I(4,4)$}] (I_4_4__Rationals) at (-16,0) {$\begin{smallmatrix}&&&3\\2&4&5&6&5&4&2\end{smallmatrix}$};
	\node[noselEEEEEEEE, label={[shift={(0.45,-0.55)}, brown]\tiny $I(4,5)$}] (I_4_5__Rationals) at (-14,-3) {$\begin{smallmatrix}&&&2\\1&2&3&4&4&3&2\end{smallmatrix}$};
	\node[noselEEEEEEEE, label={[shift={(0.45,-0.55)}, brown]\tiny $I(4,6)$}] (I_4_6__Rationals) at (-12,-6) {$\begin{smallmatrix}&&&1\\0&1&2&2&2&2&1\end{smallmatrix}$};
	\node[noselEEEEEEEE, label={[shift={(0.45,-0.55)}, brown]\tiny $I(4,7)$}] (I_4_7__Rationals) at (-10,-9) {$\begin{smallmatrix}&&&0\\0&1&1&1&1&1&1\end{smallmatrix}$};
	\node[noselEEEEEEEE, label={[shift={(0.45,-0.55)}, brown]\tiny $I(4,8)$}] (I_4_8__Rationals) at (-14,1) {$\begin{smallmatrix}&&&2\\1&2&2&3&2&2&1\end{smallmatrix}$};
	\node at (-16,3) {$\dots$};
	\node at (-16,-3) {$\dots$};
	\draw[red,->] (I_0_2__Rationals) -- (I_0_1__Rationals);
	\draw[red,->] (I_0_3__Rationals) -- (I_0_2__Rationals);
	\draw[blue,->] (I_1_1__Rationals) -- (I_0_2__Rationals);
	\draw[red,->] (I_0_4__Rationals) -- (I_0_3__Rationals);
	\draw[blue,->] (I_1_2__Rationals) -- (I_0_3__Rationals);
	\draw[blue,->] (I_1_3__Rationals) -- (I_0_4__Rationals);
	\draw[blue,->] (I_1_5__Rationals) -- (I_0_4__Rationals);
	\draw[blue,->] (I_1_8__Rationals) -- (I_0_4__Rationals);
	\draw[red,->] (I_0_4__Rationals) -- (I_0_5__Rationals);
	\draw[blue,->] (I_1_6__Rationals) -- (I_0_5__Rationals);
	\draw[red,->] (I_0_5__Rationals) -- (I_0_6__Rationals);
	\draw[blue,->] (I_1_7__Rationals) -- (I_0_6__Rationals);
	\draw[red,->] (I_0_6__Rationals) -- (I_0_7__Rationals);
	\draw[red,->] (I_0_4__Rationals) -- (I_0_8__Rationals);
	\draw[gray,dashed,->] (I_0_1__Rationals) -- (I_1_1__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_1_2__Rationals) -- (I_1_1__Rationals);
	\draw[gray,dashed,->] (I_0_2__Rationals) -- (I_1_2__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_1_3__Rationals) -- (I_1_2__Rationals);
	\draw[blue,->] (I_2_1__Rationals) -- (I_1_2__Rationals);
	\draw[gray,dashed,->] (I_0_3__Rationals) -- (I_1_3__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_1_4__Rationals) -- (I_1_3__Rationals);
	\draw[blue,->] (I_2_2__Rationals) -- (I_1_3__Rationals);
	\draw[gray,dashed,->] (I_0_4__Rationals) -- (I_1_4__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[blue,->] (I_2_3__Rationals) -- (I_1_4__Rationals);
	\draw[blue,->] (I_2_5__Rationals) -- (I_1_4__Rationals);
	\draw[blue,->] (I_2_8__Rationals) -- (I_1_4__Rationals);
	\draw[gray,dashed,->] (I_0_5__Rationals) -- (I_1_5__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_1_4__Rationals) -- (I_1_5__Rationals);
	\draw[blue,->] (I_2_6__Rationals) -- (I_1_5__Rationals);
	\draw[gray,dashed,->] (I_0_6__Rationals) -- (I_1_6__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_1_5__Rationals) -- (I_1_6__Rationals);
	\draw[blue,->] (I_2_7__Rationals) -- (I_1_6__Rationals);
	\draw[gray,dashed,->] (I_0_7__Rationals) -- (I_1_7__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_1_6__Rationals) -- (I_1_7__Rationals);
	\draw[gray,dashed,->] (I_0_8__Rationals) -- (I_1_8__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_1_4__Rationals) -- (I_1_8__Rationals);
	\draw[gray,dashed,->] (I_1_1__Rationals) -- (I_2_1__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_2_2__Rationals) -- (I_2_1__Rationals);
	\draw[gray,dashed,->] (I_1_2__Rationals) -- (I_2_2__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_2_3__Rationals) -- (I_2_2__Rationals);
	\draw[blue,->] (I_3_1__Rationals) -- (I_2_2__Rationals);
	\draw[gray,dashed,->] (I_1_3__Rationals) -- (I_2_3__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_2_4__Rationals) -- (I_2_3__Rationals);
	\draw[blue,->] (I_3_2__Rationals) -- (I_2_3__Rationals);
	\draw[gray,dashed,->] (I_1_4__Rationals) -- (I_2_4__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[blue,->] (I_3_3__Rationals) -- (I_2_4__Rationals);
	\draw[blue,->] (I_3_5__Rationals) -- (I_2_4__Rationals);
	\draw[blue,->] (I_3_8__Rationals) -- (I_2_4__Rationals);
	\draw[gray,dashed,->] (I_1_5__Rationals) -- (I_2_5__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_2_4__Rationals) -- (I_2_5__Rationals);
	\draw[blue,->] (I_3_6__Rationals) -- (I_2_5__Rationals);
	\draw[gray,dashed,->] (I_1_6__Rationals) -- (I_2_6__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_2_5__Rationals) -- (I_2_6__Rationals);
	\draw[blue,->] (I_3_7__Rationals) -- (I_2_6__Rationals);
	\draw[gray,dashed,->] (I_1_7__Rationals) -- (I_2_7__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_2_6__Rationals) -- (I_2_7__Rationals);
	\draw[gray,dashed,->] (I_1_8__Rationals) -- (I_2_8__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_2_4__Rationals) -- (I_2_8__Rationals);
	\draw[gray,dashed,->] (I_2_1__Rationals) -- (I_3_1__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_3_2__Rationals) -- (I_3_1__Rationals);
	\draw[gray,dashed,->] (I_2_2__Rationals) -- (I_3_2__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_3_3__Rationals) -- (I_3_2__Rationals);
	\draw[blue,->] (I_4_1__Rationals) -- (I_3_2__Rationals);
	\draw[gray,dashed,->] (I_2_3__Rationals) -- (I_3_3__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_3_4__Rationals) -- (I_3_3__Rationals);
	\draw[blue,->] (I_4_2__Rationals) -- (I_3_3__Rationals);
	\draw[gray,dashed,->] (I_2_4__Rationals) -- (I_3_4__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[blue,->] (I_4_3__Rationals) -- (I_3_4__Rationals);
	\draw[blue,->] (I_4_5__Rationals) -- (I_3_4__Rationals);
	\draw[blue,->] (I_4_8__Rationals) -- (I_3_4__Rationals);
	\draw[gray,dashed,->] (I_2_5__Rationals) -- (I_3_5__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_3_4__Rationals) -- (I_3_5__Rationals);
	\draw[blue,->] (I_4_6__Rationals) -- (I_3_5__Rationals);
	\draw[gray,dashed,->] (I_2_6__Rationals) -- (I_3_6__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_3_5__Rationals) -- (I_3_6__Rationals);
	\draw[blue,->] (I_4_7__Rationals) -- (I_3_6__Rationals);
	\draw[gray,dashed,->] (I_2_7__Rationals) -- (I_3_7__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_3_6__Rationals) -- (I_3_7__Rationals);
	\draw[gray,dashed,->] (I_2_8__Rationals) -- (I_3_8__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_3_4__Rationals) -- (I_3_8__Rationals);
	\draw[gray,dashed,->] (I_3_1__Rationals) -- (I_4_1__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_4_2__Rationals) -- (I_4_1__Rationals);
	\draw[gray,dashed,->] (I_3_2__Rationals) -- (I_4_2__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_4_3__Rationals) -- (I_4_2__Rationals);
	\draw[gray,dashed,->] (I_3_3__Rationals) -- (I_4_3__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_4_4__Rationals) -- (I_4_3__Rationals);
	\draw[gray,dashed,->] (I_3_4__Rationals) -- (I_4_4__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[gray,dashed,->] (I_3_5__Rationals) -- (I_4_5__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_4_4__Rationals) -- (I_4_5__Rationals);
	\draw[gray,dashed,->] (I_3_6__Rationals) -- (I_4_6__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_4_5__Rationals) -- (I_4_6__Rationals);
	\draw[gray,dashed,->] (I_3_7__Rationals) -- (I_4_7__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_4_6__Rationals) -- (I_4_7__Rationals);
	\draw[gray,dashed,->] (I_3_8__Rationals) -- (I_4_8__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_4_4__Rationals) -- (I_4_8__Rationals);
\end{tikzpicture}
\end{scaletikzpicturetowidth}
\end{center}

\end{figure}
\begin{figure}[ht]

\begin{center}
\begin{scaletikzpicturetowidth}{\textwidth}
\begin{tikzpicture}[scale=\tikzscale]
	\node at (4,3) {$\dots$};
	\node at (4,-3) {$\dots$};
	\node[noselEEEEEEEE, label={[shift={(0.45,-0.55)}, brown]\tiny $I(5,1)$}] (I_5_1__Rationals) at (6,9) {$\begin{smallmatrix}&&&1\\0&1&1&1&1&0&0\end{smallmatrix}$};
	\node[seloneEEEEEEEE, label={[shift={(0.45,-0.55)}, brown]\tiny $I(5,2)$}] (I_5_2__Rationals) at (4,6) {$\begin{smallmatrix}&&&2\\1&2&3&3&2&1&1\end{smallmatrix}$};
	\node[noselEEEEEEEE, label={[shift={(0.45,-0.55)}, brown]\tiny $I(5,3)$}] (I_5_3__Rationals) at (2,3) {$\begin{smallmatrix}&&&3\\1&3&4&5&4&3&2\end{smallmatrix}$};
	\node[noselEEEEEEEE, label={[shift={(0.45,-0.55)}, brown]\tiny $I(5,4)$}] (I_5_4__Rationals) at (0,0) {$\begin{smallmatrix}&&&4\\2&4&6&7&6&4&2\end{smallmatrix}$};
	\node[noselEEEEEEEE, label={[shift={(0.45,-0.55)}, brown]\tiny $I(5,5)$}] (I_5_5__Rationals) at (2,-3) {$\begin{smallmatrix}&&&3\\2&3&4&5&4&3&1\end{smallmatrix}$};
	\node[seloneEEEEEEEE, label={[shift={(0.45,-0.55)}, brown]\tiny $I(5,6)$}] (I_5_6__Rationals) at (4,-6) {$\begin{smallmatrix}&&&2\\1&1&2&3&3&2&1\end{smallmatrix}$};
	\node[noselEEEEEEEE, label={[shift={(0.45,-0.55)}, brown]\tiny $I(5,7)$}] (I_5_7__Rationals) at (6,-9) {$\begin{smallmatrix}&&&1\\0&0&1&1&1&1&0\end{smallmatrix}$};
	\node[noselEEEEEEEE, label={[shift={(0.45,-0.55)}, brown]\tiny $I(5,8)$}] (I_5_8__Rationals) at (2,1) {$\begin{smallmatrix}&&&1\\1&2&3&3&3&2&1\end{smallmatrix}$};
	\node[noselEEEEEEEE, label={[shift={(0.45,-0.55)}, brown]\tiny $I(6,1)$}] (I_6_1__Rationals) at (2,9) {$\begin{smallmatrix}&&&1\\1&1&2&2&1&1&1\end{smallmatrix}$};
	\node[noselEEEEEEEE, label={[shift={(0.45,-0.55)}, brown]\tiny $I(6,2)$}] (I_6_2__Rationals) at (0,6) {$\begin{smallmatrix}&&&2\\1&2&3&4&3&3&2\end{smallmatrix}$};
	\node[noselEEEEEEEE, label={[shift={(0.45,-0.55)}, brown]\tiny $I(6,3)$}] (I_6_3__Rationals) at (-2,3) {$\begin{smallmatrix}&&&3\\2&3&5&6&5&4&2\end{smallmatrix}$};
	\node[noselEEEEEEEE, label={[shift={(0.45,-0.55)}, brown]\tiny $I(6,4)$}] (I_6_4__Rationals) at (-4,0) {$\begin{smallmatrix}&&&5\\3&5&7&9&7&5&3\end{smallmatrix}$};
	\node[noselEEEEEEEE, label={[shift={(0.45,-0.55)}, brown]\tiny $I(6,5)$}] (I_6_5__Rationals) at (-2,-3) {$\begin{smallmatrix}&&&3\\2&4&5&6&5&3&2\end{smallmatrix}$};
	\node[noselEEEEEEEE, label={[shift={(0.45,-0.55)}, brown]\tiny $I(6,6)$}] (I_6_6__Rationals) at (0,-6) {$\begin{smallmatrix}&&&2\\2&3&3&4&3&2&1\end{smallmatrix}$};
	\node[noselEEEEEEEE, label={[shift={(0.45,-0.55)}, brown]\tiny $I(6,7)$}] (I_6_7__Rationals) at (2,-9) {$\begin{smallmatrix}&&&1\\1&1&1&2&2&1&1\end{smallmatrix}$};
	\node[noselEEEEEEEE, label={[shift={(0.45,-0.55)}, brown]\tiny $I(6,8)$}] (I_6_8__Rationals) at (-2,1) {$\begin{smallmatrix}&&&3\\1&2&3&4&3&2&1\end{smallmatrix}$};
	\node[noselEEEEEEEE, label={[shift={(0.45,-0.55)}, brown]\tiny $I(7,1)$}] (I_7_1__Rationals) at (-2,9) {$\begin{smallmatrix}&&&1\\0&1&1&2&2&2&1\end{smallmatrix}$};
	\node[noselEEEEEEEE, label={[shift={(0.45,-0.55)}, brown]\tiny $I(7,2)$}] (I_7_2__Rationals) at (-4,6) {$\begin{smallmatrix}&&&2\\1&2&3&4&4&3&1\end{smallmatrix}$};
	\node[noselEEEEEEEE, label={[shift={(0.45,-0.55)}, brown]\tiny $I(7,3)$}] (I_7_3__Rationals) at (-6,3) {$\begin{smallmatrix}&&&4\\2&4&5&7&6&4&2\end{smallmatrix}$};
	\node[noselEEEEEEEE, label={[shift={(0.45,-0.55)}, brown]\tiny $I(7,4)$}] (I_7_4__Rationals) at (-8,0) {$\begin{smallmatrix}&&&5\\3&6&8&10&8&6&3\end{smallmatrix}$};
	\node[noselEEEEEEEE, label={[shift={(0.45,-0.55)}, brown]\tiny $I(7,5)$}] (I_7_5__Rationals) at (-6,-3) {$\begin{smallmatrix}&&&4\\2&4&6&7&5&4&2\end{smallmatrix}$};
	\node[noselEEEEEEEE, label={[shift={(0.45,-0.55)}, brown]\tiny $I(7,6)$}] (I_7_6__Rationals) at (-4,-6) {$\begin{smallmatrix}&&&2\\1&3&4&4&3&2&1\end{smallmatrix}$};
	\node[noselEEEEEEEE, label={[shift={(0.45,-0.55)}, brown]\tiny $I(7,7)$}] (I_7_7__Rationals) at (-2,-9) {$\begin{smallmatrix}&&&1\\1&2&2&2&1&1&0\end{smallmatrix}$};
	\node[noselEEEEEEEE, label={[shift={(0.45,-0.55)}, brown]\tiny $I(7,8)$}] (I_7_8__Rationals) at (-6,1) {$\begin{smallmatrix}&&&2\\2&3&4&5&4&3&2\end{smallmatrix}$};
	\node[noselEEEEEEEE, label={[shift={(0.45,-0.55)}, brown]\tiny $I(8,1)$}] (I_8_1__Rationals) at (-6,9) {$\begin{smallmatrix}&&&1\\1&1&2&2&2&1&0\end{smallmatrix}$};
	\node[noselEEEEEEEE, label={[shift={(0.45,-0.55)}, brown]\tiny $I(8,2)$}] (I_8_2__Rationals) at (-8,6) {$\begin{smallmatrix}&&&3\\2&3&4&5&4&2&1\end{smallmatrix}$};
	\node[noselEEEEEEEE, label={[shift={(0.45,-0.55)}, brown]\tiny $I(8,3)$}] (I_8_3__Rationals) at (-10,3) {$\begin{smallmatrix}&&&4\\3&5&7&8&6&4&2\end{smallmatrix}$};
	\node[noselEEEEEEEE, label={[shift={(0.45,-0.55)}, brown]\tiny $I(8,4)$}] (I_8_4__Rationals) at (-12,0) {$\begin{smallmatrix}&&&6\\3&6&9&11&9&6&3\end{smallmatrix}$};
	\node[noselEEEEEEEE, label={[shift={(0.45,-0.55)}, brown]\tiny $I(8,5)$}] (I_8_5__Rationals) at (-10,-3) {$\begin{smallmatrix}&&&4\\2&4&6&8&7&5&3\end{smallmatrix}$};
	\node[noselEEEEEEEE, label={[shift={(0.45,-0.55)}, brown]\tiny $I(8,6)$}] (I_8_6__Rationals) at (-8,-6) {$\begin{smallmatrix}&&&3\\1&2&4&5&4&3&2\end{smallmatrix}$};
	\node[noselEEEEEEEE, label={[shift={(0.45,-0.55)}, brown]\tiny $I(8,7)$}] (I_8_7__Rationals) at (-6,-9) {$\begin{smallmatrix}&&&1\\0&1&2&2&2&1&1\end{smallmatrix}$};
	\node[noselEEEEEEEE, label={[shift={(0.45,-0.55)}, brown]\tiny $I(8,8)$}] (I_8_8__Rationals) at (-10,1) {$\begin{smallmatrix}&&&3\\1&3&4&5&4&3&1\end{smallmatrix}$};
	\node[seloneEEEEEEEE, label={[shift={(0.45,-0.55)}, brown]\tiny $I(9,1)$}] (I_9_1__Rationals) at (-10,9) {$\begin{smallmatrix}&&&2\\1&2&2&3&2&1&1\end{smallmatrix}$};
	\node[noselEEEEEEEE, label={[shift={(0.45,-0.55)}, brown]\tiny $I(9,2)$}] (I_9_2__Rationals) at (-12,6) {$\begin{smallmatrix}&&&3\\2&4&5&6&4&3&2\end{smallmatrix}$};
	\node[noselEEEEEEEE, label={[shift={(0.45,-0.55)}, brown]\tiny $I(9,3)$}] (I_9_3__Rationals) at (-14,3) {$\begin{smallmatrix}&&&5\\2&5&7&9&7&5&3\end{smallmatrix}$};
	\node[noselEEEEEEEE, label={[shift={(0.45,-0.55)}, brown]\tiny $I(9,4)$}] (I_9_4__Rationals) at (-16,0) {$\begin{smallmatrix}&&&7\\4&7&10&13&10&7&4\end{smallmatrix}$};
	\node[noselEEEEEEEE, label={[shift={(0.45,-0.55)}, brown]\tiny $I(9,5)$}] (I_9_5__Rationals) at (-14,-3) {$\begin{smallmatrix}&&&5\\3&5&7&9&7&5&2\end{smallmatrix}$};
	\node[noselEEEEEEEE, label={[shift={(0.45,-0.55)}, brown]\tiny $I(9,6)$}] (I_9_6__Rationals) at (-12,-6) {$\begin{smallmatrix}&&&3\\2&3&4&6&5&4&2\end{smallmatrix}$};
	\node[seloneEEEEEEEE, label={[shift={(0.45,-0.55)}, brown]\tiny $I(9,7)$}] (I_9_7__Rationals) at (-10,-9) {$\begin{smallmatrix}&&&2\\1&1&2&3&2&2&1\end{smallmatrix}$};
	\node[noselEEEEEEEE, label={[shift={(0.45,-0.55)}, brown]\tiny $I(9,8)$}] (I_9_8__Rationals) at (-14,1) {$\begin{smallmatrix}&&&3\\2&3&5&6&5&3&2\end{smallmatrix}$};
	\node at (-16,3) {$\dots$};
	\node at (-16,-3) {$\dots$};
	\draw[red,->] (I_5_2__Rationals) -- (I_5_1__Rationals);
	\draw[red,->] (I_5_3__Rationals) -- (I_5_2__Rationals);
	\draw[blue,->] (I_6_1__Rationals) -- (I_5_2__Rationals);
	\draw[red,->] (I_5_4__Rationals) -- (I_5_3__Rationals);
	\draw[blue,->] (I_6_2__Rationals) -- (I_5_3__Rationals);
	\draw[blue,->] (I_6_3__Rationals) -- (I_5_4__Rationals);
	\draw[blue,->] (I_6_5__Rationals) -- (I_5_4__Rationals);
	\draw[blue,->] (I_6_8__Rationals) -- (I_5_4__Rationals);
	\draw[red,->] (I_5_4__Rationals) -- (I_5_5__Rationals);
	\draw[blue,->] (I_6_6__Rationals) -- (I_5_5__Rationals);
	\draw[red,->] (I_5_5__Rationals) -- (I_5_6__Rationals);
	\draw[blue,->] (I_6_7__Rationals) -- (I_5_6__Rationals);
	\draw[red,->] (I_5_6__Rationals) -- (I_5_7__Rationals);
	\draw[red,->] (I_5_4__Rationals) -- (I_5_8__Rationals);
	\draw[gray,dashed,->] (I_5_1__Rationals) -- (I_6_1__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_6_2__Rationals) -- (I_6_1__Rationals);
	\draw[gray,dashed,->] (I_5_2__Rationals) -- (I_6_2__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_6_3__Rationals) -- (I_6_2__Rationals);
	\draw[blue,->] (I_7_1__Rationals) -- (I_6_2__Rationals);
	\draw[gray,dashed,->] (I_5_3__Rationals) -- (I_6_3__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_6_4__Rationals) -- (I_6_3__Rationals);
	\draw[blue,->] (I_7_2__Rationals) -- (I_6_3__Rationals);
	\draw[gray,dashed,->] (I_5_4__Rationals) -- (I_6_4__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[blue,->] (I_7_3__Rationals) -- (I_6_4__Rationals);
	\draw[blue,->] (I_7_5__Rationals) -- (I_6_4__Rationals);
	\draw[blue,->] (I_7_8__Rationals) -- (I_6_4__Rationals);
	\draw[gray,dashed,->] (I_5_5__Rationals) -- (I_6_5__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_6_4__Rationals) -- (I_6_5__Rationals);
	\draw[blue,->] (I_7_6__Rationals) -- (I_6_5__Rationals);
	\draw[gray,dashed,->] (I_5_6__Rationals) -- (I_6_6__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_6_5__Rationals) -- (I_6_6__Rationals);
	\draw[blue,->] (I_7_7__Rationals) -- (I_6_6__Rationals);
	\draw[gray,dashed,->] (I_5_7__Rationals) -- (I_6_7__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_6_6__Rationals) -- (I_6_7__Rationals);
	\draw[gray,dashed,->] (I_5_8__Rationals) -- (I_6_8__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_6_4__Rationals) -- (I_6_8__Rationals);
	\draw[gray,dashed,->] (I_6_1__Rationals) -- (I_7_1__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_7_2__Rationals) -- (I_7_1__Rationals);
	\draw[gray,dashed,->] (I_6_2__Rationals) -- (I_7_2__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_7_3__Rationals) -- (I_7_2__Rationals);
	\draw[blue,->] (I_8_1__Rationals) -- (I_7_2__Rationals);
	\draw[gray,dashed,->] (I_6_3__Rationals) -- (I_7_3__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_7_4__Rationals) -- (I_7_3__Rationals);
	\draw[blue,->] (I_8_2__Rationals) -- (I_7_3__Rationals);
	\draw[gray,dashed,->] (I_6_4__Rationals) -- (I_7_4__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[blue,->] (I_8_3__Rationals) -- (I_7_4__Rationals);
	\draw[blue,->] (I_8_5__Rationals) -- (I_7_4__Rationals);
	\draw[blue,->] (I_8_8__Rationals) -- (I_7_4__Rationals);
	\draw[gray,dashed,->] (I_6_5__Rationals) -- (I_7_5__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_7_4__Rationals) -- (I_7_5__Rationals);
	\draw[blue,->] (I_8_6__Rationals) -- (I_7_5__Rationals);
	\draw[gray,dashed,->] (I_6_6__Rationals) -- (I_7_6__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_7_5__Rationals) -- (I_7_6__Rationals);
	\draw[blue,->] (I_8_7__Rationals) -- (I_7_6__Rationals);
	\draw[gray,dashed,->] (I_6_7__Rationals) -- (I_7_7__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_7_6__Rationals) -- (I_7_7__Rationals);
	\draw[gray,dashed,->] (I_6_8__Rationals) -- (I_7_8__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_7_4__Rationals) -- (I_7_8__Rationals);
	\draw[gray,dashed,->] (I_7_1__Rationals) -- (I_8_1__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_8_2__Rationals) -- (I_8_1__Rationals);
	\draw[gray,dashed,->] (I_7_2__Rationals) -- (I_8_2__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_8_3__Rationals) -- (I_8_2__Rationals);
	\draw[blue,->] (I_9_1__Rationals) -- (I_8_2__Rationals);
	\draw[gray,dashed,->] (I_7_3__Rationals) -- (I_8_3__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_8_4__Rationals) -- (I_8_3__Rationals);
	\draw[blue,->] (I_9_2__Rationals) -- (I_8_3__Rationals);
	\draw[gray,dashed,->] (I_7_4__Rationals) -- (I_8_4__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[blue,->] (I_9_3__Rationals) -- (I_8_4__Rationals);
	\draw[blue,->] (I_9_5__Rationals) -- (I_8_4__Rationals);
	\draw[blue,->] (I_9_8__Rationals) -- (I_8_4__Rationals);
	\draw[gray,dashed,->] (I_7_5__Rationals) -- (I_8_5__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_8_4__Rationals) -- (I_8_5__Rationals);
	\draw[blue,->] (I_9_6__Rationals) -- (I_8_5__Rationals);
	\draw[gray,dashed,->] (I_7_6__Rationals) -- (I_8_6__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_8_5__Rationals) -- (I_8_6__Rationals);
	\draw[blue,->] (I_9_7__Rationals) -- (I_8_6__Rationals);
	\draw[gray,dashed,->] (I_7_7__Rationals) -- (I_8_7__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_8_6__Rationals) -- (I_8_7__Rationals);
	\draw[gray,dashed,->] (I_7_8__Rationals) -- (I_8_8__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_8_4__Rationals) -- (I_8_8__Rationals);
	\draw[gray,dashed,->] (I_8_1__Rationals) -- (I_9_1__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_9_2__Rationals) -- (I_9_1__Rationals);
	\draw[gray,dashed,->] (I_8_2__Rationals) -- (I_9_2__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_9_3__Rationals) -- (I_9_2__Rationals);
	\draw[gray,dashed,->] (I_8_3__Rationals) -- (I_9_3__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_9_4__Rationals) -- (I_9_3__Rationals);
	\draw[gray,dashed,->] (I_8_4__Rationals) -- (I_9_4__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[gray,dashed,->] (I_8_5__Rationals) -- (I_9_5__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_9_4__Rationals) -- (I_9_5__Rationals);
	\draw[gray,dashed,->] (I_8_6__Rationals) -- (I_9_6__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_9_5__Rationals) -- (I_9_6__Rationals);
	\draw[gray,dashed,->] (I_8_7__Rationals) -- (I_9_7__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_9_6__Rationals) -- (I_9_7__Rationals);
	\draw[gray,dashed,->] (I_8_8__Rationals) -- (I_9_8__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_9_4__Rationals) -- (I_9_8__Rationals);
\end{tikzpicture}
\end{scaletikzpicturetowidth}
\end{center}

\end{figure}
\medskip{}
\subsubsection*{Modules of the form $I(n,1)$}

Defect: $\partial I(n,1) = 1$, for $n\ge 0$.

\begin{fleqn}
\begin{align*}
I(0,1):\ & - \\
I(1,1):\ & - \\
I(2,1):\ & - \\
I(3,1):\ & \bigl(I(0,5),P(0,8)\bigr),\ \bigl(I(0,6),P(1,1)\bigr),\ \bigl(I(0,7),R_{1}^{1}(1)\bigr),\ \bigl(I(0,8),P(0,7)\bigr)\\
I(4,1):\ & \bigl(I(0,1),R_{0}^{2}(1)\bigr),\ \bigl(I(0,2),P(2,1)\bigr),\ \bigl(I(0,3),P(0,6)\bigr),\ \bigl(I(1,6),P(0,1)\bigr),\ \bigl(I(1,7),R_{1}^{4}(1)\bigr)\\
I(5,1):\ & \bigl(I(0,8),P(2,7)\bigr),\ \bigl(I(1,1),R_{0}^{1}(1)\bigr),\ \bigl(I(1,2),P(1,1)\bigr),\ \bigl(I(2,7),R_{1}^{3}(1)\bigr)\\
I(6,1):\ & \bigl(I(0,1),R_{1}^{2}(2)\bigr),\ \bigl(I(0,6),P(4,1)\bigr),\ \bigl(I(0,7),R_{\infty}^{2}(1)\bigr),\ \bigl(I(1,8),P(1,7)\bigr),\ \bigl(I(2,1),R_{0}^{3}(1)\bigr)\\
 & \bigl(I(2,2),P(0,1)\bigr),\ \bigl(I(3,7),R_{1}^{2}(1)\bigr)\\
I(7,1):\ & \bigl(I(1,1),R_{1}^{1}(2)\bigr),\ \bigl(I(1,6),P(3,1)\bigr),\ \bigl(I(1,7),R_{\infty}^{1}(1)\bigr),\ \bigl(I(2,8),P(0,7)\bigr),\ \bigl(I(3,1),R_{0}^{2}(1)\bigr)\\
 & \bigl(I(4,7),R_{1}^{1}(1)\bigr)\\
I(8,1):\ & \bigl(I(0,1),R_{0}^{1}(2)\bigr),\ \bigl(I(2,1),R_{1}^{4}(2)\bigr),\ \bigl(I(2,6),P(2,1)\bigr),\ \bigl(I(2,7),R_{\infty}^{2}(1)\bigr),\ \bigl(I(4,1),R_{0}^{1}(1)\bigr)\\
 & \bigl(I(5,7),R_{1}^{4}(1)\bigr)\\
I(9,1):\ & \bigl(I(0,7),R_{1}^{3}(3)\bigr),\ \bigl(I(1,1),R_{0}^{3}(2)\bigr),\ \bigl(I(3,1),R_{1}^{3}(2)\bigr),\ \bigl(I(3,6),P(1,1)\bigr),\ \bigl(I(3,7),R_{\infty}^{1}(1)\bigr)\\
 & \bigl(I(5,1),R_{0}^{3}(1)\bigr),\ \bigl(I(6,7),R_{1}^{3}(1)\bigr)\\
I(n,1):\ & \bigl(I(n-9,7),R_{1}^{(-n+11)\bmod 4+1}(3)\bigr),\ \bigl(I(n-8,1),R_{0}^{(-n+11)\bmod 3+1}(2)\bigr),\ \bigl(I(n-6,1),R_{1}^{(-n+11)\bmod 4+1}(2)\bigr)\\
 & \bigl(I(n-6,6),P(-n+10,1)\bigr),\ \bigl(I(n-6,7),R_{\infty}^{(-n+9)\bmod 2+1}(1)\bigr),\ \bigl(I(n-4,1),R_{0}^{(-n+11)\bmod 3+1}(1)\bigr)\\
 & \bigl(I(n-3,7),R_{1}^{(-n+11)\bmod 4+1}(1)\bigr),\ n>9\\
\end{align*}
\end{fleqn}
\subsubsection*{Modules of the form $I(n,2)$}

Defect: $\partial I(n,2) = 2$, for $n\ge 0$.

\begin{fleqn}
\begin{align*}
I(0,2):\ & \bigl(I(0,1),I(1,1)\bigr)\\
I(1,2):\ & \bigl(I(1,1),I(2,1)\bigr)\\
I(2,2):\ & \bigl(I(0,5),P(1,7)\bigr),\ \bigl(I(0,6),R_{0}^{1}(1)\bigr),\ \bigl(I(0,7),I(5,7)\bigr),\ \bigl(I(0,8),R_{1}^{2}(1)\bigr),\ \bigl(I(2,1),I(3,1)\bigr)\\
I(3,2):\ & \bigl(I(0,1),I(7,1)\bigr),\ \bigl(I(0,2),R_{1}^{1}(2)\bigr),\ \bigl(I(1,5),P(0,7)\bigr),\ \bigl(I(1,6),R_{0}^{3}(1)\bigr),\ \bigl(I(1,7),I(6,7)\bigr)\\
 & \bigl(I(1,8),R_{1}^{1}(1)\bigr),\ \bigl(I(3,1),I(4,1)\bigr)\\
I(4,2):\ & \bigl(I(1,1),I(8,1)\bigr),\ \bigl(I(1,2),R_{1}^{4}(2)\bigr),\ \bigl(I(2,6),R_{0}^{2}(1)\bigr),\ \bigl(I(2,7),I(7,7)\bigr),\ \bigl(I(2,8),R_{1}^{4}(1)\bigr)\\
 & \bigl(I(4,1),I(5,1)\bigr)\\
I(5,2):\ & \bigl(I(0,7),I(11,7)\bigr),\ \bigl(I(2,1),I(9,1)\bigr),\ \bigl(I(2,2),R_{1}^{3}(2)\bigr),\ \bigl(I(3,6),R_{0}^{1}(1)\bigr),\ \bigl(I(3,7),I(8,7)\bigr)\\
 & \bigl(I(3,8),R_{1}^{3}(1)\bigr),\ \bigl(I(5,1),I(6,1)\bigr)\\
I(n,2):\ & \bigl(I(n-5,7),I(n+6,7)\bigr),\ \bigl(I(n-3,1),I(n+4,1)\bigr),\ \bigl(I(n-3,2),R_{1}^{(-n+7)\bmod 4+1}(2)\bigr)\\
 & \bigl(I(n-2,6),R_{0}^{(-n+5)\bmod 3+1}(1)\bigr),\ \bigl(I(n-2,7),I(n+3,7)\bigr),\ \bigl(I(n-2,8),R_{1}^{(-n+7)\bmod 4+1}(1)\bigr)\\
 & \bigl(I(n,1),I(n+1,1)\bigr),\ n>5\\
\end{align*}
\end{fleqn}
\subsubsection*{Modules of the form $I(n,3)$}

Defect: $\partial I(n,3) = 3$, for $n\ge 0$.

\begin{fleqn}
\begin{align*}
I(0,3):\ & \bigl(I(0,1),I(1,2)\bigr),\ \bigl(I(0,2),I(2,1)\bigr)\\
I(1,3):\ & \bigl(I(0,5),R_{1}^{3}(1)\bigr),\ \bigl(I(0,6),I(5,1)\bigr),\ \bigl(I(0,7),I(2,8)\bigr),\ \bigl(I(0,8),I(4,7)\bigr),\ \bigl(I(1,1),I(2,2)\bigr)\\
 & \bigl(I(1,2),I(3,1)\bigr)\\
I(2,3):\ & \bigl(I(0,1),I(4,6)\bigr),\ \bigl(I(1,5),R_{1}^{2}(1)\bigr),\ \bigl(I(1,6),I(6,1)\bigr),\ \bigl(I(1,7),I(3,8)\bigr),\ \bigl(I(1,8),I(5,7)\bigr)\\
 & \bigl(I(2,1),I(3,2)\bigr),\ \bigl(I(2,2),I(4,1)\bigr)\\
I(n,3):\ & \bigl(I(n-2,1),I(n+2,6)\bigr),\ \bigl(I(n-1,5),R_{1}^{(-n+3)\bmod 4+1}(1)\bigr),\ \bigl(I(n-1,6),I(n+4,1)\bigr)\\
 & \bigl(I(n-1,7),I(n+1,8)\bigr),\ \bigl(I(n-1,8),I(n+3,7)\bigr),\ \bigl(I(n,1),I(n+1,2)\bigr)\\
 & \bigl(I(n,2),I(n+2,1)\bigr),\ n>2\\
\end{align*}
\end{fleqn}
\subsubsection*{Modules of the form $I(n,4)$}

Defect: $\partial I(n,4) = 4$, for $n\ge 0$.

\begin{fleqn}
\begin{align*}
I(0,4):\ & \bigl(I(0,1),I(1,3)\bigr),\ \bigl(I(0,2),I(2,2)\bigr),\ \bigl(I(0,3),I(3,1)\bigr),\ \bigl(I(0,5),I(3,7)\bigr),\ \bigl(I(0,6),I(2,6)\bigr)\\
 & \bigl(I(0,7),I(1,5)\bigr),\ \bigl(I(0,8),I(1,8)\bigr)\\
I(n,4):\ & \bigl(I(n,1),I(n+1,3)\bigr),\ \bigl(I(n,2),I(n+2,2)\bigr),\ \bigl(I(n,3),I(n+3,1)\bigr)\\
 & \bigl(I(n,5),I(n+3,7)\bigr),\ \bigl(I(n,6),I(n+2,6)\bigr),\ \bigl(I(n,7),I(n+1,5)\bigr)\\
 & \bigl(I(n,8),I(n+1,8)\bigr),\ n>0\\
\end{align*}
\end{fleqn}
\subsubsection*{Modules of the form $I(n,5)$}

Defect: $\partial I(n,5) = 3$, for $n\ge 0$.

\begin{fleqn}
\begin{align*}
I(0,5):\ & \bigl(I(0,6),I(2,7)\bigr),\ \bigl(I(0,7),I(1,6)\bigr)\\
I(1,5):\ & \bigl(I(0,1),I(2,8)\bigr),\ \bigl(I(0,2),I(5,7)\bigr),\ \bigl(I(0,3),R_{1}^{1}(1)\bigr),\ \bigl(I(0,8),I(4,1)\bigr),\ \bigl(I(1,6),I(3,7)\bigr)\\
 & \bigl(I(1,7),I(2,6)\bigr)\\
I(2,5):\ & \bigl(I(0,7),I(4,2)\bigr),\ \bigl(I(1,1),I(3,8)\bigr),\ \bigl(I(1,2),I(6,7)\bigr),\ \bigl(I(1,3),R_{1}^{4}(1)\bigr),\ \bigl(I(1,8),I(5,1)\bigr)\\
 & \bigl(I(2,6),I(4,7)\bigr),\ \bigl(I(2,7),I(3,6)\bigr)\\
I(n,5):\ & \bigl(I(n-2,7),I(n+2,2)\bigr),\ \bigl(I(n-1,1),I(n+1,8)\bigr),\ \bigl(I(n-1,2),I(n+4,7)\bigr)\\
 & \bigl(I(n-1,3),R_{1}^{(-n+5)\bmod 4+1}(1)\bigr),\ \bigl(I(n-1,8),I(n+3,1)\bigr),\ \bigl(I(n,6),I(n+2,7)\bigr)\\
 & \bigl(I(n,7),I(n+1,6)\bigr),\ n>2\\
\end{align*}
\end{fleqn}
\subsubsection*{Modules of the form $I(n,6)$}

Defect: $\partial I(n,6) = 2$, for $n\ge 0$.

\begin{fleqn}
\begin{align*}
I(0,6):\ & \bigl(I(0,7),I(1,7)\bigr)\\
I(1,6):\ & \bigl(I(1,7),I(2,7)\bigr)\\
I(2,6):\ & \bigl(I(0,1),I(5,1)\bigr),\ \bigl(I(0,2),R_{0}^{1}(1)\bigr),\ \bigl(I(0,3),P(1,1)\bigr),\ \bigl(I(0,8),R_{1}^{4}(1)\bigr),\ \bigl(I(2,7),I(3,7)\bigr)\\
I(3,6):\ & \bigl(I(0,6),R_{1}^{3}(2)\bigr),\ \bigl(I(0,7),I(7,7)\bigr),\ \bigl(I(1,1),I(6,1)\bigr),\ \bigl(I(1,2),R_{0}^{3}(1)\bigr),\ \bigl(I(1,3),P(0,1)\bigr)\\
 & \bigl(I(1,8),R_{1}^{3}(1)\bigr),\ \bigl(I(3,7),I(4,7)\bigr)\\
I(4,6):\ & \bigl(I(1,6),R_{1}^{2}(2)\bigr),\ \bigl(I(1,7),I(8,7)\bigr),\ \bigl(I(2,1),I(7,1)\bigr),\ \bigl(I(2,2),R_{0}^{2}(1)\bigr),\ \bigl(I(2,8),R_{1}^{2}(1)\bigr)\\
 & \bigl(I(4,7),I(5,7)\bigr)\\
I(5,6):\ & \bigl(I(0,1),I(11,1)\bigr),\ \bigl(I(2,6),R_{1}^{1}(2)\bigr),\ \bigl(I(2,7),I(9,7)\bigr),\ \bigl(I(3,1),I(8,1)\bigr),\ \bigl(I(3,2),R_{0}^{1}(1)\bigr)\\
 & \bigl(I(3,8),R_{1}^{1}(1)\bigr),\ \bigl(I(5,7),I(6,7)\bigr)\\
I(n,6):\ & \bigl(I(n-5,1),I(n+6,1)\bigr),\ \bigl(I(n-3,6),R_{1}^{(-n+5)\bmod 4+1}(2)\bigr),\ \bigl(I(n-3,7),I(n+4,7)\bigr)\\
 & \bigl(I(n-2,1),I(n+3,1)\bigr),\ \bigl(I(n-2,2),R_{0}^{(-n+5)\bmod 3+1}(1)\bigr),\ \bigl(I(n-2,8),R_{1}^{(-n+5)\bmod 4+1}(1)\bigr)\\
 & \bigl(I(n,7),I(n+1,7)\bigr),\ n>5\\
\end{align*}
\end{fleqn}
\subsubsection*{Modules of the form $I(n,7)$}

Defect: $\partial I(n,7) = 1$, for $n\ge 0$.

\begin{fleqn}
\begin{align*}
I(0,7):\ & - \\
I(1,7):\ & - \\
I(2,7):\ & - \\
I(3,7):\ & \bigl(I(0,1),R_{1}^{3}(1)\bigr),\ \bigl(I(0,2),P(1,7)\bigr),\ \bigl(I(0,3),P(0,8)\bigr),\ \bigl(I(0,8),P(0,1)\bigr)\\
I(4,7):\ & \bigl(I(0,5),P(0,2)\bigr),\ \bigl(I(0,6),P(2,7)\bigr),\ \bigl(I(0,7),R_{0}^{2}(1)\bigr),\ \bigl(I(1,1),R_{1}^{2}(1)\bigr),\ \bigl(I(1,2),P(0,7)\bigr)\\
I(5,7):\ & \bigl(I(0,8),P(2,1)\bigr),\ \bigl(I(1,6),P(1,7)\bigr),\ \bigl(I(1,7),R_{0}^{1}(1)\bigr),\ \bigl(I(2,1),R_{1}^{1}(1)\bigr)\\
I(6,7):\ & \bigl(I(0,1),R_{\infty}^{1}(1)\bigr),\ \bigl(I(0,2),P(4,7)\bigr),\ \bigl(I(0,7),R_{1}^{4}(2)\bigr),\ \bigl(I(1,8),P(1,1)\bigr),\ \bigl(I(2,6),P(0,7)\bigr)\\
 & \bigl(I(2,7),R_{0}^{3}(1)\bigr),\ \bigl(I(3,1),R_{1}^{4}(1)\bigr)\\
I(7,7):\ & \bigl(I(1,1),R_{\infty}^{2}(1)\bigr),\ \bigl(I(1,2),P(3,7)\bigr),\ \bigl(I(1,7),R_{1}^{3}(2)\bigr),\ \bigl(I(2,8),P(0,1)\bigr),\ \bigl(I(3,7),R_{0}^{2}(1)\bigr)\\
 & \bigl(I(4,1),R_{1}^{3}(1)\bigr)\\
I(8,7):\ & \bigl(I(0,7),R_{0}^{1}(2)\bigr),\ \bigl(I(2,1),R_{\infty}^{1}(1)\bigr),\ \bigl(I(2,2),P(2,7)\bigr),\ \bigl(I(2,7),R_{1}^{2}(2)\bigr),\ \bigl(I(4,7),R_{0}^{1}(1)\bigr)\\
 & \bigl(I(5,1),R_{1}^{2}(1)\bigr)\\
I(9,7):\ & \bigl(I(0,1),R_{1}^{1}(3)\bigr),\ \bigl(I(1,7),R_{0}^{3}(2)\bigr),\ \bigl(I(3,1),R_{\infty}^{2}(1)\bigr),\ \bigl(I(3,2),P(1,7)\bigr),\ \bigl(I(3,7),R_{1}^{1}(2)\bigr)\\
 & \bigl(I(5,7),R_{0}^{3}(1)\bigr),\ \bigl(I(6,1),R_{1}^{1}(1)\bigr)\\
I(n,7):\ & \bigl(I(n-9,1),R_{1}^{(-n+9)\bmod 4+1}(3)\bigr),\ \bigl(I(n-8,7),R_{0}^{(-n+11)\bmod 3+1}(2)\bigr),\ \bigl(I(n-6,1),R_{\infty}^{(-n+10)\bmod 2+1}(1)\bigr)\\
 & \bigl(I(n-6,2),P(-n+10,7)\bigr),\ \bigl(I(n-6,7),R_{1}^{(-n+9)\bmod 4+1}(2)\bigr),\ \bigl(I(n-4,7),R_{0}^{(-n+11)\bmod 3+1}(1)\bigr)\\
 & \bigl(I(n-3,1),R_{1}^{(-n+9)\bmod 4+1}(1)\bigr),\ n>9\\
\end{align*}
\end{fleqn}
\subsubsection*{Modules of the form $I(n,8)$}

Defect: $\partial I(n,8) = 2$, for $n\ge 0$.

\begin{fleqn}
\begin{align*}
I(0,8):\ & - \\
I(1,8):\ & \bigl(I(0,1),I(4,7)\bigr),\ \bigl(I(0,2),R_{1}^{2}(1)\bigr),\ \bigl(I(0,3),P(0,7)\bigr),\ \bigl(I(0,5),P(0,1)\bigr),\ \bigl(I(0,6),R_{1}^{4}(1)\bigr)\\
 & \bigl(I(0,7),I(4,1)\bigr)\\
I(2,8):\ & \bigl(I(0,8),R_{0}^{2}(1)\bigr),\ \bigl(I(1,1),I(5,7)\bigr),\ \bigl(I(1,2),R_{1}^{1}(1)\bigr),\ \bigl(I(1,6),R_{1}^{3}(1)\bigr),\ \bigl(I(1,7),I(5,1)\bigr)\\
I(3,8):\ & \bigl(I(0,1),I(8,7)\bigr),\ \bigl(I(0,7),I(8,1)\bigr),\ \bigl(I(1,8),R_{0}^{1}(1)\bigr),\ \bigl(I(2,1),I(6,7)\bigr),\ \bigl(I(2,2),R_{1}^{4}(1)\bigr)\\
 & \bigl(I(2,6),R_{1}^{2}(1)\bigr),\ \bigl(I(2,7),I(6,1)\bigr)\\
I(n,8):\ & \bigl(I(n-3,1),I(n+5,7)\bigr),\ \bigl(I(n-3,7),I(n+5,1)\bigr),\ \bigl(I(n-2,8),R_{0}^{(-n+3)\bmod 3+1}(1)\bigr)\\
 & \bigl(I(n-1,1),I(n+3,7)\bigr),\ \bigl(I(n-1,2),R_{1}^{(-n+6)\bmod 4+1}(1)\bigr),\ \bigl(I(n-1,6),R_{1}^{(-n+4)\bmod 4+1}(1)\bigr)\\
 & \bigl(I(n-1,7),I(n+3,1)\bigr),\ n>3\\
\end{align*}
\end{fleqn}
\subsubsection{Schofield pairs associated to regular exceptional modules}

\subsubsection*{The non-homogeneous tube $\mathcal{T}_{1}^{\Delta(\widetilde{\mathbb{E}}_{7})}$}

\begin{figure}[ht]

\begin{center}
\captionof{figure}{\vspace*{-10pt}$\mathcal{T}_{1}^{\Delta(\widetilde{\mathbb{E}}_{7})}$}
\begin{scaletikzpicturetowidth}{0.69999999999999996\textwidth}
\begin{tikzpicture}[scale=\tikzscale]
	\node[seltwoEEEEEEEE, label={[shift={(0.5,-0.6)}, brown]\tiny $R_{1}^{1}(1)$}] (RN_1_1_1__Rationals) at (4,0) {$\begin{smallmatrix}&&&1\\0&0&0&1&1&1&0\end{smallmatrix}$};
	\node[seltwoEEEEEEEE, label={[shift={(0.5,-0.6)}, brown]\tiny $R_{1}^{2}(1)$}] (RN_1_2_1__Rationals) at (8,0) {$\begin{smallmatrix}&&&0\\0&0&1&1&1&1&1\end{smallmatrix}$};
	\node[seltwoEEEEEEEE, label={[shift={(0.5,-0.6)}, brown]\tiny $R_{1}^{3}(1)$}] (RN_1_3_1__Rationals) at (12,0) {$\begin{smallmatrix}&&&1\\0&1&1&1&0&0&0\end{smallmatrix}$};
	\node[seltwoEEEEEEEE, label={[shift={(0.5,-0.6)}, brown]\tiny $R_{1}^{4}(1)$}] (RN_1_4_1__Rationals) at (16,0) {$\begin{smallmatrix}&&&0\\1&1&1&1&1&0&0\end{smallmatrix}$};
	\node[seltwoEEEEEEEE, label={[shift={(0.5,-0.6)}, brown]\tiny $R_{1}^{1}(1)$}] (RN_1_1_1__Rationals_repeated) at (20,0) {$\begin{smallmatrix}&&&1\\0&0&0&1&1&1&0\end{smallmatrix}$};
	\node[seltwoEEEEEEEE, label={[shift={(0.5,-0.6)}, brown]\tiny $R_{1}^{1}(2)$}] (RN_1_1_2__Rationals) at (6,2) {$\begin{smallmatrix}&&&1\\0&0&1&2&2&2&1\end{smallmatrix}$};
	\node[seltwoEEEEEEEE, label={[shift={(0.5,-0.6)}, brown]\tiny $R_{1}^{2}(2)$}] (RN_1_2_2__Rationals) at (10,2) {$\begin{smallmatrix}&&&1\\0&1&2&2&1&1&1\end{smallmatrix}$};
	\node[seltwoEEEEEEEE, label={[shift={(0.5,-0.6)}, brown]\tiny $R_{1}^{3}(2)$}] (RN_1_3_2__Rationals) at (14,2) {$\begin{smallmatrix}&&&1\\1&2&2&2&1&0&0\end{smallmatrix}$};
	\node[seltwoEEEEEEEE, label={[shift={(0.5,-0.6)}, brown]\tiny $R_{1}^{4}(2)$}] (RN_1_4_2__Rationals) at (18,2) {$\begin{smallmatrix}&&&1\\1&1&1&2&2&1&0\end{smallmatrix}$};
	\node[seltwoEEEEEEEE, label={[shift={(0.5,-0.6)}, brown]\tiny $R_{1}^{4}(3)$}] (RN_1_4_3__Rationals) at (4,4) {$\begin{smallmatrix}&&&1\\1&1&2&3&3&2&1\end{smallmatrix}$};
	\node[seltwoEEEEEEEE, label={[shift={(0.5,-0.6)}, brown]\tiny $R_{1}^{1}(3)$}] (RN_1_1_3__Rationals) at (8,4) {$\begin{smallmatrix}&&&2\\0&1&2&3&2&2&1\end{smallmatrix}$};
	\node[seltwoEEEEEEEE, label={[shift={(0.5,-0.6)}, brown]\tiny $R_{1}^{2}(3)$}] (RN_1_2_3__Rationals) at (12,4) {$\begin{smallmatrix}&&&1\\1&2&3&3&2&1&1\end{smallmatrix}$};
	\node[seltwoEEEEEEEE, label={[shift={(0.5,-0.6)}, brown]\tiny $R_{1}^{3}(3)$}] (RN_1_3_3__Rationals) at (16,4) {$\begin{smallmatrix}&&&2\\1&2&2&3&2&1&0\end{smallmatrix}$};
	\node[seltwoEEEEEEEE, label={[shift={(0.5,-0.6)}, brown]\tiny $R_{1}^{4}(3)$}] (RN_1_4_3__Rationals_repeated) at (20,4) {$\begin{smallmatrix}&&&1\\1&1&2&3&3&2&1\end{smallmatrix}$};
	\node[noselEEEEEEEE, label={[shift={(0.5,-0.6)}, brown]\tiny $R_{1}^{4}(4)$}] (RN_1_4_4__Rationals) at (6,6) {$\begin{smallmatrix}&&&2\\1&2&3&4&3&2&1\end{smallmatrix}$};
	\node[noselEEEEEEEE, label={[shift={(0.5,-0.6)}, brown]\tiny $R_{1}^{1}(4)$}] (RN_1_1_4__Rationals) at (10,6) {$\begin{smallmatrix}&&&2\\1&2&3&4&3&2&1\end{smallmatrix}$};
	\node[noselEEEEEEEE, label={[shift={(0.5,-0.6)}, brown]\tiny $R_{1}^{2}(4)$}] (RN_1_2_4__Rationals) at (14,6) {$\begin{smallmatrix}&&&2\\1&2&3&4&3&2&1\end{smallmatrix}$};
	\node[noselEEEEEEEE, label={[shift={(0.5,-0.6)}, brown]\tiny $R_{1}^{3}(4)$}] (RN_1_3_4__Rationals) at (18,6) {$\begin{smallmatrix}&&&2\\1&2&3&4&3&2&1\end{smallmatrix}$};
	\draw[blue,->] (RN_1_1_1__Rationals) -- (RN_1_1_2__Rationals);
	\draw[gray,dashed,->] (RN_1_1_1__Rationals) -- (RN_1_2_1__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[red,->] (RN_1_1_2__Rationals) -- (RN_1_2_1__Rationals);
	\draw[gray,dashed,->] (RN_1_1_2__Rationals) -- (RN_1_2_2__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[blue,->] (RN_1_1_2__Rationals) -- (RN_1_1_3__Rationals);
	\draw[blue,->] (RN_1_2_1__Rationals) -- (RN_1_2_2__Rationals);
	\draw[gray,dashed,->] (RN_1_2_1__Rationals) -- (RN_1_3_1__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[red,->] (RN_1_2_2__Rationals) -- (RN_1_3_1__Rationals);
	\draw[gray,dashed,->] (RN_1_2_2__Rationals) -- (RN_1_3_2__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[blue,->] (RN_1_2_2__Rationals) -- (RN_1_2_3__Rationals);
	\draw[blue,->] (RN_1_3_1__Rationals) -- (RN_1_3_2__Rationals);
	\draw[gray,dashed,->] (RN_1_3_1__Rationals) -- (RN_1_4_1__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[red,->] (RN_1_3_2__Rationals) -- (RN_1_4_1__Rationals);
	\draw[gray,dashed,->] (RN_1_3_2__Rationals) -- (RN_1_4_2__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[blue,->] (RN_1_3_2__Rationals) -- (RN_1_3_3__Rationals);
	\draw[gray,dashed,->] (RN_1_4_1__Rationals) -- (RN_1_1_1__Rationals_repeated) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[blue,->] (RN_1_4_1__Rationals) -- (RN_1_4_2__Rationals);
	\draw[red,->] (RN_1_4_2__Rationals) -- (RN_1_1_1__Rationals_repeated);
	\draw[gray,dashed,->,out=193,in=-13] (RN_1_4_2__Rationals) to node[midway,sloped,above]{{\tiny $\tau^{-1}$}} (RN_1_1_2__Rationals);
	\draw[blue,->] (RN_1_4_2__Rationals) -- (RN_1_4_3__Rationals_repeated);
	\draw[red,->] (RN_1_4_3__Rationals) -- (RN_1_1_2__Rationals);
	\draw[blue,->] (RN_1_4_3__Rationals) -- (RN_1_4_4__Rationals);
	\draw[gray,dashed,->] (RN_1_4_3__Rationals) -- (RN_1_1_3__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[red,->] (RN_1_4_4__Rationals) -- (RN_1_1_3__Rationals);
	\draw[gray,dashed,->] (RN_1_4_4__Rationals) -- (RN_1_1_4__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[red,->] (RN_1_1_3__Rationals) -- (RN_1_2_2__Rationals);
	\draw[blue,->] (RN_1_1_3__Rationals) -- (RN_1_1_4__Rationals);
	\draw[gray,dashed,->] (RN_1_1_3__Rationals) -- (RN_1_2_3__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[red,->] (RN_1_1_4__Rationals) -- (RN_1_2_3__Rationals);
	\draw[gray,dashed,->] (RN_1_1_4__Rationals) -- (RN_1_2_4__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[red,->] (RN_1_2_3__Rationals) -- (RN_1_3_2__Rationals);
	\draw[blue,->] (RN_1_2_3__Rationals) -- (RN_1_2_4__Rationals);
	\draw[gray,dashed,->] (RN_1_2_3__Rationals) -- (RN_1_3_3__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[red,->] (RN_1_2_4__Rationals) -- (RN_1_3_3__Rationals);
	\draw[gray,dashed,->] (RN_1_2_4__Rationals) -- (RN_1_3_4__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[red,->] (RN_1_3_3__Rationals) -- (RN_1_4_2__Rationals);
	\draw[gray,dashed,->] (RN_1_3_3__Rationals) -- (RN_1_4_3__Rationals_repeated) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[blue,->] (RN_1_3_3__Rationals) -- (RN_1_3_4__Rationals);
	\draw[red,->] (RN_1_3_4__Rationals) -- (RN_1_4_3__Rationals_repeated);
	\draw[gray,dashed,->,out=193,in=-13] (RN_1_3_4__Rationals) to node[midway,sloped,above]{{\tiny $\tau^{-1}$}} (RN_1_4_4__Rationals);
	\draw[dashed,-] (RN_1_1_1__Rationals) -- (RN_1_4_3__Rationals);
	\draw[dashed,-] (RN_1_1_1__Rationals_repeated) -- (RN_1_4_3__Rationals_repeated);
	\node (LeftTopPhantom_1) at (4,6) {};
	\draw[dashed,-] (RN_1_4_3__Rationals) -- (LeftTopPhantom_1);
	\node (RightTopPhantom_1) at (20,6) {};
	\draw[dashed,-] (RN_1_4_3__Rationals_repeated) -- (RightTopPhantom_1);
\end{tikzpicture}
\end{scaletikzpicturetowidth}
\end{center}

\end{figure}
\begin{fleqn}
\begin{align*}
R_{1}^{1}(1):\ & \bigl(I(0,8),P(0,6)\bigr),\ \bigl(I(1,6),P(0,8)\bigr),\ \bigl(I(1,7),P(1,1)\bigr)\\
R_{1}^{1}(2):\ & \bigl(R_{1}^{2}(1),R_{1}^{1}(1)\bigr),\ \bigl(I(5,7),P(0,7)\bigr),\ \bigl(I(1,6),P(1,6)\bigr),\ \bigl(I(3,1),P(2,1)\bigr),\ \bigl(I(1,7),P(4,7)\bigr)\\
R_{1}^{2}(1):\ & \bigl(I(0,5),P(0,3)\bigr),\ \bigl(I(2,1),P(0,7)\bigr),\ \bigl(I(0,6),P(1,8)\bigr),\ \bigl(I(0,7),P(2,1)\bigr)\\
R_{1}^{2}(2):\ & \bigl(R_{1}^{3}(1),R_{1}^{2}(1)\bigr),\ \bigl(I(2,2),P(0,2)\bigr),\ \bigl(I(4,7),P(1,7)\bigr),\ \bigl(I(0,6),P(2,6)\bigr),\ \bigl(I(2,1),P(3,1)\bigr)\\
 & \bigl(I(0,7),P(5,7)\bigr)\\
R_{1}^{3}(1):\ & \bigl(I(0,8),P(0,2)\bigr),\ \bigl(I(1,2),P(0,8)\bigr),\ \bigl(I(1,1),P(1,7)\bigr)\\
R_{1}^{3}(2):\ & \bigl(R_{1}^{4}(1),R_{1}^{3}(1)\bigr),\ \bigl(I(5,1),P(0,1)\bigr),\ \bigl(I(1,2),P(1,2)\bigr),\ \bigl(I(3,7),P(2,7)\bigr),\ \bigl(I(1,1),P(4,1)\bigr)\\
R_{1}^{4}(1):\ & \bigl(I(2,7),P(0,1)\bigr),\ \bigl(I(0,3),P(0,5)\bigr),\ \bigl(I(0,2),P(1,8)\bigr),\ \bigl(I(0,1),P(2,7)\bigr)\\
R_{1}^{4}(2):\ & \bigl(R_{1}^{1}(1),R_{1}^{4}(1)\bigr),\ \bigl(I(2,6),P(0,6)\bigr),\ \bigl(I(4,1),P(1,1)\bigr),\ \bigl(I(0,2),P(2,2)\bigr),\ \bigl(I(2,7),P(3,7)\bigr)\\
 & \bigl(I(0,1),P(5,1)\bigr)\\
R_{1}^{4}(3):\ & \bigl(R_{1}^{1}(2),R_{1}^{4}(1)\bigr),\ \bigl(R_{1}^{2}(1),R_{1}^{4}(2)\bigr),\ \bigl(I(8,1),P(0,7)\bigr),\ \bigl(I(6,7),P(2,1)\bigr),\ \bigl(I(4,1),P(4,7)\bigr)\\
 & \bigl(I(2,7),P(6,1)\bigr),\ \bigl(I(0,1),P(8,7)\bigr)\\
R_{1}^{1}(3):\ & \bigl(R_{1}^{2}(2),R_{1}^{1}(1)\bigr),\ \bigl(R_{1}^{3}(1),R_{1}^{1}(2)\bigr),\ \bigl(I(7,1),P(1,7)\bigr),\ \bigl(I(5,7),P(3,1)\bigr),\ \bigl(I(3,1),P(5,7)\bigr)\\
 & \bigl(I(1,7),P(7,1)\bigr)\\
R_{1}^{2}(3):\ & \bigl(R_{1}^{3}(2),R_{1}^{2}(1)\bigr),\ \bigl(R_{1}^{4}(1),R_{1}^{2}(2)\bigr),\ \bigl(I(8,7),P(0,1)\bigr),\ \bigl(I(6,1),P(2,7)\bigr),\ \bigl(I(4,7),P(4,1)\bigr)\\
 & \bigl(I(2,1),P(6,7)\bigr),\ \bigl(I(0,7),P(8,1)\bigr)\\
R_{1}^{3}(3):\ & \bigl(R_{1}^{4}(2),R_{1}^{3}(1)\bigr),\ \bigl(R_{1}^{1}(1),R_{1}^{3}(2)\bigr),\ \bigl(I(7,7),P(1,1)\bigr),\ \bigl(I(5,1),P(3,7)\bigr),\ \bigl(I(3,7),P(5,1)\bigr)\\
 & \bigl(I(1,1),P(7,7)\bigr)\\
\end{align*}
\end{fleqn}
\subsubsection*{The non-homogeneous tube $\mathcal{T}_{0}^{\Delta(\widetilde{\mathbb{E}}_{7})}$}

\begin{figure}[ht]

\begin{center}
\captionof{figure}{\vspace*{-10pt}$\mathcal{T}_{0}^{\Delta(\widetilde{\mathbb{E}}_{7})}$}
\begin{scaletikzpicturetowidth}{0.69999999999999996\textwidth}
\begin{tikzpicture}[scale=\tikzscale]
	\node[seltwoEEEEEEEE, label={[shift={(0.5,-0.6)}, brown]\tiny $R_{0}^{1}(1)$}] (RN_2_1_1__Rationals) at (4,0) {$\begin{smallmatrix}&&&1\\0&0&1&1&1&0&0\end{smallmatrix}$};
	\node[seltwoEEEEEEEE, label={[shift={(0.5,-0.6)}, brown]\tiny $R_{0}^{2}(1)$}] (RN_2_2_1__Rationals) at (8,0) {$\begin{smallmatrix}&&&0\\0&1&1&1&1&1&0\end{smallmatrix}$};
	\node[seltwoEEEEEEEE, label={[shift={(0.5,-0.6)}, brown]\tiny $R_{0}^{3}(1)$}] (RN_2_3_1__Rationals) at (12,0) {$\begin{smallmatrix}&&&1\\1&1&1&2&1&1&1\end{smallmatrix}$};
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	\node[seltwoEEEEEEEE, label={[shift={(0.5,-0.6)}, brown]\tiny $R_{0}^{2}(2)$}] (RN_2_2_2__Rationals) at (10,2) {$\begin{smallmatrix}&&&1\\1&2&2&3&2&2&1\end{smallmatrix}$};
	\node[seltwoEEEEEEEE, label={[shift={(0.5,-0.6)}, brown]\tiny $R_{0}^{3}(2)$}] (RN_2_3_2__Rationals) at (14,2) {$\begin{smallmatrix}&&&2\\1&1&2&3&2&1&1\end{smallmatrix}$};
	\node[noselEEEEEEEE, label={[shift={(0.5,-0.6)}, brown]\tiny $R_{0}^{3}(3)$}] (RN_2_3_3__Rationals) at (4,4) {$\begin{smallmatrix}&&&2\\1&2&3&4&3&2&1\end{smallmatrix}$};
	\node[noselEEEEEEEE, label={[shift={(0.5,-0.6)}, brown]\tiny $R_{0}^{1}(3)$}] (RN_2_1_3__Rationals) at (8,4) {$\begin{smallmatrix}&&&2\\1&2&3&4&3&2&1\end{smallmatrix}$};
	\node[noselEEEEEEEE, label={[shift={(0.5,-0.6)}, brown]\tiny $R_{0}^{2}(3)$}] (RN_2_2_3__Rationals) at (12,4) {$\begin{smallmatrix}&&&2\\1&2&3&4&3&2&1\end{smallmatrix}$};
	\node[noselEEEEEEEE, label={[shift={(0.5,-0.6)}, brown]\tiny $R_{0}^{3}(3)$}] (RN_2_3_3__Rationals_repeated) at (16,4) {$\begin{smallmatrix}&&&2\\1&2&3&4&3&2&1\end{smallmatrix}$};
	\node[noselEEEEEEEE, label={[shift={(0.5,-0.6)}, brown]\tiny $R_{0}^{3}(4)$}] (RN_2_3_4__Rationals) at (6,6) {$\begin{smallmatrix}&&&3\\2&3&4&6&4&3&2\end{smallmatrix}$};
	\node[noselEEEEEEEE, label={[shift={(0.5,-0.6)}, brown]\tiny $R_{0}^{1}(4)$}] (RN_2_1_4__Rationals) at (10,6) {$\begin{smallmatrix}&&&3\\1&2&4&5&4&2&1\end{smallmatrix}$};
	\node[noselEEEEEEEE, label={[shift={(0.5,-0.6)}, brown]\tiny $R_{0}^{2}(4)$}] (RN_2_2_4__Rationals) at (14,6) {$\begin{smallmatrix}&&&2\\1&3&4&5&4&3&1\end{smallmatrix}$};
	\draw[blue,->] (RN_2_1_1__Rationals) -- (RN_2_1_2__Rationals);
	\draw[gray,dashed,->] (RN_2_1_1__Rationals) -- (RN_2_2_1__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[red,->] (RN_2_1_2__Rationals) -- (RN_2_2_1__Rationals);
	\draw[gray,dashed,->] (RN_2_1_2__Rationals) -- (RN_2_2_2__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[blue,->] (RN_2_1_2__Rationals) -- (RN_2_1_3__Rationals);
	\draw[blue,->] (RN_2_2_1__Rationals) -- (RN_2_2_2__Rationals);
	\draw[gray,dashed,->] (RN_2_2_1__Rationals) -- (RN_2_3_1__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[red,->] (RN_2_2_2__Rationals) -- (RN_2_3_1__Rationals);
	\draw[gray,dashed,->] (RN_2_2_2__Rationals) -- (RN_2_3_2__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[blue,->] (RN_2_2_2__Rationals) -- (RN_2_2_3__Rationals);
	\draw[gray,dashed,->] (RN_2_3_1__Rationals) -- (RN_2_1_1__Rationals_repeated) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[blue,->] (RN_2_3_1__Rationals) -- (RN_2_3_2__Rationals);
	\draw[red,->] (RN_2_3_2__Rationals) -- (RN_2_1_1__Rationals_repeated);
	\draw[gray,dashed,->,out=200,in=-20] (RN_2_3_2__Rationals) to node[midway,sloped,above]{{\tiny $\tau^{-1}$}} (RN_2_1_2__Rationals);
	\draw[blue,->] (RN_2_3_2__Rationals) -- (RN_2_3_3__Rationals_repeated);
	\draw[red,->] (RN_2_3_3__Rationals) -- (RN_2_1_2__Rationals);
	\draw[blue,->] (RN_2_3_3__Rationals) -- (RN_2_3_4__Rationals);
	\draw[gray,dashed,->] (RN_2_3_3__Rationals) -- (RN_2_1_3__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[red,->] (RN_2_3_4__Rationals) -- (RN_2_1_3__Rationals);
	\draw[gray,dashed,->] (RN_2_3_4__Rationals) -- (RN_2_1_4__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[red,->] (RN_2_1_3__Rationals) -- (RN_2_2_2__Rationals);
	\draw[blue,->] (RN_2_1_3__Rationals) -- (RN_2_1_4__Rationals);
	\draw[gray,dashed,->] (RN_2_1_3__Rationals) -- (RN_2_2_3__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[red,->] (RN_2_1_4__Rationals) -- (RN_2_2_3__Rationals);
	\draw[gray,dashed,->] (RN_2_1_4__Rationals) -- (RN_2_2_4__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[red,->] (RN_2_2_3__Rationals) -- (RN_2_3_2__Rationals);
	\draw[gray,dashed,->] (RN_2_2_3__Rationals) -- (RN_2_3_3__Rationals_repeated) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[blue,->] (RN_2_2_3__Rationals) -- (RN_2_2_4__Rationals);
	\draw[red,->] (RN_2_2_4__Rationals) -- (RN_2_3_3__Rationals_repeated);
	\draw[gray,dashed,->,out=200,in=-20] (RN_2_2_4__Rationals) to node[midway,sloped,above]{{\tiny $\tau^{-1}$}} (RN_2_3_4__Rationals);
	\draw[dashed,-] (RN_2_1_1__Rationals) -- (RN_2_3_3__Rationals);
	\draw[dashed,-] (RN_2_1_1__Rationals_repeated) -- (RN_2_3_3__Rationals_repeated);
	\node (LeftTopPhantom_2) at (4,6) {};
	\draw[dashed,-] (RN_2_3_3__Rationals) -- (LeftTopPhantom_2);
	\node (RightTopPhantom_2) at (16,6) {};
	\draw[dashed,-] (RN_2_3_3__Rationals_repeated) -- (RightTopPhantom_2);
\end{tikzpicture}
\end{scaletikzpicturetowidth}
\end{center}

\end{figure}
\begin{fleqn}
\begin{align*}
R_{0}^{1}(1):\ & \bigl(I(2,1),P(1,1)\bigr),\ \bigl(I(2,7),P(1,7)\bigr),\ \bigl(I(0,8),P(1,8)\bigr)\\
R_{0}^{1}(2):\ & \bigl(R_{0}^{2}(1),R_{0}^{1}(1)\bigr),\ \bigl(I(5,1),P(2,1)\bigr),\ \bigl(I(5,7),P(2,7)\bigr),\ \bigl(I(2,1),P(5,1)\bigr),\ \bigl(I(2,7),P(5,7)\bigr)\\
R_{0}^{2}(1):\ & \bigl(I(1,6),P(0,2)\bigr),\ \bigl(I(1,2),P(0,6)\bigr),\ \bigl(I(1,1),P(2,1)\bigr),\ \bigl(I(1,7),P(2,7)\bigr)\\
R_{0}^{2}(2):\ & \bigl(R_{0}^{3}(1),R_{0}^{2}(1)\bigr),\ \bigl(I(7,1),P(0,1)\bigr),\ \bigl(I(7,7),P(0,7)\bigr),\ \bigl(I(4,1),P(3,1)\bigr),\ \bigl(I(4,7),P(3,7)\bigr)\\
 & \bigl(I(1,1),P(6,1)\bigr),\ \bigl(I(1,7),P(6,7)\bigr)\\
R_{0}^{3}(1):\ & \bigl(I(3,1),P(0,1)\bigr),\ \bigl(I(3,7),P(0,7)\bigr),\ \bigl(I(1,8),P(0,8)\bigr),\ \bigl(I(0,6),P(1,2)\bigr),\ \bigl(I(0,2),P(1,6)\bigr)\\
 & \bigl(I(0,1),P(3,1)\bigr),\ \bigl(I(0,7),P(3,7)\bigr)\\
R_{0}^{3}(2):\ & \bigl(R_{0}^{1}(1),R_{0}^{3}(1)\bigr),\ \bigl(I(6,1),P(1,1)\bigr),\ \bigl(I(6,7),P(1,7)\bigr),\ \bigl(I(3,1),P(4,1)\bigr),\ \bigl(I(3,7),P(4,7)\bigr)\\
 & \bigl(I(0,1),P(7,1)\bigr),\ \bigl(I(0,7),P(7,7)\bigr)\\
\end{align*}
\end{fleqn}
\subsubsection*{The non-homogeneous tube $\mathcal{T}_{\infty}^{\Delta(\widetilde{\mathbb{E}}_{7})}$}

\begin{figure}[ht]

\begin{center}
\captionof{figure}{\vspace*{-10pt}$\mathcal{T}_{\infty}^{\Delta(\widetilde{\mathbb{E}}_{7})}$}
\begin{scaletikzpicturetowidth}{0.59999999999999998\textwidth}
\begin{tikzpicture}[scale=\tikzscale]
	\node[seltwoEEEEEEEE, label={[shift={(0.5,-0.6)}, brown]\tiny $R_{\infty}^{1}(1)$}] (RN_3_1_1__Rationals) at (4,0) {$\begin{smallmatrix}&&&1\\0&1&1&2&2&1&1\end{smallmatrix}$};
	\node[seltwoEEEEEEEE, label={[shift={(0.5,-0.6)}, brown]\tiny $R_{\infty}^{2}(1)$}] (RN_3_2_1__Rationals) at (8,0) {$\begin{smallmatrix}&&&1\\1&1&2&2&1&1&0\end{smallmatrix}$};
	\node[seltwoEEEEEEEE, label={[shift={(0.5,-0.6)}, brown]\tiny $R_{\infty}^{1}(1)$}] (RN_3_1_1__Rationals_repeated) at (12,0) {$\begin{smallmatrix}&&&1\\0&1&1&2&2&1&1\end{smallmatrix}$};
	\node[noselEEEEEEEE, label={[shift={(0.5,-0.6)}, brown]\tiny $R_{\infty}^{1}(2)$}] (RN_3_1_2__Rationals) at (6,2) {$\begin{smallmatrix}&&&2\\1&2&3&4&3&2&1\end{smallmatrix}$};
	\node[noselEEEEEEEE, label={[shift={(0.5,-0.6)}, brown]\tiny $R_{\infty}^{2}(2)$}] (RN_3_2_2__Rationals) at (10,2) {$\begin{smallmatrix}&&&2\\1&2&3&4&3&2&1\end{smallmatrix}$};
	\node[noselEEEEEEEE, label={[shift={(0.5,-0.6)}, brown]\tiny $R_{\infty}^{2}(3)$}] (RN_3_2_3__Rationals) at (4,4) {$\begin{smallmatrix}&&&3\\2&3&5&6&4&3&1\end{smallmatrix}$};
	\node[noselEEEEEEEE, label={[shift={(0.5,-0.6)}, brown]\tiny $R_{\infty}^{1}(3)$}] (RN_3_1_3__Rationals) at (8,4) {$\begin{smallmatrix}&&&3\\1&3&4&6&5&3&2\end{smallmatrix}$};
	\node[noselEEEEEEEE, label={[shift={(0.5,-0.6)}, brown]\tiny $R_{\infty}^{2}(3)$}] (RN_3_2_3__Rationals_repeated) at (12,4) {$\begin{smallmatrix}&&&3\\2&3&5&6&4&3&1\end{smallmatrix}$};
	\draw[blue,->] (RN_3_1_1__Rationals) -- (RN_3_1_2__Rationals);
	\draw[gray,dashed,->] (RN_3_1_1__Rationals) -- (RN_3_2_1__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[red,->] (RN_3_1_2__Rationals) -- (RN_3_2_1__Rationals);
	\draw[gray,dashed,->] (RN_3_1_2__Rationals) -- (RN_3_2_2__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[blue,->] (RN_3_1_2__Rationals) -- (RN_3_1_3__Rationals);
	\draw[gray,dashed,->] (RN_3_2_1__Rationals) -- (RN_3_1_1__Rationals_repeated) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[blue,->] (RN_3_2_1__Rationals) -- (RN_3_2_2__Rationals);
	\draw[red,->] (RN_3_2_2__Rationals) -- (RN_3_1_1__Rationals_repeated);
	\draw[gray,dashed,->,out=219,in=-39] (RN_3_2_2__Rationals) to node[midway,sloped,above]{{\tiny $\tau^{-1}$}} (RN_3_1_2__Rationals);
	\draw[blue,->] (RN_3_2_2__Rationals) -- (RN_3_2_3__Rationals_repeated);
	\draw[red,->] (RN_3_2_3__Rationals) -- (RN_3_1_2__Rationals);
	\draw[gray,dashed,->] (RN_3_2_3__Rationals) -- (RN_3_1_3__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[red,->] (RN_3_1_3__Rationals) -- (RN_3_2_2__Rationals);
	\draw[gray,dashed,->] (RN_3_1_3__Rationals) -- (RN_3_2_3__Rationals_repeated) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[dashed,-] (RN_3_1_1__Rationals) -- (RN_3_2_3__Rationals);
	\draw[dashed,-] (RN_3_1_1__Rationals_repeated) -- (RN_3_2_3__Rationals_repeated);
\end{tikzpicture}
\end{scaletikzpicturetowidth}
\end{center}

\end{figure}
\begin{fleqn}
\begin{align*}
R_{\infty}^{1}(1):\ & \bigl(I(5,1),P(0,7)\bigr),\ \bigl(I(4,7),P(1,1)\bigr),\ \bigl(I(3,1),P(2,7)\bigr),\ \bigl(I(2,7),P(3,1)\bigr),\ \bigl(I(1,1),P(4,7)\bigr)\\
 & \bigl(I(0,7),P(5,1)\bigr)\\
R_{\infty}^{2}(1):\ & \bigl(I(5,7),P(0,1)\bigr),\ \bigl(I(4,1),P(1,7)\bigr),\ \bigl(I(3,7),P(2,1)\bigr),\ \bigl(I(2,1),P(3,7)\bigr),\ \bigl(I(1,7),P(4,1)\bigr)\\
 & \bigl(I(0,1),P(5,7)\bigr)\\
\end{align*}
\end{fleqn}
\clearpage

\subsection[Schofield pairs for the quiver $\Delta(\widetilde{\mathbb{E}}_8)$]{Schofield pairs for the quiver $\Delta(\widetilde{\mathbb{E}}_8)$ -- {\usebox{\boxEEEEEEEE}}}
\[
\xymatrix{  &   & 9\ar[d] &   &   &   &   &  \\
1\ar[r] & 2\ar[r] & 3 & 4\ar[l] & 5\ar[l] & 6\ar[l] & 7\ar[l] & 8\ar[l]}
\]
\medskip{}
\[
 C_{\Delta(\widetilde{\mathbb{E}}_{8})} = \begin{bmatrix}1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\
1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\
1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1\\
0 & 0 & 0 & 1 & 1 & 1 & 1 & 1 & 0\\
0 & 0 & 0 & 0 & 1 & 1 & 1 & 1 & 0\\
0 & 0 & 0 & 0 & 0 & 1 & 1 & 1 & 0\\
0 & 0 & 0 & 0 & 0 & 0 & 1 & 1 & 0\\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0\\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1\end{bmatrix}\qquad\Phi_{\Delta(\widetilde{\mathbb{E}}_{8})} = \begin{bmatrix}0 & 0 & -1 & 1 & 0 & 0 & 0 & 0 & 1\\
1 & 0 & -1 & 1 & 0 & 0 & 0 & 0 & 1\\
0 & 1 & -1 & 1 & 0 & 0 & 0 & 0 & 1\\
0 & 1 & -1 & 0 & 1 & 0 & 0 & 0 & 1\\
0 & 1 & -1 & 0 & 0 & 1 & 0 & 0 & 1\\
0 & 1 & -1 & 0 & 0 & 0 & 1 & 0 & 1\\
0 & 1 & -1 & 0 & 0 & 0 & 0 & 1 & 1\\
0 & 1 & -1 & 0 & 0 & 0 & 0 & 0 & 1\\
0 & 1 & -1 & 1 & 0 & 0 & 0 & 0 & 0\end{bmatrix}
\]

\medskip{}\subsubsection{Schofield pairs associated to preprojective exceptional modules}

\begin{figure}[ht]

\begin{center}
\begin{scaletikzpicturetowidth}{\textwidth}
\begin{tikzpicture}[scale=\tikzscale]
	\node[noselEEEEEEEEE, label={[shift={(0.2,-0.55)}, brown]\tiny $P(0,1)$}] (P_0_1__Rationals) at (4,6) {$\begin{smallmatrix}&&0\\1&1&1&0&0&0&0&0\end{smallmatrix}$};
	\node[noselEEEEEEEEE, label={[shift={(0.2,-0.55)}, brown]\tiny $P(0,2)$}] (P_0_2__Rationals) at (2,3) {$\begin{smallmatrix}&&0\\0&1&1&0&0&0&0&0\end{smallmatrix}$};
	\node[noselEEEEEEEEE, label={[shift={(0.2,-0.55)}, brown]\tiny $P(0,3)$}] (P_0_3__Rationals) at (0,0) {$\begin{smallmatrix}&&0\\0&0&1&0&0&0&0&0\end{smallmatrix}$};
	\node[noselEEEEEEEEE, label={[shift={(0.2,-0.55)}, brown]\tiny $P(0,4)$}] (P_0_4__Rationals) at (2,-3) {$\begin{smallmatrix}&&0\\0&0&1&1&0&0&0&0\end{smallmatrix}$};
	\node[noselEEEEEEEEE, label={[shift={(0.2,-0.55)}, brown]\tiny $P(0,5)$}] (P_0_5__Rationals) at (4,-6) {$\begin{smallmatrix}&&0\\0&0&1&1&1&0&0&0\end{smallmatrix}$};
	\node[noselEEEEEEEEE, label={[shift={(0.2,-0.55)}, brown]\tiny $P(0,6)$}] (P_0_6__Rationals) at (6,-9) {$\begin{smallmatrix}&&0\\0&0&1&1&1&1&0&0\end{smallmatrix}$};
	\node[noselEEEEEEEEE, label={[shift={(0.2,-0.55)}, brown]\tiny $P(0,7)$}] (P_0_7__Rationals) at (8,-12) {$\begin{smallmatrix}&&0\\0&0&1&1&1&1&1&0\end{smallmatrix}$};
	\node[noselEEEEEEEEE, label={[shift={(0.2,-0.55)}, brown]\tiny $P(0,8)$}] (P_0_8__Rationals) at (10,-15) {$\begin{smallmatrix}&&0\\0&0&1&1&1&1&1&1\end{smallmatrix}$};
	\node[noselEEEEEEEEE, label={[shift={(0.2,-0.55)}, brown]\tiny $P(0,9)$}] (P_0_9__Rationals) at (2,1) {$\begin{smallmatrix}&&1\\0&0&1&0&0&0&0&0\end{smallmatrix}$};
	\node[noselEEEEEEEEE, label={[shift={(0.2,-0.55)}, brown]\tiny $P(1,1)$}] (P_1_1__Rationals) at (8,6) {$\begin{smallmatrix}&&1\\0&0&1&1&0&0&0&0\end{smallmatrix}$};
	\node[noselEEEEEEEEE, label={[shift={(0.2,-0.55)}, brown]\tiny $P(1,2)$}] (P_1_2__Rationals) at (6,3) {$\begin{smallmatrix}&&1\\1&1&2&1&0&0&0&0\end{smallmatrix}$};
	\node[noselEEEEEEEEE, label={[shift={(0.2,-0.55)}, brown]\tiny $P(1,3)$}] (P_1_3__Rationals) at (4,0) {$\begin{smallmatrix}&&1\\0&1&2&1&0&0&0&0\end{smallmatrix}$};
	\node[noselEEEEEEEEE, label={[shift={(0.2,-0.55)}, brown]\tiny $P(1,4)$}] (P_1_4__Rationals) at (6,-3) {$\begin{smallmatrix}&&1\\0&1&2&1&1&0&0&0\end{smallmatrix}$};
	\node[noselEEEEEEEEE, label={[shift={(0.2,-0.55)}, brown]\tiny $P(1,5)$}] (P_1_5__Rationals) at (8,-6) {$\begin{smallmatrix}&&1\\0&1&2&1&1&1&0&0\end{smallmatrix}$};
	\node[noselEEEEEEEEE, label={[shift={(0.2,-0.55)}, brown]\tiny $P(1,6)$}] (P_1_6__Rationals) at (10,-9) {$\begin{smallmatrix}&&1\\0&1&2&1&1&1&1&0\end{smallmatrix}$};
	\node[noselEEEEEEEEE, label={[shift={(0.2,-0.55)}, brown]\tiny $P(1,7)$}] (P_1_7__Rationals) at (12,-12) {$\begin{smallmatrix}&&1\\0&1&2&1&1&1&1&1\end{smallmatrix}$};
	\node[noselEEEEEEEEE, label={[shift={(0.2,-0.55)}, brown]\tiny $P(1,8)$}] (P_1_8__Rationals) at (14,-15) {$\begin{smallmatrix}&&1\\0&1&1&0&0&0&0&0\end{smallmatrix}$};
	\node[noselEEEEEEEEE, label={[shift={(0.2,-0.55)}, brown]\tiny $P(1,9)$}] (P_1_9__Rationals) at (6,1) {$\begin{smallmatrix}&&0\\0&1&1&1&0&0&0&0\end{smallmatrix}$};
	\node[noselEEEEEEEEE, label={[shift={(0.2,-0.55)}, brown]\tiny $P(2,1)$}] (P_2_1__Rationals) at (12,6) {$\begin{smallmatrix}&&0\\0&1&1&1&1&0&0&0\end{smallmatrix}$};
	\node[noselEEEEEEEEE, label={[shift={(0.2,-0.55)}, brown]\tiny $P(2,2)$}] (P_2_2__Rationals) at (10,3) {$\begin{smallmatrix}&&1\\0&1&2&2&1&0&0&0\end{smallmatrix}$};
	\node[noselEEEEEEEEE, label={[shift={(0.2,-0.55)}, brown]\tiny $P(2,3)$}] (P_2_3__Rationals) at (8,0) {$\begin{smallmatrix}&&1\\1&2&3&2&1&0&0&0\end{smallmatrix}$};
	\node[noselEEEEEEEEE, label={[shift={(0.2,-0.55)}, brown]\tiny $P(2,4)$}] (P_2_4__Rationals) at (10,-3) {$\begin{smallmatrix}&&1\\1&2&3&2&1&1&0&0\end{smallmatrix}$};
	\node[noselEEEEEEEEE, label={[shift={(0.2,-0.55)}, brown]\tiny $P(2,5)$}] (P_2_5__Rationals) at (12,-6) {$\begin{smallmatrix}&&1\\1&2&3&2&1&1&1&0\end{smallmatrix}$};
	\node[noselEEEEEEEEE, label={[shift={(0.2,-0.55)}, brown]\tiny $P(2,6)$}] (P_2_6__Rationals) at (14,-9) {$\begin{smallmatrix}&&1\\1&2&3&2&1&1&1&1\end{smallmatrix}$};
	\node[noselEEEEEEEEE, label={[shift={(0.2,-0.55)}, brown]\tiny $P(2,7)$}] (P_2_7__Rationals) at (16,-12) {$\begin{smallmatrix}&&1\\1&2&2&1&0&0&0&0\end{smallmatrix}$};
	\node[noselEEEEEEEEE, label={[shift={(0.2,-0.55)}, brown]\tiny $P(2,8)$}] (P_2_8__Rationals) at (18,-15) {$\begin{smallmatrix}&&0\\1&1&1&1&0&0&0&0\end{smallmatrix}$};
	\node[noselEEEEEEEEE, label={[shift={(0.2,-0.55)}, brown]\tiny $P(2,9)$}] (P_2_9__Rationals) at (10,1) {$\begin{smallmatrix}&&1\\1&1&2&1&1&0&0&0\end{smallmatrix}$};
	\node[noselEEEEEEEEE, label={[shift={(0.2,-0.55)}, brown]\tiny $P(3,1)$}] (P_3_1__Rationals) at (16,6) {$\begin{smallmatrix}&&1\\1&1&2&1&1&1&0&0\end{smallmatrix}$};
	\node[noselEEEEEEEEE, label={[shift={(0.2,-0.55)}, brown]\tiny $P(3,2)$}] (P_3_2__Rationals) at (14,3) {$\begin{smallmatrix}&&1\\1&2&3&2&2&1&0&0\end{smallmatrix}$};
	\node[noselEEEEEEEEE, label={[shift={(0.2,-0.55)}, brown]\tiny $P(3,3)$}] (P_3_3__Rationals) at (12,0) {$\begin{smallmatrix}&&2\\1&2&4&3&2&1&0&0\end{smallmatrix}$};
	\node[noselEEEEEEEEE, label={[shift={(0.2,-0.55)}, brown]\tiny $P(3,4)$}] (P_3_4__Rationals) at (14,-3) {$\begin{smallmatrix}&&2\\1&2&4&3&2&1&1&0\end{smallmatrix}$};
	\node[noselEEEEEEEEE, label={[shift={(0.2,-0.55)}, brown]\tiny $P(3,5)$}] (P_3_5__Rationals) at (16,-6) {$\begin{smallmatrix}&&2\\1&2&4&3&2&1&1&1\end{smallmatrix}$};
	\node[noselEEEEEEEEE, label={[shift={(0.2,-0.55)}, brown]\tiny $P(3,6)$}] (P_3_6__Rationals) at (18,-9) {$\begin{smallmatrix}&&2\\1&2&3&2&1&0&0&0\end{smallmatrix}$};
	\node[noselEEEEEEEEE, label={[shift={(0.2,-0.55)}, brown]\tiny $P(3,7)$}] (P_3_7__Rationals) at (20,-12) {$\begin{smallmatrix}&&1\\1&1&2&2&1&0&0&0\end{smallmatrix}$};
	\node[noselEEEEEEEEE, label={[shift={(0.2,-0.55)}, brown]\tiny $P(3,8)$}] (P_3_8__Rationals) at (22,-15) {$\begin{smallmatrix}&&1\\0&0&1&1&1&0&0&0\end{smallmatrix}$};
	\node[noselEEEEEEEEE, label={[shift={(0.2,-0.55)}, brown]\tiny $P(3,9)$}] (P_3_9__Rationals) at (14,1) {$\begin{smallmatrix}&&1\\0&1&2&2&1&1&0&0\end{smallmatrix}$};
	\node[noselEEEEEEEEE, label={[shift={(0.2,-0.55)}, brown]\tiny $P(4,1)$}] (P_4_1__Rationals) at (20,6) {$\begin{smallmatrix}&&1\\0&1&2&2&1&1&1&0\end{smallmatrix}$};
	\node[noselEEEEEEEEE, label={[shift={(0.2,-0.55)}, brown]\tiny $P(4,2)$}] (P_4_2__Rationals) at (18,3) {$\begin{smallmatrix}&&2\\1&2&4&3&2&2&1&0\end{smallmatrix}$};
	\node[noselEEEEEEEEE, label={[shift={(0.2,-0.55)}, brown]\tiny $P(4,3)$}] (P_4_3__Rationals) at (16,0) {$\begin{smallmatrix}&&2\\1&3&5&4&3&2&1&0\end{smallmatrix}$};
	\node[noselEEEEEEEEE, label={[shift={(0.2,-0.55)}, brown]\tiny $P(4,4)$}] (P_4_4__Rationals) at (18,-3) {$\begin{smallmatrix}&&2\\1&3&5&4&3&2&1&1\end{smallmatrix}$};
	\node[noselEEEEEEEEE, label={[shift={(0.2,-0.55)}, brown]\tiny $P(4,5)$}] (P_4_5__Rationals) at (20,-6) {$\begin{smallmatrix}&&2\\1&3&4&3&2&1&0&0\end{smallmatrix}$};
	\node[noselEEEEEEEEE, label={[shift={(0.2,-0.55)}, brown]\tiny $P(4,6)$}] (P_4_6__Rationals) at (22,-9) {$\begin{smallmatrix}&&1\\1&2&3&3&2&1&0&0\end{smallmatrix}$};
	\node[noselEEEEEEEEE, label={[shift={(0.2,-0.55)}, brown]\tiny $P(4,7)$}] (P_4_7__Rationals) at (24,-12) {$\begin{smallmatrix}&&1\\0&1&2&2&2&1&0&0\end{smallmatrix}$};
	\node[noselEEEEEEEEE, label={[shift={(0.2,-0.55)}, brown]\tiny $P(4,8)$}] (P_4_8__Rationals) at (26,-15) {$\begin{smallmatrix}&&0\\0&1&1&1&1&1&0&0\end{smallmatrix}$};
	\node[noselEEEEEEEEE, label={[shift={(0.2,-0.55)}, brown]\tiny $P(4,9)$}] (P_4_9__Rationals) at (18,1) {$\begin{smallmatrix}&&1\\1&2&3&2&2&1&1&0\end{smallmatrix}$};
	\node at (21,3) {$\dots$};
	\node at (21,-3) {$\dots$};
	\draw[blue,->] (P_0_2__Rationals) -- (P_0_1__Rationals);
	\draw[blue,->] (P_0_3__Rationals) -- (P_0_2__Rationals);
	\draw[blue,->] (P_0_3__Rationals) -- (P_0_4__Rationals);
	\draw[blue,->] (P_0_4__Rationals) -- (P_0_5__Rationals);
	\draw[blue,->] (P_0_5__Rationals) -- (P_0_6__Rationals);
	\draw[blue,->] (P_0_6__Rationals) -- (P_0_7__Rationals);
	\draw[blue,->] (P_0_7__Rationals) -- (P_0_8__Rationals);
	\draw[blue,->] (P_0_3__Rationals) -- (P_0_9__Rationals);
	\draw[gray,dashed,->] (P_0_1__Rationals) -- (P_1_1__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[red,->] (P_1_2__Rationals) -- (P_1_1__Rationals);
	\draw[blue,->] (P_0_1__Rationals) -- (P_1_2__Rationals);
	\draw[gray,dashed,->] (P_0_2__Rationals) -- (P_1_2__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[blue,->] (P_1_3__Rationals) -- (P_1_2__Rationals);
	\draw[blue,->] (P_0_2__Rationals) -- (P_1_3__Rationals);
	\draw[gray,dashed,->] (P_0_3__Rationals) -- (P_1_3__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[blue,->] (P_0_4__Rationals) -- (P_1_3__Rationals);
	\draw[blue,->] (P_0_9__Rationals) -- (P_1_3__Rationals);
	\draw[gray,dashed,->] (P_0_4__Rationals) -- (P_1_4__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[blue,->] (P_0_5__Rationals) -- (P_1_4__Rationals);
	\draw[blue,->] (P_1_3__Rationals) -- (P_1_4__Rationals);
	\draw[gray,dashed,->] (P_0_5__Rationals) -- (P_1_5__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[blue,->] (P_0_6__Rationals) -- (P_1_5__Rationals);
	\draw[blue,->] (P_1_4__Rationals) -- (P_1_5__Rationals);
	\draw[gray,dashed,->] (P_0_6__Rationals) -- (P_1_6__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[blue,->] (P_0_7__Rationals) -- (P_1_6__Rationals);
	\draw[blue,->] (P_1_5__Rationals) -- (P_1_6__Rationals);
	\draw[gray,dashed,->] (P_0_7__Rationals) -- (P_1_7__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[blue,->] (P_0_8__Rationals) -- (P_1_7__Rationals);
	\draw[blue,->] (P_1_6__Rationals) -- (P_1_7__Rationals);
	\draw[gray,dashed,->] (P_0_8__Rationals) -- (P_1_8__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[red,->] (P_1_7__Rationals) -- (P_1_8__Rationals);
	\draw[gray,dashed,->] (P_0_9__Rationals) -- (P_1_9__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[red,->] (P_1_3__Rationals) -- (P_1_9__Rationals);
	\draw[gray,dashed,->] (P_1_1__Rationals) -- (P_2_1__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[red,->] (P_2_2__Rationals) -- (P_2_1__Rationals);
	\draw[blue,->] (P_1_1__Rationals) -- (P_2_2__Rationals);
	\draw[gray,dashed,->] (P_1_2__Rationals) -- (P_2_2__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[red,->] (P_2_3__Rationals) -- (P_2_2__Rationals);
	\draw[blue,->] (P_1_2__Rationals) -- (P_2_3__Rationals);
	\draw[gray,dashed,->] (P_1_3__Rationals) -- (P_2_3__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[blue,->] (P_1_4__Rationals) -- (P_2_3__Rationals);
	\draw[blue,->] (P_1_9__Rationals) -- (P_2_3__Rationals);
	\draw[gray,dashed,->] (P_1_4__Rationals) -- (P_2_4__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[blue,->] (P_1_5__Rationals) -- (P_2_4__Rationals);
	\draw[blue,->] (P_2_3__Rationals) -- (P_2_4__Rationals);
	\draw[gray,dashed,->] (P_1_5__Rationals) -- (P_2_5__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[blue,->] (P_1_6__Rationals) -- (P_2_5__Rationals);
	\draw[blue,->] (P_2_4__Rationals) -- (P_2_5__Rationals);
	\draw[gray,dashed,->] (P_1_6__Rationals) -- (P_2_6__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[blue,->] (P_1_7__Rationals) -- (P_2_6__Rationals);
	\draw[blue,->] (P_2_5__Rationals) -- (P_2_6__Rationals);
	\draw[gray,dashed,->] (P_1_7__Rationals) -- (P_2_7__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[blue,->] (P_1_8__Rationals) -- (P_2_7__Rationals);
	\draw[red,->] (P_2_6__Rationals) -- (P_2_7__Rationals);
	\draw[gray,dashed,->] (P_1_8__Rationals) -- (P_2_8__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[red,->] (P_2_7__Rationals) -- (P_2_8__Rationals);
	\draw[gray,dashed,->] (P_1_9__Rationals) -- (P_2_9__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[red,->] (P_2_3__Rationals) -- (P_2_9__Rationals);
	\draw[gray,dashed,->] (P_2_1__Rationals) -- (P_3_1__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[red,->] (P_3_2__Rationals) -- (P_3_1__Rationals);
	\draw[blue,->] (P_2_1__Rationals) -- (P_3_2__Rationals);
	\draw[gray,dashed,->] (P_2_2__Rationals) -- (P_3_2__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[red,->] (P_3_3__Rationals) -- (P_3_2__Rationals);
	\draw[blue,->] (P_2_2__Rationals) -- (P_3_3__Rationals);
	\draw[gray,dashed,->] (P_2_3__Rationals) -- (P_3_3__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[blue,->] (P_2_4__Rationals) -- (P_3_3__Rationals);
	\draw[blue,->] (P_2_9__Rationals) -- (P_3_3__Rationals);
	\draw[gray,dashed,->] (P_2_4__Rationals) -- (P_3_4__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[blue,->] (P_2_5__Rationals) -- (P_3_4__Rationals);
	\draw[blue,->] (P_3_3__Rationals) -- (P_3_4__Rationals);
	\draw[gray,dashed,->] (P_2_5__Rationals) -- (P_3_5__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[blue,->] (P_2_6__Rationals) -- (P_3_5__Rationals);
	\draw[blue,->] (P_3_4__Rationals) -- (P_3_5__Rationals);
	\draw[gray,dashed,->] (P_2_6__Rationals) -- (P_3_6__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[blue,->] (P_2_7__Rationals) -- (P_3_6__Rationals);
	\draw[red,->] (P_3_5__Rationals) -- (P_3_6__Rationals);
	\draw[gray,dashed,->] (P_2_7__Rationals) -- (P_3_7__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[blue,->] (P_2_8__Rationals) -- (P_3_7__Rationals);
	\draw[red,->] (P_3_6__Rationals) -- (P_3_7__Rationals);
	\draw[gray,dashed,->] (P_2_8__Rationals) -- (P_3_8__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[red,->] (P_3_7__Rationals) -- (P_3_8__Rationals);
	\draw[gray,dashed,->] (P_2_9__Rationals) -- (P_3_9__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[red,->] (P_3_3__Rationals) -- (P_3_9__Rationals);
	\draw[gray,dashed,->] (P_3_1__Rationals) -- (P_4_1__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[red,->] (P_4_2__Rationals) -- (P_4_1__Rationals);
	\draw[blue,->] (P_3_1__Rationals) -- (P_4_2__Rationals);
	\draw[gray,dashed,->] (P_3_2__Rationals) -- (P_4_2__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[red,->] (P_4_3__Rationals) -- (P_4_2__Rationals);
	\draw[blue,->] (P_3_2__Rationals) -- (P_4_3__Rationals);
	\draw[gray,dashed,->] (P_3_3__Rationals) -- (P_4_3__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[blue,->] (P_3_4__Rationals) -- (P_4_3__Rationals);
	\draw[blue,->] (P_3_9__Rationals) -- (P_4_3__Rationals);
	\draw[gray,dashed,->] (P_3_4__Rationals) -- (P_4_4__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[blue,->] (P_3_5__Rationals) -- (P_4_4__Rationals);
	\draw[blue,->] (P_4_3__Rationals) -- (P_4_4__Rationals);
	\draw[gray,dashed,->] (P_3_5__Rationals) -- (P_4_5__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[blue,->] (P_3_6__Rationals) -- (P_4_5__Rationals);
	\draw[red,->] (P_4_4__Rationals) -- (P_4_5__Rationals);
	\draw[gray,dashed,->] (P_3_6__Rationals) -- (P_4_6__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[blue,->] (P_3_7__Rationals) -- (P_4_6__Rationals);
	\draw[red,->] (P_4_5__Rationals) -- (P_4_6__Rationals);
	\draw[gray,dashed,->] (P_3_7__Rationals) -- (P_4_7__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[blue,->] (P_3_8__Rationals) -- (P_4_7__Rationals);
	\draw[red,->] (P_4_6__Rationals) -- (P_4_7__Rationals);
	\draw[gray,dashed,->] (P_3_8__Rationals) -- (P_4_8__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[red,->] (P_4_7__Rationals) -- (P_4_8__Rationals);
	\draw[gray,dashed,->] (P_3_9__Rationals) -- (P_4_9__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[red,->] (P_4_3__Rationals) -- (P_4_9__Rationals);
\end{tikzpicture}
\end{scaletikzpicturetowidth}
\end{center}

\end{figure}
\begin{figure}[ht]

\begin{center}
\begin{scaletikzpicturetowidth}{\textwidth}
\begin{tikzpicture}[scale=\tikzscale]
	\node at (-1,3) {$\dots$};
	\node at (-1,-3) {$\dots$};
	\node[noselEEEEEEEEE, label={[shift={(0.2,-0.55)}, brown]\tiny $P(5,1)$}] (P_5_1__Rationals) at (4,6) {$\begin{smallmatrix}&&1\\1&2&3&2&2&1&1&1\end{smallmatrix}$};
	\node[noselEEEEEEEEE, label={[shift={(0.2,-0.55)}, brown]\tiny $P(5,2)$}] (P_5_2__Rationals) at (2,3) {$\begin{smallmatrix}&&2\\1&3&5&4&3&2&2&1\end{smallmatrix}$};
	\node[seloneEEEEEEEEE, label={[shift={(0.2,-0.55)}, brown]\tiny $P(5,3)$}] (P_5_3__Rationals) at (0,0) {$\begin{smallmatrix}&&3\\2&4&7&5&4&3&2&1\end{smallmatrix}$};
	\node[noselEEEEEEEEE, label={[shift={(0.2,-0.55)}, brown]\tiny $P(5,4)$}] (P_5_4__Rationals) at (2,-3) {$\begin{smallmatrix}&&3\\2&4&6&4&3&2&1&0\end{smallmatrix}$};
	\node[noselEEEEEEEEE, label={[shift={(0.2,-0.55)}, brown]\tiny $P(5,5)$}] (P_5_5__Rationals) at (4,-6) {$\begin{smallmatrix}&&2\\2&3&5&4&3&2&1&0\end{smallmatrix}$};
	\node[noselEEEEEEEEE, label={[shift={(0.2,-0.55)}, brown]\tiny $P(5,6)$}] (P_5_6__Rationals) at (6,-9) {$\begin{smallmatrix}&&2\\1&2&4&3&3&2&1&0\end{smallmatrix}$};
	\node[noselEEEEEEEEE, label={[shift={(0.2,-0.55)}, brown]\tiny $P(5,7)$}] (P_5_7__Rationals) at (8,-12) {$\begin{smallmatrix}&&1\\1&2&3&2&2&2&1&0\end{smallmatrix}$};
	\node[noselEEEEEEEEE, label={[shift={(0.2,-0.55)}, brown]\tiny $P(5,8)$}] (P_5_8__Rationals) at (10,-15) {$\begin{smallmatrix}&&1\\1&1&2&1&1&1&1&0\end{smallmatrix}$};
	\node[noselEEEEEEEEE, label={[shift={(0.2,-0.55)}, brown]\tiny $P(5,9)$}] (P_5_9__Rationals) at (2,1) {$\begin{smallmatrix}&&2\\1&2&4&3&2&2&1&1\end{smallmatrix}$};
	\node[noselEEEEEEEEE, label={[shift={(0.2,-0.55)}, brown]\tiny $P(6,1)$}] (P_6_1__Rationals) at (8,6) {$\begin{smallmatrix}&&2\\1&2&3&2&1&1&0&0\end{smallmatrix}$};
	\node[noselEEEEEEEEE, label={[shift={(0.2,-0.55)}, brown]\tiny $P(6,2)$}] (P_6_2__Rationals) at (6,3) {$\begin{smallmatrix}&&3\\2&4&6&4&3&2&1&1\end{smallmatrix}$};
	\node[noselEEEEEEEEE, label={[shift={(0.2,-0.55)}, brown]\tiny $P(6,3)$}] (P_6_3__Rationals) at (4,0) {$\begin{smallmatrix}&&4\\2&5&8&6&4&3&2&1\end{smallmatrix}$};
	\node[seloneEEEEEEEEE, label={[shift={(0.2,-0.55)}, brown]\tiny $P(6,4)$}] (P_6_4__Rationals) at (6,-3) {$\begin{smallmatrix}&&3\\2&4&7&6&4&3&2&1\end{smallmatrix}$};
	\node[seloneEEEEEEEEE, label={[shift={(0.2,-0.55)}, brown]\tiny $P(6,5)$}] (P_6_5__Rationals) at (8,-6) {$\begin{smallmatrix}&&3\\1&3&6&5&4&3&2&1\end{smallmatrix}$};
	\node[noselEEEEEEEEE, label={[shift={(0.2,-0.55)}, brown]\tiny $P(6,6)$}] (P_6_6__Rationals) at (10,-9) {$\begin{smallmatrix}&&2\\1&3&5&4&3&3&2&1\end{smallmatrix}$};
	\node[noselEEEEEEEEE, label={[shift={(0.2,-0.55)}, brown]\tiny $P(6,7)$}] (P_6_7__Rationals) at (12,-12) {$\begin{smallmatrix}&&2\\1&2&4&3&2&2&2&1\end{smallmatrix}$};
	\node[noselEEEEEEEEE, label={[shift={(0.2,-0.55)}, brown]\tiny $P(6,8)$}] (P_6_8__Rationals) at (14,-15) {$\begin{smallmatrix}&&1\\0&1&2&2&1&1&1&1\end{smallmatrix}$};
	\node[noselEEEEEEEEE, label={[shift={(0.2,-0.55)}, brown]\tiny $P(6,9)$}] (P_6_9__Rationals) at (6,1) {$\begin{smallmatrix}&&2\\1&3&4&3&2&1&1&0\end{smallmatrix}$};
	\node[noselEEEEEEEEE, label={[shift={(0.2,-0.55)}, brown]\tiny $P(7,1)$}] (P_7_1__Rationals) at (12,6) {$\begin{smallmatrix}&&1\\1&2&3&3&2&1&1&0\end{smallmatrix}$};
	\node[noselEEEEEEEEE, label={[shift={(0.2,-0.55)}, brown]\tiny $P(7,2)$}] (P_7_2__Rationals) at (10,3) {$\begin{smallmatrix}&&3\\2&4&6&5&3&2&1&0\end{smallmatrix}$};
	\node[noselEEEEEEEEE, label={[shift={(0.2,-0.55)}, brown]\tiny $P(7,3)$}] (P_7_3__Rationals) at (8,0) {$\begin{smallmatrix}&&4\\3&6&9&7&5&3&2&1\end{smallmatrix}$};
	\node[noselEEEEEEEEE, label={[shift={(0.2,-0.55)}, brown]\tiny $P(7,4)$}] (P_7_4__Rationals) at (10,-3) {$\begin{smallmatrix}&&4\\2&5&8&6&5&3&2&1\end{smallmatrix}$};
	\node[noselEEEEEEEEE, label={[shift={(0.2,-0.55)}, brown]\tiny $P(7,5)$}] (P_7_5__Rationals) at (12,-6) {$\begin{smallmatrix}&&3\\2&5&7&5&4&3&2&1\end{smallmatrix}$};
	\node[noselEEEEEEEEE, label={[shift={(0.2,-0.55)}, brown]\tiny $P(7,6)$}] (P_7_6__Rationals) at (14,-9) {$\begin{smallmatrix}&&3\\2&4&6&4&3&2&2&1\end{smallmatrix}$};
	\node[noselEEEEEEEEE, label={[shift={(0.2,-0.55)}, brown]\tiny $P(7,7)$}] (P_7_7__Rationals) at (16,-12) {$\begin{smallmatrix}&&2\\1&3&4&3&2&1&1&1\end{smallmatrix}$};
	\node[noselEEEEEEEEE, label={[shift={(0.2,-0.55)}, brown]\tiny $P(7,8)$}] (P_7_8__Rationals) at (18,-15) {$\begin{smallmatrix}&&1\\1&2&2&1&1&0&0&0\end{smallmatrix}$};
	\node[noselEEEEEEEEE, label={[shift={(0.2,-0.55)}, brown]\tiny $P(7,9)$}] (P_7_9__Rationals) at (10,1) {$\begin{smallmatrix}&&2\\2&3&5&4&3&2&1&1\end{smallmatrix}$};
	\node[noselEEEEEEEEE, label={[shift={(0.2,-0.55)}, brown]\tiny $P(8,1)$}] (P_8_1__Rationals) at (16,6) {$\begin{smallmatrix}&&2\\1&2&4&3&3&2&1&1\end{smallmatrix}$};
	\node[seloneEEEEEEEEE, label={[shift={(0.2,-0.55)}, brown]\tiny $P(8,2)$}] (P_8_2__Rationals) at (14,3) {$\begin{smallmatrix}&&3\\2&4&7&6&5&3&2&1\end{smallmatrix}$};
	\node[noselEEEEEEEEE, label={[shift={(0.2,-0.55)}, brown]\tiny $P(8,3)$}] (P_8_3__Rationals) at (12,0) {$\begin{smallmatrix}&&5\\3&6&10&8&6&4&2&1\end{smallmatrix}$};
	\node[noselEEEEEEEEE, label={[shift={(0.2,-0.55)}, brown]\tiny $P(8,4)$}] (P_8_4__Rationals) at (14,-3) {$\begin{smallmatrix}&&4\\3&6&9&7&5&4&2&1\end{smallmatrix}$};
	\node[noselEEEEEEEEE, label={[shift={(0.2,-0.55)}, brown]\tiny $P(8,5)$}] (P_8_5__Rationals) at (16,-6) {$\begin{smallmatrix}&&4\\3&5&8&6&4&3&2&1\end{smallmatrix}$};
	\node[noselEEEEEEEEE, label={[shift={(0.2,-0.55)}, brown]\tiny $P(8,6)$}] (P_8_6__Rationals) at (18,-9) {$\begin{smallmatrix}&&3\\2&4&6&5&3&2&1&1\end{smallmatrix}$};
	\node[noselEEEEEEEEE, label={[shift={(0.2,-0.55)}, brown]\tiny $P(8,7)$}] (P_8_7__Rationals) at (20,-12) {$\begin{smallmatrix}&&2\\2&3&4&3&2&1&0&0\end{smallmatrix}$};
	\node[noselEEEEEEEEE, label={[shift={(0.2,-0.55)}, brown]\tiny $P(8,8)$}] (P_8_8__Rationals) at (22,-15) {$\begin{smallmatrix}&&1\\1&1&2&2&1&1&0&0\end{smallmatrix}$};
	\node[noselEEEEEEEEE, label={[shift={(0.2,-0.55)}, brown]\tiny $P(8,9)$}] (P_8_9__Rationals) at (14,1) {$\begin{smallmatrix}&&3\\1&3&5&4&3&2&1&0\end{smallmatrix}$};
	\node[noselEEEEEEEEE, label={[shift={(0.2,-0.55)}, brown]\tiny $P(9,1)$}] (P_9_1__Rationals) at (20,6) {$\begin{smallmatrix}&&2\\1&3&4&3&2&2&1&0\end{smallmatrix}$};
	\node[noselEEEEEEEEE, label={[shift={(0.2,-0.55)}, brown]\tiny $P(9,2)$}] (P_9_2__Rationals) at (18,3) {$\begin{smallmatrix}&&4\\2&5&8&6&5&4&2&1\end{smallmatrix}$};
	\node[noselEEEEEEEEE, label={[shift={(0.2,-0.55)}, brown]\tiny $P(9,3)$}] (P_9_3__Rationals) at (16,0) {$\begin{smallmatrix}&&5\\3&7&11&9&7&5&3&1\end{smallmatrix}$};
	\node[noselEEEEEEEEE, label={[shift={(0.2,-0.55)}, brown]\tiny $P(9,4)$}] (P_9_4__Rationals) at (18,-3) {$\begin{smallmatrix}&&5\\3&6&10&8&6&4&3&1\end{smallmatrix}$};
	\node[noselEEEEEEEEE, label={[shift={(0.2,-0.55)}, brown]\tiny $P(9,5)$}] (P_9_5__Rationals) at (20,-6) {$\begin{smallmatrix}&&4\\2&5&8&7&5&3&2&1\end{smallmatrix}$};
	\node[noselEEEEEEEEE, label={[shift={(0.2,-0.55)}, brown]\tiny $P(9,6)$}] (P_9_6__Rationals) at (22,-9) {$\begin{smallmatrix}&&3\\2&4&6&5&4&2&1&0\end{smallmatrix}$};
	\node[noselEEEEEEEEE, label={[shift={(0.2,-0.55)}, brown]\tiny $P(9,7)$}] (P_9_7__Rationals) at (24,-12) {$\begin{smallmatrix}&&2\\1&2&4&4&3&2&1&0\end{smallmatrix}$};
	\node[noselEEEEEEEEE, label={[shift={(0.2,-0.55)}, brown]\tiny $P(9,8)$}] (P_9_8__Rationals) at (26,-15) {$\begin{smallmatrix}&&1\\0&1&2&2&2&1&1&0\end{smallmatrix}$};
	\node[seloneEEEEEEEEE, label={[shift={(0.2,-0.55)}, brown]\tiny $P(9,9)$}] (P_9_9__Rationals) at (18,1) {$\begin{smallmatrix}&&2\\2&4&6&5&4&3&2&1\end{smallmatrix}$};
	\node at (21,3) {$\dots$};
	\node at (21,-3) {$\dots$};
	\draw[red,->] (P_5_2__Rationals) -- (P_5_1__Rationals);
	\draw[red,->] (P_5_3__Rationals) -- (P_5_2__Rationals);
	\draw[red,->] (P_5_3__Rationals) -- (P_5_4__Rationals);
	\draw[red,->] (P_5_4__Rationals) -- (P_5_5__Rationals);
	\draw[red,->] (P_5_5__Rationals) -- (P_5_6__Rationals);
	\draw[red,->] (P_5_6__Rationals) -- (P_5_7__Rationals);
	\draw[red,->] (P_5_7__Rationals) -- (P_5_8__Rationals);
	\draw[red,->] (P_5_3__Rationals) -- (P_5_9__Rationals);
	\draw[gray,dashed,->] (P_5_1__Rationals) -- (P_6_1__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[red,->] (P_6_2__Rationals) -- (P_6_1__Rationals);
	\draw[blue,->] (P_5_1__Rationals) -- (P_6_2__Rationals);
	\draw[gray,dashed,->] (P_5_2__Rationals) -- (P_6_2__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[red,->] (P_6_3__Rationals) -- (P_6_2__Rationals);
	\draw[blue,->] (P_5_2__Rationals) -- (P_6_3__Rationals);
	\draw[gray,dashed,->] (P_5_3__Rationals) -- (P_6_3__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[blue,->] (P_5_4__Rationals) -- (P_6_3__Rationals);
	\draw[blue,->] (P_5_9__Rationals) -- (P_6_3__Rationals);
	\draw[gray,dashed,->] (P_5_4__Rationals) -- (P_6_4__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[blue,->] (P_5_5__Rationals) -- (P_6_4__Rationals);
	\draw[red,->] (P_6_3__Rationals) -- (P_6_4__Rationals);
	\draw[gray,dashed,->] (P_5_5__Rationals) -- (P_6_5__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[blue,->] (P_5_6__Rationals) -- (P_6_5__Rationals);
	\draw[red,->] (P_6_4__Rationals) -- (P_6_5__Rationals);
	\draw[gray,dashed,->] (P_5_6__Rationals) -- (P_6_6__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[blue,->] (P_5_7__Rationals) -- (P_6_6__Rationals);
	\draw[red,->] (P_6_5__Rationals) -- (P_6_6__Rationals);
	\draw[gray,dashed,->] (P_5_7__Rationals) -- (P_6_7__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[blue,->] (P_5_8__Rationals) -- (P_6_7__Rationals);
	\draw[red,->] (P_6_6__Rationals) -- (P_6_7__Rationals);
	\draw[gray,dashed,->] (P_5_8__Rationals) -- (P_6_8__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[red,->] (P_6_7__Rationals) -- (P_6_8__Rationals);
	\draw[gray,dashed,->] (P_5_9__Rationals) -- (P_6_9__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[red,->] (P_6_3__Rationals) -- (P_6_9__Rationals);
	\draw[gray,dashed,->] (P_6_1__Rationals) -- (P_7_1__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[red,->] (P_7_2__Rationals) -- (P_7_1__Rationals);
	\draw[blue,->] (P_6_1__Rationals) -- (P_7_2__Rationals);
	\draw[gray,dashed,->] (P_6_2__Rationals) -- (P_7_2__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[red,->] (P_7_3__Rationals) -- (P_7_2__Rationals);
	\draw[blue,->] (P_6_2__Rationals) -- (P_7_3__Rationals);
	\draw[gray,dashed,->] (P_6_3__Rationals) -- (P_7_3__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[blue,->] (P_6_4__Rationals) -- (P_7_3__Rationals);
	\draw[blue,->] (P_6_9__Rationals) -- (P_7_3__Rationals);
	\draw[gray,dashed,->] (P_6_4__Rationals) -- (P_7_4__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[blue,->] (P_6_5__Rationals) -- (P_7_4__Rationals);
	\draw[red,->] (P_7_3__Rationals) -- (P_7_4__Rationals);
	\draw[gray,dashed,->] (P_6_5__Rationals) -- (P_7_5__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[blue,->] (P_6_6__Rationals) -- (P_7_5__Rationals);
	\draw[red,->] (P_7_4__Rationals) -- (P_7_5__Rationals);
	\draw[gray,dashed,->] (P_6_6__Rationals) -- (P_7_6__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[blue,->] (P_6_7__Rationals) -- (P_7_6__Rationals);
	\draw[red,->] (P_7_5__Rationals) -- (P_7_6__Rationals);
	\draw[gray,dashed,->] (P_6_7__Rationals) -- (P_7_7__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[blue,->] (P_6_8__Rationals) -- (P_7_7__Rationals);
	\draw[red,->] (P_7_6__Rationals) -- (P_7_7__Rationals);
	\draw[gray,dashed,->] (P_6_8__Rationals) -- (P_7_8__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[red,->] (P_7_7__Rationals) -- (P_7_8__Rationals);
	\draw[gray,dashed,->] (P_6_9__Rationals) -- (P_7_9__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[red,->] (P_7_3__Rationals) -- (P_7_9__Rationals);
	\draw[gray,dashed,->] (P_7_1__Rationals) -- (P_8_1__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[red,->] (P_8_2__Rationals) -- (P_8_1__Rationals);
	\draw[blue,->] (P_7_1__Rationals) -- (P_8_2__Rationals);
	\draw[gray,dashed,->] (P_7_2__Rationals) -- (P_8_2__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[red,->] (P_8_3__Rationals) -- (P_8_2__Rationals);
	\draw[blue,->] (P_7_2__Rationals) -- (P_8_3__Rationals);
	\draw[gray,dashed,->] (P_7_3__Rationals) -- (P_8_3__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[blue,->] (P_7_4__Rationals) -- (P_8_3__Rationals);
	\draw[blue,->] (P_7_9__Rationals) -- (P_8_3__Rationals);
	\draw[gray,dashed,->] (P_7_4__Rationals) -- (P_8_4__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[blue,->] (P_7_5__Rationals) -- (P_8_4__Rationals);
	\draw[red,->] (P_8_3__Rationals) -- (P_8_4__Rationals);
	\draw[gray,dashed,->] (P_7_5__Rationals) -- (P_8_5__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[blue,->] (P_7_6__Rationals) -- (P_8_5__Rationals);
	\draw[red,->] (P_8_4__Rationals) -- (P_8_5__Rationals);
	\draw[gray,dashed,->] (P_7_6__Rationals) -- (P_8_6__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[blue,->] (P_7_7__Rationals) -- (P_8_6__Rationals);
	\draw[red,->] (P_8_5__Rationals) -- (P_8_6__Rationals);
	\draw[gray,dashed,->] (P_7_7__Rationals) -- (P_8_7__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[blue,->] (P_7_8__Rationals) -- (P_8_7__Rationals);
	\draw[red,->] (P_8_6__Rationals) -- (P_8_7__Rationals);
	\draw[gray,dashed,->] (P_7_8__Rationals) -- (P_8_8__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[red,->] (P_8_7__Rationals) -- (P_8_8__Rationals);
	\draw[gray,dashed,->] (P_7_9__Rationals) -- (P_8_9__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[red,->] (P_8_3__Rationals) -- (P_8_9__Rationals);
	\draw[gray,dashed,->] (P_8_1__Rationals) -- (P_9_1__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[red,->] (P_9_2__Rationals) -- (P_9_1__Rationals);
	\draw[blue,->] (P_8_1__Rationals) -- (P_9_2__Rationals);
	\draw[gray,dashed,->] (P_8_2__Rationals) -- (P_9_2__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[red,->] (P_9_3__Rationals) -- (P_9_2__Rationals);
	\draw[blue,->] (P_8_2__Rationals) -- (P_9_3__Rationals);
	\draw[gray,dashed,->] (P_8_3__Rationals) -- (P_9_3__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[blue,->] (P_8_4__Rationals) -- (P_9_3__Rationals);
	\draw[blue,->] (P_8_9__Rationals) -- (P_9_3__Rationals);
	\draw[gray,dashed,->] (P_8_4__Rationals) -- (P_9_4__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[blue,->] (P_8_5__Rationals) -- (P_9_4__Rationals);
	\draw[red,->] (P_9_3__Rationals) -- (P_9_4__Rationals);
	\draw[gray,dashed,->] (P_8_5__Rationals) -- (P_9_5__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[blue,->] (P_8_6__Rationals) -- (P_9_5__Rationals);
	\draw[red,->] (P_9_4__Rationals) -- (P_9_5__Rationals);
	\draw[gray,dashed,->] (P_8_6__Rationals) -- (P_9_6__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[blue,->] (P_8_7__Rationals) -- (P_9_6__Rationals);
	\draw[red,->] (P_9_5__Rationals) -- (P_9_6__Rationals);
	\draw[gray,dashed,->] (P_8_7__Rationals) -- (P_9_7__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[blue,->] (P_8_8__Rationals) -- (P_9_7__Rationals);
	\draw[red,->] (P_9_6__Rationals) -- (P_9_7__Rationals);
	\draw[gray,dashed,->] (P_8_8__Rationals) -- (P_9_8__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[red,->] (P_9_7__Rationals) -- (P_9_8__Rationals);
	\draw[gray,dashed,->] (P_8_9__Rationals) -- (P_9_9__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[red,->] (P_9_3__Rationals) -- (P_9_9__Rationals);
\end{tikzpicture}
\end{scaletikzpicturetowidth}
\end{center}

\end{figure}
\begin{figure}[ht]

\begin{center}
\begin{scaletikzpicturetowidth}{\textwidth}
\begin{tikzpicture}[scale=\tikzscale]
	\node at (-1,3) {$\dots$};
	\node at (-1,-3) {$\dots$};
	\node[noselEEEEEEEEE, label={[shift={(0.2,-0.55)}, brown]\tiny $P(10,1)$}] (P_10_1__Rationals) at (4,6) {$\begin{smallmatrix}&&2\\2&3&5&4&3&2&2&1\end{smallmatrix}$};
	\node[noselEEEEEEEEE, label={[shift={(0.2,-0.55)}, brown]\tiny $P(10,2)$}] (P_10_2__Rationals) at (2,3) {$\begin{smallmatrix}&&4\\3&6&9&7&5&4&3&1\end{smallmatrix}$};
	\node[noselEEEEEEEEE, label={[shift={(0.2,-0.55)}, brown]\tiny $P(10,3)$}] (P_10_3__Rationals) at (0,0) {$\begin{smallmatrix}&&6\\4&8&13&10&8&6&4&2\end{smallmatrix}$};
	\node[noselEEEEEEEEE, label={[shift={(0.2,-0.55)}, brown]\tiny $P(10,4)$}] (P_10_4__Rationals) at (2,-3) {$\begin{smallmatrix}&&5\\3&7&11&9&7&5&3&2\end{smallmatrix}$};
	\node[noselEEEEEEEEE, label={[shift={(0.2,-0.55)}, brown]\tiny $P(10,5)$}] (P_10_5__Rationals) at (4,-6) {$\begin{smallmatrix}&&4\\3&6&9&7&6&4&2&1\end{smallmatrix}$};
	\node[seloneEEEEEEEEE, label={[shift={(0.2,-0.55)}, brown]\tiny $P(10,6)$}] (P_10_6__Rationals) at (6,-9) {$\begin{smallmatrix}&&3\\2&4&7&6&5&4&2&1\end{smallmatrix}$};
	\node[noselEEEEEEEEE, label={[shift={(0.2,-0.55)}, brown]\tiny $P(10,7)$}] (P_10_7__Rationals) at (8,-12) {$\begin{smallmatrix}&&2\\1&3&5&4&4&3&2&1\end{smallmatrix}$};
	\node[noselEEEEEEEEE, label={[shift={(0.2,-0.55)}, brown]\tiny $P(10,8)$}] (P_10_8__Rationals) at (10,-15) {$\begin{smallmatrix}&&1\\1&2&3&2&2&2&1&1\end{smallmatrix}$};
	\node[noselEEEEEEEEE, label={[shift={(0.2,-0.55)}, brown]\tiny $P(10,9)$}] (P_10_9__Rationals) at (2,1) {$\begin{smallmatrix}&&4\\2&4&7&5&4&3&2&1\end{smallmatrix}$};
	\node[noselEEEEEEEEE, label={[shift={(0.2,-0.55)}, brown]\tiny $P(11,1)$}] (P_11_1__Rationals) at (8,6) {$\begin{smallmatrix}&&3\\1&3&5&4&3&2&1&1\end{smallmatrix}$};
	\node[noselEEEEEEEEE, label={[shift={(0.2,-0.55)}, brown]\tiny $P(11,2)$}] (P_11_2__Rationals) at (6,3) {$\begin{smallmatrix}&&5\\3&6&10&8&6&4&3&2\end{smallmatrix}$};
	\node[noselEEEEEEEEE, label={[shift={(0.2,-0.55)}, brown]\tiny $P(11,3)$}] (P_11_3__Rationals) at (4,0) {$\begin{smallmatrix}&&7\\4&9&14&11&8&6&4&2\end{smallmatrix}$};
	\node[noselEEEEEEEEE, label={[shift={(0.2,-0.55)}, brown]\tiny $P(11,4)$}] (P_11_4__Rationals) at (6,-3) {$\begin{smallmatrix}&&6\\4&8&12&9&7&5&3&1\end{smallmatrix}$};
	\node[noselEEEEEEEEE, label={[shift={(0.2,-0.55)}, brown]\tiny $P(11,5)$}] (P_11_5__Rationals) at (8,-6) {$\begin{smallmatrix}&&5\\3&6&10&8&6&5&3&1\end{smallmatrix}$};
	\node[noselEEEEEEEEE, label={[shift={(0.2,-0.55)}, brown]\tiny $P(11,6)$}] (P_11_6__Rationals) at (10,-9) {$\begin{smallmatrix}&&4\\2&5&8&6&5&4&3&1\end{smallmatrix}$};
	\node[noselEEEEEEEEE, label={[shift={(0.2,-0.55)}, brown]\tiny $P(11,7)$}] (P_11_7__Rationals) at (12,-12) {$\begin{smallmatrix}&&3\\2&4&6&4&3&3&2&1\end{smallmatrix}$};
	\node[noselEEEEEEEEE, label={[shift={(0.2,-0.55)}, brown]\tiny $P(11,8)$}] (P_11_8__Rationals) at (14,-15) {$\begin{smallmatrix}&&2\\1&2&3&2&1&1&1&0\end{smallmatrix}$};
	\node[noselEEEEEEEEE, label={[shift={(0.2,-0.55)}, brown]\tiny $P(11,9)$}] (P_11_9__Rationals) at (6,1) {$\begin{smallmatrix}&&3\\2&5&7&6&4&3&2&1\end{smallmatrix}$};
	\node[noselEEEEEEEEE, label={[shift={(0.2,-0.55)}, brown]\tiny $P(12,1)$}] (P_12_1__Rationals) at (12,6) {$\begin{smallmatrix}&&2\\2&4&5&4&3&2&1&0\end{smallmatrix}$};
	\node[noselEEEEEEEEE, label={[shift={(0.2,-0.55)}, brown]\tiny $P(12,2)$}] (P_12_2__Rationals) at (10,3) {$\begin{smallmatrix}&&5\\3&7&10&8&6&4&2&1\end{smallmatrix}$};
	\node[noselEEEEEEEEE, label={[shift={(0.2,-0.55)}, brown]\tiny $P(12,3)$}] (P_12_3__Rationals) at (8,0) {$\begin{smallmatrix}&&7\\5&10&15&12&9&6&4&2\end{smallmatrix}$};
	\node[noselEEEEEEEEE, label={[shift={(0.2,-0.55)}, brown]\tiny $P(12,4)$}] (P_12_4__Rationals) at (10,-3) {$\begin{smallmatrix}&&6\\4&8&13&11&8&6&4&2\end{smallmatrix}$};
	\node[noselEEEEEEEEE, label={[shift={(0.2,-0.55)}, brown]\tiny $P(12,5)$}] (P_12_5__Rationals) at (12,-6) {$\begin{smallmatrix}&&5\\3&7&11&9&7&5&4&2\end{smallmatrix}$};
	\node[noselEEEEEEEEE, label={[shift={(0.2,-0.55)}, brown]\tiny $P(12,6)$}] (P_12_6__Rationals) at (14,-9) {$\begin{smallmatrix}&&4\\3&6&9&7&5&4&3&2\end{smallmatrix}$};
	\node[noselEEEEEEEEE, label={[shift={(0.2,-0.55)}, brown]\tiny $P(12,7)$}] (P_12_7__Rationals) at (16,-12) {$\begin{smallmatrix}&&3\\2&4&6&5&3&2&2&1\end{smallmatrix}$};
	\node[noselEEEEEEEEE, label={[shift={(0.2,-0.55)}, brown]\tiny $P(12,8)$}] (P_12_8__Rationals) at (18,-15) {$\begin{smallmatrix}&&1\\1&2&3&3&2&1&1&1\end{smallmatrix}$};
	\node[noselEEEEEEEEE, label={[shift={(0.2,-0.55)}, brown]\tiny $P(12,9)$}] (P_12_9__Rationals) at (10,1) {$\begin{smallmatrix}&&4\\3&5&8&6&5&3&2&1\end{smallmatrix}$};
	\node[seloneEEEEEEEEE, label={[shift={(0.2,-0.55)}, brown]\tiny $P(13,1)$}] (P_13_1__Rationals) at (16,6) {$\begin{smallmatrix}&&3\\2&3&6&5&4&3&2&1\end{smallmatrix}$};
	\node[noselEEEEEEEEE, label={[shift={(0.2,-0.55)}, brown]\tiny $P(13,2)$}] (P_13_2__Rationals) at (14,3) {$\begin{smallmatrix}&&5\\4&7&11&9&7&5&3&1\end{smallmatrix}$};
	\node[noselEEEEEEEEE, label={[shift={(0.2,-0.55)}, brown]\tiny $P(13,3)$}] (P_13_3__Rationals) at (12,0) {$\begin{smallmatrix}&&8\\5&10&16&13&10&7&4&2\end{smallmatrix}$};
	\node[noselEEEEEEEEE, label={[shift={(0.2,-0.55)}, brown]\tiny $P(13,4)$}] (P_13_4__Rationals) at (14,-3) {$\begin{smallmatrix}&&7\\4&9&14&11&9&6&4&2\end{smallmatrix}$};
	\node[noselEEEEEEEEE, label={[shift={(0.2,-0.55)}, brown]\tiny $P(13,5)$}] (P_13_5__Rationals) at (16,-6) {$\begin{smallmatrix}&&6\\4&8&12&9&7&5&3&2\end{smallmatrix}$};
	\node[noselEEEEEEEEE, label={[shift={(0.2,-0.55)}, brown]\tiny $P(13,6)$}] (P_13_6__Rationals) at (18,-9) {$\begin{smallmatrix}&&5\\3&6&9&7&5&3&2&1\end{smallmatrix}$};
	\node[noselEEEEEEEEE, label={[shift={(0.2,-0.55)}, brown]\tiny $P(13,7)$}] (P_13_7__Rationals) at (20,-12) {$\begin{smallmatrix}&&3\\2&4&6&5&4&2&1&1\end{smallmatrix}$};
	\node[noselEEEEEEEEE, label={[shift={(0.2,-0.55)}, brown]\tiny $P(13,8)$}] (P_13_8__Rationals) at (22,-15) {$\begin{smallmatrix}&&2\\1&2&3&2&2&1&0&0\end{smallmatrix}$};
	\node[noselEEEEEEEEE, label={[shift={(0.2,-0.55)}, brown]\tiny $P(13,9)$}] (P_13_9__Rationals) at (14,1) {$\begin{smallmatrix}&&4\\2&5&8&7&5&4&2&1\end{smallmatrix}$};
	\node[noselEEEEEEEEE, label={[shift={(0.2,-0.55)}, brown]\tiny $P(14,1)$}] (P_14_1__Rationals) at (20,6) {$\begin{smallmatrix}&&3\\1&4&6&5&4&3&2&1\end{smallmatrix}$};
	\node[noselEEEEEEEEE, label={[shift={(0.2,-0.55)}, brown]\tiny $P(14,2)$}] (P_14_2__Rationals) at (18,3) {$\begin{smallmatrix}&&6\\3&7&12&10&8&6&4&2\end{smallmatrix}$};
	\node[noselEEEEEEEEE, label={[shift={(0.2,-0.55)}, brown]\tiny $P(14,3)$}] (P_14_3__Rationals) at (16,0) {$\begin{smallmatrix}&&8\\5&11&17&14&11&8&5&2\end{smallmatrix}$};
	\node[noselEEEEEEEEE, label={[shift={(0.2,-0.55)}, brown]\tiny $P(14,4)$}] (P_14_4__Rationals) at (18,-3) {$\begin{smallmatrix}&&7\\5&10&15&12&9&7&4&2\end{smallmatrix}$};
	\node[noselEEEEEEEEE, label={[shift={(0.2,-0.55)}, brown]\tiny $P(14,5)$}] (P_14_5__Rationals) at (20,-6) {$\begin{smallmatrix}&&6\\4&8&12&10&7&5&3&1\end{smallmatrix}$};
	\node[noselEEEEEEEEE, label={[shift={(0.2,-0.55)}, brown]\tiny $P(14,6)$}] (P_14_6__Rationals) at (22,-9) {$\begin{smallmatrix}&&4\\3&6&9&8&6&4&2&1\end{smallmatrix}$};
	\node[noselEEEEEEEEE, label={[shift={(0.2,-0.55)}, brown]\tiny $P(14,7)$}] (P_14_7__Rationals) at (24,-12) {$\begin{smallmatrix}&&3\\2&4&6&5&4&3&1&0\end{smallmatrix}$};
	\node[noselEEEEEEEEE, label={[shift={(0.2,-0.55)}, brown]\tiny $P(14,8)$}] (P_14_8__Rationals) at (26,-15) {$\begin{smallmatrix}&&1\\1&2&3&3&2&2&1&0\end{smallmatrix}$};
	\node[noselEEEEEEEEE, label={[shift={(0.2,-0.55)}, brown]\tiny $P(14,9)$}] (P_14_9__Rationals) at (18,1) {$\begin{smallmatrix}&&4\\3&6&9&7&6&4&3&1\end{smallmatrix}$};
	\node at (21,3) {$\dots$};
	\node at (21,-3) {$\dots$};
	\draw[red,->] (P_10_2__Rationals) -- (P_10_1__Rationals);
	\draw[red,->] (P_10_3__Rationals) -- (P_10_2__Rationals);
	\draw[red,->] (P_10_3__Rationals) -- (P_10_4__Rationals);
	\draw[red,->] (P_10_4__Rationals) -- (P_10_5__Rationals);
	\draw[red,->] (P_10_5__Rationals) -- (P_10_6__Rationals);
	\draw[red,->] (P_10_6__Rationals) -- (P_10_7__Rationals);
	\draw[red,->] (P_10_7__Rationals) -- (P_10_8__Rationals);
	\draw[red,->] (P_10_3__Rationals) -- (P_10_9__Rationals);
	\draw[gray,dashed,->] (P_10_1__Rationals) -- (P_11_1__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[red,->] (P_11_2__Rationals) -- (P_11_1__Rationals);
	\draw[blue,->] (P_10_1__Rationals) -- (P_11_2__Rationals);
	\draw[gray,dashed,->] (P_10_2__Rationals) -- (P_11_2__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[red,->] (P_11_3__Rationals) -- (P_11_2__Rationals);
	\draw[blue,->] (P_10_2__Rationals) -- (P_11_3__Rationals);
	\draw[gray,dashed,->] (P_10_3__Rationals) -- (P_11_3__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[blue,->] (P_10_4__Rationals) -- (P_11_3__Rationals);
	\draw[blue,->] (P_10_9__Rationals) -- (P_11_3__Rationals);
	\draw[gray,dashed,->] (P_10_4__Rationals) -- (P_11_4__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[blue,->] (P_10_5__Rationals) -- (P_11_4__Rationals);
	\draw[red,->] (P_11_3__Rationals) -- (P_11_4__Rationals);
	\draw[gray,dashed,->] (P_10_5__Rationals) -- (P_11_5__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[blue,->] (P_10_6__Rationals) -- (P_11_5__Rationals);
	\draw[red,->] (P_11_4__Rationals) -- (P_11_5__Rationals);
	\draw[gray,dashed,->] (P_10_6__Rationals) -- (P_11_6__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[blue,->] (P_10_7__Rationals) -- (P_11_6__Rationals);
	\draw[red,->] (P_11_5__Rationals) -- (P_11_6__Rationals);
	\draw[gray,dashed,->] (P_10_7__Rationals) -- (P_11_7__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[blue,->] (P_10_8__Rationals) -- (P_11_7__Rationals);
	\draw[red,->] (P_11_6__Rationals) -- (P_11_7__Rationals);
	\draw[gray,dashed,->] (P_10_8__Rationals) -- (P_11_8__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[red,->] (P_11_7__Rationals) -- (P_11_8__Rationals);
	\draw[gray,dashed,->] (P_10_9__Rationals) -- (P_11_9__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[red,->] (P_11_3__Rationals) -- (P_11_9__Rationals);
	\draw[gray,dashed,->] (P_11_1__Rationals) -- (P_12_1__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[red,->] (P_12_2__Rationals) -- (P_12_1__Rationals);
	\draw[blue,->] (P_11_1__Rationals) -- (P_12_2__Rationals);
	\draw[gray,dashed,->] (P_11_2__Rationals) -- (P_12_2__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[red,->] (P_12_3__Rationals) -- (P_12_2__Rationals);
	\draw[blue,->] (P_11_2__Rationals) -- (P_12_3__Rationals);
	\draw[gray,dashed,->] (P_11_3__Rationals) -- (P_12_3__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[blue,->] (P_11_4__Rationals) -- (P_12_3__Rationals);
	\draw[blue,->] (P_11_9__Rationals) -- (P_12_3__Rationals);
	\draw[gray,dashed,->] (P_11_4__Rationals) -- (P_12_4__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[blue,->] (P_11_5__Rationals) -- (P_12_4__Rationals);
	\draw[red,->] (P_12_3__Rationals) -- (P_12_4__Rationals);
	\draw[gray,dashed,->] (P_11_5__Rationals) -- (P_12_5__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[blue,->] (P_11_6__Rationals) -- (P_12_5__Rationals);
	\draw[red,->] (P_12_4__Rationals) -- (P_12_5__Rationals);
	\draw[gray,dashed,->] (P_11_6__Rationals) -- (P_12_6__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[blue,->] (P_11_7__Rationals) -- (P_12_6__Rationals);
	\draw[red,->] (P_12_5__Rationals) -- (P_12_6__Rationals);
	\draw[gray,dashed,->] (P_11_7__Rationals) -- (P_12_7__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[blue,->] (P_11_8__Rationals) -- (P_12_7__Rationals);
	\draw[red,->] (P_12_6__Rationals) -- (P_12_7__Rationals);
	\draw[gray,dashed,->] (P_11_8__Rationals) -- (P_12_8__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[red,->] (P_12_7__Rationals) -- (P_12_8__Rationals);
	\draw[gray,dashed,->] (P_11_9__Rationals) -- (P_12_9__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[red,->] (P_12_3__Rationals) -- (P_12_9__Rationals);
	\draw[gray,dashed,->] (P_12_1__Rationals) -- (P_13_1__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[red,->] (P_13_2__Rationals) -- (P_13_1__Rationals);
	\draw[blue,->] (P_12_1__Rationals) -- (P_13_2__Rationals);
	\draw[gray,dashed,->] (P_12_2__Rationals) -- (P_13_2__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[red,->] (P_13_3__Rationals) -- (P_13_2__Rationals);
	\draw[blue,->] (P_12_2__Rationals) -- (P_13_3__Rationals);
	\draw[gray,dashed,->] (P_12_3__Rationals) -- (P_13_3__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[blue,->] (P_12_4__Rationals) -- (P_13_3__Rationals);
	\draw[blue,->] (P_12_9__Rationals) -- (P_13_3__Rationals);
	\draw[gray,dashed,->] (P_12_4__Rationals) -- (P_13_4__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[blue,->] (P_12_5__Rationals) -- (P_13_4__Rationals);
	\draw[red,->] (P_13_3__Rationals) -- (P_13_4__Rationals);
	\draw[gray,dashed,->] (P_12_5__Rationals) -- (P_13_5__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[blue,->] (P_12_6__Rationals) -- (P_13_5__Rationals);
	\draw[red,->] (P_13_4__Rationals) -- (P_13_5__Rationals);
	\draw[gray,dashed,->] (P_12_6__Rationals) -- (P_13_6__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[blue,->] (P_12_7__Rationals) -- (P_13_6__Rationals);
	\draw[red,->] (P_13_5__Rationals) -- (P_13_6__Rationals);
	\draw[gray,dashed,->] (P_12_7__Rationals) -- (P_13_7__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[blue,->] (P_12_8__Rationals) -- (P_13_7__Rationals);
	\draw[red,->] (P_13_6__Rationals) -- (P_13_7__Rationals);
	\draw[gray,dashed,->] (P_12_8__Rationals) -- (P_13_8__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[red,->] (P_13_7__Rationals) -- (P_13_8__Rationals);
	\draw[gray,dashed,->] (P_12_9__Rationals) -- (P_13_9__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[red,->] (P_13_3__Rationals) -- (P_13_9__Rationals);
	\draw[gray,dashed,->] (P_13_1__Rationals) -- (P_14_1__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[red,->] (P_14_2__Rationals) -- (P_14_1__Rationals);
	\draw[blue,->] (P_13_1__Rationals) -- (P_14_2__Rationals);
	\draw[gray,dashed,->] (P_13_2__Rationals) -- (P_14_2__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[red,->] (P_14_3__Rationals) -- (P_14_2__Rationals);
	\draw[blue,->] (P_13_2__Rationals) -- (P_14_3__Rationals);
	\draw[gray,dashed,->] (P_13_3__Rationals) -- (P_14_3__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[blue,->] (P_13_4__Rationals) -- (P_14_3__Rationals);
	\draw[blue,->] (P_13_9__Rationals) -- (P_14_3__Rationals);
	\draw[gray,dashed,->] (P_13_4__Rationals) -- (P_14_4__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[blue,->] (P_13_5__Rationals) -- (P_14_4__Rationals);
	\draw[red,->] (P_14_3__Rationals) -- (P_14_4__Rationals);
	\draw[gray,dashed,->] (P_13_5__Rationals) -- (P_14_5__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[blue,->] (P_13_6__Rationals) -- (P_14_5__Rationals);
	\draw[red,->] (P_14_4__Rationals) -- (P_14_5__Rationals);
	\draw[gray,dashed,->] (P_13_6__Rationals) -- (P_14_6__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[blue,->] (P_13_7__Rationals) -- (P_14_6__Rationals);
	\draw[red,->] (P_14_5__Rationals) -- (P_14_6__Rationals);
	\draw[gray,dashed,->] (P_13_7__Rationals) -- (P_14_7__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[blue,->] (P_13_8__Rationals) -- (P_14_7__Rationals);
	\draw[red,->] (P_14_6__Rationals) -- (P_14_7__Rationals);
	\draw[gray,dashed,->] (P_13_8__Rationals) -- (P_14_8__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[red,->] (P_14_7__Rationals) -- (P_14_8__Rationals);
	\draw[gray,dashed,->] (P_13_9__Rationals) -- (P_14_9__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[red,->] (P_14_3__Rationals) -- (P_14_9__Rationals);
\end{tikzpicture}
\end{scaletikzpicturetowidth}
\end{center}

\end{figure}
\begin{figure}[ht]

\begin{center}
\begin{scaletikzpicturetowidth}{\textwidth}
\begin{tikzpicture}[scale=\tikzscale]
	\node at (-1,3) {$\dots$};
	\node at (-1,-3) {$\dots$};
	\node[noselEEEEEEEEE, label={[shift={(0.2,-0.55)}, brown]\tiny $P(15,1)$}] (P_15_1__Rationals) at (4,6) {$\begin{smallmatrix}&&3\\3&5&7&5&4&3&2&1\end{smallmatrix}$};
	\node[noselEEEEEEEEE, label={[shift={(0.2,-0.55)}, brown]\tiny $P(15,2)$}] (P_15_2__Rationals) at (2,3) {$\begin{smallmatrix}&&6\\4&9&13&10&8&6&4&2\end{smallmatrix}$};
	\node[noselEEEEEEEEE, label={[shift={(0.2,-0.55)}, brown]\tiny $P(15,3)$}] (P_15_3__Rationals) at (0,0) {$\begin{smallmatrix}&&9\\6&12&19&15&12&9&6&3\end{smallmatrix}$};
	\node[noselEEEEEEEEE, label={[shift={(0.2,-0.55)}, brown]\tiny $P(15,4)$}] (P_15_4__Rationals) at (2,-3) {$\begin{smallmatrix}&&8\\5&10&16&13&10&7&5&2\end{smallmatrix}$};
	\node[noselEEEEEEEEE, label={[shift={(0.2,-0.55)}, brown]\tiny $P(15,5)$}] (P_15_5__Rationals) at (4,-6) {$\begin{smallmatrix}&&6\\4&8&13&11&9&6&4&2\end{smallmatrix}$};
	\node[noselEEEEEEEEE, label={[shift={(0.2,-0.55)}, brown]\tiny $P(15,6)$}] (P_15_6__Rationals) at (6,-9) {$\begin{smallmatrix}&&5\\3&6&10&8&7&5&3&1\end{smallmatrix}$};
	\node[seloneEEEEEEEEE, label={[shift={(0.2,-0.55)}, brown]\tiny $P(15,7)$}] (P_15_7__Rationals) at (8,-12) {$\begin{smallmatrix}&&3\\2&4&7&6&5&4&3&1\end{smallmatrix}$};
	\node[noselEEEEEEEEE, label={[shift={(0.2,-0.55)}, brown]\tiny $P(15,8)$}] (P_15_8__Rationals) at (10,-15) {$\begin{smallmatrix}&&2\\1&2&4&3&3&2&2&1\end{smallmatrix}$};
	\node[noselEEEEEEEEE, label={[shift={(0.2,-0.55)}, brown]\tiny $P(15,9)$}] (P_15_9__Rationals) at (2,1) {$\begin{smallmatrix}&&5\\3&6&10&8&6&5&3&2\end{smallmatrix}$};
	\node[noselEEEEEEEEE, label={[shift={(0.2,-0.55)}, brown]\tiny $P(16,1)$}] (P_16_1__Rationals) at (8,6) {$\begin{smallmatrix}&&4\\2&4&7&6&4&3&2&1\end{smallmatrix}$};
	\node[noselEEEEEEEEE, label={[shift={(0.2,-0.55)}, brown]\tiny $P(16,2)$}] (P_16_2__Rationals) at (6,3) {$\begin{smallmatrix}&&7\\5&9&14&11&8&6&4&2\end{smallmatrix}$};
	\node[noselEEEEEEEEE, label={[shift={(0.2,-0.55)}, brown]\tiny $P(16,3)$}] (P_16_3__Rationals) at (4,0) {$\begin{smallmatrix}&&10\\6&13&20&16&12&9&6&3\end{smallmatrix}$};
	\node[noselEEEEEEEEE, label={[shift={(0.2,-0.55)}, brown]\tiny $P(16,4)$}] (P_16_4__Rationals) at (6,-3) {$\begin{smallmatrix}&&8\\5&11&17&14&11&8&5&3\end{smallmatrix}$};
	\node[noselEEEEEEEEE, label={[shift={(0.2,-0.55)}, brown]\tiny $P(16,5)$}] (P_16_5__Rationals) at (8,-6) {$\begin{smallmatrix}&&7\\4&9&14&11&9&7&4&2\end{smallmatrix}$};
	\node[noselEEEEEEEEE, label={[shift={(0.2,-0.55)}, brown]\tiny $P(16,6)$}] (P_16_6__Rationals) at (10,-9) {$\begin{smallmatrix}&&5\\3&7&11&9&7&6&4&2\end{smallmatrix}$};
	\node[noselEEEEEEEEE, label={[shift={(0.2,-0.55)}, brown]\tiny $P(16,7)$}] (P_16_7__Rationals) at (12,-12) {$\begin{smallmatrix}&&4\\2&5&8&6&5&4&3&2\end{smallmatrix}$};
	\node[noselEEEEEEEEE, label={[shift={(0.2,-0.55)}, brown]\tiny $P(16,8)$}] (P_16_8__Rationals) at (14,-15) {$\begin{smallmatrix}&&2\\1&3&4&3&2&2&1&1\end{smallmatrix}$};
	\node[noselEEEEEEEEE, label={[shift={(0.2,-0.55)}, brown]\tiny $P(16,9)$}] (P_16_9__Rationals) at (6,1) {$\begin{smallmatrix}&&5\\3&7&10&8&6&4&3&1\end{smallmatrix}$};
	\node[noselEEEEEEEEE, label={[shift={(0.2,-0.55)}, brown]\tiny $P(17,1)$}] (P_17_1__Rationals) at (12,6) {$\begin{smallmatrix}&&3\\2&5&7&6&5&3&2&1\end{smallmatrix}$};
	\node[noselEEEEEEEEE, label={[shift={(0.2,-0.55)}, brown]\tiny $P(17,2)$}] (P_17_2__Rationals) at (10,3) {$\begin{smallmatrix}&&7\\4&9&14&12&9&6&4&2\end{smallmatrix}$};
	\node[noselEEEEEEEEE, label={[shift={(0.2,-0.55)}, brown]\tiny $P(17,3)$}] (P_17_3__Rationals) at (8,0) {$\begin{smallmatrix}&&10\\7&14&21&17&13&9&6&3\end{smallmatrix}$};
	\node[noselEEEEEEEEE, label={[shift={(0.2,-0.55)}, brown]\tiny $P(17,4)$}] (P_17_4__Rationals) at (10,-3) {$\begin{smallmatrix}&&9\\6&12&18&14&11&8&5&2\end{smallmatrix}$};
	\node[noselEEEEEEEEE, label={[shift={(0.2,-0.55)}, brown]\tiny $P(17,5)$}] (P_17_5__Rationals) at (12,-6) {$\begin{smallmatrix}&&7\\5&10&15&12&9&7&5&2\end{smallmatrix}$};
	\node[noselEEEEEEEEE, label={[shift={(0.2,-0.55)}, brown]\tiny $P(17,6)$}] (P_17_6__Rationals) at (14,-9) {$\begin{smallmatrix}&&6\\4&8&12&9&7&5&4&2\end{smallmatrix}$};
	\node[noselEEEEEEEEE, label={[shift={(0.2,-0.55)}, brown]\tiny $P(17,7)$}] (P_17_7__Rationals) at (16,-12) {$\begin{smallmatrix}&&4\\3&6&8&6&4&3&2&1\end{smallmatrix}$};
	\node[noselEEEEEEEEE, label={[shift={(0.2,-0.55)}, brown]\tiny $P(17,8)$}] (P_17_8__Rationals) at (18,-15) {$\begin{smallmatrix}&&2\\2&3&4&3&2&1&1&0\end{smallmatrix}$};
	\node[noselEEEEEEEEE, label={[shift={(0.2,-0.55)}, brown]\tiny $P(17,9)$}] (P_17_9__Rationals) at (10,1) {$\begin{smallmatrix}&&5\\4&7&11&9&7&5&3&2\end{smallmatrix}$};
	\node[noselEEEEEEEEE, label={[shift={(0.2,-0.55)}, brown]\tiny $P(18,1)$}] (P_18_1__Rationals) at (16,6) {$\begin{smallmatrix}&&4\\3&5&8&6&5&4&2&1\end{smallmatrix}$};
	\node[noselEEEEEEEEE, label={[shift={(0.2,-0.55)}, brown]\tiny $P(18,2)$}] (P_18_2__Rationals) at (14,3) {$\begin{smallmatrix}&&7\\5&10&15&12&10&7&4&2\end{smallmatrix}$};
	\node[noselEEEEEEEEE, label={[shift={(0.2,-0.55)}, brown]\tiny $P(18,3)$}] (P_18_3__Rationals) at (12,0) {$\begin{smallmatrix}&&11\\7&14&22&18&14&10&6&3\end{smallmatrix}$};
	\node[noselEEEEEEEEE, label={[shift={(0.2,-0.55)}, brown]\tiny $P(18,4)$}] (P_18_4__Rationals) at (14,-3) {$\begin{smallmatrix}&&9\\6&12&19&16&12&9&6&3\end{smallmatrix}$};
	\node[noselEEEEEEEEE, label={[shift={(0.2,-0.55)}, brown]\tiny $P(18,5)$}] (P_18_5__Rationals) at (16,-6) {$\begin{smallmatrix}&&8\\5&10&16&13&10&7&5&3\end{smallmatrix}$};
	\node[noselEEEEEEEEE, label={[shift={(0.2,-0.55)}, brown]\tiny $P(18,6)$}] (P_18_6__Rationals) at (18,-9) {$\begin{smallmatrix}&&6\\4&8&12&10&7&5&3&2\end{smallmatrix}$};
	\node[noselEEEEEEEEE, label={[shift={(0.2,-0.55)}, brown]\tiny $P(18,7)$}] (P_18_7__Rationals) at (20,-12) {$\begin{smallmatrix}&&4\\3&5&8&7&5&3&2&1\end{smallmatrix}$};
	\node[noselEEEEEEEEE, label={[shift={(0.2,-0.55)}, brown]\tiny $P(18,8)$}] (P_18_8__Rationals) at (22,-15) {$\begin{smallmatrix}&&2\\1&2&4&4&3&2&1&1\end{smallmatrix}$};
	\node[noselEEEEEEEEE, label={[shift={(0.2,-0.55)}, brown]\tiny $P(18,9)$}] (P_18_9__Rationals) at (14,1) {$\begin{smallmatrix}&&6\\3&7&11&9&7&5&3&1\end{smallmatrix}$};
	\node[noselEEEEEEEEE, label={[shift={(0.2,-0.55)}, brown]\tiny $P(19,1)$}] (P_19_1__Rationals) at (20,6) {$\begin{smallmatrix}&&4\\2&5&8&7&5&4&3&1\end{smallmatrix}$};
	\node[noselEEEEEEEEE, label={[shift={(0.2,-0.55)}, brown]\tiny $P(19,2)$}] (P_19_2__Rationals) at (18,3) {$\begin{smallmatrix}&&8\\5&10&16&13&10&8&5&2\end{smallmatrix}$};
	\node[noselEEEEEEEEE, label={[shift={(0.2,-0.55)}, brown]\tiny $P(19,3)$}] (P_19_3__Rationals) at (16,0) {$\begin{smallmatrix}&&11\\7&15&23&19&15&11&7&3\end{smallmatrix}$};
	\node[noselEEEEEEEEE, label={[shift={(0.2,-0.55)}, brown]\tiny $P(19,4)$}] (P_19_4__Rationals) at (18,-3) {$\begin{smallmatrix}&&10\\6&13&20&16&13&9&6&3\end{smallmatrix}$};
	\node[noselEEEEEEEEE, label={[shift={(0.2,-0.55)}, brown]\tiny $P(19,5)$}] (P_19_5__Rationals) at (20,-6) {$\begin{smallmatrix}&&8\\5&11&16&13&10&7&4&2\end{smallmatrix}$};
	\node[noselEEEEEEEEE, label={[shift={(0.2,-0.55)}, brown]\tiny $P(19,6)$}] (P_19_6__Rationals) at (22,-9) {$\begin{smallmatrix}&&6\\4&8&12&10&8&5&3&1\end{smallmatrix}$};
	\node[noselEEEEEEEEE, label={[shift={(0.2,-0.55)}, brown]\tiny $P(19,7)$}] (P_19_7__Rationals) at (24,-12) {$\begin{smallmatrix}&&4\\2&5&8&7&6&4&2&1\end{smallmatrix}$};
	\node[noselEEEEEEEEE, label={[shift={(0.2,-0.55)}, brown]\tiny $P(19,8)$}] (P_19_8__Rationals) at (26,-15) {$\begin{smallmatrix}&&2\\1&3&4&3&3&2&1&0\end{smallmatrix}$};
	\node[noselEEEEEEEEE, label={[shift={(0.2,-0.55)}, brown]\tiny $P(19,9)$}] (P_19_9__Rationals) at (18,1) {$\begin{smallmatrix}&&5\\4&8&12&10&8&6&4&2\end{smallmatrix}$};
	\node at (21,3) {$\dots$};
	\node at (21,-3) {$\dots$};
	\draw[red,->] (P_15_2__Rationals) -- (P_15_1__Rationals);
	\draw[red,->] (P_15_3__Rationals) -- (P_15_2__Rationals);
	\draw[red,->] (P_15_3__Rationals) -- (P_15_4__Rationals);
	\draw[red,->] (P_15_4__Rationals) -- (P_15_5__Rationals);
	\draw[red,->] (P_15_5__Rationals) -- (P_15_6__Rationals);
	\draw[red,->] (P_15_6__Rationals) -- (P_15_7__Rationals);
	\draw[red,->] (P_15_7__Rationals) -- (P_15_8__Rationals);
	\draw[red,->] (P_15_3__Rationals) -- (P_15_9__Rationals);
	\draw[gray,dashed,->] (P_15_1__Rationals) -- (P_16_1__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[red,->] (P_16_2__Rationals) -- (P_16_1__Rationals);
	\draw[blue,->] (P_15_1__Rationals) -- (P_16_2__Rationals);
	\draw[gray,dashed,->] (P_15_2__Rationals) -- (P_16_2__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[red,->] (P_16_3__Rationals) -- (P_16_2__Rationals);
	\draw[blue,->] (P_15_2__Rationals) -- (P_16_3__Rationals);
	\draw[gray,dashed,->] (P_15_3__Rationals) -- (P_16_3__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[blue,->] (P_15_4__Rationals) -- (P_16_3__Rationals);
	\draw[blue,->] (P_15_9__Rationals) -- (P_16_3__Rationals);
	\draw[gray,dashed,->] (P_15_4__Rationals) -- (P_16_4__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[blue,->] (P_15_5__Rationals) -- (P_16_4__Rationals);
	\draw[red,->] (P_16_3__Rationals) -- (P_16_4__Rationals);
	\draw[gray,dashed,->] (P_15_5__Rationals) -- (P_16_5__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[blue,->] (P_15_6__Rationals) -- (P_16_5__Rationals);
	\draw[red,->] (P_16_4__Rationals) -- (P_16_5__Rationals);
	\draw[gray,dashed,->] (P_15_6__Rationals) -- (P_16_6__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[blue,->] (P_15_7__Rationals) -- (P_16_6__Rationals);
	\draw[red,->] (P_16_5__Rationals) -- (P_16_6__Rationals);
	\draw[gray,dashed,->] (P_15_7__Rationals) -- (P_16_7__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[blue,->] (P_15_8__Rationals) -- (P_16_7__Rationals);
	\draw[red,->] (P_16_6__Rationals) -- (P_16_7__Rationals);
	\draw[gray,dashed,->] (P_15_8__Rationals) -- (P_16_8__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[red,->] (P_16_7__Rationals) -- (P_16_8__Rationals);
	\draw[gray,dashed,->] (P_15_9__Rationals) -- (P_16_9__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[red,->] (P_16_3__Rationals) -- (P_16_9__Rationals);
	\draw[gray,dashed,->] (P_16_1__Rationals) -- (P_17_1__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[red,->] (P_17_2__Rationals) -- (P_17_1__Rationals);
	\draw[blue,->] (P_16_1__Rationals) -- (P_17_2__Rationals);
	\draw[gray,dashed,->] (P_16_2__Rationals) -- (P_17_2__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[red,->] (P_17_3__Rationals) -- (P_17_2__Rationals);
	\draw[blue,->] (P_16_2__Rationals) -- (P_17_3__Rationals);
	\draw[gray,dashed,->] (P_16_3__Rationals) -- (P_17_3__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[blue,->] (P_16_4__Rationals) -- (P_17_3__Rationals);
	\draw[blue,->] (P_16_9__Rationals) -- (P_17_3__Rationals);
	\draw[gray,dashed,->] (P_16_4__Rationals) -- (P_17_4__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[blue,->] (P_16_5__Rationals) -- (P_17_4__Rationals);
	\draw[red,->] (P_17_3__Rationals) -- (P_17_4__Rationals);
	\draw[gray,dashed,->] (P_16_5__Rationals) -- (P_17_5__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[blue,->] (P_16_6__Rationals) -- (P_17_5__Rationals);
	\draw[red,->] (P_17_4__Rationals) -- (P_17_5__Rationals);
	\draw[gray,dashed,->] (P_16_6__Rationals) -- (P_17_6__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[blue,->] (P_16_7__Rationals) -- (P_17_6__Rationals);
	\draw[red,->] (P_17_5__Rationals) -- (P_17_6__Rationals);
	\draw[gray,dashed,->] (P_16_7__Rationals) -- (P_17_7__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[blue,->] (P_16_8__Rationals) -- (P_17_7__Rationals);
	\draw[red,->] (P_17_6__Rationals) -- (P_17_7__Rationals);
	\draw[gray,dashed,->] (P_16_8__Rationals) -- (P_17_8__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[red,->] (P_17_7__Rationals) -- (P_17_8__Rationals);
	\draw[gray,dashed,->] (P_16_9__Rationals) -- (P_17_9__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[red,->] (P_17_3__Rationals) -- (P_17_9__Rationals);
	\draw[gray,dashed,->] (P_17_1__Rationals) -- (P_18_1__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[red,->] (P_18_2__Rationals) -- (P_18_1__Rationals);
	\draw[blue,->] (P_17_1__Rationals) -- (P_18_2__Rationals);
	\draw[gray,dashed,->] (P_17_2__Rationals) -- (P_18_2__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[red,->] (P_18_3__Rationals) -- (P_18_2__Rationals);
	\draw[blue,->] (P_17_2__Rationals) -- (P_18_3__Rationals);
	\draw[gray,dashed,->] (P_17_3__Rationals) -- (P_18_3__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[blue,->] (P_17_4__Rationals) -- (P_18_3__Rationals);
	\draw[blue,->] (P_17_9__Rationals) -- (P_18_3__Rationals);
	\draw[gray,dashed,->] (P_17_4__Rationals) -- (P_18_4__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[blue,->] (P_17_5__Rationals) -- (P_18_4__Rationals);
	\draw[red,->] (P_18_3__Rationals) -- (P_18_4__Rationals);
	\draw[gray,dashed,->] (P_17_5__Rationals) -- (P_18_5__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[blue,->] (P_17_6__Rationals) -- (P_18_5__Rationals);
	\draw[red,->] (P_18_4__Rationals) -- (P_18_5__Rationals);
	\draw[gray,dashed,->] (P_17_6__Rationals) -- (P_18_6__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[blue,->] (P_17_7__Rationals) -- (P_18_6__Rationals);
	\draw[red,->] (P_18_5__Rationals) -- (P_18_6__Rationals);
	\draw[gray,dashed,->] (P_17_7__Rationals) -- (P_18_7__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[blue,->] (P_17_8__Rationals) -- (P_18_7__Rationals);
	\draw[red,->] (P_18_6__Rationals) -- (P_18_7__Rationals);
	\draw[gray,dashed,->] (P_17_8__Rationals) -- (P_18_8__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[red,->] (P_18_7__Rationals) -- (P_18_8__Rationals);
	\draw[gray,dashed,->] (P_17_9__Rationals) -- (P_18_9__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[red,->] (P_18_3__Rationals) -- (P_18_9__Rationals);
	\draw[gray,dashed,->] (P_18_1__Rationals) -- (P_19_1__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[red,->] (P_19_2__Rationals) -- (P_19_1__Rationals);
	\draw[blue,->] (P_18_1__Rationals) -- (P_19_2__Rationals);
	\draw[gray,dashed,->] (P_18_2__Rationals) -- (P_19_2__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[red,->] (P_19_3__Rationals) -- (P_19_2__Rationals);
	\draw[blue,->] (P_18_2__Rationals) -- (P_19_3__Rationals);
	\draw[gray,dashed,->] (P_18_3__Rationals) -- (P_19_3__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[blue,->] (P_18_4__Rationals) -- (P_19_3__Rationals);
	\draw[blue,->] (P_18_9__Rationals) -- (P_19_3__Rationals);
	\draw[gray,dashed,->] (P_18_4__Rationals) -- (P_19_4__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[blue,->] (P_18_5__Rationals) -- (P_19_4__Rationals);
	\draw[red,->] (P_19_3__Rationals) -- (P_19_4__Rationals);
	\draw[gray,dashed,->] (P_18_5__Rationals) -- (P_19_5__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[blue,->] (P_18_6__Rationals) -- (P_19_5__Rationals);
	\draw[red,->] (P_19_4__Rationals) -- (P_19_5__Rationals);
	\draw[gray,dashed,->] (P_18_6__Rationals) -- (P_19_6__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[blue,->] (P_18_7__Rationals) -- (P_19_6__Rationals);
	\draw[red,->] (P_19_5__Rationals) -- (P_19_6__Rationals);
	\draw[gray,dashed,->] (P_18_7__Rationals) -- (P_19_7__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[blue,->] (P_18_8__Rationals) -- (P_19_7__Rationals);
	\draw[red,->] (P_19_6__Rationals) -- (P_19_7__Rationals);
	\draw[gray,dashed,->] (P_18_8__Rationals) -- (P_19_8__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[red,->] (P_19_7__Rationals) -- (P_19_8__Rationals);
	\draw[gray,dashed,->] (P_18_9__Rationals) -- (P_19_9__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[red,->] (P_19_3__Rationals) -- (P_19_9__Rationals);
\end{tikzpicture}
\end{scaletikzpicturetowidth}
\end{center}

\end{figure}
\begin{figure}[ht]

\begin{center}
\begin{scaletikzpicturetowidth}{\textwidth}
\begin{tikzpicture}[scale=\tikzscale]
	\node at (-1,3) {$\dots$};
	\node at (-1,-3) {$\dots$};
	\node[noselEEEEEEEEE, label={[shift={(0.2,-0.55)}, brown]\tiny $P(20,1)$}] (P_20_1__Rationals) at (4,6) {$\begin{smallmatrix}&&4\\3&6&9&7&6&4&3&2\end{smallmatrix}$};
	\node[noselEEEEEEEEE, label={[shift={(0.2,-0.55)}, brown]\tiny $P(20,2)$}] (P_20_2__Rationals) at (2,3) {$\begin{smallmatrix}&&8\\5&11&17&14&11&8&6&3\end{smallmatrix}$};
	\node[noselEEEEEEEEE, label={[shift={(0.2,-0.55)}, brown]\tiny $P(20,3)$}] (P_20_3__Rationals) at (0,0) {$\begin{smallmatrix}&&12\\8&16&25&20&16&12&8&4\end{smallmatrix}$};
	\node[noselEEEEEEEEE, label={[shift={(0.2,-0.55)}, brown]\tiny $P(20,4)$}] (P_20_4__Rationals) at (2,-3) {$\begin{smallmatrix}&&10\\7&14&21&17&13&10&6&3\end{smallmatrix}$};
	\node[noselEEEEEEEEE, label={[shift={(0.2,-0.55)}, brown]\tiny $P(20,5)$}] (P_20_5__Rationals) at (4,-6) {$\begin{smallmatrix}&&8\\6&11&17&14&11&8&5&2\end{smallmatrix}$};
	\node[noselEEEEEEEEE, label={[shift={(0.2,-0.55)}, brown]\tiny $P(20,6)$}] (P_20_6__Rationals) at (6,-9) {$\begin{smallmatrix}&&6\\4&8&13&11&9&7&4&2\end{smallmatrix}$};
	\node[noselEEEEEEEEE, label={[shift={(0.2,-0.55)}, brown]\tiny $P(20,7)$}] (P_20_7__Rationals) at (8,-12) {$\begin{smallmatrix}&&4\\3&6&9&7&6&5&3&1\end{smallmatrix}$};
	\node[noselEEEEEEEEE, label={[shift={(0.2,-0.55)}, brown]\tiny $P(20,8)$}] (P_20_8__Rationals) at (10,-15) {$\begin{smallmatrix}&&2\\2&3&5&4&3&3&2&1\end{smallmatrix}$};
	\node[noselEEEEEEEEE, label={[shift={(0.2,-0.55)}, brown]\tiny $P(20,9)$}] (P_20_9__Rationals) at (2,1) {$\begin{smallmatrix}&&7\\4&8&13&10&8&6&4&2\end{smallmatrix}$};
	\node[noselEEEEEEEEE, label={[shift={(0.2,-0.55)}, brown]\tiny $P(21,1)$}] (P_21_1__Rationals) at (8,6) {$\begin{smallmatrix}&&5\\3&6&9&7&5&4&2&1\end{smallmatrix}$};
	\node[noselEEEEEEEEE, label={[shift={(0.2,-0.55)}, brown]\tiny $P(21,2)$}] (P_21_2__Rationals) at (6,3) {$\begin{smallmatrix}&&9\\6&12&18&14&11&8&5&3\end{smallmatrix}$};
	\node[noselEEEEEEEEE, label={[shift={(0.2,-0.55)}, brown]\tiny $P(21,3)$}] (P_21_3__Rationals) at (4,0) {$\begin{smallmatrix}&&13\\8&17&26&21&16&12&8&4\end{smallmatrix}$};
	\node[noselEEEEEEEEE, label={[shift={(0.2,-0.55)}, brown]\tiny $P(21,4)$}] (P_21_4__Rationals) at (6,-3) {$\begin{smallmatrix}&&11\\7&14&22&18&14&10&7&3\end{smallmatrix}$};
	\node[noselEEEEEEEEE, label={[shift={(0.2,-0.55)}, brown]\tiny $P(21,5)$}] (P_21_5__Rationals) at (8,-6) {$\begin{smallmatrix}&&9\\5&11&18&15&12&9&6&3\end{smallmatrix}$};
	\node[noselEEEEEEEEE, label={[shift={(0.2,-0.55)}, brown]\tiny $P(21,6)$}] (P_21_6__Rationals) at (10,-9) {$\begin{smallmatrix}&&7\\4&9&14&11&9&7&5&2\end{smallmatrix}$};
	\node[noselEEEEEEEEE, label={[shift={(0.2,-0.55)}, brown]\tiny $P(21,7)$}] (P_21_7__Rationals) at (12,-12) {$\begin{smallmatrix}&&5\\3&6&10&8&6&5&4&2\end{smallmatrix}$};
	\node[noselEEEEEEEEE, label={[shift={(0.2,-0.55)}, brown]\tiny $P(21,8)$}] (P_21_8__Rationals) at (14,-15) {$\begin{smallmatrix}&&3\\1&3&5&4&3&2&2&1\end{smallmatrix}$};
	\node[noselEEEEEEEEE, label={[shift={(0.2,-0.55)}, brown]\tiny $P(21,9)$}] (P_21_9__Rationals) at (6,1) {$\begin{smallmatrix}&&6\\4&9&13&11&8&6&4&2\end{smallmatrix}$};
	\node[noselEEEEEEEEE, label={[shift={(0.2,-0.55)}, brown]\tiny $P(22,1)$}] (P_22_1__Rationals) at (12,6) {$\begin{smallmatrix}&&4\\3&6&9&8&6&4&3&1\end{smallmatrix}$};
	\node[noselEEEEEEEEE, label={[shift={(0.2,-0.55)}, brown]\tiny $P(22,2)$}] (P_22_2__Rationals) at (10,3) {$\begin{smallmatrix}&&9\\6&12&18&15&11&8&5&2\end{smallmatrix}$};
	\node[noselEEEEEEEEE, label={[shift={(0.2,-0.55)}, brown]\tiny $P(22,3)$}] (P_22_3__Rationals) at (8,0) {$\begin{smallmatrix}&&13\\9&18&27&22&17&12&8&4\end{smallmatrix}$};
	\node[noselEEEEEEEEE, label={[shift={(0.2,-0.55)}, brown]\tiny $P(22,4)$}] (P_22_4__Rationals) at (10,-3) {$\begin{smallmatrix}&&11\\7&15&23&19&15&11&7&4\end{smallmatrix}$};
	\node[noselEEEEEEEEE, label={[shift={(0.2,-0.55)}, brown]\tiny $P(22,5)$}] (P_22_5__Rationals) at (12,-6) {$\begin{smallmatrix}&&9\\6&13&19&15&12&9&6&3\end{smallmatrix}$};
	\node[noselEEEEEEEEE, label={[shift={(0.2,-0.55)}, brown]\tiny $P(22,6)$}] (P_22_6__Rationals) at (14,-9) {$\begin{smallmatrix}&&7\\5&10&15&12&9&7&5&3\end{smallmatrix}$};
	\node[noselEEEEEEEEE, label={[shift={(0.2,-0.55)}, brown]\tiny $P(22,7)$}] (P_22_7__Rationals) at (16,-12) {$\begin{smallmatrix}&&5\\3&7&10&8&6&4&3&2\end{smallmatrix}$};
	\node[noselEEEEEEEEE, label={[shift={(0.2,-0.55)}, brown]\tiny $P(22,8)$}] (P_22_8__Rationals) at (18,-15) {$\begin{smallmatrix}&&2\\2&4&5&4&3&2&1&1\end{smallmatrix}$};
	\node[noselEEEEEEEEE, label={[shift={(0.2,-0.55)}, brown]\tiny $P(22,9)$}] (P_22_9__Rationals) at (10,1) {$\begin{smallmatrix}&&7\\5&9&14&11&9&6&4&2\end{smallmatrix}$};
	\node[noselEEEEEEEEE, label={[shift={(0.2,-0.55)}, brown]\tiny $P(23,1)$}] (P_23_1__Rationals) at (16,6) {$\begin{smallmatrix}&&5\\3&6&10&8&7&5&3&2\end{smallmatrix}$};
	\node[noselEEEEEEEEE, label={[shift={(0.2,-0.55)}, brown]\tiny $P(23,2)$}] (P_23_2__Rationals) at (14,3) {$\begin{smallmatrix}&&9\\6&12&19&16&13&9&6&3\end{smallmatrix}$};
	\node[noselEEEEEEEEE, label={[shift={(0.2,-0.55)}, brown]\tiny $P(23,3)$}] (P_23_3__Rationals) at (12,0) {$\begin{smallmatrix}&&14\\9&18&28&23&18&13&8&4\end{smallmatrix}$};
	\node[noselEEEEEEEEE, label={[shift={(0.2,-0.55)}, brown]\tiny $P(23,4)$}] (P_23_4__Rationals) at (14,-3) {$\begin{smallmatrix}&&12\\8&16&24&19&15&11&7&3\end{smallmatrix}$};
	\node[noselEEEEEEEEE, label={[shift={(0.2,-0.55)}, brown]\tiny $P(23,5)$}] (P_23_5__Rationals) at (16,-6) {$\begin{smallmatrix}&&10\\7&13&20&16&12&9&6&3\end{smallmatrix}$};
	\node[noselEEEEEEEEE, label={[shift={(0.2,-0.55)}, brown]\tiny $P(23,6)$}] (P_23_6__Rationals) at (18,-9) {$\begin{smallmatrix}&&8\\5&10&15&12&9&6&4&2\end{smallmatrix}$};
	\node[noselEEEEEEEEE, label={[shift={(0.2,-0.55)}, brown]\tiny $P(23,7)$}] (P_23_7__Rationals) at (20,-12) {$\begin{smallmatrix}&&5\\4&7&10&8&6&4&2&1\end{smallmatrix}$};
	\node[noselEEEEEEEEE, label={[shift={(0.2,-0.55)}, brown]\tiny $P(23,8)$}] (P_23_8__Rationals) at (22,-15) {$\begin{smallmatrix}&&3\\2&3&5&4&3&2&1&0\end{smallmatrix}$};
	\node[noselEEEEEEEEE, label={[shift={(0.2,-0.55)}, brown]\tiny $P(23,9)$}] (P_23_9__Rationals) at (14,1) {$\begin{smallmatrix}&&7\\4&9&14&12&9&7&4&2\end{smallmatrix}$};
	\node[noselEEEEEEEEE, label={[shift={(0.2,-0.55)}, brown]\tiny $P(24,1)$}] (P_24_1__Rationals) at (20,6) {$\begin{smallmatrix}&&5\\3&7&10&8&6&5&3&1\end{smallmatrix}$};
	\node[noselEEEEEEEEE, label={[shift={(0.2,-0.55)}, brown]\tiny $P(24,2)$}] (P_24_2__Rationals) at (18,3) {$\begin{smallmatrix}&&10\\6&13&20&16&13&10&6&3\end{smallmatrix}$};
	\node[noselEEEEEEEEE, label={[shift={(0.2,-0.55)}, brown]\tiny $P(24,3)$}] (P_24_3__Rationals) at (16,0) {$\begin{smallmatrix}&&14\\9&19&29&24&19&14&9&4\end{smallmatrix}$};
	\node[noselEEEEEEEEE, label={[shift={(0.2,-0.55)}, brown]\tiny $P(24,4)$}] (P_24_4__Rationals) at (18,-3) {$\begin{smallmatrix}&&12\\8&16&25&21&16&12&8&4\end{smallmatrix}$};
	\node[noselEEEEEEEEE, label={[shift={(0.2,-0.55)}, brown]\tiny $P(24,5)$}] (P_24_5__Rationals) at (20,-6) {$\begin{smallmatrix}&&10\\6&13&20&17&13&9&6&3\end{smallmatrix}$};
	\node[noselEEEEEEEEE, label={[shift={(0.2,-0.55)}, brown]\tiny $P(24,6)$}] (P_24_6__Rationals) at (22,-9) {$\begin{smallmatrix}&&7\\5&10&15&13&10&7&4&2\end{smallmatrix}$};
	\node[noselEEEEEEEEE, label={[shift={(0.2,-0.55)}, brown]\tiny $P(24,7)$}] (P_24_7__Rationals) at (24,-12) {$\begin{smallmatrix}&&5\\3&6&10&9&7&5&3&1\end{smallmatrix}$};
	\node[seloneEEEEEEEEE, label={[shift={(0.2,-0.55)}, brown]\tiny $P(24,8)$}] (P_24_8__Rationals) at (26,-15) {$\begin{smallmatrix}&&2\\1&3&5&5&4&3&2&1\end{smallmatrix}$};
	\node[noselEEEEEEEEE, label={[shift={(0.2,-0.55)}, brown]\tiny $P(24,9)$}] (P_24_9__Rationals) at (18,1) {$\begin{smallmatrix}&&7\\5&10&15&12&10&7&5&2\end{smallmatrix}$};
	\node at (21,3) {$\dots$};
	\node at (21,-3) {$\dots$};
	\draw[red,->] (P_20_2__Rationals) -- (P_20_1__Rationals);
	\draw[red,->] (P_20_3__Rationals) -- (P_20_2__Rationals);
	\draw[red,->] (P_20_3__Rationals) -- (P_20_4__Rationals);
	\draw[red,->] (P_20_4__Rationals) -- (P_20_5__Rationals);
	\draw[red,->] (P_20_5__Rationals) -- (P_20_6__Rationals);
	\draw[red,->] (P_20_6__Rationals) -- (P_20_7__Rationals);
	\draw[red,->] (P_20_7__Rationals) -- (P_20_8__Rationals);
	\draw[red,->] (P_20_3__Rationals) -- (P_20_9__Rationals);
	\draw[gray,dashed,->] (P_20_1__Rationals) -- (P_21_1__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[red,->] (P_21_2__Rationals) -- (P_21_1__Rationals);
	\draw[blue,->] (P_20_1__Rationals) -- (P_21_2__Rationals);
	\draw[gray,dashed,->] (P_20_2__Rationals) -- (P_21_2__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[red,->] (P_21_3__Rationals) -- (P_21_2__Rationals);
	\draw[blue,->] (P_20_2__Rationals) -- (P_21_3__Rationals);
	\draw[gray,dashed,->] (P_20_3__Rationals) -- (P_21_3__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[blue,->] (P_20_4__Rationals) -- (P_21_3__Rationals);
	\draw[blue,->] (P_20_9__Rationals) -- (P_21_3__Rationals);
	\draw[gray,dashed,->] (P_20_4__Rationals) -- (P_21_4__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[blue,->] (P_20_5__Rationals) -- (P_21_4__Rationals);
	\draw[red,->] (P_21_3__Rationals) -- (P_21_4__Rationals);
	\draw[gray,dashed,->] (P_20_5__Rationals) -- (P_21_5__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[blue,->] (P_20_6__Rationals) -- (P_21_5__Rationals);
	\draw[red,->] (P_21_4__Rationals) -- (P_21_5__Rationals);
	\draw[gray,dashed,->] (P_20_6__Rationals) -- (P_21_6__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[blue,->] (P_20_7__Rationals) -- (P_21_6__Rationals);
	\draw[red,->] (P_21_5__Rationals) -- (P_21_6__Rationals);
	\draw[gray,dashed,->] (P_20_7__Rationals) -- (P_21_7__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[blue,->] (P_20_8__Rationals) -- (P_21_7__Rationals);
	\draw[red,->] (P_21_6__Rationals) -- (P_21_7__Rationals);
	\draw[gray,dashed,->] (P_20_8__Rationals) -- (P_21_8__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[red,->] (P_21_7__Rationals) -- (P_21_8__Rationals);
	\draw[gray,dashed,->] (P_20_9__Rationals) -- (P_21_9__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[red,->] (P_21_3__Rationals) -- (P_21_9__Rationals);
	\draw[gray,dashed,->] (P_21_1__Rationals) -- (P_22_1__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[red,->] (P_22_2__Rationals) -- (P_22_1__Rationals);
	\draw[blue,->] (P_21_1__Rationals) -- (P_22_2__Rationals);
	\draw[gray,dashed,->] (P_21_2__Rationals) -- (P_22_2__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[red,->] (P_22_3__Rationals) -- (P_22_2__Rationals);
	\draw[blue,->] (P_21_2__Rationals) -- (P_22_3__Rationals);
	\draw[gray,dashed,->] (P_21_3__Rationals) -- (P_22_3__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[blue,->] (P_21_4__Rationals) -- (P_22_3__Rationals);
	\draw[blue,->] (P_21_9__Rationals) -- (P_22_3__Rationals);
	\draw[gray,dashed,->] (P_21_4__Rationals) -- (P_22_4__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[blue,->] (P_21_5__Rationals) -- (P_22_4__Rationals);
	\draw[red,->] (P_22_3__Rationals) -- (P_22_4__Rationals);
	\draw[gray,dashed,->] (P_21_5__Rationals) -- (P_22_5__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[blue,->] (P_21_6__Rationals) -- (P_22_5__Rationals);
	\draw[red,->] (P_22_4__Rationals) -- (P_22_5__Rationals);
	\draw[gray,dashed,->] (P_21_6__Rationals) -- (P_22_6__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[blue,->] (P_21_7__Rationals) -- (P_22_6__Rationals);
	\draw[red,->] (P_22_5__Rationals) -- (P_22_6__Rationals);
	\draw[gray,dashed,->] (P_21_7__Rationals) -- (P_22_7__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[blue,->] (P_21_8__Rationals) -- (P_22_7__Rationals);
	\draw[red,->] (P_22_6__Rationals) -- (P_22_7__Rationals);
	\draw[gray,dashed,->] (P_21_8__Rationals) -- (P_22_8__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[red,->] (P_22_7__Rationals) -- (P_22_8__Rationals);
	\draw[gray,dashed,->] (P_21_9__Rationals) -- (P_22_9__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[red,->] (P_22_3__Rationals) -- (P_22_9__Rationals);
	\draw[gray,dashed,->] (P_22_1__Rationals) -- (P_23_1__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[red,->] (P_23_2__Rationals) -- (P_23_1__Rationals);
	\draw[blue,->] (P_22_1__Rationals) -- (P_23_2__Rationals);
	\draw[gray,dashed,->] (P_22_2__Rationals) -- (P_23_2__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[red,->] (P_23_3__Rationals) -- (P_23_2__Rationals);
	\draw[blue,->] (P_22_2__Rationals) -- (P_23_3__Rationals);
	\draw[gray,dashed,->] (P_22_3__Rationals) -- (P_23_3__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[blue,->] (P_22_4__Rationals) -- (P_23_3__Rationals);
	\draw[blue,->] (P_22_9__Rationals) -- (P_23_3__Rationals);
	\draw[gray,dashed,->] (P_22_4__Rationals) -- (P_23_4__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[blue,->] (P_22_5__Rationals) -- (P_23_4__Rationals);
	\draw[red,->] (P_23_3__Rationals) -- (P_23_4__Rationals);
	\draw[gray,dashed,->] (P_22_5__Rationals) -- (P_23_5__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[blue,->] (P_22_6__Rationals) -- (P_23_5__Rationals);
	\draw[red,->] (P_23_4__Rationals) -- (P_23_5__Rationals);
	\draw[gray,dashed,->] (P_22_6__Rationals) -- (P_23_6__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[blue,->] (P_22_7__Rationals) -- (P_23_6__Rationals);
	\draw[red,->] (P_23_5__Rationals) -- (P_23_6__Rationals);
	\draw[gray,dashed,->] (P_22_7__Rationals) -- (P_23_7__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[blue,->] (P_22_8__Rationals) -- (P_23_7__Rationals);
	\draw[red,->] (P_23_6__Rationals) -- (P_23_7__Rationals);
	\draw[gray,dashed,->] (P_22_8__Rationals) -- (P_23_8__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[red,->] (P_23_7__Rationals) -- (P_23_8__Rationals);
	\draw[gray,dashed,->] (P_22_9__Rationals) -- (P_23_9__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[red,->] (P_23_3__Rationals) -- (P_23_9__Rationals);
	\draw[gray,dashed,->] (P_23_1__Rationals) -- (P_24_1__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[red,->] (P_24_2__Rationals) -- (P_24_1__Rationals);
	\draw[blue,->] (P_23_1__Rationals) -- (P_24_2__Rationals);
	\draw[gray,dashed,->] (P_23_2__Rationals) -- (P_24_2__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[red,->] (P_24_3__Rationals) -- (P_24_2__Rationals);
	\draw[blue,->] (P_23_2__Rationals) -- (P_24_3__Rationals);
	\draw[gray,dashed,->] (P_23_3__Rationals) -- (P_24_3__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[blue,->] (P_23_4__Rationals) -- (P_24_3__Rationals);
	\draw[blue,->] (P_23_9__Rationals) -- (P_24_3__Rationals);
	\draw[gray,dashed,->] (P_23_4__Rationals) -- (P_24_4__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[blue,->] (P_23_5__Rationals) -- (P_24_4__Rationals);
	\draw[red,->] (P_24_3__Rationals) -- (P_24_4__Rationals);
	\draw[gray,dashed,->] (P_23_5__Rationals) -- (P_24_5__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[blue,->] (P_23_6__Rationals) -- (P_24_5__Rationals);
	\draw[red,->] (P_24_4__Rationals) -- (P_24_5__Rationals);
	\draw[gray,dashed,->] (P_23_6__Rationals) -- (P_24_6__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[blue,->] (P_23_7__Rationals) -- (P_24_6__Rationals);
	\draw[red,->] (P_24_5__Rationals) -- (P_24_6__Rationals);
	\draw[gray,dashed,->] (P_23_7__Rationals) -- (P_24_7__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[blue,->] (P_23_8__Rationals) -- (P_24_7__Rationals);
	\draw[red,->] (P_24_6__Rationals) -- (P_24_7__Rationals);
	\draw[gray,dashed,->] (P_23_8__Rationals) -- (P_24_8__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[red,->] (P_24_7__Rationals) -- (P_24_8__Rationals);
	\draw[gray,dashed,->] (P_23_9__Rationals) -- (P_24_9__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[red,->] (P_24_3__Rationals) -- (P_24_9__Rationals);
\end{tikzpicture}
\end{scaletikzpicturetowidth}
\end{center}

\end{figure}
\medskip{}
\subsubsection*{Modules of the form $P(n,1)$}

Defect: $\partial P(n,1) = -2$, for $n\ge 0$.

\begin{fleqn}
\begin{align*}
P(0,1):\ & \bigl(I(0,1),P(0,2)\bigr),\ \bigl(I(0,2),P(0,3)\bigr)\\
P(1,1):\ & \bigl(I(0,9),P(0,4)\bigr),\ \bigl(I(4,8),P(0,9)\bigr)\\
P(2,1):\ & \bigl(I(3,7),P(0,2)\bigr),\ \bigl(I(1,1),P(0,5)\bigr),\ \bigl(I(3,8),P(1,9)\bigr)\\
P(3,1):\ & \bigl(R_{1}^{1}(1),P(0,1)\bigr),\ \bigl(I(5,8),P(0,6)\bigr),\ \bigl(I(8,8),P(0,9)\bigr),\ \bigl(I(2,7),P(1,2)\bigr),\ \bigl(I(0,1),P(1,5)\bigr)\\
 & \bigl(I(2,8),P(2,9)\bigr)\\
P(4,1):\ & \bigl(R_{1}^{4}(1),P(0,7)\bigr),\ \bigl(R_{1}^{2}(1),P(1,1)\bigr),\ \bigl(I(4,8),P(1,6)\bigr),\ \bigl(I(7,8),P(1,9)\bigr),\ \bigl(I(1,7),P(2,2)\bigr)\\
 & \bigl(I(1,8),P(3,9)\bigr)\\
P(5,1):\ & \bigl(R_{0}^{1}(1),P(0,1)\bigr),\ \bigl(P(7,8),P(0,8)\bigr),\ \bigl(R_{1}^{5}(1),P(1,7)\bigr),\ \bigl(R_{1}^{3}(1),P(2,1)\bigr),\ \bigl(I(3,8),P(2,6)\bigr)\\
 & \bigl(I(6,8),P(2,9)\bigr),\ \bigl(I(0,7),P(3,2)\bigr),\ \bigl(I(0,8),P(4,9)\bigr)\\
P(6,1):\ & \bigl(R_{0}^{2}(1),P(1,1)\bigr),\ \bigl(P(8,8),P(1,8)\bigr),\ \bigl(R_{1}^{1}(1),P(2,7)\bigr),\ \bigl(R_{1}^{4}(1),P(3,1)\bigr),\ \bigl(I(2,8),P(3,6)\bigr)\\
 & \bigl(I(5,8),P(3,9)\bigr)\\
P(7,1):\ & \bigl(R_{1}^{4}(2),P(0,7)\bigr),\ \bigl(R_{0}^{3}(1),P(2,1)\bigr),\ \bigl(P(9,8),P(2,8)\bigr),\ \bigl(R_{1}^{2}(1),P(3,7)\bigr),\ \bigl(R_{1}^{5}(1),P(4,1)\bigr)\\
 & \bigl(I(1,8),P(4,6)\bigr),\ \bigl(I(4,8),P(4,9)\bigr)\\
P(8,1):\ & \bigl(P(13,8),P(0,8)\bigr),\ \bigl(R_{1}^{5}(2),P(1,7)\bigr),\ \bigl(R_{0}^{1}(1),P(3,1)\bigr),\ \bigl(P(10,8),P(3,8)\bigr),\ \bigl(R_{1}^{3}(1),P(4,7)\bigr)\\
 & \bigl(R_{1}^{1}(1),P(5,1)\bigr),\ \bigl(I(0,8),P(5,6)\bigr),\ \bigl(I(3,8),P(5,9)\bigr)\\
P(9,1):\ & \bigl(P(14,8),P(1,8)\bigr),\ \bigl(R_{1}^{1}(2),P(2,7)\bigr),\ \bigl(R_{0}^{2}(1),P(4,1)\bigr),\ \bigl(P(11,8),P(4,8)\bigr),\ \bigl(R_{1}^{4}(1),P(5,7)\bigr)\\
 & \bigl(R_{1}^{2}(1),P(6,1)\bigr),\ \bigl(I(2,8),P(6,9)\bigr)\\
P(10,1):\ & \bigl(P(17,8),P(0,8)\bigr),\ \bigl(P(15,8),P(2,8)\bigr),\ \bigl(R_{1}^{2}(2),P(3,7)\bigr),\ \bigl(R_{0}^{3}(1),P(5,1)\bigr),\ \bigl(P(12,8),P(5,8)\bigr)\\
 & \bigl(R_{1}^{5}(1),P(6,7)\bigr),\ \bigl(R_{1}^{3}(1),P(7,1)\bigr),\ \bigl(I(1,8),P(7,9)\bigr)\\
P(11,1):\ & \bigl(P(18,8),P(1,8)\bigr),\ \bigl(P(16,8),P(3,8)\bigr),\ \bigl(R_{1}^{3}(2),P(4,7)\bigr),\ \bigl(R_{0}^{1}(1),P(6,1)\bigr),\ \bigl(P(13,8),P(6,8)\bigr)\\
 & \bigl(R_{1}^{1}(1),P(7,7)\bigr),\ \bigl(R_{1}^{4}(1),P(8,1)\bigr),\ \bigl(I(0,8),P(8,9)\bigr)\\
P(12,1):\ & \bigl(P(19,8),P(2,8)\bigr),\ \bigl(P(17,8),P(4,8)\bigr),\ \bigl(R_{1}^{4}(2),P(5,7)\bigr),\ \bigl(R_{0}^{2}(1),P(7,1)\bigr),\ \bigl(P(14,8),P(7,8)\bigr)\\
 & \bigl(R_{1}^{2}(1),P(8,7)\bigr),\ \bigl(R_{1}^{5}(1),P(9,1)\bigr)\\
P(13,1):\ & \bigl(P(23,8),P(0,8)\bigr),\ \bigl(P(20,8),P(3,8)\bigr),\ \bigl(P(18,8),P(5,8)\bigr),\ \bigl(R_{1}^{5}(2),P(6,7)\bigr),\ \bigl(R_{0}^{3}(1),P(8,1)\bigr)\\
 & \bigl(P(15,8),P(8,8)\bigr),\ \bigl(R_{1}^{3}(1),P(9,7)\bigr),\ \bigl(R_{1}^{1}(1),P(10,1)\bigr)\\
P(n,1):\ & \bigl(P(n+10,8),P(n-13,8)\bigr),\ \bigl(P(n+7,8),P(n-10,8)\bigr),\ \bigl(P(n+5,8),P(n-8,8)\bigr)\\
 & \bigl(R_{1}^{(n-9)\bmod 5+1}(2),P(n-7,7)\bigr),\ \bigl(R_{0}^{(n-11)\bmod 3+1}(1),P(n-5,1)\bigr),\ \bigl(P(n+2,8),P(n-5,8)\bigr)\\
 & \bigl(R_{1}^{(n-11)\bmod 5+1}(1),P(n-4,7)\bigr),\ \bigl(R_{1}^{(n-13)\bmod 5+1}(1),P(n-3,1)\bigr),\ n>13\\
\end{align*}
\end{fleqn}
\subsubsection*{Modules of the form $P(n,2)$}

Defect: $\partial P(n,2) = -4$, for $n\ge 0$.

\begin{fleqn}
\begin{align*}
P(0,2):\ & \bigl(I(1,1),P(0,3)\bigr)\\
P(1,2):\ & \bigl(P(1,1),P(0,1)\bigr),\ \bigl(I(5,8),P(0,4)\bigr),\ \bigl(P(2,8),P(0,9)\bigr),\ \bigl(I(0,1),P(1,3)\bigr)\\
P(2,2):\ & \bigl(R_{1}^{4}(1),P(0,5)\bigr),\ \bigl(P(2,1),P(1,1)\bigr),\ \bigl(I(4,8),P(1,4)\bigr),\ \bigl(P(3,8),P(1,9)\bigr)\\
P(3,2):\ & \bigl(P(4,7),P(0,1)\bigr),\ \bigl(P(7,8),P(0,6)\bigr),\ \bigl(R_{1}^{5}(1),P(1,5)\bigr),\ \bigl(P(3,1),P(2,1)\bigr),\ \bigl(I(3,8),P(2,4)\bigr)\\
 & \bigl(P(4,8),P(2,9)\bigr)\\
P(4,2):\ & \bigl(P(6,1),P(0,7)\bigr),\ \bigl(P(5,7),P(1,1)\bigr),\ \bigl(P(8,8),P(1,6)\bigr),\ \bigl(R_{1}^{1}(1),P(2,5)\bigr),\ \bigl(P(4,1),P(3,1)\bigr)\\
 & \bigl(I(2,8),P(3,4)\bigr),\ \bigl(P(5,8),P(3,9)\bigr)\\
P(5,2):\ & \bigl(P(6,9),P(0,8)\bigr),\ \bigl(P(7,1),P(1,7)\bigr),\ \bigl(P(6,7),P(2,1)\bigr),\ \bigl(P(9,8),P(2,6)\bigr),\ \bigl(R_{1}^{2}(1),P(3,5)\bigr)\\
 & \bigl(P(5,1),P(4,1)\bigr),\ \bigl(I(1,8),P(4,4)\bigr),\ \bigl(P(6,8),P(4,9)\bigr)\\
P(6,2):\ & \bigl(P(7,9),P(1,8)\bigr),\ \bigl(P(8,1),P(2,7)\bigr),\ \bigl(P(7,7),P(3,1)\bigr),\ \bigl(P(10,8),P(3,6)\bigr),\ \bigl(R_{1}^{3}(1),P(4,5)\bigr)\\
 & \bigl(P(6,1),P(5,1)\bigr),\ \bigl(I(0,8),P(5,4)\bigr),\ \bigl(P(7,8),P(5,9)\bigr)\\
P(7,2):\ & \bigl(P(8,9),P(2,8)\bigr),\ \bigl(P(9,1),P(3,7)\bigr),\ \bigl(P(8,7),P(4,1)\bigr),\ \bigl(P(11,8),P(4,6)\bigr),\ \bigl(R_{1}^{4}(1),P(5,5)\bigr)\\
 & \bigl(P(7,1),P(6,1)\bigr),\ \bigl(P(8,8),P(6,9)\bigr)\\
P(8,2):\ & \bigl(P(9,6),P(0,8)\bigr),\ \bigl(P(9,9),P(3,8)\bigr),\ \bigl(P(10,1),P(4,7)\bigr),\ \bigl(P(9,7),P(5,1)\bigr),\ \bigl(P(12,8),P(5,6)\bigr)\\
 & \bigl(R_{1}^{5}(1),P(6,5)\bigr),\ \bigl(P(8,1),P(7,1)\bigr),\ \bigl(P(9,8),P(7,9)\bigr)\\
P(n,2):\ & \bigl(P(n+1,6),P(n-8,8)\bigr),\ \bigl(P(n+1,9),P(n-5,8)\bigr),\ \bigl(P(n+2,1),P(n-4,7)\bigr)\\
 & \bigl(P(n+1,7),P(n-3,1)\bigr),\ \bigl(P(n+4,8),P(n-3,6)\bigr),\ \bigl(R_{1}^{(n-4)\bmod 5+1}(1),P(n-2,5)\bigr)\\
 & \bigl(P(n,1),P(n-1,1)\bigr),\ \bigl(P(n+1,8),P(n-1,9)\bigr),\ n>8\\
\end{align*}
\end{fleqn}
\subsubsection*{Modules of the form $P(n,3)$}

Defect: $\partial P(n,3) = -6$, for $n\ge 0$.

\begin{fleqn}
\begin{align*}
P(0,3):\ & - \\
P(1,3):\ & \bigl(P(1,1),P(0,2)\bigr),\ \bigl(P(1,8),P(0,4)\bigr),\ \bigl(P(1,9),P(0,9)\bigr)\\
P(2,3):\ & \bigl(P(2,2),P(0,1)\bigr),\ \bigl(P(2,7),P(0,5)\bigr),\ \bigl(P(2,1),P(1,2)\bigr),\ \bigl(P(2,8),P(1,4)\bigr),\ \bigl(P(2,9),P(1,9)\bigr)\\
P(3,3):\ & \bigl(P(3,6),P(0,6)\bigr),\ \bigl(P(3,2),P(1,1)\bigr),\ \bigl(P(3,7),P(1,5)\bigr),\ \bigl(P(3,1),P(2,2)\bigr),\ \bigl(P(3,8),P(2,4)\bigr)\\
 & \bigl(P(3,9),P(2,9)\bigr)\\
P(4,3):\ & \bigl(P(4,5),P(0,7)\bigr),\ \bigl(P(4,6),P(1,6)\bigr),\ \bigl(P(4,2),P(2,1)\bigr),\ \bigl(P(4,7),P(2,5)\bigr),\ \bigl(P(4,1),P(3,2)\bigr)\\
 & \bigl(P(4,8),P(3,4)\bigr),\ \bigl(P(4,9),P(3,9)\bigr)\\
P(5,3):\ & \bigl(P(5,4),P(0,8)\bigr),\ \bigl(P(5,5),P(1,7)\bigr),\ \bigl(P(5,6),P(2,6)\bigr),\ \bigl(P(5,2),P(3,1)\bigr),\ \bigl(P(5,7),P(3,5)\bigr)\\
 & \bigl(P(5,1),P(4,2)\bigr),\ \bigl(P(5,8),P(4,4)\bigr),\ \bigl(P(5,9),P(4,9)\bigr)\\
P(n,3):\ & \bigl(P(n,4),P(n-5,8)\bigr),\ \bigl(P(n,5),P(n-4,7)\bigr),\ \bigl(P(n,6),P(n-3,6)\bigr)\\
 & \bigl(P(n,2),P(n-2,1)\bigr),\ \bigl(P(n,7),P(n-2,5)\bigr),\ \bigl(P(n,1),P(n-1,2)\bigr)\\
 & \bigl(P(n,8),P(n-1,4)\bigr),\ \bigl(P(n,9),P(n-1,9)\bigr),\ n>5\\
\end{align*}
\end{fleqn}
\subsubsection*{Modules of the form $P(n,4)$}

Defect: $\partial P(n,4) = -5$, for $n\ge 0$.

\begin{fleqn}
\begin{align*}
P(0,4):\ & \bigl(I(4,8),P(0,3)\bigr)\\
P(1,4):\ & \bigl(P(3,8),P(0,2)\bigr),\ \bigl(P(1,8),P(0,5)\bigr),\ \bigl(P(2,1),P(0,9)\bigr),\ \bigl(I(3,8),P(1,3)\bigr)\\
P(2,4):\ & \bigl(P(3,9),P(0,1)\bigr),\ \bigl(P(2,7),P(0,6)\bigr),\ \bigl(P(4,8),P(1,2)\bigr),\ \bigl(P(2,8),P(1,5)\bigr),\ \bigl(P(3,1),P(1,9)\bigr)\\
 & \bigl(I(2,8),P(2,3)\bigr)\\
P(3,4):\ & \bigl(P(3,6),P(0,7)\bigr),\ \bigl(P(4,9),P(1,1)\bigr),\ \bigl(P(3,7),P(1,6)\bigr),\ \bigl(P(5,8),P(2,2)\bigr),\ \bigl(P(3,8),P(2,5)\bigr)\\
 & \bigl(P(4,1),P(2,9)\bigr),\ \bigl(I(1,8),P(3,3)\bigr)\\
P(4,4):\ & \bigl(P(4,5),P(0,8)\bigr),\ \bigl(P(4,6),P(1,7)\bigr),\ \bigl(P(5,9),P(2,1)\bigr),\ \bigl(P(4,7),P(2,6)\bigr),\ \bigl(P(6,8),P(3,2)\bigr)\\
 & \bigl(P(4,8),P(3,5)\bigr),\ \bigl(P(5,1),P(3,9)\bigr),\ \bigl(I(0,8),P(4,3)\bigr)\\
P(5,4):\ & \bigl(P(5,5),P(1,8)\bigr),\ \bigl(P(5,6),P(2,7)\bigr),\ \bigl(P(6,9),P(3,1)\bigr),\ \bigl(P(5,7),P(3,6)\bigr),\ \bigl(P(7,8),P(4,2)\bigr)\\
 & \bigl(P(5,8),P(4,5)\bigr),\ \bigl(P(6,1),P(4,9)\bigr)\\
P(6,4):\ & \bigl(P(7,2),P(0,8)\bigr),\ \bigl(P(6,5),P(2,8)\bigr),\ \bigl(P(6,6),P(3,7)\bigr),\ \bigl(P(7,9),P(4,1)\bigr),\ \bigl(P(6,7),P(4,6)\bigr)\\
 & \bigl(P(8,8),P(5,2)\bigr),\ \bigl(P(6,8),P(5,5)\bigr),\ \bigl(P(7,1),P(5,9)\bigr)\\
P(n,4):\ & \bigl(P(n+1,2),P(n-6,8)\bigr),\ \bigl(P(n,5),P(n-4,8)\bigr),\ \bigl(P(n,6),P(n-3,7)\bigr)\\
 & \bigl(P(n+1,9),P(n-2,1)\bigr),\ \bigl(P(n,7),P(n-2,6)\bigr),\ \bigl(P(n+2,8),P(n-1,2)\bigr)\\
 & \bigl(P(n,8),P(n-1,5)\bigr),\ \bigl(P(n+1,1),P(n-1,9)\bigr),\ n>6\\
\end{align*}
\end{fleqn}
\subsubsection*{Modules of the form $P(n,5)$}

Defect: $\partial P(n,5) = -4$, for $n\ge 0$.

\begin{fleqn}
\begin{align*}
P(0,5):\ & \bigl(I(3,7),P(0,3)\bigr),\ \bigl(I(3,8),P(0,4)\bigr)\\
P(1,5):\ & \bigl(R_{1}^{1}(1),P(0,2)\bigr),\ \bigl(P(1,8),P(0,6)\bigr),\ \bigl(P(4,8),P(0,9)\bigr),\ \bigl(I(2,7),P(1,3)\bigr),\ \bigl(I(2,8),P(1,4)\bigr)\\
P(2,5):\ & \bigl(P(4,1),P(0,1)\bigr),\ \bigl(P(2,7),P(0,7)\bigr),\ \bigl(R_{1}^{2}(1),P(1,2)\bigr),\ \bigl(P(2,8),P(1,6)\bigr),\ \bigl(P(5,8),P(1,9)\bigr)\\
 & \bigl(I(1,7),P(2,3)\bigr),\ \bigl(I(1,8),P(2,4)\bigr)\\
P(3,5):\ & \bigl(P(3,6),P(0,8)\bigr),\ \bigl(P(5,1),P(1,1)\bigr),\ \bigl(P(3,7),P(1,7)\bigr),\ \bigl(R_{1}^{3}(1),P(2,2)\bigr),\ \bigl(P(3,8),P(2,6)\bigr)\\
 & \bigl(P(6,8),P(2,9)\bigr),\ \bigl(I(0,7),P(3,3)\bigr),\ \bigl(I(0,8),P(3,4)\bigr)\\
P(4,5):\ & \bigl(P(4,6),P(1,8)\bigr),\ \bigl(P(6,1),P(2,1)\bigr),\ \bigl(P(4,7),P(2,7)\bigr),\ \bigl(R_{1}^{4}(1),P(3,2)\bigr),\ \bigl(P(4,8),P(3,6)\bigr)\\
 & \bigl(P(7,8),P(3,9)\bigr)\\
P(5,5):\ & \bigl(P(8,7),P(0,7)\bigr),\ \bigl(P(5,6),P(2,8)\bigr),\ \bigl(P(7,1),P(3,1)\bigr),\ \bigl(P(5,7),P(3,7)\bigr),\ \bigl(R_{1}^{5}(1),P(4,2)\bigr)\\
 & \bigl(P(5,8),P(4,6)\bigr),\ \bigl(P(8,8),P(4,9)\bigr)\\
P(6,5):\ & \bigl(P(8,9),P(0,8)\bigr),\ \bigl(P(9,7),P(1,7)\bigr),\ \bigl(P(6,6),P(3,8)\bigr),\ \bigl(P(8,1),P(4,1)\bigr),\ \bigl(P(6,7),P(4,7)\bigr)\\
 & \bigl(R_{1}^{1}(1),P(5,2)\bigr),\ \bigl(P(6,8),P(5,6)\bigr),\ \bigl(P(9,8),P(5,9)\bigr)\\
P(n,5):\ & \bigl(P(n+2,9),P(n-6,8)\bigr),\ \bigl(P(n+3,7),P(n-5,7)\bigr),\ \bigl(P(n,6),P(n-3,8)\bigr)\\
 & \bigl(P(n+2,1),P(n-2,1)\bigr),\ \bigl(P(n,7),P(n-2,7)\bigr),\ \bigl(R_{1}^{(n-6)\bmod 5+1}(1),P(n-1,2)\bigr)\\
 & \bigl(P(n,8),P(n-1,6)\bigr),\ \bigl(P(n+3,8),P(n-1,9)\bigr),\ n>6\\
\end{align*}
\end{fleqn}
\subsubsection*{Modules of the form $P(n,6)$}

Defect: $\partial P(n,6) = -3$, for $n\ge 0$.

\begin{fleqn}
\begin{align*}
P(0,6):\ & \bigl(I(2,6),P(0,3)\bigr),\ \bigl(I(2,7),P(0,4)\bigr),\ \bigl(I(2,8),P(0,5)\bigr)\\
P(1,6):\ & \bigl(I(7,8),P(0,2)\bigr),\ \bigl(P(1,8),P(0,7)\bigr),\ \bigl(R_{1}^{2}(1),P(0,9)\bigr),\ \bigl(I(1,6),P(1,3)\bigr),\ \bigl(I(1,7),P(1,4)\bigr)\\
 & \bigl(I(1,8),P(1,5)\bigr)\\
P(2,6):\ & \bigl(P(6,8),P(0,1)\bigr),\ \bigl(P(2,7),P(0,8)\bigr),\ \bigl(I(6,8),P(1,2)\bigr),\ \bigl(P(2,8),P(1,7)\bigr),\ \bigl(R_{1}^{3}(1),P(1,9)\bigr)\\
 & \bigl(I(0,6),P(2,3)\bigr),\ \bigl(I(0,7),P(2,4)\bigr),\ \bigl(I(0,8),P(2,5)\bigr)\\
P(3,6):\ & \bigl(P(7,8),P(1,1)\bigr),\ \bigl(P(3,7),P(1,8)\bigr),\ \bigl(I(5,8),P(2,2)\bigr),\ \bigl(P(3,8),P(2,7)\bigr),\ \bigl(R_{1}^{4}(1),P(2,9)\bigr)\\
P(4,6):\ & \bigl(R_{1}^{4}(2),P(0,6)\bigr),\ \bigl(P(8,8),P(2,1)\bigr),\ \bigl(P(4,7),P(2,8)\bigr),\ \bigl(I(4,8),P(3,2)\bigr),\ \bigl(P(4,8),P(3,7)\bigr)\\
 & \bigl(R_{1}^{5}(1),P(3,9)\bigr)\\
P(5,6):\ & \bigl(P(13,8),P(0,7)\bigr),\ \bigl(R_{1}^{5}(2),P(1,6)\bigr),\ \bigl(P(9,8),P(3,1)\bigr),\ \bigl(P(5,7),P(3,8)\bigr),\ \bigl(I(3,8),P(4,2)\bigr)\\
 & \bigl(P(5,8),P(4,7)\bigr),\ \bigl(R_{1}^{1}(1),P(4,9)\bigr)\\
P(6,6):\ & \bigl(P(9,1),P(0,8)\bigr),\ \bigl(P(14,8),P(1,7)\bigr),\ \bigl(R_{1}^{1}(2),P(2,6)\bigr),\ \bigl(P(10,8),P(4,1)\bigr),\ \bigl(P(6,7),P(4,8)\bigr)\\
 & \bigl(I(2,8),P(5,2)\bigr),\ \bigl(P(6,8),P(5,7)\bigr),\ \bigl(R_{1}^{2}(1),P(5,9)\bigr)\\
P(7,6):\ & \bigl(P(10,1),P(1,8)\bigr),\ \bigl(P(15,8),P(2,7)\bigr),\ \bigl(R_{1}^{2}(2),P(3,6)\bigr),\ \bigl(P(11,8),P(5,1)\bigr),\ \bigl(P(7,7),P(5,8)\bigr)\\
 & \bigl(I(1,8),P(6,2)\bigr),\ \bigl(P(7,8),P(6,7)\bigr),\ \bigl(R_{1}^{3}(1),P(6,9)\bigr)\\
P(8,6):\ & \bigl(P(11,1),P(2,8)\bigr),\ \bigl(P(16,8),P(3,7)\bigr),\ \bigl(R_{1}^{3}(2),P(4,6)\bigr),\ \bigl(P(12,8),P(6,1)\bigr),\ \bigl(P(8,7),P(6,8)\bigr)\\
 & \bigl(I(0,8),P(7,2)\bigr),\ \bigl(P(8,8),P(7,7)\bigr),\ \bigl(R_{1}^{4}(1),P(7,9)\bigr)\\
P(9,6):\ & \bigl(P(12,1),P(3,8)\bigr),\ \bigl(P(17,8),P(4,7)\bigr),\ \bigl(R_{1}^{4}(2),P(5,6)\bigr),\ \bigl(P(13,8),P(7,1)\bigr),\ \bigl(P(9,7),P(7,8)\bigr)\\
 & \bigl(P(9,8),P(8,7)\bigr),\ \bigl(R_{1}^{5}(1),P(8,9)\bigr)\\
P(10,6):\ & \bigl(P(14,7),P(0,8)\bigr),\ \bigl(P(13,1),P(4,8)\bigr),\ \bigl(P(18,8),P(5,7)\bigr),\ \bigl(R_{1}^{5}(2),P(6,6)\bigr),\ \bigl(P(14,8),P(8,1)\bigr)\\
 & \bigl(P(10,7),P(8,8)\bigr),\ \bigl(P(10,8),P(9,7)\bigr),\ \bigl(R_{1}^{1}(1),P(9,9)\bigr)\\
P(n,6):\ & \bigl(P(n+4,7),P(n-10,8)\bigr),\ \bigl(P(n+3,1),P(n-6,8)\bigr),\ \bigl(P(n+8,8),P(n-5,7)\bigr)\\
 & \bigl(R_{1}^{(n-6)\bmod 5+1}(2),P(n-4,6)\bigr),\ \bigl(P(n+4,8),P(n-2,1)\bigr),\ \bigl(P(n,7),P(n-2,8)\bigr)\\
 & \bigl(P(n,8),P(n-1,7)\bigr),\ \bigl(R_{1}^{(n-10)\bmod 5+1}(1),P(n-1,9)\bigr),\ n>10\\
\end{align*}
\end{fleqn}
\subsubsection*{Modules of the form $P(n,7)$}

Defect: $\partial P(n,7) = -2$, for $n\ge 0$.

\begin{fleqn}
\begin{align*}
P(0,7):\ & \bigl(I(1,5),P(0,3)\bigr),\ \bigl(I(1,6),P(0,4)\bigr),\ \bigl(I(1,7),P(0,5)\bigr),\ \bigl(I(1,8),P(0,6)\bigr)\\
P(1,7):\ & \bigl(I(2,1),P(0,2)\bigr),\ \bigl(P(1,8),P(0,8)\bigr),\ \bigl(I(6,8),P(0,9)\bigr),\ \bigl(I(0,5),P(1,3)\bigr),\ \bigl(I(0,6),P(1,4)\bigr)\\
 & \bigl(I(0,7),P(1,5)\bigr),\ \bigl(I(0,8),P(1,6)\bigr)\\
P(2,7):\ & \bigl(R_{1}^{4}(1),P(0,1)\bigr),\ \bigl(I(1,1),P(1,2)\bigr),\ \bigl(P(2,8),P(1,8)\bigr),\ \bigl(I(5,8),P(1,9)\bigr)\\
P(3,7):\ & \bigl(I(4,7),P(0,5)\bigr),\ \bigl(R_{1}^{5}(1),P(1,1)\bigr),\ \bigl(I(0,1),P(2,2)\bigr),\ \bigl(P(3,8),P(2,8)\bigr),\ \bigl(I(4,8),P(2,9)\bigr)\\
P(4,7):\ & \bigl(I(9,8),P(0,6)\bigr),\ \bigl(I(3,7),P(1,5)\bigr),\ \bigl(R_{1}^{1}(1),P(2,1)\bigr),\ \bigl(P(4,8),P(3,8)\bigr),\ \bigl(I(3,8),P(3,9)\bigr)\\
P(5,7):\ & \bigl(R_{1}^{1}(2),P(0,1)\bigr),\ \bigl(R_{0}^{2}(1),P(0,7)\bigr),\ \bigl(I(8,8),P(1,6)\bigr),\ \bigl(I(2,7),P(2,5)\bigr),\ \bigl(R_{1}^{2}(1),P(3,1)\bigr)\\
 & \bigl(P(5,8),P(4,8)\bigr),\ \bigl(I(2,8),P(4,9)\bigr)\\
P(6,7):\ & \bigl(P(11,8),P(0,8)\bigr),\ \bigl(R_{1}^{2}(2),P(1,1)\bigr),\ \bigl(R_{0}^{3}(1),P(1,7)\bigr),\ \bigl(I(7,8),P(2,6)\bigr),\ \bigl(I(1,7),P(3,5)\bigr)\\
 & \bigl(R_{1}^{3}(1),P(4,1)\bigr),\ \bigl(P(6,8),P(5,8)\bigr),\ \bigl(I(1,8),P(5,9)\bigr)\\
P(7,7):\ & \bigl(P(12,8),P(1,8)\bigr),\ \bigl(R_{1}^{3}(2),P(2,1)\bigr),\ \bigl(R_{0}^{1}(1),P(2,7)\bigr),\ \bigl(I(6,8),P(3,6)\bigr),\ \bigl(I(0,7),P(4,5)\bigr)\\
 & \bigl(R_{1}^{4}(1),P(5,1)\bigr),\ \bigl(P(7,8),P(6,8)\bigr),\ \bigl(I(0,8),P(6,9)\bigr)\\
P(8,7):\ & \bigl(P(13,8),P(2,8)\bigr),\ \bigl(R_{1}^{4}(2),P(3,1)\bigr),\ \bigl(R_{0}^{2}(1),P(3,7)\bigr),\ \bigl(I(5,8),P(4,6)\bigr),\ \bigl(R_{1}^{5}(1),P(6,1)\bigr)\\
 & \bigl(P(8,8),P(7,8)\bigr)\\
P(9,7):\ & \bigl(R_{1}^{4}(3),P(0,7)\bigr),\ \bigl(P(14,8),P(3,8)\bigr),\ \bigl(R_{1}^{5}(2),P(4,1)\bigr),\ \bigl(R_{0}^{3}(1),P(4,7)\bigr),\ \bigl(I(4,8),P(5,6)\bigr)\\
 & \bigl(R_{1}^{1}(1),P(7,1)\bigr),\ \bigl(P(9,8),P(8,8)\bigr)\\
P(10,7):\ & \bigl(P(19,8),P(0,8)\bigr),\ \bigl(R_{1}^{5}(3),P(1,7)\bigr),\ \bigl(P(15,8),P(4,8)\bigr),\ \bigl(R_{1}^{1}(2),P(5,1)\bigr),\ \bigl(R_{0}^{1}(1),P(5,7)\bigr)\\
 & \bigl(I(3,8),P(6,6)\bigr),\ \bigl(R_{1}^{2}(1),P(8,1)\bigr),\ \bigl(P(10,8),P(9,8)\bigr)\\
P(11,7):\ & \bigl(P(20,8),P(1,8)\bigr),\ \bigl(R_{1}^{1}(3),P(2,7)\bigr),\ \bigl(P(16,8),P(5,8)\bigr),\ \bigl(R_{1}^{2}(2),P(6,1)\bigr),\ \bigl(R_{0}^{2}(1),P(6,7)\bigr)\\
 & \bigl(I(2,8),P(7,6)\bigr),\ \bigl(R_{1}^{3}(1),P(9,1)\bigr),\ \bigl(P(11,8),P(10,8)\bigr)\\
P(12,7):\ & \bigl(P(21,8),P(2,8)\bigr),\ \bigl(R_{1}^{2}(3),P(3,7)\bigr),\ \bigl(P(17,8),P(6,8)\bigr),\ \bigl(R_{1}^{3}(2),P(7,1)\bigr),\ \bigl(R_{0}^{3}(1),P(7,7)\bigr)\\
 & \bigl(I(1,8),P(8,6)\bigr),\ \bigl(R_{1}^{4}(1),P(10,1)\bigr),\ \bigl(P(12,8),P(11,8)\bigr)\\
P(13,7):\ & \bigl(P(22,8),P(3,8)\bigr),\ \bigl(R_{1}^{3}(3),P(4,7)\bigr),\ \bigl(P(18,8),P(7,8)\bigr),\ \bigl(R_{1}^{4}(2),P(8,1)\bigr),\ \bigl(R_{0}^{1}(1),P(8,7)\bigr)\\
 & \bigl(I(0,8),P(9,6)\bigr),\ \bigl(R_{1}^{5}(1),P(11,1)\bigr),\ \bigl(P(13,8),P(12,8)\bigr)\\
P(14,7):\ & \bigl(P(23,8),P(4,8)\bigr),\ \bigl(R_{1}^{4}(3),P(5,7)\bigr),\ \bigl(P(19,8),P(8,8)\bigr),\ \bigl(R_{1}^{5}(2),P(9,1)\bigr),\ \bigl(R_{0}^{2}(1),P(9,7)\bigr)\\
 & \bigl(R_{1}^{1}(1),P(12,1)\bigr),\ \bigl(P(14,8),P(13,8)\bigr)\\
P(15,7):\ & \bigl(P(29,8),P(0,8)\bigr),\ \bigl(P(24,8),P(5,8)\bigr),\ \bigl(R_{1}^{5}(3),P(6,7)\bigr),\ \bigl(P(20,8),P(9,8)\bigr),\ \bigl(R_{1}^{1}(2),P(10,1)\bigr)\\
 & \bigl(R_{0}^{3}(1),P(10,7)\bigr),\ \bigl(R_{1}^{2}(1),P(13,1)\bigr),\ \bigl(P(15,8),P(14,8)\bigr)\\
P(n,7):\ & \bigl(P(n+14,8),P(n-15,8)\bigr),\ \bigl(P(n+9,8),P(n-10,8)\bigr),\ \bigl(R_{1}^{(n-11)\bmod 5+1}(3),P(n-9,7)\bigr)\\
 & \bigl(P(n+5,8),P(n-6,8)\bigr),\ \bigl(R_{1}^{(n-15)\bmod 5+1}(2),P(n-5,1)\bigr),\ \bigl(R_{0}^{(n-13)\bmod 3+1}(1),P(n-5,7)\bigr)\\
 & \bigl(R_{1}^{(n-14)\bmod 5+1}(1),P(n-2,1)\bigr),\ \bigl(P(n,8),P(n-1,8)\bigr),\ n>15\\
\end{align*}
\end{fleqn}
\subsubsection*{Modules of the form $P(n,8)$}

Defect: $\partial P(n,8) = -1$, for $n\ge 0$.

\begin{fleqn}
\begin{align*}
P(0,8):\ & \bigl(I(0,4),P(0,3)\bigr),\ \bigl(I(0,5),P(0,4)\bigr),\ \bigl(I(0,6),P(0,5)\bigr),\ \bigl(I(0,7),P(0,6)\bigr),\ \bigl(I(0,8),P(0,7)\bigr)\\
P(1,8):\ & \bigl(I(0,9),P(0,2)\bigr),\ \bigl(I(1,1),P(0,9)\bigr)\\
P(2,8):\ & \bigl(I(4,8),P(0,1)\bigr),\ \bigl(I(0,2),P(0,4)\bigr),\ \bigl(I(0,1),P(1,9)\bigr)\\
P(3,8):\ & \bigl(I(0,9),P(0,5)\bigr),\ \bigl(I(3,7),P(0,9)\bigr),\ \bigl(I(3,8),P(1,1)\bigr)\\
P(4,8):\ & \bigl(I(2,6),P(0,2)\bigr),\ \bigl(I(1,1),P(0,6)\bigr),\ \bigl(I(2,7),P(1,9)\bigr),\ \bigl(I(2,8),P(2,1)\bigr)\\
P(5,8):\ & \bigl(I(7,8),P(0,1)\bigr),\ \bigl(I(5,8),P(0,7)\bigr),\ \bigl(I(3,1),P(0,9)\bigr),\ \bigl(I(1,6),P(1,2)\bigr),\ \bigl(I(0,1),P(1,6)\bigr)\\
 & \bigl(I(1,7),P(2,9)\bigr),\ \bigl(I(1,8),P(3,1)\bigr)\\
P(6,8):\ & \bigl(R_{1}^{4}(1),P(0,8)\bigr),\ \bigl(I(6,8),P(1,1)\bigr),\ \bigl(I(4,8),P(1,7)\bigr),\ \bigl(I(2,1),P(1,9)\bigr),\ \bigl(I(0,6),P(2,2)\bigr)\\
 & \bigl(I(0,7),P(3,9)\bigr),\ \bigl(I(0,8),P(4,1)\bigr)\\
P(7,8):\ & \bigl(I(9,8),P(0,1)\bigr),\ \bigl(R_{1}^{5}(1),P(1,8)\bigr),\ \bigl(I(5,8),P(2,1)\bigr),\ \bigl(I(3,8),P(2,7)\bigr),\ \bigl(I(1,1),P(2,9)\bigr)\\
P(8,8):\ & \bigl(I(4,7),P(0,6)\bigr),\ \bigl(I(8,8),P(1,1)\bigr),\ \bigl(R_{1}^{1}(1),P(2,8)\bigr),\ \bigl(I(4,8),P(3,1)\bigr),\ \bigl(I(2,8),P(3,7)\bigr)\\
 & \bigl(I(0,1),P(3,9)\bigr)\\
P(9,8):\ & \bigl(I(9,8),P(0,7)\bigr),\ \bigl(I(3,7),P(1,6)\bigr),\ \bigl(I(7,8),P(2,1)\bigr),\ \bigl(R_{1}^{2}(1),P(3,8)\bigr),\ \bigl(I(3,8),P(4,1)\bigr)\\
 & \bigl(I(1,8),P(4,7)\bigr)\\
P(10,8):\ & \bigl(I(12,8),P(0,1)\bigr),\ \bigl(R_{0}^{2}(1),P(0,8)\bigr),\ \bigl(I(8,8),P(1,7)\bigr),\ \bigl(I(2,7),P(2,6)\bigr),\ \bigl(I(6,8),P(3,1)\bigr)\\
 & \bigl(R_{1}^{3}(1),P(4,8)\bigr),\ \bigl(I(2,8),P(5,1)\bigr),\ \bigl(I(0,8),P(5,7)\bigr)\\
P(11,8):\ & \bigl(I(11,8),P(1,1)\bigr),\ \bigl(R_{0}^{3}(1),P(1,8)\bigr),\ \bigl(I(7,8),P(2,7)\bigr),\ \bigl(I(1,7),P(3,6)\bigr),\ \bigl(I(5,8),P(4,1)\bigr)\\
 & \bigl(R_{1}^{4}(1),P(5,8)\bigr),\ \bigl(I(1,8),P(6,1)\bigr)\\
P(12,8):\ & \bigl(R_{1}^{4}(2),P(0,8)\bigr),\ \bigl(I(10,8),P(2,1)\bigr),\ \bigl(R_{0}^{1}(1),P(2,8)\bigr),\ \bigl(I(6,8),P(3,7)\bigr),\ \bigl(I(0,7),P(4,6)\bigr)\\
 & \bigl(I(4,8),P(5,1)\bigr),\ \bigl(R_{1}^{5}(1),P(6,8)\bigr),\ \bigl(I(0,8),P(7,1)\bigr)\\
P(13,8):\ & \bigl(R_{1}^{5}(2),P(1,8)\bigr),\ \bigl(I(9,8),P(3,1)\bigr),\ \bigl(R_{0}^{2}(1),P(3,8)\bigr),\ \bigl(I(5,8),P(4,7)\bigr),\ \bigl(I(3,8),P(6,1)\bigr)\\
 & \bigl(R_{1}^{1}(1),P(7,8)\bigr)\\
P(14,8):\ & \bigl(I(14,8),P(0,7)\bigr),\ \bigl(R_{1}^{1}(2),P(2,8)\bigr),\ \bigl(I(8,8),P(4,1)\bigr),\ \bigl(R_{0}^{3}(1),P(4,8)\bigr),\ \bigl(I(4,8),P(5,7)\bigr)\\
 & \bigl(I(2,8),P(7,1)\bigr),\ \bigl(R_{1}^{2}(1),P(8,8)\bigr)\\
P(15,8):\ & \bigl(R_{\infty}^{1}(1),P(0,8)\bigr),\ \bigl(I(13,8),P(1,7)\bigr),\ \bigl(R_{1}^{2}(2),P(3,8)\bigr),\ \bigl(I(7,8),P(5,1)\bigr),\ \bigl(R_{0}^{1}(1),P(5,8)\bigr)\\
 & \bigl(I(3,8),P(6,7)\bigr),\ \bigl(I(1,8),P(8,1)\bigr),\ \bigl(R_{1}^{3}(1),P(9,8)\bigr)\\
P(16,8):\ & \bigl(R_{\infty}^{2}(1),P(1,8)\bigr),\ \bigl(I(12,8),P(2,7)\bigr),\ \bigl(R_{1}^{3}(2),P(4,8)\bigr),\ \bigl(I(6,8),P(6,1)\bigr),\ \bigl(R_{0}^{2}(1),P(6,8)\bigr)\\
 & \bigl(I(2,8),P(7,7)\bigr),\ \bigl(I(0,8),P(9,1)\bigr),\ \bigl(R_{1}^{4}(1),P(10,8)\bigr)\\
P(17,8):\ & \bigl(R_{\infty}^{1}(1),P(2,8)\bigr),\ \bigl(I(11,8),P(3,7)\bigr),\ \bigl(R_{1}^{4}(2),P(5,8)\bigr),\ \bigl(I(5,8),P(7,1)\bigr),\ \bigl(R_{0}^{3}(1),P(7,8)\bigr)\\
 & \bigl(I(1,8),P(8,7)\bigr),\ \bigl(R_{1}^{5}(1),P(11,8)\bigr)\\
P(18,8):\ & \bigl(R_{1}^{4}(3),P(0,8)\bigr),\ \bigl(R_{\infty}^{2}(1),P(3,8)\bigr),\ \bigl(I(10,8),P(4,7)\bigr),\ \bigl(R_{1}^{5}(2),P(6,8)\bigr),\ \bigl(I(4,8),P(8,1)\bigr)\\
 & \bigl(R_{0}^{1}(1),P(8,8)\bigr),\ \bigl(I(0,8),P(9,7)\bigr),\ \bigl(R_{1}^{1}(1),P(12,8)\bigr)\\
P(19,8):\ & \bigl(R_{1}^{5}(3),P(1,8)\bigr),\ \bigl(R_{\infty}^{1}(1),P(4,8)\bigr),\ \bigl(I(9,8),P(5,7)\bigr),\ \bigl(R_{1}^{1}(2),P(7,8)\bigr),\ \bigl(I(3,8),P(9,1)\bigr)\\
 & \bigl(R_{0}^{2}(1),P(9,8)\bigr),\ \bigl(R_{1}^{2}(1),P(13,8)\bigr)\\
P(20,8):\ & \bigl(R_{0}^{2}(2),P(0,8)\bigr),\ \bigl(R_{1}^{1}(3),P(2,8)\bigr),\ \bigl(R_{\infty}^{2}(1),P(5,8)\bigr),\ \bigl(I(8,8),P(6,7)\bigr),\ \bigl(R_{1}^{2}(2),P(8,8)\bigr)\\
 & \bigl(I(2,8),P(10,1)\bigr),\ \bigl(R_{0}^{3}(1),P(10,8)\bigr),\ \bigl(R_{1}^{3}(1),P(14,8)\bigr)\\
P(21,8):\ & \bigl(R_{0}^{3}(2),P(1,8)\bigr),\ \bigl(R_{1}^{2}(3),P(3,8)\bigr),\ \bigl(R_{\infty}^{1}(1),P(6,8)\bigr),\ \bigl(I(7,8),P(7,7)\bigr),\ \bigl(R_{1}^{3}(2),P(9,8)\bigr)\\
 & \bigl(I(1,8),P(11,1)\bigr),\ \bigl(R_{0}^{1}(1),P(11,8)\bigr),\ \bigl(R_{1}^{4}(1),P(15,8)\bigr)\\
P(22,8):\ & \bigl(R_{0}^{1}(2),P(2,8)\bigr),\ \bigl(R_{1}^{3}(3),P(4,8)\bigr),\ \bigl(R_{\infty}^{2}(1),P(7,8)\bigr),\ \bigl(I(6,8),P(8,7)\bigr),\ \bigl(R_{1}^{4}(2),P(10,8)\bigr)\\
 & \bigl(I(0,8),P(12,1)\bigr),\ \bigl(R_{0}^{2}(1),P(12,8)\bigr),\ \bigl(R_{1}^{5}(1),P(16,8)\bigr)\\
P(23,8):\ & \bigl(R_{0}^{2}(2),P(3,8)\bigr),\ \bigl(R_{1}^{4}(3),P(5,8)\bigr),\ \bigl(R_{\infty}^{1}(1),P(8,8)\bigr),\ \bigl(I(5,8),P(9,7)\bigr),\ \bigl(R_{1}^{5}(2),P(11,8)\bigr)\\
 & \bigl(R_{0}^{3}(1),P(13,8)\bigr),\ \bigl(R_{1}^{1}(1),P(17,8)\bigr)\\
P(24,8):\ & \bigl(R_{1}^{4}(4),P(0,8)\bigr),\ \bigl(R_{0}^{3}(2),P(4,8)\bigr),\ \bigl(R_{1}^{5}(3),P(6,8)\bigr),\ \bigl(R_{\infty}^{2}(1),P(9,8)\bigr),\ \bigl(I(4,8),P(10,7)\bigr)\\
 & \bigl(R_{1}^{1}(2),P(12,8)\bigr),\ \bigl(R_{0}^{1}(1),P(14,8)\bigr),\ \bigl(R_{1}^{2}(1),P(18,8)\bigr)\\
P(n,8):\ & \bigl(R_{1}^{(n-21)\bmod 5+1}(4),P(n-24,8)\bigr),\ \bigl(R_{0}^{(n-22)\bmod 3+1}(2),P(n-20,8)\bigr),\ \bigl(R_{1}^{(n-20)\bmod 5+1}(3),P(n-18,8)\bigr)\\
 & \bigl(R_{\infty}^{(n-23)\bmod 2+1}(1),P(n-15,8)\bigr),\ \bigl(I(-n+28,8),P(n-14,7)\bigr),\ \bigl(R_{1}^{(n-24)\bmod 5+1}(2),P(n-12,8)\bigr)\\
 & \bigl(R_{0}^{(n-24)\bmod 3+1}(1),P(n-10,8)\bigr),\ \bigl(R_{1}^{(n-23)\bmod 5+1}(1),P(n-6,8)\bigr),\ n>24\\
\end{align*}
\end{fleqn}
\subsubsection*{Modules of the form $P(n,9)$}

Defect: $\partial P(n,9) = -3$, for $n\ge 0$.

\begin{fleqn}
\begin{align*}
P(0,9):\ & \bigl(I(0,9),P(0,3)\bigr)\\
P(1,9):\ & \bigl(I(4,8),P(0,2)\bigr),\ \bigl(I(1,1),P(0,4)\bigr)\\
P(2,9):\ & \bigl(P(3,8),P(0,1)\bigr),\ \bigl(I(5,8),P(0,5)\bigr),\ \bigl(R_{1}^{5}(1),P(0,9)\bigr),\ \bigl(I(3,8),P(1,2)\bigr),\ \bigl(I(0,1),P(1,4)\bigr)\\
P(3,9):\ & \bigl(R_{1}^{4}(1),P(0,6)\bigr),\ \bigl(P(4,8),P(1,1)\bigr),\ \bigl(I(4,8),P(1,5)\bigr),\ \bigl(R_{1}^{1}(1),P(1,9)\bigr),\ \bigl(I(2,8),P(2,2)\bigr)\\
P(4,9):\ & \bigl(P(9,8),P(0,1)\bigr),\ \bigl(P(7,8),P(0,7)\bigr),\ \bigl(R_{1}^{5}(1),P(1,6)\bigr),\ \bigl(P(5,8),P(2,1)\bigr),\ \bigl(I(3,8),P(2,5)\bigr)\\
 & \bigl(R_{1}^{2}(1),P(2,9)\bigr),\ \bigl(I(1,8),P(3,2)\bigr)\\
P(5,9):\ & \bigl(P(6,1),P(0,8)\bigr),\ \bigl(P(10,8),P(1,1)\bigr),\ \bigl(P(8,8),P(1,7)\bigr),\ \bigl(R_{1}^{1}(1),P(2,6)\bigr),\ \bigl(P(6,8),P(3,1)\bigr)\\
 & \bigl(I(2,8),P(3,5)\bigr),\ \bigl(R_{1}^{3}(1),P(3,9)\bigr),\ \bigl(I(0,8),P(4,2)\bigr)\\
P(6,9):\ & \bigl(P(7,1),P(1,8)\bigr),\ \bigl(P(11,8),P(2,1)\bigr),\ \bigl(P(9,8),P(2,7)\bigr),\ \bigl(R_{1}^{2}(1),P(3,6)\bigr),\ \bigl(P(7,8),P(4,1)\bigr)\\
 & \bigl(I(1,8),P(4,5)\bigr),\ \bigl(R_{1}^{4}(1),P(4,9)\bigr)\\
P(7,9):\ & \bigl(P(8,7),P(0,8)\bigr),\ \bigl(P(8,1),P(2,8)\bigr),\ \bigl(P(12,8),P(3,1)\bigr),\ \bigl(P(10,8),P(3,7)\bigr),\ \bigl(R_{1}^{3}(1),P(4,6)\bigr)\\
 & \bigl(P(8,8),P(5,1)\bigr),\ \bigl(I(0,8),P(5,5)\bigr),\ \bigl(R_{1}^{5}(1),P(5,9)\bigr)\\
P(8,9):\ & \bigl(P(9,7),P(1,8)\bigr),\ \bigl(P(9,1),P(3,8)\bigr),\ \bigl(P(13,8),P(4,1)\bigr),\ \bigl(P(11,8),P(4,7)\bigr),\ \bigl(R_{1}^{4}(1),P(5,6)\bigr)\\
 & \bigl(P(9,8),P(6,1)\bigr),\ \bigl(R_{1}^{1}(1),P(6,9)\bigr)\\
P(9,9):\ & \bigl(P(12,1),P(0,8)\bigr),\ \bigl(P(10,7),P(2,8)\bigr),\ \bigl(P(10,1),P(4,8)\bigr),\ \bigl(P(14,8),P(5,1)\bigr),\ \bigl(P(12,8),P(5,7)\bigr)\\
 & \bigl(R_{1}^{5}(1),P(6,6)\bigr),\ \bigl(P(10,8),P(7,1)\bigr),\ \bigl(R_{1}^{2}(1),P(7,9)\bigr)\\
P(n,9):\ & \bigl(P(n+3,1),P(n-9,8)\bigr),\ \bigl(P(n+1,7),P(n-7,8)\bigr),\ \bigl(P(n+1,1),P(n-5,8)\bigr)\\
 & \bigl(P(n+5,8),P(n-4,1)\bigr),\ \bigl(P(n+3,8),P(n-4,7)\bigr),\ \bigl(R_{1}^{(n-5)\bmod 5+1}(1),P(n-3,6)\bigr)\\
 & \bigl(P(n+1,8),P(n-2,1)\bigr),\ \bigl(R_{1}^{(n-8)\bmod 5+1}(1),P(n-2,9)\bigr),\ n>9\\
\end{align*}
\end{fleqn}
\subsubsection{Schofield pairs associated to preinjective exceptional modules}

\begin{figure}[ht]

\begin{center}
\begin{scaletikzpicturetowidth}{\textwidth}
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	\node[noselEEEEEEEEE, label={[shift={(0.2,-0.55)}, brown]\tiny $I(3,2)$}] (I_3_2__Rationals) at (-10,3) {$\begin{smallmatrix}&&1\\1&2&2&2&2&2&1&0\end{smallmatrix}$};
	\node[noselEEEEEEEEE, label={[shift={(0.2,-0.55)}, brown]\tiny $I(3,3)$}] (I_3_3__Rationals) at (-12,0) {$\begin{smallmatrix}&&2\\2&3&4&4&4&3&2&1\end{smallmatrix}$};
	\node[noselEEEEEEEEE, label={[shift={(0.2,-0.55)}, brown]\tiny $I(3,4)$}] (I_3_4__Rationals) at (-10,-3) {$\begin{smallmatrix}&&2\\1&2&3&3&3&2&2&1\end{smallmatrix}$};
	\node[seloneEEEEEEEEE, label={[shift={(0.2,-0.55)}, brown]\tiny $I(3,5)$}] (I_3_5__Rationals) at (-8,-6) {$\begin{smallmatrix}&&1\\1&2&2&2&2&1&1&1\end{smallmatrix}$};
	\node[noselEEEEEEEEE, label={[shift={(0.2,-0.55)}, brown]\tiny $I(3,6)$}] (I_3_6__Rationals) at (-6,-9) {$\begin{smallmatrix}&&1\\1&1&1&1&1&0&0&0\end{smallmatrix}$};
	\node[noselEEEEEEEEE, label={[shift={(0.2,-0.55)}, brown]\tiny $I(3,7)$}] (I_3_7__Rationals) at (-4,-12) {$\begin{smallmatrix}&&0\\0&0&0&1&1&0&0&0\end{smallmatrix}$};
	\node[noselEEEEEEEEE, label={[shift={(0.2,-0.55)}, brown]\tiny $I(3,8)$}] (I_3_8__Rationals) at (-2,-15) {$\begin{smallmatrix}&&0\\0&0&0&0&1&0&0&0\end{smallmatrix}$};
	\node[noselEEEEEEEEE, label={[shift={(0.2,-0.55)}, brown]\tiny $I(3,9)$}] (I_3_9__Rationals) at (-10,1) {$\begin{smallmatrix}&&1\\1&1&2&2&2&2&1&1\end{smallmatrix}$};
	\node[noselEEEEEEEEE, label={[shift={(0.2,-0.55)}, brown]\tiny $I(4,1)$}] (I_4_1__Rationals) at (-12,6) {$\begin{smallmatrix}&&1\\0&1&1&1&1&1&0&0\end{smallmatrix}$};
	\node[seloneEEEEEEEEE, label={[shift={(0.2,-0.55)}, brown]\tiny $I(4,2)$}] (I_4_2__Rationals) at (-14,3) {$\begin{smallmatrix}&&2\\1&2&3&3&3&2&1&1\end{smallmatrix}$};
	\node[noselEEEEEEEEE, label={[shift={(0.2,-0.55)}, brown]\tiny $I(4,3)$}] (I_4_3__Rationals) at (-16,0) {$\begin{smallmatrix}&&3\\2&4&5&5&4&3&2&1\end{smallmatrix}$};
	\node[noselEEEEEEEEE, label={[shift={(0.2,-0.55)}, brown]\tiny $I(4,4)$}] (I_4_4__Rationals) at (-14,-3) {$\begin{smallmatrix}&&2\\2&3&4&4&3&3&2&1\end{smallmatrix}$};
	\node[noselEEEEEEEEE, label={[shift={(0.2,-0.55)}, brown]\tiny $I(4,5)$}] (I_4_5__Rationals) at (-12,-6) {$\begin{smallmatrix}&&2\\1&2&3&3&2&2&2&1\end{smallmatrix}$};
	\node[noselEEEEEEEEE, label={[shift={(0.2,-0.55)}, brown]\tiny $I(4,6)$}] (I_4_6__Rationals) at (-10,-9) {$\begin{smallmatrix}&&1\\1&2&2&2&1&1&1&1\end{smallmatrix}$};
	\node[noselEEEEEEEEE, label={[shift={(0.2,-0.55)}, brown]\tiny $I(4,7)$}] (I_4_7__Rationals) at (-8,-12) {$\begin{smallmatrix}&&1\\1&1&1&1&0&0&0&0\end{smallmatrix}$};
	\node[noselEEEEEEEEE, label={[shift={(0.2,-0.55)}, brown]\tiny $I(4,8)$}] (I_4_8__Rationals) at (-6,-15) {$\begin{smallmatrix}&&0\\0&0&0&1&0&0&0&0\end{smallmatrix}$};
	\node[noselEEEEEEEEE, label={[shift={(0.2,-0.55)}, brown]\tiny $I(4,9)$}] (I_4_9__Rationals) at (-14,1) {$\begin{smallmatrix}&&1\\1&2&2&2&2&1&1&0\end{smallmatrix}$};
	\node at (-17,3) {$\dots$};
	\node at (-17,-3) {$\dots$};
	\draw[red,->] (I_0_2__Rationals) -- (I_0_1__Rationals);
	\draw[red,->] (I_0_3__Rationals) -- (I_0_2__Rationals);
	\draw[blue,->] (I_1_1__Rationals) -- (I_0_2__Rationals);
	\draw[blue,->] (I_1_2__Rationals) -- (I_0_3__Rationals);
	\draw[blue,->] (I_1_4__Rationals) -- (I_0_3__Rationals);
	\draw[blue,->] (I_1_9__Rationals) -- (I_0_3__Rationals);
	\draw[red,->] (I_0_3__Rationals) -- (I_0_4__Rationals);
	\draw[blue,->] (I_1_5__Rationals) -- (I_0_4__Rationals);
	\draw[red,->] (I_0_4__Rationals) -- (I_0_5__Rationals);
	\draw[blue,->] (I_1_6__Rationals) -- (I_0_5__Rationals);
	\draw[red,->] (I_0_5__Rationals) -- (I_0_6__Rationals);
	\draw[blue,->] (I_1_7__Rationals) -- (I_0_6__Rationals);
	\draw[red,->] (I_0_6__Rationals) -- (I_0_7__Rationals);
	\draw[blue,->] (I_1_8__Rationals) -- (I_0_7__Rationals);
	\draw[red,->] (I_0_7__Rationals) -- (I_0_8__Rationals);
	\draw[red,->] (I_0_3__Rationals) -- (I_0_9__Rationals);
	\draw[gray,dashed,->] (I_0_1__Rationals) -- (I_1_1__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_1_2__Rationals) -- (I_1_1__Rationals);
	\draw[gray,dashed,->] (I_0_2__Rationals) -- (I_1_2__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_1_3__Rationals) -- (I_1_2__Rationals);
	\draw[blue,->] (I_2_1__Rationals) -- (I_1_2__Rationals);
	\draw[gray,dashed,->] (I_0_3__Rationals) -- (I_1_3__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[blue,->] (I_2_2__Rationals) -- (I_1_3__Rationals);
	\draw[blue,->] (I_2_4__Rationals) -- (I_1_3__Rationals);
	\draw[blue,->] (I_2_9__Rationals) -- (I_1_3__Rationals);
	\draw[gray,dashed,->] (I_0_4__Rationals) -- (I_1_4__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_1_3__Rationals) -- (I_1_4__Rationals);
	\draw[blue,->] (I_2_5__Rationals) -- (I_1_4__Rationals);
	\draw[gray,dashed,->] (I_0_5__Rationals) -- (I_1_5__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_1_4__Rationals) -- (I_1_5__Rationals);
	\draw[blue,->] (I_2_6__Rationals) -- (I_1_5__Rationals);
	\draw[gray,dashed,->] (I_0_6__Rationals) -- (I_1_6__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_1_5__Rationals) -- (I_1_6__Rationals);
	\draw[blue,->] (I_2_7__Rationals) -- (I_1_6__Rationals);
	\draw[gray,dashed,->] (I_0_7__Rationals) -- (I_1_7__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_1_6__Rationals) -- (I_1_7__Rationals);
	\draw[blue,->] (I_2_8__Rationals) -- (I_1_7__Rationals);
	\draw[gray,dashed,->] (I_0_8__Rationals) -- (I_1_8__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_1_7__Rationals) -- (I_1_8__Rationals);
	\draw[gray,dashed,->] (I_0_9__Rationals) -- (I_1_9__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_1_3__Rationals) -- (I_1_9__Rationals);
	\draw[gray,dashed,->] (I_1_1__Rationals) -- (I_2_1__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_2_2__Rationals) -- (I_2_1__Rationals);
	\draw[gray,dashed,->] (I_1_2__Rationals) -- (I_2_2__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_2_3__Rationals) -- (I_2_2__Rationals);
	\draw[blue,->] (I_3_1__Rationals) -- (I_2_2__Rationals);
	\draw[gray,dashed,->] (I_1_3__Rationals) -- (I_2_3__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[blue,->] (I_3_2__Rationals) -- (I_2_3__Rationals);
	\draw[blue,->] (I_3_4__Rationals) -- (I_2_3__Rationals);
	\draw[blue,->] (I_3_9__Rationals) -- (I_2_3__Rationals);
	\draw[gray,dashed,->] (I_1_4__Rationals) -- (I_2_4__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_2_3__Rationals) -- (I_2_4__Rationals);
	\draw[blue,->] (I_3_5__Rationals) -- (I_2_4__Rationals);
	\draw[gray,dashed,->] (I_1_5__Rationals) -- (I_2_5__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_2_4__Rationals) -- (I_2_5__Rationals);
	\draw[blue,->] (I_3_6__Rationals) -- (I_2_5__Rationals);
	\draw[gray,dashed,->] (I_1_6__Rationals) -- (I_2_6__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_2_5__Rationals) -- (I_2_6__Rationals);
	\draw[blue,->] (I_3_7__Rationals) -- (I_2_6__Rationals);
	\draw[gray,dashed,->] (I_1_7__Rationals) -- (I_2_7__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_2_6__Rationals) -- (I_2_7__Rationals);
	\draw[blue,->] (I_3_8__Rationals) -- (I_2_7__Rationals);
	\draw[gray,dashed,->] (I_1_8__Rationals) -- (I_2_8__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_2_7__Rationals) -- (I_2_8__Rationals);
	\draw[gray,dashed,->] (I_1_9__Rationals) -- (I_2_9__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_2_3__Rationals) -- (I_2_9__Rationals);
	\draw[gray,dashed,->] (I_2_1__Rationals) -- (I_3_1__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_3_2__Rationals) -- (I_3_1__Rationals);
	\draw[gray,dashed,->] (I_2_2__Rationals) -- (I_3_2__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_3_3__Rationals) -- (I_3_2__Rationals);
	\draw[blue,->] (I_4_1__Rationals) -- (I_3_2__Rationals);
	\draw[gray,dashed,->] (I_2_3__Rationals) -- (I_3_3__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[blue,->] (I_4_2__Rationals) -- (I_3_3__Rationals);
	\draw[blue,->] (I_4_4__Rationals) -- (I_3_3__Rationals);
	\draw[blue,->] (I_4_9__Rationals) -- (I_3_3__Rationals);
	\draw[gray,dashed,->] (I_2_4__Rationals) -- (I_3_4__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_3_3__Rationals) -- (I_3_4__Rationals);
	\draw[blue,->] (I_4_5__Rationals) -- (I_3_4__Rationals);
	\draw[gray,dashed,->] (I_2_5__Rationals) -- (I_3_5__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_3_4__Rationals) -- (I_3_5__Rationals);
	\draw[blue,->] (I_4_6__Rationals) -- (I_3_5__Rationals);
	\draw[gray,dashed,->] (I_2_6__Rationals) -- (I_3_6__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_3_5__Rationals) -- (I_3_6__Rationals);
	\draw[blue,->] (I_4_7__Rationals) -- (I_3_6__Rationals);
	\draw[gray,dashed,->] (I_2_7__Rationals) -- (I_3_7__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_3_6__Rationals) -- (I_3_7__Rationals);
	\draw[blue,->] (I_4_8__Rationals) -- (I_3_7__Rationals);
	\draw[gray,dashed,->] (I_2_8__Rationals) -- (I_3_8__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_3_7__Rationals) -- (I_3_8__Rationals);
	\draw[gray,dashed,->] (I_2_9__Rationals) -- (I_3_9__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_3_3__Rationals) -- (I_3_9__Rationals);
	\draw[gray,dashed,->] (I_3_1__Rationals) -- (I_4_1__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_4_2__Rationals) -- (I_4_1__Rationals);
	\draw[gray,dashed,->] (I_3_2__Rationals) -- (I_4_2__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_4_3__Rationals) -- (I_4_2__Rationals);
	\draw[gray,dashed,->] (I_3_3__Rationals) -- (I_4_3__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[gray,dashed,->] (I_3_4__Rationals) -- (I_4_4__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_4_3__Rationals) -- (I_4_4__Rationals);
	\draw[gray,dashed,->] (I_3_5__Rationals) -- (I_4_5__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_4_4__Rationals) -- (I_4_5__Rationals);
	\draw[gray,dashed,->] (I_3_6__Rationals) -- (I_4_6__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_4_5__Rationals) -- (I_4_6__Rationals);
	\draw[gray,dashed,->] (I_3_7__Rationals) -- (I_4_7__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_4_6__Rationals) -- (I_4_7__Rationals);
	\draw[gray,dashed,->] (I_3_8__Rationals) -- (I_4_8__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_4_7__Rationals) -- (I_4_8__Rationals);
	\draw[gray,dashed,->] (I_3_9__Rationals) -- (I_4_9__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_4_3__Rationals) -- (I_4_9__Rationals);
\end{tikzpicture}
\end{scaletikzpicturetowidth}
\end{center}

\end{figure}
\begin{figure}[ht]

\begin{center}
\begin{scaletikzpicturetowidth}{\textwidth}
\begin{tikzpicture}[scale=\tikzscale]
	\node at (5,3) {$\dots$};
	\node at (5,-3) {$\dots$};
	\node[noselEEEEEEEEE, label={[shift={(0.2,-0.55)}, brown]\tiny $I(5,1)$}] (I_5_1__Rationals) at (4,6) {$\begin{smallmatrix}&&1\\1&1&2&2&2&1&1&1\end{smallmatrix}$};
	\node[noselEEEEEEEEE, label={[shift={(0.2,-0.55)}, brown]\tiny $I(5,2)$}] (I_5_2__Rationals) at (2,3) {$\begin{smallmatrix}&&2\\2&3&4&4&3&2&2&1\end{smallmatrix}$};
	\node[noselEEEEEEEEE, label={[shift={(0.2,-0.55)}, brown]\tiny $I(5,3)$}] (I_5_3__Rationals) at (0,0) {$\begin{smallmatrix}&&4\\3&5&7&6&5&4&3&2\end{smallmatrix}$};
	\node[noselEEEEEEEEE, label={[shift={(0.2,-0.55)}, brown]\tiny $I(5,4)$}] (I_5_4__Rationals) at (2,-3) {$\begin{smallmatrix}&&3\\2&4&5&4&4&3&2&1\end{smallmatrix}$};
	\node[noselEEEEEEEEE, label={[shift={(0.2,-0.55)}, brown]\tiny $I(5,5)$}] (I_5_5__Rationals) at (4,-6) {$\begin{smallmatrix}&&2\\2&3&4&3&3&3&2&1\end{smallmatrix}$};
	\node[noselEEEEEEEEE, label={[shift={(0.2,-0.55)}, brown]\tiny $I(5,6)$}] (I_5_6__Rationals) at (6,-9) {$\begin{smallmatrix}&&2\\1&2&3&2&2&2&2&1\end{smallmatrix}$};
	\node[noselEEEEEEEEE, label={[shift={(0.2,-0.55)}, brown]\tiny $I(5,7)$}] (I_5_7__Rationals) at (8,-12) {$\begin{smallmatrix}&&1\\1&2&2&1&1&1&1&1\end{smallmatrix}$};
	\node[noselEEEEEEEEE, label={[shift={(0.2,-0.55)}, brown]\tiny $I(5,8)$}] (I_5_8__Rationals) at (10,-15) {$\begin{smallmatrix}&&1\\1&1&1&0&0&0&0&0\end{smallmatrix}$};
	\node[seloneEEEEEEEEE, label={[shift={(0.2,-0.55)}, brown]\tiny $I(5,9)$}] (I_5_9__Rationals) at (2,1) {$\begin{smallmatrix}&&2\\1&2&3&3&2&2&1&1\end{smallmatrix}$};
	\node[noselEEEEEEEEE, label={[shift={(0.2,-0.55)}, brown]\tiny $I(6,1)$}] (I_6_1__Rationals) at (0,6) {$\begin{smallmatrix}&&1\\1&2&2&2&1&1&1&0\end{smallmatrix}$};
	\node[noselEEEEEEEEE, label={[shift={(0.2,-0.55)}, brown]\tiny $I(6,2)$}] (I_6_2__Rationals) at (-2,3) {$\begin{smallmatrix}&&3\\2&4&5&4&3&3&2&1\end{smallmatrix}$};
	\node[noselEEEEEEEEE, label={[shift={(0.2,-0.55)}, brown]\tiny $I(6,3)$}] (I_6_3__Rationals) at (-4,0) {$\begin{smallmatrix}&&4\\3&6&8&7&6&5&4&2\end{smallmatrix}$};
	\node[noselEEEEEEEEE, label={[shift={(0.2,-0.55)}, brown]\tiny $I(6,4)$}] (I_6_4__Rationals) at (-2,-3) {$\begin{smallmatrix}&&3\\2&4&6&6&5&4&3&2\end{smallmatrix}$};
	\node[noselEEEEEEEEE, label={[shift={(0.2,-0.55)}, brown]\tiny $I(6,5)$}] (I_6_5__Rationals) at (0,-6) {$\begin{smallmatrix}&&2\\1&3&4&4&4&3&2&1\end{smallmatrix}$};
	\node[noselEEEEEEEEE, label={[shift={(0.2,-0.55)}, brown]\tiny $I(6,6)$}] (I_6_6__Rationals) at (2,-9) {$\begin{smallmatrix}&&1\\1&2&3&3&3&3&2&1\end{smallmatrix}$};
	\node[noselEEEEEEEEE, label={[shift={(0.2,-0.55)}, brown]\tiny $I(6,7)$}] (I_6_7__Rationals) at (4,-12) {$\begin{smallmatrix}&&1\\0&1&2&2&2&2&2&1\end{smallmatrix}$};
	\node[noselEEEEEEEEE, label={[shift={(0.2,-0.55)}, brown]\tiny $I(6,8)$}] (I_6_8__Rationals) at (6,-15) {$\begin{smallmatrix}&&0\\0&1&1&1&1&1&1&1\end{smallmatrix}$};
	\node[noselEEEEEEEEE, label={[shift={(0.2,-0.55)}, brown]\tiny $I(6,9)$}] (I_6_9__Rationals) at (-2,1) {$\begin{smallmatrix}&&2\\2&3&4&3&3&2&2&1\end{smallmatrix}$};
	\node[noselEEEEEEEEE, label={[shift={(0.2,-0.55)}, brown]\tiny $I(7,1)$}] (I_7_1__Rationals) at (-4,6) {$\begin{smallmatrix}&&2\\1&2&3&2&2&2&1&1\end{smallmatrix}$};
	\node[noselEEEEEEEEE, label={[shift={(0.2,-0.55)}, brown]\tiny $I(7,2)$}] (I_7_2__Rationals) at (-6,3) {$\begin{smallmatrix}&&3\\2&4&6&5&5&4&3&2\end{smallmatrix}$};
	\node[noselEEEEEEEEE, label={[shift={(0.2,-0.55)}, brown]\tiny $I(7,3)$}] (I_7_3__Rationals) at (-8,0) {$\begin{smallmatrix}&&5\\3&6&9&8&7&6&4&2\end{smallmatrix}$};
	\node[noselEEEEEEEEE, label={[shift={(0.2,-0.55)}, brown]\tiny $I(7,4)$}] (I_7_4__Rationals) at (-6,-3) {$\begin{smallmatrix}&&4\\3&5&7&6&5&4&3&1\end{smallmatrix}$};
	\node[noselEEEEEEEEE, label={[shift={(0.2,-0.55)}, brown]\tiny $I(7,5)$}] (I_7_5__Rationals) at (-4,-6) {$\begin{smallmatrix}&&3\\2&3&5&5&4&3&2&1\end{smallmatrix}$};
	\node[noselEEEEEEEEE, label={[shift={(0.2,-0.55)}, brown]\tiny $I(7,6)$}] (I_7_6__Rationals) at (-2,-9) {$\begin{smallmatrix}&&2\\1&2&3&3&3&2&1&0\end{smallmatrix}$};
	\node[noselEEEEEEEEE, label={[shift={(0.2,-0.55)}, brown]\tiny $I(7,7)$}] (I_7_7__Rationals) at (0,-12) {$\begin{smallmatrix}&&1\\1&1&2&2&2&2&1&0\end{smallmatrix}$};
	\node[noselEEEEEEEEE, label={[shift={(0.2,-0.55)}, brown]\tiny $I(7,8)$}] (I_7_8__Rationals) at (2,-15) {$\begin{smallmatrix}&&1\\0&0&1&1&1&1&1&0\end{smallmatrix}$};
	\node[noselEEEEEEEEE, label={[shift={(0.2,-0.55)}, brown]\tiny $I(7,9)$}] (I_7_9__Rationals) at (-6,1) {$\begin{smallmatrix}&&2\\1&3&4&4&3&3&2&1\end{smallmatrix}$};
	\node[noselEEEEEEEEE, label={[shift={(0.2,-0.55)}, brown]\tiny $I(8,1)$}] (I_8_1__Rationals) at (-8,6) {$\begin{smallmatrix}&&1\\1&2&3&3&3&2&2&1\end{smallmatrix}$};
	\node[noselEEEEEEEEE, label={[shift={(0.2,-0.55)}, brown]\tiny $I(8,2)$}] (I_8_2__Rationals) at (-10,3) {$\begin{smallmatrix}&&3\\2&4&6&6&5&4&3&1\end{smallmatrix}$};
	\node[noselEEEEEEEEE, label={[shift={(0.2,-0.55)}, brown]\tiny $I(8,3)$}] (I_8_3__Rationals) at (-12,0) {$\begin{smallmatrix}&&5\\4&7&10&9&8&6&4&2\end{smallmatrix}$};
	\node[noselEEEEEEEEE, label={[shift={(0.2,-0.55)}, brown]\tiny $I(8,4)$}] (I_8_4__Rationals) at (-10,-3) {$\begin{smallmatrix}&&4\\3&6&8&7&6&5&3&2\end{smallmatrix}$};
	\node[noselEEEEEEEEE, label={[shift={(0.2,-0.55)}, brown]\tiny $I(8,5)$}] (I_8_5__Rationals) at (-8,-6) {$\begin{smallmatrix}&&3\\3&5&6&5&4&3&2&1\end{smallmatrix}$};
	\node[seloneEEEEEEEEE, label={[shift={(0.2,-0.55)}, brown]\tiny $I(8,6)$}] (I_8_6__Rationals) at (-6,-9) {$\begin{smallmatrix}&&2\\2&3&4&4&3&2&1&1\end{smallmatrix}$};
	\node[noselEEEEEEEEE, label={[shift={(0.2,-0.55)}, brown]\tiny $I(8,7)$}] (I_8_7__Rationals) at (-4,-12) {$\begin{smallmatrix}&&1\\1&2&2&2&2&1&0&0\end{smallmatrix}$};
	\node[noselEEEEEEEEE, label={[shift={(0.2,-0.55)}, brown]\tiny $I(8,8)$}] (I_8_8__Rationals) at (-2,-15) {$\begin{smallmatrix}&&0\\1&1&1&1&1&1&0&0\end{smallmatrix}$};
	\node[noselEEEEEEEEE, label={[shift={(0.2,-0.55)}, brown]\tiny $I(8,9)$}] (I_8_9__Rationals) at (-10,1) {$\begin{smallmatrix}&&3\\2&3&5&4&4&3&2&1\end{smallmatrix}$};
	\node[noselEEEEEEEEE, label={[shift={(0.2,-0.55)}, brown]\tiny $I(9,1)$}] (I_9_1__Rationals) at (-12,6) {$\begin{smallmatrix}&&2\\1&2&3&3&2&2&1&0\end{smallmatrix}$};
	\node[noselEEEEEEEEE, label={[shift={(0.2,-0.55)}, brown]\tiny $I(9,2)$}] (I_9_2__Rationals) at (-14,3) {$\begin{smallmatrix}&&4\\3&5&7&6&5&4&2&1\end{smallmatrix}$};
	\node[noselEEEEEEEEE, label={[shift={(0.2,-0.55)}, brown]\tiny $I(9,3)$}] (I_9_3__Rationals) at (-16,0) {$\begin{smallmatrix}&&6\\4&8&11&10&8&6&4&2\end{smallmatrix}$};
	\node[noselEEEEEEEEE, label={[shift={(0.2,-0.55)}, brown]\tiny $I(9,4)$}] (I_9_4__Rationals) at (-14,-3) {$\begin{smallmatrix}&&5\\3&6&9&8&7&5&4&2\end{smallmatrix}$};
	\node[noselEEEEEEEEE, label={[shift={(0.2,-0.55)}, brown]\tiny $I(9,5)$}] (I_9_5__Rationals) at (-12,-6) {$\begin{smallmatrix}&&4\\2&5&7&6&5&4&3&2\end{smallmatrix}$};
	\node[noselEEEEEEEEE, label={[shift={(0.2,-0.55)}, brown]\tiny $I(9,6)$}] (I_9_6__Rationals) at (-10,-9) {$\begin{smallmatrix}&&3\\2&4&5&4&3&2&2&1\end{smallmatrix}$};
	\node[noselEEEEEEEEE, label={[shift={(0.2,-0.55)}, brown]\tiny $I(9,7)$}] (I_9_7__Rationals) at (-8,-12) {$\begin{smallmatrix}&&2\\1&2&3&3&2&1&1&1\end{smallmatrix}$};
	\node[noselEEEEEEEEE, label={[shift={(0.2,-0.55)}, brown]\tiny $I(9,8)$}] (I_9_8__Rationals) at (-6,-15) {$\begin{smallmatrix}&&1\\0&1&1&1&1&0&0&0\end{smallmatrix}$};
	\node[noselEEEEEEEEE, label={[shift={(0.2,-0.55)}, brown]\tiny $I(9,9)$}] (I_9_9__Rationals) at (-14,1) {$\begin{smallmatrix}&&2\\2&4&5&5&4&3&2&1\end{smallmatrix}$};
	\node at (-17,3) {$\dots$};
	\node at (-17,-3) {$\dots$};
	\draw[red,->] (I_5_2__Rationals) -- (I_5_1__Rationals);
	\draw[red,->] (I_5_3__Rationals) -- (I_5_2__Rationals);
	\draw[blue,->] (I_6_1__Rationals) -- (I_5_2__Rationals);
	\draw[blue,->] (I_6_2__Rationals) -- (I_5_3__Rationals);
	\draw[blue,->] (I_6_4__Rationals) -- (I_5_3__Rationals);
	\draw[blue,->] (I_6_9__Rationals) -- (I_5_3__Rationals);
	\draw[red,->] (I_5_3__Rationals) -- (I_5_4__Rationals);
	\draw[blue,->] (I_6_5__Rationals) -- (I_5_4__Rationals);
	\draw[red,->] (I_5_4__Rationals) -- (I_5_5__Rationals);
	\draw[blue,->] (I_6_6__Rationals) -- (I_5_5__Rationals);
	\draw[red,->] (I_5_5__Rationals) -- (I_5_6__Rationals);
	\draw[blue,->] (I_6_7__Rationals) -- (I_5_6__Rationals);
	\draw[red,->] (I_5_6__Rationals) -- (I_5_7__Rationals);
	\draw[blue,->] (I_6_8__Rationals) -- (I_5_7__Rationals);
	\draw[red,->] (I_5_7__Rationals) -- (I_5_8__Rationals);
	\draw[red,->] (I_5_3__Rationals) -- (I_5_9__Rationals);
	\draw[gray,dashed,->] (I_5_1__Rationals) -- (I_6_1__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_6_2__Rationals) -- (I_6_1__Rationals);
	\draw[gray,dashed,->] (I_5_2__Rationals) -- (I_6_2__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_6_3__Rationals) -- (I_6_2__Rationals);
	\draw[blue,->] (I_7_1__Rationals) -- (I_6_2__Rationals);
	\draw[gray,dashed,->] (I_5_3__Rationals) -- (I_6_3__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[blue,->] (I_7_2__Rationals) -- (I_6_3__Rationals);
	\draw[blue,->] (I_7_4__Rationals) -- (I_6_3__Rationals);
	\draw[blue,->] (I_7_9__Rationals) -- (I_6_3__Rationals);
	\draw[gray,dashed,->] (I_5_4__Rationals) -- (I_6_4__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_6_3__Rationals) -- (I_6_4__Rationals);
	\draw[blue,->] (I_7_5__Rationals) -- (I_6_4__Rationals);
	\draw[gray,dashed,->] (I_5_5__Rationals) -- (I_6_5__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_6_4__Rationals) -- (I_6_5__Rationals);
	\draw[blue,->] (I_7_6__Rationals) -- (I_6_5__Rationals);
	\draw[gray,dashed,->] (I_5_6__Rationals) -- (I_6_6__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_6_5__Rationals) -- (I_6_6__Rationals);
	\draw[blue,->] (I_7_7__Rationals) -- (I_6_6__Rationals);
	\draw[gray,dashed,->] (I_5_7__Rationals) -- (I_6_7__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_6_6__Rationals) -- (I_6_7__Rationals);
	\draw[blue,->] (I_7_8__Rationals) -- (I_6_7__Rationals);
	\draw[gray,dashed,->] (I_5_8__Rationals) -- (I_6_8__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_6_7__Rationals) -- (I_6_8__Rationals);
	\draw[gray,dashed,->] (I_5_9__Rationals) -- (I_6_9__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_6_3__Rationals) -- (I_6_9__Rationals);
	\draw[gray,dashed,->] (I_6_1__Rationals) -- (I_7_1__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_7_2__Rationals) -- (I_7_1__Rationals);
	\draw[gray,dashed,->] (I_6_2__Rationals) -- (I_7_2__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_7_3__Rationals) -- (I_7_2__Rationals);
	\draw[blue,->] (I_8_1__Rationals) -- (I_7_2__Rationals);
	\draw[gray,dashed,->] (I_6_3__Rationals) -- (I_7_3__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[blue,->] (I_8_2__Rationals) -- (I_7_3__Rationals);
	\draw[blue,->] (I_8_4__Rationals) -- (I_7_3__Rationals);
	\draw[blue,->] (I_8_9__Rationals) -- (I_7_3__Rationals);
	\draw[gray,dashed,->] (I_6_4__Rationals) -- (I_7_4__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_7_3__Rationals) -- (I_7_4__Rationals);
	\draw[blue,->] (I_8_5__Rationals) -- (I_7_4__Rationals);
	\draw[gray,dashed,->] (I_6_5__Rationals) -- (I_7_5__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_7_4__Rationals) -- (I_7_5__Rationals);
	\draw[blue,->] (I_8_6__Rationals) -- (I_7_5__Rationals);
	\draw[gray,dashed,->] (I_6_6__Rationals) -- (I_7_6__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_7_5__Rationals) -- (I_7_6__Rationals);
	\draw[blue,->] (I_8_7__Rationals) -- (I_7_6__Rationals);
	\draw[gray,dashed,->] (I_6_7__Rationals) -- (I_7_7__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_7_6__Rationals) -- (I_7_7__Rationals);
	\draw[blue,->] (I_8_8__Rationals) -- (I_7_7__Rationals);
	\draw[gray,dashed,->] (I_6_8__Rationals) -- (I_7_8__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_7_7__Rationals) -- (I_7_8__Rationals);
	\draw[gray,dashed,->] (I_6_9__Rationals) -- (I_7_9__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_7_3__Rationals) -- (I_7_9__Rationals);
	\draw[gray,dashed,->] (I_7_1__Rationals) -- (I_8_1__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_8_2__Rationals) -- (I_8_1__Rationals);
	\draw[gray,dashed,->] (I_7_2__Rationals) -- (I_8_2__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_8_3__Rationals) -- (I_8_2__Rationals);
	\draw[blue,->] (I_9_1__Rationals) -- (I_8_2__Rationals);
	\draw[gray,dashed,->] (I_7_3__Rationals) -- (I_8_3__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[blue,->] (I_9_2__Rationals) -- (I_8_3__Rationals);
	\draw[blue,->] (I_9_4__Rationals) -- (I_8_3__Rationals);
	\draw[blue,->] (I_9_9__Rationals) -- (I_8_3__Rationals);
	\draw[gray,dashed,->] (I_7_4__Rationals) -- (I_8_4__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_8_3__Rationals) -- (I_8_4__Rationals);
	\draw[blue,->] (I_9_5__Rationals) -- (I_8_4__Rationals);
	\draw[gray,dashed,->] (I_7_5__Rationals) -- (I_8_5__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_8_4__Rationals) -- (I_8_5__Rationals);
	\draw[blue,->] (I_9_6__Rationals) -- (I_8_5__Rationals);
	\draw[gray,dashed,->] (I_7_6__Rationals) -- (I_8_6__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_8_5__Rationals) -- (I_8_6__Rationals);
	\draw[blue,->] (I_9_7__Rationals) -- (I_8_6__Rationals);
	\draw[gray,dashed,->] (I_7_7__Rationals) -- (I_8_7__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_8_6__Rationals) -- (I_8_7__Rationals);
	\draw[blue,->] (I_9_8__Rationals) -- (I_8_7__Rationals);
	\draw[gray,dashed,->] (I_7_8__Rationals) -- (I_8_8__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_8_7__Rationals) -- (I_8_8__Rationals);
	\draw[gray,dashed,->] (I_7_9__Rationals) -- (I_8_9__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_8_3__Rationals) -- (I_8_9__Rationals);
	\draw[gray,dashed,->] (I_8_1__Rationals) -- (I_9_1__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_9_2__Rationals) -- (I_9_1__Rationals);
	\draw[gray,dashed,->] (I_8_2__Rationals) -- (I_9_2__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_9_3__Rationals) -- (I_9_2__Rationals);
	\draw[gray,dashed,->] (I_8_3__Rationals) -- (I_9_3__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[gray,dashed,->] (I_8_4__Rationals) -- (I_9_4__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_9_3__Rationals) -- (I_9_4__Rationals);
	\draw[gray,dashed,->] (I_8_5__Rationals) -- (I_9_5__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_9_4__Rationals) -- (I_9_5__Rationals);
	\draw[gray,dashed,->] (I_8_6__Rationals) -- (I_9_6__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_9_5__Rationals) -- (I_9_6__Rationals);
	\draw[gray,dashed,->] (I_8_7__Rationals) -- (I_9_7__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_9_6__Rationals) -- (I_9_7__Rationals);
	\draw[gray,dashed,->] (I_8_8__Rationals) -- (I_9_8__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_9_7__Rationals) -- (I_9_8__Rationals);
	\draw[gray,dashed,->] (I_8_9__Rationals) -- (I_9_9__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_9_3__Rationals) -- (I_9_9__Rationals);
\end{tikzpicture}
\end{scaletikzpicturetowidth}
\end{center}

\end{figure}
\begin{figure}[ht]

\begin{center}
\begin{scaletikzpicturetowidth}{\textwidth}
\begin{tikzpicture}[scale=\tikzscale]
	\node at (5,3) {$\dots$};
	\node at (5,-3) {$\dots$};
	\node[seloneEEEEEEEEE, label={[shift={(0.2,-0.55)}, brown]\tiny $I(10,1)$}] (I_10_1__Rationals) at (4,6) {$\begin{smallmatrix}&&2\\2&3&4&3&3&2&1&1\end{smallmatrix}$};
	\node[noselEEEEEEEEE, label={[shift={(0.2,-0.55)}, brown]\tiny $I(10,2)$}] (I_10_2__Rationals) at (2,3) {$\begin{smallmatrix}&&4\\3&6&8&7&6&4&3&2\end{smallmatrix}$};
	\node[noselEEEEEEEEE, label={[shift={(0.2,-0.55)}, brown]\tiny $I(10,3)$}] (I_10_3__Rationals) at (0,0) {$\begin{smallmatrix}&&7\\5&9&13&11&9&7&5&3\end{smallmatrix}$};
	\node[noselEEEEEEEEE, label={[shift={(0.2,-0.55)}, brown]\tiny $I(10,4)$}] (I_10_4__Rationals) at (2,-3) {$\begin{smallmatrix}&&5\\4&7&10&9&7&6&4&2\end{smallmatrix}$};
	\node[noselEEEEEEEEE, label={[shift={(0.2,-0.55)}, brown]\tiny $I(10,5)$}] (I_10_5__Rationals) at (4,-6) {$\begin{smallmatrix}&&4\\3&5&8&7&6&5&4&2\end{smallmatrix}$};
	\node[noselEEEEEEEEE, label={[shift={(0.2,-0.55)}, brown]\tiny $I(10,6)$}] (I_10_6__Rationals) at (6,-9) {$\begin{smallmatrix}&&3\\2&4&6&5&4&4&3&2\end{smallmatrix}$};
	\node[noselEEEEEEEEE, label={[shift={(0.2,-0.55)}, brown]\tiny $I(10,7)$}] (I_10_7__Rationals) at (8,-12) {$\begin{smallmatrix}&&2\\2&3&4&3&2&2&2&1\end{smallmatrix}$};
	\node[noselEEEEEEEEE, label={[shift={(0.2,-0.55)}, brown]\tiny $I(10,8)$}] (I_10_8__Rationals) at (10,-15) {$\begin{smallmatrix}&&1\\1&1&2&2&1&1&1&1\end{smallmatrix}$};
	\node[noselEEEEEEEEE, label={[shift={(0.2,-0.55)}, brown]\tiny $I(10,9)$}] (I_10_9__Rationals) at (2,1) {$\begin{smallmatrix}&&4\\2&4&6&5&4&3&2&1\end{smallmatrix}$};
	\node[noselEEEEEEEEE, label={[shift={(0.2,-0.55)}, brown]\tiny $I(11,1)$}] (I_11_1__Rationals) at (0,6) {$\begin{smallmatrix}&&2\\1&3&4&4&3&2&2&1\end{smallmatrix}$};
	\node[noselEEEEEEEEE, label={[shift={(0.2,-0.55)}, brown]\tiny $I(11,2)$}] (I_11_2__Rationals) at (-2,3) {$\begin{smallmatrix}&&5\\3&6&9&8&6&5&4&2\end{smallmatrix}$};
	\node[noselEEEEEEEEE, label={[shift={(0.2,-0.55)}, brown]\tiny $I(11,3)$}] (I_11_3__Rationals) at (-4,0) {$\begin{smallmatrix}&&7\\5&10&14&12&10&8&6&3\end{smallmatrix}$};
	\node[noselEEEEEEEEE, label={[shift={(0.2,-0.55)}, brown]\tiny $I(11,4)$}] (I_11_4__Rationals) at (-2,-3) {$\begin{smallmatrix}&&6\\4&8&11&9&8&6&4&2\end{smallmatrix}$};
	\node[noselEEEEEEEEE, label={[shift={(0.2,-0.55)}, brown]\tiny $I(11,5)$}] (I_11_5__Rationals) at (0,-6) {$\begin{smallmatrix}&&4\\3&6&8&7&6&5&3&1\end{smallmatrix}$};
	\node[noselEEEEEEEEE, label={[shift={(0.2,-0.55)}, brown]\tiny $I(11,6)$}] (I_11_6__Rationals) at (2,-9) {$\begin{smallmatrix}&&3\\2&4&6&5&5&4&3&1\end{smallmatrix}$};
	\node[noselEEEEEEEEE, label={[shift={(0.2,-0.55)}, brown]\tiny $I(11,7)$}] (I_11_7__Rationals) at (4,-12) {$\begin{smallmatrix}&&2\\1&3&4&3&3&3&2&1\end{smallmatrix}$};
	\node[noselEEEEEEEEE, label={[shift={(0.2,-0.55)}, brown]\tiny $I(11,8)$}] (I_11_8__Rationals) at (6,-15) {$\begin{smallmatrix}&&1\\1&2&2&1&1&1&1&0\end{smallmatrix}$};
	\node[noselEEEEEEEEE, label={[shift={(0.2,-0.55)}, brown]\tiny $I(11,9)$}] (I_11_9__Rationals) at (-2,1) {$\begin{smallmatrix}&&3\\3&5&7&6&5&4&3&2\end{smallmatrix}$};
	\node[noselEEEEEEEEE, label={[shift={(0.2,-0.55)}, brown]\tiny $I(12,1)$}] (I_12_1__Rationals) at (-4,6) {$\begin{smallmatrix}&&3\\2&3&5&4&3&3&2&1\end{smallmatrix}$};
	\node[noselEEEEEEEEE, label={[shift={(0.2,-0.55)}, brown]\tiny $I(12,2)$}] (I_12_2__Rationals) at (-6,3) {$\begin{smallmatrix}&&5\\4&7&10&8&7&6&4&2\end{smallmatrix}$};
	\node[noselEEEEEEEEE, label={[shift={(0.2,-0.55)}, brown]\tiny $I(12,3)$}] (I_12_3__Rationals) at (-8,0) {$\begin{smallmatrix}&&8\\5&10&15&13&11&9&6&3\end{smallmatrix}$};
	\node[noselEEEEEEEEE, label={[shift={(0.2,-0.55)}, brown]\tiny $I(12,4)$}] (I_12_4__Rationals) at (-6,-3) {$\begin{smallmatrix}&&6\\4&8&12&11&9&7&5&3\end{smallmatrix}$};
	\node[noselEEEEEEEEE, label={[shift={(0.2,-0.55)}, brown]\tiny $I(12,5)$}] (I_12_5__Rationals) at (-4,-6) {$\begin{smallmatrix}&&5\\3&6&9&8&7&5&3&2\end{smallmatrix}$};
	\node[noselEEEEEEEEE, label={[shift={(0.2,-0.55)}, brown]\tiny $I(12,6)$}] (I_12_6__Rationals) at (-2,-9) {$\begin{smallmatrix}&&3\\2&4&6&6&5&4&2&1\end{smallmatrix}$};
	\node[noselEEEEEEEEE, label={[shift={(0.2,-0.55)}, brown]\tiny $I(12,7)$}] (I_12_7__Rationals) at (0,-12) {$\begin{smallmatrix}&&2\\1&2&4&4&4&3&2&1\end{smallmatrix}$};
	\node[noselEEEEEEEEE, label={[shift={(0.2,-0.55)}, brown]\tiny $I(12,8)$}] (I_12_8__Rationals) at (2,-15) {$\begin{smallmatrix}&&1\\0&1&2&2&2&2&1&1\end{smallmatrix}$};
	\node[noselEEEEEEEEE, label={[shift={(0.2,-0.55)}, brown]\tiny $I(12,9)$}] (I_12_9__Rationals) at (-6,1) {$\begin{smallmatrix}&&4\\2&5&7&6&5&4&3&1\end{smallmatrix}$};
	\node[noselEEEEEEEEE, label={[shift={(0.2,-0.55)}, brown]\tiny $I(13,1)$}] (I_13_1__Rationals) at (-8,6) {$\begin{smallmatrix}&&2\\2&4&5&4&4&3&2&1\end{smallmatrix}$};
	\node[noselEEEEEEEEE, label={[shift={(0.2,-0.55)}, brown]\tiny $I(13,2)$}] (I_13_2__Rationals) at (-10,3) {$\begin{smallmatrix}&&5\\3&7&10&9&8&6&4&2\end{smallmatrix}$};
	\node[noselEEEEEEEEE, label={[shift={(0.2,-0.55)}, brown]\tiny $I(13,3)$}] (I_13_3__Rationals) at (-12,0) {$\begin{smallmatrix}&&8\\6&11&16&14&12&9&6&3\end{smallmatrix}$};
	\node[noselEEEEEEEEE, label={[shift={(0.2,-0.55)}, brown]\tiny $I(13,4)$}] (I_13_4__Rationals) at (-10,-3) {$\begin{smallmatrix}&&7\\5&9&13&11&9&7&5&2\end{smallmatrix}$};
	\node[noselEEEEEEEEE, label={[shift={(0.2,-0.55)}, brown]\tiny $I(13,5)$}] (I_13_5__Rationals) at (-8,-6) {$\begin{smallmatrix}&&5\\4&7&10&9&7&5&4&2\end{smallmatrix}$};
	\node[noselEEEEEEEEE, label={[shift={(0.2,-0.55)}, brown]\tiny $I(13,6)$}] (I_13_6__Rationals) at (-6,-9) {$\begin{smallmatrix}&&4\\3&5&7&6&5&3&2&1\end{smallmatrix}$};
	\node[noselEEEEEEEEE, label={[shift={(0.2,-0.55)}, brown]\tiny $I(13,7)$}] (I_13_7__Rationals) at (-4,-12) {$\begin{smallmatrix}&&2\\2&3&4&4&3&2&1&0\end{smallmatrix}$};
	\node[noselEEEEEEEEE, label={[shift={(0.2,-0.55)}, brown]\tiny $I(13,8)$}] (I_13_8__Rationals) at (-2,-15) {$\begin{smallmatrix}&&1\\1&1&2&2&2&1&1&0\end{smallmatrix}$};
	\node[noselEEEEEEEEE, label={[shift={(0.2,-0.55)}, brown]\tiny $I(13,9)$}] (I_13_9__Rationals) at (-10,1) {$\begin{smallmatrix}&&4\\3&5&8&7&6&5&3&2\end{smallmatrix}$};
	\node[noselEEEEEEEEE, label={[shift={(0.2,-0.55)}, brown]\tiny $I(14,1)$}] (I_14_1__Rationals) at (-12,6) {$\begin{smallmatrix}&&3\\1&3&5&5&4&3&2&1\end{smallmatrix}$};
	\node[noselEEEEEEEEE, label={[shift={(0.2,-0.55)}, brown]\tiny $I(14,2)$}] (I_14_2__Rationals) at (-14,3) {$\begin{smallmatrix}&&6\\4&7&11&10&8&6&4&2\end{smallmatrix}$};
	\node[noselEEEEEEEEE, label={[shift={(0.2,-0.55)}, brown]\tiny $I(14,3)$}] (I_14_3__Rationals) at (-16,0) {$\begin{smallmatrix}&&9\\6&12&17&15&12&9&6&3\end{smallmatrix}$};
	\node[noselEEEEEEEEE, label={[shift={(0.2,-0.55)}, brown]\tiny $I(14,4)$}] (I_14_4__Rationals) at (-14,-3) {$\begin{smallmatrix}&&7\\5&10&14&12&10&8&5&3\end{smallmatrix}$};
	\node[noselEEEEEEEEE, label={[shift={(0.2,-0.55)}, brown]\tiny $I(14,5)$}] (I_14_5__Rationals) at (-12,-6) {$\begin{smallmatrix}&&6\\4&8&11&9&7&6&4&2\end{smallmatrix}$};
	\node[noselEEEEEEEEE, label={[shift={(0.2,-0.55)}, brown]\tiny $I(14,6)$}] (I_14_6__Rationals) at (-10,-9) {$\begin{smallmatrix}&&4\\3&6&8&7&5&4&3&2\end{smallmatrix}$};
	\node[seloneEEEEEEEEE, label={[shift={(0.2,-0.55)}, brown]\tiny $I(14,7)$}] (I_14_7__Rationals) at (-8,-12) {$\begin{smallmatrix}&&3\\2&4&5&4&3&2&1&1\end{smallmatrix}$};
	\node[noselEEEEEEEEE, label={[shift={(0.2,-0.55)}, brown]\tiny $I(14,8)$}] (I_14_8__Rationals) at (-6,-15) {$\begin{smallmatrix}&&1\\1&2&2&2&1&1&0&0\end{smallmatrix}$};
	\node[noselEEEEEEEEE, label={[shift={(0.2,-0.55)}, brown]\tiny $I(14,9)$}] (I_14_9__Rationals) at (-14,1) {$\begin{smallmatrix}&&4\\3&6&8&7&6&4&3&1\end{smallmatrix}$};
	\node at (-17,3) {$\dots$};
	\node at (-17,-3) {$\dots$};
	\draw[red,->] (I_10_2__Rationals) -- (I_10_1__Rationals);
	\draw[red,->] (I_10_3__Rationals) -- (I_10_2__Rationals);
	\draw[blue,->] (I_11_1__Rationals) -- (I_10_2__Rationals);
	\draw[blue,->] (I_11_2__Rationals) -- (I_10_3__Rationals);
	\draw[blue,->] (I_11_4__Rationals) -- (I_10_3__Rationals);
	\draw[blue,->] (I_11_9__Rationals) -- (I_10_3__Rationals);
	\draw[red,->] (I_10_3__Rationals) -- (I_10_4__Rationals);
	\draw[blue,->] (I_11_5__Rationals) -- (I_10_4__Rationals);
	\draw[red,->] (I_10_4__Rationals) -- (I_10_5__Rationals);
	\draw[blue,->] (I_11_6__Rationals) -- (I_10_5__Rationals);
	\draw[red,->] (I_10_5__Rationals) -- (I_10_6__Rationals);
	\draw[blue,->] (I_11_7__Rationals) -- (I_10_6__Rationals);
	\draw[red,->] (I_10_6__Rationals) -- (I_10_7__Rationals);
	\draw[blue,->] (I_11_8__Rationals) -- (I_10_7__Rationals);
	\draw[red,->] (I_10_7__Rationals) -- (I_10_8__Rationals);
	\draw[red,->] (I_10_3__Rationals) -- (I_10_9__Rationals);
	\draw[gray,dashed,->] (I_10_1__Rationals) -- (I_11_1__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_11_2__Rationals) -- (I_11_1__Rationals);
	\draw[gray,dashed,->] (I_10_2__Rationals) -- (I_11_2__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_11_3__Rationals) -- (I_11_2__Rationals);
	\draw[blue,->] (I_12_1__Rationals) -- (I_11_2__Rationals);
	\draw[gray,dashed,->] (I_10_3__Rationals) -- (I_11_3__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[blue,->] (I_12_2__Rationals) -- (I_11_3__Rationals);
	\draw[blue,->] (I_12_4__Rationals) -- (I_11_3__Rationals);
	\draw[blue,->] (I_12_9__Rationals) -- (I_11_3__Rationals);
	\draw[gray,dashed,->] (I_10_4__Rationals) -- (I_11_4__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_11_3__Rationals) -- (I_11_4__Rationals);
	\draw[blue,->] (I_12_5__Rationals) -- (I_11_4__Rationals);
	\draw[gray,dashed,->] (I_10_5__Rationals) -- (I_11_5__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_11_4__Rationals) -- (I_11_5__Rationals);
	\draw[blue,->] (I_12_6__Rationals) -- (I_11_5__Rationals);
	\draw[gray,dashed,->] (I_10_6__Rationals) -- (I_11_6__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_11_5__Rationals) -- (I_11_6__Rationals);
	\draw[blue,->] (I_12_7__Rationals) -- (I_11_6__Rationals);
	\draw[gray,dashed,->] (I_10_7__Rationals) -- (I_11_7__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_11_6__Rationals) -- (I_11_7__Rationals);
	\draw[blue,->] (I_12_8__Rationals) -- (I_11_7__Rationals);
	\draw[gray,dashed,->] (I_10_8__Rationals) -- (I_11_8__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_11_7__Rationals) -- (I_11_8__Rationals);
	\draw[gray,dashed,->] (I_10_9__Rationals) -- (I_11_9__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_11_3__Rationals) -- (I_11_9__Rationals);
	\draw[gray,dashed,->] (I_11_1__Rationals) -- (I_12_1__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_12_2__Rationals) -- (I_12_1__Rationals);
	\draw[gray,dashed,->] (I_11_2__Rationals) -- (I_12_2__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_12_3__Rationals) -- (I_12_2__Rationals);
	\draw[blue,->] (I_13_1__Rationals) -- (I_12_2__Rationals);
	\draw[gray,dashed,->] (I_11_3__Rationals) -- (I_12_3__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[blue,->] (I_13_2__Rationals) -- (I_12_3__Rationals);
	\draw[blue,->] (I_13_4__Rationals) -- (I_12_3__Rationals);
	\draw[blue,->] (I_13_9__Rationals) -- (I_12_3__Rationals);
	\draw[gray,dashed,->] (I_11_4__Rationals) -- (I_12_4__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_12_3__Rationals) -- (I_12_4__Rationals);
	\draw[blue,->] (I_13_5__Rationals) -- (I_12_4__Rationals);
	\draw[gray,dashed,->] (I_11_5__Rationals) -- (I_12_5__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_12_4__Rationals) -- (I_12_5__Rationals);
	\draw[blue,->] (I_13_6__Rationals) -- (I_12_5__Rationals);
	\draw[gray,dashed,->] (I_11_6__Rationals) -- (I_12_6__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_12_5__Rationals) -- (I_12_6__Rationals);
	\draw[blue,->] (I_13_7__Rationals) -- (I_12_6__Rationals);
	\draw[gray,dashed,->] (I_11_7__Rationals) -- (I_12_7__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_12_6__Rationals) -- (I_12_7__Rationals);
	\draw[blue,->] (I_13_8__Rationals) -- (I_12_7__Rationals);
	\draw[gray,dashed,->] (I_11_8__Rationals) -- (I_12_8__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_12_7__Rationals) -- (I_12_8__Rationals);
	\draw[gray,dashed,->] (I_11_9__Rationals) -- (I_12_9__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_12_3__Rationals) -- (I_12_9__Rationals);
	\draw[gray,dashed,->] (I_12_1__Rationals) -- (I_13_1__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_13_2__Rationals) -- (I_13_1__Rationals);
	\draw[gray,dashed,->] (I_12_2__Rationals) -- (I_13_2__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_13_3__Rationals) -- (I_13_2__Rationals);
	\draw[blue,->] (I_14_1__Rationals) -- (I_13_2__Rationals);
	\draw[gray,dashed,->] (I_12_3__Rationals) -- (I_13_3__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[blue,->] (I_14_2__Rationals) -- (I_13_3__Rationals);
	\draw[blue,->] (I_14_4__Rationals) -- (I_13_3__Rationals);
	\draw[blue,->] (I_14_9__Rationals) -- (I_13_3__Rationals);
	\draw[gray,dashed,->] (I_12_4__Rationals) -- (I_13_4__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_13_3__Rationals) -- (I_13_4__Rationals);
	\draw[blue,->] (I_14_5__Rationals) -- (I_13_4__Rationals);
	\draw[gray,dashed,->] (I_12_5__Rationals) -- (I_13_5__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_13_4__Rationals) -- (I_13_5__Rationals);
	\draw[blue,->] (I_14_6__Rationals) -- (I_13_5__Rationals);
	\draw[gray,dashed,->] (I_12_6__Rationals) -- (I_13_6__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_13_5__Rationals) -- (I_13_6__Rationals);
	\draw[blue,->] (I_14_7__Rationals) -- (I_13_6__Rationals);
	\draw[gray,dashed,->] (I_12_7__Rationals) -- (I_13_7__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_13_6__Rationals) -- (I_13_7__Rationals);
	\draw[blue,->] (I_14_8__Rationals) -- (I_13_7__Rationals);
	\draw[gray,dashed,->] (I_12_8__Rationals) -- (I_13_8__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_13_7__Rationals) -- (I_13_8__Rationals);
	\draw[gray,dashed,->] (I_12_9__Rationals) -- (I_13_9__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_13_3__Rationals) -- (I_13_9__Rationals);
	\draw[gray,dashed,->] (I_13_1__Rationals) -- (I_14_1__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_14_2__Rationals) -- (I_14_1__Rationals);
	\draw[gray,dashed,->] (I_13_2__Rationals) -- (I_14_2__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_14_3__Rationals) -- (I_14_2__Rationals);
	\draw[gray,dashed,->] (I_13_3__Rationals) -- (I_14_3__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[gray,dashed,->] (I_13_4__Rationals) -- (I_14_4__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_14_3__Rationals) -- (I_14_4__Rationals);
	\draw[gray,dashed,->] (I_13_5__Rationals) -- (I_14_5__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_14_4__Rationals) -- (I_14_5__Rationals);
	\draw[gray,dashed,->] (I_13_6__Rationals) -- (I_14_6__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_14_5__Rationals) -- (I_14_6__Rationals);
	\draw[gray,dashed,->] (I_13_7__Rationals) -- (I_14_7__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_14_6__Rationals) -- (I_14_7__Rationals);
	\draw[gray,dashed,->] (I_13_8__Rationals) -- (I_14_8__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_14_7__Rationals) -- (I_14_8__Rationals);
	\draw[gray,dashed,->] (I_13_9__Rationals) -- (I_14_9__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_14_3__Rationals) -- (I_14_9__Rationals);
\end{tikzpicture}
\end{scaletikzpicturetowidth}
\end{center}

\end{figure}
\begin{figure}[ht]

\begin{center}
\begin{scaletikzpicturetowidth}{\textwidth}
\begin{tikzpicture}[scale=\tikzscale]
	\node at (5,3) {$\dots$};
	\node at (5,-3) {$\dots$};
	\node[noselEEEEEEEEE, label={[shift={(0.2,-0.55)}, brown]\tiny $I(15,1)$}] (I_15_1__Rationals) at (4,6) {$\begin{smallmatrix}&&3\\3&4&6&5&4&3&2&1\end{smallmatrix}$};
	\node[noselEEEEEEEEE, label={[shift={(0.2,-0.55)}, brown]\tiny $I(15,2)$}] (I_15_2__Rationals) at (2,3) {$\begin{smallmatrix}&&6\\5&9&12&10&8&6&4&2\end{smallmatrix}$};
	\node[noselEEEEEEEEE, label={[shift={(0.2,-0.55)}, brown]\tiny $I(15,3)$}] (I_15_3__Rationals) at (0,0) {$\begin{smallmatrix}&&10\\7&13&19&16&13&10&7&4\end{smallmatrix}$};
	\node[noselEEEEEEEEE, label={[shift={(0.2,-0.55)}, brown]\tiny $I(15,4)$}] (I_15_4__Rationals) at (2,-3) {$\begin{smallmatrix}&&8\\5&10&15&13&11&8&6&3\end{smallmatrix}$};
	\node[noselEEEEEEEEE, label={[shift={(0.2,-0.55)}, brown]\tiny $I(15,5)$}] (I_15_5__Rationals) at (4,-6) {$\begin{smallmatrix}&&6\\4&8&12&10&9&7&5&3\end{smallmatrix}$};
	\node[noselEEEEEEEEE, label={[shift={(0.2,-0.55)}, brown]\tiny $I(15,6)$}] (I_15_6__Rationals) at (6,-9) {$\begin{smallmatrix}&&5\\3&6&9&7&6&5&4&2\end{smallmatrix}$};
	\node[noselEEEEEEEEE, label={[shift={(0.2,-0.55)}, brown]\tiny $I(15,7)$}] (I_15_7__Rationals) at (8,-12) {$\begin{smallmatrix}&&3\\2&4&6&5&4&3&3&2\end{smallmatrix}$};
	\node[noselEEEEEEEEE, label={[shift={(0.2,-0.55)}, brown]\tiny $I(15,8)$}] (I_15_8__Rationals) at (10,-15) {$\begin{smallmatrix}&&2\\1&2&3&2&2&1&1&1\end{smallmatrix}$};
	\node[noselEEEEEEEEE, label={[shift={(0.2,-0.55)}, brown]\tiny $I(15,9)$}] (I_15_9__Rationals) at (2,1) {$\begin{smallmatrix}&&5\\3&6&9&8&6&5&3&2\end{smallmatrix}$};
	\node[noselEEEEEEEEE, label={[shift={(0.2,-0.55)}, brown]\tiny $I(16,1)$}] (I_16_1__Rationals) at (0,6) {$\begin{smallmatrix}&&3\\2&5&6&5&4&3&2&1\end{smallmatrix}$};
	\node[noselEEEEEEEEE, label={[shift={(0.2,-0.55)}, brown]\tiny $I(16,2)$}] (I_16_2__Rationals) at (-2,3) {$\begin{smallmatrix}&&7\\4&9&13&11&9&7&5&3\end{smallmatrix}$};
	\node[noselEEEEEEEEE, label={[shift={(0.2,-0.55)}, brown]\tiny $I(16,3)$}] (I_16_3__Rationals) at (-4,0) {$\begin{smallmatrix}&&10\\7&14&20&17&14&11&8&4\end{smallmatrix}$};
	\node[noselEEEEEEEEE, label={[shift={(0.2,-0.55)}, brown]\tiny $I(16,4)$}] (I_16_4__Rationals) at (-2,-3) {$\begin{smallmatrix}&&8\\6&11&16&14&11&9&6&3\end{smallmatrix}$};
	\node[noselEEEEEEEEE, label={[shift={(0.2,-0.55)}, brown]\tiny $I(16,5)$}] (I_16_5__Rationals) at (0,-6) {$\begin{smallmatrix}&&6\\4&8&12&11&9&7&5&2\end{smallmatrix}$};
	\node[noselEEEEEEEEE, label={[shift={(0.2,-0.55)}, brown]\tiny $I(16,6)$}] (I_16_6__Rationals) at (2,-9) {$\begin{smallmatrix}&&4\\3&6&9&8&7&6&4&2\end{smallmatrix}$};
	\node[noselEEEEEEEEE, label={[shift={(0.2,-0.55)}, brown]\tiny $I(16,7)$}] (I_16_7__Rationals) at (4,-12) {$\begin{smallmatrix}&&3\\2&4&6&5&4&4&3&1\end{smallmatrix}$};
	\node[noselEEEEEEEEE, label={[shift={(0.2,-0.55)}, brown]\tiny $I(16,8)$}] (I_16_8__Rationals) at (6,-15) {$\begin{smallmatrix}&&1\\1&2&3&3&2&2&2&1\end{smallmatrix}$};
	\node[noselEEEEEEEEE, label={[shift={(0.2,-0.55)}, brown]\tiny $I(16,9)$}] (I_16_9__Rationals) at (-2,1) {$\begin{smallmatrix}&&5\\4&7&10&8&7&5&4&2\end{smallmatrix}$};
	\node[noselEEEEEEEEE, label={[shift={(0.2,-0.55)}, brown]\tiny $I(17,1)$}] (I_17_1__Rationals) at (-4,6) {$\begin{smallmatrix}&&4\\2&4&7&6&5&4&3&2\end{smallmatrix}$};
	\node[noselEEEEEEEEE, label={[shift={(0.2,-0.55)}, brown]\tiny $I(17,2)$}] (I_17_2__Rationals) at (-6,3) {$\begin{smallmatrix}&&7\\5&9&14&12&10&8&6&3\end{smallmatrix}$};
	\node[noselEEEEEEEEE, label={[shift={(0.2,-0.55)}, brown]\tiny $I(17,3)$}] (I_17_3__Rationals) at (-8,0) {$\begin{smallmatrix}&&11\\7&14&21&18&15&12&8&4\end{smallmatrix}$};
	\node[noselEEEEEEEEE, label={[shift={(0.2,-0.55)}, brown]\tiny $I(17,4)$}] (I_17_4__Rationals) at (-6,-3) {$\begin{smallmatrix}&&9\\6&12&17&14&12&9&6&3\end{smallmatrix}$};
	\node[noselEEEEEEEEE, label={[shift={(0.2,-0.55)}, brown]\tiny $I(17,5)$}] (I_17_5__Rationals) at (-4,-6) {$\begin{smallmatrix}&&7\\5&9&13&11&9&7&4&2\end{smallmatrix}$};
	\node[noselEEEEEEEEE, label={[shift={(0.2,-0.55)}, brown]\tiny $I(17,6)$}] (I_17_6__Rationals) at (-2,-9) {$\begin{smallmatrix}&&5\\3&6&9&8&7&5&3&1\end{smallmatrix}$};
	\node[noselEEEEEEEEE, label={[shift={(0.2,-0.55)}, brown]\tiny $I(17,7)$}] (I_17_7__Rationals) at (0,-12) {$\begin{smallmatrix}&&3\\2&4&6&5&5&4&2&1\end{smallmatrix}$};
	\node[noselEEEEEEEEE, label={[shift={(0.2,-0.55)}, brown]\tiny $I(17,8)$}] (I_17_8__Rationals) at (2,-15) {$\begin{smallmatrix}&&2\\1&2&3&2&2&2&1&0\end{smallmatrix}$};
	\node[noselEEEEEEEEE, label={[shift={(0.2,-0.55)}, brown]\tiny $I(17,9)$}] (I_17_9__Rationals) at (-6,1) {$\begin{smallmatrix}&&5\\3&7&10&9&7&6&4&2\end{smallmatrix}$};
	\node[noselEEEEEEEEE, label={[shift={(0.2,-0.55)}, brown]\tiny $I(18,1)$}] (I_18_1__Rationals) at (-8,6) {$\begin{smallmatrix}&&3\\3&5&7&6&5&4&3&1\end{smallmatrix}$};
	\node[noselEEEEEEEEE, label={[shift={(0.2,-0.55)}, brown]\tiny $I(18,2)$}] (I_18_2__Rationals) at (-10,3) {$\begin{smallmatrix}&&7\\5&10&14&12&10&8&5&2\end{smallmatrix}$};
	\node[noselEEEEEEEEE, label={[shift={(0.2,-0.55)}, brown]\tiny $I(18,3)$}] (I_18_3__Rationals) at (-12,0) {$\begin{smallmatrix}&&11\\8&15&22&19&16&12&8&4\end{smallmatrix}$};
	\node[noselEEEEEEEEE, label={[shift={(0.2,-0.55)}, brown]\tiny $I(18,4)$}] (I_18_4__Rationals) at (-10,-3) {$\begin{smallmatrix}&&9\\6&12&18&16&13&10&7&4\end{smallmatrix}$};
	\node[noselEEEEEEEEE, label={[shift={(0.2,-0.55)}, brown]\tiny $I(18,5)$}] (I_18_5__Rationals) at (-8,-6) {$\begin{smallmatrix}&&7\\5&10&14&12&10&7&5&3\end{smallmatrix}$};
	\node[noselEEEEEEEEE, label={[shift={(0.2,-0.55)}, brown]\tiny $I(18,6)$}] (I_18_6__Rationals) at (-6,-9) {$\begin{smallmatrix}&&5\\4&7&10&9&7&5&3&2\end{smallmatrix}$};
	\node[noselEEEEEEEEE, label={[shift={(0.2,-0.55)}, brown]\tiny $I(18,7)$}] (I_18_7__Rationals) at (-4,-12) {$\begin{smallmatrix}&&3\\2&4&6&6&5&3&2&1\end{smallmatrix}$};
	\node[noselEEEEEEEEE, label={[shift={(0.2,-0.55)}, brown]\tiny $I(18,8)$}] (I_18_8__Rationals) at (-2,-15) {$\begin{smallmatrix}&&1\\1&2&3&3&3&2&1&1\end{smallmatrix}$};
	\node[noselEEEEEEEEE, label={[shift={(0.2,-0.55)}, brown]\tiny $I(18,9)$}] (I_18_9__Rationals) at (-10,1) {$\begin{smallmatrix}&&6\\4&7&11&9&8&6&4&2\end{smallmatrix}$};
	\node[noselEEEEEEEEE, label={[shift={(0.2,-0.55)}, brown]\tiny $I(19,1)$}] (I_19_1__Rationals) at (-12,6) {$\begin{smallmatrix}&&4\\2&5&7&6&5&4&2&1\end{smallmatrix}$};
	\node[noselEEEEEEEEE, label={[shift={(0.2,-0.55)}, brown]\tiny $I(19,2)$}] (I_19_2__Rationals) at (-14,3) {$\begin{smallmatrix}&&8\\5&10&15&13&11&8&5&3\end{smallmatrix}$};
	\node[noselEEEEEEEEE, label={[shift={(0.2,-0.55)}, brown]\tiny $I(19,3)$}] (I_19_3__Rationals) at (-16,0) {$\begin{smallmatrix}&&12\\8&16&23&20&16&12&8&4\end{smallmatrix}$};
	\node[noselEEEEEEEEE, label={[shift={(0.2,-0.55)}, brown]\tiny $I(19,4)$}] (I_19_4__Rationals) at (-14,-3) {$\begin{smallmatrix}&&10\\7&13&19&16&13&10&7&3\end{smallmatrix}$};
	\node[noselEEEEEEEEE, label={[shift={(0.2,-0.55)}, brown]\tiny $I(19,5)$}] (I_19_5__Rationals) at (-12,-6) {$\begin{smallmatrix}&&8\\5&10&15&13&10&8&6&3\end{smallmatrix}$};
	\node[noselEEEEEEEEE, label={[shift={(0.2,-0.55)}, brown]\tiny $I(19,6)$}] (I_19_6__Rationals) at (-10,-9) {$\begin{smallmatrix}&&6\\4&8&11&9&7&5&4&2\end{smallmatrix}$};
	\node[noselEEEEEEEEE, label={[shift={(0.2,-0.55)}, brown]\tiny $I(19,7)$}] (I_19_7__Rationals) at (-8,-12) {$\begin{smallmatrix}&&4\\3&5&7&6&4&3&2&1\end{smallmatrix}$};
	\node[noselEEEEEEEEE, label={[shift={(0.2,-0.55)}, brown]\tiny $I(19,8)$}] (I_19_8__Rationals) at (-6,-15) {$\begin{smallmatrix}&&2\\1&2&3&3&2&1&1&0\end{smallmatrix}$};
	\node[noselEEEEEEEEE, label={[shift={(0.2,-0.55)}, brown]\tiny $I(19,9)$}] (I_19_9__Rationals) at (-14,1) {$\begin{smallmatrix}&&5\\4&8&11&10&8&6&4&2\end{smallmatrix}$};
	\node at (-17,3) {$\dots$};
	\node at (-17,-3) {$\dots$};
	\draw[red,->] (I_15_2__Rationals) -- (I_15_1__Rationals);
	\draw[red,->] (I_15_3__Rationals) -- (I_15_2__Rationals);
	\draw[blue,->] (I_16_1__Rationals) -- (I_15_2__Rationals);
	\draw[blue,->] (I_16_2__Rationals) -- (I_15_3__Rationals);
	\draw[blue,->] (I_16_4__Rationals) -- (I_15_3__Rationals);
	\draw[blue,->] (I_16_9__Rationals) -- (I_15_3__Rationals);
	\draw[red,->] (I_15_3__Rationals) -- (I_15_4__Rationals);
	\draw[blue,->] (I_16_5__Rationals) -- (I_15_4__Rationals);
	\draw[red,->] (I_15_4__Rationals) -- (I_15_5__Rationals);
	\draw[blue,->] (I_16_6__Rationals) -- (I_15_5__Rationals);
	\draw[red,->] (I_15_5__Rationals) -- (I_15_6__Rationals);
	\draw[blue,->] (I_16_7__Rationals) -- (I_15_6__Rationals);
	\draw[red,->] (I_15_6__Rationals) -- (I_15_7__Rationals);
	\draw[blue,->] (I_16_8__Rationals) -- (I_15_7__Rationals);
	\draw[red,->] (I_15_7__Rationals) -- (I_15_8__Rationals);
	\draw[red,->] (I_15_3__Rationals) -- (I_15_9__Rationals);
	\draw[gray,dashed,->] (I_15_1__Rationals) -- (I_16_1__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_16_2__Rationals) -- (I_16_1__Rationals);
	\draw[gray,dashed,->] (I_15_2__Rationals) -- (I_16_2__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_16_3__Rationals) -- (I_16_2__Rationals);
	\draw[blue,->] (I_17_1__Rationals) -- (I_16_2__Rationals);
	\draw[gray,dashed,->] (I_15_3__Rationals) -- (I_16_3__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[blue,->] (I_17_2__Rationals) -- (I_16_3__Rationals);
	\draw[blue,->] (I_17_4__Rationals) -- (I_16_3__Rationals);
	\draw[blue,->] (I_17_9__Rationals) -- (I_16_3__Rationals);
	\draw[gray,dashed,->] (I_15_4__Rationals) -- (I_16_4__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_16_3__Rationals) -- (I_16_4__Rationals);
	\draw[blue,->] (I_17_5__Rationals) -- (I_16_4__Rationals);
	\draw[gray,dashed,->] (I_15_5__Rationals) -- (I_16_5__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_16_4__Rationals) -- (I_16_5__Rationals);
	\draw[blue,->] (I_17_6__Rationals) -- (I_16_5__Rationals);
	\draw[gray,dashed,->] (I_15_6__Rationals) -- (I_16_6__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_16_5__Rationals) -- (I_16_6__Rationals);
	\draw[blue,->] (I_17_7__Rationals) -- (I_16_6__Rationals);
	\draw[gray,dashed,->] (I_15_7__Rationals) -- (I_16_7__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_16_6__Rationals) -- (I_16_7__Rationals);
	\draw[blue,->] (I_17_8__Rationals) -- (I_16_7__Rationals);
	\draw[gray,dashed,->] (I_15_8__Rationals) -- (I_16_8__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_16_7__Rationals) -- (I_16_8__Rationals);
	\draw[gray,dashed,->] (I_15_9__Rationals) -- (I_16_9__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_16_3__Rationals) -- (I_16_9__Rationals);
	\draw[gray,dashed,->] (I_16_1__Rationals) -- (I_17_1__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_17_2__Rationals) -- (I_17_1__Rationals);
	\draw[gray,dashed,->] (I_16_2__Rationals) -- (I_17_2__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_17_3__Rationals) -- (I_17_2__Rationals);
	\draw[blue,->] (I_18_1__Rationals) -- (I_17_2__Rationals);
	\draw[gray,dashed,->] (I_16_3__Rationals) -- (I_17_3__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[blue,->] (I_18_2__Rationals) -- (I_17_3__Rationals);
	\draw[blue,->] (I_18_4__Rationals) -- (I_17_3__Rationals);
	\draw[blue,->] (I_18_9__Rationals) -- (I_17_3__Rationals);
	\draw[gray,dashed,->] (I_16_4__Rationals) -- (I_17_4__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_17_3__Rationals) -- (I_17_4__Rationals);
	\draw[blue,->] (I_18_5__Rationals) -- (I_17_4__Rationals);
	\draw[gray,dashed,->] (I_16_5__Rationals) -- (I_17_5__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_17_4__Rationals) -- (I_17_5__Rationals);
	\draw[blue,->] (I_18_6__Rationals) -- (I_17_5__Rationals);
	\draw[gray,dashed,->] (I_16_6__Rationals) -- (I_17_6__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_17_5__Rationals) -- (I_17_6__Rationals);
	\draw[blue,->] (I_18_7__Rationals) -- (I_17_6__Rationals);
	\draw[gray,dashed,->] (I_16_7__Rationals) -- (I_17_7__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_17_6__Rationals) -- (I_17_7__Rationals);
	\draw[blue,->] (I_18_8__Rationals) -- (I_17_7__Rationals);
	\draw[gray,dashed,->] (I_16_8__Rationals) -- (I_17_8__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_17_7__Rationals) -- (I_17_8__Rationals);
	\draw[gray,dashed,->] (I_16_9__Rationals) -- (I_17_9__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_17_3__Rationals) -- (I_17_9__Rationals);
	\draw[gray,dashed,->] (I_17_1__Rationals) -- (I_18_1__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_18_2__Rationals) -- (I_18_1__Rationals);
	\draw[gray,dashed,->] (I_17_2__Rationals) -- (I_18_2__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_18_3__Rationals) -- (I_18_2__Rationals);
	\draw[blue,->] (I_19_1__Rationals) -- (I_18_2__Rationals);
	\draw[gray,dashed,->] (I_17_3__Rationals) -- (I_18_3__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[blue,->] (I_19_2__Rationals) -- (I_18_3__Rationals);
	\draw[blue,->] (I_19_4__Rationals) -- (I_18_3__Rationals);
	\draw[blue,->] (I_19_9__Rationals) -- (I_18_3__Rationals);
	\draw[gray,dashed,->] (I_17_4__Rationals) -- (I_18_4__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_18_3__Rationals) -- (I_18_4__Rationals);
	\draw[blue,->] (I_19_5__Rationals) -- (I_18_4__Rationals);
	\draw[gray,dashed,->] (I_17_5__Rationals) -- (I_18_5__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_18_4__Rationals) -- (I_18_5__Rationals);
	\draw[blue,->] (I_19_6__Rationals) -- (I_18_5__Rationals);
	\draw[gray,dashed,->] (I_17_6__Rationals) -- (I_18_6__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_18_5__Rationals) -- (I_18_6__Rationals);
	\draw[blue,->] (I_19_7__Rationals) -- (I_18_6__Rationals);
	\draw[gray,dashed,->] (I_17_7__Rationals) -- (I_18_7__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_18_6__Rationals) -- (I_18_7__Rationals);
	\draw[blue,->] (I_19_8__Rationals) -- (I_18_7__Rationals);
	\draw[gray,dashed,->] (I_17_8__Rationals) -- (I_18_8__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_18_7__Rationals) -- (I_18_8__Rationals);
	\draw[gray,dashed,->] (I_17_9__Rationals) -- (I_18_9__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_18_3__Rationals) -- (I_18_9__Rationals);
	\draw[gray,dashed,->] (I_18_1__Rationals) -- (I_19_1__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_19_2__Rationals) -- (I_19_1__Rationals);
	\draw[gray,dashed,->] (I_18_2__Rationals) -- (I_19_2__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_19_3__Rationals) -- (I_19_2__Rationals);
	\draw[gray,dashed,->] (I_18_3__Rationals) -- (I_19_3__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[gray,dashed,->] (I_18_4__Rationals) -- (I_19_4__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_19_3__Rationals) -- (I_19_4__Rationals);
	\draw[gray,dashed,->] (I_18_5__Rationals) -- (I_19_5__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_19_4__Rationals) -- (I_19_5__Rationals);
	\draw[gray,dashed,->] (I_18_6__Rationals) -- (I_19_6__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_19_5__Rationals) -- (I_19_6__Rationals);
	\draw[gray,dashed,->] (I_18_7__Rationals) -- (I_19_7__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_19_6__Rationals) -- (I_19_7__Rationals);
	\draw[gray,dashed,->] (I_18_8__Rationals) -- (I_19_8__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_19_7__Rationals) -- (I_19_8__Rationals);
	\draw[gray,dashed,->] (I_18_9__Rationals) -- (I_19_9__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_19_3__Rationals) -- (I_19_9__Rationals);
\end{tikzpicture}
\end{scaletikzpicturetowidth}
\end{center}

\end{figure}
\begin{figure}[ht]

\begin{center}
\begin{scaletikzpicturetowidth}{\textwidth}
\begin{tikzpicture}[scale=\tikzscale]
	\node at (5,3) {$\dots$};
	\node at (5,-3) {$\dots$};
	\node[noselEEEEEEEEE, label={[shift={(0.2,-0.55)}, brown]\tiny $I(20,1)$}] (I_20_1__Rationals) at (4,6) {$\begin{smallmatrix}&&4\\3&5&8&7&6&4&3&2\end{smallmatrix}$};
	\node[noselEEEEEEEEE, label={[shift={(0.2,-0.55)}, brown]\tiny $I(20,2)$}] (I_20_2__Rationals) at (2,3) {$\begin{smallmatrix}&&8\\6&11&16&14&11&8&6&3\end{smallmatrix}$};
	\node[noselEEEEEEEEE, label={[shift={(0.2,-0.55)}, brown]\tiny $I(20,3)$}] (I_20_3__Rationals) at (0,0) {$\begin{smallmatrix}&&13\\9&17&25&21&17&13&9&5\end{smallmatrix}$};
	\node[noselEEEEEEEEE, label={[shift={(0.2,-0.55)}, brown]\tiny $I(20,4)$}] (I_20_4__Rationals) at (2,-3) {$\begin{smallmatrix}&&10\\7&14&20&17&14&11&7&4\end{smallmatrix}$};
	\node[noselEEEEEEEEE, label={[shift={(0.2,-0.55)}, brown]\tiny $I(20,5)$}] (I_20_5__Rationals) at (4,-6) {$\begin{smallmatrix}&&8\\6&11&16&13&11&9&6&3\end{smallmatrix}$};
	\node[noselEEEEEEEEE, label={[shift={(0.2,-0.55)}, brown]\tiny $I(20,6)$}] (I_20_6__Rationals) at (6,-9) {$\begin{smallmatrix}&&6\\4&8&12&10&8&7&5&3\end{smallmatrix}$};
	\node[noselEEEEEEEEE, label={[shift={(0.2,-0.55)}, brown]\tiny $I(20,7)$}] (I_20_7__Rationals) at (8,-12) {$\begin{smallmatrix}&&4\\3&6&8&6&5&4&3&2\end{smallmatrix}$};
	\node[noselEEEEEEEEE, label={[shift={(0.2,-0.55)}, brown]\tiny $I(20,8)$}] (I_20_8__Rationals) at (10,-15) {$\begin{smallmatrix}&&2\\2&3&4&3&2&2&1&1\end{smallmatrix}$};
	\node[noselEEEEEEEEE, label={[shift={(0.2,-0.55)}, brown]\tiny $I(20,9)$}] (I_20_9__Rationals) at (2,1) {$\begin{smallmatrix}&&7\\4&8&12&10&8&6&4&2\end{smallmatrix}$};
	\node[noselEEEEEEEEE, label={[shift={(0.2,-0.55)}, brown]\tiny $I(21,1)$}] (I_21_1__Rationals) at (0,6) {$\begin{smallmatrix}&&4\\3&6&8&7&5&4&3&1\end{smallmatrix}$};
	\node[noselEEEEEEEEE, label={[shift={(0.2,-0.55)}, brown]\tiny $I(21,2)$}] (I_21_2__Rationals) at (-2,3) {$\begin{smallmatrix}&&9\\6&12&17&14&11&9&6&3\end{smallmatrix}$};
	\node[noselEEEEEEEEE, label={[shift={(0.2,-0.55)}, brown]\tiny $I(21,3)$}] (I_21_3__Rationals) at (-4,0) {$\begin{smallmatrix}&&13\\9&18&26&22&18&14&10&5\end{smallmatrix}$};
	\node[noselEEEEEEEEE, label={[shift={(0.2,-0.55)}, brown]\tiny $I(21,4)$}] (I_21_4__Rationals) at (-2,-3) {$\begin{smallmatrix}&&11\\7&14&21&18&15&11&8&4\end{smallmatrix}$};
	\node[noselEEEEEEEEE, label={[shift={(0.2,-0.55)}, brown]\tiny $I(21,5)$}] (I_21_5__Rationals) at (0,-6) {$\begin{smallmatrix}&&8\\5&11&16&14&12&9&6&3\end{smallmatrix}$};
	\node[noselEEEEEEEEE, label={[shift={(0.2,-0.55)}, brown]\tiny $I(21,6)$}] (I_21_6__Rationals) at (2,-9) {$\begin{smallmatrix}&&6\\4&8&12&10&9&7&5&2\end{smallmatrix}$};
	\node[noselEEEEEEEEE, label={[shift={(0.2,-0.55)}, brown]\tiny $I(21,7)$}] (I_21_7__Rationals) at (4,-12) {$\begin{smallmatrix}&&4\\2&5&8&7&6&5&4&2\end{smallmatrix}$};
	\node[noselEEEEEEEEE, label={[shift={(0.2,-0.55)}, brown]\tiny $I(21,8)$}] (I_21_8__Rationals) at (6,-15) {$\begin{smallmatrix}&&2\\1&3&4&3&3&2&2&1\end{smallmatrix}$};
	\node[noselEEEEEEEEE, label={[shift={(0.2,-0.55)}, brown]\tiny $I(21,9)$}] (I_21_9__Rationals) at (-2,1) {$\begin{smallmatrix}&&6\\5&9&13&11&9&7&5&3\end{smallmatrix}$};
	\node[noselEEEEEEEEE, label={[shift={(0.2,-0.55)}, brown]\tiny $I(22,1)$}] (I_22_1__Rationals) at (-4,6) {$\begin{smallmatrix}&&5\\3&6&9&7&6&5&3&2\end{smallmatrix}$};
	\node[noselEEEEEEEEE, label={[shift={(0.2,-0.55)}, brown]\tiny $I(22,2)$}] (I_22_2__Rationals) at (-6,3) {$\begin{smallmatrix}&&9\\6&12&18&15&13&10&7&4\end{smallmatrix}$};
	\node[noselEEEEEEEEE, label={[shift={(0.2,-0.55)}, brown]\tiny $I(22,3)$}] (I_22_3__Rationals) at (-8,0) {$\begin{smallmatrix}&&14\\9&18&27&23&19&15&10&5\end{smallmatrix}$};
	\node[noselEEEEEEEEE, label={[shift={(0.2,-0.55)}, brown]\tiny $I(22,4)$}] (I_22_4__Rationals) at (-6,-3) {$\begin{smallmatrix}&&11\\8&15&22&19&15&12&8&4\end{smallmatrix}$};
	\node[noselEEEEEEEEE, label={[shift={(0.2,-0.55)}, brown]\tiny $I(22,5)$}] (I_22_5__Rationals) at (-4,-6) {$\begin{smallmatrix}&&9\\6&11&17&15&12&9&6&3\end{smallmatrix}$};
	\node[noselEEEEEEEEE, label={[shift={(0.2,-0.55)}, brown]\tiny $I(22,6)$}] (I_22_6__Rationals) at (-2,-9) {$\begin{smallmatrix}&&6\\4&8&12&11&9&7&4&2\end{smallmatrix}$};
	\node[noselEEEEEEEEE, label={[shift={(0.2,-0.55)}, brown]\tiny $I(22,7)$}] (I_22_7__Rationals) at (0,-12) {$\begin{smallmatrix}&&4\\3&5&8&7&6&5&3&1\end{smallmatrix}$};
	\node[noselEEEEEEEEE, label={[shift={(0.2,-0.55)}, brown]\tiny $I(22,8)$}] (I_22_8__Rationals) at (2,-15) {$\begin{smallmatrix}&&2\\1&2&4&4&3&3&2&1\end{smallmatrix}$};
	\node[noselEEEEEEEEE, label={[shift={(0.2,-0.55)}, brown]\tiny $I(22,9)$}] (I_22_9__Rationals) at (-6,1) {$\begin{smallmatrix}&&7\\4&9&13&11&9&7&5&2\end{smallmatrix}$};
	\node[noselEEEEEEEEE, label={[shift={(0.2,-0.55)}, brown]\tiny $I(23,1)$}] (I_23_1__Rationals) at (-8,6) {$\begin{smallmatrix}&&4\\3&6&9&8&7&5&4&2\end{smallmatrix}$};
	\node[noselEEEEEEEEE, label={[shift={(0.2,-0.55)}, brown]\tiny $I(23,2)$}] (I_23_2__Rationals) at (-10,3) {$\begin{smallmatrix}&&9\\6&12&18&16&13&10&7&3\end{smallmatrix}$};
	\node[noselEEEEEEEEE, label={[shift={(0.2,-0.55)}, brown]\tiny $I(23,3)$}] (I_23_3__Rationals) at (-12,0) {$\begin{smallmatrix}&&14\\10&19&28&24&20&15&10&5\end{smallmatrix}$};
	\node[noselEEEEEEEEE, label={[shift={(0.2,-0.55)}, brown]\tiny $I(23,4)$}] (I_23_4__Rationals) at (-10,-3) {$\begin{smallmatrix}&&12\\8&16&23&19&16&12&8&4\end{smallmatrix}$};
	\node[noselEEEEEEEEE, label={[shift={(0.2,-0.55)}, brown]\tiny $I(23,5)$}] (I_23_5__Rationals) at (-8,-6) {$\begin{smallmatrix}&&9\\7&13&18&15&12&9&6&3\end{smallmatrix}$};
	\node[noselEEEEEEEEE, label={[shift={(0.2,-0.55)}, brown]\tiny $I(23,6)$}] (I_23_6__Rationals) at (-6,-9) {$\begin{smallmatrix}&&7\\5&9&13&11&9&6&4&2\end{smallmatrix}$};
	\node[noselEEEEEEEEE, label={[shift={(0.2,-0.55)}, brown]\tiny $I(23,7)$}] (I_23_7__Rationals) at (-4,-12) {$\begin{smallmatrix}&&4\\3&6&8&7&6&4&2&1\end{smallmatrix}$};
	\node[noselEEEEEEEEE, label={[shift={(0.2,-0.55)}, brown]\tiny $I(23,8)$}] (I_23_8__Rationals) at (-2,-15) {$\begin{smallmatrix}&&2\\2&3&4&3&3&2&1&0\end{smallmatrix}$};
	\node[noselEEEEEEEEE, label={[shift={(0.2,-0.55)}, brown]\tiny $I(23,9)$}] (I_23_9__Rationals) at (-10,1) {$\begin{smallmatrix}&&7\\5&9&14&12&10&8&5&3\end{smallmatrix}$};
	\node[noselEEEEEEEEE, label={[shift={(0.2,-0.55)}, brown]\tiny $I(24,1)$}] (I_24_1__Rationals) at (-12,6) {$\begin{smallmatrix}&&5\\3&6&9&8&6&5&3&1\end{smallmatrix}$};
	\node[noselEEEEEEEEE, label={[shift={(0.2,-0.55)}, brown]\tiny $I(24,2)$}] (I_24_2__Rationals) at (-14,3) {$\begin{smallmatrix}&&10\\7&13&19&16&13&10&6&3\end{smallmatrix}$};
	\node[noselEEEEEEEEE, label={[shift={(0.2,-0.55)}, brown]\tiny $I(24,3)$}] (I_24_3__Rationals) at (-16,0) {$\begin{smallmatrix}&&15\\10&20&29&25&20&15&10&5\end{smallmatrix}$};
	\node[noselEEEEEEEEE, label={[shift={(0.2,-0.55)}, brown]\tiny $I(24,4)$}] (I_24_4__Rationals) at (-14,-3) {$\begin{smallmatrix}&&12\\8&16&24&21&17&13&9&5\end{smallmatrix}$};
	\node[noselEEEEEEEEE, label={[shift={(0.2,-0.55)}, brown]\tiny $I(24,5)$}] (I_24_5__Rationals) at (-12,-6) {$\begin{smallmatrix}&&10\\6&13&19&16&13&10&7&4\end{smallmatrix}$};
	\node[noselEEEEEEEEE, label={[shift={(0.2,-0.55)}, brown]\tiny $I(24,6)$}] (I_24_6__Rationals) at (-10,-9) {$\begin{smallmatrix}&&7\\5&10&14&12&9&7&5&3\end{smallmatrix}$};
	\node[noselEEEEEEEEE, label={[shift={(0.2,-0.55)}, brown]\tiny $I(24,7)$}] (I_24_7__Rationals) at (-8,-12) {$\begin{smallmatrix}&&5\\3&6&9&8&6&4&3&2\end{smallmatrix}$};
	\node[seloneEEEEEEEEE, label={[shift={(0.2,-0.55)}, brown]\tiny $I(24,8)$}] (I_24_8__Rationals) at (-6,-15) {$\begin{smallmatrix}&&2\\1&3&4&4&3&2&1&1\end{smallmatrix}$};
	\node[noselEEEEEEEEE, label={[shift={(0.2,-0.55)}, brown]\tiny $I(24,9)$}] (I_24_9__Rationals) at (-14,1) {$\begin{smallmatrix}&&7\\5&10&14&12&10&7&5&2\end{smallmatrix}$};
	\node at (-17,3) {$\dots$};
	\node at (-17,-3) {$\dots$};
	\draw[red,->] (I_20_2__Rationals) -- (I_20_1__Rationals);
	\draw[red,->] (I_20_3__Rationals) -- (I_20_2__Rationals);
	\draw[blue,->] (I_21_1__Rationals) -- (I_20_2__Rationals);
	\draw[blue,->] (I_21_2__Rationals) -- (I_20_3__Rationals);
	\draw[blue,->] (I_21_4__Rationals) -- (I_20_3__Rationals);
	\draw[blue,->] (I_21_9__Rationals) -- (I_20_3__Rationals);
	\draw[red,->] (I_20_3__Rationals) -- (I_20_4__Rationals);
	\draw[blue,->] (I_21_5__Rationals) -- (I_20_4__Rationals);
	\draw[red,->] (I_20_4__Rationals) -- (I_20_5__Rationals);
	\draw[blue,->] (I_21_6__Rationals) -- (I_20_5__Rationals);
	\draw[red,->] (I_20_5__Rationals) -- (I_20_6__Rationals);
	\draw[blue,->] (I_21_7__Rationals) -- (I_20_6__Rationals);
	\draw[red,->] (I_20_6__Rationals) -- (I_20_7__Rationals);
	\draw[blue,->] (I_21_8__Rationals) -- (I_20_7__Rationals);
	\draw[red,->] (I_20_7__Rationals) -- (I_20_8__Rationals);
	\draw[red,->] (I_20_3__Rationals) -- (I_20_9__Rationals);
	\draw[gray,dashed,->] (I_20_1__Rationals) -- (I_21_1__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_21_2__Rationals) -- (I_21_1__Rationals);
	\draw[gray,dashed,->] (I_20_2__Rationals) -- (I_21_2__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_21_3__Rationals) -- (I_21_2__Rationals);
	\draw[blue,->] (I_22_1__Rationals) -- (I_21_2__Rationals);
	\draw[gray,dashed,->] (I_20_3__Rationals) -- (I_21_3__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[blue,->] (I_22_2__Rationals) -- (I_21_3__Rationals);
	\draw[blue,->] (I_22_4__Rationals) -- (I_21_3__Rationals);
	\draw[blue,->] (I_22_9__Rationals) -- (I_21_3__Rationals);
	\draw[gray,dashed,->] (I_20_4__Rationals) -- (I_21_4__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_21_3__Rationals) -- (I_21_4__Rationals);
	\draw[blue,->] (I_22_5__Rationals) -- (I_21_4__Rationals);
	\draw[gray,dashed,->] (I_20_5__Rationals) -- (I_21_5__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_21_4__Rationals) -- (I_21_5__Rationals);
	\draw[blue,->] (I_22_6__Rationals) -- (I_21_5__Rationals);
	\draw[gray,dashed,->] (I_20_6__Rationals) -- (I_21_6__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_21_5__Rationals) -- (I_21_6__Rationals);
	\draw[blue,->] (I_22_7__Rationals) -- (I_21_6__Rationals);
	\draw[gray,dashed,->] (I_20_7__Rationals) -- (I_21_7__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_21_6__Rationals) -- (I_21_7__Rationals);
	\draw[blue,->] (I_22_8__Rationals) -- (I_21_7__Rationals);
	\draw[gray,dashed,->] (I_20_8__Rationals) -- (I_21_8__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_21_7__Rationals) -- (I_21_8__Rationals);
	\draw[gray,dashed,->] (I_20_9__Rationals) -- (I_21_9__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_21_3__Rationals) -- (I_21_9__Rationals);
	\draw[gray,dashed,->] (I_21_1__Rationals) -- (I_22_1__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_22_2__Rationals) -- (I_22_1__Rationals);
	\draw[gray,dashed,->] (I_21_2__Rationals) -- (I_22_2__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_22_3__Rationals) -- (I_22_2__Rationals);
	\draw[blue,->] (I_23_1__Rationals) -- (I_22_2__Rationals);
	\draw[gray,dashed,->] (I_21_3__Rationals) -- (I_22_3__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[blue,->] (I_23_2__Rationals) -- (I_22_3__Rationals);
	\draw[blue,->] (I_23_4__Rationals) -- (I_22_3__Rationals);
	\draw[blue,->] (I_23_9__Rationals) -- (I_22_3__Rationals);
	\draw[gray,dashed,->] (I_21_4__Rationals) -- (I_22_4__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_22_3__Rationals) -- (I_22_4__Rationals);
	\draw[blue,->] (I_23_5__Rationals) -- (I_22_4__Rationals);
	\draw[gray,dashed,->] (I_21_5__Rationals) -- (I_22_5__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_22_4__Rationals) -- (I_22_5__Rationals);
	\draw[blue,->] (I_23_6__Rationals) -- (I_22_5__Rationals);
	\draw[gray,dashed,->] (I_21_6__Rationals) -- (I_22_6__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_22_5__Rationals) -- (I_22_6__Rationals);
	\draw[blue,->] (I_23_7__Rationals) -- (I_22_6__Rationals);
	\draw[gray,dashed,->] (I_21_7__Rationals) -- (I_22_7__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_22_6__Rationals) -- (I_22_7__Rationals);
	\draw[blue,->] (I_23_8__Rationals) -- (I_22_7__Rationals);
	\draw[gray,dashed,->] (I_21_8__Rationals) -- (I_22_8__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_22_7__Rationals) -- (I_22_8__Rationals);
	\draw[gray,dashed,->] (I_21_9__Rationals) -- (I_22_9__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_22_3__Rationals) -- (I_22_9__Rationals);
	\draw[gray,dashed,->] (I_22_1__Rationals) -- (I_23_1__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_23_2__Rationals) -- (I_23_1__Rationals);
	\draw[gray,dashed,->] (I_22_2__Rationals) -- (I_23_2__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_23_3__Rationals) -- (I_23_2__Rationals);
	\draw[blue,->] (I_24_1__Rationals) -- (I_23_2__Rationals);
	\draw[gray,dashed,->] (I_22_3__Rationals) -- (I_23_3__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[blue,->] (I_24_2__Rationals) -- (I_23_3__Rationals);
	\draw[blue,->] (I_24_4__Rationals) -- (I_23_3__Rationals);
	\draw[blue,->] (I_24_9__Rationals) -- (I_23_3__Rationals);
	\draw[gray,dashed,->] (I_22_4__Rationals) -- (I_23_4__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_23_3__Rationals) -- (I_23_4__Rationals);
	\draw[blue,->] (I_24_5__Rationals) -- (I_23_4__Rationals);
	\draw[gray,dashed,->] (I_22_5__Rationals) -- (I_23_5__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_23_4__Rationals) -- (I_23_5__Rationals);
	\draw[blue,->] (I_24_6__Rationals) -- (I_23_5__Rationals);
	\draw[gray,dashed,->] (I_22_6__Rationals) -- (I_23_6__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_23_5__Rationals) -- (I_23_6__Rationals);
	\draw[blue,->] (I_24_7__Rationals) -- (I_23_6__Rationals);
	\draw[gray,dashed,->] (I_22_7__Rationals) -- (I_23_7__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_23_6__Rationals) -- (I_23_7__Rationals);
	\draw[blue,->] (I_24_8__Rationals) -- (I_23_7__Rationals);
	\draw[gray,dashed,->] (I_22_8__Rationals) -- (I_23_8__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_23_7__Rationals) -- (I_23_8__Rationals);
	\draw[gray,dashed,->] (I_22_9__Rationals) -- (I_23_9__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_23_3__Rationals) -- (I_23_9__Rationals);
	\draw[gray,dashed,->] (I_23_1__Rationals) -- (I_24_1__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_24_2__Rationals) -- (I_24_1__Rationals);
	\draw[gray,dashed,->] (I_23_2__Rationals) -- (I_24_2__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_24_3__Rationals) -- (I_24_2__Rationals);
	\draw[gray,dashed,->] (I_23_3__Rationals) -- (I_24_3__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[gray,dashed,->] (I_23_4__Rationals) -- (I_24_4__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_24_3__Rationals) -- (I_24_4__Rationals);
	\draw[gray,dashed,->] (I_23_5__Rationals) -- (I_24_5__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_24_4__Rationals) -- (I_24_5__Rationals);
	\draw[gray,dashed,->] (I_23_6__Rationals) -- (I_24_6__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_24_5__Rationals) -- (I_24_6__Rationals);
	\draw[gray,dashed,->] (I_23_7__Rationals) -- (I_24_7__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_24_6__Rationals) -- (I_24_7__Rationals);
	\draw[gray,dashed,->] (I_23_8__Rationals) -- (I_24_8__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_24_7__Rationals) -- (I_24_8__Rationals);
	\draw[gray,dashed,->] (I_23_9__Rationals) -- (I_24_9__Rationals) node[midway,sloped,above]{{\tiny $\tau$}};
	\draw[red,->] (I_24_3__Rationals) -- (I_24_9__Rationals);
\end{tikzpicture}
\end{scaletikzpicturetowidth}
\end{center}

\end{figure}
\medskip{}
\subsubsection*{Modules of the form $I(n,1)$}

Defect: $\partial I(n,1) = 2$, for $n\ge 0$.

\begin{fleqn}
\begin{align*}
I(0,1):\ & - \\
I(1,1):\ & - \\
I(2,1):\ & \bigl(I(0,4),P(0,9)\bigr),\ \bigl(I(0,5),P(1,1)\bigr),\ \bigl(I(0,6),P(3,8)\bigr),\ \bigl(I(0,7),R_{1}^{1}(1)\bigr),\ \bigl(I(0,8),I(7,8)\bigr)\\
 & \bigl(I(0,9),P(0,8)\bigr)\\
I(3,1):\ & \bigl(I(0,1),R_{1}^{2}(1)\bigr),\ \bigl(I(0,2),P(0,7)\bigr),\ \bigl(I(1,5),P(0,1)\bigr),\ \bigl(I(1,6),P(2,8)\bigr),\ \bigl(I(1,7),R_{1}^{5}(1)\bigr)\\
 & \bigl(I(1,8),I(8,8)\bigr)\\
I(4,1):\ & \bigl(I(0,9),P(4,8)\bigr),\ \bigl(I(1,1),R_{1}^{1}(1)\bigr),\ \bigl(I(2,6),P(1,8)\bigr),\ \bigl(I(2,7),R_{1}^{4}(1)\bigr),\ \bigl(I(2,8),I(9,8)\bigr)\\
I(5,1):\ & \bigl(I(0,1),R_{0}^{1}(1)\bigr),\ \bigl(I(0,7),R_{1}^{5}(2)\bigr),\ \bigl(I(0,8),I(13,8)\bigr),\ \bigl(I(1,9),P(3,8)\bigr),\ \bigl(I(2,1),R_{1}^{5}(1)\bigr)\\
 & \bigl(I(3,6),P(0,8)\bigr),\ \bigl(I(3,7),R_{1}^{3}(1)\bigr),\ \bigl(I(3,8),I(10,8)\bigr)\\
I(6,1):\ & \bigl(I(1,1),R_{0}^{3}(1)\bigr),\ \bigl(I(1,7),R_{1}^{4}(2)\bigr),\ \bigl(I(1,8),I(14,8)\bigr),\ \bigl(I(2,9),P(2,8)\bigr),\ \bigl(I(3,1),R_{1}^{4}(1)\bigr)\\
 & \bigl(I(4,7),R_{1}^{2}(1)\bigr),\ \bigl(I(4,8),I(11,8)\bigr)\\
I(7,1):\ & \bigl(I(0,8),I(17,8)\bigr),\ \bigl(I(2,1),R_{0}^{2}(1)\bigr),\ \bigl(I(2,7),R_{1}^{3}(2)\bigr),\ \bigl(I(2,8),I(15,8)\bigr),\ \bigl(I(3,9),P(1,8)\bigr)\\
 & \bigl(I(4,1),R_{1}^{3}(1)\bigr),\ \bigl(I(5,7),R_{1}^{1}(1)\bigr),\ \bigl(I(5,8),I(12,8)\bigr)\\
I(8,1):\ & \bigl(I(1,8),I(18,8)\bigr),\ \bigl(I(3,1),R_{0}^{1}(1)\bigr),\ \bigl(I(3,7),R_{1}^{2}(2)\bigr),\ \bigl(I(3,8),I(16,8)\bigr),\ \bigl(I(4,9),P(0,8)\bigr)\\
 & \bigl(I(5,1),R_{1}^{2}(1)\bigr),\ \bigl(I(6,7),R_{1}^{5}(1)\bigr),\ \bigl(I(6,8),I(13,8)\bigr)\\
I(9,1):\ & \bigl(I(2,8),I(19,8)\bigr),\ \bigl(I(4,1),R_{0}^{3}(1)\bigr),\ \bigl(I(4,7),R_{1}^{1}(2)\bigr),\ \bigl(I(4,8),I(17,8)\bigr),\ \bigl(I(6,1),R_{1}^{1}(1)\bigr)\\
 & \bigl(I(7,7),R_{1}^{4}(1)\bigr),\ \bigl(I(7,8),I(14,8)\bigr)\\
I(10,1):\ & \bigl(I(0,8),I(23,8)\bigr),\ \bigl(I(3,8),I(20,8)\bigr),\ \bigl(I(5,1),R_{0}^{2}(1)\bigr),\ \bigl(I(5,7),R_{1}^{5}(2)\bigr),\ \bigl(I(5,8),I(18,8)\bigr)\\
 & \bigl(I(7,1),R_{1}^{5}(1)\bigr),\ \bigl(I(8,7),R_{1}^{3}(1)\bigr),\ \bigl(I(8,8),I(15,8)\bigr)\\
I(n,1):\ & \bigl(I(n-10,8),I(n+13,8)\bigr),\ \bigl(I(n-7,8),I(n+10,8)\bigr),\ \bigl(I(n-5,1),R_{0}^{(-n+11)\bmod 3+1}(1)\bigr)\\
 & \bigl(I(n-5,7),R_{1}^{(-n+14)\bmod 5+1}(2)\bigr),\ \bigl(I(n-5,8),I(n+8,8)\bigr),\ \bigl(I(n-3,1),R_{1}^{(-n+14)\bmod 5+1}(1)\bigr)\\
 & \bigl(I(n-2,7),R_{1}^{(-n+12)\bmod 5+1}(1)\bigr),\ \bigl(I(n-2,8),I(n+5,8)\bigr),\ n>10\\
\end{align*}
\end{fleqn}
\subsubsection*{Modules of the form $I(n,2)$}

Defect: $\partial I(n,2) = 4$, for $n\ge 0$.

\begin{fleqn}
\begin{align*}
I(0,2):\ & \bigl(I(0,1),I(1,1)\bigr)\\
I(1,2):\ & \bigl(I(0,4),P(1,8)\bigr),\ \bigl(I(0,5),R_{1}^{4}(1)\bigr),\ \bigl(I(0,6),I(9,8)\bigr),\ \bigl(I(0,7),I(4,1)\bigr),\ \bigl(I(0,8),I(2,9)\bigr)\\
 & \bigl(I(0,9),I(6,8)\bigr),\ \bigl(I(1,1),I(2,1)\bigr)\\
I(2,2):\ & \bigl(I(0,1),I(6,7)\bigr),\ \bigl(I(1,4),P(0,8)\bigr),\ \bigl(I(1,5),R_{1}^{3}(1)\bigr),\ \bigl(I(1,6),I(10,8)\bigr),\ \bigl(I(1,7),I(5,1)\bigr)\\
 & \bigl(I(1,8),I(3,9)\bigr),\ \bigl(I(1,9),I(7,8)\bigr),\ \bigl(I(2,1),I(3,1)\bigr)\\
I(3,2):\ & \bigl(I(1,1),I(7,7)\bigr),\ \bigl(I(2,5),R_{1}^{2}(1)\bigr),\ \bigl(I(2,6),I(11,8)\bigr),\ \bigl(I(2,7),I(6,1)\bigr),\ \bigl(I(2,8),I(4,9)\bigr)\\
 & \bigl(I(2,9),I(8,8)\bigr),\ \bigl(I(3,1),I(4,1)\bigr)\\
I(4,2):\ & \bigl(I(0,8),I(7,6)\bigr),\ \bigl(I(2,1),I(8,7)\bigr),\ \bigl(I(3,5),R_{1}^{1}(1)\bigr),\ \bigl(I(3,6),I(12,8)\bigr),\ \bigl(I(3,7),I(7,1)\bigr)\\
 & \bigl(I(3,8),I(5,9)\bigr),\ \bigl(I(3,9),I(9,8)\bigr),\ \bigl(I(4,1),I(5,1)\bigr)\\
I(n,2):\ & \bigl(I(n-4,8),I(n+3,6)\bigr),\ \bigl(I(n-2,1),I(n+4,7)\bigr),\ \bigl(I(n-1,5),R_{1}^{(-n+4)\bmod 5+1}(1)\bigr)\\
 & \bigl(I(n-1,6),I(n+8,8)\bigr),\ \bigl(I(n-1,7),I(n+3,1)\bigr),\ \bigl(I(n-1,8),I(n+1,9)\bigr)\\
 & \bigl(I(n-1,9),I(n+5,8)\bigr),\ \bigl(I(n,1),I(n+1,1)\bigr),\ n>4\\
\end{align*}
\end{fleqn}
\subsubsection*{Modules of the form $I(n,3)$}

Defect: $\partial I(n,3) = 6$, for $n\ge 0$.

\begin{fleqn}
\begin{align*}
I(0,3):\ & \bigl(I(0,1),I(1,2)\bigr),\ \bigl(I(0,2),I(2,1)\bigr),\ \bigl(I(0,4),I(5,8)\bigr),\ \bigl(I(0,5),I(4,7)\bigr),\ \bigl(I(0,6),I(3,6)\bigr)\\
 & \bigl(I(0,7),I(2,5)\bigr),\ \bigl(I(0,8),I(1,4)\bigr),\ \bigl(I(0,9),I(1,9)\bigr)\\
I(n,3):\ & \bigl(I(n,1),I(n+1,2)\bigr),\ \bigl(I(n,2),I(n+2,1)\bigr),\ \bigl(I(n,4),I(n+5,8)\bigr)\\
 & \bigl(I(n,5),I(n+4,7)\bigr),\ \bigl(I(n,6),I(n+3,6)\bigr),\ \bigl(I(n,7),I(n+2,5)\bigr)\\
 & \bigl(I(n,8),I(n+1,4)\bigr),\ \bigl(I(n,9),I(n+1,9)\bigr),\ n>0\\
\end{align*}
\end{fleqn}
\subsubsection*{Modules of the form $I(n,4)$}

Defect: $\partial I(n,4) = 5$, for $n\ge 0$.

\begin{fleqn}
\begin{align*}
I(0,4):\ & \bigl(I(0,5),I(4,8)\bigr),\ \bigl(I(0,6),I(3,7)\bigr),\ \bigl(I(0,7),I(2,6)\bigr),\ \bigl(I(0,8),I(1,5)\bigr)\\
I(1,4):\ & \bigl(I(0,1),I(2,9)\bigr),\ \bigl(I(0,2),I(7,8)\bigr),\ \bigl(I(0,9),I(3,1)\bigr),\ \bigl(I(1,5),I(5,8)\bigr),\ \bigl(I(1,6),I(4,7)\bigr)\\
 & \bigl(I(1,7),I(3,6)\bigr),\ \bigl(I(1,8),I(2,5)\bigr)\\
I(2,4):\ & \bigl(I(0,8),I(3,2)\bigr),\ \bigl(I(1,1),I(3,9)\bigr),\ \bigl(I(1,2),I(8,8)\bigr),\ \bigl(I(1,9),I(4,1)\bigr),\ \bigl(I(2,5),I(6,8)\bigr)\\
 & \bigl(I(2,6),I(5,7)\bigr),\ \bigl(I(2,7),I(4,6)\bigr),\ \bigl(I(2,8),I(3,5)\bigr)\\
I(n,4):\ & \bigl(I(n-2,8),I(n+1,2)\bigr),\ \bigl(I(n-1,1),I(n+1,9)\bigr),\ \bigl(I(n-1,2),I(n+6,8)\bigr)\\
 & \bigl(I(n-1,9),I(n+2,1)\bigr),\ \bigl(I(n,5),I(n+4,8)\bigr),\ \bigl(I(n,6),I(n+3,7)\bigr)\\
 & \bigl(I(n,7),I(n+2,6)\bigr),\ \bigl(I(n,8),I(n+1,5)\bigr),\ n>2\\
\end{align*}
\end{fleqn}
\subsubsection*{Modules of the form $I(n,5)$}

Defect: $\partial I(n,5) = 4$, for $n\ge 0$.

\begin{fleqn}
\begin{align*}
I(0,5):\ & \bigl(I(0,6),I(3,8)\bigr),\ \bigl(I(0,7),I(2,7)\bigr),\ \bigl(I(0,8),I(1,6)\bigr)\\
I(1,5):\ & \bigl(I(1,6),I(4,8)\bigr),\ \bigl(I(1,7),I(3,7)\bigr),\ \bigl(I(1,8),I(2,6)\bigr)\\
I(2,5):\ & \bigl(I(0,1),I(4,1)\bigr),\ \bigl(I(0,2),R_{1}^{1}(1)\bigr),\ \bigl(I(0,9),I(8,8)\bigr),\ \bigl(I(2,6),I(5,8)\bigr),\ \bigl(I(2,7),I(4,7)\bigr)\\
 & \bigl(I(2,8),I(3,6)\bigr)\\
I(3,5):\ & \bigl(I(0,7),I(8,7)\bigr),\ \bigl(I(0,8),I(4,9)\bigr),\ \bigl(I(1,1),I(5,1)\bigr),\ \bigl(I(1,2),R_{1}^{5}(1)\bigr),\ \bigl(I(1,9),I(9,8)\bigr)\\
 & \bigl(I(3,6),I(6,8)\bigr),\ \bigl(I(3,7),I(5,7)\bigr),\ \bigl(I(3,8),I(4,6)\bigr)\\
I(n,5):\ & \bigl(I(n-3,7),I(n+5,7)\bigr),\ \bigl(I(n-3,8),I(n+1,9)\bigr),\ \bigl(I(n-2,1),I(n+2,1)\bigr)\\
 & \bigl(I(n-2,2),R_{1}^{(-n+7)\bmod 5+1}(1)\bigr),\ \bigl(I(n-2,9),I(n+6,8)\bigr),\ \bigl(I(n,6),I(n+3,8)\bigr)\\
 & \bigl(I(n,7),I(n+2,7)\bigr),\ \bigl(I(n,8),I(n+1,6)\bigr),\ n>3\\
\end{align*}
\end{fleqn}
\subsubsection*{Modules of the form $I(n,6)$}

Defect: $\partial I(n,6) = 3$, for $n\ge 0$.

\begin{fleqn}
\begin{align*}
I(0,6):\ & \bigl(I(0,7),I(2,8)\bigr),\ \bigl(I(0,8),I(1,7)\bigr)\\
I(1,6):\ & \bigl(I(1,7),I(3,8)\bigr),\ \bigl(I(1,8),I(2,7)\bigr)\\
I(2,6):\ & \bigl(I(2,7),I(4,8)\bigr),\ \bigl(I(2,8),I(3,7)\bigr)\\
I(3,6):\ & \bigl(I(0,1),I(9,8)\bigr),\ \bigl(I(0,2),P(3,8)\bigr),\ \bigl(I(0,9),R_{1}^{5}(1)\bigr),\ \bigl(I(3,7),I(5,8)\bigr),\ \bigl(I(3,8),I(4,7)\bigr)\\
I(4,6):\ & \bigl(I(0,6),R_{1}^{4}(2)\bigr),\ \bigl(I(0,7),I(14,8)\bigr),\ \bigl(I(0,8),I(6,1)\bigr),\ \bigl(I(1,1),I(10,8)\bigr),\ \bigl(I(1,2),P(2,8)\bigr)\\
 & \bigl(I(1,9),R_{1}^{4}(1)\bigr),\ \bigl(I(4,7),I(6,8)\bigr),\ \bigl(I(4,8),I(5,7)\bigr)\\
I(5,6):\ & \bigl(I(1,6),R_{1}^{3}(2)\bigr),\ \bigl(I(1,7),I(15,8)\bigr),\ \bigl(I(1,8),I(7,1)\bigr),\ \bigl(I(2,1),I(11,8)\bigr),\ \bigl(I(2,2),P(1,8)\bigr)\\
 & \bigl(I(2,9),R_{1}^{3}(1)\bigr),\ \bigl(I(5,7),I(7,8)\bigr),\ \bigl(I(5,8),I(6,7)\bigr)\\
I(6,6):\ & \bigl(I(2,6),R_{1}^{2}(2)\bigr),\ \bigl(I(2,7),I(16,8)\bigr),\ \bigl(I(2,8),I(8,1)\bigr),\ \bigl(I(3,1),I(12,8)\bigr),\ \bigl(I(3,2),P(0,8)\bigr)\\
 & \bigl(I(3,9),R_{1}^{2}(1)\bigr),\ \bigl(I(6,7),I(8,8)\bigr),\ \bigl(I(6,8),I(7,7)\bigr)\\
I(7,6):\ & \bigl(I(3,6),R_{1}^{1}(2)\bigr),\ \bigl(I(3,7),I(17,8)\bigr),\ \bigl(I(3,8),I(9,1)\bigr),\ \bigl(I(4,1),I(13,8)\bigr),\ \bigl(I(4,9),R_{1}^{1}(1)\bigr)\\
 & \bigl(I(7,7),I(9,8)\bigr),\ \bigl(I(7,8),I(8,7)\bigr)\\
I(8,6):\ & \bigl(I(0,8),I(13,7)\bigr),\ \bigl(I(4,6),R_{1}^{5}(2)\bigr),\ \bigl(I(4,7),I(18,8)\bigr),\ \bigl(I(4,8),I(10,1)\bigr),\ \bigl(I(5,1),I(14,8)\bigr)\\
 & \bigl(I(5,9),R_{1}^{5}(1)\bigr),\ \bigl(I(8,7),I(10,8)\bigr),\ \bigl(I(8,8),I(9,7)\bigr)\\
I(n,6):\ & \bigl(I(n-8,8),I(n+5,7)\bigr),\ \bigl(I(n-4,6),R_{1}^{(-n+12)\bmod 5+1}(2)\bigr),\ \bigl(I(n-4,7),I(n+10,8)\bigr)\\
 & \bigl(I(n-4,8),I(n+2,1)\bigr),\ \bigl(I(n-3,1),I(n+6,8)\bigr),\ \bigl(I(n-3,9),R_{1}^{(-n+12)\bmod 5+1}(1)\bigr)\\
 & \bigl(I(n,7),I(n+2,8)\bigr),\ \bigl(I(n,8),I(n+1,7)\bigr),\ n>8\\
\end{align*}
\end{fleqn}
\subsubsection*{Modules of the form $I(n,7)$}

Defect: $\partial I(n,7) = 2$, for $n\ge 0$.

\begin{fleqn}
\begin{align*}
I(0,7):\ & \bigl(I(0,8),I(1,8)\bigr)\\
I(1,7):\ & \bigl(I(1,8),I(2,8)\bigr)\\
I(2,7):\ & \bigl(I(2,8),I(3,8)\bigr)\\
I(3,7):\ & \bigl(I(3,8),I(4,8)\bigr)\\
I(4,7):\ & \bigl(I(0,1),R_{1}^{4}(1)\bigr),\ \bigl(I(0,2),P(1,1)\bigr),\ \bigl(I(0,9),P(2,8)\bigr),\ \bigl(I(4,8),I(5,8)\bigr)\\
I(5,7):\ & \bigl(I(0,5),P(2,7)\bigr),\ \bigl(I(0,6),P(7,8)\bigr),\ \bigl(I(0,7),R_{0}^{2}(1)\bigr),\ \bigl(I(0,8),I(11,8)\bigr),\ \bigl(I(1,1),R_{1}^{3}(1)\bigr)\\
 & \bigl(I(1,2),P(0,1)\bigr),\ \bigl(I(1,9),P(1,8)\bigr),\ \bigl(I(5,8),I(6,8)\bigr)\\
I(6,7):\ & \bigl(I(1,5),P(1,7)\bigr),\ \bigl(I(1,6),P(6,8)\bigr),\ \bigl(I(1,7),R_{0}^{1}(1)\bigr),\ \bigl(I(1,8),I(12,8)\bigr),\ \bigl(I(2,1),R_{1}^{2}(1)\bigr)\\
 & \bigl(I(2,9),P(0,8)\bigr),\ \bigl(I(6,8),I(7,8)\bigr)\\
I(7,7):\ & \bigl(I(0,1),R_{1}^{1}(2)\bigr),\ \bigl(I(2,5),P(0,7)\bigr),\ \bigl(I(2,6),P(5,8)\bigr),\ \bigl(I(2,7),R_{0}^{3}(1)\bigr),\ \bigl(I(2,8),I(13,8)\bigr)\\
 & \bigl(I(3,1),R_{1}^{1}(1)\bigr),\ \bigl(I(7,8),I(8,8)\bigr)\\
I(8,7):\ & \bigl(I(1,1),R_{1}^{5}(2)\bigr),\ \bigl(I(3,6),P(4,8)\bigr),\ \bigl(I(3,7),R_{0}^{2}(1)\bigr),\ \bigl(I(3,8),I(14,8)\bigr),\ \bigl(I(4,1),R_{1}^{5}(1)\bigr)\\
 & \bigl(I(8,8),I(9,8)\bigr)\\
I(9,7):\ & \bigl(I(0,7),R_{1}^{4}(3)\bigr),\ \bigl(I(0,8),I(19,8)\bigr),\ \bigl(I(2,1),R_{1}^{4}(2)\bigr),\ \bigl(I(4,6),P(3,8)\bigr),\ \bigl(I(4,7),R_{0}^{1}(1)\bigr)\\
 & \bigl(I(4,8),I(15,8)\bigr),\ \bigl(I(5,1),R_{1}^{4}(1)\bigr),\ \bigl(I(9,8),I(10,8)\bigr)\\
I(10,7):\ & \bigl(I(1,7),R_{1}^{3}(3)\bigr),\ \bigl(I(1,8),I(20,8)\bigr),\ \bigl(I(3,1),R_{1}^{3}(2)\bigr),\ \bigl(I(5,6),P(2,8)\bigr),\ \bigl(I(5,7),R_{0}^{3}(1)\bigr)\\
 & \bigl(I(5,8),I(16,8)\bigr),\ \bigl(I(6,1),R_{1}^{3}(1)\bigr),\ \bigl(I(10,8),I(11,8)\bigr)\\
I(11,7):\ & \bigl(I(2,7),R_{1}^{2}(3)\bigr),\ \bigl(I(2,8),I(21,8)\bigr),\ \bigl(I(4,1),R_{1}^{2}(2)\bigr),\ \bigl(I(6,6),P(1,8)\bigr),\ \bigl(I(6,7),R_{0}^{2}(1)\bigr)\\
 & \bigl(I(6,8),I(17,8)\bigr),\ \bigl(I(7,1),R_{1}^{2}(1)\bigr),\ \bigl(I(11,8),I(12,8)\bigr)\\
I(12,7):\ & \bigl(I(3,7),R_{1}^{1}(3)\bigr),\ \bigl(I(3,8),I(22,8)\bigr),\ \bigl(I(5,1),R_{1}^{1}(2)\bigr),\ \bigl(I(7,6),P(0,8)\bigr),\ \bigl(I(7,7),R_{0}^{1}(1)\bigr)\\
 & \bigl(I(7,8),I(18,8)\bigr),\ \bigl(I(8,1),R_{1}^{1}(1)\bigr),\ \bigl(I(12,8),I(13,8)\bigr)\\
I(13,7):\ & \bigl(I(4,7),R_{1}^{5}(3)\bigr),\ \bigl(I(4,8),I(23,8)\bigr),\ \bigl(I(6,1),R_{1}^{5}(2)\bigr),\ \bigl(I(8,7),R_{0}^{3}(1)\bigr),\ \bigl(I(8,8),I(19,8)\bigr)\\
 & \bigl(I(9,1),R_{1}^{5}(1)\bigr),\ \bigl(I(13,8),I(14,8)\bigr)\\
I(14,7):\ & \bigl(I(0,8),I(29,8)\bigr),\ \bigl(I(5,7),R_{1}^{4}(3)\bigr),\ \bigl(I(5,8),I(24,8)\bigr),\ \bigl(I(7,1),R_{1}^{4}(2)\bigr),\ \bigl(I(9,7),R_{0}^{2}(1)\bigr)\\
 & \bigl(I(9,8),I(20,8)\bigr),\ \bigl(I(10,1),R_{1}^{4}(1)\bigr),\ \bigl(I(14,8),I(15,8)\bigr)\\
I(n,7):\ & \bigl(I(n-14,8),I(n+15,8)\bigr),\ \bigl(I(n-9,7),R_{1}^{(-n+17)\bmod 5+1}(3)\bigr),\ \bigl(I(n-9,8),I(n+10,8)\bigr)\\
 & \bigl(I(n-7,1),R_{1}^{(-n+17)\bmod 5+1}(2)\bigr),\ \bigl(I(n-5,7),R_{0}^{(-n+15)\bmod 3+1}(1)\bigr),\ \bigl(I(n-5,8),I(n+6,8)\bigr)\\
 & \bigl(I(n-4,1),R_{1}^{(-n+17)\bmod 5+1}(1)\bigr),\ \bigl(I(n,8),I(n+1,8)\bigr),\ n>14\\
\end{align*}
\end{fleqn}
\subsubsection*{Modules of the form $I(n,8)$}

Defect: $\partial I(n,8) = 1$, for $n\ge 0$.

\begin{fleqn}
\begin{align*}
I(0,8):\ & - \\
I(1,8):\ & - \\
I(2,8):\ & - \\
I(3,8):\ & - \\
I(4,8):\ & - \\
I(5,8):\ & \bigl(I(0,1),P(1,8)\bigr),\ \bigl(I(0,2),P(0,9)\bigr),\ \bigl(I(0,9),P(0,1)\bigr)\\
I(6,8):\ & \bigl(I(0,4),P(0,2)\bigr),\ \bigl(I(0,5),P(1,9)\bigr),\ \bigl(I(0,6),P(2,1)\bigr),\ \bigl(I(0,7),P(4,8)\bigr),\ \bigl(I(0,8),R_{1}^{2}(1)\bigr)\\
 & \bigl(I(1,1),P(0,8)\bigr)\\
I(7,8):\ & \bigl(I(0,9),P(0,7)\bigr),\ \bigl(I(1,5),P(0,9)\bigr),\ \bigl(I(1,6),P(1,1)\bigr),\ \bigl(I(1,7),P(3,8)\bigr),\ \bigl(I(1,8),R_{1}^{1}(1)\bigr)\\
I(8,8):\ & \bigl(I(0,1),P(4,8)\bigr),\ \bigl(I(0,2),P(0,6)\bigr),\ \bigl(I(2,6),P(0,1)\bigr),\ \bigl(I(2,7),P(2,8)\bigr),\ \bigl(I(2,8),R_{1}^{5}(1)\bigr)\\
I(9,8):\ & \bigl(I(0,9),P(2,1)\bigr),\ \bigl(I(1,1),P(3,8)\bigr),\ \bigl(I(3,7),P(1,8)\bigr),\ \bigl(I(3,8),R_{1}^{4}(1)\bigr)\\
I(10,8):\ & \bigl(I(0,1),P(6,8)\bigr),\ \bigl(I(0,6),P(3,7)\bigr),\ \bigl(I(0,7),P(8,8)\bigr),\ \bigl(I(0,8),R_{0}^{3}(1)\bigr),\ \bigl(I(1,9),P(1,1)\bigr)\\
 & \bigl(I(2,1),P(2,8)\bigr),\ \bigl(I(4,7),P(0,8)\bigr),\ \bigl(I(4,8),R_{1}^{3}(1)\bigr)\\
I(11,8):\ & \bigl(I(1,1),P(5,8)\bigr),\ \bigl(I(1,6),P(2,7)\bigr),\ \bigl(I(1,7),P(7,8)\bigr),\ \bigl(I(1,8),R_{0}^{2}(1)\bigr),\ \bigl(I(2,9),P(0,1)\bigr)\\
 & \bigl(I(3,1),P(1,8)\bigr),\ \bigl(I(5,8),R_{1}^{2}(1)\bigr)\\
I(12,8):\ & \bigl(I(0,8),R_{1}^{1}(2)\bigr),\ \bigl(I(2,1),P(4,8)\bigr),\ \bigl(I(2,6),P(1,7)\bigr),\ \bigl(I(2,7),P(6,8)\bigr),\ \bigl(I(2,8),R_{0}^{1}(1)\bigr)\\
 & \bigl(I(4,1),P(0,8)\bigr),\ \bigl(I(6,8),R_{1}^{1}(1)\bigr)\\
I(13,8):\ & \bigl(I(0,1),P(9,8)\bigr),\ \bigl(I(1,8),R_{1}^{5}(2)\bigr),\ \bigl(I(3,1),P(3,8)\bigr),\ \bigl(I(3,6),P(0,7)\bigr),\ \bigl(I(3,7),P(5,8)\bigr)\\
 & \bigl(I(3,8),R_{0}^{3}(1)\bigr),\ \bigl(I(7,8),R_{1}^{5}(1)\bigr)\\
I(14,8):\ & \bigl(I(1,1),P(8,8)\bigr),\ \bigl(I(2,8),R_{1}^{4}(2)\bigr),\ \bigl(I(4,1),P(2,8)\bigr),\ \bigl(I(4,7),P(4,8)\bigr),\ \bigl(I(4,8),R_{0}^{2}(1)\bigr)\\
 & \bigl(I(8,8),R_{1}^{4}(1)\bigr)\\
I(15,8):\ & \bigl(I(0,7),P(13,8)\bigr),\ \bigl(I(0,8),R_{\infty}^{1}(1)\bigr),\ \bigl(I(2,1),P(7,8)\bigr),\ \bigl(I(3,8),R_{1}^{3}(2)\bigr),\ \bigl(I(5,1),P(1,8)\bigr)\\
 & \bigl(I(5,7),P(3,8)\bigr),\ \bigl(I(5,8),R_{0}^{1}(1)\bigr),\ \bigl(I(9,8),R_{1}^{3}(1)\bigr)\\
I(16,8):\ & \bigl(I(1,7),P(12,8)\bigr),\ \bigl(I(1,8),R_{\infty}^{2}(1)\bigr),\ \bigl(I(3,1),P(6,8)\bigr),\ \bigl(I(4,8),R_{1}^{2}(2)\bigr),\ \bigl(I(6,1),P(0,8)\bigr)\\
 & \bigl(I(6,7),P(2,8)\bigr),\ \bigl(I(6,8),R_{0}^{3}(1)\bigr),\ \bigl(I(10,8),R_{1}^{2}(1)\bigr)\\
I(17,8):\ & \bigl(I(2,7),P(11,8)\bigr),\ \bigl(I(2,8),R_{\infty}^{1}(1)\bigr),\ \bigl(I(4,1),P(5,8)\bigr),\ \bigl(I(5,8),R_{1}^{1}(2)\bigr),\ \bigl(I(7,7),P(1,8)\bigr)\\
 & \bigl(I(7,8),R_{0}^{2}(1)\bigr),\ \bigl(I(11,8),R_{1}^{1}(1)\bigr)\\
I(18,8):\ & \bigl(I(0,8),R_{1}^{5}(3)\bigr),\ \bigl(I(3,7),P(10,8)\bigr),\ \bigl(I(3,8),R_{\infty}^{2}(1)\bigr),\ \bigl(I(5,1),P(4,8)\bigr),\ \bigl(I(6,8),R_{1}^{5}(2)\bigr)\\
 & \bigl(I(8,7),P(0,8)\bigr),\ \bigl(I(8,8),R_{0}^{1}(1)\bigr),\ \bigl(I(12,8),R_{1}^{5}(1)\bigr)\\
I(19,8):\ & \bigl(I(1,8),R_{1}^{4}(3)\bigr),\ \bigl(I(4,7),P(9,8)\bigr),\ \bigl(I(4,8),R_{\infty}^{1}(1)\bigr),\ \bigl(I(6,1),P(3,8)\bigr),\ \bigl(I(7,8),R_{1}^{4}(2)\bigr)\\
 & \bigl(I(9,8),R_{0}^{3}(1)\bigr),\ \bigl(I(13,8),R_{1}^{4}(1)\bigr)\\
I(20,8):\ & \bigl(I(0,8),R_{0}^{2}(2)\bigr),\ \bigl(I(2,8),R_{1}^{3}(3)\bigr),\ \bigl(I(5,7),P(8,8)\bigr),\ \bigl(I(5,8),R_{\infty}^{2}(1)\bigr),\ \bigl(I(7,1),P(2,8)\bigr)\\
 & \bigl(I(8,8),R_{1}^{3}(2)\bigr),\ \bigl(I(10,8),R_{0}^{2}(1)\bigr),\ \bigl(I(14,8),R_{1}^{3}(1)\bigr)\\
I(21,8):\ & \bigl(I(1,8),R_{0}^{1}(2)\bigr),\ \bigl(I(3,8),R_{1}^{2}(3)\bigr),\ \bigl(I(6,7),P(7,8)\bigr),\ \bigl(I(6,8),R_{\infty}^{1}(1)\bigr),\ \bigl(I(8,1),P(1,8)\bigr)\\
 & \bigl(I(9,8),R_{1}^{2}(2)\bigr),\ \bigl(I(11,8),R_{0}^{1}(1)\bigr),\ \bigl(I(15,8),R_{1}^{2}(1)\bigr)\\
I(22,8):\ & \bigl(I(2,8),R_{0}^{3}(2)\bigr),\ \bigl(I(4,8),R_{1}^{1}(3)\bigr),\ \bigl(I(7,7),P(6,8)\bigr),\ \bigl(I(7,8),R_{\infty}^{2}(1)\bigr),\ \bigl(I(9,1),P(0,8)\bigr)\\
 & \bigl(I(10,8),R_{1}^{1}(2)\bigr),\ \bigl(I(12,8),R_{0}^{3}(1)\bigr),\ \bigl(I(16,8),R_{1}^{1}(1)\bigr)\\
I(23,8):\ & \bigl(I(3,8),R_{0}^{2}(2)\bigr),\ \bigl(I(5,8),R_{1}^{5}(3)\bigr),\ \bigl(I(8,7),P(5,8)\bigr),\ \bigl(I(8,8),R_{\infty}^{1}(1)\bigr),\ \bigl(I(11,8),R_{1}^{5}(2)\bigr)\\
 & \bigl(I(13,8),R_{0}^{2}(1)\bigr),\ \bigl(I(17,8),R_{1}^{5}(1)\bigr)\\
I(24,8):\ & \bigl(I(0,8),R_{1}^{4}(4)\bigr),\ \bigl(I(4,8),R_{0}^{1}(2)\bigr),\ \bigl(I(6,8),R_{1}^{4}(3)\bigr),\ \bigl(I(9,7),P(4,8)\bigr),\ \bigl(I(9,8),R_{\infty}^{2}(1)\bigr)\\
 & \bigl(I(12,8),R_{1}^{4}(2)\bigr),\ \bigl(I(14,8),R_{0}^{1}(1)\bigr),\ \bigl(I(18,8),R_{1}^{4}(1)\bigr)\\
I(n,8):\ & \bigl(I(n-24,8),R_{1}^{(-n+27)\bmod 5+1}(4)\bigr),\ \bigl(I(n-20,8),R_{0}^{(-n+24)\bmod 3+1}(2)\bigr),\ \bigl(I(n-18,8),R_{1}^{(-n+27)\bmod 5+1}(3)\bigr)\\
 & \bigl(I(n-15,7),P(-n+28,8)\bigr),\ \bigl(I(n-15,8),R_{\infty}^{(-n+25)\bmod 2+1}(1)\bigr),\ \bigl(I(n-12,8),R_{1}^{(-n+27)\bmod 5+1}(2)\bigr)\\
 & \bigl(I(n-10,8),R_{0}^{(-n+24)\bmod 3+1}(1)\bigr),\ \bigl(I(n-6,8),R_{1}^{(-n+27)\bmod 5+1}(1)\bigr),\ n>24\\
\end{align*}
\end{fleqn}
\subsubsection*{Modules of the form $I(n,9)$}

Defect: $\partial I(n,9) = 3$, for $n\ge 0$.

\begin{fleqn}
\begin{align*}
I(0,9):\ & - \\
I(1,9):\ & \bigl(I(0,1),I(6,8)\bigr),\ \bigl(I(0,2),P(0,8)\bigr),\ \bigl(I(0,4),P(0,1)\bigr),\ \bigl(I(0,5),P(2,8)\bigr),\ \bigl(I(0,6),R_{1}^{5}(1)\bigr)\\
 & \bigl(I(0,7),I(8,8)\bigr),\ \bigl(I(0,8),I(3,1)\bigr)\\
I(2,9):\ & \bigl(I(0,9),R_{1}^{2}(1)\bigr),\ \bigl(I(1,1),I(7,8)\bigr),\ \bigl(I(1,5),P(1,8)\bigr),\ \bigl(I(1,6),R_{1}^{4}(1)\bigr),\ \bigl(I(1,7),I(9,8)\bigr)\\
 & \bigl(I(1,8),I(4,1)\bigr)\\
I(3,9):\ & \bigl(I(0,1),I(12,8)\bigr),\ \bigl(I(0,8),I(7,7)\bigr),\ \bigl(I(1,9),R_{1}^{1}(1)\bigr),\ \bigl(I(2,1),I(8,8)\bigr),\ \bigl(I(2,5),P(0,8)\bigr)\\
 & \bigl(I(2,6),R_{1}^{3}(1)\bigr),\ \bigl(I(2,7),I(10,8)\bigr),\ \bigl(I(2,8),I(5,1)\bigr)\\
I(4,9):\ & \bigl(I(1,1),I(13,8)\bigr),\ \bigl(I(1,8),I(8,7)\bigr),\ \bigl(I(2,9),R_{1}^{5}(1)\bigr),\ \bigl(I(3,1),I(9,8)\bigr),\ \bigl(I(3,6),R_{1}^{2}(1)\bigr)\\
 & \bigl(I(3,7),I(11,8)\bigr),\ \bigl(I(3,8),I(6,1)\bigr)\\
I(5,9):\ & \bigl(I(0,8),I(9,1)\bigr),\ \bigl(I(2,1),I(14,8)\bigr),\ \bigl(I(2,8),I(9,7)\bigr),\ \bigl(I(3,9),R_{1}^{4}(1)\bigr),\ \bigl(I(4,1),I(10,8)\bigr)\\
 & \bigl(I(4,6),R_{1}^{1}(1)\bigr),\ \bigl(I(4,7),I(12,8)\bigr),\ \bigl(I(4,8),I(7,1)\bigr)\\
I(n,9):\ & \bigl(I(n-5,8),I(n+4,1)\bigr),\ \bigl(I(n-3,1),I(n+9,8)\bigr),\ \bigl(I(n-3,8),I(n+4,7)\bigr)\\
 & \bigl(I(n-2,9),R_{1}^{(-n+8)\bmod 5+1}(1)\bigr),\ \bigl(I(n-1,1),I(n+5,8)\bigr),\ \bigl(I(n-1,6),R_{1}^{(-n+5)\bmod 5+1}(1)\bigr)\\
 & \bigl(I(n-1,7),I(n+7,8)\bigr),\ \bigl(I(n-1,8),I(n+2,1)\bigr),\ n>5\\
\end{align*}
\end{fleqn}
\subsubsection{Schofield pairs associated to regular exceptional modules}

\subsubsection*{The non-homogeneous tube $\mathcal{T}_{1}^{\Delta(\widetilde{\mathbb{E}}_{8})}$}

\begin{figure}[ht]

\begin{center}
\captionof{figure}{\vspace*{-10pt}$\mathcal{T}_{1}^{\Delta(\widetilde{\mathbb{E}}_{8})}$}
\begin{scaletikzpicturetowidth}{1.\textwidth}
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	\draw[red,->] (RN_1_2_6__Rationals) -- (RN_1_3_5__Rationals);
	\draw[gray,dashed,->] (RN_1_2_6__Rationals) -- (RN_1_3_6__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[red,->] (RN_1_3_5__Rationals) -- (RN_1_4_4__Rationals);
	\draw[gray,dashed,->] (RN_1_3_5__Rationals) -- (RN_1_4_5__Rationals_repeated) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[blue,->] (RN_1_3_5__Rationals) -- (RN_1_3_6__Rationals);
	\draw[red,->] (RN_1_3_6__Rationals) -- (RN_1_4_5__Rationals_repeated);
	\draw[gray,dashed,->,out=190,in=-10] (RN_1_3_6__Rationals) to node[midway,sloped,above]{{\tiny $\tau^{-1}$}} (RN_1_4_6__Rationals);
	\draw[dashed,-] (RN_1_1_1__Rationals) -- (RN_1_5_3__Rationals);
	\draw[dashed,-] (RN_1_1_1__Rationals_repeated) -- (RN_1_5_3__Rationals_repeated);
	\draw[dashed,-] (RN_1_5_3__Rationals) -- (RN_1_4_5__Rationals);
	\draw[dashed,-] (RN_1_5_3__Rationals_repeated) -- (RN_1_4_5__Rationals_repeated);
	\node (LeftTopPhantom_1) at (4,10) {};
	\draw[dashed,-] (RN_1_4_5__Rationals) -- (LeftTopPhantom_1);
	\node (RightTopPhantom_1) at (24,10) {};
	\draw[dashed,-] (RN_1_4_5__Rationals_repeated) -- (RightTopPhantom_1);
\end{tikzpicture}
\end{scaletikzpicturetowidth}
\end{center}

\end{figure}
\begin{fleqn}
\begin{align*}
R_{1}^{1}(1):\ & \bigl(I(0,9),P(0,6)\bigr),\ \bigl(I(2,6),P(0,9)\bigr),\ \bigl(I(2,7),P(1,1)\bigr),\ \bigl(I(2,8),P(3,8)\bigr)\\
R_{1}^{1}(2):\ & \bigl(R_{1}^{2}(1),R_{1}^{1}(1)\bigr),\ \bigl(I(4,1),P(0,7)\bigr),\ \bigl(I(2,6),P(1,6)\bigr),\ \bigl(I(2,7),P(4,1)\bigr),\ \bigl(I(7,8),P(4,8)\bigr)\\
 & \bigl(I(2,8),P(9,8)\bigr)\\
R_{1}^{2}(1):\ & \bigl(I(1,5),P(0,2)\bigr),\ \bigl(I(1,1),P(0,7)\bigr),\ \bigl(I(1,6),P(1,9)\bigr),\ \bigl(I(1,7),P(2,1)\bigr),\ \bigl(I(1,8),P(4,8)\bigr)\\
R_{1}^{2}(2):\ & \bigl(R_{1}^{3}(1),R_{1}^{2}(1)\bigr),\ \bigl(I(6,7),P(0,1)\bigr),\ \bigl(I(11,8),P(0,8)\bigr),\ \bigl(I(3,1),P(1,7)\bigr),\ \bigl(I(1,6),P(2,6)\bigr)\\
 & \bigl(I(1,7),P(5,1)\bigr),\ \bigl(I(6,8),P(5,8)\bigr),\ \bigl(I(1,8),P(10,8)\bigr)\\
R_{1}^{3}(1):\ & \bigl(I(2,1),P(0,1)\bigr),\ \bigl(I(5,8),P(0,8)\bigr),\ \bigl(I(1,9),P(0,9)\bigr),\ \bigl(I(0,5),P(1,2)\bigr),\ \bigl(I(0,1),P(1,7)\bigr)\\
 & \bigl(I(0,6),P(2,9)\bigr),\ \bigl(I(0,7),P(3,1)\bigr),\ \bigl(I(0,8),P(5,8)\bigr)\\
R_{1}^{3}(2):\ & \bigl(R_{1}^{4}(1),R_{1}^{3}(1)\bigr),\ \bigl(I(5,7),P(1,1)\bigr),\ \bigl(I(10,8),P(1,8)\bigr),\ \bigl(I(2,1),P(2,7)\bigr),\ \bigl(I(0,6),P(3,6)\bigr)\\
 & \bigl(I(0,7),P(6,1)\bigr),\ \bigl(I(5,8),P(6,8)\bigr),\ \bigl(I(0,8),P(11,8)\bigr)\\
R_{1}^{4}(1):\ & \bigl(I(1,1),P(1,1)\bigr),\ \bigl(I(4,8),P(1,8)\bigr),\ \bigl(I(0,9),P(1,9)\bigr)\\
R_{1}^{4}(2):\ & \bigl(R_{1}^{5}(1),R_{1}^{4}(1)\bigr),\ \bigl(I(4,7),P(2,1)\bigr),\ \bigl(I(9,8),P(2,8)\bigr),\ \bigl(I(1,1),P(3,7)\bigr),\ \bigl(I(4,8),P(7,8)\bigr)\\
R_{1}^{5}(1):\ & \bigl(I(3,7),P(0,1)\bigr),\ \bigl(I(0,2),P(0,5)\bigr),\ \bigl(I(0,1),P(2,1)\bigr),\ \bigl(I(3,8),P(2,8)\bigr)\\
R_{1}^{5}(2):\ & \bigl(R_{1}^{1}(1),R_{1}^{5}(1)\bigr),\ \bigl(I(3,6),P(0,6)\bigr),\ \bigl(I(3,7),P(3,1)\bigr),\ \bigl(I(8,8),P(3,8)\bigr),\ \bigl(I(0,1),P(4,7)\bigr)\\
 & \bigl(I(3,8),P(8,8)\bigr)\\
R_{1}^{5}(3):\ & \bigl(R_{1}^{1}(2),R_{1}^{5}(1)\bigr),\ \bigl(R_{1}^{2}(1),R_{1}^{5}(2)\bigr),\ \bigl(I(8,7),P(0,7)\bigr),\ \bigl(I(13,8),P(4,8)\bigr),\ \bigl(I(3,7),P(5,7)\bigr)\\
 & \bigl(I(8,8),P(9,8)\bigr),\ \bigl(I(3,8),P(14,8)\bigr)\\
R_{1}^{5}(4):\ & \bigl(R_{1}^{1}(3),R_{1}^{5}(1)\bigr),\ \bigl(R_{1}^{2}(2),R_{1}^{5}(2)\bigr),\ \bigl(R_{1}^{3}(1),R_{1}^{5}(3)\bigr),\ \bigl(I(23,8),P(0,8)\bigr),\ \bigl(I(18,8),P(5,8)\bigr)\\
 & \bigl(I(13,8),P(10,8)\bigr),\ \bigl(I(8,8),P(15,8)\bigr),\ \bigl(I(3,8),P(20,8)\bigr)\\
R_{1}^{1}(3):\ & \bigl(R_{1}^{2}(2),R_{1}^{1}(1)\bigr),\ \bigl(R_{1}^{3}(1),R_{1}^{1}(2)\bigr),\ \bigl(I(17,8),P(0,8)\bigr),\ \bigl(I(7,7),P(1,7)\bigr),\ \bigl(I(12,8),P(5,8)\bigr)\\
 & \bigl(I(2,7),P(6,7)\bigr),\ \bigl(I(7,8),P(10,8)\bigr),\ \bigl(I(2,8),P(15,8)\bigr)\\
R_{1}^{1}(4):\ & \bigl(R_{1}^{2}(3),R_{1}^{1}(1)\bigr),\ \bigl(R_{1}^{3}(2),R_{1}^{1}(2)\bigr),\ \bigl(R_{1}^{4}(1),R_{1}^{1}(3)\bigr),\ \bigl(I(22,8),P(1,8)\bigr),\ \bigl(I(17,8),P(6,8)\bigr)\\
 & \bigl(I(12,8),P(11,8)\bigr),\ \bigl(I(7,8),P(16,8)\bigr),\ \bigl(I(2,8),P(21,8)\bigr)\\
R_{1}^{2}(3):\ & \bigl(R_{1}^{3}(2),R_{1}^{2}(1)\bigr),\ \bigl(R_{1}^{4}(1),R_{1}^{2}(2)\bigr),\ \bigl(I(16,8),P(1,8)\bigr),\ \bigl(I(6,7),P(2,7)\bigr),\ \bigl(I(11,8),P(6,8)\bigr)\\
 & \bigl(I(1,7),P(7,7)\bigr),\ \bigl(I(6,8),P(11,8)\bigr),\ \bigl(I(1,8),P(16,8)\bigr)\\
R_{1}^{2}(4):\ & \bigl(R_{1}^{3}(3),R_{1}^{2}(1)\bigr),\ \bigl(R_{1}^{4}(2),R_{1}^{2}(2)\bigr),\ \bigl(R_{1}^{5}(1),R_{1}^{2}(3)\bigr),\ \bigl(I(21,8),P(2,8)\bigr),\ \bigl(I(16,8),P(7,8)\bigr)\\
 & \bigl(I(11,8),P(12,8)\bigr),\ \bigl(I(6,8),P(17,8)\bigr),\ \bigl(I(1,8),P(22,8)\bigr)\\
R_{1}^{3}(3):\ & \bigl(R_{1}^{4}(2),R_{1}^{3}(1)\bigr),\ \bigl(R_{1}^{5}(1),R_{1}^{3}(2)\bigr),\ \bigl(I(15,8),P(2,8)\bigr),\ \bigl(I(5,7),P(3,7)\bigr),\ \bigl(I(10,8),P(7,8)\bigr)\\
 & \bigl(I(0,7),P(8,7)\bigr),\ \bigl(I(5,8),P(12,8)\bigr),\ \bigl(I(0,8),P(17,8)\bigr)\\
R_{1}^{3}(4):\ & \bigl(R_{1}^{4}(3),R_{1}^{3}(1)\bigr),\ \bigl(R_{1}^{5}(2),R_{1}^{3}(2)\bigr),\ \bigl(R_{1}^{1}(1),R_{1}^{3}(3)\bigr),\ \bigl(I(20,8),P(3,8)\bigr),\ \bigl(I(15,8),P(8,8)\bigr)\\
 & \bigl(I(10,8),P(13,8)\bigr),\ \bigl(I(5,8),P(18,8)\bigr),\ \bigl(I(0,8),P(23,8)\bigr)\\
R_{1}^{4}(3):\ & \bigl(R_{1}^{5}(2),R_{1}^{4}(1)\bigr),\ \bigl(R_{1}^{1}(1),R_{1}^{4}(2)\bigr),\ \bigl(I(14,8),P(3,8)\bigr),\ \bigl(I(4,7),P(4,7)\bigr),\ \bigl(I(9,8),P(8,8)\bigr)\\
 & \bigl(I(4,8),P(13,8)\bigr)\\
R_{1}^{4}(4):\ & \bigl(R_{1}^{5}(3),R_{1}^{4}(1)\bigr),\ \bigl(R_{1}^{1}(2),R_{1}^{4}(2)\bigr),\ \bigl(R_{1}^{2}(1),R_{1}^{4}(3)\bigr),\ \bigl(I(19,8),P(4,8)\bigr),\ \bigl(I(14,8),P(9,8)\bigr)\\
 & \bigl(I(9,8),P(14,8)\bigr),\ \bigl(I(4,8),P(19,8)\bigr)\\
\end{align*}
\end{fleqn}
\subsubsection*{The non-homogeneous tube $\mathcal{T}_{0}^{\Delta(\widetilde{\mathbb{E}}_{8})}$}

\begin{figure}[ht]

\begin{center}
\captionof{figure}{\vspace*{-10pt}$\mathcal{T}_{0}^{\Delta(\widetilde{\mathbb{E}}_{8})}$}
\begin{scaletikzpicturetowidth}{0.69999999999999996\textwidth}
\begin{tikzpicture}[scale=\tikzscale]
	\node[seltwoEEEEEEEEE, label={[shift={(0.2,-0.6)}, brown]\tiny $R_{0}^{1}(1)$}] (RN_2_1_1__Rationals) at (4,0) {$\begin{smallmatrix}&&1\\0&1&2&2&2&1&1&1\end{smallmatrix}$};
	\node[seltwoEEEEEEEEE, label={[shift={(0.2,-0.6)}, brown]\tiny $R_{0}^{2}(1)$}] (RN_2_2_1__Rationals) at (8,0) {$\begin{smallmatrix}&&1\\1&2&2&1&1&1&0&0\end{smallmatrix}$};
	\node[seltwoEEEEEEEEE, label={[shift={(0.2,-0.6)}, brown]\tiny $R_{0}^{3}(1)$}] (RN_2_3_1__Rationals) at (12,0) {$\begin{smallmatrix}&&1\\1&1&2&2&1&1&1&0\end{smallmatrix}$};
	\node[seltwoEEEEEEEEE, label={[shift={(0.2,-0.6)}, brown]\tiny $R_{0}^{1}(1)$}] (RN_2_1_1__Rationals_repeated) at (16,0) {$\begin{smallmatrix}&&1\\0&1&2&2&2&1&1&1\end{smallmatrix}$};
	\node[seltwoEEEEEEEEE, label={[shift={(0.2,-0.6)}, brown]\tiny $R_{0}^{1}(2)$}] (RN_2_1_2__Rationals) at (6,2) {$\begin{smallmatrix}&&2\\1&3&4&3&3&2&1&1\end{smallmatrix}$};
	\node[seltwoEEEEEEEEE, label={[shift={(0.2,-0.6)}, brown]\tiny $R_{0}^{2}(2)$}] (RN_2_2_2__Rationals) at (10,2) {$\begin{smallmatrix}&&2\\2&3&4&3&2&2&1&0\end{smallmatrix}$};
	\node[seltwoEEEEEEEEE, label={[shift={(0.2,-0.6)}, brown]\tiny $R_{0}^{3}(2)$}] (RN_2_3_2__Rationals) at (14,2) {$\begin{smallmatrix}&&2\\1&2&4&4&3&2&2&1\end{smallmatrix}$};
	\node[noselEEEEEEEEE, label={[shift={(0.2,-0.6)}, brown]\tiny $R_{0}^{3}(3)$}] (RN_2_3_3__Rationals) at (4,4) {$\begin{smallmatrix}&&3\\2&4&6&5&4&3&2&1\end{smallmatrix}$};
	\node[noselEEEEEEEEE, label={[shift={(0.2,-0.6)}, brown]\tiny $R_{0}^{1}(3)$}] (RN_2_1_3__Rationals) at (8,4) {$\begin{smallmatrix}&&3\\2&4&6&5&4&3&2&1\end{smallmatrix}$};
	\node[noselEEEEEEEEE, label={[shift={(0.2,-0.6)}, brown]\tiny $R_{0}^{2}(3)$}] (RN_2_2_3__Rationals) at (12,4) {$\begin{smallmatrix}&&3\\2&4&6&5&4&3&2&1\end{smallmatrix}$};
	\node[noselEEEEEEEEE, label={[shift={(0.2,-0.6)}, brown]\tiny $R_{0}^{3}(3)$}] (RN_2_3_3__Rationals_repeated) at (16,4) {$\begin{smallmatrix}&&3\\2&4&6&5&4&3&2&1\end{smallmatrix}$};
	\node[noselEEEEEEEEE, label={[shift={(0.2,-0.6)}, brown]\tiny $R_{0}^{3}(4)$}] (RN_2_3_4__Rationals) at (6,6) {$\begin{smallmatrix}&&4\\3&5&8&7&5&4&3&1\end{smallmatrix}$};
	\node[noselEEEEEEEEE, label={[shift={(0.2,-0.6)}, brown]\tiny $R_{0}^{1}(4)$}] (RN_2_1_4__Rationals) at (10,6) {$\begin{smallmatrix}&&4\\2&5&8&7&6&4&3&2\end{smallmatrix}$};
	\node[noselEEEEEEEEE, label={[shift={(0.2,-0.6)}, brown]\tiny $R_{0}^{2}(4)$}] (RN_2_2_4__Rationals) at (14,6) {$\begin{smallmatrix}&&4\\3&6&8&6&5&4&2&1\end{smallmatrix}$};
	\draw[blue,->] (RN_2_1_1__Rationals) -- (RN_2_1_2__Rationals);
	\draw[gray,dashed,->] (RN_2_1_1__Rationals) -- (RN_2_2_1__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[red,->] (RN_2_1_2__Rationals) -- (RN_2_2_1__Rationals);
	\draw[gray,dashed,->] (RN_2_1_2__Rationals) -- (RN_2_2_2__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[blue,->] (RN_2_1_2__Rationals) -- (RN_2_1_3__Rationals);
	\draw[blue,->] (RN_2_2_1__Rationals) -- (RN_2_2_2__Rationals);
	\draw[gray,dashed,->] (RN_2_2_1__Rationals) -- (RN_2_3_1__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[red,->] (RN_2_2_2__Rationals) -- (RN_2_3_1__Rationals);
	\draw[gray,dashed,->] (RN_2_2_2__Rationals) -- (RN_2_3_2__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[blue,->] (RN_2_2_2__Rationals) -- (RN_2_2_3__Rationals);
	\draw[gray,dashed,->] (RN_2_3_1__Rationals) -- (RN_2_1_1__Rationals_repeated) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[blue,->] (RN_2_3_1__Rationals) -- (RN_2_3_2__Rationals);
	\draw[red,->] (RN_2_3_2__Rationals) -- (RN_2_1_1__Rationals_repeated);
	\draw[gray,dashed,->,out=200,in=-20] (RN_2_3_2__Rationals) to node[midway,sloped,above]{{\tiny $\tau^{-1}$}} (RN_2_1_2__Rationals);
	\draw[blue,->] (RN_2_3_2__Rationals) -- (RN_2_3_3__Rationals_repeated);
	\draw[red,->] (RN_2_3_3__Rationals) -- (RN_2_1_2__Rationals);
	\draw[blue,->] (RN_2_3_3__Rationals) -- (RN_2_3_4__Rationals);
	\draw[gray,dashed,->] (RN_2_3_3__Rationals) -- (RN_2_1_3__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[red,->] (RN_2_3_4__Rationals) -- (RN_2_1_3__Rationals);
	\draw[gray,dashed,->] (RN_2_3_4__Rationals) -- (RN_2_1_4__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[red,->] (RN_2_1_3__Rationals) -- (RN_2_2_2__Rationals);
	\draw[blue,->] (RN_2_1_3__Rationals) -- (RN_2_1_4__Rationals);
	\draw[gray,dashed,->] (RN_2_1_3__Rationals) -- (RN_2_2_3__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[red,->] (RN_2_1_4__Rationals) -- (RN_2_2_3__Rationals);
	\draw[gray,dashed,->] (RN_2_1_4__Rationals) -- (RN_2_2_4__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[red,->] (RN_2_2_3__Rationals) -- (RN_2_3_2__Rationals);
	\draw[gray,dashed,->] (RN_2_2_3__Rationals) -- (RN_2_3_3__Rationals_repeated) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[blue,->] (RN_2_2_3__Rationals) -- (RN_2_2_4__Rationals);
	\draw[red,->] (RN_2_2_4__Rationals) -- (RN_2_3_3__Rationals_repeated);
	\draw[gray,dashed,->,out=200,in=-20] (RN_2_2_4__Rationals) to node[midway,sloped,above]{{\tiny $\tau^{-1}$}} (RN_2_3_4__Rationals);
	\draw[dashed,-] (RN_2_1_1__Rationals) -- (RN_2_3_3__Rationals);
	\draw[dashed,-] (RN_2_1_1__Rationals_repeated) -- (RN_2_3_3__Rationals_repeated);
	\node (LeftTopPhantom_2) at (4,6) {};
	\draw[dashed,-] (RN_2_3_3__Rationals) -- (LeftTopPhantom_2);
	\node (RightTopPhantom_2) at (16,6) {};
	\draw[dashed,-] (RN_2_3_3__Rationals_repeated) -- (RightTopPhantom_2);
\end{tikzpicture}
\end{scaletikzpicturetowidth}
\end{center}

\end{figure}
\begin{fleqn}
\begin{align*}
R_{0}^{1}(1):\ & \bigl(I(9,8),P(0,8)\bigr),\ \bigl(I(3,7),P(1,7)\bigr),\ \bigl(I(2,1),P(2,1)\bigr),\ \bigl(I(6,8),P(3,8)\bigr),\ \bigl(I(0,7),P(4,7)\bigr)\\
 & \bigl(I(3,8),P(6,8)\bigr),\ \bigl(I(0,8),P(9,8)\bigr)\\
R_{0}^{1}(2):\ & \bigl(R_{0}^{2}(1),R_{0}^{1}(1)\bigr),\ \bigl(I(18,8),P(1,8)\bigr),\ \bigl(I(15,8),P(4,8)\bigr),\ \bigl(I(12,8),P(7,8)\bigr),\ \bigl(I(9,8),P(10,8)\bigr)\\
 & \bigl(I(6,8),P(13,8)\bigr),\ \bigl(I(3,8),P(16,8)\bigr),\ \bigl(I(0,8),P(19,8)\bigr)\\
R_{0}^{2}(1):\ & \bigl(I(4,1),P(0,1)\bigr),\ \bigl(I(8,8),P(1,8)\bigr),\ \bigl(I(2,7),P(2,7)\bigr),\ \bigl(I(1,1),P(3,1)\bigr),\ \bigl(I(5,8),P(4,8)\bigr)\\
 & \bigl(I(2,8),P(7,8)\bigr)\\
R_{0}^{2}(2):\ & \bigl(R_{0}^{3}(1),R_{0}^{2}(1)\bigr),\ \bigl(I(17,8),P(2,8)\bigr),\ \bigl(I(14,8),P(5,8)\bigr),\ \bigl(I(11,8),P(8,8)\bigr),\ \bigl(I(8,8),P(11,8)\bigr)\\
 & \bigl(I(5,8),P(14,8)\bigr),\ \bigl(I(2,8),P(17,8)\bigr)\\
R_{0}^{3}(1):\ & \bigl(I(4,7),P(0,7)\bigr),\ \bigl(I(3,1),P(1,1)\bigr),\ \bigl(I(7,8),P(2,8)\bigr),\ \bigl(I(1,7),P(3,7)\bigr),\ \bigl(I(0,1),P(4,1)\bigr)\\
 & \bigl(I(4,8),P(5,8)\bigr),\ \bigl(I(1,8),P(8,8)\bigr)\\
R_{0}^{3}(2):\ & \bigl(R_{0}^{1}(1),R_{0}^{3}(1)\bigr),\ \bigl(I(19,8),P(0,8)\bigr),\ \bigl(I(16,8),P(3,8)\bigr),\ \bigl(I(13,8),P(6,8)\bigr),\ \bigl(I(10,8),P(9,8)\bigr)\\
 & \bigl(I(7,8),P(12,8)\bigr),\ \bigl(I(4,8),P(15,8)\bigr),\ \bigl(I(1,8),P(18,8)\bigr)\\
\end{align*}
\end{fleqn}
\subsubsection*{The non-homogeneous tube $\mathcal{T}_{\infty}^{\Delta(\widetilde{\mathbb{E}}_{8})}$}

\begin{figure}[ht]

\begin{center}
\captionof{figure}{\vspace*{-10pt}$\mathcal{T}_{\infty}^{\Delta(\widetilde{\mathbb{E}}_{8})}$}
\begin{scaletikzpicturetowidth}{0.59999999999999998\textwidth}
\begin{tikzpicture}[scale=\tikzscale]
	\node[seltwoEEEEEEEEE, label={[shift={(0.2,-0.6)}, brown]\tiny $R_{\infty}^{1}(1)$}] (RN_3_1_1__Rationals) at (4,0) {$\begin{smallmatrix}&&2\\1&2&3&2&2&1&1&0\end{smallmatrix}$};
	\node[seltwoEEEEEEEEE, label={[shift={(0.2,-0.6)}, brown]\tiny $R_{\infty}^{2}(1)$}] (RN_3_2_1__Rationals) at (8,0) {$\begin{smallmatrix}&&1\\1&2&3&3&2&2&1&1\end{smallmatrix}$};
	\node[seltwoEEEEEEEEE, label={[shift={(0.2,-0.6)}, brown]\tiny $R_{\infty}^{1}(1)$}] (RN_3_1_1__Rationals_repeated) at (12,0) {$\begin{smallmatrix}&&2\\1&2&3&2&2&1&1&0\end{smallmatrix}$};
	\node[noselEEEEEEEEE, label={[shift={(0.2,-0.6)}, brown]\tiny $R_{\infty}^{1}(2)$}] (RN_3_1_2__Rationals) at (6,2) {$\begin{smallmatrix}&&3\\2&4&6&5&4&3&2&1\end{smallmatrix}$};
	\node[noselEEEEEEEEE, label={[shift={(0.2,-0.6)}, brown]\tiny $R_{\infty}^{2}(2)$}] (RN_3_2_2__Rationals) at (10,2) {$\begin{smallmatrix}&&3\\2&4&6&5&4&3&2&1\end{smallmatrix}$};
	\node[noselEEEEEEEEE, label={[shift={(0.2,-0.6)}, brown]\tiny $R_{\infty}^{2}(3)$}] (RN_3_2_3__Rationals) at (4,4) {$\begin{smallmatrix}&&4\\3&6&9&8&6&5&3&2\end{smallmatrix}$};
	\node[noselEEEEEEEEE, label={[shift={(0.2,-0.6)}, brown]\tiny $R_{\infty}^{1}(3)$}] (RN_3_1_3__Rationals) at (8,4) {$\begin{smallmatrix}&&5\\3&6&9&7&6&4&3&1\end{smallmatrix}$};
	\node[noselEEEEEEEEE, label={[shift={(0.2,-0.6)}, brown]\tiny $R_{\infty}^{2}(3)$}] (RN_3_2_3__Rationals_repeated) at (12,4) {$\begin{smallmatrix}&&4\\3&6&9&8&6&5&3&2\end{smallmatrix}$};
	\draw[blue,->] (RN_3_1_1__Rationals) -- (RN_3_1_2__Rationals);
	\draw[gray,dashed,->] (RN_3_1_1__Rationals) -- (RN_3_2_1__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[red,->] (RN_3_1_2__Rationals) -- (RN_3_2_1__Rationals);
	\draw[gray,dashed,->] (RN_3_1_2__Rationals) -- (RN_3_2_2__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[blue,->] (RN_3_1_2__Rationals) -- (RN_3_1_3__Rationals);
	\draw[gray,dashed,->] (RN_3_2_1__Rationals) -- (RN_3_1_1__Rationals_repeated) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[blue,->] (RN_3_2_1__Rationals) -- (RN_3_2_2__Rationals);
	\draw[red,->] (RN_3_2_2__Rationals) -- (RN_3_1_1__Rationals_repeated);
	\draw[gray,dashed,->,out=219,in=-39] (RN_3_2_2__Rationals) to node[midway,sloped,above]{{\tiny $\tau^{-1}$}} (RN_3_1_2__Rationals);
	\draw[blue,->] (RN_3_2_2__Rationals) -- (RN_3_2_3__Rationals_repeated);
	\draw[red,->] (RN_3_2_3__Rationals) -- (RN_3_1_2__Rationals);
	\draw[gray,dashed,->] (RN_3_2_3__Rationals) -- (RN_3_1_3__Rationals) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[red,->] (RN_3_1_3__Rationals) -- (RN_3_2_2__Rationals);
	\draw[gray,dashed,->] (RN_3_1_3__Rationals) -- (RN_3_2_3__Rationals_repeated) node[midway,sloped,above]{{\tiny $\tau^{-1}$}};
	\draw[dashed,-] (RN_3_1_1__Rationals) -- (RN_3_2_3__Rationals);
	\draw[dashed,-] (RN_3_1_1__Rationals_repeated) -- (RN_3_2_3__Rationals_repeated);
\end{tikzpicture}
\end{scaletikzpicturetowidth}
\end{center}

\end{figure}
\begin{fleqn}
\begin{align*}
R_{\infty}^{1}(1):\ & \bigl(I(13,8),P(1,8)\bigr),\ \bigl(I(11,8),P(3,8)\bigr),\ \bigl(I(9,8),P(5,8)\bigr),\ \bigl(I(7,8),P(7,8)\bigr),\ \bigl(I(5,8),P(9,8)\bigr)\\
 & \bigl(I(3,8),P(11,8)\bigr),\ \bigl(I(1,8),P(13,8)\bigr)\\
R_{\infty}^{2}(1):\ & \bigl(I(14,8),P(0,8)\bigr),\ \bigl(I(12,8),P(2,8)\bigr),\ \bigl(I(10,8),P(4,8)\bigr),\ \bigl(I(8,8),P(6,8)\bigr),\ \bigl(I(6,8),P(8,8)\bigr)\\
 & \bigl(I(4,8),P(10,8)\bigr),\ \bigl(I(2,8),P(12,8)\bigr),\ \bigl(I(0,8),P(14,8)\bigr)\\
\end{align*}
\end{fleqn}
\clearpage

Abstract: Let $k$ be a field and consider the path algebra $kQ$ of the quiver $Q$. A
pair of indecomposable $kQ$-modules $(Y,X)$ is called an orthogonal exceptional
pair if the modules are exceptional and
$\operatorname{Hom}(X,Y)=\operatorname{Hom}(Y,X)=\operatorname{Ext}^{1}(X,Y)=0$.
Denote by $\mathcal{F}(X,Y)$ the full subcategory of objects having filtration
with factors $X$ and $Y$. By the theorem of Schofield if $Z$ is exceptional but
not simple, then $Z\in\mathcal{F}(X,Y)$ for some orthogonal exceptional pair
$(Y,X)$, and $Z$ is not a simple object in $\mathcal{F}(X,Y)$. In fact, there
are precisely $s(Z)-1$ such pairs, where $s(Z)$ is the support of $Z$ (i.e the
number of nonzero components in ${\underline\dim}Z$). Whereas it is easy to
construct $Z$ given $X$ and $Y$, there is no convenient procedure yet to
determine the possible modules $X$ (called Schofield submodules of $Z$) and
then $Y$ (called Schofield factors of $Z$), when $Z$ is given. We present such
an explicit procedure in the tame case, i.e when $Q$ is Euclidean.