\label{th1}
Let $p>3$.
Assume that $\Gamma$ is a simple closed smooth curve with unit length in $\Omega$, and it is also a non-degenerate critical point of the functional
\begin{equation}\label{e91}
\mathcal{K}(\Gamma)=\int_\Gamma \mathbf{\Psi}^{\frac{p+3}{2p}}dvol_{\mathfrak{g}},
\end{equation}
where $\mathfrak{g}(X,Y)=\langle A^*X,Y\rangle$ is a Riemannian metric on $\mathbb R^2$ and $A^*$ is the adjoint matrix for $A$. We also assume that the following inequality holds on $\Gamma$:
\begin{eqnarray*}
\Upsilon_0&=&-\alpha^{1-p}\left[ 2\alpha^{-1}\alpha'a_{22}+\beta^{-1}\beta'b_{11}+b_{22}-a_{33}+\frac{p+3}2\alpha^{p-2}\beta^{-2}\mathbf{q}_{tt}+\frac{p+2}{2(p+3)}\alpha^{1-p} \beta^2 (a_{32})^2\right. \\
&&\left.-\frac{p+1}{p+3}\alpha^{1-p}\beta^2 a_{32}b_{21}-\frac{2}{p+3}\alpha^{1-p}\beta^2(b_{21})^2 +a_{11}\left(\beta^{-1}\beta''+2\alpha^{-1}\beta^{-1}\alpha'\beta'-\beta^{-2}(\beta')^2\right)\right]>0,
\end{eqnarray*}
where $a_{ij}$'s and $b_{ij}$'s is defined in Lemma \ref{lm8}, $\alpha$ and $\beta$ are defined in \eqref{80}, and $\mathbf{q}$ is defined in \eqref{e88}.
Then for each  integer $N>0$, there exists a sequence of $\varepsilon$, \textit{i.e.}, $\{\varepsilon_l\}$ converging to $0$ such that \eqref{1} has a positive solutions $u_{\varepsilon_l}$ with exactly $N$ concentration layers at mutual distance $O(\varepsilon_l|\log\varepsilon_l|)$, the center of mass of $N$ concentration layers collapses to $\Gamma$ at speed $\varepsilon_l^{1+\mu}$ for small positive constant $\mu\in(0,1/2)$. More precisely $u_{\varepsilon_l}$ has the form
\begin{eqnarray*}
u_{\varepsilon_l}(y_1,y_2)\approx -\mathbf{\Psi}^{\frac1p}(y_1,y_2)+ \mathbf{\Psi}^{\frac1p}(\gamma(\theta))\sum_{k=1}^N w\left(\left[\frac{\mathbf{\Psi}^{\frac{p-1}p}(1-\langle\gamma',n\rangle)}{\langle A^*n,n\rangle}\right]^{\frac12}\frac{t-\varepsilon_l f_k}{\varepsilon_l}\right),
\end{eqnarray*}
where $\gamma$ is a natural parametrization of $\Gamma$, $n$ is the unit vector defined in \eqref{e102} and $w$ is  the unique solution of the following problem
\begin{equation}\label{29}
-w''=|1-w|^p -1, \qquad w>0 \quad \mathrm{in} \quad \mathbb R, \qquad w'(0)=w(\pm \infty).
\end{equation}
In the expression above, the functions $f_j$'s satisfy
\[\|f_j\|_{L^\infty(0,1)}\le C|\log\varepsilon_l|^2,\qquad \sum_{j=1}^N f_j =O(\frac1{|\log\varepsilon_l|^{\frac32}}),\]
\[\min_{1\le j\le N}(f_{j+1}-f_j)\approx\frac2{\sqrt{p}}|\log\varepsilon_l|\left[\frac{\langle A^*n,n\rangle}{\Psi^{\frac{p-1}p}(1-\langle\gamma',n\rangle)}\right]^{\frac12}\]
and solves the Jacobi-Toda system for $j=1,2,\cdots,N$
\begin{eqnarray*}
\nonumber&&\varepsilon^2\alpha^{1-p}\beta\left\{-a_{11}f_j''+\left[a_{22}-b_{11}-a_{11}\left(\beta^{-1}\beta'+2\alpha^{-1}\alpha'\right)\right]f_j'+\left[a_{22}(\beta^{-1}\beta'+2\alpha^{-1}\alpha') +b_{22}-a_{33}\right]f_j\right. \\
\nonumber&&+\left[\frac{p+3}2\alpha^{p-2}\beta^{-2} \mathbf{q}_{tt}+\frac{p+2}{2(p+3)}\alpha^{1-p}\beta^2(a_{32})^2-\frac{p+1}{p+3}\alpha^{1-p}\beta^2 a_{32}b_{21}-\frac2{p+3}\alpha^{1-p}\beta^2 (b_{21})^2\right]f_j \\
&&C_0 p\alpha_p\left[e^{-\sqrt{p}\beta(f_j-f_{j-1})}-e^{-\sqrt{p}\beta (f_{j+1}-f_j)}\right]\approx0.
\end{eqnarray*}

\begin{theorem}\label{th1}
Let $p>3$.
Assume that $\Gamma$ is a simple closed smooth curve with unit length in $\Omega$, and it is also a non-degenerate critical point of the functional
\begin{equation}\label{e91}
\mathcal{K}(\Gamma)=\int_\Gamma \mathbf{\Psi}^{\frac{p+3}{2p}}dvol_{\mathfrak{g}},
\end{equation}
where $\mathfrak{g}(X,Y)=\langle A^*X,Y\rangle$ is a Riemannian metric on $\R^2$ and $A^*$ is the adjoint matrix for $A$. We also assume that the following inequality holds on $\Gamma$:
\begin{eqnarray*}
\Upsilon_0&=&-\alpha^{1-p}\left[ 2\alpha^{-1}\alpha'a_{22}+\beta^{-1}\beta'b_{11}+b_{22}-a_{33}+\frac{p+3}2\alpha^{p-2}\beta^{-2}\mathbf{q}_{tt}+\frac{p+2}{2(p+3)}\alpha^{1-p} \beta^2 (a_{32})^2\right. \\
&&\left.-\frac{p+1}{p+3}\alpha^{1-p}\beta^2 a_{32}b_{21}-\frac{2}{p+3}\alpha^{1-p}\beta^2(b_{21})^2 +a_{11}\left(\beta^{-1}\beta''+2\alpha^{-1}\beta^{-1}\alpha'\beta'-\beta^{-2}(\beta')^2\right)\right]>0,
\end{eqnarray*}
where $a_{ij}$'s and $b_{ij}$'s is defined in Lemma \ref{lm8}, $\alpha$ and $\beta$ are defined in \eqref{80}, and $\mathbf{q}$ is defined in \eqref{e88}.
Then for each  integer $N>0$, there exists a sequence of $\varepsilon$, \textit{i.e.}, $\{\varepsilon_l\}$ converging to $0$ such that \eqref{1} has a positive solutions $u_{\varepsilon_l}$ with exactly $N$ concentration layers at mutual distance $O(\varepsilon_l|\log\varepsilon_l|)$, the center of mass of $N$ concentration layers collapses to $\Gamma$ at speed $\varepsilon_l^{1+\mu}$ for small positive constant $\mu\in(0,1/2)$. More precisely $u_{\varepsilon_l}$ has the form
\begin{eqnarray*}
u_{\varepsilon_l}(y_1,y_2)\approx -\mathbf{\Psi}^{\frac1p}(y_1,y_2)+ \mathbf{\Psi}^{\frac1p}(\gamma(\theta))\sum_{k=1}^N w\left(\left[\frac{\mathbf{\Psi}^{\frac{p-1}p}(1-\langle\gamma',n\rangle)}{\langle A^*n,n\rangle}\right]^{\frac12}\frac{t-\varepsilon_l f_k}{\varepsilon_l}\right),
\end{eqnarray*}
where $\gamma$ is a natural parametrization of $\Gamma$, $n$ is the unit vector defined in \eqref{e102} and $w$ is  the unique solution of the following problem
\begin{equation}\label{29}
-w''=|1-w|^p -1, \qquad w>0 \quad \mathrm{in} \quad \R, \qquad w'(0)=w(\pm \infty).
\end{equation}
In the expression above, the functions $f_j$'s satisfy
\[\|f_j\|_{L^\infty(0,1)}\le C|\log\varepsilon_l|^2,\qquad \sum_{j=1}^N f_j =O(\frac1{|\log\varepsilon_l|^{\frac32}}),\]
\[\min_{1\le j\le N}(f_{j+1}-f_j)\approx\frac2{\sqrt{p}}|\log\varepsilon_l|\left[\frac{\langle A^*n,n\rangle}{\Psi^{\frac{p-1}p}(1-\langle\gamma',n\rangle)}\right]^{\frac12}\]
and solves the Jacobi-Toda system for $j=1,2,\cdots,N$
\begin{eqnarray*}
\nonumber&&\varepsilon^2\alpha^{1-p}\beta\left\{-a_{11}f_j''+\left[a_{22}-b_{11}-a_{11}\left(\beta^{-1}\beta'+2\alpha^{-1}\alpha'\right)\right]f_j'+\left[a_{22}(\beta^{-1}\beta'+2\alpha^{-1}\alpha') +b_{22}-a_{33}\right]f_j\right. \\
\nonumber&&+\left[\frac{p+3}2\alpha^{p-2}\beta^{-2} \mathbf{q}_{tt}+\frac{p+2}{2(p+3)}\alpha^{1-p}\beta^2(a_{32})^2-\frac{p+1}{p+3}\alpha^{1-p}\beta^2 a_{32}b_{21}-\frac2{p+3}\alpha^{1-p}\beta^2 (b_{21})^2\right]f_j \\
&&C_0 p\alpha_p\left[e^{-\sqrt{p}\beta(f_j-f_{j-1})}-e^{-\sqrt{p}\beta (f_{j+1}-f_j)}\right]\approx0.
\end{eqnarray*}
\end{theorem}

\begin{equation}\label{e55}
-\varepsilon^2 \Delta u+V(y)u=u^p, \qquad \mathrm{in} \qquad \R^\mathcal{N}.
\end{equation}

The Ambrosetti-Prodi problem is of the following form:
\begin{equation}\label{e51}
\left\{\begin{array}{ll}
-\Delta u=\zeta(u)-t\mathbf{\Psi}_0 (x), &\mbox{in $\Omega$,} \\
u=0,&\mbox{on $\partial \Omega$},
\end{array}\right.
\end{equation}
where $t>0$, $\Omega\subset \mathbb R^\mathcal{N}$ is a smooth bounded domain,  $\mathbf{\Psi}_0$ is the an eigenfunction of $-\Delta$ subject to Dirichlet boundary condition corresponding to the first eigenvalue, the function $\zeta(t)$ satisfies
\[-\infty\le\mu=\lim_{t\to +\infty} \frac{\zeta(t)}{t}>\lim_{t\to +\infty} \frac{\zeta(t)}{t}=\nu\le +\infty,\]
and the interval $(\mu, \nu)$ contains some eigenvalues of $-\Delta$ subject to Dirichlet boundary condition.

Problem \eqref{e51} was first studied by Ambrosetti and Prodi \cite{Ambrosetti_Prodi1972}. It was widely discussed in 1980's( see \cite{deFigueiredo1984,deFigueiredo_Solimini1984,Lazer_McKenna1981,Lazer_McKenna1984,Lazer_McKenna1988,Ruf_Srikanth986a,Ruf_Srikanth1986,Solimini1985,Ruf_Solimini1986} for example). The main results in these literature are that if $g(t)$ grows subcritical at infinity, \eqref{e51} has at least two solutions: one is local minimizer of the Euler-Lagrange functional, the other is the mountain-pass solution. Breuer, McKenna and Plum \cite{Breuer_McKenna_Plum2003} considered the case that $\zeta(t)=t^2$ and $\Omega$ is a unit square in $\mathbb R^2$. Using a computer assisted proof, they showed  that \eqref{e51} has at least 4 solutions.  By comparing Morse index of mountain pass solution in different spaces, de Figueiredo, Srikanth and Santra \cite{deFigueiredo_Srikanth_Santra2005}  found a non-radial solution of \eqref{e51} under the condition $\Omega$ is a unit ball and $\zeta(t)=t^2$.

However, in the case of $\zeta(t)=|t|^p$, where $1<p<\frac{\mathcal{N}+2}{\mathcal{N}-2}$ for $\mathcal{N}\ge3$ and $p>1$ for $\mathcal{N}=2$, \eqref{e51} becomes the following problem
\begin{equation}\label{e52}
\left\{\begin{array}{ll}
-\Delta u=|u|^p-t\mathbf{\Psi}_0 (x), &\mbox{in $\Omega$,} \\
u=0,&\mbox{on $\partial \Omega$}.
\end{array}\right.
\end{equation}
Dancer and Yan \cite{Dancer_Yan2005} constructed arbitrary many peak solutions of \eqref{e52} for $t>0$ large enough.  It shows that Lazer-McKenna conjecture \cite{Lazer_McKenna1981} holds in this case. Dancer and Yan \cite{Dancer_Yan2005} also proved that the mountain pass solution of \eqref{e51} has a sharp peak near the boundary for $t$ large enough. As a conclusion,   \eqref{e51} has solutions concentrating at some points in $\Omega$ or the  ones on $\partial \Omega$ as $t\to \infty$. These results have been extended to different kind of nonlinearities( see \cite{Dancer_Yan2005a, Dancer_Santra2007, delPino_Munoz2006, Li_Yan_Yang2007, Li_Yan_Yang2006, Molle_Passaseo2010, Wei_Yan2007} for instance and references therein).

However, these results concern only point concentrating solutions. Based on some numerical evidence, Hollman and McKenna \cite{Hollman_McKenna2011} asked \eqref{e52} that  whether there exist other types of concentrations  for the solutions to \eqref{e52} as $t\to+\infty$. For this problem, Manna and Santra \cite{Manna_Santra2016} considered \eqref{e52} under the condition $\Omega\subset \mathbb R^2$ and $p>2$. They proved that \eqref{e52} has a family of solutions clustering along a closed curve $\Gamma\subset \Omega$, where $\Gamma$ is a non-degenerate critical point of the functional
\begin{equation}\label{e54}
\mathcal{K}_0(\Gamma)=\int_{\Gamma} \mathbf{\Psi}_0^{\frac{p+3}{2p}}(x)dvol.
\end{equation}
Later, this result was extended to high dimensional case by Khemiri, Mahmoudi and Messaoudi  \cite{Khemiri_Mahmoudi_Messaoudi2017}. Under the condition that $\mathcal{N}\ge 3$ and $\Omega$ contain a $k$-dimensional compact submanifold $\Gamma$, which is a non-degenerate critical point of the functional
\[\mathcal{K}_1(\Gamma)=\int_{\Gamma} \mathbf{\Psi}_0^{\left(1-\frac1p\right)\left(\frac{p+1}{p-1}-\frac{n-k}2\right)}(x)dvol,\]
They proved there exists a sequence $t=t_j\to \infty$, and  solutions  $u_{t_j}$ to \eqref{e52}, which have  concentration layers concentrating near $\Gamma$.

Baraket \textit{et. al.} \cite{Baraket_Khemiri_Mahmoudi_Messaoudi2018} considered the following Neumann problem:
\begin{equation}\label{e53}
\left\{\begin{array}{ll}
-\Delta u=|u|^p-t\psi (x), &\mbox{in $\Omega$,} \\
\frac{\partial u}{\partial \mathbf{n}}=0,&\mbox{on $\partial \Omega$},
\end{array}\right.
\end{equation}
where  $\Omega\subset\mathbb R^\mathcal{N}$ is a smooth bound domain and $\mathbf{n}$ is the unit outward normal vector of $\partial\Omega$.
Under the condition $\mathcal{N}=2$, they proved that \eqref{e53} has a solution $u_t$ concentrating along a curve $\Gamma\subset \bar{\Omega}$ with $t>0$ large enough. The curve $\Gamma$ intersects $\partial\Omega$ with right angle and divides $\Omega$ into two part. Moreover $\Gamma$ is a non-degenerate critical point of the functional $\mathcal{K}_2(\Gamma)=\int_{\Gamma} \psi^{\frac{p+3}{2p}}(x)dvol$.  However, under the condition that  $\psi\equiv1$ and  $\mathcal{N}\ge 2$, Bendahou, Khemiri and Mahmoudi \cite{Bendahou_Khemiri_Mahmoudi2020} constructed a family of new solutions to \eqref{e53}, which has large number of spikes concentrating along an interior  straight line in $\Omega$ for $t\to+\infty$. Using the similar method, Ao, Fu and Liu \cite{Ao_Fu_Liu2022} constructed a similar type of  solutions  which concentrate along a segment of boundary $\partial \Omega$ in the two dimensional case.

From these results, we recognize that the high dimensional concentration behaviours of Ambrosetti-Prodi type problem is similar to that of Nonlinear Schr\"{o}dinger equation:
\begin{equation}\label{e55}
-\varepsilon^2 \Delta u+V(y)u=u^p, \qquad \mathrm{in} \qquad \mathbb R^\mathcal{N}.
\end{equation}
In 2003, Ambrosetti, Malchiodi and Ni \cite{Ambrosetti_Malchiodi_Ni2003} raised the following conjecture:
\begin{conjecture}
Let $\Gamma$ be a $k$-dimensional submanifold in $\mathbb R^\mathcal{N}$ and a nondegenerate critical point of the following functional
\[\mathcal{K}(\Gamma)=\int_\Gamma V^{\frac{p+1}{p-1}-\frac12(\mathcal{N}-k)}dvol,\]
where $1<p<\frac{n+2-k}{n-2+k}$ for $\mathcal{N}\ge 3$ and $p>1$ for $\mathcal{N}=2$.
Then there exists a family of solutions to \eqref{e55} concentrating along $\Gamma$ at least for a subsequence $\varepsilon=\varepsilon_j\to 0$.
\end{conjecture}
del Pino, Kowalczyk and Wei \cite{delPino_Kowalczyk_Wei2007} first showed that this conjecture holds in the case of $\mathcal{N}=2$ and $k=1$.  Wang, Wei and Yang \cite{Wang_Wei_Yang2011} proved that this conjecture in the case of $\mathcal{N}\ge3$ and $k=\mathcal{N}-1$.  And Mahmoudi, Sanchez and Yao \cite{Mahmoudi_Sanchez_Yao2015} proved this conjecture is valid for all cases.

However, the following more general Neumann version of \eqref{e55} also has some solutions concentrating on high dimensional set:
\begin{equation}\label{e90}
\left\{\begin{array}{ll}
\varepsilon^2 \Div(\nabla_{\mathfrak{a}(y)}u)-V(y)u+u^p=0, \qquad u>0, &\mbox{in $\Omega$,}\\
\nabla_{\mathfrak{a}(y)} u\cdot \mathbf{n}=0,  &\mbox{on $\partial\Omega$,}
\end{array}\right.
\end{equation}
where $\Omega\subset \mathbb R^2$ is a smooth bounded domain, $\mathbf{n}$ is the unit outward normal of $\partial \Omega$,
\[\mathfrak{a}(y)=(\mathfrak{a}_1(y),\mathfrak{a}_2(y), \cdots, \mathfrak{a}_N(y))\]
and
\[\nabla_{\mathfrak{a}(y)}u=(\mathfrak{a}_1(y)u_{y_1}, \mathfrak{a}_2(y)u_{y_2}, \cdots, \mathfrak{a}_N(y)u_{y_N}).\]
In the case of $\mathfrak{a}\equiv 1$ and $V(y)\equiv 1$, Wei and Yang \cite{Wei_Yang2008} constructed a sequence of solutions concentrating near a segment $\Gamma_2$ in $\Omega$. The segment intersects $\partial \Omega$ with right angle and separates $\Omega$ into two parts. Unlike the solutions constructed in \cite{delPino_Kowalczyk_Wei2007,Wang_Wei_Yang2011,Mahmoudi_Sanchez_Yao2015}, the solutions constructed in \cite{Wei_Yang2008}  have multiple  concentration layers.
Wei, Xu and Yang \cite{Wei_Xu_Yang2018} considered the case that $\mathfrak{a}(y)\equiv 1$. Let $\Gamma\subset\bar{\Omega}$ be a curve intersecting $\partial \Omega$ with right angle and separating $\Omega$ into two parts. Provided $\Gamma$ is a nondegenerate critical point of $\int_\Gamma V^{\frac{p+1}{p-1}-\frac12}$, they constructed a  sequence of solutions to \eqref{e90} clustering along $\Gamma$. Recently, Wei and Yang \cite{Wei_Yang2021} extended the result in \cite{Wei_Yang2008} into the general case. They  constructed a sequence of solutions to \eqref{e90} with multiple
concentration layers which concentrate near a closed curve $\Gamma_0\subset\Omega$ or a curve $\Gamma_1$ intersecting $\partial \Omega$ with right angle and separating $\Omega$ into two parts. Moreover,  $\Gamma_0$ and $\Gamma_1$ are all the nondegenerate geodesics embedded into the Riemannian manifold $\mathbb R^2$ with the metric $V^{\frac{2(p+1)}{p-1}-1}[\mathfrak{a}_2(y)dy_1^2+\mathfrak{a}_1(y)dy_2^2]$.

Inspired by \cite{Wei_Yang2021,Wei_Yang2008}, we consider the following problem
\begin{equation}\label{2}
\left\{\begin{array}{ll}
-\Div (A(x)\nabla u)=|u|^p-t\mathbf{\Psi}(x), &\mbox{in $\Omega$,} \\
u=0, & \mbox{on $\partial \Omega$},
\end{array}
\right.
\end{equation}
where $\Omega \subset \mathbb R^2$ is a smooth bounded domain, $t>0$ is a constant, and  $\mathbf{\Psi}(x)$ is an eigenfunction corresponding to the first Dirichlet eigenvalue of the operator $\mathfrak{L}(u)=-\Div (A(x)\nabla u)$ on $\Omega$.
Moreover, $A(x)=\{A_{ij}(x)\}_{2\times 2}$ is a symmetric positive defined matrix function, satisfying
\begin{equation}\label{e98}
\lambda |\alpha|^2 \leq \langle A(x)\alpha, \alpha \rangle \leq \Lambda |\alpha|^2,\qquad \mathrm{for} \qquad x,\alpha \in \mathbb R^n.
\end{equation}
We notice $\Div(\nabla_{\mathfrak{a}(y)}u)$ is a special form of $\Div (A(x)\nabla u)$. Based on previous work, we suspect whether \eqref{2} has a similar solution to that constructed in \cite{Wei_Yang2021,Wei_Yang2008}.

Our motivation for writing this paper is twofold. First, we plan to construct solutions to \eqref{2} with multiple concentration layers. Second, we explore the influence of the matrix $A(y)$ to the high dimensional concentration behaviours of the solutions to \eqref{2}.

Let $\varepsilon^2=t^{-(p-1)/p}$. It is easy to get that $u$ is a solution of \eqref{2} if and only if $t^{-\frac1p}u$ is the solution of the following problem
\begin{equation}\label{1}
\left\{\begin{array}{ll}
-\varepsilon^2\Div (A(x)\nabla u)=|u|^p-\mathbf{\Psi}(x), &\mbox{in $\Omega$,} \\
u=0, & \mbox{on $\partial \Omega$},
\end{array}
\right.
\end{equation}
Then we get the following theorem.

Baraket \textit{et. al.} \cite{Baraket_Khemiri_Mahmoudi_Messaoudi2018} considered the following Neumann problem:
\begin{equation}\label{e53}
\left\{\begin{array}{ll}
-\Delta u=|u|^p-t\psi (x), &\mbox{in $\Omega$,} \\
\frac{\partial u}{\partial \mathbf{n}}=0,&\mbox{on $\partial \Omega$},
\end{array}\right.
\end{equation}
where $\Omega\subset\mathbb R^\mathcal{N}$ is a smooth bound domain and $\mathbf{n}$ is the unit outward normal vector of $\partial\Omega$.
Under the condition $\mathcal{N}=2$, they proved that \eqref{e53} has a solution $u_t$ concentrating along a curve $\Gamma\subset \bar{\Omega}$ with $t>0$ large enough. The curve $\Gamma$ intersects $\partial\Omega$ with right angle and divides $\Omega$ into two part. Moreover $\Gamma$ is a non-degenerate critical point of the functional $\mathcal{K}_2(\Gamma)=\int_{\Gamma} \psi^{\frac{p+3}{2p}}(x)dvol$. However, under the condition that $\psi\equiv1$ and $\mathcal{N}\ge 2$, Bendahou, Khemiri and Mahmoudi \cite{Bendahou_Khemiri_Mahmoudi2020} constructed a family of new solutions to \eqref{e53}, which has large number of spikes concentrating along an interior straight line in $\Omega$ for $t\to+\infty$. Using the similar method, Ao, Fu and Liu \cite{Ao_Fu_Liu2022} constructed a similar type of solutions which concentrate along a segment of boundary $\partial \Omega$ in the two dimensional case.

However, the following more general Neumann version of \eqref{e55} also has some solutions concentrating on high dimensional set:
\begin{equation}\label{e90}
\left\{\begin{array}{ll}
\varepsilon^2 \Div(\nabla_{\mathfrak{a}(y)}u)-V(y)u+u^p=0, \qquad u>0, &\mbox{in $\Omega$,}\\
\nabla_{\mathfrak{a}(y)} u\cdot \mathbf{n}=0, &\mbox{on $\partial\Omega$,}
\end{array}\right.
\end{equation}
where $\Omega\subset \mathbb R^2$ is a smooth bounded domain, $\mathbf{n}$ is the unit outward normal of $\partial \Omega$,
\[\mathfrak{a}(y)=(\mathfrak{a}_1(y),\mathfrak{a}_2(y), \cdots, \mathfrak{a}_N(y))\]
and
\[\nabla_{\mathfrak{a}(y)}u=(\mathfrak{a}_1(y)u_{y_1}, \mathfrak{a}_2(y)u_{y_2}, \cdots, \mathfrak{a}_N(y)u_{y_N}).\]
In the case of $\mathfrak{a}\equiv 1$ and $V(y)\equiv 1$, Wei and Yang \cite{Wei_Yang2008} constructed a sequence of solutions concentrating near a segment $\Gamma_2$ in $\Omega$. The segment intersects $\partial \Omega$ with right angle and separates $\Omega$ into two parts. Unlike the solutions constructed in \cite{delPino_Kowalczyk_Wei2007,Wang_Wei_Yang2011,Mahmoudi_Sanchez_Yao2015}, the solutions constructed in \cite{Wei_Yang2008} have multiple concentration layers.
Wei, Xu and Yang \cite{Wei_Xu_Yang2018} considered the case that $\mathfrak{a}(y)\equiv 1$. Let $\Gamma\subset\bar{\Omega}$ be a curve intersecting $\partial \Omega$ with right angle and separating $\Omega$ into two parts. Provided $\Gamma$ is a nondegenerate critical point of $\int_\Gamma V^{\frac{p+1}{p-1}-\frac12}$, they constructed a sequence of solutions to \eqref{e90} clustering along $\Gamma$. Recently, Wei and Yang \cite{Wei_Yang2021} extended the result in \cite{Wei_Yang2008} into the general case. They constructed a sequence of solutions to \eqref{e90} with multiple
concentration layers which concentrate near a closed curve $\Gamma_0\subset\Omega$ or a curve $\Gamma_1$ intersecting $\partial \Omega$ with right angle and separating $\Omega$ into two parts. Moreover, $\Gamma_0$ and $\Gamma_1$ are all the nondegenerate geodesics embedded into the Riemannian manifold $\mathbb R^2$ with the metric $V^{\frac{2(p+1)}{p-1}-1}[\mathfrak{a}_2(y)dy_1^2+\mathfrak{a}_1(y)dy_2^2]$.

Inspired by \cite{Wei_Yang2021,Wei_Yang2008}, we consider the following problem
\begin{equation}\label{2}
\left\{\begin{array}{ll}
-\Div (A(x)\nabla u)=|u|^p-t\mathbf{\Psi}(x), &\mbox{in $\Omega$,} \\
u=0, & \mbox{on $\partial \Omega$},
\end{array}
\right.
\end{equation}
where $\Omega \subset \mathbb R^2$ is a smooth bounded domain, $t>0$ is a constant, and $\mathbf{\Psi}(x)$ is an eigenfunction corresponding to the first Dirichlet eigenvalue of the operator $\mathfrak{L}(u)=-\Div (A(x)\nabla u)$ on $\Omega$.
Moreover, $A(x)=\{A_{ij}(x)\}_{2\times 2}$ is a symmetric positive defined matrix function, satisfying
\begin{equation}\label{e98}
\lambda |\alpha|^2 \leq \langle A(x)\alpha, \alpha \rangle \leq \Lambda |\alpha|^2,\qquad \mathrm{for} \qquad x,\alpha \in \mathbb R^n.
\end{equation}
We notice $\Div(\nabla_{\mathfrak{a}(y)}u)$ is a special form of $\Div (A(x)\nabla u)$. Based on previous work, we suspect whether \eqref{2} has a similar solution to that constructed in \cite{Wei_Yang2021,Wei_Yang2008}.

Our motivation for writing this paper is twofold. First, we plan to construct solutions to \eqref{2} with multiple concentration layers. Second, we explore the influence of the matrix $A(y)$ to the high dimensional concentration behaviours of the solutions to \eqref{2}.

Let $\varepsilon^2=t^{-(p-1)/p}$. It is easy to get that $u$ is a solution of \eqref{2} if and only if $t^{-\frac1p}u$ is the solution of the following problem
\begin{equation}\label{1}
\left\{\begin{array}{ll}
-\varepsilon^2\Div (A(x)\nabla u)=|u|^p-\mathbf{\Psi}(x), &\mbox{in $\Omega$,} \\
u=0, & \mbox{on $\partial \Omega$},
\end{array}
\right.
\end{equation}
Then we get the following theorem.

Our motivation for writing this paper is twofold. First, we plan to construct solutions to \eqref{2} with multiple concentration layers. Second, we explore the influence of the matrix $A(y)$ to the high dimensional concentration behaviours of the solutions to \eqref{2}.

Multiplying the both sides of \eqref{e71} by $w_{ji,x}$ and $Z_{ji}$, respectively and integrating, we get
\[a_{11}\tilde{\alpha}^{1-p}\Lambda_1''=\int_\R hw_{ji,x}dx+c_{ji}\int_\R w_x^2dx\]
and
\[a_{11}\tilde{\alpha}^{1-p}\Lambda_2''+\lambda_0 \Lambda_2=\int_\R hZ_{ji}dx+d_{ji}.\]
Then we get
\[\|c_{ji}\|_{L^2(0,1/\varepsilon)}\le C\|\Lambda_1\|_{H^2(0,1/\varepsilon)}+\|h\|_{L^2(\mathfrak{S})}\]
and
\[\|d_{ji}\|_{L^2(0,1/\varepsilon)}\le C\|\Lambda_2\|_{H^2(0,1/\varepsilon)}+\|h\|_{L^2(\mathfrak{S})}.\]
From this fact, we have
\begin{eqnarray*}
&&\|c_{j2}(w_{j1,x}-w_{j2,x})+d_{j2}(Z_{j1}-Z_{j2})\|_{L^2(\mathfrak{S})} \\
&\le& \left[\|c_{ji}\|_{L^2(0,1/\varepsilon)}+\|d_{ji}\|_{L^2(0,1/\varepsilon)}\right] \|\mathbf{f}_1-\mathbf{f}_2\|_{H^2(0,1)} \\
&\le& C\|\mathbf{f}_1-\mathbf{f}_2\|_{H^2(0,1)}\left[\|h\|_{L^2(\mathfrak{S})}+\|\Lambda_1\|_{H^2(0, 1/\varepsilon)}+\|\Lambda_2\|_{H^2(0, 1/\varepsilon)}\right].
\end{eqnarray*}
Denote
\[\tilde{\Lambda}_1(z)=-\int_\R \phi_2(z,x)(w_{j1,x}-w_{j2,x})dx, \quad \mathrm{and} \quad \tilde{\Lambda}_2(z)=-\int_\R \phi_2(z,x)(Z_{j1}-Z_{j2})dx\]
From direct computation, we get
\[\|\tilde{\Lambda}_1\|_{H^2(0,1/\varepsilon)}+\|\tilde{\Lambda}_2\|_{H^2(0,1/\varepsilon)}\le C\|\phi_2\|_{H^2(\mathfrak{S})}\|\mathbf{f}_1-\mathbf{f}_2\|_{H^2(0,1)}\]
and
\[\|p[|w_{j1}-1|^{p-2}(w_{j1}-1)-|w_{j2}-1|^{p-2}(w_{j2}-1)]\phi_2\|_{L^2(\mathfrak{S})}\le C\|f_1-f_2\|_{H^2(0,1)}\|\phi_2\|_{H^2(\mathfrak{S})}.\]
From the priori estimate in \eqref{e21}, we arrive at
\[\|\phi^*\|_{H^2(S)}\le C\|\mathbf{f}_1-\mathbf{f}_2\|_{H^2(0,1)}\left[\|h\|_{L^2(\mathfrak{S})}+\|\Lambda_1\|_{H^2(0, 1/\varepsilon)}+\|\Lambda_2\|_{H^2(0, 1/\varepsilon)}\right].\]
Hence \eqref{e72} holds.
\end{proof}
\begin{proposition}\label{pre1}
Problem \eqref{48} has a unique solution $\phi=\tilde{T}(h)$, where $\tilde{T}$ is a linear operator and $\phi$ satisfies the following estimate
\begin{equation}\label{e74}
\|\phi\|_{H^2(\mathfrak{S})}\leq C\|h\|_{L^2(\mathfrak{S})}.
\end{equation}
Moreover, $\tilde{T}$ is Lipschitz continuous on $\mathbf{f}$ and $\mathbf{e}$:
\begin{equation}\label{e73}
\|\tilde{T}_{\mathbf{f}_1,\mathbf{e}_1}-\tilde{T}_{\mathbf{f}_2,\mathbf{e}_2}\|\le C\left[ \|\mathbf{f}_1-\mathbf{f}_2\|_{H^2(\mathfrak{S})}+\|\mathbf{e}_1-\mathbf{e}_2\|_*\right].
\end{equation}
\end{proposition}
\begin{proof}
According to the argument above, we only need to solve the system \eqref{50}-\eqref{52} and the problem \eqref{54}.
With Lemma \ref{lme4}, we first consider problem \eqref{50}-\eqref{52} for fixed function $\tilde{\psi}$. It is written into the following problem
\begin{eqnarray}\label{e22}
\nonumber \tilde{\phi}_j&=&T_j\left[\tilde{\chi}_j h+p\left[|w_j-1|^{p-2}(w_j-1)-|\mathcal{V}-1|^{p-2}(\mathcal{V}-1)\right]\tilde{\chi}_j \tilde{\phi}_j \right. \\ &&\left.-p\left[|\mathcal{V}-1|^{p-2}(\mathcal{V}-1)+1\right]\tilde{\chi}_j\tilde{\psi}, \Lambda_{j1}, \Lambda_{j2}\right], \qquad j=1,2,\cdots, N.
\end{eqnarray}
From tedious computation and Sobolev imbbeding theorem in \cite[Theorem 8.8]{Brezis2011}, we get
\[\|\Lambda_{j1}\|_{H^2(0,1/\varepsilon)}+\|\Lambda_{j2}\|_{H^2(0,1/\varepsilon)}\leq C\varepsilon^{1/2}\sum_{k=1}^N\|\tilde{\phi}_k\|_{H^2(\mathfrak{S})}+C\|\tilde{\psi}\|_{H^2(\mathfrak{S})},\]
\[\|p[|\mathcal{V}-1|^{p-2}(\mathcal{V}-1)+1]\tilde{\chi}_j\tilde{\psi}\|_{L^2(\mathfrak{S})}\le C\|\tilde{\psi}\|_{H^2(\mathfrak{S})}\]
and
\[\|p\left[|w_j-1|^{p-2}(w_j-1)-|\mathcal{V}-1|^{p-2}(\mathcal{V}-1)\right]\tilde{\chi}_j \tilde{\phi}_j\|_{L^2(\mathfrak{S})}\leq C\varepsilon^{1/2}\|\tilde{\phi}_j\|_{H^2(\mathfrak{S})}.\]
Using the fixed point theorem, we get \eqref{e22} has a unique solution $\tilde{\phi}_j=\tilde{\phi}_j(\tilde{\psi})$, $j=1,2,\cdots,N$. Moreover $\tilde{\phi}_j$ satisfies the following estimate
\begin{equation}\label{e24}
\|\tilde{\phi}_j\|_{H^2(\mathfrak{S})}\leq C\left[\|h\|_{L^2(\mathfrak{S})}+\|\tilde{\psi}\|_{H^2(\mathfrak{S})}\right].
\end{equation}
It is easy to get $\phi_j$ is Lipschitz dependent on $\psi$:
\begin{equation}\label{e70}
\|\tilde{\phi}_j(\tilde{\psi}_1)-\tilde{\phi}_j(\tilde{\psi}_2)\|_{H^2(\mathfrak{S})}\le C\|\tilde{\psi}_1-\tilde{\psi}_2\|_{H^2(\mathfrak{S})}.
\end{equation}

For the other terms in \eqref{77}, we define
\[\Lambda_1=\int_\R p\eta_\delta^\varepsilon \left[|\mathcal{V}-\tilde{\alpha}^{-1}\bar{\mathbf{q}}|^{p-2}(\mathcal{V}-\tilde{\alpha}^{-1}\bar{\mathbf{q}})+|\tilde{\alpha}^{-1}\bar{\mathbf{q}}|^{p-2}(\tilde{\alpha}^{-1}\bar{\mathbf{q}}) \right]\hat{\psi}( \hat{\varphi})w_{j,x}dx\]
and
\[\Lambda_2=\int_\R \eta_\delta^\varepsilon \hat{N}(\hat{\varphi}+\hat{\psi}(\hat{\varphi}))w_{j,x}dx.\]
From direct computation, we get
\[\|\Lambda_1\|_{L^2(0,1)}\leq C\|\psi(\hat{\varphi})\|_{L^\infty }\le C(\varepsilon^{3/2-\mu})^{\min\{p,2\}}\le C\varepsilon^{\frac52+\mu_1}\]
and
\[\|\Lambda_2\|_{L^2(0,1)}\leq C\varepsilon^{1/2}(\varepsilon^{3/2-\mu})^{\min\{p,2\}}+(\varepsilon^{3/2-\mu})^{2\min\{p,2\}}\le C\varepsilon^{\frac52+\mu_1}.\]\
From \eqref{e94}, we rewrite \eqref{77} into the following form
\begin{eqnarray}\label{73}
\nonumber&&\varepsilon^2\alpha^{1-p}\beta\left\{-a_{11}f_j''+\left[a_{22}-b_{11}-a_{11}\left(\beta^{-1}\beta'+2\alpha^{-1}\alpha'\right)\right]f_j'+\left[a_{22}(\beta^{-1}\beta'+2\alpha^{-1}\alpha') +b_{22}-a_{33}\right]f_j\right. \\
\nonumber&&+\left[\frac{p+3}2\alpha^{p-2}\beta^{-2} \mathbf{q}_{tt}+\frac{p+2}{2(p+3)}\alpha^{1-p}\beta^2(a_{32})^2-\frac{p+1}{p+3}\alpha^{1-p}\beta^2 a_{32}b_{21}-\frac2{p+3}\alpha^{1-p}\beta^2 (b_{21})^2\right]f_j \\
\nonumber&&\left.+\left[-2\varepsilon a_{11}e_j' f_j'+2\varepsilon a_{22}e_j' f_j\right]\left(\int_\R w_x^2 dx\right)^{-1}\int_\R Z_x w_xdx\right\} \\
&&C_0 p\alpha_p\left[e^{-\sqrt{p}\beta(f_j-f_{j-1})}-e^{-\sqrt{p}\beta (f_{j+1}-f_j)}\right]=\varepsilon^2 M_{1j\varepsilon}.
\end{eqnarray}
Using the same procedure and \eqref{e95}, we get \eqref{82} is written into the following form
\begin{equation}\label{e80}
\varepsilon(\varepsilon^2a_{11}\alpha^{1-p}e_j''+\lambda_0 e_j) +p\alpha_p C_1\left[e^{-\sqrt{p}\beta(f_j-f_{j-1})}+e^{-\sqrt{p}\beta(f_{j+1}-f_j)}\right]=\varepsilon^2 M_{2j\varepsilon}.
\end{equation}
In the expression \eqref{73} and \eqref{e80}, $M_{ij\varepsilon}=A_{ij\varepsilon}+K_{ij\varepsilon}$, $i=1,2$, where $K_{ij\varepsilon}$ is a compact operator and $A_{ij\varepsilon}$ is a Lipschitz operator. It satisfies the following estimates
\[\|A_{ij\varepsilon}(\mathbf{f}_1,\mathbf{e}_1)-A_{ij\varepsilon}(\mathbf{f}_2,\mathbf{e}_2)\|_{L^2(0,1)}\leq C\varepsilon^{\frac12+\mu_1}\left[\|\mathbf{f}_1-\mathbf{f}_2\|_{H^2(0,1)}+\|\mathbf{e}_1-\mathbf{e}_2\|_*\right]\]
and
\begin{equation}\label{e49}
\|A_{ij\varepsilon}\|_{L^2(0,1)}\leq C\varepsilon^{\frac12+\mu_1}, \qquad \|K_{ij\varepsilon}\|_{L^2(0,1)}\leq C\varepsilon^{\frac12+\mu_1}.
\end{equation}
\section{Proof of Theorem}\label{sect8}
To prove our main theorem, we only need to find a solution to \eqref{73}-\eqref{e80}. To do job, we need some priori estimate.

\begin{remark}

The condition that  $\Gamma$ is a non-degenerate critical point of the functional $\mathcal{K}(\Gamma)$ is equivalent to the condition that $\Gamma$ is a non-degenerate geodesic embedded into the Riemannian manifold $(\mathbb R^2,\tilde{\mathfrak{g}})$, where $\tilde{\mathfrak{g}}(y)=\mathbf{\Psi}^{\frac{p+3}p}[A_{22}dy_1^2-2A_{12}(y)dy_1dy_2+A_{11}dy_2^2]$.
\end{remark}
We will use the infinite dimensional reduction method developed in \cite{delPino_Kowalczyk_Wei2007,delPino_Kowalczyk_Wei2008,Wei_Yang2021} to prove Theorem \ref{th1}. Infinite dimensional reduction method is used to find solutions to elliptic partial differential equations concentrating on high dimensional sets. To construct a solution to \eqref{1}, we first investigate the negative solution of \eqref{1} and its property in Section \ref{sectA}. However, we need to overcome the difficulty that $-\Div (A(x)\nabla u)$ is not a symmetric operator( see Lemma \ref{lm1} for detail).  To get a local approximate solution to \eqref{1}, we need to expand the operator $\Div(A(\varepsilon y)\nabla v)$ in a proper way. Wei and Yang \cite{Wei_Yang2021} developed a method to expand $\Div(\nabla_{\mathfrak{a}}(y)u)$. However we find a easier way to expand this operator. Similar to \cite{Wei_Yang2021}, to construct local approximate solution with $N$ concentration layers, we need a fine asymptotic estimate of the function $w$. But $w$ does not have the explicit expression(comparing \cite[(3.1)]{Wei_Yang2021}). To overcome this difficulty, we use the method in \cite{Gidas_Ni_Nirenberg1981} to get the asymptotic estimate of the function $w$ in the Section \ref{sectB}.

This paper is organized as follows. In Section \ref{sect1}, we set up a new modified Fermi coordinate in some neighborhood of $\Gamma$ and write down a local form of the operator $\Div (A(\varepsilon y)\nabla u)$ in the stretched modified Fermi coordinate. In Section \ref{sect2}, we find a local approximate solution to \eqref{1}. Then we conduct the gluing procedure in Section \ref{sect5} and the outer problem is solved. To solve the inner problem, we construct linear and nonlinear theory in Section \ref{sect6} and Section \ref{sect9}, respectively. From some involved calculation in Section \ref{sect4}, we reduce the problem into some partial differential equations in Section \ref{sect7}.  At last, Theorem \ref{th1} is proven in Section \ref{sect8}.

To prove Theorem~\ref{th1}, the authors "use the infinite dimensional reduction method" from \cite{delPino_Kowalczyk_Wei2007,delPino_Kowalczyk_Wei2008,Wei_Yang2021} to construct solutions of \eqref{1} concentrating on (high-dimensional) sets.

Key ingredients/steps described in the introduction:
\begin{itemize}
\item "First investigate the negative solution of \eqref{1} and its property" (Section~\ref{sectA}), while overcoming that $-\Div(A(x)\nabla u)$ "is not a symmetric operator" (Lemma~\ref{lm1}).
\item To build a local approximation, one must "expand the operator $\Div(A(\varepsilon y)\nabla v)$ in a proper way"; although Wei--Yang \cite{Wei_Yang2021} expand $\Div(\nabla_{\mathfrak{a}}(y)u)$, the authors state they "find a easier way to expand this operator."
\item As in \cite{Wei_Yang2021}, constructing a local approximate solution with $N$ layers requires "a fine asymptotic estimate of the function $w$"; since $w$ has no explicit formula, they "use the method in \cite{Gidas_Ni_Nirenberg1981} to get the asymptotic estimate of the function $w$" (Section~\ref{sectB}).
\item Paper roadmap implementing the reduction/gluing scheme:
  \begin{itemize}
  \item Section~\ref{sect1}: set up a "new modified Fermi coordinate" near $\Gamma$ and derive a local form of $\Div(A(\varepsilon y)\nabla u)$ in stretched coordinates.
  \item Section~\ref{sect2}: construct a local approximate solution to \eqref{1}.
  \item Section~\ref{sect5}: perform the "gluing procedure" and solve the outer problem.
  \item Sections~\ref{sect6} and \ref{sect9}: develop linear and nonlinear theory to solve the inner problem.
  \item Using "involved calculation" in Section~\ref{sect4}, "reduce the problem into some partial differential equations" in Section~\ref{sect7}.
  \item "At last, Theorem~\ref{th1} is proven in Section~\ref{sect8}."
  \end{itemize}
\end{itemize}

To prove the main theorem, the authors “use the infinite dimensional reduction method” from \cite{delPino_Kowalczyk_Wei2007,delPino_Kowalczyk_Wei2008,Wei_Yang2021} to construct solutions of
\begin{equation}\label{1}
\left\{\begin{array}{ll}
-\varepsilon^2\Div (A(x)\nabla u)=|u|^p-\mathbf{\Psi}(x), &\mbox{in $\Omega$,} \\
u=0, & \mbox{on $\partial \Omega$},
\end{array}
\right.
\end{equation}
concentrating on (high-dimensional) sets.

Key ingredients/steps described in the introduction:
\begin{itemize}
\item First they “investigate the negative solution” of the boundary value problem
\begin{equation}\label{1}
\left\{\begin{array}{ll}
-\varepsilon^2\Div (A(x)\nabla u)=|u|^p-\mathbf{\Psi}(x), &\mbox{in $\Omega$,} \\
u=0, & \mbox{on $\partial \Omega$},
\end{array}
\right.
\end{equation}
and its properties (later in the paper), while overcoming that $-\Div(A(x)\nabla u)$ “is not a symmetric operator.” In particular, they use the following auxiliary lemma.

\begin{lemma}\label{lm1}
If $a\ge c$ and $u_\varepsilon$ is the minimizer of problem \eqref{6},  $u_\varepsilon(x)\to a$ on compact sets in $B_1(0)$, as $\varepsilon \to 0$.
\end{lemma}

\item To build a local approximation, one must “expand the operator $\Div(A(\varepsilon y)\nabla v)$ in a proper way”; although Wei--Yang \cite{Wei_Yang2021} expand $\Div(\nabla_{\mathfrak{a}}(y)u)$, the authors state they “find a easier way to expand this operator.”

\item As in \cite{Wei_Yang2021}, constructing a local approximate solution with $N$ layers requires “a fine asymptotic estimate of the function $w$”; since $w$ has no explicit formula, they “use the method in \cite{Gidas_Ni_Nirenberg1981} to get the asymptotic estimate of the function $w$” (later in the paper).

\item Paper roadmap implementing the reduction/gluing scheme:
  \begin{itemize}
  \item Next they set up a “new modified Fermi coordinate” near $\Gamma$ and derive a local form of $\Div(A(\varepsilon y)\nabla u)$ in stretched coordinates.
  \item Next they construct a local approximate solution to the equation above.
  \item Next they perform the “gluing procedure” and solve the outer problem.
  \item Later they develop linear and nonlinear theory to solve the inner problem.
  \item Using “involved calculation” earlier, they “reduce the problem into some partial differential equations” later.
  \item This completes the proof of the main theorem.
  \end{itemize}
\end{itemize}

\label{th1}
Let $p>3$.
Assume that $\Gamma$ is a simple closed smooth curve with unit length in $\Omega$, and it is also a non-degenerate critical point of the functional
\begin{equation}\label{e91}
\mathcal{K}(\Gamma)=\int_\Gamma \mathbf{\Psi}^{\frac{p+3}{2p}}dvol_{\mathfrak{g}},
\end{equation}
where $\mathfrak{g}(X,Y)=\langle A^*X,Y\rangle$ is a Riemannian metric on $\mathbb R^2$ and $A^*$ is the adjoint matrix for $A$. We also assume that the following inequality holds on $\Gamma$:
\begin{eqnarray*}
\Upsilon_0&=&-\alpha^{1-p}\left[ 2\alpha^{-1}\alpha'a_{22}+\beta^{-1}\beta'b_{11}+b_{22}-a_{33}+\frac{p+3}2\alpha^{p-2}\beta^{-2}\mathbf{q}_{tt}+\frac{p+2}{2(p+3)}\alpha^{1-p} \beta^2 (a_{32})^2\right. \\
&&\left.-\frac{p+1}{p+3}\alpha^{1-p}\beta^2 a_{32}b_{21}-\frac{2}{p+3}\alpha^{1-p}\beta^2(b_{21})^2 +a_{11}\left(\beta^{-1}\beta''+2\alpha^{-1}\beta^{-1}\alpha'\beta'-\beta^{-2}(\beta')^2\right)\right]>0,
\end{eqnarray*}
where the coefficients $a_{ij}$ and $b_{ij}$ are given by the following lemma.

\begin{lemma}\label{lm8}
Under the stretched modified Fermi coordinate defined by $\Phi_\varepsilon(z,s)$, we get
\begin{eqnarray*}
\nonumber\Div(A(\varepsilon y)\nabla v)&=& a_{11}(\varepsilon z)v_{zz}+2\varepsilon s a_{22}(\varepsilon z)v_{sz} +\left(a_{31}(\varepsilon z)+\varepsilon s a_{32}(\varepsilon z) +\varepsilon^2 s^2 a_{33}(\varepsilon z)\right)v_{ss} \\
&& +\varepsilon b_{11}(\varepsilon z)v_z +\left(\varepsilon b_{21}(\varepsilon z)+\varepsilon^2 s b_{22}(\varepsilon z)\right) v_s +B_0(v),
\end{eqnarray*}
where
\[a_{11}(\theta)=\frac{\langle{A^*}n,n\rangle}{1-\langle\gamma',n\rangle^2},\qquad a_{22}(\theta)=-\frac{\langle{A^*}n,n'\rangle}{1-\langle\gamma',n\rangle^2}-\frac{\langle{A^*_t}n,\gamma'\rangle}{1-\langle\gamma',n\rangle^2}, \]
\[a_{31}(\theta)=\frac{\langle{A^*}\gamma',\gamma'\rangle}{1-\langle\gamma',n\rangle^2}, \qquad a_{32}=\frac{\langle{A^*_t}\gamma',\gamma'\rangle}{1-\langle\gamma',n\rangle^2}+ \frac{2\langle{A^*}\gamma',n'\rangle}{1-\langle\gamma',n\rangle^2}-\frac{2\langle\gamma',n'\rangle\langle{A^*}\gamma',\gamma'\rangle}{\left(1-\langle\gamma',n\rangle^2\right)^2}, \]
\begin{eqnarray*}
a_{33}(\theta)&=& \frac1{1-\langle\gamma',n\rangle^2}\left(\langle{A^*}n',n'\rangle +2\langle{A^*_t}\gamma',n'\rangle+\frac12\langle{A^*_{tt}}\gamma',\gamma'\rangle\right)  \\
&&-\frac{2\langle\gamma',n'\rangle}{\left(1-\langle\gamma',n\rangle^2\right)^2}\left(\langle{A^*_t} \gamma',\gamma'\rangle +2\langle{A^*}\gamma',n'\rangle\right)  \\
&&+\left(\frac{4\langle\gamma',n'\rangle^2}{\left(1-\langle\gamma',n\rangle^2\right)^3}- \frac{\langle n',n'\rangle}{\left(1-\langle\gamma',n\rangle^2\right)^2}\right)\langle{A^*}\gamma',\gamma'\rangle,
\end{eqnarray*}
\[b_{11}(\theta)=\left[\frac{\langle{A^*}n,n\rangle}{1-\langle\gamma',n\rangle^2}\right]_{\theta}-\frac12\left(\frac1{1-\langle\gamma',n\rangle^2}\right)_\theta \langle{A^*}n,n\rangle -\frac{\langle{A^*}n,n'\rangle}{1-\langle\gamma',n\rangle^2}-\frac{\langle{A^*_t}n,\gamma'\rangle}{1-\langle\gamma',n\rangle^2},\]
\[b_{21}(\theta)=\frac{\langle\gamma',n'\rangle}{1-\langle\gamma',n\rangle^2}a_{31}+a_{32},\]
\[b_{22}(\theta)=\frac12\frac{(1-\langle\gamma',n\rangle^2)_\theta}{1-\langle\gamma',n\rangle^2}a_{22}+\left[\frac{\langle n',n'\rangle}{1-\langle\gamma',n\rangle^2}-\frac{2\langle\gamma',n'\rangle^2}{(1-\langle\gamma',n\rangle^2)^2}\right]a_{31}+\frac{\langle\gamma',n'\rangle}{1-\langle\gamma', n\rangle^2}a_{32} +\partial_\theta a_{22}+2a_{33}\]
and
\[B_0(v)=\varepsilon a_{12}(\varepsilon z,s)v_{zz}+\varepsilon^2 a_{23}(\varepsilon z.s)v_{zs}+\varepsilon^3a_{33}(\varepsilon z,s)v_{ss} +\varepsilon^2 b_{12}(\varepsilon z,s) v_z +\varepsilon^3 b_{23}(\varepsilon z,s)v_s.\]
In the expression above, funtions $a_{12}$, $a_{23}$, $a_{33}$, $b_{12}$ and $b_{23}$ are smooth functions satisfying the following estimate
\[|a_{12}(\varepsilon z, s)|\le C(1+|s|), \quad |a_{23}(\varepsilon z, s)|\le C(1+|s|^2), \quad |a_{33}(\varepsilon z, s)|\le C(1+|s|^3),\]
and
\[|b_{12}(\varepsilon z, s)|\le C(1+|s|), \quad |b_{23}(\varepsilon z, s)|\le C(1+|s|^2).\]
\end{lemma}

Moreover,
\begin{equation}\label{80}
\alpha(\theta)=\mathbf{q}(\theta,0), \qquad \beta(\theta)=\frac{\left[\mathbf{q}(\theta,0)\right]^{\frac{p-1}2}}{\left[a_{31}(\theta)\right]^{\frac12}}.
\end{equation}
and
\begin{equation}\label{e88}
\mathbf{q}(\theta, t)=\mathbf{\Psi}^{\frac1p}(\gamma(\theta)+tn(\theta)).
\end{equation}

Then for each  integer $N>0$, there exists a sequence of $\varepsilon$, \textit{i.e.}, $\{\varepsilon_l\}$ converging to $0$ such that
\begin{equation}\label{1}
\left\{\begin{array}{ll}
-\varepsilon^2\Div (A(x)\nabla u)=|u|^p-\mathbf{\Psi}(x), &\mbox{in $\Omega$,} \\
u=0, & \mbox{on $\partial \Omega$},
\end{array}
\right.
\end{equation}
has a positive solutions $u_{\varepsilon_l}$ with exactly $N$ concentration layers at mutual distance $O(\varepsilon_l|\log\varepsilon_l|)$, the center of mass of $N$ concentration layers collapses to $\Gamma$ at speed $\varepsilon_l^{1+\mu}$ for small positive constant $\mu\in(0,1/2)$. More precisely $u_{\varepsilon_l}$ has the form
\begin{eqnarray*}
u_{\varepsilon_l}(y_1,y_2)\approx -\mathbf{\Psi}^{\frac1p}(y_1,y_2)+ \mathbf{\Psi}^{\frac1p}(\gamma(\theta))\sum_{k=1}^N w\left(\left[\frac{\mathbf{\Psi}^{\frac{p-1}p}(1-\langle\gamma',n\rangle)}{\langle A^*n,n\rangle}\right]^{\frac12}\frac{t-\varepsilon_l f_k}{\varepsilon_l}\right),
\end{eqnarray*}
where $\gamma$ is a natural parametrization of $\Gamma$, $n$ is the unit vector defined by
\begin{equation}\label{e102}
n(\theta)=\frac{A(\gamma(\theta))\nu(\theta)}{|A(\gamma(\theta))\nu(\theta)|}.
\end{equation}
and $w$ is  the unique solution of the following problem
\begin{equation}\label{29}
-w''=|1-w|^p -1, \qquad w>0 \quad \mathrm{in} \quad \mathbb R, \qquad w'(0)=w(\pm \infty).
\end{equation}
In the expression above, the functions $f_j$'s satisfy
\[\|f_j\|_{L^\infty(0,1)}\le C|\log\varepsilon_l|^2,\qquad \sum_{j=1}^N f_j =O(\frac1{|\log\varepsilon_l|^{\frac32}}),\]
\[\min_{1\le j\le N}(f_{j+1}-f_j)\approx\frac2{\sqrt{p}}|\log\varepsilon_l|\left[\frac{\langle A^*n,n\rangle}{\Psi^{\frac{p-1}p}(1-\langle\gamma',n\rangle)}\right]^{\frac12}\]
and solves the Jacobi-Toda system for $j=1,2,\cdots,N$
\begin{eqnarray*}
\nonumber&&\varepsilon^2\alpha^{1-p}\beta\left\{-a_{11}f_j''+\left[a_{22}-b_{11}-a_{11}\left(\beta^{-1}\beta'+2\alpha^{-1}\alpha'\right)\right]f_j'+\left[a_{22}(\beta^{-1}\beta'+2\alpha^{-1}\alpha') +b_{22}-a_{33}\right]f_j\right. \\
\nonumber&&+\left[\frac{p+3}2\alpha^{p-2}\beta^{-2} \mathbf{q}_{tt}+\frac{p+2}{2(p+3)}\alpha^{1-p}\beta^2(a_{32})^2-\frac{p+1}{p+3}\alpha^{1-p}\beta^2 a_{32}b_{21}-\frac2{p+3}\alpha^{1-p}\beta^2 (b_{21})^2\right]f_j \\
&&C_0 p\alpha_p\left[e^{-\sqrt{p}\beta(f_j-f_{j-1})}-e^{-\sqrt{p}\beta (f_{j+1}-f_j)}\right]\approx0.
\end{eqnarray*}

Question: Let Ω⊂R2 be a smooth bounded domain, let A(x) be a smooth symmetric positive-definite 2×2 matrix field, and let Ψ be the first Dirichlet eigenfunction of L(u)=-Div(A(x)∇u) on Ω. Let Γ⊂Ω be a simple closed smooth curve of unit length with natural parametrization γ(θ), let ν(θ) be its Euclidean unit normal, and define n(θ)=A(γ(θ))ν(θ)/|A(γ(θ))ν(θ)|, q(θ,t)=Ψ(γ(θ)+tn(θ))^(1/p), α(θ)=q(θ,0), β(θ)=q(θ,0)^((p-1)/2)/a31(θ)^(1/2), and the Riemannian metric g(X,Y)=⟨A*X,Y⟩. Assume p>3, Γ is a nondegenerate critical point of K(Γ)=∫_Γ Ψ^((p+3)/(2p)) dvol_g, and on Γ one has Υ0>0, where Υ0=-α^(1-p)[2α^(-1)α'a22+β^(-1)β'b11+b22-a33+((p+3)/2)α^(p-2)β^(-2)qtt+((p+2)/(2(p+3)))α^(1-p)β^2(a32)^2-((p+1)/(p+3))α^(1-p)β^2 a32 b21-(2/(p+3))α^(1-p)β^2(b21)^2+a11(β^(-1)β''+2α^(-1)β^(-1)α'β'-β^(-2)(β')^2)], and a11,a22,a31,a32,a33,b11,b21,b22 are the coefficients arising from the stretched modified Fermi-coordinate expansion of Div(A(εy)∇v) along Γ. Under these assumptions, which statement holds for the Dirichlet problem -ε2Div(A(x)∇u)=|u|^p-Ψ(x) in Ω, u=0 on ∂Ω?

For each integer N>0, there exists a sequence {ε_l} with ε_l→0 such that the problem -ε_l2Div(A(x)∇u)=|u|^p-Ψ(x) in Ω, u=0 on ∂Ω, has a positive solution u_{ε_l} with exactly N concentration layers at mutual distance O(ε_l|log ε_l|), and the center of mass of the N layers collapses to Γ at speed ε_l^(1+μ) for some small constant μ∈(0,1/2). More precisely, in the normal coordinates y=γ(θ)+tn(θ), one has u_{ε_l}(y1,y2)≈-Ψ(y1,y2)^(1/p)+Ψ(γ(θ))^(1/p)∑_{k=1}^N w(( [Ψ(γ(θ))^((p-1)/p)(1-⟨γ',n⟩)/⟨A*n,n⟩]^(1/2) (t-ε_l f_k)/ε_l )), where w is the unique solution of -w''=|1-w|^p-1, w>0 in R, w'(0)=w(±∞). In addition, the functions f_j satisfy ||f_j||_{L^∞(0,1)}≤C|log ε_l|^2, ∑_{j=1}^N f_j=O(1/|log ε_l|^(3/2)), and min_{1≤j≤N}(f_{j+1}-f_j)≈(2/√p)|log ε_l|[⟨A*n,n⟩/(Ψ^((p-1)/p)(1-⟨γ',n⟩))]^(1/2); moreover, for j=1,2,...,N they solve the Jacobi-Toda system ε_l2α^(1-p)β{ -a11 f_j'' + [a22-b11-a11(β^(-1)β'+2α^(-1)α')]f_j' + [a22(β^(-1)β'+2α^(-1)α')+b22-a33]f_j + [((p+3)/2)α^(p-2)β^(-2)qtt + ((p+2)/(2(p+3)))α^(1-p)β^2(a32)^2 - ((p+1)/(p+3))α^(1-p)β^2 a32 b21 - (2/(p+3))α^(1-p)β^2(b21)^2]f_j } + C0 p α_p [e^(-√p β(f_j-f_{j-1})) - e^(-√p β(f_{j+1}-f_j))]≈0.

Choices:
B.
There exists varepsilon_0>0 such that for every integer N>0 and every varepsilon
a0,varepsilon_0), the problem -varepsilon^2Div(A(x)u)=|u|^p-si(x) in 9, u=0 on 99, has a positive solution u_varepsilon with exactly N concentration layers at mutual distance O(varepsilon|log varepsilon|), whose center of mass collapses to at speed varepsilon^(1+) for some small constant (0,1/2). More precisely, in the normal coordinates y=()+tn(), one has u_varepsilon(y_1,y_2)- si(y_1,y_2)^(1/p)+si(())^(1/p)_{k=1}^N w(([ si(())^((p-1)/p)(1-',n)/A*n,n]^(1/2)(t-varepsilon f_k)/varepsilon)), and the functions f_j satisfy the same bounds and JacobiToda system as in option A.
C. For each integer N>0, there exists a sequence {\varepsilon_l} with \varepsilon_l\to0 such that the problem -\varepsilon_l^2\Div(A(x)\nabla u)=|u|^p-\Psi(x) in \Omega, u=0 on \partial\Omega, has a positive solution u_{\varepsilon_l} with exactly N concentration layers whose center of mass collapses to \Gamma. More precisely, in the normal coordinates y=\gamma(\theta)+tn(\theta), one has u_{\varepsilon_l}(y_1,y_2)\approx-\Psi(y_1,y_2)^{1/p}+\Psi(\gamma(\theta))^{1/p}\sum_{k=1}^N w\!\left(\left[\frac{\Psi(\gamma(\theta))^{(p-1)/p}(1-\langle\gamma',n\rangle)}{\langle A^*n,n\rangle}\right]^{1/2}\frac{t-\varepsilon_l f_k}{\varepsilon_l}\right),
D. For each integer N>0, there exists a sequence {\varepsilon_l} with \varepsilon_l\to0 such that the problem -\varepsilon_l^2\Div(A(x)\nabla u)=|u|^p-\Psi(x) in \Omega, u=0 on \partial\Omega, has a positive solution u_{\varepsilon_l} with exactly N concentration layers at mutual distance O(\varepsilon_l), and the center of mass of the N layers collapses to \Gamma at speed \varepsilon_l^{1+\mu} for some small constant \mu\in(0,1/2). More precisely, in the normal coordinates y=\gamma(\theta)+tn(\theta), one has the same leading-order profile as in option A, while the functions f_j satisfy \|f_j\|_{L^\infty(0,1)}\le C|\log\varepsilon_l|^2, \sum_{j=1}^N f_j=O(1/|\log\varepsilon_l|^{3/2}), and \min_{1\le j\le N}(f_{j+1}-f_j)\approx \frac{2}{\sqrt p}\left[\frac{\langle A^*n,n\rangle}{\Psi^{(p-1)/p}(1-\langle\gamma',n\rangle)}\right]^{1/2}; moreover, for j=1,2,\dots,N they solve the same Jacobi-Toda system as in option A.
E. For each integer N>0, there exists a sequence {\varepsilon_l} with \varepsilon_l\to0 such that the problem -\varepsilon_l^2\Div(A(x)\nabla u)=|u|^p-\Psi(x) in \Omega, u=0 on \partial\Omega, has a positive solution u_{\varepsilon_l} with exactly N concentration layers at mutual distance O(\varepsilon_l|\log \varepsilon_l|), and the center of mass of the N layers collapses to \Gamma at speed \varepsilon_l^{1+\mu} for some small constant \mu\in(0,1/2). More precisely, in the normal coordinates y=\gamma(\theta)+tn(\theta), one has u_{\varepsilon_l}(y_1,y_2)\approx-\Psi(y_1,y_2)^{1/p}+\Psi(\gamma(\theta))^{1/p}\sum_{k=1}^N w\!\left(\left[\frac{\Psi(\gamma(\theta))^{(p-1)/p}(1-\langle\gamma',n\rangle)}{\langle A^*n,n\rangle}\right]^{1/2}\frac{t-\varepsilon_l f_k}{\varepsilon_l}\right), where w is the unique solution of -w''=|1-w|^p-1, w>0 in \mathbb R, w'(0)=w(\pm\infty). In addition, the functions f_j satisfy the same bounds as in option A, and for j=1,2,\dots,N they solve the Jacobi-Toda system obtained from option A after replacing the interaction term \;C_0 p\alpha_p\big[e^{-\sqrt p\beta(f_j-f_{j-1})}-e^{-\sqrt p\beta(f_{j+1}-f_j)}\big]\; by \;C_0 p\alpha_p\big[e^{-\sqrt p\beta(f_j-f_{j-1})}+e^{-\sqrt p\beta(f_{j+1}-f_j)}\big]\approx0.

dropped quantitative layer-spacing speed and Jacobi-Toda details

The stem gives assumptions and asks for the resulting existence statement, but it does not itself reveal the conclusion. There is no explicit answer leakage beyond framing the theorem setup.

This is essentially a theorem-recall item: the stem lists the full hypotheses and asks which conclusion holds. The correct option is basically the theorem conclusion restated in full.

Some reasoning/attention is needed to distinguish subtle variants such as sequence vs. all small ε, exact spacing scale, and anisotropic geometric factors. However, the task mainly rewards recognition of the exact theorem statement rather than genuine derivation.

The distractors are mathematically plausible and target realistic failure modes: overstrong quantifiers, weakened conclusions, incorrect spacing scale, and incorrect geometric normalization/profile scaling.

Technically well-constructed with strong distractors, but it is mostly a direct theorem restatement and therefore only weakly tests generative mathematical reasoning.


\label{thm:main}
For a skew-symmetrizable matrix $B$, the following are equivalent:
\begin{itemize}
\item[$(1)$] $B$ is of finite type.
\item[$(2)$] The $g$-fan $\mathcal{F}(B)$ is complete, that is, its support $|\mathcal{F}(B)|=\mathbb{R}^n$.
\item[$(3)$] The support $|\mathcal{F}(B)|$ contains all lattice points in $\mathbb{Z}^n$.
\end{itemize}

Cluster algebras \cite{FZ02} have been studied from many points of view. One of the basic objects in this theory is the notion of $g$-vectors \cite{FZ07}.

Let $B$ be a skew-symmetrizable matrix. The $g$-vectors associated with $B$ span simplicial cones in $\mathbb{R}^n$. These cones form a simplicial fan, which is called the $g$-fan of $B$ (see \cite{GHKK18,Re14}). The geometry of this fan encodes the structure of the associated cluster algebra.

A skew-symmetrizable matrix $B$ is said to be of \emph{finite type} if it has only finitely many $g$-vectors, or equivalently, finitely many cluster variables. Finite type matrices are well understood and play an important role in cluster theory.

It is then natural to ask how the finite type property of $B$ is reflected in the geometry of its $g$-fan. In particular, one may ask whether the $g$-fan covers the whole space $\mathbb{R}^n$, or whether it can be characterized by the property of containing all lattice points in $\mathbb{Z}^n$. This question is also motivated by \cite[Question~3.49]{D17} in representation theory. In this paper, we give a complete answer to this question in the framework of $g$-fans arising from skew-symmetrizable matrices. More precisely, we prove the following theorem.

\begin{abstract}
We study the $g$-fan associated with a skew-symmetrizable matrix in the sense of cluster algebras. We show that a skew-symmetrizable matrix is of finite type if and only if its $g$-fan is complete; equivalently (as we show), its support contains all lattice points.
\end{abstract}

It is then natural to ask how the finite type property of $B$ is reflected in the geometry of its $g$-fan. In particular, one may ask whether the $g$-fan covers the whole space $\mathbb{R}^n$, or whether it can be characterized by the property of containing all lattice points in $\mathbb{Z}^n$. This question is also motivated by \cite[Question~3.49]{D17} in representation theory. In this paper, we give a complete answer to this question in the framework of $g$-fans arising from skew-symmetrizable matrices. More precisely, we prove the following theorem.

The implication $(1)\Rightarrow(2)$ is known (Theorem~\ref{thm:Reading}). For skew-symmetric matrices, the implication $(2)\Rightarrow(1)$ was proved in \cite{HY25} via categorification using Jacobian algebras. In \cite{Y23}, we suggested that this implication might be proved by methods similar to those in \cite[Proposition~4.9]{A21} in general, but no complete proof was available. In this paper, we give a complete proof using scattering diagrams and their pull-back constructions.

An $n\times n$ skew-symmetrizable matrix $B$ is called
\begin{itemize}
\item \emph{mutation equivalent} to a matrix $B'$ if it is obtained from $B'$ by a finite sequence of mutations;
\item \emph{$2$-finite} if for any matrix $B'=(b'_{ij})$ mutation equivalent to $B$, $|b'_{ij}b'_{ji}|\le 3$ for all $i,j$.
\end{itemize}

\begin{definition}
A \emph{$g$-vector seed} is a pair $(C,(\mathbf{g}_1,\ldots,\mathbf{g}_n))$ consisting of the following data:
\begin{itemize}
\item[$(1)$] $C$ is a $2n\times n$ integer matrix whose upper part is skew-symmetrizable.
\item[$(2)$] $\mathbf{g}_1,\ldots,\mathbf{g}_n\in\mathbb Z^n$.
\end{itemize}
The tuple $(\mathbf{g}_1,\ldots,\mathbf{g}_n)$ is called the \emph{$g$-vector tuple}.
\end{definition}

\begin{example}\label{exam:rank 2 g-fan}
Any nonzero $2 \times 2$ skew-symmetrizable matrix is given by
\[
B_{b,c}:=\begin{bmatrix}0 & c \\ -b & 0\end{bmatrix},
\]
where $b$ and $c$ are integers and $bc > 0$. The $g$-fan $\mathcal{F}(B_{b,c})$ is well-known (see e.g. \cite[Example~1.15]{GHKK18} or \cite[Section~2]{N24}). In fact, for $bc \ge 4$, $\mathcal{F}(B_{b,c})$ contains infinitely many rays converging to the rays $r_{\pm}$ of slope $(-bc \pm \sqrt{bc(bc-4)})/2c$ (see Figure~\ref{fig:rank 2 g-vector fans}). In this case, the ray of slope $-b/2$ is contained in $(\mathbb{R}^2\setminus|\mathcal{F}(B_{b,c})|)\cup\{0\}$. In particular, the lattice point $(-2,b)$ is contained in $\mathbb{Z}^n\setminus|\mathcal{F}(B_{b,c})|$.
\end{example}

\begin{theorem}[{\cite[Theorem~10.6]{Re14}}]\label{thm:Reading}
If $B$ is of finite type, then $|\mathcal{F}(B)|=\mathbb{R}^n$.
\end{theorem}

\begin{proof}[Proof of Theorem~\ref{thm:main}]
The implication $(1)\Rightarrow(2)$ follows from Theorem~\ref{thm:Reading}, and $(2)\Rightarrow(3)$ clearly holds. Finally, we prove $(3)\Rightarrow(1)$. By Theorem~\ref{thm:fin 2-fin}, if $B$ is not of finite type, it is mutation equivalent to $B'$ such that $B'_I=B_{b,c}$ with $bc\ge 4$ for some subset $I \subset \{1,\ldots,n\}$. By Proposition~\ref{prop:rank 2 no lattice pt}, $\mathbb{Z}^n\setminus|\mathcal{F}(B')|\neq\emptyset$, which implies that $\mathbb{Z}^n\setminus|\mathcal{F}(B)|\neq\emptyset$ by Corollary~\ref{cor:preserve lattice pt}. This completes the proof.
\end{proof}

The post-theorem text indicates: $(1)\Rightarrow(2)$ is already known (Theorem~\ref{thm:Reading}). For skew-symmetric matrices, $(2)\Rightarrow(1)$ was proved in \cite{HY25} "via categorification using Jacobian algebras." A possible approach "by methods similar to those in \cite[Proposition~4.9]{A21}" had been suggested in \cite{Y23} but lacked a complete proof. The paper claims to provide a complete proof "using scattering diagrams and their pull-back constructions."

The post-theorem text indicates: $(1)\Rightarrow(2)$ is already known. \begin{theorem}[{\cite[Theorem~10.6]{Re14}}]\label{thm:Reading}
If $B$ is of finite type, then $|\mathcal{F}(B)|=\mathbb{R}^n$.
\end{theorem}
For skew-symmetric matrices, $(2)\Rightarrow(1)$ was proved in Assem and Yurikusa, “via categorification using Jacobian algebras.” A possible approach “by methods similar to those in \cite[Proposition~4.9]{A21}” had been suggested in Yurikusa, but lacked a complete proof. The paper claims to provide a complete proof “using scattering diagrams and their pull-back constructions.”

Biconditional or Equivalence
Classification or Bijection

Question: Let $B$ be a skew-symmetrizable $n\times n$ matrix. Its associated $g$-vectors span simplicial cones in $\mathbb{R}^n$, forming the $g$-fan $\mathcal{F}(B)$, and $|\mathcal{F}(B)|$ denotes the support of this fan (the union of all its cones). A matrix $B$ is called of finite type if it has only finitely many $g$-vectors. Which statement gives the exact characterization of when $B$ is of finite type in terms of its $g$-fan?

$B$ is of finite type if and only if the $g$-fan $\mathcal{F}(B)$ is complete, i.e. $|\mathcal{F}(B)|=\mathbb{R}^n$; equivalently, if and only if $|\mathcal{F}(B)|$ contains every lattice point of $\mathbb{Z}^n$.

Choices:
B. $B$ is of finite type if and only if the support $|\mathcal{F}(B)|$ contains every nonzero lattice point of $\mathbb{Z}^n$; equivalently, the only lattice point that may fail to lie in $|\mathcal{F}(B)|$ is the origin.
C. If $B$ is of finite type, then the $g$-fan $\mathcal{F}(B)$ is complete, i.e. $|\mathcal{F}(B)|=\mathbb{R}^n$.
D. $B$ is of finite type if and only if for every lattice point $\mathbf{m}\in\mathbb{Z}^n$ there exists a matrix $B'$ mutation equivalent to $B$ such that $\mathbf{m}\in |\mathcal{F}(B')|$.
E. $B$ is of finite type if and only if the support $|\mathcal{F}(B)|$ has nonempty interior in $\mathbb{R}^n$; equivalently, if and only if $|\mathcal{F}(B)|$ contains infinitely many lattice points of $\mathbb{Z}^n$.

The stem partially leaks the correct direction by explicitly defining completeness as |F(B)| = R^n, which immediately suggests that any equivalent condition should at least imply containment of all lattice points. This makes choice A stand out, though it does not fully settle equivalence.

The item is not a verbatim restatement, but the correct answer is very close to a weakened reformulation of completeness ('all of R^n' to 'all lattice points in Z^n'). So it avoids full tautology, but only mildly.

Some reasoning is required to distinguish exact equivalence from nearby statements such as 'all but finitely many lattice points,' 'every nonzero lattice point,' or 'density.' However, the explicit completeness condition makes the intended answer comparatively easy to infer.

The distractors are plausible and mathematically meaningful: B tests finite-exception weakening, C tests omission of the origin, D tests quantifier/mutation confusion, and E tests density versus equality/completeness. These align well with common failure modes.

A reasonably well-constructed MCQ with strong distractors, but the stem gives a substantial hint toward the correct answer and the correct option is close to a reformulation of the stated completeness condition rather than a fully independent conclusion.


Question: Let $B$ be a skew-symmetrizable $n\times n$ matrix. Its associated $g$-vectors span simplicial cones in $\mathbb{R}^n$, forming the $g$-fan $\mathcal{F}(B)$, and $|\mathcal{F}(B)|$ denotes the support of this fan (the union of all its cones). A matrix $B$ is called of finite type if it has only finitely many $g$-vectors. Which statement gives the exact characterization of when $B$ is of finite type in terms of its $g$-fan?

Choices:
B. $B$ is of finite type if and only if the support $|\mathcal{F}(B)|$ contains every nonzero lattice point of $\mathbb{Z}^n$; equivalently, the only lattice point that may fail to lie in $|\mathcal{F}(B)|$ is the origin.
C. If $B$ is of finite type, then the $g$-fan $\mathcal{F}(B)$ is complete, i.e. $|\mathcal{F}(B)|=\mathbb{R}^n$.
D. $B$ is of finite type if and only if for every lattice point $\mathbf{m}\in\mathbb{Z}^n$ there exists a matrix $B'$ mutation equivalent to $B$ such that $\mathbf{m}\in |\mathcal{F}(B')|$.
E. $B$ is of finite type if and only if the support $|\mathcal{F}(B)|$ has nonempty interior in $\mathbb{R}^n$; equivalently, if and only if $|\mathcal{F}(B)|$ contains infinitely many lattice points of $\mathbb{Z}^n$.

The stem does not explicitly state the characterization. It defines the objects involved and asks for the exact finite-type criterion, but the correct equivalence is not leaked.

The item is quite close to a direct theorem-recall question: it asks for the exact characterization of finite type in terms of the g-fan. However, it is not pure verbatim restatement because the options introduce nearby variants and weakened/altered formulations.

Some reasoning is needed to distinguish the exact iff statement from a weaker true implication and from quantifier/property confusions. Still, the task is mainly recognition of a known theorem rather than substantial generative mathematical reasoning.

The distractors are strong: one is a weaker true statement, and the others reflect plausible mathematical errors involving lattice-point coverage, mutation-equivalence, and confusing completeness with nonempty interior. They are distinct and diagnostically useful.

A solid theorem-recognition MCQ with high-quality distractors and no major answer leakage, but it primarily tests precise recall/recognition rather than deep generative reasoning.


\begin{theorem}\label{Thm}
Let $\Ga$ be a connected tetravalent graph, let $(M, H)$ be a maximal $(\frac{1}{2}, t)$-pair of $\Ga$ with $t\geq1$, and let $K=\mathrm{Core}_H(M)$. Take $u\in V(\Ga)$. Then one of the followings holds.
\begin{enumerate}[\rm(a)]
\item\label{Thmcase:t=3} $t=3$, $(H/K, M/K, H_u, M_u)=(A_{72}, A_{71}, S_4\times S_3, \mz_2)$, and $\Ga$ is a cover of $\Ga_K$.
\item\label{Thmcase:t=2} $t=2$, $(H/K, M/K, H_u, M_u)= (S_5, \AGL_1(5), A_4, \mz_2)$, and $\Ga$ is a cover of $\Ga_K\cong \K_{5, 5}-5\K_2$.
\item\label{Thmcase:t=1} $t=1$, and either $M$ is normal in $H$, or $M_u^{\Ga(u)}\cong\mz_2$ is non-normal in $H_u^{\Ga(u)}\cong D_8$ such that $K$ is semiregular on $V(\Ga)$ with $|V(\Ga_K)|\neq2^\ell$ for any integer $\ell\geq0$ and one of the followings holds:
\begin{enumerate}[\rm (c1)]
\item\label{Thmcase:t=1_cover} $K$ is the kernel of $H$ acting on $V(\Ga_K)$, $\Ga$ is a cover of $\Ga_K$, and the pair $(M/K, H/K)$ is a maximal $(\frac{1}{2}, 1)$-pair of $\Ga_K$;
\item\label{Thmcase:t=1_cycle} $K$ is the kernel of $M$ acting on $V(\Ga_K)$, $\Ga_K\cong C_r$ with $r=|V(\Ga_K)|$, and $M/K\cong D_{2r}$.
\end{enumerate}
\end{enumerate}
Moreover, examples for each of the above cases exist.
\end{theorem}

All graphs considered in this paper are finite, undirected, regular and simple. Let $\Ga$ be a graph. For a positive integer $s$, a sequence $(u_0, u_1, \ldots, u_s)$ of vertices of $\Ga$ is said to be an \emph{$s$-arc} if $u_i$ is adjacent to $u_{i+1}$ for $0\leq i\leq s-1$ and $u_{j-1}\neq u_{j+1}$ for $1\leq j\leq s-1$. A $1$-arc is simply an \emph{arc}. We use $V(\Gamma)$, $E(\Gamma)$, $A(\Ga)$ and $\Aut(\Gamma)$ to denote its vertex set, edge set, arc set and (full) automorphism group, respectively.

Let $G$ be a subgroup of $\Aut(\Ga)$. If $G$ is transitive on $V(\Ga)$ and on $E(\Ga)$ but not on $A(\Ga)$, then we say that $\Ga$ is \emph{ $G$-half-arc-transitive}. In the literature, ``half-arc-transitive'' is also abbreviated as ``$\frac{1}{2}$-arc-transitive'' or ``HAT''. When $\Ga$ is $\Aut(\Ga)$-half-arc-transitive, we omit the prefix and simply say that $\Ga$ is half-arc-transitive (or HAT).
The study of HAT graphs was initiated by Tutte in \cite{Tutte1966}. Over the past half-century, substantial progress has been made, and we refer the reader to the survey articles \cite{CPS2015,Marusic1998} for an overview. Within this broader development, considerable attention has been devoted to tetravalent HAT graphs; see, for example, \cite{M-DM,MM1999,MN-JGT,MP1999,MX1997,PW-JCTB2007,RSparl2019,Spiga-Xia2021,Xia-2021,Zhou-JACO,Zhou-PAMS}.

For a positive integer $t$, we say that $\Ga$ is \emph{$(G,t)$-arc-transitive} if $G$ is transitive on the set of $t$-arcs of $\Ga$. If such a subgroup $G$ exists, then a $(G,t)$-arc-transitive graph will be simply called an \emph{$t$-arc-transitive graph}.
For $s\in\mathbb{N}\cup\{\frac{1}{2}\}$, if $\Ga$ is $(G,s)$-arc-transitive but not $(G,\lfloor s+1\rfloor)$-arc-transitive, then we say that $\Ga$ is \emph{$(G,s)$-transitive}. A pair $(M, H)$ of subgroups of $\Aut(\Ga)$ is called an \emph{$(s,t)$-pair} of $\Ga$ if $M\leq H\leq\Aut(\Ga)$ and $\Ga$ is $(M,s)$-transitive and $(H,t)$-transitive.
Such a pair $(M, H)$ is called \emph{maximal} if $M$ is a maximal subgroup of $H$.
Note that, $\Gamma$ admits a maximal $\bigl(\frac{1}{2},t\bigr)$-pair for some positive integer $t$ if and only if $\Gamma$ is $M$-half-arc-transitive and $H$-arc-transitive for some groups $M<H$ of automorphisms.
The following problem is both natural and of considerable interest.

\begin{problem}\label{prob}
Classify maximal $(s,t)$-pairs for all $s\in\mathbb{N}\cup\{\tfrac{1}{2}\}$ and $t\in\mathbb{N}$ with $s<t$.
\end{problem}

In this paper we address Problem~\ref{prob} for tetravalent graphs in the case $s=\tfrac{1}{2}$.
Our motivation is to better understand the automorphism groups of tetravalent $G$-HAT graphs $\Gamma$.
Knowing whether a given graph $\Gamma$ is HAT is often difficult, largely because determining $\Aut(\Gamma)$ is hard in general.
If $\Gamma$ is not half-arc-transitive, then $\Aut(\Gamma)$ is arc-transitive; consequently, $\Gamma$ admits a maximal $\bigl(\frac{1}{2},t\bigr)$-pair $(M,H)$ for some positive integer $t$ with $G\le M$.

Another motivation comes from studying the symmetry of the graph of alternating cycles associated with a $G$-HAT graph.
Let $\Ga$ be a tetravalent $M$-HAT graph for some $M\leq\Aut(\Ga)$. It is easy to see that the action of
$M$ on the arc set of $\Ga$ has two orbits, denoted by $O_M^+(\Ga)$ and $O_M^-(\Ga)$. For each $\epsilon\in \{+,-\}$,
and each edge of $\Ga$, the orbit $O_M^\epsilon(\Ga)$ contains exactly one of the two arcs corresponding to this edge.
Hence $O_M^+(\Ga)$ and $O_M^-(\Ga)$ give two opposite orientations of the edges of $\Ga$ and hence give two digraphs with the same vertex set as $\Ga$, denoted by $D_M^+(\Ga)$ and $D_M^-(\Ga)$.
A cycle of $\Ga$ is called an \emph{$M$-alternating cycle} if consecutive edges along the cycle have opposite orientations in $D_M^+(\Ga)$ (and hence also opposite orientations in $D_M^-(\Ga)$). In a seminal paper~\cite{Marusic1998} in 1998, Maru\v si\v c proved that all alternating cycles of the tetravalent $M$-HAT graph $\Ga$ have the same length, half of which is called the \emph{$M$-radius} of $\Ga$, and any two alternating cycles of $\Ga$ with nonempty intersection share the same number of vertices. This number is called the \emph{$M$-attachment number} of $\Ga$, denoted by $\att_M(\Ga)$.

The \emph{graph of $M$-alternating cycles} of $\Ga$, denoted $\Alt_M(\Ga)$, is the graph whose vertices are the $M$-alternating cycles of $\Ga$, with two such cycles adjacent whenever they have a non-empty intersection (see~\cite{MP1999,PSparl2017}). In~\cite{RSparl2019}, Rivera and \v{S}parl initiated the study of the automorphism group of $\Alt_M(\Ga)$ for tetravalent $M$-HAT graphs $\Ga$.
They proved that, if either $\att_M(\Ga)$ equals the $M$-radius or $\Aut(\Ga)$ contains an element interchanging $O_M^+(\Ga)$ and $O_M^-(\Ga)$, then $\Alt_M(\Ga)$ is arc-transitive (see~\cite[Proposition~5.7]{RSparl2019}). However, a tetravalent arc-transitive $M$-HAT graph need not admit an automorphism interchanging $O_M^+(\Ga)$ and $O_M^-(\Ga)$, and so this criterion does not always apply to assert the arc-transitivity of $\Alt_M(\Ga)$. Despite this, all computed examples in~\cite{RSparl2019} still have $\Alt_M(\Ga)$ arc-transitive. This led the authors to ask the following question for potential weaker conditions ensuring the arc-transitivity of $\Alt_M(\Ga)$, and similarly for the related quotient graph $\Ga_{\mathcal{B}(M)}$ (with respect to a certain $M$-invariant partition $\mathcal{B}(M)$ of $V(\Ga)$; see~\cite{MP1999} or~\cite{RSparl2019} for definition). In particular, $\Ga_{\mathcal{B}(M)}=\Ga$ whenever $\att_M(\Ga)=1$.

\begin{question}[{\cite[Question~5.8]{RSparl2019}}]\label{que:Sparl}
Is it true that, if $\Ga$ is a tetravalent arc-transitive graph which is $M$-HAT for some $M\leq\Aut(\Ga)$, the graphs $\Alt_M(\Ga)$ and $\Ga_{\mathcal{B}(M)}$ are both arc-transitive?
\end{question}

Note that any larger HAT group $N>M$ has the same orbits on $A(\Gamma)$ with $M$, hence induces the same graph of alternating cycles. We may let $M$ be maximal subject to that $\Ga$ is $M$-HAT. Take $H\leq\Aut(\Ga)$ such that $M$ is maximal in $H$. Then $H$ is arc-transitive on $\Ga$, and so $(M, H)$ is a maximal $(\frac{1}{2}, t)$-pair. If $M$ is normal in $H$, then $H$ preserves $\{O_M^+(\Ga),O_M^-(\Ga)\}$,  and so we deduce from~\cite[Proposition~5.7]{RSparl2019} that $\Alt_M(\Ga)$ is arc-transitive. Thus, the question is reduced to the case that $M$ is non-normal in $H$.

Our main theorem characterizes tetravalent graphs admitting a maximal $(\frac{1}{2}, t)$-pair with $t\geq1$. Recall that, for a subgroup $X$ of a group $Y$, the \emph{core} of $X$ in $Y$, denoted by $\mathrm{Core}_Y(X)$, is the largest normal subgroup of $Y$ contained in $X$.
Let $\Ga$ be a graph. Assume that $G\leq\Aut(\Ga)$ is such that $G$ is transitive on the vertex set of $\Ga$. Let $N$ be a normal subgroup of $G$ such that $N$ is intransitive on $V(\Ga)$. The \emph{normal quotient graph\/} $\Ga_N$ of $\Ga$ relative to $N$ is defined as the graph with vertices the orbits of $N$ on $V(\Ga)$ and with two different orbits adjacent if there exists an edge in $\Ga$ between the vertices lying in those two orbits. If $\Ga_N$ and $\Ga$ have the same valency, then $\Ga$ is a \emph{cover} of $\Ga_N$.

For a positive integer $t$, we say that $\Ga$ is \emph{$(G,t)$-arc-transitive} if $G$ is transitive on the set of $t$-arcs of $\Ga$. If such a subgroup $G$ exists, then a $(G,t)$-arc-transitive graph will be simply called an \emph{$t$-arc-transitive graph}.
For $s\in\mathbb{N}\cup\{\frac{1}{2}\}$, if $\Ga$ is $(G,s)$-arc-transitive but not $(G,\lfloor s+1\rfloor)$-arc-transitive, then we say that $\Ga$ is \emph{$(G,s)$-transitive}. A pair $(M, H)$ of subgroups of $\Aut(\Ga)$ is called an \emph{$(s,t)$-pair} of $\Ga$ if $M\leq H\leq\Aut(\Ga)$ and $\Ga$ is $(M,s)$-transitive and $(H,t)$-transitive.
Such a pair $(M, H)$ is called \emph{maximal} if $M$ is a maximal subgroup of $H$.
Note that, $\Gamma$ admits a maximal $\bigl(\frac{1}{2},t\bigr)$-pair for some positive integer $t$ if and only if $\Gamma$ is $M$-half-arc-transitive and $H$-arc-transitive for some groups $M<H$ of automorphisms.
The following problem is both natural and of considerable interest.

Another motivation comes from studying the symmetry of the graph of alternating cycles associated with a $G$-HAT graph.
Let $\Ga$ be a tetravalent $M$-HAT graph for some $M\leq\Aut(\Ga)$. It is easy to see that the action of
$M$ on the arc set of $\Ga$ has two orbits, denoted by $O_M^+(\Ga)$ and $O_M^-(\Ga)$. For each $\epsilon\in \{+,-\}$,
and each edge of $\Ga$, the orbit $O_M^\epsilon(\Ga)$ contains exactly one of the two arcs corresponding to this edge.
Hence $O_M^+(\Ga)$ and $O_M^-(\Ga)$ give two opposite orientations of the edges of $\Ga$ and hence give two digraphs with the same vertex set as $\Ga$, denoted by $D_M^+(\Ga)$ and $D_M^-(\Ga)$.
A cycle of $\Ga$ is called an \emph{$M$-alternating cycle} if consecutive edges along the cycle have opposite orientations in $D_M^+(\Ga)$ (and hence also opposite orientations in $D_M^-(\Ga)$). In a seminal paper~\cite{Marusic1998} in 1998, Maru\v si\v c proved that all alternating cycles of the tetravalent $M$-HAT graph $\Ga$ have the same length, half of which is called the \emph{$M$-radius} of $\Ga$, and any two alternating cycles of $\Ga$ with nonempty intersection share the same number of vertices. This number is called the \emph{$M$-attachment number} of $\Ga$, denoted by $\att_M(\Ga)$.

The \emph{graph of $M$-alternating cycles} of $\Ga$, denoted $\Alt_M(\Ga)$, is the graph whose vertices are the $M$-alternating cycles of $\Ga$, with two such cycles adjacent whenever they have a non-empty intersection (see~\cite{MP1999,PSparl2017}). In~\cite{RSparl2019}, Rivera and \v{S}parl initiated the study of the automorphism group of $\Alt_M(\Ga)$ for tetravalent $M$-HAT graphs $\Ga$.
They proved that, if either $\att_M(\Ga)$ equals the $M$-radius or $\Aut(\Ga)$ contains an element interchanging $O_M^+(\Ga)$ and $O_M^-(\Ga)$, then $\Alt_M(\Ga)$ is arc-transitive (see~\cite[Proposition~5.7]{RSparl2019}). However, a tetravalent arc-transitive $M$-HAT graph need not admit an automorphism interchanging $O_M^+(\Ga)$ and $O_M^-(\Ga)$, and so this criterion does not always apply to assert the arc-transitivity of $\Alt_M(\Ga)$. Despite this, all computed examples in~\cite{RSparl2019} still have $\Alt_M(\Ga)$ arc-transitive. This led the authors to ask the following question for potential weaker conditions ensuring the arc-transitivity of $\Alt_M(\Ga)$, and similarly for the related quotient graph $\Ga_{\mathcal{B}(M)}$ (with respect to a certain $M$-invariant partition $\mathcal{B}(M)$ of $V(\Ga)$; see~\cite{MP1999} or~\cite{RSparl2019} for definition). In particular, $\Ga_{\mathcal{B}(M)}=\Ga$ whenever $\att_M(\Ga)=1$.

\begin{question}[{\cite[Question~5.8]{RSparl2019}}]\label{que:Sparl}
Is it true that, if $\Ga$ is a tetravalent arc-transitive graph which is $M$-HAT for some $M\leq\Aut(\Ga)$, the graphs $\Alt_M(\Ga)$ and $\Ga_{\mathcal{B}(M)}$ are both arc-transitive?
\end{question}

Note that any larger HAT group $N>M$ has the same orbits on $A(\Gamma)$ with $M$, hence induces the same graph of alternating cycles. We may let $M$ be maximal subject to that $\Ga$ is $M$-HAT. Take $H\leq\Aut(\Ga)$ such that $M$ is maximal in $H$. Then $H$ is arc-transitive on $\Ga$, and so $(M, H)$ is a maximal $(\frac{1}{2}, t)$-pair. If $M$ is normal in $H$, then $H$ preserves $\{O_M^+(\Ga),O_M^-(\Ga)\}$, and so we deduce from~\cite[Proposition~5.7]{RSparl2019} that $\Alt_M(\Ga)$ is arc-transitive. Thus, the question is reduced to the case that $M$ is non-normal in $H$.

Our main theorem characterizes tetravalent graphs admitting a maximal $(\frac{1}{2}, t)$-pair with $t\geq1$. Recall that, for a subgroup $X$ of a group $Y$, the \emph{core} of $X$ in $Y$, denoted by $\mathrm{Core}_Y(X)$, is the largest normal subgroup of $Y$ contained in $X$.
Let $\Ga$ be a graph. Assume that $G\leq\Aut(\Ga)$ is such that $G$ is transitive on the vertex set of $\Ga$. Let $N$ be a normal subgroup of $G$ such that $N$ is intransitive on $V(\Ga)$. The \emph{normal quotient graph\/} $\Ga_N$ of $\Ga$ relative to $N$ is defined as the graph with vertices the orbits of $N$ on $V(\Ga)$ and with two different orbits adjacent if there exists an edge in $\Ga$ between the vertices lying in those two orbits. If $\Ga_N$ and $\Ga$ have the same valency, then $\Ga$ is a \emph{cover} of $\Ga_N$.

Our main theorem characterizes tetravalent graphs admitting a maximal $(\frac{1}{2}, t)$-pair with $t\geq1$. Recall that, for a subgroup $X$ of a group $Y$, the \emph{core} of $X$ in $Y$, denoted by $\mathrm{Core}_Y(X)$, is the largest normal subgroup of $Y$ contained in $X$.
Let $\Ga$ be a graph. Assume that $G\leq\Aut(\Ga)$ is such that $G$ is transitive on the vertex set of $\Ga$. Let $N$ be a normal subgroup of $G$ such that $N$ is intransitive on $V(\Ga)$. The \emph{normal quotient graph\/} $\Ga_N$ of $\Ga$ relative to $N$ is defined as the graph with vertices the orbits of $N$ on $V(\Ga)$ and with two different orbits adjacent if there exists an edge in $\Ga$ between the vertices lying in those two orbits. If $\Ga_N$ and $\Ga$ have the same valency, then $\Ga$ is a \emph{cover} of $\Ga_N$.

Summarized in Section~\ref{sec:sum}, the proof of Theorem~\ref{Thm} is developed in Sections~\ref{sec:proofofmain} and~\ref{sec:examples}. More precisely, we first prove in Section~\ref{sec:proofofmain} that one of~\eqref{Thmcase:t=3}--\eqref{Thmcase:t=1} must occur, and then in Section~\ref{sec:examples} we construct examples for each of the cases~\eqref{Thmcase:t=3},~\eqref{Thmcase:t=2},~\eqref{Thmcase:t=1_cover} and~\eqref{Thmcase:t=1_cycle}. The example for~\eqref{Thmcase:t=1_cover} yields the following corollary, which provides a negative answer to Question~\ref{que:Sparl} of Rivera and \v{S}parl~\cite{RSparl2019}.

\begin{lemma}[{\cite[Theorem~1.1]{AAMPS2016}}]\label{lem:4HAT}
Let $\Ga$ be a tetravalent $G$-HAT graph, and let $N\trianglelefteq G$ such that $N$ has at least three orbits on $V(\Ga)$. Then one of the following holds:
\begin{enumerate}[{\rm (a)}]
 \item $\Ga$ is a cover of $\Ga_N$, $N$ is semiregular on $V(\Ga)$ and is the kernel of $G$ acting on $V(\Ga_N)$, and $\Ga_N$ is $(G/N)$-HAT.
 \item $\Ga_N$ is an $r$-cycle with $r=|V(\Ga_N)|$, and the induced permutation group of $G$ on $V(\Ga_N)$ is $\mz_r$ or $D_{2r}$.
\end{enumerate}
\end{lemma}

\begin{lemma}\label{lem:hat-2at}
Let $\Ga$ be a tetravalent graph, let $u\in V(\Ga)$, let $(M, H)$ be a maximal $(\frac{1}{2}, t)$-pair of $\Ga$ with $t\geq2$, and let $K=\mathrm{Core}_H(M)$. Then $\Ga$ is a cover of $\Ga_K$, and one of the following holds:
\begin{enumerate}[\rm (a)]
\item $t=2$, $(H/K, M/K, H_{u}, M_{u})=(S_5, \AGL_1(5), A_4, \mz_2)$, and $\Gamma_K\cong \K_{5, 5}-5\K_2$;
\item $t=3$, and $(H/K, M/K, H_{u}, M_{u})=(A_{72}, A_{71},S_4\times S_3, \mz_2)$.
\end{enumerate}
\end{lemma}

\begin{lemma}\label{lem:stab:=2}
Let $\Ga$ be a tetravalent graph, let $u\in V(\Ga)$, let $(M, H)$ be a maximal $(\frac{1}{2},1)$-pair of $\Ga$ such that $M_u^{\Ga(u)}\cong\mz_2$ and $H_u^{\Ga(u)}\cong D_8$, and let $K=\mathrm{Core}_H(M)$. Then $M_u^{\Ga(u)}\ntrianglelefteq H_u^{\Ga(u)}$, $K$ is semiregular on $V(\Ga)$ with $|V(\Ga_K)|\neq2^\ell$ for any integer $\ell\geq0$, and one of the following holds:
\begin{enumerate}[\rm (a)]
\item\label{lem:stab:=2-a} $K$ is the kernel of $H$ acting on $V(\Ga_K)$, the pair $(M/K, H/K)$ is a maximal $(\frac{1}{2}, 1)$-pair of $\Ga_K$, and $\Ga$ is a cover of $\Ga_K$;
\item\label{lem:stab:=2-b} $\Ga_K\cong C_r$ with $r=|V(\Ga_K)|$, $M/K\cong D_{2r}$, and $K$ is the kernel of $M$ acting on $V(\Ga_K)$.
\end{enumerate}
\end{lemma}

\begin{example}\label{exam:hat_in_3-trans_non-normal}
Let $M=\langle a,b\mid a^2=b^3=(ab)^2=1\rangle\times\langle c,d\mid c^2=d^3=(cd)^4=1\rangle\cong S_3\times S_4$, let $R$ be the faithful action of $M$ by right multiplication on $V:=[M\,{:}\,\langle a(cd)^2\rangle]$, let $X=\mathrm{Alt}(V)\cong A_{72}$, and let $Y=R(M'\rtimes\langle ac\rangle)\cong \mz_3\rtimes S_4$. Then $Y$ is a regular subgroup of $X$. Let $Z=R((\langle b\rangle\times\langle d\rangle)\rtimes\langle ac^{dc}\rangle)\cong \mz_3^2\rtimes \mz_2$. Computer searching shows that there exists an element $x$ of order $4$ in $\N_X(Z)\cap\mathbf{C}_X(ac^{dc})$ satisfying the following conditions:
\begin{itemize}
\item $x^2=ac^{dc}\in Z$;
\item $Y\cap Y^x=Z$;
\item $X=\langle Y,x\rangle$;
\item $YxY=YS$ for some $S=\{g,g^{-1},g^h,(g^h)^{-1}\}$, where $g\in X_v$ for some $v\in V$ and $h$ is an involution in $X_v$. \end{itemize}

Corollaries~\ref{cor-1} and~\ref{cor-2} follow from Examples~\ref{exam:hat_in_1-trans_altgra} and~\ref{exam:hat_in_3-trans_non-normal}, respectively. Now we prove Theorem~\ref{Thm}.
Let $\Ga$ be a connected tetravalent graph, let $(M, H)$ be a maximal $(\frac{1}{2}, t)$-pair of $\Ga$ with $t\geq1$, let $K=\mathrm{Core}_H(M)$, and let $u\in V(\Ga)$.
If $t\geq2$, then Lemma~\ref{lem:hat-2at} asserts that~\eqref{Thmcase:t=3} or~\eqref{Thmcase:t=2} holds.
Assume that $t=1$. If $M_u^{\Ga(u)}\cong\mz_2$ and $H_u^{\Ga(u)}\cong D_8$, then by Lemma~\ref{lem:stab:=2}, $M_u^{\Ga(u)}\cong\mz_2$ is non-normal in $H_u^{\Ga(u)}\cong D_8$ such that $K$ is semiregular on $V(\Ga)$ with $|V(\Ga_K)|\neq2^\ell$ for any integer $\ell\geq0$ and we have~\eqref{Thmcase:t=1_cover} or~\eqref{Thmcase:t=1_cycle}. If either $M_u^{\Ga(u)}\ncong\mz_2$ or $H_u^{\Ga(u)}\ncong D_8$, then since $(M, H)$ is a $(\frac{1}{2}, 1)$-pair of $\Ga$, it follows that $M_u^{\Ga(u)}$ is normal in
$H_u^{\Ga(u)}$ with $|H_u^{\Ga(u)}:M_u^{\Ga(u)}|=2$. In this case, Lemma~\ref{lem:hat-1t}\,\eqref{lem:hat-1t-c} implies that $M$ is normal in $H$, as in case~\eqref{Thmcase:t=1} of Theorem~\ref{Thm}. Moreover, according to Examples~\ref{exam:hat_in_1-trans_altgra},~\ref{exam:hat_in_3-trans_non-normal},~\ref{exam:hat_in_2-trans} and~\ref{exam:hat_in_1-trans_cycle}, examples for each of the cases~\eqref{Thmcase:t=3},~\eqref{Thmcase:t=2},~\eqref{Thmcase:t=1_cover} and~\eqref{Thmcase:t=1_cycle} of Theorem~\ref{Thm} exist. This completes the proof.

Summarized in Section~\ref{sec:sum}, the proof of Theorem~\ref{Thm} is developed in Sections~\ref{sec:proofofmain} and~\ref{sec:examples}. More precisely, we first prove in Section~\ref{sec:proofofmain} that one of~\eqref{Thmcase:t=3}--\eqref{Thmcase:t=1} must occur, and then in Section~\ref{sec:examples} we construct examples for each of the cases~\eqref{Thmcase:t=3},~\eqref{Thmcase:t=2},~\eqref{Thmcase:t=1_cover} and~\eqref{Thmcase:t=1_cycle}. The example for~\eqref{Thmcase:t=1_cover} yields the following corollary, which provides a negative answer to Question~\ref{que:Sparl} of Rivera and \v{S}parl~\cite{RSparl2019}.

\begin{corollary}\label{cor-1}
There exists a connected tetravalent graph $\Ga$ admiting a maximal $(\frac{1}{2}, 1)$-pair $(M,\Aut(\Gamma))$ such that $\Alt_M(\Ga)$ is half-arc-transitive.
\end{corollary}

Another byproduct of proving Theorem~\ref{Thm} is the discovery of the first example of a tetravalent normal-edge-transitive but non-normal Cayley graph on a nonabelian simple group.
Given a group $G$ and an inverse-closed generating subset $S\subseteq G\setminus\{1\}$, the \emph{Cayley graph} $\Cay(G,S)$ on $G$ is the graph with vertex set $G$ and edge set $\{\{g, sg\} \mid g\in G, s\in S\}$. Denote by $R_G$ the right multiplication action of $G$ on itself, and denote
\[
\Aut(G, S)=\{\a\in\Aut(G)\mid S^\a=S\}.
\]
It is well known~\cite{Godsil1981} that the normalizer of $R_G(G)$ in $\Aut(\Cay(G,S))$ is $R_G(G)\rtimes\Aut(G,S)$. We call $\Cay(G,S)$ \emph{normal-edge-transitive} if this normalizer is transitive on $E(\Cay(G,S))$, and \emph{normal} if this normalizer equals $\Aut(\Cay(G,S))$.
In 1999, Praeger~\cite{Praeger1999} proved that every connected normal-edge-transitive Cayley graph of valency $3$ on a nonabelian simple group is normal. This result was recently extended to all prime valencies in~\cite{Zhang-F-Z-Y}, where it was also shown that there exist connected normal-edge-transitive non-normal Cayley graphs of valency $8$ on nonabelian simple groups. However, it has remained open whether such graphs exist at smaller valencies, particularly valency $4$, the smallest possible valency, despite substantial work on tetravalent edge-transitive Cayley graphs on nonabelian simple groups~\cite{FangLiXu,FangWZ}. Our example for Theorem~\ref{Thm}\,\eqref{Thmcase:t=3} resolves this question by establishing the following:

\begin{corollary}\label{cor-2}
There exists a connected tetravalent normal-edge-transitive non-normal Cayley graph on $A_{71}$.
\end{corollary}

The post-theorem introduction says that “the proof of Theorem~\ref{Thm} is developed in Sections~\ref{sec:proofofmain} and~\ref{sec:examples}.” More precisely: (1) “we first prove in Section~\ref{sec:proofofmain} that one of~\eqref{Thmcase:t=3}--\eqref{Thmcase:t=1} must occur,” and then (2) “in Section~\ref{sec:examples} we construct examples for each of the cases~\eqref{Thmcase:t=3},~\eqref{Thmcase:t=2},~\eqref{Thmcase:t=1_cover} and~\eqref{Thmcase:t=1_cycle}.”

The post-theorem introduction says that “the proof of the main theorem is developed later.” More precisely: (1) “we first prove later that one of the following must occur: \begin{enumerate}[\rm(a)]
\item\label{Thmcase:t=3} $t=3$, $(H/K, M/K, H_u, M_u)=(A_{72}, A_{71}, S_4\times S_3, \mz_2)$, and $\Ga$ is a cover of $\Ga_K$.
\item\label{Thmcase:t=2} $t=2$, $(H/K, M/K, H_u, M_u)= (S_5, \AGL_1(5), A_4, \mz_2)$, and $\Ga$ is a cover of $\Ga_K\cong \K_{5, 5}-5\K_2$.
\item\label{Thmcase:t=1} $t=1$, and either $M$ is normal in $H$, or $M_u^{\Ga(u)}\cong\mz_2$ is non-normal in $H_u^{\Ga(u)}\cong D_8$ such that $K$ is semiregular on $V(\Ga)$ with $|V(\Ga_K)|\neq2^\ell$ for any integer $\ell\geq0$ and one of the followings holds:
\begin{enumerate}[\rm (c1)]
\item\label{Thmcase:t=1_cover} $K$ is the kernel of $H$ acting on $V(\Ga_K)$, $\Ga$ is a cover of $\Ga_K$, and the pair $(M/K, H/K)$ is a maximal $(\frac{1}{2}, 1)$-pair of $\Ga_K$;
\item\label{Thmcase:t=1_cycle} $K$ is the kernel of $M$ acting on $V(\Ga_K)$, $\Ga_K\cong C_r$ with $r=|V(\Ga_K)|$, and $M/K\cong D_{2r}$.
\end{enumerate}\end{enumerate},” and then (2) “later we construct examples for each of the cases given by the items \ref{Thmcase:t=3} and \ref{Thmcase:t=2} above, and by the subcases \begin{enumerate}[\rm (c1)]
\item\label{Thmcase:t=1_cover} $K$ is the kernel of $H$ acting on $V(\Ga_K)$, $\Ga$ is a cover of $\Ga_K$, and the pair $(M/K, H/K)$ is a maximal $(\frac{1}{2}, 1)$-pair of $\Ga_K$;
\item\label{Thmcase:t=1_cycle} $K$ is the kernel of $M$ acting on $V(\Ga_K)$, $\Ga_K\cong C_r$ with $r=|V(\Ga_K)|$, and $M/K\cong D_{2r}$.
\end{enumerate}.”

\label{Thm}
Let $\Ga$ be a connected tetravalent graph, let $(M, H)$ be a maximal $(\frac{1}{2}, t)$-pair of $\Ga$ with $t\geq1$, and let $K=\mathrm{Core}_H(M)$. Take $u\in V(\Ga)$. Then one of the followings holds.
\begin{enumerate}[\rm(a)]
\item\label{Thmcase:t=3} $t=3$, $(H/K, M/K, H_u, M_u)=(A_{72}, A_{71}, S_4\times S_3, \mz_2)$, and $\Ga$ is a cover of $\Ga_K$.
\item\label{Thmcase:t=2} $t=2$, $(H/K, M/K, H_u, M_u)= (S_5, \AGL_1(5), A_4, \mz_2)$, and $\Ga$ is a cover of $\Ga_K\cong \K_{5, 5}-5\K_2$.
\item\label{Thmcase:t=1} $t=1$, and either $M$ is normal in $H$, or $M_u^{\Ga(u)}\cong\mz_2$ is non-normal in $H_u^{\Ga(u)}\cong D_8$ such that $K$ is semiregular on $V(\Ga)$ with $|V(\Ga_K)|\neq2^\ell$ for any integer $\ell\geq0$ and one of the followings holds:
\begin{enumerate}[\rm (c1)]
\item\label{Thmcase:t=1_cover} $K$ is the kernel of $H$ acting on $V(\Ga_K)$, $\Ga$ is a cover of $\Ga_K$, and the pair $(M/K, H/K)$ is a maximal $(\frac{1}{2}, 1)$-pair of $\Ga_K$;
\item\label{Thmcase:t=1_cycle} $K$ is the kernel of $M$ acting on $V(\Ga_K)$, $\Ga_K\cong C_r$ with $r=|V(\Ga_K)|$, and $M/K\cong D_{2r}$.
\end{enumerate}
\end{enumerate}
Moreover, examples for each of the above cases exist.,

Question: Let \(\Gamma\) be a connected tetravalent graph. Suppose \(M\le H\le \operatorname{Aut}(\Gamma)\), \(M\) is a maximal subgroup of \(H\), \(\Gamma\) is \(M\)-half-arc-transitive (that is, \(M\) is transitive on vertices and edges but not on arcs), and \(\Gamma\) is \((H,t)\)-transitive for some integer \(t\ge 1\); equivalently, \((M,H)\) is a maximal \((\tfrac12,t)\)-pair. Let \(K=\operatorname{Core}_H(M)\), the largest normal subgroup of \(H\) contained in \(M\), and fix \(u\in V(\Gamma)\). Write \(\Gamma(u)\) for the neighbor set of \(u\), \(H_u\) and \(M_u\) for the stabilizers of \(u\), and \(G_u^{\Gamma(u)}\) for the permutation group induced by \(G_u\) on \(\Gamma(u)\). Since \(K\trianglelefteq H\), let \(\Gamma_K\) be the normal quotient graph whose vertices are the \(K\)-orbits on \(V(\Gamma)\), with two \(K\)-orbits adjacent whenever some edge of \(\Gamma\) joins them. A subgroup is semiregular on \(V(\Gamma)\) if no nonidentity element fixes a vertex, and saying that \(\Gamma\) is a cover of \(\Gamma_K\) means the natural quotient map is a covering projection. Which statement holds for every such \(\Gamma,M,H,K,u\)?

One of the following holds. (a) \(t=3\), \((H/K,\,M/K,\,H_u,\,M_u)=(A_{72},\,A_{71},\,S_4\times S_3,\,\mathbb Z_2)\), and \(\Gamma\) is a cover of \(\Gamma_K\). (b) \(t=2\), \((H/K,\,M/K,\,H_u,\,M_u)=(S_5,\,\operatorname{AGL}_1(5),\,A_4,\,\mathbb Z_2)\), and \(\Gamma\) is a cover of \(\Gamma_K\cong K_{5,5}-5K_2\) (the graph obtained from \(K_{5,5}\) by deleting a perfect matching). (c) \(t=1\), and either \(M\trianglelefteq H\), or else \(M_u^{\Gamma(u)}\cong\mathbb Z_2\) is non-normal in \(H_u^{\Gamma(u)}\cong D_8\), \(K\) is semiregular on \(V(\Gamma)\), \(|V(\Gamma_K)|\neq 2^\ell\) for every integer \(\ell\ge 0\), and in addition either (c1) \(K\) is the kernel of the action of \(H\) on \(V(\Gamma_K)\), \(\Gamma\) is a cover of \(\Gamma_K\), and \((M/K,H/K)\) is a maximal \((\tfrac12,1)\)-pair of \(\Gamma_K\); or (c2) \(K\) is the kernel of the action of \(M\) on \(V(\Gamma_K)\), \(\Gamma_K\cong C_r\) with \(r=|V(\Gamma_K)|\), and \(M/K\) is dihedral of order \(2r\). Moreover, examples realizing each listed case exist.

Choices:
B. One of the following holds. (a) \(t=3\), \((H/K,\,M/K,\,H_u,\,M_u)=(A_{72},\,A_{71},\,S_4\times S_3,\,\mathbb Z_2)\), and \(\Gamma\) is a cover of \(\Gamma_K\). (b) \(t=2\), \((H/K,\,M/K,\,H_u,\,M_u)=(S_5,\,\operatorname{AGL}_1(5),\,A_4,\,\mathbb Z_2)\), and \(\Gamma\) is a cover of \(\Gamma_K\cong K_{5,5}-5K_2\). (c) \(t=1\), and either \(M\trianglelefteq H\), or else \(M_u^{\Gamma(u)}\cong\mathbb Z_2\) is non-normal in \(H_u^{\Gamma(u)}\cong D_8\), \(K\) is semiregular on \(V(\Gamma)\), \(|V(\Gamma_K)|\neq 2^\ell\) for every integer \(\ell\ge 1\), and in addition either (c1) \(K\) is the kernel of the action of \(H\) on \(V(\Gamma_K)\), \(\Gamma\) is a cover of \(\Gamma_K\), and \((M/K,H/K)\) is a maximal \((\tfrac12,1)\)-pair of \(\Gamma_K\); or (c2) \(K\) is the kernel of the action of \(M\) on \(V(\Gamma_K)\), \(\Gamma_K\cong C_r\) with \(r=|V(\Gamma_K)|\), and \(M/K\) is dihedral of order \(2r\). Moreover, examples realizing each listed case exist.
C. One of the following holds. (a) \(t=3\), \((H/K,\,M/K,\,H_u,\,M_u)=(A_{72},\,A_{71},\,S_4\times S_3,\,\mathbb Z_2)\), and \(\Gamma\) is a cover of \(\Gamma_K\). (b) \(t=2\), \((H/K,\,M/K,\,H_u,\,M_u)=(S_5,\,\operatorname{AGL}_1(5),\,A_4,\,\mathbb Z_2)\), and \(\Gamma\) is a cover of \(\Gamma_K\cong K_{5,5}-5K_2\). (c) \(t=1\), and either \(M\trianglelefteq H\), or else \(M_u^{\Gamma(u)}\cong\mathbb Z_2\) is non-normal in \(H_u^{\Gamma(u)}\cong D_8\), \(K\) is semiregular on \(V(\Gamma)\), and in addition either (c1) \(K\) is the kernel of the action of \(H\) on \(V(\Gamma_K)\), \(\Gamma\) is a cover of \(\Gamma_K\), and \((M/K,H/K)\) is a maximal \((\tfrac12,1)\)-pair of \(\Gamma_K\); or (c2) \(K\) is the kernel of the action of \(M\) on \(V(\Gamma_K)\), \(\Gamma_K\cong C_r\) with \(r=|V(\Gamma_K)|\), and \(M/K\) is dihedral of order \(2r\).
D. One of the following holds. (a) \(t=3\), \((H/K,\,M/K,\,H_u,\,M_u)=(A_{72},\,A_{71},\,S_4\times S_3,\,\mathbb Z_2)\), and \(\Gamma_K\) is a cover of \(\Gamma\). (b) \(t=2\), \((H/K,\,M/K,\,H_u,\,M_u)=(S_5,\,\operatorname{AGL}_1(5),\,A_4,\,\mathbb Z_2)\), and \(\Gamma_K\cong K_{5,5}-5K_2\) with \(\Gamma_K\) a cover of \(\Gamma\). (c) \(t=1\), and either \(M\trianglelefteq H\), or else \(M_u^{\Gamma(u)}\cong\mathbb Z_2\) is non-normal in \(H_u^{\Gamma(u)}\cong D_8\), \(K\) is semiregular on \(V(\Gamma)\), \(|V(\Gamma_K)|\neq 2^\ell\) for every integer \(\ell\ge 0\), and in addition either (c1) \(K\) is the kernel of the action of \(H\) on \(V(\Gamma_K)\), \(\Gamma_K\) is a cover of \(\Gamma\), and \((M/K,H/K)\) is a maximal \((\tfrac12,1)\)-pair of \(\Gamma_K\); or (c2) \(K\) is the kernel of the action of \(M\) on \(V(\Gamma_K)\), \(\Gamma_K\cong C_r\) with \(r=|V(\Gamma_K)|\), and \(M/K\) is dihedral of order \(2r\). Moreover, examples realizing each listed case exist.
E. One of the following holds. (a) \(t=3\), \((H/K,\,M/K,\,H_u,\,M_u)=(A_{72},\,A_{71},\,S_4\times S_3,\,\mathbb Z_2)\), and \(\Gamma\) is a cover of \(\Gamma_K\). (b) \(t=2\), \((H/K,\,M/K,\,H_u,\,M_u)=(S_5,\,\operatorname{AGL}_1(5),\,A_4,\,\mathbb Z_2)\), and \(\Gamma\) is a cover of \(\Gamma_K\cong K_{5,5}-5K_2\). (c) \(t=1\), and either \(M\trianglelefteq H\), or else \(M_u^{\Gamma(u)}\cong\mathbb Z_2\) is normal in \(H_u^{\Gamma(u)}\cong D_8\), \(K\) is semiregular on \(V(\Gamma)\), \(|V(\Gamma_K)|\neq 2^\ell\) for every integer \(\ell\ge 0\), and in addition either (c1) \(K\) is the kernel of the action of \(H\) on \(V(\Gamma_K)\), \(\Gamma\) is a cover of \(\Gamma_K\), and \((M/K,H/K)\) is a maximal \((\tfrac12,1)\)-pair of \(\Gamma_K\); or (c2) \(K\) is the kernel of the action of \(M\) on \(V(\Gamma_K)\), \(\Gamma_K\cong C_r\) with \(r=|V(\Gamma_K)|\), and \(H/K\) is dihedral of order \(2r\). Moreover, examples realizing each listed case exist.

dropped condition |V(\Gamma_K)|\neq 2^\ell for all \ell\ge 0

direction of covering projection to normal quotient

non-normal local action and quotient group in cycle case

The stem does not reveal the correct option; it only sets up the hypotheses and asks for the full classification. There is no explicit answer leakage, though the highly specialized notation makes it clear this is a theorem-recall item.

This is essentially a direct theorem-statement recognition question: the task is to identify the exact classification theorem rather than derive a conclusion from competing mathematical principles.

Some reasoning is needed to compare subtle logical differences among the options (off-by-one condition, omitted hypotheses, swapped kernels, disjunction vs conjunction). However, this is mostly precise recall and statement verification, not genuine generative mathematical problem solving.

The distractors are strong: they are close to the target statement, differ in mathematically meaningful ways, and reflect realistic failure modes such as weakened statements, boundary mistakes, and swapped subgroup-action roles.

Well-constructed as a theorem-statement discrimination item with high-quality distractors and no answer leakage, but it is largely a tautological recall question rather than a test of generative reasoning.