\begin{align}\label{eq+1}
\sum_{\substack{bm\equiv \pm an\ppmod d\\(mn,q)=1}}\lambda(m) \tau(n)W\left(\frac mM\right)W\left(\frac {n}{N}\right),
\end{align}

\begin{align}
&E_{M,N}\ll q^{-1+\ve}(MN)^{\frac12}+q^\ve(M/N)^{\frac12}, \label{eqtbA}\\
&E_{M,N}\ll M^{\theta_f} \left(q^{-1+\ve}(MN)^{\frac12}+q^\ve(N/M)^{\frac12}\r). \label{eqtbB}
\end{align}

\begin{remark}\label{rem1}
Observe that $\ve(f,\chi)$ is independent of $\chi$ when $f$ is a Maa\ss~form.
When $\ve(f,\chi)=\ve(f)=-1$, summing both sides of equation \eqref{eqfe} over all primitive characters leads to the vanishing of $M_{f,E}(q;a,b)$ by symmetry. For holomorphic $f$, the root number $\ve(f,\chi)$ depends on $\chi$ at most though its parity $\chi(-1)$. The symmetry also implies that the total contribution of such characters $\chi$ satisfying $\ve(f,\chi)=-1$ to $M_{f,E}(q;a,b)$ is zero. Therefore, we restrict our consideration to the following cases:
\begin{itemize}
  \item for a Maa\ss~form $f$, we take $\ve(f)=1$,
  \item for a holomorphic $f$, we consider only characters with $\chi(-1)=\ve(f)$.
\end{itemize}
In both cases, it holds that
\[
\ve(f,\chi)=1,
\]
which we assume henceforth.
\end{remark}

\begin{theorem}\label{themain}
Let $f$ be a cuspidal Hecke eigenform of level $1$ with Hecke eigenvalues $\lambda(n)\ll \tau(n)n^{\theta_f}$. If $f$ is Maa\ss~form we further assume  that its root number satisfies $\ve(f)=1$. For $q\nequiv 2 \pmod4$ and $1\le a,b\le q$ satisfying $(a,b)=(ab,q)=1$, we have
\begin{align*}
M_{f,E}(q;a,b)=&c_f\prod_{p\mid qab}\left(1-\frac{\lambda(p)}{p}+\frac1{p^2}\r)\left(1-\frac1{p^2}\r)^{-2}\\
&\times\frac{c_{a,b}+c_{b,a}}{(ab)^{\frac12}}\frac{L(1,f)^2}{\zeta(2)} +O\left(q^{-\frac{1}{20}+\ve}a^{\frac{3}{10}}+q^{-\frac{1-2\theta_f}{22+16\theta_f}+\ve}b^{\frac{3+2\theta_f }{11+8\theta_f}}\r),
\end{align*}
where the coefficients $c_f$, $c_{a,b}$ are defined by
\begin{align}
c_f=
\begin{cases}
1/2,  &\text{if} \ f\ \text{is holomorphic},\\
1, &\text{if}\ f\ \text{is a Maa\ss\ form},
\end{cases}
\quad c_{a,b}=\sum_{a_1\mid a^\infty}\sum_{b_1\mid b^\infty} \frac{\lambda(aa_1b_1)\tau(ba_1b_1)}{a_1b_1}. \label{eqc1c2}
\end{align}
\end{theorem}

\subsection{Moments of twisted $L$-functions}
The study of moments in families of $L$-functions constitutes a central problem in modern number theory. As emphasized in the elegant work of Young \cite{You11}, these moments are not only pivotal for their wide-ranging applications but also serve as fundamental objects that unveil deep structural properties and inherent symmetries within the family.

Asymptotic formulae for moments of $L$-function with a power-saving error term are particularly crucial, as they play vital roles in amplification techniques, mollification methods, and resonance techniques. The complexity of such moment calculations can be quantified by the ratio $r=\log \mathcal{C}/\log |\mathcal{F}|$, where $|\mathcal{F}|$ denotes the size of the family and $\mathcal{C}$ its analytic conductor. Notably, computational difficulty increases with growing complexity. The threshold $r=4$ is the precise boundary, where most current analytic techniques fall just short of producing an asymptotic formula. In the few successful cases, some deep input is typically indispensable; see \cite{CLMR24} \cite{CIS12} \cite{IS00} \cite{Kha12} \cite{KMV00} \cite{Li24} for example.

Let $f$ and $g$ denote two fixed (holomorphic or non-holomorphic) Hecke eigenforms of level $1$, not necessarily cuspidal. The following second moment is defined in the context $r=4$: for $q\nequiv 2\pmod4$,
\begin{align}\label{eqgmoment}
M_{f,g}(q)=\frac1{\vp^*(q)}\ssum_{\chi\!\!\!\!\!\pmod q}L\left(\tfrac12,f\otimes\chi\r)\ol{L\left(\tfrac12,g\otimes\chi\r)},
\end{align}
where the sum runs over all primitive characters modulo $q$, and $\vp^*(q)$ denotes the number of such characters. Asymptotics for the moment \eqref{eqgmoment} with a power-saving error term are crucial to amplification and related analytic techniques in problems such as subconvexity, nonvanishing, and extreme values of $L$-functions; see \cite{BFKMMS23} for a rich sample of applications.

When both $f$ and $g$ correspond to non-cuspidal Eisenstein series, the moment \eqref{eqgmoment} reduces to
\[
M_{E,E}(q)=\frac1{\vp^*(q)}\ssum_{\chi\!\!\!\!\!\pmod q}\left|L\left(\tfrac12,\chi\r)\right|^4,
\]
the fourth moment of Dirichlet $L$-functions, which has a very long history and has been extensively studied; see \cite{BFK+17a} \cite{HB81} \cite{Sou07} \cite{Wu23} \cite{You11} for example. The first asymptotic formula with a power-savings error term was established by Young \cite{You11} for prime moduli. Wu \cite{Wu23} proved the asymptotic formula for all admissible moduli, and the best-known error term to date is $O\left(q^{-\frac1{20}+\ve}\r)$; see \cite{BFK+17b} \cite{GWZ25}.

When both $f$ and $g$ are cuspidal (holomorphic or Maa\ss) in \eqref{eqgmoment}, some special cases (ie. $f=g$) were studied earlier by Stefanicki \cite{Ste96} and Gao, Khan, and  Ricotta \cite{GKR09}. Blomer and Mili\'cevi\'c \cite{BM15} established a power-saving asymptotic for most moduli, specifically when $q$ is not close to a prime or to a product of two primes of comparable size. The main obstacle in these cases has been the lack of power-saving estimates for bilinear forms involving Kloosterman sums in the P\'olya-Vinogradov range.
The case of prime moduli was later addressed by Kowalski, Michel, and Sawin \cite{KMS17}, who proved the asymptotic with a power-saving error term $O\left(p^{-\frac1{144}+\ve}\r)$. The remaining case was recently resolved by  Mili\'cevi\'c, Qin, and Wu \cite{MQW25} and independently by  Pascadi~\cite{Pascadi2025}. Both works \cite{MQW25} and \cite{Pascadi2025} established a power-saving asymptotic for all admissible moduli $q$, with the former also removing the dependence on the Ramanujan--Petersson conjecture and achieving a sharper error term $O\left(q^{-\frac1{216}+\ve}\r)$.

These are termed mixed moments when one eigenform is cuspidal while the other corresponds to a non-cuspidal Eisenstein series. The first power-saving asymptotic formula for this case was established by Blomer, Fouvry, Kowalski, Michel, and Mili\'cevi\'c \cite{BFK+17a}, who proved that for prime moduli $p$
\[
M_{f,E}(p)=\frac1{\vp^*(p)}\ssum_{\chi\!\!\!\!\!\pmod p}L\left(\tfrac12,f\otimes\chi\r)\ol{L\left(\tfrac12,\chi\r)}^2=\frac{L(1,f)^2}{\zeta(2)}+O\left(p^{-\frac1{68}+\ve}\r).
\]
The error term was subsequently improved to $O\left(p^{-\frac1{64}+\ve}\r)$ by Shparlinski \cite{Shp19}, and more recently to $O\left(p^{-\frac1{22}+\ve}\r)$ for holomorphic $f$ and to $O\left(p^{-\frac5{152}+\ve}\r)$ for Maa\ss\ forms $f$ by Khan and Zhang \cite{KZ23}.

In this paper, we investigate the mixed moment for arbitrary moduli. More precisely, we study the following twisted mixed moment
\[
M_{f,E}(q;a,b)=\frac1{\varphi^*(q)}\ssum_{\chi\!\!\!\!\!\pmod q} L\left(\tfrac12,f\otimes\chi\r)L\left(\tfrac12,\ol\chi\r)^2\chi(\ol{a}b),
\]
for any integers $a,b\le q$ satisfying $(a,b)=(ab,q)=1$.

Asymptotic formulae for moments of $L$-function with a power-saving error term are particularly crucial, as they play vital roles in amplification techniques, mollification methods, and resonance techniques. The complexity of such moment calculations can be quantified by the ratio $r=\log \mathcal{C}/\log |\mathcal{F}|$, where $|\mathcal{F}|$ denotes the size of the family and $\mathcal{C}$ its analytic conductor. Notably, computational difficulty increases with growing complexity. The threshold $r=4$ is the precise boundary, where most current analytic techniques fall just short of producing an asymptotic formula. In the few successful cases, some deep input is typically indispensable; see \cite{CLMR24} \cite{CIS12} \cite{IS00} \cite{Kha12} \cite{KMV00} \cite{Li24} for example.

Let $f$ and $g$ denote two fixed (holomorphic or non-holomorphic) Hecke eigenforms of level $1$, not necessarily cuspidal. The following second moment is defined in the context $r=4$: for $q\nequiv 2\pmod4$,
\begin{align}\label{eqgmoment}
M_{f,g}(q)=\frac1{\vp^*(q)}\ssum_{\chi\!\!\!\!\!\pmod q}L\left(\tfrac12,f\otimes\chi\r)\ol{L\left(\tfrac12,g\otimes\chi\r)},
\end{align}
where the sum runs over all primitive characters modulo $q$, and $\vp^*(q)$ denotes the number of such characters. Asymptotics for the moment \eqref{eqgmoment} with a power-saving error term are crucial to amplification and related analytic techniques in problems such as subconvexity, nonvanishing, and extreme values of $L$-functions; see \cite{BFKMMS23} for a rich sample of applications.

When both $f$ and $g$ correspond to non-cuspidal Eisenstein series, the moment \eqref{eqgmoment} reduces to
\[
M_{E,E}(q)=\frac1{\vp^*(q)}\ssum_{\chi\!\!\!\!\!\pmod q}\left|L\left(\tfrac12,\chi\r)\right|^4,
\]
the fourth moment of Dirichlet $L$-functions, which has a very long history and has been extensively studied; see \cite{BFK+17a} \cite{HB81} \cite{Sou07} \cite{Wu23} \cite{You11} for example. The first asymptotic formula with a power-savings error term was established by Young \cite{You11} for prime moduli. Wu \cite{Wu23} proved the asymptotic formula for all admissible moduli, and the best-known error term to date is $O\left(q^{-\frac1{20}+\ve}\r)$; see \cite{BFK+17b} \cite{GWZ25}.

When both $f$ and $g$ are cuspidal (holomorphic or Maa\ss) in \eqref{eqgmoment}, some special cases (ie. $f=g$) were studied earlier by Stefanicki \cite{Ste96} and Gao, Khan, and Ricotta \cite{GKR09}. Blomer and Mili\'cevi\'c \cite{BM15} established a power-saving asymptotic for most moduli, specifically when $q$ is not close to a prime or to a product of two primes of comparable size. The main obstacle in these cases has been the lack of power-saving estimates for bilinear forms involving Kloosterman sums in the P\'olya-Vinogradov range.
The case of prime moduli was later addressed by Kowalski, Michel, and Sawin \cite{KMS17}, who proved the asymptotic with a power-saving error term $O\left(p^{-\frac1{144}+\ve}\r)$. The remaining case was recently resolved by Mili\'cevi\'c, Qin, and Wu \cite{MQW25} and independently by Pascadi~\cite{Pascadi2025}. Both works \cite{MQW25} and \cite{Pascadi2025} established a power-saving asymptotic for all admissible moduli $q$, with the former also removing the dependence on the Ramanujan--Petersson conjecture and achieving a sharper error term $O\left(q^{-\frac1{216}+\ve}\r)$.

These are termed mixed moments when one eigenform is cuspidal while the other corresponds to a non-cuspidal Eisenstein series. The first power-saving asymptotic formula for this case was established by Blomer, Fouvry, Kowalski, Michel, and Mili\'cevi\'c \cite{BFK+17a}, who proved that for prime moduli $p$
\[
M_{f,E}(p)=\frac1{\vp^*(p)}\ssum_{\chi\!\!\!\!\!\pmod p}L\left(\tfrac12,f\otimes\chi\r)\ol{L\left(\tfrac12,\chi\r)}^2=\frac{L(1,f)^2}{\zeta(2)}+O\left(p^{-\frac1{68}+\ve}\r).
\]
The error term was subsequently improved to $O\left(p^{-\frac1{64}+\ve}\r)$ by Shparlinski \cite{Shp19}, and more recently to $O\left(p^{-\frac1{22}+\ve}\r)$ for holomorphic $f$ and to $O\left(p^{-\frac5{152}+\ve}\r)$ for Maa\ss\ forms $f$ by Khan and Zhang \cite{KZ23}.

In this paper, we investigate the mixed moment for arbitrary moduli. More precisely, we study the following twisted mixed moment
\[
M_{f,E}(q;a,b)=\frac1{\varphi^*(q)}\ssum_{\chi\!\!\!\!\!\pmod q} L\left(\tfrac12,f\otimes\chi\r)L\left(\tfrac12,\ol\chi\r)^2\chi(\ol{a}b),
\]
for any integers $a,b\le q$ satisfying $(a,b)=(ab,q)=1$.

\begin{align}\label{eqgmoment}
M_{f,g}(q)=\frac1{\vp^*(q)}\ssum_{\chi\ppmod q}L\left(\tfrac12,f\otimes\chi\r)\ol{L\left(\tfrac12,g\otimes\chi\r)},
\end{align}

\subsection{Twisted $L$-functions}
We review some basic facts on twisted $L$-functions in this section; for further details, see \cite[Section 2.3]{BFK+17a}.
Let $\chi$ be a primitive Dirichlet character modulo $q$ and let $f$ be a cuspidal Hecke eigenform with Hecke eigenvalues $\lambda(n)$. The twisted form $f\otimes\chi$ remains cuspidal for the congruence subgroup $\Gamma_0(q^2)$ with nebentypus $\chi^2$ (see e.g \cite[Propositions 14.19 \& 14.20]{IK04}). Its $L$-function is given by
\[
L(s,f\otimes\chi)=\sum_{n\ge1}\frac{\lambda(n)\chi(n)}{n^{s}}=\prod_p\left(1-\frac{\lambda(p)\chi(p)}{p^s}+\frac{\chi^2(p)}{p^{2s}}\r)^{-1},\ \ \text{for}\ \ \re(s)>1,
\]
where the product is over primes $p$.
The completed $L$-function is defined by
\begin{align*}
\Lambda(s,f\otimes\chi)=q^s L_\infty(s,f\otimes\chi)L(s,f\otimes\chi),
\end{align*}
with
\begin{align*}
L_\infty(s,f\otimes\chi)=\begin{cases}
(2\pi)^{-\frac{k-1}2-s}\Gamma\left(\frac{k-1}2+s\r), &\text{if} \ f\ \text{is holomorphic of weight}\ k,\\
\pi^{-s-\ma}\Gamma\left(\frac{s+i\kappa+\ma}2\r)\Gamma\left(\frac{s-i\kappa+\ma}2\r), &\text{if}\ f\ \text{is a Maa\ss\ form with eigenvalue}\ \tfrac14+\kappa^2.
\end{cases}
\end{align*}
This completed $L$-function admits analytic continuation to $\mathbb{C}$ and satisfies the functional equation (see e.g \cite[Theorem 14.17, Proposition 14.20]{IK04}):
\[
\Lambda(s,f\otimes\chi)=\ve(f\otimes\chi)\Lambda(1-s,f\otimes\ol{\chi}),
\]
where the root number is given by
\begin{align*}
\ve(f\otimes\chi)=\begin{cases}
\ve(f)\ve_\chi^2, &\text{if} \ f\ \text{is holomorphic},\\
\chi(-1)\ve(f)\ve_\chi^2, &\text{if}\ f\ \text{is a Maa\ss\ form},
\end{cases}
\end{align*}
and $\ve(f)=\pm 1$ denotes the root number of $L(s,f)$.

Using the identity
\[
S(hq\ol{b}, m;a_1k)=e\left(-m\frac{\ol{a}_1}{b}\right)S_{\infty,1/a_1}(hq,m;\gammaup),
\]
where $\gammaup=a_1\ssqrt{b}k$ and $Q=a_1b$, we derive from \eqref{eqdefmS} that
\begin{align*}
\mS(a_1,r)=&\frac{\sqrt{b}}{aN_1}\sum_{h\sim H}\sum_{m\sim M^*} \lambda(m)e\left(-m\frac{\ol{a}_1}{b}\right)\\
&\times\sum_{\gammaup}\frac 1{\gammaup}S_{\infty,1/a_1}(hq, m;\gammaup)W\left(\frac {r\gammaup}{a_1\ssqrt{b}N_1}\r)\mathring{V}_{+}\left(\frac{bm}{\gammaup^2},h\right).
\end{align*}
Performing the variable substitution $(bx-hq)/N\rightarrow x$ in \eqref{eqV0} yields
\[
\mathring{V}_{+}\left(\frac{bm}{\gammaup^2},h\right)=\frac{N}{b}\Omega\Bigg(\frac{4\pi\ssqrt{hqm}}{\gammaup},\frac{4\pi\ssqrt{mN}}{\gammaup},h,m\Bigg),
\]
where
\[
\Omega(y,z,h,m)=\int_{0}^\infty W(x)W\left(\frac {hq+xN}M\r) \mJ_{+}\left(\ssqrt{y^2+xz^2}\right)\d x
\]
with
\begin{equation}\label{eqyz}
y\asymp Y:=\frac{r\ssqrt{MM^*}}{a_1\ssqrt{b}N_1}\gg q^\ve,\quad z\asymp \frac{r\ssqrt{M^*N}}{a_1\ssqrt{b}N_1},\quad h\asymp H,\quad m\asymp M^*.
\end{equation}
For $M\gg q^\ve N$, it is easy to see from \eqref{eqlargeM} and \eqref{eqyz} that
\[
y\gg z\quad \text{and}\quad y\gg z^2q^{-\ve}.
\]
Then Lemma \ref{lemdecomJ} yields the decomposition
\[
\Omega(y,z,h,m)=\Omega_+(y,z,h,m)e^{iy}+\Omega_-(y,z,h,m)e^{-iy}+O_A\left(q^{-A}\r),
\]
where the functions $\Omega_{\pm}$ satisfy
\begin{align}\label{eqWpmd}
y^{j_1}z^{j_2}h^{j_3}m^{j_4}\frac{\partial^{j_1+j_2+j_3+j_4}}{\partial y^{j_1}\partial z^{j_2}\partial h^{j_3}\partial m^{j_4}}\Omega_{\pm}(y,z,h,m)\ll_{j_1,j_2,j_3,j_4}q^{(j_1+j_2+j_3+j_4)\ve} Y^{-\frac12}.
\end{align}
After separating variables in $\Omega_\pm$ via the Mellin transform, we bound $\mS(a_1,r)$ by sums of the form
\begin{equation*}
\mS(a_1,r)\ll q^\ve\frac{N Y^{-\frac12}}{a\ssqrt{b}N_1}\B|\sum_{H<h\le 2H}\alpha_h\sum_{M^*<m\le 2M^*} \beta_m\lambda(m)\sum_{\gammaup}\frac1{\gammaup}S_{\infty,1/a_1}(hq,m;\gammaup)\Theta \B(\frac{4\pi\ssqrt{hqm}}{\gammaup}\B)\B|
\end{equation*}
for some coefficients satisfying $\alpha_h\ll h^\ve$, $\beta_m\ll m^\ve$ and a test function
\[
\Theta (y)=W\B(\frac yY\B)\exp(\pm iy)y^{-u}
\]
with $\re(u)=\ve$.

An application of the Kuznetsov formula (Lemma \ref{lemKf}) followed by spectral truncation at parameter $Y^{\frac12+\ve}$ ( Lemma \ref{lemBe}) produces
\begin{equation*}
\mS(a_1,r)\ll q^\ve\frac{N Y^{-\frac12}}{a\ssqrt{b}N_1}\B|\sum_{H<h\le 2H}\alpha_h\sum_{M^*<m\le 2M^*} \beta_m\lambda(m)\left(\mH+\mM+\mE\r)\B|,
\end{equation*}
where the spectral contributions decompose as
\[
\mH=\sum_{\substack{2\le k\le Y^{\frac12+\ve}\\ k ~\text{even}}}\sum_{f\in \mathcal{B}_k(Q)}\tilde{\Theta} (k)\ssqrt{hqm}~ \bar\rho_f(hq)~\rho_f(a_1^{-1},m),
\]
\[
\mM=\sum_{\substack{f\in \mathcal{B}(Q)\\ \kappa_f\ll Y^{\frac12+\ve}}}\hat{\Theta} (\kappa_f)\frac{\ssqrt{hqm}~ \bar\rho_f(hq)~\rho_f(a_1^{-1},m)}{\cosh(\pi \kappa_f)},
\]
\[
\mE=\frac1{4\pi}\sum_{\mc}\int_{|\kappa|\ll Y^{\frac12+\ve}}\hat{\Theta} (\kappa)\ssqrt{hqm} ~\frac{\bar\rho_{\mc}(hq,\kappa)~\rho_{1/{a_1},\mc}(m,\kappa)}{\cosh(\pi\kappa)}\d \kappa,
\]
representing the holomorphic cusp forms, Maa\ss\ cusp forms, and the Eisenstein series respectively.
We foucus our detailed analysis on the Maa\ss\ case (the holomorphic and Eisenstein cases are identical). Since $\hat{\Theta} (\kappa_f)\ll Y^{-\frac12+\ve}$ (see \eqref{eqphi1}), the contribution of the Maa\ss\ case is bounded by
\[
\ll q^\ve\frac{NY^{-1}}{a\ssqrt{b}N_1}\sum_{\substack{f\in \mathcal{B}(Q)\\ \kappa_f\ll Y^{\frac12+\ve}}}\B|\sum_{H<h\le 2H}\alpha_h\sum_{M^*<m\le 2M^*} \beta_m\lambda(m)\frac{\ssqrt{hqm} ~\bar\rho_f(hq)~\rho_f(a_1^{-1},m)}{\cosh(\pi \kappa_f)}\B|,
\]
which, after applying the Cauchy-Schwarz inequality, can be bounded by
\begin{align*}
\ll q^\ve\frac{NY^{-1}}{a\ssqrt{b}N_1}&\B(\sum_{\substack{f\in \mathcal{B}(Q)\\ \kappa_f\ll Y^{\frac12+\ve}}}\frac{1}{\cosh(\pi \kappa_f)}\B|\sum_{H<h\le 2H}\alpha_h \ssqrt{hq} ~\rho_f(hq)\B|^2\B)^{\frac12}\\
&\times \B(\sum_{\substack{f\in \mathcal{B}(Q)\\ \kappa_f\ll Y^{\frac12+\ve}}}\frac{1}{\cosh(\pi \kappa_f)}\B|\sum_{M^*<m\le 2M^*}\beta_m\lambda(m)\ssqrt{m} ~\rho_f(a_1^{-1},m)\B|^2\B)^{\frac12}.
\end{align*}
We may handle the second spectral sum via Lemma \ref{lemsls}, but for the first spectral sum, an application of Lemma \ref{lemsls} leads to an extra factor of $q$ to the final estimate. To address this issue, we apply the following estimate, established in \cite[Theorems 11.3-11.6]{GWZ25}.
\begin{lemma}\label{thmsls}
Let $Q,s\in\mathbb{N}$, $K, N\ge 1$, and let $(a_n)_{n\in[N,2N]}$ be complex coefficients. Then we have
\[
\sum_{\substack{2\le k\le K\\ k~ \text{even}}}\Gamma(k)\sum_{f\in\mathcal{B}_k(Q)}\B|\sum_{n}a_n\ssqrt{sn}~\rho_f(sn)\B|^2, \ \ \ \sum_{\mc}\int_{-K}^K\frac1{\cosh(\pi\kappa)}\B|\sum_n a_n\ssqrt{sn}~\rho_{\mc}(\pm sn,\kappa)\B|^2\d\kappa,
\]
\[
\sum_{\substack{f\in\mathcal{B}(Q)\\ |\kappa_f|\le K}}\frac1{\cosh(\pi\kappa_f)}\B|\sum_{n}a_n\ssqrt{sn}~\rho_f(sn)\B|^2
\]
are bounded by
\[
\ll (sQKN)^\ve\B(K+\frac{s^{\frac12}}{Q^{\frac12}}\B)\B(K+\frac{(s,Q)^{\frac12}N}{Q^{\frac12}}\B)N\|a_n\|_\infty.
\]
\end{lemma}
Applying Lemma \ref{thmsls} to the first spectral sum and Lemma \ref{lemsls} to the second one reduces the Maa\ss\ contribution to
\begin{equation*}
\ll q^\ve\frac{NY^{-1}}{a\ssqrt{b}N_1}\B(H^{\frac12}{M^*}^{\frac12}\left(Y+\frac{q}{a_1b}\r)^{\frac14} \left(Y+\frac{(a_1b,q)H^2}{a_1b}\r)^{\frac14}\left(Y+\frac{M^*}{a_1b}\r)^{\frac12}\B).
\end{equation*}
Taking
\[
 H=\frac{M}q,\ \ \ M^*= q^\ve\frac{a_1^2N_1^2}{r^2}\frac{bM}{N^2}, \ \ \ Y=\frac{r\ssqrt{MM^*}}{a_1\ssqrt{b}N_1}=\frac{M}{N},
\]
and observing that
$\frac{M^*}{a_1b}\ll Y$ ( since $N_1^2\ll N/a$), we obtain
for the Maa\ss\ contribution
\begin{align*}
&\ll q^\ve\frac{N}{r\ssqrt{q}}\left(Y+\frac{q}{ab}\r)^{\frac14} \left(Y+\frac{(ab,q)H^2}{ab}\r)^{\frac14}Y^{\frac12}\\
&\ll q^\ve r^{-1}\B(\frac{M}{q^{\frac12}}+\frac{(ab,q)^{\frac14}M^{\frac 54}N^{\frac14}}{(ab)^{\frac14}q} +\frac{M^{\frac 34}N^{\frac14}}{(ab)^{\frac14}q^{\frac14}}+\frac{(ab,q)^{\frac14}MN^{\frac12}}{(ab)^{\frac12}q^{\frac 34}}\B).
\end{align*}

\begin{remark}
The dependence on  $\theta_f$ in the error term of Theorem \ref{themain} arises from the trivial estimate \eqref{eqtbB}, which constitutes the sole component requiring an upper bound for the Hecke eigenvalues of $f$. The assumption $\ve(f)=1$ for a Maa\ss~form is natural, as otherwise $M_{f,E}(q;a,b)=0$ for parity reasons (see Remark \ref{rem1}). In the Maa\ss~case, the coefficient $c_f$ addresses a literature discrepancy originating from a typographical oversight in \cite[Theorem 1.2]{BFK+17a} and was subsequently replicated in \cite[Thoerem 1.1]{KZ23}.
\end{remark}

Setting $a=b=1$ in Theorem \ref{themain} results in a power-saving asymptotic for $M_{f,E}(q)$ across all admissible moduli $q$.

\begin{corollary}\label{cor}
Let $f$ be a cuspidal Hecke eigenform of level $1$ with Hecke eigenvalues $\lambda(n)\ll \tau(n)n^{\theta_f}$. If $f$ is Maa\ss~form we further assume that its root number satisfies $\ve(f)=1$. For $q\nequiv 2 \pmod4$, we have
\begin{align*}
M_{f,E}(q)=c_f\prod_{p\mid q}\left(1-\frac{\lambda(p)}{p}+\frac1{p^2}\r)\left(1-\frac1{p^2}\r)^{-2}\frac{L(1,f)^2}{\zeta(2)}+O\left(q^{-\frac{1-2\theta_f }{22+16\theta_f }+\ve}\r),
\end{align*}
where $c_f$ is the same constant as previously defined.
\end{corollary}
For the holomorphic case, Deligne's well-known work \cite{Del74} gives $\theta_f=0$, whereas for the Maa\ss~form we may take $\theta_f= 7/{64}$ due to Kim--Sarnak \cite{Kim03}. This yields an error term as sharp as the best bound \cite{KZ23} known for prime moduli.

Through the functional equations of $L$-functions and the orthogonality of characters, the evaluation of the moment $M_{f,E}(q,a,b)$ reduces to analyzing the shifted convolution sum
\begin{align}\label{eq+1}
\sum_{\substack{bm\equiv \pm an\!\!\!\!\!\pmod d\\(mn,q)=1}}\lambda(m) \tau(n)W\left(\frac mM\right)W\left(\frac {n}{N}\right),
\end{align}
where $a, b,M,N\ge1$ and $W$ is smooth test functions. The classical approach to such shifted convolution sum involves either the $\delta$-method or Jutila's circle method to eliminate the congruence condition (see \cite{BM15}). For \eqref{eq+1}, a more efficient approach is to expand the divisor function using the multiplicative decomposition $\tau(n)=\sum_{n_1n_2=n}1$, which decomposes the $n$-sum into two sums with smooth coefficients, making them amenable to Fourier-analytic techniques.

The diagonal terms (where $bm=an$) in \eqref{eq+1} contribute to the main term, which can be evaluated explicitly using standard techniques. The primary challenge arise from the off-diagonal terms, which we classify into two distinct cases based on the relative magnitudes of $bM$ and $aN$: the balanced case and the unbalanced case. These two cases demand fundamentally different approaches.

In the balanced case where $bM$ and $aN$ are of comparable size, we first remove the coprimality condition and reduce the problem to estimating the sum
\begin{align}\label{eqAdef}
A_q(a,b,M,N)=\sum_{\substack{bm\equiv \pm an \!\!\!\!\!\pmod q\\ bm\neq an}}\lambda(m) \tau(n) W\left(\frac {bm}M\r) W\left(\frac {an}N\r),
\end{align}
where $W$ are some smooth functions.
For $M\ge N$, we establish the bound
\[
A_q(a,b,M,N)\ll  q^\ve\Bigg(\frac{M}{q^{\frac12}}+\frac{(ab,q)^{\frac14}M^{\frac54}N^{\frac14}}{(ab)^{\frac14}q} +\frac{M^{\frac34}N^{\frac14}}{(ab)^{\frac14}q^{\frac14}} +\frac{(ab,q)^{\frac14}MN^{\frac12}}{(ab)^{\frac12}q^{\frac34}}\Bigg)
\]
with an analogous result holding for $N\ge M$. Compared to the bound in \cite[(8.13)]{BM15}, our result achieves two key improvements: reducing the influence of the factor $(ab,q)^{\frac12}$ and recovering a saving $(ab)^{-\frac14}$ in the third term. This improvements stem from two innovation: the elimination of dependence on the Ramanujan--Petersson conjecture and the minimization of the impact of $a, b$ through new upper bounds for sums involving Kloosterman sums, as established in the recent work of Gao, Wu, and Zhao \cite{GWZ25}.

For the unbalanced case where $BM$ and $AN$ differ significantly in size, the theory of algebraic trace functions becomes inapplicable due to its restriction to prime moduli. To address this, we employ the multiplicative decomposition of the divisor function $\tau(n)$, which allows us to reformulate the problem as analyzing a bilinear sum involving incomplete Kloosterman sum. The estimate of this bilinear form relies primarily on a recent result by Kerr, Shparlinski, Wu, and Xi \cite{KSWX23} concerning arbitrary moduli.
The approach also requires the Weil bound for Kloosterman sums and several Fourier-analytic tools, including the Poisson summation formula and the Voronoi summation formula.

\subsection{Notation}
We adopt the standard convention that $\ve$ denotes an arbitrarily small positive constant whose values may vary from one occurrence to another. All implied constants in our estimates may depend on $\ve$ as well as on the fixed cusp form $f$.
Throughout this paper, $W(x)$ denotes a smooth complex-valued function with compact support in $[1/2,3]$, satisfying
\begin{align}\label{eqW}
W^{(j)}(x)\ll_{j}1,
\end{align}
for all integers $j\ge0$. The function $W(x)$ may also vary at each occurrence. We call a modulus $q$ admissible if $q\nequiv 2\pmod4$, which is a natural assumption since no primitive characters exist modulo $q$ otherwise.

Through the functional equations of $L$-functions and the orthogonality of characters, the evaluation of the moment $M_{f,E}(q,a,b)$ (hence Theorem~\ref{themain}) “reduces to analyzing the shifted convolution sum”
\[
\sum_{\substack{bm\equiv \pm an\!\!\!\!\!\pmod d\\(mn,q)=1}}\lambda(m) \, \tau(n)\,W\!\left(\frac mM\right)W\!\left(\frac {n}{N}\right).
\]
Instead of using “the $\delta$-method or Jutila's circle method,” the argument uses “a more efficient approach” by expanding the divisor function via “the multiplicative decomposition $\tau(n)=\sum_{n_1n_2=n}1$,” which splits the $n$-sum into two smooth sums “amenable to Fourier-analytic techniques.”

The “diagonal terms (where $bm=an$)” give the main term and “can be evaluated explicitly using standard techniques.” The remaining “off-diagonal terms” are treated by splitting into two regimes “based on the relative magnitudes of $bM$ and $aN$”:

1) **Balanced case** ($bM$ comparable to $aN$): “we first remove the coprimality condition and reduce the problem” to bounding
\[
A_q(a,b,M,N)=\sum_{\substack{bm\equiv \pm an \!\!\!\!\!\pmod q\\ bm\neq an}}\lambda(m)\tau(n) W\!\left(\frac {bm}M\right) W\!\left(\frac {an}N\right),
\]
and establish the stated upper bound for $A_q(a,b,M,N)$ (and its analogue when $N\ge M$). The text attributes the improvement to “two innovation: the elimination of dependence on the Ramanujan--Petersson conjecture and the minimization of the impact of $a, b$ through new upper bounds for sums involving Kloosterman sums” from \cite{GWZ25}.

2) **Unbalanced case** ($BM$ and $AN$ very different): since “the theory of algebraic trace functions becomes inapplicable due to its restriction to prime moduli,” one again uses the decomposition of $\tau(n)$ to “reformulate the problem as analyzing a bilinear sum involving incomplete Kloosterman sum.” This bilinear form is estimated using \cite{KSWX23} (for “arbitrary moduli”), together with “the Weil bound for Kloosterman sums” and Fourier-analytic tools including “the Poisson summation formula and the Voronoi summation formula.”

Finally, the remark notes that the $\theta_f$-dependence in the error term “arises from the trivial estimate \eqref{eqtbB},” which is “the sole component requiring an upper bound for the Hecke eigenvalues of $f$.”

Through the functional equations of $L$-functions and the orthogonality of characters, the evaluation of the moment $M_{f,E}(q,a,b)$ (hence, in establishing the main theorem) “reduces to analyzing the shifted convolution sum”
\[
\sum_{\substack{bm\equiv \pm an\!\!\!\!\pmod d\\(mn,q)=1}}\lambda(m) \, \tau(n)\,W\!\left(\frac mM\right)W\!\left(\frac {n}{N}\right).
\]
Instead of using “the $\delta$-method or Jutila's circle method,” the argument uses “a more efficient approach” by expanding the divisor function via “the multiplicative decomposition $\tau(n)=\sum_{n_1n_2=n}1$,” which splits the $n$-sum into two smooth sums “amenable to Fourier-analytic techniques.”

The “diagonal terms (where $bm=an$)” give the main term and “can be evaluated explicitly using standard techniques.” The remaining “off-diagonal terms” are treated by splitting into two regimes “based on the relative magnitudes of $bM$ and $aN$”:

1) **Balanced case** ($bM$ comparable to $aN$): “we first remove the coprimality condition and reduce the problem” to bounding
\[
A_q(a,b,M,N)=\sum_{\substack{bm\equiv \pm an \!\!\!\!\pmod q\\ bm\neq an}}\lambda(m)\tau(n) W\!\left(\frac {bm}M\right) W\!\left(\frac {an}N\right),
\]
and establish the stated upper bound for $A_q(a,b,M,N)$ (and its analogue when $N\ge M$). The text attributes the improvement to “two innovation: the elimination of dependence on the Ramanujan--Petersson conjecture and the minimization of the impact of $a, b$ through new upper bounds for sums involving Kloosterman sums” from \cite{GWZ25}.

2) **Unbalanced case** ($BM$ and $AN$ very different): since “the theory of algebraic trace functions becomes inapplicable due to its restriction to prime moduli,” one again uses the decomposition of $\tau(n)$ to “reformulate the problem as analyzing a bilinear sum involving incomplete Kloosterman sum.” This bilinear form is estimated using \cite{KSWX23} (for “arbitrary moduli”), together with “the Weil bound for Kloosterman sums” and Fourier-analytic tools including “the Poisson summation formula and the Voronoi summation formula.”

Finally, the remark notes that the $\theta_f$-dependence in the error term “arises from the trivial estimate
\begin{align}
&E_{M,N}\\ll q^{-1+\ve}(MN)^{\frac12}+q^\ve(M/N)^{\frac12}, \label{eqtbA}\\
&E_{M,N}\\ll M^{\theta_f} \left(q^{-1+\ve}(MN)^{\frac12}+q^\ve(N/M)^{\frac12}\r). \label{eqtbB}
\end{align}
,” which is “the sole component requiring an upper bound for the Hecke eigenvalues of $f$.”

Question:
Let f be a cuspidal Hecke eigenform of level 1 with Hecke eigenvalues λ(n) satisfying λ(n) ≪ τ(n)n^{θ_f}, where τ(n) is the divisor function. If f is a Maaß form, also assume that its root number satisfies ε(f)=1. For an integer q with q ≢ 2 (mod 4), and integers 1 ≤ a,b ≤ q satisfying (a,b)=(ab,q)=1, define the twisted mixed moment
M_{f,E}(q;a,b) = (1/φ*(q)) ∑_{χ primitive mod q} L(1/2,f⊗χ) L(1/2,χ̄)^2 χ(āb),
where φ*(q) is the number of primitive Dirichlet characters modulo q and ā denotes the inverse of a modulo q. Also define
c_f = 1/2 if f is holomorphic, and c_f = 1 if f is a Maaß form,
and
c_{a,b} = ∑_{a_1|a^∞} ∑_{b_1|b^∞} λ(aa_1b_1)τ(ba_1b_1)/(a_1b_1),
with c_{b,a} defined by swapping a and b. Which statement holds for every such f, q, a, and b?

One has
M_{f,E}(q;a,b) = c_f · ∏_{p∣qab} (1 − λ(p)/p + 1/p^2)(1 − 1/p^2)^(−2) · ((c_{a,b}+c_{b,a})/(ab)^(1/2)) · (L(1,f)^2/ζ(2)) + O(q^(−1/20+ε)a^(3/10) + q^(−(1−2θ_f)/(22+16θ_f)+ε) b^((3+2θ_f)/(11+8θ_f))).

Choices:
B.
One has
M_{f,E}(q;a,b) = c_f \cdot \prod_{p\mid qab} (1 - \lambda(p)/p + 1/p^2)(1 - 1/p^2)^{(-2)} \cdot \frac{c_{a,b}+c_{b,a}}{(ab)^{1/2}} \cdot \frac{L(1,f)^2}{\zeta(2)} + O\!\left(q^{-1/20+\varepsilon}(ab)^{3/10} + q^{-(1-2\theta_f)/(22+16\theta_f)+\varepsilon}(ab)^{(3+2\theta_f)/(11+8\theta_f)}\right).
C.
One has
M_{f,E}(q;a,b) = c_f \cdot \prod_{p\mid qab} (1 - \lambda(p)/p + 1/p^2)(1 - 1/p^2)^{(-2)} \cdot \frac{c_{a,b}+c_{b,a}}{(ab)^{1/2}} \cdot \frac{L(1,f)^2}{\zeta(2)} + O\!\left(q^{-1/20+\varepsilon}a^{3/10} + q^{\varepsilon} b^{(3+2\theta_f)/(11+8\theta_f)}\right).
D.
One has
M_{f,E}(q;a,b) = c_f \cdot \prod_{p\mid qab} (1 - \lambda(p)/p + 1/p^2)(1 - 1/p^2)^{(-2)} \cdot \frac{c_{a,b}+c_{b,a}}{(ab)^{1/2}} \cdot \frac{L(1,f)^2}{\zeta(2)} + O\!\left(q^{-1/20+\varepsilon}a^{3/10} + q^{-(1-2\theta_f)/(22+16\theta_f)+\varepsilon} a^{(3+2\theta_f)/(11+8\theta_f)}\right).
E.
One has
M_{f,E}(q;a,b) = c_f \cdot \prod_{p\mid qab} (1 - \lambda(p)/p + 1/p^2)(1 - 1/p^2)^{(-2)} \cdot \frac{c_{a,b}}{(ab)^{1/2}} \cdot \frac{L(1,f)^2}{\zeta(2)} + O\!\left(q^{-1/20+\varepsilon}a^{3/10} + q^{-(1-2\theta_f)/(22+16\theta_f)+\varepsilon} b^{(3+2\theta_f)/(11+8\theta_f)}\right).

separate balanced/unbalanced dependence on a and b in the error term

dropped the power-saving q-decay in the second error term

assignment of the second-regime parameter dependence to b rather than a

two-sided diagonal contribution c_{a,b}+c_{b,a}

The stem does not explicitly reveal the correct formula; it only states the hypotheses and definitions. The correct choice must be identified from the options rather than being directly implied by wording in the question.

This is essentially a direct recall of a specific theorem's full asymptotic statement. The question asks which exact asymptotic formula holds, so it is very close to restating the result itself.

There is some discrimination required among subtle variants of the main term and error terms, but the task is mostly theorem recognition/recall rather than genuine generative mathematical reasoning.

The distractors are mathematically plausible and target realistic failure modes: weakening explicit exponents, dropping the symmetric term, or conflating the separate dependence on a and b. They are distinct and nontrivial.

A technically well-constructed recall question with strong distractors and no answer leakage, but it is largely a theorem-restatement item rather than a genuine reasoning-based MCQ.


Question:
Let f be a cuspidal Hecke eigenform of level 1 with Hecke eigenvalues λ(n) satisfying λ(n) ≪ τ(n)n^{θ_f}, where τ(n) is the divisor function. If f is a Maaß form, also assume that its root number satisfies ε(f)=1. For an integer q with q ≢ 2 (mod 4), and integers 1 ≤ a,b ≤ q satisfying (a,b)=(ab,q)=1, define the twisted mixed moment
M_{f,E}(q;a,b) = (1/φ*(q)) ∑_{χ primitive mod q} L(1/2,f⊗χ) L(1/2,χ̄)^2 χ(āb),
where φ*(q) is the number of primitive Dirichlet characters modulo q and ā denotes the inverse of a modulo q. Also define
c_f = 1/2 if f is holomorphic, and c_f = 1 if f is a Maaß form,
and
c_{a,b} = ∑_{a_1|a^∞} ∑_{b_1|b^∞} λ(aa_1b_1)τ(ba_1b_1)/(a_1b_1),
with c_{b,a} defined by swapping a and b. Which statement holds for every such f, q, a, and b?

Choices:
B.
One has
M_{f,E}(q;a,b) = c_f \cdot \prod_{p\mid qab} (1 - \lambda(p)/p + 1/p^2)(1 - 1/p^2)^{(-2)} \cdot \frac{c_{a,b}+c_{b,a}}{(ab)^{1/2}} \cdot \frac{L(1,f)^2}{\zeta(2)} + O\!\left(q^{-1/20+\varepsilon}(ab)^{3/10} + q^{-(1-2\theta_f)/(22+16\theta_f)+\varepsilon}(ab)^{(3+2\theta_f)/(11+8\theta_f)}\right).
C.
One has
M_{f,E}(q;a,b) = c_f \cdot \prod_{p\mid qab} (1 - \lambda(p)/p + 1/p^2)(1 - 1/p^2)^{(-2)} \cdot \frac{c_{a,b}+c_{b,a}}{(ab)^{1/2}} \cdot \frac{L(1,f)^2}{\zeta(2)} + O\!\left(q^{-1/20+\varepsilon}a^{3/10} + q^{\varepsilon} b^{(3+2\theta_f)/(11+8\theta_f)}\right).
D.
One has
M_{f,E}(q;a,b) = c_f \cdot \prod_{p\mid qab} (1 - \lambda(p)/p + 1/p^2)(1 - 1/p^2)^{(-2)} \cdot \frac{c_{a,b}+c_{b,a}}{(ab)^{1/2}} \cdot \frac{L(1,f)^2}{\zeta(2)} + O\!\left(q^{-1/20+\varepsilon}a^{3/10} + q^{-(1-2\theta_f)/(22+16\theta_f)+\varepsilon} a^{(3+2\theta_f)/(11+8\theta_f)}\right).
E.
One has
M_{f,E}(q;a,b) = c_f \cdot \prod_{p\mid qab} (1 - \lambda(p)/p + 1/p^2)(1 - 1/p^2)^{(-2)} \cdot \frac{c_{a,b}}{(ab)^{1/2}} \cdot \frac{L(1,f)^2}{\zeta(2)} + O\!\left(q^{-1/20+\varepsilon}a^{3/10} + q^{-(1-2\theta_f)/(22+16\theta_f)+\varepsilon} b^{(3+2\theta_f)/(11+8\theta_f)}\right).

The stem gives hypotheses and definitions but does not reveal the conclusion. The correct asymptotic formula is not stated or strongly hinted at in the prompt itself.

This is essentially a theorem-identification question: the setup is followed by 'Which statement holds?'. Although the options introduce nearby alternatives, the task is still largely selecting the exact theorem conclusion rather than resolving a broader mathematical issue.

Some reasoning or precise recall is needed because the choices differ in subtle but meaningful ways (symmetry in c_{a,b}+c_{b,a}, dependence on a versus b, and strength of q-decay). However, it does not strongly induce generative reasoning from the hypotheses alone.

The distractors are mathematically plausible and target realistic failure modes: overstrong combined error terms, weakened q-saving, swapped variable dependence, and omission of the symmetric c_{b,a} term.

A solid theorem-recognition MCQ with strong distractors and no answer leakage, but it leans more toward precise recall/discrimination than genuine generative reasoning.


\label{lem:anest}
Let $(M_i,g_i)$ be Riemannian manifolds of dimensions $m_i$ ($i=1,2$), and set $g=g_1\times g_2$. Then the following hold for all measurable $S\subset M_1\times M_2$ and $(m_1+m_2-1)$–rectifiable $\Sigma$:
\begin{enumerate}
\item[(i)] For every set $S \subset M_1 \times M_2$,
\begin{equation}\label{eq:anest1}
  \vv_{g_+^t}(S) = t^{m_2} \,\vv_g(S).
\end{equation}
\item[(ii)] If $\Sigma \subset M_1 \times M_2$ is $(m_1+m_2-1)$-rectifiable, then for $t \leq 1$,
\begin{equation}\label{eq:anest2}
  \hh_{g_+^t}^{m_1+m_2-1}(\Sigma)
  \leq t^{m_2-1}\,\hh_g^{m_1+m_2-1}(\Sigma).
\end{equation}
\item[(iii)] Equality holds in \eqref{eq:anest2} if and only if the $g$-normal of $\Sigma$ is tangent to $M_2$, up to a $\hh_g^{m_1+m_2-1}$-null set.
\end{enumerate}

\begin{proposition}
\label{prp:prpc}
\mbox{}
Under the hypotheses of Proposition~\ref{prp:pr} there holds
\begin{equation}
\label{eq:prpc}
\ptl E_i\to \ptl E\quad\text{in the Hausdorff topology}.
\end{equation}
\end{proposition}

\begin{proposition}
\label{Thm:sts}
Let $\Sigma\subset N$ be a smooth constant–mean–curvature hypersurface.
If $\Sigma$ is stable, then $tM\times\Sigma$ is stable for all sufficiently small $t>0$.
\end{proposition}

\begin{proposition}
\label{prp:reduce}
If Theorem \ref{thm:main} is true when the first factor is a round sphere, then it holds in general, i.e, when the first factor is any compact Riemannian manifold.
\end{proposition}

\begin{theorem}
\label{thm:main}
Let $M,N$ be compact Riemannian manifolds and $\be\in (0,1)$. Assume that the boundaries of the isoperimetric regions in $N$ of volume fraction $\be$ are of class $C^{2,\alpha}$.
\begin{enum}
\item Then there exists $t_0=t_0(\be)>0$ so that  for every $0<t\le t_0$
every isoperimetric region of volume $\be\vv(t M \times N)$ in $t M \times N$, is of the form  $ M \times S$, where $S$ is an isoperimetric region in $N$.

By scaling isotropically, we get equivalently.
\item
For every $s\ge s_0$
every isoperimetric region of volume $\be \vv(M \times s N)$ in $M \times s N$, is of the form  $M \times S$, where $S$ is an isoperimetric region in $s N$ and $s_0=t_0^{-1}$.
\end{enum}
\end{theorem}

\begin{equation}\label{eq:stab}
- \int_\Sg u \,\big(\Delta u + \ric(\nu,\nu)u + |\sg|^2 u\big) \,\geq 0,
\end{equation}

The classical isoperimetric problem seeks to determine, within a given ambient space, the subsets of prescribed volume that minimize boundary measure.
It is one of the most fundamental variational problems in geometry.

In products of Euclidean and hyperbolic spaces, W.–T. and W.–Y. Hsiang~\cite{hsiang} provided a complete description of isoperimetric hypersurfaces.
In products of a circle with a model space, isoperimetric regions were classified by Pedrosa and Ritoré \cite{peri}.
Morgan~\cite{Morpr} established lower bounds for the isoperimetric profile of a Riemannian product in terms of concave lower bounds for the profiles of the factors.
In the spherical case, Pedrosa~\cite{Pedrosa2004} classified the isoperimetric regions in the spherical cylinder $\mathbb{S}^n\times\mathbb{R}$.
In Riemannian cylinders $M\times\mathbb{R}$, Duzaar and Steffen~\cite{du-st} proved that large-volume minimizers are slabs.
This was later extended to $M\times\mathbb{R}^k$, where large isoperimetric regions are products of the first factor with geodesic balls in the second factor, see~\cite{manstr}.

A natural question was originally proposed by M. Hutchings — as reported by F. Morgan in~\cite{morpol} — who conjectured that under anisotropic scaling, the isoperimetric regions should align with a cylindrical product structure. More concretely, if one rescales one factor of a product, do minimizers eventually acquire a cylindrical structure of the form $M \times S$, where $S$ is an isoperimetric region in the other factor, once the scaling parameter is sufficiently small (or large)?

In this paper, we address this conjecture under the assumption that the isoperimetric hypersurfaces of the second factor \(N\) are of class \(C^{2,\alpha}\), a condition that holds, for example, when \(\dim N\leqslant 8\) or for small volumes (and their complements).

We begin by recalling the scaling identities for Hausdorff measure, volume, and perimeter, and fix notation for anisotropic homotheties on product manifolds.
In the Appendix we adapt the Ros–Morgan symmetrization on horizontal slices, showing that it suffices to treat the case $\mathbb{S}^m\times N$ (Proposition~\ref{prp:reduce})
\,(see also the related symmetrization results of Morgan--Howe--Harman~\cite{mosym}).
We then analyze \emph{cylinderoids} $M_1\times S$ and symmetrized competitors in $\mathbb{S}^m\times N$: compactness and slice estimates imply that, under the anisotropic deformation, isoperimetric sets subconverge (in both $L^1$ and Hausdorff topologies) to cylinderoids whose projections onto $N$ are isoperimetric (Propositions~\ref{prp:pr} and~\ref{prp:prpc}). Moreover, by standard regularity theory for perimeter minimizers,
all relevant competitors have $C^{2,\alpha}$ boundaries.
This allows us to restrict the analysis to $C^{2,\alpha}$ graphical
perturbations of $\Sigma_t$.
Next, exploiting the product spectral splitting and the scaling of eigenvalues, we establish a stability inheritance result: if $\Sigma\subset N$ is a stable constant mean curvature hypersurface, then $tM\times\Sigma$ is stable for all sufficiently small $t>0$ (Proposition~\ref{Thm:sts}).
Finally, assuming the $C^{2,\alpha}$ regularity of isoperimetric hypersurfaces in $N$, we adapt the Lyapunov–Schmidt reduction framework developed in~\cite{rqi} to the present product setting around $\Sg_t:=tM\times\Sigma$.
This analytic decomposition into a Jacobi–kernel component and its $L^2$–orthogonal complement, combined with stability estimates, yields that all nearby stationary graphs are parametrized by a finite-dimensional family of kernel deformations. This rigidity ultimately forces minimizers close to $\Sg_t$ to be cylinderoids (Theorem~\ref{thm:main}).

We begin by recalling the scaling relations for volume and perimeter, and by introducing anisotropic product scalings. We also summarize standard results on existence, regularity, and stability of isoperimetric regions in compact Riemannian manifolds.

Let $(M,g)$ be a Riemannian manifold. For a measurable set $S \subset M$, we denote by
\[
  \vv_g(S) \quad \text{and} \quad \pp_g(S)
\]
the Riemannian volume and perimeter, respectively. The $\alpha$-dimensional Hausdorff measure with respect to $g$ is written $\hh^\alpha_g(S)$. A direct computation from the definitions yields the following scaling relations:
\begin{equation}\label{eq:est}
  \hh_{t^2 g}^{\alpha}(S) = t^{\alpha}\,\hh_g^{\alpha}(S),
  \qquad \vv_{t^2 g}(S) = t^m\,\vv_g(S),
  \qquad \pp_{t^2 g}(S) = t^{m-1}\,\pp_g(S),
\end{equation}
where $m = \dim M$.
Throughout the manuscript, whenever no ambiguity arises, we shall drop subscripts and superscripts in the notation. We write $tM$ for the Riemannian manifold $(M,t^2 g)$, and refer to it as the \emph{$t$-homothety} of $(M,g)$.

Let $(M_i,g_i)$, $i=1,2$, be Riemannian manifolds, and set $g=g_1 \times g_2$.
We define two anisotropic product scalings:
\begin{itemize}
  \item The \emph{right anisotropic $t$-homothety} is
  \[
    (M_1 \times M_2, g_1 \times t^2 g_2),
  \]
  denoted by $(M_1 \times M_2, g_+^t)$ or simply $M_1 \times tM_2$.
  \item The \emph{left anisotropic $t$-homothety} is
  \[
    (M_1 \times M_2, t^2 g_1 \times g_2),
  \]
  denoted by $(M_1 \times M_2, g_-^t)$ or simply $tM_1 \times M_2$.
\end{itemize}
These anisotropic deformations play a central role in our analysis, as they geometrically encode the effect of collapsing or expanding one factor of a product manifold. We record the following scaling behavior for volume and perimeter under right anisotropic homotheties.

In products of Euclidean and hyperbolic spaces, W.–T. and W.–Y. Hsiang~\cite{hsiang} provided a complete description of isoperimetric hypersurfaces.
In products of a circle with a model space, isoperimetric regions were classified by Pedrosa and Ritoré \cite{peri}.
Morgan~\cite{Morpr} established lower bounds for the isoperimetric profile of a Riemannian product in terms of concave lower bounds for the profiles of the factors.
In the spherical case, Pedrosa~\cite{Pedrosa2004} classified the isoperimetric regions in the spherical cylinder $\mathbb{S}^n\times\mathbb{R}$.
In Riemannian cylinders $M\times\mathbb{R}$, Duzaar and Steffen~\cite{du-st} proved that large-volume minimizers are slabs.
This was later extended to $M\times\mathbb{R}^k$, where large isoperimetric regions are products of the first factor with geodesic balls in the second factor, see~\cite{manstr}.

In this paper, we address this conjecture under the assumption that the isoperimetric hypersurfaces of the second factor \(N\) are of class \(C^{2,\alpha}\), a condition that holds, for example, when \(\dim N\leqslant 8\) or for small volumes (and their complements).

We begin by recalling the scaling identities for Hausdorff measure, volume, and perimeter, and fix notation for anisotropic homotheties on product manifolds.
In the Appendix we adapt the Ros–Morgan symmetrization on horizontal slices, showing that it suffices to treat the case $\mathbb{S}^m\times N$ (Proposition~\ref{prp:reduce})
\,(see also the related symmetrization results of Morgan--Howe--Harman~\cite{mosym}).
We then analyze \emph{cylinderoids} $M_1\times S$ and symmetrized competitors in $\mathbb{S}^m\times N$: compactness and slice estimates imply that, under the anisotropic deformation, isoperimetric sets subconverge (in both $L^1$ and Hausdorff topologies) to cylinderoids whose projections onto $N$ are isoperimetric (Propositions~\ref{prp:pr} and~\ref{prp:prpc}). Moreover, by standard regularity theory for perimeter minimizers,
all relevant competitors have $C^{2,\alpha}$ boundaries.
This allows us to restrict the analysis to $C^{2,\alpha}$ graphical
perturbations of $\Sigma_t$.
Next, exploiting the product spectral splitting and the scaling of eigenvalues, we establish a stability inheritance result: if $\Sigma\subset N$ is a stable constant mean curvature hypersurface, then $tM\times\Sigma$ is stable for all sufficiently small $t>0$ (Proposition~\ref{Thm:sts}).
Finally, assuming the $C^{2,\alpha}$ regularity of isoperimetric hypersurfaces in $N$, we adapt the Lyapunov–Schmidt reduction framework developed in~\cite{rqi} to the present product setting around $\Sg_t:=tM\times\Sigma$.
This analytic decomposition into a Jacobi–kernel component and its $L^2$–orthogonal complement, combined with stability estimates, yields that all nearby stationary graphs are parametrized by a finite-dimensional family of kernel deformations. This rigidity ultimately forces minimizers close to $\Sg_t$ to be cylinderoids (Theorem~\ref{thm:main}).

We begin by recalling the scaling relations for volume and perimeter, and by introducing anisotropic product scalings. We also summarize standard results on existence, regularity, and stability of isoperimetric regions in compact Riemannian manifolds.

Let $(M,g)$ be a Riemannian manifold. For a measurable set $S \subset M$, we denote by
\[
 \vv_g(S) \quad \text{and} \quad \pp_g(S)
\]
the Riemannian volume and perimeter, respectively. The $\alpha$-dimensional Hausdorff measure with respect to $g$ is written $\hh^\alpha_g(S)$. A direct computation from the definitions yields the following scaling relations:
\begin{equation}\label{eq:est}
 \hh_{t^2 g}^{\alpha}(S) = t^{\alpha}\,\hh_g^{\alpha}(S),
 \qquad \vv_{t^2 g}(S) = t^m\,\vv_g(S),
 \qquad \pp_{t^2 g}(S) = t^{m-1}\,\pp_g(S),
\end{equation}
where $m = \dim M$.
Throughout the manuscript, whenever no ambiguity arises, we shall drop subscripts and superscripts in the notation. We write $tM$ for the Riemannian manifold $(M,t^2 g)$, and refer to it as the \emph{$t$-homothety} of $(M,g)$.

Let $(M_i,g_i)$, $i=1,2$, be Riemannian manifolds, and set $g=g_1 \times g_2$.
We define two anisotropic product scalings:
\begin{itemize}
 \item The \emph{right anisotropic $t$-homothety} is
 \[
 (M_1 \times M_2, g_1 \times t^2 g_2),
 \]
 denoted by $(M_1 \times M_2, g_+^t)$ or simply $M_1 \times tM_2$.
 \item The \emph{left anisotropic $t$-homothety} is
 \[
 (M_1 \times M_2, t^2 g_1 \times g_2),
 \]
 denoted by $(M_1 \times M_2, g_-^t)$ or simply $tM_1 \times M_2$.
\end{itemize}
These anisotropic deformations play a central role in our analysis, as they geometrically encode the effect of collapsing or expanding one factor of a product manifold. We record the following scaling behavior for volume and perimeter under right anisotropic homotheties.

\begin{proof}
Part (i) follows directly from computing the Jacobian of the identity map from $(M_1 \times M_2,g)$ to $(M_1 \times M_2,g_+^t)$.
For (ii), at a regular point of $\Sg$ the $g$-unit exterior normal vector $\nu$ can be decomposed as $\nu = a\nu_1 + b\nu_2$ so that $a^2+b^2=1$ and $\nu_1$, $\nu_2$ unit vectors tangent to $M_1$ and $M_2$, respectively. The Jacobian of the identity map from $(M_1 \times M_2,g)$ to $(M_1 \times M_2,g_+^t)$ restricted to $\Sg$ then equals $t^{m_2-1}(t^2a^2+b^2)^{1/2}$. This yields inequality \eqref{eq:anest2}.
Assertion (iii) follows from the fact that $\nu$ is tangent to $M_2$ precisely when $a=0$ almost everywhere.

\end{proof}
The following local isoperimetric inequalities are classical, see Duzaar and Steffen \cite{du-st}.
\begin{lemma}
\label{lem:ine}
Let $M$ be a compact $m$-dimensional Riemannian manifold. Given $0<v_0<\hh^m(M)$, there exist positive constants $\ga_1(M,v_0),\ga_2(M,v_0)$ such that for any set $S\subset M$ with $(m-1)$-rectifiable boundary and $0<\hh^m(S)<v_0$ the following isoperimetric inequalities hold
\begin{equation}
\label{eq:Ine1}
\hh^{m-1}(\ptl S)\ge \ga_1(M,v_0)\,\hh^m(S)
\end{equation}
and
\begin{equation}
\label{eq:Ine2}
\hh^{m-1}(\ptl S)\ge \ga_2 (M,v_0)\,\hh^m(S)^{(m-1)/m},
\end{equation}
\end{lemma}

We are now ready to prove the main results of the section.
\begin{proposition}
\label{prp:pr}
\mbox{}
Assume $0<\be<1$, $t_i\uparrow\infty\,$ and $\{E_i\}_{i\in\nn}$ be a sequence of compact symmetrized isoperimetric sets in $ \esf\times t_i N$ of volume $\vv_{g_+^{t_i}}(E_i)=\be \vv_{g_+^{t_i}}(\esf\times N)$. Let $t_0>0$ and $s_i=t_0\,t_i^{-1}$.
\begin{enum}
\item
Then, possibly passing to a non-relabeling subsequence, $\{E_i\}_{i\in\nn}$ converges both in $L^1$ and Hausdorff topology of $\esf\times t_0N$, to a finite perimeter set $E$ which is a cylinderoid, i.e $E=\esf\times \pi_2(E)$.
\item Moreover $\pi_2(E)$ is an isoperimetric set in $t_0N$, of volume fraction $\be$.
\item Furthermore
\begin{equation}
\label{eq:ane00}
\vv_{g_+^{t_0}}(E_i)=\vv_{g_+^{t_0}}(\esf\times \pi_2(E)) \,\,\text{and}\,\,\pp_{g_+^{t_0}}(E_i)\le\pp_{g_+^{t_0}}(\esf\times \pi_2(E))
\end{equation}
\end{enum}
\end{proposition}
\begin{proof}
Owing to \eqref{eq:anest1} we get
\begin{equation}
\label{eq:thm0}
\vv_{g_+^{t_0}}(E_i)=\be\vv_{g_+^{t_0}}(\esf\times N) :=v_0\quad\text{for every}\,\,i\in\nn.
\end{equation}
Since $N$ is compact, there exists an isoperimetric region $S\subset N$ of volume $v_0\,\vv(\esf)^{-1}$. As $E_i$ are isoperimetric by assumption, we have
\begin{equation}
\label{eq:cona}
\vv_{g_+^{t_i}}(\esf\times S)= \vv_{g_+^{t_i}}(E_i) \,\,\text{and}\,\,\pp_{g_+^{t_i}}(E_i)\le\pp_{g_+^{t_i}}(\esf\times S)
\end{equation}
Thanks to (ii) and (iii) of Lemma \ref{lem:anest}, we deduce
\begin{equation}
\label{eq:conb}
\frac{\pp_{g_+^{t_0}}(E_i)}{\pp_{g_+^{t_0}}(\esf\times S)}=\frac{\pp_{g_+^{s_it_i}}(E_i)}{\pp_{g_+^{s_it_i}}(\esf\times S)}\le\frac{s_i^{n-1}\pp_{g_+^{t_i}}(E_i)}{s_i^{n-1}\pp_{g_+^{t_i}}(\esf\times S)}\le 1
\end{equation}
Consequently, $\{\pp_{g_+^{t_0}}(E_i)\}_{i\in\nn}$ is bounded and so, possibly passing to a subsequence, $E_i\to E$ in the $L^1(\esf\times t_0 N)$ topology. Hence, by the semi-continuity of the perimeter, we get
\begin{equation}
\label{eq:con1}
\pp_{g_+^{t_0}}(E)\le\liminf\pp_{g_+^{t_0}}(E_i)\quad\text{and}\,\,\vv_{g_+^{t_0}}(E)=\be\vv_{g_+^{t_0}}(\esf\times N) :=v_0
\end{equation}
Owing to Lemma \ref{lem:h=l}, possibly passing to a non-relabeling subsequence, we get $E_i\to E$ in the Hausdorff topology of $\esf\times t_0 N$. We argue by contradiction, assume that $E\not=\esf\times \pi_2(E)$. Then by Lemma \ref{lem:ww} there are $\de>0$ and $w_0\in (0,\vv(\esf))$ so that
\begin{equation}
\label{eq:con2}
\vv_{g_+^{t_0}}(E_i{\{w_0\}})>\de\quad\text{for sufficiently large}\,\,i\in\nn.
\end{equation}
Hence, by \eqref{eq:anest1}, we get
\begin{equation}
\label{eq:con3}
\frac{\vv_{g_+^{t_i}}(E_i{\{w_0\}})}{\vv_{g_+^{t_i}}(E_i)}=\frac{\vv_{g_+^{t_0}}(E_i{\{w_0\}})}{\vv_{{g_+^{t_0}}}(E_i)}= \frac{\vv_{{g_+^{t_0}}}(E_i{\{w_0\}})}{\be\vv_{{g_+^{t_0}}}(\esf\times N)}\ge \frac{\de}{\be\vv_{{g_+^{t_0}}}(\esf\times N)} :=c_1>0,
\end{equation}
for sufficiently large $i\in\nn$. Observe that since $ E_i\{w_0\}$ is symmetrized there holds
\begin{equation}
\label{eq:3a}
\pi_2(\ptl E_i\cap E_i\{w_0\})=\pi_2( E_i\{w_0\}) := \Xi_i.
\end{equation}
Now owing to \eqref{eq:con3} and Fubini's Theorem we obtain
\begin{equation}
\label{eq:con4}
\vv_{g_+^{t_i}}(E_i)\le c_1^{-1}\int_{\Xi_i}\hh_{g_1}^m( E_i\cap \pi_2^{-1}(y))\,d\hh_{{t_i}^2g_2}^n
\end{equation}

\begin{proof}
Let $\bar{g}=g_1\times\bar{g}_2$, where $\bar{g}_2=g_2|_{T\Sg\times T\Sg}$.
Note that the outer unit normal $\nu$ on $tM\times \Sg$ is tangent to the second factor. Consequently,
\begin{equation}
\label{eq:curvs}
\ric^{g_-^{t}}(\nu,\nu)=\ric^{\,g_2}(\nu,\nu),
\qquad
\sg^{\bar{g}_-^{t}}=\sg^{\,\bar{g}_2},
\end{equation}
where $\ric$ denotes Ricci tensor and $\sg$ second fundamental form. Hence
\begin{equation}
\label{eq:sts00}
q(x,\vsg):=\ric^{g_-^{t}}(\nu,\nu)+|\sg^{\bar{g}_-^{t}}|^2
= \ric^{\,g_2}(\nu,\nu)+|\sg^{\,\bar{g}_2}|^2 :=q(\vsg).
\end{equation}
Set
\begin{equation}
\label{eq:sts0}
t_0^2=\frac{\la_1(g_1)}{\|q\|_\infty}, \qquad t<t_0.
\end{equation}
As a consequence of the product spectral decomposition, it suffices to show that
\begin{equation}
\label{eq:sts1}
-\int_{tM\times\Sg} f\big(\Delta f+qf\big)\geq0 \qquad \text{for all smooth } f \text{ with } \int_{tM\times\Sg} f=0,
\end{equation}
where $f(x,\vsg)=u(x)v(\vsg)$, $x\in M$, $\vsg\in \Sg$ and $\Delta=\Delta^{\bar{g}_-^{t}}$. Owing to \eqref{eq:delprt}
\begin{equation}
\label{eq:sts2}
\begin{split}
-\int_{tM\times\Sg}f(\Delta f+qf)
&=-\int_{tM\times\Sg} v^2 u\Delta^{t^2g_1} u
 -\int_{tM\times\Sg} u^2 v \Delta^{\bar{g}_2} v
 -\int_{tM\times\Sg} q u^2 v^2.
\end{split}
\end{equation}
Applying Stokes’ theorem:
\begin{equation}
\label{eq:sts3}
-\int_{tM\times\Sg} v^2 u \Delta^{t^2g_1}u
= \int_{tM} |\nabla^{t^2g_1}u|^2_{t^2g_1}\,\int_{\Sg} v^2,
\end{equation}
and
\begin{equation}
\label{eq:sts4}
-\int_{tM\times\Sg} u^2 v \Delta^{\bar{g}_2}v
= \int_{tM} u^2 \,\int_{\Sg} |\nabla^{\bar{g}_2}v|^2_{\bar{g}_2}.
\end{equation}

\section{Appendix}
In this appendix we recall the fibrewise rearrangement used throughout the paper.
We consider here a symmetrization first introduced by Ros~\cite{ros} and later generalized by Morgan~\cite{morpol} for warped product manifolds with density.
Let $M,N$ be compact Riemannian manifolds. Let $\esf$ be a round sphere with the same volume and dimension as $M$ and with isoperimetric profile not exceeding that of $M$. Fix a point $p_0\in \esf$ and consider the foliation of the geodesic balls centered at $p_0$.
Let $E\subset M\times N$ be a set of finite perimeter. Replace each non-empty horizontal slice with the unique geodesic ball in the foliation centred at $p_0$ having the same volume. We denote by $\sym E \subset \esf\times N$ this set, which we call the symmetrized set of~$E$. Then, there holds
\begin{equation}
\label{eq:sym1}
\vv_{\esf\times N}(\sym E)=\vv_{M\times N} (E)\quad \text{and}\ \quad \pp_{\esf\times N}(\sym E)\le \pp_{M\times N}(E).
\end{equation}
Consequently,
\begin{equation}
\label{eq:sym2}
I_{M\times N}\ge I_{\esf\times N}
\end{equation}

\begin{proof}
Part (i) follows directly from computing the Jacobian of the identity map from $(M_1 \times M_2,g)$ to $(M_1 \times M_2,g_+^t)$.
For (ii), at a regular point of $\Sigma$ the $g$-unit exterior normal vector $\nu$ can be decomposed as $\nu = a\nu_1 + b\nu_2$ so that $a^2+b^2=1$ and $\nu_1$, $\nu_2$ unit vectors tangent to $M_1$ and $M_2$, respectively. The Jacobian of the identity map from $(M_1 \times M_2,g)$ to $(M_1 \times M_2,g_+^t)$ restricted to $\Sigma$ then equals $t^{m_2-1}(t^2a^2+b^2)^{1/2}$. This yields inequality \eqref{eq:anest2}.
Assertion (iii) follows from the fact that $\nu$ is tangent to $M_2$ precisely when $a=0$ almost everywhere.

\end{proof}
The following local isoperimetric inequalities are classical, see Duzaar and Steffen \cite{du-st}.
\begin{lemma}
\label{lem:ine}
Let $M$ be a compact $m$-dimensional Riemannian manifold. Given $0<v_0<\hh^m(M)$, there exist positive constants  $\ga_1(M,v_0),\ga_2(M,v_0)$ such that for any set $S\subset M$ with $(m-1)$-rectifiable boundary and $0<\hh^m(S)<v_0$ the following isoperimetric inequalities hold
\begin{equation}
\label{eq:Ine1}
\hh^{m-1}(\partial S)\geqslant \ga_1(M,v_0)\,\hh^m(S)
\end{equation}
and
\begin{equation}
\label{eq:Ine2}
\hh^{m-1}(\partial S)\geqslant \ga_2 (M,v_0)\,\hh^m(S)^{(m-1)/m},
\end{equation}
\end{lemma}

The \emph{isoperimetric profile} of a Riemannian manifold $(M,g)$ is defined by
\[
  I_g(v) = \inf \big\{ \pp_g(E) : \vv_g(E) = v \big\}.
\]
A set $E \subset M$ is called an \emph{isoperimetric region} if $\pp_g(E) = I_g(\vv_g(E))$. Existence of isoperimetric regions in compact manifolds is guaranteed by standard compactness arguments in geometric measure theory, see \cite{Maggi}. Moreover, classical regularity theory implies that the boundary of an isoperimetric
region is a smooth hypersurface except for a singular set of Hausdorff codimension at least $8$,  as proved by Morgan \cite{moreg} and by Gonzales–Massari–Tamanini \cite{gomato}. In Chapter 13.2 of \cite{Morgan2016}, Morgan, following methods of Smale \cite{Smale1999},  describes constructions of irregular isoperimetric hypersurfaces in manifolds of dimension greater than seven. Furthermore, Morgan–Johnson \cite{morjohn} and Nardulli \cite{nard} show that, for small volumes (and their complements), isoperimetric hypersurfaces are smooth.

Isoperimetric regions are considered up to sets of measure zero.
In particular, every isoperimetric set is equivalent to its closure, and its perimeter can be computed on the topological boundary:
\begin{equation}\label{eq:ww4}
\mathcal{P}(E) = \mathcal{P}(\overline{E}) = \hh^{m-1}(\partial E).
\end{equation}

Let $\Sigma\subset M$ be an isoperimetric hypersurface. Standard first and second variation computations
(see for instance Barbosa--do Carmo--Eschenburg~\cite{bd})
imply that the regular part $\reg(\Sigma)$ has constant mean curvature and satisfies a stability inequality.
Specifically, for every smooth compactly supported function $u:\reg(\Sigma)\to\mathbb{R}$ with zero mean one has
\begin{equation}\label{eq:stab}
- \int_\Sigma u \,\big(\Delta u + \ric(\nu,\nu)u + |\sigma|^2 u\big) \,\geq 0,
\end{equation}

where $\nu$ is the exterior unit normal to $\Sigma$, $\Delta$ is the Laplace operator on $\Sigma$, $|\sigma|^2$ is the squared norm of the second fundamental form, and $\ric$ denotes the Ricci tensor of the ambient manifold $M$.
We denote by $J_\Sigma$ the Jacobi operator, which acts on smooth functions $u$ as follows:
\[
  J_\Sigma u = \Delta u + \ric(\nu,\nu)\,u + |\sigma|^2 u.
\]
If the inequality in \eqref{eq:stab} is strict for all nontrivial $u$, we say that $\sigma$ is \emph{strictly stable}.

Part (i) is obtained by “computing the Jacobian of the identity map from $(M_1 \times M_2,g)$ to $(M_1 \times M_2,g_+^t)$,” giving \eqref{eq:anest1}.

For (ii), “at a regular point of $\Sigma$ the $g$-unit exterior normal vector $\nu$ can be decomposed as $\nu=a\nu_1+b\nu_2$ so that $a^2+b^2=1$ and $\nu_1$, $\nu_2$ unit vectors tangent to $M_1$ and $M_2$, respectively.” The Jacobian of the identity map restricted to $\Sigma$ “equals $t^{m_2-1}(t^2a^2+b^2)^{1/2}$,” which “yields inequality \eqref{eq:anest2}” for $t\le 1$.

For (iii), “$\nu$ is tangent to $M_2$ precisely when $a=0$ almost everywhere,” which characterizes when equality holds in \eqref{eq:anest2}.

Part (i) is obtained by “computing the Jacobian of the identity map from $(M_1 \times M_2,g)$ to $(M_1 \times M_2,g_+^t)$,” giving \eqref{eq:anest1}.

For (ii), “at a regular point of $\Sigma$ the $g$-unit exterior normal vector $\nu$ can be decomposed as $\nu=a\nu_1+b\nu_2$ so that $a^2+b^2=1$ and $\nu_1$, $\nu_2$ unit vectors tangent to $M_1$ and $M_2$, respectively.” The Jacobian of the identity map restricted to $\Sigma$ “equals $t^{m_2-1}(t^2a^2+b^2)^{1/2}$,” which for $t\le 1$ yields
\begin{equation}\label{eq:anest2}
  \hh_{g_+^t}^{m_1+m_2-1}(\Sg)
  \leq t^{m_2-1}\,\hh_g^{m_1+m_2-1}(\Sg).
\end{equation}

For (iii), “$\nu$ is tangent to $M_2$ precisely when $a=0$ almost everywhere,” which characterizes when equality holds in the equation above.

\label{lem:anest}
Let $(M_i,g_i)$ be Riemannian manifolds of dimensions $m_i$ ($i=1,2$), and set $g=g_1\times g_2$. Then the following hold for all measurable $S\subset M_1\times M_2$ and $(m_1+m_2-1)$–rectifiable $\Sigma$:
\begin{enumerate}
\item[(i)] For every set $S \subset M_1 \times M_2$,
\begin{equation}\label{eq:anest1}
  \vv_{g_+^t}(S) = t^{m_2} \,\vv_g(S).
\end{equation}
\item[(ii)] If $\Sigma \subset M_1 \times M_2$ is $(m_1+m_2-1)$-rectifiable, then for $t \leq 1$,
\begin{equation}\label{eq:anest2}
  \hh_{g_+^t}^{m_1+m_2-1}(\Sg)
  \leq t^{m_2-1}\,\hh_g^{m_1+m_2-1}(\Sg).
\end{equation}
\item[(iii)] Equality holds in the equation above if and only if the $g$-normal of $\Sigma$ is tangent to $M_2$, up to a $\hh_g^{m_1+m_2-1}$-null set.
\end{enumerate}

Inequality or Bound
Biconditional or Equivalence

Question: Let \((M_i,g_i)\) for \(i=1,2\) be Riemannian manifolds of dimensions \(m_i\), let \(g=g_1\times g_2\) on \(M_1\times M_2\), and for \(t>0\) define the right anisotropic \(t\)-homothety metric by \(g_+^t=g_1\times t^2 g_2\). For a measurable set \(S\subset M_1\times M_2\), write \(\mathrm{vol}_{g}(S)\) for its Riemannian volume, and for an \((m_1+m_2-1)\)-rectifiable set \(\Sigma\subset M_1\times M_2\), write \(\mathcal H^{m_1+m_2-1}_{g}(\Sigma)\) for its \((m_1+m_2-1)\)-dimensional Hausdorff measure with respect to \(g\). Which quantitative statement holds for volume and hypersurface measure under \(g_+^t\)?

For every measurable set \(S\subset M_1\times M_2\), one has
\[
\mathrm{vol}_{g_+^t}(S)=t^{m_2}\,\mathrm{vol}_{g}(S).
\]
Moreover, if \(\Sigma\subset M_1\times M_2\) is \((m_1+m_2-1)\)-rectifiable and \(t\le 1\), then
\[
\mathcal H^{m_1+m_2-1}_{g_+^t}(\Sigma)
\le t^{m_2-1}\,\mathcal H^{m_1+m_2-1}_{g}(\Sigma).
\]
Equality in this last inequality holds if and only if the \(g\)-normal to \(\Sigma\) is tangent to the \(M_2\)-factor, up to a \(\mathcal H^{m_1+m_2-1}_{g}\)-null set.

Choices:
B.
For every measurable set \(S\subset M_1\times M_2\), one has
\[
\mathrm{vol}_{g_+^t}(S)=t^{m_2}\,\mathrm{vol}_{g}(S).
\]
Moreover, if \(\Sigma\subset M_1\times M_2\) is \((m_1+m_2-1)\)-rectifiable, then for every \(t>0\),
\[
\mathcal H^{m_1+m_2-1}_{g_+^t}(\Sigma)
\le t^{m_2-1}\,\mathcal H^{m_1+m_2-1}_{g}(\Sigma).
\]
Equality in this last inequality holds if and only if the \(g\)-normal to \(\Sigma\) is tangent to the \(M_2\)-factor, up to a \(\mathcal H^{m_1+m_2-1}_{g}\)-null set.
C.
For every measurable set \(S\subset M_1\times M_2\), one has
\[
\mathrm{vol}_{g_+^t}(S)=t^{m_2}\,\mathrm{vol}_{g}(S).
\]
Moreover, if \(\Sigma\subset M_1\times M_2\) is \((m_1+m_2-1)\)-rectifiable and \(t\le 1\), then
\[
\mathcal H^{m_1+m_2-1}_{g_+^t}(\Sigma)
\le t^{m_2-1}\,\mathcal H^{m_1+m_2-1}_{g}(\Sigma).
\]
D.
For every measurable set \(S\subset M_1\times M_2\), one has
\[
\mathrm{vol}_{g_+^t}(S)=t^{m_1}\,\mathrm{vol}_{g}(S).
\]
Moreover, if \(\Sigma\subset M_1\times M_2\) is \((m_1+m_2-1)\)-rectifiable and \(t\le 1\), then
\[
\mathcal H^{m_1+m_2-1}_{g_+^t}(\Sigma)
\le t^{m_1-1}\,\mathcal H^{m_1+m_2-1}_{g}(\Sigma).
\]
Equality in this last inequality holds if and only if the \(g\)-normal to \(\Sigma\) is tangent to the \(M_1\)-factor, up to a \(\mathcal H^{m_1+m_2-1}_{g}\)-null set.
E.
For every measurable set \(S\subset M_1\times M_2\), one has
\[
\mathrm{vol}_{g_+^t}(S)=t^{m_2}\,\mathrm{vol}_{g}(S).
\]
Moreover, if \(\Sigma\subset M_1\times M_2\) is \((m_1+m_2-1)\)-rectifiable and \(t\le 1\), then
\[
\mathcal H^{m_1+m_2-1}_{g_+^t}(\Sigma)
= t^{m_2-1}\,\mathcal H^{m_1+m_2-1}_{g}(\Sigma).
\]
In particular, the hypersurface measure always scales exactly by \(t^{m_2-1}\), independently of the orientation of the \(g\)-normal.

The stem does not explicitly state or strongly hint at the correct option; it asks for an equivalent characterization of the equality case without revealing which factor or which quantifier structure is right.

This is essentially a direct recall/restatement question: it asks which statement is equivalent to the equality case, and the correct option is the theorem-style characterization itself.

Some reasoning is needed to distinguish 'if and only if' from one-way implication, an a.e. null-set condition from pointwise regularity, and the correct factor direction. But it mostly tests precise recall of the equality characterization rather than deeper derivation.

The distractors are mathematically plausible and target common failure modes: confusing the factor, weakening to a one-way implication, overstrengthening to every regular point, and replacing the exact condition by a misleading constant-angle condition.

Well-constructed distractors and no answer leakage, but the item is largely theorem-recall and only moderately tests generative reasoning.


[Theorem~\ref{thm:mainthm} \& Corollary~\ref{cor:spec}]
\label{thm:1}
Let $G$ be simply connected. The coordinate ring of the Vinberg monoid $\textnormal{Env } G$ is a partially compactified upper cluster algebra where all frozen variables are not invertible.

The map $\pi$ coincides with the specialization of certain frozen variables. In particular, the asymptotic semigroup is obtained by specializing certain frozen variables to 0.

\label{sec:intro}
In this paper, all varieties are defined over $\BC$.

\subsection{The Vinberg monoid}
\label{sec:intro:monoid}
An \emph{algebraic monoid} is an affine variety with a monoid structure such that the multiplication map is algebraic. An algebraic monoid $M$ is called \emph{reductive} if its group of units $G(M)$ is reductive. A systematic study of reductive monoids starts from Putcha \cite{Put84} and Renner \cite{Ren85}, and it has attracted many attentions since then.

In the work of \cite{Vin95}, Vinberg classified reductive monoids using a representation theoretic approach. In particular, associated with any semisimple group $G$, Vinberg defined a universal object $\textnormal{Env } G$, in a family of flat reductive monoids, which is now called the \emph{Vinberg monoid}. The Vinberg monoid $\textnormal{Env } G$ is a reductive monoid whose group of units has the commutator group isomorphic to $G$, and any flat reductive monoid with this property can be obtained from $\textnormal{Env } G$ by a base change.

Moreover, suppose $r$ is the rank of $G$. Then there is a canonical flat map
\[
\pi: \textnormal{Env } G\longrightarrow \mathbb{C}^r
\]
onto the affine space, where the generic fibres are isomorphic to the group $G$, and the special fibre $\pi^{-1}(\mathbf{0})$ is the \emph{asymptotic semigroup} considered by Vinberg \cite{Vin95a}.

The Vinberg monoid together with the map $\pi$ have found significant applications in recent developments of the geometric representation theory, see \citelist{\cite{BNS16}\cite{FKM20}} for example.

\subsection{Cluster algebras}
\label{sec:intro:cluster}

The theory of cluster algebras was developed by Fomin and Zelevinsky \cite{FZ02} to study total positivity and dual canonical bases in semisimple groups. The coordinate rings of many varieties related to algebraic groups have shown to be (upper) cluster algebras (See \citelist{\cite{BFZ05}\cite{Wil13}\cite{Sco06}\cite{CGG+25}\cite{GLSS}} for example.).

In contrast, cluster algebra structures on algebraic monoids have received little attention. It seems that the only nontrivial example is the cluster algebra structure on the space of $n\times n$ matrices from \cite{FWZ20}*{Theorem~6.6.1}.

The existence of a cluster structure on a variety has a wide range of consequences. For example, it provides a total positivity structure, Poisson structure \cite{GSV10}, and a cluster theoretic canonical basis on its coordinate ring \cite{GHKK18}.

In this paper, we will focus on upper cluster algebras where not all frozen variables are inverted. We borrow the name \emph{partially compactified (upper) cluster algebras} from \cite{GHKK18} to highlight this choice of convention.

\subsection{Main results}
\label{sec:intro:main}

The main result of this paper is the follows.

\subsection{The Vinberg monoid}
\label{sec:intro:monoid}
An \emph{algebraic monoid} is an affine variety with a monoid structure such that the multiplication map is algebraic. An algebraic monoid $M$ is called \emph{reductive} if its group of units $G(M)$ is reductive. A systematic study of reductive monoids starts from Putcha \cite{Put84} and Renner \cite{Ren85}, and it has attracted many attentions since then.

In the work of \cite{Vin95}, Vinberg classified reductive monoids using a representation theoretic approach. In particular, associated with any semisimple group $G$, Vinberg defined a universal object $\textnormal{Env } G$, in a family of flat reductive monoids, which is now called the \emph{Vinberg monoid}. The Vinberg monoid $\textnormal{Env } G$ is a reductive monoid whose group of units has the commutator group isomorphic to $G$, and any flat reductive monoid with this property can be obtained from $\textnormal{Env } G$ by a base change.

Moreover, suppose $r$ is the rank of $G$. Then there is a canonical flat map
\[
\pi: \textnormal{Env } G\longrightarrow \mathbb{C}^r
\]
onto the affine space, where the generic fibres are isomorphic to the group $G$, and the special fibre $\pi^{-1}(\mathbf{0})$ is the \emph{asymptotic semigroup} considered by Vinberg \cite{Vin95a}.

In this paper, we will focus on upper cluster algebras where not all frozen variables are inverted. We borrow the name \emph{partially compactified (upper) cluster algebras} from \cite{GHKK18} to highlight this choice of convention.

\subsection{Main results}
\label{sec:intro:main}

The main result of this paper is the follows.

\begin{cor}\label{cor:spec}
The diagram
\begin{equation}
    \begin{tikzcd}
    \Env G \arrow[dr,"\pi"'] \arrow[rr,"\sim"] && \Spec \overline{\CU}(\wt{\bfs}(\bfi)) \arrow[dl,"\pi^{I'}"] \\ & \BC^I
    \end{tikzcd}
\end{equation}
commutes.
\end{cor}

\begin{theorem}
\label{thm:mainthm}
Let $\bfi$ be a double reduced word for $(w_0,w_0)$. Set $\Sigma=\wt{J}_\fro\backslash I'$. The isomorphism $\Psi_\bfi$ restricts to isomorphisms
\[
\overline{\CU}^\Sigma(\wt{\bfs}(\bfi))\overset{\sim}{\rightarrow}\BC[G\x ^Z T],\qquad \overline{\CU}(\wt{\bfs}(\bfi))\overset{\sim}{\rightarrow}\BC[\Env G].
\]
In other words, we have the following commutative diagram
\begin{equation}\label{eq:cr}
    \begin{tikzcd}
        & \CU(\wt{\bfs}(\bfi)) \arrow[r,"\sim"] & \BC[G^{w_0,w_0}\x ^Z T] \\
        & \overline{\CU}^\Sigma(\wt{\bfs}(\bfi)) \arrow[r,"\sim"] \arrow[u,hook] & \BC[G\x^Z T] \arrow[u,hook] \\
        & \overline{\CU}(\wt{\bfs}(\bfi)) \arrow[u,hook] \arrow[r,"\sim"] & \BC[\Env G] \arrow[u,hook]
    \end{tikzcd}
\end{equation}
\end{theorem}

The main result of this paper is the follows.

Combining with the universal property of the Vinberg monoid and the coefficient specialization of cluster algebras, Theorem \ref{thm:1} allows us to construct cluster algebra structures on a large family of reductive monoids.

\begin{thmintro}
[Theorem~\ref{thm:affineAbelianCluster}]
\label{thm:3}
Let $M$ be a flat reductive monoid. Suppose that the semisimple part of $M$ is simply connected and the abelianization of $M$ is isomorphic to an affine space $\mathbb{C}^k$ (See Section \ref{sec:prelim:Vinberg} for definitions). Then the coordinate ring of $M$ is a partially compactified upper cluster algebra with all frozen variables not invertible.
\end{thmintro}

Let us remark that the assumptions on the reductive monoid $M$ are reasonable. Any set of frozen variables comes with a map $M\to \BC^k$. This should be compatible with the abelianization for a natural cluster structure on $M$. On the other hand, our assumptions on $M$ is flexible enough to cover interesting examples. For example, they include the case where $M$ is the monoid of $n\times n$ matrices. In this case, Theorem \ref{thm:3} recovers the well-known cluster structures, presented in \cite{FWZ20}*{Theorem~6.6.1}.

\begin{thmintro}[Theorem~\ref{thm:reductiveMonoidCluster}]
\label{thm:2}
Let $H$ be a connected reductive group with simply connected commutator group. Then the coordinate ring of $H$ is a partially compactified upper cluster algebra, whose further partially compactification along all invertible frozen variables is the coordinate ring of a flat reductive monoid with unit group $H$.
\end{thmintro}

\begin{theorem}
\label{thm:reductiveMonoidCluster}
Let $H$ be a connected reductive group with simply connected derived group. Then the coordinate ring of $H$ is a partially compactified upper cluster algebra $\overline{\mathcal{U}}^\Sigma$, whose further partially compactification $\overline{\mathcal{U}}$ along all frozen variables is the coordinate ring of a flat reductive monoid with unit group $H$.
\end{theorem}

\begin{theorem}\label{thm:affineAbelianCluster}
Let $M$ be a flat reductive monoid. Suppose that the semisimple part of $M$ is simply connected and the abelianization of $M$ isomorphic to an affine space $\mathbb{C}^n$. Then the coordinate ring $\BC[M]$ is a partially compactified upper cluster algebra with all frozen variables not invertible.
\end{theorem}

\begin{example}
Let $M=M_n$ be the monoid of $n\times n$ matrices, so $G(M)=GL_n$ and $G_M=SL_n$. In this case, the subalgebra $\BC[M]^{G_M\times G_M}$ of invariant functions is isomorphic to the polynomial algebra in one variable, generated by the determinant. Hence the abelianization $A(M)$ is isomorphic to the affine line. Theorem~\ref{thm:affineAbelianCluster} gives a partially compactified upper cluster algebra structure on $\BC[M_n]$, which coincides with the cluster structure in \cite{FWZ20}*{Theorem 6.6.1}.
\end{example}

\begin{cor}\label{cor:spec}
The diagram
\begin{equation}
    \begin{tikzcd}
    \Env G \arrow[dr,"\pi"'] \arrow[rr,"\sim"] && \Spec \overline{\CU}(\wt{\bfs}(\bfi)) \arrow[dl,"\pi^{I'}"] \\ & \BC^I
    \end{tikzcd}
\end{equation}
commutes.
\end{cor}

\begin{thmintro}[Theorem~\ref{thm:mainthm} \& Corollary~\ref{cor:spec}]
\label{thm:1}
Let $G$ be simply connected. The coordinate ring of the Vinberg monoid $\Env G$ is a partially compactified upper cluster algebra where all frozen variables are not invertible.

The map $\pi$ coincides with the specialization of certain frozen variables. In particular, the asymptotic semigroup is obtained by specializing certain frozen variables to 0.
\end{thmintro}

\begin{theorem}
\label{thm:mainthm}
Let $\bfi$ be a double reduced word for $(w_0,w_0)$. Set $\Sigma=\wt{J}_\fro\backslash I'$. The isomorphism $\Psi_\bfi$ restricts to isomorphisms
\[
\overline{\CU}^\Sigma(\wt{\bfs}(\bfi))\overset{\sim}{\rightarrow}\BC[G\x ^Z T],\qquad \overline{\CU}(\wt{\bfs}(\bfi))\overset{\sim}{\rightarrow}\BC[\Env G].
\]
In other words, we have the following commutative diagram
\begin{equation}\label{eq:cr}
    \begin{tikzcd}
        & \CU(\wt{\bfs}(\bfi)) \arrow[r,"\sim"] & \BC[G^{w_0,w_0}\x ^Z T] \\
        & \overline{\CU}^\Sigma(\wt{\bfs}(\bfi)) \arrow[r,"\sim"] \arrow[u,hook] & \BC[G\x^Z T] \arrow[u,hook] \\
        & \overline{\CU}(\wt{\bfs}(\bfi)) \arrow[u,hook] \arrow[r,"\sim"] & \BC[\Env G] \arrow[u,hook]
    \end{tikzcd}
\end{equation}
\end{theorem}

Combining with the universal property of the Vinberg monoid and the coefficient specialization of cluster algebras, Theorem \ref{thm:1} allows us to construct cluster algebra structures on a large family of reductive monoids.

\begin{thmintro}
[Theorem~\ref{thm:affineAbelianCluster}]
\label{thm:3}
Let $M$ be a flat reductive monoid. Suppose that the semisimple part of $M$ is simply connected and the abelianization of $M$ is isomorphic to an affine space $\mathbb{C}^k$ (See Section \ref{sec:prelim:Vinberg} for definitions). Then the coordinate ring of $M$ is a partially compactified upper cluster algebra with all frozen variables not invertible.
\end{thmintro}

Let us remark that the assumptions on the reductive monoid $M$ are reasonable. Any set of frozen variables comes with a map $M\to \BC^k$. This should be compatible with the abelianization for a natural cluster structure on $M$. On the other hand, our assumptions on $M$ is flexible enough to cover interesting examples. For example, they include the case where $M$ is the monoid of $n\times n$ matrices. In this case, Theorem \ref{thm:3} recovers the well-known cluster structures, presented in \cite{FWZ20}*{Theorem~6.6.1}.

By inverting the appropriate frozen variables, we obtain cluster algebra structures on any connected reductive group.

\begin{thmintro}[Theorem~\ref{thm:reductiveMonoidCluster}]
\label{thm:2}
Let $H$ be a connected reductive group with simply connected commutator group. Then the coordinate ring of $H$ is a partially compactified upper cluster algebra, whose further partially compactification along all invertible frozen variables is the coordinate ring of a flat reductive monoid with unit group $H$.
\end{thmintro}

In the case when $H$ is semisimple, Theorem \ref{thm:2} was obtained by Qin and Yakimov \cite{QY25}. Independently, Oya \cite{Oya25} proved the same result under the further condition that $H$ is not of type $F_4$. In the semisimple case, there are no additional invertible frozen variables. This is also reflected by the fact that a reductive monoid with semisimple unit group is necessarily a group.

\subsection{Strategy of the proof}
\label{sec:intro:strategy}

The first key ingredient of the proof of Theorem~\ref{thm:1} is the construction of a framed group $\widetilde{G}$ of $G$. Let $I$ be the set of simple roots of $G$ with $|I|=r$. We define $\widetilde{I}$ by adding $r$ vertices labeled $i'$ and a simple edge joining $i$ and $i'$ for each $i\in I$. The framed group $\widetilde{G}$ is defined to be the corresponding Kac--Moody group. We then construct an isomorphism between the unit group of $\textnormal{Env } G$ to the standard Levi subgroup of $\widetilde{G}$ associated with $I$. Using the cluster structures on double Bruhat cells for Kac--Moody groups constructed by Williams \cite{Wil13} (See also \cite{SW21}), we obtain a cluster structure on an open part $(\textnormal{Env } G)^\circ$ of $\textnormal{Env } G$.

The second essential technique is a comparison between the cluster valuations along frozen variables and the valuation associated with the boundary divisor $\textnormal{Env } G-(\textnormal{Env } G)^\circ$. This is achieved by applying the string parametrization of the dual canonical basis.

We note that the technique of extending root data was inspired by the thickening map introduced by Bao and He \cite{BH24}. Similar ideas appeared in the recent works \citelist{\cite{Qin25}\cite{FH25}} as well.

\subsection{Future problems}
\label{sec:intro:future}

Let us close the introduction by mentioning some future directions.

Firstly, reductive monoids admit standard Poisson structures and quantization \cite{BS25}. We expect that all the results in the current paper can be extended to the quantum setting.

Secondly, we focus only on upper cluster algebras in this paper. For simply-connected simple group $G$ not of type $F_4$, Oya \cite{Oya25} proved that the upper cluster algebra structure on $\mathbb{C}[G]$ coincides with the ordinary cluster algebra in the sense of Fomin and Zelevinsky. A quantum analogue of this coincidence was recently established by Oya, Qin, and Yakimov \cite{OQY25}. It is therefore a natural question whether the upper cluster algebras studied in this paper also coincide with ordinary cluster algebras.

Thirdly, the Vinberg monoid $\textnormal{Env } G$ is isomorphic to the total coordinate ring of the wonderful compactification $\mathbf{X}$ of $G_{ad}=G/Z$. It would be interesting to explore the connections between the cluster algebra structure on $\textnormal{Env } G$ developed in the current paper with the total positivity on $\mathbf{X}$ studied by He \cite{He04}.

\vspace{.2cm}

\noindent {\bf Acknowledgment: } We are grateful to Huanchen Bao for valuable discussions and suggestions during the writing of this paper. JS is grateful to Fan Qin for helpful discussions at the early stage of the paper. JS is supported by the Glorious Sun Charity Fund.

The post-theorem introduction contains a proof strategy for Theorem~\ref{thm:1}:

1. **Construct a framed group and relate it to $\textnormal{Env }G$.** “The first key ingredient of the proof of Theorem~\ref{thm:1} is the construction of a framed group $\widetilde{G}$ of $G$.” With $I$ the simple roots, define $\widetilde{I}$ by adding vertices $i'$ and an edge between $i$ and $i'$ for each $i\in I$, and let $\widetilde{G}$ be the associated Kac--Moody group. Then “we construct an isomorphism between the unit group of $\textnormal{Env } G$ to the standard Levi subgroup of $\widetilde{G}$ associated with $I$.”

2. **Import cluster structures from Kac--Moody double Bruhat cells to an open part of $\textnormal{Env }G$.** “Using the cluster structures on double Bruhat cells for Kac--Moody groups constructed by Williams \cite{Wil13} (See also \cite{SW21}), we obtain a cluster structure on an open part $(\textnormal{Env } G)^\circ$ of $\textnormal{Env } G$.”

3. **Compare valuations to control the boundary/frozen variables via canonical basis combinatorics.** “The second essential technique is a comparison between the cluster valuations along frozen variables and the valuation associated with the boundary divisor $\textnormal{Env } G-(\textnormal{Env } G)^\circ$.” This comparison “is achieved by applying the string parametrization of the dual canonical basis.”

(Additional remark in the strategy section: “the technique of extending root data was inspired by the thickening map introduced by Bao and He \cite{BH24}.”)

The post-theorem introduction contains a proof strategy for the main theorem:

1. **Construct a framed group and relate it to $\textnormal{Env }G$.** “The first key ingredient of the proof of the main theorem is the construction of a framed group $\widetilde{G}$ of $G$.” With $I$ the simple roots, define $\widetilde{I}$ by adding vertices $i'$ and an edge between $i$ and $i'$ for each $i\in I$, and let $\widetilde{G}$ be the associated Kac--Moody group. Then “we construct an isomorphism between the unit group of $\textnormal{Env } G$ to the standard Levi subgroup of $\widetilde{G}$ associated with $I$.”

2. **Import cluster structures from Kac--Moody double Bruhat cells to an open part of $\textnormal{Env }G$.** “Using the cluster structures on double Bruhat cells for Kac--Moody groups constructed by Williams \cite{Wil13} (See also \cite{SW21}), we obtain a cluster structure on an open part $(\textnormal{Env } G)^\circ$ of $\textnormal{Env } G$.”

3. **Compare valuations to control the boundary/frozen variables via canonical basis combinatorics.** “The second essential technique is a comparison between the cluster valuations along frozen variables and the valuation associated with the boundary divisor $\textnormal{Env } G-(\textnormal{Env } G)^\circ$.” This comparison “is achieved by applying the string parametrization of the dual canonical basis.”

(Additional remark in the strategy section: “the technique of extending root data was inspired by the thickening map introduced by Bao and He \cite{BH24}.”)

[To prove the main theorem \& we first prove the following corollary. \begin{cor}\label{cor:spec}
The diagram
\begin{equation}
    \begin{tikzcd}
    \Env G \arrow[dr,"\pi"'] \arrow[rr,"\sim"] && \Spec \overline{\CU}(\wt{\bfs}(\bfi)) \arrow[dl,"\pi^{I'}"] \\ & \BC^I
    \end{tikzcd}
\end{equation}
commutes.
\end{cor}]
\label{thm:1}
Let $G$ be simply connected. The coordinate ring of the Vinberg monoid $\textnormal{Env } G$ is a partially compactified upper cluster algebra where all frozen variables are not invertible.

The map $\pi$ coincides with the specialization of certain frozen variables. In particular, the asymptotic semigroup is obtained by specializing certain frozen variables to 0.,

Classification or Bijection
Asymptotic or Limit

Question: Let $G$ be a simply connected semisimple group, and let $\Env G$ denote its Vinberg monoid with canonical flat morphism $\pi:\Env G\to \mathbb{C}^I$. Fix the seed $\widetilde{\mathbf{s}}(\mathbf{i})$ attached to a double reduced word $\mathbf{i}$ for $(w_0,w_0)$, and let $\overline{\mathcal U}(\widetilde{\mathbf{s}}(\mathbf{i}))$ be the partially compactified upper cluster algebra in which frozen variables are not inverted. Which affine scheme, together with its map to $\mathbb{C}^I$, is identified with $\Env G$?

$\Spec \overline{\mathcal U}(\widetilde{\mathbf{s}}(\mathbf{i}))$; equivalently, $\mathbb{C}[\Env G]\cong \overline{\mathcal U}(\widetilde{\mathbf{s}}(\mathbf{i}))$ as a partially compactified upper cluster algebra with all frozen variables noninvertible, and under the induced isomorphism the morphism $\pi$ agrees with the specialization map $\pi^{I'}$ of the relevant frozen variables to $\mathbb{C}^I$, so the asymptotic semigroup is obtained by setting those frozen variables equal to $0$.

Choices:
B. $\Spec \overline{\mathcal U}^{\Sigma}(\widetilde{\mathbf{s}}(\mathbf{i}))$; equivalently, $\mathbb{C}[G\times^{Z}T]\cong \overline{\mathcal U}^{\Sigma}(\widetilde{\mathbf{s}}(\mathbf{i}))$, and under this identification the morphism $\pi$ is the restriction of the specialization map $\pi^{I'}:\Spec \overline{\mathcal U}^{\Sigma}(\widetilde{\mathbf{s}}(\mathbf{i}))\to \mathbb{C}^I$.
C. $\Spec \overline{\mathcal U}(\widetilde{\mathbf{s}}(\mathbf{i}))$; equivalently, $\mathbb{C}[\Env G]\cong \overline{\mathcal U}(\widetilde{\mathbf{s}}(\mathbf{i}))$ as a partially compactified upper cluster algebra with frozen variables not inverted.
D. $\Spec \mathcal U(\widetilde{\mathbf{s}}(\mathbf{i}))$; equivalently, the open part $(\Env G)^\circ$ is all of $\Env G$, so $\mathbb{C}[\Env G]\cong \mathcal U(\widetilde{\mathbf{s}}(\mathbf{i}))$, and the morphism $\pi$ is already given on this upper cluster algebra by specializing the relevant frozen variables to $\mathbb{C}^I$.
E. $\Spec \overline{\mathcal U}(\widetilde{\mathbf{s}}(\mathbf{i}))$; equivalently, $\mathbb{C}[\Env G]\cong \overline{\mathcal U}(\widetilde{\mathbf{s}}(\mathbf{i}))$ as a partially compactified upper cluster algebra with all frozen variables noninvertible, and under the induced isomorphism the morphism $\pi$ agrees with the specialization map obtained by setting all frozen variables equal to $0$, so the asymptotic semigroup is the whole fibrewise image of this map.

compatibility of the isomorphism with the map $\pi$ and the asymptotic-semigroup specialization statement

boundary divisor controlled by frozen-variable valuations

which frozen variables are specialized and how $\pi$ maps to $\mathbb{C}^I$

The stem does not explicitly reveal the correct option. It names the relevant objects and asks for the precise identification, but the correct choice must still be distinguished from nearby variants involving the open part, the wrong compactification, or incorrect specialization behavior.

This is essentially a direct recall/restatement of a theorem: the question asks which scheme and map are identified with the Vinberg monoid, and the correct option reproduces that theorem-level statement almost verbatim.

Some reasoning is required because several options are close and differ in subtle but mathematically meaningful ways (e.g. full monoid vs open part, compatibility with the map to \(\mathbb{C}^I\), and which frozen variables are specialized). However, it primarily tests precise recall rather than deeper generative reasoning.

The distractors are plausible and target realistic failure modes: confusing \(\Env G\) with its open part, replacing the relevant algebra by a \(\Sigma\)-variant, omitting the map-compatibility clause, or misstating the specialization to \(\mathbb{C}^I\). They are distinct and mathematically aligned with common misunderstandings.

A mathematically precise MCQ with strong distractors and little answer leakage, but it is mostly a theorem-recall item rather than a non-tautological or strongly generative reasoning question.