\begin{theorem}\label{thm:toric-deg}
Let $\pi\colon \mathcal{X}\rightarrow \mathbb{D}$ be a projective fibration.
Let $(\mathcal{X},\mathcal{X}_0+\mathcal{B})$ be a log Calabi--Yau pair over $\mathbb{D}$ for which $(\mathcal{X},\mathcal{X}_0)$ is plt.
If $(\mathcal{X}_t,\mathcal{B}_t)$ is a toric pair\footnote{Meaning that $\mathcal{X}_t$ is a projective toric variety and $\mathcal{B}_t$ is the reduced torus invariant boundary.} for $t \in \mathbb{D}^*$,
then $(\mathcal{X}_0,\mathcal{B}_0)$ is a finite quotient of a toric pair.
\end{theorem}

\begin{proposition}\label{prop:tor-deg}
Let $\mathcal{X}\rightarrow \mathbb{D}$ be a klt Fano degeneration of $\pp(1,1,n)$ with $n\geq 3$, then $\mathcal{X}_0$ is toric if and only if $\mathcal{X}_0\cong \pp(x^2,y^2,n)$ such that $x^2+y^2+n=(n+2)xy$.
\end{proposition}

\begin{definition}\label{def:pt}
{\em
A pair $(X,B)$ is said to be {\em purely terminal} if the following conditions are satisfied:
\begin{enumerate}
\item the pair $(X,B)$ is plt; and
\item for every prime component $S\subset \lfloor B\rfloor$ the pair $(S,B_S)$, obtained from adjunction of $(X,B)$ to $S$, is terminal.
\end{enumerate}
}
\end{definition}

\begin{corollary}\label{cor:markov-no-deg}
Let $(a,b,c)\in \zz_{\geq 2}^3$ be a Markov triple.
Then, the weighted projective plane $\pp(a^2,b^2,c^2)$ has no non-trivial $\qq$-Gorenstein klt degenerations.
\end{corollary}

\begin{example}\label{ex:P^2-toric-model-1}
{\em
We consider the pair
$(\pp^2,B)$ where $B$ is the sum of a quadric and a transversal line.
This example is of cluster type, as described by Gross, Hacking, and Keel~\cite{GHK15}.
The choice of a torus embedding $\mathbb{G}_m^2\hookrightarrow \pp^2\setminus B$ is equivalent to the choice of a {\em toric model}
$\phi \colon (T,B_T)\dashrightarrow (\pp^2,B)$.
More precisely, $\phi$ is a crepant birational map that only extracts log canonical places and $(T,B_T)$ is a toric pair. In this example, we demonstrate
that the choice of such a toric model, or equivalently the choice of an embedding $\mathbb{G}_m^2\hookrightarrow \pp^2\setminus B$ naturally induces a degeneration of $\pp^2$ to a toric pair.
We will consider a toric model
$(\mathbb{F}_2,B_T)$ of $(\pp^2,B)$
where $\mathbb{F}_2$ is the second Hirzebruch surface. This toric model induces an embedding $j_0\colon \mathbb{F}_2\setminus B_T \simeq \mathbb{G}_m^2 \hookrightarrow \pp^2\setminus B$.

We start with the trivial family
\begin{equation}\label{fam1}
(\mathcal{T},B_{\mathcal{T}})\rightarrow \mathbb{D},
\end{equation}
where all the fibers are isomorphic
to the second Hirzebruch surface plus its toric boundary.
In other words, we have
\[
(\mathcal{T},B_{\mathcal{T}}) \simeq
(\mathbb{F}_2,B_T)\times \mathbb{D}.
\]
Let $B_{\mathcal{T}}=S_{\mathcal{T}}+\Delta_{0,\mathcal{T}}+F_{0,\mathcal{T}}+F_{\infty,\mathcal{T}}$ where $S_{\mathcal{T}}\sim \Delta_0+2F\subset \mathbb{F}_2$ and the $f$'s are the fibers of the projection $\mathbb{F}_2\rightarrow \pp^1$. We blow-up the image of a section
$s(\mathbb{D})\subset \mathcal{T}$ whose image is contained in
$F_{0,\mathcal{T}}$ and $s(0)$
is the intersection of
$F_{0,\mathcal{T},0}$ and $\Delta_{0,\mathcal{T}}$.
This leads to a family
\begin{equation}\label{fam2}
(\mathcal{T}',B_{\mathcal{T}'})\rightarrow \mathbb{D}
\end{equation}
where the general fiber is a blow-up of $\mathbb{F}_2$ along a general point of a fiber of $\mathbb{F}_2\rightarrow \pp^1$
and the central fiber is a toric surface.
We may blow-down the strict transform of
$F_{0,\mathcal{T}}$ in $\mathcal{T}'$ to obtain a family
\begin{equation}\label{fam3}
(\mathcal{Y}^+,B_{\mathcal{Y}^+}) \rightarrow \mathbb{D}.
\end{equation}
The general fiber of this family is
isomorphic to $\mathbb{F}_1$ and
$\mathcal{B}_{Y^+,t}$ has three components;
a fiber, a section with negative self-intersection, and a section with positive self-intersection.
The central fiber of~\eqref{fam3} is $\mathbb{F}_3$ with its toric boundary.
By \cite[Corollary 3.23]{HTU17}, the family admits an anti-flip which leads to a degeneration
\begin{equation}\label{fam4}
(\mathcal{Y},B_{\mathcal{Y}})\rightarrow \mathbb{D}.
\end{equation}
The general fiber of~\eqref{fam4}
is isomorphic to the general fiber of~\eqref{fam3}.
The central fiber of~\eqref{fam4} is
isomorphic to the blow-up of
$\mathbb{P}(1,1,4)$ at a torus invariant point (see for example \cite[Section 7]{UZ24}).
The family $\mathcal{Y}\rightarrow \mathbb{D}$ admits a blow-down
$\mathcal{X}\rightarrow \mathbb{D}$.
We obtain a family of Calabi--Yau pairs
\begin{equation}\label{fam5}
(\mathcal{X},\mathcal{B})\rightarrow \mathbb{D}.
\end{equation}

Below, we give an explicit description, modulo change of coordinates, of the family~\eqref{fam5} and the blow-up
$\mathcal{Y}\rightarrow \mathcal{X}$.
The family is given by
\[
\mathcal{X}_t=\{ [x:y:z:w] \mid xy=z^2+tw = 0\} \subset \pp(1,1,1,2)
\]
for $t\in \mathbb{D}$.
It is straightforward to check that $\mathcal{X}_t\cong \pp^2$ for $t\neq 0$ and $\mathcal{X}_0\cong \pp(1,1,4)$.
Indeed, $\mathcal{X}_t$, for $t\neq 0$, is a smooth del Pezzo surface of Picard rank one. The family of curves $\mathcal{B}\to \mathbb{D}$, where
\begin{equation}
\mathcal{B}_t=\{[x:y:z:w] \mid xyw^2+tz^2w^2=0\} \subset \mathcal{X}_t \subset \pp(1,1,1,2).
\end{equation}
When $t=0$, $\mathcal{B}_0$ corresponds to the toric boundary of $\pp(1,1,4)$, while for $t\neq 0$ it follows that
\begin{equation*}
B_t=\{[x:y:z:w] \mid w^2((1+t)z^2+tw)=0\}
\end{equation*}
 which corresponds to the union of the line $L_t=\{w=0\}$ and the irreducible conic
\begin{equation*}
Q_t=\{[x:y:z:w]\mid (1+t)z^2+tw=0\}.
\end{equation*}
The intersection points of $L_t$ and $Q_t$ are $[1:0:0:0]$ and $[0:1:0:0]$ which specialize to the smooth toric invariant points of $\mathcal{X}_0$.
The family~\eqref{fam4} described above is obtained by blowing up the section $s:\mathbb{D}\to \mathcal{X}$ such that $s(t)=[1:0:0:0]$, we obtain a $\qq$-Gorenstein degeneration $\mathbb{F}_1\rightsquigarrow Bl_p\pp(1,1,4)$.

By construction, the family~\eqref{fam5} admits the embedding of a product
\[
\xymatrix{
\mathbb{D} \times \mathbb{G}_m^2 \ar@{^{(}->}[rr]\ar[rd]_-{\pi_1} & &  \mathcal{X}\setminus \mathcal{B} \ar[ld] \\
& \mathbb{D}.&
}
\]
The restriction of the previous embedding to the general fiber recovers $j_0$
while the restriction to the central fiber
is just the embedding of the open torus into the toric variety $\pp(1,1,4)$.
}
\end{example}

In this article, we study degenerations of varieties and pairs.
We focus on degenerations with central fiber having klt singularities.
These are known as {\em klt degenerations}.
The problem is particularly interesting when the general fiber $\mathcal{X}_t$ of the degeneration $\mathcal{X}\rightarrow \mathbb{D}$ is a Fano variety.
In this case, there are many possible degenerations $\mathcal{X}_0$
and these degenerations are interesting even from the combinatorial perspective.
For instance, it is known that the toric degenerations of $\mathbb{P}^2$ are given by
\begin{equation}\label{eq:markov-triple}
\mathbb{P}(a^2,b^2,c^2) \text{ with $a^2+b^2+c^2=3abc$.}
\end{equation}
These triples $(a,b,c)\in \zz_{\geq 1}^3$ are known as {\em Markov triples}.
The topic of degenerations of projective surfaces is classic in algebraic geometry.
In~\cite{HP05}, Hacking and Prokhorov proved that any degeneration of $\mathbb{P}^2$ is indeed a partial smoothing of a toric surface as in~\eqref{eq:markov-triple}.
These surfaces were studied by Manetti and are nowadays known as
{\em Manetti surfaces} (see~\cite{Man91}).
In~\cite{HP10}, Hacking and Prokhorov classify del Pezzo surfaces, with quotient singularities and Picard rank one, admitting $\mathbb{Q}$-Gorenstein smoothings.
In~\cite{Pro19}, Prokhorov studied log canonical degenerations with Picard rank one of del Pezzo surfaces in $\mathbb{Q}$-Gorenstein families.
In~\cite{UZ24}, Urzua and the second author explain how to relate
the degenerations as in~\eqref{eq:markov-triple} via birational transformations.
Given two degenerations $\mathcal{X} \rightarrow \mathbb{D}$
and $\mathcal{Y}\rightarrow \mathbb{D}$ of $\mathbb{P}^2$ into
$\mathcal{X}_0 \simeq \mathbb{P}(a,b,c)$
and $\mathcal{Y}_0\simeq \mathbb{P}(d,e,f)$,
the authors explain how to perform small birational modifications to go from
$\mathcal{X}\rightarrow \mathbb{D}$ to $\mathcal{Y}\rightarrow \mathbb{D}$
and how these small modifications reflect on the combinatorics of the Markov triples.
In higher dimensions, H\"oring and Peternell proved that a klt degeneration $X$ of $\mathbb{P}^n$ is indeed isomorphic to the $n$-dimensional projective space whenever $T_X$ is semistable (see~\cite{HP24}).
They also classify normal degenerations of $\mathbb{P}^3$ with canonical singularities.
In~\cite{HKW25}, Hausen, Kir\'aly, and Wrobel explicitly determine all log terminal,
rational, degenerations of $\mathbb{P}^2$ that admit non-trivial torus actions.
In~\cite{UZ25}, Urzua and the second author classified all Wahl singularities that appear
in the degenerations of del Pezzo surfaces of degree $d$,
extending the work of Manetti and Hacking-Prokhorov in degree $9$.
Recently in~\cite{Pen25}, Peng studied special $\mathbb{G}_m$-degenerations of del Pezzo surfaces $X$
induced by log canonical places of pairs $(X,C)$ where $C$ is a nodal curve.
Peng proved that the space of special valuations of $(X,C)$ is connected and admits a
partition, which is locally finite, and each interval corresponds to a different $\mathbb{G}_m$-degeneration of $X$.

Throughout this work, we focus on understanding the degenerations of cluster type
varieties and cluster type pairs.
A {\em cluster type pair} is a pair $(X,B)$ with mild singularities
for which $K_X+B\sim 0$ and there is an embedding in codimension one
$\mathbb{G}_m^n \dashrightarrow X\setminus B$ (see Definition~\ref{def:ct}).
A {\em cluster type variety} is a variety $X$ that admits a cluster type pair $(X,B)$ structure.
In this case, we call $B$ a cluster type boundary.
Cluster type varieties and pairs can be thought of as a generalization of toric varieties and pairs.
Indeed, any toric pair is a cluster type pair.
However, the realm of cluster type pairs is much broader.
A del Pezzo surface of degree $d\geq 2$ as well as a general del Pezzo
surface of degree $d=1$ is cluster type (see, e.g.,~\cite[Theorem 2.1 and Remark 2.2]{ALP23}).
Thus, all the examples discussed in the introduction are cluster type varieties.
Our first aim is to understand degenerations of cluster type pairs
for which the limit of the cluster type boundary still has reasonable singularities.
In the next subsection, we prove that there are not many such degenerations
for toric pairs, but quite a lot for cluster type pairs.

\subsection{Degenerations of cluster type pairs}

Our first theorem states that degenerations of toric pairs
are finite quotients of toric pairs. We impose some conditions
on the singularities of the central fiber.

\begin{definition}\label{def:ct}
{\em
Let $(X,B)$ be log Calabi--Yau pair of dimension $n$.
We say that $(X,B)$ is of {\em cluster type}
or that $(X,B)$ is a {\em cluster type pair} if there exists an embedding
in codimension one
\[
\mathbb{G}_m^n \dashrightarrow X\setminus B,
\]
i.e., there exists a closed subset of codimension at least two $Z\subset \mathbb{G}_m^n$
and an embedding $\mathbb{G}_m^n\setminus Z \hookrightarrow X\setminus B$.
}
\end{definition}

Our first theorem states that degenerations of toric pairs
are finite quotients of toric pairs. We impose some conditions
on the singularities of the central fiber.

In the previous theorem,
the boundary $\mathcal{B}_0$ on the central fiber $\mathcal{X}_0$
is defined via adjunction.
This means that $\mathcal{B}_0$ is defined via the formula.
\[
K_{\mathcal{X}_0}+\mathcal{B}_0 = (K_{\mathcal{X}}+\mathcal{B}+\mathcal{X}_0)|_{\mathcal{X}_0}.
\]
In Example~\ref{ex:non-toric-central},
we show that in general $(\mathcal{X}_0,\mathcal{B}_0)$ is not a toric pair. Indeed, the central fiber $\mathcal{X}_0$ does not need to be a rational variety under the assumptions of Theorem~\ref{thm:toric-deg}.
Our next theorem gives a similar statement for degenerations of cluster type pairs.
We need to impose stronger conditions on the singularities in this case.
In the proof of Theorem~\ref{thm:toric-deg}, we will see that $(\mathcal{X},\mathcal{B})\rightarrow \mathbb{D}$ is indeed a finite quotient of an isotrivial toric family. Thus, in the toric case, there are not many interesting degenerations for which the degeneration of the toric boundary still has lc singularities.
Now, we turn to discuss the case of cluster type pairs.

\begin{theorem}\label{thm:ct-deg}
Let $\pi\colon \mathcal{X}\rightarrow \mathbb{D}$ be a projective Fano fibration.
Let $(\mathcal{X},\mathcal{X}_0+\mathcal{B})$ be a log Calabi--Yau pair of index one over $\mathbb{D}$.
Assume that $(\mathcal{X},\mathcal{X}_0)$ is plt
and purely terminal on the complement of $\mathcal{B}$.
If $(\mathcal{X}_t,\mathcal{B}_t)$ is of cluster type for $t\in \mathbb{D}^*$, then
$(\mathcal{X}_0,\mathcal{B}_0)$ is a finite quotient of a cluster type pair.
\end{theorem}

\begin{proof}[Proof of Theorem~\ref{thm:toric-deg}]
Let $f\colon \mathcal{X'}\rightarrow \mathcal{X}$ be the finite cover given by Lemma~\ref{lem:finite-cover}.
By Lemma~\ref{lem:finite-cover}.(2), we have an induced
finite cover of log Calabi--Yau pairs $(\mathcal{X}',\mathcal{B}')\rightarrow (\mathcal{X},\mathcal{B})$.
Then, up to shrinking $\mathbb{D}$ near $\{0\}$ we have a commutative diagram
\[
\xymatrix{
(\mathcal{X},\mathcal{B})\ar[d]_-{\pi} & (\mathcal{X}',\mathcal{B}') \ar[l]_-{f}\ar[d]^-{\pi'} \\
\mathbb{D} & \mathbb{D}\ar[l]_-{f_\mathbb{D}}
}
\]
where $f_\mathbb{D}$ is simply given by $t\mapsto t^k$ for some suitable positive integer $k$.
The morphism $f'\colon \mathcal{X}'\rightarrow \mathbb{D}$ is a fibration by Lemma~\ref{lem:finite-cover}.(3).
By assumption $(\mathcal{X},\mathcal{X}_0)$ is plt so by Riemann-Hurwitz we conclude that $(\mathcal{X}',\mathcal{X}_0')$ is plt as well. In particular, the variety $\mathcal{X}_0'$ is irreducible.
By Lemma~\ref{lem:finite-cover}.(4), every component of $\mathcal{B}'_t$
is the restriction to $\mathcal{X}'_t$ of a component of $\mathcal{B}'$.
Note that $\pi$ and $\pi'$ have the same general log fibers; indeed $f$ is induced by a finite cover of $\mathbb{D}$ ramified over $\{0\}$.
Therefore, the general fiber $(\mathcal{X'},\mathcal{B'})$ is a projective toric variety of dimension $n$
and Picard rank $\rho$.
Shrinking $\mathbb{D}$ around $\{0\}$, we may assume that all the fibers over $\pi'$ are irreducible.
Therefore, we conclude that $\rho(\mathcal{X}'/\mathbb{D})\leq \rho$.
On the other hand, as every component of $\mathcal{B}'_t$ is the restriction to $\mathcal{X}'_t$ of a component of $\mathcal{B}$,
we conclude that $\mathcal{B}$ has at least $n+\rho$ components.
Thus, we can compute the relative complexity of
the log Calabi--Yau pair $(\mathcal{X}',\mathcal{B}'+\mathcal{X}_0')$ over $\{0\} \in \mathbb{D}$.
We obtain
\[
c_{\{0\}}(\mathcal{X}'/\mathbb{D},\mathcal{B}'+\mathcal{X}_0') =
\dim \mathcal{X}' + \rho(\mathcal{X}'/\mathbb{D}) - |\mathcal{B}'+\mathcal{X}_0'| \leq
n+1+\rho - (n+\rho+1).
\]
Therefore, by~\cite[Theorem 1]{MS21}, we conclude that
$(\mathcal{X}',\mathcal{B}'+\mathcal{X}'_0)$ is a formally toric morphism near $\{0\}$.
Thus, the pair $(\mathcal{X}'_0,\mathcal{B}'_0)$ obtained from adjunction
of $(\mathcal{X}',\mathcal{B}'+\mathcal{X}'_0)$ must be a projective
toric log Calabi--Yau pair.
Henceforth, we have a finite crepant morpshim
of log Calabi--Yau pairs
\[
f_0\colon (\mathcal{X}'_0,\mathcal{B}'_0)
\rightarrow (\mathcal{X}_0,\mathcal{B}).
\]
We conclude that the pair $(\mathcal{X}_0,\mathcal{B}_0)$
is a finite quotient of a toric pair.
\end{proof}

\begin{theorem}\label{thm:1-comp}
Let $\pi\colon \mathcal{X}\rightarrow \mathbb{D}$ be a Fano projective morphism where $\mathcal{X}_0$ is reduced in $\mathcal{X}$ and
$(\mathcal{X},\mathcal{X}_0)$ plt.
Assume that the following conditions hold for the general fiber $\mathcal{X}_t$ with $t\neq 0$:
\begin{enumerate}
\item $\mathcal{X}_t$ is a toric surface of Picard rank one;
\item ${\rm mld}(\mathcal{X}_t)<\frac{1}{6}$;
\item $\mathcal{X}_t$ does not have singularities in the baskets
$\mathcal{F}_1,\dots,\mathcal{F}_4,\mathcal{D}$.
\end{enumerate}
Then, the pair $(\mathcal{X},\mathcal{X}_0)$ admits a $1$-complement over $\mathbb{D}$.
\end{theorem}
\begin{proof}
We break the proof into five steps.\\

\begin{example}\label{ex:non-toric-central}
{\em
We give an example of a degeneration of a toric pair where the central fiber is not a toric variety.
Fix $n\geq 3$ and consider the variety $\mathcal{Y}:=(\pp^1)^n \times \mathbb{D}$
and let $\pi_2\colon \mathcal{Y}\rightarrow \mathbb{D}$ be the projection onto the second component.
Let $B_T$ be the torus-invariant boundary of $(\pp^1)^n$
and $\mathcal{B}_\mathcal{Y}:=B_T\times \mathbb{D}$.
Therefore, the pair $(\mathcal{Y},\mathbb{B}_{\mathcal{Y}}+\mathcal{Y}_0)$
is a log Calabi--Yau pair for which every fiber of $\pi_2$ is isomorphic to
$((\pp^1)^n,B_T)$.
Consider the group $\zz_2$ acting on $\mathcal{Y}$ via
\begin{align*}
\mu & \colon (\pp^1)^n\times \mathbb{D} \rightarrow
(\pp^1)^n \times \mathbb{D} \\
\mu & \cdot ([x_1:y_1],[x_2:y_2],\dots,[x_n:y_n],t) :=
([y_1:x_1],[y_2:x_2],\dots,[y_n:x_n],-t).
\end{align*}
The log Calabi--Yau pair $(\mathcal{Y},\mathcal{B}_{\mathcal{Y}}+\mathcal{Y}_0)$
is invariant under the $\zz_2$-action.
Let $(\mathcal{X},\mathcal{B}+\mathcal{X}_0)$ be the induced quotient, so
we obtain a commutative diagram:
\[
\xymatrix{
(\mathcal{X},\mathcal{B}+\mathcal{X}_0) \ar[d]^-{\pi} & (\mathcal{Y},\mathcal{B}_{\mathcal{Y}}+\mathcal{Y}_0) \ar[d]^-{\pi_2} \ar[l]^-{/\zz_2} \\
\mathbb{D} & \mathbb{D} \ar[l]^-{/\zz_2}
}
\]
The morphism $\pi$ is a degeneration of the toric log Calabi--Yau pair
$((\pp^1)^n,B_T)$ into $(\mathcal{X}_0,\mathcal{B}_0)=((\pp^1)^n,B_T)/\zz_2$.
We have that $\mathcal{D}(\mathcal{X}_0,\mathcal{B}_0)\simeq_{\rm PL} \mathbb{P}_\rr^{n-1}$.
Thus, the central fiber $(\mathcal{X}_0,\mathcal{B}_0)$ is not a toric pair.
}
\end{example}

In the previous theorem,
the boundary $\mathcal{B}_0$ on the central fiber $\mathcal{X}_0$
is defined via adjunction.
This means that $\mathcal{B}_0$ is defined via the formula.
\[
K_{\mathcal{X}_0}+\mathcal{B}_0 = (K_{\mathcal{X}}+\mathcal{B}+\mathcal{X}_0)|_{\mathcal{X}_0}.
\]
In Example~\ref{ex:non-toric-central},
we show that in general $(\mathcal{X}_0,\mathcal{B}_0)$ is not a toric pair. Indeed, the central fiber $\mathcal{X}_0$ does not need to be a rational variety under the assumptions of Theorem~\ref{thm:toric-deg}.
Our next theorem gives a similar statement for degenerations of cluster type pairs.
We need to impose stronger conditions on the singularities in this case.
In the proof of Theorem~\ref{thm:toric-deg}, we will see that $(\mathcal{X},\mathcal{B})\rightarrow \mathbb{D}$ is indeed a finite quotient of an isotrivial toric family. Thus, in the toric case, there are not many interesting degenerations for which the degeneration of the toric boundary still has lc singularities.
Now, we turn to discuss the case of cluster type pairs.

\begin{theorem}\label{thm:ct-deg}
Let $\pi\colon \mathcal{X}\rightarrow \mathbb{D}$ be a projective Fano fibration.
Let $(\mathcal{X},\mathcal{X}_0+\mathcal{B})$ be a log Calabi--Yau pair of index one over $\mathbb{D}$.
Assume that $(\mathcal{X},\mathcal{X}_0)$ is plt
and purely terminal on the complement of $\mathcal{B}$.
If $(\mathcal{X}_t,\mathcal{B}_t)$ is of cluster type for $t\in \mathbb{D}^*$, then
$(\mathcal{X}_0,\mathcal{B}_0)$ is a finite quotient of a cluster type pair.
\end{theorem}

We refer the reader to Definition~\ref{def:pt} for the concept of purely terminal pairs.
In contrast to the toric case, a cluster type pair may have several interesting degenerations for which the degeneration of the cluster type boundary still has lc singularities. In Example~\ref{ex:P^2-toric-model-1}, we show that some toric degenerations of $\mathbb{P}^2$ can be regarded as cluster type degenerations for different embeddings of algebraic tori
$\mathbb{G}_m^2 \hookrightarrow \mathbb{P}^2\setminus C$ where $C$ is a nodal cubic.

\subsection{Degenerations of singular surfaces}

Now, we restrict ourselves to the study of degenerations of singular toric surfaces.
One of our aims is to understand how the singularities affect the possible degenerations.
Our third theorem states that almost every weighted projective plane has no interesting degenerations.

\begin{theorem}\label{no-deg-almost}
For almost all well-formed triples $(a,b,c)\in \zz_{\geq 1}^3$
the weighted projective plane $\mathbb{P}(a,b,c)$ has no non-trivial
$\mathbb{Q}$-Gorenstein klt degenerations.
\label{weighteddeg}
\end{theorem}

In the previous theorem, when we write {\em almost all}, we mean that the statement holds up to a subset $S\subsetneq \zz_{\geq 1}^3$ that has density zero with the natural density endowed from $\mathbb{Z}^3$.
We say that a degeneration is {\em trivial} if the central fiber is isomorphic to the general fiber.
However, in the setting of Theorem~\ref{no-deg-almost}, we will prove something stronger; the family is a product near the origin of the disk.
In order to prove Theorem~\ref{weighteddeg}, we will prove Theorem~\ref{thm:1-comp}, which is a general statement about
the existence of complements for degenerations of singular toric surfaces.
This means that, under some mild conditions, we prove that given a degeneration
$\mathcal{X}\rightarrow \mathbb{D}$ of a singular toric surface,
there exists some boundary $\mathcal{B}\in |-K_{\mathcal{X}}|$ for which $(\mathcal{X}_t,\mathcal{B}_t)$ has log canonical singularities for $t$ near $\{0\}\in \mathbb{D}$.
In the setting of Theorem~\ref{weighteddeg}, in most cases, we can argue that $(\mathcal{X}_t,\mathcal{B}_t)$ is toric for every $t$ and so
the statement is similar to that of Theorem~\ref{thm:toric-deg}, which states that there are no interesting such toric degenerations.
Theorem~\ref{thm:1-comp} is rather technical and depends on some meticulous analysis of basket of singularities.
The idea of using the theory of complements to understand degenerations of del Pezzo surfaces goes back to Hacking and Prokhorov.
The situation becomes a bit more delicate when we allow the general fiber of the degeneration to have singularities.
We will argue that for a Markov triple $(a,b,c)\in \zz_{\geq 2}^3$ the triple
$(a^2,b^2,c^2)$ belongs to the complement of the subset $S\subsetneq \zz_{\geq 0}^3$ of density zero mentioned above.
Thus, we conclude the following corollary.

\begin{corollary}\label{cor:markov-no-deg}
Let $(a,b,c)\in \zz_{\geq 2}^3$ be a Markov triple.
Then, the weighted projective plane $\mathbb{P}(a^2,b^2,c^2)$ has no non-trivial $\mathbb{Q}$-Gorenstein klt degenerations.
\end{corollary}

We note that Corollary~\ref{cor:markov-no-deg} can also be concluded from the work of Hacking and Prokhorov~\cite{HP10}. Indeed, every iterated degeneration of $\mathbb{P}(a^2,b^2,c^2)$ is indeed a degeneration of $\mathbb{P}^2$.
In upcoming work~\cite{Zun25}, the second author will prove
some structural theorems about $\mathbb{Q}$-Gorenstein klt degenerations
of Hirzebruch surfaces.
This will aim to finish the classification of $\mathbb{Q}$-Gorenstein
klt degenerations of minimal smooth rational surfaces.
This motivates us to pay particular attention to the weighted projective plane
$\mathbb{P}(1,1,n)$.
In this direction, using the tools introduced above, we can give
a complete classification of klt degenrations of $\mathbb{P}(1,1,n)$ with $n\geq 3$.

\begin{theorem}\label{thm:1-1-n}
Let $\mathcal{X}\rightarrow \mathbb{D}$ be a klt Fano degeneration of $\mathbb{P}(1,1,n)$ with $n \geq 3$,
then  for $\mathcal{X}_0$ one of the following holds:
\begin{enumerate}
\item $\mathcal{X}_0$ is a weighted projective plane, or
\item $\mathcal{X}_0$ is a $\mathbb{G}_m$-surface which is not toric.
\end{enumerate}
Furthermore, in the second case $\mathcal{X}_0$ is a $\mathbb{Q}$-Gorenstein deformation of a weighted projective plane.
\end{theorem}

In Proposition~\ref{prop:tor-deg},
we give a explicit classification
of the weighted projective planes
which are $\mathbb{Q}$-Gorenstein klt degenerations of $\mathbb{P}(1,1,n)$ with $n\geq 3$.

The paper is organized as follows.
In Section~\ref{sec:prelims}, we write some preliminary results regarding cluster type pairs, theory of complements, dual complexes, T-singularities, and Wahl singularities.
In Section~\ref{sec:degen-cluster-type}, we prove Theorem~\ref{thm:toric-deg} and Theorem~\ref{thm:ct-deg} regarding degenerations of toric pairs as well as cluster type pairs.
In Section~\ref{sec:complements-degen-klt-surfaces}, we prove some general statements regarding the existence of complements for degenerations of singular toric surfaces of Picard rank one. In this section, we also prove Theorem~\ref{weighteddeg} regarding the degenerations of weighted projective planes $\mathbb{P}(a,b,c)$.
In Section~\ref{sec:degen-wps}, we classify the degenerations of weighted projective planes $\mathbb{P}(1,1,n)$ with $n\geq 3$.
Finally, in Section~\ref{sec:ex-and-quest}, we give some examples and propose some questions for further research.

\subsection*{Acknowledgements}

The authors would like to thank Audric Lebovitz, Tomoki Oda, Giancarlo Urzúa, and Jos\'e Ignacio Y\'a\~nez
for many discussions related to this article.

In the proof of the main theorem, we will see that $(\mathcal{X},\mathcal{B})\rightarrow \mathbb{D}$ is indeed a finite quotient of an isotrivial toric family. Thus, in the toric case, there are not many interesting degenerations for which the degeneration of the toric boundary still has lc singularities.,

Question: Let \(\pi\colon \mathcal{X}\to \mathbb{D}\) be a projective fibration, let \(\mathcal{X}_0=\pi^{-1}(0)\), and let \((\mathcal{X},\mathcal{X}_0+\mathcal{B})\) be a log Calabi--Yau pair over \(\mathbb{D}\) (so \(K_{\mathcal{X}}+\mathcal{X}_0+\mathcal{B}\) is trivial over \(\mathbb{D}\)). Assume that \((\mathcal{X},\mathcal{X}_0)\) is plt. For every \(t\in \mathbb{D}^*:=\mathbb{D}\setminus\{0\}\), suppose the fiber pair \((\mathcal{X}_t,\mathcal{B}_t)\) is a toric pair, meaning that \(\mathcal{X}_t\) is a projective toric variety and \(\mathcal{B}_t\) is its reduced torus-invariant boundary. Which conclusion about the central fiber pair \((\mathcal{X}_0,\mathcal{B}_0)\) holds?

The central fiber pair \((\mathcal{X}_0,\mathcal{B}_0)\) is a finite quotient of a toric pair; equivalently, there exists a toric pair \((Y,B_Y)\), with \(Y\) a projective toric variety and \(B_Y\) its reduced torus-invariant boundary, together with a finite morphism of pairs \((Y,B_Y)\to (\mathcal{X}_0,\mathcal{B}_0)\).

Choices:
B. The central fiber pair \((\mathcal{X}_0,\mathcal{B}_0)\) is itself a toric pair; equivalently, \(\mathcal{X}_0\) is a projective toric variety and \(\mathcal{B}_0\) is its reduced torus-invariant boundary.
C. The central fiber pair \((\mathcal{X}_0,\mathcal{B}_0)\) is birational to a toric pair; that is, there exists a projective toric pair \((Y,B_Y)\) and a birational map of pairs \((Y,B_Y) \dashrightarrow (\mathcal{X}_0,\mathcal{B}_0)\).
D. There exists a single projective toric pair \((Y,B_Y)\) such that, after possibly shrinking \(\mathbb{D}\), every fiber pair \((\mathcal{X}_t,\mathcal{B}_t)\) for \(t\in\mathbb{D}\) is isomorphic to \((Y,B_Y)\); in particular, the family is isotrivial and the central fiber is a toric pair isomorphic to the general fiber.
E. The central fiber pair \((\mathcal{X}_0,\mathcal{B}_0)\) admits a finite crepant morphism to a toric pair; equivalently, there exists a projective toric pair \((Y,B_Y)\) and a finite morphism of pairs \((\mathcal{X}_0,\mathcal{B}_0)\to (Y,B_Y)\).

The stem does not explicitly state or strongly hint at the key conclusion that the central fiber is a finite quotient of a toric pair. The relevant distinction only appears in the answer choices.

Although the question is theorem-driven, it is not a bare restatement: the options force the test-taker to distinguish among nearby conclusions such as toricity, birational toricity, isotriviality, and the direction of a finite morphism.

Some reasoning is required to select the strongest valid conclusion and reject stronger, weaker, or directionally incorrect variants. However, for someone who knows the theorem, the answer is largely recall-based rather than deeply generative.

The distractors are mathematically plausible and well-targeted: one is too strong, one is weaker-but-true-looking, one confuses family-level isotriviality with fiberwise toricity, and one reverses the finite morphism direction.

A strong MCQ with little answer leakage and high-quality distractors; it tests careful theorem discrimination well, though it remains somewhat recall-oriented rather than fully generative.


\begin{introtheorem}\label{intthm:character}
    If $H$ is a type B or type C Hessenberg space, there exist non-negative integers $a,b \in \mathbb{N}$, integers $c,d \in \{0,1\}$, and a subset $I \subseteq [n]$, each determined by $S(H)$, such that
    \[
     \lrep{H}_1 = a\,\mathbb{1} + \sum_{i\in I} \fh_i + b\,\fh_1 + c\,\fs + d\,\bm{\delta}
    \]
    and
    \[
     \rrep{H}_1 = \bm{\chi} + \sum_{i\in I} \left(\fh_i - \mathbb{1}\right) + b\left(\fh_1-\mathbb{1}\right) + c \left(
    \fs-\mathbb{1}\right) + d\,\bm{\delta},
    \]
    where
    \begin{itemize}
        \item $\bm{\chi}$ is the character of the defining representation of $\Wn$ on $\C^n$
        \item $\fh_i$ is the character of the action of $\Wn$ on the cosets of $\mathfrak{S}_i\times \mathfrak{W}_{n-i}$,
        \item $\fs$ is the character of the action of $\Wn$ on the cosets of $\mathfrak{W}_1\times \mathfrak{W}_{n-1}$,
        \item $\mathbb{1}$ is the character of the trivial representation, and
        \item $\bm{\delta}$ is the character $w\mapsto (-1)^{\abs{\Neg(w)}}$ where $\Neg(w) = \{w(i) \mid i \in [n],\; w(i) < 0\}$.
    \end{itemize}
    Moreover, $c$ and $d$ are never simultaneously $1$, and $d$ is always $0$ in type B.
\end{introtheorem}

\begin{proposition}\label{prop:permutohedral}
    Let $\bm{\chi}$ be the character of the defining representation of $\Wn$ (i.e. the action on $\C^n$). Let $\fh_i$ be the character of the action of $\Wn$ on the cosets of $\mathfrak{S}_i \times \mathfrak{W}_{n-i}$. Let $\mathbb{1}$ be the character of the trivial representation. Then
   \[
     \lrep{\Delta}_1 = \sum_{i=1}^n \fh_i - \bm{\chi}
    \hspace{1cm}\text{and}\hspace{1cm}
    \rrep{\Delta}_1 = \sum_{i=1}^n \fh_i - n\,\mathbb{1}.
    \]
\end{proposition}

\begin{definition}\label{def:itypes}
Consider $S(H)$ and $i \in [n]$. We call $i$\begin{itemize}
    \item \emph{uncovered} if $\begin{cases} \{t_{i-1},t_i\} \cap S(H) = \emptyset &\text{and } i \neq n-1\text{, or} \\ \{t_{n-1},t_{n-1},t_{n}\} \cap S(H) = \emptyset &\text{and } i = n-1\end{cases}$
    \item \emph{surrounded} if $i \in [n-2]$ and $\{t_{i-1},t_i\} \cap S(H) = \{t_{i-1}\}$ and $t_m \in S(H)\text{ for some } m>i$,
    \item \emph{shaded} if $t_i \in S(H)$ or if $t_n \in S(H)$ and $i=n-1$.
\end{itemize}
\end{definition}

\begin{theorem}\label{thm:character}
    Say $\Delta \subsetneq H$. Let $\bm{\chi}$ be the character of the defining representation of $\Wn$ (i.e. the action on $\C^n$). Let $\fh_i$ be the character of the action of $\Wn$ on the cosets of $\mathfrak{S}_i\times \mathfrak{W}_{n-i}$. Let $\fs$ be the character of the action of $\Wn$ on cosets of $\mathfrak{W}_1 \times \mathfrak{W}_{n-1}$. Let $\mathbb{1}$ be the character of the trivial representation and $\bm{\delta}$ be the character $w\mapsto (-1)^{\abs{\Neg(w)}}$. Then
    \begin{align*}
     \lrep{H}_1 = \abs{\{i \in [n] \mid i\text{ shaded}\,\}}\mathbb{1} &+ \sum_{i\text{ uncovered}} \fh_i \\[6pt]&+ \abs{\{i \in [n-2] \mid i\text{ surrounded}\,\}}\fh_1 \\[6pt]&+ \begin{cases}
    \fs &\text{if $t_n \notin S(H)$}\\
    \bm{\delta} &\text{if }\{t_{n-1},t_n\} \cap S(H) = \{t_{n}\}\\
    0 &\text{otherwise.}
\end{cases}
    \end{align*}
    and
        \begin{align*}
     \rrep{H}_1 = \bm{\chi} &+ \sum_{i\text{ uncovered}} \left(\fh_i - \mathbb{1}\right) \\[6pt]&+ \abs{\{i \in [n-2] \mid i\text{ surrounded}\,\}}\left(\fh_1-\mathbb{1}\right) \\[6pt]&+ \begin{cases}
    \fs-\mathbb{1} &\text{if $t_n \notin S(H)$}\\
    \bm{\delta} &\text{if }\{t_{n-1},t_n\} \cap S(H) = \{t_{n}\}\\
    0 &\text{otherwise.}
\end{cases}
    \end{align*}
\end{theorem}

\label{sec:intro}
Let $\mathfrak{W}_n$ be the group of signed permutations, which are permutations $w$ on the set $\{1,\ldots,n,-n,\ldots,-1\}$ such that $w(-i) = -w(i)$. The group $\mathfrak{W}_n$ is also the Weyl group of types B and C, and so is associated with two distinct root systems. A Hessenberg space $H$ is a particular subset of roots, which determines a set $S(H)$ of transpositions in $\mathfrak{W}_n$. Given a Hessenberg space, one can define the \emph{graded ring of splines} $\mathcal{M}_{H}$ on $\mathfrak{W}_n$. This ring has two $\C[x_\bullet]\coloneqq \C[x_1,\ldots,x_{n}]$-module structures and a $\mathfrak{W}_n$-module structure. This paper determines the algebraic and combinatorial properties of the degree-one graded piece of $\mathcal{M}_{H}$ from the combinatorics of $S(H)$.\par
Let $(i,j)$ be the unique transposition in $\mathfrak{W}_n$ that switches $i$ with $j$ and $-i$ with $-j$ (note $i=-j$ is possible). If $H$ is a Hessenberg space, then the ring of splines is
 \[
 \mathcal{M}_{H} = \bigoplus_{i \geq 0} \mathcal{M}_{H}^i \coloneqq \left\{ \left.\bm{\rho} \in \prod_{w \in \mathfrak{W}_n} \C[x_\bullet]\; \right|\; \bm{\rho}(w) - \bm{\rho}(w(i,j)) \in \left\langle x_{w(i)}-x_{w(j)} \right\rangle\text{ if } (i,j) \in S(H) \right\}\!,
 \]
with (graded) $\mathfrak{W}_n$-module structure $w\cdot \bm{\rho}(v) = w\bm{\rho}(w^{-1}v)$ and (graded) $\C[x_\bullet]$-module structure given by multiplication. \par
The ring $\mathcal{M}_{H}$ is isomorphic to the $T$-equivariant cohomology of a smooth subvariety of the full flag variety $G/B$ called a \emph{regular semisimple Hessenberg variety}. The connection between this combinatorial ring and equivariant cohomology is called \emph{GKM Theory} \cite{GKM_theory}, and it allows for the study of this cohomology in purely combinatorial and algebraic terms. \par
The $\mathfrak{W}_n$-module structure on $\mathcal{M}_{H}$ was first defined as the \emph{dot action} on equivariant cohomology by Tymoczko in \cite{tymoczko2008permutation}. There are two natural $\mathfrak{W}_n$-equivariant quotients, called the left $\mathrm{L}_{H}$ and right $\mathrm{R}_{H}$ quotients, of $\mathcal{M}_{H}$ that are in fact graded $\mathbb{C}$-vector spaces (the left quotient corresponds to ordinary cohomology). The graded $\mathfrak{W}_n$-module structure of $\mathcal{M}_{H}$ induces graded $\mathfrak{W}_n$-representations on the quotients. The primary goal of this paper is to compute the characters of the degree-one pieces of
\[
\mathrm{L}_{H} = \bigoplus_{i\geq 0} \left(\mathrm{L}_{H}\right)_i \;\; \text{ and }\;\;\mathrm{R}_{H} = \bigoplus_{i\geq 0} \left(\mathrm{R}_{H}\right)_i
\] from the combinatorial data of $S(H)$. In doing so, we compute the dot action on second (equivariant and ordinary) cohomology for all regular semisimple Hessenberg varieties in types B and C. This generalizes what is known in type A \cite{chow_linearp,chohonglee_second_cohom,ayzenberg2022second}.\par
These graded representations are of interest to algebraic combinatorists in part because they are the $\mathfrak{W}_n$-equivalents of very well-studied $\mathfrak{S}_n$-representations, which have connections to chromatic symmetric functions \cite{guaypaquet2016shar_wachs_conj,SW2016chromaticquasisymmetric,brosnan_chow_dotactn_is_chromsym} and LLT polynomials \cite{guaypaquet2016shar_wachs_conj,Ayzenberg2018isospectral, ALEXANDERSSON2018LLTchromsym}. We detail some of these connections in the Appendix. From these works, in type A there are combinatorial formulas to compute the characters of both $\mathrm{L}_{H}$ and $\mathrm{R}_{H}$ in at least one basis for the character space of $\mathfrak{S}_n$. This is \textbf{not} the case in types B or C.\par
For one specific family of $H$ (those corresponding to simple roots), the character of $\mathrm{L}_{H}$ was computed by Stembridge \cite{Stembridge_Permuto} in general type. Stembridge's results are via a connection to \emph{permutohedra}, as the associated Hessenberg varieties are in fact toric varieties. While it is possible to compute the left and right characters for general $H$ in types B and C, even partial results are rare and rely on sophisticated geometric tools \cite{BalibanuCrooks_sheaves,PrecupSommers_sheaves}. Formulas are non-combinatorial (and usually non-positive), require the computation of type B/C Green polynomials and Poincar\'e polynomials of other varieties, and proofs require the geometry of perverse sheaves. \par
In contrast, for the degree-one graded piece, the main result of this paper applies to all Hessenberg spaces $H$ and is entirely computable from the combinatorics of roots and transpositions in $\mathfrak{W}_n$. Moreover, we compute the degree-one pieces of the characters $\mathfrak{ch}\left(\mathrm{L}_{H}\right)$ and $\mathfrak{ch}\left(\mathrm{R}_{H}\right)$ in a positive manner.

\setcounter{tocdepth}{1}

Our theorem is the first type B/C result to provide an explicit non-negative expansion for a graded piece of $\lrep{H}$ or $\rrep{H}$ in terms of characters of irreducible- and/or permutation-representations for all Hessenberg spaces. It is also the first to compute the characters for any graded piece of $\lrep{H}$ and $\rrep{H}$ by directly providing a basis for the representation, and in doing so gives a family of characters with which one can attempt to expand the catalog of results and connections that exist in type A to types B and C.\par

The right character follows from the general procedure of ``removing" from any permutation representation with permutation basis $\{v_i\}$ the trivial sub-representation $\C\left\{ \sum v_i \right\}$. In particular, if $\mathfrak{p}$ is the character of a representation with permutation basis $\{v_i\}$, then the character on the complementary subspace $\C\{v_i-v_{i+1}\}$ is $\mathfrak{p}-\mathbb{1}$.
\end{proof}
In type B, $t_n \in S(H)$ implies that $t_{n-1} \in S(H)$, and so it is impossible to get the $\bm{\delta}$ term from Theorem \ref{thm:character}. In type C the opposite is true: $t_{n-1} \in S(H)$ implies that $t_n \in S(H)$, and so $\delta$ may appear. This is the ``moreover" part of Theorem \ref{intthm:character}.
\begin{example}
 The table below gives every $\lrep{H}_1$ and $\lrep{H}_1$ for $n=4$.
\begin{table}[H]
\centering
\renewcommand{\arraystretch}{1.5}
\begin{tabular}{|c|p{2in}|p{2in}|p{0.3in}|}\hline
 { $t_i \in S(H)$} & {\large $\lrep{H}_1$} & \large $\rrep{H}_1$ &\large dim\\\hline
 $\emptyset$ & $\fh_1 +\fh_2+\fh_3+\fh_4-\bchi$ & $\fh_1 +\fh_2+\fh_3+\fh_4-4\triv$ & $76$ \\\hline
 $\{t_1\}$ & $\triv + \fh_3 + \fh_4 + \fs$ &$\bchi + \fh_3 + \fh_4 + \fs -3\triv$ & $53$\\\hline
 $\{t_2\}$ & $\triv+\fh_1+\fh_4+\fs $ &$\bchi+\fh_1+\fh_4+\fs -3\triv$ &$29$\\\hline
 $\{t_3\}$ & $\triv+\fh_1+\fh_2+\fs $ &$\bchi+\fh_1+\fh_2+\fs -3\triv$ &$37$\\\hline
 $\{t_4\}$ & $2\triv+\fh_1+\fh_2+\bm{\delta}$ &$\bchi+\fh_1+\fh_2+\bm{\delta}-2\triv$ &$35$\\\hline
 $\{t_1,t_2\}$ & $2\triv+\fh_4+\fs $ &$\bchi+\fh_4+\fs -2\triv$ &$22$\\\hline
 $\{t_1,t_3\}$ & $2\triv+\fh_1+\fs $ &$\bchi+\fh_1+\fs -2\triv$ &$14$\\\hline
 $\{t_1,t_4\}$ & $3\triv+\fh_1 +\bm{\delta}$ &$\bchi+\fh_1 +\bm{\delta}-\triv$ &$12$\\\hline
 $\{t_2,t_3\}$ & $2\triv+\fh_1+\fs $ &$\bchi+\fh_1+\fs -2\triv$ &$14$\\\hline
 $\{t_2,t_4\}$ & $3\triv+\fh_1+\bm{\delta} $ &$\bchi+\fh_1+\bm{\delta} -\triv$ &$12$\\\hline
 $\{t_3,t_4\}$ & $2\triv+\fh_1+\fh_2 $ &$\bchi+\fh_1+\fh_2 -2\triv$ &$34$\\\hline
 $\{t_1,t_2,t_3\}$ &$3\triv+\fs $ &$\bchi+\fs -\triv$ &$7$\\\hline
 $\{t_1,t_2,t_4\}$ &$4\triv+\bm{\delta} $ &$\bchi+\bm{\delta} $ &$5$\\\hline
 $\{t_1,t_3,t_4\}$ &$3\triv+\fh_1$ &$\bchi+\fh_1-\triv$ &$11$\\\hline
 $\{t_2,t_3,t_4\}$ & $3\triv+\fh_1 $ &$\bchi+\fh_1 -\triv$ &$11$\\\hline
 $\{t_1,t_2,t_3,t_4\}$ & $4\mathbb{1}$ &$\bchi$ &$4$\\\hline

\clearpage
\begin{table}[H]
 \centering
 \renewcommand{\arraystretch}{2.5}
 \begin{tabular}{|p{3.5in}|p{2.3in}|}\hline
 {\large Representations or Characters of $\Wn$} & {\large Expansions in $\Lambda_n(x,y)$} \\\hline
 The trivial character $\mathbb{1}$ & $s_{(n),\emptyset} = h_{(n),\emptyset}$ \\\hline
 The sign character $\mathrm{sgn}$ & $s_{\emptyset,(1^n)}$ \\\hline
 $\bm{\delta}$ where $\bm{\delta}(w) = (-1)^{\abs{\Neg(w)}}$ & $s_{\emptyset,(n)} = h_{\emptyset,(n)}$ \\\hline
 The irreducible character $\chi^{(\lambda,\mu)}$ & $s_{\lambda,\mu}$ \\\hline
 $\mathrm{sgn} \otimes \chi^{(\lambda,\mu)} $ & $s_{\mu^t,\lambda^t}$ \\\hline
 $\bm{\delta} \otimes \chi^{(\lambda,\mu)} $ & $s_{\mu,\lambda}$ \\\hline
 The defining representation on $\C^n$ (so $\bm{\chi}$) & $h_{(n-1),(1)}$ \\\hline
 Action on cosets of $\mathfrak{S}_k\times \mathfrak{W}_{n-k}$ (so $\fh_k$) & $\ds h_{(n-k),\emptyset} \sum_{j=0}^{\lambda_i} h_{(j),(k-j)}$ \\\hline
 Action on cosets of $\mathfrak{W}_1\times \mathfrak{W}_{n-1}$ (so $\fs$) & $\ds h_{(n-1,1),\emptyset}$ \\\hline
 Action on cosets of $\mathfrak{S}_\lambda \times \mathfrak{W}_\mu$ & $\ds h_{\mu,\emptyset} \prod_{\lambda_i \in \lambda} \sum_{j=0}^{\lambda_i} h_{(j),(\lambda_i-j)}$ \\\hline
 Action on cosets of the type D${}_n$ subgroup & $\ds h_{(n),\emptyset} + h_{\emptyset,(n)} $ \\\hline
 An induced character $\left(\chi \times \theta\right)_{\mathfrak{W}_k \times \mathfrak{W}_\ell}^{\Wn}$ & $\Frob_{BC}(\chi)\Frob_{BC}(\theta)$\\\hline
 The induced character $\left(\chi^{(\lambda,\mu)} \times \chi^{(\alpha,\beta)}\right)_{\mathfrak{W}_k \times \mathfrak{W}_\ell}^{\Wn}$ & $(s_\lambda(x)\cdot s_\alpha(x))(s_\mu(y)\cdot s_\beta(y))$\\\hline
 The induced character $\left(\mathbb{1}\right)_{\Sn}^{\Wn}$ & ${\ds \sum_{k=0}^n s_{(k),(n-k)} = \sum_{k=0}^n h_{(k),(n-k)}}$ \\\hline
 The induced character $\left(\mathbb{1} \times \bm{\delta} \right)_{\mathfrak{W}_\lambda \times \mathfrak{W}_\mu}^{\Wn}$ & $\ds h_{\lambda,\mu} = \sum_{\gamma}\sum_{\nu} K_{\gamma,\lambda}K_{\nu,\mu}s_\gamma(x)s_\nu(y)$ \\\hline
 \end{tabular}\vspace*{4pt}
 \captionsetup{width=6.2in}
 \caption{Representations of $\Wn$ and Frobenius characteristics \cite{GeissKinch_hyperoctChars,Macdonald_symfuncs, Zelevinsky_reps_classicalgroups, Stembridge_Permuto, Stembridge_guide, Skandera_hyperoct}.}
 \label{tab:reps}
\end{table}
\clearpage

Theorem \ref{intthm:character} is Proposition \ref{prop:permutohedral} if $H = \Delta$ and Theorem \ref{thm:character} otherwise, which use language from Definition \ref{def:itypes} to describe the coefficients $a,b,c$, and $d$. The representation ring of $\mathfrak{W}_n$ is isomorphic to the \emph{type B/C symmetric functions} via the Frobenius characteristic map \cite{Macdonald_symfuncs}. The tools to translate Theorem \ref{intthm:character} into type B/C symmetric functions are in the Appendix. There, we make the observation that $\mathfrak{ch}\left(\mathrm{L}_{H}\right)_1$ is in fact $h_{\lambda,\mu}$-positive. This is particularly interesting, as it generalizes the behavior seen in type A, where the left graded character is conjectured to be $h_\lambda$-positive and is intimately connected to \emph{chromatic symmetric functions}. This is known as the graded Stanley-Stembridge conjecture \cite{StanleyStembridge,SW2016chromaticquasisymmetric}, and has been the subject of much research \cite{STANLEY1995chromsym, Gasharov96, GuayPaquet13,  harada2017cohomology, brosnan_chow_dotactn_is_chromsym,Dahlberg19, Abreu_Nigro20}. A proof of the ungraded conjecture was recently given by Hikita \cite{Hikita_stanstem}.\par

Our theorem is the first type B/C result to provide an explicit non-negative expansion for a graded piece of $\mathfrak{ch}\left(\mathrm{L}_{H}\right)$ or $\mathfrak{ch}\left(\mathrm{R}_{H}\right)$ in terms of characters of irreducible- and/or permutation-representations for all Hessenberg spaces. It is also the first to compute the characters for any graded piece of $\mathfrak{ch}\left(\mathrm{L}_{H}\right)$ and $\mathfrak{ch}\left(\mathrm{R}_{H}\right)$ by directly providing a basis for the representation, and in doing so gives a family of characters with which one can attempt to expand the catalog of results and connections that exist in type A to types B and C.\par

The paper is structured as follows. Section \ref{sec:background} provides some of the necessary background on signed permutations, Hessenberg spaces, and splines. Section \ref{sec:ideals_of_transpositions} shows how to translate from Hessenberg spaces $H$ to sets $S(H)$ of signed transpositions in a manner that unifies some type B and type C calculations. Section \ref{sec:trivial} reduces the $S(H)$ that one must consider in order to compute $\mathcal{M}_{H}^1$ to a much smaller collection, one that completely unifies the type B and type C calculations. Section \ref{sec:one_inversion} determines which elements of $\mathfrak{W}_n$ have exactly one $H$-inversion for each $H$. Section \ref{sec:linear_splines} defines several sets of splines contained in $\mathcal{M}_{H}^1$, provides linear relations between them, and computes the dot action on them. Section \ref{sec:generators} argues, using the elements from \S \ref{sec:one_inversion}, that the splines from \S \ref{sec:linear_splines} form a $\mathbb{C}$-generating set for $\mathcal{M}_{H}^1$, and uses the linear relations between them to reduce the size of this generating set. Section \ref{sec:bases_reps} reduces this generating set further in two different ways, one that results in a basis for $\mathcal{M}_{H}^1$ that is conducive to computing $\mathfrak{ch}\left(\mathrm{L}_{H}\right)_1$ and one that is conducive to computing $\mathfrak{ch}\left(\mathrm{R}_{H}\right)_1$. Then, we prove the main theorem. Appendix \ref{sec:symmetricfunctions} describes how one can translate this result to the language of type B/C symmetric functions.

Theorem~\ref{intthm:character} is obtained by giving an explicit basis for the relevant graded piece and computing the dot action on it: the paper “directly provid[es] a basis for the representation,” and thereby computes “the characters for any graded piece of $\mathfrak{ch}(\mathrm{L}_{H})$ and $\mathfrak{ch}(\mathrm{R}_{H})$” (in particular degree $1$ here). The outlined proof route is:
\begin{itemize}
\item Translate from Hessenberg spaces $H$ to sets $S(H)$ of signed transpositions (Section~\ref{sec:ideals_of_transpositions}).
\item Reduce the $S(H)$ that must be considered to compute $\mathcal{M}_{H}^1$ to “a much smaller collection” (Section~\ref{sec:trivial}).
\item Determine which elements of $\mathfrak{W}_n$ have “exactly one $H$-inversion for each $H$” (Section~\ref{sec:one_inversion}).
\item Define “several sets of splines contained in $\mathcal{M}_{H}^1$,” give “linear relations between them,” and “compute the dot action on them” (Section~\ref{sec:linear_splines}).
\item Use the one-$H$-inversion elements to argue these splines form a $\mathbb{C}$-generating set for $\mathcal{M}_{H}^1$, then use the linear relations to “reduce the size of this generating set” (Section~\ref{sec:generators}).
\item Further reduce to bases “conducive to computing $\mathfrak{ch}(\mathrm{L}_{H})_1$” and “conducive to computing $\mathfrak{ch}(\mathrm{R}_{H})_1$,” and then “prove the main theorem” (Section~\ref{sec:bases_reps}).
\end{itemize}
Additionally, the Appendix develops the translation to type B/C symmetric functions and notes the positivity phenomenon: “we make the observation that $\mathfrak{ch}(\mathrm{L}_{H})_1$ is in fact $h_{\lambda,\mu}$-positive.”

In establishing the main theorem, one gives an explicit basis for the relevant graded piece and computes the dot action on it: the paper “directly provid[es] a basis for the representation,” and thereby computes “the characters for any graded piece of $\mathfrak{ch}(\mathrm{L}_{H})$ and $\mathfrak{ch}(\mathrm{R}_{H})$” (in particular degree $1$ here). The outlined proof route is:
\begin{itemize}
\item Translate from Hessenberg spaces $H$ to sets $S(H)$ of signed transpositions.
\item Reduce the $S(H)$ that must be considered to compute $\mathcal{M}_{H}^1$ to “a much smaller collection”.
\item Determine which elements of $\mathfrak{W}_n$ have “exactly one $H$-inversion for each $H$”.
\item Define “several sets of splines contained in $\mathcal{M}_{H}^1$,” give “linear relations between them,” and “compute the dot action on them”.
\item Use the one-$H$-inversion elements to argue these splines form a $\mathbb{C}$-generating set for $\mathcal{M}_{H}^1$, then use the linear relations to “reduce the size of this generating set”.
\item Further reduce to bases “conducive to computing $\mathfrak{ch}(\mathrm{L}_{H})_1$” and “conducive to computing $\mathfrak{ch}(\mathrm{R}_{H})_1$,” and then prove the main theorem.
\end{itemize}
Additionally, the Appendix develops the translation to type B/C symmetric functions and notes the positivity phenomenon: “we make the observation that $\mathfrak{ch}(\mathrm{L}_{H})_1$ is in fact $h_{\lambda,\mu}$-positive.”

\label{intthm:character}
    If $H$ is a type B or type C Hessenberg space, there exist non-negative integers $a,b \in \mathbb{N}$, integers $c,d \in \{0,1\}$, and a subset $I \subseteq [n]$, each determined by $S(H)$, such that
    \[
     \mathfrak{ch}\left(\mathrm{L}_{H}\right)_1 = a\,\mathbb{1} + \sum_{i\in I} \fh_i + b\,\fh_1 + c\,\mathfrak{s} + d\,\bm{\delta}
    \]
    and
    \[
     \mathfrak{ch}\left(\mathrm{R}_{H}\right)_1 = \bm{\chi} + \sum_{i\in I} \left(\fh_i - \mathbb{1}\right) + b\left(\fh_1-\mathbb{1}\right) + c \left(
    \mathfrak{s}-\mathbb{1}\right) + d\,\bm{\delta},
    \]
    where
    \begin{itemize}
        \item $\bm{\chi}$ is the character of the defining representation of $\mathfrak{W}_n$ on $\mathbb{C}^n$
        \item $\fh_i$ is the character of the action of $\mathfrak{W}_n$ on the cosets of $\mathfrak{S}_i\times \mathfrak{W}_{n-i}$,
        \item $\mathfrak{s}$ is the character of the action of $\mathfrak{W}_n$ on the cosets of $\mathfrak{W}_1\times \mathfrak{W}_{n-1}$,
        \item $\mathbb{1}$ is the character of the trivial representation, and
        \item $\bm{\delta}$ is the character $w\mapsto (-1)^{\left| \Neg(w) \right|}$ where $\Neg(w) = \{w(i) \mid i \in [n],\; w(i) < 0\}$.
    \end{itemize}
    Moreover, $c$ and $d$ are never simultaneously $1$, and $d$ is always $0$ in type B.,

Question: Let \(\mathfrak{W}_n\) be the group of signed permutations, i.e. permutations \(w\) of \(\{1,\ldots,n,-n,\ldots,-1\}\) satisfying \(w(-i)=-w(i)\). Let \(H\) be a type B or type C Hessenberg space, and let \(S(H)\) be its associated set of transpositions in \(\mathfrak{W}_n\). Let \((\mathrm{L}_H)_1\) and \((\mathrm{R}_H)_1\) denote the degree-one graded pieces of the left and right \(\mathfrak{W}_n\)-representation quotients associated to \(H\), and let \(\mathfrak{ch}((\mathrm{L}_H)_1)\) and \(\mathfrak{ch}((\mathrm{R}_H)_1)\) be their characters. Here \(\mathbb{1}\) is the trivial character, \(\bm{\chi}\) is the character of the defining representation of \(\mathfrak{W}_n\) on \(\mathbb{C}^n\), \(\mathfrak{h}_i\) is the character of the action of \(\mathfrak{W}_n\) on the cosets of \(\mathfrak{S}_i\times \mathfrak{W}_{n-i}\), \(\mathfrak{s}\) is the character of the action of \(\mathfrak{W}_n\) on the cosets of \(\mathfrak{W}_1\times \mathfrak{W}_{n-1}\), and \(\bm{\delta}\) is the character \(w\mapsto (-1)^{|\Neg(w)|}\), where \(\Neg(w)=\{w(i)\mid i\in[n],\; w(i)<0\}\). Which of the following conclusions about these degree-one characters holds?

There exist non-negative integers \(a,b\in\mathbb{N}\), integers \(c,d\in\{0,1\}\), and a subset \(I\subseteq [n]\), each determined by \(S(H)\), such that
\[
\mathfrak{ch}\big((\mathrm{L}_H)_1\big)=a\,\mathbb{1}+\sum_{i\in I}\mathfrak{h}_i+b\,\mathfrak{h}_1+c\,\mathfrak{s}+d\,\bm{\delta},
\]
and
\[
\mathfrak{ch}\big((\mathrm{R}_H)_1\big)=\bm{\chi}+\sum_{i\in I}(\mathfrak{h}_i-\mathbb{1})+b(\mathfrak{h}_1-\mathbb{1})+c(\mathfrak{s}-\mathbb{1})+d\,\bm{\delta}.
\]
Moreover, \(c\) and \(d\) are never simultaneously equal to \(1\), and in type B one always has \(d=0\).

Choices:
B.
There exist non-negative integers \(a,b\in\mathbb{N}\), integers \(c,d\in\{0,1\}\), and a subset \(I\subseteq [n]\), each determined by \(S(H)\), such that
\[
\mathfrak{ch}\big((\mathrm{L}_H)_1\big)=a\,\mathbb{1}+\sum_{i\in I}\mathfrak{h}_i+b\,\mathfrak{h}_1+c\,\mathfrak{s}+d\,\bm{\delta},
\]
and
\[
\mathfrak{ch}\big((\mathrm{R}_H)_1\big)=\bm{\chi}+\sum_{i\in I}(\mathfrak{h}_i-\mathbb{1})+b(\mathfrak{h}_1-\mathbb{1})+c(\mathfrak{s}-\mathbb{1})+d\,\bm{\delta}.
\]
Moreover, one always has \(c=d=0\) in type B, and in type C the coefficients \(c\) and \(d\) may both be equal to \(1\).
C.
There exist non-negative integers \(a,b\in\mathbb{N}\), integers \(c,d\in\{0,1\}\), and a subset \(I\subseteq [n]\), each determined by \(S(H)\), such that
\[
\mathfrak{ch}\big((\mathrm{L}_H)_1\big)=a\,\mathbb{1}+\sum_{i\in I}\mathfrak{h}_i+b\,\mathfrak{h}_1+c\,\mathfrak{s}+d\,\bm{\delta},
\]
and
\[
\mathfrak{ch}\big((\mathrm{R}_H)_1\big)=\bm{\chi}+\sum_{i\in I}(\mathfrak{h}_i-\mathbb{1})+b(\mathfrak{h}_1-\mathbb{1})+c(\mathfrak{s}-\mathbb{1})+d\,\bm{\delta}.
\]
D.
There exist non-negative integers \(a,b\in\mathbb{N}\), integers \(c,d\in\{0,1\}\), and a subset \(I\subseteq [n]\), each determined by \(S(H)\), such that
\[
\mathfrak{ch}\big((\mathrm{L}_H)_1\big)=a\,\mathbb{1}+\sum_{i\in I}\mathfrak{h}_i+b\,\mathfrak{h}_1+c\,\mathfrak{s}+d\,\bm{\delta},
\]
and
\[
\mathfrak{ch}\big((\mathrm{R}_H)_1\big)=\bm{\chi}+\sum_{i\in I}(\mathfrak{h}_i-\mathbb{1})+b(\mathfrak{h}_1-\mathbb{1})+c(\mathfrak{s}-\mathbb{1})+d(\bm{\delta}-\mathbb{1}).
\]
Moreover, \(c\) and \(d\) are never simultaneously equal to \(1\), and in type B one always has \(d=0\).
E.
There exist non-negative integers \(a,b\in\mathbb{N}\), integers \(c,d\in\{0,1\}\), and a subset \(I\subseteq [n-1]\), each determined by \(S(H)\), such that
\[
\mathfrak{ch}\big((\mathrm{L}_H)_1\big)=a\,\mathbb{1}+\sum_{i\in I}\mathfrak{h}_i+b\,\mathfrak{h}_1+c\,\mathfrak{s}+d\,\bm{\delta},
\]
and
\[
\mathfrak{ch}\big((\mathrm{R}_H)_1\big)=\bm{\chi}+\sum_{i\in I}(\mathfrak{h}_i-\mathbb{1})+b(\mathfrak{h}_1-\mathbb{1})+c(\mathfrak{s}-\mathbb{1})+d\,\bm{\delta}.
\]
Moreover, \(c\) and \(d\) are never simultaneously equal to \(1\), and in type B one always has \(d=0\).

The stem defines notation and context but does not explicitly reveal the correct conclusion. There is no direct textual leakage of the extra conditions that distinguish the correct option.

The item is essentially a theorem-identification question: the correct choice states a full structural conclusion, and the task is to recognize the exact version. It is not a pure tautology, since alternatives differ in meaningful details, but it is close to restating a known result.

Selecting the answer requires careful comparison of subtle representation-theoretic details, but mostly tests precise recall/discrimination rather than genuine generative derivation or synthesis.

The distractors are plausible and mathematically targeted: one weakens the theorem, others alter boundary conditions or a specific character term, and they align with realistic failure modes such as dropping constraints or confusing basis terms.

A solid but theorem-recognition-heavy MCQ: little answer leakage and strong distractors, but only moderate success at avoiding tautological recall and inducing genuine generative reasoning.


\begin{theorem}\label{Thm: main1}
    Assume that $(M^n,g,E)$ is a spin generalized asymptotically flat manifold with nonnegative scalar curvature. Then we have the following mass-capacity inequality
    $$m(M,g,E)\geq 2\mathfrak c(M,g,E).$$
    If the equality holds, then $(M,g)$ is harmonically conformal  to $\mathbb R^n\setminus S$, where $S$ is a bounded closed subset of $\mathbb R^n$ with Hausdorff dimension no greater than $\frac{n-2}{2}$. In particular, we have
    $$\pi_i(M)=0\mbox{ for all }1\leq i\leq n-1- \left\lfloor\frac{n}{2}\right\rfloor.$$
\end{theorem}

The investigation of mass constitutes a central topic in  general relativity. Notably, the basic concept of mass for isolated gravitational systems was first formulated by Arnowitt-Deser-Misner \cite{Arnowitt-Deser-Misner}, with such systems modeled as asymptotically flat manifolds. Later, Bartnik \cite{Bartnik1986} and also Chru\'sciel \cite{Chru86} showed that the mass is indeed a geometric quantity of asymptotically flat manifolds. We recall the definition of asymptotically flat manifolds and mass as follows.
\begin{definition}
    A complete Riemannian $n$-manifold $(M,g)$ with dimension $n\geq 3$ is asymptotically flat if there is a compact subset $K\subset M$ such that
\begin{itemize}
\item[(i)] the complement $M-K$ consists of finitely many ends $\{E_l\}_{l=1}^k$, where each end $E_l$ is diffeomorphic to $\mathbb R^n-\bar B_1$ with $$\bar B_1=\{x\in \mathbb R^n:|x|\leq 1\},$$
\item[(ii)]  the metric $g$ on each end $E_l$ has the expression $g=g_{ij}\mathrm dx_i\otimes\mathrm dx_j$ in the Euclidean coordinate chart, where the metric components satisfy the decay condition
\begin{equation*}\label{Eq: decay}
|g_{ij}-\delta_{ij}|+|x||\partial g_{ij}|+|x|^2|\partial^2 g_{ij}|=O\left(|x|^{-\tau}\right),
\end{equation*}
as $x\to\infty$, where $\tau$ is a positive constant greater than $\frac{n-2}{2}$,
\item[(iii)] the scalar curvature $R(g)$ belongs to $L^1(M,g)$.
\end{itemize}
For convenience, each end $E_l$ will be called an asymptotically flat end of $(M,g)$.
\end{definition}

\begin{definition}
Let $(M,g)$ be an asymptotically flat manifold and $E$ be an asymptotically flat end of $(M,g)$.
    The mass of the end $E$ is defined as
    $$m(M,g,E)=\frac{1}{2n(n-1)\omega_n}\lim_{\rho\to+\infty}\int_{S_\rho}(\partial_j g_{ij}-\partial_i g_{jj})\frac{x_i}{|x|}\,\mathrm d\sigma,$$
    where $\omega_n$ denotes the volume of the Euclidean unit ball $B_1^n$, $S_\rho$ denotes the coordinate sphere $\{|x|=\rho\}$, and $\mathrm d\sigma$ is the Euclidean area element of $S_\rho$.
\end{definition}

Many important works have revealed the source of mass. The positive mass conjecture, with the view that nonnegative energy density contributes to nonnegative mass, states that asymptotically flat manifolds with nonnegative scalar curvature have nonnegative mass on each asymptotically flat end, where the mass vanishes on one end if and only if it is the Euclidean space. This conjecture was finally proved by Schoen-Yau \cite{SY79,SY17} with the minimal surface method, and also by Witten \cite{Witten81} with the harmonic spinor method under the extra assumption that the asymptotically flat manifold is spin. Recently, the three-dimensional positive mass theorem was given new proofs by Bray-Kazaras-Khuri-Stern \cite{BKKS22} using level-set method and also by Agostiniani-Mazzieri-Oronzio \cite{AMO24} using potential method. With black holes taken into consideration, the Penrose inequality states that if an asymptotically flat manifold with only one asymptotically flat end $E$ has nonnegative scalar curvature and the boundary is an outer-minimizing horizon $\Sigma$, then we have the inequality
$$
m(M,g, E)\geq \frac{1}{2}\left(\frac{|\Sigma|_g}{|\mathbb S^{n-1}|}\right)^{\frac{n-2}{n-1}},
$$
where the equality holds if and only if $(M,g)$ is isometric to the half spatial Schwarzschild manifold
$(M_m,g_m)$ with
$$M_m=\mathbb R^n\setminus B_{m/2}\mbox{ and }g_m=\left(1+\frac{m}{2}|x|^{2-n}\right)^{\frac{4}{n-2}}g_{euc}.$$
The Penrose inequality in dimension three was first proved independently by Huisken-Ilmanen \cite{HI01} and Bray \cite{Bray01} using different flow methods. Notice that the potential approach  developed by Agostiniani-Mazzieri-Oronzio in \cite{AMO24} also works for the three-dimensional Penrose ineqality. Based on Bray's original proof, Bray-Lee \cite{BL09} removed the technical use of the Gauss-Bonnet formula  and generalized the Penrose inequality up to dimension seven. In order to prove the Penrose inequality, Bray developed a mass-capacity inequality
$$ m(M,g,E)\geq \mathfrak c(M,g,\Sigma),$$
where $\mathfrak c(M,g,\Sigma)$ denotes the capacity of $\Sigma$ in $(M,g)$. For more works related to mass-capacity inequalities for asymptotically flat manifolds, the audience can refer to the works \cite{BM08, Miao24, Miao25} and the references therein.

The study of mass is closely related to conformal geometry. Based upon the fundamental work of Trudinger \cite{Trudinger68} and Aubin \cite{Aubin76}, Schoen ultimately resolved the Yamabe problem by leveraging the positive mass theorem. As another link between conformal geometry and general relativity, Schoen-Yau \cite{Schoen-Yau} reduced the Liouville theorem in conformal geometry to a consequence of the positive mass theorem for a class of generalized asymptotically flat manifolds, which was recently established in the works \cite{Lesourd-Unger-Yau,LLU23, Zhu23,BC2005,CZ2024}.

In this paper, we continue the study of mass inequalities for generalized asymptotically flat manifolds, which are initiated from Schoen-Yau's work \cite{Schoen-Yau}. Such manifolds  are also well-known in \cite{Lesourd-Unger-Yau} as {\it asymptotically flat manifolds with arbitrary ends} with the following precise definition.
\begin{definition}
    A triple $(M,g,E)$ is said to be a generalized asymptotically flat manifold if
    \begin{itemize}
        \item $(M,g)$ is a complete Riemannian $n$-manifold {\it without boundary}, whose dimension $n$ is no less than three;
        \item $E$ is an asymptotically flat end of $(M,g)$;
\item the scalar curvature $R(g)$ belongs to $L^1(E,g)$.
    \end{itemize}
\end{definition}

The positive mass theorem for generalized asymptotically flat manifolds was systematically researched in \cite{Lesourd-Unger-Yau,LLU23, Zhu23,BC2005,CZ2024}. Soon after, the Penrose inequality for generalized asymptotically flat manifolds was initiated by the second-named author \cite{Zhu2024} to include spatial slices of the {\it extreme} Reissner-Nordstr\"om spacetime into application, where the ultimate goal is to show the following {\it mass-systole conjecture}:
\begin{conjecture}
Let $(M,g, E)$ be a generalized asymptotically flat manifold with nonnegative scalar curvature. Then we have
$$
m(M,g, E)\geq \frac{1}{2}\left(\frac{\sys(M,g,E)}{|\mathbb S^{n-1}|}\right)^{\frac{n-2}{n-1}},
$$
where
\begin{equation*}
\sys(M,g,E)=\inf\left\{|\Sigma|_g:\left.
\begin{array}{c}\text{$\Sigma$ is a smoothly embedded hypersurface }\\
\text{homologous to $\partial E$ in $M$}
\end{array}\right.\right\}.
\end{equation*}
Moreover, the equality holds if and only if
\begin{itemize}
    \item $(M,g)$ is isometric to the Euclidean space;
    \item or there is a strictly outer-minimizing minimal $(n-1)$-sphere $\Sigma_h$ homologous to $\partial E$ such that the region outside $\Sigma_h$ is isometric to the half spatial Schwarzschild manifold with mass $m>0$.
\end{itemize}
\end{conjecture}
The conjecture is now only known to be true when $M$ has the underlying topology $\mathbb R^3\setminus\{O\}$ concerning the work \cite{Zhu2024}. As a further exploration, we are going to consider the mass-capacity inequality for generalized asymptotically flat manifolds.

To present our theorem, we need to introduce the {\it capacity} for generalized asymptotically flat manifolds.
Given any constant $s>1$ we shall use $E_s$ to  denote the open subset of $E$ diffeomorphic to $\mathbb R^n\setminus \bar B_s$.
Let $\eta$ denote a fixed smooth function satisfying $\eta\equiv 0$ in $E_2$ and $\eta\equiv 1$ outside $E_1$.
\begin{definition}\label{def:capacity}
    The capacity of a generalized asymptotically flat manifold $(M,g,E)$ is defined to be
    $$\mathfrak c(M,g,E)=\inf_{\phi\in\mathcal C}\frac{1}{n(n-2)\omega_n}\int_M|\nabla_g\phi|^2\,\mathrm d\mu_g,$$
    where  $\mathcal C$ denotes the collection of all smooth functions $\phi$ such that $\phi-\eta$ has compact support.
\end{definition}
\begin{remark}
  It is a standard fact that we can relax the test function $\phi$ to be any function in $W^{1,2}_{loc}(M,g)$ such that $\phi-\eta$ has compact support.
\end{remark}

In this paper, we are going to prove the following theorem.

Many important works have revealed the source of mass. The positive mass conjecture, with the view that nonnegative energy density contributes to nonnegative mass, states that asymptotically flat manifolds with nonnegative scalar curvature have nonnegative mass on each asymptotically flat end, where the mass vanishes on one end if and only if it is the Euclidean space. This conjecture was finally proved by Schoen-Yau \cite{SY79,SY17} with the minimal surface method, and also by Witten \cite{Witten81} with the harmonic spinor method under the extra assumption that the asymptotically flat manifold is spin. Recently, the three-dimensional positive mass theorem was given new proofs by Bray-Kazaras-Khuri-Stern \cite{BKKS22} using level-set method and also by Agostiniani-Mazzieri-Oronzio \cite{AMO24} using potential method. With black holes taken into consideration, the Penrose inequality states that if an asymptotically flat manifold with only one asymptotically flat end $E$ has nonnegative scalar curvature and the boundary is an outer-minimizing horizon $\Sigma$, then we have the inequality
$$
m(M,g, E)\geq \frac{1}{2}\left(\frac{|\Sigma|_g}{|\mathbb S^{n-1}|}\right)^{\frac{n-2}{n-1}},
$$
where the equality holds if and only if $(M,g)$ is isometric to the half spatial Schwarzschild manifold
$(M_m,g_m)$ with
$$M_m=\mathbb R^n\setminus B_{m/2}\mbox{ and }g_m=\left(1+\frac{m}{2}|x|^{2-n}\right)^{\frac{4}{n-2}}g_{euc}.$$
The Penrose inequality in dimension three was first proved independently by Huisken-Ilmanen \cite{HI01} and Bray \cite{Bray01} using different flow methods. Notice that the potential approach developed by Agostiniani-Mazzieri-Oronzio in \cite{AMO24} also works for the three-dimensional Penrose ineqality. Based on Bray's original proof, Bray-Lee \cite{BL09} removed the technical use of the Gauss-Bonnet formula and generalized the Penrose inequality up to dimension seven. In order to prove the Penrose inequality, Bray developed a mass-capacity inequality
$$ m(M,g,E)\geq \mathfrak c(M,g,\Sigma),$$
where $\mathfrak c(M,g,\Sigma)$ denotes the capacity of $\Sigma$ in $(M,g)$. For more works related to mass-capacity inequalities for asymptotically flat manifolds, the audience can refer to the works \cite{BM08, Miao24, Miao25} and the references therein.

In this paper, we continue the study of mass inequalities for generalized asymptotically flat manifolds, which are initiated from Schoen-Yau's work \cite{Schoen-Yau}. Such manifolds are also well-known in \cite{Lesourd-Unger-Yau} as {\it asymptotically flat manifolds with arbitrary ends} with the following precise definition.
\begin{definition}
 A triple $(M,g,E)$ is said to be a generalized asymptotically flat manifold if
 \begin{itemize}
 \item $(M,g)$ is a complete Riemannian $n$-manifold {\it without boundary}, whose dimension $n$ is no less than three;
 \item $E$ is an asymptotically flat end of $(M,g)$;
\item the scalar curvature $R(g)$ belongs to $L^1(E,g)$.
 \end{itemize}
\end{definition}

The positive mass theorem for generalized asymptotically flat manifolds was systematically researched in \cite{Lesourd-Unger-Yau,LLU23, Zhu23,BC2005,CZ2024}. Soon after, the Penrose inequality for generalized asymptotically flat manifolds was initiated by the second-named author \cite{Zhu2024} to include spatial slices of the {\it extreme} Reissner-Nordstr\"om spacetime into application, where the ultimate goal is to show the following {\it mass-systole conjecture}:
\begin{conjecture}
Let $(M,g, E)$ be a generalized asymptotically flat manifold with nonnegative scalar curvature. Then we have
$$
m(M,g, E)\geq \frac{1}{2}\left(\frac{\sys(M,g,E)}{|\mathbb S^{n-1}|}\right)^{\frac{n-2}{n-1}},
$$
where
\begin{equation*}
\sys(M,g,E)=\inf\left\{|\Sigma|_g:\left.
\begin{array}{c}\text{$\Sigma$ is a smoothly embedded hypersurface }\\
\text{homologous to $\partial E$ in $M$}
\end{array}\right.\right\}.
\end{equation*}
Moreover, the equality holds if and only if
\begin{itemize}
 \item $(M,g)$ is isometric to the Euclidean space;
 \item or there is a strictly outer-minimizing minimal $(n-1)$-sphere $\Sigma_h$ homologous to $\partial E$ such that the region outside $\Sigma_h$ is isometric to the half spatial Schwarzschild manifold with mass $m>0$.
\end{itemize}
\end{conjecture}
The conjecture is now only known to be true when $M$ has the underlying topology $\mathbb R^3\setminus\{O\}$ concerning the work \cite{Zhu2024}. As a further exploration, we are going to consider the mass-capacity inequality for generalized asymptotically flat manifolds.

To present our theorem, we need to introduce the {\it capacity} for generalized asymptotically flat manifolds.
Given any constant $s>1$ we shall use $E_s$ to denote the open subset of $E$ diffeomorphic to $\mathbb R^n\setminus \bar B_s$.
Let $\eta$ denote a fixed smooth function satisfying $\eta\equiv 0$ in $E_2$ and $\eta\equiv 1$ outside $E_1$.
\begin{definition}\label{def:capacity}
 The capacity of a generalized asymptotically flat manifold $(M,g,E)$ is defined to be
 $$\mathfrak c(M,g,E)=\inf_{\phi\in\mathcal C}\frac{1}{n(n-2)\omega_n}\int_M|\nabla_g\phi|^2\,\mathrm d\mu_g,$$
 where $\mathcal C$ denotes the collection of all smooth functions $\phi$ such that $\phi-\eta$ has compact support.
\end{definition}
\begin{remark}
 It is a standard fact that we can relax the test function $\phi$ to be any function in $W^{1,2}_{loc}(M,g)$ such that $\phi-\eta$ has compact support.
\end{remark}

In this paper, we are going to prove the following theorem.

\begin{abstract}
In the spin case, we can establish a mass-capacity inequality for generalized asymptotically flat manifolds $(M,g,E)$ with nonnegative scalar curvature, where the equality implies that $(M,g)$ is harmonically conformal to $\mathbb R^n\setminus S$ for a closed bounded subset $S$ of $\mathbb R^n$ with Hausdorff dimension no greater than $\frac{n-2}{2}$.
\end{abstract}

To present our theorem, we need to introduce the {\it capacity} for generalized asymptotically flat manifolds.
Given any constant $s>1$ we shall use $E_s$ to denote the open subset of $E$ diffeomorphic to $\mathbb R^n\setminus \bar B_s$.
Let $\eta$ denote a fixed smooth function satisfying $\eta\equiv 0$ in $E_2$ and $\eta\equiv 1$ outside $E_1$.
\begin{definition}\label{def:capacity}
 The capacity of a generalized asymptotically flat manifold $(M,g,E)$ is defined to be
 $$\mathfrak c(M,g,E)=\inf_{\phi\in\mathcal C}\frac{1}{n(n-2)\omega_n}\int_M|\nabla_g\phi|^2\,\mathrm d\mu_g,$$
 where $\mathcal C$ denotes the collection of all smooth functions $\phi$ such that $\phi-\eta$ has compact support.
\end{definition}
\begin{remark}
 It is a standard fact that we can relax the test function $\phi$ to be any function in $W^{1,2}_{loc}(M,g)$ such that $\phi-\eta$ has compact support.
\end{remark}

To include previous Bray's mass-capacity inequality as a special case, we also extend our main theorem to the setting of Riemannian manifolds with corner defined as follows.
\begin{definition}
 A triple $(M,g,\Sigma)$ will be called a Riemannian manifold with corner if $M$ is a smooth manifold equipped with a continuous metric $g$, and $\Sigma$ is a closed separating smooth hypersurface $\Sigma$ in $M$ such that the metric $g$ is actually smooth to boundary in the closure of every component of $M\setminus \Sigma$. For convenience, $\Sigma$ will be called the corner of $(M,g)$.
\end{definition}
In particular, we can consider generalized asymptotically flat manifolds with corner.
\begin{definition}
 A generalized asymptotically flat manifold $(M,g,E)$ will be called a generalized asymptotically flat manifold with corner, if there exists a closed separating smooth hypersurface $\Sigma$ such that $(M,g,\Sigma)$ is a Riemannian manifold with corner.
\end{definition}

Throughout this paper, we always take the convention that $\mathbb S^2$ has mean curvature two in $\mathbb R^3$ with respect to the outward unit normal. With extra mean-curvature condition along the corner, we can prove
\begin{theorem}\label{Thm: main2}
 Assume that $(M^n,g,E)$ is a spin generalized asymptotically flat manifold with corner, which has nonnegative scalar curvature. Assume that the sum of the mean curvatures of the corner on both sides with respect to the outward unit normal is nonnegative. Then we have the following mass-capacity inequality
 $$m(M,g,E)\geq 2\mathfrak c(M,g,E).$$
 If the equality holds, then $(M,g)$ is smooth and harmonically conformal to $\mathbb R^n\setminus S$, where $S$ is a bounded closed subset of $\mathbb R^n$ with Hausdorff dimension no greater than $\frac{n-2}{2}$. In particular, we have
 $$\pi_i(M)=0\mbox{ for all }1\leq i\leq n-1- \left\lfloor\frac{n}{2}\right\rfloor.$$
\end{theorem}
As a direct consequence, we can show
\begin{corollary}[Bray's mass capacity inequality]\label{Cor: Bray}
 Assume that $(M^n,g)$ is a spin asymptotically flat manifold with only one asymptotically flat end $E$, non-empty boundary $\Sigma$, and nonnegative scalar curvature.
Assume further that the boundary $\Sigma$ is minimal. Then we have
\begin{equation}\label{Eq: Bray's mass-capacity}
 m(M,g,E)\geq \mathfrak c(M,g,\Sigma),
\end{equation}
where $\mathfrak c(M,g,\Sigma)$ denotes the capacity of $\Sigma$ in $(M,g)$ given by
$$\mathfrak c(M,g,\Sigma)=\inf_{\phi\in\mathcal C}\frac{1}{n(n-2)\omega_n}\int_M|\nabla_g\phi|^2\,\mathrm d\mu_g,$$
 where $\mathcal C$ denotes the collection of all smooth functions $\phi$ with compact support such that $\phi\equiv 1$ on $\Sigma$.
The equality holds in \eqref{Eq: Bray's mass-capacity} if and only if $(M,g)$ is isometric to the half spatial Schwarzschild manifold.
\end{corollary}

\begin{proposition}\label{Prop: mass capacity}
 If $(M,g)$ has nonnegative scalar curvature, then we have
 $$m(M,g,E)\geq 2 \mathfrak c(M,g,E).$$
\end{proposition}
\begin{proof}
 Given any constant $\varepsilon>0$ we define
 \begin{equation}\label{Eq: approximation metric}
 g_\varepsilon=u_\varepsilon^{\frac{4}{n-2}}g\mbox{ with } u_\varepsilon=\frac{u+\varepsilon}{1+\varepsilon},
 \end{equation}
 where $u$ is the positive harmonic function from Corollary \ref{Cor: function u}.
 Note that $u_\varepsilon$ has a positive lower bound, so $(M,g_\varepsilon)$ remains to be a complete Riemannian manifold without boundary. It follows from the expansion of $u$ in Corollary \ref{Cor: function u} that $(M,g_\varepsilon,E)$ remains to be a generalized asymptotically flat manifold and that we have
$$m(M,g_\varepsilon,E)=m(M,g,E)-2(1+\varepsilon)^{-1}\mathfrak c(M,g,E).$$
 Clearly, $u_\varepsilon$ is a harmonic function on $(M,g)$ since $u$ is. We see that $(M,g_\varepsilon)$ has nonnegative scalar curvature. Then it follows from the spin positive mass theorem (see \cite{BC2005, CZ2024} for instance) for generalized asymptotically flat manifolds that we have
 $$m(M,g_\varepsilon,E)\geq 0,$$ which yields
 $$m(M,g,E)\geq 2(1+\varepsilon)^{-1}\mathfrak c(M,g,E).$$
 The proof is now completed by letting $\varepsilon\to 0$.
\end{proof}
\begin{remark}
 If one simply takes the function $u$ to be the conformal factor, then the positive mass theorem may not be applied since the Riemannian manifold after conformal deformation can be incomplete. This explains the reason why we have to use an approximation argument for the mass-capacity inequality.
\end{remark}

\begin{proof}[Proof of Theorem \ref{Thm: main2} ]
 By repeating the proof of Corollary \ref{Cor: function u} we are able to construct a positive weakly harmonic $C^{1,\alpha}$-function $u$ on $(M,g)$ such that we have the expansion
 $$u(x)=1-\mathfrak c(M,g,E)\cdot|x|^{2-n}+w,$$
 with $w=O_2(|x|^{2-n-\tau})$ as $x\to\infty$. For any constant $\varepsilon>0$, the conformal manifold
 $(M,g_\varepsilon,E)$ with $$g_\varepsilon=\left(\frac{u+\varepsilon}{1+\varepsilon}\right)^{\frac{4}{n-2}}g$$ is a spin generalized asymptotically flat manifold with corner satisfying the assumption of Proposition \ref{Prop: spin PMT with corner}. As a consequence, we have
 $$0\leq m(M,g_\varepsilon,E)=m(M,g,E)-2(1+\varepsilon)^{-1}\mathfrak c(M,g,E).$$
 Letting $\varepsilon\to 0$, we obtain the desired mass-capacity inequality
 $$m(M,g,E)\geq 2\mathfrak c(M,g,E).$$
 If the equality holds, then the same argument in the proof of Proposition \ref{Prop: spin PMT with corner} yields that $(M,\bar g)$ with $\bar g=u^{\frac{4}{n-2}}g$ is smooth and flat up to a change of the smooth structure. Notice that $u^{-1}$ is a weakly harmonic function on $(M,\bar g)$. We conclude that $u$ is smooth and so $g$ is also smooth. Now we return to the smooth case and the desired conclusions come from Theorem \ref{Thm: main1}.
\end{proof}

\begin{remark}
In the non-spin case, the mass-capacity inequality is still true from the exactly same proof assuming the validity of the positive mass theorem. However, the rigidity seems out of reach with current techniques due to the issue of incompleteness.
\end{remark}

To include previous Bray's mass-capacity inequality as a special case, we also extend our main theorem to the setting of Riemannian manifolds with corner defined as follows.
\begin{definition}
    A triple $(M,g,\Sigma)$ will be called a Riemannian manifold with corner if $M$ is a smooth manifold equipped with a continuous metric $g$, and $\Sigma$ is a closed separating smooth hypersurface $\Sigma$ in $M$ such that the metric $g$ is actually smooth to boundary in the closure of every component of $M\setminus \Sigma$. For convenience, $\Sigma$ will be called the corner of $(M,g)$.
\end{definition}
In particular, we can consider generalized asymptotically flat manifolds with corner.
\begin{definition}
    A generalized asymptotically flat manifold $(M,g,E)$ will be called a generalized asymptotically flat manifold with corner, if there exists a closed separating smooth hypersurface $\Sigma$ such that $(M,g,\Sigma)$ is a Riemannian manifold with corner.
\end{definition}

Throughout this paper, we always take the convention that $\mathbb S^2$ has mean curvature two in $\mathbb R^3$ with respect to the outward unit normal. With extra mean-curvature condition along the corner, we can prove
\begin{theorem}\label{Thm: main2}
  Assume that $(M^n,g,E)$ is a spin generalized asymptotically flat manifold with corner, which has nonnegative scalar curvature. Assume that the sum of the mean curvatures of the corner on both sides with respect to the outward unit normal is nonnegative.  Then we have the following mass-capacity inequality
    $$m(M,g,E)\geq 2\mathfrak c(M,g,E).$$
    If the equality holds, then $(M,g)$ is smooth and harmonically conformal  to $\mathbb R^n\setminus S$, where $S$ is a bounded closed subset of $\mathbb R^n$ with Hausdorff dimension no greater than $\frac{n-2}{2}$. In particular, we have
    $$\pi_i(M)=0\mbox{ for all }1\leq i\leq n-1- \left\lfloor\frac{n}{2}\right\rfloor.$$
\end{theorem}
As a direct consequence, we can show
\begin{corollary}[Bray's mass capacity inequality]\label{Cor: Bray}
    Assume that $(M^n,g)$ is a spin asymptotically flat manifold with only one asymptotically flat end $E$, non-empty boundary $\Sigma$, and nonnegative scalar curvature.
Assume further that the boundary $\Sigma$ is minimal. Then we have
\begin{equation}\label{Eq: Bray's mass-capacity}
    m(M,g,E)\geq \mathfrak c(M,g,\Sigma),
\end{equation}
where $\mathfrak c(M,g,\Sigma)$ denotes the capacity of $\Sigma$ in $(M,g)$ given by
$$\mathfrak c(M,g,\Sigma)=\inf_{\phi\in\mathcal C}\frac{1}{n(n-2)\omega_n}\int_M|\nabla_g\phi|^2\,\mathrm d\mu_g,$$
    where  $\mathcal C$ denotes the collection of all smooth functions $\phi$ with compact support such that $\phi\equiv 1$ on $\Sigma$.
The equality holds in \eqref{Eq: Bray's mass-capacity} if and only if $(M,g)$ is isometric to the half spatial Schwarzschild manifold.
\end{corollary}

\subsection*{Arrangements}
The remaining part of this paper is arranged as follows.
Sections \ref{Sec: inequality} and \ref{Sec: rigidity} are devoted to proving Theorem \ref{Thm: main1}. In Section \ref{Sec: inequality}, we prove the mass-capacity inequality based on the conformal method. In Section \ref{Sec: rigidity}, we establish the rigidity in the equality case of the mass-capacity inequality by constructing sufficiently many parallel spinors from an approximation procedure. In Section \ref{Sec: corner}, we show necessary modifications in the corner case and present the proofs of Theorem \ref{Thm: main2} and Corollary \ref{Cor: Bray}.

\bigskip

Sections \ref{Sec: inequality} and \ref{Sec: rigidity} are devoted to proving Theorem~\ref{Thm: main1}. In Section \ref{Sec: inequality}, the mass-capacity inequality is proved "based on the conformal method." In Section \ref{Sec: rigidity}, the rigidity in the equality case is established "by constructing sufficiently many parallel spinors from an approximation procedure."

Next we prove the main theorem. First, we establish the mass-capacity inequality, “based on the conformal method.” Then we establish the rigidity in the equality case, “by constructing sufficiently many parallel spinors from an approximation procedure.”

Question: Let $(M^n,g,E)$ be a spin generalized asymptotically flat manifold with $n\ge 3$, meaning that $(M,g)$ is a complete boundaryless Riemannian $n$-manifold, $E$ is an asymptotically flat end of $(M,g)$, and the scalar curvature satisfies $R(g)\in L^1(E,g)$. Assume moreover that the scalar curvature is nonnegative. Let $m(M,g,E)$ denote the ADM mass of the end $E$, and let $\mathfrak c(M,g,E)$ denote the capacity of the end $E$ (equivalently, the constant appearing in the asymptotic expansion $u(x)=1-\mathfrak c(M,g,E)|x|^{2-n}+o(|x|^{2-n})$ on $E$ for the associated positive harmonic function tending to $1$ at infinity). Which of the following conclusions is valid under these hypotheses?

One has the mass-capacity inequality $$m(M,g,E)\ge 2\mathfrak c(M,g,E).$$ If equality holds, then $(M,g)$ is harmonically conformal to $\mathbb R^n\setminus S$, meaning that for some bounded closed set $S\subset \mathbb R^n$ there is a conformal identification with Euclidean space off $S$ whose conformal factor is positive and harmonic; moreover $S$ has Hausdorff dimension at most $\frac{n-2}{2}$. In particular, $$\pi_i(M)=0\quad\text{for all }1\le i\le n-1-\left\lfloor\frac n2\right\rfloor.$$

Choices:
B. One has the mass-capacity inequality $$m(M,g,E)\ge \mathfrak c(M,g,E).$$ If equality holds, then $(M,g)$ is harmonically conformal to $\mathbb R^n\setminus S$ for some bounded closed set $S\subset \mathbb R^n$ with Hausdorff dimension at most $\frac{n-2}{2}$, and in particular $$\pi_i(M)=0\quad\text{for all }1\le i\le n-1-\left\lfloor\frac n2\right\rfloor.$$
C. One has the mass-capacity inequality $$m(M,g,E)\ge 2\mathfrak c(M,g,E).$$
D. One has the mass-capacity inequality $$m(M,g,E)\ge 2\mathfrak c(M,g,E).$$ If equality holds, then $(M,g)$ is globally conformal to $\mathbb R^n\setminus S$ for some bounded closed set $S\subset \mathbb R^n$ with Hausdorff dimension at most $\frac{n-2}{2}$; moreover the conformal factor extends smoothly across $S$. In particular, $$\pi_i(M)=0\quad\text{for all }1\le i\le n-1.$$
E. There exists a bounded closed set $S\subset \mathbb R^n$ with Hausdorff dimension at most $\frac{n-2}{2}$, depending only on $n$, such that every spin generalized asymptotically flat manifold $(M^n,g,E)$ with nonnegative scalar curvature and satisfying $$m(M,g,E)=2\mathfrak c(M,g,E)$$ is harmonically conformal to $\mathbb R^n\setminus S$. In particular, $$\pi_i(M)=0\quad\text{for all }1\le i\le n-1-\left\lfloor\frac n2\right\rfloor.$$

sharp factor 2 from conformal mass formula

entire equality-case rigidity and homotopy conclusion dropped

equality gives harmonic conformality off a singular set, not smooth extension across it nor full homotopy vanishing

dependence of the singular set on the manifold; no universal set S depending only on n

The stem states only the hypotheses and definitions; it does not explicitly reveal the sharp mass-capacity inequality, the equality case, or the topological consequences. There is no direct answer leakage.

This is very close to a theorem-recall item: the correct choice is essentially the full theorem statement under its exact hypotheses. However, the alternatives differ in sharp constants, rigidity strength, and quantifier structure, so it is not a pure verbatim restatement.

Some reasoning is required to distinguish the strongest valid conclusion from a weaker true statement and from subtly overstated false variants. Still, success depends largely on recalling the theorem rather than generating a conclusion from first principles.

The distractors are strong: one is a weaker true statement, others modify the sharp factor, overstate regularity/topological rigidity, or introduce an incorrect universal quantifier. These are plausible and mathematically meaningful failure modes.

A solid MCQ with no answer leakage and strong distractors, but it leans toward theorem recognition rather than deep generative reasoning.