\label{thm:simply}
  Let $S$ be a surface of infinite-type with non-planar ends. Then $\Gamma_k(S)$ is connected and simply connected for each $k\in \mathbb{N}$.

\label{aut}
In this section we prove Theorem \ref{thm:geometric} and henceforth assume that $S$ has no planar ends and boundary. We state the results proved in \cite{irmak2004} without proof whenever

\begin{definition}
  Let $S$ be an infinite-type surface with non-planar ends, $k\in \N$, and let $\Gamma^0_k(S)$ be the collection of cut systems on $S$ with exactly $k$ curves. Let $\Gamma^1_k(S)$ be the graph obtained by attaching a 1-cell between two cut systems in $\Gamma^0_k(S)$ whenever they are related by an elementary move. The generalized Hatcher--Thurston complex, $\Gamma_k(S)$, is obtained by attaching 2-cells to every cycle of type $1,\ 2,$ or $3$.
\end{definition}
Note that $\Gamma^1_1(S)$ is just the Schmutz graph $ \mathcal{G}(S)$. Since the Schmutz graph is connected and $\text{diam}( \mathcal{G}(S))\leq 4$ we conclude that $\Gamma^1_1(S)$ is connected with finite diameter. This will be useful in the proof of connectedness of $\Gamma_k(S)$.

\section{Connectedness and Simple Connectedness of $\Gamma_k(S)$}\label{con}
In this section we prove Theorem \ref{thm:simply}.
\begin{lemma}\label{lem:common}
  Let $v,w$ be cut systems in $\Gamma_k(S)$ so that $v$ and $w$ have all but one curve in common. Then there is a path of length at most $4$ from $v$ to $w$ in $\Gamma^1_1(S)$.
\end{lemma}
\begin{proof}
  Let $v-w = \{a\}$ and $w-v=\{b\}$ and let $R$ be the surface obtained by cutting $S$ along the representatives of all the common curves, $c_1,\cdots, c_{k-1}$, of $v$ and $w$. There is a path $a\lr d_1\lr\cdots\lr d_{n}\lr b$ in $\Gamma^1_1(R) = \mathcal{G}(R)$ with $n\leq 3$. Since $d_i$ are non-separating curves in $R$ it follows that $v_i = (d_i, c_1,\cdots, c_{k-1})$ is a cut system. Thus, $v\lr v_1\lr\cdots \lr v_n\lr w$ is the desired path.
\end{proof}
\begin{proof}[Proof of first half of Theorem \ref{thm:simply}]
  Let $v,w\in \Gamma^0_k(S)$. We use induction on the number of curves $\ell = |v-v\cap w|$. Lemma \ref{lem:common} shows that for the base case $\ell=1$ there is a path from $v$ to $w$. Suppose now that $v-w = \{a_0,\cdots, a_{\ell-1}\}$, $w-v=\{b_0,\cdots, b_{\ell-1}\}$, and $v\cap w = \{c_0,\cdots, c_{k-\ell-1}\}$. Let $K\subset S$ be a compact sub-surface containing all the curves in $v$ and $w$ and let $d$ be a non-separating curve in some component of $S-K$. Then, using induction, there is a path from $v' = (d,a_1,\cdots,a_{\ell-1},c_0,\cdots,c_{k-\ell-1})$ to $w'=(d,b_1,\cdots,b_{\ell-1},c_0,\cdots,c_{k-\ell-1})$. Using Lemma \ref{lem:common}, there is a path from $v$ to $v'$ and $w$ to $w'$.
\begin{figure}[h]
\[\begin{tikzcd}
	v && w \\
	\\
	{v'} && {w'} & {}
	\arrow[squiggly, tail reversed, from=1-1, to=3-1]
	\arrow[squiggly, tail reversed, from=1-3, to=3-3]
	\arrow[squiggly, tail reversed, from=3-1, to=3-3]
\end{tikzcd}\]
\caption{Constructing the path from $v$ to $w$.}
\end{figure}
Using the construction above and the fact that the length of the path in Lemma \ref{lem:common} is at most $4$, it follows that the diameter of $\Gamma^1_k(S)$ is at most $8k-4$.
\end{proof}
If $v,w$ are two cut systems in $\Gamma_k(S)$ so that all of the curves in $v\cup w$ are distinct then any path in $\Gamma^1_k(S)$ between $v$ and $w$ has length at least $k$. This means that the diameter of $\Gamma^1_k(S)$ has a lower bound $k$.\\

In order to prove that $\Gamma_k(S)$ is simply-connected we define the following notion of radius of a path, and segments.
\begin{definition}
  The radius of a path $p = v_0 \lr \cdots \lr v_n$ in $\Gamma^1_k(S)$ with respect to a non-separating curve, $a\in v_j$ for some $j$, is defined as $\max_{v_i}\l(\min_{b\in v_i}i(a,b)\r)$. If $a_0,\cdots, a_n$ are non-separating curves in $S$ then $p$ is a $a_0,\cdots,\ a_n-$segment if $a_0,\cdots,a_n\in v_i$ for each vertex $v_i$ in the path $p$.
\end{definition}

\begin{proposition}\label{prop:base-case}
  Every closed path $p=a_0\lr\cdots\lr a_{n-1}\lr a_0$ in $\Gamma_1(S)$ is null-homotopic.
\end{proposition}

\label{pre}
\textbf{Ends of a Surface.} Let $S$ be a surface of infinite-type and let $\{K_i\}_{i\in \N}$ be a strictly increasing chain of compact sets ordered by inclusion which covers $S$. Then an

\begin{ntheorem}\label{thm:geometric}
  Let $S$ be a surface of infinite-type with no planar ends and empty boundary. Then $\text{Aut}(\Gamma_k(S))$ is isomorphic to $\map^*(S)$ for each $k\in \N$.
\end{ntheorem}

\begin{ntheorem}\label{thm:simply}
  Let $S$ be a surface of infinite-type with non-planar ends. Then $\Gamma_k(S)$ is connected and simply connected for each $k\in \N$.
\end{ntheorem}

\begin{ncorollary}\label{cor:iso}
  Let $S,S'$ be surfaces of infinite-type with no planar ends and empty boundary. Let $\vp:\Gamma_k(S)\to \Gamma_k(S')$ be an isomorphism. Then $S$ is homeomorphic to $S'$.
\end{ncorollary}

\label{intro}
\addcontentsline{toc}{section}{Introduction}
All surfaces in this paper will be connected, orientable, without any punctures or marked points, and possibly with non-empty compact boundary. The mapping class group, $\text{Map}(S)$, is the group of all orientation preserving homeomorphisms of the surface $S$ up to isotopy. The extended mapping class group, $\text{Map}^*(S)$ is the group of all homeomorphisms of $S$ up to isotopy. When the fundamental group $\pi_1(S)$ of the surface $S$ is finitely generated, we say that $S$ is of finite-type; otherwise $S$ is infinite-type.\\

Recently, there has been an effort in the literature to generalize results about curve graphs and complexes from the finite-type case to the infinite-type case. In \cite{bavard}, the authors show that the automorphism group of the curve complex, $ \mathcal{C}(S)$, of an infinite-type surface with empty boundary is geometric, that is, every automorphism of $ \mathcal{C}(S)$ is induced by the action of $\text{Map}^*(S)$, generalizing the result of \cites{ivanov, luo1999, korkmaz} for finite-type surfaces. Independently, \cite{hernandez} also proved that the automorphism groups of the curve complex, non-separating curve complex, and the Schmutz graph are all isomorphic to each other and to $\text{Map}^*(S)$ for surfaces of infinite-type with no planar ends and finitely many boundary components. Moreover, \cite{hernandez, bavard} show that these complexes can differentiate infinite-type surfaces. In \cite{branman}, the author generalizes the result that automorphism group of the pants complex is isomorphic to the extended mapping class group, a result originally proved in \cite{margalit04} for finite-type surfaces. \\

In this paper, we generalize the Hatcher--Thurston complex to the setting of infinite-type surfaces with possible compact boundary and non-planar ends. The Hatcher--Thurston complex or Cut-system complex, $ \mathcal{HT}(S)$, is a 2-dimensional CW-complex associated to surfaces of finite-type in which the vertices are maximal cut-systems of the surface, edges correspond to an elementary move between cut-systems, and 2-cells are attached to triangles, rectangles, and pentagons as described in Section \ref{pre}. This cut-system complex was first introduced in \cite{hatcher} to obtain a presentation for the mapping class group of finite-type surfaces. In \cite{hatcher}, the authors use a Morse-theoretic proof to show that the complex is connected and simply connected. Later, Wajnryb gave a combinatorial proof of this in \cite{wajnryb}. In \cite{irmak2004}, the automorphisms of the Hatcher--Thurston complex were shown to be geometric. In this paper, the complex $ \mathcal{HT}(S)$ is generalized to a countable collection of complexes $\left(\Gamma_k(S)\right)_{k\in \mathbb{N}}$ whose vertices are given by cut-systems with exactly $k$ non-separating curves, as opposed to maximal cut-systems, with $1-$cells and $2-$cells defined in a similar fashion to $ \mathcal{HT}(S)$. In case that $S$ is a finite-type surface of genus $g$, the collection $\{\Gamma_k(S)\}$ is finite with $1\leq k \leq g$ and $\Gamma_g(S) = \mathcal{HT}(S)$. Precise definitions of these complexes are given in Section \ref{pre}. Since the construction of $1$-cells and $2-$cells of the pants complex in \cite{branman} is similar to that of the Hatcher-Thurston complex, generalization of the pants complex to infinite-type surfaces leads to similar issues with connectedness as the generalization of the Hatcher-Thurston complex. In \cite{branman} these issues are resolved by introducing a family of new topologies on the vertex space and extending it to the enitre pants space. As will be discussed in Section \ref{pre}, we resolve this issue, among others, by instead defining the sequence $(\Gamma_k(S))_{k\in \mathbb{N}}$ of complexes discussed above. In Section \ref{con}, we show the following result.

\textbf{Remark.} At multiple points throughout the proof of Theorem \ref{thm:simply} the existence of at least one non-planar end is invoked, leading one to ask whether this is a necessary condition. In other words, is $\Gamma_k(S)$ simply connected for all $1\leq k \leq g$ when $S$ is a genus $g$ surface? Since $\Gamma_g(S) = \mathcal{HT}(S)$ for finite-genus surfaces, as mentioned earlier, it follows that $\Gamma_g(S)$ is simply-connected. But the proof of this, in \cite{wajnryb, hatcher}, makes use of the maximality of the cut-systems with $g$ curves in a genus $g$ surface. To the author's knowledge, this question does not seem to be answered in the literature.\\

Now we describe the construction of the curves $\b_i$. Let $R$ be a subsurface of $S$ with compact boundary and at least one non-planar end so that all the curves $\a_0,\cdots, \a_{n-1}$ are contained in $R$. Let $\delta$ be a separating curve in $R$ disjoint from all $\a_i$ so that the two surfaces, $R_1$ and $R_2$, obtained by cutting $R$ along $\delta$ satisfy the following: $R_1$ is compact and contains all $\a_i$ and $R_2$ is of infinite-type with a single compact boundary component corresponding to the curve $\delta$ in $R$ and has non-planar ends. Let $\tau_0$ and $\tau_1$ be homotopically distinct arcs in $R_2$ with their end points lying on the boundary of $R_2$ so that $\tau_0\cap \tau_1 = \emptyset$. Since $\delta$ and $\a_0$ are disjoint and $i(a_0,a_1)=1$, it can be shown that there exist disjoint essential arcs, $\sigma_0,\ \sigma_1$, with end points on the boundary of $R_1$ corresponding to $\delta$ and $|\sigma_0\cap \a_0| = 1$ and $|\sigma_1\cap \a_1|=1$. Gluing $R_1$ and $R_2$ together along the boundary corresponding to $\delta$ in such a way that $\beta_0 = \sigma_0\cup \tau_0$ and $\beta_1 = \sigma_1\cup \tau_1$ are closed curves. Then by construction we have $i(b_0,a_0) = i(b_1,a_1) = 1$ and that $b_i$ are non-separating. Let $\b_2$ be a non-separating curve in $R_2$ which intersects $\tau_0$ and $\tau_1$ exactly once. Thus we have our desired curves $b_0,\ b_1,$ and $b_2$.
\end{proof}
The above proposition implies that $\Gamma_1(S)$ is simply-connected when $S$ has non-planar ends. We use this as our base case for induction on $k$, to show that $\Gamma_k(S)$ is also simply-connected. From here on we assume the induction hypothesis: for every surface $S$ the complex $\Gamma_j(S)$ is simply connected for all $j<k$. We prove the simple connectedness now by also inducting on radius of the path. We start by a special case of radius 0 curves.
\begin{proposition}\label{prop:sp-radius-0}
 Suppose that $p = v_0 \lr \cdots \lr v_{n-1}$ is a closed path in $\Gamma_k(S)$ so that all the vertices $v_i$ contain a common curve $c$. Then $p$ is null-homotopic.
\end{proposition}
\begin{proof}
 Let $p'$ be the path in $\Gamma_{k-1}(S)$ where the vertices are $v_i- \{c\}$. By the induction hypothesis, it follows that $p'$ is null-homotopic, meaning that it can be split into paths of type 1, 2, and 3. Since adding $c$ back to each vertex in $p'$ does not remove any edge connections therefore $p$ is null-homotopic.
\end{proof}
\begin{proposition}[Lemma 13, \cite{wajnryb}]\label{prop:hex-path}
 Let $a_0, a_1, a_2$ be disjoint non-separating curves so that:
 \begin{enumerate}
 \item all unions $\a_i\cup \a_j$ are non-separating in $S$.
 \item the union $\a_0\cup \a_1\cup \a_2$ separates $S$.
 \end{enumerate}
 Then there exists disjoint non-separating curves $b_0, b_1, b_2$ so that $i(a_i,b_i) = 1$, $i(a_{i+1}, b_{i}) = 1$ (indices modulo $3$), and $i(a_i, b_{i+1}) = 0$ for all other $i,j$. Moreover, if $(a_0,a_1,c_2,\cdots, c_{k-1})$ is a vertex in $\Gamma_k(S)$ then the closed path $p = v_0 \lr \cdots \lr v_5 \lr v_0$ defined as
 \begin{align*}
 (a_0, a_1, c_2,\cdots, c_{k-1})\lr &(a_0, b_1,c_2,\cdots, c_{k-1}) \lr (a_0, a_2,c_2,\cdots, c_{k-1}) \lr (b_0, a_2,c_2,\cdots, c_{k-1})\\
 &\ \lr (a_1,a_2,c_2,\cdots, c_{k-1})\lr (a_1, b_2,c_2,\cdots, c_{k-1})\lr (a_0, a_1,c_2,\cdots, c_{k-1})
 \end{align*}
 is null-homotopic in $\Gamma_k(S)$.
\end{proposition}
\begin{proof}
 Due to conditions on $a_0,\ a_1,\ a_2$ it follows that $S - \bigcup_{i}\a_i$ has exactly two connected components, $R_1$ and $R_2$, both with at least three boundary components corresponding to the curves $\a_0,\ \a_1,$ and $\a_2$. Let $\delta_i$ be disjoint arcs in $R_1$ with endpoints on the boundary components corresponding to $\a_i$ and $\a_{i+1}$ (indices taken modulo 3). Similarly define disjoint arcs $\delta'_i$ on $R_2$. Defining $\b_i = \delta_i\cup \delta'_i$ we get disjoint non-separating curves $b_0, b_1, b_2$ with the desired intersections with $a_i$. In order to prove the second part of the proposition, define $c = T_{a_1}(b_0)$. Then it follows from Proposition \ref{prop:dehn-int} that $i(c, b_0) = i(c, a_1) = i(c,b_1) = i(c,a_0) = 1$ and $i(c,b_2) = i(c, a_2) = 0$. If $(a_0, a_1, c_2,\cdots, c_{k-1})$ is a vertex in $\Gamma_k(S)$ then define the vertices $w_0 = (b_1,b_2, c_2,\cdots, c_{k-1}), w_1 = (c,b_2,c_2,\cdots, c_{k-1}), w_2 = (c,a_2,c_2,\cdots, c_{k-1})$. Then the path $p = v_0 \lr \cdots \lr v_5 \lr v_0$ as defined above can be split into paths of type 1, 2, and 3 as in Figure \ref{fig:hex-path}.
 \begin{figure}
 \[\begin{tikzcd}
 {v_0} && {v_1} && {v_2} && {v_3} \\
 \\
 && {w_0} && {w_2} \\
 &&& {w_1} \\
 {v_5} &&&&&& {v_4}
 \arrow[tail reversed, from=1-1, to=1-3]
 \arrow[tail reversed, from=1-3, to=1-5]
 \arrow[tail reversed, from=1-5, to=1-7]
 \arrow[tail reversed, from=1-7, to=5-7]
 \arrow[tail reversed, from=3-3, to=1-3]
 \arrow[tail reversed, from=3-3, to=4-4]
 \arrow[tail reversed, from=3-3, to=5-1]
 \arrow[tail reversed, from=3-5, to=1-5]
 \arrow[tail reversed, from=3-5, to=1-7]
 \arrow[tail reversed, from=3-5, to=5-7]
 \arrow[tail reversed, from=4-4, to=3-5]
 \arrow[tail reversed, from=4-4, to=5-1]
 \arrow[tail reversed, from=5-1, to=1-1]
 \arrow[tail reversed, from=5-7, to=5-1]
\end{tikzcd}\]
 \caption{Splitting the path $p$ into paths of type $1$, $2$, and $3$.}
 \label{fig:hex-path}
 \end{figure}
\end{proof}
Before beginning the proof of the next lemma, note that if $v$ and $w$ are vertices in $\Gamma_k(S)$ with common curves $(a_0, \cdots, a_m)$ then there is an $(a_0,\cdots, a_m)-$segment connecting $v$ and $w$. This follows directly from the proof of connectedness. This fact will be repeatedly used in the proof of the next lemma.
\begin{lemma}\label{lem:radius-zero}
 Let $p$ be a path in $\Gamma_k(S)$ of radius $0$ with respect to some curve $a_0$. Then $p$ is null-homotopic.
\end{lemma}
\begin{proof}
 We prove this by induction on the number of segments in the path $p$. Suppose that $a_0\in v_0$, where $v_0$ is a vertex in $p$. We call the maximal $a_0-$segment containing $v_0$ the first segment. From Proposition \ref{prop:sp-radius-0} it follows that if $p$ is a $a_0-$segment then $p$ is null homotopic. Suppose this is not the case and $v_1$ is the terminal vertex of the first segment. Let $a_1\in v_1$ be a curve distinct from $a_0$ which will serve as the common curve of the next segment, which we call the second segment. Let $v_2$ be the terminal element of second segment. If $p$ is just made up of these two segments then both $v_1$ and $v_2$ contain $a_0$ and $a_1$, thus there is a $(a_0, a_1)-$segment from $v_1$ to $v_2$, therefore implying that $p$ can be split into a $a_0-$segment and a $a_1-$segment, both of which are null-homotopic due to Proposition \ref{prop:sp-radius-0}. Suppose now that $p$ has more than $2$ segments. Then let $a_2$ be a curve in the vertex $v$ which follows $v_2$, so that $i(a_0, a_2)=0$. Let $v_3$ be the terminal vertex in the $a_2-$segment. Now we try to reduce the number of segments.\\

The proof of this result closely follows the combinatorial proof of \cite{wajnryb} with alterations made when required for the infinite-type case. The proof proceeds by induction on $k$. We first show that the $1-$skeleton, $\Gamma^1_k(S)$, is connected and that its diameter is bounded between $k$ and $8k-4$ using induction on $k$. To further prove that $\Gamma_k(S)$ is simply connected, we introduce the notions of radius of a path in $\Gamma_k(S)$ and segments, following the proof by Wajnryb. The remaining proof can then be divided into three steps. It is first shown in Proposition \ref{prop:base-case} that $\Gamma_1(S)$ is simply-connected by first showing that closed paths with radius $1$ are contractible. Note that this argument uses the assumption that $S$ has at least one non-planar end. Continuing the induction on $k$, the second step shows that any closed 0-radius path is contractible in $\Gamma_k(S)$, see Proposition \ref{lem:radius-zero}, by induction on the number of segments in the path. Finally, again using the result about radius $0$ paths and the assumption of a non-planar end the theorem statement follows.\\

\textbf{Remark.} At multiple points throughout the proof of Theorem \ref{thm:simply} the existence of at least one non-planar end is invoked, leading one to ask whether this is a necessary condition. In other words, is $\Gamma_k(S)$ simply connected for all $1\leq k \leq g$ when $S$ is a genus $g$ surface? Since $\Gamma_g(S) = \mathcal{HT}(S)$ for finite-genus surfaces, as mentioned earlier, it follows that $\Gamma_g(S)$ is simply-connected. But the proof of this, in \cite{wajnryb, hatcher}, makes use of the maximality of the cut-systems with $g$ curves in a genus $g$ surface. To the author's knowledge, this question does not seem to be answered in the literature.\\

In Section \ref{aut}, we show that automorphisms of $\Gamma_k(S)$ are geometric:
\begin{ntheorem}\label{thm:geometric}
  Let $S$ be a surface of infinite-type with no planar ends and empty boundary. Then $\text{Aut}(\Gamma_k(S))$ is isomorphic to $\text{Map}^*(S)$ for each $k\in \mathbb{N}$.
\end{ntheorem}
We prove this by constructing an action of $\text{Aut}(\Gamma_k(S))$ on the Schmutz graph, $ \mathcal{G}(S)$, as done by \cite{irmak2004}. The proof in \cite{irmak2004} is almost entirely transferable to the infinite-type case, but parts of it are reproduced here for the sake of completeness. As a direct corollary to Theorem \ref{thm:geometric} we have the following:
\begin{ncorollary}\label{cor:iso}
  Let $S,S'$ be surfaces of infinite-type with no planar ends and empty boundary. Let $\varphi:\Gamma_k(S)\to \Gamma_k(S')$ be an isomorphism. Then $S$ is homeomorphic to $S'$.
\end{ncorollary}
The Hatcher--Thurston complex is an important tool in the study of mapping class group of finite-type surfaces. By studying the stabilizers of vertices, edges, and cells in $ \mathcal{HT}(S)$ of the action of $\text{Map}(S)$ Wajnryb gave a simple presentation of $\text{Map}(S)$ in \cite{wajnryb2}, which was inspired by a similar approach to the problem by Harer in \cite{harer}. Harer in the same paper uses a simpler version of the Hatcher--Thurston complex to compute the second cohomology of the mapping class group. These methods are not directly transferable to big mapping class group as they are not finitely generated, in fact they are not even countably generated. In \cite{patel}, the authors give topological generators for the pure mapping class group $\text{PMap}(S)$, the subgroup of $\text{Map}(S)$ which are end-preserving, in terms of Dehn Twists and handle shift maps. For infinite-type surfaces with countably many non-planar ends it may be possible to use the results presented in this paper to determine relations in $\text{PMap}(S)$ between Dehn Twists and handle shift maps by applying the techniques of Wajnryb and Harer.
\vspace{1em}
\newline
\textbf{Outline.} In Section \ref{pre} the main objects of this paper are defined and conventions and notations which are used throughout the paper are discussed. Section \ref{con} is dedicated to the proof of Theorem \ref{thm:simply} and Section \ref{aut} to the proof of Theorem \ref{thm:geometric} and Corollary \ref{cor:iso}.
\vspace{1em}
\newline
\textbf{Acknowledgements.} The author would like to thank Sumanta Das for posing the question which led to this paper and his comments for improving the quality of the paper. The author also extends gratitude to Dan Margalit and Tyrone Ghaswala, whose comments helped improve the paper.

The proof of Theorem~\ref{thm:simply} “closely follows the combinatorial proof of \cite{wajnryb} with alterations made when required for the infinite-type case” and “proceeds by induction on $k$.” First, one shows “that the $1$-skeleton, $\Gamma^1_k(S)$, is connected” and that “its diameter is bounded between $k$ and $8k-4$ using induction on $k$.” For simple connectedness, the proof “introduce[s] the notions of radius of a path in $\Gamma_k(S)$ and segments, following the proof by Wajnryb,” and then splits into three steps: (1) in Proposition~\ref{prop:base-case}, prove “$\Gamma_1(S)$ is simply-connected by first showing that closed paths with radius $1$ are contractible,” using the assumption that “$S$ has at least one non-planar end”; (2) for general $k$, show “that any closed $0$-radius path is contractible in $\Gamma_k(S)$, see Proposition~\ref{lem:radius-zero}, by induction on the number of segments in the path”; (3) “finally, again using the result about radius $0$ paths and the assumption of a non-planar end the theorem statement follows.”

Question: Let $S$ be a connected, orientable surface with no punctures or marked points and possibly nonempty compact boundary. Assume that $S$ is of infinite type, meaning that $\pi_1(S)$ is not finitely generated, and that $S$ has non-planar ends. For each $k\in\mathbb{N}$, let $\Gamma_k(S)$ denote the generalized Hatcher--Thurston $2$-dimensional CW-complex whose vertices are cut-systems consisting of exactly $k$ non-separating curves on $S$, with $1$-cells and $2$-cells defined analogously to the Hatcher--Thurston complex. Which statement holds for every such surface $S$ and every $k\in\mathbb{N}$?

For every $k\in\mathbb{N}$, the complex $\Gamma_k(S)$ is connected and simply connected.

Choices:
B. For every $k\in\mathbb{N}$, the complex $\Gamma_k(S)$ is connected, and it is simply connected whenever $k$ is sufficiently large.
C. For every $k\in\mathbb{N}$, the complex $\Gamma_k(S)$ is connected.
D. There exists $N\in\mathbb{N}$ such that for every $k\ge N$, the complex $\Gamma_k(S)$ is connected and simply connected.
E. For every $k\in\mathbb{N}$, the complex $\Gamma_k(S)$ is path connected and contractible.

simple-connectedness strengthened to contractibility

The stem does not explicitly reveal the correct choice or embed the conclusion in the wording. It only states the hypotheses and asks for the valid conclusion.

This is essentially a direct theorem-recall item: the correct answer appears to restate the main result under the given hypotheses, with distractors formed by weakening or strengthening that statement.

Some reasoning is needed to distinguish the exact theorem statement from nearby variants such as mere connectedness, eventual simple connectivity, or contractibility. However, the item mainly tests recognition of the precise theorem rather than substantial generative mathematical reasoning.

The distractors are plausible and mathematically meaningful: one is a weaker true statement, others alter quantifiers or overstrengthen the conclusion. These reflect common failure modes in theorem selection.

A solid theorem-identification MCQ with strong distractors and no answer leakage, but it is largely a direct restatement of a result and only moderately tests genuine reasoning.


\label{main}
Let $H$ be a split classical group over $F$,
and let $\phi \in \Phi_\mathrm{temp}(H(F))$.
Suppose that one of the following holds:
\begin{description}
\item[(B)]
$H = \SO_{2n+1}$, $p > 6n+3$ and $\phi$ is arbitrary;
\item[(C)]
$H = \Sp_{2n}$, $p > 6n$ and $\phi$ satisfies $(\star)$;
\item[(D)]
$H = \SO_{2n}$, $p > 6n$ and $\phi$ satisfies $(\star)$.
\end{description}
Set $\psi = \widehat\phi$.

\begin{enumerate}
\item
There exists $\pi \in \Pi_\psi$ such that
\[
d_{H^\vee}(\mathrm{Ad}(H^\vee)N_\phi) \in \bar\mathcal{N}(\pi)^{\max}.
\]
\item
For $\pi \in \Pi_\psi$ and for $\oo_H^\mathrm{st} \in \bar\mathcal{N}(\pi)$,
if $\oo_H^\mathrm{st} \not= d_{H^\vee}(\mathrm{Ad}(H^\vee)N_\phi)$, then
\[
\dim(\oo_H^\mathrm{st}) < \dim(d_{H^\vee}(\mathrm{Ad}(H^\vee)N_\phi)).
\]
Here, $\dim(\oo_H^\mathrm{st})$ is the Zariski dimension of the orbit as an algebraic variety.
\item
If we further assume Hypothesis \ref{hypo} below, then Conjecture \ref{upper} is true for $\phi$, i.e.,
\[
\bar\mathcal{N}(\pi)^{\max} \leq d_{H^\vee}(\mathrm{Ad}(H^\vee)N_\phi)
\]
for any $\pi \in \Pi_\psi$.
\end{enumerate}

\begin{thm}\label{main}
Let $H$ be a split classical group over $F$,
and let $\phi \in \Phi_\temp(H(F))$.
Suppose that one of the following holds:
\begin{description}
\item[(B)]
$H = \SO_{2n+1}$, $p > 6n+3$ and $\phi$ is arbitrary;
\item[(C)]
$H = \Sp_{2n}$, $p > 6n$ and $\phi$ satisfies $(\star)$;
\item[(D)]
$H = \SO_{2n}$, $p > 6n$ and $\phi$ satisfies $(\star)$.
\end{description}
Set $\psi = \widehat\phi$.

\begin{enumerate}
\item
There exists $\pi \in \Pi_\psi$ such that
\[
d_{H^\vee}(\Ad(H^\vee)N_\phi) \in \bar\NN(\pi)^{\max}.
\]
\item
For $\pi \in \Pi_\psi$ and for $\oo_H^\st \in \bar\NN(\pi)$,
if $\oo_H^\st \not= d_{H^\vee}(\Ad(H^\vee)N_\phi)$, then
\[
\dim(\oo_H^\st) < \dim(d_{H^\vee}(\Ad(H^\vee)N_\phi)).
\]
Here, $\dim(\oo_H^\st)$ is the Zariski dimension of the orbit as an algebraic variety.
\item
If we further assume Hypothesis \ref{hypo} below, then Conjecture \ref{upper} is true for $\phi$, i.e.,
\[
\bar\NN(\pi)^{\max} \leq d_{H^\vee}(\Ad(H^\vee)N_\phi)
\]
for any $\pi \in \Pi_\psi$.
\end{enumerate}
\end{thm}

This work is motivated by the elusive relation
between the irreducible admissible Harish-Chandra characters of a reductive $p$-adic group
and the Langlands-Arthur parameters.
Arguably the best-known instance of this relation is the Hiraga--Ichino--Ikeda formal degree formula
for square-integrable characters in terms of adjoint $\gamma$-factors \cite{HII}.
The formal degree concerns the coefficient of the distribution attached to the zero nilpotent orbit
in the Harish-Chandra--Howe local character expansion.
At the opposite end,
the largest orbits that contribute to the character expansion determine the wavefront set of the character distribution.
They are the focus of the present paper.
Under a technical assumption, which is unnecessary for $\SO_{2n+1}$, and for $p$ large (see Theorem \ref{main} for the precise conditions), we prove that
the largest-dimensional orbits in the union of the geometric wavefront sets of the representations
in the Aubert--Zelevinsky dual of a tempered $L$-packet for a split classical $p$-adic group
{\it equal} the Spaltenstein dual of the corresponding tempered nilpotent Langlands parameter.
\par

A relation of this sort, Conjecture \ref{upper},  was proposed in \cite{CK,HLLS},
motivated by the results of \cite{CMBO1,CMBO2} for unipotent representations of split (and `inner-to-split') $p$-adic groups.
The fact that such a link should exist had already been apparent from
the foundational work of Adams--Barbasch--Vogan on the definition and construction of microlocal Arthur packets \cite{ABV}.
In Theorem \ref{main}, we also show that the full Conjecture \ref{upper} follows by further assuming Hypothesis \ref{hypo}.
\vskip 10pt

We fix a non-archimedean local field $F$ of characteristic zero and of residue characteristic $p \gg 0$.
Let $\overline{F}$ be a fixed algebraic closure of $F$.
In this section, we state our main theorem (Theorem \ref{main})
and explain our idea for the proof.

\subsection{Wavefront set}
Let $H$ be a connected reductive group over $F$ with the Lie algebra $\mathfrak{h}$.
We denote by $\mathcal{N}(\mathfrak{h}(F))$ (resp.~ $\mathcal{N}(\mathfrak{h}(\overline{F}))$)
the set of nilpotent $\mathrm{Ad}(H(F))$-orbits (resp.~ $\mathrm{Ad}(H(\overline{F}))$-orbits)
in $\mathfrak{h}(F)$ (resp.~ $\mathfrak{h}(\overline{F})$).
For an irreducible admissible representation $\pi$ of $H(F)$,
we denote its distribution character by $\Theta_\pi$.
The Harish-Chandra--Howe local character expansion (\cite[Theorem 16.2]{HC}) states that
\[
\Theta_\pi(\exp(X)) = \sum_{\oo_H \in \mathcal{N}(\mathfrak{h}(F))} c_{\oo_H}(\pi) \hat\mu_{\oo_H}(\exp(X))
\]
for $X \in \mathfrak{h}(F)$ sufficiently near to $0$,
where $\hat\mu_{\oo_H}$ is the Fourier transform of the orbital integral,
and $c_{\oo_H}(\pi)$ is a constant.
Set $\mathcal{N}(\pi) = \{ \oo_H \in \mathcal{N}(\mathfrak{h}(F)) \;|\; c_{\oo_H}(\pi)\not= 0\}$
and let $\bar\mathcal{N}(\pi)$ be the set of $\oo_H^\mathrm{st} \in \mathcal{N}(\mathfrak{h}(\overline{F}))$
such that $\oo_H^\mathrm{st}$ contains a rational orbit $\oo_H$ belonging to $\mathcal{N}(\pi)$.
We denote by $\bar\mathcal{N}(\pi)^{\max}$ the subset of $\bar\mathcal{N}(\pi)$ consisting of maximal orbits
with respect to the closure ordering
\[
\oo_H^\mathrm{st} \leq \oo_H'^\mathrm{st} \overset{\text{def}}{\iff} \oo_H^\mathrm{st} \subset \overline{\oo_H'^\mathrm{st}}.
\]
We call $\bar\mathcal{N}(\pi)^{\max}$ the \emph{(geometric) wavefront set} of $\pi$.

\subsection{$L$-parameters}
Denote by $H^\vee$ the complex Langlands dual group,
and by $\mathfrak{h}^\vee$ the Lie algebra of $H^\vee$.
Let $W_F$ be the Weil group of $F$.
An \emph{$L$-parameter} for $H(F)$ is a homomorphism
\[
\phi \colon W_F \times \SL_2(\mathbb{C}) \rightarrow H^\vee
\]
such that $\phi(W_F)$ consists of semisimple elements, $\phi|_{W_F}$ is smooth and $\phi|_{\SL_2(\mathbb{C})}$ is algebraic.
It is called \emph{tempered} if $\phi(W_F)$ is bounded.
Let $\Phi(H(F))$ be the set of $H^\vee$-conjugacy classes of $L$-parameters for $H(F)$,
and let $\Phi_\mathrm{temp}(H(F))$ be its subset consisting of tempered $L$-parameters.
For $\phi \in \Phi(H(F))$, by the derivative of $\phi|_{\SL_2(\mathbb{C})}$,
we have a linear map
\[
d\phi|_{\SL_2(\mathbb{C})} \colon \sl_2(\mathbb{C}) \rightarrow \mathfrak{h}^\vee.
\]
We denote by $N_{\phi} \in \mathfrak{h}^\vee$ the image of
$\begin{pmatrix}
0 & 1 \\ 0 & 0
\end{pmatrix}$ under this map.
The $\mathrm{Ad}(H^\vee)$-orbit of $N_\phi$ is denoted by $\mathrm{Ad}(H^\vee) N_\phi$.
\par

Let $\mathcal{N}(\mathfrak{h}^\vee)$ be the set of nilpotent $\mathrm{Ad}(H^\vee)$-orbits in $\mathfrak{h}^\vee$.
We denote by
\[
d_{H} \colon \mathcal{N}(\mathfrak{h}(\overline{F})) \rightarrow \mathcal{N}(\mathfrak{h}^\vee), \quad
d_{H^\vee} \colon \mathcal{N}(\mathfrak{h}^\vee) \rightarrow \mathcal{N}(\mathfrak{h}(\overline{F}))
\]
the Spaltenstein duality maps (\cite[Section 10]{Sp}).
The images of these maps are the sets of \emph{special nilpotent conjugacy classes}.
Moreover, $\oo_H^\mathrm{st} \leq d_{H^\vee}(d_H(\oo_H^\mathrm{st}))$,
and the equality holds if and only if $\oo_H^\mathrm{st}$ is special.
See \cite[Appendix A]{BV} for more details.
An $L$-parameter $\phi \in \Phi(H(F))$ gives
a nilpotent orbit
\[
d_{H^\vee}(\mathrm{Ad}(H^\vee) N_\phi) \in \mathcal{N}(\mathfrak{h}(\overline{F})).
\]

\subsection{$L$-packets and $A$-packets}
Now suppose that $H$ is a split classical group over $F$.
Namely, $H$ is a symplectic group or a special orthogonal group.
Let $\mathrm{Irr}(H(F))$ be the set of equivalence classes of irreducible representations of $H(F)$,
and let $\Irr_\mathrm{temp}(H(F))$ be its subset consisting of tempered representations.
The local Langlands conjecture proven by Arthur \cite[Theorem 2.2.1]{Ar} gives a canonical surjective map
\[
\mathrm{Irr}(H(F)) \rightarrow \Phi(H(F))
\]
with finite fibers, and preserving the temperedness.
The fiber of $\phi \in \Phi(H(F))$ is denoted by $\Pi_\phi$
and is called the \emph{$L$-packet} associated to $\phi$.
\par

We say that an irreducible representation $\pi$ of $H(F)$ is \emph{co-tempered}
if its Zelevinsky--Aubert dual $\hat\pi$ is tempered.
For $\phi \in \Phi_\mathrm{temp}(H(F))$, its \emph{dual $A$-parameter}
\[
\psi = \widehat\phi \colon W_F \times \SL_2(\mathbb{C}) \times \SL_2(\mathbb{C}) \rightarrow H^\vee
\]
is defined by $\psi(w,g_1,g_2) = \phi(w,g_2)$.
Its \emph{$A$-packet} is given by
\[
\Pi_\psi = \{\pi \in \mathrm{Irr}(H(F)) \;|\; \hat\pi \in \Pi_\phi\}.
\]

\subsection{Conjecture and the main result}
The following conjecture was independently formulated by Kim and the second author \cite[Conjectures 1.1, 1.9]{CK}
and Hazeltine--Liu--Lo--Shahidi \cite[Conjecture 1.1]{HLLS}.

\begin{conj}\label{upper}
Let $\phi \in \Phi(H(F))$.
Then for any $\pi \in \Pi_\phi$, we have
\[
\bar\mathcal{N}(\hat\pi)^{\max} \leq d_{H^\vee}(\mathrm{Ad}(H^\vee)N_\phi).
\]
Moreover, if $\phi$ is tempered,
then the equality holds for some $\pi \in \Pi_\phi$.
\end{conj}

This is relevant for any connected reductive group $H$
provided the local Langlands correspondence for $H(F)$ is assumed.
This conjecture is known in the following cases.

\begin{itemize}
\item
$H=\GL_m$, \cite{MW}, see \cite[Theorem 1.4]{CK}.

\item
$\pi$ is any depth-zero simple supercuspidal representation of a classical group (\cite[Theorem 1.8]{CK}).

\item
{$H$ is an inner form of a split group and}
$\pi$ is any unipotent representation of $H(F)$ with real infinitesimal character
(\cite[Theorem 1.4.1]{CMBO2}).

\item
$H$ is an exceptional group $G_2$ (\cite[Theorem 1.10]{CK}).
\end{itemize}

As explained in \cite[Theorems 1.5, 1.8]{HLLS},
Conjecture \ref{upper} implies
a part of Jiang's conjecture and a part of the Generalized Shahidi Conjecture on local ABV packets
(see \cite[Conjectures 1.4, 1.7]{HLLS}).

\begin{rem}
An analogue of Conjecture \ref{upper} for covering groups
has been formulated by Gao--Liu--Lo--Shahidi \cite[Conjecture 1.2]{GLLS}, and proved for the Kazhdan--Patterson cover of $\GL_m$ in \cite[Theorem 1.3]{GLLS}.
\end{rem}

In this paper, we give a result toward Conjecture \ref{upper} for split classical groups under some conditions.
We focus on the case where $\phi$ is tempered
since Conjecture \ref{upper} is reduced to this case by \cite[Proposition 2.2]{CK} or \cite[Theorem 1.2]{HLLS}.
For $\phi \in \Phi_\mathrm{temp}(H(F))$, we consider the following condition:
\begin{description}
\item[($\star$)]
for any irreducible self-dual subrepresentation $\phi_0 \subset \phi$,
its determinant $\det(\phi_0)$ is trivial on the inertia subgroup $I_F$ of $W_F$.
\end{description}
Our main result is stated as follows.

\subsection{Wavefront set}
Let $H$ be a connected reductive group over $F$ with the Lie algebra $\mathfrak{h}$.
We denote by $\mathcal{N}(\mathfrak{h}(F))$ (resp.~ $\mathcal{N}(\mathfrak{h}(\overline{F}))$)
the set of nilpotent $\mathrm{Ad}(H(F))$-orbits (resp.~ $\mathrm{Ad}(H(\overline{F}))$-orbits)
in $\mathfrak{h}(F)$ (resp.~ $\mathfrak{h}(\overline{F})$).
For an irreducible admissible representation $\pi$ of $H(F)$,
we denote its distribution character by $\Theta_\pi$.
The Harish-Chandra--Howe local character expansion (\cite[Theorem 16.2]{HC}) states that
\[
\Theta_\pi(\exp(X)) = \sum_{\oo_H \in \mathcal{N}(\mathfrak{h}(F))} c_{\oo_H}(\pi) \hat\mu_{\oo_H}(\exp(X))
\]
for $X \in \mathfrak{h}(F)$ sufficiently near to $0$,
where $\hat\mu_{\oo_H}$ is the Fourier transform of the orbital integral,
and $c_{\oo_H}(\pi)$ is a constant.
Set $\mathcal{N}(\pi) = \{ \oo_H \in \mathcal{N}(\mathfrak{h}(F)) \;|\; c_{\oo_H}(\pi)\not= 0\}$
and let $\bar\mathcal{N}(\pi)$ be the set of $\oo_H^\mathrm{st} \in \mathcal{N}(\mathfrak{h}(\overline{F}))$
such that $\oo_H^\mathrm{st}$ contains a rational orbit $\oo_H$ belonging to $\mathcal{N}(\pi)$.
We denote by $\bar\mathcal{N}(\pi)^{\max}$ the subset of $\bar\mathcal{N}(\pi)$ consisting of maximal orbits
with respect to the closure ordering
\[
\oo_H^\mathrm{st} \leq \oo_H'^\mathrm{st} \overset{\text{def}}{\iff} \oo_H^\mathrm{st} \subset \overline{\oo_H'^\mathrm{st}}.
\]
We call $\bar\mathcal{N}(\pi)^{\max}$ the \emph{(geometric) wavefront set} of $\pi$.

\subsection{$L$-parameters}
Denote by $H^\vee$ the complex Langlands dual group,
and by $\mathfrak{h}^\vee$ the Lie algebra of $H^\vee$.
Let $W_F$ be the Weil group of $F$.
An \emph{$L$-parameter} for $H(F)$ is a homomorphism
\[
\phi \colon W_F \times \SL_2(\mathbb{C}) \rightarrow H^\vee
\]
such that $\phi(W_F)$ consists of semisimple elements, $\phi|_{W_F}$ is smooth and $\phi|_{\SL_2(\mathbb{C})}$ is algebraic.
It is called \emph{tempered} if $\phi(W_F)$ is bounded.
Let $\Phi(H(F))$ be the set of $H^\vee$-conjugacy classes of $L$-parameters for $H(F)$,
and let $\Phi_\mathrm{temp}(H(F))$ be its subset consisting of tempered $L$-parameters.
For $\phi \in \Phi(H(F))$, by the derivative of $\phi|_{\SL_2(\mathbb{C})}$,
we have a linear map
\[
d\phi|_{\SL_2(\mathbb{C})} \colon \sl_2(\mathbb{C}) \rightarrow \mathfrak{h}^\vee.
\]
We denote by $N_{\phi} \in \mathfrak{h}^\vee$ the image of
$\begin{pmatrix}
0 & 1 \\ 0 & 0
\end{pmatrix}$ under this map.
The $\mathrm{Ad}(H^\vee)$-orbit of $N_\phi$ is denoted by $\mathrm{Ad}(H^\vee) N_\phi$.
\par

Let $\mathcal{N}(\mathfrak{h}^\vee)$ be the set of nilpotent $\mathrm{Ad}(H^\vee)$-orbits in $\mathfrak{h}^\vee$.
We denote by
\[
d_{H} \colon \mathcal{N}(\mathfrak{h}(\overline{F})) \rightarrow \mathcal{N}(\mathfrak{h}^\vee), \quad
d_{H^\vee} \colon \mathcal{N}(\mathfrak{h}^\vee) \rightarrow \mathcal{N}(\mathfrak{h}(\overline{F}))
\]
the Spaltenstein duality maps (\cite[Section 10]{Sp}).
The images of these maps are the sets of \emph{special nilpotent conjugacy classes}.
Moreover, $\oo_H^\mathrm{st} \leq d_{H^\vee}(d_H(\oo_H^\mathrm{st}))$,
and the equality holds if and only if $\oo_H^\mathrm{st}$ is special.
See \cite[Appendix A]{BV} for more details.
An $L$-parameter $\phi \in \Phi(H(F))$ gives
a nilpotent orbit
\[
d_{H^\vee}(\mathrm{Ad}(H^\vee) N_\phi) \in \mathcal{N}(\mathfrak{h}(\overline{F})).
\]

\begin{conj}\label{upper}
Let $\phi \in \Phi(H(F))$.
Then for any $\pi \in \Pi_\phi$, we have
\[
\bar\mathcal{N}(\hat\pi)^{\max} \leq d_{H^\vee}(\mathrm{Ad}(H^\vee)N_\phi).
\]
Moreover, if $\phi$ is tempered,
then the equality holds for some $\pi \in \Pi_\phi$.
\end{conj}

\begin{rem}
An analogue of Conjecture \ref{upper} for covering groups
has been formulated by Gao--Liu--Lo--Shahidi \cite[Conjecture 1.2]{GLLS}, and proved for the Kazhdan--Patterson cover of $\GL_m$ in \cite[Theorem 1.3]{GLLS}.
\end{rem}

In this paper, we give a result toward Conjecture \ref{upper} for split classical groups under some conditions.
We focus on the case where $\phi$ is tempered
since Conjecture \ref{upper} is reduced to this case by \cite[Proposition 2.2]{CK} or \cite[Theorem 1.2]{HLLS}.
For $\phi \in \Phi_\mathrm{temp}(H(F))$, we consider the following condition:
\begin{description}
\item[($\star$)]
for any irreducible self-dual subrepresentation $\phi_0 \subset \phi$,
its determinant $\det(\phi_0)$ is trivial on the inertia subgroup $I_F$ of $W_F$.
\end{description}
Our main result is stated as follows.

\begin{hyp}\label{hypo}
Let $\oo_H^\st \in \NN(\h(\overline{F}))$ be a special nilpotent orbit.
Define $\oo_G^\st \in \NN(\g^\theta(\overline{F}))$ by
\[
\oo_G^\st = d_{G^\vee}(\Ad(G^\vee)d_{H}(\oo_H^\st)).
\]
Then for any constants $\{c_{\oo_H} \;|\; \oo_H \subset \oo_H^\st\}$ such that
$\sum_{\oo_H \subset \oo_H^\st} c_{\oo_H} \hat\mu_{\oo_H}$ is stable,
there exist constants $\{a_{\oo_G} \;|\; \oo_G \subset \oo_G^\st\}$ such that
\[
\sum_{\oo_G \subset \oo_G^\st} a_{\oo_G} \hat\mu_{\oo_G}
\quad\text{is equal to the transfer of}\quad
\sum_{\oo_H \subset \oo_H^\st} c_{\oo_H} \hat\mu_{\oo_H}
\]
on a small enough neighborhood of $\theta \in G(F) \rtimes \theta$.
\end{hyp}

\begin{conj}\label{upper}
Let $\phi \in \Phi(H(F))$.
Then for any $\pi \in \Pi_\phi$, we have
\[
\bar\NN(\hat\pi)^{\max} \leq d_{H^\vee}(\Ad(H^\vee)N_\phi).
\]
Moreover, if $\phi$ is tempered,
then the equality holds for some $\pi \in \Pi_\phi$.
\end{conj}

\begin{thm}\label{twist}
Let $\phi \colon W_F \times \SL_2(\C) \rightarrow G^\vee$
be a self-dual tempered $L$-parameter for $G(F) = \GL_m(F)$,
and let $\pi_\psi$ be the Zelevinsky dual of the irreducible tempered representation
$\pi_\phi$ corresponding to $\phi$.
Then
\[
\bar\NN_\theta(\tl\pi_\psi)^{\max} = \{d_{G^\vee}(\Ad(G^\vee) N_\phi)\}.
\]
Moreover, $c_{\oo_G}(\tl\pi_\psi) = c_{\oo_G'}(\tl\pi_\psi)$
for any rational orbits $\oo_G$ and $\oo_G'$ contained in $d_{G^\vee}(\Ad(G^\vee) N_\phi)$.
\end{thm}
\begin{proof}
Recall that $\bar\NN(\pi_\psi)^{\max} = \{d_{G^\vee}(\Ad(G^\vee) N_\phi)\}$
by M{\oe}glin--Waldspurger \cite{MW} as explained in \cite[Theorem 1.4]{CK}.
Then Varma showed in \cite[Lemma 5.29]{V} that
for any $\oo_G^\st \in \bar\NN_\theta(\tl\pi_\psi)^{\max}$,
the corresponding degenerate Whittaker model is nonzero.
Hence we have
\[
\oo_G^\st \leq d_{G^\vee}(\Ad(G^\vee) N_\phi).
\]
On the other hand, by Konno \cite[Theorem 4.1]{K}
together with the multiplicity one result of degenerate Whittaker models by M{\oe}glin--Waldspurger \cite{MW}
(cf., see \cite[Proposition 4.4]{K}),
we have
\[
d_{G^\vee}(\Ad(G^\vee) N_\phi) \in \bar\NN_\theta(\tl\pi_\psi)^{\max}.
\]
Hence we obtain the first assertion.
The last assertion follows from \cite[Theorem 4.1(2), Remark 4.2]{K}.
\end{proof}

As a consequence of Theorem \ref{transfer}, we obtain the following.
\begin{cor}\label{cor1}
We have
\[
d_{H^\vee}(\Ad(H^\vee) N_\phi) \in \bar\NN(\psi).
\]
Moreover, for $\oo_H^\st \in \bar\NN(\psi)$,
if $\oo_H^\st \not= d_{H^\vee}(\Ad(H^\vee) N_\phi)$, then
\[
\dim(\oo_H^\st) < \dim(d_{H^\vee}(\Ad(H^\vee) N_\phi)).
\]
\end{cor}
\begin{proof}
Set $d = \dim(d_{H^\vee}(\Ad(H^\vee) N_\phi))$.
As in the proof of Theorem \ref{transfer},
consider the unique $L$-parameter $\phi_0 \in \Phi_\temp(H(F))$
such that $\phi_0$ is trivial on $W_F$ and
\[
\Ad(H^\vee) N_{\phi_0} = \Ad(H^\vee) N_\phi.
\]
Set $\psi_0 = \widehat\phi_0$.
For a constant $e \in \C^\times$
the transfer of $S\Theta_\psi - eS\Theta_{\psi_0}$ is
\[
\Theta_{\tl\pi_\psi}-e\Theta_{\tl\pi_{\psi_0}}
= \sum_{\oo_G \in \NN(\g^\theta(F))} \left( c_{\oo_G}(\tl\pi_\psi)-e c_{\oo_G}(\tl\pi_{\psi_0}) \right) \hat\mu_{\oo_G}.
\]
Since $d_{G^\vee}(\Ad(G^\vee) N_\phi) = d_{G^\vee}(\Ad(G^\vee) N_{\phi_0})$,
by Theorem \ref{twist}, we can choose $e \in \C^\times$ such that
for $\oo_G^\st \in \NN(\g^\theta(\overline{F}))$,
if $c_{\oo_G}(\tl\pi_\psi)-e c_{\oo_G}(\tl\pi_{\psi_0}) \not= 0$ for some $\oo_G \subset \oo_G^\st$,
then $\oo_G^\st < d_{G^\vee}(\Ad(G^\vee) N_\phi)$.
This together with Theorem \ref{transfer} implies that
\[
\sum_{\substack{\oo_H^\st \in \NN(\h(\overline{F})) \\ \dim(\oo_H^\st) \geq d}}
\left( \sum_{\oo_H \subset \oo_H^\st} c_{\oo_H}(\psi) \hat\mu_{\oo_H}\right)
=
e\sum_{\substack{\oo_H^\st \in \NN(\h(\overline{F})) \\ \dim(\oo_H^\st) \geq d}}
\left( \sum_{\oo_H \subset \oo_H^\st} c_{\oo_H}(\psi_0) \hat\mu_{\oo_H}\right)
\]
By \cite[Theorem 1.4.1]{CMBO2},
in the right-hand side, only $\oo_H^\st = d_{H^\vee}(\Ad(H^\vee) N_\phi)$ can contribute.
By the linear independence of $\hat\mu_{\oo_H}$ (\cite[Theorem 5.11]{HC}),
the same holds for the left-hand side.
\end{proof}

\begin{prop}\label{refine}
Assume Hypothesis \ref{hypo}.
Then for any $\oo_H^\st \in \bar\NN(\psi)$,
we have
\[
\oo_H^\st \leq d_{H^\vee}(\Ad(H^\vee) N_\phi).
\]
\end{prop}
\begin{proof}
For each special orbit $\oo_H^\st \in \NN(\h(\overline{F}))$,
let $\{a_{\oo_G}(\psi) \;|\; \oo_G \subset d_{G^\vee}(\Ad(G^\vee)d_{H}(\oo_H^\st))\}$
be the constants in Hypothesis \ref{hypo}
given by $\{c_{\oo_H}(\psi) \;|\; \oo_H \subset \oo_H^\st\}$.
Then by considering the transfer of $S\Theta_\psi$,
we see that the local character expansion of $\Theta_{\tl\pi_\psi}$ is given by
\[
\Theta_{\tl\pi_\psi} = \sum_{\oo_H^\st}
\sum_{\oo_G \subset \oo_G^\st} a_{\oo_G}(\psi) \hat\mu_{\oo_G},
\]
where $\oo_H^\st$ runs over special nilpotent orbits in $\NN(\h(\overline{F}))$
and $\oo_G$ runs over rational orbits contained in
$\oo_G^\st = d_{G^\vee}(\Ad(G^\vee)d_{H}(\oo_H^\st))$.
In particular, if $\oo_H^\st \in \bar\NN(\psi)$,
then $d_{G^\vee}(\Ad(G^\vee)d_{H}(\oo_H^\st)) \in \bar\NN_\theta(\tl\pi_\psi)$.
By Theorem \ref{twist}, we have
\[
d_{G^\vee}(\Ad(G^\vee)d_{H}(\oo_H^\st)) \leq d_{G^\vee}(\Ad(G^\vee) N_\phi).
\]
Hence $\Ad(G^\vee)d_{H}(\oo_H^\st) \geq \Ad(G^\vee) N_\phi$.
Note that
\[
\Ad(G^\vee)d_{H}(\oo_H^\st) \cap H^\vee = d_{H}(\oo_H^\st),
\quad
\Ad(G^\vee) N_\phi \cap H^\vee = \Ad(H^\vee) N_\phi.
\]
(Here, if $H=\SO_{2n}$, we consider $\O_{2n}(\C)$-orbits actually. See Remark \ref{OvsSO}.)
Hence $d_{H}(\oo_H^\st) \geq \Ad(H^\vee) N_\phi$,
and we conclude that
$\oo_H^\st \leq d_{H^\vee}(d_{H}(\oo_H^\st)) \leq d_{H^\vee}(\Ad(H^\vee) N_\phi)$.
\end{proof}

Now we assume that we have
\begin{itemize}
\item
$\oo_H^\st \in \NN(\h(\overline{F}))$
with $\dim(\oo_H^\st) \geq \dim(d_{H^\vee}(\Ad(H^\vee)N_\phi))$;
\item
$\oo_H \in \NN(\h(F))$ with $\oo_H \subset \oo_H^\st$;
\item
$\pi_0 \in \Pi_\psi$ with $\psi = \widehat\phi$
\end{itemize}
such that $c_{\oo_H}(\pi_0) \not= 0$.
The goal is to show that $\oo_H^\st = d_{H^\vee}(\Ad(H^\vee)N_\phi)$.
\par

By the induction hypothesis, we have
$\dim(\oo_{H_i}^\st) \leq \dim (d_{H_i^\vee}(\Ad(H_i^\vee)N_{\phi_i}))$ for $i \in \{1,2\}$.
Hence by Proposition \ref{WS} and Lemma \ref{dimW},
we have
\begin{align*}
\dim(\oo_H^\st) &= \dim\left(W(\oo_{H_1}^\st, \oo_{H_2}^\st)\right)
\\&\leq \dim\left(
W(d_{H_1^\vee}(\Ad(H_1^\vee)N_{\phi_1}), d_{H_2^\vee}(\Ad(H_2^\vee)N_{\phi_2}))
\right)
\\&\leq
\dim\left(
d_{H^\vee}(\Ad(H^\vee)\xi(
d_{H_1}(d_{H_1^\vee}(\Ad(H_1^\vee)N_{\phi_1})), d_{H_2}(d_{H_2^\vee}(\Ad(H_2^\vee)N_{\phi_2}))
))
\right)
\\&\leq
\dim\left(
d_{H^\vee}(\Ad(H^\vee)\xi(\Ad(H_1^\vee)N_{\phi_1}, \Ad(H_2^\vee)N_{\phi_2}))
\right)
\\&=
\dim\left(d_{H^\vee}(\Ad(H^\vee)N_{\phi})\right).
\end{align*}
Here, we used the facts
that $d_{H_i}(d_{H_i^\vee}(\Ad(H_i^\vee)N_{\phi_i})) \geq \Ad(H_i^\vee)N_{\phi_i}$
and that $d_{H^\vee}$ is an order-reversing map.
Since we assume that $\dim(\oo_H^\st) \geq \dim\left(d_{H^\vee}(\Ad(H^\vee)N_{\phi})\right)$,
this must be an equality.
Moreover, it together with Lemma \ref{dimW} implies that
$\oo_{H_i}^\st = d_{H_i^\vee}(\Ad(H_i^\vee)N_{\phi_i})$ for $i \in \{1,2\}$.
Then by proposition \ref{WS}, we have
\begin{align*}
\oo_{H^\st}
&\leq d_{H^\vee}(\Ad(H^\vee)\xi(d_{H_1}(\oo_{H_1}^\st), d_{H_2}(\oo_{H_2}^\st)))
\\&\leq d_{H^\vee}(\Ad(H^\vee)\xi(\Ad(H_1^\vee)N_{\phi_1}, \Ad(H_2^\vee)N_{\phi_2}))
\\&= d_{H^\vee}(\Ad(H^\vee)N_{\phi}).
\end{align*}
Since $\dim(\oo_H^\st) = \dim\left(d_{H^\vee}(\Ad(H^\vee)N_{\phi})\right)$,
we conclude that $\oo_H^\st = d_{H^\vee}(\Ad(H^\vee)N_\phi)$,
as desired.
\par

The proof of Theorem \ref{main} (3) is similar.
Suppose that $\oo_H^\st \in \NN(\h(\overline{F}))$ contains a rational orbit $\oo_H$
such that $c_{\oo_H}(\pi_0) \not= 0$ for some $\pi_0 \in \Pi_\psi$.
Consider $s \in \Sc_\psi$ satisfying ($\ast$).
If $s = 1$, then Proposition \ref{refine} implies that $\oo_H^\st \leq d_{H^\vee}(\Ad(H^\vee)N_\phi)$.
Otherwise, we can write $\oo_H^\st = W(\oo_{H_1}^\st, \oo_{H_2}^\st)$ as above.
Then $\oo_{H_i}^\st \leq d_{H_i^\vee}(\Ad(H_i^\vee)N_{\phi_i})$ by the induction hypothesis.
By Proposition \ref{WS} and Lemma \ref{Worder} below, we have
\begin{align*}
\oo_H^\st &= W(\oo_{H_1}^\st, \oo_{H_2}^\st)
\\&\leq W(d_{H_1^\vee}(\Ad(H_1^\vee)N_{\phi_1}), d_{H_2^\vee}(\Ad(H_2^\vee)N_{\phi_2}))
\\&\leq
d_{H^\vee}(\Ad(H^\vee)\xi(
d_{H_1}(d_{H_1^\vee}(\Ad(H_1^\vee)N_{\phi_1})), d_{H_2}(d_{H_2^\vee}(\Ad(H_2^\vee)N_{\phi_2}))
))
\\&\leq
d_{H^\vee}(\Ad(H^\vee)\xi(\Ad(H_1^\vee)N_{\phi_1}, \Ad(H_2^\vee)N_{\phi_2}))
\\&=
d_{H^\vee}(\Ad(H^\vee)N_{\phi}).
\end{align*}
This completes the proof of Theorem \ref{main}.
\end{proof}

\subsection{Outline of the proof}\label{sec.outline}
Throughout the paper, we assume that $p \gg 0$ as in Theorem \ref{main}.
The proof of Theorem \ref{main} is similar to the proofs of the Generic Packet Conjecture
by Konno \cite{K} and Varma \cite{V}.
The main idea is to compare local character expansions via
\emph{endoscopic character identities}.
\par

Fix $\phi \in \Phi_\mathrm{temp}(H(F))$ and set $\psi = \widehat\phi$.
Set $G = \GL_m$, where $m$ is the dimension of the standard representation $H^\vee \rightarrow G^\vee = \GL_m(\mathbb{C})$.
By composing with this map, we can regard $\psi$ as an $A$-parameter for $G(F)$,
and obtain an irreducible self-dual representation $\pi_\psi$ of $G(F)$.
Define an involution $\theta$ on $G(F)$ by
\[
\theta(g) = J{}^tg^{-1}J^{-1},
\quad
J = \begin{pmatrix}
&&&1 \\
&&-1&\\
&\iddots&& \\
(-1)^{m-1}&&
\end{pmatrix} \in G(F).
\]
We can extend $\pi_\psi$ to an irreducible representation $\tl\pi_\psi$ of $G \rtimes \langle \theta \rangle$.
The twisted character of $\tl\pi_\psi$ is denoted by $\Theta_{\tl\pi_\psi}$.
\par

Let $\Sc_\psi$ be the component group for $\psi$,
and let $\widehat\Sc_\psi$ be its Pontryagin dual.
Recall that Arthur defined a map
\[
\Pi_\psi \rightarrow \widehat\Sc_\psi, \;
\pi \mapsto \langle \cdot, \pi \rangle_\psi.
\]
Notice that this is bijective since $\psi = \widehat\phi$.
Set $s_\psi = \psi(\1_{W_F}, \mathbf{1}, -\mathbf{1}) = \phi(\1_{W_F}, -\mathbf{1})$.
In this setting, \cite[Theorem 2.2.1]{Ar} asserts that
\begin{description}
\item[(T1)]
the distribution
\[
S\Theta_{\psi} = \sum_{\pi \in \Pi_\psi} \langle s_\psi, \pi \rangle_\psi \Theta_\pi
\]
is stable, and $\Theta_{\tl\pi_\psi}$ is the transfer of $S\Theta_\psi$;
\item[(T2)]
Each $s \in \Sc_\psi$ with $s \not= 1$ gives an elliptic endoscopic group $H_1 \times H_2$ of $H$
and an $A$-parameter $\psi_i$ for $H_i$
(of the form $\psi_i = \widehat\phi_i$ for some $\phi_i \in \Phi_\mathrm{temp}(H_i)$) such that
\[
S\Theta_{\psi, s} = \sum_{\pi \in \Pi_\psi} \langle s \cdot s_\psi, \pi \rangle_\psi \Theta_\pi
\]
is the transfer of $S\Theta_{\psi_1} \otimes S\Theta_{\psi_2}$.
\end{description}
\par

The first step of the proof of Theorem \ref{main} is to establish an analogue for the twisted $\GL_m$ case.
By the twisted local character expansion established by Clozel \cite[Theorem 3]{C},
we can expand
\[
\Theta_{\tl\pi_\psi}
= \sum_{\oo_G \in \mathcal{N}(\mathfrak{g}^\theta(F))} c_{\oo_G}(\tl\pi_\psi) \hat\mu_{\oo_G}.
\]
See Section \ref{sec.TLCE} below for the notations.
We define $\NN_\theta(\tl\pi_\psi) \subset \mathcal{N}(\mathfrak{g}^\theta(F))$
and $\bar\NN_\theta(\tl\pi_\psi) \subset \mathcal{N}(\mathfrak{g}^\theta(\overline{F}))$
similar to $\mathcal{N}(\pi)$ and $\bar\mathcal{N}(\pi)$, respectively.
In Theorem \ref{twist} below, we will show that
\[
\bar\NN_\theta(\tl\pi_\psi)^{\max} = \{d_{G^\vee}(\mathrm{Ad}(G^\vee) N_\phi)\}.
\]
This is an immediate consequence of
results of Konno \cite{K}, Varma \cite{V}, and M{\oe}glin--Waldspurger \cite{MW}.
Moreover, Konno's result \cite[Theorem 4.1(2)]{K} says that
the constant $c_{\oo_G}(\tl\pi_\psi)$ for $\oo_G \subset d_{G^\vee}(\mathrm{Ad}(G^\vee) N_\phi)$
does not depend on the choice of the rational orbit.
\par

Set $\oo_H^\mathrm{st} = d_{H^\vee}(\mathrm{Ad}(H^\vee) N_\phi)$ and $\oo_G^\mathrm{st} = d_{G^\vee}(\mathrm{Ad}(G^\vee) N_\phi)$.
The second step of the proof of Theorem \ref{main} is to show that
there exists a constant $\gamma_{\oo_H}$ for each rational orbit $\oo_H \subset \oo_H^\mathrm{st}$ such that
\[
\sum_{\oo_G \subset \oo_G^\mathrm{st}} \hat\mu_{\oo_G}
\quad
\text{is equal to the transfer of}
\quad
\sum_{\oo_H \subset \oo_H^\mathrm{st}} \gamma_{\oo_H}\hat\mu_{\oo_H}
\]
on a small enough neighborhood of $\theta \in G(F) \rtimes \theta$.
See Theorem \ref{transfer} below.
Hypothesis \ref{hypo} is an analogue of this result.
These claims could be proven by similar arguments to the standard case explained in the next paragraph,
but there is no explicit reference in the literature.
Thus we will prove Theorem \ref{transfer} in this paper.
The proof uses Conjecture \ref{upper} in the case established in \cite{CMBO2},
which requires that $H$ is split over $F$.
\par

An analogous result for the standard endoscopy with unramified endoscopic groups
is known by Waldspurger \cite{W_pave}.
Let $H_1 \times H_2$ be an unramified endoscopic group of $H$.
For a special nilpotent orbit $\oo_{H_i}^\mathrm{st} \in \mathcal{N}(\h_i(\overline{F}))$,
Waldspurger explicitly defined a nilpotent orbit
\[
\oo_H^\mathrm{st} = W(\oo_{H_1}^\mathrm{st}, \oo_{H_2}^\mathrm{st}) \in \mathcal{N}(\mathfrak{h}(\overline{F})),
\]
and proved a certain transfer result in \cite[XII.9 Th\'eor\`eme]{W_pave}.
The third step of the proof of Theorem \ref{main} is to relate the Waldspurger map $W(\oo_{H_1}^\mathrm{st}, \oo_{H_2}^\mathrm{st})$
and the Spaltenstein duality (Proposition \ref{WS}).
\par

Combining the two transfer results, we can relate the local character expansion of $S\Theta_\psi$ (resp.~ $S\Theta_{\psi,s}$)
with the one of $\Theta_{\tl\pi_\psi}$ (resp.~ $S\Theta_{\psi_1} \otimes S\Theta_{\psi_2}$).
Then by induction on $m$,
we can deduce Theorem \ref{main} in Section \ref{sec.proof}.
To use \cite[XII.9 Th\'eor\`eme]{W_pave}, we need the condition ($\star$) in Cases (C) and (D),
which is equivalent to saying that all endoscopic groups $H_1 \times H_2$ arising from $s \in \Sc_\psi$ are unramified.

\begin{rem}\label{OvsSO}
The local Langlands correspondence for $H = \SO_{2n}$ as proven by Arthur,
is given up to $\O_{2n}(F)$-conjugation.
Thus, in turn, we should replace $\mathrm{Ad}(H^\vee)N_\phi$ by $\mathrm{Ad}(\O_{2n}(\mathbb{C}))N_\phi$ in our results.
We will ignore this discrepancy since it is not an important difference for the geometric wavefront set results.
By replacing $\SO_{2n}$ with $\O_{2n}$, we can use the following properties:
\begin{enumerate}
\item
the distributions on $\SO_{2n}(F)$ considered in the paper are all $\O_{2n}(F)$-invariant;
\item
the transfer of stable distributions on $H(F)$ to $G(F) \rtimes \theta$ is injective;
\item
$\mathrm{Ad}(G^\vee)\oo_{H^\vee} \cap H^\vee = \oo_{H^\vee}$ for $\oo_{H^\vee} \in \mathcal{N}(\mathfrak{h}^\vee)$.
\end{enumerate}
Note that (2) and (3) hold when $H = \SO_{2n+1}$ or $H = \Sp_{2n}$.
\end{rem}

The proof of Theorem~\ref{main} follows Konno~\cite{K} and Varma~\cite{V}: the "main idea is to compare local character expansions via \emph{endoscopic character identities}." Fix $\phi\in\Phi_\mathrm{temp}(H(F))$, set $\psi=\widehat\phi$, and embed into $G=\GL_m$ via the standard map $H^\vee\to G^\vee$. This yields a self-dual $\pi_\psi$ of $G(F)$, extended to $\tilde\pi_\psi$ on $G\rtimes\langle\theta\rangle$, whose twisted character $\Theta_{\tilde\pi_\psi}$ matches Arthur’s stable distribution
\[
S\Theta_{\psi}=\sum_{\pi\in\Pi_\psi}\langle s_\psi,\pi\rangle_\psi\,\Theta_\pi
\]
by endoscopic character identities (Arthur’s (T1)), and similarly for $S\Theta_{\psi,s}$ (Arthur’s (T2)).

(1) "The first step" is an analogue in the twisted $\GL_m$ case: using Clozel’s twisted local character expansion
\[
\Theta_{\tilde\pi_\psi}=\sum_{\oo_G\in\mathcal N(\mathfrak g^\theta(F))} c_{\oo_G}(\tilde\pi_\psi)\,\hat\mu_{\oo_G},
\]
one defines $\bar\mathcal N_\theta(\tilde\pi_\psi)^{\max}$ and proves (Theorem~\ref{twist})
\[
\bar\mathcal N_\theta(\tilde\pi_\psi)^{\max}=\{d_{G^\vee}(\mathrm{Ad}(G^\vee)N_\phi)\},
\]
using results of Konno, Varma, and M\oe glin--Waldspurger; additionally, Konno’s result implies the constants $c_{\oo_G}(\tilde\pi_\psi)$ for $\oo_G\subset d_{G^\vee}(\mathrm{Ad}(G^\vee)N_\phi)$ do not depend on the rational orbit.

(2) Setting $\oo_H^{\mathrm{st}}=d_{H^\vee}(\mathrm{Ad}(H^\vee)N_\phi)$ and $\oo_G^{\mathrm{st}}=d_{G^\vee}(\mathrm{Ad}(G^\vee)N_\phi)$, "the second step" is a transfer statement (Theorem~\ref{transfer}): for each rational orbit $\oo_H\subset\oo_H^{\mathrm{st}}$ there is a constant $\gamma_{\oo_H}$ such that on a small neighborhood of $\theta$,
\[
\sum_{\oo_G\subset\oo_G^{\mathrm{st}}}\hat\mu_{\oo_G}\quad\text{equals the transfer of}\quad\sum_{\oo_H\subset\oo_H^{\mathrm{st}}}\gamma_{\oo_H}\hat\mu_{\oo_H}.
\]
Hypothesis~\ref{hypo} is described as an analogue of this transfer result.

(3) "The third step" relates Waldspurger’s explicit map for standard endoscopy,
\[
\oo_H^{\mathrm{st}}=W(\oo_{H_1}^{\mathrm{st}},\oo_{H_2}^{\mathrm{st}}),
\]
to Spaltenstein duality (Proposition~\ref{WS}).

Finally, "combining the two transfer results," one relates the local character expansions of $S\Theta_\psi$ and $S\Theta_{\psi,s}$ to those of $\Theta_{\tilde\pi_\psi}$ and $S\Theta_{\psi_1}\otimes S\Theta_{\psi_2}$, and then "by induction on $m$" deduces Theorem~\ref{main} (in Section~\ref{sec.proof}). The condition $(\star)$ in Cases (C) and (D) is needed to apply Waldspurger’s theorem, equivalently ensuring the relevant endoscopic groups $H_1\times H_2$ are unramified.

The proof of Theorem~\ref{main} follows Konno~\cite{K} and Varma~\cite{V}: the "main idea is to compare local character expansions via \emph{endoscopic character identities}." Fix $\phi\in\Phi_\mathrm{temp}(H(F))$, set $\psi=\widehat\phi$, and embed into $G=\GL_m$ via the standard map $H^\vee\to G^\vee$. This yields a self-dual $\pi_\psi$ of $G(F)$, extended to $\tilde\pi_\psi$ on $G\rtimes\langle\theta\rangle$, whose twisted character $\Theta_{\tilde\pi_\psi}$ matches Arthur’s stable distribution
\[
S\Theta_{\psi}=\sum_{\pi\in\Pi_\psi}\langle s_\psi,\pi\rangle_\psi\,\Theta_\pi
\]
by endoscopic character identities (Arthur’s (T1)), and similarly for $S\Theta_{\psi,s}$ (Arthur’s (T2)).

(1) "The first step" is an analogue in the twisted $\GL_m$ case: using Clozel’s twisted local character expansion
\[
\Theta_{\tilde\pi_\psi}=\sum_{\oo_G\in\mathcal N(\mathfrak g^\theta(F))} c_{\oo_G}(\tilde\pi_\psi)\,\hat\mu_{\oo_G},
\]
one defines $\bar\mathcal N_\theta(\tilde\pi_\psi)^{\max}$ and proves the following theorem.
\begin{thm}\label{twist}
Let $\phi \colon W_F \times \SL_2(\C) \rightarrow G^\vee$
be a self-dual tempered $L$-parameter for $G(F) = \GL_m(F)$,
and let $\pi_\psi$ be the Zelevinsky dual of the irreducible tempered representation
$\pi_\phi$ corresponding to $\phi$.
Then
\[
\bar\NN_\theta(\tl\pi_\psi)^{\max} = \{d_{G^\vee}(\Ad(G^\vee) N_\phi)\}.
\]
Moreover, $c_{\oo_G}(\tl\pi_\psi) = c_{\oo_G'}(\tl\pi_\psi)$
for any rational orbits $\oo_G$ and $\oo_G'$ contained in $d_{G^\vee}(\Ad(G^\vee) N_\phi)$.
\end{thm}
This uses results of Konno, Varma, and M\oe glin--Waldspurger.

(2) Setting $\oo_H^{\mathrm{st}}=d_{H^\vee}(\mathrm{Ad}(H^\vee)N_\phi)$ and $\oo_G^{\mathrm{st}}=d_{G^\vee}(\mathrm{Ad}(G^\vee)N_\phi)$, "the second step" is the following transfer statement.
\begin{thm}\label{transfer}
Let $\oo_H^\st \in \NN(\h(\overline{F}))$ be a special nilpotent orbit.
Define $\oo_G^\st \in \NN(\g^\theta(\overline{F}))$ by
\[
\oo_G^\st = d_{G^\vee}(\Ad(G^\vee)d_{H}(\oo_H^\st)).
\]
Then for each rational orbit $\oo_H \subset \oo_H^\st$,
there exists a constant $\gamma_{\oo_H} \in \C$ such that
\[
\sum_{\oo_G \subset \oo_G^\st} \hat\mu_{\oo_G}
\quad\text{is equal to the transfer of }\quad
\sum_{\oo_H \subset \oo_H^\st} \gamma_{\oo_H} \hat\mu_{\oo_H}
\]
on a small enough neighborhood of $\theta \in G(F) \rtimes \theta$.
\end{thm}
The following hypothesis is described as an analogue of this transfer result.
\begin{hyp}\label{hypo}
Let $\oo_H^\st \in \NN(\h(\overline{F}))$ be a special nilpotent orbit.
Define $\oo_G^\st \in \NN(\g^\theta(\overline{F}))$ by
\[
\oo_G^\st = d_{G^\vee}(\Ad(G^\vee)d_{H}(\oo_H^\st)).
\]
Then for any constants $\{c_{\oo_H} \;|\; \oo_H \subset \oo_H^\st\}$ such that
$\sum_{\oo_H \subset \oo_H^\st} c_{\oo_H} \hat\mu_{\oo_H}$ is stable,
there exist constants $\{a_{\oo_G} \;|\; \oo_G \subset \oo_G^\st\}$ such that
\[
\sum_{\oo_G \subset \oo_G^\st} a_{\oo_G} \hat\mu_{\oo_G}
\quad\text{is equal to the transfer of}\quad
\sum_{\oo_H \subset \oo_H^\st} c_{\oo_H} \hat\mu_{\oo_H}
\]
on a small enough neighborhood of $\theta \in G(F) \rtimes \theta$.
\end{hyp}

(3) "The third step" relates Waldspurger’s explicit map for standard endoscopy,
\[
\oo_H^{\mathrm{st}}=W(\oo_{H_1}^{\mathrm{st}},\oo_{H_2}^{\mathrm{st}}),
\]
to Spaltenstein duality using the proposition below.
\begin{prop}\label{WS}
For $i \in \{1,2\}$, let $\oo_{H_i}^\st \in \NN(\h_i(\overline{F}))$ be a special nilpotent orbit.
Then
\[
W(\oo_{H_1}^\st, \oo_{H_2}^\st)
\leq d_{H^\vee}(\Ad(H^\vee)\xi(d_{H_1}(\oo_{H_1}^\st), d_{H_2}(\oo_{H_2}^\st))).
\]
\end{prop}

Finally, "combining the two transfer results," one relates the local character expansions of $S\Theta_\psi$ and $S\Theta_{\psi,s}$ to those of $\Theta_{\tilde\pi_\psi}$ and $S\Theta_{\psi_1}\otimes S\Theta_{\psi_2}$, and then "by induction on $m$" establishes the main theorem (later in the paper). The condition $(\star)$ in Cases (C) and (D) is needed to apply Waldspurger’s theorem, equivalently ensuring the relevant endoscopic groups $H_1\times H_2$ are unramified.

\label{main}
Let $H$ be a split classical group over $F$,
and let $\phi \in \Phi_\mathrm{temp}(H(F))$.
Suppose that one of the following holds:
\begin{description}
\item[(B)]
$H = \SO_{2n+1}$, $p > 6n+3$ and $\phi$ is arbitrary;
\item[(C)]
$H = \Sp_{2n}$, $p > 6n$ and $\phi$ satisfies $(\star)$;
\item[(D)]
$H = \SO_{2n}$, $p > 6n$ and $\phi$ satisfies $(\star)$.
\end{description}
Set $\psi = \widehat\phi$.

\begin{enumerate}
\item
There exists $\pi \in \Pi_\psi$ such that
\[
d_{H^\vee}(\mathrm{Ad}(H^\vee)N_\phi) \in \bar\mathcal{N}(\pi)^{\max}.
\]
\item
For $\pi \in \Pi_\psi$ and for $\oo_H^\mathrm{st} \in \bar\mathcal{N}(\pi)$,
if $\oo_H^\mathrm{st} \not= d_{H^\vee}(\mathrm{Ad}(H^\vee)N_\phi)$, then
\[
\dim(\oo_H^\mathrm{st}) < \dim(d_{H^\vee}(\mathrm{Ad}(H^\vee)N_\phi)).
\]
Here, $\dim(\oo_H^\mathrm{st})$ is the Zariski dimension of the orbit as an algebraic variety.
\item
If we further assume the following hypothesis, then the corresponding upper-bound conjecture holds for $\phi$, i.e.,
\[
\bar\mathcal{N}(\pi)^{\max} \leq d_{H^\vee}(\mathrm{Ad}(H^\vee)N_\phi)
\]
for any $\pi \in \Pi_\psi$.

\begin{hyp}\label{hypo}
Let $\oo_H^\st \in \NN(\h(\overline{F}))$ be a special nilpotent orbit.
Define $\oo_G^\st \in \NN(\g^\theta(\overline{F}))$ by
\[
\oo_G^\st = d_{G^\vee}(\Ad(G^\vee)d_{H}(\oo_H^\st)).
\]
Then for any constants $\{c_{\oo_H} \;|\; \oo_H \subset \oo_H^\st\}$ such that
$\sum_{\oo_H \subset \oo_H^\st} c_{\oo_H} \hat\mu_{\oo_H}$ is stable,
there exist constants $\{a_{\oo_G} \;|\; \oo_G \subset \oo_G^\st\}$ such that
\[
\sum_{\oo_G \subset \oo_G^\st} a_{\oo_G} \hat\mu_{\oo_G}
\quad\text{is equal to the transfer of}\quad
\sum_{\oo_H \subset \oo_H^\st} c_{\oo_H} \hat\mu_{\oo_H}
\]
on a small enough neighborhood of $\theta \in G(F) \rtimes \theta$.
\end{hyp}

\begin{conj}\label{upper}
Let $\phi \in \Phi(H(F))$.
Then for any $\pi \in \Pi_\phi$, we have
\[
\bar\NN(\hat\pi)^{\max} \leq d_{H^\vee}(\Ad(H^\vee)N_\phi).
\]
Moreover, if $\phi$ is tempered,
then the equality holds for some $\pi \in \Pi_\phi$.
\end{conj}
\end{enumerate}

Question:
Let $H$ be a split classical group over $F$ with Lie algebra $\mathfrak h$, and let
$\phi: W_F\times \mathrm{SL}_2(\mathbb C)\to H^\vee$ be a tempered $L$-parameter. Assume one of the following:
(B) $H=\mathrm{SO}_{2n+1}$, $p>6n+3$, and $\phi$ is arbitrary;
(C) $H=\mathrm{Sp}_{2n}$, $p>6n$, and for every irreducible self-dual subrepresentation $\phi_0\subset \phi$, one has $\det(\phi_0)|_{I_F}=1$;
(D) $H=\mathrm{SO}_{2n}$, $p>6n$, and for every irreducible self-dual subrepresentation $\phi_0\subset \phi$, one has $\det(\phi_0)|_{I_F}=1$.
Set $\psi=\widehat\phi$, and let $\Pi_\psi$ be the packet attached to $\psi$. Let $N_\phi\in \mathfrak h^\vee$ be the image of $\begin{pmatrix}0&1\\0&0\end{pmatrix}$ under $d\phi|_{\mathrm{SL}_2(\mathbb C)}$, and let $d_H:\mathcal N(\mathfrak h(\overline F))\to \mathcal N(\mathfrak h^\vee)$ and $d_{H^\vee}:\mathcal N(\mathfrak h^\vee)\to \mathcal N(\mathfrak h(\overline F))$ be the Spaltenstein duality maps. For an irreducible admissible representation $\pi$ of $H(F)$, write
$\Theta_\pi(\exp X)=\sum_{\mathcal O_H} c_{\mathcal O_H}(\pi)\,\widehat\mu_{\mathcal O_H}(\exp X)$
near $0$, define $\mathcal N(\pi)=\{\mathcal O_H: c_{\mathcal O_H}(\pi)\neq 0\}$, let $\bar{\mathcal N}(\pi)$ be the set of nilpotent $\mathrm{Ad}(H(\overline F))$-orbits containing some rational orbit in $\mathcal N(\pi)$, and let $\bar{\mathcal N}(\pi)^{\max}$ be the maximal elements of $\bar{\mathcal N}(\pi)$ for the closure order $\mathcal O\le \mathcal O'$ iff $\mathcal O\subset \overline{\mathcal O'}$. Under these assumptions, which statement is valid?

All of the following hold:
(1) There exists $\pi\in \Pi_\psi$ such that
\[
d_{H^\vee}(\mathrm{Ad}(H^\vee)N_\phi)\in \bar{\mathcal N}(\pi)^{\max}.
\]
(2) For every $\pi\in \Pi_\psi$ and every $\mathcal O_H^{\mathrm{st}}\in \bar{\mathcal N}(\pi)$, if
$\mathcal O_H^{\mathrm{st}}\neq d_{H^\vee}(\mathrm{Ad}(H^\vee)N_\phi)$, then
\[
\dim(\mathcal O_H^{\mathrm{st}})<\dim\bigl(d_{H^\vee}(\mathrm{Ad}(H^\vee)N_\phi)\bigr),
\]
where $\dim$ is the Zariski dimension of the orbit.
(3) If one further assumes the following transfer hypothesis: for every special nilpotent orbit $\mathcal O_H^{\mathrm{st}}\in \mathcal N(\mathfrak h(\overline F))$, after defining
\[
\mathcal O_G^{\mathrm{st}}=d_{G^\vee}(\mathrm{Ad}(G^\vee)d_H(\mathcal O_H^{\mathrm{st}})),
\]
any stable distribution of the form $\sum_{\mathcal O_H\subset \mathcal O_H^{\mathrm{st}}} c_{\mathcal O_H}\widehat\mu_{\mathcal O_H}$ is transferred, on a sufficiently small neighborhood of $\theta\in G(F)\rtimes \theta$, to some distribution $\sum_{\mathcal O_G\subset \mathcal O_G^{\mathrm{st}}} a_{\mathcal O_G}\widehat\mu_{\mathcal O_G}$, then for every $\pi\in \Pi_\psi$ one has
\[
\bar{\mathcal N}(\pi)^{\max}\le d_{H^\vee}(\mathrm{Ad}(H^\vee)N_\phi).
\]

Choices:
B.
All of the following hold:
(1) For every $\pi\in \Pi_\psi$ such that $d_{H^\vee}(\mathrm{Ad}(H^\vee)N_\phi)\in \bar{\mathcal N}(\pi)$, one has
\[
d_{H^\vee}(\mathrm{Ad}(H^\vee)N_\phi)\in \bar{\mathcal N}(\pi)^{\max}.
\]
(2) For every $\pi\in \Pi_\psi$ and every $\mathcal O_H^{\mathrm{st}}\in \bar{\mathcal N}(\pi)$, if
$\mathcal O_H^{\mathrm{st}}\neq d_{H^\vee}(\mathrm{Ad}(H^\vee)N_\phi)$, then
\[
\dim(\mathcal O_H^{\mathrm{st}})\le \dim\bigl(d_{H^\vee}(\mathrm{Ad}(H^\vee)N_\phi)\bigr),
\]
where $\dim$ is the Zariski dimension of the orbit.
(3) If one further assumes the transfer hypothesis stated in the theorem, then for every $\pi\in \Pi_\psi$ one has
\[
\bar{\mathcal N}(\pi)\le d_{H^\vee}(\mathrm{Ad}(H^\vee)N_\phi).
\]
C.
Both of the following hold:
(1) There exists $\pi\in \Pi_\psi$ such that
\[
d_{H^\vee}(\mathrm{Ad}(H^\vee)N_\phi)\in \bar{\mathcal N}(\pi)^{\max}.
\]
(2) For every $\pi\in \Pi_\psi$ and every $\mathcal O_H^{\mathrm{st}}\in \bar{\mathcal N}(\pi)$, if
$\mathcal O_H^{\mathrm{st}}\neq d_{H^\vee}(\mathrm{Ad}(H^\vee)N_\phi)$, then
\[
\dim(\mathcal O_H^{\mathrm{st}})<\dim\bigl(d_{H^\vee}(\mathrm{Ad}(H^\vee)N_\phi)\bigr).
\]
D.
All of the following hold:
(1) There exists $\pi\in \Pi_\psi$ such that
\[
d_{H^\vee}(\mathrm{Ad}(H^\vee)N_\phi)= \bar{\mathcal N}(\pi)^{\max}.
\]
(2) For every $\pi\in \Pi_\psi$ and every $\mathcal O_H^{\mathrm{st}}\in \bar{\mathcal N}(\pi)$, one has
\[
\mathcal O_H^{\mathrm{st}}\le d_{H^\vee}(\mathrm{Ad}(H^\vee)N_\phi).
\]
(3) In particular, without any additional transfer hypothesis, for every $\pi\in \Pi_\psi$ one has
\[
\bar{\mathcal N}(\pi)^{\max}\le d_{H^\vee}(\mathrm{Ad}(H^\vee)N_\phi).
\]
E.
All of the following hold:
(1) There exists $\pi\in \Pi_\psi$ such that
\[
d_{H^\vee}(\mathrm{Ad}(H^\vee)N_\phi)\in \bar{\mathcal N}(\pi)^{\max}.
\]
(2) For every $\pi\in \Pi_\psi$ and every $\mathcal O_H^{\mathrm{st}}\in \bar{\mathcal N}(\pi)$, if
$\mathcal O_H^{\mathrm{st}}\neq d_{H^\vee}(\mathrm{Ad}(H^\vee)N_\phi)$, then
\[
\dim(\mathcal O_H^{\mathrm{st}})<\dim\bigl(d_{H^\vee}(\mathrm{Ad}(H^\vee)N_\phi)\bigr).
\]
(3) If one further assumes the following transfer hypothesis: for every special nilpotent orbit $\mathcal O_H^{\mathrm{st}}\in \mathcal N(\mathfrak h(\overline F))$, after defining
\[
\mathcal O_G^{\mathrm{st}}=d_{G^\vee}(\mathrm{Ad}(G^\vee)d_H(\mathcal O_H^{\mathrm{st}})),
\]
there exist constants $a_{\mathcal O_G}$ such that the specific distribution
\[
\sum_{\mathcal O_G\subset \mathcal O_G^{\mathrm{st}}}\widehat\mu_{\mathcal O_G}
\]
is equal, on a sufficiently small neighborhood of $\theta\in G(F)\rtimes \theta$, to the transfer of every stable distribution of the form $\sum_{\mathcal O_H\subset \mathcal O_H^{\mathrm{st}}} c_{\mathcal O_H}\widehat\mu_{\mathcal O_H}$; then for every $\pi\in \Pi_\psi$ one has
\[
\bar{\mathcal N}(\pi)^{\max}\le d_{H^\vee}(\mathrm{Ad}(H^\vee)N_\phi).
\]

maximal-elements conclusion strengthened to all orbits and strict inequality weakened to non-strict

dropped the additional upper-bound conclusion under the extra transfer hypothesis

removed dependence on the extra transfer hypothesis and replaced existence/membership by equality of maximal set

quantification in the transfer hypothesis replaced by a single fixed transferred distribution independent of the stable coefficients

The stem itself does not reveal the correct option; the correct statement must be identified by comparing nuanced alternatives. Although the source JSON contains an annotated correct choice, that is external metadata rather than leakage in the posed MCQ.

The item is very close to theorem recall: one option essentially reproduces the intended theorem, while others are slight quantifier or hypothesis modifications. It is not purely tautological, but it is only a mild reformulation rather than a genuinely new application.

Selecting the answer requires some careful reasoning about logical strength, strict vs. non-strict inequalities, maximal elements vs. all elements, and the role of the extra transfer hypothesis. Still, the task mainly tests precise recognition of a theorem statement rather than substantial generative mathematical reasoning.

The distractors are strong: they are plausible theorem-like variants involving common failure modes such as overstrengthening conclusions, weakening hypotheses, or altering quantification. They are distinct and mathematically aligned with realistic misunderstandings.

A well-constructed theorem-discrimination MCQ with strong distractors and no stem-level answer leakage, but it primarily tests precise recall/logical parsing of a theorem rather than deeper generative reasoning.


\begin{theorem}\label{thm}
Assume $1\leq p <2$, $d\geq 1$ and the pressure $P$ satisfies $P'(\bar{\rho})>0$ for $\bar{\rho}>0$. There is a small constant $\delta_1>0$ such that if
$
\|(n_0,\bv_0)\|_{\dB_{p,1}^{\frac{d}{p}}}^{\ell,J_\varepsilon}
+\varepsilon\|(n_0,\bv_0)\|_{\dB_{2,1}^{\frac{d}{2}+1}}^{h,J_\varepsilon}
\leq \delta_1,
$
then the Cauchy problem \eqref{1.2}-\eqref{1.2initial} admits a uniform global unique solution in $E_\infty^{J_\varepsilon}$ for all $\varepsilon>0$.
\end{theorem}

\begin{equation}\label{DefHLB}
\|f\|_{\dB_{q_1,1}^{s_1}}^{h,J_{\varepsilon}}\triangleq \sum_{j\geq J_{\varepsilon}}
2^{s_1j}\|\ddj f\|_{q_1}\quad \mbox{and}\quad
\|f\|_{\dB_{q_2,1}^{s_2}}^{\ell,J_{\varepsilon}}\triangleq \sum_{j\leq J_{\varepsilon}-1}
2^{s_2j}\|\ddj f\|_{q_2}.
\end{equation}

This work studies the compressible Euler equations with linear damping, a model describing gas flow through a porous medium where the solid matrix exerts a frictional force proportional to and opposite to the fluid momentum. The governing system is given by
\begin{eqnarray}\label{1.1}
\left\{\aligned
&\d_t \rho+\mbox{\rm div}\;\! (\rho \bu)=0, \\
&\d_t (\rho \bu)+\nabla \cdot (\rho \bu\otimes \bu)+\nabla P(\rho)+\alpha\rho \bu=0
\endaligned\right.
\end{eqnarray}
for $t\geq0,\ \bm{x}\in \R^d(d\geq 1)$ and a damping coefficient $\alpha>0$
. Here $\rho\in\mathbb{R}^{+}$, $\bu=(u^1, u^2,\cdots, u^d)^\top$ ($\top$ represents transpose) denote the density and velocity of fluid flow, respectively. The pressure function $P=P(\rho)$ is assumed to be smooth around the constant reference density $\bar{\rho}>0$.

There are lots of mathematical results about the existence and asymptotic behavior of system \eqref{1.1} in sobolev space (see \cite{HL,HuangFM2,Wangdehua,Sui,Tan3,XY} ). In critical Besov space, Fang and Xu \cite{FXu} (with improvements in \cite{Jiu}) studied the existence and asymptotic behavior of classical solutions. Later, Xu and Wang \cite{XuWang} justified the relaxation convergence from \eqref{1.1} to the porous medium equation. And also, there are some results on hyperbolic system for balance laws including \eqref{1.1} (see \cite{XuKawaARM14,XuKawaARM15}). Recently, Crin-Barat and Danchin \cite{BaratDanchin,BaratDanchin2,BaratDanchin3} improved those results \cite{XuKawaARM14,XuKawaARM15,XuWang} in the $L^2$-$L^p$ hybrid Besov spaces with $2\leq p\leq\max\{4,\frac{2d}{d+2}\}$, where the low frequencies are bounded in $L^p$-type spaces and the high frequencies in $L^2$-type spaces with a specific linearity assumption. And then, the first author and Xu \cite{XuZhang} remove the assumption and to show that the results of \cite{BaratDanchin,BaratDanchin2} hold true for the general pressure function. Very recently, Crin-Barat and Song \cite{BSong} extend the work of \cite{BaratDanchin,BaratDanchin2,XuZhang} to the case $p\in[2,\infty)$. And a natural question is how about $p<2$?

Introducing a new unknown called ``enthalpy"
$n(\rho)\triangleq \int_1^{\rho} \frac{P'(s)}{s} ds.$ System \eqref{1.1} can be rewritten as
\begin{eqnarray}\label{1.2}
\left\{\aligned
&\d_t n + \bv \cdot \nabla n + P'(\bar{\rho})\mbox{\rm div}\;\! \bv + G(n)\mbox{\rm div}\;\! \bv = 0 \\
&\d_t \bv+ \bv\cdot \nabla \bv+\nabla n= -\frac{1}{\varepsilon} \bv
\endaligned\right.
\end{eqnarray}
with the initial data
\begin{eqnarray}\label{1.2initial}
 (n,\bv)|_{t=0}=(n_0, \bv_0).
\end{eqnarray}
Here $P'(\bar{\rho})>0$ for $\bar{\rho}>0$, $\bar{\rho}=P'(\bar{\rho})=1$ and the composite function $G(n)$ is smooth.

Define $J_\varepsilon=-[\log_2\varepsilon]+k$ being the threshold between high and low frequencies with a suitable integer $k$ is to be determined later, and the space $E_T^{J_\varepsilon}$ as follows
\begin{equation*}
\begin{split}
&E_T^{J_\varepsilon}=\bigg\{(n,\bv)|(n^{\ell,J_\varepsilon},\bv^{\ell,J_\varepsilon})\in \cC_b([0,T];\dB_{p,1}^{\frac{d}{p}}),\
\varepsilon n^{\ell,J_\varepsilon}\in \tL_T^1(\dB_{p,1}^{\frac{d}{p}+2}),\
\bv^{\ell,J_\varepsilon}\in \tL_T^1(\dB_{p,1}^{\frac{d}{p}+1}),
\varepsilon^{-\frac{1}{2}} \bv\in \tL_T^2(\dB_{p,1}^{\frac{d}{p}}),\\
&\qquad\qquad
(\varepsilon n^{h,J_\varepsilon}, \varepsilon\bv^{h,J_\varepsilon})\in \cC_b([0,T];\dB_{2,1}^{\frac{d}{2}+1}),\
(n^{h,J_\varepsilon},\bv^{h,J_\varepsilon}) \in \tL_T^1(\dB_{2,1}^{\frac{d}{2}+1}),
\varepsilon^{-1}\bv+\nabla n\in \tL_T^1(\dB_{p,1}^{\frac{d}{p}})
\bigg\}.
\end{split}
\end{equation*}
When $T=\infty$ we also use $E_\infty^{J_\varepsilon}$ for convenience.
For the reader's convenience, We introduce the notations $\|\cdot\|_{\dB_{q_1,1}^s}^{h,J_{\varepsilon}}$ and
$\|\cdot\|_{\dB_{q_2,1}^s}^{\ell,J_{\varepsilon}}$ to denote Besov semi-norms with respect to the threshold $J_{\varepsilon}$, that is,
\begin{equation}\label{DefHLB}
\|f\|_{\dB_{q_1,1}^{s_1}}^{h,J_{\varepsilon}}\triangleq \sum_{j\geq J_{\varepsilon}}
2^{s_1j}\|\dot \Delta_j f\|_{q_1}\quad \mbox{and}\quad
\|f\|_{\dB_{q_2,1}^{s_2}}^{\ell,J_{\varepsilon}}\triangleq \sum_{j\leq J_{\varepsilon}-1}
2^{s_2j}\|\dot \Delta_j f\|_{q_2}.
\end{equation}
It is not difficult to deduce that for all $\sigma_0>0$,
\begin{equation}\label{HLEst}
\|f\|_{\dB_{q_1,1}^{s_1}}^{h,J_{\varepsilon}}
\leq
2^{-\sigma_0J_{\varepsilon}}\|f\|_{\dB_{q_1,1}^{s_1+\sigma_0}}^{h,J_{\varepsilon}}\quad \mbox{and}\quad
\|f\|_{\dB_{q_1,1}^{s_1}}^{\ell,J_{\varepsilon}}
\leq
2^{\sigma_0J_{\varepsilon}}\|f\|_{\dB_{q_1,1}^{s_1-\sigma_0}}^{\ell,J_{\varepsilon}}.
\end{equation}

The main goal of this paper is to broaden the assumption on $p$ and to show that the results of \cite{BaratDanchin,BaratDanchin2,XuZhang} hold true for the case $p<2$. Our main result is stated as follows.

\begin{eqnarray}\label{1.1}
\left\{\aligned
&\d_t \rho+\div (\rho \bu)=0, \\
&\d_t (\rho \bu)+\nabla \cdot (\rho \bu\otimes \bu)+\nabla P(\rho)+\alpha\rho \bu=0
\endaligned\right.
\end{eqnarray}

\begin{eqnarray}\label{1.2}
\left\{\aligned
&\d_t n + \bv \cdot  \nabla n + P'(\bar{\rho})\div \bv + G(n)\div \bv = 0 \\
&\d_t \bv+ \bv\cdot \nabla \bv+\nabla n= -\frac{1}{\varepsilon} \bv
\endaligned\right.
\end{eqnarray}

Introducing a new unknown called ``enthalpy"
$n(\rho)\triangleq \int_1^{\rho} \frac{P'(s)}{s} ds.$ System \eqref{1.1} can be rewritten as
\begin{eqnarray}\label{1.2}
\left\{\aligned
&\d_t n + \bv \cdot \nabla n + P'(\bar{\rho})\div \bv + G(n)\div \bv = 0 \\
&\d_t \bv+ \bv\cdot \nabla \bv+\nabla n= -\frac{1}{\varepsilon} \bv
\endaligned\right.
\end{eqnarray}
with the initial data
\begin{eqnarray}\label{1.2initial}
 (n,\bv)|_{t=0}=(n_0, \bv_0).
\end{eqnarray}
Here $P'(\bar{\rho})>0$ for $\bar{\rho}>0$, $\bar{\rho}=P'(\bar{\rho})=1$ and the composite function $G(n)$ is smooth.

\begin{remark}
Based on Theorem \ref{thm}, the relaxation limit for the case $p<2$ can be performed by the same procedure as \cite{BaratDanchin2}.
\end{remark}

\section{Global a priori estimates}
In this section, we only give the key a priori estimate for the case $p<2$, which lead to the global existence and uniqueness of solutions in $E_\infty^{J_\varepsilon}$. See \cite{BShouZhang} for more details. For convenience, we use $ f\lesssim g $ to denote that there exists a generic constant $C>0$ independent on $\varepsilon$ such that $f\leq C g$ in this section.

Finally, by classical remainder estimates (see \cite[Theorem 2.85]{chemin}) we can directly obtain
\begin{equation}\nonumber
\begin{aligned}
\|R[a,b]\|_{\dB_{p,1}^{s_1}}^{\ell,J_\varepsilon}
\lesssim
\|b\|_{\dB_{\frac{2p}{2-p},1}^{s_1-\frac{d}{2}}}\|a\|_{\dB_{2,1}^{\frac{d}{2}}}
\lesssim
\left(\|b\|_{\dB_{p,1}^{s_1}}^{\ell,J_\varepsilon}+ 2^{(s_1-\frac{d}{p})J_\varepsilon}\|b\|_{\dB_{2,1}^{\frac{d}{2}}}^{h,J_\varepsilon} \right)\|a\|_{\dB_{2,1}^{\frac{d}{2}}}.
\end{aligned}
\end{equation}
Adding above three inequality together, we can finally obtain \eqref{newproduct}.
\end{proof}
For simplicity, we define $\cX(T):=\|(n,\bv)\|_{E_T^{J\varepsilon}}$. The proof of Theorem \ref{thm} reduces to establishing a global-in-time a priori estimate. Specifically, we claim that if
\begin{equation*}\begin{split}
\|n\|_{L^\infty}+\|\bv\|_{L^\infty} \ll 1 \quad \text{on} \quad [0,T],
\end{split}
\end{equation*}
then there exists a constant $C$, independent of $T$ and $\varepsilon$, such that
\begin{equation}\label{APrioriEstimate}
\cX(T)\leq C\big( \cX(0)+\cX^2(T)\big) \qquad\text{for all } \varepsilon>0.
\end{equation}

Next, we will deal with the nonlinear part. For the first nonlinear term, it follows the Lemma \ref{NonClassicalProLaw1} that
\begin{equation*}
\begin{split}
\|\bv\cdot \nabla n\|_{\tL_T^1(\dB_{p,1}^{\frac{d}{p}})}^{\ell,J_\varepsilon}
&\lesssim
\left(
\|\nabla n\|_{\tL_T^2(\dB_{p,1}^{\frac{d}{p}})}^{\ell,J_\varepsilon}
+ \|\nabla n\|_{\tL_T^2(\dB_{2,1}^{\frac{d}{2}})}^{h,J_\varepsilon}
\right)
\|\bv\|_{\tL_T^2(\dB_{2,1}^{\frac{d}{2}})}.
\end{split}
\end{equation*}
Then the interpolation inequalities (see Lemma 3.2 in \cite{XuZhang}) implies
\begin{equation}\label{NL1}
\begin{split}
\|\bv\cdot \nabla n\|_{\tL_T^1(\dB_{p,1}^{\frac{d}{p}})}^{\ell,J_\varepsilon}
&\lesssim
\left(\left(\| n\|_{\tL_T^\infty(\dB_{p,1}^{\frac{d}{p}})}^{\ell,J_\varepsilon}
\|n\|_{\tL_T^1(\dB_{p,1}^{\frac{d}{p}+2})}^{\ell,J_\varepsilon}\right)^{\frac{1}{2}}
+\left(\|n\|_{\tL_T^\infty(\dB_{2,1}^{\frac{d}{2}+1})}^{h,J_\varepsilon}
\|n\|_{\tL_T^1(\dB_{2,1}^{\frac{d}{2}+1})}^{h,J_\varepsilon}\right)^{\frac{1}{2}}
\right)\\
&\quad
\times\left(\|\bv\|_{\tL_T^2(\dB_{p,1}^{\frac{d}{p}})}^{\ell,J_\varepsilon}
+\varepsilon\left(\|\bv\|_{\tL_T^\infty(\dB_{2,1}^{\frac{d}{2}+1})}^{h,J_\varepsilon}
\|\bv\|_{\tL_T^1(\dB_{2,1}^{\frac{d}{2}+1})}^{h,J_\varepsilon}\right)^{\frac{1}{2}}\right)\\
&\lesssim
\cX^2(T)+\cX(T)\varepsilon^{-\frac{1}{2}}\|\bv\|_{\tL_T^2(\dB_{p,1}^{\frac{d}{p}})}^{\ell,J_\varepsilon}.
\end{split}
\end{equation}
And similarly, for the convection term we can deduce
\begin{equation}\label{NL2}
\begin{split}
\|\bv\cdot \nabla \bv\|_{\tL_T^1(\dB_{p,1}^{\frac{d}{p}})}^{\ell,J_\varepsilon}
\lesssim
\cX^2(T).
\end{split}
\end{equation}
For the last nonlinear term, the inequality \eqref{newproduct} implies
\begin{equation*}
\begin{split}
\|G(n)\div \bv\|_{\tL_T^1(\dB_{p,1}^{\frac{d}{p}})}^{\ell,J_\varepsilon}
&\lesssim
\|G(n)\|_{\tL_T^\infty(\dB_{2,1}^{\frac{d}{2}})}
\left(\|\div \bv\|_{\tL_T^1(\dB_{p,1}^{\frac{d}{p}})}^{\ell,J_\varepsilon}
+ \|\div \bv\|_{\tL_T^1(\dB_{2,1}^{\frac{d}{2}})}^{h,J_\varepsilon}\right).
\end{split}
\end{equation*}
By the classical composition estimate (see \cite{chemin}), we can obtain
\begin{equation}\label{NL3}
\begin{split}
\|G(n)\div \bv\|_{\tL_T^1(\dB_{p,1}^{\frac{d}{p}})}^{\ell,J_\varepsilon}
\lesssim
\|n\|_{\tL_T^\infty(\dB_{2,1}^{\frac{d}{2}})}
\left(\|\bv\|_{\tL_T^1(\dB_{p,1}^{\frac{d}{p}+1})}^{\ell,J_\varepsilon} +\|\bv\|_{\tL_T^1(\dB_{2,1}^{\frac{d}{2}+1})}^{h,J_\varepsilon}\right)
\lesssim
\cX^2(T).
\end{split}
\end{equation}

Inserting \eqref{NL1}-\eqref{NL3} into \eqref{LowEst3}, we can obtain
\begin{equation*}
\begin{split}
&\|(n, \varepsilon \bz)\|_{\tL_T^\infty(\dot B_{p,1}^{\frac{d}{p}})}^{\ell,J_\varepsilon}
+\bigg(\varepsilon\|n\|_{\tL_T^1(\dot B_{p,1}^{\frac{d}{p}+2})}^{\ell,J_\varepsilon}
+\|\bz\|_{\tL_T^1(\dot B_{p,1}^{\frac{d}{p}})}^{\ell,J_\varepsilon}\bigg)\\
&\quad\leq C\bigg(
\|(n_0, \varepsilon \bz_0)\|_{\dot B_{p,1}^{\frac{d}{p}}}^{\ell,J_\varepsilon}
+\cX^2(T)+\cX(T)\varepsilon^{-\frac{1}{2}}
\|\bv\|_{\tL_T^2(\dB_{p,1}^{\frac{d}{p}})}^{\ell,J_\varepsilon}\bigg) ,
\end{split}
\end{equation*}
which eventually leads to
\begin{equation}\label{LowEst4}
\begin{split}
&\|n\|_{\tL_T^\infty(\dot B_{p,1}^{\frac{d}{p}})}^{\ell,J_\varepsilon}
+\|\bv\|_{\tL_T^\infty(\dot B_{p,1}^{\frac{d}{p}})}^{\ell,J_\varepsilon}
+\bigg(\varepsilon\|n\|_{\tL_T^1(\dot B_{p,1}^{\frac{d}{p}+2})}^{\ell,J_\varepsilon}+\|\varepsilon^{-1}\bv+\nabla n\|_{\tL_T^1(\dot B_{p,1}^{\frac{d}{p}})}^{\ell,J_\varepsilon}
+\|\bv\|_{\tL_T^1(\dot B_{p,1}^{\frac{d}{p}+1})}^{\ell,J_\varepsilon}
+\varepsilon^{-\frac{1}{2}}
\|\bv\|_{\tL_T^2(\dB_{p,1}^{\frac{d}{p}})}^{\ell,J_\varepsilon}\bigg)\\
&\quad\lesssim
\cX(0)+\cX^2(T).
\end{split}
\end{equation}

Since $2^{J_\varepsilon}\approx 2^{k}\varepsilon^{-1}$, by the spatial embedding
$\dB_{p,1}^{\frac{d}{p}}(\R^d)\hookrightarrow \dB_{2,1}^{\frac{d}{2}}(\R^d)\hookrightarrow L^\infty(\R^d)$, the classical estimation on smooth functions (see Corollary 2.65 in \cite{chemin}) and \eqref{HLEst} we can obtain that
\begin{equation}\label{HNonLinEst1}
\begin{split}
\varepsilon\|(\nabla G(n),\nabla \bv)\|_{L_T^\infty(L^\infty)}
\lesssim
\varepsilon\|(n, \bv)\|_{\tL_T^\infty(\dB_{2,1}^{\frac{d}{2}+1})}
\lesssim\cX(T).
\end{split}
\end{equation}
By the continuity equation, it is not difficult to get
\begin{equation}\label{HNonLinEst2}
\begin{split}
\varepsilon\|\d_t G(n)\|_{L_T^\infty(L^\infty)}
&\lesssim
\varepsilon \|\div \bv+\bv \cdot \nabla n
+ G(n)\div\bv\|_{\tL_T^\infty(\dB_{2,1}^{\frac{d}{2}})}\\
&\lesssim
\varepsilon\|\bv\|_{\tL_T^\infty(\dB_{2,1}^{\frac{d}{2}+1})}
+\|\bv\|_{\tL_T^\infty(\dB_{2,1}^{\frac{d}{2}})}\varepsilon \|\nabla n\|_{\tL_T^\infty(\dB_{2,1}^{\frac{d}{2}})}\
+\|n\|_{\tL_T^\infty(\dB_{2,1}^{\frac{d}{2}})}\varepsilon \|\div \bv\|_{\tL_T^\infty(\dB_{2,1}^{\frac{d}{2}})}\\
&\lesssim
\cX(T)+\cX^2(T).
\end{split}
\end{equation}

\begin{remark}
Note that $\dot{B}_{p,1}^{\frac{d}{p}}\hookrightarrow \dot{B}_{2,1}^{\frac{d}{2}}$ when $p<2$, that means the existence and uniqueness of solutions hold in a smaller space rather than escaping to the complement $\dot{B}_{2,1}^{\frac{d}{2}}\setminus \dot{B}_{p,1}^{\frac{d}{p}}$. On the other hand, \eqref{DefHLB} implies that a smaller space for $p$ corresponds to more singular and concentrated data, which naturally arises in many physical problems (such as point sources and vortex filaments).
\end{remark}

\begin{remark}
Based on Theorem \ref{thm}, the relaxation limit for the case $p<2$ can be performed by the same procedure as \cite{BaratDanchin2}.
\end{remark}

The major difficulty of proof of Theorem \ref{thm} lies in dealing with the nonlinear terms. To achieve it, a new product estimation will be developed, see Lemma \ref{NonClassicalProLaw1} below. As a matter of fact, the new tool could be applied to investigate other systems, such as the hyperbolic-parabolic chemotaxis system (see \cite{BaratShou2}) in $L^2-L^p$ framework and Navier-Stokes equations (see \cite{Danchin} ) in $L^p-L^2$ framework for $p<2$.

The post-theorem introduction only indicates that "[t]he major difficulty of proof of Theorem \ref{thm} lies in dealing with the nonlinear terms" and that, "[t]o achieve it, a new product estimation will be developed, see Lemma \ref{NonClassicalProLaw1} below." No further proof steps are described.

The post-theorem introduction only indicates that "[t]he major difficulty of proof of the main theorem lies in dealing with the nonlinear terms" and that, "[t]o achieve it, a new product estimation will be developed," namely the following lemma.

\begin{lemma}\label{NonClassicalProLaw1}
Let $0<s_1<\frac{d}{2}$, $1\leq p<2$. Then, we have the following inequality
\begin{align}\label{newproduct}
\|ab\|_{\dB_{p,1}^{s_1}}^{\ell,J_\varepsilon}
\leq C
\left(\|b\|_{\dB_{p,1}^{s_1}}^{\ell,J_\varepsilon}+ 2^{(s_1-\frac{d}{p})J_\varepsilon}\|b\|_{\dB_{2,1}^{\frac{d}{2}}}^{h,J_\varepsilon} \right)\|a\|_{\dB_{2,1}^{\frac{d}{2}}}
\end{align}
with $C$ independent on $\varepsilon$.
\end{lemma}

No further proof steps are described.

\label{thm}
Assume $1\leq p <2$, $d\geq 1$ and the pressure $P$ satisfies $P'(\bar{\rho})>0$ for $\bar{\rho}>0$. There is a small constant $\delta_1>0$ such that if
$
\|(n_0,\bv_0)\|_{\dB_{p,1}^{\frac{d}{p}}}^{\ell,J_\varepsilon}
+\varepsilon\|(n_0,\bv_0)\|_{\dB_{2,1}^{\frac{d}{2}+1}}^{h,J_\varepsilon}
\leq \delta_1,
$
then the Cauchy problem
\begin{eqnarray}\label{1.2}
\left\{\aligned
&\d_t n + \bv \cdot  \nabla n + P'(\bar{\rho})\div \bv + G(n)\div \bv = 0 \\
&\d_t \bv+ \bv\cdot \nabla \bv+\nabla n= -\frac{1}{\varepsilon} \bv
\endaligned\right.
\end{eqnarray}
with initial data
\begin{eqnarray}\label{1.2initial}
 (n,\bv)|_{t=0}=(n_0, \bv_0).
\end{eqnarray}
admits a uniform global unique solution in $E_\infty^{J_\varepsilon}$ for all $\varepsilon>0$.

Question:
Let $1\le p<2$, $d\ge1$, and let the pressure law satisfy $P'(\bar\rho)>0$ for $\bar\rho>0$. For a given $\varepsilon>0$, consider the Cauchy problem
$$
\begin{cases}
\partial_t n+\mathbf v\cdot\nabla n+P'(\bar\rho)\,\operatorname{div}\mathbf v+G(n)\operatorname{div}\mathbf v=0,\\
\partial_t \mathbf v+\mathbf v\cdot\nabla \mathbf v+\nabla n=-\dfrac1\varepsilon\mathbf v,
\end{cases}
$$
with initial data $(n,\mathbf v)|_{t=0}=(n_0,\mathbf v_0)$, where $G(n)$ is the smooth function appearing in the enthalpy formulation. Let $J_\varepsilon=-[\log_2\varepsilon]+k$ for a suitable integer $k$, and define the low/high-frequency Besov seminorms by
$$
\|f\|_{\dot B_{q,1}^s}^{\ell,J_\varepsilon}=\sum_{j\le J_\varepsilon-1}2^{sj}\|\dot\Delta_j f\|_{L^q},
\qquad
\|f\|_{\dot B_{q,1}^s}^{h,J_\varepsilon}=\sum_{j\ge J_\varepsilon}2^{sj}\|\dot\Delta_j f\|_{L^q}.
$$
Write $n^{\ell,J_\varepsilon},\mathbf v^{\ell,J_\varepsilon}$ and $n^{h,J_\varepsilon},\mathbf v^{h,J_\varepsilon}$ for the corresponding low- and high-frequency parts. Also, $E_\infty^{J_\varepsilon}$ means the space obtained by taking $T=\infty$ in
$$
\begin{aligned}
E_T^{J_\varepsilon}=\Big\{(n,\mathbf v):&\ (n^{\ell,J_\varepsilon},\mathbf v^{\ell,J_\varepsilon})\in C_b([0,T];\dot B_{p,1}^{d/p}),\ \varepsilon n^{\ell,J_\varepsilon}\in \widetilde L_T^1(\dot B_{p,1}^{d/p+2}),\\
&\ \mathbf v^{\ell,J_\varepsilon}\in \widetilde L_T^1(\dot B_{p,1}^{d/p+1}),\ \varepsilon^{-1/2}\mathbf v\in \widetilde L_T^2(\dot B_{p,1}^{d/p}),\\
&\ (\varepsilon n^{h,J_\varepsilon},\varepsilon\mathbf v^{h,J_\varepsilon})\in C_b([0,T];\dot B_{2,1}^{d/2+1}),\ (n^{h,J_\varepsilon},\mathbf v^{h,J_\varepsilon})\in \widetilde L_T^1(\dot B_{2,1}^{d/2+1}),\\
&\ \varepsilon^{-1}\mathbf v+\nabla n\in \widetilde L_T^1(\dot B_{p,1}^{d/p})\Big\}.
\end{aligned}
$$
Assume there is a sufficiently small constant $\delta_1>0$ such that the initial data satisfy
$$
\|(n_0,\mathbf v_0)\|_{\dot B_{p,1}^{d/p}}^{\ell,J_\varepsilon}+\varepsilon\|(n_0,\mathbf v_0)\|_{\dot B_{2,1}^{d/2+1}}^{h,J_\varepsilon}\le \delta_1.
$$
Which conclusion about the solution holds under these hypotheses?

For every $\varepsilon>0$, the above Cauchy problem admits a uniform global unique solution $(n,\mathbf v)$ belonging to $E_\infty^{J_\varepsilon}$.

Choices:
B. For every $\varepsilon>0$, the above Cauchy problem admits a uniform global unique solution $(n,\mathbf v)$ belonging to $E_\infty^{J_\varepsilon}$, provided in addition that $0<\frac{d}{p}<\frac d2$ so that the low-frequency nonlinear estimates close.
C. For every $\varepsilon>0$, the above Cauchy problem admits a global unique solution $(n,\mathbf v)$ belonging to $E_\infty^{J_\varepsilon}$.
D. There exists $\varepsilon_0>0$ such that for every $0<\varepsilon\le \varepsilon_0$, the above Cauchy problem admits a global unique solution $(n,\mathbf v)$ belonging to $E_\infty^{J_\varepsilon}$.
E.
For every $\varepsilon>0$, the above Cauchy problem admits a uniform global unique solution $(n,\mathbf v)$ belonging to $E_\infty^{J_\varepsilon}$ whenever the smallness assumption is strengthened to
$$
\|(n_0,\mathbf v_0)\|_{\dot B_{p,1}^{d/p}}^{\ell,J_\varepsilon}+\|(n_0,\mathbf v_0)\|_{\dot B_{2,1}^{d/2+1}}^{h,J_\varepsilon}\le \delta_1.
$$

critical $\varepsilon$-weight on high frequencies

The stem does not explicitly reveal choice A. It states the hypotheses in full but does not directly state the conclusion, so there is no overt answer leakage.

This is essentially a theorem-recall item: the stem lays out the full assumptions and asks for the theorem's conclusion. The correct option is a near-direct restatement of the intended result rather than a genuinely new inference.

There is some reasoning pressure because the options differ in subtle ways (uniformity in ε, extra assumptions, stronger smallness, restricted ε-range). However, the task is still mostly recognition/matching of the exact theorem statement rather than substantial mathematical generation.

The distractors are plausible and mathematically meaningful: they alter quantifiers, uniformity, admissible parameter ranges, or the ε-weighted smallness condition. These reflect realistic failure modes in reading or recalling PDE well-posedness results.

A technically well-constructed theorem-recognition MCQ with strong distractors, but it is largely tautological and tests recall of the exact statement more than generative reasoning.