\label{thm:GWP}
Let $T>0$ and $\alpha\in[\alpha_{0},\,2]$, where $1.1108\approx\alpha_{0}\in (1,\frac{6}{5})$ is the root of the cubic equation $2\alpha^3+3\alpha^2-4\alpha-2=0.$

Assume that the initial data $u_0$ satisfies
\begin{align}\label{eq:init}
& u_0 \in \mathsf{H} \triangleq \Big\{ f\in L^{2}(\Omega): \int_{0}^{1}f(x,z)\,\mathrm{d} z=0,\,\,\forall\,x\in \mathbb{T}\Big\}, \quad  \Lambda_{h}^{\delta_m}u_0 \in L^{2}(\Omega),\nonumber\\
& \omega_{0}\triangleq\partial_{z}u_0 \in L^{\infty}(\Omega),\quad \Lambda_{h}^{\frac{3-2\alpha}{2}}\omega_{0} \in L^{2}(\Omega),\quad \text{and}\quad \partial_{z}\omega_{0} \in L^{2}(\Omega),
\end{align}
where
\begin{equation}\label{eq:deltam}
	\delta_m \triangleq \max \Big\{\frac{\alpha}{2},\,\frac{2(-\alpha^2-\alpha+3)}{3-\alpha}\Big\}.
\end{equation}
Then, the system \eqref{eq:main} with initial data $u_0$ has a unique global strong solution on $[0,T]$, with
\begin{align}\label{eq:sol}
& u\in C\big([0,T], \mathsf{H}\big),\quad \Lambda_{h}^{\delta_m}u \in L^\infty\big([0,T], L^{2}(\Omega)\big),\quad  \Lambda_{h}^{\delta_m + \frac\alpha2}u \in L^2\big([0,T], L^2(\Omega)\big),\nonumber\\
& \omega\triangleq\partial_{z}u \in L^\infty\big([0,T], L^\infty(\Omega)\big),\quad \Lambda_{h}^{\frac{3-2\alpha}{2}}\omega \in L^\infty\big([0,T], L^{2}(\Omega)\big),\quad  \Lambda_{h}^{\frac{3-\alpha}{2}}\omega \in L^2\big([0,T], L^2(\Omega)\big),\nonumber\\
& \partial_{z}\omega \in L^\infty\big([0,T], L^{2}(\Omega)\big),\quad \text{and}\quad \Lambda_{h}^{\frac{\alpha}{2}}\partial_{z}\omega \in L^2\big([0,T], L^2(\Omega)\big).
\end{align}

\begin{equation}\label{eq:c}
 c \triangleq \min_{i=1,\ldots,8}\Big(\frac{\nu}{16C_i}\Big)^{\frac{1}{\mu_i}}X_0^{-\frac{1-\mu_i}{2\mu_i}}>0.
\end{equation}

\begin{theorem}\label{thm:GWP}
Let $T>0$ and $\alpha\in[\alpha_{0},\,2]$, where $1.1108\approx\alpha_{0}\in (1,\frac{6}{5})$ is the root of the cubic equation $2\alpha^3+3\alpha^2-4\alpha-2=0.$

Assume that the initial data $u_0$ satisfies
\begin{align}\label{eq:init}
& u_0 \in \mathsf{H} \triangleq \Big\{ f\in L^{2}(\Omega): \int_{0}^{1}f(x,z)\,\dd z=0,\,\,\forall\,x\in \mathbb{T}\Big\}, \quad  \Lambda_{h}^{\delta_m}u_0 \in L^{2}(\Omega),\nonumber\\
& \omega_{0}\triangleq\partial_{z}u_0 \in L^{\infty}(\Omega),\quad \Lambda_{h}^{\frac{3-2\alpha}{2}}\omega_{0} \in L^{2}(\Omega),\quad \text{and}\quad \partial_{z}\omega_{0} \in L^{2}(\Omega),
\end{align}
where
\begin{equation}\label{eq:deltam}
	\delta_m \triangleq \max \Big\{\frac{\alpha}{2},\,\frac{2(-\alpha^2-\alpha+3)}{3-\alpha}\Big\}.
\end{equation}
Then, the system \eqref{eq:main} with initial data $u_0$ has a unique global strong solution on $[0,T]$, with
\begin{align}\label{eq:sol}
& u\in C\big([0,T], \mathsf{H}\big),\quad \Lambda_{h}^{\delta_m}u \in L^\infty\big([0,T], L^{2}(\Omega)\big),\quad  \Lambda_{h}^{\delta_m + \frac\alpha2}u \in L^2\big([0,T], L^2(\Omega)\big),\nonumber\\
& \omega\triangleq\partial_{z}u \in L^\infty\big([0,T], L^\infty(\Omega)\big),\quad \Lambda_{h}^{\frac{3-2\alpha}{2}}\omega \in L^\infty\big([0,T], L^{2}(\Omega)\big),\quad  \Lambda_{h}^{\frac{3-\alpha}{2}}\omega \in L^2\big([0,T], L^2(\Omega)\big),\nonumber\\
& \partial_{z}\omega \in L^\infty\big([0,T], L^{2}(\Omega)\big),\quad \text{and}\quad \Lambda_{h}^{\frac{\alpha}{2}}\partial_{z}\omega \in L^2\big([0,T], L^2(\Omega)\big).
\end{align}
\end{theorem}

\label{sec:intro}
The primitive equations constitute one of the fundamental models in planetary oceanic and atmospheric dynamics; see, for instance, the books \cite{majda2003introduction, vallis2017atmospheric}. These equations are derived from the (rotating) incompressible Navier-Stokes equations under the hydrostatic balance assumption for the vertical pressure component. In geophysical fluid dynamics, the horizontal turbulent mixing is typically much stronger than the vertical one, so that the horizontal viscosity dominates while the vertical viscosity is comparatively weak and is often neglected.

In this paper, we consider the two-dimensional primitive equations with fractional horizontal dissipation:
\begin{equation}\label{eq:main}
\left\{\begin{array}{l}
\partial_t u + u\partial_{x}u+\widetilde{w}\partial_{z}u + \partial_{x}p+\nu\Lambda_{h}^{\alpha}u= 0,\\[3pt]
\partial_{x}u+ \partial_{z}\widetilde{w}=0, \\[3pt]
\partial_{z}p=0,
\end{array}\right.
\end{equation}
defined on a 2D periodic channel
\[
 \Omega \triangleq \big\{(x,z):\ x\in \mathbb{T},\,z\in [0,1]\big\},
\]
where $(x,z)$ denote the horizontal and vertical variables, $(u,\widetilde{w})$ represent the horizontal and vertical velocity components, respectively, and $p$ is the pressure.
Here $\mathbb{T}$ denotes the 1D periodic domain of length one, and $\Lambda_{h}^{\alpha}=(-\partial_{x}^{2})^{\frac{\alpha}{2}}$ denotes the horizontal fractional Laplacian, defined by
\[
\Lambda_{h}^{\alpha}u(x,z) \triangleq c_\alpha\,{\rm p.v.}\int_{\mathbb R} \frac{u(x,z)-u(y,z)}{|x-y|^{1+\alpha}}\,\mathrm{d} y,\quad c_\alpha = \tfrac{2^\alpha\Gamma(\frac{\alpha+1}{2})}{\sqrt\pi\,|\Gamma(-\frac\alpha2)|}, \quad \alpha\in(0,2),
\]
where $u$ is viewed as a $1$-periodic function in the $x$-variable and ${\rm p.v.}$ denotes the principal value. The fractional Laplacian provides a continuous interpolation between the inviscid system (as $\alpha \to 0$) and the viscous system (as $\alpha \to 2$).
The equation \eqref{eq:main}$_2$ enforces incompressibility, while \eqref{eq:main}$_3$ expresses the hydrostatic pressure balance.

The system \eqref{eq:main} can be regarded as the hydrostatic limit of the Navier-Stokes equations with fractional horizontal dissipation, and the derivation follows analogously to \cite{azerad2001mathematical,li2019primitive}.

We further impose the initial condition
\[
u(x,z,0) = u_0(x,z),
\]
and the boundary condition
\[
\widetilde{w}(x,0,t)=\widetilde{w}(x,1,t)=0.
\]
Together with \eqref{eq:main}$_2$,  this implies that the vertical velocity $\widetilde{w}$ is uniquely determined by $u$ through
\[
\widetilde{w}(x,z,t)=-\int_{0}^{z}\partial_{x}u(x,\tilde{z},t)\,\mathrm{d}\tilde{z}.
\]
This introduces a loss of one horizontal derivative in $\widetilde{w}$ compared to the Navier-Stokes equations, a distinctive characteristic of the primitive equations.

\vskip 1em
The first systematic mathematical studies of the primitive equations were carried out in the 1990s by Lions, Temam, and Wang \cite{lions1992equations,lions1992new}, who established the global existence of weak solutions. However, the uniqueness of weak solutions remains an open problem, even in the two-dimensional setting; see \cite{bresch2003uniqueness,kukavica2014primitive,li2017existence,medjo2010uniqueness} for several partial results in this direction.

The global existence and uniqueness of strong solutions to the three-dimensional primitive equations with full viscosity have been known since the breakthrough work of Cao and Titi \cite{cao2007global}, which exploits the anisotropic structure of the system. See also \cite{hieber2016global,kobelkov2006existence,kukavica2007regularity} for further generalizations. The well-posedness theory was later extended to the primitive equations with only horizontal viscosity in a series of works \cite{cao2016global,cao2017strong,cao2020global}.

On the other hand, for the inviscid primitive equations, the loss of a horizontal derivative in the vertical velocity renders the system ill-posed in Sobolev spaces, even locally in time; see \cite{han2016ill,renardy2009ill}. Local well-posedness can be recovered under additional structural assumptions, such as the local Rayleigh condition \cite{brenier1999homogeneous,kukavica2014local,masmoudi2012h}, or by assuming real-analytic initial data \cite{kukavica2011local}. Moreover, smooth solutions may develop singularities in finite time; see \cite{cao2015finite,collot2023stable,wong2015blowup}.

These contrasting behaviors highlight that horizontal viscosity plays a crucial role in stabilizing the flow. To understand the transition between the viscous and inviscid regimes, Abdo, Lin, and Tan \cite{abdo2025well} introduced a family of primitive equations with fractional horizontal dissipation \eqref{eq:main}, which interpolate between the two systems. They identified a sharp transition between local well-posedness and ill-posedness at the critical dissipation exponent $\alpha = 1$. Specifically, the system \eqref{eq:main} is ill-posed in Sobolev spaces when $\alpha < 1$, analogous to the inviscid case, while it is locally well-posed when $\alpha > 1$, analogous to the viscous case. In the critical regime $\alpha = 1$, the well-posedness of solutions depends on the interplay between the horizontal viscosity coefficient $\nu$ and the size of the initial data.

Furthermore, they established a Beale-Kato-Majda type regularity criterion
\begin{equation}\label{eq:BKM}
\int_0^T\|\Lambda_{h}^{\frac{3-\alpha}{2}}\omega(t)\|_{L^2}^2\mathrm{d} t<\infty,
\end{equation}
under which a local solution can be extended up to time $T$, where $\omega \triangleq \partial_z u$ denotes the hydrostatic vorticity. The criterion \eqref{eq:BKM} was verified for $\alpha\geq1$ under a smallness assumption on the initial data, leading to a global well-posedness theory for small initial data. For general initial data, global well-posedness was established for $\alpha \geq \tfrac{6}{5}$. They further posed the following intriguing open problem:
\begin{quote}
\emph{The global well-posedness problem for the system \eqref{eq:main} with general initial data in the range $\alpha \in (1, \tfrac{6}{5})$ remains open.}
\end{quote}

\vskip 1em
In this paper, we address and partially resolve this problem. In particular, we establish the global well-posedness of \eqref{eq:main} for $\alpha \geq \alpha_0$, where $\alpha_0 \approx 1.1108 < \tfrac{6}{5}$. The precise statement of our main theorem is as follows.

In this paper, we consider the two-dimensional primitive equations with fractional horizontal dissipation:
\begin{equation}\label{eq:main}
\left\{\begin{array}{l}
\partial_t u + u\partial_{x}u+\widetilde{w}\partial_{z}u + \partial_{x}p+\nu\Lambda_{h}^{\alpha}u= 0,\\[3pt]
\partial_{x}u+ \partial_{z}\widetilde{w}=0, \\[3pt]
\partial_{z}p=0,
\end{array}\right.
\end{equation}
defined on a 2D periodic channel
\[
 \Omega \triangleq \big\{(x,z):\ x\in \mathbb{T},\,z\in [0,1]\big\},
\]
where $(x,z)$ denote the horizontal and vertical variables, $(u,\widetilde{w})$ represent the horizontal and vertical velocity components, respectively, and $p$ is the pressure.
Here $\mathbb{T}$ denotes the 1D periodic domain of length one, and $\Lambda_{h}^{\alpha}=(-\partial_{x}^{2})^{\frac{\alpha}{2}}$ denotes the horizontal fractional Laplacian, defined by
\[
\Lambda_{h}^{\alpha}u(x,z) \triangleq c_\alpha\,{\rm p.v.}\int_{\mathbb R} \frac{u(x,z)-u(y,z)}{|x-y|^{1+\alpha}}\,\mathrm{d} y,\quad c_\alpha = \tfrac{2^\alpha\Gamma(\frac{\alpha+1}{2})}{\sqrt\pi\,|\Gamma(-\frac\alpha2)|}, \quad \alpha\in(0,2),
\]
where $u$ is viewed as a $1$-periodic function in the $x$-variable and ${\rm p.v.}$ denotes the principal value. The fractional Laplacian provides a continuous interpolation between the inviscid system (as $\alpha \to 0$) and the viscous system (as $\alpha \to 2$).
The equation \eqref{eq:main}$_2$ enforces incompressibility, while \eqref{eq:main}$_3$ expresses the hydrostatic pressure balance.

The system \eqref{eq:main} can be regarded as the hydrostatic limit of the Navier-Stokes equations with fractional horizontal dissipation, and the derivation follows analogously to \cite{azerad2001mathematical,li2019primitive}.

We further impose the initial condition
\[
u(x,z,0) = u_0(x,z),
\]
and the boundary condition
\[
\widetilde{w}(x,0,t)=\widetilde{w}(x,1,t)=0.
\]
Together with \eqref{eq:main}$_2$, this implies that the vertical velocity $\widetilde{w}$ is uniquely determined by $u$ through
\[
\widetilde{w}(x,z,t)=-\int_{0}^{z}\partial_{x}u(x,\tilde{z},t)\,\mathrm{d}\tilde{z}.
\]
This introduces a loss of one horizontal derivative in $\widetilde{w}$ compared to the Navier-Stokes equations, a distinctive characteristic of the primitive equations.

These contrasting behaviors highlight that horizontal viscosity plays a crucial role in stabilizing the flow. To understand the transition between the viscous and inviscid regimes, Abdo, Lin, and Tan \cite{abdo2025well} introduced a family of primitive equations with fractional horizontal dissipation \eqref{eq:main}, which interpolate between the two systems. They identified a sharp transition between local well-posedness and ill-posedness at the critical dissipation exponent $\alpha = 1$. Specifically, the system \eqref{eq:main} is ill-posed in Sobolev spaces when $\alpha < 1$, analogous to the inviscid case, while it is locally well-posed when $\alpha > 1$, analogous to the viscous case. In the critical regime $\alpha = 1$, the well-posedness of solutions depends on the interplay between the horizontal viscosity coefficient $\nu$ and the size of the initial data.

Furthermore, they established a Beale-Kato-Majda type regularity criterion
\begin{equation}\label{eq:BKM}
\int_0^T\|\Lambda_{h}^{\frac{3-\alpha}{2}}\omega(t)\|_{L^2}^2\mathrm{d} t<\infty,
\end{equation}
under which a local solution can be extended up to time $T$, where $\omega \triangleq \partial_z u$ denotes the hydrostatic vorticity. The criterion \eqref{eq:BKM} was verified for $\alpha\geq1$ under a smallness assumption on the initial data, leading to a global well-posedness theory for small initial data. For general initial data, global well-posedness was established for $\alpha \geq \tfrac{6}{5}$. They further posed the following intriguing open problem:
\begin{quote}
\emph{The global well-posedness problem for the system \eqref{eq:main} with general initial data in the range $\alpha \in (1, \tfrac{6}{5})$ remains open.}
\end{quote}

\vskip 1em
In this paper, we address and partially resolve this problem. In particular, we establish the global well-posedness of \eqref{eq:main} for $\alpha \geq \alpha_0$, where $\alpha_0 \approx 1.1108 < \tfrac{6}{5}$. The precise statement of our main theorem is as follows.

\begin{equation}\label{eq:BKM}
\int_0^T\|\Lambda_{h}^{\frac{3-\alpha}{2}}\omega(t)\|_{L^2}^2\dd t<\infty,
\end{equation}

\vskip 1em
In this paper, we address and partially resolve this problem. In particular, we establish the global well-posedness of \eqref{eq:main} for $\alpha \geq \alpha_0$, where $\alpha_0 \approx 1.1108 < \tfrac{6}{5}$. The precise statement of our main theorem is as follows.

For $N_{22}$, $\|\partial_xu\|_{L_{z}^{2}L_{x}^{\frac{2p}{p-2}}}$ can be estimated by \eqref{eq:ux}, namely
\[
\|\partial_{x}u\|_{L_{z}^{2}L_{x}^{\frac{2p}{p-2}}} \leq C\|\Lambda_{h}^{1+\frac{1}{p}}u\|_{L^{2}}
\leq C\|\Lambda_{h}^{\delta}u\|_{L^{2}}^{1-\theta_{21}} \|\Lambda_{h}^{\delta+\frac{\alpha}{2}}u\|_{L^{2}}^{\theta_{21}},\quad \text{where}\quad \theta_{21} = \tfrac{2-2\delta+\frac{2}{p}}{\alpha},
\]
with a different $p$ defined in \eqref{eq:p2}.
 We use \eqref{eq:interp2} and interpolate $\|\Lambda_{h}^{2\rho-\frac{\alpha}{2}}\omega\|_{L_{x}^{p}}$ by $\|\omega\|_{L^\infty_x}$ and $\|\Lambda_{h}^{\rho+\frac{\alpha}{2}}\omega\|_{L_{x}^{2}}$. Note that this offers further enhancement compared with \eqref{eq:omegaenhance}. We choose
\begin{equation}\label{eq:p2}
 p \triangleq \frac{4\rho+2\alpha}{4\rho-\alpha}\in(2,\infty), \quad \text{so that}\quad
 \theta_{22} = \frac2p = \frac{2\rho-\frac\alpha2}{\rho+\frac\alpha2},
\end{equation}
and obtain
\begin{equation}\label{eq:omegaenhance2}
 \|\Lambda_{h}^{2\rho-\frac{\alpha}{2}}\omega\|_{L^p_x}\leq C \|\omega\|_{L^\infty_x}^{1-\theta_{22}}\|\Lambda_{h}^{\rho+\frac{\alpha}{2}}\omega\|_{L^2_x}^{\theta_{22}}
 = C \|\omega\|_{L^\infty_x}^{\frac{2\alpha-2\rho}{2\rho+\alpha}}\|\Lambda_{h}^{\rho+\frac{\alpha}{2}}\omega\|_{L^2_x}^{\frac{4\rho-\alpha}{2\rho+\alpha}}.
\end{equation}
Consequently, we apply Young's inequality and deduce
\begin{align}\label{eq:N22}
N_{22} &\leq C \|\omega_{0}\|_{L^{\infty}}^{1-\theta_{22}}\|\Lambda_{h}^{\delta}u\|_{L^{2}}^{1-\theta_{21}} \|\Lambda_{h}^{\delta+\frac{\alpha}{2}}u\|_{L^{2}}^{\theta_{21}} \|\Lambda_{h}^{\frac{\alpha}{2}}\partial_{z}\omega\|_{L^{2}} \|\Lambda_{h}^{\rho+\frac{\alpha}{2}}\omega\|_{L^2_x}^{\theta_{22}} \nonumber\\
&\leq \frac{\nu}{4}\|\Lambda_{h}^{\rho+\frac{\alpha}{2}}\omega\|_{L^{2}}^{2}
+C(\|\Lambda_{h}^{\delta+\frac{\alpha}{2}} u \|_{L^{2}}^{\frac{2\theta_{21}}{1-\theta_{22}}}+\|\Lambda_{h}^{\frac{\alpha}{2}}\partial_{z}\omega\|_{L^{2}}^{2}),
\end{align}
where we absorb $\|\omega_{0}\|_{L^{\infty}}$ and $\|\Lambda_{h}^{\delta}u\|_{L^{2}}$ (thanks to \eqref{eq:uimprove}) to the constant $C$. Since $\|\Lambda_{h}^{\delta+\frac\alpha2}u\|_{L^{2}}\in L^2(0,T)$, we require
\begin{equation}\label{eq:rhocond}
\frac{2\theta_{21}}{1-\theta_{22}} = \frac{(8-4\delta)\rho+(1-2\delta)\alpha}{\alpha(\alpha-\rho)}\leq 2,\quad \Longleftrightarrow\quad
\rho\leq \frac{\alpha(2\alpha+2\delta-1)}{2(\alpha-2\delta+4)}.
\end{equation}
From the two conditions \eqref{eq:rhocond1} and \eqref{eq:rhocond}, we define
\begin{equation}\label{eq:g}
 g(\delta) \triangleq \min\Big\{\frac{\alpha(2\alpha+2\delta-1)}{2(\alpha-2\delta+4)},\,\frac{\delta+\alpha-1}{2}\Big\} =
 \begin{cases}
 \frac{\alpha(2\alpha+2\delta-1)}{2(\alpha-2\delta+4)}, &\delta\in[\frac{2-\alpha}{2}, 2-\alpha],\\
 \frac{\delta+\alpha-1}{2},&\text{otherwise.}
 \end{cases}
\end{equation}
Taking $\delta = \delta_1$ in \eqref{eq:delta1}, the conditions \eqref{eq:rhocond1} and \eqref{eq:rhocond} hold when $\rho\leq g(\delta_1)=\rho_1$, where $\rho_1$ is defined in \eqref{eq:rho1}.

\begin{lemma}\label{lem:uimprove2}
Let $T>0$ and $\alpha\in[\alpha_0,\alpha_1)$. Then we have the bound \eqref{eq:uimprove}, namely
\[
\|\Lambda_{h}^{\delta}u(t)\|^2_{L^2} + \nu\int_0^t\|\Lambda_{h}^{\delta+\frac{\alpha}{2}}u(\tau)\|_{L^2}^2 \dd\tau\leq C,\quad\forall\, t\in[0,T],
\]
for any $\delta\in (\delta_1,\delta_2]$, where
\begin{align}\label{eq:delta2}
\delta_2 \triangleq \frac{\alpha(2\alpha-1)(2-\alpha)}{2(4-\alpha-2\alpha^2)},
\end{align}
and $C$ depends on $\|\Lambda_{h}^{\delta}u_0\|_{L^2}$, $\|\Lambda_{h}^{\delta_*}u_0\|_{L^2}$, $\|\Lambda_{h}^{\rho_1}\omega_0\|_{L^{2}}$, $\|\partial_z\omega_0\|_{L^{2}}$, $\|\omega_0\|_{L^\infty}$ and $T$.
\end{lemma}
\begin{proof}
 The proof follows from Lemma \ref{lem:uimprove}. We improve the bound \eqref{eq:omegaenhance}, replacing $\|\Lambda_{h}^{\frac{\alpha}{2}}\omega\|_{L^2_x}$ with $\|\Lambda_{h}^{\rho+\frac{\alpha}{2}}\omega\|_{L^2_x}$ in the interpolation:
 \begin{equation}\label{eq:omegaenhance3}
 \|\Lambda_{h}^{\delta-\frac{\alpha}{2}}\omega\|_{L^p_x}\leq C \|\omega\|_{L^\infty_x}^{1-\theta_{12}}\|\Lambda_{h}^{\rho+\frac{\alpha}{2}}\omega\|_{L^2_x}^{\theta_{12}}
 = C \|\omega\|_{L^\infty_x}^{\frac{2\alpha+2\rho-2\delta}{2\rho+\alpha}}\|\Lambda_{h}^{\rho+\frac{\alpha}{2}}\omega\|_{L^2_x}^{\frac{2\delta-\alpha}{2\rho+\alpha}},
 \end{equation}
 where $p$ and $\theta_{12}$ are redefined by
 \[
 p \triangleq \frac{2(2\rho+\alpha)}{2\delta-\alpha}\in (2,\infty),\quad \text{so that} \quad
 \theta_{12}=\frac{2}{p} = \frac{\delta-\frac\alpha2}{\rho+\frac\alpha2},
 \]
 which is analogous to \eqref{eq:omegaenhance2}. Together with \eqref{eq:ux}, the estimate on $U_{22}$ becomes
 \begin{align}\label{eq:U22e}
 U_{22} & \leq C \|\omega_0\|_{L^\infty}^{1-\theta_{12}}\|\Lambda_{h}^{\rho+\frac{\alpha}{2}}\omega\|_{L^2}^{\theta_{12}}\|\Lambda_{h}^{\delta}u\|_{L^{2}}^{1-\theta_{11}} \|\Lambda_{h}^{\delta+\frac{\alpha}{2}}u\|_{L^{2}}^{1+\theta_{11}} \nonumber\\
 &\leq \frac{\nu}{8} \|\Lambda_{h}^{\delta+\frac{\alpha}{2}}u\|_{L^2}^2 + C\|\omega_0\|_{L^{\infty}}^{\frac{2(1-\theta_{12})}{1-\theta_{11}}}\|\Lambda_{h}^{\rho+\frac{\alpha}{2}}\omega\|_{L^{2}}^{\frac{2\theta_{12}}{1-\theta_{11}}}\|\Lambda_{h}^{\delta}u\|_{L^{2}}^{2}.
\end{align}
To apply Gr\"onwall inequality, we require
\begin{equation}\label{eq:deltacond}
 \frac{2\theta_{12}}{1-\theta_{11}}=\frac{2\alpha(2\delta-\alpha)}{(\alpha-1)(2\delta+\alpha)+2\rho(\alpha+2\delta-2)}\leq2,\quad\Longleftrightarrow\quad
 \delta\leq\frac{2\alpha^2-\alpha + 2\rho(\alpha-2)}{2(1-2\rho)}\triangleq f(\rho).
\end{equation}
Note that if we take $\rho=0$, \eqref{eq:deltacond} reduces to the bound \eqref{eq:deltabound} in Lemma \ref{lem:uimprove} with $\delta_1 = f(0)$.
Thanks to the a priori bound on $\|\Lambda_{h}^{\rho_1}\omega\|_{L^2}$, we plug in $\rho=\rho_1$ and deduce that
\[\delta\leq f(\rho_1) = \delta_2,\]
where we note that $\rho_1 = \frac{2\alpha^2-\alpha}{8-4\alpha}$ when $\alpha\in[\alpha_0,\alpha_1)$.
The rest of the argument is analogous to Lemma \ref{lem:uimprove}, and we conclude with the bound \eqref{eq:uimprove}.
\end{proof}

Finally, we determine the minimal regularity requirement on $u_0$ that ensures global regularity. In the last iteration $k+1$, such that $\rho_k \leq \rho^* < \rho_{k+1}$, we apply Lemma \ref{lem:omegaimprove} with $\rho = \rho^*$. In this case, the condition \eqref{eq:rhocond} becomes
\[
\rho^*\leq \frac{\alpha(2\alpha+2\delta-1)}{2(\alpha-2\delta+4)},\quad \Longleftrightarrow\quad
\delta\geq \frac{2(-\alpha^2-\alpha+3)}{3-\alpha}\triangleq \delta_{**}.
\]
Hence, we may replace $\delta_{k+1}$ by the (smaller) quantity $\delta_{**}$ in Lemma \ref{lem:omegaimprove} (see the right panel of Figure \ref{fig1}). Consequently, the regularity criterion \eqref{eq:BKM} holds if the initial data satisfy
\[
\Lambda_{h}^{\delta_*}u_0\in L^2,\quad \Lambda_{h}^{\delta_{**}}u_0\in L^2,\quad \Lambda_{h}^{\frac{3-2\alpha}{2}}\omega_0\in L^2,\quad \partial_z\omega_0\in L^2,\quad \text{and}\quad \omega_0\in L^\infty.
\]
Define
\[
 \delta_m \triangleq \max \{\delta_*,\, \delta_{**}\} = \max \Big\{\frac{\alpha}{2},\,\frac{-2\alpha^2+2\alpha+1}{\alpha},\, \frac{2(-\alpha^2-\alpha+3)}{3-\alpha}\Big\}.
\]
Then, $\delta_m$ is the minimum initial regularity on $u_0$ that guarantees global well-posedness of the system \eqref{eq:main}. Observe that
\[
 \frac{-2\alpha^2+2\alpha+1}{\alpha}\leq \max \Big\{\frac{\alpha}{2},\, \frac{2(-\alpha^2-\alpha+3)}{3-\alpha}\Big\},\quad \forall\,\alpha\in[\alpha_0,\tfrac65).
\]
Therefore, $\delta_m$ can be expressed as in \eqref{eq:deltam}. The proof of Theorem \ref{thm:GWP} is thus complete.

\vskip 1em

\begin{remark}
A strong solution of \eqref{eq:main} is defined as (see \cite[Definition 1]{abdo2025well})
\begin{align*}
& u\in C\big([0,T], \mathsf{H}\big),\quad \Lambda_{h}^{\frac\alpha2}u \in L^\infty\big([0,T], L^{2}(\Omega)\big),\quad  \Lambda_{h}^{\alpha}u \in L^2\big([0,T], L^2(\Omega)\big),\\
& \omega \in L^\infty\big([0,T], L^\infty(\Omega)\big),\quad \quad  \Lambda_{h}^{\frac{\alpha}{2}}\omega \in L^2\big([0,T], L^2(\Omega)\big).
\end{align*}
It is known that such a strong solution may not be unique. A sufficient condition ensuring uniqueness is the regularity criterion \eqref{eq:BKM}. Theorem \ref{thm:GWP} establishes \eqref{eq:BKM} and therefore guarantees uniqueness of the strong solution.

The parameter $\delta_m$ in \eqref{eq:deltam} represents the minimal regularity required of $u_0$ for global well-posedness. Note that $\delta_m = \tfrac{\alpha}{2}$ when $\alpha \geq \tfrac{-7+\sqrt{193}}{6} \approx 1.149$. In this case, no additional regularity on $u_0$ is needed to guarantee uniqueness of the strong solution. Nonetheless, additional regularity is still required for $\omega_0$ and $\partial_z \omega_0$.

In \cite{abdo2025well}, it was shown that the regularity criterion \eqref{eq:BKM} also ensures global well-posedness for classical and smooth solutions. Consequently, if the initial data $u_0$ possess additional regularity, Theorem \ref{thm:GWP} yields global well-posedness for classical and smooth solutions.
\end{remark}

The key idea underlying the proof of Theorem \ref{thm:GWP} is to make full use of the a priori $L^\infty$ bound on the hydrostatic vorticity $\omega$ (see \eqref{eq:omegaMP}). This bound serves as the foundation for deriving a hierarchy of enhanced a priori energy estimates, which significantly strengthen the regularity control of the solution. We develop an iterative procedure to obtain successive a priori bounds at higher levels of regularity. As a result, we are able to relax the global well-posedness assumption from $\alpha \geq \tfrac{6}{5}$ to the improved threshold $\alpha \geq \alpha_0$, thereby extending the range of admissible dissipation exponents.

\vskip 1em

Our next result focuses on the global well-posedness of \eqref{eq:main} for $\alpha\in[1,\alpha_0)$ under the assumption of small initial data.
\begin{theorem}\label{thm:small}
Let $T>0$ and $\alpha\in[1,\alpha_0)$. Assume that the initial data $u_0$ satisfies \eqref{eq:init}. Then, there exists a small constant $c>0$, such that if
\[ \|\omega_0\|_{L^\infty}<c, \]
then the system \eqref{eq:main} with initial data $u_0$ has a unique global strong solution on $[0,T]$, satisfying \eqref{eq:sol}.
\end{theorem}

\begin{remark}
A small-data global well-posedness result was established in \cite{abdo2025well}, requiring
\[
 \|\omega_0\|_{L^\infty}<c,\quad\text{and}\quad
 \|\Lambda_{h}^\alpha u_{0}\|^2_{L^2}+\|\Lambda_{h}^{\frac{\alpha}{2}}\omega_{0}\|_{L^{2}}^{2}+ \|\partial_{z}\omega_{0}\|^2_{L^2}<c.
\]
In contrast, our result does not impose any smallness condition on $\|\Lambda_{h}^\alpha u_{0}\|^2_{L^2}+\|\Lambda_{h}^{\frac{\alpha}{2}}\omega_{0}\|_{L^{2}}^{2}+ \|\partial_{z}\omega_{0}\|^2_{L^2}$. In fact, this quantity need not even be bounded.
Under the assumption \eqref{eq:init}, we only require
\[
X_0 = \|\Lambda_{h}^{\delta_m} u_{0}\|^2_{L^2}+\|\Lambda_{h}^{\frac{3-2\alpha}{2}}\omega_{0}\|_{L^{2}}^{2}+ \|\partial_{z}\omega_{0}\|^2_{L^2}<\infty.
\]
The small constant c appearing in \eqref{eq:c} is given explicitly and may depend on $\nu$ and $X_0$.
Crucially, however, smallness of $X_0$ itself is not required.
Hence, our result substantially relaxes the smallness assumption compared to \cite{abdo2025well}.

Moreover, our result extends to the critical regime, when $\alpha = 1$.
Together with the ill-posedness result for $\|\omega_0\|_{L^\infty} \gg \nu$ in the critical regime (see \cite[Theorem 5.2]{abdo2025well}), this demonstrates a sharp transition in the solution behavior: from ill-posedness to well-posedness.
\end{remark}

The proof of Theorem \ref{thm:small} utilizes the same a priori energy estimates used in the proof of Theorem \ref{thm:GWP}. Consequently, the regularity assumptions on the initial data are consistent across the two results.

\vskip 1em
The rest of the paper is organized as follows. In Section \ref{sec:prelim}, we present preliminary results, including several a priori bounds and functional inequalities. Section \ref{sec:apriori} is devoted to establishing enhanced a priori energy estimates. Building on these estimates, we prove our main results, Theorems \ref{thm:GWP} and \ref{thm:small}, in Sections \ref{sec:GWP} and \ref{sec:GWPsmall}, respectively.

\subsection*{Notations}
We use the symbol $C$ to denote a generic positive constant, which may vary from line to line. The dependence of $C$ on specific parameters will be clear from the context and explicitly indicated whenever necessary.

We write $L^p_x = L^p_x(\mathbb{T})$ and $L^p_z = L^p_z([0,1])$ for the Lebesgue spaces in the $x$- and $z$-variables, respectively.
For a function $f = f(x,z)$, we define the mixed norm
\[
\|f\|_{L_{z}^{q}L_{x}^{p}}\triangleq \Big\|\|f\|_{L_{x}^{p}}\Big\|_{L_{z}^{q}}
= \bigg(\int_0^1\Big(\int_\mathbb{T} f(x,z)^p\,\mathrm{d} x\Big)^{\frac{q}{p}}\,\mathrm{d} z\bigg)^{\frac{1}{q}},
\]
and denote $L^p = L^p(\Omega) = L^p_zL^p_x$.

The post-theorem introduction gives the following proof idea for Theorem~\ref{thm:GWP}:

- The “key idea” is to “make full use of the a priori $L^\infty$ bound on the hydrostatic vorticity $\omega$ (see \eqref{eq:omegaMP}).”
- This $L^\infty$ bound “serves as the foundation for deriving a hierarchy of enhanced a priori energy estimates,” which “significantly strengthen the regularity control of the solution.”
- The authors “develop an iterative procedure to obtain successive a priori bounds at higher levels of regularity.”
- “As a result,” they “relax the global well-posedness assumption from $\alpha \geq \tfrac{6}{5}$ to the improved threshold $\alpha \geq \alpha_0$,” thereby “extending the range of admissible dissipation exponents.”

Additionally, the remark explains the uniqueness mechanism: Theorem~\ref{thm:GWP} “establishes \eqref{eq:BKM} and therefore guarantees uniqueness of the strong solution,” since a “sufficient condition ensuring uniqueness is the regularity criterion \eqref{eq:BKM}.”

The post-theorem introduction gives the following proof idea for the main theorem:

- The “key idea” is to “make full use of the a priori $L^\infty$ bound on the hydrostatic vorticity $\omega$,” namely
\begin{equation}\label{eq:omegaMP}
\|\omega(t)\|_{L^{\infty}} \leq\|\omega_{0}\|_{L^{\infty}}.
\end{equation}
- This $L^\infty$ bound “serves as the foundation for deriving a hierarchy of enhanced a priori energy estimates,” which “significantly strengthen the regularity control of the solution.”
- The authors “develop an iterative procedure to obtain successive a priori bounds at higher levels of regularity.”
- “As a result,” they “relax the global well-posedness assumption from $\alpha \geq \tfrac{6}{5}$ to the improved threshold $\alpha \geq \alpha_0$,” thereby “extending the range of admissible dissipation exponents.”

Additionally, the remark explains the uniqueness mechanism: the main theorem “establishes
\begin{equation}\label{eq:BKM}
\int_0^T\|\Lambda_{h}^{\frac{3-\alpha}{2}}\omega(t)\|_{L^2}^2\dd t<\infty,
\end{equation}
and therefore guarantees uniqueness of the strong solution,” since a “sufficient condition ensuring uniqueness is the regularity criterion given by the equation above.”

\label{thm:GWP}
Let $T>0$ and $\alpha\in[\alpha_{0},\,2]$, where $1.1108\approx\alpha_{0}\in (1,\frac{6}{5})$ is the root of the cubic equation $2\alpha^3+3\alpha^2-4\alpha-2=0.$

Assume that the initial data $u_0$ satisfies
\begin{align}\label{eq:init}
& u_0 \in \mathsf{H} \triangleq \Big\{ f\in L^{2}(\Omega): \int_{0}^{1}f(x,z)\,\mathrm{d} z=0,\,\,\forall\,x\in \mathbb{T}\Big\}, \quad  \Lambda_{h}^{\delta_m}u_0 \in L^{2}(\Omega),\nonumber\\
& \omega_{0}\triangleq\partial_{z}u_0 \in L^{\infty}(\Omega),\quad \Lambda_{h}^{\frac{3-2\alpha}{2}}\omega_{0} \in L^{2}(\Omega),\quad \text{and}\quad \partial_{z}\omega_{0} \in L^{2}(\Omega),
\end{align}
where
\begin{equation}\label{eq:deltam}
\t\delta_m \triangleq \max \Big\{\frac{\alpha}{2},\,\frac{2(-\alpha^2-\alpha+3)}{3-\alpha}\Big\}.
\end{equation}
Then, the system
\begin{equation}\label{eq:main}
\left\{\begin{array}{l}
\partial_t u + u\partial_{x}u+\widetilde{w}\partial_{z}u + \partial_{x}p+\nu\Lambda_{h}^{\alpha}u= 0,\\[3pt]
\partial_{x}u+ \partial_{z}\widetilde{w}=0, \\[3pt]
\partial_{z}p=0,
\end{array}\right.
\end{equation}
with initial data $u_0$ has a unique global strong solution on $[0,T]$, with
\begin{align}\label{eq:sol}
& u\in C\big([0,T], \mathsf{H}\big),\quad \Lambda_{h}^{\delta_m}u \in L^\infty\big([0,T], L^{2}(\Omega)\big),\quad  \Lambda_{h}^{\delta_m + \frac\alpha2}u \in L^2\big([0,T], L^2(\Omega)\big),\nonumber\\
& \omega\triangleq\partial_{z}u \in L^\infty\big([0,T], L^\infty(\Omega)\big),\quad \Lambda_{h}^{\frac{3-2\alpha}{2}}\omega \in L^\infty\big([0,T], L^{2}(\Omega)\big),\quad  \Lambda_{h}^{\frac{3-\alpha}{2}}\omega \in L^2\big([0,T], L^2(\Omega)\big),\nonumber\\
& \partial_{z}\omega \in L^\infty\big([0,T], L^{2}(\Omega)\big),\quad \text{and}\quad \Lambda_{h}^{\frac{\alpha}{2}}\partial_{z}\omega \in L^2\big([0,T], L^2(\Omega)\big).
\end{align},

Question:
Let \(\Omega=\mathbb T\times[0,1]\), where \(\mathbb T\) is the one-dimensional periodic domain of length \(1\), and let \(\Lambda_h^\beta=(-\partial_x^2)^{\beta/2}\) denote the horizontal fractional Laplacian. Consider the 2D primitive equations
\[
\left\{\begin{array}{l}
\partial_t u+u\partial_xu+\widetilde w\partial_zu+\partial_xp+\nu\Lambda_h^\alpha u=0,\\[2pt]
\partial_xu+\partial_z\widetilde w=0,\\[2pt]
\partial_zp=0,
\end{array}\right.
\]
on \(\Omega\), with initial data \(u(x,z,0)=u_0(x,z)\) and boundary conditions \(\widetilde w(x,0,t)=\widetilde w(x,1,t)=0\). Fix \(T>0\) and \(\alpha\in[\alpha_0,2]\), where \(\alpha_0\approx1.1108\) is the root of \(2\alpha^3+3\alpha^2-4\alpha-2=0\). Define
\[
\mathsf H=\Big\{f\in L^2(\Omega): \int_0^1 f(x,z)\,dz=0\ \text{for all }x\in\mathbb T\Big\},
\qquad \omega_0=\partial_z u_0,
\]
and
\[
\delta_m=\max\left\{\frac\alpha2,\frac{2(-\alpha^2-\alpha+3)}{3-\alpha}\right\}.
\]
Assume
\[
u_0\in \mathsf H,\qquad \Lambda_h^{\delta_m}u_0\in L^2(\Omega),\qquad \omega_0\in L^\infty(\Omega),\qquad \Lambda_h^{\frac{3-2\alpha}{2}}\omega_0\in L^2(\Omega),\qquad \partial_z\omega_0\in L^2(\Omega).
\]
Which existence-and-uniqueness statement holds for this problem?

There exists exactly one global strong solution on \([0,T]\) to the above system with initial data \(u_0\). Writing \(\omega=\partial_z u\), the solution satisfies
\[
u\in C([0,T],\mathsf H),\qquad \Lambda_h^{\delta_m}u\in L^\infty([0,T],L^2(\Omega)),\qquad \Lambda_h^{\delta_m+\frac\alpha2}u\in L^2([0,T],L^2(\Omega)),
\]
\[
\omega\in L^\infty([0,T],L^\infty(\Omega)),\qquad \Lambda_h^{\frac{3-2\alpha}{2}}\omega\in L^\infty([0,T],L^2(\Omega)),\qquad \Lambda_h^{\frac{3-\alpha}{2}}\omega\in L^2([0,T],L^2(\Omega)),
\]
\[
\partial_z\omega\in L^\infty([0,T],L^2(\Omega)),\qquad \Lambda_h^{\frac\alpha2}\partial_z\omega\in L^2([0,T],L^2(\Omega)).
\]

Choices:
B.
There exists exactly one global strong solution on \([0,T]\) to the above system with initial data \(u_0\). Writing \(\omega=\partial_z u\), the solution satisfies
\[
u\in C([0,T],\mathsf H),\qquad \Lambda_h^{\delta_m}u\in L^\infty([0,T],L^2(\Omega)),\qquad \Lambda_h^{\delta_m+\frac\alpha2}u\in L^2([0,T],L^2(\Omega)),
\]
\[
\omega\in L^\infty([0,T],L^\infty(\Omega)),\qquad \Lambda_h^{\frac{3-2\alpha}{2}}\omega\in L^\infty([0,T],L^2(\Omega)),\qquad \Lambda_h^{\frac{3-\alpha}{2}}\omega\in L^\infty([0,T],L^2(\Omega)),
\]
\[
\partial_z\omega\in L^\infty([0,T],L^2(\Omega)),\qquad \Lambda_h^{\frac\alpha2}\partial_z\omega\in L^2([0,T],L^2(\Omega)).
\]
C.
There exists a global strong solution on \([0,T]\) to the above system with initial data \(u_0\). Writing \(\omega=\partial_z u\), the solution satisfies
\[
u\in C([0,T],\mathsf H),\qquad \Lambda_h^{\delta_m}u\in L^\infty([0,T],L^2(\Omega)),\qquad \Lambda_h^{\delta_m+\frac\alpha2}u\in L^2([0,T],L^2(\Omega)),
\]
\[
\omega\in L^\infty([0,T],L^\infty(\Omega)),\qquad \Lambda_h^{\frac{3-2\alpha}{2}}\omega\in L^\infty([0,T],L^2(\Omega)),\qquad \Lambda_h^{\frac{3-\alpha}{2}}\omega\in L^2([0,T],L^2(\Omega)),
\]
\[
\partial_z\omega\in L^\infty([0,T],L^2(\Omega)),\qquad \Lambda_h^{\frac\alpha2}\partial_z\omega\in L^2([0,T],L^2(\Omega)).
\]
D.
For every \(\alpha\in(1,2]\), there exists exactly one global strong solution on \([0,T]\) to the above system with initial data \(u_0\). Writing \(\omega=\partial_z u\), the solution satisfies
\[
u\in C([0,T],\mathsf H),\qquad \Lambda_h^{\delta_m}u\in L^\infty([0,T],L^2(\Omega)),\qquad \Lambda_h^{\delta_m+\frac\alpha2}u\in L^2([0,T],L^2(\Omega)),
\]
\[
\omega\in L^\infty([0,T],L^\infty(\Omega)),\qquad \Lambda_h^{\frac{3-2\alpha}{2}}\omega\in L^\infty([0,T],L^2(\Omega)),\qquad \Lambda_h^{\frac{3-\alpha}{2}}\omega\in L^2([0,T],L^2(\Omega)),
\]
\[
\partial_z\omega\in L^\infty([0,T],L^2(\Omega)),\qquad \Lambda_h^{\frac\alpha2}\partial_z\omega\in L^2([0,T],L^2(\Omega)).
\]
E.
There exists exactly one global strong solution on \([0,T]\) to the above system with initial data \(u_0\). Writing \(\omega=\partial_z u\), the solution satisfies
\[
u\in C([0,T],\mathsf H),\qquad \Lambda_h^{\delta_m}u\in L^\infty([0,T],L^2(\Omega)),\qquad \Lambda_h^{\delta_m+\frac\alpha2}u\in L^2([0,T],L^2(\Omega)),
\]
\[
\omega\in L^\infty([0,T],L^\infty(\Omega)),\qquad \Lambda_h^{\frac{3-2\alpha}{2}}\omega\in L^\infty([0,T],L^2(\Omega)),\qquad \Lambda_h^{\frac{3-\alpha}{2}}u\in L^2([0,T],L^2(\Omega)),
\]
\[
\partial_z\omega\in L^\infty([0,T],L^2(\Omega)),\qquad \Lambda_h^{\frac\alpha2}\partial_z\omega\in L^2([0,T],L^2(\Omega)).
\]

The stem does not explicitly reveal the correct option. It gives the PDE, hypotheses, and asks which theorem-style conclusion holds; the correct answer is not stated verbatim in the stem.

This is essentially a direct recall/restatement of a specific existence-and-uniqueness theorem under the listed assumptions. The options differ only by small perturbations of the theorem statement rather than competing mathematical conclusions derived from reasoning.

Some attention is needed to distinguish uniqueness vs. mere existence, the sharp alpha threshold, and the exact regularity spaces. However, the task mainly tests recognition of the exact theorem statement rather than generating a conclusion from the PDE and assumptions.

The distractors are plausible and targeted: one strengthens a norm improperly, one drops uniqueness, one alters the sharp parameter range, and one swaps the controlled quantity. These reflect realistic failure modes in reading or recalling PDE regularity theorems.

A technically well-constructed theorem-recall MCQ with strong distractors, but it is largely tautological and tests precise recall more than genuine generative mathematical reasoning.


For a positive integer $n$, let $[n]:=\{1,2,\ldots,n\}$. A {\em set composition} $\phi$ of $[n]$,
denoted by $\phi\vDash[n]$, is a list of mutually disjoint nonempty subsets
$\phi_1/\phi_2/\ldots/\phi_{\ell}$ of $[n]$ whose union is $[n]$.
Each $\phi_i$ is called a {\em block} of $\phi$ and the number of blocks of $\phi$ is denoted
by $\ell(\phi)$.
A set composition $\phi$ is said to be {\em odd} if each block of $\phi$ consists of odd number
of elements. For instance, $\phi=7/148/9/6/235$ is an odd set composition of $[9]$ with $\ell(\phi)=5$.
Let
$$
O(n,k):=|\{\phi\vDash[n]: \phi\text{ odd}, \ell(\phi)=k\}|.
$$
Note that $O(n,k)=0$ if $n$ and $k$ have different parity. By the compositional formula for
the exponential generating functions (see~\cite[Proposition~5.1.3]{St}), we have
\begin{align*}
&\quad\sum_{n,k} O(n,k)t^k\frac{x^n}{n!}=\frac{t\sinh(x)}{1-t\sinh(x)}\\
&=tx+2t^2\frac{x^2}{2!}+(t+6t^3)\frac{x^3}{3!}+(8t^2+24t^4)\frac{x^3}{3!}+
(t+60t^3+120t^5)\frac{x^5}{5!}+\cdots.
\end{align*}

The {\em Catalan numbers}
$$
\left(C_n=\frac{1}{n+1}{2n\choose n}\right)_{n\geq0}=(1,1,2,5,14,42,132,\ldots)
$$
and the {\em tangent numbers} $\{T_{2n+1}\}_{n\geq0}$ defined by the Taylor expansion
$$
\tan(x)=\sum_{n\geq0}T_{2n+1}\frac{x^{2n+1}}{(2n+1)!}=
x+2\frac{x^3}{3!}+16\frac{x^5}{5!}+272\frac{x^7}{7!}+7936\frac{x^9}{9!}+\cdots
$$
are well-known combinatorial sequences in enumerative combinatorics.

The following identity, which connects Catalan numbers with tangent numbers via $O(n,k)$,
was found by Aliniaeifard and Li~\cite{AL} in their study of non-commutative peak algebra
and later proved by Zhao, Lin and Zang~\cite{ZLZ} using generating functions.

The following identity, which connects Catalan numbers with tangent numbers via $O(n,k)$,
was found by Aliniaeifard and Li~\cite{AL} in their study of non-commutative peak algebra
and later proved by Zhao, Lin and Zang~\cite{ZLZ} using generating functions.

For instance, when $n=2$ identity~\eqref{eq:Cat-Tan} reads
$$
1\cdot2^4C_0-60\cdot2^2C_1+120\cdot2^0C_2=16.
$$
Although Catalan numbers and tangent numbers have been extensively studied in combinatorics
(see~\cite[Exercise~6.19]{St} and~\cite{St10}), there seems to be no combinatorial identity relating
them other than~\eqref{eq:Cat-Tan}. As observed by Aliniaeifard and Li~\cite{AL},
combining~\cite[Corollary~11.5]{AL} and
Theorem~\ref{thm:Cat-Tan} results in the following intriguing identity for tangent numbers
\begin{equation}\label{eq:Genocchi}
\sum_{k=0}^{n-1}(-1)^{k}{2n\choose 2k+1}2^{2n-2k}T_{2k+1}=2^{2n+1}.
\end{equation}
A generating function proof of Theorem~\ref{thm:Cat-Tan} and a combinatorial proof of~\eqref{eq:Genocchi},
together with its surprising arithmetic applications, were given in~\cite{ZLZ}.
However, a combinatorial proof of Theorem~\ref{thm:Cat-Tan} remains elusive.

The main objective of this paper is to provide a combinatorial involution proof of
Theorem~\ref{thm:Cat-Tan}.
In the course, we find a new combinatorial identity similar to~\eqref{eq:Genocchi}:
\begin{equation}\label{eq:Tan2}
\sum_{k=0}^{n}(-1)^{k}{2n+1\choose 2k}2^{2n-2k}T_{2k+1}=(-1)^nT_{2n+1}.
\end{equation}
This identity is also reminiscent of the one proved by Andrews and Gessel~\cite{AG}:
\begin{equation}\label{eq:TanAG}
T_{2n+1}+\sum_{k=1}^{n}(-1)^{k}{2n+1\choose 2k}2^{2k-1}T_{2n-2k+1}=(-1)^n2^{2n}.
\end{equation}

Let $\LB_{2n+1}$ be the set of all labeled binary trees on $[2n+1]$ and let $\IB_{2n+1}$ be the set of
all complete increasing binary trees on $[2n+1]$. It is clear that $\IB_{2n+1}\subseteq\LB_{2n+1}$.
For a labeled binary tree $T\in\LB_{2n+1}$, let $\h(T)$ be half the number of edges of $T$.
The following lemma shows that the left-hand side of~\eqref{eq:Cat-Tan} is
a signed counting of labeled binary trees on $[2n+1]$.
\begin{lemma}\label{lem:sign}
For $n\geq0$, we have
\begin{equation}\label{eq:sign}
\sum_{T\in\LB_{2n+1}}(-1)^{\h(T)}=\sum_{k=0}^n (-1)^kO(2n+1,2k+1)2^{2n-2k}C_k.
\end{equation}
\end{lemma}
\begin{proof}
Each labeled binary tree in $\LB_{2n+1}$ with $2k$ edges can be constructed by
\begin{itemize}
\item choosing a complete binary tree $T$ with $2k+1$ nodes,
\item choosing a set composition $\phi$ of $[2n+1]$ with $2k+1$ blocks,
\item forming an odd unimodal permutation using letters in $\phi_i$ for the label of the $i$-th node
(under the in-order traversal) of $T$ for each $1\leq i\leq 2k+1$.
\end{itemize}
There are $C_k$ choices for $T$, $O(2n+1,2k+1)$ choices for $\phi$, and $2^{2n+1-(2k+1)}$ ways to form
the labels of $T$ from $\phi$, whence~\eqref{eq:sign} follows.
\end{proof}

A permutation $\pi=\pi_1\pi_2\cdots\pi_n$ of $[n]$ is {\em down-up} (or {\em alternating}) if
$$
\pi_1>\pi_2<\pi_3>\pi_4<\cdots.
$$
If the above inequalities are reversed, then $\pi$ is said to be {\em up-down}.
Let $\A_n$ be the set of all down-up permutations of $[n]$.
A result attributed to Andr\'e~\cite{An} asserts that
\begin{equation}\label{eul:egf}
1+\sum_{n\geq1}|\A_n|\frac{x^n}{n!}=\sec(x)+\tan(x).
\end{equation}
Thus, $T_{2n+1}=|\A_{2n+1}|$.
It is a classical result (see~\cite{St10}) that $T_{2n+1}(q)$ has the combinatorial interpretation
\begin{equation}\label{int:qtan}
T_{2n+1}(q)=\sum_{\pi\in\A_{2n+1}}q^{\inv(\pi)},
\end{equation}
where $\inv(\pi):=|\{(i,j)\in[n]^2: i<j,\pi_i>\pi_j\}|$ is the {\em inversion number} of a permutation $\pi$.
The following is our first $q$-analog of~\eqref{eq:Tan2}.
\begin{theorem}\label{thm:q-analog1}
For $n\geq0$, we have
\begin{equation}\label{q1:Tan2}
\sum_{k=0}^{n}(-1)^{k}{2n+1\brack 2k}(-q;q)_{2n-2k}\widetilde T_{2k+1}(q)=(-1)^nT_{2n+1}(q),
\end{equation}
where $\widetilde T_{1}(q)=1$ and
\begin{equation}
\widetilde T_{2k+1}(q)=\sum_{i=0}^{k-1} {2k\brack 2i+1} T_{2i + 1}(q) T_{2k - 2i - 1}(q) \quad (k \geq 1).
\end{equation}
\end{theorem}

For a labeled tree $T\in\LB_{2n+1}$ with $2k+1$ nodes, consider the word concatenation
$w(T)=\alpha_1\alpha_2\cdots\alpha_{2k+1}$, where $\alpha_i$ is the labeling of the unimodal
permutation of the $i$-th node (under the in-order traversal) of $T$. For instance,
if $T$ is the first tree in Fig.~\ref{ex:lbt}, then $w(T)=467381952$.
It is clear that $w(T)$ is a permutation of $[2n+1]$. The {\em inversion number of $T$},
denoted by $\inv(T)$, is thus defined as
$$
\inv(T):=\inv(w(T)).
$$
The following interpretation of signed $q$-tangent numbers is a consequence of
Lemma~\ref{lem:invo}.
\begin{lemma}\label{lem:newint}
For $n\geq0$, we have
\begin{equation}\label{new:tan}
\sum_{T\in\LB_{2n+1}}(-1)^{\h(T)}q^{\inv(T)}=(-1)^nT_{2n+1}(q).
\end{equation}
\end{lemma}
\begin{proof}It is plain to check that our involution $\kappa$ has the following feature:
$$
\inv(T)=\inv(\kappa(T)) \quad\text{for any $T\in\LB_{2n+1}$},
$$
that is, it preserves the inversion numbers of labeled trees. The result then follows from Lemma~\ref{lem:invo}, interpretation~\eqref{int:qtan} and the known fact that $T\mapsto w(T)$ establishes a one-to-one correspondence between $\IB_{2n+1}$ and $\A_{2n+1}$ preserving inversion numbers.
\end{proof}
 We are ready for the proof of Theorem~\ref{thm:q-analog1}.
\begin{proof}[{\bf Proof of Theorem~\ref{thm:q-analog1}}]
Recall the well-known combinatorial interpretation for the $q$-binomial coefficients:
\begin{equation}\label{eq:qmul}
{n\brack k}=\sum_{({\mathcal A}, {\mathcal B})}q^{\inv({\mathcal A}, {\mathcal B})},
\end{equation}
summed over all set compositions $({\mathcal A}, {\mathcal B})$ of $[n]$ such that $|{\mathcal A}|=k$, and
$$\inv({\mathcal A}, {\mathcal B}):=|\{(a,b)\in{\mathcal A}\times{\mathcal B}: a>b\}|.$$
A labeled binary tree $T\in\LB_{2n+1}$ can be decomposed as $(T_g,r,T_d)$, where $r$ is the root and $T_g$ and $T_d$ (possibly empty) are left branch and right branch of $r$, respectively. It follows from this decomposition, Lemma~\ref{lem:unimodal}, the interpretation~\eqref{new:tan} of $q$-tangent numbers and the interpretation~\eqref{eq:qmul} of $q$-binomial coefficients that
$$
(-1)^nT_{2n+1}(q)=(-q;q)_{2n}+\sum_{k=1}^{n}(-1)^{k}{2n+1\brack 2k}(-q;q)_{2n-2k}\sum_{i=0}^{k-1}{2k\brack 2i+1}T_{2i+1}(q)T_{2k-2i-1}(q),
$$
where $(-q;q)_{2n}$ counts single-node labeled binary trees on $[2n+1]$ by inversion numbers.
This completes the proof of the theorem.
\end{proof}

In view of~\eqref{eul:egf}, the number $|\A_{2n}|:=S_{2n}$ is known as a {\em secant number}.
Consider the {\em$q$-secant number }
$$
S_{2n}(q):=\sum_{\pi\in\A'_{2n}}q^{\inv(\pi)},
$$
where $\A'_n$ denotes the set of all up-down permutations of $[n]$. For convenience, set $S_0(q)=1$. It was known (see~\cite{FH10}) that
$$
\sum_{n\geq0}S_{2n}(q)\frac{x^{2n}}{(q;q)_{2n}}=\frac{1}{\cos_q(x)}.
 $$
 The following is our second $q$-analog of~\eqref{eq:Tan2}.
\begin{theorem}\label{thm:q-analog2}
For $n\geq0$, we have
\begin{equation}\label{q2:Tan2}
\sum_{k=0}^{n}(-1)^{k}{2n+1\brack 2k}(-q;q)_{2n-2k} T_{2k+1}(q)=(-1)^n\widehat T_{2n+1}(q),
\end{equation}
where
\begin{equation}
(-1)^n\widehat T_{2n+1}(q)=\sum_{k=0}^{n} (-1)^{k} {2n+1\brack 2k}q^{2k} S_{2k}(q).
\end{equation}
Equivalently,
$$
\sum_{k=0}^{n}(-1)^{k}{2n+1\brack 2k}\bigl((-q;q)_{2n-2k} T_{2k+1}(q)-q^{2k} S_{2k}(q)\bigr)=0.
$$
\end{theorem}

\begin{theorem}\label{thm:q-sec:tan}
For $n\geq0$, we have
\begin{equation}\label{eq:q-sec:tan}
\sum_{k=0}^n(-1)^k{2n+1\brack 2k}S_{2k}(q)=(-1)^nT_{2n+1}(q).
\end{equation}
\end{theorem}

\begin{equation}\label{eq:Cat-Tan}
\sum_{k=0}^n (-1)^kO(2n+1,2k+1)2^{2n-2k}C_k=(-1)^nT_{2n+1}.
\end{equation}

\begin{theorem}\label{thm:Cat-Tan}
For $n\geq0$, we have
\begin{equation}\label{eq:Cat-Tan}
\sum_{k=0}^n (-1)^kO(2n+1,2k+1)2^{2n-2k}C_k=(-1)^nT_{2n+1}.
\end{equation}
\end{theorem}

For instance, when $n=2$ identity~\eqref{eq:Cat-Tan} reads
$$
1\cdot2^4C_0-60\cdot2^2C_1+120\cdot2^0C_2=16.
$$
Although Catalan numbers and tangent numbers have been extensively studied in combinatorics
(see~\cite[Exercise~6.19]{St} and~\cite{St10}), there seems to be no combinatorial identity relating
them other than~\eqref{eq:Cat-Tan}. As observed by Aliniaeifard and Li~\cite{AL},
combining~\cite[Corollary~11.5]{AL} and
Theorem~\ref{thm:Cat-Tan} results in the following intriguing identity for tangent numbers
\begin{equation}\label{eq:Genocchi}
\sum_{k=0}^{n-1}(-1)^{k}{2n\choose 2k+1}2^{2n-2k}T_{2k+1}=2^{2n+1}.
\end{equation}
A generating function proof of Theorem~\ref{thm:Cat-Tan} and a combinatorial proof of~\eqref{eq:Genocchi},
together with its surprising arithmetic applications, were given in~\cite{ZLZ}.
However, a combinatorial proof of Theorem~\ref{thm:Cat-Tan} remains elusive.

The main objective of this paper is to provide a combinatorial involution proof of
Theorem~\ref{thm:Cat-Tan}.
In the course, we find a new combinatorial identity similar to~\eqref{eq:Genocchi}:
\begin{equation}\label{eq:Tan2}
\sum_{k=0}^{n}(-1)^{k}{2n+1\choose 2k}2^{2n-2k}T_{2k+1}=(-1)^nT_{2n+1}.
\end{equation}
This identity is also reminiscent of the one proved by Andrews and Gessel~\cite{AG}:
\begin{equation}\label{eq:TanAG}
T_{2n+1}+\sum_{k=1}^{n}(-1)^{k}{2n+1\choose 2k}2^{2k-1}T_{2n-2k+1}=(-1)^n2^{2n}.
\end{equation}

The rest of this paper is organized as follows. In Section~\ref{sec:2}, we introduce the combinatorial
structure of complete binary trees labeled by odd unimodal permutations and then construct
a sign-reversing involution on them in Section~\ref{sec:3} to prove Theorem~\ref{thm:Cat-Tan}
combinatorially. In Section~\ref{sec:4}, we prove two different $q$-analogs of~\eqref{eq:Tan2}
using our involution on labeled binary trees and some other involutions on permutation pairs.

To prove Theorem~\ref{thm:Cat-Tan} combinatorially, the paper (i) introduces in Section~\ref{sec:2} “the combinatorial structure of complete binary trees labeled by odd unimodal permutations,” and then (ii) “construct[s] a sign-reversing involution on them in Section~\ref{sec:3} to prove Theorem~\ref{thm:Cat-Tan} combinatorially.”

To prove the main theorem combinatorially, the paper (i) introduces in the next section “the combinatorial structure of complete binary trees labeled by odd unimodal permutations,” and then (ii) “construct[s] a sign-reversing involution on them in a later section to prove the main theorem combinatorially.”

Question:
For a positive integer $m$, let $[m]=\{1,2,\ldots,m\}$. A set composition $\phi\vDash [m]$ is an ordered list $\phi_1/\phi_2/\cdots/\phi_\ell$ of pairwise disjoint nonempty subsets whose union is $[m]$. Such a set composition is called odd if every block $\phi_i$ has odd cardinality. Let
$$
O(m,k):=|\{\phi\vDash [m]: \phi \text{ is odd and } \ell(\phi)=k\}|.
$$
Also let the Catalan numbers be $C_k=\frac{1}{k+1}\binom{2k}{k}$, and let the tangent numbers $T_{2r+1}$ be defined by
$$
\tan(x)=\sum_{r\ge 0} T_{2r+1}\frac{x^{2r+1}}{(2r+1)!}.
$$
Which statement holds for every integer $n\ge 0$?

For every $n\ge 0$,
$$
\sum_{k=0}^n (-1)^k\,O(2n+1,2k+1)\,2^{2n-2k}C_k = (-1)^n T_{2n+1}.
$$

Choices:
B.
For every $n\ge 0$,
$$
\sum_{k=0}^n (-1)^k\,O(2n+1,2k+1)\,2^{2n-2k}C_k = T_{2n+1}.
$$
C.
For every $n\ge 0$,
$$
\sum_{k=0}^n (-1)^k\,O(2n+1,2k+1)\,2^{2n-2k}C_k
$$
is an integer multiple of $T_{2n+1}$.
D.
For every $n\ge 0$,
$$
\sum_{k=0}^n (-1)^k\,O(2n+1,2k+1)\,2^{2n-2k}C_k = (-1)^n C_n.
$$
E.
For every $n\ge 0$,
$$
\sum_{k=0}^n (-1)^k\,O(2n+1,k)\,2^{2n-k}C_k = (-1)^n T_{2n+1}.
$$

global sign from sign-reversing involution

target sequence identified by surviving fixed points

odd-block/odd-index restriction in the tree labeling construction

The stem gives only definitions of set compositions, Catalan numbers, and tangent numbers. It does not state or strongly hint at the exact identity, sign, or indexing pattern appearing in the correct choice.

The item is essentially a theorem-identification question: the correct option is the exact target identity, while the others are small perturbations of it. This makes it very close to a direct restatement rather than a problem with an independently derived conclusion.

There is some reasoning pressure because the choices differ in subtle but meaningful ways (global sign, exact equality vs weaker divisibility, odd-index restriction, target sequence). However, solving it still mainly amounts to recognizing or recalling the theorem rather than generating a conclusion from scratch.

The distractors are plausible and mathematically targeted: one drops the sign, one weakens the statement, one swaps the target sequence, and one alters the summation/indexing conditions. These align with realistic failure modes and are clearly distinct.

A reasonably strong theorem-recognition MCQ with good distractors and no answer leakage, but it is largely a near-verbatim identity-selection item rather than a genuinely generative reasoning question.


The notion of a \textit{quasi-Einstein metric} generalizes Einstein metrics by equipping a Riemannian manifold $(M,g)$ with a smooth vector field $X$.

\begin{defn}
    Let $(M,g)$ be a Riemannian manifold, and let $X$ be a smooth vector field defined on $X$.  The triple $(M,g,X)$ is said to be \textbf{quasi-Einstein} if
        \begin{align} \label{qe}
            \mathrm{Ric} + \frac{1}{2}\mathcal{L}_Xg - \frac{1}{m}X^* \otimes X^* = \lambda g
        \end{align}
    where $m \neq 0$ and $\lambda$ are constants, $X^*$ is one form dual to the vector field $X$ induced by $g$, and $\mathcal{L}$ is the Lie derivative.
\end{defn}

\noindent  In this paper, we will assume $M$ has no boundary.  Observe that the above definition reduces to the Einstein condition when $X$ is identically zero (we will call such solutions \textit{trivial}).

The tensor on the left-hand side of (\ref{qe}) is known as the \textit{Bakry-\'Emery} Ricci tensor.  Bakry and \'Emery studied this tensor as it pertains to diffusion processes in \cite{Bakry85}.  Qian studied Comparison geometry for this tensor in \cite{Qian97}.

When $m=2$, solutions to (\ref{qe}) are known as \textit{near-horizon geometries}, and are therefore of interest in general relativity.  See \cite{Kunduri09} for a discussion.  In the limiting case, when $m=\infty$, solutions to (\ref{qe}) as known as \textit{Ricci solitons}, which are self-similar solutions to the Ricci flow.

Case, Shu, and Wei studied solutions to (\ref{qe}) in the gradient case (i.e. where $X = \nabla \phi$ for some smooth function $\phi$ defined on $M$) in \cite{Case08}.  Among other things, they showed that there do not exist any non-trivial, compact quasi-Einstein metrics of constant scalar curvature.  It is natural to ask whether or not we can obtain solutions if we no longer require $X$ to be a gradient field.  If the $X$ is gradient condition is relaxed to the requirement that $dX^*=0$, work done in \cite{Bahuaud22} and \cite{Chrusciel05} shows that there are still no non-trivial solutions with constant scalar curvature when $\lambda \geq 0$, and when $\lambda < 0$. There is only a simple class of non-trivial examples in this case, namely those where $M = S^1 \times N$ is a product, and $X$ is constant length and tangent to the fibers of $S^1$ (see \cite{Wylie23}).

The story is more interesting if we no longer assume $dX^*=0$.  In \cite{Chen16}, Chen-Liang-Zhu showed that most compact simple Lie groups admit non-trivial quasi-Einstein metrics.  In \cite{Lim22}, Lim studied and classified quasi-Einstein metrics on locally homogeneous 3-manifolds.  We recall the definition of locally homogeneous now:

\begin{defn}
    A Riemannian manifold $(M,g)$ is called \textbf{locally homogeneous} if around any points $x, y \in M$, there exist neighborhoods $U_x$ of $x$ and $U_y$ of $y$ such that there exists an isometry $\Phi$ mapping $U_x$ onto $U_y$ such that $\Phi(x) = y$.
\end{defn}

Since locally homogeneous manifolds always have constant scalar curvature, Lim's examples show that removing the $dX^* = 0$ assumption allows for a number of more interesting examples.  Among these are circle bundles over $S^2$ known as \textit{Berger spheres}.

In the quest to find additional examples of compact quasi-Einstein metrics of constant scalar curvature, it is natural to ask whether or not there exist examples of quasi-Einstein metrics which have constant scalar curvature, but are not locally locally homogeneous.  This would provide a new class of examples in addition to the ones studied by Chen-Liang-Zhu and Lim mentioned above.  In higher dimensions, the answer to this question is ``yes".  A class of examples of such metrics can be realized as circle bundles over a K\"ahler-Einstein base, which are studied by Kunduri and Lucietti in \cite{Kunduri12}.

However, one of the main results in this paper is that the answer to the above question is ``no" in the three dimensional case.

\begin{defn}
 Let $(M,g)$ be a Riemannian manifold, and let $X$ be a smooth vector field defined on $X$. The triple $(M,g,X)$ is said to be \textbf{quasi-Einstein} if
 \begin{align} \label{qe}
 \mathrm{Ric} + \frac{1}{2}\mathcal{L}_Xg - \frac{1}{m}X^* \otimes X^* = \lambda g
 \end{align}
 where $m \neq 0$ and $\lambda$ are constants, $X^*$ is one form dual to the vector field $X$ induced by $g$, and $\mathcal{L}$ is the Lie derivative.
\end{defn}

Case, Shu, and Wei studied solutions to (\ref{qe}) in the gradient case (i.e. where $X = \nabla \phi$ for some smooth function $\phi$ defined on $M$) in \cite{Case08}. Among other things, they showed that there do not exist any non-trivial, compact quasi-Einstein metrics of constant scalar curvature. It is natural to ask whether or not we can obtain solutions if we no longer require $X$ to be a gradient field. If the $X$ is gradient condition is relaxed to the requirement that $dX^*=0$, work done in \cite{Bahuaud22} and \cite{Chrusciel05} shows that there are still no non-trivial solutions with constant scalar curvature when $\lambda \geq 0$, and when $\lambda < 0$. There is only a simple class of non-trivial examples in this case, namely those where $M = S^1 \times N$ is a product, and $X$ is constant length and tangent to the fibers of $S^1$ (see \cite{Wylie23}).

The story is more interesting if we no longer assume $dX^*=0$. In \cite{Chen16}, Chen-Liang-Zhu showed that most compact simple Lie groups admit non-trivial quasi-Einstein metrics. In \cite{Lim22}, Lim studied and classified quasi-Einstein metrics on locally homogeneous 3-manifolds. We recall the definition of locally homogeneous now:

Since locally homogeneous manifolds always have constant scalar curvature, Lim's examples show that removing the $dX^* = 0$ assumption allows for a number of more interesting examples. Among these are circle bundles over $S^2$ known as \textit{Berger spheres}.

In the quest to find additional examples of compact quasi-Einstein metrics of constant scalar curvature, it is natural to ask whether or not there exist examples of quasi-Einstein metrics which have constant scalar curvature, but are not locally locally homogeneous. This would provide a new class of examples in addition to the ones studied by Chen-Liang-Zhu and Lim mentioned above. In higher dimensions, the answer to this question is ``yes". A class of examples of such metrics can be realized as circle bundles over a K\"ahler-Einstein base, which are studied by Kunduri and Lucietti in \cite{Kunduri12}.

However, one of the main results in this paper is that the answer to the above question is ``no" in the three dimensional case.

\begin{align} \label{qe}
            \Ric + \frac{1}{2}\mathcal{L}_Xg - \frac{1}{m}X^* \otimes X^* = \lambda g
        \end{align}

However, one of the main results in this paper is that the answer to the above question is ``no" in the three dimensional case.

To obtain the above result, we use the fact that a compact quasi-Einstein metric of constant scalar curvature must have $X$ Killing. This result was first shown by Ghosh in Theorem 4.2 of \cite{Ghosh20}. In fact, it follows from equation (5.18) of \cite{Ghosh20} that the converse is also true, and so the conditions of $(M,g)$ having constant scalar curvature and $X$ being Killing are equivalent in the compact case (see also \cite{Cochran2024} and \cite{CostaFilho24}, whose authors were unaware of Ghosh's results at the time of publication). As shown in \cite{Bahuaud24}, it also follows that $X$ has constant length and that the integral curves of $X$ are geodesics, in addition to being Killing. Thus, a compact quasi-Einstein metric $(M,g,X)$ of constant scalar curvature necessarily admits a constant length Killing field (namely $X$). In Section 2, we outline how, in the three dimensional case, this shows that there exists a Sasakian structure of constant $\phi$-sectional curvature. We then invoke a classification due to Tanno in \cite{Tanno69} to conclude that Theorem \ref{thm:1} holds.

\begin{thm} \label{thm:2}
 Let $(B, \check{g})$ be a compact Einstein manifold and let $p: P \to B$ be a principle $S^1$-bundle on $B$ with fibers of length $2\pi$, classified by $\alpha \in H^2(B, \mathbb{Z})$. Suppose further that $X$ is a smooth vector field on $P$ tangent to the fibers of $\pi$. If $(P,g,X)$ is an $S^1$-invariant quasi-Einstein metric of constant scalar curvature, then either:
 \begin{itemize}
 \item[(a)] $\check{\lambda} + \frac{|X|^2}{m} = 0$, $\alpha = 0$, and $P$ is the Riemannian product $B \times S^1$ (i.e. the circle bundle is trivial), or
 \item[(b)] $\check{\lambda} + \frac{|X|^2}{m} > 0$ and there exists an almost K\"ahler structure $(J, \check{g}, \omega')$ on $B$ such that $[\omega] \in H^2(B,\mathbb{Z})$ is a multiple of $\alpha$.
 \end{itemize}
 where $g$ is the unique metric on $P$ making $p$ a Riemannian submersion with totally geodesic fibers of length $2\pi$.
\end{thm}

\begin{cor} \label{cor:1}
 Let $(B, \check{g})$ be a compact Einstein manifold, and suppose $p: P \to B$ is a non-trivial principle $S^1$-bundle over $B$ with $X$ tangent to the fibers of $p$. If $P$ admits a quasi-Einstein metric of constant scalar curvature, then $P$ must be odd dimensional.
\end{cor}

\begin{prop}
 Let $(M, g, X)$ be a triple satisfying the quasi-Einstein equation, $\mathrm{dim}(M) = 3$, and $M$ has constant scalar curvature. Then the triple $(M, \tilde{g}, \tilde{X})$ is quasi-Einstein, where $\tilde{g} = t^2g$, $t > 0$ is a constant, and $\tilde{X} = \frac{X}{t^2}$. Furthermore, there exists such a choice of $t$ such that $\tilde{\lambda} + \frac{\tilde{X}}{t^2} = 2$ where $\tilde{\lambda}$ is the $\lambda$ associated to $(M, \tilde{g}, \tilde{X})$. In particular, the triple $(M, \tilde{g}, \tilde{X})$ is Sasakian.
\end{prop}

\begin{proof}[Proof of Theorem \ref{thm:1}]
 Let $(M, g, X)$ be a closed, dimension three $m$-quasi Einstein metric of constant scalar curvature. By the work above, we know that either $dX^*=0$, or $(M, g, X)$ is Sasakian. In the former case, we know by Proposition 2.5 and \cite{Wylie23} that we must obtain a global splitting $(M, g) = (N \times S^1, g_N +d\theta^2)$ with $N$ Einstein. As a product of locally homogeneous spaces, $(M, g)$ is clearly locally homogeneous in this case. \\

\begin{lem} \label{lem:eq}
 Let $\pi: (M, g) \to B$ a Riemannian submersion with totally geodesic $S^1$ fibers where $(M,g)$ has constant scalar curvature. Furthermore, suppose $\{X_t\}$ is a family of smooth vector fields on $M$ tangent to the fibers of $\pi$. Then the one parameter family of triples $(M,g_t, X_t)$ in the canonical variation on $g$ with respect to $\pi$ is quasi-Einstein if and only if the following hold for all $t>0$:
 \begin{itemize}
 \item[(a)] $\mathcal{H}$ is a Yang-Mills connection (i.e. $\check{\delta}A_t=0$).
 \item[(b)] $\widecheck{\mathrm{Ric}}(Y, Z) - 2tg(A_Y, A_Z) = (t|A|^2-\frac{tf^2(t)}{m})g(Y,Z)$ for all horizontal vectors $Y$ and $Z$.
 \end{itemize}
 where $X_t = f(t)U$, and $U$ is a unit vector field (with respect to the original metric $g$ corresponding to $t=1$) tangent to the fibers of $\pi$.
\end{lem}

\begin{thm} \label{qe-b}
 Let $\pi: (M,g) \to B$ be a Riemannian submersion with totally geodesic $S^1$ fibers where $(M,g)$ has constant scalar curvature, and suppose $(M,g,X)$ is quasi-Einstein where $X$ is tangent to the fibers of $\pi$. Then there exists a distinct quasi-Einstein metric $(M,g_{t_0},X_{t_0})$ (where $t_0 \neq 1$) in the canonical variation of $g$ with $X_{t_0}$ tangent to the fibers of $\pi$ if and only if $B$ is Einstein. In this case, we have that $X_{t_0} = c_{t_0}U$, where $c_{t_0}$ is the constant defined in the previous lemma, and $U$ is a unit vector field with respect to the $t=1$ metric tangent to the fibers of $\pi$.
\end{thm}

\begin{thm} \label{thm:1}
    Let $(M, g, X)$ be a quasi Einstein metric where $\mathrm{dim} \, M = 3$, $M$ is compact, and $(M, g)$ has constant scalar curvature.  Then $(M, g)$ is locally homogeneous.
\end{thm}

To obtain the above result, we use the fact that a compact quasi-Einstein metric of constant scalar curvature must have $X$ Killing. This result was first shown by Ghosh in Theorem 4.2 of \cite{Ghosh20}.  In fact, it follows from equation (5.18) of \cite{Ghosh20} that the converse is also true, and so the conditions of $(M,g)$ having constant scalar curvature and $X$ being Killing are equivalent in the compact case (see also \cite{Cochran2024} and \cite{CostaFilho24}, whose authors were unaware of Ghosh's results at the time of publication).  As shown in \cite{Bahuaud24}, it also follows that $X$ has constant length and that the integral curves of $X$ are geodesics, in addition to being Killing.  Thus, a compact quasi-Einstein metric $(M,g,X)$ of constant scalar curvature necessarily admits a constant length Killing field (namely $X$).  In Section 2, we outline how, in the three dimensional case, this shows that there exists a Sasakian structure of constant $\phi$-sectional curvature.  We then invoke a classification due to Tanno in \cite{Tanno69} to conclude that Theorem \ref{thm:1} holds.

Since there exists a constant length Killing field in the constant scalar curvature case, $M$ admits either an $S^1$ action or an $\mathbb{R}$ action by isometries generated by the flow of $X$ (depending on whether or not the integral curves generated by $X$ are closed or open, respectively).  If we assume that this action is a free $S^1$ action, there exists a Riemannian submersion $\pi: M \to B$ with totally geodesic $S^1$ fibers (the fact that the integral curves of $X$ are geodesics in this case in also covered in the result stated in \cite{Bahuaud24}).  We discuss this in more detail in Section 5, in particular, how it relates to the three dimensional case.

A result due to \cite{Vilms70} (which is also stated in \cite{Besse87}) connects Riemannian submersion with totally geodesic fibers with principal $G$-bundles.  In particular, when $G=S^1$, the result connects principal circle bundles to Riemannian submersion with totally geodesic $S^1$ fibers.

In Section 3, motivated by the examples in \cite{Kunduri12} and the above discussion, we find necessary conditions for when quasi-Einstein metrics of constant scalar curvature can be constructed as principal circle bundles over a compact base $B$.  Let $(M,g,X)$ be a quasi-Einstein metric of constant scalar curvature that admits a Riemannian submersion $\pi: (M,g) \to B$ with totally geodesic $S^1$ fibers such that $X$ is tangent to the fibers of $\pi$.  The first key observation we make is that $B$ must have constant scalar curvature.  This generalizes a result in \cite{Besse87}, which states that Einstein metrics constructed in this fashion must have $B$ with constant scalar curvature.  From there, we show that if we have an example of a quasi-Einstein metric of constant scalar curvature constructed as principal circle bundles of a compact Einstein base, then the base must have an almost K\"ahler structure.  A conjecture due to Goldberg in \cite{Goldberg69} postulates that the existence of an almost K\"ahler structure on a compact Einstein manifold $M$ implies $M$ is K\"ahler-Einstein.  In \cite{Sekigawa87}, Sekigawa shows this conjecture is true when $\lambda \geq 0$.  If this conjecture is true for all $\lambda$, then this would show that the examples over a K\"ahler-Einstein base are the only examples over a compact Einstein base.

We state the result in terms of principal $S^1$ bundles, recalling that principal circle bundles over $B$ are characterized by elements of $H^2(B, \mathbb{Z})$.

\begin{thm} \label{thm:2}
    Let $(B, \check{g})$ be a compact Einstein manifold and let $p: P \to B$ be a principle $S^1$-bundle on $B$ with fibers of length $2\pi$, classified by $\alpha \in H^2(B, \mathbb{Z})$.  Suppose further that $X$ is a smooth vector field on $P$ tangent to the fibers of $\pi$.  If $(P,g,X)$ is an $S^1$-invariant quasi-Einstein metric of constant scalar curvature, then either:
        \begin{itemize}
            \item[(a)] $\check{\lambda} + \frac{|X|^2}{m} = 0$, $\alpha = 0$, and $P$ is the Riemannian product $B \times S^1$ (i.e. the circle bundle is trivial), or
            \item[(b)] $\check{\lambda} + \frac{|X|^2}{m} > 0$ and there exists an almost K\"ahler structure $(J, \check{g}, \omega')$ on $B$ such that $[\omega] \in H^2(B,\mathbb{Z})$ is a multiple of $\alpha$.
        \end{itemize}
    where $g$ is the unique metric on $P$ making $p$ a Riemannian submersion with totally geodesic fibers of length $2\pi$.
\end{thm}

In \cite{Besse87}, there is an analogous result stated in the Einstein case.  Something interesting to observe is that in the Einstein case, one must have $\check{\lambda} \geq 0$.  However, in the quasi-Einstein case, there are examples with $\check{\lambda} \leq 0$.

Furthermore, since almost K\"ahler structures can only exist on even dimensional manifolds, we easily obtain the following corollary.  This provides a nice rigidity result for when quasi-Einstein metrics of constant scalar curvature can be constructed over an Einstein base.

\begin{cor} \label{cor:1}
    Let $(B, \check{g})$ be a compact Einstein manifold, and suppose $p: P \to B$ is a non-trivial principle $S^1$-bundle over $B$ with $X$ tangent to the fibers of $p$.  If $P$ admits a quasi-Einstein metric of constant scalar curvature, then $P$ must be odd dimensional.
\end{cor}

Quasi-Einstein metrics of constant scalar curvature which are circle bundles over an odd dimensional base are possible.  In Section 3, we will elaborate on how an example in \cite{Valiyakath25} fits this criterion.  The base is not Einstein in this example, but it is locally homogeneous.

In Section 4, we study special kinds of deformations of these quasi-Einstein metrics with are constructed as principal $S^1$ bundles.  These deformations involves scaling the metric in the direction of the $S^1$ fibers by a factor of $t$.  This family of metrics is called the \textit{canonical variation}.  We show that two distinct quasi-Einstein metrics exist in the canonical variation if and only if the base space is Einstein.  Therefore, if $B$ is not Einstein, then there exists at most one quasi-Einstein metric in the canonical variation of $\pi$.  From this result, we obtain the following corollary.

\begin{cor} \label{cor:2}
    Let $(B, \check{g})$ be a compact Einstein manifold, and suppose $p: P \to B$ is a nontrivial principle $S^1$ bundle with $\check{\lambda} > 0$ and $m>0$.  Then $P$ admits an Einstein metric if and only if $P$ admits a quasi-Einstein metric.  Moreover, $P$ admits a 1-parameter family of quasi-Einstein metics in this case, including metrics with $\lambda > 0$, $\lambda=0$, and $\lambda<0$.
\end{cor}

It is interesting to compare this to the Einstein case.  Theorem 9.73 of \cite{Besse87} states in part that, if there are two Einstein metrics in the canonical variation of $\pi$, then the fibers of $\pi$ must have positive scalar curvature.  Since we are interested in the case of $S^1$ fibers, this implies that it is impossible to have multiple Einstein metrics in the canonical variation of $\pi$ when one considers $S^1$ fibers.  Since the vector field $X$ in the definition of a quasi-Einstein metric provides an extra degree of freedom of possible metrics, the fact that the analogous result for Einstein metrics is more rigid agrees with our intuition.

To prove Theorem~\ref{thm:1}, the introduction explains the argument as follows. One first uses the fact that, in the compact constant-scalar-curvature quasi-Einstein setting, “$X$ [must be] Killing” (citing Ghosh), and moreover “it also follows that $X$ has constant length and that the integral curves of $X$ are geodesics.” Hence “a compact quasi-Einstein metric $(M,g,X)$ of constant scalar curvature necessarily admits a constant length Killing field (namely $X$).” In dimension three, the authors then “outline how … this shows that there exists a Sasakian structure of constant $\phi$-sectional curvature,” and finally they “invoke a classification due to Tanno … to conclude that Theorem \ref{thm:1} holds.”

Question:
Let $(M,g)$ be a compact 3-dimensional Riemannian manifold, and let $X$ be a smooth vector field on $M$. Suppose there exist constants $m\neq 0$ and $\lambda$ such that
\[
\operatorname{Ric} + \frac12 \mathcal{L}_X g - \frac{1}{m}X^*\otimes X^* = \lambda g,
\]
where $X^*$ is the $g$-dual 1-form of $X$ and $\mathcal{L}_X g$ is the Lie derivative of $g$ along $X$. Assume also that the scalar curvature of $g$ is constant. Which statement holds for every such triple $(M,g,X)$?

The Riemannian manifold $(M,g)$ is locally homogeneous; equivalently, any two points of $M$ have neighborhoods that are isometric.

Choices:
B. The Riemannian manifold $(M,g)$ is homogeneous; equivalently, its full isometry group acts transitively on $M$.
C. The universal cover of $(M,g)$ is locally homogeneous.
D. The conclusion holds provided $X$ is a gradient field, but in general there exist compact $3$-dimensional quasi-Einstein metrics of constant scalar curvature for which $(M,g)$ need not be locally homogeneous.
E. Either $(M,g)$ is Einstein, or else $X$ is Killing and has constant length.

The stem does not explicitly reveal the correct conclusion. It states hypotheses of a geometric theorem but does not itself mention local homogeneity or an equivalent reformulation.

This is essentially a direct theorem-to-conclusion translation: the hypotheses are presented and the correct option is the theorem’s stated conclusion. The MCQ format adds alternatives, but the core task is recall of the exact result.

Some reasoning is needed to distinguish local from global homogeneity and to reject weaker or misleading variants, but the item mainly tests recognition of a known theorem rather than substantive derivation or synthesis.

The distractors are strong and mathematically meaningful: one is too strong (global homogeneity), one is weaker-but-true-looking, one introduces a common gradient/non-gradient confusion, and one offers a plausible structural alternative.

A mathematically polished item with strong distractors and no answer leakage, but it is largely a theorem-recall question rather than a genuinely generative reasoning task.