\begin{lemma}[see \cite{lorenz}]\label{lemalorenz}
	Let $a>0$, $\sigma> 1$ and $s\in[0,\frac{3}{2})$. Then, there exists a positive constant $C_{a,\sigma,s}$ such that, for all $f,g\in \dot{H}_{a,\sigma}^{s}(\mathbb{R}^3)$, we have
	$$\|fg\|_{\dot{H}_{a,\sigma}^{s}}\leq C_{a,\sigma,s}	\|f\|_{\dot{H}_{a,\sigma}^{s}}\|g\|_{\dot{H}_{a,\sigma}^{s}}.$$
\end{lemma}

\begin{theorem}\label{teoremaexistenciaB}
Assume that $a>0$, $\sigma>1$, $\alpha>\frac{1}{2}$, $\beta>\frac{1}{2}$ and $0\leq s<\frac{3}{2}$. Let $(u_0,\theta_0)\in \dot{H}_{a,\sigma}^s$ such that $\hbox{\emph{div}}\,u_0=0$. Then,
  there exist an instant $T=T(a,\sigma,s,\alpha,\beta,u_0,\theta_0)>0$ and a unique mild solution $(u,\theta)\in C_T(\dot{H}_{a,\sigma}^s)$ for the Boussinesq equations \emph{(\ref{micropolar})} that satisfies
    \begin{align*}
  &\|(u,\theta)\|_{L^\infty_T(\dot{H}_{a,\sigma}^{s})}
\lesssim \frac{1-T}{T^{1-\frac{1}{2\alpha}}+T^{1-\frac{1}{2\beta}}}.
  \end{align*}
 \end{theorem}

\begin{lemma}[see \cite{pontofixo}]\label{pontofixo}
		Let $(X,\|\cdot\|)$ be a Banach space, $L:X\rightarrow X$ a continuous linear operator and
 $B:X\times X\rightarrow X$ a continuous bilinear operator, i.e., there exist positive constants $C_1$ and $C_2$ such that
 $$\|L(x)\|\leq C_1 \|x\|,\quad \|B(x,y)\|\leq C_2 \|x\|\|y\|,\quad \forall x,y\in X.$$
 Then, for each $C_1\in (0,1)$ and $x_0\in X$ that satisfy $4C_2\|x_0\| <(1-C_1)^2$, one has that the equation
 $$a=x_0+B(a,a)+L(a), \quad a\in X,$$
 admits solution $x\in X$. Moreover, $x$ satisfies the inequality $\|x\|\leq \frac{2\|x_0\|}{1-C_1}$ and this is the only one such that $\|x\|\leq  \frac{1-C_1}{2C_2}.$
	\end{lemma}

\begin{align}\label{lwfinal}
\displaystyle\|L(w,v)\|_{X}\leq  T \|(w,v)\|_{X},\quad\forall (w,v)\in X.
\end{align}

\begin{lemma}[see \cite{robert}]\label{lemanovow1}
Let $a\geq0$, $\sigma\geq1$, $s\in\mathbb{R}$ and $\theta\geq1$. The following inequality holds:
$$\|f\|_{\dot{H}_{a,\sigma}^{s+1}}\leq \|f\|_{\dot{H}_{a,\sigma}^{s}}^{1-\frac{1}{\theta}}\|f\|_{\dot{H}_{a,\sigma}^{s+\theta}}^{\frac{1}{\theta}}.$$
  \end{lemma}

\begin{align}\label{wilber17}
\nonumber\displaystyle\|B_1((w,v),(\gamma,\phi))(t)\|_{\dot{H}_{a,\sigma}^s}&\leq  \int_0^t\Big(\int_{\mathbb{R}^3} |\xi|^{2}e^{-2(t-\tau)|\xi|^{2\alpha}}|\xi|^{2s} e^{2a|\xi|^{\frac{1}{\sigma}}} |\mathcal{F}(\gamma\otimes w)(\xi)|^2\,d\xi\Big)^{\frac{1}{2}}\,d\tau\\
\nonumber&\leq C_{\alpha} \int_0^t (t-\tau)^{-\frac{1}{2\alpha}}\Big(\int_{\mathbb{R}^3} |\xi|^{2s} e^{2a|\xi|^{\frac{1}{\sigma}}} |\mathcal{F}(\gamma\otimes w)(\xi)|^2\,d\xi\Big)^{\frac{1}{2}}\,d\tau\\
&\leq C_{\alpha} \int_0^t (t-\tau)^{-\frac{1}{2\alpha}} \|(\gamma\otimes w)(\tau)\|_{\dot{H}_{a,\sigma}^{s}}\,d\tau.
\end{align}

\begin{align}\label{wilber25}
 \frac{1}{2}\partial_t|\widehat{u}(t)|^2+|\xi|^{2\alpha}|\widehat{ u}|^2 &\leq
|\widehat{u}\cdot \widehat{u\cdot\nabla \displaystyle u}|+|\widehat{u}_3 \widehat{\theta}|.
\end{align}

\begin{align}
\nonumber\displaystyle\|B_1((w,v),(\gamma,\phi))(t)\|_{\dot{H}_{a,\sigma}^s}
&\leq C_{a,\sigma,s,\alpha}\|\gamma\|_{L_T^{\infty}(\dot{H}_{a,\sigma}^{s})} \|w\|_{L_T^{\infty}(\dot{H}_{a,\sigma}^{s})}\int_0^t (t-\tau)^{-\frac{1}{2\alpha}} d\tau\\
\label{w1}&\leq C_{a,\sigma,s,\alpha}T^{1-\frac{1}{2\alpha}}\|(w,v)\|_{X} \|(\gamma,\phi)\|_{X},
\end{align}

\begin{align}\label{wilber19}
\nonumber &I_2=(I_{21}^{(1)}+I_{21}^{(2)},I_{22}):= \Big(\int_{0}^{\kappa_m}[e^{- (\kappa_m-\tau)(-\Delta)^\alpha}-e^{- (\kappa_n-\tau)(-\Delta)^\alpha}] \mathbb{P}[u\cdot \nabla u]\,d\tau+\\ &
\int_{0}^{\kappa_m}[e^{-  (\kappa_n-\tau)(-\Delta)^\alpha}-e^{- (\kappa_m-\tau)(-\Delta)^\alpha}] \mathbb{P}[\theta e_3]\,d\tau,\int_{0}^{\kappa_m}[e^{- (\kappa_m-\tau)(-\Delta)^\beta}-e^{- (\kappa_n-\tau)(-\Delta)^\beta}] (u\cdot \nabla \theta)\,d\tau\Big),
\end{align}

\begin{align}\label{3}
\frac{1}{2}\frac{d}{dt}\|u(t)\|_{\dot{H}_{a,\sigma}^s}^2+\|u(t)\|_{\dot{H}_{a,\sigma}^{s+\alpha}}^2 &\leq
|\langle u, u\cdot\nabla u\rangle_{\dot{H}_{a,\sigma}^s}|+  |\langle u_3,\theta\rangle_{\dot{H}_{a,\sigma}^s} |.
\end{align}

\hspace{0.5cm}
In this paper, we consider the following three-dimensional Boussinesq equations:
\begin{equation}\label{micropolar}
\left\{
\begin{array}{l}
u_t
\;\!+\,
u \cdot \nabla u
\,+\,
\nabla \;\!p \:\!\,+\,  \:\!(-\Delta)^\alpha u
\;=\;
\theta e_3, \quad x\in \mathbb{R}^3, \quad t > 0,\\
\theta_t
\;\!+\,
u \cdot \nabla \theta\,+\, \,(-\Delta)^\beta \theta
\;=\;
0, \quad x\in \mathbb{R}^3, \quad t > 0,\\
\mbox{div}\:u  \;=\; 0, \quad x\in \mathbb{R}^3, \quad t > 0,\\
u(x,0) \,=\, u_0(x),
\;\,
\theta(x,0) \,=\, \theta_0(x), \quad x\in \mathbb{R}^3,
\end{array}
\right.
\end{equation}
where $u(x,t) = \left(u_{1}(x,t), u_{2}(x,t), u_{3}(x,t)\right) \in \mathbb{R}^3$ denotes the incompressible velocity field, and
$\theta(x,t) \in \mathbb{R}$ represents the temperature of the fluid. The parameters $\alpha$ and $\beta$ belong to
$(\tfrac{1}{2},\infty)$, and we set $e_{3} = (0,0,1)$. The initial velocity field $u_{0}$ in \eqref{micropolar} is assumed to be divergence-free, that is,
$$
\operatorname{div} u_{0} = 0,
$$
and we denote by $(-\Delta)^{\gamma}$ the fractional Laplacian, defined for a suitable function $f$ by
$$
\mathcal{F}[(-\Delta)^{\gamma} f](\xi) = |\xi|^{2\gamma} \widehat{f}(\xi) .
$$

Observe that, in the case $\theta \equiv 0$ and $\alpha = 1$, the system \eqref{micropolar} reduces to the classical incompressible Navier-Stokes equations
\begin{equation}\label{NS}
\left\{
\begin{array}{l}
u_t
\;\!+\,
u \cdot \nabla u
\,+\,
\nabla \;\!p
\;=\;
\Delta u, \quad x\in \mathbb{R}^3, \quad t > 0,\\

\mbox{div}\:u  \;=\; 0, \quad x\in \mathbb{R}^3, \quad t > 0.

\end{array}
\right.
\end{equation}

The Boussinesq equations (\ref{micropolar}) are simplified yet powerful models widely used in the study of oceanic and atmospheric dynamics. By incorporating buoyancy effects while filtering out sound waves, they provide a mathematically tractable framework for describing stratified fluid flows under the influence of gravity. These equations also arise in several other areas of Physics, including thermal convection, geophysical fluid dynamics, and plasma modeling; see, for instance, \cite{MR2245751,MR2540168} for more details.

It is worth noting, however, that although \eqref{micropolar} with fractional dissipation may initially appear to be a purely mathematical generalization, there are geophysical scenarios in which the Boussinesq equations  with a fractional Laplacian (\ref{micropolar}) naturally arise. A typical example occurs in the middle atmosphere, where upward-travelling flows experience changes due to variations in atmospheric properties, even though the incompressibility and Boussinesq approximations remain valid. In this regime, the effects of kinematic and thermal diffusion are attenuated by the thinning of the atmosphere, an anomalous behavior that can be effectively modeled using a spatial fractional Laplacian. See \cite{MR2379269, Gill1982} for further discussions.

There is an extensive literature, in both two and three dimensions, concerning the well-posedness and qualitative properties of the system \eqref{micropolar} in a variety of functional settings (see, for instance, \cite{MR4884564,MR3503190,MR4795499,MR3808344,MR3008326,MR3759571,cilon1} and the references therein).

In this work, we are concerned with the time evolution of solutions. More precisely, we aim to establish finite-time blow-up criteria for mild solutions of \eqref{micropolar} in Sobolev-Gevrey type spaces. For $a \ge 0$, $\sigma \ge 1$, and $s \in \mathbb{R}$, the (homogeneous) Sobolev-Gevrey space is defined by
$$
\dot{H}_{a,\sigma}^s=\dot{H}_{a,\sigma}^s(\mathbb{R}^3)
    = \left\{ f \in \mathcal{S}' : \widehat{f} \in L^1_{\operatorname{loc}}\,\, \text{and} \,\, \
       \|f\|_{\dot{H}_{a,\sigma}^s}:=\Big[\int_{\mathbb{R}^3} |\xi|^{2s} e^{2a |\xi|^{1/\sigma}} |\widehat{f}(\xi)|^2\, d\xi\Big]^{\frac{1}{2}} < \infty
      \right\},
$$
where $\widehat{f}$ denotes the Fourier transform
$$
\mathcal{F}(f)(\xi)=\widehat{f}(\xi):=\int_{\mathbb{R}^3}e^{-i\xi\cdot x}f(x)\,dx.
$$
Essentially, $\dot{H}_{a,\sigma}^s \equiv L^{2}\!(|\xi|^{s} e^{a|\xi|^{1/\sigma}}\, d\xi)$ and, in particular, $\dot{H}_{0,\sigma}^s \equiv \dot{H}^s$, \textit{i.e.}, the classical (homogeneous) Sobolev space.  This class of functions plays a crucial role because, according to  Paley--Wiener Theorem (see \cite{HormanderALPDO1}, Chapter~7, for further details), for $\sigma=1$ a function $ f $ belongs to the space $ \dot{H}_{a,1}^s(\mathbb{R}^d) $ if, and only if, it admits a holomorphic extension $ F $ to the strip

$$
S_a = \{ x + i y  \in \mathbb{C}^d: x, y \in \mathbb{R}^d,\ |y| < a \},
$$
such that
$$
    \sup_{|y| < a} \| F(x + i y) \|_{\dot{H}^s} < \infty.
$$
In other words, the parameter $ a \geq 0 $ determines the width of the complex strip to which
functions in $ \dot{H}_{a,1}^s(\mathbb{R}^d) $ can be analytically extended. For $\sigma>1$, the regularity falls into a non-analytic Gevrey regime, meaning that derivatives grow in a controlled but non-analytic manner, which is equivalently reflected by a subexponential decay at high frequencies.

Consequently, the study of
differential equations in such analyticity-based function classes has attracted significant attention in recent years. See, for instance, \cite{MR3504420, MR4632081, MR4369830, MR2169876, MR1026858, MR2265624} and references therein for more details on this topic.

The first main result of this work establishes the local well-posedness of the Cauchy problem \eqref{micropolar} in  Sobolev-Gevrey class. Recall that, we say that $(u,\theta)=(u,\theta)(x,t)$ is a mild solution to the Boussinesq equations (\ref{micropolar}) if this application satisfies the associated integral formulations (\ref{wilber15}) and (\ref{wilber16}), which are established via the fractional heat semigroup.

\hspace{0.5cm}
In this paper, we consider the following three-dimensional Boussinesq equations:
\begin{equation}\label{micropolar}
\left\{
\begin{array}{l}
u_t
\;\!+\,
u \cdot \nabla u
\,+\,
\nabla \;\!p \:\!\,+\, \:\!(-\Delta)^\alpha u
\;=\;
\theta e_3, \quad x\in \mathbb{R}^3, \quad t > 0,\\
\theta_t
\;\!+\,
u \cdot \nabla \theta\,+\, \,(-\Delta)^\beta \theta
\;=\;
0, \quad x\in \mathbb{R}^3, \quad t > 0,\\
\mbox{div}\:u \;=\; 0, \quad x\in \mathbb{R}^3, \quad t > 0,\\
u(x,0) \,=\, u_0(x),
\;\,
\theta(x,0) \,=\, \theta_0(x), \quad x\in \mathbb{R}^3,
\end{array}
\right.
\end{equation}
where $u(x,t) = \left(u_{1}(x,t), u_{2}(x,t), u_{3}(x,t)\right) \in \mathbb{R}^3$ denotes the incompressible velocity field, and
$\theta(x,t) \in \mathbb{R}$ represents the temperature of the fluid. The parameters $\alpha$ and $\beta$ belong to
$(\tfrac{1}{2},\infty)$, and we set $e_{3} = (0,0,1)$. The initial velocity field $u_{0}$ in \eqref{micropolar} is assumed to be divergence-free, that is,
$$
\operatorname{div} u_{0} = 0,
$$
and we denote by $(-\Delta)^{\gamma}$ the fractional Laplacian, defined for a suitable function $f$ by
$$
\mathcal{F}[(-\Delta)^{\gamma} f](\xi) = |\xi|^{2\gamma} \widehat{f}(\xi) .
$$

Observe that, in the case $\theta \equiv 0$ and $\alpha = 1$, the system \eqref{micropolar} reduces to the classical incompressible Navier-Stokes equations
\begin{equation}\label{NS}
\left\{
\begin{array}{l}
u_t
\;\!+\,
u \cdot \nabla u
\,+\,
\nabla \;\!p
\;=\;
\Delta u, \quad x\in \mathbb{R}^3, \quad t > 0,\\

In this work, we are concerned with the time evolution of solutions. More precisely, we aim to establish finite-time blow-up criteria for mild solutions of \eqref{micropolar} in Sobolev-Gevrey type spaces. For $a \ge 0$, $\sigma \ge 1$, and $s \in \mathbb{R}$, the (homogeneous) Sobolev-Gevrey space is defined by
$$
\dot{H}_{a,\sigma}^s=\dot{H}_{a,\sigma}^s(\mathbb{R}^3)
 = \left\{ f \in \mathcal{S}' : \widehat{f} \in L^1_{\operatorname{loc}}\,\, \text{and} \,\, \
 \|f\|_{\dot{H}_{a,\sigma}^s}:=\Big[\int_{\mathbb{R}^3} |\xi|^{2s} e^{2a |\xi|^{1/\sigma}} |\widehat{f}(\xi)|^2\, d\xi\Big]^{\frac{1}{2}} < \infty
 \right\},
$$
where $\widehat{f}$ denotes the Fourier transform
$$
\mathcal{F}(f)(\xi)=\widehat{f}(\xi):=\int_{\mathbb{R}^3}e^{-i\xi\cdot x}f(x)\,dx.
$$
Essentially, $\dot{H}_{a,\sigma}^s \equiv L^{2}\!(|\xi|^{s} e^{a|\xi|^{1/\sigma}}\, d\xi)$ and, in particular, $\dot{H}_{0,\sigma}^s \equiv \dot{H}^s$, \textit{i.e.}, the classical (homogeneous) Sobolev space. This class of functions plays a crucial role because, according to Paley--Wiener Theorem (see \cite{HormanderALPDO1}, Chapter~7, for further details), for $\sigma=1$ a function $ f $ belongs to the space $ \dot{H}_{a,1}^s(\mathbb{R}^d) $ if, and only if, it admits a holomorphic extension $ F $ to the strip

$$
S_a = \{ x + i y \in \mathbb{C}^d: x, y \in \mathbb{R}^d,\ |y| < a \},
$$
such that
$$
 \sup_{|y| < a} \| F(x + i y) \|_{\dot{H}^s} < \infty.
$$
In other words, the parameter $ a \geq 0 $ determines the width of the complex strip to which
functions in $ \dot{H}_{a,1}^s(\mathbb{R}^d) $ can be analytically extended. For $\sigma>1$, the regularity falls into a non-analytic Gevrey regime, meaning that derivatives grow in a controlled but non-analytic manner, which is equivalently reflected by a subexponential decay at high frequencies.

Consequently, the study of
differential equations in such analyticity-based function classes has attracted significant attention in recent years. See, for instance, \cite{MR3504420, MR4632081, MR4369830, MR2169876, MR1026858, MR2265624} and references therein for more details on this topic.

The first main result of this work establishes the local well-posedness of the Cauchy problem \eqref{micropolar} in Sobolev-Gevrey class. Recall that, we say that $(u,\theta)=(u,\theta)(x,t)$ is a mild solution to the Boussinesq equations (\ref{micropolar}) if this application satisfies the associated integral formulations (\ref{wilber15}) and (\ref{wilber16}), which are established via the fractional heat semigroup.

\begin{equation}\label{micropolar}
\left\{
\begin{array}{l}
u_t
\;\!+\,
u \cdot \nabla u
\,+\,
\nabla \;\!p \:\!\,+\,  \:\!(-\Delta)^\alpha u
\;=\;
\theta e_3, \quad x\in \mathbb{R}^3, \quad t > 0,\\
\theta_t
\;\!+\,
u \cdot \nabla \theta\,+\, \,(-\Delta)^\beta \theta
\;=\;
0, \quad x\in \mathbb{R}^3, \quad t > 0,\\
\mbox{div}\:u  \;=\; 0, \quad x\in \mathbb{R}^3, \quad t > 0,\\
u(x,0) \,=\, u_0(x),
\;\,
\theta(x,0) \,=\, \theta_0(x), \quad x\in \mathbb{R}^3,
\end{array}
\right.
\end{equation}

\begin{align}\label{wilber15}
u(t)= e^{- t(-\Delta)^{\alpha}}u_0 - \int_{0}^te^{- (t-\tau)(-\Delta)^{\alpha}} \mathbb{P}(u\cdot\nabla u)\,d\tau + \int_{0}^te^{- (t-\tau)(-\Delta)^{\alpha}} \mathbb{P}(\theta e_3)\,d\tau,\quad\forall t>0.
\end{align}

\begin{align}\label{wilber16}
\theta(t)= e^{- t(-\Delta)^{\beta}}\theta_0 - \int_{0}^te^{-  (t-\tau)(-\Delta)^{\beta}} [u\cdot \nabla\theta]\,d\tau ,\quad\forall t>0.
\end{align}

The first main result of this work establishes the local well-posedness of the Cauchy problem \eqref{micropolar} in Sobolev-Gevrey class. Recall that, we say that $(u,\theta)=(u,\theta)(x,t)$ is a mild solution to the Boussinesq equations (\ref{micropolar}) if this application satisfies the associated integral formulations (\ref{wilber15}) and (\ref{wilber16}), which are established via the fractional heat semigroup.

Next, we investigate the qualitative behavior of the local solution provided by Theorem \ref{teoremaexistenciaB}, assuming that its maximal lifespan is finite.

First of all, apply Helmholtz's projector $\mathbb{P}$, defined via Fourier transform (see \cite{PN} for more details)
$$
\mathcal{F}[\mathbb{P}f](\xi) := \widehat{f}(\xi) - \frac{\widehat{f}(\xi)\cdot \xi}{|\xi|^2}\, \xi,
$$
the heat semigroup $e^{-(t-\tau)(-\Delta)^{\alpha}}$ to the first equation in (\ref{micropolar}), and integrate the result obtained over $[0,t]$ to obtain
\begin{align}\label{wilber15}
u(t)= e^{- t(-\Delta)^{\alpha}}u_0 - \int_{0}^te^{- (t-\tau)(-\Delta)^{\alpha}} \mathbb{P}(u\cdot\nabla u)\,d\tau + \int_{0}^te^{- (t-\tau)(-\Delta)^{\alpha}} \mathbb{P}(\theta e_3)\,d\tau,\quad\forall t>0.
\end{align}
Similarly, by using the heat semigroup $e^{- (t-\tau)(-\Delta)^{\beta}}$ in the second equation of (\ref{micropolar}), and integrating over $[0,t]$, we deduce
\begin{align}\label{wilber16}
\theta(t)= e^{- t(-\Delta)^{\beta}}\theta_0 - \int_{0}^te^{- (t-\tau)(-\Delta)^{\beta}} [u\cdot \nabla\theta]\,d\tau ,\quad\forall t>0.
\end{align}
Hence, we are able to write the following equality:
\begin{align}\label{estimativaprojecao3}
(u,\theta)(t)&= (e^{- t(-\Delta)^{\alpha}}u_0,e^{- t (-\Delta)^{\beta}}\theta_0) + B((u,\theta),(u,\theta))(t)+L(u,\theta)(t),\quad\forall t>0.
\end{align}
where
\begin{align}\label{bilinearB}
B((w,v),(\gamma,\phi))(t)=(B_1((w,v),(\gamma,\phi)),B_2((w,v),(\gamma,\phi)))(t),\quad\forall t>0,
\end{align}
and also
\begin{align}\label{linearL}
L(w,v)(t)=(L_1(w,v)(t),L_2(w,v)(t)),\quad\forall t>0,
\end{align}
with
\begin{align}\label{bilinearB1}
B_1((w,v),(\gamma,\phi))(t)= - \int_{0}^te^{- (t-\tau)(-\Delta)^{\alpha}} \mathbb{P}(w\cdot\nabla \gamma)\,d\tau,
\end{align}
\begin{align}\label{bilinearB2}
B_2((w,v),(\gamma,\phi))(t)= - \int_{0}^te^{- (t-\tau)(-\Delta)^{\beta}} [w\cdot \nabla \phi]\,d\tau,
\end{align}
\begin{align}\label{linearL1eL2}
L_1(w,v)(t)= \int_{0}^te^{- (t-\tau)(-\Delta)^{\alpha}} \mathbb{P}[ve_3]\,d\tau \quad \hbox{ and }\quad L_2(w,v)(t)= 0,
\end{align}
for all $w,v,\gamma,\phi\in C_T(\dot{H}_{a,\sigma}^s)$ ($T>0$ will be revealed below). Denote $X= [C_T (\dot{H}_{a,\sigma}^{s})]^3$ $\times C_T (\dot{H}_{a,\sigma}^{s}) (\equiv C_T (\dot{H}_{a,\sigma}^{s})$ throughout this work)$\footnote{Here, $\|(f,g)\|_{X}:=[\|f\|^2_{L^\infty_T(\dot{H}_{a,\sigma}^s)}+\|g\|^2_{L^\infty_T(\dot{H}_{a,\sigma}^s)}]^{\frac{1}{2}}$}$ and notice that $B$ and $L$ are bilinear and linear operators on $X\times X$ and $X$, respectively.

At last, notice that
\begin{align}\label{mudanca1}
\displaystyle\|e^{- t(-\Delta)^{\alpha}} u_0\|^2_{\dot{H}_{a,\sigma}^s}&=\int_{\mathbb{R}^3} e^{-2 t|\xi|^{2\alpha}}|\xi|^{2s} e^{2a|\xi|^{\frac{1}{\sigma}}} |\widehat{u}_0(\xi)|^2\,d\xi\leq \|u_0\|_{\dot{H}_{a,\sigma}^{s}}^2,\quad\forall t\in [0,T],
\end{align}
and also that
\begin{align}\label{mudanca2}
\displaystyle\|e^{- t(-\Delta)^{\beta}} \theta_0\|_{\dot{H}_{a,\sigma}^s}\leq \|\theta_0\|_{\dot{H}_{a,\sigma}^{s}},\quad\forall t\in [0,T].
\end{align}
From (\ref{mudanca1}) and (\ref{mudanca2}), it follows that
\begin{align}\label{dadoincial}
\|(e^{-t(-\Delta)^{\alpha}}u_0,e^{-t (-\Delta)^{\beta}}\theta_0)\|_{X}\leq\|(u_0,\theta_0)\|_{\dot{H}_{a,\sigma}^{s}}.
\end{align}
Thus, we are able to determine $T>0$ as follows:
$$T< \min\Big\{\Big[\sqrt{8C_{a,\sigma,s,\alpha,\beta}\|(u_0,\theta_0)\|_{\dot{H}_{a,\sigma}^{s}}}+1\Big]^{-\frac{4\alpha}{2\alpha-1}},\Big[\sqrt{8C_{a,\sigma,s,\alpha,\beta}\|(u_0,\theta_0)\|_{\dot{H}_{a,\sigma}^{s}}}+1\Big]^{-\frac{4\beta}{2\beta-1}}\Big\},$$
where $C_{a,\sigma,s,\alpha,\beta}$ is established in (\ref{w5}). Therefore, by Lemma \ref{pontofixo}, there exists a unique solution $(u,\theta)\in X$ for the equation (\ref{estimativaprojecao3}) (it is enough to take a look at (\ref{lwfinal}), (\ref{w5}) and (\ref{dadoincial})) that satisfies the following inequality:
$$\|(u,\theta)\|_{X}\leq \frac{1-T}{2C_{a,\sigma,s,\alpha,\beta}[T^{1-\frac{1}{2\alpha}}+T^{1-\frac{1}{2\beta}}]}.$$

Now, let us assume that $\displaystyle \limsup_{t\nearrow T^*} \|(u,\theta)(t)\|_{\dot{H}_{a,\sigma}^{s}} < \infty$. Thus, apply Theorem \ref{teoremaexistenciaB} to obtain a positive constant $C_{a,\sigma,s}$ such that
\begin{align}\label{limitacao}
\displaystyle \|(u,\theta)(t)\|_{\dot{H}_{a,\sigma}^{s}} \leq C_{a,\sigma,s},\quad\forall\,t\in[0,T^*).
\end{align}
Thereby, by integrating over $[0,t]$ the inequality (\ref{esqueci}) and using (\ref{limitacao}), we can write the following inequality:
\begin{align*}
&\nonumber\|(u,\theta)(t)\|_{\dot{H}_{a,\sigma}^s}^2+\int_0^t\|u(\tau)\|_{\dot{H}_{a,\sigma}^{s+\alpha}}^2+\int_0^t\|\theta(\tau)\|_{\dot{H}_{a,\sigma}^{s+\beta}}^2\,d\tau \leq \|(u_0,\theta_0)\|_{\dot{H}_{a,\sigma}^s}^2+C_{a,\sigma,s,\alpha,\beta}T^*,
\end{align*}
for all $t\in[0,T^*).$ As a result, particularly, one has
\begin{align}\label{e4}
\int_0^t\|u(\tau)\|_{\dot{H}_{a,\sigma}^{s+\alpha}}^2d\tau+\int_0^t\|\theta(\tau)\|_{\dot{H}_{a,\sigma}^{s+\beta}}^2\,d\tau \leq C_{a,\sigma,s,\alpha,\beta,u_0,\theta_0,T^*},\quad\forall\,t\in[0,T^*).
\end{align}

On the other hand, take $L^2$-inner product of the Boussinesq equations (\ref{micropolar}), with $u$ and $\theta$, respectively, and integrate the results obtained over $[0,t]$ in order to be capable of writing the following inequality:
\begin{align}\label{wilber32}
\frac{1}{2}\frac{d}{dt}\|(u,\theta)(t)\|_{L^2}^2+\|(-\Delta)^{\frac{\alpha}{2}}u(t)\|_{L^2}^2+\|(-\Delta)^{\frac{\beta}{2}}\theta(t)\|_{L^2}^2 &\leq \|(u,\theta)(t)\|_{L^2}^{2},\quad\forall t\in [0,T^*).
\end{align}
By Gronwall's Lemma, we have
\begin{align}\label{normal2}
 \|(u,\theta)(t)\|_{L^2}\leq e^{T^*}\|(u_0,\theta_0)\|_{L^2},\quad\forall t\in [0,T^*).
\end{align}
Thereby, by using (\ref{wilber12}) and (\ref{normal2}), one infers
\begin{align*}
\frac{C_{s,\alpha,u_0,\theta_0,T^*}}{[e^{C_\alpha(T^*-t)}-1]^{\frac{2s(2\alpha-1)}{3\alpha}}}\left(\frac{C'_{\sigma,s,\alpha,u_0,\theta_0,T^*}}{[e^{C_\alpha(T^*-t)}-1]^{\frac{2\alpha-1}{3\alpha\sigma}}}\right)^k \leq \|(u,\theta)(t)\|_{\dot{H}^{s+\frac{k}{2\sigma}}}^2,\quad\forall t\in [0,T^*).
\end{align*}
As a consequence, we have the inequality below:
\begin{align}\label{wnovo1}
\frac{C_{s,\alpha,u_0,\theta_0,T^*}}{[e^{C_\alpha(T^*-t)}-1]^{\frac{2s(2\alpha-1)}{3\alpha}}}\frac{\left(\frac{2aC'_{\sigma,s,\alpha,u_0,\theta_0,T^*}}{[e^{C_\alpha(T^*-t)}-1]^{\frac{2\alpha-1}{3\alpha\sigma}}}\right)^k }{k!} &\leq \int_{\mathbb{R}^3} \frac{(2a|\xi|^{\frac{1}{\sigma}})^k}{k!}|\xi|^{2s}|(\widehat{u},\widehat{\theta})(t)|^{2}\,d\xi,\quad\forall t\in [0,T^*).
\end{align}
Hence, by using Monotone Convergence Theorem in the inequality (\ref{wnovo1}), we deduce
\begin{align}\label{wilber14}
\nonumber\frac{C_{s,\alpha,u_0,\theta_0,T^*}}{[e^{C_\alpha(T^*-t)}-1]^{\frac{2s(2\alpha-1)}{3\alpha}}}\sum_{k=2\sigma_0+1}^{\infty}\frac{\left(\frac{2aC'_{\sigma,s,\alpha,u_0,\theta_0,T^*}}{[e^{C_\alpha(T^*-t)}-1]^{\frac{2\alpha-1}{3\alpha\sigma}}}\right)^k }{k!} &\leq \int_{\mathbb{R}^3} \sum_{k=2\sigma_0+1}^{\infty}\frac{(2a|\xi|^{\frac{1}{\sigma}})^k}{k!}|\xi|^{2s}|(\widehat{u},\widehat{\theta})(t)|^{2}\,d\xi\\
&\leq \|(u,\theta)(t)\|_{\dot{H}_{a,\sigma}^s}^2,
\end{align}
for all $t\in [0,T^*)$, where $2\sigma_0$ is the integer part of $2\sigma\mu$.

The proof of this result relies on a fixed-point argument. However, for both the classical and the fractional Boussinesq equations, a major difficulty arises in handling the coupling terms $\theta e_{3}$ and $(u \cdot \nabla)\theta$, which demand delicate estimates within the chosen functional framework.

Next, we investigate the qualitative behavior of the local solution provided by Theorem \ref{teoremaexistenciaB}, assuming that its maximal lifespan is finite.

\begin{theorem}\label{teoremaB}
Assume that $a>0$, $\sigma>1$, $\alpha\geq 1, \beta \geq 1, 0\leq s<\frac{3}{2}$ and $n\in \mathbb{N}$. Let $(u_0,\theta_0)\in \dot{H}_{a,\sigma}^s$ such that $\hbox{\emph{div}}\,u_0=0$. Consider that $(u,\theta)\in C([0,T^*);\dot{H}_{a,\sigma}^s)$ is the  solution
for the Boussinesq equations \emph{(\ref{micropolar})} in the maximal time interval $0\leq t < T^*$  given in Theorem \emph{\ref{teoremaexistenciaB}}. If $T^*<\infty$, then the following statements hold:
      \begin{enumerate}
  \item[\textbf{\emph{i)}}]  $\displaystyle \limsup_{t\nearrow T^*} \|(u,\theta)(t)\|_{\dot{H}_{\frac{a}{(\sqrt{\sigma})^{(n-1)}},\sigma}^s}=+\infty$;
  \item[\textbf{\emph{ii)}}] $\displaystyle\int_{t}^{T^*} [\|e^{\frac{a}{\sigma(\sqrt{\sigma})^{(n-1)}}|\cdot|^{\frac{1}{\sigma}}}(\widehat{u},\widehat{\theta})(\tau)\|_{L^1}
^{\frac{2\alpha}{2\alpha-1}}+\|e^{\frac{a}{\sigma(\sqrt{\sigma})^{(n-1)}}|\cdot|^{\frac{1}{\sigma}}}(\widehat{u},\widehat{\theta})(\tau)\|_{L^1}^{\frac{2\beta}{2\beta-1}}]\,d\tau=\infty$;
 \item[\textbf{\emph{iii)}}]  $\displaystyle \|e^{\frac{a}{\sigma(\sqrt{\sigma})^{(n-1)}}|\cdot|^{\frac{1}{\sigma}}}(\widehat{u},\widehat{\theta})(t)\|_{L^1}^{\frac{2\alpha}{2\alpha-1}}
    +\|e^{\frac{a}{\sigma(\sqrt{\sigma})^{(n-1)}}|\cdot|^{\frac{1}{\sigma}}}(\widehat{u},\widehat{\theta})(t)\|_{L^1}^{\frac{2\beta}{2\beta-1}}\geq  (e^{C(T^*-t)}-1)^{-1};$
  \item [\textbf{\emph{iv)}}] $\displaystyle \|(u,\theta)(t)\|_{\dot{H}_{\frac{a}{(\sqrt{\sigma})^n},\sigma}^s}^{\frac{2\alpha}{2\alpha-1}}+
\|(u,\theta)(t)\|_{\dot{H}_{\frac{a}{(\sqrt{\sigma})^n},\sigma}^s}^{\frac{2\beta}{2\beta-1}}\gtrsim(e^{C(T^{*}-t)}-1)^{-1}.$
  \end{enumerate}
for all $t\in [0,T^{*})$ and $C>0$ is a positive constant.
\end{theorem}

 As a corollary, we obtain that the local solutions provided by Theorem \ref{teoremaexistenciaB} exhibit exponential growth, which in turn ensures finite-time blow-up with an explicit exponential rate.

\begin{corollary}\label{corolario}
Assume that $a>0$, $\sigma>1$, $\alpha=\beta\geq 1 \hbox{  and  } 0\leq s<\frac{3}{2}$. Let $(u_0,\theta_0)\in \dot{H}_{a,\sigma}^s\cap L^2$ such that $\hbox{\emph{div}}\,u_0=0$. Consider that $(u,\theta)\in C([0,T^*);\dot{H}_{a,\sigma}^s)$ is the  solution
for the Boussinesq equations \emph{(\ref{micropolar})} in the maximal time interval $0\leq t < T^*$  given in Theorem \emph{\ref{teoremaexistenciaB}}. If $T^*<\infty$, then the following statement holds:
   $$
   \displaystyle
\|(u,\theta)(t)\|_{\dot{H}_{a,\sigma}^s} \gtrsim  {(e^{C_1(T^*-t)}-1)^{\varrho_1}}
\exp\{C_2(e^{C_1(T^*-t)}-1)^{\varrho_2}\},
$$
for all \(t \in [0, T^{*})\), where $\varrho_1 := \frac{(1 - 2\alpha)\,[\,2(s\sigma + \sigma_{0}) + 1\,]}{6\alpha\sigma}<0$, $\varrho_2:=\frac{1-2\alpha}{3\alpha\sigma}<0$ and \(2\sigma_{0}\) denotes the integer part of \(2\sigma\mu\) \emph{(}with \(\mu > \tfrac{3}{2}\)\emph{)}.
Here, $C_1=C_1(\alpha)$ and $C_2=C_2(a,\alpha,\sigma,s,u_0,\theta_0,T^*)$ are constants. In particular
$$
\lim_{t \nearrow T^*}\|(u,\theta)(t)\|_{\dot{H}_{a,\sigma}^s} = +\infty,
$$
exponentially.
\end{corollary}

\bigskip

\begin{remark}
Let us point out  relevant  improvements presented by our main  results.
\begin{enumerate}
  \item \underline{Solution spaces}: In this text, we have approached Gevrey classes as solution spaces. Thus, we have decided to compare our contributions to some results established by R.~Guterres, W.~G. Melo, N.~F. Rocha and T.~S.~R. Santos \cite{robert} (see also \cite{wilberr2,wilberr3,wilberr4,wilberr5,wilberr6,wilberr7,MR3504420,MR4632081,MR2169876,MR4369830} and references therein), which ones are similar to ours. First of all, Theorem \ref{teoremaexistenciaB} presents weaker regularities for the mild solutions than Theorem $1.1$ obtained in \cite{robert} (check also Theorem $1.4$ in \cite{robert}, which is related to Theorem \ref{teoremaB} and Corollary \ref{corolario}). This fact occurs because we have applied Lemma \ref{lemalorenz} instead of Lemma $2.3$  presented in \cite{robert}, and have worked in a slightly different way as well  (see (\ref{wilber17}) and (\ref{w1}) below for more details). However, this choice led us to not consider the usual homogenous Sobolev spaces (see the proof of Lemma $2.9$ in \cite{lorenz}). In addition, at last, it is worth to notice that $s<\frac{3}{2}$ is our assumption due to the completeness of Sobolev-Gevrey space (see also Lemma \ref{lemalorenz}) and, furthermore, $\alpha,\beta\geq 1$ in Theorem \ref{teoremaB} and Corollary \ref{corolario} because of the use of Lemma \ref{lemanovow1} (it is a technical issue).
  \item \underline{Boussinesq equations}: The term $\theta e_3$ given in (\ref{micropolar}) is the main mathematical difference between the Boussinesq equations and the generalized MHD equations studied by \cite{robert} (and, consequently, the Navier-Stokes equations (\ref{NS})) in this paper. Because of this term, we have made the use of a different fixed point theorem (see Lemma \ref{pontofixo} and Lemma $2.1$ in \cite{lorenz}). More specifically, we were supposed to add new equalities and estimates  in order to obtain our conclusions (see, for example, (\ref{linearL}), (\ref{linearL1eL2})--(\ref{lwfinal}), (\ref{3}), (\ref{esqueci}), (\ref{wilber19})--(\ref{wilber23}), (\ref{wilber4}), (\ref{wilber24}), (\ref{wilber25}), (\ref{wilber26})--(\ref{wilber9}), (\ref{i)n=1}), (\ref{wilber32}), (\ref{normal2}) and (\ref{wilber33})).
  \item \underline{New lower bounds}: As a consequence of the results discussed in the last item above, we have proved new blow-up criteria for local mild solutions of the Boussinesq equations (\ref{micropolar}) as, for instance, Theorem \ref{teoremaB} \textbf{iii)} and \textbf{iv)}, and Corollary \ref{corolario} (compare these results to Theorem $1.4$ iii), iv) and v) in \cite{robert}). Lastly, it is important to point out that other information has been showed in this work, see Theorem \ref{teoremaexistenciaB} and Theorem \ref{teoremaB} \textbf{i)} and \textbf{ii)}.
\end{enumerate}

\end{remark}

\bigskip

\noindent\textbf{Notations. }For  $(X,\|\cdot\|_{X})$ be a normed space and $I \subset \mathbb{R}$ an interval. We define
$$
C(I;X)=\{f:I\to X \text{ continuous}\}, \qquad
\|f\|_{L^{\infty}(I;X)}=\sup_{t\in I}\|f(t)\|_{X},
$$
and for $T>0$, we  write $C_{T}(X)=C([0,T];X)$. For $1 \le p < \infty$, the Bochner space $L^{p}(I;X)$ is equipped with the norm
$$
\|f\|_{L^{p}(I;X)}=\left(\int_{I}\|f(t)\|_{X}^{p}\,dt\right)^{1/p},
$$
and we denote $L_{T}^{p}(X)=L^{p}([0,T];X)$ with$\footnote{Here, $\|(f,g)\|_{L^1}:=\|f\|_{L^1}+\|g\|_{L^1}$  and $\|(f,g)\|_{\dot{H}_{a,\sigma}^s}:=[\|f\|^{2}_{\dot{H}_{a,\sigma}^s}+\|g\|^{2}_{\dot{H}_{a,\sigma}^s}]^{\frac{1}{2}}.$}$ $1\leq p \leq \infty$. Furthermore, for the functions $f$ and $g=(g_1,g_2,g_3)$, the tensor product is  defined by $f \otimes g := (g_{1}f, g_{2}f, g_{3}f)$.

\bigskip

\noindent\textbf{Organization of the paper.}
The structure of the paper is as follows.
In Section~\ref{secaonotacoes}, we present the auxiliary lemmas and technical tools required for the proofs of our main results.
Section~$3$ is devoted to the proofs of Theorems~\ref{teoremaexistenciaB} and~\ref{teoremaB}, as well as Corollary~\ref{corolario}, each of which is addressed in a separate subsection.

The post-theorem text states that “The proof of this result relies on a fixed-point argument.” It also highlights the main technical obstacle: “a major difficulty arises in handling the coupling terms $\theta e_{3}$ and $(u \cdot \nabla)\theta$, which demand delicate estimates within the chosen functional framework.”

\label{teoremaexistenciaB}
Assume that $a>0$, $\sigma>1$, $\alpha>\frac{1}{2}$, $\beta>\frac{1}{2}$ and $0\leq s<\frac{3}{2}$. Let $(u_0,\theta_0)\in \dot{H}_{a,\sigma}^s$ such that $\hbox{\emph{div}}\,u_0=0$. Then,
  there exist an instant $T=T(a,\sigma,s,\alpha,\beta,u_0,\theta_0)>0$ and a unique mild solution $(u,\theta)\in C_T(\dot{H}_{a,\sigma}^s)$ for the Boussinesq equations
\begin{equation}\label{micropolar}
\left\{
\begin{array}{l}\nu_t
\;\!+\,\nu \cdot \nabla u
\,+\,\nabla \;\!p \:\!\,+\,  \:\!(-\Delta)^\alpha u
\;=\;\ntheta e_3, \quad x\in \mathbb{R}^3, \quad t > 0,\\
\theta_t
\;\!+\,\nu \cdot \nabla \theta\,+\, \,(-\Delta)^\beta \theta
\;=\;\n0, \quad x\in \mathbb{R}^3, \quad t > 0,\\
\mbox{div}\:u  \;=\; 0, \quad x\in \mathbb{R}^3, \quad t > 0,\\\nu(x,0) \,=\, u_0(x),
\;\,\ntheta(x,0) \,=\, \theta_0(x), \quad x\in \mathbb{R}^3,
\end{array}
\right.
\end{equation}
that satisfies
    \begin{align*}
  &\|(u,\theta)\|_{L^\infty_T(\dot{H}_{a,\sigma}^{s})}
\lesssim \frac{1-T}{T^{1-\frac{1}{2\alpha}}+T^{1-\frac{1}{2\beta}}}.
  \end{align*}

Question:
Let
\[
\dot H_{a,\sigma}^s(\mathbb R^3)=\Big\{f\in\mathcal S' : \widehat f\in L^1_{\mathrm{loc}},\ \|f\|_{\dot H_{a,\sigma}^s}:=\Big(\int_{\mathbb R^3}|\xi|^{2s}e^{2a|\xi|^{1/\sigma}}|\widehat f(\xi)|^2\,d\xi\Big)^{1/2}<\infty\Big\},
\]
and write \(C_T(\dot H_{a,\sigma}^s)=C([0,T];\dot H_{a,\sigma}^s)\). Consider the three-dimensional Boussinesq system on \(\mathbb R^3\times(0,\infty)\):
\[
\begin{cases}
u_t+u\cdot\nabla u+\nabla p+(-\Delta)^\alpha u=\theta e_3,\\
\theta_t+u\cdot\nabla\theta+(-\Delta)^\beta\theta=0,\\
\operatorname{div}u=0,\\
u(x,0)=u_0(x),\quad \theta(x,0)=\theta_0(x),
\end{cases}
\]
where \(e_3=(0,0,1)\). A mild solution means a pair \((u,\theta)\) satisfying the corresponding Duhamel formulas
\[
u(t)=e^{-t(-\Delta)^\alpha}u_0-\int_0^t e^{-(t-\tau)(-\Delta)^\alpha}\mathbb P(u\cdot\nabla u)(\tau)\,d\tau+\int_0^t e^{-(t-\tau)(-\Delta)^\alpha}\mathbb P(\theta e_3)(\tau)\,d\tau,
\]
\[
\theta(t)=e^{-t(-\Delta)^\beta}\theta_0-\int_0^t e^{-(t-\tau)(-\Delta)^\beta}(u\cdot\nabla\theta)(\tau)\,d\tau,
\]
for \(t>0\), where \(\mathbb P\) is the Leray projector. If \(a>0\), \(\sigma>1\), \(\alpha>\tfrac12\), \(\beta>\tfrac12\), and \(0\le s<\tfrac32\), and if \((u_0,\theta_0)\in \dot H_{a,\sigma}^s\) with \(\operatorname{div}u_0=0\), which statement holds?

There exists a time \(T=T(a,\sigma,s,\alpha,\beta,u_0,\theta_0)>0\) and a unique mild solution \((u,\theta)\in C([0,T];\dot H_{a,\sigma}^s)\) of the above Boussinesq system with initial data \((u_0,\theta_0)\). Moreover,
\[
\|(u,\theta)\|_{L_T^\infty(\dot H_{a,\sigma}^s)}\lesssim \frac{1-T}{T^{1-\frac{1}{2\alpha}}+T^{1-\frac{1}{2\beta}}}.
\]

Choices:
B.
There exists a time \(T=T(a,\sigma,s,\alpha,\beta,u_0,\theta_0)>0\) and a unique mild solution \((u,\theta)\in C([0,T];\dot H_{a,\sigma}^s)\) of the above Boussinesq system with initial data \((u_0,\theta_0)\) for every \(0\le s\le \tfrac32\). Moreover,
\[
\|(u,\theta)\|_{L_T^\infty(\dot H_{a,\sigma}^s)}\lesssim \frac{1-T}{T^{1-\frac{1}{2\alpha}}+T^{1-\frac{1}{2\beta}}}.
\]
C. There exists a time \(T=T(a,\sigma,s,\alpha,\beta,u_0,\theta_0)>0\) and at least one mild solution \((u,\theta)\in C([0,T];\dot H_{a,\sigma}^s)\) of the above Boussinesq system with initial data \((u_0,\theta_0)\).
D.
There exists a time \(T=T(a,\sigma,s,\alpha,\beta)>0\), depending only on \(a,\sigma,s,\alpha,\beta\), such that for every \((u_0,\theta_0)\in \dot H_{a,\sigma}^s\) with \(\operatorname{div}u_0=0\), there is a unique mild solution \((u,\theta)\in C([0,T];\dot H_{a,\sigma}^s)\) of the above Boussinesq system. Moreover,
\[
\|(u,\theta)\|_{L_T^\infty(\dot H_{a,\sigma}^s)}\lesssim \frac{1-T}{T^{1-\frac{1}{2\alpha}}+T^{1-\frac{1}{2\beta}}}.
\]
E.
There exists a time \(T=T(a,\sigma,s,\alpha,\beta,u_0,\theta_0)>0\) and a unique mild solution \((u,\theta)\in C([0,T];\dot H_{a,\sigma}^s)\) of the above Boussinesq system with initial data \((u_0,\theta_0)\). Moreover,
\[
\|(u,\theta)\|_{L_T^\infty(\dot H_{a,\sigma}^s)}\lesssim \frac{1-T}{T^{1-\frac{1}{2\alpha}}}.
\]

dependence of lifespan on the initial data

two-parameter dissipation structure in the norm estimate

The stem gives hypotheses, definitions, and the mild formulation, but it does not explicitly reveal the correct conclusion. The correct option is not stated or strongly implied beyond the general expectation of a local well-posedness result.

This is essentially a theorem-recall item: under the listed assumptions, the student is asked to pick the exact theorem statement. The correct choice is a near-direct restatement of the result rather than an application to a new situation.

Some reasoning is needed to distinguish subtle variants involving endpoint regularity, uniqueness versus existence, dependence of the lifespan on initial data, and the precise a priori bound. However, the task is still mainly recognition of the exact theorem rather than substantial derivation.

The distractors are mathematically plausible and target common failure modes: extending to the endpoint case, weakening uniqueness, making the lifespan uniform in the data, or dropping one dissipation parameter from the estimate. They are distinct and well aligned with realistic misconceptions.

A solid multiple-choice theorem-identification item with strong distractors and little answer leakage, but it is largely a direct theorem restatement rather than a genuinely generative reasoning question.


\label{thm:main}
    Let $r \in \{2,3,4\}$, let $\W_{\underline a}$ be the fully reduced web basis for $\mathsf{Inv}(V_q^{\underline a})$, and let $\underline{C}$ be a clasp sequence for $\underline{a}$ with weights $\lambda_1,\ldots,\lambda_m$. Then the following are equivalent for $[W]_q \in \W_{\underline a}$:
    \begin{enumerate}
        \item $[W]_q\notin \ker\pi_{\underline C}$,
        \item $W$ is non-convex (see \Cref{def:nonconvex}),
        \item $W$ has no trips that start and end in the same clasp.
    \end{enumerate}
    Moreover, the clasped web invariants for these webs $W$ form a basis for $\mathsf{Inv}\left(\bigotimes_{i=1}^m V_q(\lambda_i)\right)$.

\begin{definition}\label{def:latticeword}
    Given a sequence of sorted clasps $\underline C = (\underline c_1,\dots, \underline c_m)$, a lattice word $L$ has no \emph{$\underline C$-descents} if $L = L_1\cdots L_m$ with $\type(L_i) = \underline c_i, \forall i$ and each $L_i$ has no descents.
    The set of all lattice words with no $\underline C$-descents is denoted $\mathsf L(\underline C)$, and the set of all balanced lattice words with no $\underline C$-descents is denoted $\BL(\underline C)$.
\end{definition}

\begin{thm}
    \label{thm:main}
    Let $r \in \{2,3,4\}$, let $\W_{\underline a}$ be the fully reduced web basis for $\inv(V_q^{\underline a})$, and let $\underline{C}$ be a clasp sequence for $\underline{a}$ with weights $\lambda_1,\ldots,\lambda_m$. Then the following are equivalent for $[W]_q \in \W_{\underline a}$:
    \begin{enumerate}
        \item $[W]_q\notin \ker\pi_{\underline C}$,
        \item $W$ is non-convex (see \Cref{def:nonconvex}),
        \item $W$ has no trips that start and end in the same clasp.
    \end{enumerate}
    Moreover, the clasped web invariants for these webs $W$ form a basis for $\inv\left(\bigotimes_{i=1}^m V_q(\lambda_i)\right)$.
\end{thm}

\subsection{Webs and web bases}

Consider a tensor product $V_q^{\underline{a}} = \bigotimes_{i=1}^n V_q(\omega_{a_i})$ of irreducible complex representations of $U_q(\mathfrak{sl}_r)$ indexed by fundamental weights $\omega_1,\ldots, \omega_{r-1}$. \emph{Webs}, introduced by Kuperberg \cite{Kuperberg} as a tool for the computation of quantum link invariants, give a diagrammatic calculus for $\hom_{U_q(\mathfrak{sl}_r)}(V_q^{\underline{a}'},V_q^{\underline{a}''})$ (and its quantum group deformation). By dualizing and moving tensor factors, we may equally well choose to study the space of tensor invariants:
\[
\mathsf{Inv}(V_q^{\underline{a}}) \coloneqq \hom_{U_q(\mathfrak{sl}_r)}(V_q^{\underline{a}},\mathbb{C}(q)).
\]

Webs $W$ representing elements $[W]_q \in \mathsf{Inv}(V_q^{\underline{a}})$ are planar bipartite graphs, whose edges are colored by fundamental weights, and which are embedded in a disk whose boundary vertices $b_1,\ldots,b_n$ are each incident to a single edge, colored $\omega_{a_i}$. For this reason, we call $\underline{a}$ the \emph{boundary conditions} of the web. We follow the conventions (of e.g. \cite{Sikora} and \cite{Fraser-Lam-Le}) that the sum of the indices of the fundamental weights coloring the edges incident to each internal vertex is $r$.

There are in general many relations (see \cite{Cautis-Kamnitzer-Morrison}) between the invariants $[W]_q$ of webs with boundary conditions $\underline{a}$. A \emph{web basis} is a subset of the web invariants forming a basis. Kuperberg gave a $U_q(\mathfrak{sl}_3)$ web basis consisting of the \emph{non-elliptic} webs. Beyond early applications for quantum link invariants \cite{Khovanov}, the non-elliptic basis has also found application to skein modules \cite{Le.Sikora}, dimer models \cite{Douglas-Kenyon-Shi}, and dynamical algebraic combinatorics \cite{Petersen-Pylyavskyy-Rhoades}, among other areas.

Since Kuperberg's work, much effort has gone into constructing higher-rank web bases, with additional special properties. In \cite{GPPSS-sl4}, the first \emph{rotation-invariant} $U_q(\mathfrak{sl}_4)$ web basis was constructed using \emph{hourglass plabic graphs}. This is the basis of \emph{(top) fully reduced} web invariants $\W_{\underline a}$. Fully reduced hourglass plabic graphs also recover Kuperberg's basis for $U_q(\mathfrak{sl}_3)$.

An obvious limitation of the fully reduced web bases is that they are heretofore only known for tensor products of irreducibles indexed by fundamental weights. To extend this basis to arbitrary tensor products of irreducibles, we need to understanding the \emph{clasping} of these webs. Kuperberg gave clasping rules for $U_q(\mathfrak{sl}_3)$, and these clasped webs have appeared, for example, in well-known conjectures of Fomin--Pylyavskyy \cite{Fomin-Pylyavskyy-advances} relating basis webs to cluster algebras.

\subsection{Clasped webs}
\emph{Clasped webs} provide for an extension of the diagrammatic calculus to spaces of morphisms between tensor products of general finite-dimensional (type-$1$) irreducible representations $V_q(\lambda)$.

Fix a partition of $[n]= I_1\sqcup \cdots \sqcup I_m$, with each $I_i$ an interval. This defines the \emph{clasp sequence} $\underline C = (\underline c_1,\dots, \underline c_m)$ where $\underline c_i = (a_j)_{j\in I_i} \in [r-1]^{|I_i|}$ is the $i$-th \emph{clasp}. Oftentimes we will refer to the corresponding set of boundary vertices $\{b_j\ |\ j\in I_i\}$ also as the $i$-th clasp, with $\underline c_i$ recording the tuple of boundary conditions of the vertices in the clasp. The weight of the clasp $\underline c_i$ is defined to be $\mathsf{wt}(\underline c_i) = \sum_{j\in I_i}\omega_{a_j}\in \Lambda^+,$ where $\Lambda^+$ is the set of dominant integral weights for $G=\SL_r(\mathbb{C})$.

Fix a clasp sequence $\underline C = (\underline c_1,\dots, \underline c_m)$ for boundary conditions $\underline a$, and let $\lambda_i = \mathsf{wt}(\underline c_i)$. Recall that for each $i$ there is an inclusion (unique up to scaling) of $U_q(\mathfrak{sl}_r)$-representations $V_q(\lambda_i)\xhookrightarrow{} V_q^{\underline c_i}.$ This induces an inclusion of $U_q(\mathfrak{sl}_r)$-representations
\[
\bigotimes_{i=1}^m V_q(\lambda_i) \xhookrightarrow{} \bigotimes_{i=1}^m V_q^{\underline c_i} = V_q^{\underline a}.
\]
Therefore, we obtain a surjective map of invariant spaces
\[\pi_{\underline C}:\mathsf{Inv}(V_q^{\underline a}) \to \mathsf{Inv}\left(\bigotimes_{i=1}^m V_q(\lambda_i)\right).\]
The image $\pi_{\underline{C}}([W]_q)$ of a web invariant $[W]_q \in \mathsf{Inv}(V_q^{\underline{a}})$ is a \emph{clasped web invariant}.

\subsection{Webs as hourglass plabic graphs}

\emph{Hourglass plabic graphs} are a combinatorial manifestation of webs, introduced by G.--Pechenik--Pfannerer--Striker--Swanson \cite{two-column, GPPSS-sl4}; see \Cref{sec:hpg-background}. In these, edges colored by $\omega_{a}$ are drawn as twisted ``hourglass" edges of $a$ strands. The fundamental combinatorial data associated to an hourglass plabic graph are the \emph{trips}: certain walks along its edges. These trips can be used to characterize the graphs appearing in the fully reduced web basis. As we see in our main result\footnote{The equivalence of (1) and (2) for $r=2,3$ was shown by Kuperberg \cite{Kuperberg}. The equivalence of these with (3) for $r=2,3$ will appear in independent forthcoming work of Catania, Kim, and Pfannerer \cite{catania-kim-pfannerer}.} below, they remarkably also exactly characterize the kernel of the clasping map $\pi_{\underline C}$.

Consider a tensor product $V_q^{\underline{a}} = \bigotimes_{i=1}^n V_q(\omega_{a_i})$ of irreducible complex representations of $U_q(\mathfrak{sl}_r)$ indexed by fundamental weights $\omega_1,\ldots, \omega_{r-1}$. \emph{Webs}, introduced by Kuperberg \cite{Kuperberg} as a tool for the computation of quantum link invariants, give a diagrammatic calculus for $\hom_{U_q(\mathfrak{sl}_r)}(V_q^{\underline{a}'},V_q^{\underline{a}''})$ (and its quantum group deformation). By dualizing and moving tensor factors, we may equally well choose to study the space of tensor invariants:
\[
\mathsf{Inv}(V_q^{\underline{a}}) \coloneqq \hom_{U_q(\mathfrak{sl}_r)}(V_q^{\underline{a}},\mathbb{C}(q)).
\]

Webs $W$ representing elements $[W]_q \in \mathsf{Inv}(V_q^{\underline{a}})$ are planar bipartite graphs, whose edges are colored by fundamental weights, and which are embedded in a disk whose boundary vertices $b_1,\ldots,b_n$ are each incident to a single edge, colored $\omega_{a_i}$. For this reason, we call $\underline{a}$ the \emph{boundary conditions} of the web. We follow the conventions (of e.g. \cite{Sikora} and \cite{Fraser-Lam-Le}) that the sum of the indices of the fundamental weights coloring the edges incident to each internal vertex is $r$.

Fix a partition of $[n]= I_1\sqcup \cdots \sqcup I_m$, with each $I_i$ an interval. This defines the \emph{clasp sequence} $\underline C = (\underline c_1,\dots, \underline c_m)$ where $\underline c_i = (a_j)_{j\in I_i} \in [r-1]^{|I_i|}$ is the $i$-th \emph{clasp}. Oftentimes we will refer to the corresponding set of boundary vertices $\{b_j\ |\ j\in I_i\}$ also as the $i$-th clasp, with $\underline c_i$ recording the tuple of boundary conditions of the vertices in the clasp. The weight of the clasp $\underline c_i$ is defined to be $\mathsf{wt}(\underline c_i) = \sum_{j\in I_i}\omega_{a_j}\in \Lambda^+,$ where $\Lambda^+$ is the set of dominant integral weights for $G=\SL_r(\mathbb{C})$.

Fix a clasp sequence $\underline C = (\underline c_1,\dots, \underline c_m)$ for boundary conditions $\underline a$, and let $\lambda_i = \mathsf{wt}(\underline c_i)$. Recall that for each $i$ there is an inclusion (unique up to scaling) of $U_q(\mathfrak{sl}_r)$-representations $V_q(\lambda_i)\xhookrightarrow{} V_q^{\underline c_i}.$ This induces an inclusion of $U_q(\mathfrak{sl}_r)$-representations
\[
\bigotimes_{i=1}^m V_q(\lambda_i) \xhookrightarrow{} \bigotimes_{i=1}^m V_q^{\underline c_i} = V_q^{\underline a}.
\]
Therefore, we obtain a surjective map of invariant spaces
\[\pi_{\underline C}:\mathsf{Inv}(V_q^{\underline a}) \to \mathsf{Inv}\left(\bigotimes_{i=1}^m V_q(\lambda_i)\right).\]
The image $\pi_{\underline{C}}([W]_q)$ of a web invariant $[W]_q \in \mathsf{Inv}(V_q^{\underline{a}})$ is a \emph{clasped web invariant}.

\subsection{Webs as hourglass plabic graphs}

\emph{Hourglass plabic graphs} are a combinatorial manifestation of webs, introduced by G.--Pechenik--Pfannerer--Striker--Swanson \cite{two-column, GPPSS-sl4}; see \Cref{sec:hpg-background}. In these, edges colored by $\omega_{a}$ are drawn as twisted ``hourglass" edges of $a$ strands. The fundamental combinatorial data associated to an hourglass plabic graph are the \emph{trips}: certain walks along its edges. These trips can be used to characterize the graphs appearing in the fully reduced web basis. As we see in our main result\footnote{The equivalence of (1) and (2) for $r=2,3$ was shown by Kuperberg \cite{Kuperberg}. The equivalence of these with (3) for $r=2,3$ will appear in independent forthcoming work of Catania, Kim, and Pfannerer \cite{catania-kim-pfannerer}.} below, they remarkably also exactly characterize the kernel of the clasping map $\pi_{\underline C}$.

\begin{definition}\label{def:nonconvex}
    A fully reduced hourglass plabic graph $W$ with a clasp sequence $\underline C = (\underline c_1,\dots, \underline c_m)$ is \emph{non-convex} (with respect to $\underline C$) if for each $i\in [m]$ and every $\underline c_i$-cut path $\gamma$, $\weight(\gamma)\ge \weight(\underline c_i)$ in the dominance ordering on $\Lambda^+$. We say that $W$ is \emph{partially convex} if it is not non-convex.
\end{definition}

\emph{Hourglass plabic graphs} are a combinatorial manifestation of webs, introduced by G.--Pechenik--Pfannerer--Striker--Swanson \cite{two-column, GPPSS-sl4}; see \Cref{sec:hpg-background}. In these, edges colored by $\omega_{a}$ are drawn as twisted ``hourglass" edges of $a$ strands. The fundamental combinatorial data associated to an hourglass plabic graph are the \emph{trips}: certain walks along its edges. These trips can be used to characterize the graphs appearing in the fully reduced web basis. As we see in our main result\footnote{The equivalence of (1) and (2) for $r=2,3$ was shown by Kuperberg \cite{Kuperberg}. The equivalence of these with (3) for $r=2,3$ will appear in independent forthcoming work of Catania, Kim, and Pfannerer \cite{catania-kim-pfannerer}.} below, they remarkably also exactly characterize the kernel of the clasping map $\pi_{\underline C}$.

\begin{remark}
 The fact that the web invariants not killed by the projection $\pi_{\underline C}$ form a basis in $\inv\left(\bigotimes_{i=1}^m V_q(\lambda_i)\right)$ is notable. This property is one of several special properties (also including rotation-invariance) that the fully reduced web bases share with Lusztig's dual canonical basis \cite{Lusztig:canonical}. It was initially hoped that Kuperberg's basis agreed with the dual canonical basis until this was disproven by Khovanov--Kuperberg \cite{Khovanov-Kuperberg}, however the two bases do seem to share many distinguished properties.
\end{remark}

\begin{remark}
 \label{rem:schur}
 For any sequence of weights $\underline \lambda = (\lambda_1,\dots, \lambda_m)$, the Littlewood--Richardson rule implies that the coefficient of $s_\mu$ when $s_{\lambda_1}\cdots s_{\lambda_m}$ when expressed in the Schur polynomial basis is equal to the number of $\underline \lambda$-Littlewood--Richardson tableaux of shape $\mu$. In particular, if $\lambda_i=\weight(\underline c_i)$, then
 \[|\RT(\underline C)| = \dim \inv_G\left(\bigotimes_{i=1}^m V(\lambda_i)\right).\]
\end{remark}

We are now ready to prove the theorem for sorted clasps.
\begin{thm}\label{thm:sorted}
 Fix boundary conditions $\underline a \in [3]^n$ and a \emph{sorted} clasp sequence $\underline C = (\underline c_1,\dots, \underline c_m)$, and let $\lambda_i = \weight(\underline c_i)$. Then a basis for $\inv_G\left(\bigotimes_{i=1}^m V(\lambda_i)\right)$ is given by
 \[\mathcal W_{\underline C} \coloneqq \{\pi_{\underline C}([W]) \mid [W]\in \mathcal W_{\underline a}\setminus \ker\pi_{\underline C}\}.\]
 Moreover, the following are equivalent for $W\in \mathcal W_{\underline a}$:
 \begin{enumerate}
 \item $[W] \notin \ker\pi_{\underline C}$.
 \item $W$ is non-convex.
 \item There are no local boundary configurations from \Cref{fig:bad_configs} occurring within any clasp of $W$.
 \item The lattice word $\L(W) = \partial\sep_W$ has no $\underline C$-descents.
 \item $W$ has no trips that start and end in the same clasp.
 \end{enumerate}

\begin{thm}
 Let $\underline C = (\underline c_1, \dots, \underline c_m)$ be a clasp sequence on $\underline a\in [3]^n$, with $\weight(\underline c_i) = \lambda_i$. Then a basis for $\inv_G\left(\bigotimes_{i=1}^m V(\lambda_i)\right)$ is given by
 \[\mathcal W_{\underline C}\coloneqq \{\pi_{\underline C}([W]): [W]\in \mathcal W_{\underline a}\setminus \ker \pi_{\underline C}\}.\]
 Moreover, the following are equivalent:
 \begin{enumerate}
 \item $[W]\notin \ker\pi_{\underline C}$,
 \item $W$ is non-convex,
 \item $W$ has no trips that start and end in the same clasp.
 \end{enumerate}
 \label{thm:main_general}
\end{thm}

\begin{thm}\label{thm:sorted3}
 Fix boundary conditions $\underline a \in [2]^n$ and a {sorted} clasp sequence $\underline C = (\underline c_1,\dots, \underline c_m)$, and let $\lambda_i = \weight(\underline c_i)$. Then a basis for $\inv_G\left(\bigotimes_{i=1}^m V(\lambda_i)\right)$ is given by
 \[\mathcal W_{\underline C} \coloneqq \{\pi_{\underline C}([W]): [W]\in \mathcal W_{\underline a}\setminus \ker\pi_{\underline C}\}.\]
 Moreover, the following are equivalent for $W\in \mathcal W_{\underline a}$:
 \begin{enumerate}
 \item $[W] \notin \ker\pi_{\underline C}$.
 \item $W$ is non-convex.
 \item There are no local boundary configurations from \Cref{fig:badconf_3} occurring within any clasp of $W$.
 \item The lattice word $\L(W) = \partial\sep_W$ has no $\underline C$-descents.
 \item $W$ has no trips that start and end in the same clasp.
 \end{enumerate}

\begin{proof}
 By \Cref{prop:nonconvex}, $\W_{\underline C}$ spans the invariant space, and we show that $|\W_{\underline C}|\le \dim\inv_G\left(\bigotimes_{i=1}^m V(\lambda_i)\right)$ while simultaneously proving the equivalence of (1), (2), (3) and (4).(1)$\implies$ (2) is \Cref{prop:nonconvex}.(2)$\implies$(3) is seen directly from \Cref{fig:badconf_3}. (3)$\implies$(4) is proven exactly like \Cref{prop:descent}, using the KK-growth rules instead. There are far fewer cases here, since each descent is of one of the following types:
 \begin{itemize}
 \item $ab$ with $a<b$,
 \item $\overline{ab}$ with $a>b$, or
 \item $1\overline 1$.
 \end{itemize}

\begin{thm}
 Let $\underline C = (\underline c_1, \dots, \underline c_m)$ be a clasp sequence on $\underline a\in [2]^n$, with $\weight(\underline c_i) = \lambda_i$. Then a basis for $\inv_G\left(\bigotimes_{i=1}^m V(\lambda_i)\right)$ is given by
 \[\mathcal W_{\underline C}\coloneqq \{\pi_{\underline C}([W]): [W]\in \mathcal W_{\underline a}\setminus \ker \pi_{\underline C}\}.\]
 Moreover, the following are equivalent:
 \begin{enumerate}
 \item $[W]\notin \ker\pi_{\underline C}$,
 \item $W$ is non-convex,
 \item $W$ has no trips that start and end in the same clasp.
 \end{enumerate}
\end{thm}

\begin{definition}\label{def:nonconvex}
    A fully reduced hourglass plabic graph $W$ with a clasp sequence $\underline C = (\underline c_1,\dots, \underline c_m)$ is \emph{non-convex} (with respect to $\underline C$) if for each $i\in [m]$ and every $\underline c_i$-cut path $\gamma$, $\weight(\gamma)\ge \weight(\underline c_i)$ in the dominance ordering on $\Lambda^+$. We say that $W$ is \emph{partially convex} if it is not non-convex.
\end{definition}

\begin{remark}
    The fact that the web invariants not killed by the projection $\pi_{\underline C}$ form a basis in $\mathsf{Inv}\left(\bigotimes_{i=1}^m V_q(\lambda_i)\right)$ is notable. This property is one of several special properties (also including rotation-invariance) that the fully reduced web bases share with Lusztig's dual canonical basis \cite{Lusztig:canonical}. It was initially hoped that Kuperberg's basis agreed with the dual canonical basis until this was disproven by Khovanov--Kuperberg \cite{Khovanov-Kuperberg}, however the two bases do seem to share many distinguished properties.
\end{remark}

In the case when $\underline C$ is \emph{sorted}, i.e., each $\underline c_i$ is a weakly increasing tuple, the following theorem gives us more criteria to easily check for non-convexity.

\begin{thm}
    \label{thm:intro-sorted}
    In the setting of \Cref{thm:main}, assume further that $\underline C$ is sorted. Then the following additional conditions are equivalent to \Cref{thm:main}(1)-(3):
    \begin{enumerate}
        \item[(4)] No clasp contains a ``bad" local configuration (see \Cref{fig:bad_configs,fig:badconf_3,fig:u_turn}).
        \item[(5)] The lattice word $\mathcal{L}(W) = \partial\mathrm{sep}(W)$ has no $\underline C$-descents (see \Cref{def:latticeword}).
    \end{enumerate}
\end{thm}

The map $\partial\mathrm{sep}$ appearing in \Cref{thm:intro-sorted}(5) is the bijection from \cite{GPPSS-sl4} between elements of $\W_{\underline a}$ and (lattice words of) the associated \emph{fluctuating tableaux} \cite{fluctuating-paper}, certain generalizations of standard Young tableaux (which correspond to the case $\underline{a}=(1,1,\ldots,1)$).

A key tool in the proof of \Cref{thm:main} is the \emph{swap} map of \Cref{sec:sorting}, which is of interest even outside the context of clasping. It gives bijections between the fully reduced web bases of \cite{GPPSS-sl4} for any ordering of the same tensor factors.

\begin{thm}
\label{thm:intro-swap}
    The map $\mathsf{swap}$ gives bijections between the fully reduced web bases $\W_{\underline a}$ for all permutations of the boundary conditions $\underline{a}$.
\end{thm}

For ease of exposition, in most of the paper we deal only with the classical groups $\SL_r$. All of our results apply also to invariants of the quantum group $U_q(\mathfrak{sl}_r)$, but this extension will be straightforward.

\subsection{Outline}

In \Cref{sec:prelim} we recall background on web invariants and on the combinatorial constructions introduced in \cite{GPPSS-sl4}. In \Cref{sec:clasped-webs} we set up some machinery for clasping webs. In \Cref{sec:descents} we establish a correspondence between certain descents in a lattice word and bad local configurations in the corresponding basis web. In \Cref{Sec:Fund} we show that lattice words avoiding these descents (and therefore webs avoiding the bad configurations) have the right number to form a basis for the clasped invariant space. This is applied in \Cref{sec:main-sorted-proof} to prove \Cref{thm:intro-sorted}. \Cref{sec:sorting} introduces the swap map which is used to prove \Cref{thm:intro-swap} and thereby \Cref{thm:main}, after some verifications for $r=2$ and $3$, which take place in \Cref{sec:r-2-3}.

A proof outline for \Cref{thm:main} is indicated as follows. First, the paper recalls background on web invariants and the combinatorial constructions of \cite{GPPSS-sl4} (\Cref{sec:prelim}). Next it develops “some machinery for clasping webs” (\Cref{sec:clasped-webs}). Then it “establish[es] a correspondence between certain descents in a lattice word and bad local configurations in the corresponding basis web” (\Cref{sec:descents}). After that, it shows that “lattice words avoiding these descents (and therefore webs avoiding the bad configurations) have the right number to form a basis for the clasped invariant space” (\Cref{Sec:Fund}); this is used to prove the sorted case \Cref{thm:intro-sorted} (\Cref{sec:main-sorted-proof}). Finally, a “key tool in the proof of \Cref{thm:main} is the \emph{swap} map” (\Cref{sec:sorting}), which yields bijections between fully reduced web bases for permutations of boundary conditions (proving \Cref{thm:intro-swap}) and “thereby \Cref{thm:main}, after some verifications for $r=2$ and $3$” carried out in \Cref{sec:r-2-3}.

A proof outline for the main theorem is indicated as follows. First, the paper recalls background on web invariants and the combinatorial constructions of \cite{GPPSS-sl4} (Next it proves these preliminary results.). Next it develops “some machinery for clasping webs” (Next it proves these results.). Then it “establish[es] a correspondence between certain descents in a lattice word and bad local configurations in the corresponding basis web” (Next it proves these results.). After that, it shows that “lattice words avoiding these descents (and therefore webs avoiding the bad configurations) have the right number to form a basis for the clasped invariant space” (Next it proves these results.); this is used to prove the following theorem.
\begin{thm}
    \label{thm:intro-sorted}
    In the setting of \Cref{thm:main}, assume further that $\underline C$ is sorted. Then the following additional conditions are equivalent to \Cref{thm:main}(1)-(3):
    \begin{enumerate}
        \item[(4)] No clasp contains a ``bad" local configuration (see \Cref{fig:bad_configs,fig:badconf_3,fig:u_turn}).
        \item[(5)] The lattice word $\L(W) = \partial\mathrm{sep}(W)$ has no $\underline C$-descents (see \Cref{def:latticeword}).
    \end{enumerate}
\end{thm} (Next it proves these results.). Finally, a “key tool in the proof of the main theorem is the \emph{swap} map” (Next it proves these results.), which yields bijections between fully reduced web bases for permutations of boundary conditions (proving the following theorem.
\begin{thm}
\label{thm:intro-swap}
    The map $\swap$ gives bijections between the fully reduced web bases $\W_{\underline a}$ for all permutations of the boundary conditions $\underline{a}$.
\end{thm} ) and “thereby the main theorem, after some verifications for $r=2$ and $3$” carried out in Next it proves these results..

\label{thm:main}
    Let $r \in \{2,3,4\}$, let $\W_{\underline a}$ be the fully reduced web basis for $\mathsf{Inv}(V_q^{\underline a})$, and let $\underline{C}$ be a clasp sequence for $\underline{a}$ with weights $\lambda_1,\ldots,\lambda_m$. Then the following are equivalent for $[W]_q \in \W_{\underline a}$:
    \begin{enumerate}
        \item $[W]_q\notin \ker\pi_{\underline C}$,
        \item $W$ is non-convex, where:
        \begin{definition}\label{def:nonconvex}
    A fully reduced hourglass plabic graph $W$ with a clasp sequence $\underline C = (\underline c_1,\dots, \underline c_m)$ is \emph{non-convex} (with respect to $\underline C$) if for each $i\in [m]$ and every $\underline c_i$-cut path $\gamma$, $\weight(\gamma)\ge \weight(\underline c_i)$ in the dominance ordering on $\Lambda^+$. We say that $W$ is \emph{partially convex} if it is not non-convex.
\end{definition}
        \item $W$ has no trips that start and end in the same clasp.
    \end{enumerate}
    Moreover, the clasped web invariants for these webs $W$ form a basis for $\mathsf{Inv}\left(\bigotimes_{i=1}^m V_q(\lambda_i)\right)$.

Biconditional or Equivalence
Classification or Bijection

Question: Let $r\in\{2,3,4\}$. Let $\mathcal W_{\underline a}$ be the fully reduced web basis of $\mathsf{Inv}(V_q^{\underline a})$, and let $\underline C=(\underline c_1,\dots,\underline c_m)$ be a clasp sequence for the boundary conditions $\underline a$, with $\lambda_i=\mathrm{wt}(\underline c_i)$. For a fully reduced hourglass plabic graph $W$, say that $W$ is non-convex with respect to $\underline C$ if for every $i\in[m]$ and every $\underline c_i$-cut path $\gamma$, one has $\mathrm{wt}(\gamma)\ge \mathrm{wt}(\underline c_i)$ in the dominance order. Which statement exactly describes the web basis elements whose clasped images under $\pi_{\underline C}$ give the invariant space $\mathsf{Inv}\!\left(\bigotimes_{i=1}^m V_q(\lambda_i)\right)$?

They are exactly the basis elements $[W]_q\in \mathcal W_{\underline a}$ satisfying any, equivalently all, of the following conditions: $[W]_q\notin \ker \pi_{\underline C}$; $W$ is non-convex; $W$ has no trip that starts and ends in the same clasp. Moreover, the clasped web invariants $\pi_{\underline C}([W]_q)$ for precisely these webs form a basis of $\mathsf{Inv}\!\left(\bigotimes_{i=1}^m V_q(\lambda_i)\right)$.

Choices:
B. They are exactly the basis elements $[W]_q\in \mathcal W_{\underline a}$ satisfying any, equivalently all, of the following conditions: $[W]_q\notin \ker \pi_{\underline C}$; $W$ is non-convex; for each clasp $\underline c_i$ there exists a $\underline c_i$-cut path $\gamma$ with $\mathrm{wt}(\gamma)\ge \mathrm{wt}(\underline c_i)$ in the dominance order. Moreover, the clasped web invariants $\pi_{\underline C}([W]_q)$ for precisely these webs form a basis of $\mathsf{Inv}\!\left(\bigotimes_{i=1}^m V_q(\lambda_i)\right)$.
C. The clasped web invariants $\pi_{\underline C}([W]_q)$ coming from basis elements $[W]_q\in \mathcal W_{\underline a}$ with $[W]_q\notin \ker \pi_{\underline C}$ span $\mathsf{Inv}\!\left(\bigotimes_{i=1}^m V_q(\lambda_i)\right)$; in particular, every web satisfying the equivalent conditions $[W]_q\notin \ker \pi_{\underline C}$, non-convexity, and having no trip that starts and ends in the same clasp contributes such a spanning vector.
D. They are exactly the basis elements $[W]_q\in \mathcal W_{\underline a}$ satisfying any, equivalently all, of the following conditions: $[W]_q\notin \ker \pi_{\underline C}$; $W$ is non-convex; $W$ has no trip whose two endpoints lie in adjacent clasps. Moreover, the clasped web invariants $\pi_{\underline C}([W]_q)$ for precisely these webs form a basis of $\mathsf{Inv}\!\left(\bigotimes_{i=1}^m V_q(\lambda_i)\right)$.
E. They are exactly the basis elements $[W]_q\in \mathcal W_{\underline a}$ satisfying any, equivalently all, of the following conditions: $[W]_q\notin \ker \pi_{\underline C}$; $W$ is partially convex; $W$ has no trip that starts and ends in the same clasp. Moreover, the clasped web invariants $\pi_{\underline C}([W]_q)$ for precisely these webs form a basis of $\mathsf{Inv}\!\left(\bigotimes_{i=1}^m V_q(\lambda_i)\right)$.

forall-cut-path quantifier in non-convexity

basis conclusion weakened to spanning conclusion

trip obstruction changed from same clasp to adjacent clasps

The stem does not explicitly reveal the correct option. It states the hypotheses and asks for an equivalent condition plus the resulting basis statement, without naming the key conclusion about trips in the same clasp.

This is essentially a direct theorem-recall item: the stem presents the setup and asks for the statement equivalent to those conditions and the accompanying basis result. It is very close to restating the theorem rather than posing an independent problem.

Some reasoning is needed to distinguish the exact theorem statement from nearby variants such as 'spans' versus 'forms a basis' or altered endpoint conditions. However, the task is mainly precise recall/recognition rather than substantive generative mathematical reasoning.

The distractors are plausible and mathematically targeted: they vary quantifier dependence, weaken 'basis' to 'spans,' or subtly alter the forbidden-trip condition. These reflect realistic failure modes in recalling or parsing the theorem.

A technically strong but theorem-recall-heavy MCQ. It avoids direct answer leakage and uses good distractors, but it is largely tautological and only moderately tests genuine reasoning.


Question: Let $r\in\{2,3,4\}$. Let $\mathcal W_{\underline a}$ be the fully reduced web basis of $\mathsf{Inv}(V_q^{\underline a})$, and let $\underline C=(\underline c_1,\dots,\underline c_m)$ be a clasp sequence for the boundary conditions $\underline a$, with $\lambda_i=\mathrm{wt}(\underline c_i)$. For a fully reduced hourglass plabic graph $W$, say that $W$ is non-convex with respect to $\underline C$ if for every $i\in[m]$ and every $\underline c_i$-cut path $\gamma$, one has $\mathrm{wt}(\gamma)\ge \mathrm{wt}(\underline c_i)$ in the dominance order. Which statement exactly describes the web basis elements whose clasped images under $\pi_{\underline C}$ give the invariant space $\mathsf{Inv}\!\left(\bigotimes_{i=1}^m V_q(\lambda_i)\right)$?

Choices:
B. They are exactly the basis elements $[W]_q\in \mathcal W_{\underline a}$ satisfying any, equivalently all, of the following conditions: $[W]_q\notin \ker \pi_{\underline C}$; $W$ is non-convex; for each clasp $\underline c_i$ there exists a $\underline c_i$-cut path $\gamma$ with $\mathrm{wt}(\gamma)\ge \mathrm{wt}(\underline c_i)$ in the dominance order. Moreover, the clasped web invariants $\pi_{\underline C}([W]_q)$ for precisely these webs form a basis of $\mathsf{Inv}\!\left(\bigotimes_{i=1}^m V_q(\lambda_i)\right)$.
C. The clasped web invariants $\pi_{\underline C}([W]_q)$ coming from basis elements $[W]_q\in \mathcal W_{\underline a}$ with $[W]_q\notin \ker \pi_{\underline C}$ span $\mathsf{Inv}\!\left(\bigotimes_{i=1}^m V_q(\lambda_i)\right)$; in particular, every web satisfying the equivalent conditions $[W]_q\notin \ker \pi_{\underline C}$, non-convexity, and having no trip that starts and ends in the same clasp contributes such a spanning vector.
D. They are exactly the basis elements $[W]_q\in \mathcal W_{\underline a}$ satisfying any, equivalently all, of the following conditions: $[W]_q\notin \ker \pi_{\underline C}$; $W$ is non-convex; $W$ has no trip whose two endpoints lie in adjacent clasps. Moreover, the clasped web invariants $\pi_{\underline C}([W]_q)$ for precisely these webs form a basis of $\mathsf{Inv}\!\left(\bigotimes_{i=1}^m V_q(\lambda_i)\right)$.
E. They are exactly the basis elements $[W]_q\in \mathcal W_{\underline a}$ satisfying any, equivalently all, of the following conditions: $[W]_q\notin \ker \pi_{\underline C}$; $W$ is partially convex; $W$ has no trip that starts and ends in the same clasp. Moreover, the clasped web invariants $\pi_{\underline C}([W]_q)$ for precisely these webs form a basis of $\mathsf{Inv}\!\left(\bigotimes_{i=1}^m V_q(\lambda_i)\right)$.

The stem does not explicitly reveal option A. It introduces the definition of non-convexity, but several options reuse or modify that notion, so the correct answer is not leaked in a trivial way.

The item is essentially asking for the exact statement of a theorem characterizing which webs survive under clasping and form a basis. This is very close to direct theorem restatement rather than a genuinely new inference task.

There is some reasoning pressure because the distractors alter quantifiers, weaken 'basis' to 'span,' or change the trip obstruction. However, the task mainly tests precise recall/recognition of the theorem rather than generating the conclusion from underlying principles.

The distractors are strong: they are mathematically close, distinct, and target common errors such as quantifier weakening, confusing basis with spanning, and replacing the correct obstruction by a nearby but false one.

A solid theorem-recognition MCQ with high-quality distractors and little answer leakage, but it is mostly a near-verbatim recall item rather than a non-tautological, generative reasoning question.


[\autoref{thm_main}]
	\label{thmA}
	Let $X$ be an equidimensional projective scheme over a field $\mathbb{k}$.
	Let $\mathscr{L}$ be an ample line bundle on $X$ and $W \subseteq Z$ be two closed subschemes of $X$.
	Then the following three conditions are equivalent:
	\begin{enumerate}[\;\;\rm (a)]
		\item $\II_W$ is integral over $\II_Z$.
		\item $s(Z, X) = s(W, X)$ viewed in $A_*(X)$.
		\item $\deg_\mathscr{L}\left(s^i(Z, X)\right) = \deg_\mathscr{L}\left(s^i(W, X)\right)$ for all $i \ge 0$.
	\end{enumerate}

\begin{headcor}[\autoref{thm_main_zeta}]
	\label{corB}
	Let $I \subseteq J \subset R$ be two homogeneous ideals in a polynomial ring $R = \kk[x_0, \ldots,x_n]$.
	Then the following two conditions are equivalent:
	\begin{enumerate}[\;\;\rm (a)]
		\item $J$ is integral over $I$.
		\item $\zeta_I(t) = \zeta_J(t)$.
	\end{enumerate}
\end{headcor}

\begin{headthm}[\autoref{thm_main}]
	\label{thmA}
	Let $X$ be an equidimensional projective scheme over a field $\kk$.
	Let $\LL$ be an ample line bundle on $X$ and $W \subseteq Z$ be two closed subschemes of $X$.
	Then the following three conditions are equivalent:
	\begin{enumerate}[\;\;\rm (a)]
		\item $\II_W$ is integral over $\II_Z$.
		\item $s(Z, X) = s(W, X)$ viewed in $A_*(X)$.
		\item $\deg_\LL\left(s^i(Z, X)\right) = \deg_\LL\left(s^i(W, X)\right)$ for all $i \ge 0$.
	\end{enumerate}
\end{headthm}

\begin{corollary}
	\label{thm_main_zeta}
	Let $J \subset R = \kk[x_0,\ldots,x_n]$ be a homogeneous ideal such that $J \supseteq I$.
	Then the following two conditions are equivalent:
	\begin{enumerate}[\;\;\rm (a)]
		\item $J$ is integral over $I$.
		\item $\zeta_I(t) = \zeta_J(t)$.
	\end{enumerate}
\end{corollary}

The \emph{Segre class} associated with an embedding of schemes plays a central role in Fulton--MacPherson's intersection theory and underlies numerous applications.
A detailed account of Segre classes and their applications can be found in \cite{FULTON_INTER,ALUFFI_SURVEY}.

A central and defining property of Segre classes is their \emph{birational invariance}.
Indeed, if $Z$ is a proper closed subvariety of a variety $X$ over a field $\mathbb{k}$, this invariance yields
$$
s(Z, X) \;=\; \eta_*\left(s(E, \mathcal{B})\right) \;=\; \eta_*\left(c\left(N_E\mathcal{B}\right)^{-1} \smallfrown [E]\right) \;=\; \eta_*\left(\frac{[E]}{1+E}\right) \;\in\; A_*(Z),
$$
where $\pi : \mathcal{B} = \Bl_Z(X) \rightarrow X$ is the blow-up of $X$ along $Z$, $\eta : E = E_ZX \rightarrow Z$ is the exceptional divisor, and $N_E\mathcal{B}$ is the normal bundle to $E$ in $\mathcal{B}$ (see \cite[\S 4.2]{FULTON_INTER}).
Thus, by birational invariance, computing Segre classes reduces to the case of effective Cartier divisors, whose Segre classes admit an explicit formula since they are regularly embedded.

The birational invariance of Segre classes also implies that they depend only on the \emph{integral closure} of the underlying ideal sheaf.
More precisely, if we let $Z'= V\left(\overline{\II_Z}\right) \subset X$ where $\overline{\II_Z} \subset \OO_X$ is the integral closure of the ideal sheaf $\II_Z$ of $Z$, then the Segre class $s(Z, X) \in A_*(Z)$ of $Z$ equals the Segre class $s(Z', X) \in A_*(Z')$ of $Z'$ (since $Z'_{\rm red} = Z_{\rm red}$, we can identify their Chow groups).

\smallskip

Our main result demonstrates that Segre classes, in fact, determine integral dependence, leading to the following criterion:

\begin{theorem}
	\label{thm_main}
	Assume \autoref{setup_segre_int_dep}.
	Then the following five conditions are equivalent:
	\begin{enumerate}[\;\;\rm (a)]
		\item $\II_W$ is integral over $\II_Z$.
		\item ${j_Z}_*\left(s(Z, X)\right) = {j_W}_*\left(s(W, X)\right)$ in $A_*(X)$.
		\item $\deg_\LL\left(s^i(Z, X)\right) = \deg_\LL\left(s^i(W, X)\right)$ for all $i \ge 0$.
		\item $\deg_\LL\left(\nu^i(\underline{\sigma}, X)\right) = \deg_\LL\left(\nu^i(\underline{\eta}, X)\right)$ for all $i \ge 0$.
		\item $\deg_\LL\left(\beta^i(\underline{\sigma}, X)\right) = \deg_\LL\left(\beta^i(\underline{\eta}, X)\right)$ for all $i \ge 0$.
	\end{enumerate}
\end{theorem}

\begin{corollary}
 \label{cor_numerical_snapper}
 The following statements hold:
 \begin{enumerate}[\rm (i)]
 \item The terms of $P_{\LL, Z}(m, n)$ of degree $d$ are given as follows
 $$
 P_{\LL, Z}(m, n) \;=\; \sum_{i=0}^d \frac{\deg_\LL\left(\beta^i(\underline{\sigma}, X)\right)}{i!(d-i)!} \,m^{d-i} n^{i} \;+\; \text{{\rm(}lower degree terms{\rm)}}.
 $$
 \item For all $m \in \ZZ$ and $n \gg 0$, we have the equality
 $$
 P_{\LL, Z}(m, n) \;=\; \chi\big(\II_Z^n \otimes \LL^{\otimes (m+n)}\big).
 $$
 \item If $\LL$ is very ample on $X$, we get the equality
 $$
 P_{\LL, Z}(m, n) \;=\; h^0\big(X, \II_Z^n \otimes \LL^{\otimes (m+n)}\big)
 $$
 for all $m \gg 0$ and $n \gg 0$.
 \end{enumerate}
\end{corollary}
\begin{proof}
 (i) By \cite[Theorem B.7, Definition B.8]{KLEIMAN_PICARD}, we have
 $$
 \chi\left(\pi^*(\LL)^{\otimes m} \otimes \OO_{\mathcal{B}^\LL}(n)\right) \;=\; \sum_{i=0}^d \frac{a_{i}}{i!(d-i)!} \,m^{d-i} n^{i} \;+\; \text{(lower degree terms)}
 $$
 where $a_{i}= \int c_1\left(\pi^*(\LL)\right)^{d-i} c_1\left(\OO_{\mathcal{B}^\LL}(1)\right)^i \smallfrown \left[\mathcal{B}\right]$.
 Then the equalities
 $$
 a_i \;=\; \int c_1\left(\LL\right)^{d-i} \smallfrown \pi_*\left( c_1\left(\OO_{\mathcal{B}^\LL}(1)\right)^{i} \smallfrown \left[\mathcal{B}\right]\right) \;=\; \deg_\LL\left(\beta^i(\underline{\sigma}, X)\right)
 $$
 follow from the projection formula and \autoref{prop_blow_up_formula}.

\begin{lemma}
 \label{lem_num_equiv}
 The following conditions are equivalent:
 \begin{enumerate}[\;\;\rm (a)]
 \item $\deg_\LL\left(s^i(Z, X)\right) = \deg_\LL\left(s^i(W, X)\right)$ for all $i \ge 0$.
 \item $\deg_\LL\left(\nu^i(\underline{\sigma}, X)\right) = \deg_\LL\left(\nu^i(\underline{\eta}, X)\right)$ for all $i \ge 0$.
 \item $\deg_\LL\left(\beta^i(\underline{\sigma}, X)\right) = \deg_\LL\left(\beta^i(\underline{\eta}, X)\right)$ for all $i \ge 0$.
 \end{enumerate}
\end{lemma}
\begin{proof}
 (a) $\Rightarrow$ (b):
 By utilizing \autoref{eq_vogel_Z}, we obtain
 \begin{align*}
 \deg_\LL\left(\nu^i(\underline{\sigma}, X)\right) &\;=\; \int c_1(\LL)^{d-i} \smallfrown \nu^i(\underline{\sigma}, X) \\
 &\;=\; \sum_{j=0}^i\binom{i-1}{j-1} p^{i-j} \int c_1(\LL)^{d-i} c_1(\LL)^{i-j} \smallfrown s^j(Z, X) \\
 &\;=\; \sum_{j=0}^i\binom{i-1}{j-1} p^{i-j} \deg_\LL\left(s^j(Z, X)\right).
 \end{align*}
 Similarly, \autoref{eq_vogel_W} yields
 $$
 \deg_\LL\left(\nu^i(\underline{\eta}, X)\right) \;=\; \sum_{j=0}^i\binom{i-1}{j-1} p^{i-j} \deg_\LL\left(s^j(W, X)\right).
 $$
 So the implication (a) $\Rightarrow$ (b) follows.

\begin{theorem}
 \label{thm_main}
 Assume \autoref{setup_segre_int_dep}.
 Then the following five conditions are equivalent:
 \begin{enumerate}[\;\;\rm (a)]
 \item $\II_W$ is integral over $\II_Z$.
 \item ${j_Z}_*\left(s(Z, X)\right) = {j_W}_*\left(s(W, X)\right)$ in $A_*(X)$.
 \item $\deg_\LL\left(s^i(Z, X)\right) = \deg_\LL\left(s^i(W, X)\right)$ for all $i \ge 0$.
 \item $\deg_\LL\left(\nu^i(\underline{\sigma}, X)\right) = \deg_\LL\left(\nu^i(\underline{\eta}, X)\right)$ for all $i \ge 0$.
 \item $\deg_\LL\left(\beta^i(\underline{\sigma}, X)\right) = \deg_\LL\left(\beta^i(\underline{\eta}, X)\right)$ for all $i \ge 0$.
 \end{enumerate}
\end{theorem}
\begin{proof}
 (a) $\Rightarrow$ (b): This implication is a consequence of the birational invariance of Segre classes.
 For completeness, we include a short argument.
 Assume that $\overline{\II_Z} = \overline{\II_W}$.
 Thus $Z$ and $W$ have the same support, and so we may identify their Chow groups.
 Let $X_1,\ldots,X_k$ be the irreducible components of $X$ and $m_i$ be the geometric multiplicity of $X_i$ in $X$.
 Let $Z_i = Z \cap X_i$ and $W_i = W \cap X_i$.
 By \cite[Lemma 4.2]{FULTON_INTER}, we have $s(Z, X) = \sum_{i=1}^k m_i s(Z_i, X_i) \in A_*(Z)$ and $s(W, X) = \sum_{i=1}^k m_i s(W_i, X_i) \in A_*(W)$.
 Thus to show the equality $s(Z, X) = s(W, X) \in A_*(Z) = A_*(W)$, we may assume that $X$ is irreducible.

\begin{proposition}
 \label{prop_hard}
 Assume that $\Rees(\II_W)$ is not of finite type as a module over $\Rees(\II_Z)$.
 Then we have the following strict inequality
 $$
 \lim_{n \to \infty} \frac{h^0\left(X, \II_W^{n}/\II_Z^{n} \otimes \LL^{\otimes 2pn} \right)}{n^d} \;>\; 0.
 $$
\end{proposition}
\begin{proof}
 To simplify notation, let
 $$
 \sC \;:=\; \Rees^{\LL^{\otimes p}}(\II_Z) \;=\; \bigoplus_{n\ge 0} \II_Z^n \otimes \LL^{\otimes pn} \quad \text{ and } \quad \sD \;:=\; \Rees^{\LL^{\otimes p}}(\II_W) \;=\; \bigoplus_{n\ge 0} \II_W^n \otimes \LL^{\otimes pn}.
 $$
 Let $\pi : \sY = \fProj_X\left(\sD\right) \rightarrow X$ be the natural projection.
 We cannot have that $W_{\rm red} = X_{\rm red}$, because otherwise the ideal sheaves $\II_Z \subseteq \II_W$ would be nilpotent, and so they would have the same integral closure (equal to the nilradical ideal sheaf of $X$).
 It then follows that $\sY$ is equidimensional of dimension $d$ (see \autoref{rem_blowup_irr_comp}).
 By assumption, we have that $\fProj_X\left(\sD/\sC_+\sD\right) \neq \varnothing$, where $\sC_+\sD \subset \sD$ denotes the graded ideal in $\sD$ generated by $\sC_+ = \bigoplus_{n\ge 1} \sC_n$.
 Let $y \in \sY$ be a closed point lying in $\fProj_X\left(\sD/\sC_+\sD\right)$.
 Let $\II_y \subset \OO_\sY$ be the ideal sheaf of $y$. Let $\aaa \subset \sD$ be a graded ideal in $\sD$ whose sheafification is $\II_y = \widetilde{\aaa}$ and such that $\aaa \supset \sC_+\sD$.
 Since $\pi$ is a closed morphism, the closed point $y \in \sY$ is mapped to a closed point $x = \pi(y)$ in $X$.

\begin{example}[Line bundles that are big and nef do not suffice in \autoref{thm_main}]
 \label{examp}
 Assume $\kk$ is algebraically closed.
 Consider the following two successive blow-ups:
 \begin{enumerate}[\rm (i)]
 \item Let $\pi_1 : Y = \Bl_{p}\left(\PP_\kk^3\right) \rightarrow \PP_\kk^3$ be the blow-up of $\PP_{\kk}^3$ along a closed point $p \in \PP_\kk^3$.
 Notice that the exceptional divisor $E_p\PP_\kk^3$ of $Y=\Bl_{p}\left(\PP_\kk^3\right)$ is isomorphic to $\PP_{\kk}^2$.
 \item Choose a line $C$ in $\PP_\kk^2 \cong E_p\PP_\kk^3$.
 Let $\pi_2 : X = \Bl_C(Y) \rightarrow Y$ be the blow-up of $Y$ along $C$.
 Notice that $X$ is a smooth threefold.
 Let $E = E_CY$ be the exceptional divisor of $X = \Bl_C(Y)$.
 \end{enumerate}
 Let $\pi = \pi_1 \circ \pi_2 : X \rightarrow \PP_\kk^3$ be the natural projection.
 Let $\mathcal{L} = \pi^*\big(\OO_{\PP_\kk^3}(1)\big)$ be the line bundle on $X$ given as the pullback of $\OO_{\PP_\kk^3}(1)$.
 Notice that $\mathcal{L}$ is big and nef.
 Let $k \ge 2$ and consider the effective Cartier divisor $kE$.
 As closed subschemes, we have $E \subsetneq kE \subset X$ (although they have the same support).
 We also obtain that their Segre classes are different
 $$
 s(E, X) \;=\; E - E^2 + E^3 \;\neq\; kE -k^2E^2 + k^3E^3 \;=\; s(kE, X);
 $$
 here $E^i$ denotes the self-intersection of $E$
 (see \cite[Proposition 4.1(a)]{FULTON_INTER}).
 However, we can compute the vanishings
 $$
 \deg_{\mathcal{L}}\left(E\right) \;=\; \deg_{\mathcal{L}}\left(E^2\right) \;=\; \deg_{\mathcal{L}}\left(E^3\right) \;=\; 0.
 $$
 This shows that the big and nef line bundle $\mathcal{L}$ cannot be used in \autoref{thm_main}.
\end{example}
\begin{proof}
 We only need to show the claimed vanishings.
 The projection formula yields
 $$
 \deg_\mathcal{L}(E) \;=\; \int c_1\big(\OO_{\PP_\kk^3}(1)\big)^2 \smallfrown \pi_*\left([E]\right) \quad \text{ and } \quad \deg_\mathcal{L}(E^2) \;=\; \int c_1\big(\OO_{\PP_\kk^3}(1)\big) \smallfrown \pi_*\left(E \cdot [E]\right).
 $$
 By construction, $E$ is mapped onto the point $p$ via $\pi$.
 Since $[E] \in A_2(X)$ and $E\cdot [E] \in A_1(X)$, by dimension reasons and the definition of proper pushforward, we get $\pi_*([E]) = 0$ and $\pi_*(E \cdot [E]) = 0$ (see \cite[\S 1.4]{FULTON_INTER}).
 It then follows that $\deg_{\mathcal{L}}\left(E\right) = \deg_{\mathcal{L}}\left(E^2\right) = 0$.

In \autoref{thmA}, even if one is not interested in integral dependence, the equivalence (b) $\Leftrightarrow$ (c) seems interesting in its own right.
We note that \autoref{thmA} is sharp in the sense that the ampleness hypothesis appears to be indispensable.
\autoref{examp} illustrates this by exhibiting a big and nef line bundle for which the conclusion of \autoref{thmA} does not hold.

The proof of \autoref{thmA} relies on van Gastel’s foundational result \cite{VanGastel}, which establishes the connection between Fulton–MacPherson’s intersection theory \cite{FULTON_INTER} and St\"uckrad–Vogel’s intersection theory \cite{VOGEL,FOV}.
Indeed, our starting point is van Gastel’s result expressing Segre classes in terms of Vogel cycles, and conversely expressing Vogel cycles in terms of Segre classes.
An advantage of considering the Vogel cycle is that it opens the door to applying technical positivity results, including the one established in \autoref{prop_hard}.

\smallskip

As an application of \autoref{thmA}, we show that Aluffi’s Segre zeta function \cite{aluffi2017segre} furnishes a criterion for the integral dependence of homogeneous ideals in a polynomial ring.
This power series records information about the Segre classes that arise when the ideal is extended to projective spaces of arbitrarily large dimension (see \autoref{sect_zeta} for further details).
We obtain the following corollary:

\begin{headcor}[\autoref{thm_main_zeta}]
	\label{corB}
	Let $I \subseteq J \subset R$ be two homogeneous ideals in a polynomial ring $R = \mathbb{k}[x_0, \ldots,x_n]$.
	Then the following two conditions are equivalent:
	\begin{enumerate}[\;\;\rm (a)]
		\item $J$ is integral over $I$.
		\item $\zeta_I(t) = \zeta_J(t)$.
	\end{enumerate}
\end{headcor}

The idea of detecting integral dependence through numerical invariants originates in Rees’ seminal work \cite{REES}.
Rees proved that in an equidimensional and universally catenary Noetherian local ring $(R, \mm)$, two $\mm$-primary ideals $I \subseteq J$ have the same integral closure if and only if they share the same Hilbert–Samuel multiplicity.
Since then, the search for numerical criteria to characterize integral dependence has been an important research topic in algebraic geometry, commutative algebra and singularity
theory.
Much effort has gone into extending Rees’ theorem to arbitrary ideals, modules, and more broadly, to algebras (see \cite{BOEGER,TEISSIER_CYC,TEISSIER_RES2, KLEIMAN_THORUP_GEOM,KLEIMAN_THORUP_MIXED, GG, FLENNER_MANARESI,SUV_MULT, GAFFNEY, CIU, UV_CRIT_MOD,     UV_NUM_CRIT,   PTUV, cid2023relative, cidruiz2024polar, CRPU2, BLQ, das2024numerical}).
In this direction, two notable results are:
\begin{itemize}[\;--]
	\item Teissier's Principle of Specialization of Integral Dependence (PSID) \cite{TEISSIER_CYC}, \cite[Appendice I]{TEISSIER_RES2}.
	Given a map of germs of analytic spaces,  the PSID asserts that
	for a family of zero-dimensional ideals with constant Hilbert-Samuel multiplicity, a section is integrally
	dependent on the total family if and only if it is integrally dependent along the fibers corresponding to a dense open  subset of the base.
	\item An extension of Rees’ theorem to arbitrary ideals (in an equidimensional and universally catenary Noetherian local ring $(R, \mm)$) was established by Polini, Trung, Ulrich, and Validashti \cite{PTUV}, using the multiplicity sequence introduced by Achilles and Manaresi \cite{ACH_MANA}. (In the analytic setting, an analogous result had been obtained earlier by Gaffney and Gassler \cite{GG}.)
\end{itemize}
Note that these criteria are local in nature, whereas \autoref{thmA} furnishes a global criterion.

\medskip

  \noindent
\textbf{Outline.}
The structure of the paper is as follows.
In \autoref{sect_prelim}, we introduce the notation used throughout and recall the necessary preliminary results.
\autoref{sect_van_Gastel} develops the relation between Segre classes and Vogel cycles and shows how this connection can be applied.
The proof of \autoref{thmA} is given in \autoref{sect_main}.
Finally, \autoref{sect_zeta} contains the proof of \autoref{corB}.

The post-theorem introduction indicates that the proof of \autoref{thmA} \emph{relies on van Gastel’s foundational result} \cite{VanGastel}, which "establishes the connection between Fulton–MacPherson’s intersection theory" \cite{FULTON_INTER} and "St\"uckrad–Vogel’s intersection theory" \cite{VOGEL,FOV}. The stated \emph{starting point} is "van Gastel’s result expressing Segre classes in terms of Vogel cycles, and conversely expressing Vogel cycles in terms of Segre classes." The introduction further explains the strategy/advantage: "considering the Vogel cycle" allows one to apply "technical positivity results, including the one established in \autoref{prop_hard}." (The full proof is deferred: "The proof of \autoref{thmA} is given in \autoref{sect_main}.")

The post-theorem introduction indicates that, in establishing the main theorem, the proof \emph{relies on van Gastel’s foundational result} \cite{VanGastel}, which "establishes the connection between Fulton–MacPherson’s intersection theory" \cite{FULTON_INTER} and "St\"uckrad–Vogel’s intersection theory" \cite{VOGEL,FOV}. The stated \emph{starting point} is "van Gastel’s result expressing Segre classes in terms of Vogel cycles, and conversely expressing Vogel cycles in terms of Segre classes." The introduction further explains the strategy/advantage: "considering the Vogel cycle" allows one to apply "technical positivity results, including the one established in the following proposition." \begin{proposition}
	\label{prop_hard}
	Assume that $\Rees(\II_W)$ is not of finite type as a module over $\Rees(\II_Z)$.
	 Then we have the following strict inequality
	 $$
	 \lim_{n \to \infty} \frac{h^0\left(X, \II_W^{n}/\II_Z^{n} \otimes \LL^{\otimes 2pn} \right)}{n^d} \;>\; 0.
	 $$
\end{proposition} (The full proof is deferred: the proof of the main theorem is given later.)

To prove the main theorem, let $X$ be an equidimensional projective scheme over a field $\mathbb{k}$.
	\label{thmA}
	Let $\mathscr{L}$ be an ample line bundle on $X$ and $W \subseteq Z$ be two closed subschemes of $X$.
	Then the following three conditions are equivalent:
	\begin{enumerate}[\;\;\rm (a)]
		\item $\II_W$ is integral over $\II_Z$.
		\item $s(Z, X) = s(W, X)$ viewed in $A_*(X)$.
		\item $\deg_\mathscr{L}\left(s^i(Z, X)\right) = \deg_\mathscr{L}\left(s^i(W, X)\right)$ for all $i \ge 0$.
	\end{enumerate}

Question:
Let $X$ be an equidimensional projective scheme over a field $\mathbb{k}$, let $\mathscr L$ be an ample line bundle on $X$, and let $W\subseteq Z\subseteq X$ be closed subschemes. Write $s(Z,X)$ and $s(W,X)$ for their Segre classes, viewed in the Chow group $A_*(X)$, and let $s^i(Z,X)$ and $s^i(W,X)$ denote the codimension-$i$ components. For such a component $\alpha$, define its $\mathscr L$-degree by
$$
\deg_{\mathscr L}(\alpha):=\int c_1(\mathscr L)^{\dim X-i}\smallfrown \alpha.
$$
Which statement is equivalent to the ideal sheaf $\mathcal I_W$ being integral over $\mathcal I_Z$?

It is equivalent to either of the following, and hence to both: the Segre classes agree in $A_*(X)$, $$s(Z,X)=s(W,X),$$ and, equivalently, their $\mathscr L$-degrees agree in every codimension, $$\deg_{\mathscr L}(s^i(Z,X))=\deg_{\mathscr L}(s^i(W,X))\quad\text{for all } i\ge 0.$$

Choices:
B. It is equivalent to the equality of Segre classes after pushforward to numerical degrees: $$\deg_{\mathscr L}(s^i(Z,X))=\deg_{\mathscr L}(s^i(W,X))\quad\text{for all }0\le i\le \dim X-1,$$ and hence also to $$s(Z,X)=s(W,X)\text{ in }A_*(X).$$
C. It implies, and is in particular sufficient for, the equality of all $\mathscr L$-degrees of the Segre components: $$\deg_{\mathscr L}(s^i(Z,X))=\deg_{\mathscr L}(s^i(W,X))\quad\text{for all } i\ge 0.$$
D. It is equivalent to requiring that for every ample line bundle $\mathscr M$ on $X$, the numerical equalities $$\deg_{\mathscr M}(s^i(Z,X))=\deg_{\mathscr M}(s^i(W,X))\quad\text{for all } i\ge 0$$ hold; equivalently, $$s(Z,X)=s(W,X)\text{ in }A_*(X).$$
E. It is equivalent to the equality of the reduced Segre classes, namely that $$s(Z_{\mathrm{red}},X)=s(W_{\mathrm{red}},X)\text{ in }A_*(X),$$ and equivalently that $$\deg_{\mathscr L}(s^i(Z_{\mathrm{red}},X))=\deg_{\mathscr L}(s^i(W_{\mathrm{red}},X))\quad\text{for all } i\ge 0.$$

equivalence dropped to one-way implication from integrality to degree equalities

fixed ample line bundle replaced by uniformity over all ample line bundles

scheme-structure sensitivity of Segre classes replaced by reduced-subscheme equality

The stem does not state the equivalence outright; it only introduces the relevant objects and notation. While the setup strongly signals that Segre classes and their degrees are central, it does not explicitly leak which precise equivalence is correct.

This is very close to a direct restatement of the underlying theorem: integrality of ideal sheaves is equivalent to equality of Segre classes, equivalently equality of the corresponding degrees. The correct choice essentially reproduces that theorem verbatim.

There is some reasoning required to distinguish the exact theorem from nearby variants such as one-way implication, strengthened quantification, or reduced-subscheme replacement. However, for a student who knows the theorem, the correct answer is largely recall rather than substantial generation or synthesis.

The distractors are mathematically plausible and target realistic failure modes: truncating the codimension range, weakening equivalence to implication, over-strengthening to all ample line bundles, and incorrectly passing to reduced subschemes. They are distinct and well aligned with conceptual misunderstandings.

A mathematically well-constructed recall-style MCQ with strong distractors, but it is highly theorem-restatement-based and thus only moderately effective at testing genuine generative reasoning.