\begin{theorem}\label{Teo1}
	Assume that the potential $W$ satisfies \hyperlink{W1}{$(W_1)$}-\hyperlink{W4}{$(W_4)$}. Then, for every $\varepsilon > 0$ and $a_0 > 0$, there exists $\delta > 0$ such that for all $a \in (0, a_0]$, and any solution $u_a$ of \eqref{Equation} with initial data satisfying
	$$
	\|u_a(0)-u_*\| + \|\dot{u}_a(0)\| < \delta,
	$$
	the following hold:
	\begin{itemize}
		\item[\emph{(a)}] $\displaystyle \sup_{t \ge 0} \left( \|u_a(t)-u_*\| + \|\dot{u}_a(t)\| \right) \le \varepsilon$;
		\item[\emph{(b)}] $\displaystyle \lim_{t \to +\infty} u_a(t) =u_*$ and $\displaystyle\lim_{t \to +\infty} \dot{u}_a(t) = 0$.
	\end{itemize}
	If, additionally, \hyperlink{W2}{$(W_2)$} and \hyperlink{W3}{$(W_3)$} hold with $\delta = \lambda = +\infty$, then there exists $R > 0$ such that, for all $a \in (0, a_0]$, and any solution $u_a$ of \eqref{Equation},
	\begin{itemize}
		\item[\emph{(c)}] $\displaystyle \sup_{t \ge 0} \left( \|u_a(t)-u_*\| + \|\dot{u}_a(t)\| \right) \le R$;
		\item[\emph{(d)}] $\displaystyle \lim_{t \to +\infty} u_a(t) =u_*$ and  $\displaystyle\lim_{t \to +\infty} \dot{u}_a(t) = 0$.
	\end{itemize}
	Moreover, under the additional assumption \hyperlink{W5}{$(W_5)$}, for each $a >0$ there exist $\gamma(a) > 0$ and $C(a)>0$ such that the corresponding solution $u_a$ satisfies the exponential decay estimate
	\begin{equation*}
		\|u_a(t)-u_*\| + \|\dot{u}_a(t)\| \le C(a) e^{-\gamma(a) t}, \quad \forall\, t \ge 0.
	\end{equation*}
	In particular, if $0<a_1<a_2$ and $a\in[a_1,a_2]$, then there exist $C_*>0$ and $\gamma_*>0$, depending only on $a_1,a_2$ such that
	\begin{equation*}
		\|u_a(t)-u_*\| + \|\dot{u}_a(t)\| \le C_* e^{-\gamma_* t},\quad \forall\, t \ge 0,\ \forall\, a\in[a_1,a_2].
	\end{equation*}
\end{theorem}

Second-order dissipative systems of the form
\begin{equation}\label{Equation}
	\ddot{u}(t) + a \dot{u}(t) + \nabla W(u(t)) = 0,
\end{equation}
arise naturally in mechanics, optimization, and nonlinear dynamics. A classical example is the \emph{heavy-ball method with friction}, in which a particle moves on the graph of a potential $W$ under the action of inertia and a linear damping force with coefficient $a>0$. The heavy-ball method was introduced by Polyak in 1964 to speed up the convergence of iterative schemes \cite{Polyak}, and was formalized by Attouch, Goudou, and Redont \cite{AttouchGoudouRedont}, who analyzed the frictional equation to explore the minima of potential $W$. Beyond heavy-ball method with friction, the prototype equation \eqref{Equation} also models the motion of a \emph{damped simple pendulum}.
Let $u=\theta$ be the pendulum angle, take $W(\theta)=\frac{g}{\ell}(1-\cos\theta)$, then
$$
\ddot{\theta}(t)+ a\,\dot{\theta}(t)+\dfrac{g}{\ell}\sin\theta(t)=0,
$$
which is the standard damped pendulum equation. This modeling is classical in nonlinear dynamics and the term $a\theta'$ represents the damping force, with $a > 0$. Depending on the initial energy, this system exhibits oscillatory motions, convergence to stable equilibria, and heteroclinic trajectories connecting different critical points of $W$. A comprehensive analysis of the pendulum dynamics can be found in \cite{Mawhin}. In fact, beyond these classical examples, a number of other mechanical and computational systems reduce to the general dissipative structure \eqref{Equation}, which reinforces the broad applicability and theoretical interest of the problem.

Second-order equations with constant damping also appear in the study of generalized gradient systems. Recent works have achieved significant progress in the asymptotic analysis of such systems. Jendoubi and collaborators showed that an equation of the type \eqref{Equation} can be interpreted as a quasi-gradient system via small deformations of the total energy. They proved that the desingularizing function in the Kurdyka-Łojasiewicz inequality satisfies $\varphi(t) \ge c\sqrt{t}$ when the potential $W$ is definable and of class $C^2$ \cite{Jendoubi1}. This result implies that every trajectory of a quasi-gradient system either converges to a critical point or its norm diverges to infinity. Another relevant development concerns Newton-type systems with {\it Hessian-driven damping}, combining the continuous Newton method with the heavy-ball system. Boţ and Csetnek in \cite{BotCsetnek} observed that such systems are second-order in both time and space, due to the presence of the acceleration term and the Hessian, and can be seen as a mixture of Newton’s method with the heavy-ball system. This hybrid approach inherits advantages from both methods, as shown in the optimization context by Alvarez et al. \cite{Alvarez}. We also mention that several recent works have focused on nonhomogeneous variants of \eqref{Equation} and on models where the damping parameter depends on time, such as \cite{1,2,3,4,6,7}.

The term $a\dot{u}$ in \eqref{Equation} represents a damping force, while $\nabla W(u)$ acts as a restoring force derived from the potential energy $W(u)$. In this work we focus on the second-order dissipative systems \eqref{Equation}, where $a > 0$ is a damping coefficient, $u\colon \mathbb{R} \to \mathbb{R}^N$ is the unknown trajectory, and $W\colon \mathbb{R}^N \to \mathbb{R}$ is a potential. Throughout the paper we impose the following assumptions on $W$:
\begin{itemize}
	\item[\hypertarget{W1}{($W_1$)}] $W\in C^2(\mathbb{R}^N; \mathbb{R})$.

	\item[\hypertarget{W2}{($W_2$)}] There exists $\delta > 0$ and $u_* \in \mathbb{R}^{N}$ such that
	$$
	W(u_*) = 0 \quad \text{and} \quad W(u) > 0 \quad \text{for all} \quad u \in B_\delta(u_*) \setminus \{u_*\}.
	$$

	\item[\hypertarget{W3}{($W_3$)}] There exists $\lambda > 0$ such that
	$$
	\langle \nabla W(u), u - u_* \rangle > 0 \quad \text{for all} \quad u \in B_\lambda(u_*) \setminus \{u_*\}.
	$$

	\item[\hypertarget{W4}{($W_4$)}] $W$ is coercive, that is,
	$$
	\lim_{\|u\| \to +\infty} W(u) = +\infty.
	$$

	\item[\hypertarget{W5}{($W_5$)}] There exist constants $\alpha, \beta, \mu > 0$ such that, for every $x \in B_\lambda(u_*)$,
	$$
	\alpha\|x - u_*\|^2 \le W(x) \le \beta\|x - u_*\|^2 \quad \text{and} \quad \langle \nabla W(x), x - u_* \rangle \ge \mu\|x - u_*\|^2.
	$$
\end{itemize}

A simple example satisfying \hyperlink{W1}{$(W_1)$}-\hyperlink{W4}{$(W_5)$} with $u_*=0$ is the harmonic potential
$$
W(u)=\frac{1}{2}\|u\|^2.
$$
In this case, \eqref{Equation} reduces to the classical damped harmonic oscillator. This model describes a {\it mass-spring-damper system}, where $a$ represents the viscous damping and $u$ is the Hookean restoring force. Solutions exhibit decaying oscillations and converge to the equilibrium $u=0$. Such systems appear  not only in mechanics but also in electrical circuits and other dissipative models driven by an energy potential, see \cite{Edwards,Sunday1,Sunday2}. Beyond the quadratic case, the class of potentials satisfying assumptions \hyperlink{W1}{$(W_1)$}-\hyperlink{W4}{$(W_4)$} is broad and includes many models from mechanics, nonlinear dynamics, and phase-transition theory. A general and important family is obtained by taking
$$
W(u)=\Phi(u-u_*),
$$
where $\Phi\colon\mathbb{R}^N\to\mathbb{R}$ is a convex, positive function with a global minimum at the origin. Such potentials arise naturally in the study of quasilinear Allen-Cahn type equations, see, for instance, \cite{Isneri}. In particular, coercive polynomial potentials, such as $W(u)=\|u-u_*\|^p$, with $p\ge 2$, and the exponential potential
$$
W(u)=\frac12\big(e^{\|u-u_*\|^2}-1\big),
$$
which is convex and coercive, fits perfectly into this model. Another classical example is the Ginzburg-Landau potential
$$
W(u)=\frac14\left(\|u\|^2-1\right)^2,
$$
whose set of global minima is the unit sphere $\{u\in\mathbb{R}^N : \|u\| = 1\}$. This potential play a central role in the theory of phase transitions and superconductivity, see \cite{Allen,5}. A further relevant example, frequently appearing in the study of dissipative hyperbolic PDEs, is the quartic double-well potential
$$
W(u)=c_1\|u\|^{4}-c_2\|u\|^{2}+c_3,\quad c_1,c_2,c_3>0,
$$
whose structure is extensively used in nonlinear wave equations with damping. Such models constitute a natural infinite-dimensional counterpart of \eqref{Equation}, see, for instance, \cite{8}. These examples illustrate that the hypotheses imposed on $W$ encompass both linear and nonlinear restoring forces, as in the damped mass-spring system, as well as potentials arising in gradient flows, optimization, and models of phase separation.

With these examples in mind, we are ready to present our main stability theorem, which shows that the qualitative behavior of \eqref{Equation} is remarkably robust and yields uniform control of all trajectories near the equilibrium.

Second-order dissipative systems of the form
\begin{equation}\label{Equation}
 \ddot{u}(t) + a \dot{u}(t) + \nabla W(u(t)) = 0,
\end{equation}
arise naturally in mechanics, optimization, and nonlinear dynamics. A classical example is the \emph{heavy-ball method with friction}, in which a particle moves on the graph of a potential $W$ under the action of inertia and a linear damping force with coefficient $a>0$. The heavy-ball method was introduced by Polyak in 1964 to speed up the convergence of iterative schemes \cite{Polyak}, and was formalized by Attouch, Goudou, and Redont \cite{AttouchGoudouRedont}, who analyzed the frictional equation to explore the minima of potential $W$. Beyond heavy-ball method with friction, the prototype equation \eqref{Equation} also models the motion of a \emph{damped simple pendulum}.
Let $u=\theta$ be the pendulum angle, take $W(\theta)=\frac{g}{\ell}(1-\cos\theta)$, then
$$
\ddot{\theta}(t)+ a\,\dot{\theta}(t)+\dfrac{g}{\ell}\sin\theta(t)=0,
$$
which is the standard damped pendulum equation. This modeling is classical in nonlinear dynamics and the term $a\theta'$ represents the damping force, with $a > 0$. Depending on the initial energy, this system exhibits oscillatory motions, convergence to stable equilibria, and heteroclinic trajectories connecting different critical points of $W$. A comprehensive analysis of the pendulum dynamics can be found in \cite{Mawhin}. In fact, beyond these classical examples, a number of other mechanical and computational systems reduce to the general dissipative structure \eqref{Equation}, which reinforces the broad applicability and theoretical interest of the problem.

The term $a\dot{u}$ in \eqref{Equation} represents a damping force, while $\nabla W(u)$ acts as a restoring force derived from the potential energy $W(u)$. In this work we focus on the second-order dissipative systems \eqref{Equation}, where $a > 0$ is a damping coefficient, $u\colon \mathbb{R} \to \mathbb{R}^N$ is the unknown trajectory, and $W\colon \mathbb{R}^N \to \mathbb{R}$ is a potential. Throughout the paper we impose the following assumptions on $W$:
\begin{itemize}
 \item[\hypertarget{W1}{($W_1$)}] $W\in C^2(\mathbb{R}^N; \mathbb{R})$.

\item[\hypertarget{W2}{($W_2$)}] There exists $\delta > 0$ and $u_* \in \mathbb{R}^{N}$ such that
 $$
 W(u_*) = 0 \quad \text{and} \quad W(u) > 0 \quad \text{for all} \quad u \in B_\delta(u_*) \setminus \{u_*\}.
 $$

\item[\hypertarget{W5}{($W_5$)}] There exist constants $\alpha, \beta, \mu > 0$ such that, for every $x \in B_\lambda(u_*)$,
 $$
 \alpha\|x - u_*\|^2 \le W(x) \le \beta\|x - u_*\|^2 \quad \text{and} \quad \langle \nabla W(x), x - u_* \rangle \ge \mu\|x - u_*\|^2.
 $$
\end{itemize}

A simple example satisfying \hyperlink{W1}{$(W_1)$}-\hyperlink{W4}{$(W_5)$} with $u_*=0$ is the harmonic potential
$$
W(u)=\frac{1}{2}\|u\|^2.
$$
In this case, \eqref{Equation} reduces to the classical damped harmonic oscillator. This model describes a {\it mass-spring-damper system}, where $a$ represents the viscous damping and $u$ is the Hookean restoring force. Solutions exhibit decaying oscillations and converge to the equilibrium $u=0$. Such systems appear not only in mechanics but also in electrical circuits and other dissipative models driven by an energy potential, see \cite{Edwards,Sunday1,Sunday2}. Beyond the quadratic case, the class of potentials satisfying assumptions \hyperlink{W1}{$(W_1)$}-\hyperlink{W4}{$(W_4)$} is broad and includes many models from mechanics, nonlinear dynamics, and phase-transition theory. A general and important family is obtained by taking
$$
W(u)=\Phi(u-u_*),
$$
where $\Phi\colon\mathbb{R}^N\to\mathbb{R}$ is a convex, positive function with a global minimum at the origin. Such potentials arise naturally in the study of quasilinear Allen-Cahn type equations, see, for instance, \cite{Isneri}. In particular, coercive polynomial potentials, such as $W(u)=\|u-u_*\|^p$, with $p\ge 2$, and the exponential potential
$$
W(u)=\frac12\big(e^{\|u-u_*\|^2}-1\big),
$$
which is convex and coercive, fits perfectly into this model. Another classical example is the Ginzburg-Landau potential
$$
W(u)=\frac14\left(\|u\|^2-1\right)^2,
$$
whose set of global minima is the unit sphere $\{u\in\mathbb{R}^N : \|u\| = 1\}$. This potential play a central role in the theory of phase transitions and superconductivity, see \cite{Allen,5}. A further relevant example, frequently appearing in the study of dissipative hyperbolic PDEs, is the quartic double-well potential
$$
W(u)=c_1\|u\|^{4}-c_2\|u\|^{2}+c_3,\quad c_1,c_2,c_3>0,
$$
whose structure is extensively used in nonlinear wave equations with damping. Such models constitute a natural infinite-dimensional counterpart of \eqref{Equation}, see, for instance, \cite{8}. These examples illustrate that the hypotheses imposed on $W$ encompass both linear and nonlinear restoring forces, as in the damped mass-spring system, as well as potentials arising in gradient flows, optimization, and models of phase separation.

With these examples in mind, we are ready to present our main stability theorem, which shows that the qualitative behavior of \eqref{Equation} is remarkably robust and yields uniform control of all trajectories near the equilibrium.

\begin{equation}\label{Equation}
	\ddot{u}(t) + a \dot{u}(t) + \nabla W(u(t)) = 0,
\end{equation}

With these examples in mind, we are ready to present our main stability theorem, which shows that the qualitative behavior of \eqref{Equation} is remarkably robust and yields uniform control of all trajectories near the equilibrium.

In this section we discuss the dynamics of the conservative system
\begin{equation}\label{EqConservative}
 \ddot{u}(t) + \nabla W(u(t)) = 0,
\end{equation}
which arises as the limit of \eqref{Equation} when the damping parameter $a$ tends to zero. Throughout this section, we continue to assume that $W$ satisfies \hyperlink{W1}{$(W_1)$}-\hyperlink{W4}{$(W_4)$} and, without loss of generality, that the global minimum of $W$ is located at $u_*=0$. In contrast with the dissipative
case, the conservative dynamics do not generate asymptotic stability, since no decay mechanism is present. To analyze the qualitative behavior of \eqref{EqConservative}, it is convenient to rewrite it as a first-order system in $\mathbb{R}^{2N}$ by setting $x=u$ and $y=\dot{u}$:
\begin{equation}\label{S0}
 \begin{cases}
 \dot{x} = y,\\[2mm]
 \dot{y} = -\nabla W(x).
 \end{cases}
\end{equation}
The natural energy associated with \eqref{EqConservative},
$$
E(x,y) := \frac12\|y\|^2 + W(x), \quad (x,y)\in B_\eta(0)\times \mathbb{R}^N,
$$
where $\eta=\min\{\delta,\lambda\}$ with $\delta$ and $\lambda$ given in \hyperlink{W2}{$(W_2)$}-\hyperlink{W3}{$(W_3)$}, acts as a Lyapunov function for the equilibrium $(0,0)$. Indeed, if $(x(t),y(t))$ is a solution of \eqref{S0}, then deriving
$E(x(t),y(t))$ along the trajectory yields
$$
\frac{d}{dt}E(x(t),y(t))= \langle y(t),\dot{y}(t)\rangle + \langle \nabla W(x(t)),\dot{x}(t)\rangle.
$$
Using $\dot{x}=y$ and $\dot{y}=-\nabla W(x)$, we immediately obtain
$$
\frac{d}{dt}E(x(t),y(t))= -\langle y(t),\nabla W(x(t))\rangle+ \langle \nabla W(x(t)),y(t)\rangle= 0.
$$
Thus, if $u(t)$ is any solution of \eqref{EqConservative}, then the energy is conserved in time, that is,
\begin{equation}\label{EnergyConservation}
 \frac12\|\dot{u}(t)\|^2 + W(u(t)) = \tau,\quad \forall\, t\in\mathbb{R},
\end{equation}
for some $\tau\ge 0$ determined by the initial condition. From the Lyapunov Criterion, conservation of energy implies that
$(0,0)$ is stable: for every $\varepsilon>0$ there exists
$\delta>0$ such that
$$
\|u(0)\| + \|\dot{u}(0)\| < \delta\quad\Longrightarrow\quad\sup_{t\ge 0} \big( \|u(t)\| + \|\dot{u}(t)\| \big) \le \varepsilon.
$$
However, stability cannot be asymptotic. Indeed, if a nontrivial solution $(u(t),\dot{u}(t))$ were to converge to $(0,0)$ as $t\to+\infty$, then \eqref{EnergyConservation} would force $\tau=0$, which implies that $u(t)\equiv 0$ identically. Hence the only trajectory converging to the equilibrium is the constant solution.

We now turn to the relation between the conservative regime and the dissipative dynamics studied in Section \ref{Sec2}. Assume that $W$ satisfies \hyperlink{W1}{$(W_1)$}-\hyperlink{W5}{$(W_5)$}. The conservative system \eqref{EqConservative} arises as the limit of the dissipative equation \eqref{Equation} when the damping parameter $a$ tends to zero. In the dissipative regime, Section \ref{Sec2} established that solutions are asymptotically stable and even converge exponentially fast to the equilibrium with decay rate $\gamma(a)>0$ depending on $a$. A natural question is whether this behavior persists, in some sense, in the limit as $a\to 0^{+}$. The answer is negative: the limit dynamics is not asymptotically stable, and the passage $a\to 0^{+}$ is singular. This phenomenon can be described rigorously through a perturbation argument. Let $(a_n)$ be a sequence of positive numbers with $a_n \to 0$, and let $u_n$ be the solution of the perturbed problem
\begin{equation}\label{EqDamped-n}
 \ddot u_n(t) + a_n \dot u_n(t) + \nabla W(u_n(t)) = 0,
\end{equation}
with fixed initial data $(u_n(0),\dot u_n(0)) = (z_0,z_1)$, where $(z_0,z_1)\neq(0,0)$ and $\|(z_0,z_1)\|<\sigma$. Here $\sigma>0$ depends on a prescribed $\varepsilon\in(0,\delta)$, with $\delta>0$ given in \hyperlink{W2}{\((W_2)\)}, and is provided by the uniform stability result of Lemma \ref{L5}. In particular, Lemma \ref{L5} ensures that
$$
\lim_{t\to+\infty} (u_n(t),\dot u_n(t)) = (0,0)\quad\text{and}\quad\|u_n(t)\| + \|\dot u_n(t)\| \le \varepsilon \quad\text{for all } t\ge 0\text{ and }n\in\mathbb{N}.
$$
Moreover, by Lemma \ref{L7}, each $u_n$ satisfies the exponential decay estimate
\begin{equation}\label{Exp}
 \|u_n(t)\| + \|\dot u_n(t)\|\le C e^{-\gamma(a_n) t}, \qquad t\ge 0,
\end{equation}
where $C>0$ is a constant independent of $a_n$. A compactness argument shows that, up to a subsequence, $u_n$ converges in $C^2_{\mathrm{loc}}([0,+\infty))$ to a limit
function $u_0$, which solves the conservative system
$$
\ddot u_0 + \nabla W(u_0)=0
$$
with initial data $u_0(0)=z_0$ and $\dot u_0(0)=z_1$, and satisfying
\begin{equation*}
 \sup_{t\ge 0}\left(\|u_0(t)\|+\|\dot{u}_0(t)\|\right) \leq \epsilon.
\end{equation*}
Although each $u_n$ converges asymptotically to the equilibrium, the limit solution $u_0$ does not. Indeed, suppose by contradiction that
$$
\lim_{t\to+\infty} u_0(t)=0\quad\text{and}\quad
\lim_{t\to+\infty} \dot u_0(t)=0.
$$
Since $W$ is continuous and $W(0)=0$, it follows that $W(u_0(t))\to 0$ and $\|\dot u_0(t)\|\to 0$. Inserting these limits into the conservation law \eqref{EnergyConservation}, we obtain $\tau=0$. But $\tau=0$ implies
$$
\dot u_0(t)\equiv 0\quad\text{and}\quad W(u_0(t))\equiv 0,
$$
so that $u_0(t)= 0$ for all $t\in\mathbb{R}$. This contradicts the fact that $(u_0(0),\dot u_0(0))\neq(0,0)$. Therefore,
$$
\lim_{t \to +\infty} (u_0(t),\dot{u}_0(t)) \neq (0,0).
$$
This argument shows that although each damped solution converges to the equilibrium, the limit solution does not. Equivalently, the exponential decay rate $\gamma(a)$ obtained for $a>0$ must necessarily satisfy $\gamma(a)\to 0$ as $a\to 0^+$, and no uniform decay estimate can hold on an interval of the form $(0,a_0]$. The transition from $a>0$ to $a=0$ represents a loss of asymptotic stability and a qualitative change in the global dynamics.

\begin{lemma}\label{asd}
 Let $B\subset \mathbb R^{2N}$ be any bounded set containing the initial data $z_0$.
 Then there exists a bounded set $K\subset \mathbb{R}^{2N}$ such that
 $$
 T(t) B \subset K,\quad \forall\, t\ge 0.
 $$
 In other words, $K$ serves as a absorbing set for the family of semigroup $\{T(t)\}_{t\ge 0}$.
\end{lemma}
\begin{proof}
Fix $a>0$ and let $z_0=(x_0,y_0)\in B$. Since $B$ is bounded in $\mathbb{R}^{2N}$ and $W$ is continuous by \hyperlink{W1}{$(W_1)$}, there exists a $R_B>0$ such that
$$
\sup_{z_0\in B} E(z_0)\le R_B.
$$
Therefore, by the monotonicity property of the energy in \eqref{decreasing-property}, we obtain
$$
W(x_a(t))\leq R_B\quad\text{and}\quad\|y_a(t)\|^2\leq 2R_B\quad\forall \, t\geq 0,
$$
where $x_a$ and $y_a$ denote the components of the trajectory $\varphi_a(t;z_0)$. Next, by the coercivity assumption \hyperlink{W4}{$(W_4)$}, the sublevel set
$$
\mathcal{S}_{B} \coloneqq \{ x \in \mathbb{R}^{N}:W(x)\le R_B\}
$$
is bounded. Consequently, we may define the positive real number
$$
C_B \coloneqq \sup \left\{\|x\|^{2} : x \in \mathcal{S}_{B}\right\} < \infty.
$$
In this way, we have
$$
\Vert T(t)z_0 \Vert_{\ast}^2 = \Vert (x_a(t), y_a(t))\Vert_{\ast}^2 = \Vert x_a(t) \Vert^2 + \Vert y_a(t) \Vert^2 \leq C_B + 2R_B,\quad \forall\, t\ge 0,
$$
Now, defining
$$
K\coloneqq\Big\{(x,y)\in\mathbb{R}^{2N}:\|(x,y)\|_{\ast}^{2}\le \sqrt{C_B + 2R_B}\Big\}
$$
it follows the inclusion $T(t)B \subset K$ for all $t\geq 0$, which finishes our proof.
\end{proof}

The Theorem \ref{Teo1} asserts that the equilibrium $(u_*,0)$ of the system \eqref{Equation} is uniformly asymptotically stable with respect to the parameter $a\in(0,a_0]$: every solution starting sufficiently close to $(u_*,0)$ remains in an arbitrarily small neighborhood of the equilibrium and converges to it as $t\to +\infty$, with bounds that are uniform in $a$. The fundamental point of our study is the construction of a strict Lyapunov functional for the first-order formulation of the dissipative system
\begin{equation}\label{S-intro}
	\begin{cases}
		\dot{x}=y,\\[1mm]
		\dot{y}=-a\,y-\nabla W(x),
	\end{cases}
\end{equation}
which encodes the full dynamics of \eqref{Equation} in the phase space $\mathbb{R}^{2N}$. To this end, we introduce the Lyapunov functional
\begin{equation}
	V_a(x,y):=\frac{1}{2}\|y\|^2+2W(x)+\frac{1}{2}\|y+a(x-u_*)\|^2,
\end{equation}
which can be interpreted as a smooth perturbation of the mechanical energy
$$
E(x,y):=\frac12\|y\|^2+W(x).
$$
The functional $V_a$ allows us to obtain stability estimates that are uniform with respect to $a$. The assumptions \hyperlink{W1}{($W_1$)}-\hyperlink{W4}{($W_4$)} play a crucial role in ensuring that $V_a$ is strictly decreasing along non-trivial trajectories. The condition \hyperlink{W5}{($W_5$)} is only required to derive exponential decay of solutions, this decay cannot be uniform as $a\to 0^{+}$, since the corresponding rate $\gamma(a)$ necessarily satisfies $\gamma(a)\to 0$. Furthermore, the introduction of the Lyapunov functional $V_a$ therefore provides a robust framework for establishing the uniform asymptotic stability of \eqref{Equation}. Furthermore, we believe that $V_a$ highlights a mechanism that can be extended to non-homogeneous cases, to time-dampened systems, and to more general configurations such as trajectories over Hilbert space.

We emphasize that the conclusions of Theorem \ref{Teo1} are not restricted to a specific equilibrium. In fact, the result applies to any global minimizer $u_*$ of the potential $W$ satisfying assumptions \hyperlink{W1}{$(W_1)$}-\hyperlink{W4}{$(W_4)$}, yielding uniform asymptotic stability of the corresponding equilibrium $(u_*,0)$. Moreover, when the additional condition \hyperlink{W5}{$(W_5)$} holds, the convergence toward $(u_*,0)$ is exponential. In this sense, Theorem \ref{Teo1} provides a robust stability result around arbitrary admissible minima of the potential.

When $a=0$, the equation \eqref{Equation} reduces to the conservative system
$$
\ddot u(t) + \nabla W(u(t))=0,
$$
corresponding to the motion of a particle on $W$ without friction, so that mechanical energy is preserved. Here, $u(t)$ is the position of the particle, $\ddot{u}(t)$ its acceleration, and $\nabla W(u)$ the force derived from $W$. Under assumptions \hyperlink{W1}{($W_1$)}-\hyperlink{W4}{($W_4$)}, and in particular the coercivity of $W$, every solution remains globally bounded in $\mathbb{R}^{N}$, that is,
$$
\sup_{t\in\mathbb{R}}\left(\|u(t)\|+\|\dot u(t)\|\right)<\infty.
$$
Thus blow-up cannot occur and no trajectory can converge asymptotically to the equilibrium. In this Hamiltonian-type regime, the combination of confinement and energy conservation leads to the typical behaviors of conservative dynamics: periodic orbits and transition solutions (heteroclinic or homoclinic). Hence, the case $a=0$ produces a qualitative change in the dynamics, yielding globally bounded but non-convergent trajectories.

In addition to the study of the asymptotic stability of \eqref{Equation}, two additional components complement our study. In Section \ref{Sec4}, we revisit the dynamics from the point of view of nonlinear semigroup theory, aiming for a comprehensive qualitative description of the dissipative flux generated by the first-order system \eqref{Equation}. This approach shows, in particular, the existence of a global attractor $\mathcal{A}$ for the corresponding semigroup, which is a compact, invariant subset of the phase space that attracts all bounded sets in forward time. Moreover, $\mathcal{A}$ contains the entire set of equilibria of the system, that is,
$$\{(x,0)\in\mathbb{R}^{2N}:\nabla W (x)=0\}\subset \mathcal{A}.
$$
For readers interested in the general theory of global attractors and nonlinear semigroups, we refer to the classical monographs \cite{Livro-Alexandre,Hale,Temam}. Finally, Section \ref{Sec5} presents numerical simulations for some cases of \eqref{Equation} involving quadratic, exponential, and double-well potentials implemented in a computational environment based on \emph{R}. The simulations rely on a functional-programming framework supported by \texttt{tidyverse} (data manipulation), \texttt{deSolve} (time-integration of differential equations), and \texttt{ggplot2} (phase-plane visualization and graphical output). This computational component has a dual purpose: to geometrically illustrate the results developed in the previous sections and to reveal characteristics of the dynamics that remain inaccessible to our methods. For instance, when the potential $W$ possesses multiple global minima, such as $W(u)=(u^2-1)^2/4$, numerical evidence suggests that local maxima separating successive minima act as unstable equilibria. In the conservative regime $a=0$, simulations indicate that trajectories in the phase plane $\mathbb{R}^{2N}$ form closed periodic orbits. Furthermore, when $W(u)=u^2/2$, equation \eqref{Equation} is linear and exhibits the classical critical damping threshold at $a=2$. Our numerical experiments indicate that the same phenomenon persists in nonlinear settings: for instance, when $W(u)=(e^{u^{2}}-1)/2$, simulations suggest that $a=2$ again separates oscillatory (underdamped) dynamics from non-oscillatory (overdamped) decay. This numerical evidence points to a broader conjectural picture: nonlinear potentials with a single global minimizer may still inherit a critical damping regime analogous to the linear case. We have included complete executable \emph{R} scripts at the end of Section \ref{Sec5}. The reader can inspect, modify, or compile them to visualize trajectories, replicate phase plane geometry, and experiment with potentials or additional parameter regimes. Therefore, numerical simulation serves as a gateway to more in-depth mathematical and computational investigations.

The plan of the paper is as follows. Section \ref{Sec2} contains the proof of Theorem \ref{Teo1}, based on the construction of a strict Lyapunov functional adapted to the dissipative system. In Section \ref{Sec3} we briefly discuss the conservative case $a=0$ and the singular nature of the limit as $a\to 0^{+}$. Section \ref{Sec4} examines the global dynamics of nonlinear semigroup theory, providing a complementary viewpoint to the Lyapunov analysis of Section \ref{Sec2}. Finally, Section \ref{Sec5} presents numerical simulations for representative potentials, including quadratic, exponential, and double-well examples.

To prove Theorem~\ref{Teo1} the key idea is the “construction of a strict Lyapunov functional for the first-order formulation” of \eqref{Equation} written as the phase-space system \eqref{S-intro}. The argument introduces
\[
V_a(x,y):=\tfrac12\|y\|^2+2W(x)+\tfrac12\|y+a(x-u_*)\|^2,
\]
viewed as a “smooth perturbation of the mechanical energy” \(E(x,y)=\frac12\|y\|^2+W(x)\). The functional \(V_a\) is used to obtain “stability estimates that are uniform with respect to \(a\)”. The assumptions \hyperlink{W1}{\((W_1)\)}–\hyperlink{W4}{\((W_4)\)} are used to ensure that \(V_a\) is “strictly decreasing along non-trivial trajectories,” which yields uniform asymptotic stability (solutions starting sufficiently close remain close and converge as \(t\to+\infty\)). Under the additional condition \hyperlink{W5}{\((W_5)\)}, the same Lyapunov framework is used to “derive exponential decay of solutions,” with a non-uniform rate as \(a\to0^+\) (the text notes “\(\gamma(a)\to0\)”).

To prove the main theorem, the key idea is the “construction of a strict Lyapunov functional for the first-order formulation” of
\begin{equation}\label{Equation}
	\ddot{u}(t) + a \dot{u}(t) + \nabla W(u(t)) = 0,
\end{equation}
written as the phase-space system \eqref{S-intro}. The argument introduces
\[
V_a(x,y):=\tfrac12\|y\|^2+2W(x)+\tfrac12\|y+a(x-u_*)\|^2,
\]
viewed as a “smooth perturbation of the mechanical energy” \(E(x,y)=\frac12\|y\|^2+W(x)\). The functional \(V_a\) is used to obtain “stability estimates that are uniform with respect to \(a\)”. The assumptions \hyperlink{W1}{\((W_1)\)}–\hyperlink{W4}{\((W_4)\)} are used to ensure that \(V_a\) is “strictly decreasing along non-trivial trajectories,” which yields uniform asymptotic stability (solutions starting sufficiently close remain close and converge as \(t\to+\infty\)). Under the additional condition \hyperlink{W5}{\((W_5)\)}, the same Lyapunov framework is used to “derive exponential decay of solutions,” with a non-uniform rate as \(a\to0^+\) (the text notes “\(\gamma(a)\to0\)”).

\label{Teo1}
	Assume that the potential $W$ satisfies \hyperlink{W1}{$(W_1)$}-\hyperlink{W4}{$(W_4)$}. Then, for every $\varepsilon > 0$ and $a_0 > 0$, there exists $\delta > 0$ such that for all $a \in (0, a_0]$, and any solution $u_a$ of
	\begin{equation}\label{Equation}
	\ddot{u}(t) + a \dot{u}(t) + \nabla W(u(t)) = 0,
\end{equation}
	with initial data satisfying
	$$
	\|u_a(0)-u_*\| + \|\dot{u}_a(0)\| < \delta,
	$$
	the following hold:
	\begin{itemize}
		\item[\emph{(a)}] $\displaystyle \sup_{t \ge 0} \left( \|u_a(t)-u_*\| + \|\dot{u}_a(t)\| \right) \le \varepsilon$;
		\item[\emph{(b)}] $\displaystyle \lim_{t \to +\infty} u_a(t) =u_*$ and $\displaystyle\lim_{t \to +\infty} \dot{u}_a(t) = 0$.
	\end{itemize}
	If, additionally, \hyperlink{W2}{$(W_2)$} and \hyperlink{W3}{$(W_3)$} hold with $\delta = \lambda = +\infty$, then there exists $R > 0$ such that, for all $a \in (0, a_0]$, and any solution $u_a$ of the equation above,
	\begin{itemize}
		\item[\emph{(c)}] $\displaystyle \sup_{t \ge 0} \left( \|u_a(t)-u_*\| + \|\dot{u}_a(t)\| \right) \le R$;
		\item[\emph{(d)}] $\displaystyle \lim_{t \to +\infty} u_a(t) =u_*$ and  $\displaystyle\lim_{t \to +\infty} \dot{u}_a(t) = 0$.
	\end{itemize}
	Moreover, under the additional assumption \hyperlink{W5}{$(W_5)$}, for each $a >0$ there exist $\gamma(a) > 0$ and $C(a)>0$ such that the corresponding solution $u_a$ satisfies the exponential decay estimate
	\begin{equation*}
		\|u_a(t)-u_*\| + \|\dot{u}_a(t)\| \le C(a) e^{-\gamma(a) t}, \quad \forall\, t \ge 0.
	\end{equation*}
	In particular, if $0<a_1<a_2$ and $a\in[a_1,a_2]$, then there exist $C_*>0$ and $\gamma_*>0$, depending only on $a_1,a_2$ such that
	\begin{equation*}
		\|u_a(t)-u_*\| + \|\dot{u}_a(t)\| \le C_* e^{-\gamma_* t},\quad \forall\, t \ge 0,\ \forall\, a\in[a_1,a_2].
	\end{equation*}

Universal–Existential
Asymptotic or Limit

Question:
Let $u_a:\mathbb{R}\to\mathbb{R}^N$ solve the damped second-order system
$$
\ddot u(t)+a\dot u(t)+\nabla W(u(t))=0,
$$
where $a>0$ and $W:\mathbb{R}^N\to\mathbb{R}$ is a potential. Assume $W\in C^2(\mathbb{R}^N)$, there exist $u_*\in\mathbb{R}^N$ and radii $\delta,\lambda>0$ such that $W(u_*)=0$, $W(x)>0$ for every $x\in B_\delta(u_*)\setminus\{u_*\}$, $\langle \nabla W(x),x-u_*\rangle>0$ for every $x\in B_\lambda(u_*)\setminus\{u_*\}$, and $W$ is coercive. In addition, when needed, assume the stronger local estimate
$$
\alpha\|x-u_*\|^2\le W(x)\le \beta\|x-u_*\|^2,
\qquad
\langle \nabla W(x),x-u_*\rangle\ge \mu\|x-u_*\|^2
$$
for all $x\in B_\lambda(u_*)$, for some $\alpha,\beta,\mu>0$. Which statement holds about the stability and decay of solutions?

For every $\varepsilon>0$ and $a_0>0$, there exists $\rho>0$ such that for every $a\in(0,a_0]$, any solution $u_a$ with
$$
\|u_a(0)-u_*\|+\|\dot u_a(0)\|<\rho
$$
satisfies
$$
\sup_{t\ge0}\bigl(\|u_a(t)-u_*\|+\|\dot u_a(t)\|\bigr)\le \varepsilon,
$$
and
$$
\lim_{t\to+\infty}u_a(t)=u_*,\qquad \lim_{t\to+\infty}\dot u_a(t)=0.
$$
If, moreover, the positivity condition $W(x)>0$ for $x\neq u_*$ and the directional monotonicity condition $\langle \nabla W(x),x-u_*\rangle>0$ for $x\neq u_*$ hold globally (that is, with $\delta=\lambda=+\infty$), then there exists $R>0$ such that for every $a\in(0,a_0]$ and every solution $u_a$,
$$
\sup_{t\ge0}\bigl(\|u_a(t)-u_*\|+\|\dot u_a(t)\|\bigr)\le R,
$$
and again
$$
\lim_{t\to+\infty}u_a(t)=u_*,\qquad \lim_{t\to+\infty}\dot u_a(t)=0.
$$
Moreover, under the additional quadratic/monotonicity assumption above, for each $a>0$ there exist constants $\gamma(a)>0$ and $C(a)>0$ such that
$$
\|u_a(t)-u_*\|+\|\dot u_a(t)\|\le C(a)e^{-\gamma(a)t}\qquad\forall t\ge0.
$$
In particular, if $0<a_1<a_2$ and $a\in[a_1,a_2]$, then there exist constants $C_*>0$ and $\gamma_*>0$, depending only on $a_1,a_2$, such that
$$
\|u_a(t)-u_*\|+\|\dot u_a(t)\|\le C_*e^{-\gamma_* t}\qquad\forall t\ge0,\ \forall a\in[a_1,a_2].
$$

Choices:
B.
For every $\varepsilon>0$ and $a_0>0$, there exists $\rho>0$ such that for every $a\in(0,a_0]$, any solution $u_a$ with
$$
\|u_a(0)-u_*\|+\|\dot u_a(0)\|<\rho
$$
satisfies
$$
\sup_{t\ge0}\bigl(\|u_a(t)-u_*\|+\|\dot u_a(t)\|\bigr)\le \varepsilon,
$$
and
$$
\lim_{t\to+\infty}u_a(t)=u_*,\qquad \lim_{t\to+\infty}\dot u_a(t)=0.
$$
If, moreover, the positivity condition $W(x)>0$ for $x\neq u_*$ and the directional monotonicity condition $\langle \nabla W(x),x-u_*\rangle>0$ for $x\neq u_*$ hold globally, then there exist constants $R>0$, $\gamma_*>0$, and $C_*>0$, depending only on $a_0$, such that for every $a\in(0,a_0]$ and every solution $u_a$,
$$
\|u_a(t)-u_*\|+\|\dot u_a(t)\|\le C_*e^{-\gamma_* t}\qquad\forall t\ge0.
$$
C.
For every $\varepsilon>0$ and $a_0>0$, there exists $\rho>0$ such that for every $a\in(0,a_0]$, any solution $u_a$ with
$$
\|u_a(0)-u_*\|+\|\dot u_a(0)\|<\rho
$$
satisfies
$$
\sup_{t\ge0}\bigl(\|u_a(t)-u_*\|+\|\dot u_a(t)\|\bigr)\le \varepsilon,
$$
and
$$
\lim_{t\to+\infty}u_a(t)=u_*,\qquad \lim_{t\to+\infty}\dot u_a(t)=0.
$$
Moreover, under the additional quadratic/monotonicity assumption above, for each fixed $a>0$ there exist constants $\gamma(a)>0$ and $C(a)>0$ such that
$$
\|u_a(t)-u_*\|+\|\dot u_a(t)\|\le C(a)e^{-\gamma(a)t}\qquad\forall t\ge0.
$$
D.
For every $\varepsilon>0$ there exists $\rho>0$ such that for every $a>0$, any solution $u_a$ with
$$
\|u_a(0)-u_*\|+\|\dot u_a(0)\|<\rho
$$
satisfies
$$
\sup_{t\ge0}\bigl(\|u_a(t)-u_*\|+\|\dot u_a(t)\|\bigr)\le \varepsilon,
$$
and
$$
\lim_{t\to+\infty}u_a(t)=u_*,\qquad \lim_{t\to+\infty}\dot u_a(t)=0.
$$
Moreover, under the additional quadratic/monotonicity assumption above, there exist constants $\gamma_*>0$ and $C_*>0$, independent of $a>0$, such that
$$
\|u_a(t)-u_*\|+\|\dot u_a(t)\|\le C_*e^{-\gamma_* t}\qquad\forall t\ge0,\ \forall a>0.
$$
E.
For every $\varepsilon>0$ and $a_0>0$, there exists $\rho>0$ such that for every $a\in(0,a_0]$, any solution $u_a$ with
$$
\|u_a(0)-u_*\|+\|\dot u_a(0)\|<\rho
$$
satisfies
$$
\sup_{t\ge0}\bigl(\|u_a(t)-u_*\|+\|\dot u_a(t)\|\bigr)\le \varepsilon.
$$
If, moreover, the positivity condition $W(x)>0$ for $x\neq u_*$ and the directional monotonicity condition $\langle \nabla W(x),x-u_*\rangle>0$ for $x\neq u_*$ hold globally, then there exists $R>0$ such that for every $a\in(0,a_0]$ and every solution $u_a$,
$$
\sup_{t\ge0}\bigl(\|u_a(t)-u_*\|+\|\dot u_a(t)\|\bigr)\le R,
$$
and every bounded trajectory converges to some equilibrium point of $W$ as $t\to+\infty$.
$$
$$
Moreover, under the additional quadratic/monotonicity assumption above, for each $a>0$ there exist constants $\gamma(a)>0$ and $C(a)>0$ such that
$$
\|u_a(t)-u_*\|+\|\dot u_a(t)\|\le C(a)e^{-\gamma(a)t}\qquad\forall t\ge0.
$$

extra global exponential estimate without (W5)

dropped the global-positivity/global-monotonicity conclusion and the uniform-in-$[a_1,a_2]$ exponential constants

uniformity in $a$ down to $a=0$ for stability radius and exponential rate

limit identified only as some equilibrium rather than the distinguished $u_*$ under global assumptions

The stem does not explicitly state the conclusion; it gives hypotheses and asks which asymptotic claim follows. Although the assumptions are detailed, they do not directly reveal the exact correct choice, especially because the options differ in subtle quantifier and uniformity claims.

The item is close to a theorem-recall question: the correct option is essentially a full theorem statement under the listed assumptions. Still, it is not a pure restatement, since the distractors alter important logical features such as convergence clauses, global hypotheses, and uniformity in the damping parameter.

Selecting the correct option requires meaningful reasoning about which conclusions are justified: local vs global attraction, dependence on bounds for a, and when exponential decay can be uniform. The solver must distinguish valid conclusions from stronger but false variants and a weaker true variant.

The distractors are mathematically plausible and reflect common failure modes: overclaiming uniformity in a, dropping necessary radial monotonicity for global attraction, omitting convergence, or extending exponential estimates to a→0. They are distinct and well-targeted.

A strong, technically sophisticated MCQ with high-quality distractors and genuine logical discrimination, though it remains somewhat theorem-statement-driven rather than fully generative.


Question:
Let $u_a:\mathbb{R}\to\mathbb{R}^N$ solve the damped second-order system
$$
\ddot u(t)+a\dot u(t)+\nabla W(u(t))=0,
$$
where $a>0$ and $W:\mathbb{R}^N\to\mathbb{R}$ is a potential. Assume $W\in C^2(\mathbb{R}^N)$, there exist $u_*\in\mathbb{R}^N$ and radii $\delta,\lambda>0$ such that $W(u_*)=0$, $W(x)>0$ for every $x\in B_\delta(u_*)\setminus\{u_*\}$, $\langle \nabla W(x),x-u_*\rangle>0$ for every $x\in B_\lambda(u_*)\setminus\{u_*\}$, and $W$ is coercive. In addition, when needed, assume the stronger local estimate
$$
\alpha\|x-u_*\|^2\le W(x)\le \beta\|x-u_*\|^2,
\qquad
\langle \nabla W(x),x-u_*\rangle\ge \mu\|x-u_*\|^2
$$
for all $x\in B_\lambda(u_*)$, for some $\alpha,\beta,\mu>0$. Which statement holds about the stability and decay of solutions?

Choices:
B.
For every $\varepsilon>0$ and $a_0>0$, there exists $\rho>0$ such that for every $a\in(0,a_0]$, any solution $u_a$ with
$$
\|u_a(0)-u_*\|+\|\dot u_a(0)\|<\rho
$$
satisfies
$$
\sup_{t\ge0}\bigl(\|u_a(t)-u_*\|+\|\dot u_a(t)\|\bigr)\le \varepsilon,
$$
and
$$
\lim_{t\to+\infty}u_a(t)=u_*,\qquad \lim_{t\to+\infty}\dot u_a(t)=0.
$$
If, moreover, the positivity condition $W(x)>0$ for $x\neq u_*$ and the directional monotonicity condition $\langle \nabla W(x),x-u_*\rangle>0$ for $x\neq u_*$ hold globally, then there exist constants $R>0$, $\gamma_*>0$, and $C_*>0$, depending only on $a_0$, such that for every $a\in(0,a_0]$ and every solution $u_a$,
$$
\|u_a(t)-u_*\|+\|\dot u_a(t)\|\le C_*e^{-\gamma_* t}\qquad\forall t\ge0.
$$
C.
For every $\varepsilon>0$ and $a_0>0$, there exists $\rho>0$ such that for every $a\in(0,a_0]$, any solution $u_a$ with
$$
\|u_a(0)-u_*\|+\|\dot u_a(0)\|<\rho
$$
satisfies
$$
\sup_{t\ge0}\bigl(\|u_a(t)-u_*\|+\|\dot u_a(t)\|\bigr)\le \varepsilon,
$$
and
$$
\lim_{t\to+\infty}u_a(t)=u_*,\qquad \lim_{t\to+\infty}\dot u_a(t)=0.
$$
Moreover, under the additional quadratic/monotonicity assumption above, for each fixed $a>0$ there exist constants $\gamma(a)>0$ and $C(a)>0$ such that
$$
\|u_a(t)-u_*\|+\|\dot u_a(t)\|\le C(a)e^{-\gamma(a)t}\qquad\forall t\ge0.
$$
D.
For every $\varepsilon>0$ there exists $\rho>0$ such that for every $a>0$, any solution $u_a$ with
$$
\|u_a(0)-u_*\|+\|\dot u_a(0)\|<\rho
$$
satisfies
$$
\sup_{t\ge0}\bigl(\|u_a(t)-u_*\|+\|\dot u_a(t)\|\bigr)\le \varepsilon,
$$
and
$$
\lim_{t\to+\infty}u_a(t)=u_*,\qquad \lim_{t\to+\infty}\dot u_a(t)=0.
$$
Moreover, under the additional quadratic/monotonicity assumption above, there exist constants $\gamma_*>0$ and $C_*>0$, independent of $a>0$, such that
$$
\|u_a(t)-u_*\|+\|\dot u_a(t)\|\le C_*e^{-\gamma_* t}\qquad\forall t\ge0,\ \forall a>0.
$$
E.
For every $\varepsilon>0$ and $a_0>0$, there exists $\rho>0$ such that for every $a\in(0,a_0]$, any solution $u_a$ with
$$
\|u_a(0)-u_*\|+\|\dot u_a(0)\|<\rho
$$
satisfies
$$
\sup_{t\ge0}\bigl(\|u_a(t)-u_*\|+\|\dot u_a(t)\|\bigr)\le \varepsilon.
$$
If, moreover, the positivity condition $W(x)>0$ for $x\neq u_*$ and the directional monotonicity condition $\langle \nabla W(x),x-u_*\rangle>0$ for $x\neq u_*$ hold globally, then there exists $R>0$ such that for every $a\in(0,a_0]$ and every solution $u_a$,
$$
\sup_{t\ge0}\bigl(\|u_a(t)-u_*\|+\|\dot u_a(t)\|\bigr)\le R,
$$
and every bounded trajectory converges to some equilibrium point of $W$ as $t\to+\infty$.
$$
$$
Moreover, under the additional quadratic/monotonicity assumption above, for each $a>0$ there exist constants $\gamma(a)>0$ and $C(a)>0$ such that
$$
\|u_a(t)-u_*\|+\|\dot u_a(t)\|\le C(a)e^{-\gamma(a)t}\qquad\forall t\ge0.
$$

The stem does not state the conclusion outright. It gives hypotheses and asks which global/local stability and decay statement is valid; the correct answer is not leaked explicitly, though the extra quadratic assumptions do hint that some exponential decay statement may be relevant.

This is close to theorem recognition: the correct option is essentially a full theorem statement assembled from the listed assumptions. However, it is not a pure verbatim restatement, since the alternatives differ in subtle quantifiers, global-vs-local assumptions, and uniformity claims.

Moderate reasoning is required to compare the strength of the claims, especially the dependence on a, the need for global positivity/monotonicity, and when exponential rates can be uniform. Still, the task is mainly discriminating among theorem variants rather than generating a conclusion from scratch.

The distractors are mathematically plausible and target realistic failure modes: unjustified uniformity as a -> 0, overly strong global exponential decay, omission of global conclusions, and convergence only to some equilibrium. They are distinct and well aligned with common misunderstandings.

A solid MCQ with strong distractors and little answer leakage, but it remains largely a theorem-identification item rather than a deeply generative reasoning question.


\label{sec.Intro}
Aperiodic tilings of Euclidean space, a topic once regarded as an esoteric pursuit, are now widely recognized as a subject of central interest. Beginning with work of Hao Wang and Robert Berger in the 1960's studying a relationship between aperiodic tilings and the Halting problem for Turing machines, results surrounding aperiodic tilings have had substantial impacts in the last century within mathematics, computer science, the natural sciences, and even the liberal arts. Excellent summaries of many of these results, up until about 2010, can be found in \cite[Chapter 1]{Gard1997}, \cite[Chapters 1, 2]{Sene1995}, \cite[Forward by Roger Penrose]{BaakGrim2013}, and \cite[Chapter 1]{BaakGrim2013}. There has also been significant and highly publicized progress in this subject in the last decade.

One landmark, which has also become an inspiration in art and architecture, was Roger Penrose's discovery of a pair of polygonal tiles which could tile the plane, but could only do so aperiodically. Penrose published a first version of his tilings in 1974 \cite{Penr1974} (see also \cite{Penr1979}). Soon after, they were introduced to the general public in an article by Martin Gardner in the Scientific American \cite{Gard1977}.

\begin{figure}[h]
 \caption{Tiles used to construct rhombic Penrose tilings.\\}\label{fig.PenTiles}
 \centering
 \vspace*{-30bp}
 \includegraphics[width=0.75\textwidth]{PenTiles1mod}
\end{figure}

There are several related collections of tilings which are referred to as Penrose tilings. In this article we will consider \textit{rhombic Penrose tilings} (RPTs), which are tilings of the plane by copies of two rhombuses with angles $72^\circ$ and $36^\circ$, pictured in Figure \ref{fig.PenTiles}. The rhombuses are allowed to be rotated, but they must fit together edge to edge, and the matching rules indicated by the arrows must also be enforced. A small patch of an RPT is pictured in Figure \ref{fig.PenPatch} (see also \cite[Figures 1.2, 6.44]{BaakGrim2013}). When the rhombuses are chosen to have side length 1, we call the resulting tilings \textit{unit length RPTs}.

The main problem that we wish to investigate in this paper is that of estimating the number of vertices of an RPT which lie in $B_R$, the closed Euclidean ball of radius $R$, centered at some point in the plane which we fix to be the origin. RPTs can be defined using primitive inflation rules (see \cite[Section 6.2]{BaakGrim2013}), which guarantee existence of limiting frequencies of tile types in large regions. Although it is not obvious, it is not too difficult to deduce (see for example \cite[Remark 5]{SchmTrev2018}) that if $V$ is the vertex set of a unit length RPT then, for $R\ge 1$,
\begin{equation}\label{eqn.PenVertTrivEst}
 \#(V\cap B_R)=\pi C_P R^2 + O(R),
\end{equation}
where
\[C_P=\frac{\phi}{5}\sqrt{10+2\sqrt{5}}\approx 1.231,\]
with $\phi=(1+\sqrt{5})/2$. The error term on the right hand side of \eqref{eqn.PenVertTrivEst} comes from counting tiles which intersect the boundary of $B_R$.
Since there are in fact $\gg R$ such tiles, it may appear that this error term is essentially best possible (cf. results of Solomon from \cite{Solo2011} on counting tiles in \textit{supertiles} of primitive substitutions). However, we will show that for this special class of tilings, one can actually do better.

\begin{equation}\label{eqn.PenVertTrivEst}
	\#(V\cap B_R)=\pi C_P R^2 + O(R),
\end{equation}

\begin{figure}[h]
	\caption{A patch of an RPT constructed using the cut and project method detailed in Section \ref{sec.ModelSets}. The 4 colors of vertices are translates of the cut and project sets produced using the different windows.\\}\label{fig.PenPatch}
	\centering
	\includegraphics[width=0.75\textwidth]{PenPatch2}
\end{figure}

\begin{figure}[h]
	\caption{Tiles used to construct rhombic Penrose tilings.\\}\label{fig.PenTiles}
	\centering
	\vspace*{-30bp}
	\includegraphics[width=0.75\textwidth]{PenTiles1mod}
\end{figure}

There are several related collections of tilings which are referred to as Penrose tilings. In this article we will consider \textit{rhombic Penrose tilings} (RPTs), which are tilings of the plane by copies of two rhombuses with angles $72^\circ$ and $36^\circ$, pictured in Figure \ref{fig.PenTiles}. The rhombuses are allowed to be rotated, but they must fit together edge to edge, and the matching rules indicated by the arrows must also be enforced. A small patch of an RPT is pictured in Figure \ref{fig.PenPatch} (see also \cite[Figures 1.2, 6.44]{BaakGrim2013}). When the rhombuses are chosen to have side length 1, we call the resulting tilings \textit{unit length RPTs}.

\begin{figure}[h]
 \caption{A patch of an RPT constructed using the cut and project method detailed in Section \ref{sec.ModelSets}. The 4 colors of vertices are translates of the cut and project sets produced using the different windows.\\}\label{fig.PenPatch}
 \centering
 \includegraphics[width=0.75\textwidth]{PenPatch2}
\end{figure}

The work of de Bruijn, et al., mentioned at the beginning of this section (see references listed above), implies the following result.
\begin{theorem}\label{thm.PenVerts}
Every non-singular point set $\Lambda_\omega$, constructed as above, is the set of vertices of a unit length rhombic Penrose tiling. Furthermore, the vertex set of any unit length rhombic Penrose tiling can be obtained, after rotation, as the limit in an appropriate topology (e.g. the Chabauty-Fell topology) of a sequence of non-singular point sets $\Lambda_\omega$.
\end{theorem}
An example of a RPT obtained using the cut and project method described above is shown in Figure \ref{fig.PenPatch}. Theorem \ref{thm.PenVerts} will allow us to derive Theorem \ref{thm.Pen} from the following result.
\begin{theorem}\label{thm.PenCPVersion}
For the cut and project sets $\Lambda_{m,\omega}$ defined above, there is a constant $C>0$ with the property that, for any $\omega\in\C$, any integer $1\le m\le 4$, and any $R\ge 2$,
\[\left|\#\left(\Lambda_{m,\omega}\cap B_R\right)-\pi R^2\mr{dens}(\cL)\left|W_{m,\omega}\right|\right|\le CR^{2/3}(\log R)^{2/3},\]
where $|W_{m,\omega}|$ denotes the area of $W_{m,\omega}$, and $\mr{dens}(\cL)$ is the reciprocal of the volume of a fundamental domain for $\R^4/\cL$.
\end{theorem}
An important point is that the error term in this theorem is uniform in $\omega$, which allows us to take limits of non-singular point sets, to account for all possible vertex sets of RPTs, in order to establish Theorem \ref{thm.Pen}. There are a few other small details that have to be ironed out, but they are not difficult to justify, so we postpone their discussion until the end of Section \ref{sec.PenProof}. The majority of the remainder of the paper will be focused on proving Theorem \ref{thm.PenCPVersion}.

If $W\subseteq\R^2$ then we write $\chi_W$ for the indicator function of $W$. The similarity in notation to $\chi_R$, as defined above, will not cause confusion in what follows. Observe that
\[\widehat{\chi}_W(y)=\int_We(-x\cdot y)\mr{d}x=\frac{-1}{(2\pi|y|)^2}\int_W\nabla_x^2\left( e(-x\cdot y)\right)\mr{d}x,\]
where $\nabla_x^2=\nabla_x\cdot\nabla_x$ is the Laplacian. By the Divergence Theorem, under reasonable hypotheses on $W$, this allows us to express the Fourier transform as a line integral over the boundary of $W$. Applying this argument in the case when $W$ is a polygon leads to a particularly nice formula for the Fourier transform, which we reproduce here in the form given in \cite{BranColzTrav1997}.
\begin{proposition}\label{prop.FourTranPoly}
 Let $P\subseteq\R^2$ be a polygon with counterclockwise oriented vertices $\{a_j\}_{j=1}^m$. Denote by $\sigma_j$ the unit vector parallel to the side $[a_j,a_{j+1}]$ and by $v_j$ the outward unit normal to this side. Then, with $a_{m+1}:=a_1$, we have
 \[\widehat{\chi}_P(y)=\frac{-1}{(2\pi|y|)^2}\sum_{j=1}^m\left(\frac{e(-y\cdot a_{j+1})-e(-y\cdot a_{j})}{y\cdot \sigma_j}\right)y\cdot v_j.\]
\end{proposition}
This proposition is proved as \cite[Lemma 2.2]{BranColzTrav1997}. The quantity in parenthesis in the sum is interpreted to equal
\[-\left(2\pi i |a_{j+1}-a_j|\right)e\left(\frac{-y\cdot(a_j+a_{j+1})}{2}\right),\]
when $y\cdot\sigma_j=0$. Proposition \ref{prop.FourTranPoly} implies that
\begin{equation}\label{eqn.PolyFourEst}
 \widehat{\chi}_P(y)\ll_P \frac{1}{|y|}\sum_{j=1}^m\min\left\{1,\frac{1}{|y\cdot\sigma_j|}\right\}.
\end{equation}
One thing to note is that these Fourier coefficients decay in most directions like $|y|^{-2}$. However, in directions perpendicular to the sides of $P$, they decay only like $|y|^{-1}$. This turns out to be a source of some difficulty, which is given special attention in the proofs below.

\noindent\emph{Bound for $E_2$:} To estimate $E_2$, first write
\[\mc{S}_{m,n}=\left\{\theta\in L^\circledast : \frac{2^{m-1}}{R}<|\theta|\le\frac{2^m}{R},~\frac{nR}{M2^m}\le|\theta^\star|<\frac{(n+1)R}{M2^m}\right\},\]
with $M$ chosen as in \eqref{eqn.MChoice}. Then we have that
\begin{align}
 E_2&\ll R^{1/2}\sum_{m=1}^{\lceil \log_2R\rceil}\sum_{n=1}^\infty\sum_{\theta\in\mc{S}_{m,n}}\frac{1}{|\theta|^{3/2}}|\widehat{\chi}_W(\theta^\star)|\left(1+\delta^2|\theta^\star|^2\right)^{-N}\nonumber\\
 &\ll R^2 \sum_{m=1}^{\lceil\log_2R\rceil}2^{-3m/2}\sum_{n=1}^\infty\left(1+\frac{\delta^2n^2R^2}{2^{2m}}\right)^{-N}\sum_{\theta\in\mc{S}_{m,n}}|\widehat{\chi}_W(\theta^\star)|.\label{eqn.E2Bound1}
\end{align}
Here we will use the following lemma.
\begin{lemma}\label{lem.ChiWEst2}
 With notation as above, for $m,n\in\N$ with $m\le\lceil \log_2R\rceil$, we have that
 \[\sum_{\theta\in\mc{S}_{m,n}}|\widehat{\chi}_W(\theta^\star)|\ll\frac{2^{2m}\log n}{nR^2}.\]
\end{lemma}
\begin{proof}
 Let us write
 \[\tilde{\mc{S}}=\left\{\lambda\in \frac{1}{25}L : \frac{2^{m-1}}{R}<|\lambda|\le\frac{2^m}{R},~\frac{nR}{M2^m}\le|\lambda^\star|<\frac{(n+1)R}{M2^m},~|\mr{arg}(\lambda^\star)|\le \frac{\pi}{5}\right\}.\]
 By the argument used in the proof of Lemma \ref{lem.ChiWEst1} leading up to equation \eqref{eqn.ChiWEst1}, it is sufficient for this lemma to prove that
 \begin{equation*}
 \sum_{\lambda\in\tilde{\mc{S}}}\frac{1}{|\lambda^\star|^2|\sin(\mathrm{arg}(\lambda^\star))|}\ll\frac{2^{2m}\log n}{nR^2}.
 \end{equation*}
 Label the elements of $\star(\tilde{\mc{S}})$ as
 \[\lambda_j^\star=\rho_je^{it_j},\]
 for $1\le j\le J:=|\tilde{\mc{S}}|$, with each $\rho_j\in\R^+$ and with $0< |t_1|\le |t_2|\le\cdots \le |t_J|\le\pi/5$. Also, label the element of $\tilde{\mc{S}}$ corresponding to each $\lambda_j^\star$ as $\lambda_j$.

\noindent\emph{Proof of Theorem \ref{thm.Pen}:} Let $V$ be the vertex set of a unit length RPT. If $V$ can be constructed as a cut and project set, as $V=\Lambda_\omega$ for some $\omega\in\C$, then the result of Theorem \ref{thm.Pen} follows immediately from Theorem \ref{thm.PenCPVersion}. Otherwise, by Theorem \ref{thm.PenVerts}, there is a sequence of complex numbers $\{\omega_j\}_{j=1}^\infty$ with the property that, after a possible rotation,
\[V=\lim_{j\rar\infty}\Lambda_{\omega_j},\]
where the limit is taken in the Chabauty-Fell topology. For $R\ge 2$, the number of points in $\R^2$ which lie on the boundary of $B_R$, and which are the limits of sequences of points $\lambda_j\in\Lambda_{\omega_j}$, as $j\rar\infty,$ must be $\ll R^{2/3}(\log R)^{2/3}$. This follows from the uniformity over $\omega\in \C$ of the error term in Theorem \ref{thm.PenCPVersion}. It then follows, again by uniformity, that
\begin{align}
 \#(V\cap B_R)&=\lim_{j\rar\infty}\#\left(\Lambda_{\omega_j}\cap B_R\right)+O(R^{2/3}(\log R)^{2/3})\nonumber\\
 &=\pi R^2\mr{dens}(\cL)\sum_{m=1}^4|W_{m,\omega+m}|+O( R^{2/3}(\log R)^{2/3}).\label{eqn.ChabLim1}
\end{align}
Finally, we compute that
\[\mr{dens}(\cL)=\mr{disc}(L)^{-1/2}=\frac{4}{25\sqrt{5}}.\]
Substituting this in \eqref{eqn.ChabLim1}, together with the areas of the pentagonal windows, completes the proof of Theorem \ref{thm.Pen}.

Readers who are familiar with the analogous counting problem for integer lattice points in the plane, called the Gauss circle problem, may at this point recognize some similarities. Let us write
\[\#(\mathbb{Z}^2\cap B_R)=\pi R^2 + E(R).\]
Gauss himself observed that $|E(R)|\le 2\sqrt{2}\pi R$, and he conjectured that it should be possible to improve the exponent on $R$  to obtain an error term of $\ll R^{1/2}$. The next substantial result was published in 1906 by Sierpi\'nski \cite{Sier1906}, who showed that $E(R)\ll R^{2/3}$ (see also \cite{Schi1972}). Subsequent improvements have been made by many authors (the introduction of \cite{Huxl2003} contains a list), and currently the best known upper bound, due to Martin Huxley \cite{Huxl2003}, is $E(R)\ll R^{131/208}(\log R)^\tau$, with $\tau=18627/8320$.

The important link which allows us to employ techniques used in the Gauss circle problem in order to study vertices of Penrose tilings is a remarkable observation of de Bruijn, that the vertex set of any RPT can be obtained from a cut and project set, by projecting `slices' of lattice points in a higher dimensional space onto a 2 dimensional subspace. We describe this connection in detail in Section \ref{sec.ModelSets}. With this as a general framework, we are able in Sections \ref{sec.Fourier}-\ref{sec.PenProof} to bring in tools from Fourier analysis and to approach our point counting problem through the lens of the Poisson summation formula (PSF). Results of a similar flavor, estimating discrepancies of points in growing patches of cut and projects sets, have recently been explored by Koivusalo and Lagac{\'e} in \cite{KoivLagacBjorHart2025}, and by R\"uhr, Smilansky, and Weiss in \cite{RuhrSmilWeis2024}.

We remark that our proof, although structurally similar, is not a na\"ive application of known methods (e.g. Huxley's Fourier analytic approach to lattice point counting in regions with smooth boundary, using Van der Corput's method). Here, there are two new difficulties which are not encountered in the classical versions of the Gauss circle problem. The first is that, for the cut and project sets that we consider, the problem amounts to counting points in a lattice in a $4$-dimensional cylinder which is only growing radially. Since the height of the cylinder stays fixed, the smoothing necessary to apply the PSF must shrink faster in the direction of the height than in the radial direction. Secondly, the lattices from which we derive RPTs are irrational. This has the effect that, depending on how well approximable the lattice points are by rational points, smoothing our regions may pick up points which are unusually close to their boundaries. This is a real phenomenon, which can potentially cause a blow up in the Fourier coefficients after using the PSF. It is one of the main issues that we must confront using the specific geometric and algebraic data which define RPTs.

\begin{figure}[h]
	\caption{A patch of an RPT constructed using the cut and project method detailed in Section \ref{sec.ModelSets}. The 4 colors of vertices are translates of the cut and project sets produced using the different windows.\\}\label{fig.PenPatch}
	\centering
	\includegraphics[width=0.75\textwidth]{PenPatch2}
\end{figure}

Finally, there is the question of how these results could be generalized. Several parts of the proofs below can be adapted quite well to deal with much wider classes of cut and project sets. However, there are some crucial details in the proof, in particular those contained in Section \ref{sec.PenProof}, which form a narrow gate that any more general result would need to pass through. The cut and project sets used to define RPTs are extremely special, both in the algebraic properties of the lattice involved and in the exact shape, symmetry, and orientation of the windows involved, with respect to the specific underlying lattice. Any attempt to modify the lattice, or change the windows (even by a small rotation) will destroy important features of the argument and make it difficult or even impossible to control the sizes of the Fourier coefficients involved.

On the other hand, it seems likely that the arguments given here should apply equally well, with the appropriate modifications, to vertex sets of Ammann-Beenker tilings (see \cite[Section 10.4]{GrunShep1987} and \cite{Been1982}). We leave it to the interested reader to work out the details.

To prove Theorem~\ref{thm.Pen}, the authors connect vertex counting in a unit length RPT to the Gauss circle problem by using “a remarkable observation of de Bruijn, that the vertex set of any RPT can be obtained from a cut and project set, by projecting `slices' of lattice points in a higher dimensional space onto a 2 dimensional subspace” (detailed in Section~\ref{sec.ModelSets}). With this framework, they “bring in tools from Fourier analysis” and “approach [the] point counting problem through the lens of the Poisson summation formula (PSF)” (Sections~\ref{sec.Fourier}--\ref{sec.PenProof}).

They emphasize two main technical obstacles (and thus key components of the argument): (1) the counting reduces to “counting points in a lattice in a $4$-dimensional cylinder which is only growing radially,” so because “the height of the cylinder stays fixed, the smoothing necessary to apply the PSF must shrink faster in the direction of the height than in the radial direction”; and (2) the relevant lattices are “irrational,” so “depending on how well approximable the lattice points are by rational points, smoothing our regions may pick up points which are unusually close to their boundaries,” which “can potentially cause a blow up in the Fourier coefficients after using the PSF.” Controlling this requires “the specific geometric and algebraic data which define RPTs,” and the authors note that “crucial details in the proof, in particular those contained in Section~\ref{sec.PenProof}, … make it difficult or even impossible to control the sizes of the Fourier coefficients involved” under general perturbations of lattice/windows.

To prove the main theorem, the authors connect vertex counting in a unit length RPT to the Gauss circle problem by using “a remarkable observation of de Bruijn, that the vertex set of any RPT can be obtained from a cut and project set, by projecting `slices' of lattice points in a higher dimensional space onto a 2 dimensional subspace” (detailed later). With this framework, they “bring in tools from Fourier analysis” and “approach [the] point counting problem through the lens of the Poisson summation formula (PSF)” (proved later).

They emphasize two main technical obstacles (and thus key components of the argument): (1) the counting reduces to “counting points in a lattice in a $4$-dimensional cylinder which is only growing radially,” so because “the height of the cylinder stays fixed, the smoothing necessary to apply the PSF must shrink faster in the direction of the height than in the radial direction”; and (2) the relevant lattices are “irrational,” so “depending on how well approximable the lattice points are by rational points, smoothing our regions may pick up points which are unusually close to their boundaries,” which “can potentially cause a blow up in the Fourier coefficients after using the PSF.” Controlling this requires “the specific geometric and algebraic data which define RPTs,” and the authors note that “crucial details in the proof, in particular those contained in the later argument, … make it difficult or even impossible to control the sizes of the Fourier coefficients involved” under general perturbations of lattice/windows.

\label{thm.Pen}
Let $V$ be the set of vertices of any unit length RPT. Then, for $R\ge 2$,
\[\#(V\cap B_R)=\pi C_P R^2 + O(R^{2/3}(\log R)^{2/3}).\],

Question:
Let $V$ be the vertex set of any unit length rhombic Penrose tiling, meaning a tiling of the plane by copies of the two Penrose rhombi of side length $1$ with angles $72^\circ$ and $36^\circ$, placed edge-to-edge and satisfying the usual matching rules. Let $B_R$ denote the closed Euclidean ball of radius $R$ centered at the origin, and define
\[
C_P=\frac{\phi}{5}\sqrt{10+2\sqrt{5}}, \qquad \phi=\frac{1+\sqrt{5}}{2}.
\]
Which quantitative estimate holds for the number of vertices of the tiling in $B_R$ when $R\ge 2$?

For every such $V$ and all $R\ge 2$,
\[
\#(V\cap B_R)=\pi C_P R^2 + O\!\left(R^{2/3}(\log R)^{2/3}\right).
\]

Choices:
B.
For every such $V$ and all $R\ge 2$,
\[
\#(V\cap B_R)=\pi C_P R^2 + O\!\left(R^{1/2}(\log R)^{2/3}\right).
\]
C.
For every such $V$ and all $R\ge 2$,
\[
\#(V\cap B_R)=\pi C_P R^2 + o(R).
\]
D.
For every such $V$ there exists a constant $K_V>0$ such that for all $R\ge 2$,
\[
\#(V\cap B_R)=\pi C_P R^2 + O\!\left(K_V R^{2/3}(\log R)^{2/3}\right).
\]
E.
For every such $V$ and all $R\ge 2$,
\[
\#(V\cap B_R)=\pi C_P R^2 + O\!\left(R^{2/3}\right).
\]

The stem defines the setting and the constant C_P but does not reveal the asymptotic error term, the logarithmic factor, or the quantifier structure. There is no explicit or trivial cue pointing to choice A.

The item is largely a theorem-identification question: one option states the precise target estimate, and the task is to pick the correct formulation among close variants. This is better than a pure restatement, but still only a mild reformulation rather than a genuinely new conclusion derived from the setup.

There is some reasoning pressure because the choices differ in subtle but meaningful ways: stronger false bounds, a weaker true statement, a quantifier-dependent variant, and omission of a log factor. However, the item primarily tests precise theorem recall/recognition rather than substantial generative mathematical reasoning from first principles.

The distractors are strong: B is an over-strong error term, C is a weaker true statement, D probes uniformity versus tiling-dependent constants, and E tests whether the log factor is necessary. These are plausible and mathematically aligned with common failure modes.

A solid MCQ with no answer leakage and high-quality distractors, but it functions mainly as precise theorem recognition rather than a deeply generative reasoning task.


\begin{thm}
  \label{main1}
  Let $X$ be a tropical variety in $\mathbb R^n$.
  Then $\mathbf E(|X|)$ is finitely generated if and only if $|X|$ is a rational affine linear subspace of $\mathbb R^n$.
\end{thm}

Tropical geometry is a subject of mathematics developed in recent decades, which studies geometric objects called tropical varieties.
Sometimes tropical geometry is called \textquotedblleft the algebraic geometry over $\mathbb T$\textquotedblright, where $\mathbb T$ is a semifield called the tropical semifield.
Since $\mathbb T$ is not a field, tropical geometry is not included in classical algebraic geometry.
However, for many albedo-geometric theories, their tropical versions are studied.
For instance, the tropical intersection theory \cite{allermann2010first,jensen2013computing,jensen2016stable,katz2012tropical,rau2016intersections}, the theory of divisors on tropical curves \cite{amini2013reduced,an2014canonical,haase2012linear,hladky2013rank,len2014brill}, and the theory of tropical homology \cite{amini2020hodge,gross2019sheaf,itenberg2019tropical,ruddat2021homology} have been developed.

One of the basic objects in algebraic geometry is the ideals of rings.
In tropical geometry, there are several analogous objects.
The simplest one is the concept of ideal of a semiring, which is defined as a submodule of the semiring itself.
Another one is the concept of tropical ideal, which is defined by Maclagan and Rinc\'{o}n in \cite{maclagan2018tropical}.
It is an analogue of an ideal of a ring since the quotient semiring $R/E$ is defined.
Moreover, there are several specific class of congruences.
In \cite{giansiracusa2016equations}, J. Giansiracusa and N. Giansiracusa define \textit{bend congruences} to develop the theory of scheme theoretic tropicalization.
Another important class is the congruences of the form $\mathbf{E}(Z)$ for some $Z \subset \mathbb R^n$.
Here,
$$\mathbf{E}(Z) = \{ (f,g) \in \mathbb T[x_1, \ldots, x_n] \ | \ f|_Z = g|_Z \}.$$
The semiring $\mathbb T[x_1, \ldots, x_n]$ is sometimes replaced by another one such as the Laurent polynomial semiring $\mathbb T[x_1^{\pm}, \ldots, x_n^{\pm}]$, the rational function semifield $\mathbb T(x_1, \ldots, x_n)$, the polynomial ring $\mathbb B[x_1, \ldots, x_n]$ over the Boolean semifield, or the polynomial function semiring $\mathbb T[x_1, \ldots, x_n]_{\mathrm{fcn}}$.
This is a simple analogue of the concept of vanishing ideal in algebraic geometry.
The congruences of this form are studied by Bertram and Easton in \cite{bertram2017tropical}, Jo\'{o} and Mincheva in \cite{joo2018prime}, Song in \cite{song2024congruences}, and the author in \cite{ito2022local}.
Also, \textit{prime congruence} are defined in \cite{joo2018prime}, which are of course analogues of the prime ideals of rings.

Almost all rings appeared in classical algebraic geometry are Noetherian, and their ideals are finitely generated.
This fact is used to develop the dimension theory.
Hence it is natural that we consider whether a given ideal (or congruence) is finitely generated.
There are few known results.
In \cite[Example 3.10]{maclagan2018tropical}, an example of tropical ideal which is not finitely generated is given.
Also, it holds by definition that a bend relation defined by a polynomial is finitely generated.
In this paper, we consider the congruences on $\mathbb T[x_1^{\pm}, \ldots, x_n^{\pm}]$ of the form $\mathbf E(Z)$.
In particular, we consider the case that $Z$ is the support of a tropical variety.
The following is our main result.

Almost all rings appeared in classical algebraic geometry are Noetherian, and their ideals are finitely generated.
This fact is used to develop the dimension theory.
Hence it is natural that we consider whether a given ideal (or congruence) is finitely generated.
There are few known results.
In \cite[Example 3.10]{maclagan2018tropical}, an example of tropical ideal which is not finitely generated is given.
Also, it holds by definition that a bend relation defined by a polynomial is finitely generated.
In this paper, we consider the congruences on $\mathbb T[x_1^{\pm}, \ldots, x_n^{\pm}]$ of the form $\mathbf E(Z)$.
In particular, we consider the case that $Z$ is the support of a tropical variety.
The following is our main result.

Thus, for almost all tropical variety $X$, $\mathbf E(X)$ is not finitely generated.
The only if part of this theorem is a consequence of the following more general result.

\begin{thm}
 \label{main}
 Let $X$ be a tropical variety in $\mathbb R^n$.
 Then $\mathbf E(X)$ is finitely generated if and only if $|X|$ is a rational affine linear subspace of $\mathbb R^n$.
\end{thm}

Hence, for a subset $Z \subset W$, in order to show that $\mathbf E(Z)$ is not finitely generated, it is enough to show that $\mathbf E(\iota^{-1}(Z))$ is not finitely generated.
Thus we may assume that $Z$ is not included in any proper rational affine linear subspace of $\mathbb R^n$.
Note that if $Z$ is the support of a tropical variety in $\mathbb R^n$, then so is $\iota^{-1}(Z)$ by Lemma \ref{trop var reduction}.

Let $X$ be a tropical variety in $\mathbb R^n$.
Assume that $W := |X|$ is a $d$-dimensional rational affine linear subspace of $\mathbb R^n$.
By Lemma \ref{translation}, we may assume that $\mathbf 0 \in W$, i.e., $W$ is a linear subspace of $\mathbb R^n$.
Since $W$ is rational, the orthogonal complement $W'$ (with respect to the standard inner product) of $W$ is also rational.
We use the notations $\Lambda(W) = W \cap \mathbb Z^n$ and $\Lambda(W') = W' \cap \mathbb Z^n$.
Let $\{ \bm q_1, \ldots, \bm q_{n-d} \}$ be a basis of $\Lambda(W')$.
We see that the congruence $\mathbf E(W)$ is generated by the finite set $S := \left\{ \left(\overline{\bm x^{\bm q_i}}, 0 \right) \ | \ i=1, \ldots, n-d \right\}$.

\begin{prop}
 \label{trop unbounded}
 Let $X$ be a tropical variety in $\mathbb R^n$.
 Assume that $|X|$ is not included in a proper rational affine linear subspace of $\mathbb R^n$.
 Then $|X|$ is unbounded in all rational directions.
\end{prop}

\begin{proof}[Proof of \textquotedblleft Proposition \ref{trop unbounded in f} $\Longrightarrow$ Proposition \ref{trop unbounded}\textquotedblright]
 Assume that Proposition \ref{trop unbounded in f} holds.
 Let $X$ be a tropical variety in $\mathbb R^n$ whose support is not included in a proper rational affine linear subspace of $\mathbb R^n$.
 Take any $\bm p \in \mathbb Z^n \setminus \{ \mathbf 0 \}$.
 If the function $f: |X| \to \mathbb R, \bm a \mapsto \bm a \cdot \bm p$ is constant with the value $k$, then $|X|$ is included in the set $\{ \bm a \in \mathbb R^n \ | \ \bm a \cdot \bm p = k \}$, which is a proper rational affine linear subspace of $\mathbb R^n$.
 This contradicts to the assumption, hence $f$ is not constant.
 Thus $f$ is upper unbounded.
 Since $\bm p$ is an arbitrary element of $\mathbb Z^n \setminus \{ \mathbf 0 \}$, $|X|$ is unbounded in all rational directions.
\end{proof}

\begin{proof}[Proof of the only if part of Theorem \ref{main}]
 Let $X$ be a tropical variety such that $|X|$ is not a rational affine linear subspace of $\mathbb R^n$.
 We may assume that $|X|$ is not included in any proper rational affine linear subspace of $\mathbb R^n$ by Lemma \ref{fg reduction} and the observation after it.
 Hence, by Proposition \ref{trop unbounded}, $|X|$ is unbounded in all rational directions.
 In order to use Theorem \ref{not fg}, we check that $n \geq 2$.
 If $n =0$, $|X| = \{ \mathbf 0 \}$.
 It contradicts to the assumption that $|X|$ is not an affine linear subspace.
 If $n =1$, $|X|$ is a singleton or the whole space $\mathbb R$ (by Lemma \ref{n in n}).
 It also contradicts to the assumption.
 Thus $n \geq 2$, and hence by Theorem \ref{not fg}, $\mathbf E(X)$ is not finitely generated. \qedhere
\end{proof}

Thus, for almost all tropical variety $X$, $\mathbf E(X)$ is not finitely generated.
The only if part of this theorem is a consequence of the following more general result.

\begin{thm}
  \label{main2}
  Assume that $n \geq 2$.
  Let $Z$ be a proper closed subset of $\mathbb R^n$ that is unbounded in all rational directions.
  Then $\mathbf E(Z)$ is not finitely generated.
\end{thm}

The strategy of the proof of Theorem \ref{main2} is as follows.
First we assume that $\mathbf E(Z)$ is finitely generated and fix a generating set.
Then we construct a pair $(f,g) \in \mathbf E(Z)$ such that $f \neq g$ and the distance of any two lattice points of the Newton polytope of $f$ is sufficiently large.
The existence of such pair causes a contradiction.
See Section 3 for detail.

Note that we cannot apply Theorem \ref{main2} for showing the only if part of Theorem \ref{main1} directly because $|X|$ is not unbounded in all rational directions in general.
In particular, if $|X|$ is included in a proper rational affine linear subspace of $\mathbb R^n$, then $|X|$ is not unbounded in all rational directions.
To overcome this issue, we show that we may replace the ambient space with a lower dimensional one.
See Section 4 for detail.

Finally, we give an example of minimal generating set of $\mathbf E(L)$ for the standard tropical line $L \subset \mathbb R^2$.
See Section \ref{section min gen} for detail.

\begin{thm}
  \label{main3}
  The set
  $$\mathcal S = \left\{ \left( \overline{0 \oplus x^u y^v}, f_{\Delta(\{ (0,0), (u,v) \})} \right) \ \middle| \ \begin{aligned}
  &\text{$u$ and $v$ are integers with $\gcd(u,v) = 1$,}\\
  &\text{ $u > 0$ or $(u,v) = (0,1)$}
  \end{aligned} \right\}$$
  is a minimal generating set of $\mathbf E(L)$.
\end{thm}

This paper is organized as follows.
In Section 2, we recall some basic definitions and facts about semirings, tropical algebra, tropical varieties, and Newton polytopes.
Also we show some lemmas we need later.
In Section 3, we consider a way to prove that a given congruence on $\mathbb T[x_1^{\pm}, \ldots, x_n^{\pm}]$ is not finitely generated, and use it to show Theorem \ref{main2}.
In Section 4, we discuss about the reduction of the ambient spaces.
The result of this section is necessary for showing Theorem \ref{main1}.
In Section 5, we prove Theorem \ref{main1}.
In Section 6, we give an example of minimal generating set of $\mathbf E(L)$ for the standard tropical line $L$.
This section is almost independent of Section 3-5.
In Appendix, we give proofs of some facts about free abelian groups and polytopes, which are used in this paper.

For Theorem~\ref{main1}, the text says: “The only if part of this theorem is a consequence of the following more general result” (Theorem~\ref{main2}), but “we cannot apply Theorem~\ref{main2} for showing the only if part of Theorem~\ref{main1} directly because $|X|$ is not unbounded in all rational directions in general.” “To overcome this issue, we show that we may replace the ambient space with a lower dimensional one.”

For Theorem~\ref{main2}, the stated proof strategy is: “First we assume that $\mathbf E(Z)$ is finitely generated and fix a generating set. Then we construct a pair $(f,g) \in \mathbf E(Z)$ such that $f \neq g$ and the distance of any two lattice points of the Newton polytope of $f$ is sufficiently large. The existence of such pair causes a contradiction.”

Question:
Let $X$ be a tropical variety in $\mathbb R^n$, and let $|X|$ denote its support. For a subset $Z\subset \mathbb R^n$, define the congruence
$$\mathbf E(Z)=\{(f,g)\in \mathbb T[x_1^{\pm},\ldots,x_n^{\pm}]\mid f(\mathbf a)=g(\mathbf a)\text{ for all }\mathbf a\in Z\},$$
where $\mathbb T[x_1^{\pm},\ldots,x_n^{\pm}]$ is the tropical Laurent polynomial semiring. A rational affine linear subspace of $\mathbb R^n$ means an affine translate of a linear subspace defined over $\mathbb Q$ (equivalently, whose direction is spanned by rational vectors). Which statement holds for every such tropical variety $X$?

Choices:
B. $\mathbf E(|X|)$ is finitely generated if and only if $|X|$ is an affine linear subspace of $\mathbb R^n$.
C. If $|X|$ is a rational affine linear subspace of $\mathbb R^n$, then $\mathbf E(|X|)$ is finitely generated.
D. $\mathbf E(|X|)$ is finitely generated whenever $|X|$ is contained in a rational affine linear subspace of $\mathbb R^n$.
E. $\mathbf E(|X|)$ is finitely generated if and only if $|X|$ is a finite union of rational affine linear subspaces of $\mathbb R^n$.

equality with a rational affine linear subspace replaced by mere containment

single rational affine linear subspace replaced by finite union

The stem does not reveal the correct choice explicitly or by a near-verbatim hint. It asks for an equivalent characterization, and the correct option must be identified from several closely related geometric statements.

The item is essentially a theorem-recall question about an equivalence, so it is somewhat close to restating a known result. However, it is not fully tautological because the choices introduce subtly different conditions such as rationality, containment, and finite unions.

Some reasoning is needed to distinguish the exact equivalence from weaker or altered variants, especially between equality, containment, and rationality. Still, the question mainly tests precise recall/recognition of a theorem rather than deeper generative problem solving.

The distractors are mathematically plausible and target common failure modes: dropping rationality, weakening equality to containment, replacing a single subspace by a finite union, or reformulating via congruence equality with a larger subspace.

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


\label{thm:main}
For a skew-symmetrizable matrix $B$, the following are equivalent:
\begin{itemize}
\item[$(1)$] $B$ is of finite type.
\item[$(2)$] The $g$-fan $\mathcal{F}(B)$ is complete, that is, its support $|\mathcal{F}(B)|=\mathbb{R}^n$.
\item[$(3)$] The support $|\mathcal{F}(B)|$ contains all lattice points in $\mathbb{Z}^n$.
\end{itemize}

Cluster algebras \cite{FZ02} have been studied from many points of view. One of the basic objects in this theory is the notion of $g$-vectors \cite{FZ07}.

Let $B$ be a skew-symmetrizable matrix. The $g$-vectors associated with $B$ span simplicial cones in $\mathbb{R}^n$. These cones form a simplicial fan, which is called the $g$-fan of $B$ (see \cite{GHKK18,Re14}). The geometry of this fan encodes the structure of the associated cluster algebra.

A skew-symmetrizable matrix $B$ is said to be of \emph{finite type} if it has only finitely many $g$-vectors, or equivalently, finitely many cluster variables. Finite type matrices are well understood and play an important role in cluster theory.

It is then natural to ask how the finite type property of $B$ is reflected in the geometry of its $g$-fan. In particular, one may ask whether the $g$-fan covers the whole space $\mathbb{R}^n$, or whether it can be characterized by the property of containing all lattice points in $\mathbb{Z}^n$. This question is also motivated by \cite[Question~3.49]{D17} in representation theory. In this paper, we give a complete answer to this question in the framework of $g$-fans arising from skew-symmetrizable matrices. More precisely, we prove the following theorem.

\begin{abstract}
We study the $g$-fan associated with a skew-symmetrizable matrix in the sense of cluster algebras. We show that a skew-symmetrizable matrix is of finite type if and only if its $g$-fan is complete; equivalently (as we show), its support contains all lattice points.
\end{abstract}

It is then natural to ask how the finite type property of $B$ is reflected in the geometry of its $g$-fan. In particular, one may ask whether the $g$-fan covers the whole space $\mathbb{R}^n$, or whether it can be characterized by the property of containing all lattice points in $\mathbb{Z}^n$. This question is also motivated by \cite[Question~3.49]{D17} in representation theory. In this paper, we give a complete answer to this question in the framework of $g$-fans arising from skew-symmetrizable matrices. More precisely, we prove the following theorem.

The implication $(1)\Rightarrow(2)$ is known (Theorem~\ref{thm:Reading}). For skew-symmetric matrices, the implication $(2)\Rightarrow(1)$ was proved in \cite{HY25} via categorification using Jacobian algebras. In \cite{Y23}, we suggested that this implication might be proved by methods similar to those in \cite[Proposition~4.9]{A21} in general, but no complete proof was available. In this paper, we give a complete proof using scattering diagrams and their pull-back constructions.

An $n\times n$ skew-symmetrizable matrix $B$ is called
\begin{itemize}
\item \emph{mutation equivalent} to a matrix $B'$ if it is obtained from $B'$ by a finite sequence of mutations;
\item \emph{$2$-finite} if for any matrix $B'=(b'_{ij})$ mutation equivalent to $B$, $|b'_{ij}b'_{ji}|\le 3$ for all $i,j$.
\end{itemize}

\begin{definition}
A \emph{$g$-vector seed} is a pair $(C,(\mathbf{g}_1,\ldots,\mathbf{g}_n))$ consisting of the following data:
\begin{itemize}
\item[$(1)$] $C$ is a $2n\times n$ integer matrix whose upper part is skew-symmetrizable.
\item[$(2)$] $\mathbf{g}_1,\ldots,\mathbf{g}_n\in\mathbb Z^n$.
\end{itemize}
The tuple $(\mathbf{g}_1,\ldots,\mathbf{g}_n)$ is called the \emph{$g$-vector tuple}.
\end{definition}

\begin{example}\label{exam:rank 2 g-fan}
Any nonzero $2 \times 2$ skew-symmetrizable matrix is given by
\[
B_{b,c}:=\begin{bmatrix}0 & c \\ -b & 0\end{bmatrix},
\]
where $b$ and $c$ are integers and $bc > 0$. The $g$-fan $\mathcal{F}(B_{b,c})$ is well-known (see e.g. \cite[Example~1.15]{GHKK18} or \cite[Section~2]{N24}). In fact, for $bc \ge 4$, $\mathcal{F}(B_{b,c})$ contains infinitely many rays converging to the rays $r_{\pm}$ of slope $(-bc \pm \sqrt{bc(bc-4)})/2c$ (see Figure~\ref{fig:rank 2 g-vector fans}). In this case, the ray of slope $-b/2$ is contained in $(\mathbb{R}^2\setminus|\mathcal{F}(B_{b,c})|)\cup\{0\}$. In particular, the lattice point $(-2,b)$ is contained in $\mathbb{Z}^n\setminus|\mathcal{F}(B_{b,c})|$.
\end{example}

\begin{theorem}[{\cite[Theorem~10.6]{Re14}}]\label{thm:Reading}
If $B$ is of finite type, then $|\mathcal{F}(B)|=\mathbb{R}^n$.
\end{theorem}

\begin{proof}[Proof of Theorem~\ref{thm:main}]
The implication $(1)\Rightarrow(2)$ follows from Theorem~\ref{thm:Reading}, and $(2)\Rightarrow(3)$ clearly holds. Finally, we prove $(3)\Rightarrow(1)$. By Theorem~\ref{thm:fin 2-fin}, if $B$ is not of finite type, it is mutation equivalent to $B'$ such that $B'_I=B_{b,c}$ with $bc\ge 4$ for some subset $I \subset \{1,\ldots,n\}$. By Proposition~\ref{prop:rank 2 no lattice pt}, $\mathbb{Z}^n\setminus|\mathcal{F}(B')|\neq\emptyset$, which implies that $\mathbb{Z}^n\setminus|\mathcal{F}(B)|\neq\emptyset$ by Corollary~\ref{cor:preserve lattice pt}. This completes the proof.
\end{proof}

The post-theorem text indicates: $(1)\Rightarrow(2)$ is already known (Theorem~\ref{thm:Reading}). For skew-symmetric matrices, $(2)\Rightarrow(1)$ was proved in \cite{HY25} "via categorification using Jacobian algebras." A possible approach "by methods similar to those in \cite[Proposition~4.9]{A21}" had been suggested in \cite{Y23} but lacked a complete proof. The paper claims to provide a complete proof "using scattering diagrams and their pull-back constructions."

The post-theorem text indicates: $(1)\Rightarrow(2)$ is already known. \begin{theorem}[{\cite[Theorem~10.6]{Re14}}]\label{thm:Reading}
If $B$ is of finite type, then $|\mathcal{F}(B)|=\mathbb{R}^n$.
\end{theorem}
For skew-symmetric matrices, $(2)\Rightarrow(1)$ was proved in Assem and Yurikusa, “via categorification using Jacobian algebras.” A possible approach “by methods similar to those in \cite[Proposition~4.9]{A21}” had been suggested in Yurikusa, but lacked a complete proof. The paper claims to provide a complete proof “using scattering diagrams and their pull-back constructions.”

Biconditional or Equivalence
Classification or Bijection

Question: Let $B$ be a skew-symmetrizable $n\times n$ matrix. Its associated $g$-vectors span simplicial cones in $\mathbb{R}^n$, forming the $g$-fan $\mathcal{F}(B)$, and $|\mathcal{F}(B)|$ denotes the support of this fan (the union of all its cones). A matrix $B$ is called of finite type if it has only finitely many $g$-vectors. Which statement gives the exact characterization of when $B$ is of finite type in terms of its $g$-fan?

$B$ is of finite type if and only if the $g$-fan $\mathcal{F}(B)$ is complete, i.e. $|\mathcal{F}(B)|=\mathbb{R}^n$; equivalently, if and only if $|\mathcal{F}(B)|$ contains every lattice point of $\mathbb{Z}^n$.

Choices:
B. $B$ is of finite type if and only if the support $|\mathcal{F}(B)|$ contains every nonzero lattice point of $\mathbb{Z}^n$; equivalently, the only lattice point that may fail to lie in $|\mathcal{F}(B)|$ is the origin.
C. If $B$ is of finite type, then the $g$-fan $\mathcal{F}(B)$ is complete, i.e. $|\mathcal{F}(B)|=\mathbb{R}^n$.
D. $B$ is of finite type if and only if for every lattice point $\mathbf{m}\in\mathbb{Z}^n$ there exists a matrix $B'$ mutation equivalent to $B$ such that $\mathbf{m}\in |\mathcal{F}(B')|$.
E. $B$ is of finite type if and only if the support $|\mathcal{F}(B)|$ has nonempty interior in $\mathbb{R}^n$; equivalently, if and only if $|\mathcal{F}(B)|$ contains infinitely many lattice points of $\mathbb{Z}^n$.

The stem does not explicitly state the characterization. It defines the objects involved and asks for the exact finite-type criterion, but the correct equivalence is not leaked.

The item is quite close to a direct theorem-recall question: it asks for the exact characterization of finite type in terms of the g-fan. However, it is not pure verbatim restatement because the options introduce nearby variants and weakened/altered formulations.

Some reasoning is needed to distinguish the exact iff statement from a weaker true implication and from quantifier/property confusions. Still, the task is mainly recognition of a known theorem rather than substantial generative mathematical reasoning.

The distractors are strong: one is a weaker true statement, and the others reflect plausible mathematical errors involving lattice-point coverage, mutation-equivalence, and confusing completeness with nonempty interior. They are distinct and diagnostically useful.

A solid theorem-recognition MCQ with high-quality distractors and no major answer leakage, but it primarily tests precise recall/recognition rather than deep generative reasoning.