\begin{theorem}\label{thm:main}
Consider any $T \ge 1$ and $L,\alpha>0$. Then there exist an $\alpha$-strongly convex set $S$ and an $L$-smooth, $L$-strongly convex function $f$ in dimension $d=2(T+1)$ such that every \texttt{FO-LMO} method applied to $(f, S)$ with $x_0 = 0$ has $x_T$ infeasible or
\[
f(x_T) - \min_{x\in S}f(x) \geq \frac{1}{528}\frac{L\, \mathrm{diam}(S)^2}{(T+1)^2}.
\]
Consequently, for any $\varepsilon>0$, there exist problem instances where $T\geq \sqrt{\frac{L\, \mathrm{diam}(S)^2}{528\varepsilon}} - 1$ iterations are required for any \texttt{FO-LMO} method to reach a suboptimality of $f(x_T) - \min_{x\in S}f(x) \leq \varepsilon$.
\end{theorem}

In this work, we consider convex constrained optimization problems of the form
$$ \begin{cases}
\min & f(x)\\
\mathrm{s.t.\ } & x\in S \subseteq \mathbb{R}^d.
\end{cases}$$
In particular, we consider high-dimensional problems where $d$ may be arbitrarily large.
Frank-Wolfe methods and the broader family of ``projection-free'' algorithms using a linear minimization subroutine have found renewed interest due to their scalability. See the survey~\cite{braun2025conditionalgradientmethods} and references therein. In a basic form, these methods iterate from an initialization $x_0$, producing for $k=0,1,\dots$
\begin{align*}
    p_k &= \nabla f(x_k),\\
    z_{k+1} & \in\argmin_{x\in S}\langle p_k, x\rangle,\\
    x_{k+1} & \in \operatorname{conv}\{x_k, z_{k+1}\}.
\end{align*}
As examples, an exact line search implementation of Frank-Wolfe would set $x_{k+1}$ as the minimizer of $f$ on the segment $[x_k,z_{k+1}]$. Similarly, a fixed ``open-loop'' stepsize implementation would fix $x_{k+1} = \theta_k x_k + (1-\theta_k)z_{k+1}$ for predetermined $\theta_k\in [0,1]$.
The two key computational oracles assumed here are access to gradients of the objective $f$ to compute $p_k$ and access to a Linear Minimization Oracle (LMO) to produce $z_{k+1}$. As a shorthand, we denote $\mathtt{LMO}_S(p) \in \argmin_{x\in S}\langle p, x\rangle$ as an oracle providing a selection of this minimizer (potentially selected adversarially).

In this work, we provide complexity lower bounds for the family of all deterministic first-order methods using a linear minimization oracle, containing the above Frank-Wolfe methods. Our focus is on lower bounds for problem classes where the constraint set $S$ possesses structural properties like strong convexity or smoothness. Below, we formalize these classes of problems and algorithms.

\paragraph{The Families of Smooth and Strongly Convex Constraint Sets and Functions.} A problem instance is defined by a function $f\colon \mathbb{R}^d\rightarrow \mathbb{R}$ and a set $S\subseteq\mathbb{R}^d$ as well as an initialization $x_0$, which we will set as $x_0=0$ without loss of generality. A problem class is defined by a set of allowable $f$ and $S$ values. We consider the standard family of convex objective functions parameterized by $0\leq\mu\leq L$, defined as
\begin{align*}
    \text{$f$ is $\mu$-strongly convex if\quad }& f(\lambda x + (1-\lambda)y)\leq \lambda f(x) + (1-\lambda)f(y) - \frac{\lambda(1-\lambda)\mu}{2}\|y-x\|_2^2\\
    &\hskip7.0cm \forall x,y\in \mathbb{R}^d, \lambda\in[0,1], \\
    \text{$f$ is $L$-smooth if\quad } & \|\nabla f(y)-\nabla f(x)\|_2\leq L\|y-x\|_2 \quad \forall x,y\in \mathbb{R}^d.
\end{align*}
If $L=\mu$, then $f$ must be a quadratic function of the form $\frac{L}{2}\|x-x_\star\|_2^2$ up to an additive constant. Such simple quadratics will suffice for our theoretical development.

We consider compact convex sets $S$ with diameter $\mathrm{diam}(S)=\max_{x,y\in S}\|x-y\|_2$ with at least one of the natural parallel notions of strong convexity and smoothness for constraint sets. We say
$$ \text{$S$ is $\alpha$-strongly convex if\quad } \lambda x + (1-\lambda)y + B\left(0,\ \frac{\lambda(1-\lambda)\alpha}{2}\|y-x\|_2^2\right) \subseteq S $$
for all $x,y\in S, \lambda\in [0,1]$ where $B(0,r)=\{x\in\mathbb{R}^d \mid \|x\|_2\leq r\}$ denotes the closed ball of radius $r$. Normal vectors $n \in N_S(x) := \{n \mid \langle n, y-x\rangle \leq 0\ \forall y\in S\}$ at each $x\in\operatorname{bdry}S$ serve as the analogues of gradients. Considering any unit length normal vectors, we define
\begin{align*}
    \text{$S$ is $\beta$-smooth if\quad } \|n_y - n_x\|_2 \leq \beta \|y-x\|_2 \quad \forall &x,y\in\operatorname{bdry}S,\\
    &n_y\in \{n\in N_S(y) \mid \|n\|_2=1\},\\
    &n_x\in \{n\in N_S(x) \mid \|n\|_2=1\}.
\end{align*}
Note that the above definition requires that smooth sets have a unique unit normal vector at each boundary point.
For a classical reference providing equivalent characterizations of the strong convexity of a set, see~\cite{vial1982strong}. A modernized treatment of smoothness and strong convexity of sets was given by~\cite{liu2023gauges}. Therein, parallels to the smoothness and strong convexity of functions are explored.

\paragraph{The Family of First-Order Linear Minimization Oracle Methods (\texttt{FO-LMO}).}
For a given problem instance $(f,S)$ and initialization $x_0 \in S$, a \texttt{FO-LMO} method generates sequences of search directions $p_k\in\mathbb{R}^d$, linear minimization solutions $z_{k+1} = \mathtt{LMO}_S(p_k)\in\argmin_{x\in S}\langle p_k, x\rangle$, and iterates $x_{k+1}$. We require that $p_k$ is a deterministic function of the oracle responses $\{(f(x_i),\nabla f(x_i))\}^k_{i=0}$ and $\{z_{i}\}^k_{i=1}$ so far and that $x_{k+1}$ is a deterministic function of the responses $\{(f(x_i),\nabla f(x_i))\}^k_{i=0}$ and $\{z_{i}\}^{k+1}_{i=1}$. Note that we do not require that $x_{k+1}$ lie in the convex hull of $x_0$ and $\{z_{i}\}_{i=1}^{k+1}$. However, when outside this convex hull, the iterates may fail to be feasible.
This \texttt{FO-LMO} family of methods includes, for example, the Away-step Frank–Wolfe methods~\cite{lacoste2015global,beck2017linearly}, Pairwise Frank–Wolfe methods~\cite{lacoste2015global,tsuji2022pairwise}, Fully Corrective Frank–Wolfe methods~\cite{holloway1974extension,lacoste2015global}, and Blended Pairwise Frank–Wolfe methods~\cite{braun2019blended}.

In this model, each iteration of the algorithm can make one call to a first-order oracle, returning $(f(x_k), \nabla f(x_k))$, and one call to the LMO. Our theory then bounds iteration complexity to measure the minimum number of such pairs of oracle calls needed to reach a target suboptimality.

For the minimization of an $L$-smooth convex function $f$ over a compact convex set $S$, Frank-Wolfe methods~\cite{frank1956algorithm,jaggi2013revisiting} are known to provide convergence rates of the following form
$$ f(x_T) - \min_{x\in S}f(x) \leq \mathcal{O}\left(\frac{L\, \mathrm{diam}(S)^2}{T}\right). $$
This provides an upper bound on the iteration complexity of computing a point with $\varepsilon$-suboptimality (i.e., $f(x_T)-\min_{x\in S}f(x)\leq\varepsilon$) of $\mathcal{O}(L\, \mathrm{diam}(S)^2/\varepsilon)$. A matching complexity lower bound was provided by Lan~\cite{lan2013complexity}, establishing this as the order of the optimal LMO complexity.
Garber and Hazan~\cite{garber2015faster} showed that given the additional structure that $f$ and $S$ are strongly convex, this rate can be accelerated to
$$ f(x_T) - \min_{x\in S}f(x) \leq \mathcal{O}\left(\frac{L\, \mathrm{diam}(S)^2}{T^2}\right) $$
giving an iteration complexity of $\mathcal{O}(\sqrt{L\, \mathrm{diam}(S)^2/\varepsilon})$. This accelerated result poses the natural question of whether further acceleration is possible or if this complexity is the optimal strongly convex order.

Providing recent progress on this question, Halbey et al.~\cite{halbey2026lower} showed that two particular variants of Frank-Wolfe (exact line search and the short stepsize procedure) cannot improve upon this $\mathcal{O}(1/T^2)$ rate. However, these results do not preclude the possibility of other \texttt{FO-LMO} methods exceeding this rate.
Similar limitations in unconstrained minimization were overcome by the foundational work~\cite{nemirovski1983problem}, establishing information/oracle complexity lower bounds against the whole family of gradient methods. This was accomplished by the design of hard functions whose gradient reveals only one new coordinate of information per step. This property, known as a ``zero-chain'' property, can prevent {\it any gradient-span method} from having made substantial progress until dimension-many steps have been taken. By combining this with an adversarial ``resisting oracle'', lower bounds against {\it all deterministic gradient methods} were achieved.

\paragraph{Our Contributions.} We extend the classical zero-chain approach to LMOs to derive complexity lower bounds on any \texttt{FO-LMO} method over the problem classes corresponding to minimizing a quadratic $f(x)=\frac{L}{2}\|x-x_\star\|_2^2$ over structured sets, possessing either strong convexity or smoothness. Our primary result shows that over $\alpha$-strongly convex constraints, no \texttt{FO-LMO} method can guarantee $\varepsilon$-suboptimality in fewer than $\Omega(\sqrt{L\, \mathrm{diam}(S)^2/\varepsilon})$ iterations. By matching the upper bounding guarantee of~\cite{garber2015faster}, this establishes $\Theta(\sqrt{L\, \mathrm{diam}(S)^2/\varepsilon})$ as the optimal order of complexity for this class. Formally, we prove the following in Section~\ref{sec:general}.

\paragraph{The Families of Smooth and Strongly Convex Constraint Sets and Functions.} A problem instance is defined by a function $f\colon \mathbb{R}^d\rightarrow \mathbb{R}$ and a set $S\subseteq\mathbb{R}^d$ as well as an initialization $x_0$, which we will set as $x_0=0$ without loss of generality. A problem class is defined by a set of allowable $f$ and $S$ values. We consider the standard family of convex objective functions parameterized by $0\leq\mu\leq L$, defined as
\begin{align*}
 \text{$f$ is $\mu$-strongly convex if\quad }& f(\lambda x + (1-\lambda)y)\leq \lambda f(x) + (1-\lambda)f(y) - \frac{\lambda(1-\lambda)\mu}{2}\|y-x\|_2^2\\
 &\hskip7.0cm \forall x,y\in \mathbb{R}^d, \lambda\in[0,1], \\
 \text{$f$ is $L$-smooth if\quad } & \|\nabla f(y)-\nabla f(x)\|_2\leq L\|y-x\|_2 \quad \forall x,y\in \mathbb{R}^d.
\end{align*}
If $L=\mu$, then $f$ must be a quadratic function of the form $\frac{L}{2}\|x-x_\star\|_2^2$ up to an additive constant. Such simple quadratics will suffice for our theoretical development.

We consider compact convex sets $S$ with diameter $\mathrm{diam}(S)=\max_{x,y\in S}\|x-y\|_2$ with at least one of the natural parallel notions of strong convexity and smoothness for constraint sets. We say
$$ \text{$S$ is $\alpha$-strongly convex if\quad } \lambda x + (1-\lambda)y + B\left(0,\ \frac{\lambda(1-\lambda)\alpha}{2}\|y-x\|_2^2\right) \subseteq S $$
for all $x,y\in S, \lambda\in [0,1]$ where $B(0,r)=\{x\in\mathbb{R}^d \mid \|x\|_2\leq r\}$ denotes the closed ball of radius $r$. Normal vectors $n \in N_S(x) := \{n \mid \langle n, y-x\rangle \leq 0\ \forall y\in S\}$ at each $x\in\operatorname{bdry}S$ serve as the analogues of gradients. Considering any unit length normal vectors, we define
\begin{align*}
 \text{$S$ is $\beta$-smooth if\quad } \|n_y - n_x\|_2 \leq \beta \|y-x\|_2 \quad \forall &x,y\in\operatorname{bdry}S,\\
 &n_y\in \{n\in N_S(y) \mid \|n\|_2=1\},\\
 &n_x\in \{n\in N_S(x) \mid \|n\|_2=1\}.
\end{align*}
Note that the above definition requires that smooth sets have a unique unit normal vector at each boundary point.
For a classical reference providing equivalent characterizations of the strong convexity of a set, see~\cite{vial1982strong}. A modernized treatment of smoothness and strong convexity of sets was given by~\cite{liu2023gauges}. Therein, parallels to the smoothness and strong convexity of functions are explored.

For the minimization of an $L$-smooth convex function $f$ over a compact convex set $S$, Frank-Wolfe methods~\cite{frank1956algorithm,jaggi2013revisiting} are known to provide convergence rates of the following form
$$ f(x_T) - \min_{x\in S}f(x) \leq \mathcal{O}\left(\frac{L\, \mathrm{diam}(S)^2}{T}\right). $$
This provides an upper bound on the iteration complexity of computing a point with $\varepsilon$-suboptimality (i.e., $f(x_T)-\min_{x\in S}f(x)\leq\varepsilon$) of $\mathcal{O}(L\, \mathrm{diam}(S)^2/\varepsilon)$. A matching complexity lower bound was provided by Lan~\cite{lan2013complexity}, establishing this as the order of the optimal LMO complexity.
Garber and Hazan~\cite{garber2015faster} showed that given the additional structure that $f$ and $S$ are strongly convex, this rate can be accelerated to
$$ f(x_T) - \min_{x\in S}f(x) \leq \mathcal{O}\left(\frac{L\, \mathrm{diam}(S)^2}{T^2}\right) $$
giving an iteration complexity of $\mathcal{O}(\sqrt{L\, \mathrm{diam}(S)^2/\varepsilon})$. This accelerated result poses the natural question of whether further acceleration is possible or if this complexity is the optimal strongly convex order.

Providing recent progress on this question, Halbey et al.~\cite{halbey2026lower} showed that two particular variants of Frank-Wolfe (exact line search and the short stepsize procedure) cannot improve upon this $\mathcal{O}(1/T^2)$ rate. However, these results do not preclude the possibility of other \texttt{FO-LMO} methods exceeding this rate.
Similar limitations in unconstrained minimization were overcome by the foundational work~\cite{nemirovski1983problem}, establishing information/oracle complexity lower bounds against the whole family of gradient methods. This was accomplished by the design of hard functions whose gradient reveals only one new coordinate of information per step. This property, known as a ``zero-chain'' property, can prevent {\it any gradient-span method} from having made substantial progress until dimension-many steps have been taken. By combining this with an adversarial ``resisting oracle'', lower bounds against {\it all deterministic gradient methods} were achieved.

\paragraph{Our Contributions.} We extend the classical zero-chain approach to LMOs to derive complexity lower bounds on any \texttt{FO-LMO} method over the problem classes corresponding to minimizing a quadratic $f(x)=\frac{L}{2}\|x-x_\star\|_2^2$ over structured sets, possessing either strong convexity or smoothness. Our primary result shows that over $\alpha$-strongly convex constraints, no \texttt{FO-LMO} method can guarantee $\varepsilon$-suboptimality in fewer than $\Omega(\sqrt{L\, \mathrm{diam}(S)^2/\varepsilon})$ iterations. By matching the upper bounding guarantee of~\cite{garber2015faster}, this establishes $\Theta(\sqrt{L\, \mathrm{diam}(S)^2/\varepsilon})$ as the optimal order of complexity for this class. Formally, we prove the following in Section~\ref{sec:general}.

As a secondary result, we provide a partial generalization to $\beta$-smooth sets. These bounds are meaningful in the regime of only modestly smooth sets, having $\beta=\Omega(1/\sqrt{\varepsilon})$. In this regime, no \texttt{LMO-span} method (see Section~\ref{sec:SC} for a formal definition) can improve past the optimal compact convex set complexity of $\Theta(L\, \mathrm{diam}(S)^2/\varepsilon)$ or over the optimal $\alpha$-strongly convex set complexity of $\Theta(\sqrt{L\, \mathrm{diam}(S)^2/\varepsilon})$. Theorems~\ref{thm:smooth} and~\ref{thm:mixed} formalize these limits on acceleration due to smoothness.

First, Section~\ref{subsec:Construction-SC} constructs our candidate hard problem instance for each dimension and verifies its validity (computing its strong convexity constant and diameter). Then Section~\ref{subsec:zero-chain-SC} establishes a key ``zero-chain'' property of these hard instances, showing that any \texttt{LMO-span} method applied will only discover one new coordinate per iteration. Finally, by leveraging this property, Section~\ref{subsec:proof-SC} proves this section's main result, Theorem~\ref{thm:SC}, showing that no method in $T$ steps can guarantee a suboptimality less than $\Omega(L\, \mathrm{diam}(S)^2/T^2)$. Combined with the upper bound of~\cite{garber2015faster}, this establishes the optimal iteration complexity for strongly convex constrained \texttt{LMO-span} optimization as $\Theta(\sqrt{L\, \mathrm{diam}(S)^2/\varepsilon})$.

Any considered \texttt{FO-LMO} method must set $x_T$ deterministically as a function of the history of oracle responses. Crucially, while the algorithm must fix $(x_T)_i$ for $i \in U$, the resisting oracle remains free to select any completion of the permutation $\pi$ over $U$. Let $\mathcal{P}_U\subseteq \mathcal{P}_d$ denote all such permutations. If some completion $\pi\in\mathcal{P}_U$ has $x_T\not\in S_\pi$, then the adversarial oracle can select that $\pi$, making $x_T$ infeasible and the theorem hold trivially. Hence, we can assume for all $\pi\in \mathcal{P}_U$ that $x_T\in S_\pi$. From the strong convexity of $f$ and optimality of $x^{(\pi)}_\star$ guaranteeing $\langle \nabla f(x^{(\pi)}_\star), x_T - x^{(\pi)}_\star\rangle\geq 0$ since $x_T\in S_\pi$, the suboptimality for any $\pi\in\mathcal{P}_U$ is bounded by
\begin{equation}\label{eq:sc-lower-bound}
 f(x_T) - f(x_\star^{(\pi)}) \ge \frac{1}{2}\|x_T - x_\star^{(\pi)}\|_2^2 \ge \frac{1}{2} \sum_{j\in U} \left( (x_T)_{j} - (x^{(\pi)}_\star)_{j} \right)^2.
\end{equation}
Let $\pi_\star \in \mathcal{P}_U$ denote the permutation choice maximizing this lower bound. Then, we have
\begin{align*}
 f(x_T) - f(x_\star^{(\pi_\star)}) &\geq \frac{1}{2} \sum_{j\in U} \left( (x_T)_{j} - (x^{(\pi_\star)}_\star)_{j} \right)^2\\
 &\geq \frac{1}{|\mathcal{P}_U|}\sum_{\pi\in\mathcal{P}_U} \left(\frac{1}{2}\sum_{j\in U} \left( (x_T)_{j} - (x^{(\pi)}_\star)_{j} \right)^2\right)\\
 & = \frac{1}{2}\sum_{j\in U} \frac{1}{|\mathcal{P}_U|}\sum_{\pi\in\mathcal{P}_U} \left( (x_T)_{j} - (x^{(\pi)}_\star)_{j} \right)^2\\
 &\geq \frac{m}{2}\mathrm{Var}\left\{ \rho(w_d - w_k) \right\}_{k = d-m+1}^d \\
 &=\frac{\rho^2 m}{2} \mathrm{Var}\{w_{k}\}_{k = d-m+1}^d\\
 &= \rho^2 m C^2 \mathrm{Var}\{\sqrt{k}\}_{k = d-m+1}^d
\end{align*}
where the first inequality is by~\eqref{eq:sc-lower-bound}, the second inequality lower bounds this maximal $\pi_\star$ by the average lower bound over $\mathcal{P}_U$, and the third inequality notes that each inner sum is minimized when $(x_T)_{j}$ is the average of $\{\rho(w_d - w_{d-m+1}),\dots, \rho(w_d - w_{d})\}$ yielding the stated variance lower bound.

\begin{theorem}\label{thm:smooth}
For any $d \ge 1$ and $\beta > 0$, there exist a convex $\beta$-smooth set $S_\beta$ and a $1$-smooth, $1$-strongly convex function $f_\beta$ such that every \texttt{LMO-span} method applied to $(f_\beta, S_\beta)$ starting from $x_0=0$ has for all $t\leq d-1$
 \begin{equation}
 f_\beta(x_t) - \min_{x\in S_\beta}f_\beta(x) \geq \max \left(0, \sqrt{\frac{d-t}{2}}\left(\frac{\mathrm{diam}(S_\beta) - 2/\beta}{\sqrt{2}d}+\frac{1}{\beta\sqrt{d}}\right)-\frac{1}{\sqrt{2}\beta}\right)^2 .
 \end{equation}
 In particular, for any fixed budget $T \ge 1$, there exist $S_\beta$ and $f_\beta$ in dimension $d=2T$ such that
 \[
 f_\beta(x_T) - \min_{x\in S_\beta}f_\beta(x) \geq \max\left(0,\frac{\mathrm{diam}(S_\beta) - 2/\beta}{4\sqrt{T}} - \frac{\sqrt{2}-1}{2\beta }\right)^2 .
 \]
\end{theorem}
\begin{theorem}\label{thm:mixed}
 For any $d \ge 1$ and $\beta > 0$, there exist a $\beta$-smooth, $1/(1+1/\beta)$-strongly convex set $S_\beta$ and a $1$-smooth, $1$-strongly convex function $f_\beta$ such that every \texttt{LMO-span} method applied to $(f_\beta, S_\beta)$ starting from $x_0 = 0$ has for all $t\leq d-1$
 \begin{equation}
 f_\beta(x_t) - \min_{x\in S_\beta}f_\beta(x) \geq \max\left(0,\sqrt{\frac{2}{5}\frac{(d-t)(\mathrm{diam}(S_\beta) - 2/\beta)^2}{(d+2)^3}} - \frac{1}{\sqrt{2}\beta}\right)^2 .
 \end{equation}
 In particular, for any fixed budget $T \ge 1$, there exist $S_\beta$ and $f_\beta$ in dimension $d=2(T+1)$ such that
 \[
 f_\beta(x_T) - \min_{x\in S_\beta}f_\beta(x) \geq \max\left(0,\frac{\mathrm{diam}(S_\beta) - 2/\beta}{\sqrt{20}(T+2)} - \frac{1}{\sqrt{2}\beta }\right)^2 .
 \]
\end{theorem}
\noindent In both of these theorems, the lower bounds are only meaningful in the regime of $\beta=\Omega(1/\sqrt{\varepsilon})$ as otherwise, the two-term maximums above will take value zero, making the lower bound vacuous.

Inductively applying Lemma~\ref{lem:approx-chain}, the iterate $x_t \in \operatorname{conv}\{x_0, z_1, \dots, z_t\}$ must have final $d-t$ coordinates $i\geq t+1$ equal with value bounded by $|(x_t)_i| \leq \max_{1 \le k \le t} |\delta_k| \le \frac{1}{\beta \sqrt{d-t+1}}$. Then the suboptimality $f_\beta(x_t) - f_\beta(x_\star) = \frac{1}{2}\|x_t-x_\star\|^2_2$ is lower bounded by
\begin{align*}
 f_\beta(x_t) - f_\beta(x_\star) &\ge \frac{1}{2} \sum_{i=t+1}^d \left((x_t)_i - \left(\frac{1}{d}+\frac{1}{\beta \sqrt{d}}\right)\right)^2 \\
 &\ge \frac{d-t}{2}\max \left(0, \frac{1}{d}+\frac{1}{\beta\sqrt{d}}-\frac{1}{\beta\sqrt{d-t+1}}\right)^2 \\
 &\ge \max \left(0, \sqrt{\frac{d-t}{2}}\left(\frac{\mathrm{diam}(S_\beta) - 2/\beta}{\sqrt{2}d}+\frac{1}{\beta\sqrt{d}}\right)-\frac{1}{\sqrt{2}\beta}\right)^2.
\end{align*}
Specializing to $t=T$ and $d=2T$, this provides a lower bound of
$$ f_\beta(x_T) - f_\beta(x_\star) \geq \max\left(0,\frac{\mathrm{diam}(S_\beta) - 2/\beta}{4\sqrt{T}} - \frac{\sqrt{2}-1}{2\beta }\right)^2 . $$
\subsection{Proof of Theorem~\ref{thm:mixed}}
Given any $d>0$, consider the set $S$ previously defined in~\eqref{eq:S-simple} and the parameter $\nu$ defined by~\eqref{eq:nu-def}. Fix any $\beta > 0$ and define
$$ S_\beta := S + B(0,1/\beta) . $$
From Lemma~\ref{lem:diam-SC}, it is immediate that the diameter of $S_\beta$ is $\mathrm{diam}(S_\beta) = \mathrm{diam}(S)+2/\beta = 2C(2-\sqrt{2}) + 2/\beta$. Further, $S_\beta$ is $\beta$-smooth and $\alpha = 1/(1+1/\beta)$-strongly convex by the calculus rules for Minkowski sums~\cite[Lemma 9]{liu2023gauges}.

While Theorem~\ref{thm:main} applies for any $L,\alpha>0$, it does not allow a free selection of $\mathrm{diam}(S)$. Rather, our constructed hard ``zero-chain'' set $S$ has $\mathrm{diam}(S) = \Theta(1/\alpha d)$. One cannot arbitrarily select $\alpha$ and $\mathrm{diam}(S)$ as they must satisfy, for example, that $\mathrm{diam}(S)\leq 2/\alpha$. In the limit where $\mathrm{diam}(S)= 2/\alpha$, the set $S$ is forced to be (up to translation) the ball $B(0,1/\alpha)$. Such forced structure enables faster algorithms\footnote{For example, when $\mathrm{diam}(S)= 2/\alpha$ and $S = B(0,1/\alpha)$, an LMO can be used to explicitly compute orthogonal projections onto the feasible region $S$. Then the class of \texttt{FO-LMO} methods includes projected gradient methods, which are known to converge linearly for smooth, strongly convex objectives.}. The development of hard instances for any selection of $\mathrm{diam}(S) \leq 2/\alpha$ and resulting more nuanced complexity bounds is left as an important future direction.

As a secondary result, we provide a partial generalization to $\beta$-smooth sets. These bounds are meaningful in the regime of only modestly smooth sets, having $\beta=\Omega(1/\sqrt{\varepsilon})$. In this regime, no \texttt{LMO-span} method (see Section~\ref{sec:SC} for a formal definition) can improve past the optimal compact convex set complexity of $\Theta(L\, \mathrm{diam}(S)^2/\varepsilon)$ or over the optimal $\alpha$-strongly convex set complexity of $\Theta(\sqrt{L\, \mathrm{diam}(S)^2/\varepsilon})$. Theorems~\ref{thm:smooth} and~\ref{thm:mixed} formalize these limits on acceleration due to smoothness.

Since our constructions throughout use an $L$-smooth, $L$-strongly convex objective, all of our lower bounds apply against the wider class of problems with smooth, strongly convex objectives having any $0 \leq \mu \leq L$. Hence, our theory highlights a fundamental difficulty of constrained optimization via LMOs: {\it Even on perfectly conditioned objective functions, hard adversarial constraint sets constitute a barrier to linear convergence.} This stands in contrast to gradient methods with an orthogonal projection oracle where convergence is dominated by objective function conditioning.

\paragraph{Outline.} Section~\ref{sec:SC} first derives a lower bound via a novel construction of a strongly convex feasible region that is hard for all \texttt{FO-LMO} methods with an additional span restriction. Section~\ref{sec:general} removes this restriction, proving Theorem~\ref{thm:main}. This construction is the main technical innovation of our work. Section~\ref{sec:smooth} then provides results extending lower bounds to the specialized setting of sets with modest levels of smoothness.

\subsection{Related Work}
Our Theorem~\ref{thm:main} differs from the previously mentioned lower bounding result of~\cite{halbey2026lower} in two aspects, making the result complementary. In terms of algorithmic scope, our bound provides a wider guarantee, establishing a universal lower bound against all \texttt{FO-LMO} methods. In contrast,~\cite{halbey2026lower} provides a hard instance for two standard implementations of Frank-Wolfe, namely those fixing $p_k=\nabla f(x_k)$ and using an exact line search or the short stepsize procedure to select $x_{k+1}$. In terms of problem class scope, our construction requires a nonsmooth feasible region and a large problem dimension $d$ (linear in the number of iterations to be run). Such a high dimensionality assumption (i.e., $d>T$) is classical and widespread in the optimization complexity literature~\cite{nemirovski1983problem}. In contrast,~\cite{halbey2026lower} is able to provide a hard instance using a smooth ball in $d=2$ dimensions. As a result, they provide a stronger illustration of the limitations of the two methods that their theory covers.

As mentioned above, most classical lower bounding results in unconstrained first-order optimization rely on setting the problem dimension larger than the number of iterations to be conducted. This enables ``zero-chain'' arguments where the objective function is designed to reveal only one new coordinate to the given gradient-span algorithm at each iteration. The main technical innovation of our work is the design of a hard strongly convex set where the LMO possesses a similar zero-chain property to these classical hard objective constructions. Such constructions open the possibility to extend proof techniques and insights from existing unconstrained optimization lower bounds to constrained LMO/projection-free settings.

For example, although our strongly convex convergence rate lower bound matches the upper bound of~\cite{garber2015faster} in order, the two differ by constant factors: our lower bound has a universal constant of $1/528$ and their upper bound has $9/2$. This leaves open the question of determining exactly minimax optimal algorithms and hard problem instances. In settings of unconstrained first-order minimization, the Performance Estimation Problem (PEP) techniques pioneered by~\cite{drori2014performance,Interpolation,Interpolation2} have provided such theory. For example, in smooth convex optimization, these facilitated the identification of the Optimized Gradient Method~\cite{kim2016optimized} and an exactly matching lower bound~\cite{drori2017exact}. PEP was extended to cover structured smooth and strongly convex sets by Luner and Grimmer~\cite{luner2024performance}. As a result, future work may leverage PEP to similarly close the constant-factor gap left for \texttt{FO-LMO} methods.

Another important property of most Frank-Wolfe methods is that their trajectory is independent of the choice of inner product used. This differs from projected-gradient methods, where the choice of inner product and notion of orthogonality for projections affects the algorithm trajectory, making good preconditioning important for practical success. The line of work~\cite{Pena2023,Wirth2025,Wirth2026} provided Frank-Wolfe with ``affine-covariant'' convergence theory, matching the method's affine covariant nature by avoiding notions like smoothness and strong convexity defined in terms of a fixed inner product. Developing lower bounding theory in affine-covariant terms is an interesting future direction.

One can view the oracle model of an LMO as assuming first-order access to the support function $p \mapsto \sup\{\langle p, x\rangle \mid x\in S\}$. For sets with $0\in S$, a polar transformation establishes this as dual to assuming first-order access to the Minkowski gauge $y \mapsto \inf\{ \gamma>0 \mid y/\gamma \in S\}$. This gauge model has been studied by the works~\cite{Renegar2016,Grimmer2021-part1,Grimmer2021-part2,Zakaria2022,Lu2023} as an alternative projection-free framework. The works~\cite{liu2023gauges} and~\cite{samakhoana2024scalable} developed accelerated $\mathcal{O}(1/T^2)$ convergence guarantees for gauge methods over smooth sets, complementing the accelerated LMO rates for strongly convex sets. Identifying any structural relationships between the complexity of these dual oracle models and problem settings is another interesting direction.

Question:
Let \(\mathrm{diam}(S)=\max_{x,y\in S}\|x-y\|_2\). A compact convex set \(S\subseteq\mathbb{R}^d\) is \(\alpha\)-strongly convex if for all \(x,y\in S\) and \(\lambda\in[0,1]\),
\[
\lambda x+(1-\lambda)y+B\!\left(0,\frac{\lambda(1-\lambda)\alpha}{2}\|y-x\|_2^2\right)\subseteq S,
\]
where \(B(0,r)=\{u\in\mathbb{R}^d:\|u\|_2\le r\}\). A differentiable function \(f\colon\mathbb{R}^d\to\mathbb{R}\) is \(L\)-smooth if \(\|\nabla f(y)-\nabla f(x)\|_2\le L\|y-x\|_2\) for all \(x,y\), and is \(L\)-strongly convex if
\[
f(\lambda x+(1-\lambda)y)\le \lambda f(x)+(1-\lambda)f(y)-\frac{\lambda(1-\lambda)L}{2}\|y-x\|_2^2
\]
for all \(x,y\in\mathbb{R}^d\), \(\lambda\in[0,1]\). A FO-LMO method is an algorithm that uses first-order information on \(f\) together with a linear minimization oracle over \(S\), and it is started from \(x_0=0\). For arbitrary \(T\ge 1\) and \(L,\alpha>0\), which quantitative lower-bound statement holds in dimension \(d=2(T+1)\)?

There exist an \(\alpha\)-strongly convex set \(S\subseteq\mathbb{R}^{2(T+1)}\) and a function \(f\colon\mathbb{R}^{2(T+1)}\to\mathbb{R}\) that is both \(L\)-smooth and \(L\)-strongly convex such that every FO-LMO method applied to \((f,S)\) with \(x_0=0\) either produces an infeasible iterate \(x_T\notin S\) or satisfies
\[
f(x_T)-\min_{x\in S} f(x)\ge \frac{1}{528}\,\frac{L\,\mathrm{diam}(S)^2}{(T+1)^2}.
\]
Consequently, for every \(\varepsilon>0\), there exist such problem instances for which any FO-LMO method requires at least
\[
\sqrt{\frac{L\,\mathrm{diam}(S)^2}{528\varepsilon}}-1
\]
iterations to obtain a point with \(f(x_T)-\min_{x\in S}f(x)\le \varepsilon\).

Choices:
B.
There exist an \(\alpha\)-strongly convex set \(S\subseteq\mathbb{R}^{2(T+1)}\) and a function \(f\colon\mathbb{R}^{2(T+1)}\to\mathbb{R}\) that is both \(L\)-smooth and \(L\)-strongly convex such that every FO-LMO method applied to \((f,S)\) with \(x_0=0\) either produces an infeasible iterate \(x_T\notin S\) or satisfies
\[
f(x_T)-\min_{x\in S} f(x)\ge \frac{1}{528}\,\frac{L\,\mathrm{diam}(S)^2}{T+1}.
\]
Consequently, for every \(\varepsilon>0\), there exist such problem instances for which any FO-LMO method requires at least
\[
\frac{L\,\mathrm{diam}(S)^2}{528\varepsilon}-1
\]
iterations to obtain a point with \(f(x_T)-\min_{x\in S}f(x)\le \varepsilon\).
C.
There exist an \(\alpha\)-strongly convex set \(S\subseteq\mathbb{R}^{2(T+1)}\) and a function \(f\colon\mathbb{R}^{2(T+1)}\to\mathbb{R}\) that is both \(L\)-smooth and \(L\)-strongly convex such that every FO-LMO method applied to \((f,S)\) with \(x_0=0\) satisfies
\[
f(x_T)-\min_{x\in S} f(x)\ge \frac{1}{528}\,\frac{L\,\mathrm{diam}(S)^2}{(T+1)^2}.
\]
D.
There exist an \(\alpha\)-strongly convex set \(S\subseteq\mathbb{R}^{2(T+1)}\) and a function \(f\colon\mathbb{R}^{2(T+1)}\to\mathbb{R}\) that is both \(L\)-smooth and \(L\)-strongly convex such that every FO-LMO method applied to \((f,S)\) with \(x_0=0\) either produces an infeasible iterate \(x_T\notin S\) or satisfies
\[
f(x_T)-\min_{x\in S} f(x)\ge \frac{1}{528}\,\frac{L\,\mathrm{diam}(S)}{(T+1)^2}.
\]
Consequently, for every \(\varepsilon>0\), there exist such problem instances for which any FO-LMO method requires at least
\[
\sqrt{\frac{L\,\mathrm{diam}(S)}{528\varepsilon}}-1
\]
iterations to obtain a point with \(f(x_T)-\min_{x\in S}f(x)\le \varepsilon\).
E.
For every \(\alpha\)-strongly convex set \(S\subseteq\mathbb{R}^{2(T+1)}\) and every function \(f\colon\mathbb{R}^{2(T+1)}\to\mathbb{R}\) that is both \(L\)-smooth and \(L\)-strongly convex, every FO-LMO method applied to \((f,S)\) with \(x_0=0\) either produces an infeasible iterate \(x_T\notin S\) or satisfies
\[
f(x_T)-\min_{x\in S} f(x)\ge \frac{1}{528}\,\frac{L\,\mathrm{diam}(S)^2}{(T+1)^2}.
\]
Consequently, for every \(\varepsilon>0\), any FO-LMO method requires at least
\[
\sqrt{\frac{L\,\mathrm{diam}(S)^2}{528\varepsilon}}-1
\]
iterations to obtain a point with \(f(x_T)-\min_{x\in S}f(x)\le \varepsilon\).

The stem does not reveal the correct option explicitly. It gives definitions and setup, but the exact lower-bound form, quantifiers, infeasibility alternative, and iteration complexity corollary must still be identified from the choices.

This is essentially a theorem-identification item: the task is to recognize the exact quantitative statement of a lower-bound result. It tests recall of the theorem statement more than selection among independently reasoned mathematical conclusions.

There is some reasoning pressure because the student must compare subtle differences in rate exponent, diameter scaling, existential vs. universal quantification, and omission of the infeasibility clause. However, the item mainly rewards precise recall rather than generating a conclusion from first principles.

The distractors are plausible and mathematically meaningful: they alter the rate from 1/(T+1)^2 to 1/(T+1), drop a necessary caveat, change diam(S)^2 to diam(S), or switch existential to universal quantification. These reflect realistic failure modes in reading theorem statements.

A moderately strong MCQ with no answer leakage and high-quality distractors, but it is largely a theorem-recall/restatement question rather than a genuinely generative reasoning task.


\begin{thm}\label{thm-main11}
    For any complex curve in a K\"ahler surface $M^4$, the first eigenvalue $\Lambda_1$ of its area Jacobi operator $\mathcal{L}$ satisfies
    $$\Lambda_1\geq 2\,\mathfrak{Ric}.$$
\end{thm}

The first eigenvalue of the Laplace-Beltrami operator on a compact Riemannian manifold encodes crucial geometric information, including curvature bounds. A fundamental result in this direction is the Lichnerowicz theorem~\cite{Lichnerowicz}, which states that if the Ricci curvature of an \(n\)-dimensional compact manifold satisfies \(\mathrm{Ric} \ge (n-1)K\) for some \(K>0\), then the first positive eigenvalue \(\lambda_1\) obeys \(\lambda_1 \ge nK\). This estimate is sharp, and the Obata theorem~\cite{Obata1962} characterizes the equality case: \(\lambda_1 = nK\) holds iff the manifold is isometric to a sphere of constant sectional curvature \(K\). In the K\"ahler setting, Futaki~\cite{Futaki} proved that the first eigenvalue of the complex Laplacian is bounded below by \(k\) for a compact K\"ahler manifold satisfying \(\mathrm{Ric} \ge k\) with \(k>0\). The equality case was later characterized by Tam and Yu~\cite{Tam-Yu}.

The area Jacobi operator $\mathcal{L}$, which governs the second variation of the area functional for minimal submanifolds, is a second-order, self-adjoint elliptic operator. As such, it can be regarded as the extrinsic counterpart to the intrinsic Laplace–Beltrami operator. Unlike the latter, it is not positive (or even nonnegative) in general. We denote by $\Lambda_1$ the smallest nonzero eigenvalue of $\mathcal{L}$, referred to as the first eigenvalue. Since $\mathcal{L}$ serves as the stability operator, $\Lambda_1$ governs the infinitesimal behavior of the minimal submanifold under deformations. A positive $\Lambda_1$ means the submanifold is locally minimizing, whereas a negative one indicates instability.
In recent decades, considerable attention has been devoted to estimating the first eigenvalue of the area Jacobi operator for minimal surfaces in various ambient spaces. In \cite{Perdomo}, Perdomo proved that if $\Sigma^2$ is a compact oriented minimal surface which is not totally geodesic in $\mathbb{S}^3$, then the first eigenvalue of its area Jacobi operator satisfies $\Lambda_1 \leq -4$, with equality if and only if $\Sigma^2$ is isometric to the Clifford torus. This result sparked a series of further estimates for $\Lambda_1$ on closed hypersurfaces of constant mean curvature or conatnt scalar curvature (see \cite{Alias,Ambrozio,BatistaCavalcanteMelo,Batista,Bray,Chen-Cheng,Cheng,Dung,Li-Wang,Merono,Zhu} and references therein).
Micallef and Wolfson~\cite{MW} obtained fundamental results on the stability of minimal surfaces in 4-dimensional manifolds by considering a complex version of the second variation of area. Subsequently, for superminimal surfaces with negative spin in self-dual Einstein $4$-dimensional manifolds, Montiel and Urbano~\cite{MU} proved that the first eigenvalue of the area Jacobi operator satisfies $\Lambda_1\geq-2\mathfrak{c}$, where $\mathfrak{c}$ denotes the Einstein constant of the ambient space, and they computed exactly the index explicitly in terms of the genus and area of the surface.

This paper focuses on another important class of codimension-2 minimal surfaces, i.e., complex curves in Kähler surfaces, which are superminimal surfaces of positive spin. It is a classical fact that such surfaces are absolutely area‑minimizing and thus stable. Consequently, their associated area Jacobi operator $\mathcal{L}$ is a non‑negative self‑adjoint elliptic operator, exactly analogous to the Laplace–Beltrami operator on a Riemannian manifold. While the non-negativity of $\mathcal{L}$ is well understood, obtaining sharp quantitative estimates for its spectrum in terms of the ambient curvature is a more subtle problem, and the focus of our investigation. We established a universal lower bound for the first eigenvalue $\Lambda_1$ of $\mathcal{L}$. To state our result, we denote by
\[
\mathfrak{Ric} \triangleq \inf_{\substack{p \in \Sigma, \\ u \in UT_p M^4}} \operatorname{Ric}(u, u),
\]
the infimum of the Ricci curvature over all unit tangent vectors at all points of $M^4$.

If we restrict the ambient space to a Kähler–Einstein surface with positive scalar curvature, then the estimate established in Theorem \ref{thm-main11} is sharp. In this setting, the curvature invariant $\mathfrak{Ric}$ simplifies to the Einstein constant $\mathfrak{c}>0$. We explore the equality case and are able to compute exactly the dimension of the first eigenspace. This is accomplished by relating it to the space of Jacobi fields of $\mathcal{W}^+$, which corresponds precisely to the preimage of the space of holomorphic sections of a certain line bundle, and then applying the Riemann-Roch theorem in conjunction with the Dolbeault theorem.
\begin{thm}\label{thm-main2}
 Let \(M^{4}\) be a K\"ahler-Einstein surface with Einstein constant \(\mathfrak{c}>0\), and let \(x\colon\Sigma \rightarrow M^{4}\) be a complex curve. Then
 \begin{itemize}
 \item[{\rm(1)}] the first eigenvalue $\Lambda_1$ of its area Jacobi operator is at least $2\mathfrak{c}$,
 \item[{\rm(2)}] if $\Sigma$ has genus $g\leq 1$, then $\Lambda_1=2\mathfrak{c}$ and the corresponding eigenspace has dimension
 \[
 \mathfrak{c}\frac{\operatorname{Area}}{2\pi}+1-g+\dim H^{\,0}(N^*_\Sigma\otimes K_\Sigma^2),
 \]
 where $N^*_\Sigma$ is the dual of the normal bundle, $K_\Sigma$ is the canonical line bundle of $\Sigma$, and $H^{\,0}(N^*_\Sigma\otimes K_\Sigma^2)$ denotes the space of holomorphic sections of $N^*_\Sigma\otimes K_\Sigma^2$.
\end{itemize}
\end{thm}

Now we assume that $x$ is a complex curve, then \eqref{eq-sfm1} yields
$$ a \equiv 0,~~~ b \equiv 0,$$
which implies $ x $ is a superminimal surface with positive spin. Consequently, $x$ minimizes the functional $\mathcal{W}^+$ within its homotopy class. Without loss of generality, we assume $\lambda\equiv 1$ and $\mu\equiv 0$. Denote by $\theta_{12}$ (resp. $\theta_{34}$) the connection form of the tangent (resp. normal) bundle of $x$, It follows from \eqref{eq-pull}$\sim$\eqref{eq-conn} that
\begin{equation}\label{eq-o1100}
 x^*\omega_{1\bar{1}} = -i\theta_{12}, ~~~x^*\omega_{0\bar{0}} = -i\theta_{34}.
\end{equation}
Differentiating \eqref{eq-o1100} yields the Gauss-Ricci equation
\begin{equation}\label{eq-Gauss1}
 (R_{1212}+R_{1234})\,\theta\wedge\bar\theta = -2x^*(\Omega_{0\bar0}+\Omega_{1\bar1}),
\end{equation}
where \(R_{1212}\) (resp. \(R_{1234}\)) denotes the curvature of the tangent (resp. normal) bundle, defined by
\[
d\theta_{12} = \frac{R_{1212}}{2i}\,\theta\wedge\bar\theta, \qquad
d\theta_{34} = \frac{R_{1234}}{2i}\,\theta\wedge\bar\theta.
\]
Note that
$-i(\Omega_{0\bar0}+ \Omega_{1\bar1})$ is exactly the Ricci form of $M^4$, which implies
\begin{equation}\label{eq-Ric}
x^*(\Omega_{0\bar0}+ \Omega_{1\bar1})=-\mathrm{Ric}(e_1, \bar e_1)\theta\wedge\bar\theta.
\end{equation}
Therefore, we can also reformulate the Gauss-Ricci equation \eqref{eq-Gauss1} as
\begin{equation}\label{eq-Gauss2}
 R_{1212}+R_{1234}=2\mathrm{Ric}(e_1,\bar e_1).
\end{equation}

Using \eqref{eq-W+3} and \eqref{eq-lambda1}, we obtain that along the variation determined by $V$,
\begin{equation}\label{eq-varW}
 \frac{\partial^{2}}{\partial t^{2}} \bigg|_{t=0} \mathcal{W}^{+}
\leq 8 \int_{\Sigma} \Bigl( |\nu_{\bar11}|^2-\frac{1}{2}\mathfrak{Ric}\,|\nu_{\bar1}|^2\Bigr) dA \le \frac{1}{2} \int_{\Sigma} \bigl( \Lambda_{1}^{2} - 2 \Lambda_{1}\mathfrak{Ric} \bigr) |\nu|^{2} dA,
\end{equation}
where the first inequality follows from
 $$\operatorname{Ric}(e_{1}, \bar e_{1})=\operatorname{Ric}(\mathrm{Re}(e_{1}), \mathrm{Re}(e_{1}))+\operatorname{Ric}(\mathrm{Im}(e_{1}), \mathrm{Im}(e_{1}))\geq \frac{1}{2}\mathfrak{Ric},$$
with $\mathrm{Re}(e_{1})$ and $\mathrm{Im}(e_{1})$ denoting the real and imaginary parts of $e_1$. The fact that any complex curve minimizes $\mathcal{W}^+$ implies the left-hand side of \eqref{eq-varW} is nonnegative. Together with $\Lambda_1>0$, this yields the lower bound
$$\Lambda_1\geq 2\,\mathfrak{Ric}.$$
\end{proof}
\section{The proof of Theorem 2}\label{sec5}
We denote by ${E}_0^{\mathcal{W}^+}$ (resp. $E_0^{\mathcal{A}}$) the space of Jacobi fields associated with $\mathcal{W}^+$ (resp. $\mathcal{A}$). It follows from \eqref{eq-Area} and \eqref{eq-Will} that
\begin{equation}\label{eq-Jaco}
 E_0^{\mathcal{W}^+}=\{V=\nu e_0+\bar\nu\bar e_0 \mid \nu_{\bar1\bar1}=0\},~~~~~~E_0^{\mathcal{A}}=\{V=\nu e_0+\bar\nu\bar e_0\mid \nu_{\bar1}=0\}.
\end{equation}
Obviously, there holds
$$E_0^{\mathcal{A}}\subset E_0^{\mathcal{W}^+}.$$

{\bf Claim 1.} {\em If $E_0^{\mathcal{W}^+}\setminus E_0^{\mathcal{A}}\neq\emptyset$, then $\Lambda_1=2\mathfrak{c}$ and $E_0^{\mathcal{W}^+}\setminus E_0^{\mathcal{A}}$ is exactly the first eigenspace of $\mathcal{L}$}.
\vskip 0.2cm
\noindent To prove this claim, first note that $\mathcal{L}$ is a nonnegative self-adjoint operator. Hence, for each eigenvalue $\Lambda_j$ of $\mathcal{L}$, we can choose a basis $\{\mathrm{V}_{j_1}, \mathrm{V}_{j_2}, \cdots \mathrm{V}_{j_{k_j}}\}$ of the corresponding eigenspace $E_j^{\mathcal{A}}$ such that the collection
$$\{\mathrm{V}_{j_l}\mid 1\leq l\leq {k_j}, j\in \mathbb{Z}^{\geq 0}\}$$
forms a complete orthonormal system of \( L^2(\Gamma(N_\Sigma)) \).
Let $V\in E_0^{\mathcal{W}^+}\setminus E_0^{\mathcal{A}}$ be a Jacobi field of $\mathcal{W}^+$. Write
\[V = \sum_{j=0}^{+\infty}\sum_{l=1}^{k_j} \varepsilon_{j_l} \mathrm{V}_{j_l}.\]
Then we have
\[\mathcal{L}(V) = -\sum_{j=1}^{+\infty}\sum_{l=1}^{k_j} \Lambda_j \varepsilon_{j_l} \mathrm{V}_{j_l}.\]
and then
\[\int_{\Sigma}|\mathcal{L}(V)|^2 dA = \sum_{j=1}^{+\infty}\sum_{l=1}^{k_j} \varepsilon_{j_l}^2\Lambda_j^2, \quad\quad \int_{\Sigma}\langle \mathcal{L}(V), V \rangle dA= -\sum_{j=1}^{+\infty}\sum_{l=1}^{k_j} \varepsilon_{j_l}^2 \Lambda_j.\]
Substituting these equalities into \eqref{eq-W+5}, we derive
\[\sum_{j=1}^{+\infty}\sum_{l=1}^{k_j} \varepsilon_{j_l}^2 \Lambda_j(\Lambda_j - 2\mathfrak{c}) = 0.\]
It follows from
$$\sum_{j=1}^{+\infty}\sum_{l=1}^{k_j}\varepsilon_{j_l}^2\neq0,~~\text{and}~~\Lambda_{j}>\Lambda_1\geq 2\mathfrak{c},~j\geq 2$$
that
$$\Lambda_1=2c,~~\text{and}~~\varepsilon_{j_l}=0,~j\geq 2,$$
which implies $V$ is an eigensection associated with the first eigenvalue $\Lambda_1=2\mathfrak{c}$. Therefore, we obtain
$$E_0^{\mathcal{W}^+}\setminus E_0^{\mathcal{A}}\subset E_1^{\mathcal{A}}.$$
When $\Lambda_1=2\mathfrak{c}$, the converse inclusion follows directly from \eqref{eq-W+5}. Thus Claim $1$ is proved.

From \eqref{eq-E0W} and Claim $2$, we derive
$$\dim E_0^{\mathcal{W}^+}=\dim \ker{\bar\partial_1}+\dim \ker{\bar\partial}= \dim H^0(\Sigma, N_\Sigma \otimes \overline{K_\Sigma})+\dim E_0^{\mathcal{A}}.$$
By Riemann-Roch Theorem, we have
$$\begin{aligned}\dim H^0(\Sigma, N_\Sigma \otimes \overline{K_\Sigma})=&\deg(N_\Sigma \otimes \overline{K_\Sigma})+1-g+\dim H^0(\Sigma, N_\Sigma^* \otimes K_\Sigma^2)\\
=&\,\mathfrak{c}\frac{\mathrm{Area}(\Sigma)}{2\pi}+1-g+\dim H^0(\Sigma, N_\Sigma^* \otimes K_\Sigma^2).
\end{aligned}$$
So when $g\leq 1$, there holds
$$\dim E_0^{\mathcal{W}^+}-\dim E_0^{\mathcal{A}}=\dim H^0(\Sigma, N_\Sigma \otimes \overline{K_\Sigma})\geq 1.$$
Together with Claim 1, this yields the second conclusion of this theorem.
\end{proof}
\begin{remark}
 It follows from the proof of Claim 1 that for complex curves in K\"ahler-Einstein surfaces of nonpositive scalar curvature, $${E}^{\mathcal{W}^+}_0={E}^{\mathcal{A}}_0,$$
 i.e., the Jacobi fields of $\mathcal{W}^+$ are exactly the same as those of $\mathcal{A}$.
\end{remark}

Somewhat surprisingly, the proof of this theorem relies on computing the second variation of the following  conformally invariant functional
$$\mathcal{W}^+(x) \triangleq \int_{\Sigma} (|\vec H|^2 + K(\mathfrak{e}_1, \mathfrak{e}_2, \mathfrak{e_1}, \mathfrak{e_2}) + K(\mathfrak{e}_1, \mathfrak{e}_2, \mathfrak{e_3}, \mathfrak{e_4}))dA,$$
introduced by Montiel and Urbano in \cite{Montiel-Urbano} for closed oriented surfaces in an oriented $4$-dimensional Riemannian manifold. Here $\vec H$ represents the mean curvature vector, $K$ denotes the ambient  curvature tensor, and $\{\mathfrak{e}_1, \mathfrak{e}_2, \mathfrak{e}_3, \mathfrak{e}_4\}$ is an orthonormal basis with $\{\mathfrak{e}_1, \mathfrak{e}_2\}$ tangent to the surface $x$ and $\{\mathfrak{e}_3, \mathfrak{e}_4\}$ spanning its normal space. Associated with $\mathcal{W}^+$, Montiel and Urbano \cite{Montiel-Urbano} also introduced
$$\mathcal{W}^-(x) \triangleq \int_{\Sigma} (|\vec H|^2 + K(\mathfrak{e}_1, \mathfrak{e}_2, \mathfrak{e_1}, \mathfrak{e_2}) - K(\mathfrak{e}_1, \mathfrak{e}_2, \mathfrak{e_3}, \mathfrak{e_4}))dA.$$
The average of these two functionals is precisely the Willmore functional,
$$\mathcal{W}(x) = \int_{\Sigma} (|\vec H|^2 + K_{1212}) dA,$$
an important global conformal invariant for surfaces in general curved Riemannian manifolds that has attracted considerable attention recently (see \cite{Marques1,Mondino,Mondino1,Wang-Xie} and the references therein). In \cite{Montiel-Urbano}, Montiel and Urbano studied $\mathcal{W}^+$ and $\mathcal{W}^-$ from the perspective of twistor theory and showed that a superminimal surface with positive (resp. negative) spin minimize $\mathcal{W}^+$ (resp. $\mathcal{W}^-$) among its homotopy class. In \cite{Wang-Xie}, we investigated these functionals from the viewpoint of conformal geometry and derived the following expressions,
\begin{equation}\label{eq-W+}
\mathcal{W}^+(x) = 4 \int_{\Sigma} |\psi|^2 dA + 2\pi (\chi + \chi^{\perp}),~~~\mathcal{W}^-(x) = 4 \int_{\Sigma} |\phi|^2 dA + 2\pi (\chi - \chi^{\perp}),
\end{equation}
where
$\psi$ and $\phi$ are two local conformal invariants arising from the normal-bundle-valued Hopf differential, and $\chi$ (resp. $\chi^{\perp}$) denotes the Euler characteristics of the tangent bundle (resp. normal bundle). When the ambient space is a Kähler surface, complex curves are superminimal surfaces with positive spin and hence are minimizers of $\mathcal{W}^+$.

If we restrict the ambient space to a Kähler–Einstein surface with positive scalar curvature, then the estimate established in Theorem \ref{thm-main11} is sharp. In this setting, the curvature invariant $\mathfrak{Ric}$ simplifies to the Einstein constant $\mathfrak{c}>0$. We explore the equality case and are able to compute exactly the dimension of the first eigenspace. This is accomplished by relating it to the space of Jacobi fields of $\mathcal{W}^+$, which corresponds precisely to the preimage of the space of holomorphic sections of a certain line bundle, and then applying the Riemann-Roch theorem in conjunction with the Dolbeault theorem.
\begin{thm}\label{thm-main2}
 Let \(M^{4}\) be a K\"ahler-Einstein surface with Einstein constant \(\mathfrak{c}>0\), and let \(x\colon\Sigma \rightarrow M^{4}\) be a complex curve. Then
 \begin{itemize}
    \item[{\rm(1)}] the first eigenvalue $\Lambda_1$ of its area Jacobi operator is at least $2\mathfrak{c}$,
    \item[{\rm(2)}] if $\Sigma$ has genus $g\leq 1$, then $\Lambda_1=2\mathfrak{c}$ and the corresponding eigenspace has dimension
    \[
    \mathfrak{c}\frac{\operatorname{Area}}{2\pi}+1-g+\dim H^{\,0}(N^*_\Sigma\otimes K_\Sigma^2),
    \]
    where $N^*_\Sigma$ is the dual of the normal bundle, $K_\Sigma$ is the canonical line bundle of $\Sigma$, and $H^{\,0}(N^*_\Sigma\otimes K_\Sigma^2)$ denotes  the space of holomorphic sections of $N^*_\Sigma\otimes K_\Sigma^2$.
\end{itemize}
\end{thm}

This paper is organized as follows. In Section~\ref{sec2}, we review the necessary background on surface theory in  Kähler surfaces. Section~\ref{sec3} is devoted to compute the variation of geometric quantities for a complex curve in a K\"ahler surface. Theorem~\ref{thm-main11} is established in Section~\ref{sec4}. Finally, in Section~\ref{sec5}, we focus on the Kähler-Einstein setting and prove Theorem \ref{thm-main2}.

The post-theorem introduction says that, “Somewhat surprisingly, the proof of this theorem relies on computing the second variation of the following conformally invariant functional”
\[
\mathcal{W}^+(x) \triangleq \int_{\Sigma} (|\vec H|^2 + K(\mathfrak{e}_1, \mathfrak{e}_2, \mathfrak{e_1}, \mathfrak{e_2}) + K(\mathfrak{e}_1, \mathfrak{e}_2, \mathfrak{e_3}, \mathfrak{e_4}))\,dA.
\]
It recalls that in a K"ahler surface “complex curves are superminimal surfaces with positive spin and hence are minimizers of $\mathcal{W}^+$.” It then points to where the argument is carried out: “Theorem~\ref{thm-main11} is established in Section~\ref{sec4},” after “Section~\ref{sec3} is devoted to compute the variation of geometric quantities for a complex curve in a K"ahler surface.”

The post-theorem introduction says that, “Somewhat surprisingly, the proof of this theorem relies on computing the second variation of the following conformally invariant functional”
\[
\mathcal{W}^+(x) \triangleq \int_{\Sigma} (|\vec H|^2 + K(\mathfrak{e}_1, \mathfrak{e}_2, \mathfrak{e_1}, \mathfrak{e_2}) + K(\mathfrak{e}_1, \mathfrak{e}_2, \mathfrak{e_3}, \mathfrak{e_4}))\,dA.
\]
It recalls that in a K"ahler surface “complex curves are superminimal surfaces with positive spin and hence are minimizers of $\mathcal{W}^+$.” It then points to where the argument is carried out: to prove the main theorem, the paper first devotes the next part to computing the variation of geometric quantities for a complex curve in a K"ahler surface, and later carries out the proof.

Question:
Let \(M^4\) be a K\"ahler surface, and let \(x\colon \Sigma \to M^4\) be a complex curve in \(M^4\). Let \(\mathcal L\) denote the area Jacobi operator of \(\Sigma\) (the self-adjoint elliptic operator governing the second variation of area), and let \(\Lambda_1\) be its first eigenvalue. Define
\[
\mathfrak{Ric}:=\inf\{\operatorname{Ric}(u,u): p\in M^4,\ u\in T_pM^4,\ |u|=1\},
\]
the infimum of the ambient Ricci curvature over all unit tangent vectors of \(M^4\). Which statement holds for every such complex curve?

The first eigenvalue of the area Jacobi operator satisfies \(\Lambda_1 \ge 2\,\mathfrak{Ric}\).

Choices:
B. The first eigenvalue of the area Jacobi operator satisfies \(\Lambda_1 \ge \mathfrak{Ric}\).
C. The first eigenvalue of the area Jacobi operator satisfies \(\Lambda_1 \ge 0\).
D. The first eigenvalue of the area Jacobi operator satisfies \(\Lambda_1 \ge 2\,\operatorname{Ric}(u,u)\) for every unit tangent vector \(u\in T_pM^4\) at every point \(p\in M^4\).
E. If \(\mathfrak{Ric}>0\), then the first eigenvalue of the area Jacobi operator satisfies \(\Lambda_1 > 2\,\mathfrak{Ric}\).

sharp factor 2 from second-variation identity

ambient-curvature-dependent lower bound dropped

global infimum over Ricci replaced by pointwise uniform lower bound

non-strict lower bound strengthened to strict inequality

The stem does not explicitly reveal the correct estimate or its sharp constant; it only sets up notation and asks which bound holds.

This is essentially a direct recall of a specific theorem/result: the task is to identify the stated estimate itself rather than infer a new conclusion from premises.

There is some reasoning pressure because the options vary by sharp constant, quantifier strength, and weaker-vs-stronger formulations, but success still depends mainly on knowing the exact theorem statement.

The distractors are plausible and mathematically meaningful: they test confusion about the coefficient, a merely weaker true statement, an overly strong pointwise quantifier, and an unjustified stronger bound.

A mathematically well-constructed recall-style MCQ with strong distractors, but it is largely theorem restatement rather than a genuinely generative reasoning question.


\begin{thm}\label{thm:ubiquity:antiraysRays}
    Let $D$ be a digraph.
    If there is a sequence $(\cX_n)_{n\in\N}$ such that, for every $n\in\N$, the set $\cX_n$ consists of $n$ rays and $n$ anti-rays of~$D$ that are all pairwise disjoint and such that the sequence $(\cX_n)_{n\in\N}$ does not concentrate in any end of~$D$, then $D$ contains a set of infinitely many rays and infinitely many anti-rays every two of which are disjoint.
\end{thm}

\label{sec:intro}

The property of a graph $H$ to be \emph{ubiquitous} describes the concept that the existence of $n$ pairwise disjoint copies for all $n\in\ensuremath{\mathbb N}$ of a graph $H$ in a graph $G$ implies that $G$ already contains $\aleph_0$ many pairwise disjoint copies of~$H$.
Originally, this was considered with respect to the subgraph relation and Halin showed that the ray is ubiquitous~\cite{Halin65}*{Satz 1} and that the double ray is ubiquitous~\cite{Halin70}*{Satz 3}.
Later, counterexamples \cites{A77, A02} regarding being ubiquitous were found, which even include trees.
Hence, ubiquity was also considered with respect to other relations such as the minor and topological minor relation, and we refer to \cites{A79,A80,BEEGHPTI,BEEGHPTII,BEEGHPTIII} for ubiquity results regarding these notions.
Many of these results leverage that finite graphs (resp.~trees) are well-quasi ordered under the minor (resp.~topological minor) relation.

It should be noted that Andreae~\cite{A02} posed the major and still open conjecture whether all locally finite connected graphs are ubiquitous under the minor relation. In the same paper he also provided an example of an uncountable non-ubiquitous graph under the minor relation, whose construction exploits that uncountable graphs are not well-quasi ordered under minors.

Further modified questions regarding ubiquity have been raised, e.g.~where disjointness is replaced by edge-disjointness.
In this setting it turned out to be complicated to even verify that double rays are ubiquitous regarding the subgraph relation~\cite{BCP15}.

For digraphs, much less is known, even for the subdigraph relation, and for the rest of this article, we shall only consider the property of being ubiquitous under that relation.
While Zuther~\cite{Z1997}*{Theorem 2.17} proved that the consistently oriented ray is ubiquitous, Gut et al.~\cite{GKR2024}*{Theorem 1.1} proved the same for all possible orientations of the undirected ray.
Later, Gut et al.~\cite{GKR2025}*{Theorem 1.3} classified almost all orientations of the undirected double ray that are ubiquitous.
The only orientation they left out was the consistently oriented one.
Due to that, they posed the following problem.

\begin{problem}\cite{GKR2024}*{Problem 1.3}\label{prob:ubiquity:doubleRays}
    Is the consistently oriented double ray ubiquitous?
\end{problem}

Our main result (Theorem~\ref{thm:main:intro:ReductionToOneEnd:doubleRays}) states that it suffices to verify Problem~\ref{prob:ubiquity:doubleRays} for one-ended digraphs.
(We refer to Section~\ref{sec:prelims} for the definition of ends of digraphs.)

\label{sec:intro}

The property of a graph $H$ to be \emph{ubiquitous} describes the concept that the existence of $n$ pairwise disjoint copies for all $n\in\ensuremath{\mathbb N}$ of a graph $H$ in a graph $G$ implies that $G$ already contains $\aleph_0$ many pairwise disjoint copies of~$H$.
Originally, this was considered with respect to the subgraph relation and Halin showed that the ray is ubiquitous~\cite{Halin65}*{Satz 1} and that the double ray is ubiquitous~\cite{Halin70}*{Satz 3}.
Later, counterexamples \cites{A77, A02} regarding being ubiquitous were found, which even include trees.
Hence, ubiquity was also considered with respect to other relations such as the minor and topological minor relation, and we refer to \cites{A79,A80,BEEGHPTI,BEEGHPTII,BEEGHPTIII} for ubiquity results regarding these notions.
Many of these results leverage that finite graphs (resp.~trees) are well-quasi ordered under the minor (resp.~topological minor) relation.

It should be noted that Andreae~\cite{A02} posed the major and still open conjecture whether all locally finite connected graphs are ubiquitous under the minor relation. In the same paper he also provided an example of an uncountable non-ubiquitous graph under the minor relation, whose construction exploits that uncountable graphs are not well-quasi ordered under minors.

For digraphs, much less is known, even for the subdigraph relation, and for the rest of this article, we shall only consider the property of being ubiquitous under that relation.
While Zuther~\cite{Z1997}*{Theorem 2.17} proved that the consistently oriented ray is ubiquitous, Gut et al.~\cite{GKR2024}*{Theorem 1.1} proved the same for all possible orientations of the undirected ray.
Later, Gut et al.~\cite{GKR2025}*{Theorem 1.3} classified almost all orientations of the undirected double ray that are ubiquitous.
The only orientation they left out was the consistently oriented one.
Due to that, they posed the following problem.

\begin{problem}\cite{GKR2024}*{Problem 1.3}\label{prob:ubiquity:doubleRays}
 Is the consistently oriented double ray ubiquitous?
\end{problem}

Our main result (Theorem~\ref{thm:main:intro:ReductionToOneEnd:doubleRays}) states that it suffices to verify Problem~\ref{prob:ubiquity:doubleRays} for one-ended digraphs.
(We refer to Section~\ref{sec:prelims} for the definition of ends of digraphs.)

\begin{problem}\cite{GKR2024}*{Problem 1.3}\label{prob:ubiquity:doubleRays}
    Is the consistently oriented double ray ubiquitous?
\end{problem}

\begin{theorem}\label{thm:main:intro:ReductionToOneEnd:doubleRays}
    The consistently oriented double ray is ubiquitous if and only if it is ubiquitous restricted to the class of one-ended digraphs.
\end{theorem}

\begin{abstract}
We prove that the consistently oriented double ray is ubiquitous if and only if it is ubiquitous restricted to the class of one-ended digraphs.
Additionally, we prove the same equivalence for the disjoint union of a consistently forward and a consistently backward oriented ray.
Furthermore, we discuss the connection between these two ubiquity problems.
\end{abstract}

For digraphs, much less is known, even for the subdigraph relation, and for the rest of this article, we shall only consider the property of being ubiquitous under that relation.
While Zuther~\cite{Z1997}*{Theorem 2.17} proved that the consistently oriented ray is ubiquitous, Gut et al.~\cite{GKR2024}*{Theorem 1.1} proved the same for all possible orientations of the undirected ray.
Later, Gut et al.~\cite{GKR2025}*{Theorem 1.3} classified almost all orientations of the undirected double ray that are ubiquitous.
The only orientation they left out was the consistently oriented one.
Due to that, they posed the following problem.

\begin{problem}\cite{GKR2024}*{Problem 1.3}\label{prob:ubiquity:doubleRays}
 Is the consistently oriented double ray ubiquitous?
\end{problem}

Our main result (Theorem~\ref{thm:main:intro:ReductionToOneEnd:doubleRays}) states that it suffices to verify Problem~\ref{prob:ubiquity:doubleRays} for one-ended digraphs.
(We refer to Section~\ref{sec:prelims} for the definition of ends of digraphs.)

As a major step in the proof of Theorem~\ref{thm:main:intro:ReductionToOneEnd:doubleRays}, we verify Problem~\ref{prob:ubiquity:doubleRays} for a large, but technical, class of digraphs, see Theorem~\ref{thm:ubiquity:doubleRays}.

\begin{problem}\label{prob:ubiquity:antiraysRays}
 Is the disjoint union of a consistently forward and a consistently backward oriented ray ubiquitous?
\end{problem}

\begin{theorem}\label{thm:main:intro:ReductionToOneEnd:RaysAntiRays}
 The disjoint union of a consistently forward and a consistently backward oriented ray is ubiquitous if and only if it is ubiquitous restricted to the class of one-ended digraphs.
\end{theorem}

\begin{proof}[Proof of Theorem~\ref{thm:main:intro:ReductionToOneEnd:doubleRays}.]
 Let us assume that Problem~\ref{prob:ubiquity:doubleRays} holds for the class of one-ended digraphs and let $D$ be a digraphs and $(\cR_n)_{n\in\N}$ be a valid sequence in~~$D$.
 By Theorem~\ref{thm:ubiquity:doubleRays}, we may assume that $(\cR_n)_{n\in\N}$ concentrates in finitely many ends.
 Then there is a valid sequence $(\cQ_n)_{n\in\N}$ induced by $(\cR_n)_{n\in\N}$ such that all tails from all elements of $\bigcup_{n\in\N}\cQ_n$ are equivalent.
 By Lemma~\ref{lem:reduction:doubleRays:oneEnd}, there exists a one-ended subdigraph $D'$ of~$D$ and a valid sequence $(\cP_n)_{n\in\N}$ in~$D'$ induced by $(\cQ_n)_{n\in\N}$.
 Thus, there are infinitely many pairwise disjoint double rays in~$D'$ and hence in~$D$.
\end{proof}

\begin{problem}\cite{GKR2024}*{Problem 1.3}\label{prob:ubiquity:doubleRays}
    Is the consistently oriented double ray ubiquitous?
\end{problem}

\begin{theorem}\label{thm:main:intro:ReductionToOneEnd:doubleRays}
    The consistently oriented double ray is ubiquitous if and only if it is ubiquitous restricted to the class of one-ended digraphs.
\end{theorem}

\begin{thm}\label{thm:ubiquity:doubleRays}
    Let $D$ be a digraph.
    If there is a valid sequence $(\cR_n)_{n\in\N}$ that does not concentrate in finitely many ends, then $D$ contains infinitely many pairwise disjoint double rays.
\end{thm}

As a major step in the proof of Theorem~\ref{thm:main:intro:ReductionToOneEnd:doubleRays}, we verify Problem~\ref{prob:ubiquity:doubleRays} for a large, but technical, class of digraphs, see Theorem~\ref{thm:ubiquity:doubleRays}.

Additionally, we consider the following more general version of \cite{HH2024+}*{Problem 4.2}.

\begin{problem}\label{prob:ubiquity:antiraysRays}
    Is the disjoint union of a consistently forward and a consistently backward oriented ray ubiquitous?
\end{problem}

We also verify Problem~\ref{prob:ubiquity:antiraysRays} for a large, but technical, class of digraphs (Theorem~\ref{thm:ubiquity:antiraysRays}) and show that Problem~\ref{prob:ubiquity:antiraysRays} holds if it holds for all one-ended digraphs.

\begin{theorem}\label{thm:main:intro:ReductionToOneEnd:RaysAntiRays}
    The disjoint union of a consistently forward and a consistently backward oriented ray is ubiquitous if and only if it is ubiquitous restricted to the class of one-ended digraphs.
\end{theorem}

\medskip

This paper is structured as follows.
After introducing some terminology in Section~\ref{sec:prelims}, we prove Theorem~\ref{thm:main:intro:ReductionToOneEnd:RaysAntiRays} in Section~\ref{sec:ubiquity:antiraysRays}.
In Section~\ref{sec:ubiquity:DoubleRays}, we prove Theorem~\ref{thm:main:intro:ReductionToOneEnd:doubleRays}.
In Section~\ref{sec:equivalence}, we discuss connections between Problems~\ref{prob:ubiquity:doubleRays} and \ref{prob:ubiquity:antiraysRays}.

As a major step in the proof of Theorem~\ref{thm:main:intro:ReductionToOneEnd:doubleRays}, the authors "verify Problem~\ref{prob:ubiquity:doubleRays} for a large, but technical, class of digraphs" (see Theorem~\ref{thm:ubiquity:doubleRays}).

Question: A digraph $H$ is called ubiquitous under the subdigraph relation if, whenever a digraph $G$ contains $n$ pairwise disjoint copies of $H$ for every $n\in\mathbb{N}$, the digraph $G$ already contains countably infinitely many pairwise disjoint copies of $H$. Let $C$ be the consistently oriented double ray, that is, the two-way infinite directed path $\cdots \to v_{-1} \to v_0 \to v_1 \to \cdots$. Which of the following statements is equivalent to saying that $C$ is ubiquitous under the subdigraph relation?

The consistently oriented double ray $C$ is ubiquitous when restricted to one-ended digraphs; that is, for every one-ended digraph $G$ (a digraph with exactly one end), if $G$ contains $n$ pairwise disjoint copies of $C$ for every $n\in\mathbb{N}$, then $G$ contains countably infinitely many pairwise disjoint copies of $C$.

Choices:
B. The consistently oriented double ray $C$ is ubiquitous when restricted to digraphs with at most one end; that is, for every digraph $G$ with at most one end, if $G$ contains $n$ pairwise disjoint copies of $C$ for every $n\in\mathbb{N}$, then $G$ contains countably infinitely many pairwise disjoint copies of $C$.
C. For every one-ended digraph $G$, if $G$ contains countably infinitely many pairwise disjoint copies of $C$, then in particular $G$ contains $n$ pairwise disjoint copies of $C$ for every $n\in\mathbb{N}$.
D. There exists a one-ended digraph $G$ such that, whenever $G$ contains $n$ pairwise disjoint copies of $C$ for every $n\in\mathbb{N}$, every digraph containing $G$ as a subdigraph contains countably infinitely many pairwise disjoint copies of $C$.
E. The consistently oriented double ray $C$ is ubiquitous when restricted to finitely-ended digraphs; that is, for every digraph $G$ with finitely many ends, if $G$ contains $n$ pairwise disjoint copies of $C$ for every $n\in\mathbb{N}$, then $G$ contains countably infinitely many pairwise disjoint copies of $C$.

The stem defines ubiquity and the digraph C but does not explicitly or implicitly point to the correct restriction class ('one-ended digraphs'). The correct answer is not leaked by wording in the prompt.

The item is not a direct restatement of the definition of ubiquity. It asks for an equivalent reformulation involving a nontrivial restriction to a specific class of digraphs, so the student must distinguish among genuinely different statements.

The question requires some reasoning or theorem recognition to separate exact equivalence from nearby variants (at most one end, finitely many ends, converse implication, existential quantifier). However, it mainly tests precise recall/recognition rather than deep generative derivation.

The distractors are plausible and mathematically meaningful: they vary boundary conditions, weaken/alter quantifiers, or reverse implication. These reflect common failure modes in reading equivalence statements and theorem hypotheses.

A strong MCQ: it avoids answer leakage, is not tautological, and uses high-quality distractors. Its main limitation is that it leans more toward theorem recognition than substantial generative reasoning.