\label{examples}
In this section, we provide a concrete example to show how specific  high  rank  elliptic curve of the form
$y^2=x(x^2+B)$  with a known set of rational points correspond to a single

\begin{thm} \label{main_theorem}
		Let $k$ be a number field containing a primitive $s$-th root of unity. Assume the strong version of Lang's conjecture (Conjecture \ref{conj1}) holds for varieties defined over $k$. Then the rank $r(J_C(k))$ is uniformly bounded as $J_C$ varies over the Jacobian varieties of all smooth curves $C$ defined over $k$ by equation \eqref{curve_main} with genus $g_{r,s}(C) \geq 1$.
	\end{thm}

A central theme in arithmetic geometry concerns the structure of the group of rational points on algebraic varieties defined over number fields. For an abelian variety $A$ over a number field $k$, the celebrated Mordell-Weil theorem asserts that the group $A(k)$ is finitely generated \cite{Silverman1986}. Thus, $A(k) \cong A(k)_{\text{tors}} \oplus \mathbb{Z}^r$, where $A(k)_{\text{tors}}$ is the finite torsion subgroup and $r = \rk A(k)$ is the Mordell-Weil rank. Understanding the behavior of this rank, particularly for Jacobians $J_C$ of algebraic curves $C$, remains a profound challenge. A fundamental open question asks whether $r(J_C(k))$ is uniformly bounded as $C$ varies over all curves of a fixed genus $g \geq 1$ defined over $k$.

Even for elliptic curves ($g=1$) over $k=\mathbb{Q}$, this question is unresolved. While computational evidence reveals curves of high rank, theoretical heuristics suggest boundedness, possibly with rank exceeding 21 being rare \cite{Park2019}. Deep conjectures in Diophantine geometry provide conditional evidence for rank boundedness. Notably, Lang's conjectures on rational points on varieties of general type \cite{Lang1986, Lang1991} imply, via the work of Caporaso, Harris, and Mazur \cite{Caporaso1997, caporaso2022update}, strong uniformity statements about the number of rational points on curves. Furthermore, Pasten demonstrated that conjectures on Diophantine approximation imply Honda's conjecture on rank boundedness for elliptic curves \cite{Pasten2019}. The link between point counts and rank was solidified by Dimitrov, Gao, and Habegger \cite{Dimitrov2021}, making uniform point boundedness equivalent to uniform rank boundedness.

An alternative approach, pioneered by Yamagishi \cite{Yamagishi2003} for the family $y^2=ax^4+bx^2+c$, employs geometric methods. By constructing parameter spaces for curves with multiple rational points and demonstrating that these spaces (or relevant subvarieties) become varieties of general type, one can invoke Lang's conjectures to deduce finiteness and uniformity properties, ultimately leading to rank bounds. This geometric strategy was adapted by the author in \cite{Salami2025toappear} to study the uniformity of Mordell-Weil rank of Jacobian variety of the family of curves
 $ y^s = ax^r+b$.

This paper applies the same geometric methodology to investigate the closely related two-parameter family of curves defined by the affine equation
 \begin{equation} \label{curve_main}
 C_{a,b}: y^s=x(ax^r+b),
 \end{equation}
 where $r \geq 1, s \geq 2$ are fixed integers, and $k$ is a number field containing $\zeta_s$, a primitive $s$-th root of unity. We focus on smooth curves $C_{a,b}$ with genus $g_{r,s}(C_{a,b}) \geq 1$. Our main result, conditional on Lang's conjecture, establishes uniform rank boundedness for this family.

\begin{conj}[Lang, \cite{Lang1986, Lang1991}]
		\label{conj1}
		Let $k$ be a number field and $X$ a smooth variety of general type over $k$.
		\begin{enumerate}
			\item[(a)] \textbf{(Weak)} The set of $k$-rational points $X(k)$ is not Zariski dense in $X$.
			\item[(b)] \textbf{(Strong)} There exists a proper Zariski-closed subset $Z \subset X$ such that for any finite extension $K$ of $k$, the set of $K$-rational points on $X \setminus Z$ is finite.
		\end{enumerate}
	\end{conj}

\begin{equation} \label{curve_main}
		C_{a,b}: y^s=x(ax^r+b),
	\end{equation}

\begin{abstract}
 Let $k$ be a number field. We investigate the Mordell-Weil ranks of Jacobian varieties $J_C$ associated with algebraic curves $C$ of genus $g \geq 1$ defined by affine equations of the form $y^s=x(ax^r+b)$, where $a, b \in k$ ($ab \neq 0$), and $r \geq 1, s \geq 2$ are fixed integers. Assuming the strong version of Lang's conjecture concerning rational points on varieties of general type, we establish that the ranks $r(J_C(k))$ are uniformly bounded as $C$ varies within this family.
 Our methodology builds upon the geometric approach employed by H. Yamagishi and subsequently adapted by the author for the family $y^s=ax^r+b$. We construct a parameter space $\Wc_n$ for curves possessing $n+1$ specified rational points and analyze its birational model $\Xc_n$, a complete intersection variety. The geometric properties of the fibers of $\Xc_n \to \Sym^{n+1}(\Pp^1)$, specifically their genus and gonality, are studied. Combining these geometric insights with Faltings' theorem, uniformity conjectures stemming from Lang's work, and recent results connecting rank with the number of rational points, we deduce the main boundedness result. In the case of genus one curves $C$, it
 states that the rank of elliptic curves $y^2=x (x^2+B)$
 is uniformly bounded subject to the strong version of Lang's conjecture.
\end{abstract}

A central theme in arithmetic geometry concerns the structure of the group of rational points on algebraic varieties defined over number fields. For an abelian variety $A$ over a number field $k$, the celebrated Mordell-Weil theorem asserts that the group $A(k)$ is finitely generated \cite{Silverman1986}. Thus, $A(k) \cong A(k)_{\text{tors}} \oplus \mathbb{Z}^r$, where $A(k)_{\text{tors}}$ is the finite torsion subgroup and $r = \rk A(k)$ is the Mordell-Weil rank. Understanding the behavior of this rank, particularly for Jacobians $J_C$ of algebraic curves $C$, remains a profound challenge. A fundamental open question asks whether $r(J_C(k))$ is uniformly bounded as $C$ varies over all curves of a fixed genus $g \geq 1$ defined over $k$.

This paper applies the same geometric methodology to investigate the closely related two-parameter family of curves defined by the affine equation
 \begin{equation} \label{curve_main}
 C_{a,b}: y^s=x(ax^r+b),
 \end{equation}
 where $r \geq 1, s \geq 2$ are fixed integers, and $k$ is a number field containing $\zeta_s$, a primitive $s$-th root of unity. We focus on smooth curves $C_{a,b}$ with genus $g_{r,s}(C_{a,b}) \geq 1$. Our main result, conditional on Lang's conjecture, establishes uniform rank boundedness for this family.

The main result and its consequences provide new geometric evidence for rank boundedness conjectures for elliptic curves similar to that given in author's previous work \cite{Salami2025toappear}
different from that of given in \cite{Pasten2019}.
To be explicit, let us consider
the case $g_{r,s}(C) = 1$ and $k=\Q$, where the curve $C $ is an elliptic curves of the form $y^2= x(x^2+ b)$ with $\Z$.
By Theorem\ref{main_theorem}, one can conclude that the rank of elliptic curves
 is uniformly bounded subject to the strong version of Lang's conjecture, see Section \ref{sec:applications}.
 We note that the record of rank in this family is 14 due to M. Watkins obtained in 2002, according to \cite{DujellaRankRecord}.
 As a particular case of the above family, is the congruent number elliptic curves. There are only 27 congruent number $N$ for which the rank of corresponding elliptic curves $y^2=x (x^2-N^2)$ is equal to 7. It is also suspected that there does not exist any congruent number elliptic curve of rank 8, see \cite{Watkins2014}.
Theorem\ref{main_theorem} implies the uniformity of rank in family of congruent number elliptic curves, subject to the strong version of Lang's conjecture, see Section \ref{sec:applications}.

\begin{thm} \label{uniformity_theorem}
 Assume the weak version of Lang's conjecture (Conjecture \ref{conj1}) holds over $k$. Let $r \geq 1, s \geq 2$ such that $g_{r,s}(C) \geq 1$, and let $n_0$ be as in Theorem \ref{finiteness_theorem}. Then for $n \geq n_0$, there exists a uniform bound $M_0 = M_0(k, r, s, n)$ such that $\#\Cc_{\ba_n}(k) < M_0$ for all valid choices of $\ba_n$. Furthermore, if $g_{r,s}(C) \geq 2$, there exists an integer $m_0 > n_0$ such that $\Cc_{\ba_m}(k) = \emptyset$ for all $m \geq m_0$ and all valid $\ba_m$. (The strong Lang conjecture implies $M_0$ and $m_0$ can be chosen independently of $k$.)
 \end{thm}

\begin{proof}[Proof of Theorem \ref{main_theorem}]
 Let $k$ be a number field, $[k:\Q]=d$. Let $\mathcal{F}$ be the family of smooth curves $C: y^s=x(ax^r+b)$ over $k$ with genus $g=g_{r,s}(C) \geq 1$. Assume, for contradiction, that the rank $r(J_C(k))$ is unbounded for $C \in \mathcal{F}$.
 Then there exists a sequence of non-isomorphic curves $C_j \in \mathcal{F}$ such that $r_j = r(J_{C_j}(k)) \to \infty$ as $j \to \infty$.
 By the result of Dimitrov, Gao, and Habegger (Theorem \ref{dgh}), there is a constant $c=c(g,d)$ such that $\#C_j(k) \leq c^{1+r_j}$. Since $r_j \to \infty$, it follows that $\#C_j(k) \to \infty$.

Now, assume the Strong Lang Conjecture (Conjecture \ref{conj1}). This implies the strong version of Theorem \ref{uniformity_theorem}: there exists an integer $m_0$ (depending only on $r, s$, possibly $g$, but not $k$) such that for any $m \geq m_0$, $\Cc_{\ba_m}(K) = \emptyset$ for any number field $K$ and any valid $\ba_m \in \Uc_{r,m}(K)$. This means no smooth curve in the family \eqref{curve_main} over any number field $K$ can possess $m+1$ points $(\alpha_i, \beta_i)$ where the $\alpha_i$ are distinct, non-zero, and satisfy $\alpha_i^r \neq \alpha_j^r$, provided $m \ge m_0$.

\begin{thm}
 \label{CN-rank1}
 Assume the strong version of Lang's conjecture (Conjecture \ref{conj1}) holds. Then the Mordell-Weil ranks
$r(E_B(\Q))$ of elliptic curves $E_B: y^2=x (x^2+B)$ are uniformly bounded as $B$ varies over all integers.
\end{thm}

The main  result and its consequences provide new geometric evidence for rank boundedness  conjectures  for elliptic curves  similar to  that given in author's previous work \cite{Salami2025toappear}
different from that of given in \cite{Pasten2019}.
To be explicit, let us consider
the case $g_{r,s}(C) = 1$ and $k=\mathbb{Q}$, where the curve $C $ is an elliptic curves of the form  $y^2= x(x^2+ b)$ with  $\mathbb{Z}$.
By  Theorem\ref{main_theorem}, one can conclude that the  rank of elliptic curves
 is  uniformly bounded subject to the strong version of Lang's conjecture,   see Section \ref{sec:applications}.
    We note that  the record of rank in this family is 14 due to  M. Watkins obtained in 2002, according to \cite{DujellaRankRecord}.
 As a particular case of the above family, is the congruent number elliptic curves. There are only 27 congruent number $N$ for which the rank of corresponding elliptic curves $y^2=x (x^2-N^2)$ is equal to 7. It is also suspected that there does not exist any congruent number elliptic  curve of rank 8, see   \cite{Watkins2014}.
Theorem\ref{main_theorem}  implies the uniformity of rank  in family of congruent number elliptic  curves, subject to the strong version of Lang's conjecture,   see Section \ref{sec:applications}.

	The proof of Theorem\ref{main_theorem} relies crucially on studying the set $\Cc_{\ba_n}(k)$ consisting of smooth curves $C$ in the family \eqref{curve_main} that pass through a given set of $n+1$ points with $x$-coordinates $\alpha_i \in k$ ($i=0,\dots,n$) where $\ba_n$ represents the collection $\{\alpha_0, \dots, \alpha_n\}$ of distinct, non-zero elements of $k$ satisfying $\alpha_i^r \neq \alpha_j^r$ for $i \neq j$. We establish the following finiteness and uniformity results for these sets.

	\begin{thm} \label{finiteness_theorem}
		Let $k$ be a number field containing $\zeta_s$. Let $n_0=4$ if $s=2$ and $n_0=3$ otherwise. For any choice of $\ba_n = \{\alpha_0, \ldots, \alpha_n\}$ consisting of $n+1$ distinct, non-zero elements in $k$ with $\alpha_i^r \neq \alpha_j^r$ for $i \neq j$, and for integers $r \geq 1, s \geq 2$ such that $g_{r,s}(C) \geq 1$, the set $\Cc_{\ba_n}(k)$ is infinite if $n < n_0$ and finite if $n \geq n_0$. Consequently, for a fixed $\ba_n$ with $n \geq n_0$, the rank $r(J_C(k))$ is bounded for $C \in \Cc_{\ba_n}(k)$.
	\end{thm}

We use the uniformity results given this non-conditional result to discuss  a solution for the problem of arithmetic progression  on the points on the elliptic curves $y^2=x (x^2+B)$ as well as the congruent number elliptic curves, in Section~\ref{sec:applications}.

	Using the uniformity theorems on the number of $k$-rational points on curve of genus $\geq 2$ gibem in \cite{Caporaso1997}, as
	consequences of weak version of Lang's conjecture, we showed the cardinal number of $\Cc_{\ba_n}(k)$ does not depends on the choice of ${\ba_n}$ and  is   uniformly bounded and eventually empty for large  n.

	\begin{thm} \label{uniformity_theorem}
		Assume the weak version of Lang's conjecture (Conjecture \ref{conj1}) holds over $k$. Let $r \geq 1, s \geq 2$ such that $g_{r,s}(C) \geq 1$, and let $n_0$ be as in Theorem \ref{finiteness_theorem}. Then for $n \geq n_0$, there exists a uniform bound $M_0 = M_0(k, r, s, n)$ such that $\#\Cc_{\ba_n}(k) < M_0$ for all valid choices of $\ba_n$. Furthermore, if $g_{r,s}(C) \geq 2$, there exists an integer $m_0 > n_0$ such that $\Cc_{\ba_m}(k) = \emptyset$ for all $m \geq m_0$ and all valid $\ba_m$. (The strong Lang conjecture implies $M_0$ and $m_0$ can be chosen independently of $k$.)
	\end{thm}

	The structure of the paper mirrors the logical progression of the proof. Section \ref{sec_geom_constr} details the construction of the parameter space $\Wc_n$ using twists and its birational model $\Xc_n$, a complete intersection variety, establishing their relationship. Section \ref{sec_fiber_analysis} focuses on the geometry of the fibers $\Xc_{\ba_n}$ of $\Xc_n \to \Sym^{n+1}(\mathbb{P}^1)$, calculating their genus and gonality, analyzing rational points in low-genus cases, and invoking the theory of towers of curves for the general case, integrating necessary background results. Section \ref{sec_finiteness_uniformity_proofs} provides proofs for Theorems \ref{finiteness_theorem} and \ref{uniformity_theorem}, incorporating Faltings' theorem and Lang's conjectures. In Section \ref{sec_main_theorem_proof}, we prove  the main result, Theorem \ref{main_theorem}, using the uniformity theorem and the work of Dimitrov, Gao, and Habegger.
	 Throughout, we cite analogous arguments from \cite{Salami2025toappear} where applicable.
	In Section~\ref{sec:applications}, we  have provided consequences of our results for   elliptic curves of the form $y^2=x(x^2+B)$ and  in particular the congruent number elliptic curves.
	 Finally, in   Section \ref{examples}, we have provided a computational example of the correspondence between
	the set of curves  $\Cc_{\ba_n}$ and rational points on the fiber $\Xc_{\ba_n}$.

The post-theorem introduction gives the following proof outline for Theorem\ref{main_theorem}.

- The proof \emph{“relies crucially on studying the set $\Cc_{\ba_n}(k)$ consisting of smooth curves $C$ in the family \eqref{curve_main} that pass through a given set of $n+1$ points with $x$-coordinates $\alpha_i\in k$”}, where $\ba_n=\{\alpha_0,\dots,\alpha_n\}$ are distinct nonzero elements with $\alpha_i^r\neq \alpha_j^r$ for $i\neq j$.

- One first \emph{“establish[es]… finiteness and uniformity results for these sets”} via Theorems \ref{finiteness_theorem} and \ref{uniformity_theorem} (finiteness for $n\ge n_0$; uniform bounds $\#\Cc_{\ba_n}(k)<M_0$ for all valid $\ba_n$, and eventual emptiness for large $n$ when $g_{r,s}(C)\ge 2$). The proofs of these results are said to be in Section~\ref{sec_finiteness_uniformity_proofs}, \emph{“incorporating Faltings' theorem and Lang's conjectures.”}

- Geometric input is developed beforehand: Section~\ref{sec_geom_constr} \emph{“details the construction of the parameter space $\Wc_n$ using twists and its birational model $\Xc_n$, a complete intersection variety”}; Section~\ref{sec_fiber_analysis} \emph{“focuses on the geometry of the fibers $\Xc_{\ba_n}$…, calculating their genus and gonality, analyzing rational points in low-genus cases, and invoking the theory of towers of curves for the general case.”}

- Finally, \emph{“In Section \ref{sec_main_theorem_proof}, we prove the main result, Theorem \ref{main_theorem}, using the uniformity theorem and the work of Dimitrov, Gao, and Habegger.”}

\label{main_theorem}
		Let $k$ be a number field containing a primitive $s$-th root of unity. Assume the strong version of Lang's conjecture holds for varieties defined over $k$, namely:
		\begin{conj}[Lang, \cite{Lang1986, Lang1991}]
		\label{conj1}
		Let $k$ be a number field and $X$ a smooth variety of general type over $k$.
		\begin{enumerate}
			\item[(a)] \textbf{(Weak)} The set of $k$-rational points $X(k)$ is not Zariski dense in $X$.
			\item[(b)] \textbf{(Strong)} There exists a proper Zariski-closed subset $Z \subset X$ such that for any finite extension $K$ of $k$, the set of $K$-rational points on $X \setminus Z$ is finite.
		\end{enumerate}
	\end{conj}
		Then the rank $r(J_C(k))$ is uniformly bounded as $J_C$ varies over the Jacobian varieties of all smooth curves $C$ defined over $k$ by the equation
		\begin{equation} \label{curve_main}
		C_{a,b}: y^s=x(ax^r+b),
	\end{equation}
		with genus $g_{r,s}(C) \geq 1%.

Question:
Let $k$ be a number field containing a primitive $s$-th root of unity, and fix integers $r\ge 1$ and $s\ge 2$. Assume the strong version of Lang's conjecture over $k$: for every smooth variety $X$ of general type over $k$, there exists a proper Zariski-closed subset $Z\subset X$ such that for every finite extension $K/k$, the set $(X\setminus Z)(K)$ is finite. For each smooth curve $C=C_{a,b}$ over $k$ given by
\[
C_{a,b}: y^s=x(ax^r+b),
\]
with genus $g_{r,s}(C)\ge 1$, let $J_C$ be its Jacobian and let $r(J_C(k))$ denote the Mordell-Weil rank of the finitely generated abelian group $J_C(k)$. Which conclusion about these ranks is valid under these hypotheses?

The Mordell-Weil ranks are uniformly bounded in this family: there exists a constant $M$ such that for every smooth curve $C$ over $k$ of the form $y^s=x(ax^r+b)$ with $g_{r,s}(C)\ge 1$, one has
\[
r(J_C(k))\le M.
\]
Equivalently, as $J_C$ varies over the Jacobians of all such curves, the ranks $r(J_C(k))$ remain bounded by a single constant depending only on the fixed data of the family.

Choices:
B.
The Mordell-Weil ranks are uniformly bounded in this family after allowing finite extensions: there exists a constant $M$ such that for every smooth curve $C$ over $k$ of the form $y^s=x(ax^r+b)$ with $g_{r,s}(C)\ge 1$, and for every finite extension $K/k$, one has
\[
r(J_C(K))\le M.
\]
Equivalently, as $J_C$ varies over the Jacobians of all such curves and $K$ varies over finite extensions of $k$, the ranks remain bounded by a single constant depending only on the fixed data of the family.
C. For each smooth curve $C$ over $k$ of the form $y^s=x(ax^r+b)$ with $g_{r,s}(C)\ge 1$, the Mordell-Weil group $J_C(k)$ is finitely generated, so in particular the rank $r(J_C(k))$ is finite for every individual curve.
D.
The Mordell-Weil ranks are uniformly bounded in this family only for the subfamily of curves with genus $g_{r,s}(C)\ge 2$: there exists a constant $M$ such that every smooth curve $C$ over $k$ of the form $y^s=x(ax^r+b)$ with $g_{r,s}(C)\ge 2$ satisfies
\[
r(J_C(k))\le M,
\]
while no conclusion is asserted for the genus-one members of the family.
E.
There exists a constant $M$ such that for every smooth curve $C$ over $k$ of the form $y^s=x(ax^r+b)$ with $g_{r,s}(C)\ge 1$, one has a uniform bound on rational points
\[
\#C(k)\le M.
\]
Equivalently, the family has uniformly bounded numbers of $k$-rational points, and hence the Jacobian ranks are bounded as well.

The stem does not explicitly reveal the correct conclusion. It states the hypotheses and asks which rank statement follows, without embedding the uniform boundedness claim itself.

The item is close to theorem recall: the correct option is essentially the natural conclusion one would expect from the cited hypotheses. However, the alternatives introduce meaningful quantifier and scope variations, so it is not a pure verbatim restatement.

Some reasoning is needed to distinguish the exact strength of the conclusion from nearby statements: uniformity over k versus all finite extensions, pointwise finiteness versus uniform boundedness, and genus-1 versus genus-≥2 scope. Still, the task mainly tests recognition of the theorem's precise conclusion rather than deep derivation.

The distractors are plausible and mathematically structured: B overstates uniformity across extensions, C weakens to individual finite generation, D alters the genus range, and E confuses rational-point bounds with rank bounds. These reflect realistic failure modes.

A solid MCQ with strong distractors and little answer leakage, but it leans more toward precise theorem recall than genuinely generative mathematical reasoning.


\begin{theorem}
\label{thm1.1}
    Let $n\geq 3$ and let $u$ be a smooth solution to the special Lagrangian equation (\ref{SLE}) in $B_2(0)$ with the phase constant $\theta \geq 0$. Suppose that $\lambda(D^2 u) \subset  \overline{\Gamma^{\frac{1}{2}(n-2)}_{n-1}}$, then
    \begin{equation}
        |D^2u(0)| \leq C(n, \|u\|_{C^{0,1}(B_1(0))},\theta).
    \end{equation}
\end{theorem}

\begin{proof}
    of Theorem \ref{thm1.2}. The proof of Theorem \ref{thm1.2} is exactly the same with Theorem \ref{thm1.1} except that we use the doubling inequality in Proposition \ref{doubling2} instead of the one in Proposition \ref{doubling1}. And the constant $C$ depends on $\varepsilon$ as well.
\end{proof}

\section{Appendix}
\label{appendix}
\begin{theorem}
\label{grad_yuan}
    (Gradient estimate when $|\theta| \geq \frac{n-2}{2}\pi$). (\cite{private_note} Theorem 5.1.1).
    Let $n\geq 3$. Suppose that $u$ is a smooth solution to the special Lagrangian equation (\ref{SLE}) on $B_{R}(0)$ with $|\theta| \geq \frac{n-2}{2}\pi$, then
\begin{equation}
    |Du(0)| \leq C(n)\frac{osc_{B_{R}(0)}u}{R}.
\end{equation}

\end{theorem}
\begin{proof}
    We may assume that $R= 1 $ by rescaling $u(Rx)/R^2$. Note that
\[D^2(\frac{u(Rx)}{R^2}) = D^2u(Rx).\]
Then $\frac{u(Rx)}{R^2}$ satisfies the special Lagrangian equation with the same phase constant $\theta$.
Suppose we have the estimate
\begin{equation*}
    |D(\frac{u(R  x)}{R^2})(0)| \leq C(n)osc_{B_1(0)}(\frac{u(Rx)}{R^2}),
\end{equation*}
then we can simplify the equation and obtain the desired estimate. We may also assume $\theta \geq \frac{n-2}{2}\pi$ by symmetry.

Let \[M:=osc_{B_1(0)} u.\]
We may assume $M>0$. We may replace $u$ by $\Tilde{u}:= u-\min_{B_1(0)}u+M$. We then have $M\leq u \leq 2M$ in $B_1(0)$.
Consider the test function
\begin{equation*}
    w = (1-|x|^2)|Du| + \frac{n}{M}u^2.
\end{equation*}
Write $\eta := 1-|x|^2$.
Suppose that the maximum of $w$ lies on the boundary, say at $x^*$ where $|x^*| = 1$, then
\begin{equation*}
    2nM \geq w(x^*) = \frac{n}{M}u^2(x^*) \geq w(0) = |Du(0)|+\frac{n}{M}u^2(0) \geq |Du(0)|.
\end{equation*}

If the maximum point $x^*$ lies in the interior, we may assume $|Du(x^*)|>0$, since otherwise we can obtain the desired bound. We may further assume there exists a $k \in{\{1,\dots,n\}}$ such that $u_k \geq \frac{|Du|}{\sqrt{n}}$ at $x^*$. If not, we may consider the function \[\Tilde{u}(x_1,\dots,x_k,\dots,x_n) = u(x_1,\dots,-x_k,\dots,x_n),\] which satisfies the special Lagrangian equation with the same phase constant $\theta$. At $x^*$, we have
\begin{equation}
\label{2_3}
    0 = w_i = \eta|Du|_i+\eta_i|Du|+2\frac{n}{M}uu_i.
    \end{equation}
A direct computation gives
\begin{equation*}
    |Du|_i = \frac{\sum_j u_j u_{ij}}{|Du|}.
\end{equation*}
Assume that $D^2u$ is diagonalized at $x^*$, we have
\begin{equation}
\label{2_4}
    \frac{u_i u_{ii}}{|Du|} = |Du|_i = -\frac{2\eta_i|Du|+2uu_i\frac{n}{M}}{\eta}.
\end{equation}
In particular, for the $k\in \{1,\dots,n\}$ chosen just now,
\begin{align*}
     2\eta_k|Du|+2uu_k\frac{n}{M}&\geq -2|Du|+2u u_k \frac{n}{M}\\
     &\geq -2|Du| + 2M\frac{|Du|}{\sqrt{n}}{\frac{n}{M}} =2(\sqrt{n}-1)|Du| >0.
\end{align*}
Using (\ref{2_4}) and the fact that $u_k >0 $,
we have $\lambda_k = u_{kk} <0 $. Also, by Lemma 2.1 of \cite{Wang2014}, $|\lambda_n| \leq \lambda_i$ for $i \in \{1,\dots,n-1\}$. Hence, we have $\lambda_k = \lambda_n$.

At the maximum point $x^*$, we have
\begin{equation}
\label{2_5}
    0 \geq \Delta_g w = \Delta_g \eta |Du| + 2g^{\alpha\beta} \eta_a |Du|_\beta+ \eta \Delta_g |Du| + 2\frac{n}{M}g^{\alpha \beta}u_\alpha u_\beta+ 2\frac{n}{M}u\Delta_g u.
\end{equation}
We look at each term separately. We will use diagonal coordinates at $x^*$. Consider the first term, we have
\begin{equation*}
    \Delta_g \eta = -2\sum_\alpha g^{\alpha\alpha} \geq -2ng^{nn},
\end{equation*}
where we use the fact that $|\lambda_n|\leq \lambda_i$ for any $1 \leq i \leq n-1$. Hence,
\begin{equation}
\label{2_6}
    \Delta_g \eta |Du| \geq -2ng^{nn}|Du|.
\end{equation}
Using (\ref{2_4}), the inequality $|\lambda_n|\leq \lambda_i$ for $1\leq i \leq n-1$, and the bound $M\leq u \leq 2M$ for the second term, we have
\begin{equation}
\label{secondterm2}
\begin{split}
\sum_{\alpha,\beta}2g^{\alpha\beta} \eta_a |Du|_\beta &= \sum_\alpha2g^{\alpha\alpha}(-2x_\alpha)(\frac{2x_\alpha|Du|+2\frac{n}{M}uu_\alpha}{\eta}) \\
&\geq -8g^{nn}(\sum_\alpha x_\alpha^2)\frac{|Du|}{\eta}-8n^2 g^{nn}\frac{|Du|}{\eta}\\
&\geq -8g^{nn}\frac{|Du|}{\eta}-8n^2 g^{nn}\frac{|Du|}{\eta}.
\end{split}
\end{equation}
Directly computing the third term, we have
\begin{align*}
    \Delta_g |Du| &= \sum_{\alpha,\beta} g^{\alpha\beta}\partial_\alpha\partial_\beta|Du|\\
    &= \sum_{\alpha,\beta} g^{\alpha \beta} (\frac{\sum_i u_{i\alpha}u_{\beta i}-\sum_i u_{\alpha\beta i}u_i}{|Du|}-\frac{\sum_i u_i u_{\beta i} \sum_j u_j u_{\alpha j}}{|Du|^3}) \\
    &= \sum_{\alpha,\beta} g^{\alpha \beta} (\frac{\sum_i u_{i\alpha}u_{\beta i}}{|Du|}-\frac{\sum_i u_i u_{\beta i} \sum_j u_j u_{\alpha j}}{|Du|^3}),
\end{align*}
where we use the fact that $\sum_{\alpha,\beta}g^{\alpha\beta}u_{\alpha\beta i} = 0$.
Suppose that, at the point $x^*$, $D^2u$ is diagonal, then
\begin{align*}
    \Delta_g |Du| &= \sum_{\alpha} g^{\alpha \alpha} \frac{ u_{\alpha\alpha}^2|Du|^2- u_\alpha^2 u_{\alpha \alpha}^2 }{|Du|^3} \geq 0.
\end{align*}
Hence,
\begin{equation}
\label{thirdterm}
    \eta \Delta_g |Du| \geq 0.
\end{equation}
Recall that $u_n \geq \frac{|Du|}{\sqrt{n}}$ at $x^*$ and $|\lambda_n| \leq \lambda_i$ for $i \in \{1,\dots,n-1\}$, we can estimate the fourth term by
\begin{equation}
\label{fourthterm}
    \sum_\alpha\frac{2n}{M}g^{\alpha\alpha}u_\alpha^2 \geq \frac{2n}{M}g^{nn}u_n^2 \geq \frac{2n}{M}g^{nn}\frac{|Du|^2}{n} = \frac{2}{M}g^{nn}|Du|^2.
\end{equation}
For the fifth term, using the fact that $\lambda_i\geq 0$ for $0\leq i\leq n-1$ and (\ref{2_4}), we have
\begin{equation}
\label{fifthterm2}
\begin{split}
    \Delta_g u=  \sum_\alpha g^{\alpha\alpha}u_{\alpha\alpha} \geq g^{nn}u_{nn}= -g^{nn}\frac{|Du|(\eta_n|Du|+2\frac{n}{M}u u_n)}{u_n\eta} \geq -g^{nn}\frac{6n|Du|}{\eta}.
\end{split}
\end{equation}
Applying the estimates (\ref{2_6}), (\ref{secondterm2}), (\ref{thirdterm}), (\ref{fourthterm}) and (\ref{fifthterm2}) to (\ref{2_5}), we have
\begin{equation*}
    0 \geq g^{nn}(-C|Du|-\frac{C}{\eta}|Du|+\frac{C}{M}|Du|^2).
\end{equation*}
Hence, we have
\begin{equation*}
    \eta(x^*)|Du(x^*)| \leq CM,
\end{equation*}
and
\begin{equation*}
    |Du(0)| \leq w(0) \leq w(x^*) = \eta|Du(x^*)|+\frac{n}{M}u^2(x^*)
    \leq CM.
\end{equation*}
\end{proof}

\begin{lemma}
\label{grad_est}
Let $u$ be a smooth solution to the special Lagrangian equation (\ref{SLE}) with $\theta\geq 0$ on $B_{R}(0) \subset \mathbb{R}^n$.
\begin{enumerate} [(a)]
    \item (Gradient estimate for Theorem \ref{thm1.1}.) Suppose that $n\geq 3$ and $\lambda(D^2 u) \subset  \overline{\Gamma^{\frac{1}{2}(n-2)}_{n-1}}$, then we have
\begin{equation}
    |Du(0)| \leq C(n)\frac{osc_{B_{R}(0)}u}{R}.
\end{equation}
\item  (Gradient estimate for Theorem \ref{thm1.2}.) Suppose that $n = 3$ and $\sigma_2(D^2u) \geq (\frac{3}{5}-\varepsilon) \lambda_2 \lambda_3$ for some constant $\varepsilon >0$ where $\lambda_1 \geq \lambda_2 \geq \lambda_3 $ are eigenvalues of $D^2 u$, then we have
\begin{equation}
    |Du(0)| \leq C(\varepsilon)\frac{osc_{B_{R}(0)}u}{R}.
\end{equation}
\end{enumerate}

\end{lemma}

In this paper, we prove the Hessian estimate for the special Lagrangian equation in general phases with constraints, first introduced by Zhou \cite{Zhou2021}. Our result extends Shankar's argument \cite{Shankar2024HessianEF} which also employs the doubling method.

Let $u$ be a smooth solution of the special Lagrangian equation in dimension $n\geq 3$ with a phase constant $\theta \geq 0$ given by
\begin{equation}
\label{SLE}
    \arctan \lambda_1 + \arctan \lambda_2 + \dots + \arctan \lambda_n = \theta,
\end{equation}
where $\lambda_i $ are the eigenvalues of $D^2u$.

Hessian estimates for the special Lagrangian equation have been extensively studied and established in various contexts. In dimension three, Warren-Yuan \cite{WarrenYuan2009} obtained the estimate for the critical phase, and later Warren-Yuan \cite{a359f720-81a7-3f4b-b3f0-6e19654fb57a} further extended the results to both the critical and supercritical phases. Wang–Yuan \cite{Wang2014} further developed the estimates for the critical and supercritical phases for general dimensions $n\geq 3$. Chen–Warren–Yuan \cite{Chen2009} proved the estimate for convex solutions. Warren-Yuan \cite{WarrenYuan2008} also derived the Hessian estimate under the assumptions
\[3+(1-\varepsilon)\lambda_i^2+2\lambda_i\lambda_j \geq 0,\text{ } 1\leq i,j\leq n,\] for some $\varepsilon >0$, and $|Du(x)|\leq \delta(n) |x|$. Ding \cite{Ding2023} later established the Hessian estimate with the assumption
\[(3+\varepsilon)+(1+\varepsilon)\lambda_i^2+2\lambda_i\lambda_j \geq 0, \text{ } 1\leq i,j\leq n,\] for some $\varepsilon >0$. Finally, Zhou \cite{Zhou2021} obtained the Hessian estimate on general phases with constraints. Counterexamples below the critical phase were constructed by Mooney-Savin \cite{MooneySavin2024}, Nadirashvili-Vl{\u{a}}du{\c{t}} \cite{NadirashviliVladut2010}, and Wang-Yuan \cite{WangYuan2013}.

A natural question is whether we can find a pointwise argument for the a priori Hessian estimates of the special Lagrangian equation. In fact, Chen–Warren–\\Yuan \cite{Chen2009}, Wang–Yuan \cite{Wang2014} and Zhou \cite{Zhou2021} made use of the Michael-Simon mean value inequality \cite{MichaelSimon1973} and the Jacobi inequality to show the Hessian estimates for the special Lagrangian equation. The study of Hessian estimates for the special Lagrangian equation is analogous to that of the gradient estimate for minimal surfaces.

Bombieri–De Giorgi–Miranda \cite{Bombieri1969} established an a priori gradient estimate for graphical minimal hypersurfaces, which was later simplified by Trudinger \cite{Trudinger1972}. The key ingredients are the Jacobi inequality $\Delta a \geq |\nabla a|^2$ and the Michael-Simon mean value inequality. Korevaar \cite{Korevaar1987} subsequently established a pointwise argument for the gradient estimate without using the Michael-Simon mean-value inequality. A natural question is whether such an argument can be adapted for higher codimension minimal surfaces. An example is that Dimler \cite{Dimler2023} obtained the gradient estimate for classical solutions of minimal surface system assuming the area decreasing condition and that all but one component have small $L^\infty$ norm.

The doubling method provides an alternative approach to establishing Hessian estimates without using the Michael-Simon mean value inequality and can also be used to derive interior regularity results. The first step is to prove an Alexandrov-type theorem. Next, the uniform norm of the solution is flattened so that Savin's small perturbation theorem (\cite{Savin2007}, Theorem 1.3) can be applied to obtain Hessian estimates in a small ball. Finally, by employing the Jacobi inequality, one can derive the doubling inequality, which allows the Hessian estimate in a larger ball to be controlled by the estimate in the smaller ball.

Several works have constructed doubling inequalities to establish Hessian estimates and interior regularity results. Qiu \cite{Qiu2024} derived the Hessian estimate of the sigma-2 equation in dimension $3$. Shankar-Yuan \cite{10.4007/annals.2025.201.2.4} established the Hessian estimate for the sigma-2 equation when $n=4$. Shankar-Yuan \cite{ShankarYuan2024} proved the interior regularity of strictly convex solutions of the
Monge-Amp$\grave{e}$re equation. Shankar \cite{Shankar2024HessianEF} further developed a doubling argument to establish Hessian estimates for the special Lagrangian equation in several settings, without using the Michael-Simon mean value inequality. In particular, the argument applies to solutions with the critical and supercritical phases, and to convex solutions. It also applies to semi-convex solutions satisfying the Jacobi inequality. In this article, we adapt a doubling method to establish our results.

In related work, Trudinger \cite{Trudinger1980} established a doubling inequality in the context of the Harnack inequality for weak solutions $u \in W^{2,n}$ of general second order elliptic quasilinear equations. Caffarelli-Wang \cite{CaffarelliWang1993} made use of a Harnack inequality approach to give an alternative proof of the $C^{1,\alpha}$ regularity of Lipschitz solutions to
\[F(v,II) = 0,\]
where $II$ is the second fundamental form of a hypersurface $S$ in $\mathbb{R}^{n+1}$ and $v$ is its normal.

We say that $\lambda(D^2u) \subset A$ if the eigenvalues $(\lambda_1,\dots, \lambda_n)$ of $D^2u$ belong to $A \subset \mathbb{R}^n$. Let
\begin{equation}
\label{cone}
    \Gamma_{n-1}^c = \{\Lambda \in \mathbb{R}^n| \text{ }\sigma_{n-1} > c\lambda_2 \cdot \lambda_3\cdot ... \cdot \lambda_n, \text{ } \lambda_{n-1} >0 \},
\end{equation}
where $(\lambda_1,\dots,\lambda_n)$ is an arrangement of $\Lambda$ such that $\lambda_1 \geq \lambda_2 \geq \dots \geq \lambda_n$ and $c$ is a constant. We denote the closure of the cone $\Gamma_{n-1}^c$ by $\overline{\Gamma_{n-1}^c}$. From now on, we will always assume $\lambda_1 \geq \lambda_2 \geq  \dots \geq \lambda_n$.

The following theorems were proved by Zhou \cite{Zhou2021} using the Michael-Simon mean value inequality. We establish a compactness argument without using it.

Let $u$ be a smooth solution of the special Lagrangian equation in dimension $n\geq 3$ with a phase constant $\theta \geq 0$ given by
\begin{equation}
\label{SLE}
 \arctan \lambda_1 + \arctan \lambda_2 + \dots + \arctan \lambda_n = \theta,
\end{equation}
where $\lambda_i $ are the eigenvalues of $D^2u$.

Hessian estimates for the special Lagrangian equation have been extensively studied and established in various contexts. In dimension three, Warren-Yuan \cite{WarrenYuan2009} obtained the estimate for the critical phase, and later Warren-Yuan \cite{a359f720-81a7-3f4b-b3f0-6e19654fb57a} further extended the results to both the critical and supercritical phases. Wang–Yuan \cite{Wang2014} further developed the estimates for the critical and supercritical phases for general dimensions $n\geq 3$. Chen–Warren–Yuan \cite{Chen2009} proved the estimate for convex solutions. Warren-Yuan \cite{WarrenYuan2008} also derived the Hessian estimate under the assumptions
\[3+(1-\varepsilon)\lambda_i^2+2\lambda_i\lambda_j \geq 0,\text{ } 1\leq i,j\leq n,\] for some $\varepsilon >0$, and $|Du(x)|\leq \delta(n) |x|$. Ding \cite{Ding2023} later established the Hessian estimate with the assumption
\[(3+\varepsilon)+(1+\varepsilon)\lambda_i^2+2\lambda_i\lambda_j \geq 0, \text{ } 1\leq i,j\leq n,\] for some $\varepsilon >0$. Finally, Zhou \cite{Zhou2021} obtained the Hessian estimate on general phases with constraints. Counterexamples below the critical phase were constructed by Mooney-Savin \cite{MooneySavin2024}, Nadirashvili-Vl{\u{a}}du{\c{t}} \cite{NadirashviliVladut2010}, and Wang-Yuan \cite{WangYuan2013}.

A natural question is whether we can find a pointwise argument for the a priori Hessian estimates of the special Lagrangian equation. In fact, Chen–Warren–\\Yuan \cite{Chen2009}, Wang–Yuan \cite{Wang2014} and Zhou \cite{Zhou2021} made use of the Michael-Simon mean value inequality \cite{MichaelSimon1973} and the Jacobi inequality to show the Hessian estimates for the special Lagrangian equation. The study of Hessian estimates for the special Lagrangian equation is analogous to that of the gradient estimate for minimal surfaces.

Several works have constructed doubling inequalities to establish Hessian estimates and interior regularity results. Qiu \cite{Qiu2024} derived the Hessian estimate of the sigma-2 equation in dimension $3$. Shankar-Yuan \cite{10.4007/annals.2025.201.2.4} established the Hessian estimate for the sigma-2 equation when $n=4$. Shankar-Yuan \cite{ShankarYuan2024} proved the interior regularity of strictly convex solutions of the
Monge-Amp$\grave{e}$re equation. Shankar \cite{Shankar2024HessianEF} further developed a doubling argument to establish Hessian estimates for the special Lagrangian equation in several settings, without using the Michael-Simon mean value inequality. In particular, the argument applies to solutions with the critical and supercritical phases, and to convex solutions. It also applies to semi-convex solutions satisfying the Jacobi inequality. In this article, we adapt a doubling method to establish our results.

We say that $\lambda(D^2u) \subset A$ if the eigenvalues $(\lambda_1,\dots, \lambda_n)$ of $D^2u$ belong to $A \subset \mathbb{R}^n$. Let
\begin{equation}
\label{cone}
 \Gamma_{n-1}^c = \{\Lambda \in \mathbb{R}^n| \text{ }\sigma_{n-1} > c\lambda_2 \cdot \lambda_3\cdot ... \cdot \lambda_n, \text{ } \lambda_{n-1} >0 \},
\end{equation}
where $(\lambda_1,\dots,\lambda_n)$ is an arrangement of $\Lambda$ such that $\lambda_1 \geq \lambda_2 \geq \dots \geq \lambda_n$ and $c$ is a constant. We denote the closure of the cone $\Gamma_{n-1}^c$ by $\overline{\Gamma_{n-1}^c}$. From now on, we will always assume $\lambda_1 \geq \lambda_2 \geq \dots \geq \lambda_n$.

The following theorems were proved by Zhou \cite{Zhou2021} using the Michael-Simon mean value inequality. We establish a compactness argument without using it.

\begin{equation}
\label{SLE}
    \arctan \lambda_1 + \arctan \lambda_2 + \dots + \arctan \lambda_n = \theta,
\end{equation}

This paper follows the outline of Shankar \cite{Shankar2024HessianEF}. In our settings, Jacobi inequalities have already been established by Zhou \cite{Zhou2021}. The new contribution of this paper is the proof of the Alexandrov-type theorem. Then we can apply the doubling inequality argument developed by Shankar \cite{Shankar2024HessianEF}, with only minor modifications, to obtain the Hessian estimates.

\begin{lemma}
\label{grad_est}
Let $u$ be a smooth solution to the special Lagrangian equation (\ref{SLE}) with $\theta\geq 0$ on $B_{R}(0) \subset \mathbb{R}^n$.
\begin{enumerate} [(a)]
 \item (Gradient estimate for Theorem \ref{thm1.1}.) Suppose that $n\geq 3$ and $\lambda(D^2 u) \subset \overline{\Gamma^{\frac{1}{2}(n-2)}_{n-1}}$, then we have
\begin{equation}
 |Du(0)| \leq C(n)\frac{osc_{B_{R}(0)}u}{R}.
\end{equation}
\item (Gradient estimate for Theorem \ref{thm1.2}.) Suppose that $n = 3$ and $\sigma_2(D^2u) \geq (\frac{3}{5}-\varepsilon) \lambda_2 \lambda_3$ for some constant $\varepsilon >0$ where $\lambda_1 \geq \lambda_2 \geq \lambda_3 $ are eigenvalues of $D^2 u$, then we have
\begin{equation}
 |Du(0)| \leq C(\varepsilon)\frac{osc_{B_{R}(0)}u}{R}.
\end{equation}
\end{enumerate}

We restate the Jacobi inequalities established by Zhou for our settings.
\begin{lemma}
\label{jacobi1}
 (Jacobi inequality for $n\geq 3).$ (\cite{Zhou2021} Lemma 4.3).
 Consider the same setting as Theorem \ref{thm1.1}. Let $n\geq 3$ and let $u$ be a smooth solution to the special Lagrangian equation (\ref{SLE}) with $\theta\geq 0$ and $\lambda_1 = \lambda_2 = \dots = \lambda_m = \lambda > \lambda_{m+1} $ at a point $p$. Suppose that $\lambda(D^2 u) \subset \overline{\Gamma_{n-1}^{\frac{1}{2}(n-2)}}$, then there exist constants $\alpha,\delta>0 $ depending only on $n$ such that if $\lambda>\delta$, then $b_m:= \frac{1}{m}\sum_{i=1}^m \lambda_i$ is smooth near $p$ and
 \begin{equation}
 \Delta_g b_m(p) \geq (1+ \alpha) \frac{|\nabla_g b_m|^2}{b_m}(p).
 \end{equation}

\begin{proposition}
\label{partial_regularity}
Let $n\geq 3$ and let $u_k \in C^\infty(B_2(0))$ be a sequence of smooth solutions to the special Lagrangian equation (\ref{SLE}) in $B_2(0)$ with the phase constant $\theta \geq 0$ and
\[\|u_k\|_{C^{0,1}(B_1(0))} \leq A.\]
Suppose that a subsequence of $u_k$ converges uniformly in $B_1(0)$ to a continuous function $u \in C^0(B_1(0))$, and either
\begin{enumerate}
[(i)]
 \item $\lambda(D^2 u_k) \subset \overline{\Gamma^{\frac{1}{2}(n-2)}_{n-1}}$ for all $k \in \mathbb{N}$, or
 \item $n =3$ and there exists a constant $\varepsilon > 0$ such that
 \[\sigma_2(D^2 u_k) \geq (\frac{3}{5}-\varepsilon) \lambda_2(D^2u_k)\lambda_3(D^2u_k) \text{ for all } k \in \mathbb{N},\]
\end{enumerate}
then $u$ is twice differentiable almost everywhere in $B_1(0)$, and for almost every $x \in B_1(0)$, there is a quadratic polynomial $Q$ such that
\begin{equation}
 \sup_{y\in B_r(x)} |u(y)-Q(y)| = o(r^2).
\end{equation}
\end{proposition}
\begin{proof}
 Without loss of generality, we may assume that $u_k$ converges to $u$ uniformly. We can immediately see that $u$ is Lipschitz:
 \begin{equation}
 \label{lip}
 \|u\|_{C^{0,1}(B_1(0))} \leq A.
 \end{equation}
 Moreover, by Rademacher's theorem, $u$ is almost everywhere differentiable.

\section{Doubling inequality}
\label{section doubling inequality}
In this section, we establish the doubling inequalities for our settings. The argument in \cite{Shankar2024HessianEF} Section 4 applies here with minor modifications. First, if there exists a point $p$ such that $\lambda_{\max}(D^2u(p))$ is not sufficiently large, the Jacobi inequalities in Lemma \ref{jacobi1} and Lemma \ref{jacobi2} may not hold at $p$. However, $\lambda_{\max}(D^2u(p))$ is bounded in this case. Another difference is that, in the setting of Theorem \ref{thm1.2}, there could be two negative eigenvalues at a point $p$. However, by Lemma \ref{n=3eigenvalues}, $|D^2u(p)|$ is bounded in this case.
\begin{proposition}
\label{doubling1}
Let $n\geq 3$ and let $u$ be a smooth solution to the special Lagrangian equation (\ref{SLE}) in $B_2(0)$ with the phase constant $\theta \geq 0$. Suppose that $\lambda(D^2 u) \subset \overline{\Gamma^{\frac{1}{2}(n-2)}_{n-1}}$, then for any $y \in B_{\frac{1}{2}}(0)$ and $r<\frac{1}{4}$, we have
\begin{equation*}
 \sup_{B_\frac{1}{4}(y)} \lambda_{\max}(D^2u) \leq C \sup_{B_r(y)} \lambda_{\max}(D^2u)+C,
\end{equation*}
where $C$ is a constant depending on $r,n$ and $\|u\|_{C^{0,1}(B_1(0))}$.
\end{proposition}
\begin{proof}
 Let $\alpha, h^{-1} \gg 1$ be two constants to be chosen later. We choose the same cutoff function $\eta(x)$ as in \cite{Shankar2024HessianEF} Section 5:
 \begin{equation}
 \eta(x) = ( e^{(S- \varphi(x))/h} - 1)_+,
 \end{equation} where
 \begin{equation}
 \varphi = (x-y)\cdot Du - u +u(y) - \frac{\alpha^{-1}2^\alpha}{|x-y|^{2\alpha}},
 \end{equation}
 and
 \begin{equation}
 S = -1 - \|(x-y)\cdot Du - u+u(y)\|_{L^\infty(B_{\frac{1}{2}}(y))} - \alpha^{-1}2^{3\alpha}.
 \end{equation}
 The choice of cutoff function is motivated by Korevaar exponential type cutoff function \cite{Korevaar1987} and Guan-Qiu \cite{GuanQiu2019} type radial derivative $(x-y)\cdot Du - u$. Observe that $S-\varphi<0$ on $\partial B_{\frac{1}{2}}(y)$ and $S-\varphi>0$ on $B_{\frac{1}{4}}(y)$ if $\alpha$ is large enough.

\begin{proposition}
\label{doubling2}
 Let $n=3$ and let $u$ be a smooth solution to the special Lagrangian equation (\ref{SLE}) in $B_2(0)$ with the phase constant $\theta \geq 0$. Suppose that $\sigma_2(D^2u) \geq (\frac{3}{5}-\varepsilon) \lambda_2 \lambda_3$ for some constant $\varepsilon >0$ where $\lambda_1 \geq \lambda_2 \geq \lambda_3 $ are eigenvalues of $D^2 u$, then for any $y \in B_{\frac{1}{2}}(0)$ and $r<\frac{1}{4}$,
\begin{equation*}
 \sup_{B_\frac{1}{4}(y)} \lambda_{\max}(D^2u) \leq C \sup_{B_r(y)} \lambda_{\max}(D^2u)+C,
\end{equation*}
where $C$ is a constant depending on $r, \|u\|_{C^{0,1}(B_1(0))}$ and $\varepsilon$.
\end{proposition}
\begin{proof}
The proof is nearly identical to that of Proposition \ref{doubling1} except we have an extra case $\lambda_2(p) < 0$. By Lemma \ref{n=3eigenvalues}, we can obtain the bound $|D^2u(p)| \leq \max(1,\varepsilon)$. Hence, the same argument as in Case (a) in Proposition \ref{doubling1} works after replacing the constant $\delta$ by $\max(1,\varepsilon)$.
\end{proof}
\section{Proof of Theorem \ref{thm1.1} and \ref{thm1.2}}
\label{section main theorem proof}
\begin{proof} of Theorem \ref{thm1.1}. Now we have established the Alexandrov-type theorems in Proposition \ref{partial_regularity} and the doubling inequality in Proposition \ref{doubling1}. The proof of Theorem \ref{thm1.1} follows the same argument as in \cite{Shankar2024HessianEF} Section 6, except that we have an additional constant term in the doubling inequality. (6.5) in \cite{Shankar2024HessianEF} will be modified to
\begin{equation}
 \begin{split}
 \sup_{B_R(y)}|D^2u_k| &\leq C(r,n,A,\sup_{B_r(y)}|D^2u_k|) + C(r,n, A) \\
 &\leq C(r,n,A,C(n,Q,\sigma)) + C(r,n, A)\\
 &\leq C(n,A,Q,\sigma),
 \end{split}
\end{equation}
where $A$ is the uniform upper bound of $\|u_k\|_{C^{0,1}(B_1(0))}$ for any $k \in \mathbb{N}$, $Q$ is the quadratic polynomial given by Proposition \ref{partial_regularity} for the uniform limit $u$ at a point $y$ where $u$ is twice differentiable at, and $\sigma$ satisfies $|u(x)-Q(x)|\leq \sigma(|x-y|) $ with $\sigma(r) =o(r^2)$ as $r\rightarrow 0$. And the rest of the proof is the same.
\end{proof}
\begin{proof}
 of Theorem \ref{thm1.2}. The proof of Theorem \ref{thm1.2} is exactly the same with Theorem \ref{thm1.1} except that we use the doubling inequality in Proposition \ref{doubling2} instead of the one in Proposition \ref{doubling1}. And the constant $C$ depends on $\varepsilon$ as well.
\end{proof}

In dimension three, the result can be refined to the following:
\begin{theorem}
\label{thm1.2}
    Let $n =3$ and let $u$ be a smooth solution to the special Lagrangian equation (\ref{SLE}) in $B_2(0)$ with the phase constant $\theta \geq 0$. Suppose that $\sigma_2(D^2u) \geq (\frac{3}{5}-\varepsilon) \lambda_2 \lambda_3$ for some constant $\varepsilon >0$, where $\lambda_1 \geq \lambda_2 \geq \lambda_3 $ are eigenvalues of $D^2 u$, then we have
    \begin{equation}
        |D^2u(0)| \leq C( \|u\|_{C^{0,1}(B_1(0))},\theta,\varepsilon).
    \end{equation}
\end{theorem}
Note that $\lambda_2 $ can be negative in this case (only if $\varepsilon>\frac{3}{5}$).

This paper follows the outline of Shankar \cite{Shankar2024HessianEF}. In our settings, Jacobi inequalities have already been established by Zhou \cite{Zhou2021}. The new contribution of this paper is the proof of the Alexandrov-type theorem. Then we can apply the doubling inequality argument developed by Shankar \cite{Shankar2024HessianEF}, with only minor modifications, to obtain the Hessian estimates.

\textbf{Outline:} The structure of the paper is as follows. In Section \ref{section preliminary}, we show some preliminary results. In particular, we show the gradient estimates for our settings in Lemma \ref{grad_est} in which we modify the proof of the gradient estimate for large phase by Yuan \cite{private_note}. In Section \ref{section partial regularity}, we establish the Alexandrov-type theorem for our settings. In Section \ref{section doubling inequality}, we establish the doubling inequalities for our settings. In fact, the argument of Shankar \cite{Shankar2024HessianEF} works with minor adjustments. In Section \ref{section main theorem proof}, we complete the proofs of Theorem \ref{thm1.1} and Theorem \ref{thm1.2}. In Section \ref{appendix}, we restate the proof of the gradient estimate of Yuan \cite{private_note}.

The paper says it “follows the outline of Shankar \cite{Shankar2024HessianEF}.” In the authors’ setting, “Jacobi inequalities have already been established by Zhou \cite{Zhou2021}.” The “new contribution of this paper is the proof of the Alexandrov-type theorem.” With that in hand, they “apply the doubling inequality argument developed by Shankar \cite{Shankar2024HessianEF}, with only minor modifications, to obtain the Hessian estimates” (i.e., the bounds in Theorem~\ref{thm1.1}).

More concretely via the stated outline: (1) Section~\ref{section preliminary} proves preliminary results, “in particular” a “gradient estimates” lemma (Lemma~\ref{grad_est}) by “modif[ying] the proof … by Yuan \cite{private_note}”; (2) Section~\ref{section partial regularity} “establish[es] the Alexandrov-type theorem”; (3) Section~\ref{section doubling inequality} “establish[es] the doubling inequalities,” where “the argument of Shankar … works with minor adjustments”; and (4) Section~\ref{section main theorem proof} “complete[s] the proofs of Theorem~\ref{thm1.1} and Theorem~\ref{thm1.2}.”

\label{thm1.1}
    Let $n\geq 3$ and let $u$ be a smooth solution to the special Lagrangian equation
    \begin{equation}
\label{SLE}
    \arctan \lambda_1 + \arctan \lambda_2 + \dots + \arctan \lambda_n = \theta,
\end{equation}
    in $B_2(0)$ with the phase constant $\theta \geq 0$. Suppose that $\lambda(D^2 u) \subset  \overline{\Gamma^{\frac{1}{2}(n-2)}_{n-1}}$, then
    \begin{equation}
        |D^2u(0)| \leq C(n, \|u\|_{C^{0,1}(B_1(0))},\theta).
    \end{equation},

Question:
Let $n\ge 3$, and let $u$ be a smooth solution on $B_2(0)\subset \mathbb{R}^n$ of the special Lagrangian equation
\[
\arctan \lambda_1+\arctan \lambda_2+\cdots+\arctan \lambda_n=\theta,
\]
with phase constant $\theta\ge 0$, where $\lambda_1\ge \lambda_2\ge \cdots \ge \lambda_n$ are the eigenvalues of $D^2u$. Assume that for every point of $B_2(0)$, the eigenvalue vector $\lambda(D^2u)$ belongs to the closure $\overline{\Gamma_{n-1}^{\frac12(n-2)}}$ of the cone
\[
\Gamma_{n-1}^c=\{\Lambda=(\lambda_1,\dots,\lambda_n)\in \mathbb{R}^n:\ \sigma_{n-1}(\Lambda)>c\,\lambda_2\lambda_3\cdots \lambda_n,\ \lambda_{n-1}>0\},
\]
where $\sigma_{n-1}$ denotes the $(n-1)$st elementary symmetric polynomial. Which of the following conclusions about the Hessian at the origin holds under these assumptions?

There is a constant $C$, depending only on $n$, $\|u\|_{C^{0,1}(B_1(0))}$, and $\theta$, such that
\[
|D^2u(0)|\le C(n,\|u\|_{C^{0,1}(B_1(0))},\theta).
\]

Choices:
B.
There is a constant $C$, depending only on $n$ and $\theta$, such that
\[
|D^2u(0)|\le C(n,\theta).
\]
C.
There is a constant $C$, depending only on $n$, $\|u\|_{C^{0,1}(B_1(0))}$, and $\theta$, such that
\[
\lambda_{\max}(D^2u(0))\le C(n,\|u\|_{C^{0,1}(B_1(0))},\theta).
\]
D.
There is a constant $C$, depending only on $n$, $\|u\|_{C^{0,1}(B_2(0))}$, and $\theta$, such that
\[
\sup_{B_1(0)}|D^2u|\le C(n,\|u\|_{C^{0,1}(B_2(0))},\theta).
\]
E.
There is a constant $C$, depending only on $n$, $\|u\|_{C^{0,1}(B_1(0))}$, and $\theta$, such that whenever $\lambda(D^2u(0))\in \overline{\Gamma_{n-1}^{\frac12(n-2)}}$ one has
\[
|D^2u(0)|\le C(n,\|u\|_{C^{0,1}(B_1(0))},\theta).
\]

The stem does not state the conclusion explicitly or give away choice A verbatim; it only provides the hypotheses and asks for the valid estimate.

This is essentially a theorem-recall item: the hypotheses in the stem closely match the theorem, and the correct option is basically the theorem's stated conclusion.

Some reasoning is needed to distinguish the exact sharp conclusion from stronger, weaker, or malformed variants, but the task is mainly recognition of the precise theorem statement rather than genuine derivation.

The distractors are plausible and mathematically meaningful: they test missing dependence on the Lipschitz norm, weakening to only the top eigenvalue, overstrengthening to an interior supremum estimate, and replacing a global cone assumption with a pointwise one.

A mathematically well-constructed theorem-identification MCQ with strong distractors, but it is too close to direct theorem restatement to strongly test generative reasoning.


\label{thm_main}(Unique determination)
Let $\Omega\subset\mathbb{R}^n$ be a bounded open set with smooth
boundary $\partial\Omega$. Let $s\in \mathbb{R}^+\setminus \mathbb{Z}$, $\lfloor s \rfloor>\max\{m,\,n/2\}$.
Suppose that $q_j\in L^\infty(\Omega)$ satisfies $q_j\geq 0$ in $\Omega$ for $j=1,\,2$.
Let $\mathbf{P}_1$ and $\mathbf{P}_2$ be nonlinearities of the form \eqref{DEF:nonlinear P} with coefficients $a_{\sigma,k}^{(1)},\, a_{\sigma,k}^{(2)}(x) \in C (\overline\Omega)$. Let $W_1,\, W_2\subset\Omega_e$ be two arbitrary nonempty bounded open sets. If the DN-maps $\Lambda_{\mathbf{P}_j}: \mathcal{X}_{\varepsilon_0}(W_1)\rightarrow (H^{-s}(\Omega_e))^*$ satisfy
\[
\Lambda_{\mathbf{P}_1}f|_{W_2} = \Lambda_{\mathbf{P}_2}f|_{W_2} \quad \mbox{ for all } f \in \mathcal{X}_{\varepsilon_0}(W_1),
\]
then for all multi-indices $\sigma$ such that $0\le |\sigma|\le m$ and for all $k=1,\ldots, K-1$ it holds
$$
q_1=q_2\quad\hbox{and}\quad	a_{\sigma,k}^{(1)} = a_{\sigma,k}^{(2)} \quad \hbox{ in }\Omega,
$$
and thus the nonlinearities $\mathbf P_1$ and $\mathbf P_2$ coincide.

Let $\Omega$ be a bounded open set in $\mathbb{R}^n$, $n\geq 1$, with smooth boundary $\partial\Omega$, and
$\Omega_e := \mathbb{R}^n\setminus \overline\Omega$ be the exterior of $\Omega$.
Let $m\geq 0$ be an integer and $s\in \mathbb{R}^+\setminus \mathbb{Z}$, where $\mathbb{R}^+:=(0,\infty)$ and $\mathbb{Z}$ denotes the set of all integers in $\mathbb{R}$.
We consider inverse problems for the nonlinear fractional Schr\"odinger equation (NLFSE):
\begin{equation}\label{eq_1_IBVP}
\begin{cases}
(-\Delta)^su + q(x)u +\mathbf{P}(u)= 0 &\hbox{ in } \Omega,\\
u = f &\hbox{ in } \Omega_e,
\end{cases}
\end{equation}
where the fractional Laplacian $(-\Delta)^s$ is the nonlocal pseudo-differential operator defined by $(-\Delta)^su: = \mathcal{F}^{-1}(|\cdot|^{2s}\mathcal{F}(u))$. Here $\mathcal{F} (u)$ represents the Fourier transform of $u$.
The local nonlinear perturbation $\mathbf{P} (u)$ is defined as
\begin{align}\label{DEF:nonlinear P}
\mathbf{P} (u) := uP_1(x,D)u + u^2 P_2(x,D)u+\ldots+u^{K-1}P_{K-1}(x,D)u,
\end{align}
where $K\geq 2$ is an integer, and the linear differential operators $P_k(x,D)$ of order $m$ are of the form
$$
P_k(x,D):=\sum_{\sigma: \,|\sigma|\leq m}a_{\sigma,k}(x)D^\sigma,\quad k=1,\ldots,K-1,
$$
with scalar coefficients $a_{\sigma,k}$ depending only on $x$. Here $\sigma=(\sigma_1,\ldots,\sigma_n)$ is multi-index of length $|\sigma|=\sigma_1+\ldots+\sigma_n=\ell$ for $\ell\leq m$, and therefore $D^\sigma u := \frac{\partial^{\sigma_1}}{\partial x_1^{\sigma_1}}\ldots\frac{\partial^{\sigma_n}}{\partial x_n^{\sigma_n}} u$. \\

We show in Theorem \ref{thm_wellposdness} that \eqref{eq_1_IBVP} is well-posed in $H^s(\mathbb{R}^n)$ for any data $f$ in a small ball $\mathcal{X}_{\varepsilon_0}(\Omega_e)$ in $C^\infty_c(\Omega_e)$, where for any open set $O\subset \mathbb R^n$ we define
\[
\mathcal{X}_{\varepsilon_0}(O):=\{f\in C^\infty_c(O):\,\|f\|_{C^\infty_c(O)}\le\varepsilon_0\}.
\]
This well-posedness result for \eqref{eq_1_IBVP} allows us to define the corresponding Dirichlet-to-Neumann (DN) map
$$
\Lambda_{\mathbf{P}}: H^{s}(\Omega_e)\rightarrow (H^{-s}(\Omega_e))^*,\quad f\mapsto (-\Delta)^s u_f|_{\Omega_e}.
$$

In this framework, we ask the following inverse problem: Does the map $\Lambda_{\mathbf{P}}$ uniquely determine the nonlinearity $\mathbf P$? The answer is affirmative, and we give a full description of the results in the following Theorem~\ref{thm_main}:

\begin{theorem}\label{thm_main}(Unique determination)
Let $\Omega\subset\R^n$ be a bounded open set with smooth
boundary $\p\Omega$. Let $s\in \R^+\setminus \mathbb{Z}$, $\floor{s}>\max\{m,\,n/2\}$.
Suppose that $q_j\in L^\infty(\Omega)$ satisfies $q_j\geq 0$ in $\Omega$ for $j=1,\,2$.
Let $\mathbf{P}_1$ and $\mathbf{P}_2$ be nonlinearities of the form \eqref{DEF:nonlinear P} with coefficients $a_{\sigma,k}^{(1)},\, a_{\sigma,k}^{(2)}(x) \in C (\overline\Omega)$. Let $W_1,\, W_2\subset\Omega_e$ be two arbitrary nonempty bounded open sets. If the DN-maps $\Lambda_{\mathbf{P}_j}: \mathcal{X}_{\varepsilon_0}(W_1)\rightarrow (H^{-s}(\Omega_e))^*$ satisfy
\[
\Lambda_{\mathbf{P}_1}f|_{W_2} = \Lambda_{\mathbf{P}_2}f|_{W_2} \quad \mbox{ for all } f \in \mathcal{X}_{\varepsilon_0}(W_1),
\]
then for all multi-indices $\sigma$ such that $0\le |\sigma|\le m$ and for all $k=1,\ldots, K-1$ it holds
$$
q_1=q_2\quad\hbox{and}\quad	a_{\sigma,k}^{(1)} = a_{\sigma,k}^{(2)} \quad \hbox{ in }\Omega,
$$
and thus the nonlinearities $\mathbf P_1$ and $\mathbf P_2$ coincide.
\end{theorem}

\begin{theorem}[Well-posedness for the nonlinear equation]
\label{thm_wellposdness}
Let $\Omega\subset\R^n$ be a bounded open domain with smooth
boundary $\p\Omega$. Let $s\in \R^+\setminus \mathbb{Z}$ and $m\in \mathbb{N}$ satisfy $\floor{s}>\max\{m,\,n/2\}$.
Suppose that $q\in L^\infty(\Omega)$ satisfies $q\geq 0$ in $\Omega$, and the coefficients in $\mathbf{P}(u)$ satisfy $a_{\sigma,k}\in C (\overline\Omega)$.
Then there exists a sufficiently small $\varepsilon_0>0$ such that for any $f\in \mathcal{X}_{\varepsilon_0}(\Omega_e)$,
the Dirichlet problem \eqref{eq_1_IBVP} has a unique solution $u\in H^s(\R^n)\cap C^s(\R^n)$. Moreover, there exists a constant $C>0$, independent of $u$ and $f$, such that
\[
\|u\|_{C^{s}(\R^n)}+ \|u\|_{H^{s}(\R^n)}\le C\|f\|_{C^\infty_c(\Omega_e)}.
\]
\end{theorem}

\subsection{Motivation and connection to the literature}
This work addresses a natural generalization of the inverse problems for linear fractional elliptic equations, such as the fractional Schr\"odinger equation $(-\Delta)^s+q$ for $s\in (0,1)$. Our unique determination result of Theorem \ref{thm_main} particularly finds its place within the context of the very active field of inverse problems of fractional Calder\'on problem, known as the fractional analogue of Calder\'on problem. First introduced in the seminal work \cite{GSU20}, the fractional Calder\'on problem asks to uniquely determine a scalar potential $q$ in the fractional Schr\"odinger equation from the related DN-map. It corresponds to our problem \eqref{eq_1_IBVP} in the case $\mathbf P \equiv 0$ for $s\in (0,1)$. Already in the first work \cite{GSU20}, the authors were able to provide a uniqueness result from partial data for $L^\infty$ potentials.

Because the DN-map acts on a quotient space, we have to make sure that it is well-defined as indicated above. We also need to show that it is indeed a bounded linear operator. In what follows, we will use $f$ rather than $[f]$ in order to simplify the notation.
\begin{prop}
Let $\Omega\subset\R^n$ be a bounded open domain with smooth
boundary $\p\Omega$. Let $s\in \R^+\setminus \mathbb{Z}$ and $m\in \mathbb{N}$ satisfy $\floor{s}>\max\{m,\,n/2\}$.
Suppose that $q\in L^\infty(\Omega)$ satisfies $q\geq 0$ in $\Omega$, and the coefficients in $\mathbf{P}(u)$ satisfy $a_{\sigma,k}\in C (\overline\Omega)$.
Then the DN-map $\Lambda_\mathbf{P}$ is well-defined and bounded.
\end{prop}
\begin{proof}
We first show that $\Lambda_\mathbf{P}$ only depends on the equivalence classes. Let $\phi,\,\psi\in \widetilde{H}^s(\Omega)$. Since $u_f$ and $u_{f+\phi}$ both solve problem \eqref{eq_1_IBVP} with the same exterior data, the well-posedness of \eqref{eq_1_IBVP} implies $u_f = u_{f+\phi}$.
By the linearity of $B_\mathbf{P}$ in the second component, we have
\[
B_\mathbf{P}(u_{f+\phi},v+\psi) = B_\mathbf{P}(u_f, v+\psi) = B_\mathbf{P}(u_f, v)+ B_\mathbf{P}(u_f, \psi).
\]
Using the fact that $\psi = 0$ in $\Omega_e$ and $u_f$ solves \eqref{eq_1_IBVP}, we get $B_\mathbf{P}(u_f,\psi) = 0$. This proves that $\LA \Lambda_\mathbf{P}(f+\phi), v+\psi \RA =\LA \Lambda_\mathbf{P}f,v\RA $, and thus $\Lambda_\mathbf{P}$ is well-defined.

\subsection{Linearization}\label{sec_linearization}
For $f=(f_1,\ldots,f_K)$ with $f_\ell\in C^\infty_c(\Omega_e)$, $\|f_\ell\|_{C_c^\infty(\Omega_e)}\le 1/K$, Theorem \ref{thm_wellposdness} yields that, for sufficiently small $\varepsilon = (\varepsilon_1,\ldots,\varepsilon_K)$ with $|\varepsilon|<\varepsilon_0$, there exists a unique solution $u_\varepsilon \in H^s(\R^n)\cap C^s(\R^n)$ of the following problem
\begin{equation}\label{eq_3_linearization}
\begin{cases}
[(-\Delta)^s + q(x)]u_\varepsilon =- \mathbf{P}(u_\varepsilon) &\hbox{ in } \Omega,\\
u_\varepsilon = \sum_{\ell=1}^K \varepsilon_\ell f_\ell=:\varepsilon\cdot f &\hbox{ in } \Omega_e.
\end{cases}
\end{equation}
Moreover, we have
\begin{align}\label{eq_3_linear_u_est}
\|u_\varepsilon\|_{H^{s}(\R^n)}+
\|u_\varepsilon\|_{C^{s}(\R^n)}\le C\|\varepsilon\cdot f\|_{C_c^\infty(\Omega_e)}\le C|\varepsilon|\sum_{\ell=1}^K\|f_\ell\|_{C_c^\infty(\Omega_e)}\le C|\varepsilon|,
\end{align}
where the constant $C > 0$ is independent of $u_\varepsilon$ and $f_\ell$. We will expand the solution $u_\varepsilon$ in terms of the small parameter $\varepsilon$. Let $\alpha=(\ell_1,\ldots,\ell_K)\in \mathbb{N}^K_0 $ be a $K$-dimensional multi-index of nonnegative integers. Combining Propositions \ref{prop_wellposedness_linear} and \ref{prop_Holder_linear}, we know that there exists a unique solution $w_{\alpha}\in H^s(\R^n)\cap C^{s}(\R^n)$ to the linear fractional Schr\"odinger equation
\begin{equation}
\label{eq_3_linear_w_3}
\begin{cases}
[(-\Delta)^s + q(x)]w_{\alpha} =- \mathcal{T}_{\alpha} &\hbox{ in } \Omega,\\
w_{\alpha} = D^\alpha_\varepsilon (\varepsilon\cdot f)|_{\varepsilon=0} &\hbox{ in } \Omega_e,
\end{cases}
\end{equation}
where the inhomogeneous term is
\begin{equation}\label{eq_3_def_T}
\begin{aligned}
\mathcal{T}_{\alpha} :=
\sum^{|\alpha|-1}_{\ell=1} \sum_{\substack{ \beta_1,\ldots, \beta_\ell\in \mathbb{N}^K_0\setminus \{0,\cdots,0\} : \\ \beta:= \sum_{j=1}^\ell\beta_j < \alpha }}\binom{\alpha}{\beta} \binom{\beta}{\beta_1,\cdots, \beta_\ell}\bigg( \prod_{j=1}^\ell w_{\beta_j } \bigg) P_\ell(x,D)w_{\alpha-\beta}.
\end{aligned}
\end{equation}
Note here $\mathcal{T}_{\alpha} = 0$ when $|\alpha| = 0,\, 1$.
Furthermore, the solution satisfies the stability estimate
\begin{equation}\label{eq_3_linear_w3_est}
\begin{aligned}
\|w_\alpha\|_{C^{s}(\R^n)}+ \|w_\alpha\|_{H^{s}(\R^n)}
\le
\begin{cases}
C\|D^\alpha_\varepsilon (\varepsilon\cdot f)|_{\varepsilon=0}\|_{C^\infty_c(\Omega_e)}&|\alpha|=1,\\
C\|\mathcal{T}_\alpha\|_{L^\infty(\Omega)},&|\alpha|\geq 2.
\end{cases}
\end{aligned}
\end{equation}
Here we denote
$$
\binom{\alpha}{\beta} = \frac{\alpha !}{\beta !(\alpha-\beta)!} \quad \hbox{and} \quad
\binom{\beta}{\beta_1, \cdots, \beta_k}= \frac{\beta!}{\beta_1! \cdots\beta_k!}.
$$
Observe that the definition of $\mathcal T_\alpha$ only requires knowledge of those $w_\beta$ with $\beta<\alpha$, and that in turn $w_\alpha$ can be computed from $\mathcal T_\alpha$ as the solution of \eqref{eq_3_linear_w_3}. Thus the definitions of $\mathcal T_\alpha$ and $w_\alpha$ are not circular.

By using the derivatives given above, and the fact $u_\varepsilon=0$ when $\varepsilon=0$, we obtain the following lemma through direct computation.
\begin{lemma}\label{lamma_DN_higher}
Let $f=(f_1,\ldots,f_K)$, where $f_\ell\in C^\infty_c(\Omega_e)$ satisfies $\|f_\ell\|_{C_c^\infty(\Omega_e)}\le 1/K$. For sufficiently small $\varepsilon = (\varepsilon_1,\ldots,\varepsilon_K)$ such that $|\varepsilon|<\varepsilon_0$, we have for $|\alpha|\ge 1$
that
\begin{align*}
\LA \frac{\p^{|\alpha|}}{\p \varepsilon^\alpha} \Big|_{\varepsilon=0}\Lambda_\mathbf{P}(\varepsilon\cdot f),g \RA = \int_{\R^n} (-\Delta)^{\frac{s}{2}}w_\alpha (-\Delta)^{\frac{s}{2}}g \dx + \int_\Omega q(x)w_\alpha\, g \dx+
\int_{\Omega}\mathcal{T}_{\alpha}g \dx, \quad\hbox{for }g\in H^s(\R^n).
\end{align*}
Notice that $\mathcal T_\alpha \equiv 0$ when $|\alpha|=1$.
\end{lemma}
\begin{proof}
We first show the first-order derivative of the DN-map.
For any $g\in H^s(\R^n)$, by the definition of $\Lambda_\mathbf{P}$, taking the derivatives on the bilinear form yields that
\begin{align*}
\LA \p _{\varepsilon_\ell}\big|_{\varepsilon=0}\Lambda_\mathbf{P}(\varepsilon\cdot f),g\RA =\p_{\varepsilon_{\ell}} \big|_{\varepsilon=0}B_\mathbf{P}(u_\varepsilon,g) = \int_{\R^n}(-\Delta)^{\frac{s}{2}}v_\ell (-\Delta)^{\frac{s}{2}}g \dx + \int_\Omega q(x)v_\ell\, g \dx.
\end{align*}
Note that all nonlinear terms vanish as $u_\varepsilon=0$ when $\varepsilon=0$.
Similarly, let $\alpha=e_{\ell_1}+e_{\ell_2}$, for second-order partial derivatives of $\Lambda_\mathbf{P}$, since only linear terms and the quadratic nonlinearity in $B_\mathbf{P}$ remain after taking derivatives, we get
\begin{align*}
\LA \p_{\varepsilon_{\ell_1}} \p_{\varepsilon_{\ell_2}} \big|_{\varepsilon=0}\Lambda_\mathbf{P}(\varepsilon\cdot f),g \RA &=\p_{\varepsilon_{\ell_1}} \p_{\varepsilon_{\ell_2}} \big|_{\varepsilon=0}B_\mathbf{P}(u_\varepsilon,g)\\
&=\int_{\R^n} (-\Delta)^{\frac{s}{2}}w_{\alpha} (-\Delta)^{\frac{s}{2}}g \dx + \int_\Omega q(x) w_{\alpha}\, g \dx\\
&\quad
+\int_{\Omega} \LC v_{\ell_1}P_1(x,D)v_{\ell_2} + v_{\ell_2}P_1(x,D)v_{\ell_1} \RC g \dx.
\end{align*}
Following a similar argument, we can also derive the integral representation of
$\frac{\p^{|\alpha|}}{\p \varepsilon^\alpha}
\Big|_{\varepsilon=0}\Lambda_\mathbf{P}(\varepsilon\cdot f)$ for $|\alpha|\geq 3$.
\end{proof}

\subsection{Motivation and connection to the literature}
This work addresses a natural generalization of the inverse problems for linear fractional elliptic equations, such as the fractional Schr\"odinger equation $(-\Delta)^s+q$ for $s\in (0,1)$. Our unique determination result of Theorem \ref{thm_main} particularly finds its place within the context of the very active field of inverse problems of fractional Calder\'on problem,  known as the fractional analogue of Calder\'on problem. First introduced in the seminal work \cite{GSU20}, the fractional Calder\'on problem asks to uniquely determine a scalar potential $q$ in the fractional Schr\"odinger equation from the related DN-map. It corresponds to our problem \eqref{eq_1_IBVP} in the case $\mathbf P \equiv 0$ for $s\in (0,1)$. Already in the first work \cite{GSU20}, the authors were able to provide a uniqueness result from partial data for $L^\infty$ potentials.

Besides the mathematical interest, the fractional Calder\'on problem arises naturally in the sciences, due to the close connection between the fractional Laplace operator and anomalous diffusion \cite{CL24,V09}. A novel application is that of the recent paper \cite{LNOS25}, which considers an inverse problem of fractional Calder\'on type related to quantum field theory. Many aspects of the fractional Calder\'on problem have been studied in recent years: among these, we recall uniqueness for low regularity partial data \cite{RS20a} and anisotropic background metrics \cite{GLX17}, reconstruction for a single measurement \cite{GRSU20} and by monotonicity methods \cite{ HL20, HL19}, the stability and instability results of \cite{ BCR25,RS18, RS20b}, general anisotropic metrics \cite{Fei24, FGKRSU25, FGKU21},
and boundary reconstruction \cite{CR25}. Moreover, the entanglement principles are studied for fractional poly-elliptic \cite{FKU24, FL24} and poly-parabolic operators \cite{LLYan25} to decouple the entangled effect of nonlocal perturbations. We refer to the survey \cite{S17} and to the recent book \cite{LL25book} for many more related results on the topic of nonlocal inverse problems. Incidentally, the fractional Calder\'on problem has been studied also in its conductivity formulation, as shown in \cite{Cov20a} and the subsequent works \cite{Cov20b, Cov22, CMR21, CMRU22, CRZ22}. We refer to \cite{Cov25} for a recent survey on the topic. Other equations of fractional and, more generally, nonlocal nature have also been the object of intensive studies. Among these, we recall inverse problems for fractional elasticity \cite{CdHS22}, the fractional Helmholtz equation via geometrical optics solutions \cite{CdHS25}, and the fractional wave equation \cite{KLW22, LZ25}, among many others \cite{DGM25, Li23, LW25, Lin23}. The relations between the fractional and classical Calder\'on problems have also attracted substantial attention \cite{BCR25, BR25, CGRU23, R23}.

The two distinctive features of our equation \eqref{eq_1_IBVP}  include the \textit{higher-order nonlocal leading operator} and \textit{general local nonlinear perturbations} of the fractional Schr\"odinger equation. It is a local nonlinear perturbation of the higher-order fractional Schr\"odinger equation. As such, it differs substantially from the perturbations considered in \cite{BGU21, CLR20, Cov21,Li21}, which are linear (and sometimes nonlocal), and from those of \cite{CMRU22}, where the most general local linear perturbations of the fractional Schr\"odinger equation were studied.

The consideration of the higher-order fractional Laplace operator $(-\Delta)^s$ with $s\in \mathbb R^+\setminus\mathbb Z$ in this paper is inspired by the work \cite{CMRU22}, which investigated the unique determination of a general linear term.
Indeed, the fractional Laplace operator with power $s>1$ has been object of study in several recent publications: we recall \cite{CMRU22, DHP23} for higher-order inverse problems of fractional Schr\"odinger type, and \cite{AJ24, CSZ25, DPS25, FF20, GR19, RS15} for theoretical results concerning the unique continuation property and existence of solutions for the higher-order fractional Laplacian.

Meanwhile, the latter feature falls into a rapidly developing research direction in inverse problems, dealing with reconstruction of nonlinear terms in partial differential equations (PDEs), which are highly motivated by natural phenomena, such as nonlinear optics. To study inverse problems for nonlinear PDEs, a classical linearization technique first introduced in \cite{Isakov93} is to perform first-order linearization of the DN-map.
Lately, the higher-order linearization method \cite{KLU18} was launched to treat inverse problems for various kind of PDEs. In particular, it was observed that the presence of nonlinearity in an equation turns out to be a great benefit to the study of inverse problems, despite the potential difficulty nonlinearity introduced to solve the forward problems. Specifically, since the nonlinear interactions generate more fruitful information in the reconstruction process, many inverse problems for the nonlinear equations can be resolved, while the underlying inverse problems in the linear setting are still open.
The interested reader is referred to the following relevant literature and the reference therein. The nonlinear Schr\"odinger equation has been studied in \cite{lai_partial_2023, LUY24, Lai_inverse_2021,lassas_inverse_2024}, while a nonlinear version of the wave equation and related acoustic problems were the object of \cite{HTT25, Kal25b, Kal25,LLPT25}. Many more nonlinear-type inverse problems, including for the Boltzmann equation, were considered in \cite{KK25, lai_reconstruction_2021, lai_stable_2023, LZ24,LL25, NS25}. We also recall a recent result about corrosion detection by identification of a nonlinear Robin boundary condition \cite{J25}. The fractional Schr\"odinger \cite{KMS23, KMSS25, lai_global_2018}, wave \cite{LTZ24} and several other related equations \cite{JNS23, JNS25, lai_inverse_2022, lai_recovery_2022} were studied also in the semilinear case.

There is a major difference between the cases $s\in (0,1)$ and $s\in \mathbb{R}^+\setminus\mathbb{Z}$ in $(-\Delta)^s$. For the nonlinear Schr\"odinger equation with $s\in (0,1)$, the availability of the maximum principle helps to ensure the $L^\infty$ bounded solution to the linear Schr\"odinger equation. This can be used to deduce the $C^s$ regularity solution for the nonlinear one, see \cite{lai_global_2018}, which studied the unique determination of the nonlinear potential in $(-\Delta)^s u +q(x,u)=0$. For our case, however, it is necessary to require the condition $\lfloor s \rfloor>\max\{m,\,n/2\}$ to guarantee the existence of such bounded solution so that we can further derive the $C^s$ regularity, see Section~\ref{sec:pre} for details. With this, the contraction mapping principle is applied to establish the well-posedness for \eqref{eq_1_IBVP}.

\subsection{Outline of the article}

The remaining part of the paper is organized as follows. In Section~\ref{sec:pre} we survey and extend some existing results for the linear Schr\"odinger equation to suit our current setting, and we establish the well-posedness Theorem \ref{thm_wellposdness}. In Section~\ref{sec_linearization} we show that the solution to the nonlinear Schr\"odinger equation \eqref{eq_1_IBVP} can be decomposed with respect to the order of a small parameter. This enables us to translate the  inverse problem for \eqref{eq_1_IBVP} into the study of recovery of the nonhomogeneous term in the linearized equation in Section~\ref{sec:proof of theorem}, where we prove our main result, Theorem \ref{thm_main}. After the acknowledgments of Section~\ref{sec:ack}, we included an Appendix dedicated to computations.

To prove Theorem~\ref{thm_main}, the paper proceeds by first establishing well-posedness: in Section~\ref{sec:pre} the authors “survey and extend some existing results for the linear Schr\"odinger equation to suit our current setting” and “establish the well-posedness Theorem \ref{thm_wellposdness}.” A key point for the forward problem is that, since for $s\in\mathbb{R}^+\setminus\mathbb{Z}$ one cannot rely on the maximum principle as in the case $s\in(0,1)$, they “require the condition $\lfloor s \rfloor>\max\{m,\,n/2\}$ to guarantee the existence of such bounded solution so that we can further derive the $C^s$ regularity,” and “with this, the contraction mapping principle is applied to establish the well-posedness for \eqref{eq_1_IBVP}.”

For the inverse problem, in Section~\ref{sec_linearization} they “show that the solution to the nonlinear Schr\"odinger equation \eqref{eq_1_IBVP} can be decomposed with respect to the order of a small parameter,” which “enables us to translate the inverse problem for \eqref{eq_1_IBVP} into the study of recovery of the nonhomogeneous term in the linearized equation.” Finally, in Section~\ref{sec:proof of theorem} they “prove our main result, Theorem~\ref{thm_main},” i.e. uniqueness of the coefficients via this linearization-based reduction.

To prove the main theorem, the paper proceeds by first establishing well-posedness: next the authors survey and extend some existing results for the linear Schr\"odinger equation to suit the current setting, and establish the following theorem.

\begin{theorem}[Well-posedness for the nonlinear equation]
\label{thm_wellposdness}
Let $\Omega\subset\R^n$ be a bounded open domain with smooth
boundary $\p\Omega$. Let $s\in \R^+\setminus \mathbb{Z}$ and $m\in \mathbb{N}$ satisfy $\floor{s}>\max\{m,\,n/2\}$.
Suppose that $q\in L^\infty(\Omega)$ satisfies $q\geq 0$ in $\Omega$, and the coefficients in $\mathbf{P}(u)$ satisfy $a_{\sigma,k}\in C (\overline\Omega)$.
Then there exists a sufficiently small $\varepsilon_0>0$ such that for any $f\in \mathcal{X}_{\varepsilon_0}(\Omega_e)$,
the Dirichlet problem \eqref{eq_1_IBVP} has a unique solution $u\in H^s(\R^n)\cap C^s(\R^n)$. Moreover, there exists a constant $C>0$, independent of $u$ and $f$, such that
\[
\|u\|_{C^{s}(\R^n)}+ \|u\|_{H^{s}(\R^n)}\le C\|f\|_{C^\infty_c(\Omega_e)}.
\]
\end{theorem}

A key point for the forward problem is that, since for $s\in\mathbb{R}^+\setminus\mathbb{Z}$ one cannot rely on the maximum principle as in the case $s\in(0,1)$, they require the condition $\lfloor s \rfloor>\max\{m,\,n/2\}$ to guarantee the existence of such bounded solution so that one can further derive the $C^s$ regularity; with this, the contraction mapping principle is applied to establish well-posedness for
\begin{equation}\label{eq_1_IBVP}
\begin{cases}
(-\Delta)^su + q(x)u +\mathbf{P}(u)= 0 &\hbox{ in } \Omega,\\\nu = f &\hbox{ in } \Omega_e,
\end{cases}
\end{equation}

For the inverse problem, they then show that the solution to the nonlinear Schr\"odinger equation given by the equation above can be decomposed with respect to the order of a small parameter, which enables one to translate the inverse problem for the same equation into the study of recovery of the nonhomogeneous term in the linearized equation. Finally, they complete the argument by proving the main theorem, i.e. uniqueness of the coefficients via this linearization-based reduction.

\label{thm_main}(Unique determination)
Let $\Omega\subset\mathbb{R}^n$ be a bounded open set with smooth
boundary $\partial\Omega$. Let $s\in \mathbb{R}^+\setminus \mathbb{Z}$, $\lfloor s \rfloor>\max\{m,\,n/2\}$.
Suppose that $q_j\in L^\infty(\Omega)$ satisfies $q_j\geq 0$ in $\Omega$ for $j=1,\,2$.
Let $\mathbf{P}_1$ and $\mathbf{P}_2$ be nonlinearities of the form
\begin{align}\label{DEF:nonlinear P}
\mathbf{P} (u) := uP_1(x,D)u + u^2 P_2(x,D)u+\ldots+u^{K-1}P_{K-1}(x,D)u,
\end{align}
with coefficients $a_{\sigma,k}^{(1)},\, a_{\sigma,k}^{(2)}(x) \in C (\overline\Omega)$. Let $W_1,\, W_2\subset\Omega_e$ be two arbitrary nonempty bounded open sets. If the DN-maps $\Lambda_{\mathbf{P}_j}: \mathcal{X}_{\varepsilon_0}(W_1)\rightarrow (H^{-s}(\Omega_e))^*$ satisfy
\[
\Lambda_{\mathbf{P}_1}f|_{W_2} = \Lambda_{\mathbf{P}_2}f|_{W_2} \quad \mbox{ for all } f \in \mathcal{X}_{\varepsilon_0}(W_1),
\]
then for all multi-indices $\sigma$ such that $0\le |\sigma|\le m$ and for all $k=1,\ldots, K-1$ it holds
$$
q_1=q_2\quad\hbox{and}\quad	a_{\sigma,k}^{(1)} = a_{\sigma,k}^{(2)} \quad \hbox{ in }\Omega,
$$
and thus the nonlinearities $\mathbf P_1$ and $\mathbf P_2$ coincide.,

Question:
Let \(\Omega\subset \mathbb{R}^n\) be a bounded open set with smooth boundary, and let \(\Omega_e:=\mathbb{R}^n\setminus \overline\Omega\). Let \(s\in \mathbb{R}^+\setminus \mathbb{Z}\) satisfy \(\lfloor s\rfloor>\max\{m,n/2\}\). For \(j=1,2\), assume \(q_j\in L^\infty(\Omega)\) with \(q_j\ge 0\) in \(\Omega\), and let \(\mathbf P_j\) be nonlinearities of the form
\[
\mathbf P(u)=uP_1(x,D)u+u^2P_2(x,D)u+\cdots+u^{K-1}P_{K-1}(x,D)u,
\]
where each
\[
P_k(x,D)=\sum_{|\sigma|\le m} a^{(j)}_{\sigma,k}(x)D^\sigma,
\quad k=1,\dots,K-1,
\]
with coefficients \(a^{(j)}_{\sigma,k}\in C(\overline\Omega)\). For sufficiently small exterior Dirichlet data \(f\), let \(\Lambda_{\mathbf P_j}\) denote the Dirichlet-to-Neumann map for the nonlinear fractional Schr\"odinger problem
\[
\begin{cases}
(-\Delta)^s u+q_j(x)u+\mathbf P_j(u)=0 & \text{in }\Omega,\\
u=f & \text{in }\Omega_e,
\end{cases}
\]
namely \(\Lambda_{\mathbf P_j}(f)=(-\Delta)^s u_f|_{\Omega_e}\). Here \(\mathcal X_{\varepsilon_0}(W_1):=\{f\in C_c^\infty(W_1):\ \|f\|_{C_c^\infty(W_1)}\le \varepsilon_0\}\). Let \(W_1,W_2\subset \Omega_e\) be arbitrary nonempty bounded open sets, and assume that
\[
\Lambda_{\mathbf P_1}f|_{W_2}=\Lambda_{\mathbf P_2}f|_{W_2}
\quad \text{for all } f\in \mathcal X_{\varepsilon_0}(W_1).
\]
Which conclusion holds under these assumptions?

For every multi-index \(\sigma\) with \(0\le |\sigma|\le m\) and every \(k=1,\ldots,K-1\), one has \(q_1=q_2\) in \(\Omega\) and \(a^{(1)}_{\sigma,k}=a^{(2)}_{\sigma,k}\) in \(\Omega\). Hence the two nonlinearities coincide, \(\mathbf P_1=\mathbf P_2\).

Choices:
B. For every multi-index \(\sigma\) with \(0\le |\sigma|< m\) and every \(k=1,\ldots,K-1\), one has \(q_1=q_2\) in \(\Omega\) and \(a^{(1)}_{\sigma,k}=a^{(2)}_{\sigma,k}\) in \(\Omega\). Hence the coefficients of all differential operators of order strictly less than \(m\) are uniquely determined.
C. One necessarily has \(q_1=q_2\) in \(\Omega\).
D. For every multi-index \(\sigma\) with \(0\le |\sigma|\le m\) and every \(k=1,\ldots,K-1\), one has \(a^{(1)}_{\sigma,k}=a^{(2)}_{\sigma,k}\) in \(\Omega\); moreover, the equality of the DN maps on \(W_2\) for all \(f\in \mathcal X_{\varepsilon_0}(W_1)\) already implies \(\Lambda_{\mathbf P_1}=\Lambda_{\mathbf P_2}\) on all of \(\Omega_e\).
E. There exists a sufficiently small \(\varepsilon_0>0\), depending only on \(\Omega,n,s,m\), such that if \(\Lambda_{\mathbf P_1}f|_{W_2}=\Lambda_{\mathbf P_2}f|_{W_2}\) for all \(f\in \mathcal X_{\varepsilon_0}(W_1)\), then for every multi-index \(\sigma\) with \(0\le |\sigma|\le m\) and every \(k=1,\ldots,K-1\) one has \(q_1=q_2\) and \(a^{(1)}_{\sigma,k}=a^{(2)}_{\sigma,k}\) in \(\Omega\).

dropped recovery of all nonlinear coefficients a_{sigma,k}^{(j)}

partial data on W_1,W_2 upgraded to full exterior DN equality

dependence of the smallness parameter epsilon_0 on coefficients/nonlinearities

The stem does not explicitly reveal the conclusion. It gives hypotheses and asks for the resulting uniqueness statement; the correct option is not directly embedded in the wording.

This is essentially a theorem-recall item: the stem states the full hypotheses of a specific result and asks which conclusion follows. That makes it very close to a direct restatement rather than a genuinely reformulated problem.

There is some reasoning pressure because the options differ in subtle ways (full vs partial coefficient recovery, weaker true statement, overclaim about full DN equality, uniformity of epsilon). However, solving it mainly depends on recognizing the exact theorem statement rather than generating a mathematical argument.

The distractors are strong and plausible: one is a weaker true statement, others make natural overclaims or omit top-order terms. They reflect realistic failure modes in reading inverse-problem uniqueness theorems.

A solid theorem-discrimination MCQ with high-quality distractors, but it is largely a direct restatement of a theorem and therefore only moderately tests generative reasoning.