\label{thm:deg}
Consider a projective variety $X \subset {\mathbb P}^n$ (over an algebraically closed field ${\mathbb K}$ of characteristic zero),
together with a (reduced) simple normal crossings (snc) divisor $E \subset {\mathbb P}^n$.
Assume that the degrees of both $X$ and $E$ are at most $d$.
Then there is a pair $(n',d')$ which can be explicitly computed in terms of $(n,d)$, such that $(X,E)$ has a log resolution of singularities
$(X',E')$, where $(X',E')$ can be embedded in ${\mathbb P}^{n'}$ and both $X'$ and $E'$ have degrees at most $d'$ in ${\mathbb P}^{n'}$.

\begin{remark}[\bf Summary remark]\label{rem:summaryremark}
The data involved in resolution of singularities of a marked ideal $\ucI = (Z,X,E,\cI,\mu)$
with initial data $\ga = (0,n,m,d,l,q,\mu)$, have bounds given by the recursive functions above, with final data
vector majorized by $\Ga^{(m)}(\ga)$.

As in \cite{BGMW}, we can also keep track of the Grzegorczyk complexity classes of the resolution data.
Beginning with Step II above, $M(n,d) \in \cE^3$ (cf. Lemma \ref{lem:mult}; we recall that, in general, $\cE^1$ contains
all linear functions, $\cE^2$ all polynomials, and $\cE^3$ all towers of exponential functions$,\ldots$). In Step IB, then $\De_1(\ga)
\in \cE^3$ and, by induction on $m$, $\Ga^{(m-1)}(\De_\mathrm{I}(\ga)) \in \cE^{m+2}$, and $\Ga_{\mathrm{I}}^{(m)}(\ga) \in \cE^{m+2}$.
In Step IIA, $\Ga_{\mathrm{IIB}}^{(m)}(\ga) \in \cE^{m+3}$, from Lemmas \ref{lem:blupsummary} and \ref{lem:mult},
and finally $\Ga^{(m)}(\ga) \in \cE^{m+3}$,
in Step IIB. For more details, see \cite{BGMW}.
\end{remark}

\begin{theorem}\label{thm:deg}
Consider a projective variety $X \subset \IP^n$ (over an algebraically closed field $\IK$ of characteristic zero),
together with a (reduced) simple normal crossings (snc) divisor $E \subset \IP^n$.
Assume that the degrees of both $X$ and $E$ are at most $d$.
Then there is a pair $(n',d')$ which can be explicitly computed in terms of $(n,d)$, such that $(X,E)$ has a log resolution of singularities
$(X',E')$, where $(X',E')$ can be embedded in $\IP^{n'}$ and both $X'$ and $E'$ have degrees at most $d'$ in $\IP^{n'}$.
\end{theorem}

\label{sec:intro}

The \emph{degree} $\deg X$ of an irreducible projective variety $X \subset {\mathbb P}^n$ (over an algebraically closed field ${\mathbb K}$)
denotes the number of intersection points of $X$ with a generic linear subspace $L$ of complementary dimension ($\dim L =
n - \dim X$). In general, the \emph{degree} of $X \subset {\mathbb P}^n$ is the sum of the degrees of the irreducible components of $X$
of all dimensions. The \emph{degree} of a divisor $E \subset {\mathbb P}^n$ likewise means the sum of the degrees of its (reduced) components (which
are not necessarily irreducible).

For example, if $X$ is a hypersurface and $F$ generates the ideal of homogeneous polynomials vanishing on $X$, then
$\deg X$ equals the degree of the homogeneous polynomial $F$.

The purpose of this article is to give a proof of the following result that Caucher Birkar posed to us as a question.

\begin{abstract}
Consider a projective variety $X \subset \IP^n$ (over an algebraically closed field of characteristic zero),
together with a (reduced) simple normal crossings divisor $E \subset \IP^n$, where
the degrees of both $X$ and $E$ are at most $d$. We show
there is a pair $(n',d')$ which can be explicitly computed in terms of $(n,d)$, such that $(X,E)$ has a log resolution of singularities
$(X',E')$, where $(X',E')$ can be embedded in $\IP^{n'}$ and both $X'$ and $E'$ have degrees at most $d'$ in $\IP^{n'}$.
\end{abstract}

The \emph{degree} $\deg X$ of an irreducible projective variety $X \subset \IP^n$ (over an algebraically closed field $\IK$)
denotes the number of intersection points of $X$ with a generic linear subspace $L$ of complementary dimension ($\dim L =
n - \dim X$). In general, the \emph{degree} of $X \subset \IP^n$ is the sum of the degrees of the irreducible components of $X$
of all dimensions. The \emph{degree} of a divisor $E \subset \IP^n$ likewise means the sum of the degrees of its (reduced) components (which
are not necessarily irreducible).

The purpose of this article is to give a proof of the following result that Caucher Birkar posed to us as a question.

An snc divisor $E$ transforms by a blowing-up $\s$ with smooth centre $C$ that is snc with $E$, to an snc divisor defined
by the strict (or birational) transform of $E$ plus the exceptional divisor of $\s$.
\emph{Log resolution} of $(X,E)$ means a resolution of singularities $X' \to X$ given by a composite of smooth blowings-up
as above, such that $X', E'$ have only simple normal crossings, where $E'$ is the final transform of $E$.
If we assume that $E$ is ordered, then $E'$ is also ordered following the sequence of blowings-up.

\begin{theorem}\label{thm:gendeg}
Consider a projective variety $X \subset \IP^n$ (over an algebraically closed field $\IK$ of characteristic zero),
together with an snc divisor $E \subset \IP^n$.
We assume that $X$ and $E$
can be defined by homogeneous polynomials of degrees at most $d$. Then there is a pair
$(n',d')$ which can be explicitly computed in terms of $(n,d)$, such that $(X,E)$ has a log resolution of singularities
$(X',E')$, where $(X',E')$ can be embedded in $\IP^{n'}$ and defined in the latter by homogeneous polynomials
of degrees at most $d'$.
\end{theorem}

Theorem \ref{thm:gendeg} is a consequence of estimates given in \cite{BGMW} or in Sections \ref{sec:affmarkedideal}--\ref{sec:summaryest}
below, in the context of log resolution of
singularities of an embedded algebraic variety $X \hookrightarrow Z$ ($Z$ smooth) together with an ordered snc divisor $E \subset Z$.
The triple $(Z,X,E)$ can be defined locally by polynomial data in affine spaces $\IA^n$ with rational transition mappings,
and there is a log resolution of singularities $(X',E') \subset Z'$ which can be defined by data with effective bounds
on local affine embedding dimension, degrees of all polynomials involved, number of affine charts, number of blowings-up
needed, etc., in terms of bounds on local data needed to define $(X,E) \subset Z$.
Example \ref{ex:proj} below shows how to apply these effective bounds to obtain Theorem \ref{thm:gendeg}.

\begin{example}\label{ex:proj} \emph{Case of a projective variety and snc divisor $(X,E) \subset \IP^n$.} We begin with
a marked ideal $\cI$ as in Example \ref{ex:princ,desing}(2), with $Z=\IP^n$, and we can reduce to the case
that $E=\emptyset$ by replacing $(X,E)$ by $(X\cup E,\,\emptyset)$. Then there is an \emph{affine marked ideal} $\ucT$
(a certain collection of resolution data associated to $\ucI$) as in
Section \ref{sec:affmarkedideal} below, where the affine charts
$(\IK^{n_\al})_\al$ are affine charts of $\IP^n$ (each $n_\al = n$). See Definition \ref{def:affmarkedideal} and Remark
\ref{rem:affmarkedideal}. In the initial data given by Definition \ref{notn:affmarkedideal}, all terms are bounded in a
simple way in terms of the given degree bounds of Theorem \ref{thm:gendeg}.

\begin{definition}\label{def:affmarkedideal}
An \emph{affine marked ideal} $\ucT$ is a collection of tuples together with an associated order $\mu$,
\begin{equation}\label{eq:affmarkedideal}
\ucT = \left(\left\{U_{\al\be}, X_{\al\be}, E_{\al\be}, \cI_{\al\be}, \left(\IK^{n_\al}\right)_\al: \al \in A,\,\be\in B_\al\right\}, \mu\right),
\end{equation}
where $A$ and the $B_\al$ are finite index sets, and
\begin{enumerate}
\item $(\IK^{n_\al})_\al \cong \IK^{n_\al}$, with affine coordinates $x_\al = (x_{\al 1},\ldots,x_{\al,n_\al})$;
\item $\{U_{\al\be}: \be\in B_\al\}$ is an open covering of $(\IK^{n_\al})_\al$, where $U_{\al\be} \subset (\IK^{n_\al})_\al$
is the complement of the zero set of a polynomial $f_{\al\be} \in \IK[x_\al]$;
\item $E_{\al\be}$ is a collection of smooth divisors in $(\IK^{n_\al})_\al$, each given by an equation $x_{\al j} = 0$,
for some $j=1,\ldots,n_\al$;
\item $X_{\al\be} \subset (\IK^{n_\al})_\al$ is a closed subset, where $X_{\al\be} \cap U_{\al\be}$ is smooth; moreover,
there is a set of parameters (coordinates) on $U_{\al\be}$,
$$
u_{\al\be, 1},\ldots,u_{\al\be,n_\al} \in \IK[x_\al],
$$
where each $u_{\al\be, i}$ is either a coordinate $x_{\al j}$ describing an exceptional divisor (i.e., an element of $E_{\al\be}$)
or is transverse to $E_{\al\be}$ over $U_{\al\be}$,
and $\cI_{X_{\al\be}}$ is the ideal
$$
\cI_{X_{\al\be}} = (u_{\al\be, 1},\ldots,u_{\al\be,n_\al-m}) \subset \IK[x_\al],
$$
with $u_{\al\be, 1},\ldots,u_{\al\be,n_\al-m}$ all transverse to $E_{\al\be}$;
\item $\cI_{\al\be}$ is an ideal $(g_{\al\be,1},\ldots,g_{\al\be,\overline{j}}) \subset \IK[x_\al]$,
and
$$
\cosupp (\cI_{\al\be},\mu) \cap U_{\al\be} \cap U_{\al\be'} = \cosupp (\cI_{\al\be'},\mu) \cap U_{\al\be} \cap U_{\al\be'},
$$
where $(\cI_{\al\be},\mu)$ denotes the marked ideal $(U_{\al\be},X_{\al\be}\cap U_{\al\be}, E_{\al\be}, \cI_{\al\be},\mu)$;
\item for all $\al_1,\al_2 \in A$, $\be_1\in B_{\al_1}$, $\be_2\in B_{\al_2}$,
there exist
\begin{align*}
v_{\al_1\be_1\al_2\be_2,1},\ldots,v_{\al_1\be_1\al_2\be_2,n_{\al_2}} &\in \IK[x_{\al_1}],\\
w_{\al_1\be_1\al_2\be_2,1},\ldots,w_{\al_1\be_1\al_2\be_2,n_{\al_2}} &\in \IK[x_{\al_1}],
\end{align*}
such that
$$
x_{\al_1} \mapsto \left(\frac{v_{\al_1\be_1\al_2\be_2,1}}{w_{\al_1\be_1\al_2\be_2,1}},\ldots,\frac{v_{\al_1\be_1\al_2\be_2,n_{\al_2}}}{w_{\al_1\be_1\al_2\be_2,n_{\al_2}}}\right)(x_{\al_1})
$$
induces a birational mapping $i_{\al_1\be_1\al_2\be_2}: X_{\al_1\be_1} \dashrightarrow X_{\al_2\be_2}$;
moreover, the birational mappings $i_{\al_1\be_1\al_2\be_2}$ determine a variety $X_{\ucT}$, unique up to isomorphism,
together with open embeddings $j_{\al\be}: X_{\al\be} \cap U_{\al\be} \hookrightarrow X_{\ucT}$ defining an open covering
of $X_{\ucT}$, such that $j_{\al_2\be_2}^{-1}\circ j_{\al_1\be_1} = i_{\al_1\be_1\al_2\be_2}$;
\item $\mu$ is a nonnegative integer.
\end{enumerate}
\end{definition}

An snc divisor $E$ transforms by a blowing-up ${\sigma}$ with smooth centre $C$ that is snc with $E$, to an snc divisor defined
by the strict (or birational) transform of $E$ plus the exceptional divisor of ${\sigma}$.
\emph{Log resolution} of $(X,E)$ means a resolution of singularities $X' \to X$ given by a composite of smooth blowings-up
as above, such that $X', E'$ have only simple normal crossings, where $E'$ is the final transform of $E$.
If we assume that $E$ is ordered, then $E'$ is also ordered following the sequence of blowings-up.

We prove, in fact, the following variant of Theorem \ref{thm:deg}; the two assertions are equivalent because of the degree
bounds in Section \ref{sec:deg} below

\begin{theorem}\label{thm:gendeg}
Consider a projective variety $X \subset {\mathbb P}^n$ (over an algebraically closed field ${\mathbb K}$ of characteristic zero),
together with an snc divisor $E \subset {\mathbb P}^n$.
We assume that $X$ and $E$
can be defined by homogeneous polynomials of degrees at most $d$. Then there is a pair
$(n',d')$ which can be explicitly computed in terms of $(n,d)$, such that $(X,E)$ has a log resolution of singularities
$(X',E')$, where $(X',E')$ can be embedded in ${\mathbb P}^{n'}$ and defined in the latter by homogeneous polynomials
of degrees at most $d'$.
\end{theorem}

Theorem \ref{thm:gendeg} is a consequence of estimates given in \cite{BGMW} or in Sections \ref{sec:affmarkedideal}--\ref{sec:summaryest}
below, in the context of log resolution of
singularities of an embedded algebraic variety $X \hookrightarrow Z$ ($Z$ smooth) together with an ordered snc divisor $E \subset Z$.
The triple $(Z,X,E)$ can be defined locally by polynomial data in affine spaces ${\mathbb A}^n$ with rational transition mappings,
and there is a log resolution of singularities $(X',E') \subset Z'$ which can be defined by data with effective bounds
on local affine embedding dimension, degrees of all polynomials involved, number of affine charts, number of blowings-up
needed, etc., in terms of bounds on local data needed to define $(X,E) \subset Z$.
Example \ref{ex:proj} below shows how to apply these effective bounds to obtain Theorem \ref{thm:gendeg}.

The estimates in the article show that the contribution of the dimension $n$ to $(n',d')$ in the theorems above
dwarfs that of the degree $d$. This is highlighted in \cite{BGMW} by a complexity bound in terms of
Grzegorczyk complexity classes ${\mathcal E}^l$, $l\geq 0$, of primitive recursive (integer) functions,
where the functions in each ${\mathcal E}^l$ require at most $l$ nested primitive recursions \cite{Grz}, \cite{WW}. The number of nested
recursions involved in the desingularization algorithm for $(X,E) \subset {\mathbb P}^n$, is bounded by $n + 3$ (cf.
Remark \ref{rem:summaryremark}).

We use the algorithm for functorial resolution of singularities as presented in \cite{BMinv}, \cite{BMfunct} (the version
in \cite{Wlodar} was used in \cite{BGMW}). Log resolution of singularities
of an embedded pair $(X,E) \subset Z$, or of an ideal ${\mathcal I} \subset \cO_Z$ together with an snc divisor $E\subset Z$,
follows from resolution of singularities of a collection of resolution data called a \emph{marked ideal}. (Non-embedded)
desingularization of a pair $(X,E)$ follows from functoriality of the embedded case (applied locally).

There are several articles in the literature on implementation of algorithms for resolution of singularities (e.g., \cite{BS},
\cite{F-KP}). Sections \ref{sec:affmarkedideal}-\ref{sec:bounds} below can be compared with explicit computation of marked ideals
in \cite{BS}, which raises the challenge of ``super-exponential growth of exponents'' for generators of coefficient marked ideals.
The purpose of our methods is to provide effective bounds on measures of the complexity of the algorithm.

The post-theorem text does not give step-by-step “main steps” for proving Theorem~\ref{thm:deg}, but it does indicate the proof strategy via a variant and where the needed bounds come from: one proves the variant Theorem~\ref{thm:gendeg}, and “the two assertions are equivalent because of the degree bounds in Section~\ref{sec:deg} below.” Then “Theorem~\ref{thm:gendeg} is a consequence of estimates given in \cite{BGMW} or in Sections~\ref{sec:affmarkedideal}--\ref{sec:summaryest} below,” namely effective bounds “in the context of log resolution of singularities of an embedded algebraic variety $X\hookrightarrow Z$ ($Z$ smooth) together with an ordered snc divisor $E\subset Z$.” The argument is described as: define $(Z,X,E)$ locally by “polynomial data in affine spaces ${\mathbb A}^n$ with rational transition mappings,” apply the functorial resolution algorithm (“as presented in \cite{BMinv}, \cite{BMfunct}”) and the reduction to “a collection of resolution data called a \emph{marked ideal},” obtaining a log resolution $(X',E')\subset Z'$ “defined by data with effective bounds on local affine embedding dimension, degrees of all polynomials involved, number of affine charts, number of blowings-up needed, etc., in terms of bounds on local data needed to define $(X,E)\subset Z$.” Finally, “Example~\ref{ex:proj} below shows how to apply these effective bounds to obtain Theorem~\ref{thm:gendeg},” and the non-embedded case follows since “(Non-embedded) desingularization of a pair $(X,E)$ follows from functoriality of the embedded case (applied locally).”

The post-theorem text does not give step-by-step “main steps” for proving the main theorem, but it does indicate the proof strategy via a variant and where the needed bounds come from: one proves the variant
\begin{theorem}\label{thm:gendeg}
Consider a projective variety $X \subset \IP^n$ (over an algebraically closed field $\IK$ of characteristic zero),
together with an snc divisor $E \subset \IP^n$.
We assume that $X$ and $E$
can be defined by homogeneous polynomials of degrees at most $d$. Then there is a pair
$(n',d')$ which can be explicitly computed in terms of $(n,d)$, such that $(X,E)$ has a log resolution of singularities
$(X',E')$, where $(X',E')$ can be embedded in $\IP^{n'}$ and defined in the latter by homogeneous polynomials
of degrees at most $d'$.
\end{theorem}
(and “the two assertions are equivalent because of the degree bounds in Section~\ref{sec:deg} below.”). Then “Theorem~\ref{thm:gendeg} is a consequence of estimates given in \cite{BGMW} or in Sections~\ref{sec:affmarkedideal}--\ref{sec:summaryest} below,” namely effective bounds “in the context of log resolution of singularities of an embedded algebraic variety $X\hookrightarrow Z$ ($Z$ smooth) together with an ordered snc divisor $E\subset Z$.” The argument is described as: define $(Z,X,E)$ locally by “polynomial data in affine spaces ${\mathbb A}^n$ with rational transition mappings,” apply the functorial resolution algorithm (“as presented in \cite{BMinv}, \cite{BMfunct}”) and the reduction to “a collection of resolution data called a \emph{marked ideal},” obtaining a log resolution $(X',E')\subset Z'$ “defined by data with effective bounds on local affine embedding dimension, degrees of all polynomials involved, number of affine charts, number of blowings-up needed, etc., in terms of bounds on local data needed to define $(X,E)\subset Z$.” Finally, the following example shows how to apply these effective bounds to obtain Theorem~\ref{thm:gendeg}:

\begin{example}\label{ex:proj} \emph{Case of a projective variety and snc divisor $(X,E) \subset \IP^n$.} We begin with
a marked ideal $\cI$ as in Example \ref{ex:princ,desing}(2), with $Z=\IP^n$, and we can reduce to the case
that $E=\emptyset$ by replacing $(X,E)$ by $(X\cup E,\,\emptyset)$. Then there is an \emph{affine marked ideal} $\ucT$
(a certain collection of resolution data associated to $\ucI$) as in
Section \ref{sec:affmarkedideal} below, where the affine charts
$(\IK^{n_\al})_\al$ are affine charts of $\IP^n$ (each $n_\al = n$). See Definition \ref{def:affmarkedideal} and Remark
\ref{rem:affmarkedideal}. In the initial data given by Definition \ref{notn:affmarkedideal}, all terms are bounded in a
simple way in terms of the given degree bounds of Theorem \ref{thm:gendeg}.

If $\s: Z' \to Z$ is a blowing-up of $Z$ with smooth closed centre $C$, then $Z'$ is embedded in $\IP^n\times\IP^r$,
$r = \codim C - 1$, and the affine marked ideal $\ucT'$ defined over $\ucT$ by the blowing-up (Section \ref{sec:affmarkedideal})
has the affine charts of $\IP^n\times\IP^r$; $\ucT'$ corresponds to a marked ideal on the strict transform $Z'$ of $Z$
in $\IP^n\times\IP^r$.

The final affine marked ideal $\ucT_*$ given by the resolution algorithm corresponds to a marked ideal $\ucI_*$
on the final strict transform of $Z$ in a finite product $\IP^n\times\IP^{r_1}\times\IP^{r_2}\times\cdots$, where the codimensions
of the successive centres of blowing up are $r_1+1,\, r_2+1,\ldots$ in the successive products of projective spaces.
We can embed the final product in $\IP^{n'}$, for suitable $n'$, using the Segre embedding, and effective estimates
for $(X',E')$ follow from those of Section \ref{sec:summaryest}.
\end{example}

and the non-embedded case follows since “(Non-embedded) desingularization of a pair $(X,E)$ follows from functoriality of the embedded case (applied locally).”

Question: Let $X \subset \mathbb P^n$ be a projective variety over an algebraically closed field $\mathbb K$ of characteristic zero, and let $E \subset \mathbb P^n$ be a reduced simple normal crossings divisor. Here the degree of a projective variety is the sum of the degrees of its irreducible components, and for an irreducible component it is the number of intersection points with a generic linear subspace of complementary dimension; the degree of a divisor is the sum of the degrees of its reduced components. Assume that both $X$ and $E$ have degree at most $d$. A log resolution of $(X,E)$ means a resolution of singularities obtained by a sequence of smooth blowings-up such that the final variety $X'$ and the final transform $E'$ are both simple normal crossings. Which statement holds for $(X,E)$ under these assumptions?

There exists a pair $(n',d')$, explicitly computable in terms of $(n,d)$, such that $(X,E)$ admits a log resolution of singularities $(X',E')$ with $(X',E')$ embeddable in $\mathbb P^{n'}$, and in that embedding both $X'$ and $E'$ have degree at most $d'$.

Choices:
B. There exists a pair $(n',d')$, explicitly computable in terms of $(n,d)$, such that $(X,E)$ admits a log resolution of singularities $(X',E')$ with $(X',E')$ embeddable in $\mathbb P^{n'}$, and in that embedding both $X'$ and $E'$ can be defined by homogeneous polynomials of degrees at most $d'$.
C. $(X,E)$ admits a log resolution of singularities $(X',E')$ such that $(X',E')$ is embeddable in some projective space $\mathbb P^{N}$.
D. There exists a pair $(n',d')$, explicitly computable in terms of $(n,d)$, such that for every log resolution of singularities $(X',E')$ of $(X,E)$, one can embed $(X',E')$ in $\mathbb P^{n'}$ so that both $X'$ and $E'$ have degree at most $d'$ in $\mathbb P^{n'}$.
E. There exists a pair $(n',d')$, explicitly computable in terms of $(n,d)$, such that $(X,E)$ admits a log resolution of singularities $(X',E')$ with $(X',E')$ embeddable in $\mathbb P^{n'}$, and in that embedding both $X'$ and $E'$ have degrees exactly at most $d'$ independently of the number and codimensions of the blowings-up used to obtain the resolution.

degree-bounded conclusion replaced by defining-equations-bounded conclusion

dropped explicit computability and uniform bounds on ambient dimension and degrees

existential choice of one bounded resolution strengthened to all log resolutions

dependence on successive blow-up codimensions and Segre-embedding growth suppressed


The present work characterizes, under the model-theoretic hypothesis of finite-dimensionality,
irreducible bi-modules, i.e., abelian groups together with two commuting subrings of endomorphisms, and which are "irreducible" for the bi-action.\\
This result is well-known for groups of finite Morley rank \cite{zilber1984some}. It is also known for $o$-minimal theories (see, for example \cite{peterzil2000simple}). The main problem is that the two proofs use techniques proper to the two families of theories involved (in the finite Morley rank case, the indecomposability theorem is necessary). A natural question arises: to which generality can we extend the linearization?\\
An interesting family of theories is finite-dimensional theories, in the sense of \cite{wagner2020dimensional}.
\begin{defn}
    A theory $T$ is said to be \emph{finite-dimensional} if there exists a function $\operatorname{dim}$ from the class of all interpretable subset in any model $\mathcal{M}$ of $T$ into $\omega\cup\{-\infty\}$ such that for any $\phi(x,y)$ formula, $X,Y$ interpretable sets in $T$ and $f$ interpretable function from $X$ to $Y$,
\begin{itemize}
    \item If $a,a'$ have same type over $\emptyset$, $\operatorname{dim}(\phi(x,a))=\operatorname{dim}(\phi(x,a'))$
    \item $\operatorname{dim}(\emptyset)=-\infty$ and $\operatorname{dim}(X)=0$ if and only if $X$ is finite;
    \item $\operatorname{dim}(X\cup Y)=\max\{\operatorname{dim}(X),\operatorname{dim}(Y)\}$;
    \item If $\operatorname{dim}(f^{-1}(y))\geq k$ if for any $y\in Y$, then $\operatorname{dim}(X)\geq \operatorname{dim}(Y)+k$;
    \item If $\operatorname{dim}(f^{-1}(y))\leq k$ for any $y\in Y$, then $\operatorname{dim}(X)\leq \operatorname{dim}(Y)+k$.
\end{itemize}
\end{defn}
Examples of finite-dimensional theories are $o$-minimal theories, supersimple theories of finite Lascar rank, and supperosy theories of finite $U^{\text{\thorn}}$-rank.\\
 A first proof of Zilber's Field Theorem in this context has been given by Deloro in \cite{deloro2024zilber}. An important hypothesis of the theorem is that the module on which the ring of endomorphisms acts is connected. This clearly holds in the finite Morley rank and $o$-minimal context, but in general is not true: it is sufficient to take a supersimple theory of finite Lascar rank. Therefore, it remains unclear what happens for non virtually connected modules. The aim of this article is exactly to analyse this case.\\
 In reality, we will work with two more general notions of endomorphisms: endogenies and quasi-endomorphisms. An endogeny $\phi$ of an abelian group $A$ is a subgroup of $A\times A$ such that $\pi_1(\phi)=A$ and $\{a\in A:\ (0,a)\in \phi\}$ is finite. The latter subgroup is called the \emph{katakernel} of $\phi$. A \emph{quasi-endomorphism} $\phi$ is a subgroup of $A$ such that $\pi_1(\phi)$, that is called the domain of $\phi$, is of finite index in $A$ and the katakernel is finite. Therefore, an endogeny is a quasi-endomorphism that is total.\\
Endogenies and quasi-endomorphisms arise naturally in the finite-dimensional context. Indeed, let $H\geq H_1$ be definable subgroups of the definable abelian group $G$ and $\Gamma\leq \operatorname{End}(G)$. Assume that $H$ and $H_1$ are \emph{$\Gamma$-almost invariant} \hbox{i.e.} $\gamma(H)\cap H$ is of finite index in $\gamma(H)$ for any $\gamma\in \Gamma $. Then, $\Gamma$ acts on $G/H$ by endogenies (with katakernel of $\gamma$ equal to $\gamma(H)+H/H$) and $\Gamma$ acts by quasi-endomorphisms on $H/H_1$ with domain $\{h\in H:\ \gamma(h)\in H\}$. In the finite Morley rank context and in the $o$-minimal one, we can avoid working with endogenies or quasi-endomorphisms since, if $H$ is $\Gamma$-almost invariant, the connected component $H^0$ is $\Gamma$-invariant and the action of $\Gamma$ on $G/H^0$ is again by endomorphisms.\\
The article follows the ideas of \cite{WagnerDeloro}, which proves the linearization theorem for endogenies in the connected case.\\
In the first section, we introduce all the definitions that we need in the article, and we verify some easy lemmas.\\
Then, we proceed to prove the following result.
\begin{thmA}
In a finite-dimensional theory, let $A$ be an abelian definable group and $\Gamma,\Delta$ two invariant pre-rings of definable endogenies such that:
\begin{itemize}
    \item $\Gamma,\Delta$ sharply commute;
    \item $A$ is absolutely $(\Gamma,\Delta)$-minimal;
    \item $\Gamma$ is essentially unbounded and $\Delta$ is essentially infinite or vice-versa.
\end{itemize}
    There is a finite $\Gamma$ and $\Delta$-invariant subgroup $F$ such that $\Gamma/{\sim}$ and $\Delta/{\sim}$ act by endomorphisms on $A/F$.
\end{thmA}
Moreover, $F$ will be given explicitly.\\
In the second section, we verify the easy case in which both $\Gamma$ and $\Delta$ act by endogenies with finite kernel or finite image.\\
The third section analyzes the case in which the unbounded pre-ring of endogenies has an endogeny with infinite kernel and infinite image. Finally, we complete the proof in the fourth section.\\
The main result of the fifth section is the following linearization theorem.
\begin{thmB}
    Let $A$ be an abelian definable group and $\Gamma,\Delta$ two invariant pre-rings of definable endogenies such that
    \begin{itemize}
        \item $C^{\#}(\Gamma)=\Delta$ and $C^{\#}(\Delta)=\Gamma$;
        \item $A$ is absolutely $(\Gamma,\Delta)$-minimal;
        \item $\Gamma$ is essentially unbounded and $\Delta$ is essentially infinite or vice-versa;
        \item $A_0=\Kat(\Gamma,\Delta)$.
    \end{itemize}
    Then, there exists $K$ a definable infinite field such that $A/A_0$ is a finite-dimensional $K$-vector space contained into $\big(\Gamma/{\sim}\big)\cap \big(\Delta/{\sim}\big)$ and $\Gamma/{\sim}$ and $\Delta/{\sim}$ act $K$-linearly on $A/A_0$.
\end{thmB}
\textbf{Remark}:\\
These two theorems are not direct extensions of the Theorems in \cite{WagnerDeloro}. Indeed, we do not assume $A$ minimal but \emph{absolutely minimal} \hbox{i.e.} there exists no almost invariant infinite definable subgroups of infinite index. In reality, if we assume connectivity, our proofs can also be applied in the minimal case and so, in this sense, the results are an extension of the connected case.\\
In the sixth section, we introduce some basic results for quasi-endomorphisms. Then, in sections $7,8,9$, we prove an extension of Theorem $A$ in the case one of the two pre-rings $\Gamma,\Delta$ is of quasi-endomorphisms and the other of endogenies.\\
In section $10$, we prove a version of Theorem $A$ and $B$ in the case that $A$ has finite $n$-torsion for every $n<\omega$. Finally, in section $11$, we prove a generalization of Zilber's Field Theorem in finite-dimensional theories.
\subsection{Finite-dimensional groups}
In this subsection, we recall the main results on finite-dimensional groups, in particular Lemma \ref{boundedind}, which will be fundamental in the proof of Theorem A and Theorem B.

mptyset)=-\infty$ and $\operatorname{dim}(X)=0$ if and only if $X$ is finite;
 \item $\operatorname{dim}(X\cup Y)=\max\{\operatorname{dim}(X),\operatorname{dim}(Y)\}$;
 \item If $\operatorname{dim}(f^{-1}(y))\geq k$ if for any $y\in Y$, then $\operatorname{dim}(X)\geq \operatorname{dim}(Y)+k$;
 \item If $\operatorname{dim}(f^{-1}(y))\leq k$ for any $y\in Y$, then $\operatorname{dim}(X)\leq \operatorname{dim}(Y)+k$.
\end{itemize}
\end{defn}
Examples of finite-dimensional theories are $o$-minimal theories, supersimple theories of finite Lascar rank, and supperosy theories of finite $U^{\text{\thorn}}$-rank.\\
 A first proof of Zilber's Field Theorem in this context has been given by Deloro in \cite{deloro2024zilber}. An important hypothesis of the theorem is that the module on which the ring of endomorphisms acts is connected. This clearly holds in the finite Morley rank and $o$-minimal context, but in general is not true: it is sufficient to take a supersimple theory of finite Lascar rank. Therefore, it remains unclear what happens for non virtually connected modules. The aim of this article is exactly to analyse this case.\\
 In reality, we will work with two more general notions of endomorphisms: endogenies and quasi-endomorphisms. An endogeny $\phi$ of an abelian group $A$ is a subgroup of $A\times A$ such that $\pi_1(\phi)=A$ and $\{a\in A:\ (0,a)\in \phi\}$ is finite. The latter subgroup is called the \emph{katakernel} of $\phi$. A \emph{quasi-endomorphism} $\phi$ is a subgroup of $A$ such that $\pi_1(\phi)$, that is called the domain of $\phi$, is of finite index in $A$ and the katakernel is finite. Therefore, an endogeny is a quasi-endomorphism that is total.\\
Endogenies and quasi-endomorphisms arise naturally in the finite-dimensional context. Indeed, let $H\geq H_1$ be definable subgroups of the definable abelian group $G$ and $\Gamma\leq \operatorname{End}(G)$. Assume that $H$ and $H_1$ are \emph{$\Gamma$-almost invariant} \hbox{i.e.} $\gamma(H)\cap H$ is of finite index in $\gamma(H)$ for any $\gamma\in \Gamma $. Then, $\Gamma$ acts on $G/H$ by endogenies (with katakernel of $\gamma$ equal to $\gamma(H)+H/H$) and $\Gamma$ acts by quasi-endomorphisms on $H/H_1$ with domain $\{h\in H:\ \gamma(h)\in H\}$. In the finite Morley rank context and in the $o$-minimal one, we can avoid working with endogenies or quasi-endomorphisms since, if $H$ is $\Gamma$-almost invariant, the connected component $H^0$ is $\Gamma$-invariant and the action of $\Gamma$ on $G/H^0$ is again by endomorphisms.\\
The article follows the ideas of \cite{WagnerDeloro}, which proves the linearization theorem for endogenies in the connected case.\\
In the first section, we introduce all the definitions that we need in the article, and we verify some easy lemmas.\\
Then, we proceed to prove the following result.
\begin{thmA}
In a finite-dimensional theory, let $A$ be an abelian definable group and $\Gamma,\Delta$ two invariant pre-rings of definable endogenies such that:
\begin{itemize}
 \item $\Gamma,\Delta$ sharply commute;
 \item $A$ is absolutely $(\Gamma,\Delta)$-minimal;
 \item $\Gamma$ is essentially unbounded and $\Delta$ is essentially infinite or vice-versa.
\end{itemize}
 There is a finite $\Gamma$ and $\Delta$-invariant subgroup $F$ such that $\Gamma/{\sim}$ and $\Delta/{\sim}$ act by endomorphisms on $A/F$.
\end{thmA}
Moreover, $F$ will be given explicitly.\\
In the second section, we verify the easy case in which both $\Gamma$ and $\Delta$ act by endogenies with finite kernel or finite image.\\
The third section analyzes the case in which the unbounded pre-ring of endogenies has an endogeny with infinite kernel and infinite image. Finally, we complete the proof in the fourth section.\\
The main result of the fifth section is the following linearization theorem.
\begin{thmB}
 Let $A$ be an abelian definable group and $\Gamma,\Delta$ two invariant pre-rings of definable endogenies such that
 \begin{itemize}
 \item $C^{\#}(\Gamma)=\Delta$ and $C^{\#}(\Delta)=\Gamma$;
 \item $A$ is absolutely $(\Gamma,\Delta)$-minimal;
 \item $\Gamma$ is essentially unbounded and $\Delta$ is essentially infinite or vice-versa;
 \item $A_0=\Kat(\Gamma,\Delta)$.
 \end{itemize}
 Then, there exists $K$ a definable infinite field such that $A/A_0$ is a finite-dimensional $K$-vector space contained into $\big(\Gamma/{\sim}\big)\cap \big(\Delta/{\sim}\big)$ and $\Gamma/{\sim}$ and $\Delta/{\sim}$ act $K$-linearly on $A/A_0$.
\end{thmB}
\textbf{Remark}:\\
These two theorems are not direct extensions of the Theorems in \cite{WagnerDeloro}. Indeed, we do not assume $A$ minimal but \emph{absolutely minimal} \hbox{i.e.} there exists no almost invariant infinite definable subgroups of infinite index. In reality, if we assume connectivity, our proofs can also be applied in the minimal case and so, in this sense, the results are an extension of the connected case.\\
In the sixth section, we introduce some basic results for quasi-endomorphisms. Then, in sections $7,8,9$, we prove an extension of Theorem $A$ in the case one of the two pre-rings $\Gamma,\Delta$ is of quasi-endomorphisms and the other of endogenies.\\
In section $10$, we prove a version of Theorem $A$ and $B$ in the case that $A$ has finite $n$-torsion for every $n<\omega$. Finally, in section $11$, we prove a generalization of Zilber's Field Theorem in finite-dimensional theories.
\subsection{Finite-dimensional groups}
In this subsection, we recall the main results on finite-dimensional groups, in particular Lemma \ref{boundedind}, which will be fundamental in the proof of Theorem A and Theorem B.

\begin{lemma}\label{boundedind}
    Let $G$ be a definable group of finite dimension and $\{H_i\}_{i\in I}$ a family of uniformly definable subgroups. Then, there exists $n,d<\omega$ such that there is no $J=\{j_1,...,j_n\}\subseteq I$ of cardinality $n$ such that $|\bigcap_{i=1}^kH_{j_i}:\bigcap_{i=1}^{k+1} H_{j_i}|$ has index greater than $d$ for any $k\leq n-1$.
\end{lemma}

\def\notind#1#2{#1\setbox0=\hbox{$#1x$}\kern\wd0
\hbox to 0pt{\mathchardef\nn=12854\hss$#1\nn$\kern1.4\wd0\hss}
\hbox to 0pt{\hss$#1\mid$\hss}\lower.9\ht0 \hbox to 0pt{\hss$#1\smile$\hss}\kern\wd0}

\end{theorem}
\begin{proof}
 We prove the base case. Given $\Gamma$ by hypothesis essentialy unbounded then it not admits an ascending chain of finite $\Gamma$-invariant subgroups. Therefore let $A_0$ this subgroup, it contains the katakernel of $\Delta$, that therefore is finite.\\
 Being the katakernel finite, there exists only boundedly many elements for $\Delta$ or $\Delta$ is unbounded (and then the proof follows using the previous idea).\\
 Then we can take a representative with maximal image and we have to prove that
\end{proof}
\begin{theorem}

\begin{proof}
Let $n=\dim(G)$. We will work in a sufficiently saturated structure $\mathfrak{M}$ in which $G$ is definable and assume that $\phi(x,y)$ is the formula defining the family. Assume, by contrary, that the conclusion is false then for $N=n+2$ and for any $k\in \omega$, there exists $g_1,...,g_N$ such that $|\bigcap_{j\leq k-1} \phi(\mathfrak{M},g_j)/\bigcap_{j\leq k} \phi(\mathfrak{M},g_j)|\geq k$. Therefore, the partial type given by the formulas
$$\exists a^1_1,...,a^1_k,...,a^N_1,...,a_k^N: a^i_j\in \bigcap_{j=1}^i \phi(\mathfrak{M},x_j)\wedge a^i_j{a^i_k}^{-1}\not\in \phi(\mathfrak{M},x_j)\wedge \forall_{i\leq N}\ \phi(\mathfrak{M},x_i)\leq G(\mathfrak{M})$$
for all $k<\omega$, is finitely satisfable. By compactness and saturation, there exist subgroups $\{H_i\}_{i\leq N}$ such that $|\bigcap_{i<j}H_i/\bigcap_{i\leq j}H_i|$ is infinite for every $i$. This would imply that the dimension of $G$ is strictly greater than $n$, a contradiction.
\end{proof}
If we apply these results to the centralizers of elements in $G$, we obtain that a finite-dimensional definable group has the hereditary $\widetilde{\mathfrak{M}}_c$-property.
\begin{defn}
    A group $G$ is \emph{hereditarily-$\widetilde{\mathfrak{M}}_c$} if for any definable subgroups $H,N$, such that $N$ is normalised by $H$, there exist natural numbers $n_{HN}$ and $d_{HN}$ such that any sequence of centralisers
    $$C_H(a_0/N)\geq C_H(a_0,a_1/N)\geq...\geq C_H(a_0,a_1,...,a_n/N)\geq...$$
    with each centraliser of index at least $d_{HN}$ in the previous one, has length at most $n_{HN}$.
\end{defn}
The notions of almost containment and commensurability are fundamental in the study of groups definable in finite-dimensional theories.
\begin{defn}
\begin{itemize}
    \item  Let $G$ be a group and $A, B\leq G$ subgroups. $A$ \emph{almost contains} $B$, denoted $A\apprge B$, if $|B:B\cap A|$ is finite.
    \item  If $A\apprle B$ and $B\apprle A$, $A$ and $B$ are \emph{commensurable}.
    \item Given a family of subgroups $\{G_i\}_{i\in I}$ in $G$, the family $\{G_i\}_{i\in I}$ is \emph{uniformly commensurable} if there exists $n<\omega$ such that $|G_i/G_i\cap G_j|\leq n$ for every $i,j\in I$.
\end{itemize}
  \end{defn}

Assume towards a contradiction that the conclusion of Theorem~\ref{boundedind} fails. Let $n=\dim(G)$ and set $N=n+2$. Working in a sufficiently saturated structure $\mathfrak{M}$ where the uniformly definable family is given by $\phi(x,y)$, the failure yields that for every $k\in\omega$ there exist parameters $g_1,\dots,g_N$ such that
\[
\left|\bigcap_{j\leq k-1}\phi(\mathfrak{M},g_j)\,\Big/\,\bigcap_{j\leq k}\phi(\mathfrak{M},g_j)\right|\ge k.
\]
From this, one writes a partial type asserting (via witnesses $a^i_j$) that each successive intersection properly decreases in a way producing many distinct cosets, and that each $\phi(\mathfrak{M},x_i)$ is a subgroup of $G(\mathfrak{M})$. This partial type is "finitely satisfiable". By compactness and saturation, obtain subgroups $\{H_i\}_{i\le N}$ such that for every $i$,
\[
\left|\bigcap_{i<j}H_i\,\Big/\,\bigcap_{i\le j}H_i\right|\text{ is infinite}.
\]
Such an infinite drop at each step would force $\dim(G)$ to be strictly greater than $n$, contradicting $n=\dim(G)$. Hence the desired bounds $n,d<\omega$ exist.

Assume towards a contradiction that the conclusion of the main theorem fails. Let $n=\dim(G)$ and set $N=n+2$. Working in a sufficiently saturated structure $\mathfrak{M}$ where the uniformly definable family is given by $\phi(x,y)$, the failure yields that for every $k\in\omega$ there exist parameters $g_1,\dots,g_N$ such that
\[
\left|\bigcap_{j\leq k-1}\phi(\mathfrak{M},g_j)\,\Big/\,\bigcap_{j\leq k}\phi(\mathfrak{M},g_j)\right|\ge k.
\]
From this, one writes a partial type asserting (via witnesses $a^i_j$) that each successive intersection properly decreases in a way producing many distinct cosets, and that each $\phi(\mathfrak{M},x_i)$ is a subgroup of $G(\mathfrak{M})$. This partial type is "finitely satisfiable". By compactness and saturation, obtain subgroups $\{H_i\}_{i\le N}$ such that for every $i$,
\[
\left|\bigcap_{i<j}H_i\,\Big/\,\bigcap_{i\le j}H_i\right|\text{ is infinite}.
\]
Such an infinite drop at each step would force $\dim(G)$ to be strictly greater than $n$, contradicting $n=\dim(G)$. This completes the proof of the main theorem.

Question: Let $G$ be a definable group of finite dimension, and let $\{H_i\}_{i\in I}$ be a uniformly definable family of subgroups of $G$ (that is, all $H_i$ are given by instances of a single defining formula). Which statement holds about finite chains of successive intersections of members of this family?

There exist natural numbers $n,d<\omega$ such that no subset $J=\{j_1,\dots,j_n\}\subseteq I$ of cardinality $n$ satisfies
\[
\left|\bigcap_{i=1}^k H_{j_i} : \bigcap_{i=1}^{k+1} H_{j_i}\right|>d
\]
for every $k\le n-1$. Equivalently, for any $n$ distinct indices $j_1,\dots,j_n$, at least one successive index
\[
\left|\bigcap_{i=1}^k H_{j_i} : \bigcap_{i=1}^{k+1} H_{j_i}\right|
\]
is at most $d$.

Choices:
B.
There exists a natural number $d<\omega$ such that for every natural number $n<\omega$, no subset $J=\{j_1,\dots,j_n\}\subseteq I$ of cardinality $n$ satisfies
\[
\left|\bigcap_{i=1}^k H_{j_i} : \bigcap_{i=1}^{k+1} H_{j_i}\right|>d
\]
for every $k\le n-1$.
C.
There exists a natural number $n<\omega$ such that no subset $J=\{j_1,\dots,j_n\}\subseteq I$ of cardinality $n$ satisfies
\[
\left|\bigcap_{i=1}^k H_{j_i} : \bigcap_{i=1}^{k+1} H_{j_i}\right|\text{ is infinite}
\]
for every $k\le n-1$.
D.
There exist natural numbers $n,d<\omega$ such that for every subset $J=\{j_1,\dots,j_n\}\subseteq I$ of cardinality $n$ and every $k\le n-1$,
\[
\left|\bigcap_{i=1}^k H_{j_i} : \bigcap_{i=1}^{k+1} H_{j_i}\right|\le d.
\]
E.
There exist natural numbers $n,d<\omega$ such that no subset $J=\{j_1,\dots,j_n\}\subseteq I$ of cardinality $n$ satisfies
\[
\left|H_{j_k}:H_{j_k}\cap H_{j_{k+1}}\right|>d
\]
for every $k\le n-1$.

simultaneous existence of both bounds n and d

replace uniform finite bound by mere finiteness of some drop

at least one bounded drop versus all drops bounded

successive intersections replaced by pairwise intersection indices


Every solitary capillary-gravity water wave $\bar{u}(c)=(\bar{\eta}(c), \bar{\xi}(c))$ with wave speed $c = \frac{1}{\sqrt{1+\varepsilon^2}}$ for $\varepsilon \in (0,\varepsilon_0)$ is conditionally orbitally stable in the following sense: For every $R > 0$ and $\rho > 0$, there exists $\rho_0 > 0$ such that if
		$u=(\eta,\xi) \colon [0,T) \to \mathscr{F}_R$ is a continuous solution of the Hamiltonian system \eqref{eq:water-wave-bd} that preserves the energy $\mathcal{H}$ and momentum $\mathcal{P}$, and if the initial data satisfy
		\[
		\|\eta(0) - \bar{\eta}(c)\|_{H^1(\mathbb{R}^2)} + \|\xi(0) - \bar{\xi}(c)\|_{H^{1/2}_{*}(\mathbb{R}^2)} < \rho_0,
		\]
		then for all $t \in [0,T)$,
		\begin{equation*}
			\inf_{(x_0,y_0) \in \mathbb{R}^2} \big( \|\eta(t, \cdot - (x_0,y_0)) - \bar{\eta}(c)\|_{H^1} + \|\xi(t, \cdot - (x_0,y_0)) - \bar{\xi}(c)\|_{H^{1/2}_{*}} \big) < \rho.
		\end{equation*}

In this article, we investigate the stability of traveling wave solutions for capillary-gravity water waves in a three-dimensional fluid domain of finite depth. The physical system is governed by the incompressible Euler equations,
	\begin{equation}
		\label{eq:euler1}
		\frac{\partial \mathbf{u}}{\partial t} + (\mathbf{u} \cdot \nabla) \mathbf{u} + \nabla P = g \mathbf{e}_3,
	\end{equation}
	on the evolving fluid domain
	\begin{equation}
		\label{eq:euler2}
		\Omega(t)=\{(x_1, x_2, x_3) \in \mathbb{R}^3 : -1< x_3< \eta(t, x_1, x_2)\}.
	\end{equation}
	Here, the water density is normalized to unity, $\mathbf{u}(t, \cdot): \Omega(t) \to \mathbb{R}^3$ denotes the velocity field, $P(t, \cdot): \Omega(t) \to \mathbb{R}$ is the pressure, $g > 0$ is the gravitational constant, and $\mathbf{e}_3 = (0,0,1)$. The fluid is bounded below by a rigid flat bed at $\{x_3 = -1\}$, while the upper boundary—described by the graph of $\eta$—represents a free interface with the air (modeled as a vacuum). A key feature of this free-boundary problem is that the free surface elevation $\eta$ is an unknown of the system. For solitary waves, we require $\eta(t, x_1, x_2) \to 0$ as $r = \sqrt{x_1^2 + x_2^2} \to \infty$, so the asymptotic fluid depth is normalized to 1.

	The boundary conditions for the system are specified as follows. At the rigid bottom \(x_3 = -1\), the kinematic condition of impermeability requires
	\begin{equation}
		\label{eq:impermeability}
		\mathbf{u} \cdot \mathbf{n} = 0,
	\end{equation}
	where \(\mathbf{n}\) is the unit outward normal to the fluid domain. On the free surface \(x_3 = \eta(t, x_1, x_2)\), the kinematic boundary condition is given by
	\begin{equation}
		\label{eq:kinematic}
		\partial_t \eta = \sqrt{1 + (\partial_{x_1} \eta)^2 + (\partial_{x_2} \eta)^2}  \, \mathbf{u} \cdot \mathbf{n},
	\end{equation}
	which states that the surface moves with the fluid. Dynamically, the pressure at the interface satisfies the Young–Laplace law:
	\begin{equation}
		\label{eq:pressure}
		P = \sigma \, \mathrm{div} \left( \frac{\nabla_{\!\perp} \eta}{\sqrt{1 + |\nabla_{\!\perp} \eta|^2}} \right),
	\end{equation}
	where \(\nabla_{\!\perp} = (\partial_{x_1}, \partial_{x_2})\) and \(\sigma > 0\) is the surface tension coefficient. Together with the gravitational acceleration \(g > 0\) in the bulk, the presence of surface tension at the interface leads to the terminology {\em capillary-gravity waves} for solutions of system \eqref{eq:euler1}--\eqref{eq:pressure}.

	Under the assumption of irrotational flow, the velocity field can be expressed as \(\mathbf{u} = \nabla \Phi\), and the system can be reformulated in terms of the velocity potential \(\Phi\). The resulting boundary value problem for the capillary-gravity water-wave system is:
	\begin{equation}
		\label{eq:water-wave-full}
		\begin{cases}
			\Delta \Phi = 0 & \text{in } \Omega(t), \\
			\partial_n \Phi = 0 & \text{on } x_3 = -1, \\
			\partial_t \eta = \partial_{x_3} \Phi - \nabla_{\perp} \eta \cdot \nabla_{\perp} \Phi & \text{on } x_3 = \eta(x_1, x_2, t), \\
			\partial_t \Phi = -\dfrac{1}{2} |\nabla \Phi|^2 - g \eta + \sigma \, \mathrm{div} \left( \dfrac{\nabla_{\perp} \eta}{\sqrt{1 + |\nabla_{\perp} \eta|^2}} \right) & \text{on } x_3 = \eta(x_1, x_2, t).
		\end{cases}
	\end{equation}

	A fundamental tool in the modern analysis of water waves is the Dirichlet-to-Neumann operator (DNO), which allows for a reduction of the problem's dimension by exploiting the harmonicity of the velocity potential. Defining the surface velocity potential as $\xi(x_1,x_2,t) = \Phi(x_1,x_2,\eta(x_1,x_2,t),t)$, the DNO $G(\eta)$ is defined by
	\begin{equation}
		G(\eta)\xi = \partial_{x_3}\Phi - \nabla_{\perp}\eta \cdot \nabla_{\perp}\Phi,
	\end{equation}
	where all quantities are evaluated at the free surface and $\nabla_\perp = (\partial_{x_1}, \partial_{x_2})$. In terms of the DNO, the water wave system \eqref{eq:water-wave-full} transforms into the Hamiltonian formulation \cite{Craig1,Craig2,Craig3}:
	\begin{equation}
		\label{eq:water-wave-bd}
		\begin{cases}
			\eta_t = G(\eta)\xi, \\
			\xi_t = \frac{1}{2(1+|\nabla_{\perp}\eta|^2)}\left[(G(\eta)\xi+\nabla_{\perp}\eta\cdot\nabla_{\perp}\xi)^2-|\nabla_{\perp}\xi|^2|\nabla_\perp\eta|^2-|\nabla_\perp\xi|^2\right] \\
			\quad\quad - g\eta + \sigma\operatorname{div} \left( \frac{\nabla_{\perp} \eta}{\sqrt{1 + |\nabla_{\perp} \eta|^2}} \right).
		\end{cases}
	\end{equation}
	This formulation reveals the Hamiltonian structure of the problem, with the energy functional given by
	\begin{equation}
		\label{1.energy-2}
		\mathcal{H}(\eta,\xi) = \int_{\mathbb{R}^2} \left( \frac12 \xi G(\eta)\xi + \frac12 g \eta^2 + \sigma \left( \sqrt{1+|\nabla_\perp \eta|^2} - 1 \right) \right) dx_1dx_2.
	\end{equation}
	The first term represents the kinetic energy, the second the gravitational potential energy, and the third the capillary energy due to surface tension.

	Due to the translation invariance of the system in the horizontal directions, there exist conserved momentum functionals. For waves propagating primarily in the $x_1$-direction, the relevant conserved quantity is the $x_1$-component of the momentum, given by
	\begin{equation}
		\label{eq:momentum}
		\mathcal{P}(\eta,\xi) = \int_{\mathbb{R}^2} \eta_{x_1} \xi  dx_1dx_2.
	\end{equation}
	This momentum functional plays a crucial role in both the existence theory and stability analysis of solitary waves \cite{Benjamin1974}.
	The DNO framework, along with the energy and momentum functionals, provides the mathematical foundation for both the existence theory and our subsequent stability analysis.

	A complete global well-posedness theory for the 3D gravity–capillary wave system in finite depth remains an open problem. (Existing global results, such as those by Germain–Masmoudi–Shatah \cite{GMS2012} and Wu \cite{Wu2009, Wu2011}, assume infinite depth and zero surface tension.) In the finite-depth setting, local well-posedness in 3D has been established by Alazard–Burq–Zuily \cite{ABZ2011} in the presence of bottom topography or for infinite depth, and by Ming–Zhang–Zhang \cite{MZZ2012} over long time scales.

	The existence theory for three-dimensional traveling waves features two main classes: doubly periodic waves and fully localized solitary waves. Doubly periodic waves, also known as short-crested waves, were first rigorously established by Reeder and Shinbrot \cite{Reed} via bifurcation theory. Earlier formal computations were given by Fuchs \cite{f1952} and Sretenskiĭ \cite{s1953}. Subsequent developments include the variational approach of Craig and Nicholls \cite{Cra2000} and spatial dynamics methods by Groves and Mielke \cite{gm2001}, Groves and Haragus \cite{gh2003}, and Nilsson \cite{n2019}. Recent advances for doubly periodic waves with vorticity include the work of Lokharu, Seth, and Wahlén \cite{lsw2020}, Groves et al. \cite{gnpw2024}, and Seth, Varholm, and Wahlén \cite{svw2024}.

	For fully localized solitary waves, where $\eta(x_1, x_2) \to 0$ as $|(x_1, x_2)| \to \infty$, two principal approaches have been successful. Variational methods seek critical points of the energy functional \eqref{1.energy-2} subject to momentum constraints, as in the work of Groves and Sun \cite{Groves} using mountain-pass arguments for the augmented energy $E_c(\eta, \xi) = \mathcal{H}(\eta, \xi) - c\mathcal{P}(\eta, \xi)$, and Buffoni et al. \cite{BGSW2013, Buffoni2018} via constrained minimization. In contrast, the waves we study were constructed by Gui, et al. \cite{GLLWY} using a non-variational Lyapunov–Schmidt reduction method, yielding small-amplitude solitary waves with speed $c = 1/\sqrt{1+\varepsilon^2}$ for $\varepsilon > 0$ sufficiently small. For a comprehensive survey of recent progress in water waves, we refer to \cite{H}.

	The primary objective of this paper is to establish the orbital stability of these fully localized three-dimensional solitary waves from \cite{GLLWY}. Crucially, these waves are not obtained as energy minimizers under momentum constraints, but rather as critical points through a reduction procedure. Consequently, their stability cannot be deduced from standard variational principles \cite{Caz, Benjamin1974} that apply to constrained minimizers \cite{Buffoni2004, Buffoni2005, Buffoni2009, Groves2010, Groves2011, Groves2015}.

	Our approach adapts the energy–momentum framework pioneered by Benjamin \cite{Benjamin} and rigorously developed by Grillakis, Shatah, and Strauss (GSS) \cite{GSS1,GSS2}. This method constructs a Lyapunov functional from a carefully chosen combination of the energy $\mathcal{H}$ and momentum $\mathcal{P}$, with stability following from the convexity properties of this functional evaluated at the wave profile. A key precedent is Mielke's work \cite{Mielke} on conditional energetic stability, which adapted the GSS method to accommodate the mismatch between well-posedness and energy spaces. While our strategy is philosophically similar, a fundamental distinction lies in the object of study: the waves we consider are not variational minimizers but arise from a non-variational construction. Consequently, we develop a framework to directly address the orbital stability of these specific wave profiles, synthesizing the GSS method with a detailed analysis of their linearized dynamics.

	Our main result establishes the conditional orbital stability of the wave profiles $\bar{u}(c)=(\bar{\eta}(c), \bar{\xi}(c))$ from \cite{GLLWY}:

The boundary conditions for the system are specified as follows. At the rigid bottom \(x_3 = -1\), the kinematic condition of impermeability requires
 \begin{equation}
 \label{eq:impermeability}
 \mathbf{u} \cdot \mathbf{n} = 0,
 \end{equation}
 where \(\mathbf{n}\) is the unit outward normal to the fluid domain. On the free surface \(x_3 = \eta(t, x_1, x_2)\), the kinematic boundary condition is given by
 \begin{equation}
 \label{eq:kinematic}
 \partial_t \eta = \sqrt{1 + (\partial_{x_1} \eta)^2 + (\partial_{x_2} \eta)^2} \, \mathbf{u} \cdot \mathbf{n},
 \end{equation}
 which states that the surface moves with the fluid. Dynamically, the pressure at the interface satisfies the Young–Laplace law:
 \begin{equation}
 \label{eq:pressure}
 P = \sigma \, \mathrm{div} \left( \frac{\nabla_{\!\perp} \eta}{\sqrt{1 + |\nabla_{\!\perp} \eta|^2}} \right),
 \end{equation}
 where \(\nabla_{\!\perp} = (\partial_{x_1}, \partial_{x_2})\) and \(\sigma > 0\) is the surface tension coefficient. Together with the gravitational acceleration \(g > 0\) in the bulk, the presence of surface tension at the interface leads to the terminology {\em capillary-gravity waves} for solutions of system \eqref{eq:euler1}--\eqref{eq:pressure}.

A fundamental tool in the modern analysis of water waves is the Dirichlet-to-Neumann operator (DNO), which allows for a reduction of the problem's dimension by exploiting the harmonicity of the velocity potential. Defining the surface velocity potential as $\xi(x_1,x_2,t) = \Phi(x_1,x_2,\eta(x_1,x_2,t),t)$, the DNO $G(\eta)$ is defined by
 \begin{equation}
 G(\eta)\xi = \partial_{x_3}\Phi - \nabla_{\perp}\eta \cdot \nabla_{\perp}\Phi,
 \end{equation}
 where all quantities are evaluated at the free surface and $\nabla_\perp = (\partial_{x_1}, \partial_{x_2})$. In terms of the DNO, the water wave system \eqref{eq:water-wave-full} transforms into the Hamiltonian formulation \cite{Craig1,Craig2,Craig3}:
 \begin{equation}
 \label{eq:water-wave-bd}
 \begin{cases}
 \eta_t = G(\eta)\xi, \\
 \xi_t = \frac{1}{2(1+|\nabla_{\perp}\eta|^2)}\left[(G(\eta)\xi+\nabla_{\perp}\eta\cdot\nabla_{\perp}\xi)^2-|\nabla_{\perp}\xi|^2|\nabla_\perp\eta|^2-|\nabla_\perp\xi|^2\right] \\
 \quad\quad - g\eta + \sigma\operatorname{div} \left( \frac{\nabla_{\perp} \eta}{\sqrt{1 + |\nabla_{\perp} \eta|^2}} \right).
 \end{cases}
 \end{equation}
 This formulation reveals the Hamiltonian structure of the problem, with the energy functional given by
 \begin{equation}
 \label{1.energy-2}
 \mathcal{H}(\eta,\xi) = \int_{\mathbb{R}^2} \left( \frac12 \xi G(\eta)\xi + \frac12 g \eta^2 + \sigma \left( \sqrt{1+|\nabla_\perp \eta|^2} - 1 \right) \right) dx_1dx_2.
 \end{equation}
 The first term represents the kinetic energy, the second the gravitational potential energy, and the third the capillary energy due to surface tension.

For fully localized solitary waves, where $\eta(x_1, x_2) \to 0$ as $|(x_1, x_2)| \to \infty$, two principal approaches have been successful. Variational methods seek critical points of the energy functional \eqref{1.energy-2} subject to momentum constraints, as in the work of Groves and Sun \cite{Groves} using mountain-pass arguments for the augmented energy $E_c(\eta, \xi) = \mathcal{H}(\eta, \xi) - c\mathcal{P}(\eta, \xi)$, and Buffoni et al. \cite{BGSW2013, Buffoni2018} via constrained minimization. In contrast, the waves we study were constructed by Gui, et al. \cite{GLLWY} using a non-variational Lyapunov–Schmidt reduction method, yielding small-amplitude solitary waves with speed $c = 1/\sqrt{1+\varepsilon^2}$ for $\varepsilon > 0$ sufficiently small. For a comprehensive survey of recent progress in water waves, we refer to \cite{H}.

Our approach adapts the energy–momentum framework pioneered by Benjamin \cite{Benjamin} and rigorously developed by Grillakis, Shatah, and Strauss (GSS) \cite{GSS1,GSS2}. This method constructs a Lyapunov functional from a carefully chosen combination of the energy $\mathcal{H}$ and momentum $\mathcal{P}$, with stability following from the convexity properties of this functional evaluated at the wave profile. A key precedent is Mielke's work \cite{Mielke} on conditional energetic stability, which adapted the GSS method to accommodate the mismatch between well-posedness and energy spaces. While our strategy is philosophically similar, a fundamental distinction lies in the object of study: the waves we consider are not variational minimizers but arise from a non-variational construction. Consequently, we develop a framework to directly address the orbital stability of these specific wave profiles, synthesizing the GSS method with a detailed analysis of their linearized dynamics.

Our main result establishes the conditional orbital stability of the wave profiles $\bar{u}(c)=(\bar{\eta}(c), \bar{\xi}(c))$ from \cite{GLLWY}:

Our main result establishes the conditional orbital stability of the wave profiles $\bar{u}(c)=(\bar{\eta}(c), \bar{\xi}(c))$ from \cite{GLLWY}:

\begin{theorem}
 Every solitary capillary-gravity water wave $\bar{u}(c)=(\bar{\eta}(c), \bar{\xi}(c))$ defined in \eqref{bareta} with wave speed $c = \frac{1}{\sqrt{1+\varepsilon^2}}$ for $\varepsilon \in (0,\varepsilon_0)$ is conditionally orbitally stable in the following sense: For every $R > 1$ and $\rho > 0$, there exists $\rho_0 > 0$ such that if
 $u=(\eta,\xi) \colon [0,T) \to \mathscr{F}_R$ is a continuous solution of \eqref{Hamilton} that preserves the functionals $\mathcal{H}$ and $\mathcal{P}$, and if the initial data satisfy
 \[
 \|\eta(0) - \bar{\eta}(c)\|_{H^1(\mathbb{R}^2)} + \|\xi(0) - \bar{\xi}(c)\|_{H^{1/2}_{*}(\mathbb{R}^2)} < \rho_0,
 \]
 then for all $t \in [0,T)$,
 \begin{equation*}
 \inf_{(x_0,y_0) \in \mathbb{R}^2} \left( \|\eta(t, \cdot - (x_0,y_0)) - \bar{\eta}(c)\|_{H^1(\mathbb{R}^2)} + \|\xi(t, \cdot - (x_0,y_0)) - \bar{\xi}(c)\|_{H^{1/2}_{*}(\mathbb{R}^2)} \right) < \rho.
 \end{equation*}
 \end{theorem}

Recall that $\mathcal{H}_c(\eta,\xi)=\mathcal{V}(\eta)+\frac{1}{2}\langle \xi, G(\eta)\xi \rangle +c\langle \xi, \partial_{x_1}\eta \rangle$. For fixed $\eta$, this can be rewritten by completing the square in $\xi$:
 $$\mathcal{H}_c(\eta, \xi) = \mathcal{V}_c^{{aug}}(\eta) + \frac{1}{2} \left\langle \xi + c G(\eta)^{-1} \partial_{x_1}\eta, G(\eta) (\xi + c G(\eta)^{-1} \partial_{x_1}\eta) \right\rangle,$$
 where $\mathcal{V}_c^{{aug}}(\eta) = \mathcal{V}(\eta) - \frac{1}{2} c^2 \int_{\mathbb{R}^2} \partial_{x_1}\eta [G(\eta)^{-1} \partial_{x_1}\eta] \, dx_1dx_2$ is the augmented potential, and the quadratic term is nonnegative (zero at the minimizing $\xi = -c G(\eta)^{-1} \partial_{x_1}\eta$). Substituting the perturbed values $\eta = \bar{\eta} + v$ and $\xi = \bar{\xi} + w$ gives
 $$\mathcal H_c(\bar{\eta} + v, \bar{\xi} + w) = \mathcal V_c^{\mathrm{aug}}(\bar{\eta} + v) + \frac{1}{2} \left\langle w + \mathcal B(v), G(\bar{\eta} + v) (w + \mathcal B(v)) \right\rangle,$$
 where $\mathcal B(v) = c G(\bar{\eta} + v)^{-1} \partial_{x_1}(\bar{\eta} + v) - c G(\bar{\eta})^{-1} \partial_{x_1}\bar{\eta}$ (noting $\bar{\xi} = -c G(\bar{\eta})^{-1} \partial_{x_1}\bar{\eta}$, so $\mathcal{B}(v) = \bar{\xi} + c G(\bar{\eta} + v)^{-1} \partial_{x_1}(\bar{\eta} + v)$).
 Then we derive
 $$\mathcal H_c(\bar u + z) - \mathcal H_c(\bar u) = \left[ \mathcal V_c^{\mathrm{aug}}(\bar{\eta} + v) - \mathcal V_c^{\mathrm{aug}}(\bar{\eta}) \right] + \frac{1}{2} \left\langle w + \mathcal{B}(v), G(\bar{\eta} + v) (w + \mathcal{B}(v)) \right\rangle.$$
 The second variation operator $\mathcal{A} = D^2 \mathcal H_c(\bar{u})$ has quadratic form
 $$\frac{1}{2} \langle\langle \mathcal{A} z, z \rangle \rangle= \frac{1}{2} \langle \bar{C} v, v \rangle + \frac{1}{2} \left\langle w + \bar{B} v, \bar{G} (w + \bar{B} v) \right\rangle,$$
 where $\bar{C} = D^2 \mathcal V_c^{{aug}}(\bar{\eta})$, $\bar{G} = G(\bar{\eta})$, and $\bar{B} v = G(\bar{\eta})^{-1} \mathcal L_{\bar{u}(c)} v$ with $\mathcal L_{\bar{u}(c)} v = c \partial_{x_1}v + DG(\bar{\eta})[v] \bar{\xi}$. We have
 \[
 K\rho_0^2\ge \mathcal{H}_c(\bar u+z)-\mathcal{H}_c(\bar u)=: J_1+J_2+J_3+J_4+J_5,
 \]
 where
 \begin{align*}
 J_1 &= \tfrac{1}{2} \langle\langle \mathcal{A} z, z \rangle \rangle,\\
 J_2 &= \mathcal V_c^{\mathrm{aug}}(\bar{\eta} + v) - \mathcal V_c^{\mathrm{aug}}(\bar{\eta})-\tfrac{1}{2} \langle \bar{C} v, v \rangle, \\
 J_3 &= \tfrac{1}{2} \langle G(\bar \eta+v)(\mathcal{B}(v)- \bar B v), w+\mathcal{B}(v)\rangle, \\
 J_4 &=\tfrac{1}{2} \langle (G(\bar \eta+v)-G(\bar \eta))(w+\bar B v), w+\bar B v\rangle,\\
 J_5 &=\tfrac{1}{2} \langle \mathcal{B}(v)-\bar B v, G(\bar \eta+v)(w+\bar B v) \rangle.
 \end{align*}
 For the term $J_1$, we derive from Proposition \ref{lem5-3} and $\alpha_3=O(\|z\|_{\mathscr{F}}^2)$ that
 \[
 J_1\ge \frac{1}{2} \alpha_{+}\|z_+\|_{\mathscr{F}}^2 -O(\|z\|_{\mathscr{F}}^4).
 \]
 For the second term $J_2$, we have $J_2=O(\|v\|^3_{H^1(\mathbb R^2)})$. For the term $J_3$, we notice from the positive definiteness of $ G(\bar \eta +v)$ on $H^{1/2}_{*}(\mathbb R^2)$ and Lemma \ref{DNO bound} that the inner product $\langle G a, b \rangle$ satisfies the Cauchy-Schwarz inequality in the $ H^{1/2}_{*} $ norm:
 \begin{align*}
 &|\langle G(\bar \eta +v) (\mathcal{B}(v)- \bar B v), w+\mathcal{B}(v) \rangle|\\
 &\leq \sqrt{\langle G (\mathcal{B}(v)- \bar B v), (\mathcal{B}(v)- \bar B v) \rangle} \cdot \sqrt{\langle G (w+\mathcal{B}(v)), (w+\mathcal{B}(v)) \rangle}\\
 &\le \|\mathcal{B}(v)- \bar B v\|_{*,1/2} \cdot \|w+\mathcal{B}(v)\|_{*,1/2}.
 \end{align*}
 We note that
 \begin{align*}
 \partial_{x_1}(\mathcal{B}(v)- \bar B v)&=-c\left(K(\bar \eta+v)[\bar \eta+v]-K(\bar \eta)[\bar \eta]\right)-\partial_{x_1}\bar B v\\
 &=-c (K(\bar \eta+v)-K(\bar \eta))[v]\\
 &\quad -c\big( K(\bar \eta+v)[\bar \eta]-K(\bar \eta)[\bar \eta]- DK(\bar \eta)[v] \bar \eta \big).
 \end{align*}
 Then, we obtain from Proposition \ref{lemA5} that
 \begin{align*}
 \|\partial_{x_1}(\mathcal{B}(v)- \bar{B} v)\|_{H^{-1}(\mathbb R^2)}&\le C\left(\|K(\bar \eta+v)-K(\bar \eta)\|_{\mathcal{L}(H^1, H^{-1})}\|v\|_{H^1}+\|v\|_{H^1}^2\right)\\
 &\le C\left(\|K(\bar \eta+v)-K(\bar \eta)\|_{\mathcal{L}(H^{1/2}, H^{-1/2})}\|v\|_{H^1}+\|v\|_{H^1}^2\right) \\
 &\le C\|v\|_{H^1}^{1+\alpha}.
 \end{align*}
 Similarly, by Proposition \ref{lemA4}, we have
 \begin{align*}
 \|\partial_{x_2}(\mathcal{B}(v)- \bar B v)\|_{H^{-1}(\mathbb R^2)}&\le C\left(\|L(\bar \eta+v)-L(\bar \eta)\|_{\mathcal{L}(H^1, H^{-1})}\|v\|_{H^1}+\|v\|_{H^1}^2\right)\\
 &\le C\left(\|L(\bar \eta+v)-L(\bar \eta)\|_{\mathcal{L}(H^{1/2}, H^{-1/2})}\|v\|_{H^1}+\|v\|_{H^1}^2\right) \\
 &\le C\|v\|_{H^1}^{1+\alpha}.
 \end{align*}
 Thus, we have
 \begin{align*}
 \|\mathcal{B}(v)- \bar B v\|_{H^{1/2}_{*}(\mathbb R^2)}\le C\|\nabla(\mathcal{B}(v)- \bar B v)\|_{H^{-1}(\mathbb R^2)} \le C\|v\|_{H^1}^{1+\alpha}.
 \end{align*}
 For $\|w+\mathcal{B}(v)\|_{*,1/2}$, we obtain
 \[
 \|w+\mathcal{B}(v)\|_{*,1/2}\le \|w\|_{*,1/2}+\|\mathcal{B}(v)\|_{*,1/2}.
 \]
 Since $\bar B \in \mathcal{L}(H^1(\mathbb R^2), H_{*}^{1/2}(\mathbb R^2))$, we derive
 \begin{align*}
 \|\mathcal{B}(v)\|_{*,1/2}&\le \|\mathcal{B}(v)-\bar B v\|_{*,1/2}+\|\bar B v\|_{*,1/2}\le C\|v\|_{H^1}.
 \end{align*}
 Then we have
 \[
 |J_3|\le C\|v\|_{H^1}^{1+\alpha}\|z\|_{\mathscr{F}}.
 \]
 For the term $J_4$, we derive from the analyticity of $G(\cdot)$ and similar argument in the proof of Proposition \ref{lemA4} that
 \begin{align*}
 \left|J_4\right| & =\tfrac{1}{2}|\langle(G(\bar{\eta}+v)-\bar{G})(w+\bar{B} v), w+\bar{B} v\rangle| \\
 & \leq \tfrac{1}{2}\|G(\bar{\eta}+v)-\bar{G}\|_{H_*^{1 / 2} \rightarrow H_{*}^{-1 / 2}} \cdot\|w+\bar{B} v\|_{H_*^{1 / 2}}^2\\
 & \le C\|v\|_{H^1}^\alpha \|z\|_\mathscr{F}^2.
 \end{align*}
 For the term $J_5$, we obtain
 \begin{align*}
 |J_5| &=\tfrac{1}{2} |\langle \mathcal{B}(v)-\bar B v, G(\bar \eta+v)(w+\bar B v) \rangle|\\
 &\le C \sqrt{\langle (\mathcal{B}(v)- \bar B v), G(\bar \eta+v)(\mathcal{B}(v)- \bar B v) \rangle} \cdot \sqrt{\langle (w+\bar{B} v), G(\bar \eta+v)(w+\bar{B} v) \rangle}\\
 &\le C\|\mathcal{B}(v)- \bar B v\|_{*,1/2} \cdot \|w+\bar{B} v\|_{*,1/2}\\
 &\le C\|v\|_{H^1}^{1+\alpha}\|z\|_{\mathscr{F}}.
 \end{align*}
 Combining the above estimates for $J_1-J_5$, we have
 \begin{align*}
 K\rho_0^2\ge \frac{1}{2} \alpha_{+}\|z_+\|_{\mathscr{F}}^2 -O(\|z\|_{\mathscr{F}}^4)+O(\|v\|^3_{H^1(\mathbb R^2)})+O(\|v\|_{H^1}^{1+\alpha}\|z\|_{\mathscr{F}})+O(\|v\|_{H^1}^\alpha \|z\|_\mathscr{F}^2).
 \end{align*}
 As $\|z_+\|_{\mathscr{F}}\le \|z\|_{\mathscr{F}}\le 2\|z_{+}\|_{\mathscr{F}}$ in a suitable neighbourhood of $z=0$, we obtain $K\rho_0^2\ge \alpha_{+}/9\|z\|_{\mathscr{F}}^2$ as long as $\|z\|_{\mathscr{F}}\le \rho_1$.
 This implies that for all $t \in [0, T_*)$:
 \[
 \|z(t)\|_\mathscr{F} \leq \sqrt{\frac{9K}{\alpha_+}} \rho_0,
 \]
 provided we choose $\rho_0 < (\alpha_{+}/9K)^{1/2}\rho_1$.

This theorem affirms the stability of specific wave profiles from \cite{GLLWY}, as opposed to the stability of a set of minimizers. Our proof carefully accounts for the distinctive structure of these reduction-based waves while handling the technical challenges posed by the functional setting. The conservation of both energy $\mathcal{H}$ and momentum $\mathcal{P}$ plays an essential role in the analysis, allowing us to construct an appropriate Lyapunov functional despite the non-variational origin of the waves.

	The paper is organized as follows. Section 2 introduces the Hamiltonian formulation via the Dirichlet-to-Neumann operator and establishes the appropriate functional setting for our analysis. Section 3 reviews the construction and properties of the solitary waves from \cite{GLLWY}, with particular emphasis on their asymptotic behavior and the spectral structure of the linearized operator. Section 4 completes the proof of our main stability theorem by implementing a refined version of the GSS method within Mielke's framework, carefully addressing the challenges posed by the non-variational nature of our waves. Technical estimates and auxiliary results are collected in the Appendix.

Our proof "carefully accounts for the distinctive structure of these reduction-based waves" and uses that "[t]he conservation of both energy $\mathcal{H}$ and momentum $\mathcal{P}$ plays an essential role in the analysis, allowing us to construct an appropriate Lyapunov functional despite the non-variational origin of the waves." The argument proceeds by: (i) introducing "the Hamiltonian formulation via the Dirichlet-to-Neumann operator" and setting up "the appropriate functional setting"; (ii) reviewing the solitary waves from \cite{GLLWY}, emphasizing "their asymptotic behavior and the spectral structure of the linearized operator"; and (iii) completing the stability proof by "implementing a refined version of the GSS method within Mielke's framework," while "carefully addressing the challenges posed by the non-variational nature of our waves."

Every solitary capillary-gravity water wave $\bar{u}(c)=(\bar{\eta}(c), \bar{\xi}(c))$ with wave speed $c = \frac{1}{\sqrt{1+\varepsilon^2}}$ for $\varepsilon \in (0,\varepsilon_0)$ is conditionally orbitally stable in the following sense: For every $R > 0$ and $\rho > 0$, there exists $\rho_0 > 0$ such that if
		$u=(\eta,\xi) \colon [0,T) \to \mathscr{F}_R$ is a continuous solution of the Hamiltonian system
	\begin{equation}
		\label{eq:water-wave-bd}
		\begin{cases}
			\eta_t = G(\eta)\xi, \\
			\xi_t = \frac{1}{2(1+|\nabla_{\perp}\eta|^2)}\left[(G(\eta)\xi+\nabla_{\perp}\eta\cdot\nabla_{\perp}\xi)^2-|\nabla_{\perp}\xi|^2|\nabla_\perp\eta|^2-|\nabla_\perp\xi|^2\right] \\
			\quad\quad - g\eta + \sigma\operatorname{div} \left( \frac{\nabla_{\perp} \eta}{\sqrt{1 + |\nabla_{\perp} \eta|^2}} \right).
		\end{cases}
	\end{equation}
		that preserves the energy $\mathcal{H}$ and momentum $\mathcal{P}$, and if the initial data satisfy
		\[
		\|\eta(0) - \bar{\eta}(c)\|_{H^1(\mathbb{R}^2)} + \|\xi(0) - \bar{\xi}(c)\|_{H^{1/2}_{*}(\mathbb{R}^2)} < \rho_0,
		\]
		then for all $t \in [0,T)$,
		\begin{equation*}
			\inf_{(x_0,y_0) \in \mathbb{R}^2} \big( \|\eta(t, \cdot - (x_0,y_0)) - \bar{\eta}(c)\|_{H^1} + \|\xi(t, \cdot - (x_0,y_0)) - \bar{\xi}(c)\|_{H^{1/2}_{*}} \big) < \rho.
		\end{equation*},

Question:
Let \(\bar u(c)=(\bar\eta(c),\bar\xi(c))\) be any solitary capillary-gravity water wave with speed \(c=\frac{1}{\sqrt{1+\varepsilon^2}}\) for some \(\varepsilon\in(0,\varepsilon_0)\). Consider the Hamiltonian water-wave system for \(u=(\eta,\xi)\):
\[
\begin{cases}
\eta_t = G(\eta)\xi, \\
\xi_t = \frac{1}{2(1+|\nabla_{\perp}\eta|^2)}\left[(G(\eta)\xi+\nabla_{\perp}\eta\cdot\nabla_{\perp}\xi)^2-|\nabla_{\perp}\xi|^2|\nabla_{\perp}\eta|^2-|\nabla_{\perp}\xi|^2\right]
- g\eta + \sigma\operatorname{div}\!\left( \frac{\nabla_{\perp} \eta}{\sqrt{1 + |\nabla_{\perp} \eta|^2}} \right),
\end{cases}
\]
where \(G(\eta)\) is the Dirichlet-to-Neumann operator defined by \(G(\eta)\xi=\partial_{x_3}\Phi-\nabla_{\perp}\eta\cdot\nabla_{\perp}\Phi\) at the free surface. Suppose that for some \(R>0\), \(u=(\eta,\xi):[0,T)\to \mathscr F_R\) is a continuous solution that preserves the energy \(\mathcal H\) and momentum \(\mathcal P\). Which statement holds for arbitrary \(\rho>0\) when the initial data are sufficiently close to \((\bar\eta(c),\bar\xi(c))\) in \(H^1(\mathbb R^2)\times H_*^{1/2}(\mathbb R^2)\)?

For every \(R>0\) and \(\rho>0\), there exists \(\rho_0>0\) such that if
\[
\|\eta(0)-\bar\eta(c)\|_{H^1(\mathbb R^2)}+\|\xi(0)-\bar\xi(c)\|_{H_*^{1/2}(\mathbb R^2)}<\rho_0,
\]
then for every \(t\in[0,T)\),
\[
\inf_{(x_0,y_0)\in\mathbb R^2}\Big(\|\eta(t,\cdot-(x_0,y_0)) - \bar\eta(c)\|_{H^1}+\|\xi(t,\cdot-(x_0,y_0)) - \bar\xi(c)\|_{H_*^{1/2}}\Big)<\rho.
\]
Equivalently, every such solitary wave \(\bar u(c)\) is conditionally orbitally stable modulo spatial translations.

Choices:
B.
For every \(R>0\) and \(\rho>0\), there exists \(\rho_0>0\) such that if
\[
\|\eta(0)-\bar\eta(c)\|_{H^1(\mathbb R^2)}+\|\xi(0)-\bar\xi(c)\|_{H_*^{1/2}(\mathbb R^2)}<\rho_0,
\]
then for every \(t\in[0,T)\),
\[
\|\eta(t)-\bar\eta(c)\|_{H^1}+\|\xi(t)-\bar\xi(c)\|_{H_*^{1/2}}<\rho.
\]
That is, the wave is conditionally orbitally stable without taking the infimum over spatial translations.
C.
For every \(R>0\) and \(\rho>0\), there exists \(\rho_0>0\) such that if
\[
\|\eta(0)-\bar\eta(c)\|_{H^1(\mathbb R^2)}+\|\xi(0)-\bar\xi(c)\|_{H_*^{1/2}(\mathbb R^2)}<\rho_0,
\]
then for every \(t\in[0,T)\), there exists \((x_0,y_0)\in\mathbb R^2\) for which
\[
\|\eta(t,\cdot-(x_0,y_0)) - \bar\eta(c)\|_{H^1}<\rho.
\]
D.
There exists \(\rho_0>0\), depending only on \(\rho>0\) and not on \(R\), such that whenever
\[
\|\eta(0)-\bar\eta(c)\|_{H^1(\mathbb R^2)}+\|\xi(0)-\bar\xi(c)\|_{H_*^{1/2}(\mathbb R^2)}<\rho_0,
\]
for any continuous solution \(u=(\eta,\xi):[0,T)\to \mathscr F_R\) of the Hamiltonian system preserving \(\mathcal H\) and \(\mathcal P\), one has for every \(t\in[0,T)\),
\[
\inf_{(x_0,y_0)\in\mathbb R^2}\Big(\|\eta(t,\cdot-(x_0,y_0)) - \bar\eta(c)\|_{H^1}+\|\xi(t,\cdot-(x_0,y_0)) - \bar\xi(c)\|_{H_*^{1/2}}\Big)<\rho.
\]
E.
For every \(R>0\) and \(\rho>0\), there exists \(\rho_0>0\) such that if
\[
\|\eta(0)-\bar\eta(c)\|_{H^1(\mathbb R^2)}+\|\xi(0)-\bar\xi(c)\|_{H_*^{1/2}(\mathbb R^2)}<\rho_0,
\]
then for every \(t\in[0,T)\),
\[
\inf_{(x_0,y_0)\in\mathbb R^2}\Big(\|\eta(t,\cdot-(x_0,y_0)) - \bar\eta(c)\|_{H^1}+\|\xi(t,\cdot-(x_0,y_0)) - \bar\xi(c)\|_{H_*^{1/2}}\Big)<\rho,
\]
for every continuous solution \(u=(\eta,\xi):[0,T)\to \mathscr F_R\) of the Hamiltonian system with sufficiently small initial data, even without assuming preservation of both \(\mathcal H\) and \(\mathcal P\).


[Convergence of general test functions on $A$] \label{thm:1} Let $f$ be as above and $N_{\mathbb R}(0,1)$ be a standard Gaussian random variable. Moreover, let $S$ denote the random variable with density $\varrho_{sc}(x)\deq \frac{1}{2\pi}\sqrt{(4-x^2)_+}$. Then
\[\frac{f_{ii}(A)-\mathbb E[f_{ii}(A)]}{\sqrt{V_{ii}(f)}}\xrightarrow{d} N_{\mathbb R}(0,1)\]
as $N \to \infty$, where $V_{ii}(f)$ is defined as
\begin{align*}
&\,
\frac{2}{N}\biggl(\int_{-2}^2 f(x)^2\varrho_{sc}(x)\mathrm{d}x-\biggl(\int_{-2}^2 f(x)\varrho_{sc}(x)\mathrm{d}x\biggr)^2\biggr)+N \mathcal{C}_{4}(H_{12})\biggl(\int_{-2}^2 f(x)(1-x^2)\varrho_{sc}(x)\mathrm{d}x \biggr)^2
\\
 =&\,\frac{2}{N}\mathrm{Var}(f(S)) +N\mathcal C_4(H_{12}) \big(\mathbb E [f(S)(1-S^2)]\big)^2
\end{align*}
and
\[\mathbb E [f_{ii}(A)] = \int_{-2}^2 f(x)\varrho_{sc}(x)\mathrm{d}x + O(N^{-\tau/100}V_{ii}(f)^{1/2})\,.\]

Fix small $\tau>0$. In this paper, we consider the following class of random matrices.
\begin{definition} [Sparse matrix] \label{def:sparse} Let $q\in [N^{\tau/2},N^{1/2-\tau/2}]$. Consider a real-symmetric $N\times N$ matrix $H$ whose entries $H_{ij}$ satisfy the following conditions.
 \begin{enumerate}
 \item[(i)] The upper-triangular entries ($H_{ij}\col 1 \leqslant i \leqslant j\leqslant N$) are independent.
 \item[(ii)] We have $\mathbb E H_{ij}=0$, $ \mathbb E H_{ij}^2=(1+O(\delta_{ij}))/N$, and $\mathbb E H_{ij}^4\asymp 1/(Nq^2)$ for all $i,j$.
 \item[(iii)] For any $k\geqslant 3$, we have
 $\mathbb E|H_{ij}|^k \leqslant C_k/ (Nq^{k-2})$ for all $i,j$.
 \end{enumerate}
 We define the random matrix
 $$
 A = H + f \mathrm{e} \mathrm{e}^*\,,
 $$
 where $\mathrm{e} \deq N^{-1/2} (1,1,\dots,1)^*$, and $f \asymp q$.
\end{definition}
One major motivation for Definition \ref{def:sparse} is the sparse Erd\H{o}s-R\'enyi graph $\mathcal G(N,p)$. More precisely, it is an undirected graph on $N$ vertices, and each edge is connected with probability $p$, independent from any other edges. Let $\mathcal A$ denote the adjacency matrix of the graph. Explicitly, $\mathcal A = (\mathcal A_{ij})_{i,j = 1}^N$ is a symmetric $N\times N$ matrix with independent upper triangular entries $(\mathcal A_{ij} \col i \leqslant j)$ satisfying
\begin{equation*}
 {\mathcal A}_{ij}=\begin{cases}
 1 & \txt{with probability } p
 \\
 0 & \txt{with probability } 1-p\,.
 \end{cases}
\end{equation*}
We introduce the normalized adjacency matrix
\begin{equation} \label{1.11}
 A\deq \sqrt{\frac{1}{p(1-p)N}}\,\mathcal A\,,
\end{equation}
where the normalization is chosen so that the eigenvalues of $A$ are typically of order one. More precisely, let $\lambda_1\geqslant \cdots \geqslant \lambda_N$ be the eigenvalues of $A$. It can be shown that the empirical eigenvalue density of $A$ satisfies
\begin{equation}
 \mu(x)\deq \frac{1}{N}\sum_{i=1}^N\delta_{\lambda_i}(x) \overset{w}{\longrightarrow}\varrho_{sc}(x)\deq \frac{1}{2\pi} \sqrt{(4-x^2)_+}
\end{equation}
almost surely as $N\to \infty$. It is easy to check that when $N^{-1+\tau} \leqslant p\leqslant N^{-\tau}$, the rescaled adjacency matrix $A$ satisfies Definition \ref{def:sparse} with $q\deq\sqrt{Np}$.

The matrix $A$ has typically $N^2p$ nonzero entries, and hence $A$ is \textit{sparse} whenever $p\to 0$ as $N\to \infty$. The celebrated Wigner-dyson-Mehta (WDM) universality conjecture asserts that the local spectral properties of a random matrix do not depend on the explicit distribution of the matrix entries, and they are only determined by the symmetry class of the matrix. During the past decade, the
universality conjecture for sparse matrices has been established in a series of papers \cite{EKYY1,EKYY2,HLY15,LS1,HLY,HK20,Lee21,HY22} in
great generality. More precisely, it has been shown that when $p\geqslant N^{-1+o(1)}$, the averaged
$n$-point correlation functions and the distribution of a single eigenvalue gap of $A$ coincide with those of the
GOE, and the edge eigenvalues of $A$ have Tracy-Widom distributions.

Another important topic in random matrix theory is the study of linear eigenvalue statistics $\operatorname{Tr} f(A)$. When the graph is dense, $H\deq A-\mathbb EA$ is essentially a Wigner matrix, and the distribution of $\operatorname{Tr} f(H)$ was obtained both on the global \cite{LP,BWZ09} and mesoscopic scales \cite{HK}. For sparse matrices, the distribution of $\operatorname{Tr} f(H)$ was computed in \cite{ST12,H19}, where \cite{ST12} treated the global scale, and \cite{H19} handled the mesoscopic scales with the special test function $f(x)=(x-\mathrm{i})^{-1}$.

As a natural extension of the linear statistics, one can also study the fluctuations of functions (i.e. $f_{ij}(H)\deq f(H)_{ij}$) of random matrices. For Wigner matrices, the distribution of $f_{ij}(H)$ was derived both on the global \cite{ORS,ES16} and mesoscopic \cite{CEK6} scales.

In this paper, we study the fluctuation of $f_{ii}(A)$ on the sparse levels $N^{-1+\tau} \leqslant p\leqslant N^{-\tau}$. Our test functions $f\equiv f_N \in C^{\infty}(\mathbb R)$ live on the scale $\eta_*\in [N^{-1+\tau},1]$. More precisely, let $F\in C_c^{\infty}(\mathbb R)$ be a function independent of $N$ and $E\in[-2+\tau, 2-\tau]$. Then \[f(x)\deq F\biggl(\frac{x-E}{\eta_*}\biggr)\,.\]
We further require that
\[f'(x) \neq0 \text{ only if }x\in (-2+\tau, 2-\tau)\,.\]

Assuming that all off-diagonal entries of $H$ are identically distributed, we may now state our main result.

\begin{definition} [Sparse matrix] \label{def:sparse} Let $q\in [N^{\tau/2},N^{1/2-\tau/2}]$. Consider a real-symmetric $N\times N$ matrix $H$ whose entries $H_{ij}$ satisfy the following conditions.
	\begin{enumerate}
		\item[(i)] The upper-triangular entries ($H_{ij}\col 1 \leqslant i \leqslant j\leqslant N$) are independent.
		\item[(ii)] We have $\bb E H_{ij}=0$, $ \bb E H_{ij}^2=(1+O(\delta_{ij}))/N$, and $\bb E H_{ij}^4\asymp 1/(Nq^2)$ for all $i,j$.
		\item[(iii)] For any $k\geqslant 3$, we have
		$\bb E|H_{ij}|^k \leqslant C_k/ (Nq^{k-2})$ for all $i,j$.
	\end{enumerate}
	We define the random matrix
	$$
	A = H + f \e \e^*\,,
	$$
	where $\e \deq N^{-1/2} (1,1,\dots,1)^*$, and $f \asymp q$.
\end{definition}

(ii) For simplicity, here we only state the result for $f_{ii}(A)$. It can be checked that the same result also holds for $f_{ii}(H)$.
\end{remark}

\begin{align*}\label{ee}
\biggl|\int_{\Omega_{\alpha}}(\mathrm{i}y\chi(y/\eta_*)f''(x) + \mathrm{i}\eta^{-1}_*(f(x)+\mathrm{i}f'(x)y)\chi'(y/\eta_*))(\mathcal{E}-\mathcal{E}_{\alpha})\mathrm{d}x\mathrm{d}y\biggr| &\prec N^{-\tau}(\eta_*/q+(\eta_*/N)^{1/2}) \\
&\asymp N^{-\tau}\sqrt{V_{ii}(f)}\,.
\end{align*}
Hence,
\[\psi_{ii}'(\lambda) = \frac{\mathrm{i}}{2\pi \sqrt{V_{ii}(f)}}\int_{\Omega_{\alpha}}(\mathrm{i}y\chi(y/\eta_*)f''(x) + \mathrm{i}\eta^{-1}_*(f(x)+\mathrm{i}f'(x)y)\chi'(y/\eta_*))\mathcal{E}_{\alpha}(z) \mathrm{d}x\mathrm{d}y +O_{\prec}(N^{-\tau})\,.\]
Using Lemma \ref{lemma:integral_estimate}, we could estimate the derivatives of $e_{\alpha}(\lambda)$ w.r.t. entries of $H$. We have, for $k\neq i$, that
\begin{align*}
 \frac{\partial e_{\alpha}(\lambda)}{\partial H_{ki}} &= e_{\alpha}(\lambda)\biggl[ \frac{-2\mathrm{i}\lambda}{\pi(1+\delta_{ki}) \sqrt{V_{ii}(f)}}\int_{\Omega_{\alpha}}\partial_{\bar{z}}\tilde{f}(z)G_{ii}G_{ik}\mathrm{d}^2z \biggr]\\
 &\prec 1 \,,\numberthis \label{e1}\\
 \sum_{k=1}^N \frac{\partial e_{\alpha}(\lambda)}{\partial H_{ki}} &= e_{\alpha}(\lambda)\biggl[ \frac{-2\mathrm{i}\lambda}{\pi(1+\delta_{ki})\sqrt{V_{ii}(f)}}\int_{\Omega_{\alpha}}\partial_{\bar{z}}\tilde{f}(z)G_{ii}\sum_{k=1}^N G_{ik}\mathrm{d}^2z \biggr] \\
 &\prec N^{1/2}\,, \numberthis \label{e2}\\
 \frac{\partial^2 e_{\alpha}(\lambda)}{\partial H_{ki}^2} &= \begin{aligned}[t]
 e_{\alpha}(\lambda)\biggl[ &\frac{-2\mathrm{i}\lambda}{\pi(1+\delta_{ki}) \sqrt{V_{ii}(f)}}\int_{\Omega_{\alpha}} \partial_{\bar{z}}\tilde{f}(z)G_{ii}G_{ik}\mathrm{d}^2z \biggr]^2 \\ & + e_{\alpha}(\lambda)\frac{2\mathrm{i}\lambda}{\pi(1+\delta_{ki})^2 \sqrt{V_{ii}(f)}}\int_{\Omega_{\alpha}}\partial_{\bar{z}}\tilde{f}(z)(G_{ii}^2G_{kk}+3G_{ki}^2G_{ii})\mathrm{d}^2z
 \end{aligned}
 \\
 &= e_{\alpha}(\lambda)\frac{2\mathrm{i}\lambda}{\pi(1+\delta_{ki})^2 \sqrt{V_{ii}(f)}}\int_{\Omega_{\alpha}}\partial_{\bar{z}}\tilde{f}(z)m^3 \mathrm{d}^2z + O_{\prec}(1)\,, \numberthis \label{e3}
\end{align*}
where we have applied Theorem \ref{thm:semiA} for \eqref{e1} and \eqref{e3}, and Theorem \ref{thm:isoA} for \eqref{e2}. In general, we have
\begin{equation}
\label{ep}
\bigg|\frac{\partial^r e_{\alpha}(\lambda)}{\partial H_{ki}^r}\bigg| \prec \eta_*V_{ii}(f)^{-1/2}\,.
\end{equation}

Let $\alpha = \tau/100$. Using the fact that $|m(z)| \asymp 1$ and $|m'(z)| \prec |y|^{-1/2}$, the integrand above is of the order $O_{\prec}(N^{-1}(|y|^{-1}+|y'|^{-1}) + q^{-2})$. Applying Lemma \ref{lemma:integral_estimate} twice,
\begin{align*}
 &\biggl|\int_{ \Omega_{\alpha}^2}\partial_{\bar{z}}\tilde{f}(z)\partial_{\bar{z'}}\tilde{f}(z') \biggl[2N^{-1} m(z)m(z')\frac{m(z')-m(z)}{z'-z}+N\mathcal {C}_{4}(H_{12})m^3(z)m^3(z')\biggr]\mathrm{d}^2z'\mathrm{d}^2z \biggr|\\
 &\prec \eta_*/N + \eta^2_*/q^2 \\
 &\prec V_{ii}(f)\,.
\end{align*}
Hence we could replace $e_{\alpha}(\lambda)$ by $e(\lambda)$ by \eqref{er}. Next, we argue that we could replace $\Omega_{\alpha}^2$ by $\bb C^2$ in the above integral. The contributions from the regions $(\bb C\setminus\Omega_{\alpha})^2$ and $(\bb C\setminus\Omega_{\alpha})\times\Omega_{\alpha}$ could be shown to be of order $O_{\prec}(N^{-\tau/100}V_{ii}(f))$ as follows:
\begin{align*}
 &\biggl|\int_{(\bb C \setminus \Omega_{\alpha})^2}\partial_{\bar{z}}\tilde{f}(z)\partial_{\bar{z'}}\tilde{f}(z') \biggl[2N^{-1} m(z)m(z')\frac{m(z')-m(z)}{z'-z}+N\mathcal {C}_{4}(H_{12})m^3(z)m^3(z')\biggr]\mathrm{d}^2z'\mathrm{d}^2z \biggr|\\
 &\prec \int_{(\bb C \setminus \Omega_{\alpha})^2}|\partial_{\bar{z}}\tilde{f}(z)\partial_{\bar{z'}}\tilde{f}(z')| \biggl[N^{-1}(|y|^{-1} + |y'|^{-1})+q^{-2}\biggr]\mathrm{d}^2z'\mathrm{d}^2z \\
 &\prec \int_{\bb C \setminus \Omega_{\alpha}} |\partial_{\bar{z}}\tilde{f}(z)|\biggl[N^{-1}(|y|^{-1}(N^{\alpha -1})^2||f''||_{1} + N^{\alpha -1 }||f''||_{1})+q^{-2}(N^{\alpha -1})^2||f''||_{1}\biggr]\mathrm{d}^2z\\
 &\prec N^{-1}(N^{\alpha-1})^{3}||f''||_{1}^2 +q^{-2}(N^{\alpha -1})^4||f''||_{1}^2 \\
 &\prec N^{-\tau/100}V_{ii}(f)\,.
\end{align*}
Similarly,
\begin{align*}
 &\biggl|\int_{(\bb C \setminus \Omega_{\alpha})\times\Omega_{\alpha}}\partial_{\bar{z}}\tilde{f}(z)\partial_{\bar{z'}}\tilde{f}(z') \biggl[2N^{-1} m(z)m(z')\frac{m(z')-m(z)}{z'-z}+N\mathcal {C}_{4}(H_{12})m^3(z)m^3(z')\biggr]\mathrm{d}^2z'\mathrm{d}^2z \biggr|\\
 &\prec N^{-1}(N^{\alpha - 1}\eta_*||f''||_{1} + (N^{\alpha -1})^2||f''||_{1})+q^{-2}(N^{\alpha -1})^2\eta_*||f''||_{1} \\
 &\prec N^{-\tau/100}V_{ii}(f)\,,
\end{align*}
where Lemma \ref{lemma:integral_estimate} is applied once for the second estimate.
This completes the proof of Proposition \ref{prop:char}.
\subsection{Evaluation of Integrals} \label{subsection}
We now evaluate the integrals appearing in Proposition \ref{prop:char} by applying Green's Theorem, which states that for any sufficiently smooth function $F(z)$,
\begin{equation}
\label{eq:green}
\int_{\Omega}\partial_{\bar{z}}F(z)\mathrm{d}^2z=\frac{-\mathrm{i}}{2}\int_{\partial \Omega}F(z)\mathrm{d}z\,.
\end{equation}
 Now let $\eta >0$ be small and define
\[ \widetilde{\Omega}_{\eta} \deq \{ (x+\mathrm{i}y, x'+\mathrm{i}y') \in \bb C^2:|y| \geqslant \eta/2, |y'| \geqslant \eta/2\}\,.\]
The boundary $\partial\widetilde{\Omega}_{\eta}$ consists of four branches, corresponding to the possible sign combinations of the imaginary part of $z$ and $z'$. Consider first the branch $z' = x' + \mathrm{i}\eta/2$, $z= x-\mathrm{i}\eta/2$, we have
\begin{align*}
 &\quad\frac{m(z')-m(z)}{z'-z}\\ &= \frac{-\mathrm{i}\eta+x-x'+\sqrt{z'^2 -4}+\sqrt{z^2-4}}{2(x'-x+\mathrm{i}\eta)} \\
 &=\frac{(\sqrt{4-x'^2}+\sqrt{4-x^2})\eta}{2((x'-x)^2+\eta^2)} +\frac{O(\eta^2)}{(x'-x)^2+\eta^2}
 +\frac{(x'-x)(x-x'+\mathrm{i}\sqrt{4-x'^2}+\mathrm{i}\sqrt{4-x^2})}{2((x'-x)^2+\eta^2)}\,,
 \numberthis \label{1}\\ \\
 &\quad m(z')m(z)\\ &= \frac{(-x'+\mathrm{i}\sqrt{4-x'^2} + O(\eta))(-x-\mathrm{i}\sqrt{4-x^2} + O(\eta))}{4}\,.\numberthis \label{2}
\end{align*}
 As $\eta$ tends to $0$, multiplying \eqref{1} and \eqref{2} together and summing over analogous expressions from the other boundary branches yields $-2\pi\sqrt{4-x^2}\delta(x'-x)+\sqrt{4-x^2}\sqrt{4-x'^2}$. Using \eqref{eq:green},
\begin{align*}
 &\quad\int_{\bb C^2}\partial_{\bar{z}}\tilde{f}(z)\partial_{\bar{z'}}\tilde{f}(z') m(z)m(z')\frac{m(z')-m(z)}{z'-z}\mathrm{d}^2z'\mathrm{d}^2z \\ &= \lim_{\eta \to0^+} \int_{\widetilde{\Omega}_{\eta}}\partial_{\bar{z}}\tilde{f}(z)\partial_{\bar{z'}}\tilde{f}(z') m(z)m(z')\frac{m(z')-m(z)}{z'-z}\mathrm{d}^2z'\mathrm{d}^2z \\
 &= \frac{1}{2}\int_{-2}^2\int_{-2}^2f(x)f(x')\pi\sqrt{4-x^2}\delta(x'-x)\mathrm{d}x\mathrm{d}x' -\biggl( \frac{1}{2} \int_{-2}^2 f(x)\sqrt{4-x^2}\mathrm{d}x\biggr)^2\\
 &=\pi^2\int_{-2}^2 f(x)^2\varrho_{sc}(x)\mathrm{d}x -\biggl( \pi \int_{-2}^2 f(x)\varrho_{sc}(x)\mathrm{d}x\biggr)^2 \,.\numberthis \label{eq:integral1}
\end{align*}
Similarly, denoting $z= x+\mathrm{i}\eta/2$, $z' = x' + \mathrm{i}\eta/2$, we have
\begin{align*}
 &\quad m^3(z)m^3(z') - m^3(\bar{z})m^3(z')-m^3(z)m^3(\bar{z'})+m^3(\bar{z})m^3(\bar{z'})\\
 &=(m^3(z)-m^3(\bar{z}))(m^3(z')-m^3(\bar{z'}))\\
 &=-\sqrt{4-x^2}(1-x^2)\sqrt{4-x'^2}(1-x'^2) + O(\eta)\,,
\end{align*}
and hence
\begin{align*}
 \int_{\bb C^2}\partial_{\bar{z}}\tilde{f}(z)\partial_{\bar{z'}}\tilde{f}(z') m^3(z')m^3(z)\mathrm{d}^2z'\mathrm{d}^2z &= \biggl(\pi\int_{-2}^2 f(x)(1-x^2)\varrho_{sc}(x)\mathrm{d}x \biggr)^2 \,.\numberthis \label{eq:integral2}
\end{align*}
Substituting \eqref{eq:integral1} and \eqref{eq:integral2} into \eqref{eq:char}, then integrating w.r.t. $\lambda$ yields Theorem \ref{thm:1}.

Here, $\mathcal C_4$ denotes the fourth cumulant.
\begin{remark}
	(i) The first term on $V_{ii}(f)$ is of order $\eta_*/N$, while the second term is of order $\eta^2_*/q^2$. Thus, for the diagonal entries $f_{ii}(A)$, there is a phase transition of the fluctuation on the scale $\eta_*/N\asymp \eta^2_*/q^2$, i.e.\,$\eta_*\asymp q^2/N= p$.

(ii) For simplicity, here we only state the result for $f_{ii}(A)$. It can be checked that the same result also holds for $f_{ii}(H)$.
\end{remark}

Our proof begins with the strategies of \cite{HK,H,LS18}, by converting the general test function to the Green function, and then compute through the cumulant expansion formula. Notably, for sparse matrices, we do not need to remove the diagonal contribution $H_{ii}\int f(x) x \varrho_{sc}(x)\mathrm{d} x$ as in the Wigner case \cite{ES16}, since this term is always negligible in the sparse regime $p\leqslant N^{-o(1)}$. In addition, we have new terms arising from the large expectation of $A$.

\paragraph{Acknowledgment}The author is partially supported by Hong Kong RGC Grant
No.\,21300223.

Question: Fix a small \(\tau>0\). Let \(q\in [N^{\tau/2},N^{1/2-\tau/2}]\), and let \(H\) be a real-symmetric \(N\times N\) matrix whose upper-triangular entries are independent and satisfy \(\mathbb E H_{ij}=0\), \(\mathbb E H_{ij}^2=(1+O(\delta_{ij}))/N\), \(\mathbb E H_{ij}^4\asymp 1/(Nq^2)\), and for every \(k\ge 3\), \(\mathbb E|H_{ij}|^k\le C_k/(Nq^{k-2})\); assume also that all off-diagonal entries of \(H\) are identically distributed. Set \(A=H+f_0\,ee^*\), where \(e=N^{-1/2}(1,\dots,1)^*\) and \(f_0\asymp q\). Let \(\eta_*\in [N^{-1+\tau},1]\), choose \(E\in[-2+\tau,2-\tau]\) and \(F\in C_c^{\infty}(\mathbb R)\) independent of \(N\), and define the test function \(f(x)=F\bigl((x-E)/\eta_*\bigr)\), with \(f'(x)\neq 0\) only for \(x\in(-2+\tau,2-\tau)\). If \(f(A)_{ii}\) denotes the \((i,i)\)-entry of \(f(A)\), and \(S\) is a random variable with semicircle density \(\varrho_{sc}(x)=\frac{1}{2\pi}\sqrt{(4-x^2)_+}\), which statement holds as \(N\to\infty\)?

The normalized diagonal entry converges to a standard real Gaussian: \[\frac{f(A)_{ii}-\mathbb E[f(A)_{ii}]}{\sqrt{V_{ii}(f)}}\xrightarrow{d} N_{\mathbb R}(0,1),\] where \[V_{ii}(f)=\frac{2}{N}\left(\int_{-2}^2 f(x)^2\varrho_{sc}(x)\,dx-\left(\int_{-2}^2 f(x)\varrho_{sc}(x)\,dx\right)^2\right)+N\,\mathcal C_4(H_{12})\left(\int_{-2}^2 f(x)(1-x^2)\varrho_{sc}(x)\,dx\right)^2,\] equivalently \[V_{ii}(f)=\frac{2}{N}\operatorname{Var}(f(S))+N\mathcal C_4(H_{12})\bigl(\mathbb E[f(S)(1-S^2)]\bigr)^2,\] with \(\mathcal C_4(H_{12})\) the fourth cumulant of an off-diagonal entry. Moreover, \[\mathbb E[f(A)_{ii}] = \int_{-2}^2 f(x)\varrho_{sc}(x)\,dx + O\bigl(N^{-\tau/100}V_{ii}(f)^{1/2}\bigr).\]

Choices:
B. The normalized diagonal entry converges to a standard real Gaussian: \[\frac{f(A)_{ii}-\mathbb E[f(A)_{ii}]}{\sqrt{\widetilde V_{ii}(f)}}\xrightarrow{d} N_{\mathbb R}(0,1),\] where \[\widetilde V_{ii}(f)=\frac{2}{N}\left(\int_{-2}^2 f(x)^2\varrho_{sc}(x)\,dx-\left(\int_{-2}^2 f(x)\varrho_{sc}(x)\,dx\right)^2\right)+N\,\mathcal C_4(H_{12})\left(\int_{-2}^2 f(x)\,x\,\varrho_{sc}(x)\,dx\right)^2,\] and moreover \[\mathbb E[f(A)_{ii}] = \int_{-2}^2 f(x)\varrho_{sc}(x)\,dx + O\bigl(N^{-\tau/100}\widetilde V_{ii}(f)^{1/2}\bigr).\]
C. The expectation satisfies \[\mathbb E[f(A)_{ii}] = \int_{-2}^2 f(x)\varrho_{sc}(x)\,dx + O\bigl(N^{-\tau/100}V_{ii}(f)^{1/2}\bigr),\] where \[V_{ii}(f)=\frac{2}{N}\operatorname{Var}(f(S))+N\mathcal C_4(H_{12})\bigl(\mathbb E[f(S)(1-S^2)]\bigr)^2.\]
D. The normalized diagonal entry converges to a standard real Gaussian: \[\frac{f(A)_{ii}-\mathbb E[f(A)_{ii}]}{\sqrt{V^{\mathrm{univ}}_{ii}(f)}}\xrightarrow{d} N_{\mathbb R}(0,1),\] where the variance is universal and given by \[V^{\mathrm{univ}}_{ii}(f)=\frac{2}{N}\left(\int_{-2}^2 f(x)^2\varrho_{sc}(x)\,dx-\left(\int_{-2}^2 f(x)\varrho_{sc}(x)\,dx\right)^2\right),\] so the fourth-cumulant contribution is negligible for all admissible sparse matrices. Moreover, \[\mathbb E[f(A)_{ii}] = \int_{-2}^2 f(x)\varrho_{sc}(x)\,dx + O\bigl(N^{-\tau/100}(V^{\mathrm{univ}}_{ii}(f))^{1/2}\bigr).\]
E. For every fixed index \(i\), the diagonal entry itself satisfies a central limit theorem without centering: \[\frac{f(A)_{ii}-\int_{-2}^2 f(x)\varrho_{sc}(x)\,dx}{\sqrt{V_{ii}(f)}}\xrightarrow{d} N_{\mathbb R}(0,1),\] where \[V_{ii}(f)=\frac{2}{N}\operatorname{Var}(f(S))+N\mathcal C_4(H_{12})\bigl(\mathbb E[f(S)(1-S^2)]\bigr)^2.\] In particular, one may replace \(\mathbb E[f(A)_{ii}]\) by its leading semicircle term in the normalization.

negligible diagonal-term replaced by Wigner-type x-moment

dropped the CLT conclusion and kept only the expectation asymptotic

non-negligible fourth-cumulant term removed from variance

centering by exact expectation replaced by leading-order deterministic term