\label{Main theorem}
	Suppose $T$ is a Calder\'on-Zygmund operator defined as in (\ref{def of Tf}) associated with a kernel satisfying (\ref{Size condition}) and the $L_2$-mean H\"ormander condition (\ref{L2 mean Hormander condition}).
	Let $T_{\epsilon}$ be defined as in (\ref{def of T epsilon f}).
	Suppose $(T_{\epsilon})_{\epsilon>0}$ is of strong type $(p_0,p_0)$ for some $1< p_0 <\infty$, that is, for any $f\in L_{p_0}(\mN)$,
	\[
	\norm{(T_{\epsilon}f)_{\epsilon>0}}_{L_{p_0}(\mN;\ell_{\infty})} \lesssim  \left\| f \right\| _{L_{p_0}(\mN)}.
	\]
	Then, $(T_{\epsilon})_{\epsilon>0}$ is of weak type $(1,1)$, that is, for any $f\in L_1(\mN)$,
	\[
	\norm{(T_{\epsilon}f)_{\epsilon>0}}_{\Lambda_{1,\infty}(\mN;\ell_{\infty})} \lesssim  \left\| f \right\| _{L_1(\mN)}.
	\]
We refer to see Appendix \ref{Appendix: Noncommutative Lp spaces and maximal norms} for the definitions of the noncommutative maximal norm $ \left\| \cdot \right\| _{L_p(\mN; \ell_{\infty})}$ and the noncommutative weak maximal norm $ \left\| \cdot \right\| _{\Lambda_{p,\infty}(\mN;\ell_{\infty})}$.

\begin{align}\label{Hormander condition}
	\sup_{y\in\R^d,|v|>0} \int_{|x-y|\geq 2|v|} |K(x,y+v)-K(x,y)| \dif x < \infty,
    \end{align}

\begin{align} \label{L2-integral regularity condition}
	\sum_{m\geq 1} \sup_{{\begin{subarray}{c} y\in Q \\ Q\ \text{dyadic} \end{subarray}}} \Big( 2^{md}\ell(Q)^d  \int_{2^m\ell(Q)\leq|x-c(Q)|\leq2^{m+1}\ell(Q)} |K(x,y)-K(x,c(Q))|^2 \dif x \Big)^{\frac{1}{2}}
	<\infty,
    \end{align}

\begin{theorem}\label{Main theorem}
	Suppose $T$ is a Calder\'on-Zygmund operator defined as in (\ref{def of Tf}) associated with a kernel satisfying (\ref{Size condition}) and the $L_2$-mean H\"ormander condition (\ref{L2 mean Hormander condition}).
	Let $T_{\epsilon}$ be defined as in (\ref{def of T epsilon f}).
	Suppose $(T_{\epsilon})_{\epsilon>0}$ is of strong type $(p_0,p_0)$ for some $1< p_0 <\infty$, that is, for any $f\in L_{p_0}(\mN)$,
	\[
	\norm{(T_{\epsilon}f)_{\epsilon>0}}_{L_{p_0}(\mN;\ell_{\infty})} \lesssim \norm{f}_{L_{p_0}(\mN)}.
	\]
	Then, $(T_{\epsilon})_{\epsilon>0}$ is of weak type $(1,1)$, that is, for any $f\in L_1(\mN)$,
	\[
	\norm{(T_{\epsilon}f)_{\epsilon>0}}_{\Lambda_{1,\infty}(\mN;\ell_{\infty})} \lesssim \norm{f}_{L_1(\mN)}.
	\]
We refer to see Appendix \ref{Appendix: Noncommutative Lp spaces and maximal norms} for the definitions of the noncommutative maximal norm $\norm{\cdot}_{L_p(\mN; \ell_{\infty})}$ and the noncommutative weak maximal norm $\norm{\cdot}_{\Lambda_{p,\infty}(\mN;\ell_{\infty})}$.
    \end{theorem}

Calder\'on-Zygmund operators, a well-known class of singular integral operators, were introduced by Calder\'on and Zygmund \cite{Calderon1952On} in their seminal 1952 work.
The kernel of such an operator satisfies specific conditions, including the size and regularity conditions, the latter of which was refined by H\"ormander \cite{Hormander1960Estimates} in 1960.

The standard nonconvolution type Calder\'on-Zygmund operator is defined as
    \[
    Tf(x) = \int_{\R^d} K(x,y) f(y) {\rm d} y, \quad x\notin \operatorname{supp} f,
    \]
where $K:\R^d \times \R^d \setminus  \left\{ (x,x): x\in\R^d \right\}  \rightarrow \C$
is a standard nonconvolution type Calder\'on-Zygmund kernel, which is a binary function satisfying
    \begin{itemize}
	\item \textbf{Size condition:} there exists $C_1>0$ such that
	   \begin{align} \label{Size condition}
       |K(x,y)| \leq \frac{C_1}{|x-y|^d} ;
       \end{align}
	\item  \textbf{Lipschitz regularity condition:} there exists  $\gamma\in [0,1]$ and $C_2>0$ such that
	   \begin{align} \label{Lipschitz regularity condition}
	   |K(x,y)-K(x,z)|+|K(y,x)-K(z,x)| \leq \frac{C_2 |y-z|^{\gamma}}{|x-y|^{d+\gamma}}, \quad\text{if}\  |x-y|\geq 2|y-z|,
	   \end{align}
    \end{itemize}
where $x$, $y$, $z\in\R^d$.
The Lipschitz regularity condition (\ref{Lipschitz regularity condition}) can be relaxed to certain variants of smoothness conditions (see \cite{Grafakos2019Alimited, Suzuki2021TheCalderon, Watson1990Weighted}).
In the following, all Calder\'on-Zygmund kernels under consideration are of nonconvolution type.

Inspired by operator algebras, harmonic analysis, noncommutative geometry, and quantum probability (see \cite{Cadilhac2018Weak, Chen2013Harmonic, Junge2014Smooth, Junge2018Noncommutative, Mei2007Operator, Mei2009Pseudo-localization, Mei2017Free, Mei2022Mikhlin, Parcet2009Pseudo-localization}), noncommutative harmonic analysis has become an exciting field in recent years.
However, generalizing the Calder\'on-Zygmund decomposition, a key technique in harmonic analysis first introduced in \cite{Calderon1952On}, to the noncommutative setting is challenging.

In 2009, Parcet \cite{Parcet2009Pseudo-localization} rigorously constructed a kind of noncommutative Calder\'on-Zygmund decomposition by using Cuculescu's maximal weak type $(1,1)$ estimates for noncommutative martingales \cite{Cuculescu1971Martingales}.
With this tool, Parcet established the weak type $(1,1)$ estimates for operator-valued Calder\'on-Zygmund singular integrals with the kernel satisfying the Lipschitz regularity condition (\ref{Lipschitz regularity condition})
(Cadilhac later gave a shorter proof in \cite{Cadilhac2018Weak}).
Notably,  the H\"ormander condition
    \begin{align}\label{Hormander condition}
	\sup_{y\in\R^d,|v|>0} \int_{|x-y|\geq 2|v|} |K(x,y+v)-K(x,y)| {\rm d} x < \infty,
    \end{align}
 which is weaker than (\ref{Lipschitz regularity condition}), is already sufficient for weak type $(1,1)$ boundedness in classical Calder\'on-Zygmund theory.
It is therefore natural to ask whether the noncommutative weak type $(1,1)$ boundedness still holds under the H\"ormander condition (\ref{Hormander condition})?
This question remains open.

A step forward was made in 2022 by Cadilhac et al. \cite{Cadilhac2022Spectral}.
They developed a more effective noncommutative Calder\'on-Zygmund decomposition without the off-diagonal term of the good function, which allowed them to prove the noncommutative weak type $(1,1)$ estimates for Calder\'on-Zygmund operators if the kernel $K$ satisfies
    \begin{align} \label{L2-integral regularity condition}
	\sum_{m\geq 1} \sup_{{\begin{subarray}{c} y\in Q \\ Q\ \text{dyadic} \end{subarray}}} \Big( 2^{md}\ell(Q)^d  \int_{2^m\ell(Q)\leq|x-c(Q)|\leq2^{m+1}\ell(Q)} |K(x,y)-K(x,c(Q))|^2 {\rm d} x \Big)^{\frac{1}{2}}
	<\infty,
    \end{align}
 where $\ell(Q)$ and $c(Q)$ stand for the length and the center of the dyadic cube $Q$, respectively.
 More generally, for $1\leq q\leq\infty$, we say that a kernel $K$ satisfies the $L_q$-integral regularity condition if
    \begin{align} \label{sum delta < infty}
	\sum_{m\geq 1} \delta_q(m) <\infty,
    \end{align}
where $\delta_q(m)$ is defined as
     \begin{align*}
	 \sup_{{\begin{subarray}{c} y\in \R^d,\ R>0 \\ |v|<R \end{subarray}}} & \Big( (2^m R)^{d(q-1)} \int_{2^mR\leq |x-y|\leq 2^{m+1}R} |K(x,y+v)-K(x,y)|^q {\rm d} x\Big)^{\frac{1}{q}},
    \end{align*}
for $m\geq 1$.
Roughly speaking, the H\"ormander condition (\ref{Hormander condition}) corresponds to the $L_1$-integral regularity condition i.e. (\ref{sum delta < infty}) with $q=1$, while \eqref{L2-integral regularity condition} corresponds to the $L_2$-integral regularity condition, i.e., (\ref{sum delta < infty}) with $q=2$.
The fact $\delta_1(m)\lesssim_d \delta_2(m)$ shows that the $L_2$-integral regularity condition is stronger than the H\"ormander condition (\ref{Hormander condition}), but weaker than the Lipschitz regularity condition (\ref{Lipschitz regularity condition}).
By applying the noncommutative Calder\'on-Zygmund decomposition in  \cite{Cadilhac2022Spectral}, Hong, the first author, and Xu \cite{hong2023maximal} established the weak type (1,1) boundedness for the noncommutative maximal truncated Calder\'on-Zygmund operators under the $L_2$-integral regularity condition.
We refer the reader to \cite{Cadilhac2022Spectral, Lai2024Noncommutative, Mei2009Pseudo-localization} for more results on weak (1,1) boundedness theory for noncommutative Calder\'on-Zygmund operators.

In this paper, we study a kernel smoothness condition that is slightly weaker than the $L_q$-integral regularity condition (\ref{sum delta < infty}): the $L_q$-mean H\"ormander condition ($1\leq q\leq\infty$), defined by
    \begin{align} \label{Hq < infty}
	\sum_{m\geq 1} H_q (m) < \infty,
    \end{align}
where $H_q(m)$ is defined as
    \begin{align*}	 \sup_{{\begin{subarray}{} y\in \R^d, \\ R>0 \end{subarray}}} & \Big(\frac{(2^m R)^{d(q-1)}}{|B_R|} \int_{|v|\leq R}\int_{2^mR\leq |x-y|\leq 2^{m+1}R} |K(x,y+v)-K(x,y)|^q {\rm d} x {\rm d} v\Big)^{\frac{1}{q}},     \end{align*}
for $m\geq 1$.
Here, $B_R$ denotes the ball centered at the origin with radius $R>0$, and $|B_R|$ denotes its volume.
It is not hard to see that $H_q(m) \lesssim_d \delta_q(m)$, meaning the $L_q$-mean H\"ormander condition (\ref{Hq < infty}) is indeed weaker than the $L_q$-integral regularity condition (\ref{sum delta < infty}).
In what follows, we focus on the case $q=2$, i.e., the $L_2$-mean H\"ormander condition:
    \begin{align}\label{L2 mean Hormander condition}
	\sum_{m\geq 1} H_2 (m) < \infty,
    \end{align}
where $H_2(m)$ is defined as in (\ref{Hq < infty}) with $q=2$.

For two smoothness conditions $A$ and $B$, we write $A\prec B$ to mean that $A$ is weaker than $B$.
Thus, we have the following conclusion: \newline
    \begin{center}
    \scalebox{0.94}{
    $\begin{array}{c}
        \begin{array}{c}
            \textbf{\large $L_2$-mean H\"ormander} \\
            \textbf{\large condition} \\
        \end{array}
        $\scalebox{1.3}{$\prec$}$
        \begin{array}{c}
            \textbf{\large $L_2$-integral regularity} \\
            \textbf{\large condition} \\
            \end{array}
        $\scalebox{1.3}{$\prec$}$
        \begin{array}{c}
            \textbf{\large Lipschitz regularity} \\
            \textbf{\large condition}\\
        \end{array}
    \end{array}$
    } \newline
    \end{center}
This naturally leads to a question: can the $L_2$-mean H\"ormander condition (\ref{L2 mean Hormander condition}) still guarantee the noncommutative weak type $(1,1)$ boundedness of the maximal Calder\'on-Zygmund operators?
This paper gives a positive answer to this question.

Before stating the main result, we introduce some notions.
Let $\mM$ be a von Neumann algebra equipped with a normal semifinite faithful ($n. s. f. $, for short) trace $\tau$.
Let $\mN = L_{\infty}(\R^d)\bar\otimes \mM$ be the tensor von Neumann algebra with the tensor trace  $\phi = \int\otimes \tau$.
Denote the noncommutative $L_p$ space associated with $(\mN, \phi)$ by $L_p(\mN)$ for $1\leq p\leq\infty$ (see Appendix \ref{Appendix: Noncommutative Lp spaces and maximal norms} for details).
The noncommutative Calder\'on-Zygmund operator $T$ with a kernel $K$ is defined as
    \begin{align}\label{def of Tf}
	T f(x) = \int_{\R^d} K(x,y) f(y) {\rm d} y, \quad x\notin \overrightarrow{\operatorname{supp}}\ f,
    \end{align}
where $f\in L_1(\mN)\bigcap L_{\infty}(\mN)$ is compactly supported and measurable.
Here, $\overrightarrow{\text{supp}}\ f$ denotes the support of $f$ as an operator-valued function in $\R^d$ rather than the support projection as an element in a von Neumann algebra.
For any $\epsilon>0$, the associated truncated singular integral $T_{\epsilon}f$ is defined as
    \begin{align}\label{def of T epsilon f}
	T_{\epsilon}f(x) = \int_{|x-y|>\epsilon} K(x,y) f(y) {\rm d} y.
    \end{align}

The main result
is stated as follows.

In 2009, Parcet \cite{Parcet2009Pseudo-localization} rigorously constructed a kind of noncommutative Calder\'on-Zygmund decomposition by using Cuculescu's maximal weak type $(1,1)$ estimates for noncommutative martingales \cite{Cuculescu1971Martingales}.
With this tool, Parcet established the weak type $(1,1)$ estimates for operator-valued Calder\'on-Zygmund singular integrals with the kernel satisfying the Lipschitz regularity condition (\ref{Lipschitz regularity condition})
(Cadilhac later gave a shorter proof in \cite{Cadilhac2018Weak}).
Notably, the H\"ormander condition
 \begin{align}\label{Hormander condition}
 \sup_{y\in\R^d,|v|>0} \int_{|x-y|\geq 2|v|} |K(x,y+v)-K(x,y)| {\rm d} x < \infty,
 \end{align}
 which is weaker than (\ref{Lipschitz regularity condition}), is already sufficient for weak type $(1,1)$ boundedness in classical Calder\'on-Zygmund theory.
It is therefore natural to ask whether the noncommutative weak type $(1,1)$ boundedness still holds under the H\"ormander condition (\ref{Hormander condition})?
This question remains open.

A step forward was made in 2022 by Cadilhac et al. \cite{Cadilhac2022Spectral}.
They developed a more effective noncommutative Calder\'on-Zygmund decomposition without the off-diagonal term of the good function, which allowed them to prove the noncommutative weak type $(1,1)$ estimates for Calder\'on-Zygmund operators if the kernel $K$ satisfies
 \begin{align} \label{L2-integral regularity condition}
 \sum_{m\geq 1} \sup_{{\begin{subarray}{c} y\in Q \\ Q\ \text{dyadic} \end{subarray}}} \Big( 2^{md}\ell(Q)^d \int_{2^m\ell(Q)\leq|x-c(Q)|\leq2^{m+1}\ell(Q)} |K(x,y)-K(x,c(Q))|^2 {\rm d} x \Big)^{\frac{1}{2}}
 <\infty,
 \end{align}
 where $\ell(Q)$ and $c(Q)$ stand for the length and the center of the dyadic cube $Q$, respectively.
 More generally, for $1\leq q\leq\infty$, we say that a kernel $K$ satisfies the $L_q$-integral regularity condition if
 \begin{align} \label{sum delta < infty}
 \sum_{m\geq 1} \delta_q(m) <\infty,
 \end{align}
where $\delta_q(m)$ is defined as
 \begin{align*}
 \sup_{{\begin{subarray}{c} y\in \R^d,\ R>0 \\ |v|<R \end{subarray}}} & \Big( (2^m R)^{d(q-1)} \int_{2^mR\leq |x-y|\leq 2^{m+1}R} |K(x,y+v)-K(x,y)|^q {\rm d} x\Big)^{\frac{1}{q}},
 \end{align*}
for $m\geq 1$.
Roughly speaking, the H\"ormander condition (\ref{Hormander condition}) corresponds to the $L_1$-integral regularity condition i.e. (\ref{sum delta < infty}) with $q=1$, while \eqref{L2-integral regularity condition} corresponds to the $L_2$-integral regularity condition, i.e., (\ref{sum delta < infty}) with $q=2$.
The fact $\delta_1(m)\lesssim_d \delta_2(m)$ shows that the $L_2$-integral regularity condition is stronger than the H\"ormander condition (\ref{Hormander condition}), but weaker than the Lipschitz regularity condition (\ref{Lipschitz regularity condition}).
By applying the noncommutative Calder\'on-Zygmund decomposition in \cite{Cadilhac2022Spectral}, Hong, the first author, and Xu \cite{hong2023maximal} established the weak type (1,1) boundedness for the noncommutative maximal truncated Calder\'on-Zygmund operators under the $L_2$-integral regularity condition.
We refer the reader to \cite{Cadilhac2022Spectral, Lai2024Noncommutative, Mei2009Pseudo-localization} for more results on weak (1,1) boundedness theory for noncommutative Calder\'on-Zygmund operators.

In this paper, we study a kernel smoothness condition that is slightly weaker than the $L_q$-integral regularity condition (\ref{sum delta < infty}): the $L_q$-mean H\"ormander condition ($1\leq q\leq\infty$), defined by
 \begin{align} \label{Hq < infty}
 \sum_{m\geq 1} H_q (m) < \infty,
 \end{align}
where $H_q(m)$ is defined as
 \begin{align*} \sup_{{\begin{subarray}{} y\in \R^d, \\ R>0 \end{subarray}}} & \Big(\frac{(2^m R)^{d(q-1)}}{|B_R|} \int_{|v|\leq R}\int_{2^mR\leq |x-y|\leq 2^{m+1}R} |K(x,y+v)-K(x,y)|^q {\rm d} x {\rm d} v\Big)^{\frac{1}{q}}, \end{align*}
for $m\geq 1$.
Here, $B_R$ denotes the ball centered at the origin with radius $R>0$, and $|B_R|$ denotes its volume.
It is not hard to see that $H_q(m) \lesssim_d \delta_q(m)$, meaning the $L_q$-mean H\"ormander condition (\ref{Hq < infty}) is indeed weaker than the $L_q$-integral regularity condition (\ref{sum delta < infty}).
In what follows, we focus on the case $q=2$, i.e., the $L_2$-mean H\"ormander condition:
 \begin{align}\label{L2 mean Hormander condition}
 \sum_{m\geq 1} H_2 (m) < \infty,
 \end{align}
where $H_2(m)$ is defined as in (\ref{Hq < infty}) with $q=2$.

Before stating the main result, we introduce some notions.
Let $\mM$ be a von Neumann algebra equipped with a normal semifinite faithful ($n. s. f. $, for short) trace $\tau$.
Let $\mN = L_{\infty}(\R^d)\bar\otimes \mM$ be the tensor von Neumann algebra with the tensor trace $\phi = \int\otimes \tau$.
Denote the noncommutative $L_p$ space associated with $(\mN, \phi)$ by $L_p(\mN)$ for $1\leq p\leq\infty$ (see Appendix \ref{Appendix: Noncommutative Lp spaces and maximal norms} for details).
The noncommutative Calder\'on-Zygmund operator $T$ with a kernel $K$ is defined as
 \begin{align}\label{def of Tf}
 T f(x) = \int_{\R^d} K(x,y) f(y) {\rm d} y, \quad x\notin \overrightarrow{\operatorname{supp}}\ f,
 \end{align}
where $f\in L_1(\mN)\bigcap L_{\infty}(\mN)$ is compactly supported and measurable.
Here, $\overrightarrow{\text{supp}}\ f$ denotes the support of $f$ as an operator-valued function in $\R^d$ rather than the support projection as an element in a von Neumann algebra.
For any $\epsilon>0$, the associated truncated singular integral $T_{\epsilon}f$ is defined as
 \begin{align}\label{def of T epsilon f}
 T_{\epsilon}f(x) = \int_{|x-y|>\epsilon} K(x,y) f(y) {\rm d} y.
 \end{align}

The main result
is stated as follows.

\textbf{Step 1.}
Let $T_{\epsilon}$ be a noncommutative Calder\'on-Zygmund truncated operator with a complex kernel $K$ (defined as in (\ref{def of T epsilon f})).
We decompose $K$ into its real and imaginary parts: $\Re(K)$ and $\Im(K)$.
The truncated operators associated with $\Re(K)$ and $\Im(K)$ are then given by
 \begin{align*}
 \Re(T_{\epsilon})f(x) = \int_{|x-y|>\epsilon} \Re(K)(x,y) f(y) \dif y
 \end{align*}
and
 \begin{align*}
 \Im(T_{\epsilon})f(x) = \int_{|x-y|>\epsilon} \Im(K)(x,y) f(y) \dif y.
 \end{align*}
\begin{lemma}\label{reduction to real kernel}
 Let $T$ be a noncommutative Calder\'on-Zygmund operator (defined as in (\ref{def of Tf})) whose kernel $K$ satisfies (\ref{Size condition}) and the $L_2$-mean H\"ormander condition (\ref{L2 mean Hormander condition}).
 Let $T_{\epsilon}$ be defined as in (\ref{def of T epsilon f}).
 Then,
 \begin{enumerate}[(i)]
 \item Both $\Re(K)$ and $\Im(K)$ satisfy (\ref{Size condition}) and (\ref{L2 mean Hormander condition}).
 \item If $(T_{\epsilon})_{\epsilon>0}$ is of strong type $(p_0,p_0)$ for some $p_0\in (1,\infty)$, then, both $(\Re(T_{\epsilon}))_{\epsilon>0}$ and $(\Im(T_{\epsilon}))_{\epsilon>0}$ are of strong type $(p_0,p_0)$.
 \item If $(\Re(T_{\epsilon}))_{\epsilon>0}$ and $(\Im(T_{\epsilon}))_{\epsilon>0}$ are of weak type $(1,1)$, so is $(T_{\epsilon})_{\epsilon>0}$.
 \end{enumerate}
\end{lemma}
By this lemma, we may assume that the kernel $K$ is real throughout this paper.

\begin{lemma}\label{reduction to lacunary sequence}
 Let $T$ be a noncommutative Calder\'on-Zygmund operator (defined as in (\ref{def of Tf})) whose kernel $K$ satisfies (\ref{Size condition}) and the $L_2$-mean H\"ormander condition (\ref{L2 mean Hormander condition}).
 Let $T_{\epsilon}^{i_{\epsilon}}$ be defined as in (\ref{T epsilon,i epsilon f(x)}).
 Then
 \begin{enumerate}[(i)]
 \item $(T_{\epsilon}^{i_{\epsilon}})_{\epsilon>0}$ is of strong type $(p,p)$ for $1<p\leq\infty$.
 \item $(T_{\epsilon}^{i_{\epsilon}})_{\epsilon>0}$ is of weak type $(1,1)$.
 \end{enumerate}
 \end{lemma}

The preceding two lemmas imply that Theorem \ref{Main theorem} follows directly from the weak type (1,1) boundedness of $S_i$ given in (\ref{Sif(x)}).
Thus, it remains to prove
\begin{theorem}\label{Second theorem}
 Let $T$ be a noncommutative Calder\'on-Zygmund operator defined as in (\ref{def of Tf}) with the real kernel $K$ satisfying (\ref{Size condition}) and the $L_2$-mean H\"ormander condition (\ref{L2 mean Hormander condition}).
 Let $S_i$ be defined as in (\ref{Sif(x)}).
 Then, the sequence of operators $(S_i)_{i\in\Z}$ is of weak type $(1,1)$, that is, for any $f\in L_1(\mN)$,
 \[
 \norm{(S_i f)_{i\in\Z}}_{\Lambda_{1,\infty}(\mN;\ell_{\infty})} \lesssim \norm{f}_{L_1(\mN)}.
 \]
 More precisely, for any $f\in L_1(\mN)$ and $\lambda>0$, there exists a projection $e\in\mN$ such that
 \[
 \sup_{i\in\Z}\norm{e S_if e}_{L_{\infty(\mN)}} \leq \lambda \quad\text{and}\quad \phi(e^{\bot})\lesssim \lambda^{-1} \norm{f}_{L_1(\mN)} .
 \]
\end{theorem}

\subsection{Estimate for the good function (\ref{estimate of good function}) }
For any $i\in\Z$, it follows from the definitions of $S_i$ and $T_{\epsilon}^{i_{\epsilon}}$ ((\ref{Sif(x)}) and (\ref{T epsilon,i epsilon f(x)})) that there exists $\epsilon_i > 0$ such that
 \[ S_i = S_{\epsilon_i}- T_{\epsilon_i}^i. \]
Under the assumptions of Theorem \ref{Main theorem} and Lemma \ref{reduction to lacunary sequence} $(i)$, both $(S_{\epsilon_i})_{i \in \mathbb{Z}}$ and $(T_{\epsilon_i}^i)_{i \in \mathbb{Z}}$ are of strong type $(p_0, p_0)$.
Therefore, from the above equality, $(S_i)_{i \in \mathbb{Z}}$ is also of strong type $(p_0, p_0)$.
In addition, since $g$ is positive (by Theorem \ref{nc CZ decomposition with convolution}) and the kernel $K$ is real-valued (as stated in Section \ref{Section: Two reductions}), the operator $S_i g$ is self-adjoint.
These two facts imply that for any $i\in\Z$, we can find a positive element $a\in\mN$ such that for any $i\in\Z$,
 \[
 -a \leq S_i g \leq a \quad\text{and}\quad \norm{a}_{L_{p_0}(\mN)}\lesssim \norm{g}_{L_{p_0}(\mN)}.
 \]
Taking $e_1 = \chi_{(0,\lambda]}(a)$. Then we have
 \[
 -\lambda \leq -e_1 a e_1 \leq e_1 S_i g e_1 \leq e_1 a e _1 \leq \lambda.
 \]
Finally, applying the Chebyshev inequality, the H\"older inequality, along with the boundedness of $g$ in Lemma \ref{properties of nc CZ decomposition with convolution} $(i)$, we obtain
 \begin{align*}
 \phi(e_1^{\bot})
 \leq \lambda^{-p_0} \norm{a}_{L_{p_0}(\mN)}^{p_0}
 \leq \lambda^{-p_0} \norm{g}_{L_{\infty}(\mN)}^{p_0-1} \norm{g}_{L_1(\mN)}
 \lesssim \lambda^{-1} \norm{f}_{L_1(\mN)},
 \end{align*}
which yields (\ref{estimate of good function}).

To estimate $F_1$.
We first observe that by the supports of $h_{r_{n-j}}$ and $\chi_Q$, the noncommutative $L_1$-norm in $F_1$ equals
 \begin{align*} \Big\|\int_Q \Big(\int_{|u-y|\leq r_{n-j}} (K_j(x,y)-K_j(x,c(Q))) h_{r_{n-j}}(y-u)\dif y \Big)p_Q f(u) q_Q\dif u \Big\|_{L_1(\mM)}.
 \end{align*}
By using Lemma \ref{a useful lemma for bad function} (in Appendix \ref{Appendix: Noncommutative Lp spaces and maximal norms}), the above estimate is bounded by a product
 \begin{align*}
 & \Big\| \Big(\int_{Q} \Big(\int_{B_{r_{n-j}}(u)} \abso{K_i(x,y)-K_i(x,c(Q))} |h_{r_{n-j}}(y-u)|\dif y \Big)^2 p_Q f(u)p_Q \dif u\Big)^{^{\frac{1}{2}}}\Big\|_{L_{1}(\mM)} \\
 & \cdot \Big\|\Big(\int_{Q} q_Q f(u) q_Q \dif u\Big)^{\frac{1}{2}} \Big\|_{L_{\infty}(\mM)}.
 \end{align*}
For the second term, an application of \eqref{q_Q f_Q q_Q < lambda q_Q} shows that
 \begin{align*} \Big\|\Big(\int_{Q} q_Q f(u) q_Q \dif u\Big)^{\frac{1}{2}} \Big\|_{L_{\infty}(\mM)} \lesssim (|Q|\lambda)^{\frac{1}{2}}.
 \end{align*}
Hence,
 \begin{align}
 F_1
 \lesssim &\sum_{n\geq 2} \sum_{j\leq n-1} \sum_{Q\in\mQ_{n-j}} (|Q|\lambda)^{\frac{1}{2}} \notag\\
 & \label{xxxyyy}\frac{1}{|B_{r_{n-j}}|}\int_{\R^d} \Big\|\int_{Q} \Big(\int_{B_{r_{n-j}}(u)} \abso{K_i(x,y)-K_i(x,c(Q))} \dif y \Big)^2 p_Q f(u)p_Q \dif u\Big\|_{L_{\frac{1}{2}}(\mM)}^{\frac{1}{2}} \dif x.
 \end{align}
The same argument as in (\ref{B(c(Q))}) allows us to majorize (\ref{xxxyyy}) by
 \begin{align*}
 \frac{1}{|B_{r_{n-j-1}}|}\int_{\R^d} \Big\|\int_{Q} \Big(\int_{B_{r_{n-j-1}}(c(Q))} \abso{K_j(x,y)-K_j(x,c(Q))} \dif y \Big)^2 p_Q f(u)p_Q \dif u\Big\|_{L_{\frac{1}{2}}(\mM)}^{\frac{1}{2}} \dif x.
 \end{align*}
By Lemma \ref{the second lemma for bad function} $(i)$ and the $L_2$-mean H\"ormander condition \eqref{L2 mean Hormander condition}, we obtain
 \begin{align*}
 F_1 &\lesssim \sum_{n\geq 2} \sum_{j\leq n-1} \sum_{Q\in\mQ_{n-j}}
 \lbracket{ \big(H_2^2(n-1) + H_2^2(n) + 2^{-2(n-1)}\big) \tau(p_Q) \phi(fp_Q \chi_{Q}) }^{\frac{1}{2}} (|Q| \lambda)^{\frac{1}{2}} \\
 &\lesssim \sum_{n\geq 2} \lbracket{2H_2(n-1)+2^{-n+1}} \sum_{j\leq n-1} \sum_{Q\in\mQ_{n-j}}\big(\tau(p_Q) \phi(fp_Q \chi_{Q}) |Q| \lambda \big)^{\frac{1}{2}} \\
 &\lesssim \sum_{m\geq 1} \sum_{Q\in\mQ_m} \big( \tau(p_Q) \phi(fp_Q \chi_{Q}) |Q| \lambda \big)^{\frac{1}{2}}.
 \end{align*}
Finally, applying the Cauchy-Schwarz inequality twice, we estimate $F_1$ as follows:
 \begin{align*}
 F_1 \lesssim \Big(\sum_{m=1}^{\infty} \lambda \phi(p_m)\Big)^{\frac{1}{2}}
 \Big(\sum_{m=1}^{\infty} \phi(f p_m)\Big)^{\frac{1}{2}}
 = \big(\lambda \phi(1-q)\big)^{\frac{1}{2}} \big(\phi(f(1-q))\big)^{\frac{1}{2}}
 &\lesssim \norm{f}_{L_1(\mN)}.
 \end{align*}

Our main strategy in this paper is to develop a refined noncommutative Calder\'on-Zygmund decomposition, adapted to kernels satisfying the $L_2$-mean H\"ormander condition (\ref{L2 mean Hormander condition}).
The previous methods are insufficient in this context.
On one hand, Parcet's decomposition \cite{Parcet2009Pseudo-localization} requires the stronger Lipschitz regularity condition (\ref{Lipschitz regularity condition}) and may not be directly applied to the noncommutative maximal operator.
On the other hand, while the decomposition by Cadilhac et al. \cite{Cadilhac2022Spectral} can deal with the $L_2$-integral regularity condition (\ref{sum delta < infty}) with $q=2$, it does not cover our target.
To address this gap, we introduce a slightly new noncommutative Calder\'on-Zygmund decomposition achieved by further splitting of the bad function into a convolution part and a remainder part.

This paper is organized as follows.
We begin by presenting our key tool: the refined noncommutative Calder\'on-Zygmund decomposition (Theorem \ref{nc CZ decomposition with convolution}) in Section \ref{Section: Noncommutative C-Z decomposition}.
Next, Section \ref{Section: Two reductions} starts the proof of the main result (Theorem \ref{Main theorem}), reducing it to proving the weak type (1,1) estimates for the lacunary sequences with real kernels (Theorem \ref{Second theorem}).
With this reduction established, Section \ref{Proof of the main theorem} is devoted to proving Theorem \ref{Second theorem}.
Finally, some related background concepts are included in Appendix \ref{Appendix: Noncommutative Lp spaces and maximal norms}.

Now, we conclude this section by listing the notation required for our later analysis.\\

\textbf{Notation}
\begin{itemize}
	\item $\mQ_n$: the set of all dyadic cubes of side length $2^{-n}$, $n\in\Z$.
	\item $Q_{x,n}$: the unique cube in $\mQ_n$ containing $x\in\R^d$.
	\item $\mQ$: the set of all standard dyadic cubes in $\R^d$, that is, $\mQ = \bigcup\limits_n\mQ_n$.
	\item $c(Q)$: the center of the cube $Q$.
	\item $lQ$: the concentric cube sharing the center of  $Q$ such that its length is $l$ times the length of $Q$.
	\item $B_r(x)$: the ball with $x\in\R^d$ as its center and $r>0$ as its radius, and the center $x$ will be omitted when it is the origin.
	\item $|E|$: the volume of set $E\subset\R^d$.
	\item $[k]$: the integer part of $k$.

\end{itemize}

To prove Theorem~\ref{Main theorem}, the paper’s strategy is to “develop a refined noncommutative Calder\'on-Zygmund decomposition, adapted to kernels satisfying the $L_2$-mean H\"ormander condition (\ref{L2 mean Hormander condition}).” Since “previous methods are insufficient,” the authors “introduce a slightly new noncommutative Calder\'on-Zygmund decomposition achieved by further splitting of the bad function into a convolution part and a remainder part.”

The proof of Theorem~\ref{Main theorem} then proceeds via the following stated structure: (1) present the refined decomposition (Theorem~\ref{nc CZ decomposition with convolution}); (2) in Section~\ref{Section: Two reductions}, “start the proof of the main result (Theorem~\ref{Main theorem}), reducing it to proving the weak type (1,1) estimates for the lacunary sequences with real kernels (Theorem~\ref{Second theorem});” and (3) “with this reduction established,” prove Theorem~\ref{Second theorem} in Section~\ref{Proof of the main theorem}.

To prove the main theorem, the paper’s strategy is to “develop a refined noncommutative Calder\'on-Zygmund decomposition, adapted to kernels satisfying the $L_2$-mean H\"ormander condition
\begin{align}\label{L2 mean Hormander condition}
	\sum_{m\geq 1} H_2 (m) < \infty,
    \end{align}
.” Since “previous methods are insufficient,” the authors “introduce a slightly new noncommutative Calder\'on-Zygmund decomposition achieved by further splitting of the bad function into a convolution part and a remainder part.”

In establishing the main theorem, the proof then proceeds via the following stated structure: (1) present the refined decomposition. We first prove the following theorem.

\begin{theorem} \label{nc CZ decomposition with convolution}
	Fix $f\in\mN_{c,+}$, $\lambda>0$, and $s\in\N$.
	Let $(q_n)_{n\in\Z}$ and $(p_n)_{n\in\Z}$ be the sequences of projections in Lemma \ref{cuculescu's construction}, where $q_n = \sum\limits_{Q\in\mQ_n} q_Q \chi_Q$, $p_n = \sum\limits_{Q\in\mQ_n} p_Q \chi_Q$.
	Suppose $h\in C_{c}^{\infty} (\R^d)$ satisfies
    \begin{align*}
    \supp h \subset B_1,\quad \int_{\R^d} h =1, \quad h\geq 0.
    \end{align*}
    Define $h_r(x)= \frac{1}{r^d} h(\frac{x}{r})$ and $r_n= 2^{-n-1}\sqrt{d}$.
	Then, $f$ admits the decomposition:
	\begin{align*}
	f = g+ b
	  = g + \widetilde{b}_{\text{d}} + b_{\text{d}}^0 + \widetilde{b}_{\text{off}} + b_{\text{off}}^0,
	\end{align*}
	where $g$ is positive and $b$ is self-adjoint in $\mN$.
	More precisely, these components are defined as follows:
	    \[g=qfq + \sum\limits_{n\in\Z}p_n f_n p_n,\]
        \begin{gather*}
        \widetilde{b}_{\text{d}} = \sum\limits_{n\in\Z} \widetilde{b}_{\text{d},n} = \sum\limits_{n\in\Z}\sum\limits_{Q\in\mQ_n}\widetilde{b}_{\text{d},n}^Q ,\quad
        b_{\text{d}}^0 = \sum\limits_{n\in\Z} b_{\text{d},n}^0=\sum\limits_{n\in\Z} \sum\limits_{Q\in\mQ_n}(b_{\text{d},n}^Q-\widetilde{b}_{\text{d},n}^Q),\\
        \widetilde{b}_{\text{off}} = \sum\limits_{n\in\Z} \widetilde{b}_{\text{off},n} = \sum\limits_{n\in\Z} \sum\limits_{Q\in\mQ_n} \widetilde{b}_{\text{off},n}^Q ,\quad
        b_{\text{off}}^0 = \sum\limits_{n\in\Z} b_{\text{off},n}^0 = \sum\limits_{n\in\Z}\sum\limits_{Q\in\mQ_n}(b_{\text{off},n}^Q-\widetilde{b}_{\text{off},n}^Q),
        \end{gather*}
     where for each $Q\in\mQ_n$,
        \begin{gather*}
     	b_{\text{d},n}^Q = p_Q(f-f_n)p_Q\chi_Q,
     	\quad b_{\text{off},n}^Q = p_Q(f-f_n)q_Q\chi_Q + q_Q(f-f_n)p_Q\chi_Q, \\
     	\widetilde{b}_{\text{d},n}^Q = b_{\text{d},n}^Q * h_{r_n},
     	\quad \widetilde{b}_{\text{off},n}^Q = b_{\text{off},n}^Q * h_{r_n}.
        \end{gather*}
    \end{theorem}

(2) next, “start the proof of the main result, reducing it to proving the weak type (1,1) estimates for the lacunary sequences with real kernels (Theorem~\ref{Second theorem});” and (3) “with this reduction established,” prove Theorem~\ref{Second theorem} later. \label{Proof of the main theorem}
Note that every operator in a von Neumann algebra can be decomposed into a linear combination of four positive elements.
By this fact and the density of $\mN_{c,+}$ i

\label{Main theorem}
	Suppose $T$ is a Calder\'on-Zygmund operator defined as in
\begin{align}\label{def of Tf}
	T f(x) = \int_{\R^d} K(x,y) f(y) \dif y, \quad x\notin \overrightarrow{\supp}\ f,
    \end{align}
associated with a kernel satisfying
\begin{align} \label{Size condition}
       |K(x,y)| \leq \frac{C_1}{|x-y|^d} ;
       \end{align}
and the $L_2$-mean H\"ormander condition
\begin{align}\label{L2 mean Hormander condition}
	\sum_{m\geq 1} H_2 (m) < \infty,
    \end{align}
	Let $T_{\epsilon}$ be defined as in
\begin{align}\label{def of T epsilon f}
	T_{\epsilon}f(x) = \int_{|x-y|>\epsilon} K(x,y) f(y) \dif y.
    \end{align}
	Suppose $(T_{\epsilon})_{\epsilon>0}$ is of strong type $(p_0,p_0)$ for some $1< p_0 <\infty$, that is, for any $f\in L_{p_0}(\mN)$,
	\[
	\norm{(T_{\epsilon}f)_{\epsilon>0}}_{L_{p_0}(\mN;\ell_{\infty})} \lesssim  \left\| f \right\| _{L_{p_0}(\mN)}.
	\]
	Then, $(T_{\epsilon})_{\epsilon>0}$ is of weak type $(1,1)$, that is, for any $f\in L_1(\mN)$,
	\[
	\norm{(T_{\epsilon}f)_{\epsilon>0}}_{\Lambda_{1,\infty}(\mN;\ell_{\infty})} \lesssim  \left\| f \right\| _{L_1(\mN)}.
	\]
We refer the reader to
\label{Appendix: Noncommutative Lp spaces and maximal norms}
This part consists of three subsections: noncommutative $L_p$ spaces, noncommutative maximal norms, and noncommutative square functions.
Al
for the definitions of the noncommutative maximal norm $ \left\| \cdot \right\| _{L_p(\mN; \ell_{\infty})}$ and the noncommutative weak maximal norm $ \left\| \cdot \right\| _{\Lambda_{p,\infty}(\mN;\ell_{\infty})}$.,

Question:
Let \(\mathcal M\) be a von Neumann algebra with a normal semifinite faithful trace \(\tau\), and set \(\mathcal N=L_\infty(\mathbb R^d)\bar\otimes \mathcal M\) with tensor trace \(\phi=\int\otimes \tau\). Suppose \(T\) is a Calder\'on-Zygmund operator acting on \(\mathcal N\)-valued functions by
\[
Tf(x)=\int_{\mathbb R^d} K(x,y)f(y)\,dy
\]
for \(x\) outside the support of \(f\), where the kernel satisfies the size estimate
\[
|K(x,y)|\le \frac{C_1}{|x-y|^d}.
\]
Assume also that \(K\) satisfies the \(L_2\)-mean H\"ormander condition
\[
\sum_{m\ge 1} H_2(m)<\infty,
\]
where
\[
H_2(m)=\sup_{y\in\mathbb R^d,\,R>0}\left(\frac{(2^mR)^d}{|B_R|}\int_{|v|\le R}\int_{2^mR\le |x-y|\le 2^{m+1}R}|K(x,y+v)-K(x,y)|^2\,dx\,dv\right)^{1/2},
\]
and \(B_R\) is the Euclidean ball of radius \(R\) centered at the origin. For \(\varepsilon>0\), define the truncated operators
\[
T_\varepsilon f(x)=\int_{|x-y|>\varepsilon} K(x,y)f(y)\,dy.
\]
Suppose that for some \(1<p_0<\infty\), the family \((T_\varepsilon)_{\varepsilon>0}\) is of strong type \((p_0,p_0)\), i.e.
\[
\|(T_\varepsilon f)_{\varepsilon>0}\|_{L_{p_0}(\mathcal N;\ell_\infty)}\lesssim \|f\|_{L_{p_0}(\mathcal N)}
\quad\text{for all }f\in L_{p_0}(\mathcal N).
\]
Which quantitative estimate is valid under these assumptions?

The family \((T_\varepsilon)_{\varepsilon>0}\) is of weak type \((1,1)\): for every \(f\in L_1(\mathcal N)\),
\[
\|(T_\varepsilon f)_{\varepsilon>0}\|_{\Lambda_{1,\infty}(\mathcal N;\ell_\infty)}\lesssim \|f\|_{L_1(\mathcal N)}.
\]

Choices:
B.
The family \((T_\varepsilon)_{\varepsilon>0}\) is of strong type \((1,1)\): for every \(f\in L_1(\mathcal N)\),
\[
\|(T_\varepsilon f)_{\varepsilon>0}\|_{L_1(\mathcal N;\ell_\infty)}\lesssim \|f\|_{L_1(\mathcal N)}.
\]
C.
For every \(f\in L_1(\mathcal N)\) and every fixed \(\varepsilon>0\), the truncated operator \(T_\varepsilon\) is of weak type \((1,1)\):
\[
\|T_\varepsilon f\|_{\Lambda_{1,\infty}(\mathcal N)}\lesssim \|f\|_{L_1(\mathcal N)}.
\]
D.
There exists \(p_0\in(1,\infty)\) such that whenever \((T_\varepsilon)_{\varepsilon>0}\) is of strong type \((p_0,p_0)\), one has for every \(f\in L_1(\mathcal N)\),
\[
\|(T_\varepsilon f)_{\varepsilon>0}\|_{\Lambda_{1,\infty}(\mathcal N;\ell_\infty)}\lesssim \Big(\sum_{m\ge 1} H_2(m)^2\Big)^{1/2}\,\|f\|_{L_1(\mathcal N)}.
\]
E.
The family \((T_\varepsilon)_{\varepsilon>0}\) is of weak type \((1,1)\) provided the kernel satisfies only the \(L_2\)-mean H\"ormander condition and the size estimate; equivalently, for every \(f\in L_1(\mathcal N)\),
\[
\|(T_\varepsilon f)_{\varepsilon>0}\|_{\Lambda_{1,\infty}(\mathcal N;\ell_\infty)}\lesssim \|f\|_{L_1(\mathcal N)},
\]
without assuming any strong type \((p_0,p_0)\) bound for \((T_\varepsilon)_{\varepsilon>0}\).

The stem does not explicitly state the endpoint weak-(1,1) conclusion. It presents hypotheses and asks for the valid consequence, without giving away the answer directly.

This is essentially a direct theorem-recall item: the hypotheses are laid out in full and the correct option is the theorem's endpoint conclusion almost verbatim.

There is some reasoning pressure because the options distinguish maximal-family weak type, fixed-ε weak type, nonuniform bounds, and an unjustified strong-(1,1) upgrade. Still, the item mainly tests recognition/recall of the stated theorem rather than genuine derivation.

The distractors are mathematically plausible and well-targeted: one is a weaker true statement, one adds an unnecessary stronger hypothesis, one weakens uniformity, and one overstates the endpoint to strong type. These reflect realistic failure modes.

A solid recall-oriented MCQ with strong distractors and no answer leakage, but it is largely a direct restatement of a theorem and only moderately tests generative reasoning.


[Structure of $f(R)$ gravity in spherical symmetry]
\label{main-theo}
Consider the field equations in $f(R)$ gravity \eqref{Eq1-14}, coupled to a
possibly massive, real-valued scalar field $\phi$. Suppose that the defining function $f= f(R)$ and the matter potential $U=U(\phi)$ satisfy the following conditions:
\bel{hypo-Einstein}
\aligned
& (1)
\, &  \phi \, U'(\phi) &\geq&& 0,
\\
& (2)
\, &  U(\phi) &\geq&& 0,
\\
& (3)
\, & f'(R) &>&&0,
\\
& (4)
\, &
f(R) &\leq&& R \, f'(R).
\endaligned
\ee
Then, the field equations of $f(R)$ gravity for spherically symmetric spacetimes can be reduced to an \emph{integro-differential system} consisting of two first-order coupled, nonlinear hyperbolic equations, stated in Proposition~\ref{main-propo-first-order}, below.
Indeed, this system is equivalent to the full Einstein equations in the class of $C^1$ solutions that are suitably regular at the center in the sense of Definition~\ref{definition}, below. Furthermore, the Hawking mass is future non-decreasing along radial directions and non-increasing along null directions. In the limit ${f(R) \to R}$ and
$U(\phi)\to 0$, one recovers Christodoulou's formulation of the Einstein-massless scalar field system.

\label{section---1}

\subsection{Purpose of this paper}

\paragraph{Theories of modified gravity}

In recent years, new observational data suggest that extensions of Einstein's field equations may be relevant for explaining the accelerated expansion of the Universe and certain galactic-scale instabilities. Many of the proposed physical theories of modified gravity exhibit undesirable features, including lack of strong hyperbolicity in commonly used gauges, or non-uniqueness of solutions, or changes in the character of the evolution equations that may alter the system's causal structure. The development of robust numerical simulations in nonlinear, physically relevant regimes is hampered by the lack of mathematically rigorous formulations.
Indeed, only a few studies address the formulation and well-posedness of these theories \cite{Avalos,Lehner2,Lehner,Cocke,Hilditch1,Felice,Figueras,LeFlochMa17a,Reall,Reall2,Pretorius,Salgado}.
More recently, alternative, well-posed theories have been proposed; cf.~\cite{Brady-Figueras,Figueras} (and the references therein).

Among the various extensions of general relativity, $f(R)$ gravity is widely regarded as a natural and physically viable alternative to Einstein's theory \cite{Felice,Ferreira}. The $f(R)$ field equations are considerably more involved than the Einstein equations: in addition to the second-order Ricci curvature terms, they contain \emph{fourth-order} metric derivatives (in particular, second-order derivatives of the scalar curvature). This partly explains why, despite their physical importance, rigorous mathematical results to date encompass only local well-posedness~\cite{Felice,LeFlochMa17a} and the global nonlinear stability in the near-Minkowski regime \cite{LeFlochMa23}.

\paragraph{Evolution in spherical symmetry.}

In this paper, we initiate the study of the global evolution of a (massive) scalar field in $f(R)$ gravity in spherical symmetry. Christodoulou \cite{Chr,Chr2,Chr3} developed, within general relativity, a framework to analyze a massless scalar field in spherical symmetry using Bondi-type coordinates. This framework was also employed to study black-hole formation via numerical methods; cf.~\cite{Goldwirth-Piran} and the references therein. These developments inspired further investigations of gravitational collapse~\cite{Brady-etal,Hilditch2,Zhang-Lu, Michel-Moss}, as well as additional mathematical results, cf.~\cite{Chae,Costa-Mena,Costa-Duarte-Mena} (classical solutions) and \cite{LeFloch-Mena} (generalized solutions). Building on Christodoulou's subsequent work using a double-null foliation~\cite{Chr-bv,Chr-naked}, many studies have also addressed the gravitational collapse toward black holes in spherical symmetry. Next, with the growing interest in modified gravity, subsequent works include \cite{Quo-Joshi, Zhang-etal-2016, Chow}, which mostly consider quadratic curvature corrections to the Einstein--Hilbert action. Despite these advances, rigorous mathematical results on the \emph{global geometry} of Cauchy developments in $f(R)$ gravity are still lacking.

\paragraph{Aim of this paper.}

Our aim here is to advance the mathematical analysis of $f(R)$ gravity by building on the work of Christodoulou  \cite{Chr,Chr2,Chr3}.
We are also relying on the formulation proposed in LeFloch and Ma \cite{LeFlochMa17a,LeFlochMa23} to deal with the equations of $f(R)$ gravity in the near-Minkowski regime. In the present paper, we focus on spacetimes containing a (possibly massive) scalar field evolving in spherical symmetry and, drawing on Christodoulou's  insights, we provide a framework suited to addressing the global causal structure of such spacetimes.
Our companion paper \cite{LeFloch-Mena} considered the evolution of a self-gravitating massive scalar field and provided a first step toward understanding the role of a massive scalar field on the global spacetime geometry. While in \cite{LeFloch-Mena} we considered generalized solutions, in the present paper we consider solutions of class $C^1$ that satisfy regularity conditions at the center and are asymptotically Euclidean. Following Christodoulou, we use a generalization of the Bondi--Sachs coordinates and formulate the characteristic initial value problem with data imposed on a future light cone.

The new system is significantly more involved than the Einstein system, and a remarkable mathematical structure is uncovered here by introducing  an augmented formulation in which the metric and its scalar curvature are regarded as independent unknowns. We thus obtain a new integro-differential system of two coupled, first-order, nonlinear hyperbolic equations whose unknowns are the spacetime scalar curvature and, in this case, the scalar field. We then prove several consistency and regularity properties for the new system and its solutions as well as monotonicity properties of the Hawking mass in the $f(R)$ setting. In the process we also recover the general relativity limit (when ${f(R) \to R}$), as well as the special case of a massless scalar field.

\subsection{A first-order formulation of $f(R)$ gravity}

\paragraph{Action of $f(R)$ modified gravity.}

Recall first that Einstein's theory is based on the Einstein--Hilbert action
\be
\Acal_{\text{EH}}[\phi, g] := \int_M \Big( {R_g  \over 16 \pi} + L[\phi,g] \Big) \, dV_g,
\ee
associated with a $(1+3)$--dimensional spacetime $({\Mcal},g)$ with signature $(-, +,+,+)$.
Hence, the functional $\Acal_{\text{EH}}[\phi, g] $ is determined from the scalar curvature $R_g$ of the metric $g$ representing the geometry of the spacetime and
 a Lagrangian $L[\phi,g]$ describing the matter content and represented by certain fields $\phi$ defined on $\Mcal$. Here, $dV_g$ denotes the canonical volume form associated with $g$.
As is well-known, the action $\Acal_{\text{EH}}[\phi, g]$ is (formally) critical at metrics $g$
satisfying Einstein's field equations
\be
\label{Eq1-01}
G := \Ric - {R_g \over 2} \, g = 8 \pi \, T[\phi,g]
\ee
in which the right-hand side
\bel{Eq:12}
T_{ab}[\phi,g] := -2 \, {\delta L \over \delta g^{ab}} [\phi,g]  + g_{ab}\, L[\phi,g]
\ee
is the stress-energy tensor of the matter system (Latin indices $a,b,..= 0, \ldots, 3$ denoting spacetime components).
Our emphasis in this paper is on massive scalar fields $\phi$ described by
\be
\label{stress-energy}
T_{ab} = \nabla_a \phi \nabla_b \phi- \big( {1 \over 2} \nabla^c \phi \nabla_c \phi + U(\phi) \Big) g_{ab},
\ee
in which the potential function is $U=U(\phi)$, so that the field $\phi$ satisfies the Klein-Gordon equation
\be
\Box_g \phi = U'(\phi).
\end{equation}
We study an extension of Einstein's theory, defined as follows. A function $f: \RR \to \RR$ being given, we consider the modified gravity action for the $f(R)$ theory
$$
\Acal_{\text{MG}}[\phi,g] =: \int_M \Big( {f(R) \over 16 \pi} + L[\phi, g]\Big) \, dV,
$$
whose critical points satisfy the following \emph{field equations of modified gravity}
\bel{Eq1-14}
E_{ab} :=
f'(R) \, G_{ab} - \frac{1}{2} \Big( f(R) - R f'(R) \Big) g_{ab}
+  \big( g_{ab} \, \Box_g   - \nabla_a\nabla_b \big)  \big( f'(R) \big) = 8 \pi \, T_{ab}[\phi,g].
\ee
The right-hand side is still given by \eqref{Eq:12}. The modified gravity tensor $E_{ab}$ replaces
the Einstein tensor $G_{ab}$ and satisfies $\nabla^a E_{ab} = 0$,
 so that $T_{ab}$ is also divergence-free.

\paragraph{Main result.}

In local coordinates, \eqref{Eq1-14} consists of a nonlinear system of fourth-order partial differential equations, while Einstein theory in \eqref{Eq1-01} leads to second-order equations. We investigate
how to include the effect of these fourth-order terms in techniques developed earlier for the Einstein equations.
The condition $f'(R) >0$ in \eqref{hypo-Einstein}, below, is fundamental throughout the $f(R)$ theory and, for instance, we use it to be able to introduce the \emph{conformal metric} $\gt$ and the \emph{augmented variable} $\rho$ (in \eqref{equa-233}, below):
\be
\gt_{ab} := e^{\kappa \rho} g_{ab},
\qquad
\rho := {1 \over \kappa} \log f'(R).
\ee
Here, $\kappa>0$ is a parameter  for the definition of $\rho$, so that the general relativity limit corresponds to $f(R)\to R$ when ${\kappa \to 0}$. We summarize our results as follows.

\vskip.15cm

Our aim here is to advance the mathematical analysis of $f(R)$ gravity by building on the work of Christodoulou \cite{Chr,Chr2,Chr3}.
We are also relying on the formulation proposed in LeFloch and Ma \cite{LeFlochMa17a,LeFlochMa23} to deal with the equations of $f(R)$ gravity in the near-Minkowski regime. In the present paper, we focus on spacetimes containing a (possibly massive) scalar field evolving in spherical symmetry and, drawing on Christodoulou's insights, we provide a framework suited to addressing the global causal structure of such spacetimes.
Our companion paper \cite{LeFloch-Mena} considered the evolution of a self-gravitating massive scalar field and provided a first step toward understanding the role of a massive scalar field on the global spacetime geometry. While in \cite{LeFloch-Mena} we considered generalized solutions, in the present paper we consider solutions of class $C^1$ that satisfy regularity conditions at the center and are asymptotically Euclidean. Following Christodoulou, we use a generalization of the Bondi--Sachs coordinates and formulate the characteristic initial value problem with data imposed on a future light cone.

The new system is significantly more involved than the Einstein system, and a remarkable mathematical structure is uncovered here by introducing an augmented formulation in which the metric and its scalar curvature are regarded as independent unknowns. We thus obtain a new integro-differential system of two coupled, first-order, nonlinear hyperbolic equations whose unknowns are the spacetime scalar curvature and, in this case, the scalar field. We then prove several consistency and regularity properties for the new system and its solutions as well as monotonicity properties of the Hawking mass in the $f(R)$ setting. In the process we also recover the general relativity limit (when ${f(R) \to R}$), as well as the special case of a massless scalar field.

Recall first that Einstein's theory is based on the Einstein--Hilbert action
\be
\Acal_{\text{EH}}[\phi, g] := \int_M \Big( {R_g \over 16 \pi} + L[\phi,g] \Big) \, dV_g,
\ee
associated with a $(1+3)$--dimensional spacetime $({\Mcal},g)$ with signature $(-, +,+,+)$.
Hence, the functional $\Acal_{\text{EH}}[\phi, g] $ is determined from the scalar curvature $R_g$ of the metric $g$ representing the geometry of the spacetime and
 a Lagrangian $L[\phi,g]$ describing the matter content and represented by certain fields $\phi$ defined on $\Mcal$. Here, $dV_g$ denotes the canonical volume form associated with $g$.
As is well-known, the action $\Acal_{\text{EH}}[\phi, g]$ is (formally) critical at metrics $g$
satisfying Einstein's field equations
\be
\label{Eq1-01}
G := \Ric - {R_g \over 2} \, g = 8 \pi \, T[\phi,g]
\ee
in which the right-hand side
\bel{Eq:12}
T_{ab}[\phi,g] := -2 \, {\delta L \over \delta g^{ab}} [\phi,g] + g_{ab}\, L[\phi,g]
\ee
is the stress-energy tensor of the matter system (Latin indices $a,b,..= 0, \ldots, 3$ denoting spacetime components).
Our emphasis in this paper is on massive scalar fields $\phi$ described by
\be
\label{stress-energy}
T_{ab} = \nabla_a \phi \nabla_b \phi- \big( {1 \over 2} \nabla^c \phi \nabla_c \phi + U(\phi) \Big) g_{ab},
\ee
in which the potential function is $U=U(\phi)$, so that the field $\phi$ satisfies the Klein-Gordon equation
\be
\Box_g \phi = U'(\phi).
\end{equation}
We study an extension of Einstein's theory, defined as follows. A function $f: \RR \to \RR$ being given, we consider the modified gravity action for the $f(R)$ theory
$$
\Acal_{\text{MG}}[\phi,g] =: \int_M \Big( {f(R) \over 16 \pi} + L[\phi, g]\Big) \, dV,
$$
whose critical points satisfy the following \emph{field equations of modified gravity}
\bel{Eq1-14}
E_{ab} :=
f'(R) \, G_{ab} - \frac{1}{2} \Big( f(R) - R f'(R) \Big) g_{ab}
+ \big( g_{ab} \, \Box_g - \nabla_a\nabla_b \big) \big( f'(R) \big) = 8 \pi \, T_{ab}[\phi,g].
\ee
The right-hand side is still given by \eqref{Eq:12}. The modified gravity tensor $E_{ab}$ replaces
the Einstein tensor $G_{ab}$ and satisfies $\nabla^a E_{ab} = 0$,
 so that $T_{ab}$ is also divergence-free.

In local coordinates, \eqref{Eq1-14} consists of a nonlinear system of fourth-order partial differential equations, while Einstein theory in \eqref{Eq1-01} leads to second-order equations. We investigate
how to include the effect of these fourth-order terms in techniques developed earlier for the Einstein equations.
The condition $f'(R) >0$ in \eqref{hypo-Einstein}, below, is fundamental throughout the $f(R)$ theory and, for instance, we use it to be able to introduce the \emph{conformal metric} $\gt$ and the \emph{augmented variable} $\rho$ (in \eqref{equa-233}, below):
\be
\gt_{ab} := e^{\kappa \rho} g_{ab},
\qquad
\rho := {1 \over \kappa} \log f'(R).
\ee
Here, $\kappa>0$ is a parameter for the definition of $\rho$, so that the general relativity limit corresponds to $f(R)\to R$ when ${\kappa \to 0}$. We summarize our results as follows.

\vskip.15cm

Our aim here is to advance the mathematical analysis of $f(R)$ gravity by building on the work of Christodoulou \cite{Chr,Chr2,Chr3}.
We are also relying on the formulation proposed in LeFloch and Ma \cite{LeFlochMa17a,LeFlochMa23} to deal with the equations of $f(R)$ gravity in the near-Minkowski regime. In the present paper, we focus on spacetimes containing a (possibly massive) scalar field evolving in spherical symmetry and, drawing on Christodoulou's insights, we provide a framework suited to addressing the global causal structure of such spacetimes.
Our companion paper \cite{LeFloch-Mena} considered the evolution of a self-gravitating massive scalar field and provided a first step toward understanding the role of a massive scalar field on the global spacetime geometry. While in \cite{LeFloch-Mena} we considered generalized solutions, in the present paper we consider solutions of class $C^1$ that satisfy regularity conditions at the center and are asymptotically Euclidean. Following Christodoulou, we use a generalization of the Bondi--Sachs coordinates and formulate the characteristic initial value problem with data imposed on a future light cone.

\vskip.15cm

\vskip.15cm

In the modified field equations, there are terms that contain derivatives of the scalar
curvature, that is, third order derivatives of the metric functions $\nu$ and $\lambda$. This leads to an essential difficulty in dealing with the equations.
Following LeFloch and Ma~\cite{LeFlochMa17a}, we introduce an \emph{augmented system}
 where the relation
\be
\rho = \frac{1}{\kappa}\ln f'(R) \quad \text{(conformal formulation)}
\ee
is no longer imposed but $\rho$
 is regarded as an independent variable.
To clarify the notation, we introduce a new independent variable denoted by
\be
\rhoh \quad \text{(augmented conformal formulation)}
\ee
which plays the role of $\rho$ and will coincides with $\rho$ once the constraint below is enforced.
In view of Proposition~\ref{prop essential f(R) Bondi}, this leads us to the following system
\bel{Bondi system diff augmented1.5}
\aligned
\del_r(\nu+\lambda)
& = \frac{3}{4}\kappa^{2}r|\del_r\rhoh|^2 + 4\pi r e^{- \kappa\rhoh} \, |\del_r\phi|^2,
\\
\del_r\big(re^{\nu- \lambda}\big)
 & = \big(1 - r^{2}e^{-2\kappa\rhoh} \big( \Vstar(\rhoh) + 8\pi U(\phi) \big) \big)e^{\nu+\lambda},
 \\
D(\del_r(r\phi)) + {r \over 2} e^{\nu+\lambda- \kappa\rhoh}U'(\phi)
& = \frac{1}{2}e^{\nu+\lambda}\Big(1 - e^{-2\lambda} - r^2e^{-2\kappa\rhoh}\big(\Vstar(\rhoh)+8\pi U(\phi)\big)\Big)\del_r\phi \\
& \quad + \frac{\kappa r}{2}\big(D\rhoh \del_r\phi + D\phi \del_r\rhoh\big),
\\
D\big(\del_r(r\rhoh)\big) + {r \over \kappa} e^{\nu+\lambda} \Wstar(\rhoh)
& = \frac{1}{2}e^{\nu+\lambda}\Big(1 - e^{-2\lambda} - r^2e^{-2\kappa\rhoh}\big(\Vstar(\rhoh)+8\pi U(\phi)\big)\Big)\del_r\rhoh
\\
& - \frac{8\pi r}{3\kappa} e^{- \kappa\rhoh}\del_r\phi D\phi
 + {r \over \kappa} e^{\nu+\lambda} \frac{16\pi}{3}e^{-2\kappa\rhoh}U(\phi).
\endaligned
\ee
Observe that, as before, the last equation of \eqref{Bondi system diff augmented1.5} is equivalent to
\be
\label{box-tilde-rho}
\Box_{\gt}\rhoh - \frac{2}{\kappa} \Wstar(\rhoh)
= - {8\pi \over 3\kappa}e^{-2\kappa\rhoh}\big(\sigma+4 \, U(\phi)\big).
\ee
Now that we have defined the augmented system, the next question is whether a solution of \eqref{Bondi system diff augmented1.5} (with a certain regularity) is also a solution of the original system. We emphasize that to be a solution of the full set of field equations, a solution of \eqref{equa-three-equa}, \eqref{Bondi scalar}, and \eqref{eq:599}
must additionally satisfy the condition
\bel{Bondi system diff augmented1.5 constraint}
f'(R) = e^{\kappa\rhoh},
\ee
which we regard as a nonlinear differential constraint on the solutions.
 Conversely, any classical solution of the original $f(R)$ system with $\rho=\frac{1}{\kappa}\ln f'(R)$ satisfies \eqref{Bondi system diff augmented1.5} upon setting $\rhoh=\rho$; thus the two formulations are equivalent in the admissible class once \eqref{Bondi system diff augmented1.5 constraint} holds on the initial cone (and is then propagated by the evolution)

Under the regularity conditions at the center and the asymptotic flatness condition, we can reduce the field equations (cf.~Appendix~\ref{section---A2})) to the the following integro-differential system with the main unknowns $\phi$ and $\rho$:
\bel{notresysteme}
\aligned
D(\del_r(r\phi))
& = \frac{1}{2}e^{\nu+\lambda}\Big(1 - e^{-2\lambda} - r^2e^{-2\kappa\rho}\big(\Vstar(\rho)+8\pi U(\phi)\big)\Big)\del_r\phi
\\
 & \quad + \frac{\kappa r}{2}\big(D\rho \del_r\phi + D\phi \del_r\rho\big)-{r \over 2} e^{\nu+\lambda- \kappa\rho}U'(\phi),
\\
D\big(\del_r(r\rho)\big)
& = \frac{1}{2}e^{\nu+\lambda}\Big(1 - e^{-2\lambda} - r^2e^{-2\kappa\rho}\big(\Vstar(\rho)+8\pi U(\phi)\big)\Big)\del_r\rho
\\
& \quad - \frac{8\pi r}{3\kappa} e^{- \kappa\rho}\del_r\phi D\phi- {r \over \kappa} e^{\nu+\lambda}\Big(\Wstar(\rho) - \frac{16\pi}{3}e^{-2\kappa\rho}U(\phi)\Big),
\endaligned
\ee
in which the metric coefficients $\nu, \lambda$ are given by
\bel{eq:403}
\aligned
\nu + \lambda
& = - \frac{3}{4}\kappa^{2} \int_r^{+ \infty}s|\del_r\rho|^2ds - 4\pi\int_r^{+ \infty} e^{- \kappa\rho} \, s|\del_r\phi|^2 \, ds,
\\
e^{\nu- \lambda}
& = {1 \over r} \int_0^r\Big(1 - s^{2}e^{-2\kappa\rho} \big( \Vstar(\rho) + 8\pi U(\phi) \big)\Big)e^{\nu+\lambda} \, ds.
\endaligned
\ee
We refer to \eqref{notresysteme} as the \emph{augmented conformal system} of modified gravity.

Suppose now that $h$ and $l$ are solutions to the system \eqref{Bondi system int augmented3} defined on $[0,u_0]\times [0,+ \infty)$. If
\bel{center.condition.2}
h,l\in C^1\big([0,u_0]\times [0,+ \infty)\big),
\ee
then the first and second regularity conditions at the center hold. In fact, it follows from the system \eqref{Bondi system int augmented3} that
\be
\label{more-conditions-center}
\hb,\,\lb,\,\nu,\,\lambda\in C^1\big([0,u_0]\times [0,+ \infty)\big).
\ee
In view of \eqref{eq fR components} and \eqref{eq Bondi components T}, we get
\be
\aligned
& r^2{e^{-\kappa\lb}}\big(E-8\pi T\big)(D,D)
\\
& = r{e^{\nu- \lambda}}\Big(\frac{e^{\nu- \lambda}}{2}\del_r(\nu+\lambda) - 2\del_u\lambda\Big)
- \frac{3r^2\kappa^{2}{ } }{2}(D\lb)^2-8\pi r^2{ }(D\hb)^2.
\endaligned
\ee
Hence, from the regularity conditions, and taking into account \eqref{more-conditions-center}, we deduce that
\be
\lim_{r\to0}{\Big(r^2e^{-\kappa\rho}\big(E-8\pi T\big)(D,D)\Big)}= 0,
\ee
since $\nu,\lambda=O(r)$ and $D\hb,D\lb$ are bounded near $r=0$. For the second regularity condition at the center, we compute
\be
r\del_r\lb = l- \lb, \qquad r\del_r\hb = h- \hb.
\ee
 This shows that $h=\partial_r(r\phi)$ and $l=\partial_r(r\rho)$ are $C^1$ on $[0,u_0]\times[0,+\infty)$, as claimed. Conversely, if the two center regularity conditions hold and $(h,l)\in C^0$, the right-hand sides of \eqref{Bondi system int augmented3} are continuous; integrating along characteristics then yields $(h,l)\in C^1$, so the two formulations are equivalent in the admissible class.

\vskip.15cm

We point out that our assumptions are quite natural since they ensure positivity and monotonicity properties that also arise in the massless case. The condition $\phi \, U'(\phi) \geq 0$ is imposed since it guarantees that the Klein-Gordon energy is \emph{defocusing}, so that the forward evolution will not be limited by the matter model.
The conditions ${U(\phi) \geq 0}$ and ${f(R) \leq R \, f'(R)}$ are required to prove that the Hawking-mass is non-negative.

\subsection{Outline of this paper.}

In Section~\ref{section---2}, we introduce Bondi coordinates and express the field equations of $f(R)$ gravity for a scalar field. We then identify the essential equations, which imply the full set of $f(R)$ field equations, provided a mild regularity condition is assumed at the center. In Section~\ref{section---A2}, we analyze the regularity at the center. In Section~\ref{section---3}, we introduce a first-order formulation and establish that it is equivalent to the full set of $f(R)$ equations. Finally, Section~\ref{section--- 4} is devoted to the study of the Hawking mass and its monotonicity properties.

In the discussion after Theorem~\ref{main-theo}, the paper indicates the following argument structure. First, introduce Bondi coordinates and write the spherically symmetric $f(R)$-scalar field equations; then “identify the essential equations, which imply the full set of $f(R)$ field equations, provided a mild regularity condition is assumed at the center.” Next, “analyze the regularity at the center.” Then “introduce a first-order formulation and establish that it is equivalent to the full set of $f(R)$ equations,” yielding the claimed integro-differential system. Finally, “study of the Hawking mass and its monotonicity properties,” using the assumptions (notably $U(\phi)\ge 0$ and $f(R)\le R f'(R)$) which are said to be “required to prove that the Hawking-mass is non-negative,” and $\phi U'(\phi)\ge 0$ to ensure the Klein--Gordon energy is “defocusing.”

In the discussion after the main theorem, the paper indicates the following argument structure. First, introduce Bondi coordinates and write the spherically symmetric $f(R)$-scalar field equations; then “identify the essential equations, which imply the full set of $f(R)$ field equations, provided a mild regularity condition is assumed at the center.” Next, “analyze the regularity at the center.” Then “introduce a first-order formulation and establish that it is equivalent to the full set of $f(R)$ equations,” yielding the claimed integro-differential system. Finally, “study of the Hawking mass and its monotonicity properties,” using the assumptions (notably $U(\phi)\ge 0$ and $f(R)\le R f'(R)$) which are said to be “required to prove that the Hawking-mass is non-negative,” and $\phi U'(\phi)\ge 0$ to ensure the Klein--Gordon energy is “defocusing.”

[Structure of $f(R)$ gravity in spherical symmetry]
\label{main-theo}
Consider the field equations in $f(R)$ gravity \eqref{Eq1-14}, coupled to a
possibly massive, real-valued scalar field $\phi$. Suppose that the defining function $f= f(R)$ and the matter potential $U=U(\phi)$ satisfy the following conditions:
\bel{hypo-Einstein}
\aligned
& (1)
\, &  \phi \, U'(\phi) &\geq&& 0,
\\
& (2)
\, &  U(\phi) &\geq&& 0,
\\
& (3)
\, & f'(R) &>&&0,
\\
& (4)
\, &
f(R) &\leq&& R \, f'(R).
\endaligned
\ee
Then, the field equations of $f(R)$ gravity for spherically symmetric spacetimes can be reduced to an \emph{integro-differential system} consisting of two first-order coupled, nonlinear hyperbolic equations. We first state the following proposition.

\begin{proposition}[A first-order formulation of $f(R)$ gravity]
\label{main-propo-first-order}
\bse\label{Bondi system int augmented3}
The equations of $f(R)$ gravity take the form of a first-order system, in which $h$ and $l$ are the main unknowns:
\be
\aligned
& Dh = \frac{1}{2r}\big(g- \gb\big)(h- \hb) +
\frac{\kappa}{2}\big(q(h- \hb) + p(l- \lb)\big)-{r \over 2} e^{\nu+\lambda- \kappa\lb}U'(\hb),
\\
& Dl = \frac{1}{2r}\big(g- \gb\big)(l- \lb)
- \frac{8\pi}{3\kappa}e^{- \kappa\lb}(h- \hb) p-{r \over \kappa} e^{\nu+\lambda}\Big(\Wstar(\lb)- \frac{16\pi}{3}e^{-2\kappa\lb}U(\hb)\Big),
\endaligned
\ee
in which the metric coefficient $\nu + \lambda$ is computed by
\be
\aligned
& \nu + \lambda = - \frac{3\kappa^{2}}{4}\int_r^{+ \infty}\frac{|l- \lb|^2}{s} \, ds - 4\pi\int_r^{+ \infty}\frac{|h- \hb|^2}{se^{\kappa\lb}} \, ds,
\endaligned
\ee
and the auxiliary variables $p$ and $q$ are defined explicitly by
\bel{p,q}
\aligned
&p = \frac{\kappa}{2r}\int_0^r\Big((l- \lb)p + (h- \hb)q\Big) \, ds +\frac{1}{2r}\int_0^r\frac{e^{\nu- \lambda}(h- \hb)}{s} \, ds
- \frac{1}{2r}\int_0^rse^{\nu+\lambda- \kappa\lb}U'(\hb) \, ds,
\\
& \kappa q
= - \frac{8\pi}{3 r}\int_0^r\frac{(h- \hb)p}{e^{\kappa\lb}} \, ds+ \frac{\kappa}{2r}\int_0^r{\frac{e^{\nu- \lambda}(l- \lb)}{s} \, ds}
 - \frac{1}{r}\int_0^r se^{\nu+\lambda}\Big (\Wstar(\lb)-
\frac{16\pi}{3} e^{-2\kappa\lb}U(\hb)\Big) \, ds.
\endaligned
\ee
\ese
\end{proposition}

Indeed, this system is equivalent to the full Einstein equations in the class of $C^1$ solutions that are suitably regular at the center in the following sense.

\begin{definition}
\label{definition}
\emph{The \emph{first regularity condition at the center\footnote{In view of \eqref{eq fR components} and \eqref{eq Bondi components T}, this condition can be expressed in terms of first-order derivatives of the unknowns.}}} is defined as
\be
\label{regularity}
\lim_{r\to0} r^2e^{-\kappa\rho}\big(E - 8\pi T\big)(D,D) = 0.
\ee
The \emph{second regularity condition at the center} is defined as
\be
\label{regularity2}
\del_r(r\phi) \text{ and }\del_r(r\rho) \text{ are } C^1\text{ on } [0,u_0]\times [0,+ \infty).
\ee
\end{definition}

Furthermore, the Hawking mass is future non-decreasing along radial directions and non-increasing along null directions. In the limit ${f(R) \to R}$ and
$U(\phi)\to 0$, one recovers Christodoulou's formulation of the Einstein-massless scalar field system.,

Implication
Biconditional or Equivalence

Question: Consider the modified-gravity field equations E_{ab}=8π T_{ab} on a spherically symmetric spacetime, where E_{ab}:=f'(R)G_{ab}−(1/2)(f(R)−R f'(R))g_{ab}+(g_{ab}□_g−∇_a∇_b)(f'(R)) and T_{ab}=∇_aφ∇_bφ−( (1/2)∇^cφ∇_cφ+U(φ) )g_{ab} for a real-valued scalar field φ. Assume that φ U'(φ)≥0, U(φ)≥0, f'(R)>0, and f(R)≤R f'(R). Introduce ρ:=(1/κ)log f'(R) with κ>0, and in Bondi-type coordinates let h:=∂_r(rφ) and l:=∂_r(rρ). Which conclusion about the resulting spherically symmetric f(R) system holds under these assumptions?

For spherically symmetric spacetimes, the f(R) field equations admit an explicit reduction to an integro-differential system consisting of two coupled first-order nonlinear hyperbolic equations whose main unknowns are h and l (equivalently, φ and ρ=(1/κ)log f'(R)), with the remaining metric coefficients and auxiliary quantities determined by explicit radial integral formulas. In the class of C^1 solutions satisfying the two center regularity conditions lim_{r→0} r^2 e^{−κρ}(E−8πT)(D,D)=0 and ∂_r(rφ), ∂_r(rρ) ∈ C^1([0,u_0]×[0,∞))—where D is the characteristic Bondi derivative—this first-order integro-differential system is equivalent to the original spherically symmetric f(R) field equations. Moreover, the Hawking mass of the symmetry spheres is future non-decreasing along radial directions and non-increasing along null directions, and in the limit f(R)→R together with U(φ)→0 one recovers Christodoulou’s formulation of the Einstein–massless-scalar-field system.

Choices:
B. For spherically symmetric spacetimes, the f(R) field equations admit an explicit reduction to an integro-differential system consisting of two coupled first-order nonlinear hyperbolic equations whose main unknowns are h and l (equivalently, φ and ρ=(1/κ)log f'(R)), with the remaining metric coefficients and auxiliary quantities determined by explicit radial integral formulas. This first-order integro-differential system is equivalent to the original spherically symmetric f(R) field equations for every C^1 solution, without any additional regularity assumptions at the center. Moreover, the Hawking mass of the symmetry spheres is future non-decreasing along radial directions and non-increasing along null directions, and in the limit f(R)→R together with U(φ)→0 one recovers Christodoulou’s formulation of the Einstein–massless-scalar-field system.
C. For spherically symmetric spacetimes, the f(R) field equations admit an explicit reduction to an integro-differential system consisting of two coupled first-order nonlinear hyperbolic equations whose main unknowns are h and l (equivalently, φ and ρ=(1/κ)log f'(R)), with the remaining metric coefficients and auxiliary quantities determined by explicit radial integral formulas. Moreover, in the limit f(R)→R together with U(φ)→0 one recovers Christodoulou’s formulation of the Einstein–massless-scalar-field system.
D. For spherically symmetric spacetimes, the f(R) field equations admit an explicit reduction to an integro-differential system consisting of two coupled first-order nonlinear hyperbolic equations whose main unknowns are h and l (equivalently, φ and ρ=(1/κ)log f'(R)), with the remaining metric coefficients and auxiliary quantities determined by explicit radial integral formulas. In the class of C^1 solutions satisfying the two center regularity conditions lim_{r→0} r^2 e^{−κρ}(E−8πT)(D,D)=0 and ∂_r(rφ), ∂_r(rρ) ∈ C^1([0,u_0]×[0,∞))—where D is the characteristic Bondi derivative—this first-order integro-differential system is equivalent to the original spherically symmetric f(R) field equations. Moreover, the Hawking mass of the symmetry spheres is future non-decreasing both along radial directions and along null directions, and in the limit f(R)→R together with U(φ)→0 one recovers Christodoulou’s formulation of the Einstein–massless-scalar-field system.
E. For spherically symmetric spacetimes, the f(R) field equations admit an explicit reduction to an integro-differential system consisting of two coupled first-order nonlinear hyperbolic equations whose main unknowns are h and l (equivalently, φ and ρ=(1/κ)log f'(R)), with the remaining metric coefficients and auxiliary quantities determined by explicit radial integral formulas. In the class of C^1 solutions satisfying the two center regularity conditions lim_{r→0} r^2 e^{−κρ}(E−8πT)(D,D)=0 and ∂_r(rφ), ∂_r(rρ) ∈ C^1([0,u_0]×[0,∞))—where D is the characteristic Bondi derivative—this first-order integro-differential system is equivalent to the original spherically symmetric f(R) field equations. Moreover, the Hawking mass of the symmetry spheres is non-negative and therefore constant along radial directions and null directions, and in the limit f(R)→R together with U(φ)→0 one recovers Christodoulou’s formulation of the Einstein–massless-scalar-field system.

center regularity needed for equivalence

equivalence and Hawking-mass monotonicity clauses removed

direction/sign of Hawking-mass monotonicity along null directions

nonnegative Hawking mass incorrectly upgraded to constancy

The stem does not reveal the correct option directly. It states the hypotheses and asks for the equivalent formulation, but the correct answer must still be identified from several closely related alternatives.

The item is close to a theorem-recall question: the correct option is essentially the exact equivalence statement under the stated hypotheses. However, it is not a pure restatement because the distractors alter direction of implication, integration bounds, and regularity assumptions.

Some reasoning is needed to distinguish exact equivalence from weaker or tampered variants, especially regarding the admissible class and the metric integral formula. Still, the task is mainly precise recognition of a known theorem statement rather than substantial generative mathematical reasoning.

The distractors are mathematically plausible and target realistic failure modes: incorrect boundary/integration range, one-way implication instead of equivalence, dropped regularity hypotheses, and an unjustified extra conclusion. They are distinct and nontrivial.

A solid but theorem-recognition-heavy MCQ. It avoids answer leakage and uses strong distractors, but it tests precision of recall/comparison more than genuinely generative reasoning.


Question: Consider the modified-gravity field equations E_{ab}=8π T_{ab} on a spherically symmetric spacetime, where E_{ab}:=f'(R)G_{ab}−(1/2)(f(R)−R f'(R))g_{ab}+(g_{ab}□_g−∇_a∇_b)(f'(R)) and T_{ab}=∇_aφ∇_bφ−( (1/2)∇^cφ∇_cφ+U(φ) )g_{ab} for a real-valued scalar field φ. Assume that φ U'(φ)≥0, U(φ)≥0, f'(R)>0, and f(R)≤R f'(R). Introduce ρ:=(1/κ)log f'(R) with κ>0, and in Bondi-type coordinates let h:=∂_r(rφ) and l:=∂_r(rρ). Which conclusion about the resulting spherically symmetric f(R) system holds under these assumptions?

Choices:
B. For spherically symmetric spacetimes, the f(R) field equations admit an explicit reduction to an integro-differential system consisting of two coupled first-order nonlinear hyperbolic equations whose main unknowns are h and l (equivalently, φ and ρ=(1/κ)log f'(R)), with the remaining metric coefficients and auxiliary quantities determined by explicit radial integral formulas. This first-order integro-differential system is equivalent to the original spherically symmetric f(R) field equations for every C^1 solution, without any additional regularity assumptions at the center. Moreover, the Hawking mass of the symmetry spheres is future non-decreasing along radial directions and non-increasing along null directions, and in the limit f(R)→R together with U(φ)→0 one recovers Christodoulou’s formulation of the Einstein–massless-scalar-field system.
C. For spherically symmetric spacetimes, the f(R) field equations admit an explicit reduction to an integro-differential system consisting of two coupled first-order nonlinear hyperbolic equations whose main unknowns are h and l (equivalently, φ and ρ=(1/κ)log f'(R)), with the remaining metric coefficients and auxiliary quantities determined by explicit radial integral formulas. Moreover, in the limit f(R)→R together with U(φ)→0 one recovers Christodoulou’s formulation of the Einstein–massless-scalar-field system.
D. For spherically symmetric spacetimes, the f(R) field equations admit an explicit reduction to an integro-differential system consisting of two coupled first-order nonlinear hyperbolic equations whose main unknowns are h and l (equivalently, φ and ρ=(1/κ)log f'(R)), with the remaining metric coefficients and auxiliary quantities determined by explicit radial integral formulas. In the class of C^1 solutions satisfying the two center regularity conditions lim_{r→0} r^2 e^{−κρ}(E−8πT)(D,D)=0 and ∂_r(rφ), ∂_r(rρ) ∈ C^1([0,u_0]×[0,∞))—where D is the characteristic Bondi derivative—this first-order integro-differential system is equivalent to the original spherically symmetric f(R) field equations. Moreover, the Hawking mass of the symmetry spheres is future non-decreasing both along radial directions and along null directions, and in the limit f(R)→R together with U(φ)→0 one recovers Christodoulou’s formulation of the Einstein–massless-scalar-field system.
E. For spherically symmetric spacetimes, the f(R) field equations admit an explicit reduction to an integro-differential system consisting of two coupled first-order nonlinear hyperbolic equations whose main unknowns are h and l (equivalently, φ and ρ=(1/κ)log f'(R)), with the remaining metric coefficients and auxiliary quantities determined by explicit radial integral formulas. In the class of C^1 solutions satisfying the two center regularity conditions lim_{r→0} r^2 e^{−κρ}(E−8πT)(D,D)=0 and ∂_r(rφ), ∂_r(rρ) ∈ C^1([0,u_0]×[0,∞))—where D is the characteristic Bondi derivative—this first-order integro-differential system is equivalent to the original spherically symmetric f(R) field equations. Moreover, the Hawking mass of the symmetry spheres is non-negative and therefore constant along radial directions and null directions, and in the limit f(R)→R together with U(φ)→0 one recovers Christodoulou’s formulation of the Einstein–massless-scalar-field system.

The stem does not explicitly reveal the correct option. It sets up the hypotheses and notation, but the decisive details in the answer choices—especially the center regularity assumptions and the precise Hawking-mass monotonicity directions—are not leaked.

This is essentially a theorem-conclusion matching question: the stem lists hypotheses and asks for the resulting conclusion. The choices do introduce competing variants, so it is not a pure verbatim restatement, but it remains close to recall of a specific theorem rather than a genuinely new inference task.

There is some reasoning pressure because the solver must distinguish subtle differences involving regularity assumptions, equivalence claims, and monotonicity signs. However, the task is mostly precise theorem recognition, not derivation. Also, choice C is a weaker statement that is still true if A is true, which reduces the need for decisive generative reasoning.

Several distractors are plausible and reflect realistic mathematical errors: omitting needed regularity, flipping a monotonicity direction, or overstating constancy. However, choice C is a weaker true statement rather than a genuinely false distractor, making the item technically ambiguous and lowering distractor quality.

Moderately good on leakage avoidance, but only middling as an MCQ. It mainly tests theorem recall, and the presence of a weaker true option makes the question ambiguous rather than cleanly discriminative.


\label{Main thm}
Let $X/K$ be a smooth projective surface endowed with an elliptic fibration $\pi \colon X \rightarrow \mathbb{P}^1_K$ with at least one singular fibre and a section. Let $\mathcal{X}/ \Spec(\mathcal{O})$ be a smooth projective model of $X$ over some open subset of $\Spec(\Oo_{K})$. Then $\Sigma(\mathcal{X})$ is a finitely generated abelian group.

\begin{thm} \label{Main thm}
Let $X/K$ be a smooth projective surface endowed with an elliptic fibration $\pi \colon X \rightarrow \Pp^1_K$ with at least one singular fibre and a section. Let $\mathcal{X}/ \Spec(\Oo)$ be a smooth projective model of $X$ over some open subset of $\Spec(\Oo_{K})$. Then $\Sigma(\mathcal{X})$ is a finitely generated abelian group.
\end{thm}

Let $X$ be a smooth projective variety over a number field $K$. A conjecture originally attributed to Swinnerton-Dyer (see \cite{zbMATH04061376}[Section 5] and also \cite{MR1086888} and \cite{MR1159204}) says that the Chow groups $\mathrm{CH}^i(X)$ are finitely generated. At present, this conjecture is widely open, and we do not even know whether the $n$-torsion subgroup $\mathrm{CH}^2(X)[n]$ is finitely generated when $X$ is a surface, although by Rojtman's theorem \cite{MR577137} this is true over the algebraic closure of $K$. The strongest evidence concerning the finiteness of $\mathrm{CH}^2(X)_{\mathrm{tors}}$ is due to \cite{CT-R}[Theorem C], where the authors show that $\mathrm{CH}^2(X)_{\mathrm{tors}}$ is finitely generated whenever $H^2(X, \Oo_X) = 0$ (see also \cite{MR1300892}). We refer the reader to \cite{MR1744949} and \cite{OTS}, which contain surveys regarding this problem and a more comprehensive account of the literature.

In general, to study $\mathrm{CH}^2(X)$ one spreads out $X$ to a smooth projective model $\mathcal{X} / \mathcal{O}$, where $\mathcal{O}$ is a localization of the ring of integers of $K$ at finitely many places, and considers the localization sequence
$$\bigoplus_{\mathfrak{p} \subset \mathcal{O}} \mathrm{Pic}(\mathcal{X}_{\Ff_{\mathfrak{p}}}) \rightarrow \mathrm{CH}^2(\mathcal{X}) \rightarrow \mathrm{CH}^2(X) \rightarrow 0$$
(note that this can be continued to the left using $K$-theory). Assume from now on that $H^1(X, \Oo_X) = 0$, so, after further localizing $\mathcal{O}$ if necessary, we have $\mathrm{Pic}(\mathcal{X}_{\Ff_{\mathfrak{p}}}) = \operatorname{NS}(\mathcal{X}_{\Ff_{\mathfrak{p}}})$ for every $\mathfrak{p} \subset \mathcal{O}$. The difficulty when $H^2(X, \Oo_X) \neq 0$ arises from the fact that the rank of $\operatorname{NS}(\mathcal{X}_{\Ff_{\mathfrak{p}}})$ can jump for infinitely many places (as for example happens for K3 surfaces), and one is not able to control the image of $\bigoplus_{\mathfrak{p} \subset \mathcal{O}} \mathrm{NS}(\mathcal{X}_{\Ff_{\mathfrak{p}}})$ in $\mathrm{CH}^2(\mathcal{X})$.

Let us call this image $\Sigma(\mathcal{X}) \cong \ker(\mathrm{CH}^2(\mathcal{X}) \rightarrow \mathrm{CH}^2(X) )$. This group appeared in \cite{MR1177312}, where the case $X = E \times E$ is studied, with $E/ \mathbb{Q}$ an elliptic curve. In particular, they show that $\Sigma(\mathcal{X})$ is finite when $K = \mathbb{Q}$ and $E$ has complex multiplication (there are only finitely many such curves over $\mathbb{Q}$). Moreover, the case where $E$ is an arbitrary elliptic curve over $\mathbb{Q}$ is also investigated. The proof starts by interpreting the new divisors modulo $\mathfrak{p}$ as graphs of the Frobenius, and then uses the theory of modular curves and their reduction modulo $\mathfrak{p}$ developed in \cite{MR337993} to control these cycles in $\mathrm{CH}^2(\mathcal{X})$. This is in fact one of the very few instances for which $H^2(X, \Oo_X) \neq 0$ and we nevertheless understand the role of vertical cycles in $\mathrm{CH}^2(\mathcal{X})$. See also \cite{MR1417619}, where these ideas are pushed further to prove more finiteness results regarding self products of elliptic curves over $\mathbb{Q}$. Our aim in this brief article is to generalize Mildenhall's finiteness to elliptic fibrations over $\mathbb{P}^1$:

Then all but finitely many primes satisfy (*), and the Tate algorithm implies that we can resolve the singularities of the Weierstrass model $\mathcal{X}'/R$ simultaneously over $R = \Oo_{\p}$. In particular, $X/K$ has good reduction at $\p$, and all its reductions in positive characteristic maintain the same configuration of singular fibres. Thus we have in this case a smooth projective model $\mathcal{X} / \Spec(\Oo)$, where $\Oo$ is the localization of $\Oo_{K}$ at the finitely many primes which do not satisfy (*), and an elliptic fibration $\tilde{\pi} \colon \mathcal{X} \rightarrow \mathbb{P}^1_{\Oo}$. For any $\p \subset \Oo$ we denote by $\pi_{\Ff_{\p}} \colon X_{\Ff_{\p}} \rightarrow \Pp^1_{\Ff_{\p}}$ the induced elliptic fibration over the residue field of $\p$.

\begin{proof}
Pick any lift $f = f_0/f_1$ of $\bar{f} = \bar{f}_{0}/ \bar{f}_{1}$, where
$\bar{f}_{0}, \bar{f}_{1} \in \Ff_{\p}[t]$ are coprime and
$f_0,f_1 \in R[t]$ have the same degrees as $\bar{f}_{0}$ and $\bar{f}_{1}$
respectively. Then $f_0$ and $f_1$ are also coprime and
$v_{\p}(f_0) = v_{\p}(f_1) = 0$. For a local parameter $\pi$ of $\p$ we
consider lifts of the form
\[
f_\lambda = \frac{f_0 + \pi \lambda}{f_1}, \qquad \lambda \in R.
\]
Then
\[
f_1^3\,P\bigl((f_0 + \pi \lambda)/f_1\bigr)
= (f_0 + \pi \lambda)^3 + a_4 f_1^2 (f_0+\pi \lambda)+ a_6 f_1^3
\eqqcolon J_{\lambda},
\]
and the right-hand side is coprime to $f_1$ since $f_0$ and $f_1$ are coprime
modulo $\p$. Thus it is enough to show that there are infinitely many
choices of $\lambda$ such that $J_{\lambda}$ is squarefree in $\overline{K}[t]$.
If now, for a ring $S$, we denote by $S[t]_{N} \cong S^{N+1}$ the
$S$-module of polynomials of degree $\leq N$, then the subset of
squarefree polynomials in $S[t]_{N}$ is a Zariski open subset defined
over $\Z$. Hence it is enough to show that $J_{\lambda}$ is squarefree over $K[t]$
for infinitely many $\lambda$. Now note that
\[
Q(x,t) = x^3 + a_4 f_1^2 x+ a_6 f_1^3
\]
satisfies $Q(f_1 x , t) = f_1^3 P(x,t)$ and that $P(x,t)$ is irreducible
in $K[x,t]$. If, for a contradiction, we had $Q(x,t) = Q_0(x,t) Q_1(x,t)$,
then
\[
Q_0(f_1x,t)\, Q_1(f_1x,t) = f_1^3 P(x,t),
\]
which would imply, by degree considerations in $x$, that either $Q_0$ or
$Q_1$ is constant in $x$, which is impossible unless the other factor
is a unit. Thus $Q(x,t)$ is irreducible, and hence also
$Q(f_0 + \pi x,t)$ must be irreducible. Finally, by Hilbert
irreducibility, we have infinitely many $x_0 \in K$ such that
$Q(f_0 + \pi x_0,t)$ remains irreducible over $K$, and in particular, squarefree.
\end{proof}
So we can choose $L$ such that $\Spec(L)$ is geoemtrically irreducible and the generic fiber of $\mathcal{C}(D_{\Ff_{\p}})$ is a smooth projective hyperelliptic curve.
\begin{prop}
The induced map $\phi \colon \mathcal{C}(D_{\Ff_{\p}}) \rightarrow \Pp^1_{R}$ is finite (of degree $2$) and flat. The special fibre of $\mathcal{C}(D_{\Ff_{\p}})$ splits into two components $(\Pp^1_{\Ff_{\p}})^{+} \cup (\Pp^1_{\Ff_{\p}})^{-}$ meeting at the zeroes and poles of $\bar{g}$. The two components are swapped by the reduction modulo $\p$ of the hyperelliptic involution $\iota \colon \mathcal{C}(D_{\Ff_{\p}}) \xrightarrow{\sim} \mathcal{C}(D_{\Ff_{\p}})$.
\end{prop}

\subsection{Chow groups and Proof of Theorem \ref{Main thm}}
Chow groups of schemes over Dedekind domains and their functoriality are studied in Fulton's book \cite{MR1644323}[Chapter 20]. We recall the main results that we are going to use (keeping the notation from the previous section). Let $\mathcal{X}$ be a separated scheme of finite type over $\mathcal{O}$ (or $R$) with generic fibre $X$ and special fibres $X_{\Ff_{\p}}$ (both may be empty in Fulton's setting). If $Z \subset \mathcal{X}$ is a closed integral subscheme we let
\[
\dim_{\Oo}(Z) =
\begin{cases}
 \dim(Z_{K}) & \text{if } Z \text{ is flat over } \Oo, \\[4pt]
 \dim(Z) - 1 & \text{if } Z \text{ is vertical}.
\end{cases}
\]
The group $\mathrm{CH}_i(\mathcal{X} / \Oo)$ is then defined as the free abelian group generated by integral closed subschemes of relative dimension $i$ modulo rational equivalence (thus $\mathrm{CH}^2(\mathcal{X}) = \mathrm{CH}_0(\mathcal{X}/ \Oo)$ in this notation). The specialization map
\[
\sigma_p \colon \mathrm{CH}_i(X) \rightarrow \mathrm{CH}_i(X_{\Ff_{\p}})
\]
is defined in \cite{MR1644323}[Section 20.3] in full generality. Moreover, if $Z \subset X$ is a prime cycle and $\mathcal{Z} \subset \mathcal{X}$ denotes its Zariski closure, which is flat over $\Oo$, then
\[
\sigma_p(Z) = [\mathcal{Z}_{\Ff_{\p}}] \in \mathrm{CH}_i(X_{\Ff_{\p}}),
\]
i.e. $\sigma_p(Z)$ is the element of $\mathrm{CH}_i(X_{\Ff_{\p}})$ associated to the special fibre of $\mathcal{Z}$, seen as a closed subscheme of $X_{\Ff_{\p}}$. The following lemma is well-known:

\begin{proof}
Let $\pi \in \Oo$ be a local parameter for $\p$ and, if necessary, localize $\Oo$ further to obtain $\Oo'$ so that $\pi$ has only $\p$ as a simple zero on $\Spec(\Oo')$. Let now $\pi_{| \mathcal{Z}}$ be considered as a rational function on $\mathcal{Z}$. Then by the definition of rational equivalence
\[
\mathrm{div}(\pi_{| \mathcal{Z}}) = [\mathcal{Z}_{\Ff_{\p}}] = 0
\]
in the Chow group $\mathrm{CH}_{i-1}(\mathcal{Z}/ \Oo')$. Hence, by the fact that proper pushforward respects rational equivalence, we get that $[\mathcal{Z}_{\Ff_{\p}}] = 0$ also in $\mathrm{CH}_{i-1}(\mathcal{X}/ \Oo')$. Finally, this is also zero in $\mathrm{CH}_{i-1}(\mathcal{X}/ \Oo)$ by functoriality.
\end{proof}
Let now $X$ be an elliptic surface with a model $\mathcal{X} / R$ as in the previous section. Let $\psi \colon \mathcal{X}(D_{\Ff_{\p}}) \rightarrow \mathcal{X}$ be the $2\!:\!1$ covering corresponding to some new section $D_{\Ff_{\p}} \in \NS(X_{\Ff_{\p}})$. Let
\[
\psi_{\Ff_{\p}} \colon \mathcal{X}(D_{\Ff_{\p}})_{\Ff_{\p}} = (X_{\Ff_{\p}})^{+} \cup (X_{\Ff_{\p}})^{-} \longrightarrow X_{\Ff_{\p}}
\]
be the induced (finite, flat) morphism between special fibres. By construction
\[
\psi_{\Ff_{\p}}^{*}(D_{\Ff_{\p}}) = \sigma_{\p}(D^{+} + D^{-}),
\]
where $D^{+} \subset X(D_{\Ff_{\p}}) = \mathcal{X}(D_{\Ff_{\p}})_{K}$ is the generic fibre of $\mathcal{D}^{+} \subset \mathcal{X}(D_{\Ff_{\p}})$. Thus the previous lemma implies that $[D_{\Ff_{\p}}] \in \mathrm{CH}_0(\mathcal{X} / R)$ satisfies
\[
0 = \psi^{*}([D_{\Ff_{\p}}]) \in \mathrm{CH}_0(\mathcal{X}(D_{\Ff_{\p}}) / R).
\]
Hence
\[
0 = \psi_{*}(\psi^{*}[D_{\Ff_{\p}}]) = 2 \cdot [D_{\Ff_{\p}}] \in \mathrm{CH}_{0}(\mathcal{X}/R).
\]
This shows that the element $(\dots, 0, D_{\Ff_{\p}}, 0, \dots) \in \bigoplus_{\p \subset \Oo} \NS(X_{\Ff_{\p}})$ has image in $\mathrm{CH}_0(\mathcal{X}/ \Oo)[2]$. Finally, all the components of the singular fibres of $\pi \colon X \rightarrow \Pp^1_K$ are defined over some field extension $K'/K$ of degree $N$. Hence, if $(\dots, 0, F, 0, \dots) \in \bigoplus_{\p \subset \Oo} \NS(X_{\Ff_{\p}})$ describes a component of some singular fibre which is not defined over $K$, we must also have that
\[
N \cdot (\dots, 0, F, 0, \dots) = 0 \quad \text{in } \mathrm{CH}_0(\mathcal{X}/ \Oo)
\]
by a standard push–pull argument and so the image of $\bigoplus_{\p \subset \Oo} \NS(X_{\Ff_{\p}})$ is contained in $\mathrm{CH}_0(\mathcal{X}/ \Oo)[2N] = \mathrm{CH}^2(\mathcal{X})[2N]$, which is a finite group \cite{CT-R}[Theorem 1.1.]. The result then follows since there are only finitely many primes which do not satisfy (**) and since $\NS(X_{\Ff_{\p}})$ is always finitely generated.

This applies to many surfaces of interest, e.g., (some) K3 surfaces. Also note that $H^2(X, \Oo_X)$ can be arbitrarily large and that there are no restrictions on the number field $K$.
\begin{cor}
Let $X/K$ be as in the theorem above. Then $\mathrm{CH}^2(X)[n]$ is finite for every $n>0$ and the group $\mathrm{CH}^2(X)$ is finitely generated if and only if $\mathrm{CH}^2(\mathcal{X})$ is finitely generated.
\end{cor}
This follows from \cite{CT-R}[Theorem 1.1.], which says that $\mathrm{CH}^2(\mathcal{X})[n]$ is finite. To prove Theorem \ref{Main thm} we use the relation between the Néron--Severi groups and the Mordell--Weil groups of elliptic surfaces. Our idea is to realize any new section of $\pi_{\Ff_{\mathfrak{p}}} \colon \mathcal{X}_{\Ff_{\mathfrak{p}}} \rightarrow \mathbb{P}^1_{\Ff_{\mathfrak{p}}}$ as (a component of) the degeneration of a $2$-section $D \subset X$ defined over $K$. This is made precise in Section \ref{final}. Once we do this, the result will follow almost immediately from the formal properties of Chow groups, specialization maps and the aforementioned finiteness results.

To prove Theorem \ref{Main thm} we use the relation between the N\'eron--Severi groups and the Mordell--Weil groups of elliptic surfaces. The idea is to realize any new section of \(\pi_{\Ff_{\mathfrak{p}}} \colon \mathcal{X}_{\Ff_{\mathfrak{p}}} \to \mathbb{P}^1_{\Ff_{\mathfrak{p}}}\) as (a component of) the degeneration of a \(2\)-section \(D \subset X\) defined over \(K\) (made precise in Section \ref{final}). Once this is done, the result follows almost immediately from the formal properties of Chow groups, specialization maps, and the aforementioned finiteness results.

To prove the main theorem, we use the relation between the N\'eron--Severi groups and the Mordell--Weil groups of elliptic surfaces. The idea is to realize any new section of \(\pi_{\Ff_{\mathfrak{p}}} \colon \mathcal{X}_{\Ff_{\mathfrak{p}}} \to \mathbb{P}^1_{\Ff_{\mathfrak{p}}}\) as (a component of) the degeneration of a \(2\)-section \(D \subset X\) defined over \(K\) (made precise later as follows. \label{final}
Let now $R$ be any localization of $\Oo$ at to some prime $\p$ satisfying (**). We denote by $\mathcal{X}/R$ the model of $X$ over $R$ as in the previous section. We have a natural map $). Once this is done, the result follows almost immediately from the formal properties of Chow groups, specialization maps, and the aforementioned finiteness results.

Question:
Let $K$ be a number field, let $X/K$ be a smooth projective surface, and suppose $X$ is endowed with an elliptic fibration $\pi\colon X\to \mathbb{P}^1_K$ that has a section and at least one singular fibre. Let $\mathcal{X}\to \operatorname{Spec}(\mathcal{O})$ be a smooth projective model of $X$ over an open subset of $\operatorname{Spec}(\mathcal{O}_K)$. Define $\Sigma(\mathcal{X})$ to be the image of the localization map
\[
\bigoplus_{\mathfrak p\subset \mathcal O} \operatorname{Pic}(\mathcal X_{\mathbb F_{\mathfrak p}})\longrightarrow \mathrm{CH}^2(\mathcal X),
\]
so equivalently $\Sigma(\mathcal{X})\cong \ker\bigl(\mathrm{CH}^2(\mathcal X)\to \mathrm{CH}^2(X)\bigr)$. Which statement is valid under these assumptions?

Choices:
B. $\Sigma(\mathcal{X})$ is a finite abelian group.
C. $\Sigma(\mathcal{X})$ is a countable abelian group.
D. For every smooth projective model $\mathcal{X}\to \operatorname{Spec}(\mathcal O)$ of $X$ over an open subset of $\operatorname{Spec}(\mathcal O_K)$, the group $\Sigma(\mathcal{X})$ is contained in $\mathrm{CH}^2(\mathcal X)[m]$ for some integer $m\ge 1$ depending only on $X/K$.
E. If $\mathfrak p\subset \mathcal O$ is any prime of good reduction, then every class in $\operatorname{Pic}(\mathcal X_{\mathbb F_{\mathfrak p}})$ maps to a torsion element of $\mathrm{CH}^2(\mathcal X)$; hence $\Sigma(\mathcal X)$ is a torsion abelian group.

The stem does not explicitly reveal that Sigma(\mathcal{X}) is finitely generated, nor does it strongly hint at that exact conclusion. It only sets up the geometric/arithmetic context and definition.

This is essentially a direct theorem-recall item: under the stated hypotheses, the correct option is the main structural conclusion about \Sigma(\mathcal{X}). The question mostly asks the student to recognize the theorem rather than derive a new consequence.

There is some reasoning required to distinguish 'finitely generated' from nearby alternatives such as 'finite,' 'countable,' or 'torsion,' but the item still mainly tests recall of the exact theorem statement rather than substantial generative mathematical reasoning.

The distractors are mathematically plausible and target natural failure modes: confusing finitely generated with finite, selecting a weaker true statement, or overclaiming torsion/uniform boundedness. They are distinct and well-designed.

A solid recall-based theorem-identification MCQ with strong distractors and no answer leakage, but it is largely tautological and only moderately tests genuine reasoning.