\label{thm:main-intro} Let $q$ be a prime power and $N \geqslant 1$ be an integer. Let $\mathcal{A} = \mathcal{A}_{N,q}$ be a subset of monic polynomials of  $\mathbb{F}_q[t]$ with degree $N$ with size
\begin{equation} \label{main-minimum-size}
|\mathcal{A}| \gg  q^N \exp\big(-\tfrac{1}{3}\sqrt{N \log q}\big).
\end{equation}
Assume there exists a subset $\mathcal{S}=\mathcal{S}_{N,q} \subset \mathcal{A}$ satisfying
\begin{equation} \label{main-asymptotic-subset}
	|\mathcal{S}| = (1+o(1)) |\mathcal{A}| \quad \text{ as } q^N \to \infty.
\end{equation}
and also
\begin{equation} \label{main-mult-energy}
|\{(F_1,F_2,G_1,G_2)\in \mathcal{S}^4:\,\,\,F_1F_2=G_1G_2\}|=(2+o(1))|\mathcal{S}|^2 \quad \text{ as } q^N \to \infty.
\end{equation}
If $f$ is a Steinhaus multiplicative function over $\mathbb{F}_q[t]$ then
\begin{equation}   \label{eqn:CLT} \tag{$\star$}
\frac{1}{\sqrt{|\mathcal{A}|}} \sum_{F \in \mathcal{A}} f(F)  \xrightarrow{d} \mathcal{CN}(0,1) \quad \text{ as } q^N \to \infty.
\end{equation}

\begin{theorem}\label{thm:CLT-shiftedprimes}
    Let $q \geq 3$ be a prime power and let $N \geq 1$ be an integer. Let $Z \in \mathcal{M}$ satisfy $\deg Z \leq N-1$. If $f$ is a Steinhaus random multiplicative function over $\mathbb{F}_q[t]$, then \eqref{eqn:CLT} holds for
    \[
    \mathcal{A} = \{ P + Z : P \text{ monic irreducible and } \deg P = N \}
    \]
	 as $N \to \infty$.
\end{theorem}

\begin{theorem}\label{thm:main-intro} Let $q$ be a prime power and $N \geq 1$ be an integer. Let $\mathcal{A} = \mathcal{A}_{N,q}$ be a subset of monic polynomials of  $\mathbb{F}_q[t]$ with degree $N$ with size
\begin{equation} \label{main-minimum-size}
|\mathcal{A}| \gg  q^N \exp\big(-\tfrac{1}{3}\sqrt{N \log q}\big).
\end{equation}
Assume there exists a subset $\mathcal{S}=\mathcal{S}_{N,q} \subset \mathcal{A}$ satisfying
\begin{equation} \label{main-asymptotic-subset}
	|\mathcal{S}| = (1+o(1)) |\mathcal{A}| \quad \text{ as } q^N \to \infty.
\end{equation}
and also
\begin{equation} \label{main-mult-energy}
|\{(F_1,F_2,G_1,G_2)\in \mathcal{S}^4:\,\,\,F_1F_2=G_1G_2\}|=(2+o(1))|\mathcal{S}|^2 \quad \text{ as } q^N \to \infty.
\end{equation}
If $f$ is a Steinhaus multiplicative function over $\mathbb{F}_q[t]$ then
\begin{equation}   \label{eqn:CLT} \tag{$\star$}
\frac{1}{\sqrt{|\mathcal{A}|}} \sum_{F \in \mathcal{A}} f(F)  \xrightarrow{d} \mathcal{CN}(0,1) \quad \text{ as } q^N \to \infty.
\end{equation}
\end{theorem}

\begin{theorem}[Uniform Shiu bound]\label{thm:shiu} Let $q \geq 2$ be a prime power and $N \geq 1$ be an integer. Let $g : \mathcal{M} \to [0,\infty)$ be a non-negative multiplicative function on $\mathcal{M}$ satisfying $\log g(P^{\ell}) \ll \ell $  for every integer $\ell$ and monic irreducible $P$, and $g(F) \ll_{\epsilon} 2^{\epsilon \deg F}$ for every $\epsilon > 0$ and any $F \in \mathcal{M}$.  If  $A \in \mathcal{M}_N$,  $0< \beta < 1/2$, $\beta N < h \leq N-1$, and $N$ is sufficiently large depending only on $\beta$, then
    \begin{align*}
        \sum_{\substack{F\in \mathcal{I}(A,h)}}g(F)\ll_{\beta} \frac{q^{h+1}}{N}\exp\Big(\sum_{\substack{P\in\mathcal{P}_{\leq N}}}\frac{g(P)}{q^{\deg P}}\Big).
    \end{align*}
\end{theorem}

\begin{theorem}\label{thm:CLT-kprimes} Let $q \geq 2$ be a prime power and let $N \geq 1$ be an integer. Assume  $k=o(\log N)$ as $N \to \infty$. If $f$ is a Steinhaus random multiplicative function over $\mathbb{F}_q[t]$, then \eqref{eqn:CLT} holds:
\begin{enumerate}[label=(\alph*)]
	\item As $N \to \infty$, for degree $N$ polynomials with $k$ irreducible factors, namely
	\[
	\mathcal{A} = \mathcal{P}_k(N) := \{ F \in \mathcal{M}_N : \Omega(F) = k \}.
	\]
	\item As $N \to \infty$, for   degree $N$ squarefree polynomials   with $k$ irreducible factors,  namely
	\[
	\mathcal{A} = \mathcal{S}_k(N) := \{ F \in \mathcal{M}_N : \Omega(F) = \omega(F) = k\}.
	\]
	\item As $N \to \infty$, for degree $N$  polynomials   with $k$ distinct irreducible factors,  namely
	\[
	\mathcal{A} = \mathcal{D}_k(N) := \{ F \in \mathcal{M}_N : \omega(F) = k\}.
	\]
\end{enumerate}
\end{theorem}

\begin{theorem}\label{thm:CLT-intervals}
    Let $q \geq 2$ be a prime power and let $N\geq 3$ be an integer. Let $K \in \mathcal{M}_N$  and $1 \leq h \leq N-2$. If $f$ is a Steinhaus random multiplicative function over $\mathbb{F}_q[t]$, then \eqref{eqn:CLT} holds for
    \[
    \mathcal{A} = \mathcal{I}(K,h) := \{ F \in \mathcal{M}_N : \deg(F-K) \leq h\}
    \]
    in both of the following cases:
    \begin{enumerate}[label=(\alph*)]
    	\item As $q^{h} \to \infty$ provided $q^{h+1} = o(q^N/N)$.
    	\item As $h \to \infty$ provided $q^{h+1} = o(q^N/N^c)$ for some fixed $c > 2\log 2-1$.
    \end{enumerate}

\end{theorem}

\label{sec:intro}

\subsection{Number field setting} \label{subsec:intro-nf} Given a sequence $(f(p))_{p}$ indexed by primes $p$ of independent random variables uniformly distributed on the complex unit circle $\mathbb{S}^1 = \{z \in \mathbb{C}: |z| =1\}$, a \textit{Steinhaus random multiplicative function} (over $\mathbb{N}$) is a random variable $f : \mathbb{N} \to \mathbb{C}$ defined by $f(1) = 1$ and
$f(n) = f(p_1)^{a_1} \cdots f(p_r)^{a_r},$
where $n = p_1^{a_1} \cdots p_r^{a_r}$ is the prime factorization of $n \geqslant 2$.  The study of these random functions originated with Wintner \cite{Wintner-1944} in 1944 and have re-emerged as a very active area of interest in the past decades; see a recent survey by Harper \cite{Harper-2024} for related discussion.

Given any finite subset $\mathcal{A} \subseteq \mathbb{N}$, the complex random variable
$|\mathcal{A}|^{-1/2} \sum_{n \in \mathcal{A}} f(n)$
has mean 0 and variance 1. Inspired by the central limit theorem (CLT), one might ask: does this random sum converge in distribution to the standard complex normal $\mathcal{CN}(0,1)$ as $|\mathcal{A}| \to \infty$? The answer depends on the subsets $\mathcal{A}$. Central limit theorems have been established in many cases such as  integers   with few prime factors, short intervals, and polynomial values. More precisely, denoting $\omega(n)$ as the number of distinct primes dividing $n$, the random sum over $\mathcal{A}$ satisfies a CLT for
\begin{itemize}
	\item $\mathcal{A} = \{ 1 \leqslant n \leqslant x : \omega(n) = k\}$ with $k = o(\log \log x)$ as $x \to \infty$,
	\item $\mathcal{A} = \{ x-y \leqslant n \leqslant x \}$ with $y \leqslant x/(\log x)^{2\log 2-1+\varepsilon}$ as $x \to \infty$,
	\item $\mathcal{A} = \{ Q(n) : 1 \leqslant n \leqslant x \}$ as $x \to \infty$, where $Q \in \mathbb{Z}[t]$ is not of the form $a (t+b)^c$ for $a,b,c \in \mathbb{Z}$.
\end{itemize}
These results are respectively due to Harper \cite{Harper-2013},  Soundararajan--Xu \cite{SoundXu-2023}, and Klurman--Shkredov--Xu \cite{RandomChowla}. On the other hand, Harper   surprisingly showed that the natural choice
$
\mathcal{A} = [1,x] \cap \mathbb{N}
$
does \textit{not} converge to the standard complex normal \cite{Harper-2013} and, in fact, the sum converges to zero in distribution \cite{Harper-2020}; see Gorodetsky--Wong \cite{GorodetskyWong-2025-2, GorodetskyWong-2025} for recent striking progress and Atherfold--Najnudel \cite{AtherfoldNajnudel-2025}  for a closely related breakthrough. A natural question, then, is over which subsets $\mathcal{A}$ does such a central limit theorem hold?   Soundararajan and Xu \cite{SoundXu-2023} recently established a flexible criterion.

\hypertarget{customlabel}{\label{thm:SoundXu-integer}}
\begin{theorem*}[Soundararajan--Xu]
    Let $x \geqslant 10$ be large. Let $\mathcal{A}=\mathcal{A}_x$ be a subset of $[1,x] \cap \mathbb{N}$ with size
    \[
    |\mathcal{A}|\geqslant x \exp(-\tfrac13\sqrt{\log x\log\log x}).
    \]
   Assume there exists a subset $\mathcal{S}=\mathcal{S}_x\subset\mathcal{A}_x$ with size $|\mathcal{S}|=(1+o(1))|\mathcal{A}|$ as $x \to \infty$ satisfying
    \begin{align*}
        |\{(s_1,s_2,s_3,s_4)\in\mathcal{S}^4:\,\,\, s_1s_2=s_3s_4\}|=(2+o(1))|\mathcal{S}|^2 \quad \text{ as } x \to \infty.
    \end{align*}
   If $f$ is a Steinhaus random multiplicative function over $\mathbb{Z}$, then
    \begin{align*}
        \frac{1}{\sqrt{|\mathcal{A}|}}\sum_{n\in\mathcal{A}}f(n) \xrightarrow{d} \mathcal{CN}(0,1) \quad \text{ as } x \to \infty.
    \end{align*}
\end{theorem*}
\begin{remark*}
    Here and throughout, the notation $\,\xrightarrow{d}\,$ indicates convergence in distribution, and $\mathcal{CN}(0,1)$ is the standard complex normal with mean 0 and variance 1.
\end{remark*}

The principal goal of this paper is to extend this central limit theorem criterion and some applications from the \textit{number field setting} over $\mathbb{Z}$ to the \textit{function field setting} over the polynomial ring $\mathbb{F}_q[t]$, where $q$ is a prime power and $\mathbb{F}_q$ is the unique finite field of $q$ elements (see, e.g., \cite{Gran-Harp-Sound-2015} for an introduction to multiplicative functions over $\mathbb{F}_q[t]$).

\subsection{Function field setting} \label{subsec:intro-ff} Let $\mathcal{M}$ be the set of monic polynomials of $\mathbb{F}_q[t]$. Given a sequence  $(f(P))_{P}$ indexed by monic irreducible polynomials $P$ of independent random variables  uniformly distributed on $\mathbb{S}^1$, a \textit{Steinhaus random multiplicative function} (over $\mathbb{F}_q[t]$) is a random variable $f :  \mathcal{M} \to \mathbb{C}$ defined by $f(1) = 1$ and
\begin{align*}
 	f(F)= f(P_1)^{a_1} \cdots f(P_r)^{a_r},
\end{align*}
where $F = P_1^{a_1} \cdots P_r^{a_r}$ is the unique factorization of $F \in \mathcal{M}$ into powers of distinct monic irreducibles $P_1,\dots,P_r$.  Given any finite subset $\mathcal{A} \subseteq \mathcal{M}$, the complex random variable
\begin{align}\label{eq:partialsums}
    \frac{1}{\sqrt{|\mathcal{A}|}} \sum_{F\in\mathcal{A}}f(F)
\end{align}
has mean 0 and variance 1, so the same question arises: does this random sum converge in distribution to the standard complex normal $\mathcal{CN}(0,1)$ as $|\mathcal{A}| \to \infty$?  This question has strong parallels with the number field setting, but there are fewer examples over function fields.

Corresponding to Harper's result \cite{Harper-2013,Harper-2020} over $\mathbb{Z}$ from \S\ref{subsec:intro-nf}, a theorem of Soundararajan--Zaman \cite{SoundZaman-2022} implies that the random sum \eqref{eq:partialsums} with the natural choice
	\[
	\mathcal{A} = \mathcal{M}_N := \{ F \in \mathcal{M} : \deg F = N \}
	\]
	does \textit{not} converge in distribution to the standard normal (and in fact, converges to zero) when taking first $q \to \infty$ and then $N \to \infty$. The moments of this example have recently been computed by Hofmann--Hoganson--Menon--Verreault--Zaman \cite{FUSRP-2023}. A central limit theorem was established by Aggarwal--Subedi--Verreault--Zaman--Zheng \cite{FUSRP2020} for Rademacher random multiplicative functions  over polynomials with few irreducible factors, namely when
\[
\mathcal{A} = \{ F \in \mathcal{M}_N : \omega(F) = k\} \text{ with  } k = o(\log N)  \text{ as }N \to \infty.
\]
Here $\omega(F)$ is the number of distinct monic irreducible factors of $F$. This exactly matches Harper  \cite{Harper-2013} in \S\ref{subsec:intro-nf}. Otherwise, as far as we are aware, there are no other examples of CLTs over $\mathbb{F}_q[t]$.

\subsection{Results} Our main result is a function field analogue of Soundararajan and Xu's theorem.

Given any finite subset $\mathcal{A} \subseteq \mathbb{N}$, the complex random variable
$|\mathcal{A}|^{-1/2} \sum_{n \in \mathcal{A}} f(n)$
has mean 0 and variance 1. Inspired by the central limit theorem (CLT), one might ask: does this random sum converge in distribution to the standard complex normal $\mathcal{CN}(0,1)$ as $|\mathcal{A}| \to \infty$? The answer depends on the subsets $\mathcal{A}$. Central limit theorems have been established in many cases such as integers with few prime factors, short intervals, and polynomial values. More precisely, denoting $\omega(n)$ as the number of distinct primes dividing $n$, the random sum over $\mathcal{A}$ satisfies a CLT for
\begin{itemize}
 \item $\mathcal{A} = \{ 1 \leqslant n \leqslant x : \omega(n) = k\}$ with $k = o(\log \log x)$ as $x \to \infty$,
 \item $\mathcal{A} = \{ x-y \leqslant n \leqslant x \}$ with $y \leqslant x/(\log x)^{2\log 2-1+\varepsilon}$ as $x \to \infty$,
 \item $\mathcal{A} = \{ Q(n) : 1 \leqslant n \leqslant x \}$ as $x \to \infty$, where $Q \in \mathbb{Z}[t]$ is not of the form $a (t+b)^c$ for $a,b,c \in \mathbb{Z}$.
\end{itemize}
These results are respectively due to Harper \cite{Harper-2013}, Soundararajan--Xu \cite{SoundXu-2023}, and Klurman--Shkredov--Xu \cite{RandomChowla}. On the other hand, Harper surprisingly showed that the natural choice
$
\mathcal{A} = [1,x] \cap \mathbb{N}
$
does \textit{not} converge to the standard complex normal \cite{Harper-2013} and, in fact, the sum converges to zero in distribution \cite{Harper-2020}; see Gorodetsky--Wong \cite{GorodetskyWong-2025-2, GorodetskyWong-2025} for recent striking progress and Atherfold--Najnudel \cite{AtherfoldNajnudel-2025} for a closely related breakthrough. A natural question, then, is over which subsets $\mathcal{A}$ does such a central limit theorem hold? Soundararajan and Xu \cite{SoundXu-2023} recently established a flexible criterion.

\hypertarget{customlabel}{\label{thm:SoundXu-integer}}
\begin{theorem*}[Soundararajan--Xu]
 Let $x \geqslant 10$ be large. Let $\mathcal{A}=\mathcal{A}_x$ be a subset of $[1,x] \cap \mathbb{N}$ with size
 \[
 |\mathcal{A}|\geqslant x \exp(-\tfrac13\sqrt{\log x\log\log x}).
 \]
 Assume there exists a subset $\mathcal{S}=\mathcal{S}_x\subset\mathcal{A}_x$ with size $|\mathcal{S}|=(1+o(1))|\mathcal{A}|$ as $x \to \infty$ satisfying
 \begin{align*}
 |\{(s_1,s_2,s_3,s_4)\in\mathcal{S}^4:\,\,\, s_1s_2=s_3s_4\}|=(2+o(1))|\mathcal{S}|^2 \quad \text{ as } x \to \infty.
 \end{align*}
 If $f$ is a Steinhaus random multiplicative function over $\mathbb{Z}$, then
 \begin{align*}
 \frac{1}{\sqrt{|\mathcal{A}|}}\sum_{n\in\mathcal{A}}f(n) \xrightarrow{d} \mathcal{CN}(0,1) \quad \text{ as } x \to \infty.
 \end{align*}
\end{theorem*}
\begin{remark*}
 Here and throughout, the notation $\,\xrightarrow{d}\,$ indicates convergence in distribution, and $\mathcal{CN}(0,1)$ is the standard complex normal with mean 0 and variance 1.
\end{remark*}

\subsection{Function field setting} \label{subsec:intro-ff} Let $\mathcal{M}$ be the set of monic polynomials of $\mathbb{F}_q[t]$. Given a sequence $(f(P))_{P}$ indexed by monic irreducible polynomials $P$ of independent random variables uniformly distributed on $\mathbb{S}^1$, a \textit{Steinhaus random multiplicative function} (over $\mathbb{F}_q[t]$) is a random variable $f : \mathcal{M} \to \mathbb{C}$ defined by $f(1) = 1$ and
\begin{align*}
 f(F)= f(P_1)^{a_1} \cdots f(P_r)^{a_r},
\end{align*}
where $F = P_1^{a_1} \cdots P_r^{a_r}$ is the unique factorization of $F \in \mathcal{M}$ into powers of distinct monic irreducibles $P_1,\dots,P_r$. Given any finite subset $\mathcal{A} \subseteq \mathcal{M}$, the complex random variable
\begin{align}\label{eq:partialsums}
 \frac{1}{\sqrt{|\mathcal{A}|}} \sum_{F\in\mathcal{A}}f(F)
\end{align}
has mean 0 and variance 1, so the same question arises: does this random sum converge in distribution to the standard complex normal $\mathcal{CN}(0,1)$ as $|\mathcal{A}| \to \infty$? This question has strong parallels with the number field setting, but there are fewer examples over function fields.

Corresponding to Harper's result \cite{Harper-2013,Harper-2020} over $\mathbb{Z}$ from \S\ref{subsec:intro-nf}, a theorem of Soundararajan--Zaman \cite{SoundZaman-2022} implies that the random sum \eqref{eq:partialsums} with the natural choice
 \[
 \mathcal{A} = \mathcal{M}_N := \{ F \in \mathcal{M} : \deg F = N \}
 \]
 does \textit{not} converge in distribution to the standard normal (and in fact, converges to zero) when taking first $q \to \infty$ and then $N \to \infty$. The moments of this example have recently been computed by Hofmann--Hoganson--Menon--Verreault--Zaman \cite{FUSRP-2023}. A central limit theorem was established by Aggarwal--Subedi--Verreault--Zaman--Zheng \cite{FUSRP2020} for Rademacher random multiplicative functions over polynomials with few irreducible factors, namely when
\[
\mathcal{A} = \{ F \in \mathcal{M}_N : \omega(F) = k\} \text{ with } k = o(\log N) \text{ as }N \to \infty.
\]
Here $\omega(F)$ is the number of distinct monic irreducible factors of $F$. This exactly matches Harper \cite{Harper-2013} in \S\ref{subsec:intro-nf}. Otherwise, as far as we are aware, there are no other examples of CLTs over $\mathbb{F}_q[t]$.

\subsection{Results} Our main result is a function field analogue of Soundararajan and Xu's theorem.

\begin{remark} \label[remark]{rem:limit-convention}
Here and throughout, any limit of $q$ or $N$ permits the parameters to vary and depend on each other arbitrarily unless explicitly stated otherwise. For example, the limit $q^N \to \infty$ includes the cases $q$ fixed with $N \to \infty$, $N$ fixed with $q \to \infty$, and more.
\end{remark}

We refer to the subset of polynomials whose prime factors all have degree exceeding an integer $z \geq 1$ as \textit{$z$-rough polynomials}. In other words, polynomials $F$ such that
\begin{equation} \label{def:min-degree}
P^-(F) := \min\{ \deg P : P \mid F, \, P \text{ monic irreducible} \}
\end{equation}
exceeds $z$.
 As suggested by Xu \cite{Xu-2022}, one can expect a CLT when summing over sufficiently rough polynomials. As a brief final application of \cref{thm:main-intro}, we confirm this observation in \S\ref{sec:roughpolyCLT}.
\begin{theorem}\label{thm:CLT-rough}
 Let $q \geq 2$ be a prime power and $N\geq 1$ an integer. Assume $N^{1/2} \leq z \leq N-1$ is an integer satisfying $N^{1/2} = o(z)$ as $N \to \infty$. If $f$ is a Steinhaus random multiplicative function over $\mathbb{F}_q[t]$, then \eqref{eqn:CLT} holds for
 \[
 \mathcal{A} = \{ F \in \mathcal{M}_N : P^-(F) > z \}
 \]
 as $N \to \infty$.
\end{theorem}

\begin{enumerate}[label=(\roman*)]
 \item $|\mathcal{A} \backslash \mathcal{S}| = o(|\mathcal{A}|)$. \label{thmmain:i}
 \item $|\{ (F_1,F_2,G_1,G_2) \in \mathcal{S}^4 : F_1 F_2=G_1 G_2 , F_1 \neq G_1, F_1 \neq G_2 \}| = o(|\mathcal{A}|^2)$. \label{thmmain:ii}
 \item $|\{F\in \mathcal{S}:\ P^+_\prec(F)=Q\}| = o(|\mathcal{A}|)$ for every monic irreducible $Q$. \label{thmmain:iii}
\end{enumerate}
If $f$ is a Steinhaus random multiplicative function over $\mathbb{F}_q[t]$ then, as $q^N\to\infty$,
\[
Z=\frac{1}{\sqrt{|\mathcal{A}|}}\sum_{F\in \mathcal{A}} f(F)\ \xrightarrow{d}\ \mathcal{CN}(0,1).
\]
\end{theorem}
\begin{remark}\label[remark]{rem:martingale} The advantage of our refined filtration with $P_{\prec}^+(F)$ can be seen from condition (iii). Observe that
\begin{align*}
\max_{Q \in \mathcal{P}} | \{ F \in \mathcal{M}_N : P^+_{\prec}(F) = Q \} | & \ll q^N \exp\Big( - \tfrac{1}{2} \sqrt{N \log q} \Big), \\
\max_{d \geq 1} | \{ F \in \mathcal{M}_N : P^+(F) = d \} | & \gg q^N /N,
\end{align*}
upon applying \cref{lem:filtration} with $h=N-1$ for the first bound, and noting $d=N$ corresponds to the total number of monic irreducibles for the second bound. The latter filtration is based purely on the degree $P^+(F)$ and would require us to assume $|\mathcal{A}| \gg q^N/N$ in \cref{thm:main-intro}. Aggarwal--Subedi--Verreault--Zaman--Zheng \cite{FUSRP2020} successfully used this filtration in their case, because their desired application for $\mathcal{A}$ was sufficiently large. However, the constraint $|\mathcal{A}| \gg q^N/N$ is too strict for our intended applications. Our refined filtration with $P^+_{\prec}(F)$ allows us to only require $|\mathcal{A}| \gg q^N \exp(- \tfrac{1}{3}\sqrt{N \log q})$ in \cref{thm:main-intro}, which is notably more flexible. It also more closely mimics Harper \cite{Harper-2013} and his filtration based on $\{ m \in [1,n] \cap \mathbb{Z} : P^+(m) = p\}$ for any prime $p$, where $P^+(m)$ denotes the largest prime factor of $m \in \mathbb{Z}$.
\end{remark}

Fix $\epsilon > 0$ sufficiently small. As in the proof of \cref{prop:short-interval-mult-energy}, we again count off-diagonal solutions to $F_1F_2 = G_1G_2$ for $F_1,F_2,G_1,G_2 \in \mathcal{S}$ with the same GCD parametrization. Write $F_1 = GA, G_1 = GB, F_2 = HB,$ and $G_2 = HA$. Denoting $d = \deg G = \deg H$, it follows by \cref{prop:short-interval-mult-energy} and its proof that $N-h \leq d \leq h$, and we may assume $h \geq N/2$ without loss. Set $m=(1+ \epsilon)\log N$ so that $\Omega(GABH)\leq 2m$. This yields the following estimate for the off-diagonal solutions:
\begin{align}
 \mathsf{E}_\times(\mathcal{S}) - 2|\mathcal{S}|^2 + |\mathcal{S}|
 &\ll \sum_{d=N-h}^{h} \, \ssum_{\substack{G, H \in \mathcal{M}_d}} \, \ssum_{\substack{A, B \in \mathcal{M}_{N-d}\\ GA, HB, GB, HA\in \mathcal{I}(K,h)}}2^{2m-\Omega(GABH)} \nonumber\\
 &=2^{2m}\sum_{d=N-h}^{h} \sum_{\substack{G\in\mathcal{M}_d}}2^{-\Omega(G)}\ssum_{\substack{A,B\in \mathcal{I}(K,h)/G \\ A,B \in \mathcal{M}_{N-d}}}2^{-\Omega(A)-\Omega(B)}\sum_{\substack{ H\in \mathcal{I}(K,h)/A \\ H \in \mathcal{M}_d}}2^{-\Omega(H)}. \label{eqn:short-intervals-off-diag}
\end{align}
As described at the end of \S\ref{sec:intervals}, for each $A \in \mathcal{M}_{N-d}$, the interval quotient $\mathcal{I}(K,h)/A$ is \textit{equal} to the interval $\mathcal{I}(\widetilde{K},h+d-N)$ for some $\widetilde{K}$ (depending on $A$) of degree $d = N - (N-d)$ because $h+d-N \geq 0$ in our sum. Thus, by \cref{thm:shiu} with $g(H) = 2^{-\Omega(H)}$, it follows that
 \[
 \sum_{\substack{ H\in \mathcal{I}(K,h)/A \\ H \in \mathcal{M}_d}}2^{-\Omega(H)} = \sum_{\substack{ H\in \mathcal{I}(\widetilde{K},h+d-N) }}2^{-\Omega(H)} \ll \frac{q^{h+d-N+1}}{d} \exp\Big( \sum_{P \in \mathcal{P}_{\leq d} } \frac{1}{2q^{\deg(P)}} \Big) \ll \frac{q^{h+d-N+1}}{d^{1/2}}
 \]
 for each $A \in \mathcal{M}_{N-d}$, since by the prime number theorem $\sum_{P \in \mathcal{P}_{\leq d} } q^{-\deg(P)} = \log d +O(1)$. By \eqref{eqn:short-intervals-off-diag}, this implies that
 \[
\mathsf{E}_\times(\mathcal{S}) - 2|\mathcal{S}|^2 + |\mathcal{S}| \ll 2^{2m}\sum_{d=N-h}^{h} \frac{q^{h+d-N+1}}{d^{1/2}} \sum_{\substack{G\in\mathcal{M}_d}}2^{-\Omega(G)}\Big( \sum_{\substack{A\in \mathcal{I}(K,h)/G \\ A \in \mathcal{M}_{N-d}}}2^{-\Omega(A)} \Big)^2
 \]
 since the sums over $A$ and $B$ are identical. For the sum over $A$, the same argument with \cref{thm:shiu} applies. Indeed, for each $G \in \mathcal{M}_d$, the interval quotient $\mathcal{I}(K,h)/G$ is \textit{equal} to the interval $\mathcal{I}(\widetilde{K}',h-d)$ for some $\widetilde{K}'$ (depending on $G$) of degree $N-d$ because $h-d \geq 0$ in our sum. Thus, by \cref{thm:shiu} with $g(A) = 2^{-\Omega(A)}$, we similarly have that
 \[
 \sum_{\substack{A\in \mathcal{I}(K,h)/G \\ A \in \mathcal{M}_{N-d}}}2^{-\Omega(A)}
 =
 \sum_{\substack{A\in \mathcal{I}(\widetilde{K}',h-d)}}2^{-\Omega(A)} \ll \frac{q^{h-d+1}}{N-d} \exp\Big( \sum_{P \in \mathcal{P}_{\leq N-d} } \frac{1}{2 q^{\deg(P)}} \Big) \ll \frac{q^{h-d+1}}{(N-d)^{1/2}}.
 \]
Inserting this into the previous equation, we deduce that
 \[
 \mathsf{E}_\times(\mathcal{S}) - 2|\mathcal{S}|^2 + |\mathcal{S}| \ll 2^{2m}\sum_{d=N-h}^{h} \frac{q^{3h-N-d+3}}{d^{1/2} (N-d)} \sum_{\substack{G\in\mathcal{M}_d}}2^{-\Omega(G)}.
 \]
Upon noting $\mathcal{I}(G,d-1) = \mathcal{M}_d$, we again apply \cref{thm:shiu} on the sum over $G$ to see that
\[
\sum_{\substack{G\in\mathcal{M}_d}}2^{-\Omega(G)} \ll \frac{q^{d+1}}{d^{1/2}}
\]
for each $N-h \leq d \leq h$. Therefore, we may conclude that
\[
 \mathsf{E}_\times(\mathcal{S}) - 2|\mathcal{S}|^2 + |\mathcal{S}| \ll 2^{2m} q^{3h+3-N}\sum_{d=N-h}^h \frac{1}{d(N-d)} \ll 2^{2m} q^{3h+3-N} \frac{\log N}{N}
\]
since $N-d \geq h \geq N/2$ and $\sum_{d=N-h}^h \frac{1}{d} \leq \sum_{d=1}^N \frac{1}{d} \ll \log N$. Using our choice $m = (1+\epsilon) N$ and noting $|\mathcal{S}| \asymp |\mathcal{I}(K,h)| = q^{h+1}$ by \cref{lem:short-interval-hardy-ramanujan}, we have overall that
\[
2|\mathcal{S}|^2 - |\mathcal{S}| \leq \mathsf{E}_\times(\mathcal{S}) \leq 2|\mathcal{S}|^2 - |\mathcal{S}| + O\Big( |\mathcal{S}|^2 q^{h+1-N} N^{2\log 2 (1+\epsilon) - 1} \log N \Big).
\]
The error term is $o(|\mathcal{S}|^2)$ provided $q^{h+1} = o(q^N / N^{2\log 2-1+2\epsilon} )$. Our assumption states $q^{h+1} = o(q^N/N^c)$ as $h \to \infty$ for some fixed $c > 2\log 2-1$. By fixing $\epsilon = \epsilon(c) > 0$ sufficiently small, we deduce that $\mathsf{E}_\times(\mathcal{S}) = 2 |\mathcal{S}|^2(1+o(1))$ as $h \to \infty$. This verifies (ii) and establishes case (b).
\end{proof}

\begin{remark} \label[remark]{rem:limit-convention}
Here and throughout, any limit of $q$ or $N$ permits the parameters  to vary and depend on each other arbitrarily unless explicitly stated otherwise.  For example, the limit $q^N \to \infty$ includes the cases $q$ fixed with $N \to \infty$, $N$ fixed with $q \to \infty$, and more.
\end{remark}

As remarked in \cite{SoundXu-2023}, the size constraints on $|\mathcal{A}|$ and $|\mathcal{S}|$ are mild, but \eqref{main-mult-energy} is  pivotal. This condition  concerns the \textit{multiplicative energy} $\mathsf{E}_{\times}(\mathcal{S})$ of a finite set $\mathcal{S} \subseteq \mathcal{M}$, defined by
\begin{equation} \label{eqn:mult-energy}
    \mathsf{E}_{\times}(\mathcal{S}) := |\{(F_1,F_2,G_1,G_2)\in \mathcal{S}^4:F_1F_2=G_1G_2\}| = \mathbb{E} \Big| \sum_{F \in \mathcal{S} } f(F) \Big|^4.
\end{equation}
We say a family of sets $\mathcal{S}$ have \textit{asymptotically trivial multiplicative energy} when
\begin{equation}\label{trivialenergy}
\mathsf{E}_{\times}(\mathcal{S}) =(2+o(1))|\mathcal{S}|^2 \quad \text{ as } |\mathcal{S}| \to \infty.
\end{equation}
The trivial (or diagonal) solutions $(F_1, F_2, F_1, F_2)$ and $(F_1, F_2, F_2, F_1)$ for \eqref{eqn:mult-energy} always exist, so it is immediate that $\mathsf{E}_{\times}(\mathcal{S}) \geqslant 2|\mathcal{S}|^2 - |\mathcal{S}|$. Statement \eqref{trivialenergy} therefore requires that the multiplicative energy is dominated by the trivial solutions in the limit.

The proof of \cref{thm:main-intro} largely follows Soundararajan--Xu by invoking a central limit theorem for martingale difference sequences due to McLeish (\cref{thm:mcleishclt}). This approach was utilized in \cite{FUSRP2020} for function fields but they applied a coarse filtration depending only on
\begin{equation} \label{def:max-degree}
P^+(F) := \max\{ \deg P : P \mid F, P \text{ monic irreducible } \},
\end{equation}
the maximum degree of any monic irreducible dividing $F \in \mathcal{M}$. We require a more refined filtration for each individual prime $P$, necessitating a new estimate for smooth polynomials in short intervals (\cref{lem:filtration}) in addition to a related estimate of Gorodetsky \cite{OfirSmoothPolys23}; see \cref{rem:martingale} for details.

From \cref{thm:main-intro}, we deduce  four central limit theorems:
\begin{itemize}
	\item (\cref{thm:CLT-intervals})  CLT for short intervals
	\item (\cref{thm:CLT-kprimes}) CLT for restricted number of prime factors
	\item (\cref{thm:CLT-shiftedprimes}) CLT for shifted primes
	\item (\cref{thm:CLT-rough}) CLT for rough polynomials
\end{itemize}
Each application requires a variety of function field number theoretic lemmas, many of which we could not find in the literature with sufficient uniformity. We established several new results which we expect to be of independent interest: a uniform Hildebrand bound for smooth polynomials in short intervals (\cref{prop:intervalshift}), a uniform Shiu's theorem for multiplicative functions (\cref{thm:shiu}), and a uniform Chebyshev bound for rough polynomials in short intervals (\cref{lem:chebyshev}). See  \S\ref{sec:CLTs} for details on these four central limit theorems and number theoretic lemmas.

\subsection*{Organization} \cref{sec:CLTs} describes the CLT applications deduced from \cref{thm:main-intro}  and other number theoretic results over $\mathbb{F}_q[t]$. \cref{sec:prelim} collects our notation and basic terminology regarding function field intervals, and then presents a  simplified version of the Selberg sieve for function fields. \cref{sec:smoothroughpolys} develops novel bounds on the numbers of smooth and rough polynomials in an interval (\cref{prop:intervalshift,lem:chebyshev}). \cref{sec:clts} establishes \cref{thm:main-intro} by applying these lemmas.

The remaining subsections are dedicated to deducing the four central limit theorems described in \cref{sec:CLTs}.   \cref{sec:shiu} prepares a function field analogue of Shiu's theorem (\cref{thm:shiu}), so that \cref{sec:interval-CLT} can establish the short interval CLT (\cref{thm:CLT-intervals}). \cref{sec:almostprimesCLT,sec:shiftedprimesCLT} respectively establish the few prime factors CLT (\cref{thm:CLT-kprimes}) and shifted prime CLT (\cref{thm:CLT-shiftedprimes}). Finally, \cref{sec:roughpolyCLT} presents a short proof of the rough polynomial CLT (\cref{thm:CLT-rough}).

\subsection*{Acknowledgments} The authors thank Lior Bary-Soroker, Ofir Gorodetsky, Dimitris Koukoulopoulos, and Max Xu for helpful discussions and references. This research was conducted as part of the 2024 Fields Undergraduate Summer Research Program, and the authors are deeply grateful for the Fields Institute's support. AZ was partially supported by NSERC grant RGPIN-2022-04982.

The proof of \cref{thm:main-intro} is described as largely following Soundararajan--Xu by "invoking a central limit theorem for martingale difference sequences due to McLeish (\cref{thm:mcleishclt})." The authors note that prior work in function fields \cite{FUSRP2020} used "a coarse filtration depending only on" \(P^+(F)\), but here they "require a more refined filtration for each individual prime \(P\)," which in turn "necessitat[es] a new estimate for smooth polynomials in short intervals (\cref{lem:filtration})" together with "a related estimate of Gorodetsky." Finally, they state that \cref{thm:main-intro} is established in \cref{sec:clts} "by applying these lemmas."

The proof of the main theorem is described as largely following Soundararajan--Xu by invoking the following central limit theorem for martingale difference sequences due to McLeish.

\begin{theorem}[Complex-Valued McLeish CLT]\label{thm:mcleishclt}
Let $Z_1,\dots,Z_N$ be a complex-valued martingale difference sequence with $\mathbb E[|Z_n|^4]<\infty$
for every $n$, and put $S_N=\sum_{n=1}^N Z_n$. Assume
\({
\sum_{n=1}^N \mathbb E[|Z_n|^2] = 1.
}\)
Then for real $t_1,t_2$ with $t^2=(t_1^2+t_2^2)/2$,
\begin{align*}
\mathbb{E}\Big[e^{i t_1\operatorname{Re}(S_N)+i t_2\operatorname{Im}(S_N)}\Big]
= e^{-t^2/2} & + O\Big(e^{t^2}\Big(\Big(\sum_{n=1}^N\mathbb E[|Z_n|^4]\Big)^{1/4} + \Big(\mathbb E\Big[\Big(\sum_{n=1}^N |Z_n|^2 - 1\Big)^2\Big]\Big)^{1/2}\Big) \Big) \\
& \qquad + O\Big( \max_{\phi\in[0,2\pi]}\Big(\mathbb E\Big[\Big(\sum_{n=1}^N(e^{-i\phi}Z_n^2+e^{i\phi}\overline{Z_n}^2)\Big)^2\Big]\Big)^{1/2} \Big).
\end{align*}
\end{theorem}

The authors note that prior work in function fields \cite{FUSRP2020} used a coarse filtration depending only on $P^+(F)$, but here they require a more refined filtration for each individual prime $P$, which in turn necessitates a new estimate for smooth polynomials in short intervals (\cref{lem:filtration}) together with a related estimate of Gorodetsky. Finally, they state that the main theorem is established later by applying these lemmas.

Question:
Let \(\mathcal M\) be the set of monic polynomials in \(\mathbb F_q[t]\), where \(q\) is a prime power. A Steinhaus random multiplicative function over \(\mathbb F_q[t]\) is a random function \(f:\mathcal M\to\mathbb C\) such that the values \(f(P)\) on monic irreducible polynomials \(P\) are independent and uniformly distributed on the unit circle \(\mathbb S^1\), with \(f(1)=1\) and \(f(F)=\prod_i f(P_i)^{a_i}\) whenever \(F=\prod_i P_i^{a_i}\) is the factorization of \(F\) into powers of distinct monic irreducibles. Fix an integer \(N\ge 1\), and let \(\mathcal A=\mathcal A_{N,q}\) be a subset of the monic polynomials of degree \(N\) such that
\[
|\mathcal A|\gg q^N\exp\!\big(-\tfrac13\sqrt{N\log q}\big).
\]
Assume there is a subset \(\mathcal S=\mathcal S_{N,q}\subset \mathcal A\) with
\[
|\mathcal S|=(1+o(1))|\mathcal A| \quad \text{as } q^N\to\infty,
\]
and whose multiplicative energy satisfies
\[
\big|\{(F_1,F_2,G_1,G_2)\in \mathcal S^4: F_1F_2=G_1G_2\}\big|=(2+o(1))|\mathcal S|^2 \quad \text{as } q^N\to\infty.
\]
Here \(\mathcal{CN}(0,1)\) denotes the standard complex normal distribution. As \(q^N\to\infty\), which limiting statement holds?

For every Steinhaus random multiplicative function \(f\) over \(\mathbb F_q[t]\),
\[
\frac{1}{\sqrt{|\mathcal A|}}\sum_{F\in\mathcal A} f(F) \xrightarrow{d} \mathcal{CN}(0,1)
\quad \text{as } q^N\to\infty.
\]

Choices:
B.
For every Steinhaus random multiplicative function \(f\) over \(\mathbb F_q[t]\),
\[
\frac{1}{\sqrt{|\mathcal A|}}\sum_{F\in\mathcal A} f(F) \xrightarrow{d} \mathcal{CN}(0,1)
\quad \text{as } N\to\infty \text{ with } q \text{ fixed}.
\]
C.
For every Steinhaus random multiplicative function \(f\) over \(\mathbb F_q[t]\), the normalized sums
\[
\frac{1}{\sqrt{|\mathcal A|}}\sum_{F\in\mathcal A} f(F)
\]
form a tight family of complex-valued random variables with mean \(0\) and variance \(1\) as \(q^N\to\infty\).
D.
There exists a Steinhaus random multiplicative function \(f\) over \(\mathbb F_q[t]\) such that
\[
\frac{1}{\sqrt{|\mathcal A|}}\sum_{F\in\mathcal A} f(F) \xrightarrow{d} \mathcal{CN}(0,1)
\quad \text{as } q^N\to\infty.
\]
E.
For every Steinhaus random multiplicative function \(f\) over \(\mathbb F_q[t]\),
\[
\frac{1}{\sqrt{|\mathcal A|}}\sum_{F\in\mathcal A} f(F) \xrightarrow{d} \mathcal{CN}(0,1)
\quad \text{as } q^N\to\infty,
\]
provided merely that
\[
\big|\{(F_1,F_2,G_1,G_2)\in \mathcal S^4: F_1F_2=G_1G_2\}\big|\le (2+o(1))|\mathcal S|^2
\quad \text{as } q^N\to\infty.
\]


Let $\mathbb{D}$ be the unit disk, let $f \colon \mathbb{D} \to \mathbb{D}$ be
a holomorphic self map of $\mathbb{D}$, and consider the dynamical system given by its iterates $\{ f^n \}_{n\in \mathbb{N}}$.
The Denjoy-Wolff Theorem states that if $f$ is not conjugate to a rotation, then there exists a point $p \in \overline{\mathbb{D}}$ such that the orbit of every point in $\mathbb{D}$ converges to $p$, that is, $f^n(z) \to p$ as $n \to \infty$ for every $z \in \mathbb{D}$. We say that $p$ is the {\em Denjoy-Wolff point} of $f$.

In the case where $f$ is an {\em inner function}, that is, for $\lambda$-almost every $\zeta\in\partial \mathbb{D}$ the radial limit \[ f^*(\zeta)=\lim_{r\to1} f(r\zeta)\] belongs to $\partial\mathbb{D}$ (where $\lambda$ denotes the normalized Lebesgue measure on $\partial\mathbb{D}$), one can consider the dynamical system defined in the unit circle given by the radial extension
$$
f^{*} \colon \partial\mathbb{D} \longrightarrow \partial\mathbb{D}.
$$
Assuming that the Denjoy-Wolff point $p$ lies in $\partial\mathbb{D}$, a natural question to ask is whether points in the unit circle also converge to the Denjoy-Wolff point under the iteration of $f^*$.
In the seminal work of Aaronson, Doreing and Mañé \cite{Aaronson78, mane_dynamics_1991}, this question is answered by means of a complete characterization in terms of infinite sums. More precisely, $\lambda$-almost every point on $\partial\mathbb{D}$ converges to the Denjoy-Wolff point $p\in\partial \mathbb{D}$ if and only if \[\sum_{n\geq 0} 1-|f^n(0|<\infty.\]

It is well-known that {\em hyperbolic} inner functions (i.e. for which the angular derivative $f'(p)\in (0,1)$, see Section \ref{sect-hyperbolic-inner}) always satisfy the condition above. Going one step further, one may ask at which
rate do these orbits approach the Denjoy–Wolff point.

If the Denjoy-Wolff point $p\in\partial \mathbb{D}$ is not a singularity (i.e. $f$ extends as a holomorphic map around $p$), then $p$ is an attracting fixed point for $f$. Using Koenigs' linearizing coordinates, we see that $f$ behaves like the map $z \mapsto f'(p) z$ near $p$, showing geometric convergence to the fixed point for points on $\partial \mathbb{D}$ in a neighborhood of $p$.

The case when the Denjoy–Wolff point is a singularity is much more complicated, due to the lack of normal forms. In this paper, we establish an explicit rate of convergence for hyperbolic inner functions, covering the case when the Denjoy–Wolff point is a singularity. Namely, if we denote $A(p;a,b) \coloneqq \{ z \in \mathbb{C}\colon a <|z-p| < b\}$, we prove the following. \\

In the case where $f$ is an {\em inner function}, that is, for $\lambda$-almost every $\zeta\in\partial \mathbb{D}$ the radial limit \[ f^*(\zeta)=\lim_{r\to1} f(r\zeta)\] belongs to $\partial\mathbb{D}$ (where $\lambda$ denotes the normalized Lebesgue measure on $\partial\mathbb{D}$), one can consider the dynamical system defined in the unit circle given by the radial extension
$$
f^{*} \colon \partial\mathbb{D} \longrightarrow \partial\mathbb{D}.
$$
Assuming that the Denjoy-Wolff point $p$ lies in $\partial\mathbb{D}$, a natural question to ask is whether points in the unit circle also converge to the Denjoy-Wolff point under the iteration of $f^*$.
In the seminal work of Aaronson, Doreing and Mañé \cite{Aaronson78, mane_dynamics_1991}, this question is answered by means of a complete characterization in terms of infinite sums. More precisely, $\lambda$-almost every point on $\partial\mathbb{D}$ converges to the Denjoy-Wolff point $p\in\partial \mathbb{D}$ if and only if \[\sum_{n\geq 0} 1-|f^n(0|<\infty.\]

If the Denjoy-Wolff point $p\in\partial \mathbb{D}$ is not a singularity (i.e. $f$ extends as a holomorphic map around $p$), then $p$ is an attracting fixed point for $f$. Using Koenigs' linearizing coordinates, we see that $f$ behaves like the map $z \mapsto f'(p) z$ near $p$, showing geometric convergence to the fixed point for points on $\partial \mathbb{D}$ in a neighborhood of $p$.

Without being precise, we say that a shrinking target is a collection of arcs of $\partial\mathbb{D}$ shrinking in length, and the problem is to determine whether orbits hit these arcs almost surely. Indeed, the arcs $A(p;\alpha^{(1+\varepsilon)n} , \alpha^{(1-\varepsilon)n})$ in Theorem \ref{A} form a shrinking target, and we want to conclude that it is hit almost surely. Via Möbius transformations, we convert our autonomous system into a non-autonomous one fixing $0$, to be able to apply the criteria in \cite{benini_shrinking_2024} to determine that the shrinking target is hit almost surely. We make use of some facts about the rate of convergence of the orbit of $0$ to the Denjoy-Wolff point for hyperbolic inner functions.
\

In this section, we prove Theorem \ref{A}, which states that if $f \colon \mathbb{D} \to \mathbb{D}$ is a hyperbolic inner function with Denjoy-Wolff point $p \in \partial{\mathbb{D}}$, $\alpha = f'(p) \in (0,1)$, then for all $0<\varepsilon<1$, we have $$(f^*)^n(\zeta) \in A(p;\alpha^{(1+\varepsilon)n} , \alpha^{(1-\varepsilon)n}) \cap \partial\mathbb{D}$$ for $n$ large enough and $\lambda$-almost every $\zeta \in \partial\mathbb{D}$.

\begin{lemma}\label{cotas+}
 Let $f \colon \mathbb{D} \to \mathbb{D}$ be a hyperbolic self map of $\mathbb{D}$ and let $p \in \partial\mathbb{D}$ be its Denjoy-Wolff point. Let $\alpha = f'(p)$. Then, for every $\delta > 0$, there exists a real constant $C \geq 1$ such that $$\frac{1}{C} (\alpha - \delta)^n \leq |f^n(0) - p| \leq C \alpha^n,$$ for $n$ large enough.

We now prove that given $0 < \varepsilon < 1$, $$(f^*)^n(\zeta) \in D(p, \alpha^{(1-\varepsilon)n}) \cap \partial\mathbb{D}$$ for $n$ large enough and $\lambda$-almost every $\zeta \in \partial\mathbb{D}$, where $\alpha = f'(p) \in (0,1)$.

Consider the arc $J_n(\varepsilon) = D(p,\alpha^{(1-\varepsilon)n}) \cap \partial\mathbb{D}$, $n \in \mathbb{N}$, and define
 \begin{align*}
 E(\varepsilon) &\coloneqq \{ \zeta \in \partial\mathbb{D}\colon f^n(\zeta) \in J_n(\varepsilon) \text{\ for all $n$ large enough}\}\\
 &= \{ \zeta \in \partial\mathbb{D} \colon (f^n(\zeta)) \text{\ fails to hit\ } (J_n(\varepsilon)^c)\},
 \end{align*}
 where $J_n(\varepsilon)^c = \partial\mathbb{D}\setminus J_n(\varepsilon)$. We will prove that $E(\varepsilon)$ has full measure.

We finally prove that given $0 < \varepsilon < 1$, $$(f^*)^n(\zeta) \not \in D(p, \alpha^{(1+\varepsilon)n}) \cap \partial\mathbb{D}$$ for $n$ large enough and $\lambda$-almost every $\zeta \in \partial\mathbb{D}$.

\begin{teo}\label{A}
        Let $f \colon \mathbb{D} \to \mathbb{D}$ be a hyperbolic inner function with Denjoy-Wolff point $p \in \partial{\mathbb{D}}$. Let $\alpha = f'(p) \in (0,1)$. Then for all $0<\varepsilon<1$, we have $$(f^*)^n(\zeta) \in A(p;\alpha^{(1+\varepsilon)n} , \alpha^{(1-\varepsilon)n}) \cap \partial\mathbb{D}$$ for $n$ large enough and $\lambda$-almost every $\zeta \in \partial\mathbb{D}$.
\end{teo}

\
The proof of the theorem is  based on the concept of \textit{shrinking targets} of $\partial\mathbb{D}$ and their hitting properties (see Section \ref{sec:shrinking}), developed in \cite{benini_shrinking_2024} to analyze the recurrent behavior of compositions of inner functions fixing 0.

Without being precise, we say that a shrinking target is a collection of  arcs of $\partial\mathbb{D}$ shrinking in length, and the problem is to determine whether orbits hit these arcs almost surely. Indeed, the arcs  $A(p;\alpha^{(1+\varepsilon)n} , \alpha^{(1-\varepsilon)n})$ in Theorem \ref{A} form a shrinking target, and we want to conclude that it is hit almost surely. Via Möbius transformations, we convert our autonomous system into a non-autonomous one fixing $0$, to be able to apply the criteria in \cite{benini_shrinking_2024} to determine that the shrinking target is hit almost surely. We make use of some facts about the rate of convergence of the orbit of $0$ to the Denjoy-Wolff point for hyperbolic inner functions.
\

{\bf Acknowledgements. } The authors gratefully acknowledge the Barcelona Introduction to Mathematical Research (BIMR) program at the Centre de Recerca Matemàtica (CRM) for providing an excellent research environment and support during the development of this work.

The proof of Theorem~\ref{A} is "based on the concept of \textit{shrinking targets} of $\partial\mathbb{D}$ and their hitting properties" (Section~\ref{sec:shrinking}), as developed in \cite{benini_shrinking_2024}. The arcs $A(p;\alpha^{(1+\varepsilon)n},\alpha^{(1-\varepsilon)n})$ appearing in Theorem~\ref{A} "form a shrinking target", and the goal is to "conclude that it is hit almost surely" by boundary orbits. "Via Möbius transformations", the authors "convert" the autonomous system into "a non-autonomous one fixing $0$" in order to "apply the criteria in \cite{benini_shrinking_2024} to determine that the shrinking target is hit almost surely". The argument also "make[s] use of some facts about the rate of convergence of the orbit of $0$ to the Denjoy-Wolff point for hyperbolic inner functions."

The proof of Theorem~\ref{A} is "based on the concept of \textit{shrinking targets} of $\partial\mathbb{D}$ and their hitting properties" (Section~\ref{sec:shrinking}), as developed in \cite{benini_shrinking_2024}. The arcs $A(p;\alpha^{(1+\varepsilon)n},\alpha^{(1-\varepsilon)n})$ appearing in the main theorem "form a shrinking target", and the goal is to "conclude that it is hit almost surely" by boundary orbits. "Via Möbius transformations", the authors "convert" the autonomous system into "a non-autonomous one fixing $0$" in order to "apply the criteria in \cite{benini_shrinking_2024} to determine that the shrinking target is hit almost surely". The argument also "make[s] use of some facts about the rate of convergence of the orbit of $0$ to the Denjoy-Wolff point for hyperbolic inner functions."

\label{A}
        Let $f \colon \mathbb{D} \to \mathbb{D}$ be a hyperbolic inner function with Denjoy-Wolff point $p \in \partial{\mathbb{D}}$. Let $\alpha = f'(p) \in (0,1)$. Then for all $0<\varepsilon<1$, we have $$(f^*)^n(\zeta) \in A(p;\alpha^{(1+\varepsilon)n} , \alpha^{(1-\varepsilon)n}) \cap \partial\mathbb{D}$$ for $n$ large enough and $\lambda$-almost every $\zeta \in \partial\mathbb{D}$.,

Question: Let \(\mathbb D\) be the unit disk, and let \(f:\mathbb D\to\mathbb D\) be a hyperbolic inner function whose Denjoy--Wolff point is \(p\in \partial\mathbb D\). Let \(\alpha=f'(p)\in(0,1)\), and let \(f^*\) denote the radial boundary map, defined for \(\lambda\)-almost every \(\zeta\in\partial\mathbb D\) by \(f^*(\zeta)=\lim_{r\to1^-}f(r\zeta)\), where \(\lambda\) is normalized Lebesgue measure on \(\partial\mathbb D\). For \(a,b>0\), define the annulus centered at \(p\) by \(A(p;a,b)=\{z\in\mathbb C: a<|z-p|<b\}\). Under these assumptions, which asymptotic statement holds for the boundary iterates \((f^*)^n\) as \(n\to\infty\)?

For every \(\varepsilon\in(0,1)\), for \(\lambda\)-almost every \(\zeta\in\partial\mathbb D\) there exists \(N=N(\zeta,\varepsilon)\) such that for all \(n\ge N\),
\[
(f^*)^n(\zeta)\in A\bigl(p;\alpha^{(1+\varepsilon)n},\alpha^{(1-\varepsilon)n}\bigr)\cap\partial\mathbb D,
\]
equivalently,
\[
\alpha^{(1+\varepsilon)n}<\bigl|(f^*)^n(\zeta)-p\bigr|<\alpha^{(1-\varepsilon)n}
\]
for all sufficiently large \(n\).

Choices:
B.
For every \(\varepsilon\in(0,1)\), for \(\lambda\)-almost every \(\zeta\in\partial\mathbb D\) there exists \(N=N(\zeta,\varepsilon)\) such that for all \(n\ge N\),
\[
(f^*)^n(\zeta)\in A\bigl(p;\alpha^{(1-\varepsilon)n},\alpha^{(1+\varepsilon)n}\bigr)\cap\partial\mathbb D,
\]
equivalently,
\[
\alpha^{(1-\varepsilon)n}<\bigl|(f^*)^n(\zeta)-p\bigr|<\alpha^{(1+\varepsilon)n}
\]
for all sufficiently large \(n\).
C.
For \(\lambda\)-almost every \(\zeta\in\partial\mathbb D\), one has
\[
(f^*)^n(\zeta)\to p
\quad\text{as }n\to\infty.
\]
Equivalently, for \(\lambda\)-almost every \(\zeta\in\partial\mathbb D\) and every neighborhood \(U\) of \(p\) in \(\partial\mathbb D\), there exists \(N=N(\zeta,U)\) such that \((f^*)^n(\zeta)\in U\) for all \(n\ge N\).
D.
There exists \(N=N(\varepsilon)\) such that for every \(\varepsilon\in(0,1)\), for \(\lambda\)-almost every \(\zeta\in\partial\mathbb D\) and all \(n\ge N\),
\[
(f^*)^n(\zeta)\in A\bigl(p;\alpha^{(1+\varepsilon)n},\alpha^{(1-\varepsilon)n}\bigr)\cap\partial\mathbb D,
\]
equivalently,
\[
\alpha^{(1+\varepsilon)n}<\bigl|(f^*)^n(\zeta)-p\bigr|<\alpha^{(1-\varepsilon)n}
\]
for all \(n\ge N\).
E.
For every \(\varepsilon\in(0,1)\), for \(\lambda\)-almost every \(\zeta\in\partial\mathbb D\) there exists \(N=N(\zeta,\varepsilon)\) such that for all \(n\ge N\),
\[
(f^*)^n(\zeta)\in D\bigl(p,\alpha^{(1+\varepsilon)n}\bigr)\cap\partial\mathbb D,
\]
equivalently,
\[
\bigl|(f^*)^n(\zeta)-p\bigr|<\alpha^{(1+\varepsilon)n}
\]
for all sufficiently large \(n\).


\label{bubbletree}(See also Theorem 2 of \cite{HS}) 	Let $ (M^4,h)$ be a closed four-dimensional Riemannian manifold and  $A_k\in\mathfrak {U}_G^{p_k}(M^4)$ a sequence of $p_k$-Yang-Mills connection with uniformly bounded energy\footnote{It means that $\mathcal{YM}_{p_k}(A_k)$ is uniformly bounded.} and such that $p_k \underset{k\to+\infty}{\rightarrow} 2^+$. Then we have, up to a sub-sequence,
		\begin{enumerate}
			\item There exists finitely many points $\{p_1,\dots, p_N\}$ and a Yang-Mills connection $A_\infty\in \mathfrak U_G(M^4)$ such that $A_k$ converges to $A_\infty$ in $\mathfrak C^\infty_{G,\mathrm{loc}}(M\setminus \{p_1,\dots, p_N\})$.
			\item For each $i\in \{1, \dots, N\}$, there exists $N_i\in \mathbf{N}$ sequences of points $(p_k^{i,j})_k$ converging to $p_i$,  $N_i$ sequences of scalars $(\lambda_k^{i,j})_k$ converging to zero, $N_i$ non-trivial Yang-Mills connections, called bubbles,  $A_\infty^{i,j} \in\mathfrak{U}_G(S^4)$ such that,
			$$(\phi_{k}^{i,j})^*(A_k) \rightarrow \widehat{A}_\infty^{i,j}=\pi_*A_\infty^{i,j}  \text{ into } \mathfrak C^\infty_{G,\mathrm{loc}}(\mathbf{R}^4 \setminus \{\text{ finitely many points}\}),$$
			where $\phi_{k}^{i,j} (x)= p_k^{i,j} +\lambda _k^{i,j} x$ in local coordinates and $\pi$ is the stereographic projection.
			\item Moreover there is no loss of energy, i.e
			$$ \lim_{k\rightarrow +\infty}\int_{M^4} \vert F_{A_k}\vert^2_h\, \mathrm{vol}_h  = \int_{M^4} \vert F_{A_\infty}\vert^2_h\, \mathrm{vol}_h +\sum_{i=1}^N \sum_{j=1}^{N_i} \int_{S^4} \vert F_{A_\infty^{i,j}}\vert^2\, \mathrm{vol}. $$
		\end{enumerate}

\begin{align}
			\mathrm{ind}_{\mathcal{YM}_{p_k}}(A_k) &\geq \mathrm{ind}_\mathcal{YM}(A_{\infty})+ \sum_{i=1}^N\sum_{j=1}^{N_i} \mathrm{ind}_\mathcal{YM}(A_{\infty}^{i,j}), \\
			\mathrm{ind}^0_{\mathcal{YM}_{p_k}}(A_k) &\leq \mathrm{ind}^0_\mathcal{YM}(A_{\infty})+ \sum_{i=1}^N \sum_{j=1}^{N_i} \mathrm{ind}^0_\mathcal{YM}(A_{\infty}^{i,j}),
		\end{align} where $\mathrm{ind}^0_{\mathcal{YM}_{p}} (A)=\ind_{\mathcal{YM}_p}(A) + \dim \left(\ker Q_{A,p} \cap \ker \diff_A^*\right)$ is the extended index of the connection and $Q_{A,p}$ is the quadratic form associated to the second variation.
	\end{theorem}

	This result is the Yang-Mills counterpart of the same result obtained in \cite{DaRiSchla}  for Sacks Uhlenbeck relaxation of the Dirichlet energy of maps into spheres and symmetric spaces in general see \cite{Schlagen}.
\medskip

	The paper is organised as follows. After that we introduce, in section 1 and 2,  notations and main notion for the definition of indices, we perform a fine analysis in neck region, section 3 to 5, here for the sake of clarity, we will state the results  in the Euclidean framework. They remain true on small enough geodesic balls of a Riemannian manifold, up to slight modification of constant which as no effect on the desired result. Finally, in section 6, we prove our main theorem in a general setting.

	\medskip

	\noindent{\bf Acknowledgements} This work has been initiated during a one semester visit of the first author at the ETH Z\"urich. He would like to thank the mathematics department at ETH for its hospitality.
	\setcounter{tocdepth}{2}
	\tableofcontents

	\newpage
	\section{Preliminaries on weak-connections}
	\label{preli}
	We work in the context of weak-connection introduce by Petrache-Rivière \cite{PR14}, see also the introduction and the appendix of the work of the first and second author \cite{GL24}. Here, we just have to extend the theory to $W^{2,2}$-bundles with $W^{1,p}$-connection with $p>2$, which is mostly obvious since every thing which is true for $W^{1,2}$-connection will be automatically true for $W^{1,p}$-connections.
	Let $(M^4, h)$ be an arbitrary Riemannian manifold, $G$ a compact Lie subgroup of $\mathrm{SU}(n)$ with Lie algebra $\mathfrak{g}$, $p\geq 2$, a weak $W^{1,p}$-connection is a $\mathfrak{g}$-valued 1-form $A$ (called the connection form) such that
	\begin{itemize}
		\item $A\in \mathrm{L}^{4,\infty}(M,T^*M\otimes \mathfrak{g})$\footnote{See \cite{Grafakos1} for a complete introduction to Lorentz spaces.},
		\item $F_A= \diff A + A \wedge A$ (defined as a distribution) is in $\mathrm{L}^p$,
		\item locally, there exists a $\mathrm{W}^{1,(4,\infty)}$-gauge $g$ satisfying $A^g\in \mathrm{W}^{1,p}$, where $ A^g := g^{-1} A g+g^{-1}\diff g$ is the expression of $A$ after the gauge change $g$,
	\end{itemize}
	We denote by $\mathfrak A^p_G(M^4)$ the space of such connections. The main advantage of this formulation, is that the connection form is globally defined, which includes the classical case, since by \cite[theorem A]{PR14}, every $\mathrm{W}^{2,2}$-bundle is a trivial $\mathrm{W}^{1,(4,\infty)}$-bundle, see also the appendix of \cite{GL24}. Moreover, we can also define an associated notion of convergence, i.e. smooth convergence in local good gauge, we denote it by $\mathfrak{C}_{G,loc}^\infty$- convergence , see Proposition-definition A.8 of \cite{GL24} for a precise definition.

	\section{Second variation : computation and finiteness of the Morse index}
	We start by reminding the expression of the first variation of the $p$-Yang-Mills functional in our framework.

	\begin{proposition}[Section 2 of \cite{HS}] For every connection $A\in\mathfrak U_G^p(M)$,  for all $a\in \mathrm{W}^{1,p}(M, T^*M\otimes \mathfrak{g})$,
		\begin{equation} \frac{\diff}{\diff t}_{|t=0}  \mathcal{YM}_p(A + ta) = p\int_{M}(1+|F_A|_h^2)^{\frac{p}{2}-1} \langle F_{A}, \diff_{A} a
			\rangle_h \mathrm{vol}_h. \label{variationpremiere}
		\end{equation}
		Hence $A\in\mathfrak U_G^p(M)$ is a $p$-Yang-Mills connections, if it satisfies in the distributional sense the following equation
		\begin{equation} \label{equationdepYMdiv}
			\diff_A^* \left((1+|F_A|_h^2)^{p/2-1}F_A\right) =0 .
		\end{equation}
	\end{proposition}

	\begin{remark}
		Recall that  the Bianchi identity is satisfies by any connection, i.e. \begin{equation}
			\label{equationdeBianchi} \diff_A F_A =0.
		\end{equation}
	\end{remark}

	It turns out, as already proved in \cite{HS}, that solution of the $p$-Yang-Mills equation is smooth in Coulomb gauge and satisfies so-called $\varepsilon$-regularity estimates under a small energy assumption. But for sake of completeness, we give new proofs of those results  in the appendix. In particular, knowing solutions are regular, the $p$-Yang-Mills equation \eqref{equationdepYMdiv} rewrites as

	\begin{equation} \label{equationdepYM}
		\diff_A^* F_A = \frac{p-2}{2} \star \frac{\diff |F_A|_h^2\wedge \star F_A}{1+|F_A|_h^2}.
	\end{equation}
	Let us now consider the second variation.
	\begin{proposition}
		If $(A_t)_{t\in(-\varepsilon, \varepsilon)}\in \mathfrak A_G^p(M)$ is a one parameter  family of weak connections where $A:=A_0$ is a $p$-Yang-Mills connection and $a:=\displaystyle {\frac{\diff}{\diff t}}_{|t=0} A_t\in   \mathrm{W}^{1,p}(M, T^*M\otimes \mathfrak{g})$ then \begin{equation}
			\frac{\diff^2}{\diff t^2}_{|t=0}\mathcal{YM}_p(A_t) =p  \int_{M}(1+|F_A|_h^2)^{\frac{p}{2}-1} \left( |\diff_{A} a|_h^2 + \langle F_{A}, [a, a] \rangle_h + (p-2) \frac{\langle F_A, \diff_A a \rangle_h^2}{1+|F_A|_h^2} \right)\mathrm{vol}_h.  \label{variationseconde}
		\end{equation}
	\end{proposition}

	\begin{proof}
		From \[ \diff_{A + a} b = \diff_{A} b + [a, b], \] \[ F_{A + a} - F_{A} = \diff_{A} a + a \wedge a, \] and the embedding $\mathrm{W}^{1, 2} \hookrightarrow \mathrm{L}^{4,2}$, we deduce, by substituting in \eqref{variationpremiere} :
		\begin{align*}
			\mathrm{D} \mathcal{YM}_{A + a} (b) & =p\langle  (1+|F_A+d_Aa+a\wedge a|_h^2)^{\frac{p}{2}-1}( F_{A} + \diff_{A}
			a + a \wedge a), \diff_{A} b + [a, b] \rangle_h\\
			& = \mathrm{D} \mathcal{YM}_{A} (b) +p(p-2)\langle \left(1+\vert F_A\vert^2_h\right)^{\frac{p}{2}-2}\langle F_A, d_Aa\rangle_h F_A, d_Ab\rangle_h  \\
			&+p \langle  \left(1+\vert F_A\vert^2_h\right)^{\frac{p}{2}-1}F_{A}, [a, b] \rangle_h +p
			\langle  \left(1+\vert F_A\vert^2_h\right)^{\frac{p}{2}-1}\diff_{A} a, \diff_{A} b \rangle_h \\
			&+ \gdo{ \| a
				\|^2_{\mathrm{W}^{1, 2}} \| b\|_{\mathrm{W}^{1, 2}}}
		\end{align*}
		which implies the result.

	\end{proof}

	\begin{definition} $ Q_{p,A}$ is the quadratic form defined by the right hand side of \eqref{variationseconde}.
	\end{definition}

	A consequence of the gauge invariance is that $Q_{p,A}$ has a huge kernel, as shown in the following result :
	\begin{proposition}
		\[ \Imag{\diff_{A}} \subset \ker Q_{p,A}\]
	\end{proposition}

	\begin{proof}
		Identical to proposition 2.3 of \cite{GL24}.
	\end{proof}

	\noindent Recall the index of a quadratic form $q$ is the non-negative integer $\ind q$ defined as \[ \ind q = \sup \left\{ \dim W \mid q_{|W} < 0\right\}.\]

	\begin{proposition} Let $\mathfrak{Q}_{p,A}(a) = Q_{p,A}(a) +\int_{M}(1+|F_A|_h^2)^{\frac{p}{2}-1}|\diff^*_A  a|_h^2 \,\mathrm{vol}_h$. The index and kernel of $Q_{p,A}$ can be expressed from those of $\mathfrak{Q}_{p,A}$ as follows : \begin{align*}
			\ind Q_{p,A} &= \ind {\mathfrak{Q}_{p,A}}\\
			\ker Q_{p,A} &=  \ker {\mathfrak{Q}_{p,A}} \oplus  \Imag \diff_{A}
		\end{align*}
	\end{proposition}

	\begin{proof}
		Identical to proposition 2.4 of \cite{GL24}.
	\end{proof}

	\begin{propdef} \label{finitudeindice} $\ind^0\mathfrak{Q}_{p,A}:= \ind {\mathfrak{Q}_{p,A}} + \dim  \ker {\mathfrak{Q}_{p,A}}$ is finite and \[\ind^0 \mathfrak{Q}_{p,A} = \ind {Q_{p,A}} + \dim \left( \ker {Q_{p,A}}\cap {\ker \diff_{A}^*}\right).\] We call this quantity the extended index of the quadratic form $Q_{p,A}$.
	\end{propdef}

	\begin{proof}
		The equality comes from the previous proposition. We shall then prove the finiteness of $\ind^0\mathfrak{Q}_{p,A}$. Let $\mathcal{E}(\lambda)$ be the (possibly trivial) eigenspace associated with the eigenvalue $\lambda$ of the non-negative elliptic operator $\diff_{A}^* \diff_{A} + \diff_{A} \diff_{A}^*$ . For all $\Lambda\in\R$, we denote \[ \mathcal{E}^{\Lambda} = \bigoplus_{\lambda \leq \Lambda} \mathcal{E}(\lambda).
		\]
		For all $\Lambda \in \R$, $\dim \mathcal{E}^\Lambda < +\infty$. We choose $\Lambda > C \Vert F_{A} \Vert_{\mathrm{L}^\infty(M)}$ where $C$ is such that \[\left| \int_M \langle F_{A}, [a, a] \rangle_h \mathrm{vol}_h \right| \leq C  \int_M |F_{A}|_h\, |a|^2_h \mathrm{vol}_h.\]

		\noindent If $a\in  \left(\mathcal{E}^\Lambda\right)^\perp$, then
		\begin{align*}
			\mathfrak{Q}_{p,A}(a) &\geq  \int_M (1+|F_A|_h^2)^{\frac{p}{2}-1}\left( \vert d^*_Aa\vert_h^2+\vert d_Aa\vert^2_h +\langle F_A, [a,a]\rangle_h\right) \mathrm{vol}_h\\
			&\geq  \int_M \vert d^*_Aa\vert^2_h+\vert d_Aa\vert^2_h +\langle F_A, [a,a]\rangle_h \mathrm{vol}_h\\
			&\geq \Lambda \int_M  |a|^2_h \mathrm{vol}_h + \int_M \langle F_{A}, a\wedge a \rangle_h \mathrm{vol}_h\\
			&\geq \int_M \left(\Lambda - C|F_{A}|_h\right) |a|^2_h \mathrm{vol}_h.
		\end{align*} We conclude that if $W$ a subspace on which $\mathfrak{Q}_{p, A}$ is negative-definite, $W\cap \left(\mathcal{E}^\Lambda\right)^\perp = \{0\}$. Consequently, $\dim W \leq \mathrm{codim}\left(\mathcal{E}^\Lambda\right)^\perp = \dim \mathcal{E}^\Lambda <+\infty$ so $\ind \mathfrak{Q}_{p,A} < +\infty$. A similar argument can be applied to prove that $\ker \mathfrak{Q}_{p,A}$ is also finite dimensional.
	\end{proof}

	\section{$\varepsilon$-regularity, $\mathrm{L}^2$ and $\mathrm{L}^{2,1}$ quantization}
	We state here the $\varepsilon$-regularity pointwise estimate, the proof of which is to be found in the appendix. Then we give the $L^2$ quantization, also proved in \cite{HS}, but we do it in a different setting which permits us to improve it to the stronger $L^{2,1}$ quantization.
	\subsection{$\varepsilon$-regularity}
	\begin{theorem}[\cref{epsregcurvaturepointwiseproof}] \label{epsregbis}
		Let $G$ be a compact Lie group. There exist $\varepsilon_{G}>0$ and constants $C_{G},p_G>0$ such that for all $p\in[2,2+p_G]$, for all $R\in]0,1]$ and for all $1$-form $A$ in  $\mathrm{W}^{1,2}\left(\mathrm{B}_R,\Lambda^1 \R^4\otimes  \mathfrak{g}\right)$ satisfying the small energy condition:
		$$\int_{ \mathrm{B}_R}\left|F_A\right|^2 \diff x <\varepsilon_{G},
		$$ and the $p$-Yang-Mills equation \eqref{equationdepYM}, then the following estimate holds: \begin{equation}
			R^4\left\|F\right\|_{\mathrm{L}^{\infty}\left(\mathrm{B}_{R/2}\right)}^2 \leq C_{G} \int_{\mathrm{B}_R}\left|F_A\right|^2 \diff x.
		\end{equation}
	\end{theorem}

	\subsection{$\mathrm{L}^{2,\infty}$ bound on the curvature}
	Once one has $\varepsilon$-regularity, it is classical to derive the weak-$L^2$ quantization.

	\begin{lemma} There exists $ C_G> 0$ and $p_G>0$ such that for all $p\in[2,2+p_G]$ and for all $p$-Yang-Mills connection $A$ in $\mathrm{B}_R\backslash \overline{\mathrm{B}_r} $ with $0<4r<R<1$, if $A$ satisfies
		\[ \sup_{r \leq \rho \leq R /2} \| F_A \|_{\mathrm{L}^2(\mathrm{B}_{2\rho} \backslash \mathrm{B}_\rho)} \leq \varepsilon_{G},\] then \[ \|F_A\|_{\mathrm{L}^{2,\infty}(\mathrm{B}_{R}\backslash \mathrm{B}_{r})} \leq C_G \sup_{r \leq \rho \leq R/2} \| F_A \|_{\mathrm{L}^2(\mathrm{B}_{2\rho}\backslash \mathrm{B}_\rho)}.\]
	\end{lemma}
	It is well known that the dyadic energy goes to zero in the neck, in fact it is a direct consequence of the bubble-neck  decomposition, see section 3.1 of \cite{HS} or proof of theorem VII.3 of \cite{rivière2015variations} for the classical case. Then we obtain the weak-$L^2$ quantization as a corollary.\\

	\begin{proof} Denote $\varepsilon =\displaystyle \sup_{r \leq \rho \leq R/2} \| F_A \|_{\mathrm{L}^2(\mathrm{B}_{2\rho}\backslash \mathrm{B}_\rho)} $ and apply the $\varepsilon$-regularity result \eqref{epsregbis} on balls $\mathrm{B}(x,|x|/3)$ for $x\in \mathrm{B}_{R/2}\backslash \mathrm{B}_{2r}$: there exists $C>0$ such that for all $x\in \mathrm{B}_{R/2}\backslash \mathrm{B}_{2r}$, \[|x|^2|F_A|(x) \leq C \| F_A
		\|_{\mathrm{L}^2(\mathrm{B}(x,|x|/3)}\leq C \| F_A
		\|_{\mathrm{L}^2(\mathrm{B}_{4|x|/3}\backslash\mathrm{B}_{2|x|/3})}\leq C\varepsilon. \] Therefore \begin{align*}
			\|F_A\|_{\mathrm{L}^{2,\infty}(\mathrm{B}_{R/2}\backslash \mathrm{B}_{2r})} &\leq  \|C\varepsilon |x|^{-2}\|_{\mathrm{L}^{2,\infty}(\mathrm{B}_{R/2}\backslash \mathrm{B}_{2r})} \\
			&\leq C\varepsilon\||x|^{-1}\|_{\mathrm{L}^{4,\infty}(\mathrm{B}_{R/2}\backslash \mathrm{B}_{2r})} ^2 \\
			& \leq C\varepsilon\||x|^{-1}\|_{\mathrm{L}^{4,\infty}(\R^4)}^2\\
			&  \leq C|\mathrm{B}_1|^{1/2}\varepsilon.
		\end{align*}
	\end{proof}

	\subsection{$\mathrm{L}^{2,1}$ bound on the curvature}

	To have $\mathrm{L}^{2,1}$ curvature bound, we need to use a refined version of Uhlenbeck \cite[Corollary 2.2]{UhlenbeckKarenK1982CwLb} with an hypothesis of small curvature in $\mathrm{L}^{2,\infty}$ in the neck region, which is due to Rivière \cite{rivière2015variations, riviere2002interpolation}.
	\begin{theorem} There exist $\varepsilon_G, C_G> 0$ with the following property: for all $R,
		r $ with $0<2r < R <1$, if $A \in \mathrm{W}^{1, 2} (\mathrm{B}_R \backslash
		\overline{\mathrm{B}_r}, \Lambda^1 \R^4 \otimes \mathfrak{g})$
		satisfies \[\| F_A \|_{\mathrm{L}^{2, \infty}
			(\mathrm{B}_R \backslash \overline{\mathrm{B}_r})} + \| F_A \|_{\mathrm{L}^2 (\mathrm{B}_{2 r} \backslash
			\overline{\mathrm{B}_{r}})} \leq \varepsilon_G\] then there exists $g
		\in \mathrm{W}^{2, 2} (\mathrm{B}_R \backslash \overline{\mathrm{B}_r}, G)$
		such that $\diff^* A^g = 0$ and
		\[ \int_{\mathrm{B}_R\backslash
			\overline{\mathrm{B}_r}}  \left( | \nabla A^g |^2 + \frac{ | A^g |^2}{|x|^2}  \right) \diff x
		\leq C_G \int_{\mathrm{B}_R \backslash \overline{\mathrm{B}_r}} | F_A
		|^2 \diff x. \]
		\label{recollementjauge}
	\end{theorem}
	\begin{proof}
		Combine \cite[Lemma 4.4]{uhlenbeck_chern_1985} to extend the connection up to gauge in the whole ball $\mathrm{B}_R$ with curvature $\mathrm{L}^2$-small in the ball $\mathrm{B}_r$. This implies that the curvature is small in $\mathrm{L}^{2,\infty}$ in the whole ball $\mathrm{B}_R$ and \cite[Theorem IV.4]{rivière2015variations} applies.
	\end{proof}

	\begin{remark}
		In the limiting case $r=R/2$, the hypothesis reduces to \[\| F_A \|_{\mathrm{L}^2 (\mathrm{B}_{R} \backslash
			\overline{\mathrm{B}_{R/2}})} \leq \varepsilon_0\] and the conclusion is a form of Uhlenbeck theorem which applies on the dyadic annulus $\mathrm{B}_R \backslash \overline{\mathrm{B}_{R/2}}$.
	\end{remark}

	\

	\noindent We are now able to prove the crucial part of this section:

	\begin{proposition} There exists $\varepsilon_G,p_G, C_G> 0$ such that, for all $p\in[2,2+p_G]$ and all $p$-Yang-Mills connection $A\in W^{1,2} (\mathrm{B}_R\backslash \overline{\mathrm{B}_r} ,  \Lambda^1 \R^4 \otimes \mathfrak{g})$, for all $R,r$ with $0<r<4r<R<1$, if $A$ satisfies
		\[ \sup_{r \leq \rho \leq R /2} \| F_A \|_{\mathrm{L}^2(\mathrm{B}_{2\rho} \backslash \mathrm{B}_\rho)} \leq \varepsilon, \] then
		\[ \|  F_{A} \|_{\mathrm{L}^{2,1} (\mathrm{B}_{R/2} \backslash \mathrm{B}_{  2r})} \leq C \|  F_{A} \|_{\mathrm{L}^2 (\mathrm{B}_{R} \backslash \mathrm{B}_{ r})}.\]
		\label{inegalitequantificationYM}
	\end{proposition}

	\begin{proof}		The proof is inspired by \cite[lemma III.2]{riviere2002interpolation} on $\mathrm{L}^2$-quantization for Yang-Mills connections. Let's use the same notations as in \eqref{recollementjauge}, up to a gauge transformation, we can assume that the conclusion of this theorem is satisfied by $A$.

		\noindent Let $\chi \in \mathcal{C}^{\infty}_c (\R^4, [0 ; 1])$ such that $\supp \chi \subset \text{B}_{R} \backslash \text{B}_{r}$, $\chi \equiv 1$ in $\text{B}_{R/2} \backslash
		\text{B}_{2r}$ and $\| | x | \nabla \chi
		\|_{\text{L}^{\infty}} \leq C$, with $C$ independent of $R,r$.

		\noindent From Bianchi identity \eqref{equationdeBianchi}, we have $\diff (\chi F_A) = - \chi [A, F_A] + \diff
		\chi \wedge F_A$. Using Hölder inequality (Lorentz spaces version, see \cite[theorem 4.5]{Hunt1966}) and the improved Sobolev-embedding $\dot{\mathrm{W}}^{1,2}(\R^4) \hookrightarrow \mathrm{L}^{4,2}(\R^4)$ (\cite[theorem 8.1]{peetrelorentzsobolev}):

		\begin{equation}
			\label{majorationL4/3}
			\begin{split}
				\| \chi [A, F_A] \|_{\text{L}^{4 / 3, 1} (\R^4)} & =
				\| [\chi A, F_A] \|_{\text{L}^{4 / 3, 1}
					(\R^4)}\\
				& \leq C\| \chi A \|_{\text{L}^{4, 2} (\R^4)}  \|
				F_A \|_{\mathrm{L}^2 (\mathrm{B}_R \backslash \mathrm{B}_r)}\\
				& \leq C  \| \nabla(\chi A) \|_{\text{L}^{2} (\R^4)} \|
				F_A \|_{\mathrm{L}^2 (\mathrm{B}_R \backslash \mathrm{B}_r)}.
			\end{split}
		\end{equation}
		We can use the fact we choose a good gauge to prove the following estimate \begin{equation}
			\label{gradientcutoff}
			\begin{split}
				\| \nabla(\chi A) \|_{\text{L}^{2} (\R^4)}^2 &\leq C \int_{\R^4} |\chi \nabla A|^2 + |\nabla \chi|^2| A |^2 \diff x \\
				&\leq C \int_{\text{B}_{R} \backslash \text{B}_{r}} |\nabla A|^2 + 				\frac{1}{|x|^2} | A |^2 \diff x \\
				&\leq C \int_{\mathrm{B}_R \backslash
					\mathrm{B}_{r}} | F_A |^2 \diff x.
			\end{split}
		\end{equation}

		\noindent Since $\diff \chi$ has support in the annuli
		$(\mathrm{B}_{R} \backslash \mathrm{B}_{R/2})\cup (\mathrm{B}_{2r} \backslash \mathrm{B}_{r})$, using Hölder inequality we obtain
		\begin{align}
			\| \diff \chi \wedge F_A \|_{\text{L}^{4 / 3, 1}
				(\R^4)}& = \| \diff \chi \wedge F_A \|_{\text{L}^{4 / 3, 1}
				(\mathrm{B}_{R} \backslash \mathrm{B}_{r})}\\ &\leq C {\| F_A \|_{\mathrm{L}^2 (\mathrm{B}_R \backslash \mathrm{B}_{r})}}  \| \diff \chi \|_{\text{L}^{4, 2}\left(\R^4 \right)} \notag \\
			& \leq C {\| F_A \|_{\mathrm{L}^2 (\mathrm{B}_R \backslash \mathrm{B}_{r})}} \left( \| |x|^{-1}\|_{\text{L}^{4, 2}(\mathrm{B}_{R} \backslash \mathrm{B}_{R/2})} +  \| |x|^{-1}\|_{\text{L}^{4, 2}(\mathrm{B}_{2r} \backslash \mathrm{B}_{r})}\right)\notag \\
			&\leq C {\| F_A \|_{\mathrm{L}^2 (\mathrm{B}_R \backslash \mathrm{B}_{r})}} \| |x|^{-1}\|_{\text{L}^{4, 2}\left(\mathrm{B}_{1} \backslash \mathrm{B}_{1/2}\right)} \label{scalintL42}\\ &\leq C {\| F_A \|_{\mathrm{L}^2 (\mathrm{B}_R \backslash \mathrm{B}_{r})}}. \label{cutoffcourbureestimate}
		\end{align}

\begin{theorem}
		\label{bubbletree}(See also Theorem 2 of \cite{HS}) 	Let $ (M^4,h)$ be a closed four-dimensional Riemannian manifold and  $A_k\in\mathfrak {U}_G^{p_k}(M^4)$ a sequence of $p_k$-Yang-Mills connection with uniformly bounded energy\footnote{It means that $\mathcal{YM}_{p_k}(A_k)$ is uniformly bounded.} and such that $p_k \underset{k\to+\infty}{\rightarrow} 2^+$. Then we have, up to a sub-sequence,
		\begin{enumerate}
			\item There exists finitely many points $\{p_1,\dots, p_N\}$ and a Yang-Mills connection $A_\infty\in \mathfrak U_G(M^4)$ such that $A_k$ converges to $A_\infty$ in $\mathfrak C^\infty_{G,\mathrm{loc}}(M\setminus \{p_1,\dots, p_N\})$.
			\item For each $i\in \{1, \dots, N\}$, there exists $N_i\in \N$ sequences of points $(p_k^{i,j})_k$ converging to $p_i$,  $N_i$ sequences of scalars $(\lambda_k^{i,j})_k$ converging to zero, $N_i$ non-trivial Yang-Mills connections, called bubbles,  $A_\infty^{i,j} \in\mathfrak{U}_G(S^4)$ such that,
			$$(\phi_{k}^{i,j})^*(A_k) \rightarrow \widehat{A}_\infty^{i,j}=\pi_*A_\infty^{i,j}  \text{ into } \mathfrak C^\infty_{G,\mathrm{loc}}(\R^4 \setminus \{\text{ finitely many points}\}),$$
			where $\phi_{k}^{i,j} (x)= p_k^{i,j} +\lambda _k^{i,j} x$ in local coordinates and $\pi$ is the stereographic projection.
			\item Moreover there is no loss of energy, i.e
			$$ \lim_{k\rightarrow +\infty}\int_{M^4} \vert F_{A_k}\vert^2_h\, \mathrm{vol}_h  = \int_{M^4} \vert F_{A_\infty}\vert^2_h\, \mathrm{vol}_h +\sum_{i=1}^N \sum_{j=1}^{N_i} \int_{S^4} \vert F_{A_\infty^{i,j}}\vert^2\, \mathrm{vol}. $$
		\end{enumerate}
	\end{theorem}

The calculus of variations of Yang-Mills energy for connections of principal $G-$bundles over  $4-$dimensional Riemannian manifolds  where $G$ is a compact Lie Group has received a close attention of the geometric analysis community
	in the early 80's in relation with important differential topology/geometry questions.  While the main focus was concentrated on special absolute minimizing solutions called Self-Dual/Anti-self-dual solutions or {\it instantons}, the construction of non zero Morse index Yang-Mills fields was also a subject of great interest in particular in relation with the so called Atiyah-Jones conjecture. The conjecture is about to know whether the homotopy type of the Moduli space ${\mathcal M}_n$ of instantons of topological charge $n$ modulo the gauge group action, for a $SU(2)-$bundle  with second Chern class $n$ over a four dimensional Riemannian manifold $M^4$, is ``comparable'' to the homotopy type of the space of general $SU(2)-$connections ${\mathcal A}_n$   modulo gauge group action with second Chern class $n$. More specifically, a question is about to know whether,  the low degree homotopy groups of the two spaces coincide or not.
	This question is intimately related to the existence  of non zero Morse index Yang-Mills Fields. This general consideration can be sketched as follows : assuming a non zero $k-$homotopy class within the Moduli space of Instantons $\alpha\in \pi_k({\mathcal M}_n)$ would ``trivialize'' (would be mapped to zero)
	for the canonical inclusion map $\iota$ into the general space of instantons $\iota_\ast\alpha=0$ in $\pi_k({\mathcal A}_n)$, one would easily produce a {\it min-max operation} whose underlying admissible space would be made of any continuous map from the $k+1$ ball $B^{k+1}$ into ${\mathcal A}_n$ whose boundary $\partial B^{k+1}$ is mapped into the space of instantons (which happen to realise the minimal Yang-Mills energy $8\pi^2 |n|$ of the given $SU(2)$ bundle) and the restriction to the boundary would realise the class $\alpha\ne 0$ in $\pi_k({\mathcal M}_n)$. Because the minimal energy $8\pi^2\,|n|$  is characterising  the membership to the moduli space ${\mathcal M}_n$, the ``width'' of such a min-max operation would be non trivial (strictly larger than $8\pi^2\, |n|$) otherwise one would contradicts the non triviality of $\alpha$ in $\pi_k({\mathcal M}_n)$. It is then ``natural'' to expect  this width to be realised by a non-zero Morse index $k+1$ Yang-Mills Fields.   In order to produce  a rigorous proof of this statement major analytical challenges have to be solved. The main reason is that, due in particular to its conformal invariance, the Yang-Mills energy cannot fulfil the necessary requirements for implementing Palais-Smale deformation theory in infinite dimensional space as such. Numerous publications have been devoted to this difficulty that Yang-Mills energy is sharing with several famous Lagrangians such as the Dirichlet energy of maps from a surface into  closed Riemannian manifolds whose critical points are called ``harmonic maps''.  The main motivation behind the present work is to bring a new piece of understanding of how the absence of Palais-Smale property for Yang-Mills energy can be ``overcome'' while implementing general min-max operations in particular. Before describing our main result we conclude this preamble by mentioning that there has been numerous contributions in the direction of Atiyah Jones conjecture and that we will omit to report on since this is not the object of the present paper.

	In a pioneered work in concentration compactness theory (see \cite{SaU}), J.Sacks and K.Uhlenbeck, were looking for absolute minimizers of the Dirichlet energy of maps from the two sphere $S^2$ into an arbitrary closed sub-manifold $N^n$ of an euclidean space $R^N$realising a non zero free homotopy class of $\pi_2({\mathbf{N}}^n)$. Because of the absence of Sobolev embeddings of $W^{1,2}(S^2,N^n)$ into $C^0$ the existence of a minimizer was a-priori not guaranteed. To remedy to this difficulty, the two authors introduced an ``enhanced'' version of the Dirichlet energy known nowadays as ``Sacks-Uhlenbeck approximation'' :
	\[
	E_p(u):=\int_{S^2}(1+|d u|^2_{S^2})^{p/2}\ dvol_{S^2}\ .
	\]
	for $p>2$. Maps of finite $E_p$ energy are exactly elements from the Sobolev space $W^{1,p}(S^2,N^n)$ which this time, for $p>2$, embeds compactly into $C^{0}(S^2,N^n)$. Because of this last fact the existence of a minimizers in a given free homotopy class is straightforward. For similar reasons the $p-$energy is sub-critical and satisfies the Palais Smale condition and general min-max operation can be implemented in the Banach Manifold ${\frak M}:=W^{1,p}(S^2,N^n)$ equipped with the Finsler structure given simply by the $W^{1,p}$ norm restricted to the tangent space $T_u{\frak M}$ to ${\frak M}$ at every element $u\in W^{1,p}(S^2,N^n)$. Then, once the existence of  minimizers or min-max critical points to the enhanced energies have been established, for any $p>2$,  the general strategy  introduced first in \cite{SaU} consists in ``following'' the obtained critical points to $E_{p_k}$ as $p_k\rightarrow 2^{+}$ and hopefully converge to critical points to the Dirichlet energy itself. In order to pass to the limit a  delicate analysis has to be performed. The central result in this analysis is the so called $\epsilon-$regularity theorem, saying roughly that below some universal positive threshold of energy (depending only on $N^n$ and not on $p$), every norm of every critical point of the $p-$energy is controlled. This permits, modulo extraction of a subsequence, to pass to the limit away from finitely many points in the domain while at these points so called ``bubbles'' (i.e. concentrated critical points) can be formed at the limit. These bubbles are connected to each other by regions which are called ``neck regions''.

	In a similar way, in order to overcome the absence of Palais-Smale property for the Yang-Mills energy, one introduces the $p-$Yang-Mills energy of any $G-$connection $A$ (where $G$ is a comp[act Lie group) over a 4 dimensional Riemannian manifold
	$ (M^4,h)$ :
	\[
	{\mathcal Y}{\mathcal M}_p(A):=\int_{M^4}(1+|F_A|^2)^{p/2}\, \mbox{vol}_h\ .
	\]
	This energy is well defined in the space of Sobolev $W^{1,p}$ $G-$connections over $M^4$ (see \cite{FU}), in the present paper we will work in a slightly more general context of weak $W^{1,p}$ $G-$connections denoted $\mathfrak {U}_G^{p}(M^4)$, see section \ref{preli}. The $\epsilon-$regularity for $p-$Yang-Mills critical points over $4-$dimensional manifolds has been first established in \cite{HS}. For the convenience of the reader we present in the appendix A.3 of the present work an alternative proof.
	In the present work we study the passage to the limit for $p_k-$Yang-Mills critical points of uniformly bounded energy as $p_k\rightarrow 2$. Thanks to the epsilon regularity we deduce a strong convergence/bubble tree convergence property our first main result is about the absence of loss of energy in the neck regions connecting the bubbles between themselves. Precisely we have

In a pioneered work in concentration compactness theory (see \cite{SaU}), J.Sacks and K.Uhlenbeck, were looking for absolute minimizers of the Dirichlet energy of maps from the two sphere $S^2$ into an arbitrary closed sub-manifold $N^n$ of an euclidean space $R^N$realising a non zero free homotopy class of $\pi_2({\mathbf{N}}^n)$. Because of the absence of Sobolev embeddings of $W^{1,2}(S^2,N^n)$ into $C^0$ the existence of a minimizer was a-priori not guaranteed. To remedy to this difficulty, the two authors introduced an ``enhanced'' version of the Dirichlet energy known nowadays as ``Sacks-Uhlenbeck approximation'' :
 \[
 E_p(u):=\int_{S^2}(1+|d u|^2_{S^2})^{p/2}\ dvol_{S^2}\ .
 \]
 for $p>2$. Maps of finite $E_p$ energy are exactly elements from the Sobolev space $W^{1,p}(S^2,N^n)$ which this time, for $p>2$, embeds compactly into $C^{0}(S^2,N^n)$. Because of this last fact the existence of a minimizers in a given free homotopy class is straightforward. For similar reasons the $p-$energy is sub-critical and satisfies the Palais Smale condition and general min-max operation can be implemented in the Banach Manifold ${\frak M}:=W^{1,p}(S^2,N^n)$ equipped with the Finsler structure given simply by the $W^{1,p}$ norm restricted to the tangent space $T_u{\frak M}$ to ${\frak M}$ at every element $u\in W^{1,p}(S^2,N^n)$. Then, once the existence of minimizers or min-max critical points to the enhanced energies have been established, for any $p>2$, the general strategy introduced first in \cite{SaU} consists in ``following'' the obtained critical points to $E_{p_k}$ as $p_k\rightarrow 2^{+}$ and hopefully converge to critical points to the Dirichlet energy itself. In order to pass to the limit a delicate analysis has to be performed. The central result in this analysis is the so called $\epsilon-$regularity theorem, saying roughly that below some universal positive threshold of energy (depending only on $N^n$ and not on $p$), every norm of every critical point of the $p-$energy is controlled. This permits, modulo extraction of a subsequence, to pass to the limit away from finitely many points in the domain while at these points so called ``bubbles'' (i.e. concentrated critical points) can be formed at the limit. These bubbles are connected to each other by regions which are called ``neck regions''.

In a similar way, in order to overcome the absence of Palais-Smale property for the Yang-Mills energy, one introduces the $p-$Yang-Mills energy of any $G-$connection $A$ (where $G$ is a comp[act Lie group) over a 4 dimensional Riemannian manifold
 $ (M^4,h)$ :
 \[
 {\mathcal Y}{\mathcal M}_p(A):=\int_{M^4}(1+|F_A|^2)^{p/2}\, \mbox{vol}_h\ .
 \]
 This energy is well defined in the space of Sobolev $W^{1,p}$ $G-$connections over $M^4$ (see \cite{FU}), in the present paper we will work in a slightly more general context of weak $W^{1,p}$ $G-$connections denoted $\mathfrak {U}_G^{p}(M^4)$, see section \ref{preli}. The $\epsilon-$regularity for $p-$Yang-Mills critical points over $4-$dimensional manifolds has been first established in \cite{HS}. For the convenience of the reader we present in the appendix A.3 of the present work an alternative proof.
 In the present work we study the passage to the limit for $p_k-$Yang-Mills critical points of uniformly bounded energy as $p_k\rightarrow 2$. Thanks to the epsilon regularity we deduce a strong convergence/bubble tree convergence property our first main result is about the absence of loss of energy in the neck regions connecting the bubbles between themselves. Precisely we have

Let $a \in \mathrm{W}^{1,2}(M, T^*M\otimes \mathfrak{g})$, $\widehat{a} \in \mathrm{W}^{1,2}(\Sph^4, T^*\Sph^4\otimes \mathfrak{g})$. Consider $\chi \in
 \mathcal{C}^\infty_c ([0, 2[, [0, 1])$ such that $\chi_{| [0, 1]} = 1$. Introduce for $\eta>0$,
 \[ a_\eta (x) = \left( 1 - \chi \left( \frac{| x - p |}{\eta} \right) \right) a (x), \]
 and, in stereographic coordinates,
 \[\widehat{a}_{\eta}(x) =\chi \left( 2 \eta |x| \right)
 \widehat{a} \left( x \right).\] Apply \cref{approximationlemma} to $a_{\eta}$ and to $\widehat{a}_{\eta}$ (replacing $M$ by $S^4$) to get two sequences $(a_{\eta,k})_k$ and $(\widehat{a}_{\eta,k})_k$, such that
 \begin{align*}
 Q_{p_k,A_k} (a_{\eta,k}) &\underset{k\to+\infty}{\rightarrow} Q_{2, A_\infty} (a_{\eta}),\\
 Q_{p_k , \phi_k^*A_k} ( \widehat{a}_{\eta,k}) &\underset{k\to+\infty}{\rightarrow} Q_{2,\widehat{A}_\infty} (\widehat{a}_{\eta}),
 \end{align*} and for all $k$ large enough $a_{\eta,k}$ and $(\phi_k)_*\widehat{a}_{\eta,k}$ have disjoint support. We have \[
 Q_{p_k,A_k} (a_{\eta,k}+ (\phi_k)_*\widehat{a}_{\eta,k}) = Q_{p_k,A_k} (a_{\eta,k})+ Q_{p_k,A_k}( (\phi_k)_*\widehat{a}_{\eta,k}). \] By contrast with \cite[theorem 4.1]{GL24}, the problem is not conformally invariant but we still have $Q_{p_k,A_k}( (\phi_k)_*\widehat{a}_{\eta,k}) - Q_{p_k , \phi_k^*A_k} ( \widehat{a}_{\eta,k}) \underset{k\to+\infty}{\rightarrow} 0$ since \begin{align*}Q_{p_k,A_k}( (\phi_k)_*\widehat{a}_{\eta,k}) =& \int_{\mathrm{B}_{1/\eta}}(1+|F_{\phi_k^*A_k}|_{\phi_k^*h}^2)^{\frac{p_k}{2}-1} \left( |\diff_{\phi_k^*A_k} \widehat{a}_{\eta,k}|_{\phi_k^*h}^2 + \langle F_{\phi_k^*A_k}, [\widehat{a}_{\eta,k}, \widehat{a}_{\eta,k}] \rangle_{\phi_k^*h} \right)\mathrm{vol}_{\phi_k^*h}\\
 & + (p_k-2) \int_{\mathrm{B}_{1/\eta}} (1+|F_{\phi_k^*A_k}|_{\phi_k^*h}^2)^{\frac{p_k}{2}-2} \langle F_{\phi_k^*A_k}, \diff_{\phi_k^*A_k} \widehat{a}_{\eta,k} \rangle_{\phi_k^*h}^2 \mathrm{vol}_{\phi_k^*h} \end{align*} so \[ Q_{p_k,A_k} (a_{\eta,k}+ (\phi_k)_*\widehat{a}_{\eta,k}) \underset{k\to+\infty}{\rightarrow} Q_{2,A_\infty} (a_{\eta})+ Q_{2,\widehat{A}_\infty} (\widehat{a}_{\eta}).\]
 From the identity \[\diff_{A_\infty} a_{\eta} = -\diff \left(\chi\left(\frac{|\cdot - p|}{\eta}\right)\right) \wedge a + \left(1-\chi\left(\frac{|\cdot - p|}{\eta}\right)\right)\diff_{A_\infty}a \] and using the fact that $t\mapsto t \chi'(t)$ is bounded, we obtain
 \[|\diff_{A_\infty} a_{\eta}| \leq C\frac{|a|}{|x-p|} \mathbf{1}_{\mathrm{B}(p,2\eta)}+ |\diff_{A_\infty} a|.\] Using the embeddings $\mathrm{L}^{4,\infty} \cdot \mathrm{L}^{4,2} \hookrightarrow \mathrm{L}^2$ and $\mathrm{W}^{1,2}\hookrightarrow \mathrm{L}^{4,2}$, we deduce that the right-hand side is in $\mathrm{L}^2$ and by the dominated convergence theorem $\diff_{A_\infty} a_{\eta}\underset{\eta \to 0}{\rightarrow} \diff_{A_\infty} a$ in $\mathrm{L}^2$. It is straightforward that $a_{\eta}\underset{\eta \to 0}{\rightarrow} a$ in $\mathrm{L}^4$ so we deduce \[\lim_{\eta\to 0} Q_{2,A_\infty}(a_\eta)=Q_{2,A_\infty}(a). \] A similar argument show that \[\lim_{\eta\to 0} Q_{2,\widehat{A}_\infty}(\widehat{a}_\eta)=Q_{2,\widehat{A}_\infty}(\widehat{a})\] and therefore \[\lim_{\eta \to 0} \lim_{k\to+\infty} Q_{p_k,A_k} (a_{\eta,k}+ (\phi_k)_*\widehat{a}_{\eta,k}) = Q_{2,A_\infty} (a)+ Q_{2,\widehat{A}_\infty} (\widehat{a}).\] Now, if $Q_{2,{A}_{\infty}} (a) < 0$ and
 $Q_{2,\widehat{A}_{\infty}} (\widehat{a}) < 0$, for $\eta$ small enough and $k$ large enough, \[Q_{p_k,A_k}(a_{\eta,k}+(\phi_k)_* \widehat{a}_{\eta,k})<0.\]
 Let $W\subset \mathrm{W}^{1,2}(M, T^*M\otimes \mathfrak{g})$, $\widehat{W} \subset \mathrm{W}^{1,2}(\Sph^4, T^*\Sph^4\otimes \mathfrak{g})$ of finite dimension such that $Q_{2,A_\infty |W}<0$ and $Q_{2,\widehat{A}_\infty |\widehat{W}}<0$. Consider $(\phi^1,\dots,\phi^n)$ (resp. $(\psi^1,\dots,\psi^m)$) a basis of $W$ (resp. of $\widehat{W}$), orthonormal for $Q_{2,A_\infty}$ (resp. for $Q_{2,\widehat{A}_\infty }$). For $\eta>0$ and $k\in \N$, denote $\mathcal{B}_{\eta,k}$ the family whose elements are the $\phi^i_{\eta,k}$ and $\widehat{\psi^j}_{\eta,k}$ as defined above. Consider $M_{\eta,k}$ the Gram matrix for the quadratic form $Q_{p_k,A_k}$ of this family. From the earlier discussion, $\displaystyle \lim_{\eta \to 0} \lim_{k\to+\infty} M_{\eta,k}= -I$ and an implicit function argument ensures the existence, for $k$ large enough and $\eta$ small enough, of invertible matrices $(P_{\eta,k})$ such that $P_{\eta,k} ^\top M_{\eta,k}P_{\eta,k} = - I$. In particular $\Vect{\mathcal{B}_{\eta,k}}$ has dimension $n+m$ and $Q_{p_k,A_k}$ is negative definite on $\Vect{ \mathcal{B}_{\eta,k}}$. We have then proved \[\mathrm{ind}_{\mathcal{YM}_{p_k}}(A_k) \geq \mathrm{ind}_\mathcal{YM}(A_{\infty})+ \mathrm{ind}_\mathcal{YM}(\widehat{A}_{\infty}).\] \end{proof}

Define now the operators $\mathcal{L}_{\eta, \infty}$ and $\widehat{\mathcal{L}}_{\eta, \infty}$ by \begin{align*}
 \mathcal{L}_{\eta, \infty} a &= \omega_{\eta, \infty}^{-1} \left( \Delta_{A_\infty} a +
 \star [\star F_{A_\infty}, a]\right),\\
 \widehat{\mathcal{L}}_{\eta, \infty} \widehat{a} &= \widehat{\omega}_{\eta, \infty}^{-1} \left( \Delta_{\widehat{A}_\infty} \widehat{a} +
 \star[ \star F_{\widehat{A}_\infty}, \widehat{a}]\right),
 \end{align*} such that \begin{align*}
 \mathcal{Q}_{2,A_{\infty}} (a) &= \langle a, \mathcal{L}_{\eta, \infty} a \rangle_{\omega_{\eta,
 \infty}},\\
 \mathcal{Q}_{2,\widehat{A}_{\infty}} (\widehat{a}) &= \langle \widehat{a}, \widehat{\mathcal{L}}_{\eta, \infty} \widehat{a} \rangle_{\widehat{\omega}_{\eta,
 \infty}}
 \end{align*} for $a\in \mathrm{W}^{1,2}(M, T^*M\otimes \mathfrak{g})$ and $\widehat{a} \in \mathrm{W}^{1,2}(\mathrm{S}^4, T^*\mathrm{S}^4\otimes \mathfrak{g})$. The operators $\mathcal{L}_{\eta,k}$ (resp. $\mathcal{L}_{\eta,\infty}$, $\widehat{\mathcal{L}}_{\eta, \infty}$) can be diagonalized with respect to the inner products involving the weights $\omega_{\eta,k}$ (resp. $\omega_{\eta,\infty}$, $\widehat{\omega}_{\eta,
 \infty}$) according to \cref{diagonalisationofquadraticforms}. This guarantees, as stated before, that studying the index of $A_k$ can done by analysing the spectrum of $\mathcal{L}_{\eta, k}$. As for \cite[proposition 4.5]{GL24}, the proof of \cref{semicontinuitesupindex} is a direct consequence of the following result. \begin{lemma} \label{prelim1}
 If $\eta>0$ is small enough, for all $(a_k)_{k \in \N}$ such that for all
 $k \in \N$, $a_k \in W_{\eta, k}$ and $\| a_k \|_{\omega_{\eta, k}} =
 1$, there exists $a_{\infty} \in \mathrm{W}^{1, 2} (M, T^*M \otimes \mathfrak{g})$ and $\widehat{a}_{\infty}\in \mathrm{W}^{1,2}(\R^4, T^*\R^4\otimes\mathfrak{g})$ such that, up to extraction, $a_k \rightharpoonup
 a_{\infty}$ and $\widehat{a}_k \rightharpoonup
 \widehat{a}_{\infty}$ in $\mathrm{W}^{1, 2}$, where $\widehat{a}_k := \phi_k^*a_k$, and for $\delta > 0$ small enough : \begin{equation}
 \int_{M\backslash \mathrm{B}_{\delta} (p)} (| \diff_{A_{\infty}}
 a_{\infty} |^2_h + | \diff_{A_{\infty}}^{\ast} a_{\infty} |^2_h + |
 a_{\infty} |^2_h) \mathrm{vol}_h = \lim_{k \rightarrow + \infty}
 \int_{M\backslash \mathrm{B}_{\delta} (p)} \left( \left|
 \diff_{A_{_k}} a_k \right|^2_h + | \diff_{A_k}^{\ast} a_k |^2_h + |
 a_k |^2_h \right) \mathrm{vol}_h
 \end{equation} and \begin{equation}
 \int_{\mathrm{B}_{1 / \delta}} (| \diff_{\widehat{A}_{\infty}}
 \widehat{a}_{\infty} |^2 + | \diff_{\widehat{A}_{\infty}}^{\ast}
 \widehat{a}_{\infty} |^2 + | a_{\infty} |^2) \diff x = \lim_{k
 \rightarrow + \infty} \int_{\mathrm{B}_{1 / \delta}} \left( \left|
 \diff_{\widehat{A}_{_k}} \widehat{a}_k \right|^2 + | \diff_{\widehat{A}_k}^{\ast}
 \widehat{a}_k |^2 + | \widehat{a}_k |^2 \right) \diff x.
 \end{equation} Moreover $(a_{\infty},
 \widehat{a}_{\infty}) \neq (0, 0)$.
 \end{lemma}

The result as such has already been proven in \cite{HS}. Our approach in the present work for proving theorem~\ref{bubbletree} is following the general strategy  originally introduced in the seminal work \cite{RiviereLin}.
	It consists in looking for  interpolation spaces estimates in neck regions. In fact this strategy allows for an up-grade of the energy quantization result to a much sharper $\mathrm{L}^{2,1}$energy  quantization (see corollary~\ref{L2quantization}). The first $L^{2,1}-$energy quantization result has been obtained by the second and the third author in \cite{laurainriviere2014angular}. We would like to stress at this stage that theorem~\ref{bubbletree}  and even more corollary~\ref{L2quantization} should be very striking to experts in conformal geometric analysis. Indeed, the $L^2$ energy quantization and even more the $L^{2,1}$ energy quantization are notoriously known to fail in general for the Sacks Uhlenbeck approximation of the Dirichlet Energy (\cite{LiW}) (except when the target is a symmetric space see \cite{LiZhu}  and \cite{DaRiSchla}). Our result is another illustration of the fact that Yang-Mills energy is enjoying more stability properties in bubble tree analysis than the ``cousin Lagrangian'' given by the Dirichlet energy in 2 dimension. It goes in the same direction of Waldron's theorem asserting that  Yang-Mills heat flow  never blows-up in finite time \cite{Waldron}. contrary to the harmonic map flow.

	In the second part of the paper, we implement the $L^{2,1}-$energy quantization in neck regions for proving a Morse index stability result following a strategy introduced by \cite{DGR22}. The following result is generalising to $p-$Yang-Mills the previous result obtained by the two first authors in \cite{GL24}.
	\begin{theorem}
		Let $ (M^4,h)$ be a closed four-dimensional Riemannian manifold and $A_k\in\mathfrak{U}_G^{p_k}(M^4)$ a sequence of $p_k$-Yang-Mills connections with uniformly bounded energy and such that $p_k \underset{k\to+\infty}{\rightarrow }2^+$. Let $A_\infty\in \mathfrak U_G(M^4)$ and $A_\infty^{i,j} \in\mathfrak U_G(S^4) $ be its bubble-tree limit in the sense of theorem \ref{bubbletree}. Then, for $k$ large enough, we have
		\begin{align}
			\mathrm{ind}_{\mathcal{YM}_{p_k}}(A_k) &\geq \mathrm{ind}_\mathcal{YM}(A_{\infty})+ \sum_{i=1}^N\sum_{j=1}^{N_i} \mathrm{ind}_\mathcal{YM}(A_{\infty}^{i,j}), \\
			\mathrm{ind}^0_{\mathcal{YM}_{p_k}}(A_k) &\leq \mathrm{ind}^0_\mathcal{YM}(A_{\infty})+ \sum_{i=1}^N \sum_{j=1}^{N_i} \mathrm{ind}^0_\mathcal{YM}(A_{\infty}^{i,j}),
		\end{align} where $\mathrm{ind}^0_{\mathcal{YM}_{p}} (A)=\ind_{\mathcal{YM}_p}(A) + \dim \left(\ker Q_{A,p} \cap \ker \diff_A^*\right)$ is the extended index of the connection and $Q_{A,p}$ is the quadratic form associated to the second variation.
	\end{theorem}

	This result is the Yang-Mills counterpart of the same result obtained in \cite{DaRiSchla}  for Sacks Uhlenbeck relaxation of the Dirichlet energy of maps into spheres and symmetric spaces in general see \cite{Schlagen}.
\medskip

	The paper is organised as follows. After that we introduce, in section 1 and 2,  notations and main notion for the definition of indices, we perform a fine analysis in neck region, section 3 to 5, here for the sake of clarity, we will state the results  in the Euclidean framework. They remain true on small enough geodesic balls of a Riemannian manifold, up to slight modification of constant which as no effect on the desired result. Finally, in section 6, we prove our main theorem in a general setting.

	\medskip

	\noindent{\bf Acknowledgements} This work has been initiated during a one semester visit of the first author at the ETH Z\"urich. He would like to thank the mathematics department at ETH for its hospitality.
	\setcounter{tocdepth}{2}
	\tableofcontents

	\newpage

Our approach in the present work for proving theorem~\ref{bubbletree} is following the general strategy originally introduced in the seminal work \cite{RiviereLin}. It consists in looking for interpolation spaces estimates in neck regions. In fact this strategy allows for an up-grade of the energy quantization result to a much sharper $\mathrm{L}^{2,1}$ energy quantization (see corollary~\ref{L2quantization}).

Our approach in the present work for proving the main theorem is following the general strategy originally introduced in the seminal work RiviereLin. It consists in looking for interpolation spaces estimates in neck regions. In fact this strategy allows for an up-grade of the energy quantization result to a much sharper $\mathrm{L}^{2,1}$ energy quantization; we record this as follows.

\begin{corollary} \label{L2quantization}
		There exists $\varepsilon_G, C_G> 0$ and $p_G>0$ such that for all $p\in[2,2+p_G]$ and for all $p$-Yang-Mills connection $A$ in $\mathrm{B}_R\backslash \overline{\mathrm{B}_r} $, for all $R,r$ with $0<r<4r<R<1$, if $A$ satisfies
		\[ \sup_{r \leq \rho \leq R /2} \| F_A \|_{\mathrm{L}^2(\mathrm{B}_{2\rho} \backslash \mathrm{B}_\rho)} \leq \varepsilon_{G}',\] then \[ \|F_A\|_{\mathrm{L}^{2}(\mathrm{B}_{R}\backslash \mathrm{B}_{r})} \leq C  \sup_{r \leq \rho \leq R/2} \| F_A \|_{\mathrm{L}^2(\mathrm{B}_{2\rho}\backslash \mathrm{B}_\rho)}\] and
		\[ \|F_A\|_{\mathrm{L}^{2,1}(\mathrm{B}_{R/2}\backslash \mathrm{B}_{2r})} \leq C  \sup_{r \leq \rho \leq R/2} \| F_A \|_{\mathrm{L}^2(\mathrm{B}_{2\rho}\backslash \mathrm{B}_\rho)}.\]
	\end{corollary}

Existential–Universal
Asymptotic or Limit

Question:
Let $(M^4,h)$ be a closed four-dimensional Riemannian manifold, let $G$ be a compact Lie group, and let $A_k\in\mathfrak U_G^{p_k}(M^4)$ be a sequence of $p_k$-Yang--Mills connections, meaning critical points of the $p_k$-Yang--Mills functional
\[
\mathcal{YM}_{p_k}(A)=\int_{M^4}(1+|F_A|^2)^{p_k/2}\,\mathrm{vol}_h,
\]
with uniformly bounded energy $\sup_k \mathcal{YM}_{p_k}(A_k)<\infty$ and $p_k\to 2^+$. Here $F_A$ is the curvature of $A$. Using $\mathfrak C^\infty_{G,\mathrm{loc}}$ for smooth local convergence modulo gauge, and writing $\pi:S^4\setminus\{\infty\}\to\mathbf R^4$ for stereographic projection, which asymptotic bubble-tree statement holds after passing to a subsequence?

There exist finitely many points $\{p_1,\dots,p_N\}\subset M^4$ and a Yang--Mills connection $A_\infty\in\mathfrak U_G(M^4)$ such that:
1. $A_k\to A_\infty$ in $\mathfrak C^\infty_{G,\mathrm{loc}}\bigl(M\setminus\{p_1,\dots,p_N\}\bigr)$.
2. For each $i\in\{1,\dots,N\}$, there exist an integer $N_i\in\mathbf N$, sequences of points $p_k^{i,j}\to p_i$, sequences of scales $\lambda_k^{i,j}\to 0$, and nontrivial Yang--Mills connections $A_\infty^{i,j}\in\mathfrak U_G(S^4)$, for $j=1,\dots,N_i$, such that if in local coordinates near $p_i$ one defines
\[
\phi_k^{i,j}(x)=p_k^{i,j}+\lambda_k^{i,j}x,
\]
then
\[
(\phi_k^{i,j})^*A_k\to \widehat A_\infty^{i,j}:=\pi_*A_\infty^{i,j}
\quad\text{in }\mathfrak C^\infty_{G,\mathrm{loc}}\bigl(\mathbf R^4\setminus\{\text{finitely many points}\}\bigr).
\]
3. There is no loss of energy:
\[
\lim_{k\to\infty}\int_{M^4}|F_{A_k}|_h^2\,\mathrm{vol}_h
=
\int_{M^4}|F_{A_\infty}|_h^2\,\mathrm{vol}_h
+\sum_{i=1}^N\sum_{j=1}^{N_i}\int_{S^4}|F_{A_\infty^{i,j}}|^2\,\mathrm{vol}.
\]

Choices:
B.
There exist finitely many points $\{p_1,\dots,p_N\}\subset M^4$ and a Yang--Mills connection $A_\infty\in\mathfrak U_G(M^4)$ such that:
1. $A_k\to A_\infty$ in $\mathfrak C^\infty_{G,\mathrm{loc}}\bigl(M\setminus\{p_1,\dots,p_N\}\bigr)$.
2. For each $i\in\{1,\dots,N\}$, there exist an integer $N_i\in\mathbf N$, sequences of points $p_k^{i,j}\to p_i$, sequences of scales $\lambda_k^{i,j}\to 0$, and nontrivial Yang--Mills connections $A_\infty^{i,j}\in\mathfrak U_G(S^4)$, for $j=1,\dots,N_i$, such that if in local coordinates near $p_i$ one defines
\[
\phi_k^{i,j}(x)=p_k^{i,j}+\lambda_k^{i,j}x,
\]
then
\[
(\phi_k^{i,j})^*A_k\to \widehat A_\infty^{i,j}:=\pi_*A_\infty^{i,j}
\quad\text{in }\mathfrak C^\infty_{G,\mathrm{loc}}\bigl(\mathbf R^4\bigr).
\]
3. There is no loss of energy:
\[
\lim_{k\to\infty}\int_{M^4}|F_{A_k}|_h^2\,\mathrm{vol}_h
=
\int_{M^4}|F_{A_\infty}|_h^2\,\mathrm{vol}_h
+\sum_{i=1}^N\sum_{j=1}^{N_i}\int_{S^4}|F_{A_\infty^{i,j}}|^2\,\mathrm{vol}.
\]
C.
There exist finitely many points $\{p_1,\dots,p_N\}\subset M^4$, a Yang--Mills connection $A_\infty\in\mathfrak U_G(M^4)$, and for each $i\in\{1,\dots,N\}$ an integer $N_i\in\mathbf N$, sequences $p_k^{i,j}\to p_i$, scales $\lambda_k^{i,j}\to 0$, and nontrivial Yang--Mills connections $A_\infty^{i,j}\in\mathfrak U_G(S^4)$ such that:
1. $A_k\to A_\infty$ in $\mathfrak C^\infty_{G,\mathrm{loc}}\bigl(M\setminus\{p_1,\dots,p_N\}\bigr)$.
2. If in local coordinates near $p_i$ one defines
\[
\phi_k^{i,j}(x)=p_k^{i,j}+\lambda_k^{i,j}x,
\]
then
\[
(\phi_k^{i,j})^*A_k\to \widehat A_\infty^{i,j}:=\pi_*A_\infty^{i,j}
\quad\text{in }\mathfrak C^\infty_{G,\mathrm{loc}}\bigl(\mathbf R^4\setminus\{\text{finitely many points}\}\bigr).
\]
3. The Yang--Mills energy satisfies the lower bound
\[
\liminf_{k\to\infty}\int_{M^4}|F_{A_k}|_h^2\,\mathrm{vol}_h
\ge
\int_{M^4}|F_{A_\infty}|_h^2\,\mathrm{vol}_h
+\sum_{i=1}^N\sum_{j=1}^{N_i}\int_{S^4}|F_{A_\infty^{i,j}}|^2\,\mathrm{vol}.
\]
D.
There exist finitely many points $\{p_1,\dots,p_N\}\subset M^4$ and a Yang--Mills connection $A_\infty\in\mathfrak U_G(M^4)$ such that:
1. $A_k\to A_\infty$ in $\mathfrak C^\infty_{G,\mathrm{loc}}\bigl(M\setminus\{p_1,\dots,p_N\}\bigr)$.
2. For each $i\in\{1,\dots,N\}$, there exist an integer $N_i\in\mathbf N$, sequences of points $p_k^{i,j}\to p_i$, sequences of scales $\lambda_k^{i,j}\to 0$, and nontrivial Yang--Mills connections $A_\infty^{i,j}\in\mathfrak U_G(S^4)$, for $j=1,\dots,N_i$, such that if in local coordinates near $p_i$ one defines
\[
\phi_k^{i,j}(x)=p_k^{i,j}+\lambda_k^{i,j}x,
\]
then
\[
(\phi_k^{i,j})^*A_k\to \widehat A_\infty^{i,j}:=\pi_*A_\infty^{i,j}
\quad\text{in }\mathfrak C^\infty_{G,\mathrm{loc}}\bigl(\mathbf R^4\setminus\{\text{finitely many points}\}\bigr).
\]
3. There exists a constant $C<\infty$, depending only on $(M,h)$ and $G$, such that for every concentration point $p_i$ one has $N_i\le C$, and hence the total number of bubbles is bounded a priori by $CN$.
E.
There exist finitely many points $\{p_1,\dots,p_N\}\subset M^4$ and a Yang--Mills connection $A_\infty\in\mathfrak U_G(M^4)$ such that:
1. $A_k\to A_\infty$ in $\mathfrak C^\infty_{G,\mathrm{loc}}\bigl(M\setminus\{p_1,\dots,p_N\}\bigr)$.
2. For each $i\in\{1,\dots,N\}$, there exists a single sequence of points $p_k^{i}\to p_i$, a single sequence of scales $\lambda_k^{i}\to 0$, and a nontrivial Yang--Mills connection $A_\infty^{i}\in\mathfrak U_G(S^4)$ such that, with
\[
\phi_k^{i}(x)=p_k^{i}+\lambda_k^{i}x,
\]
we have
\[
(\phi_k^{i})^*A_k\to \widehat A_\infty^{i}:=\pi_*A_\infty^{i}
\quad\text{in }\mathfrak C^\infty_{G,\mathrm{loc}}\bigl(\mathbf R^4\setminus\{\text{finitely many points}\}\bigr).
\]
3. There is no loss of energy:
\[
\lim_{k\to\infty}\int_{M^4}|F_{A_k}|_h^2\,\mathrm{vol}_h
=
\int_{M^4}|F_{A_\infty}|_h^2\,\mathrm{vol}_h
+\sum_{i=1}^{N}\int_{S^4}|F_{A_\infty^{i}}|^2\,\mathrm{vol}.
\]

possible secondary bubbling in rescaled limit

exact no-loss energy identity weakened to lower-semicontinuity inequality

non-uniform control on number of bubbles per concentration point

multiple bubbles at a single concentration point