\begin{theorem}\label{main theorem}
 	Let $q \ge 3$ be fixed. Uniformly for
 $
 	\tau \le \log n + \log_2 n - \theta(n),
 	$
 	where $2\leq \theta(n)\ll \log_3 n$ is any function tending arbitrarily slowly to infinity as $n \to \infty$, we have
 	\[
 	\phi_n(\tau)
 	:= \frac{1}{|\mathcal{H}_n|}
 	\sum_{\substack{D \in \mathcal{H}_n \\ L(1, \chi_D) \ge e^{\gamma}\tau}} 1
 	= \exp\!\left(
 	-\,C_1\!\left(q^{\{\log \kappa(\tau)\}}\right)
 	\,\frac{q^{\,\tau - C_0\!\left(q^{\{\log \kappa(\tau)\}}\right)}}{\tau}
 	\bigl(1 + o_\theta(1)\bigr)
 	\right),
 	\]
 	where the constants $C_0\!\left(q^{\{\log \kappa(\tau)\}}\right)$ and $C_1\!\left(q^{\{\log \kappa(\tau)\}}\right)$ depend on $\tau$, $q$, and $\kappa(\tau)$, yet remain bounded as the argument varies between $1$ and $q$.
 	Further details on these quantities can be found in~\cite[Theorem~1.3]{Lumley}.
 \end{theorem}

\begin{lemma}\label{character sum over nonsquare}
	Let $\ell=\ell_1 \ell_2^2\neq \square\in \mathbb{F}_q[t]$ with $\ell_1$ square-free. For any $\varepsilon>0$, we have
	\[
	\sum_{D\in \mathcal{H}_n}\chi_D(\ell)\ll  q^{(1/2+\varepsilon)n}\exp\left((d(\ell_1))^{1-\varepsilon}\right)\prod_{P\mid \ell_2}\left(1+|P|^{-1/2-\varepsilon}\right).
	\]

\end{lemma}

One of the central problems in analytic number theory is to determine the value distribution and obtain sharp bounds for the maximal size of Dirichlet $L$-functions on the line $\operatorname{Re}(s)=1$. These quantities are deeply intertwined with fundamental arithmetic invariants such as class numbers of number fields, and are closely linked to estimates for character sums via Fourier analysis. In particular, the tail behavior of the value distribution plays a decisive role in determining the true order of magnitude of extreme values.

Over the past few decades, significant progress has been made toward understanding the tails of these distributions and the maximal size of $L$-functions on $\operatorname{Re}(s)=1$. This line of research encompasses the Riemann zeta function and Dirichlet $L$-functions in the modulus aspect (see~\cite{AMM, AMMP, BS3, DOKIC, GSzeta}), as well as the family of quadratic Dirichlet $L$-functions $L(1,\chi_d)$ (see~\cite{DM, GS, Lumley}), and more generally, various families of automorphic $L$-functions (see~\cite{Lamzouri}).

An important $\mathrm{GL}(1)$ family of symplectic type is formed by the quadratic Dirichlet $L$-functions $L(s,\chi_d)$, where $d$ ranges over fundamental discriminants. The analytic study of this family is notably challenging due to the lack of a natural orthogonality relation among the characters. The tail behavior of $L(1,\chi_d)$ is closely tied to the distribution of class numbers of quadratic number fields. In this direction, Granville and Soundararajan~\cite{GS} carried out a seminal analysis of these quantities, and following their approach, Dahl and Lamzouri~\cite{DL} investigated an interesting subfamily, known as the \emph{Chowla family}, corresponding to real quadratic fields.

\subsection{Distribution of $L$-functions in the hyperelliptic ensemble}\label{large values section}
We now move our discussion to the setup of $L$-functions over function fields, a topic of interest in modern research in number theory; see, e.g., \cite{AT, AK, BF, BF2, DFL, DL, FR, Floreanegative, Florea, Jung, Lumley2}.
This includes investigations of various statistical properties of these $L$-functions, such as moments, non-vanishing, and value distribution.

Before we enunciate the main theorem of this paper, we need to present
 some basic notations on function fields.
 Let $\mathbb{F}_{q}$ be a finite field of odd cardinality and $\mathbb{F}_{q}[t]$ be the polynomial ring over $\mathbb{F}_{q}$ in variable $t$. Let $D\in \mathbb{F}_q[t]$ be a monic square-free polynomial. The quadratic character $\chi_D$ attached to $D$ is defined using quadratic residue symbol for $\mathbb{F}_{q}[t]$ by $\chi_{D}(f)=\left(\frac{f}{D}\right)$ and the corresponding Dirichlet $L$-function is denoted by $L(s, \chi_D)$.
 Let $\mathcal{H}_n$
 be the family of all curves given in affine form by $C_P: y^2=D(t)$, where $D$ be a monic square-free polynomial of degree $n$. These curve are non-singular of genus $g$ given by \eqref{lambda} and \eqref{genus}.

In~\cite{Lumley}, Lumley studied the distribution of $L(1,\chi_D)$ as $D$ ranges over $\mathcal{H}_n$.
 The following theorem strengthens the range of uniformity for the tail of the distribution of $L(1,\chi_D)$ established in~\cite[Theorem~1.2]{Lumley}.
Throughout, $\log$ denotes the logarithm to base $q$, while $\ln$ is the natural logarithm, and $\log_j$ (respectively $\ln_j$) for the $j$-fold iterated logarithm.

\begin{align}\label{genus}
2g=d(D)-1-\lambda
\end{align}

\begin{align}\label{lambda}
\lambda=
\left\{
\begin{array}
[c]{ll}
1, & \;\text{if\, $d(D)$ even}, \\
0, &\; \text{if\, $d(D)$ odd},
\end{array}
\right.
\end{align}

Before we enunciate the main theorem of this paper, we need to present
 some basic notations on function fields.
 Let $\mathbb{F}_{q}$ be a finite field of odd cardinality and $\mathbb{F}_{q}[t]$ be the polynomial ring over $\mathbb{F}_{q}$ in variable $t$. Let $D\in \mathbb{F}_q[t]$ be a monic square-free polynomial. The quadratic character $\chi_D$ attached to $D$ is defined using quadratic residue symbol for $\mathbb{F}_{q}[t]$ by $\chi_{D}(f)=\left(\frac{f}{D}\right)$ and the corresponding Dirichlet $L$-function is denoted by $L(s, \chi_D)$.
 Let $\mathcal{H}_n$
 be the family of all curves given in affine form by $C_P: y^2=D(t)$, where $D$ be a monic square-free polynomial of degree $n$. These curve are non-singular of genus $g$ given by \eqref{lambda} and \eqref{genus}.

\subsection{Truncating the Euler product of $L(1, \chi_D)$}
We show that $L(1, \chi_D)$ can be well approximated by its short Euler product for almost all $D \in \mathcal{H}_n$. Let
\begin{align}\label{short euler product}
L(1,\chi_D; M):= \prod_{d(P)\le M}\left(1-\frac{\chi_D(P)}{|P|}\right)^{-1}=\sum_{\substack{f\in \mathcal{M}\\ P\mid f \implies d(P)\leq M}}\frac{\chi_D(f)}{|f|}.
\end{align}
 Let $M:=3\log n$. From \cite[Lemma 2.2]{Lumley}, for $D\in \mathcal{H}_n$, we have
 \begin{align}\label{asym for L(1)}
 L(1,\chi_{D})=L(1,\chi_D; M)\left(1+O\left(\frac{1}{n^{1/2}\log n}\right)\right).
 \end{align}
The following lemma provides a better approximation on the truncation length and the associated error chosen optimally for the proof of Theorem~\ref{main theorem}.
\begin{lemma}\label{full l to its truncation}
For sufficiently large $n$,
\[
L(1, \chi_D)=L(1, \chi_D; N)\left(1+O\left(\frac{1}{f(n)\log n}\right)\right)
\]
for all but at most $q^{\frac{7n}{10}}$ elements $D\in \mathcal{H}_n$, where $N=\log n+\log_2 n+3\log f(n)$, and $f(n) \to \infty$ arbitrarily slowly as $n \to \infty$, with $f(n) \ll \log n$.
 \end{lemma}
\begin{proof}
 By using \eqref{asym for L(1)}, we have
 \[
 L(1, \chi_D) =
 L(1, \chi_D; N)
 \exp \!\bigg(
 \sum_{\substack{N < d(P) \le M}}
 \bigg(\frac{\chi_D(P)}{|P|}
 + O\!\left( \frac{1}{|P|^2} \right)
 \bigg)\bigg)
 \bigg( 1 + O\!\bigg( \frac{1}{n^{\tfrac{1}{2}} \log n} \bigg) \bigg).
 \]
 Next, we identify a density zero subfamily of $\mathcal{H}_n$ for which
 \begin{align}\label{DP bound}
 \Bigg|\sum_{N < d(P) \le M} \frac{\chi_D(P)}{|P|}\Bigg| \ge \frac{1}{h(n)}
 \end{align}
 holds, where $
 \log n \ll h(n) \to \infty$ as $n \to \infty.$
 It is worth noting that $h(n)$ must grow faster than $\log n$ chosen appropriately later in order for the saddle point analysis to yield an asymptotic formula exhibiting double exponential decay (see Section~\ref{spa}).

The additional factor $1/\sqrt{\log n}$ in the choice of $c$ is essential for proving Theorem \ref{theorem on resonance method}.
Hence, using \eqref{asym for L(1)}, we can that there exists a $D\in \mathcal{H}_n$ such that
\[
L(1,\chi_D)\ge e^{\gamma}\left(\log n +\log_2 n+C_2(q)-\beta+o(1)\right).
\]
We use \eqref{asym for L(1)}, \eqref{ratio final}, \eqref{constant c} to obtain
\begin{align*}
&\bigg(\tau_{\beta, n}+\frac{1}{2\sqrt{\log n}}\bigg)\sum_{D\in \mathcal{H}_n}R_D^2\leq \sum_{D\in \mathcal{H}_n}L(1, \chi_D)R_D^2\\
&= \sum_{\substack{D\in \mathcal{H}_n\\ L(1, \chi_D)< \tau_{\beta, n}}}L(1, \chi_D)R_D^2+\sum_{\substack{D\in \mathcal{H}_n\\ L(1, \chi_D)\geq \tau_{\beta, n}}}L(1, \chi_D)R_D^2< \tau_{\beta, n} \sum_{D\in \mathcal{H}_n}R_D^2+ \sum_{\substack{D\in \mathcal{H}_n\\ L(1, \chi_D)\geq \tau_{\beta, n}}}L(1, \chi_D)R_D^2.
\end{align*}
From the above computation of $S_2$ together with \eqref{scx} gives us
\[
\sum_{\substack{D\in \mathcal{H}_n\\ L(1, \chi_D)\geq \tau_{\beta, n}}}L(1, \chi_D)R_D^2> \frac{e^{\gamma}}{2 \sqrt{\log n}}\sum_{D\in \mathcal{H}_n}R_D^2> \frac{e^{\gamma}}{2}\frac{q^{(1+c')n\left(1+O(\log_2 n/\log n)\right)}}{\sqrt{\log n}}.
\]
On the other hand, using \cite[Proposition 1.4]{Lumley} that
\[
\max_{D\in \mathcal{H}_n}L(1, \chi_D)\leq 2e^{\gamma}\log n+O_q(1),
\]
we have from \eqref{RD},
\begin{align*}
\sum_{\substack{D\in \mathcal{H}_n\\ L(1, \chi_D)\geq \tau_{\beta, n}}}L(1, \chi_D)R_D^2&\leq |\mathcal{H}_n| \left(\max_{D\in \mathcal{H}_n} R_D^2 \cdot \max_{D\in \mathcal{H}_n}L(1, \chi_D)\right)\phi_n(\beta)\\
&\leq \left(2e^{\gamma}\log n\right) q^{\left(1+\frac{2c\zeta_{\mathbb{A}}(2)}{\ln q}\right)\left(1+O\left(\frac{\log_2 n}{\log n}\right)\right)n} \phi_n(\beta).
\end{align*}
Comparing the above lower and upper bound, and using $c$ as expressed in \eqref{constant c}, we finally conclude that
\[
\phi_n(\beta)> \frac{1}{4(\log n)^{3/2}}q^{-\frac{c(3\ln 2-\pi/2)\zeta_{\mathbb{A}}(2)}{\ln q}\left(1+O\left(\log_2 n/\log n\right)\right)}\geq e^{-\frac{q^{-\beta}\left(1+O\left(1/\sqrt{\log n}\right)\right)\ln q}{2}}.
\]
\end{proof}

From the estimation of $\mathcal{S}_{c, N}$ and $\mathcal{R}_{c, N}$, we finally obtain
\begin{align}\label{ratio}
\frac{S_1}{S_2}\ge \frac{|\mathcal{H}_n| \mathcal{E}(N) \mathcal{E}_1(N) \prod_{d(P)< N} (1-|P|^{-1}) + O\left(q^{\left(\frac12+\frac{2c\zeta_{\mathbb{A}}(2)}{\ln q}+\frac{11\varepsilon}{6}\right)n}\right)}{|\mathcal{H}_n| \mathcal{E}(N) \mathcal{E}_2(N)\prod_{d(P)<N}h(P)^{-1} + O\left( (q^{\left(\frac12+\frac{2c\zeta_{\mathbb{A}}(2)}{\ln q}+2\varepsilon\right)n}\right)}.
\end{align}
 Noting $N=\log n+\log_2 n +\log c$, we first optimize $c$ from the ratio to the right-hand side of \eqref{ratio} by comparing the error terms. To do this, we first calculate the size of $\mathcal{E}(N)=\prod_{d(P)<N}\left(1-r_P^2 \right)^{-2} \left(1+r_P^2\right)$. Applying Abel's summation formula,
\begin{align*}
&\ln \mathcal{E}(N)=-2\sum_{n\leq N}\Pi_{q}(n)\ln\left(\tfrac{2q^n}{q^N}-\tfrac{q^{2n}}{q^{2N}}\right)
+\sum_{n\leq N}\Pi_{q}(n)\ln\left(2-\tfrac{2q^n}{q^N}+\tfrac{q^{2n}}{q^{2N}}\right)\\
&= \frac{\zeta_{\mathbb{A}}(2)}{\ln q}\left(2(2-\ln 4)+(\ln 2+\pi/2-2)\right)\frac{q^N}{N}\left(1+O\left(\frac{\log N}{N}\right)\right),
\end{align*}
where the first and second integrals are referenced from \cite[page $841$]{AMMP} and \cite[page $20$]{DM} respectively by using the prime polynomial theorem and change of variable $q^t=u$. So, from the choice of $N$, we have
\begin{align*}
\mathcal{E}(N)=q^{\widetilde{c}\left(1+O\left(\frac{\log_2 n}{\log n}\right)\right)n},
\end{align*}
where $\widetilde{c}=\left(c(2+\pi/2-3\ln 2)\zeta_{\mathbb{A}}(2)\right)/\ln q$.
Noticing that $\mathcal{E}_2(N)\asymp 1/n$, from \eqref{ratio}, we compare the first term of the denominator to the both error terms to restrict ourself
\begin{align}\label{optimzed constant}
\frac12+\frac{2c\zeta_{\mathbb{A}}(2)}{\ln q}< 1+\frac{c\left(2+\pi/2-3\ln 2\right)\zeta_{\mathbb{A}}(2)}{\ln q}\implies c<\frac{\ln q}{2(3\ln 2-\pi/2)\zeta_{\mathbb{A}}(2)}.
\end{align}
Also, from the above computations, we obtain
\begin{align}\label{scx}
\mathcal{S}_{c, N}=q^{n\widetilde{c}\left(1+O(\log_2 n/\log n)\right)}.
\end{align}
To extract the main term from the ratio on the right-hand side of \eqref{ratio}, it remains to estimate $\mathcal{E}_1(N)$, $\prod_{d(P)\le N} \left(1-|P|^{-1}\right)$, and $\mathcal{E}_2(N)\prod_{d(P)\le N} h(P)^{-1} $.
Using the classical estimate using the Euler--Maclaurin summation formula,
\[
\sum_{n\le N} \frac{1}{n} = \ln N + \gamma + \frac{1}{2N} + O\Bigl(\frac{1}{N^2}\Bigr),
\]
one readily obtains
\[
\prod_{d(P)\le N} \left(1-\frac{1}{|P|}\right)^{-1} = e^{\gamma} N \left(1 + \frac{1}{2N} + O\Bigl(\frac{1}{N^2}\Bigr)\right).
\]
Similarly, following the approach used above for evaluating Euler products, we find
\[
\mathcal{E}_1(N) = q^{\left( \left(c_2 - c_3 + O\bigl(\frac{\log N}{N}\bigr)\right) \frac{\zeta_{\mathbb{A}}(2)}{N \ln q} \right)},
\]
where the constants $c_2$ and $c_3$ are as in \cite[page 21]{DM}, and
\[
\mathcal{E}_2(N)\prod_{d(P)\le N} h(P)^{-1} = \prod_{d(P)\le N} \left( 1 + \frac{(1-r_P^2)^2}{|P|(1+r_P^2)} \right)
= q^{\left( \left(c_2 + O\bigl(\frac{\log N}{N}\bigr)\right) \frac{\zeta_{\mathbb{A}}(2)}{N \ln q} \right)}.
\]

We emphasize that Theorem~\ref{main theorem} subsequently improves the uniformity range of~\cite[Theorem~1.2]{Lumley} from
 $\log n - 2\log_2 n - \log_3 n$ to $\log n + \log_2 n - \theta(n)$, where $\theta(n)$ grows arbitrarily slowly to infinity as $n \to \infty$. A similar range of uniformity as in Lumley’s work is also found in the work of Dahl and Lamzouri~\cite[Theorem~1.2]{DL}.
 For a discussion on the significance of these uniformity ranges, see~\cite{MV}.
Motivated by the work of Granville and Soundararajan~\cite{GS}, Lumley conjectured that as $n\to \infty$,
 \[
 \max_{D\in \mathcal{H}_{n}} L(1,\chi_D) = \log n + \log_2 n + O(1).
 \]
 It is plausible that if the asymptotic formula in Theorem~\ref{main theorem} were to hold up to the boundary of the admissible range, then the above conjecture would follow.
 Our result, however, falls slightly short of this limit, requiring that $\theta(n)$ remain bounded as stated in Theorem~\ref{main theorem}.
 Nevertheless, this constitutes the first unconditional result approaching the conjectural range, comparing upon the works of~\cite{DDLL, DL, GS, GSzeta, Lamzouri, Lumley}.
 Conditionally on the Generalized Riemann Hypothesis (GRH), the corresponding result was established by Granville and Soundararajan~\cite[Theorem~4]{GS} in the number field case.

Finally, by employing a long resonator method, we provide further evidence that the double-exponential decay may persists up to the conjectured level, where $\theta(n)$ transitions from an arbitrarily slowly unbounded function to a bounded one.

\begin{theorem}\label{theorem on resonance method}
	Let $q \ge 3$ and for every $\beta > 0$, define $\tau_{\beta, n}
	= e^{\gamma}\!\left(\log n + \log_2 n + C_2(q) - \beta\right),$ where
	\[
	C_2(q)
	= \frac{1}{2}
	- \left(\frac{\pi}{4} - \frac{\ln 2}{2}\right)\frac{q}{q-1}
	+ \log\!\left(
	\frac{(q-1)\ln q}{2q(3\ln 2 - \pi/2)}
	\right).
	\]
	Then we have
	\[
\phi_n(\beta)
:= \frac{1}{|\mathcal{H}_n|}
\sum_{\substack{D \in \mathcal{H}_n \\ L(1, \chi_D) \ge \tau_{\beta, n}}} 1
	\ge \exp\!\left(
	-\frac{q^{-\beta}\ln q}{2}
	\left(1 + o(1)\right)
	\right).
	\]
\end{theorem}
The best previously known unconditional result in this direction over number fields was due to Granville and Soundararajan~\cite[Theorem~5b]{GS}, while the conditional analogue under the GRH was obtained by the author and Maiti~\cite[Theorem~2]{DM}.

During the proof of Theorem~\ref{theorem on resonance method}, we also observed that
\[
\max_{D\in \mathcal{H}_n} L(1,\chi_D)
\ge e^{\gamma}\Bigl(\log n + \log_2 n + C_2(q) + o(1)\Bigr),
\]
where \(C_2(q)\) is as defined in the statement of the theorem. This result refines~\cite[Theorem~1.6]{Lumley} by providing an explicit value for the constant \(C_2(q)\). More strikingly, one has \(C_2(q) > 0\) for \(q > 10\); for instance, \(C_2(17) \approx 0.04\). This is the first time a positive constant is observed, showing that the maximum of \(L(1,\chi_D)\) can exceed the range \(\log n + \log_2 n\) (see~\cite[Conjecture~2]{MV}).

\subsubsection{Applications} For a monic square-free polynomial $D \in \mathbb{F}_q[t]$, define
\[
h_D = \lvert \mathrm{Pic}(\mathcal{O}_D) \rvert,
\]
where $\mathrm{Pic}(\mathcal{O}_D)$ denotes the Picard group of the ring of integers $\mathcal{O}_D \subset \mathbb{F}_q[t](\sqrt{D(t)})$.
Artin~\cite{Artin} established a class number formula over the hyperelliptic ensemble that connects $h_D$ and $L(1, \chi_D)$:
\begin{equation*}
L(1, \chi_D) = \frac{\sqrt{q}}{\sqrt{|D|}}\, h_D = q^{-g} h_D
\quad \text{for } D \in \mathcal{H}_{2g+1}.
\end{equation*}
From Theorem~\ref{main theorem}, we directly extend the range of uniformity in \cite[Corollary~1.8]{Lumley}, showing that the tail of the distribution of large values of $h_D$ over $\mathcal{H}_{2g+1}$ decays doubly exponentially. An analysis towards maximal value for $h_D$ follows from Theorem~\ref{theorem on resonance method}.

For $n = 2g + 2$ and $D \in \mathcal{H}_{2g+2}$, Artin also proved that
\begin{equation*}
L(1, \chi_D) = \frac{q - 1}{\sqrt{|D|}}\, h_D R_D,
\end{equation*}
where $R_D$ denotes the regulator of $\mathcal{O}_D$ (see \cite[Chapter 14]{ROS}).
Analogous results hold for the tail of the distribution of $h_D R_D$ as $D$ varies over $\mathcal{H}_{2g+2}$, improving~\cite[Corollary~1.9]{Lumley} in a similar manner.
 \subsection{Essence of the paper} We employ two distinct methods to prove Theorems~\ref{main theorem} and~\ref{theorem on resonance method}.
 The study of the distribution of $L(1, \chi_d)$ over number fields through an underlying probabilistic model was initiated by Granville and Soundararajan~\cite{GS}.
 Lumley~\cite{Lumley} studied the tail of the distribution for $L(1, \chi_D)$ over function fields from the corresponding probabilistic model (see \cite[eq.~(1.7)]{Lumley}) and computing large moments og $L(1, \chi_D)$ over the family.
 To extend the range of uniformity stated in Theorem~\ref{main theorem}, we able to compute much higher moments for the short Euler product $L(1, \chi_D; N)$ and then utilize the saddle-point method.
 The Lemma~\ref{full l to its truncation} provides an optimal way to approximate $L(1, \chi_D)$ by $L(1, \chi_D; N)$, allowing us to taken a shorter truncation length $N$.
 This step relies crucially on a large sieve type estimate together with Chebyshev’s inequality.

 To prove Theorem~\ref{theorem on resonance method}, we employ the long resonance method for quadratic Dirichlet $L$-functions over number fields, as developed by the author and Maiti in~\cite{DM}.
 The character sum estimate in Lemma~\ref{character sum over nonsquare} plays a central role in both results: it extends the range of uniformity as in Theorem \ref{main theorem} via computing larger moments of $L(1, \chi_D; N)$ over the family, and it enables the control of long resonators in Theorem~\ref{theorem on resonance method}. These ideas appear difficult to implement for higher degree $L$-functions but may still be applicable to families of $\mathrm{GL}(1)$ $L$-functions associated with higher order characters.

To extend the range of uniformity stated in Theorem~\ref{main theorem}, we are able to compute much higher moments for the short Euler product $L(1, \chi_D; N)$ and then utilize the saddle-point method. The Lemma~\ref{full l to its truncation} provides an optimal way to approximate $L(1, \chi_D)$ by $L(1, \chi_D; N)$, allowing us to taken a shorter truncation length $N$. This step relies crucially on a large sieve type estimate together with Chebyshev’s inequality. The character sum estimate in Lemma~\ref{character sum over nonsquare} plays a central role: it extends the range of uniformity as in Theorem~\ref{main theorem} via computing larger moments of $L(1, \chi_D; N)$ over the family.

To extend the range of uniformity stated in the main theorem, we are able to compute much higher moments for the short Euler product $L(1, \chi_D; N)$ and then utilize the saddle-point method.

For sufficiently large $n$,
\[
L(1, \chi_D)=L(1, \chi_D; N)\left(1+O\left(\frac{1}{f(n)\log n}\right)\right)
\]
for all but at most $q^{\frac{7n}{10}}$ elements $D\in \mathcal{H}_n$, where $N=\log n+\log_2 n+3\log f(n)$, and $f(n) \to \infty$ arbitrarily slowly as $n \to \infty$, with $f(n) \ll \log n$. This provides an optimal way to approximate $L(1, \chi_D)$ by $L(1, \chi_D; N)$, allowing us to taken a shorter truncation length $N$. This step relies crucially on a large sieve type estimate together with Chebyshev’s inequality.

Let $\ell=\ell_1 \ell_2^2\neq \square\in \mathbb{F}_q[t]$ with $\ell_1$ square-free. For any $\varepsilon>0$, we have
\[
\sum_{D\in \mathcal{H}_n}\chi_D(\ell)\ll  q^{(1/2+\varepsilon)n}\exp\left((d(\ell_1))^{1-\varepsilon}\right)\prod_{P\mid \ell_2}\left(1+|P|^{-1/2-\varepsilon}\right).
\]
This character sum estimate plays a central role: it extends the range of uniformity in establishing the main theorem via computing larger moments of $L(1, \chi_D; N)$ over the family.

Question:
Let \(\mathbb F_q\) be a finite field of odd cardinality with fixed \(q\ge 3\), and let \(\mathcal H_n\) denote the set of monic square-free polynomials \(D\in \mathbb F_q[t]\) of degree \(n\) (equivalently, the hyperelliptic ensemble \(y^2=D(t)\)). For each \(D\in\mathcal H_n\), let \(\chi_D(f)=\left(\frac{f}{D}\right)\) be the quadratic character modulo \(D\), and let \(L(s,\chi_D)\) be the associated Dirichlet \(L\)-function. Define the tail distribution
\[
\phi_n(\tau):=\frac{1}{|\mathcal H_n|}\sum_{\substack{D\in\mathcal H_n\\ L(1,\chi_D)\ge e^{\gamma}\tau}}1,
\]
where \(\gamma\) is Euler's constant, \(\log\) denotes logarithm base \(q\), \(\log_j\) denotes the \(j\)-fold iterated base-\(q\) logarithm, and \(\{x\}\) denotes the fractional part of \(x\). For any function \(\theta(n)\) with \(2\le \theta(n)\ll \log_3 n\) and \(\theta(n)\to\infty\) arbitrarily slowly as \(n\to\infty\), which statement holds uniformly for all \(\tau\le \log n+\log_2 n-\theta(n)\)?

\[
\phi_n(\tau)=\exp\!\left(-\,C_1\!\left(q^{\{\log \kappa(\tau)\}}\right)\,\frac{q^{\,\tau-C_0\!\left(q^{\{\log \kappa(\tau)\}}\right)}}{\tau}\bigl(1+o_\theta(1)\bigr)\right),
\]
where \(C_0\!\left(q^{\{\log \kappa(\tau)\}}\right)\) and \(C_1\!\left(q^{\{\log \kappa(\tau)\}}\right)\) may depend on \(\tau\), \(q\), and \(\kappa(\tau)\), but remain bounded as their argument varies between \(1\) and \(q\).

Choices:
B.
\[
\phi_n(\tau)=\exp\!\left(-\,C_1\!\left(q^{\{\log \kappa(\tau)\}}\right)\,\frac{q^{\,\tau-C_0\!\left(q^{\{\log \kappa(\tau)\}}\right)}}{\tau}\bigl(1+o_\theta(1)\bigr)\right),
\]
where \(C_0\!\left(q^{\{\log \kappa(\tau)\}}\right)\) and \(C_1\!\left(q^{\{\log \kappa(\tau)\}}\right)\) depend only on \(q\) and remain bounded as their argument varies between \(1\) and \(q\).
C.
\[
\phi_n(\tau)=\exp\!\left(-\,C_1\!\left(q^{\{\log \kappa(\tau)\}}\right)\,\frac{q^{\,\tau-C_0\!\left(q^{\{\log \kappa(\tau)\}}\right)}}{\tau}\,(1+o(1))\right),
\]
for some bounded functions \(C_0\!\left(q^{\{\log \kappa(\tau)\}}\right)\) and \(C_1\!\left(q^{\{\log \kappa(\tau)\}}\right)\) as their argument varies between \(1\) and \(q\).
D.
\[
\phi_n(\tau)=\exp\!\left(-\,C_1\!\left(q^{\{\log \kappa(\tau)\}}\right)\,\frac{q^{\,\tau-C_0\!\left(q^{\{\log \kappa(\tau)\}}\right)}}{\tau}\bigl(1+o_\theta(1)\bigr)\right),
\]
uniformly for all \(\tau\le \log n+\log_2 n+\theta(n)\), where \(C_0\!\left(q^{\{\log \kappa(\tau)\}}\right)\) and \(C_1\!\left(q^{\{\log \kappa(\tau)\}}\right)\) may depend on \(\tau\), \(q\), and \(\kappa(\tau)\), but remain bounded as their argument varies between \(1\) and \(q\).
E.
\[
\phi_n(\tau)=\exp\!\left(-\,C_1\!\left(q^{\log \kappa(\tau)}\right)\,\frac{q^{\,\tau-C_0\!\left(q^{\log \kappa(\tau)}\right)}}{\tau}\bigl(1+o_\theta(1)\bigr)\right),
\]
where \(C_0\!\left(q^{\log \kappa(\tau)}\right)\) and \(C_1\!\left(q^{\log \kappa(\tau)}\right)\) may depend on \(\tau\), \(q\), and \(\kappa(\tau)\), yet remain bounded as the argument varies between \(1\) and \(q\).

dependence of bounded constants on \(\tau\) and \(\kappa(\tau)\)

dropped the specific \(o_\theta(1)\) uniformity/dependence information while keeping the same main asymptotic shape

upper range of uniformity in \(\tau\) beyond \(\log n+\log_2 n-\theta(n)\)

periodic dependence through fractional part \(\{\log \kappa(\tau)\}\)

The stem does not explicitly reveal the correct option. It sets up the theorem context and asks which asymptotic holds, but the exact dependence on the fractional part, the error term, and the range of uniformity are not given away.

This is very close to asking for the precise statement of a known theorem, so it is partly a restatement task. However, it is not completely tautological because the options differ in subtle but meaningful ways involving uniformity, parameter dependence, and periodicity.

Some reasoning is needed to distinguish the exact theorem statement from weaker or false variants, especially regarding the range in tau and the role of {log kappa(tau)}. Still, the item mainly tests recall/recognition of a technical result rather than substantial generative mathematical reasoning.

The distractors are strong: they are mathematically close to the correct statement, target realistic failure modes (weakened error term, incorrect dependence of constants, overextended uniformity range, and loss of periodic fractional-part structure), and are clearly distinct.

A technically well-constructed theorem-recognition MCQ with strong distractors and little answer leakage, but it primarily tests precise recall of a result rather than deeper generative reasoning.


\begin{theorem}[The method of moment from \cite{bollobas2001random}, Theorem 1.23]\label{moment}
Let $\left\{(\Omega_i,\mathbb{P}_i)\right\}_{i\in \mathbb{N}}$ be a sequence of probability spaces.
For $m\in \mathbb{N}$, let $N_{1,i},\cdots,N_{m,i}:\Omega_i\to \mathbb{N}_0$ be random variables for all $i\in \mathbb{N}$
and suppose there exist $\lambda_1,\cdots,\lambda_m\in(0,\infty)$ such that:
$$\lim_{i\to \infty}\mathbb{E}[(N_{1,i})_{k_1}\cdots(N_{m,i})_{k_m}]=\lambda_1^{k_1}\cdots\lambda_m^{k_m}$$
for all $k_1\cdots k_m\in \mathbb{N}$. Then
$$\lim_{i\to \infty}\mathbb{P}[(N_{1,i})=n_1,\cdots,(N_{m,i})=n_m]=\prod^k_{j=1}\frac{\lambda_j^{n_j} e^{-\lambda_j}}{n_j!}.$$
That is, $(N_{1,i},\cdots,N_{m,i}):\Omega_i \to \mathbb{N}^m_0$ converges jointly in distribution to a vector which is independently Poisson distributed with means
$\lambda_1,\cdots,\lambda_m$.

\end{theorem}

\section{Some claims}\label{claim}
Before the proof of the main theorem, we have to make some claims which will simplify the situation.
We consider the stratum whose zero order is $O(1)$, and it inplies $|\kappa|=O(g)$.

\begin{clm} \label{clm1}
A closed saddle connection is non-separable.
\end{clm}
\begin{proof}
For a closed connection $\gamma$ of $(X,\omega)$, if it's separable, then it cuts $X$ into $X_1$ and $X_2$, and $\partial X_1=\partial X_2=\gamma$.
By Stokes theorem, $\int_\gamma \omega=0$, but since $\gamma$ is a saddle connection its length is $|\int_\gamma \omega|$, which is a contradiction.
\end{proof}

\begin{clm} \label{clm2}
The probability measure of the subset of $\H_g(\kappa)$ which has a closed saddle connection involving $q$ cylinders of multiplicity $p$
goes to $0$ as $g\to \infty$, where $p>1$ or $q>0$.
\end{clm}

\begin{proof}
From Theorem \ref{Siegel of closed},
the Siegel-Veech constant of configuration with closed saddle connection involving $q$ cylinders of multiplicity $p$ is $O(\frac{1}{g^{2p+q-2}})$ .
Denote by $\mathcal{C}^{p,q}_g(\kappa)[\frac{a}{\sqrt{g}},\frac{b}{\sqrt{g}}]$ the subset of $\H_g(\kappa)$ consisting of surfaces which have a closed saddle connection
of length in the interval $[\frac{a}{\sqrt{g}},\frac{b}{\sqrt{g}}]$
involving $q$ cylinders of multiplicity $p$.
Then for $p\geq 2$ or $q\geq 1$, we have
 $$\V(\mathcal{C}^{p,q}_g(\kappa)[\frac{a}{\sqrt{g}},\frac{b}{\sqrt{g}}])\leq |\kappa|O(\frac{1}{g^{q}})\frac{b^2-a^2}{g}\V(\H_g(\kappa))=O(\frac{1}{g})\V(\H_g(\kappa)$$
where the coefficient $|\kappa|$ is the selection of the zero that the closed saddle connection goes through.
So when $g\to \infty$ its probability measure goes to $0$
and it suffices to consider the closed saddle connection with multiplicity $1$ and no cylinders around it.
\end{proof}

\begin{clm} \label{clm3}
The angle of a closed saddle connection at the zero which it connects is odd multiples of $\pi$.
\end{clm}

\begin{proof}
For an Abelian differential $(X,\omega)$, if it has a closed saddle connection $\gamma$ connecting a zero $p$, if its angle at $p$ is $2k\pi$,
then the holonomy of $\gamma$ and $-\gamma$ have same direction. But
$$\int_\gamma \omega=-\int_{-\gamma} \omega.$$
It implies $\int_\gamma \omega=0$, which is a contradiction.
\end{proof}

\begin{clm} \label{clm4}
If a closed saddle connection has angle $\pi$ at one side, it must have a cylinder at the side.
\end{clm}

\begin{proof}
Let $\gamma$ be a colsed saddle connection on $(X,\omega)$ which has angle $\pi$ at one side. Since $(X,\omega)$ is oriented, its normal bundle of $\gamma$ at the side is oriented.
Consider the exponential map from the bundle to $(X,\omega)$, which is well-defined on $\gamma \times [0,s]$ for some $s$ small sufficiently and the image is a cylinder.

\end{proof}

Combine claim $2,3$ and $4$, we have that for a closed saddle connection $\gamma$ on $(X,\omega)$ in principal stratum,
since its total angle around the zero is $4\pi$ and has to be divided into odd multiples of $\pi$,
it must have angle $\pi$ at one side of $\gamma$ and $3\pi$ at another.
So there is a cyliner with $\gamma$ as a boundary. As for the other boundary, if it consists of some open saddle connections,
then these saddle connections have angles $\pi$. That means there are at least two non-homologous saddle connections have angle $\pi$,
which occurs on a set of measure zero.
So we only need to consider the cylinder bounded by curves homologous to $\gamma$.
Then from Claim \ref{clm2} the probability measure of $\mathcal{C}_g(1^{2g-2})[\frac{a}{\sqrt{g}},\frac{b}{\sqrt{g}}]$ has limit
 $$\lim_{g\to \infty}\frac{\V(\mathcal{C}_g(1^{2g-2})[\frac{a}{\sqrt{g}},\frac{b}{\sqrt{g}}])}{\V(\H_g(1^{2g-2})}=O(\frac{1}{g})\to 0.$$

So we need to consider the question on stratum with higher order zeros. Next we will consider $\H_g(2^{g-1})$ firstly.

\section{Surgery}\label{surgery}
This section we will introduce some surgeries that, when combined, can collapse a closed saddle connection.
\subsection{Open up a higher-order zero}
Masur, Rafi and Randecker introduce a surgery in \cite{masur2024lengths} which is a variation of a surgery from \cite{eskin2003moduli}.
This surgery can collapse a saddle connection which is not closed and has multiplicity $1$.

Let $(X,\omega)$ be an Abelian differential, $\sigma$ be an open saddle connection with endpoints $v_1$ and $v_2$,
whose orders are $n_1$ and $n_2$ respectively.
Since the total angle at $v_1$ is $2(n_1+1)\pi$,
one can extend $\sigma$ from $v_1$ along its direction to $v_3$ and denote by $\sigma'$ the geodesic segment $v_1v_3$
such that $\ell_\omega(\sigma)=\ell_\omega(\sigma')$
and the angle between $\sigma$ and $\sigma'$ is
$2k_1\pi$ and $2k_2\pi$, where $k_1+k_2=n_1+1$.
If $\sigma'$ does not go through other zeros of $(X,\omega)$,
which means $\sigma'$ is a ray from $v_1$, the following surgery can be carried out.

Cut along $\sigma+\sigma'$ and denote the two copies of $\sigma+\sigma'$ and $v_1$ by $\sigma^{\pm}+(\sigma')^{\pm}$ and $v_1^{\pm}$.
Then glue $\sigma^+$ and $(\sigma')^+$, $\sigma^-$ and $(\sigma')^-$.
This surgery reduces the order of $v_1$ and constructs a new Abelian differential $(X',\omega')$.
On $(X',\omega')$, the total angles at $v_1^+$, $v_1^-$ and $v_2$ are $2k_1\pi$, $2k_2\pi$, and $2(n_2+2)\pi$ respectively,
that is, the orders of $v_1^+$, $v_1^-$ and $v_2$ are $k_1-1$, $k_2-1$, and $n_2+1$.
Particularly, if $v_1$ is a simple zero, the surgery collapses the saddle connection. We call this surgery \emph{collapsing surgery}.
\begin{figure}[h]
    \centering
    \includegraphics[width=0.6\linewidth]{collapsing}
    \caption{The collapsing surgery}
    \label{collapsing}
\end{figure}

In our situation, we need to use the inverse of collapsing surgery which we call \emph{opening surgery}.
From Claim \ref{clm2}, it suffices to consider the subset $\mathcal{C}^{1,0}_g(2^{g-1})[\frac{a}{\sqrt{g}},\frac{b}{\sqrt{g}}]$.
For an Abelian differential $(X,\omega)$ in $\mathcal{C}^{1,0}_g(2^{g-1})[\frac{a}{\sqrt{g}},\frac{b}{\sqrt{g}}]$,
choose a closed saddle connection $\gamma$ with length in $[\frac{a}{\sqrt{g}},\frac{b}{\sqrt{g}}]$ at a double zero $p$.
We can reverse the surgery introduced above as follows.

Since the total angle at $p$ is $6\pi$, from Claim \ref{clm3} and Claim \ref{clm4}, the angles on both side of $\gamma$ at $p$ must be $3\pi$.
Fix the orientation of $\gamma$, choose two rays $\sigma^+$ and $\sigma^-$ from $p$ such that
$\ell_\omega(\sigma^+)=\ell_\omega(\sigma^-)=\ell_\omega(\gamma)$,
and the angle between $\gamma$ and $\sigma^+$
is $\pi$, the angle between $\gamma$ and $\sigma^-$ is $-\pi$, where the sign is consistent with the orientation of the surface.
Denote by $q^+$ and $q^-$ the other endpoints of $\sigma^+$ and $\sigma^-$.
Then cut along $\sigma^+ + \sigma^-$ and glue the two copies of $\sigma^+$ and $\sigma^-$.
After the surgery, the two copies of $p$ become a regular and a simple zero, $q^+$ and $q^-$ are glued to become a new simple zero $q$.

\begin{figure}[h]
   \centering
   \includegraphics[width=0.6\linewidth]{openinglocally}
  \caption{The opening surgery}
  \label{opening}
\end{figure}

This surgery is the inverse of the collapsing surgery above: the double zero is replaced by two simple zeros,
and it can be carried out except for the surgery locus goes through some zero,
that is, it has two non-homologous saddle connections with angle $2\pi$.
From period mapping, such subset has measure zero. We call the subset on which the surgery can be carried out \emph{permissible set},
and denote it by $\mathcal{C}^{1,0}_{g,Per}(2^{g-1})[\frac{a}{\sqrt{g}},\frac{b}{\sqrt{g}}]$, from the discussion above we have
\begin{proposition}
$\mathcal{C}^{1,0}_{g,Per}(2^{g-1})[\frac{a}{\sqrt{g}},\frac{b}{\sqrt{g}}]$ is a full measure subset of $\mathcal{C}^{1,0}_g(2^{g-1})[\frac{a}{\sqrt{g}},\frac{b}{\sqrt{g}}]$.
\end{proposition}

By the surgery of collapsing this map is one-to-one and denote the opening surgery by $F_1$.
\begin{figure}[h]
    \centering
    \includegraphics[width=0.6\linewidth]{opening}
    \caption{The resulting surface}
    \label{opening surface}
\end{figure}

\begin{remark}
We can also choose $\sigma^{\pm}$ to make the angle between $\gamma$ and them are $2\pi$,
it suffices to make sure the opening operation can obtain an abelian differential and a smooth loop homotopic to $\gamma$.
\end{remark}
\subsection{Move zero along closed curve and pinch}
We have constructed the mapping $F_1$,
which takes a closed saddle connection $\gamma$ at a double zero to two saddle connections $\gamma_1$ and $\gamma_2$
sharing same endpoints $p$ and $q$ which are both simple zeros.
Moreover, the angles between the two saddle connections at the two simple zeros (on both sides) are all $2\pi$,
and if we denote by $\ell_\omega(\gamma)=L$, we have $\ell_{\omega'}(\gamma_1)=L$, $\ell_{\omega'}(\gamma_2)=2L$, where $(X',\omega')=F_1(X,\omega)$.
If we want to collapse $\gamma_1$ and $\gamma_2$ simultaneously, we have to make a surgery to move $q$ along $\gamma_2$ to adjust the length of $\gamma_2$.

\subsubsection{Move the zero locally by period mapping.}

First we need to choose a special basis of $H^1(X,\Sigma;\mathbb{C})$, where $\Sigma=\{p,q,p_2,\cdots,p_{g-1}\}$
is the zero set of $(X',\omega')=F_1(X,\omega)$.
From Claim \ref{clm1}, the homology class $[\gamma]=[\gamma_1+\gamma_2]$ is non-separable.
So we can choose a basis of $H^1(X,\mathbb{C})$
$$(\alpha_1,\beta_1,\cdots,\alpha_g,\beta_g),$$
where $\alpha_1=[\gamma]$.

Next we choose the relative homology class
$$(pq,pp_2\cdots,pp_{g-1}),$$
where $[\gamma_1]=pq$, and $pp_i$ is freely homotopic to $0$ in $H^1(X,\mathbb{C})$.
Together they compose a basis of $H^1(X,\Sigma;\mathbb{C})$,
and their holonomy gives a period mapping around $(X',\omega')$ locally.

Let $\omega'(\alpha_1)=(x_1,y_1)$, $\omega'(\beta_1)=(x_2,y_2)$, $\omega'(\alpha_2)=(x_3,y_3)$
$\omega'(\beta_2)=(x_4,y_4)$.
Now choose a relative cohomology class $\upsilon \in H^1(X,\Sigma;\mathbb{C})$ ,
which can be considered as a tangent vector in $T_{\H_g(1,1,\cdots,2)}(X',\omega')$ such that
\begin{equation}\label{moving}
\upsilon(\alpha_1)=(-x_1,-y_1), \upsilon(\alpha_2)=(-\frac{y_2x_1}{y_4},\frac{x_2y_1}{x_4})
\end{equation}
and on the other basis we assign zero to $\upsilon$.

\begin{remark}\label{explain}
Here $\upsilon(\alpha_1)$ is chosen to guarantee that along the curve $(X',\omega')+t\upsilon$ in moduli space the zero moves along $\gamma$,
and $\upsilon(\alpha_2)$ is chosen to guarantee the resulting surface has area $1$, since the area of $(X,\omega)$ can be written by
$$\int_X|\omega|^2=\frac{i}{2}\sum_i(\int_{\alpha_i}\omega \int_{\beta_i}\overline{\omega}-\int_{\beta_i}\omega \int_{\alpha_i}\overline{\omega}).$$
\end{remark}

Consider the curve in $\H_g(1,1,2,,\cdots,2)$: $(X',\omega')+t\upsilon, t\in [0,1]$.
If $\gamma_1$ and $\gamma_2$ don't degenerate along the curve, then we obtain
$$F_2:\mathcal{C}^{1,0}_{g,Per}(2^{g-1})[\frac{a}{\sqrt{g}},\frac{b}{\sqrt{g}}]  \to \H_g(1^2,2^{g-1})$$
such that $F_2[(X,\omega)]=F_1[(X,\omega)]+\upsilon$.
From the construction above the resulting surface has two saddle connections
in the relative homology class $\gamma_1$ and $\gamma_2$ with equal length.

\subsubsection{$F_2$ is well-defined almost everywhere}
We have defined a mapping if the saddle connections $\gamma_1$ and $\gamma_2$ are preserved along the curve $(X',\omega')+t\upsilon, t\in [0,1]$
 on $\mathcal{C}^{1,0}_{g,Per}(2^{g-1})$.
Next we will see the mapping can be defined for almost every translation surface in $\mathcal{C}^{1,0}_{g,Per}(2^{g-1})$.

From above we know the moving surgery can be realized except for some of $\gamma_1$ or $\gamma_2$ degenerate along $(X',\omega')+t\upsilon, t\in [0,1]$.
Suppose $T$ is the first time when the geodesic in $[\gamma_1]$ can be represented by $pp_i-p_ip_j-\cdots-p_kq$,
since the saddle connection is smooth on $(X',\omega')+t\upsilon,t\in [0,T]$, the corner of $pp_i-p_ip_j-\cdots-p_kq$ must be $\pi$.
But the holonomy of $pp_i$ is not changed on $(X',\omega')+t\upsilon,t\in [0,T]$
This implies there exists some $pp_i$ such that its holonomy on $(X',\omega')$ has the same direction with $p_0p_1$.
And by the relation of holonomy under $F_1$, this implies $p_0p_i$ has the same direction with the closed curve on $(X,\omega)$.
Under period mapping, on each local chart this is a measure-zero set.
So all such $(X,\omega)$ is a subset of measure zero in $ \H_g(2^{g-1})$, also in $\mathcal{C}^{1,0}_{g,Per}(2^{g-1})[\frac{a}{\sqrt{g}},\frac{b}{\sqrt{g}}]$.

\subsubsection{Pinching}
On the subset of $\mathcal{C}^{1,0}_{g,Per}(2^{g-1})[\frac{a}{\sqrt{g}},\frac{b}{\sqrt{g}}]$ where $F_2$ is well-defined,
we can cut along $\gamma_1+\gamma_2$ and denote the two copies of $p$ and $q$ by $q^+$, $q^-$, $p^+$, $p^-$,
the two copies of $\gamma_1$ and $\gamma_2$ by $\gamma^+_1$, $\gamma^-_1$, $\gamma^+_2$, $\gamma^-_2$.
Notice that the total angles at $q^+$, $q^-$, $p^+$, $p^-$ are all $2\pi$,
which means $\gamma^\pm_1$ have same direction with $\gamma^\pm_2$,
then glue $\gamma^+_1$ with $\gamma^+_2$, $\gamma^-_1$ with $\gamma^-_2$, which make $q^+$, $q^-$, $p^+$, $p^-$ become regular points.
Since $\gamma_1+\gamma_2$ is non-separable, this surgery reduces the genus by one, and obtains a new abelian differential
on which we mark two points $p^+$ and $p^-$ so as to find its inverse.
Considering the choice of which zero is selected after applying the inverse map, we finally obtain:
$$F_3: \mathcal{C}^{1,0}_{g,Per}(2^{g-1})[\frac{a}{\sqrt{g}},\frac{b}{\sqrt{g}}]\to M\times \H_{g-1}(2^{g-2},0,0)\times A_{[\frac{a}{\sqrt{g}},\frac{b}{\sqrt{g}}]},$$
where $M$ is the combinatorial data of choosing one from $g-1$ zeros.

\begin{figure}[h]
    \centering
    \includegraphics[width=0.6\linewidth]{movingandcut}
    \caption{Moving and cut}
    \label{moving and cut}
\end{figure}

\begin{figure}[h]
    \centering
    \includegraphics[width=0.6\linewidth]{pinching}
    \caption{Pinching}
    \label{pinching}
\end{figure}

\subsubsection{$F_3$ is measure-preserving.}
This surgery can be inverse except for some directions which has saddle connections, which is a measure zero subset $A_0(X,\omega)$
in $A_{[\frac{a}{\sqrt{g}},\frac{b}{\sqrt{g}}]}$ for every surface in $\H_{g-1}(2^{g-2},0,0)$:
choose a vector $\kappa$ in $A_{[\frac{a}{\sqrt{g}},\frac{b}{\sqrt{g}}]}$,
consider the two rays from $p^+$ and $p^-$ whose holonomy is $\kappa$, if the rays exist, then cut the surface along the two rays
to obtain two closed loci and glue them.
Let
$$\tilde{\H}=\{(X,\omega, \tau): (X,\omega)\in \H_{g-1}(2^{g-2},0,0), \tau \in A_0(X,\omega)\},$$
which is a measure zero subset of $\H_{g-1}(2^{g-2},0,0)\times A_{[\frac{a}{\sqrt{g}},\frac{b}{\sqrt{g}}]}$.
So we have
$$F_3( \mathcal{C}^{1,0}_{g,Per}(2^{g-1}))\subset [\H_{g-1}(2^{g-2},0,0)\times (A_{[\frac{a}{\sqrt{g}},\frac{b}{\sqrt{g}}]})]\setminus \tilde{\H}.$$

Moreover, under local chart determined by the basis we choose,
$$F_3(z_1,\cdots,z_n)=(z_1,z_i+\upsilon(z_i)),$$
whose Jacobian has deteminant one and thus is measure-preserving.

\section{Collapse closed saddle connections simultaneously}
From the construction above we can collapse one closed saddle connection on some $(X,\omega)\in \mathcal{C}^{1,0}_{g}(2^{g-1}))$ and obtain a new surface
$(X',\omega')\in \H_{g-1}(2^{g-2},0,0)$. if we want to collapse any $k$ closed saddle connections,
we have to ensure the locus of surgery for these saddle connections are disjoint.
For a given interval $[\frac{a}{\sqrt{g}},\frac{b}{\sqrt{g}}]$,
define the \emph{exception set} $\mathcal{C}^{exc}_{g,Per}(2^{g-1})[\frac{a}{\sqrt{g}},\frac{b}{\sqrt{g}}]$ to be the subset of
$\mathcal{C}^{1,0}_{g,Per}(2^{g-1})[\frac{a}{\sqrt{g}},\frac{b}{\sqrt{g}}]$
such that on every $(X,\omega)\in \mathcal{C}^{exc}_{g,Per}(2^{g-1})[\frac{a}{\sqrt{g}},\frac{b}{\sqrt{g}}]$ there are two closed saddle connections with length in
$[\frac{a}{\sqrt{g}},\frac{b}{\sqrt{g}}]$ whose surgery loci intersect.
First we will show that the measure of exception set goes to zero as genus goes to infinity.

\subsection{The measure of exception set}

For $(X,\omega)\in \mathcal{C}^{exc}_{g,Per}(2^{g-1})[\frac{a}{\sqrt{g}},\frac{b}{\sqrt{g}}]$, there are three situations:
\begin{itemize}
  \item[1]  There exist two closed saddle connections on $(X,\omega)$ with lengths in $[\frac{a}{\sqrt{g}},\frac{b}{\sqrt{g}}]$
            that do not share a zero. And the loci of opening surgery intersect or the two closed saddle connections intersect.
  \item[2]  There exist two closed saddle connections on $(X,\omega)$ with lengths in $[\frac{a}{\sqrt{g}},\frac{b}{\sqrt{g}}]$ sharing one zero.
\end{itemize}

Note that in the first situation, we can find a curve connecting the two zeros with length no more than $\frac{2b}{\sqrt{g}}$.

Fix $B\in \mathbb{R}^+$, define $\mathcal{C}_g^{1}(\frac{B}{\sqrt{g}})$ be the set of $(X,\omega)\in \mathcal{C}^{1,0}_{g,Per}(2^{g-1})[0,\frac{B}{\sqrt{g}}]$
such that there exist two closed saddle connections on $(X,\omega)$
that do not share a zero, moreover the lengths of the two closed saddle connections and the distance between the two zeros are less than $\frac{B}{\sqrt{g}}$.
Define $\mathcal{C}_g^{2}(\frac{B}{\sqrt{g}})$ be the set
of $(X,\omega)\in \mathcal{C}^{1,0}_{g,Per}(2^{g-1})[0,\frac{B}{\sqrt{g}}]$ such that there exists two closed saddle connections with lengths less than $\frac{B}{\sqrt{g}}$ sharing one zero.
Obviously for $B\geq 2b$, we have
$$ \mathcal{C}^{exc}_{g,Per}(2^{g-1})[\frac{a}{\sqrt{g}},\frac{b}{\sqrt{g}}]\subset \mathcal{C}_g^{1}(\frac{B}{\sqrt{g}}) \cup \mathcal{C}_g^{2}(\frac{B}{\sqrt{g}}).$$

We need to compute the measure of $\mathcal{C}_g^{1}(\frac{B}{\sqrt{g}})$ and $\mathcal{C}_g^{2}(\frac{B}{\sqrt{g}})$ when $g\to \infty$.
First consider $\mathcal{C}_g^{2}(\frac{B}{\sqrt{g}})$, we have
\begin{proposition}\label{chain}
$$\lim_{g\to \infty}\frac{\V(\mathcal{C}_g^{2}(\frac{B}{\sqrt{g}}))}{\V(\H_g(2^{g-1}))} = 0.$$
\end{proposition}

\begin{proof}
For $(X,\omega) \in \mathcal{C}_g^{2}(\frac{B}{\sqrt{g}})$,
let $p$ be a zero on $(X,\omega)$ and $\gamma_1, \gamma_2$ be the closed saddle connections with lengths less than $\frac{B}{\sqrt{g}}$ at $p$.

As above we can collapse $\gamma_1$ and get a new translation surface in $\H_{g-1}(2^{g-2},0,0)$.
After the surgery $\gamma_2$ will become a segment connecting two marked regular points which are from the pinching surgery.
So the image of mapping $F_3$ on $\mathcal{C}_g^{2}(\frac{B}{\sqrt{g}})$ is in
$\H_{g-1,\frac{B}{\sqrt{g}}}(2^{g-2},0,0)\times D(\frac{B}{\sqrt{g}})$,
where $\H_{g-1,\frac{B}{\sqrt{g}}}(2^{g-2},0,0)$ is the subset that there exists a segment connnecting the two marked points of length less than
$\frac{B}{\sqrt{g}}$ and $D(\frac{B}{\sqrt{g}})$ is the disk of radius $\frac{B}{\sqrt{g}}$.
From \cite[Theorem 1.2]{aggarwal2019large}, the Siegel-Veech constant of saddle connections connecting two fix zeros of order $m_1$ and $m_2$ is
$$c=(m_1+1)(m_2+1)(1+O(\frac{1}{g})).$$
So we have
$$\V(\mathcal{C}_g^{2}(\frac{B}{\sqrt{g}}))\leq (g-1)\frac{B^2}{g}\V(\H_{g-1,\frac{B}{\sqrt{g}}}(2^{g-2},0,0)) \leq c(g-1)\frac{B^4}{g^2}\V(\H_{g-1}(2^{g-2},0,0)),$$
where the coefficient $g-1$ is the choice of $p$.
Then by \ref{Volume}, we have
$$\lim_{g\to \infty}\frac{\V(\mathcal{C}_g^{2}(\frac{B}{\sqrt{g}}))}{\V(\H_g(2^{g-1}))}\leq \lim_{g\to \infty}3c\frac{B^4}{g}=0$$
\end{proof}

For $\mathcal{C}_g^{1}(\frac{B}{\sqrt{g}})$ we also have
\begin{proposition}\label{intersect}
$$\lim_{g\to \infty}\frac{\V(\mathcal{C}_g^{1}(\frac{B}{\sqrt{g}}))}{\V(\H_g(2^{g-1}))} = 0.$$
\end{proposition}

\begin{proof}
For $(X,\omega) \in \mathcal{C}_g^{1}(\frac{B}{\sqrt{g}})$, let $p_1, p_2$ be the two zeros on $(X,\omega)$ with distance less than $\frac{B}{\sqrt{g}}$
and $\gamma_1, \gamma_2$ be the two closed saddle connections connecting $p_1, p_2$.
For every $k\in \mathbb{N}^+$, denote by $\mathcal{C}_g^{1,k}(\frac{B}{\sqrt{g}})$ the subset of $\mathcal{C}_g^{1}(\frac{B}{\sqrt{g}})$
that the geodesic between $p_1$ and $p_2$ is a concatenation of $k$ open saddle connections.
Their lengths are less than $\frac{B}{\sqrt{g}}$.
Choose the shortest saddle connection, using the surgery in \cite{masur2024lengths} we can collapse it and get a new translation surface in
$\H_g(3,1,2^{g-3})$. This is because the locus, that is the extension of the shortest saddle connection can not intersect the whole geodesic.
Repeat the surgery until $p_1$ and $p_2$ are collapsed to one zero and we get a map
$$F:\mathcal{C}_g^{1,k}(\frac{B}{\sqrt{g}})\to \mathcal{C}_g^{2}(2+k,1^k,2^{g-2-k})(\frac{B}{\sqrt{g}})\times D^k(\frac{B}{\sqrt{g}}),$$
where $\mathcal{C}_g^{2}(2+k,1^k,2^{g-2-k})(\frac{B}{\sqrt{g}})$ is the subset of $\H_g(2+k,1^k,2^{g-2-k})$ such that there are
two closed saddle connections $\gamma_1, \gamma_2$ connecting the zero of order $2+k$ with lengths less than $\frac{B}{\sqrt{g}}$ .

Similarly one can collapse $\gamma_1$ and the other will become a saddle connection connecting two zeros.
Note that the order is larger than $2$, although the collapsing surgery can also be realized, we need to require the angles
on the two sides of the closed saddle connections collapsed, which decide the order of zeros after collapsing.
We will explain this situation in detail in Section \ref{general case}.

Let $\mathcal{C}_g^{2,b',b''}(2+k,1^k,2^{g-2-k})(\frac{B}{\sqrt{g}})$ be the subset of
$\mathcal{C}_g^{2}(2+k,1^k,2^{g-2-k})(\frac{B}{\sqrt{g}})$ such that
the two angles on both sides of $\gamma_1$ are $(2b'+1)\pi$ and $(2b''+1)\pi$, where $b'+b''=2+k$.
Then after collapsing we get a map
$$F':\mathcal{C}_g^{2,b',b''}(2+k,1^k,2^{g-2-k})(\frac{B}{\sqrt{g}}) \to \H_{g-1,\frac{B}{\sqrt{g}}}(b'-1,b''-1,1^k,2^{g-2-k})\times D(\frac{B}{\sqrt{g}}).$$

Similar to Proposition \ref{chain}, we have
$$\V(\mathcal{C}_g^{2,b',b''}(2+k,1^k,2^{g-2-k})(\frac{B}{\sqrt{g}}))\leq b'b''\frac{B^4}{g^2}\V(\H_{g-1}(b'-1,b''-1,1^k,2^{g-2-k})).$$

Sum all $k$ and $(b',b'')$ we have
$$\V(\mathcal{C}_g^{1}(\frac{B}{\sqrt{g}}))\leq (g-1)(g-2)\sum_k \sum_{(b',b'')}b'b''\frac{B^4}{g^2}\V(\H_{g-1}(b'-1,b''-1,1^k,2^{g-2-k}))(\frac{B^2}{g})^k,$$
where the coefficient $(g-1)(g-2)$ is the choice of $p_1$ and $p_2$.
Then when $g\to \infty$ we have
$$\frac{\V(\mathcal{C}_g^{1}(\frac{B}{\sqrt{g}}))}{\V(\H_g(2^{g-1}))}\leq
\sum_k \sum_{(b',b'')}b'b''B^4\frac{\V(\H_{g-1}(b'-1,b''-1,1^k,2^{g-2-k}))}{\V(\H_g(2^{g-1}))}(\frac{B^2}{g})^k=O(\frac{1}{g}).$$
\end{proof}

Combine the two proposition we have
\begin{corollary}
$$\frac{\V(\mathcal{C}^{exc}_{g,Per}(2^{g-1})[\frac{a}{\sqrt{g}},\frac{b}{\sqrt{g}}])}{\V(\H_g(2^{g-1}))}=O(\frac{1}{g}).$$
\end{corollary}

\subsection{Collapsing simultaneously}
Now fix a positive integer $K$ and $n_1,\cdots,n_k$ a partition of $K$.
We want to compute the limit of the expectation
$$(L_{g,[a_1,b_1]})_{n_1}\cdots (L_{g,[a_k,b_k]})_{n_k},$$
for which we will collapse $K$ closed saddle connections in order.

Define $\hat{\mathcal{C}}^{1,0}_{g}(2^{g-1})=(X,\omega,\Gamma_1,\cdots,\Gamma_k)$,
where $(X,\omega)\in \mathcal{C}^{1,0}_{g}(2^{g-1})$,
and $\Gamma_i$ is an ordered list of $n_i$ closed saddle connections $(\gamma_{1,i},\cdots,\gamma_{n_i,i})$
with lengths in $[\frac{a_i}{\sqrt{g}},\frac{b_i}{\sqrt{g}}]$.
Suppose $a_1\leq b_1 \leq a_2\leq b_2 \cdots \leq a_k \leq b_k$, let $B=b_k$.
Then we have
$$\bigcup_i \mathcal{C}^{exc}_{g,Per}(2^{g-1})[\frac{a_i}{\sqrt{g}},\frac{b_i}{\sqrt{g}}]\subset \mathcal{C}_g^{1}(\frac{B}{\sqrt{g}})\cup \mathcal{C}_g^{2}(\frac{B}{\sqrt{g}}).$$

Let $\mathcal{C}'_{g,Per}=\mathcal{C}^{1,0}_{g,Per}(2^{g-1})\setminus (\mathcal{C}_g^{1}(\frac{B}{\sqrt{g}})\cup \mathcal{C}_g^{2}(\frac{B}{\sqrt{g}}))$,
we will adjust the surgery from last section on $\mathcal{C}'_{g,Per}$ to collapse the closed saddle connections in
$\Gamma_1,\cdots,\Gamma_k$ in order.
The opening and pinching operations are same to the situation of one curve. The only modification is the equation \ref{moving}:
by remark \ref{explain}, we need to choose the moving vector $\upsilon$ to ensure the area unchanged,
hence we can choose a basis containing $\gamma'_1,\gamma_1\cdots,\gamma'_K,\gamma_K$,
where $\gamma'_i$ is a saddle connection from opening $\gamma_i$.
Then the moving vector $\upsilon$ can be constructed like equation \ref{moving}: after defining the value on $\gamma_i$,
we can choose another relative homotopy class and give a value of $\upsilon$ on it to ensure the area unchanged.
This means we open and move the closed saddle connections simultaneously and then pinch them.

Finally  we obtain a mapping
$$\hat{F}: \hat{\mathcal{C}}'_{g,Per}(2^{g-1}) \to M_K\cdot \tilde{\H}_{g-K}(2^{g-1-K},0^{2K}) \times \prod^k_{i=1} (A_{[\frac{a_i}{\sqrt{g}},\frac{b_i}{\sqrt{g}}]})^{n_i},$$
where $\hat{\mathcal{C}}'_{g,Per}(2^{g-1})$ is defined as $\hat{\mathcal{C}}^{1,0}_{g}(2^{g-1})$
and $M_K$ is a combinatorial data consisting of choosing $K$ ordered zeros from $g-1$ zeros to label. As the discussion above,
we have
\begin{equation*}
\begin{aligned}
&\int_{\hat{\mathcal{C}}'_{g,Per}(2^{g-1})}(L_{g,[a_1,b_1]})_{n_1}\cdots (L_{g,[a_k,b_k]})_{n_k}d\mu_{MV}\\&=\V(\hat{\mathcal{C}}'_{g}(2^{g-1})))
\\&=|M_K|\prod^k_{i=1}[\frac{\pi(b^2-a^2)}{g}]^{n_i}\V(\H_{g-k}(2^{g-1-K}))\\
&\to \prod^k_{i=1}[\pi(b_i^2-a_i^2)]^{n_i}\V(\H_{g-K}(2^{g-1-K})), g\to \infty,
\end{aligned}
\end{equation*}
where the limit is because $|M_K|=(g-1)\cdots (g-K)$ and
$$\lim_{g\to \infty}\frac{(g-1)\cdots (g-K)}{g^{n_1+\cdots+n_k}}=\frac{(g-1)\cdots (g-K)}{g^K}=1.$$

Let $L'_{g,[a_i,b_i]}$ be the restriction of $L_{g,[a_i,b_i]}$ on $\mathcal{C}'_{g,Per}(2^{g-1})$.
Then the limit of factorial moment of $(L'_{g,[a_i,b_i]})_i$ is
$$\lim_{g\to \infty}\mathbb{E}[(L'_{g,[a_1,b_1]})_{n_1}\cdots (L'_{g,[a_k,b_k]})_{n_k}]=\lim_{g\to \infty}\frac{\V(\H_{g-K}(2^{g-1-K}))}{\V(\mathcal{C}'_{g,Per}(2^{g-1}))}\prod^k_{i=1}[\pi(b_i^2-a_i^2)]^{n_i}.$$
From Proposition \ref{chain} and \ref{intersect}, we have
$$\lim_{g\to \infty}\frac{\V(\mathcal{C}'_{g,Per}(2^{g-1}))}{\V(\H_g(2^{g-1}))}=1.$$

From Theorem \ref{Volume}, we have
$$\lim_{g\to \infty}\frac{\V(\H_{g-K}(2^{g-1-K}))}{\V(\H_{g}(2^{g-1}))}=3^K.$$
So we have
$$\lim_{g\to \infty}\mathbb{E}[(L'_{g,[a_1,b_1]})_{n_1}\cdots (L'_{g,[a_k,b_k]})_{n_k}]=\prod^k_{i=1}[3\pi(b_i^2-a_i^2)]^{n_i}.$$
By theorem \ref{moment}, $(L'_{g,[a_1,b_1]},\cdots,L'_{g,[a_k,b_k]})$ converges jointly in distribution to a vector of random variables with Poisson distributions of means $\lambda_{[a_i,b_i]}$, where
$$\lambda_{[a_i,b_i]}=3\pi(b_i^2-a_i^2).$$

From the property of Poisson distribution, we have
$$\mathbb{P}(L'_{g,[a_1,b_1]}=n_1,\cdots,L'_{g,[a_k,b_k]}=n_k)=\prod_{i=1}^k\frac{\lambda_{[a_i,b_i]}^{n_i}e^{-\lambda_{[a_i,b_i]}}}{n_i!}.$$
On the other hand
$$\mathbb{P}(L'_{g,[a_1,b_1]}=n_1,\cdots,L'_{g,[a_k,b_k]}=n_k)=\frac{\V(\{(X,\omega):L'_{g,[a_i,b_i]}(X,\omega)=n_i,i=1,\cdots,k\})}{\V(\mathcal{C}'_{g,Per}(2^{g-1}))}.$$
Note that
\begin{equation*}
\begin{aligned}
&\V(\{(X,\omega):L_{g,[a_i,b_i]}(X,\omega)=n_i,i=1,\cdots,k\})-
\V(\mathcal{C}_g^{1}(\frac{B}{\sqrt{g}})\cup \mathcal{C}_g^{2}(\frac{B}{\sqrt{g}}))\\ &\leq \V(\{(X,\omega):L'_{g,[a_i,b_i]}(X,\omega)=n_i,i=1,\cdots,k\})\\&\leq
\V(\{(X,\omega):L_{g,[a_i,b_i]}(X,\omega)=n_i,i=1,\cdots,k\}).
\end{aligned}
\end{equation*}
From Proposition \ref{chain} and \ref{intersect}, we have
$$\lim_{g\to \infty}\frac{\V(\mathcal{C}'_{g,Per}(2^{g-1}))}{\V(\H_g(2^{g-1}))}=1,\\ \lim_{g\to \infty}\frac{\V(\mathcal{C}_g^{1}(\frac{B}{\sqrt{g}})\cup \mathcal{C}_g^{2}(\frac{B}{\sqrt{g}}))}{\V(\H_g(2^{g-1}))}=0.$$
So
\begin{equation*}
\begin{aligned}
&\mathbb{P}(L_{g,[a_1,b_1]}=n_1,\cdots,L_{g,[a_k,b_k]}=n_k)-
\frac{\V(\mathcal{C}_g^{1}(\frac{B}{\sqrt{g}})\cup \mathcal{C}_g^{2}(\frac{B}{\sqrt{g}}))}{\V(\H_g(2^{g-1}))}\\ &\leq \mathbb{P}(L'_{g,[a_1,b_1]}=n_1,\cdots,L'_{g,[a_k,b_k]}=n_k)\frac{\V(\mathcal{C}'_{g,Per}(2^{g-1}))}{\V(\H_g(2^{g-1}))}\\&\leq
\mathbb{P}(L_{g,[a_1,b_1]}=n_1,\cdots,L_{g,[a_k,b_k]}=n_k).
\end{aligned}
\end{equation*}

Let $g \to \infty$ we have
$$\mathbb{P}(L_{g,[a_1,b_1]}=n_1,\cdots,L_{g,[a_k,b_k]}=n_k)=\mathbb{P}(L'_{g,[a_1,b_1]}=n_1,\cdots,L'_{g,[a_k,b_k]}=n_k).$$
So $(L_{g,[a_1,b_1]},\cdots,L_{g,[a_k,b_k]})$ also converges jointly in distribution to a vector of random variables
with Poisson distributions of means $\lambda_{[a_i,b_i]}$, where
$$\lambda_{[a_i,b_i]}=3\pi(b_i^2-a_i^2).$$

Since we don't need to deal with simple zeros in the collapsing surgery,
samilarly to the proof of theorem \ref{closed}, we can prove Corollary \ref{general}.

\section{General stratum}\label{general case}
This section we consider the stratum $\H_g(m^{O(g)},1^{2g-2-mO(g)})$, where $m\geq 3$.

If a stratum has zeros of order more than $3$, the angles at both sides are not certain.
But we can consider some fixed configuration of closed saddle connection.
For a configuration $(J,b'_k,b''_k,a'_i,a''_i)$ of closed saddle connection.
We have known it suffices to consider multiplicity $1$ without cylinders, then the configuration become $(b',b'')$, where $b'+b''=m$,
which means the angles of the closed saddle connection are $(2b'+1)\pi$ on one side and $(2b''+1)\pi$ on the other side.
Now we choose all such configuration at each $m$-order zeros.

For $k$ such closed saddle connections, we can use the surgery in Section \ref{surgery} to collapse them.
Note that in this situation the opening surgery will replace the zero of order $m$ to two zeros of order $1$ and $m-1$,
and the angles at both sides of the zero of $m-1$ order are $(2b')\pi$ and $(2b'')\pi$. Then after moving and pinching,
the simple zero become two regular points and the zero of $m-1$ order become two zeros of order $b'-1$ and $b''-1$.
Finally we obtain a mapping
$$\hat{F}: \hat{\mathcal{C}}'_{g,Per}(m^{O(g)},1^{O'(g)})\to \tilde{\H}_{g-K}((b'-1)^K,(b''-1)^K,m^{O(g)-K},1^{O'(g)})\times \prod^k_{i=1} (A_{[\frac{a_i}{\sqrt{g}},\frac{b_i}{\sqrt{g}}]})^{n_i},$$
where $O'(g)=2g-2-mO(g)$.
Again we have
$$\lim_{g\to \infty}\mathbb{E}[(L'_{g,[a_1,b_1]})_{n_1}\cdots (L'_{g,[a_k,b_k]})_{n_k}]=\prod^k_{i=1}(\frac{(m+1)\pi(b_i^2-a_i^2)}{b'b''})^{n_i}.$$

And similar to the proof of theorem \ref{closed} we have
\begin{theorem}
For the stratum $\H(m^{O(g)},1^{2g-2-mO(g)})$, a configuration $(b',b'')$ and $a,b\in \mathbb{R}$,
let the random variable
$$L_{\mathcal{C},g,[a,b]}:\H(m^{O(g)},1^{2g-2-mO(g)}) \to \mathbb{N}_0$$
be the number of closed saddle connections on $(X,\omega)$ satisfying the configuration with lengths in $[\frac{a}{\sqrt{g}},\frac{b}{\sqrt{g}}]$.
Then for disjoint intervals $[a_1,b_1],\cdots,[a_k,b_k]$, when $g\to \infty$, the random variable sequence
$$(L_{\mathcal{C},g,[a_1,b_1]},\cdots,L_{\mathcal{C},g,[a_k,b_k]}):\H(m^{O(g)},1^{2g-2-mO(g)}) \to \mathbb{N}^k_0$$
converges jointly in distribution to a vector of random variables with Poisson distributions of means $\lambda_{[a_i,b_i]}$, where
$$\lambda_{[a_i,b_i]}=\frac{(m+1)\pi(b_i^2-a_i^2)}{b'b''}.$$

\end{theorem}

In 2024, Masur, Rafi and Randecker study the distribution of open saddle connections on a random translation surface in \cite{masur2024lengths}.
In this paper we study the distribution of closed saddle connections.

A translation surface can be denoted by $(X,\omega)$ where $X$ is a closed Riemann surface of genus $g$ and $\omega$ is an Abelian differential on $X$.
$(X,\omega)$ has a flat structure defined by the conical metric $|\omega|$, with singularities at the zeros of $\omega$.
If an $|\omega|$-geodesic of $(X,\omega)$ connects two zeros (or one zero to itself) of $\omega$ and has no other zeros in its interior,
we call it an open (or closed) saddle connection.

Let $\H_g$ be the moduli space of unit-area translation surfaces of genus $g$.
$\H_g$ has a natural stratification according to the zero order of Abelian differentials.
Particularly, the stratum consisting of all abelian differentials in $\H_g$ which have only simple zeros is called principal stratum,
denoted by $\H_g(1^{2g-2})$. For convenience, if a stratum consists of Abelian differentials on which the number of zeros of order $d_i$ is $\kappa_i$
we denote it by
$$\H_g(d_1^{\kappa_1},\cdots,d_k^{\kappa_k})$$
where $\sum^{k}_{i=1}\kappa_id_i=2g-2$.

Together with a finite natural measure which is introduced by Masur and Veech denoted by $\mu_{MV}$,
every stratum $\H_g(\kappa)$ of $\H_g$ becomes a probability space
with probability measure $\frac{\mu_{MV}}{\operatorname{Vol}{\H_g(\kappa)}}$, where $\operatorname{Vol}{\H_g(\kappa)}$ is the measure of total stratum,
which we call the Masur-Veech volume of $\H_g(\kappa)$.

Given $(X,\omega)\in \H_g(1^{2g-2})$ and $a,b\in \mathbb{R}^+$,
let $N_{g,[a,b]}(X,\omega)$ be the number of open saddle connections with lengths in the interval $[\frac{a}{g},\frac{b}{g}]$ on $(X,\omega)$.
Then $N_{g,[a,b]}:\H_g(1^{2g-2})\to \mathbb{N}_0$ becomes a random variable.
In \cite{masur2024lengths}, Masur, Rafi and Randecker proved :

Together with a finite natural measure which is introduced by Masur and Veech denoted by $\mu_{MV}$,
every stratum $\H_g(\kappa)$ of $\H_g$ becomes a probability space
with probability measure $\frac{\mu_{MV}}{\V{\H_g(\kappa)}}$, where $\V{\H_g(\kappa)}$ is the measure of total stratum,
which we call the Masur-Veech volume of $\H_g(\kappa)$.

Their method is based on the relationship between Poisson distribution and its factorial moment \cite{bollobas2001random},
which was used first in the work of Mirzakhani–Petri \cite{mirzakhani2019lengths} to study the distribution of closed hyperbolic geodesics on random hyperbolic surfaces in Teichm\"uller space with respect to Weil-Peterson measure.

\begin{theorem}\label{closed}
For the stratum $\H(2^{g-1})$ and disjoint intervals $[a_1,b_1],\cdots,[a_k,b_k]$, when $g\to \infty$, the random variable sequence
$$(L_{g,[a_1,b_1]},\cdots,L_{g,[a_k,b_k]}):\H(2^{g-1}) \to \mathbb{N}^k_0$$
converges jointly in distribution to a vector of random variables with Poisson distributions of means $\lambda_{[a_i,b_i]}$, where
$$\lambda_{[a_i,b_i]}=3\pi(b_i^2-a_i^2).$$
\end{theorem}
\begin{remark}
We will expalin the question is trivial for the principal stratum in Section \ref{claim}:
the probability measure of $\mathcal{C}_g(1^{2g-2})[\frac{a}{\sqrt{g}},\frac{b}{\sqrt{g}}]$ is zero when $g\to \infty$.
For principal stratum, the length interval to be considered should be $[a,b]$.
\end{remark}

Recall that a random variable $X:\Omega \to \mathbb{N}_0$ is Poisson distributed with mean $\lambda$ if
$$\mathbb{P}(X=k)=\frac{\lambda^k e^{-\lambda}}{k!}.$$
The following theorem from \cite{bollobas2001random} gives the relationship between factorial moment and distribution for Poisson distribution:
\begin{theorem}[The method of moment from \cite{bollobas2001random}, Theorem 1.23]\label{moment}
Let $\left\{(\Omega_i,\mathbb{P}_i)\right\}_{i\in \mathbb{N}}$ be a sequence of probability spaces.
For $m\in \mathbb{N}$, let $N_{1,i},\cdots,N_{m,i}:\Omega_i\to \mathbb{N}_0$ be random variables for all $i\in \mathbb{N}$
and suppose there exist $\lambda_1,\cdots,\lambda_m\in(0,\infty)$ such that:
$$\lim_{i\to \infty}\mathbb{E}[(N_{1,i})_{k_1}\cdots(N_{m,i})_{k_m}]=\lambda_1^{k_1}\cdots\lambda_m^{k_m}$$
for all $k_1\cdots k_m\in \mathbb{N}$. Then
$$\lim_{i\to \infty}\mathbb{P}[(N_{1,i})=n_1,\cdots,(N_{m,i})=n_m]=\prod^k_{j=1}\frac{\lambda_j^{n_j} e^{-\lambda_j}}{n_j!}.$$
That is, $(N_{1,i},\cdots,N_{m,i}):\Omega_i \to \mathbb{N}^m_0$ converges jointly in distribution to a vector which is independently Poisson distributed with means
$\lambda_1,\cdots,\lambda_m$.

From Theorem \ref{Volume}, we have
$$\lim_{g\to \infty}\frac{\V(\H_{g-K}(2^{g-1-K}))}{\V(\H_{g}(2^{g-1}))}=3^K.$$
So we have
$$\lim_{g\to \infty}\mathbb{E}[(L'_{g,[a_1,b_1]})_{n_1}\cdots (L'_{g,[a_k,b_k]})_{n_k}]=\prod^k_{i=1}[3\pi(b_i^2-a_i^2)]^{n_i}.$$
By theorem \ref{moment}, $(L'_{g,[a_1,b_1]},\cdots,L'_{g,[a_k,b_k]})$ converges jointly in distribution to a vector of random variables with Poisson distributions of means $\lambda_{[a_i,b_i]}$, where
$$\lambda_{[a_i,b_i]}=3\pi(b_i^2-a_i^2).$$

From the property of Poisson distribution, we have
$$\mathbb{P}(L'_{g,[a_1,b_1]}=n_1,\cdots,L'_{g,[a_k,b_k]}=n_k)=\prod_{i=1}^k\frac{\lambda_{[a_i,b_i]}^{n_i}e^{-\lambda_{[a_i,b_i]}}}{n_i!}.$$
On the other hand
$$\mathbb{P}(L'_{g,[a_1,b_1]}=n_1,\cdots,L'_{g,[a_k,b_k]}=n_k)=\frac{\V(\{(X,\omega):L'_{g,[a_i,b_i]}(X,\omega)=n_i,i=1,\cdots,k\})}{\V(\mathcal{C}'_{g,Per}(2^{g-1}))}.$$
Note that
\begin{equation*}
\begin{aligned}
&\V(\{(X,\omega):L_{g,[a_i,b_i]}(X,\omega)=n_i,i=1,\cdots,k\})-
\V(\mathcal{C}_g^{1}(\frac{B}{\sqrt{g}})\cup \mathcal{C}_g^{2}(\frac{B}{\sqrt{g}}))\\ &\leq \V(\{(X,\omega):L'_{g,[a_i,b_i]}(X,\omega)=n_i,i=1,\cdots,k\})\\&\leq
\V(\{(X,\omega):L_{g,[a_i,b_i]}(X,\omega)=n_i,i=1,\cdots,k\}).
\end{aligned}
\end{equation*}
From Proposition \ref{chain} and \ref{intersect}, we have
$$\lim_{g\to \infty}\frac{\V(\mathcal{C}'_{g,Per}(2^{g-1}))}{\V(\H_g(2^{g-1}))}=1,\\ \lim_{g\to \infty}\frac{\V(\mathcal{C}_g^{1}(\frac{B}{\sqrt{g}})\cup \mathcal{C}_g^{2}(\frac{B}{\sqrt{g}}))}{\V(\H_g(2^{g-1}))}=0.$$
So
\begin{equation*}
\begin{aligned}
&\mathbb{P}(L_{g,[a_1,b_1]}=n_1,\cdots,L_{g,[a_k,b_k]}=n_k)-
\frac{\V(\mathcal{C}_g^{1}(\frac{B}{\sqrt{g}})\cup \mathcal{C}_g^{2}(\frac{B}{\sqrt{g}}))}{\V(\H_g(2^{g-1}))}\\ &\leq \mathbb{P}(L'_{g,[a_1,b_1]}=n_1,\cdots,L'_{g,[a_k,b_k]}=n_k)\frac{\V(\mathcal{C}'_{g,Per}(2^{g-1}))}{\V(\H_g(2^{g-1}))}\\&\leq
\mathbb{P}(L_{g,[a_1,b_1]}=n_1,\cdots,L_{g,[a_k,b_k]}=n_k).
\end{aligned}
\end{equation*}

Let $g \to \infty$ we have
$$\mathbb{P}(L_{g,[a_1,b_1]}=n_1,\cdots,L_{g,[a_k,b_k]}=n_k)=\mathbb{P}(L'_{g,[a_1,b_1]}=n_1,\cdots,L'_{g,[a_k,b_k]}=n_k).$$
So $(L_{g,[a_1,b_1]},\cdots,L_{g,[a_k,b_k]})$ also converges jointly in distribution to a vector of random variables
with Poisson distributions of means $\lambda_{[a_i,b_i]}$, where
$$\lambda_{[a_i,b_i]}=3\pi(b_i^2-a_i^2).$$

And similar to the proof of theorem \ref{closed} we have
\begin{theorem}
For the stratum $\H(m^{O(g)},1^{2g-2-mO(g)})$, a configuration $(b',b'')$ and $a,b\in \mathbb{R}$,
let the random variable
$$L_{\mathcal{C},g,[a,b]}:\H(m^{O(g)},1^{2g-2-mO(g)}) \to \mathbb{N}_0$$
be the number of closed saddle connections on $(X,\omega)$ satisfying the configuration with lengths in $[\frac{a}{\sqrt{g}},\frac{b}{\sqrt{g}}]$.
Then for disjoint intervals $[a_1,b_1],\cdots,[a_k,b_k]$, when $g\to \infty$, the random variable sequence
$$(L_{\mathcal{C},g,[a_1,b_1]},\cdots,L_{\mathcal{C},g,[a_k,b_k]}):\H(m^{O(g)},1^{2g-2-mO(g)}) \to \mathbb{N}^k_0$$
converges jointly in distribution to a vector of random variables with Poisson distributions of means $\lambda_{[a_i,b_i]}$, where
$$\lambda_{[a_i,b_i]}=\frac{(m+1)\pi(b_i^2-a_i^2)}{b'b''}.$$

Their method is based on the relationship between Poisson distribution and its factorial moment \cite{bollobas2001random},
which was used first in the work of Mirzakhani–Petri \cite{mirzakhani2019lengths} to study the distribution of closed hyperbolic geodesics on random hyperbolic surfaces in Teichm\"uller space with respect to Weil-Peterson measure.

As for translation surface, to use the method,
\cite{masur2024lengths} introduces a surgery to collapse the open saddle connections to obtain a new general translation surface in a new stratum.
This surgery does not work for closed saddle connections and they proposed the distribution question in the situation of closed saddle connections,
which inspires this work.
This paper gives a new surgery to collapse a closed saddle connection, which ensures the same method can be used for closed saddle connections.

Let $L_{g,[a,b]}(X,\omega)$ be the number of closed saddle connections on $(X,\omega)$ with lengths in $[\frac{a}{\sqrt{g}},\frac{b}{\sqrt{g}}]$.
Denote by $\mathcal{C}_g(\kappa)[\frac{a}{\sqrt{g}},\frac{b}{\sqrt{g}}]$ the subset of $\H_g(\kappa)$ consisting of translation
surfaces on which there exist closed saddle connections
with lengths in $[\frac{a}{\sqrt{g}},\frac{b}{\sqrt{g}}]$.
The main result of this paper is the following theorem

\begin{theorem}\label{closed}
For the stratum $\H(2^{g-1})$ and disjoint intervals $[a_1,b_1],\cdots,[a_k,b_k]$, when $g\to \infty$, the random variable sequence
$$(L_{g,[a_1,b_1]},\cdots,L_{g,[a_k,b_k]}):\H(2^{g-1}) \to \mathbb{N}^k_0$$
converges jointly in distribution to a vector of random variables with Poisson distributions of means $\lambda_{[a_i,b_i]}$, where
$$\lambda_{[a_i,b_i]}=3\pi(b_i^2-a_i^2).$$
\end{theorem}
\begin{remark}
We will expalin the question is trivial for the principal stratum in Section \ref{claim}:
the probability measure of $\mathcal{C}_g(1^{2g-2})[\frac{a}{\sqrt{g}},\frac{b}{\sqrt{g}}]$ is zero when $g\to \infty$.
For principal stratum, the length interval to be considered should be $[a,b]$.
\end{remark}

\begin{corollary}\label{general}
Let $\H_g(2^{O(g)},1^{2g-2-2O(g)})$ be a stratum with $O(g)$ double zeros, then the result in Theorem \ref{closed} is also true.
\end{corollary}

Their method is based on the relationship between Poisson distribution and its factorial moment \cite{bollobas2001random}, which was used first in the work of Mirzakhani--Petri \cite{mirzakhani2019lengths}. For translation surfaces, \cite{masur2024lengths} introduces a surgery to collapse the open saddle connections to obtain a new general translation surface in a new stratum. "This surgery does not work for closed saddle connections"; the paper "gives a new surgery to collapse a closed saddle connection, which ensures the same method can be used for closed saddle connections."

\label{open}
Let $[a_1,b_1],\cdots,[a_k,b_k]$ be $k$ disjoint intervals. Then as $g\to \infty$, the vector of random variables
$$(N_{g,[a_1,b_1]},\cdots,N_{g,[a_k,b_k]}):\H_g(1^{2g-2})\to \mathbb{N}_0^k$$
converges jointly in distribution to a vector of random variables with Poisson distributions of means $\lambda_{[a_i,b_i]}$, where
$$\lambda_{[a_i,b_i]}=8\pi(b_i^2-a_i^2)$$
for $i=1,\cdots,k$. That is,
$$\mathbb{P}(N_{g,[a_1,b_1]}=n_1,\cdots,N_{g,[a_k,b_k]}=n_k)=\prod_{i=1}^k\frac{\lambda_{[a_i,b_i]}^{n_i}e^{-\lambda_{[a_i,b_i]}}}{n_i!}$$,

Question: Let H_g(1^{2g-2}) be the principal stratum of unit-area translation surfaces of genus g, consisting of pairs (X, omega) where X is a closed Riemann surface of genus g and omega is an Abelian differential with only simple zeros, equipped with the normalized Masur-Veech probability measure. For a,b > 0 and (X, omega) in H_g(1^{2g-2}), let N_{g,[a,b]}(X, omega) be the number of open saddle connections on (X, omega) (that is, |omega|-geodesic segments joining two zeros of omega and having no zeros in their interior) whose lengths lie in [a/g, b/g]. For fixed disjoint intervals [a_1,b_1], ..., [a_k,b_k] in R_{>0}, which statement holds as g -> infinity for the random vector (N_{g,[a_1,b_1]}, ..., N_{g,[a_k,b_k]})?

The vector (N_{g,[a_1,b_1]}, ..., N_{g,[a_k,b_k]}) converges jointly in distribution to a vector of independent Poisson random variables with means lambda_[a_i,b_i] = 8*pi*(b_i^2 - a_i^2) for i = 1, ..., k. Equivalently, for every n_1, ..., n_k in N_0, lim_{g -> infinity} P(N_{g,[a_1,b_1]} = n_1, ..., N_{g,[a_k,b_k]} = n_k) = product_{i=1}^k (lambda_[a_i,b_i]^{n_i} e^{-lambda_[a_i,b_i]} / n_i!).

Choices:
B. The vector (N_{g,[a_1,b_1]}, ..., N_{g,[a_k,b_k]}) converges jointly in distribution to a vector of independent Poisson random variables with means lambda_[a_i,b_i] = 8*pi*(b_i - a_i) for i = 1, ..., k. Equivalently, for every n_1, ..., n_k in N_0, lim_{g -> infinity} P(N_{g,[a_1,b_1]} = n_1, ..., N_{g,[a_k,b_k]} = n_k) = product_{i=1}^k (lambda_[a_i,b_i]^{n_i} e^{-lambda_[a_i,b_i]} / n_i!).
C. For each fixed i = 1, ..., k, the random variable N_{g,[a_i,b_i]} converges in distribution, as g -> infinity, to a Poisson random variable with mean 8*pi*(b_i^2 - a_i^2).
D. The vector (N_{g,[a_1,b_1]}, ..., N_{g,[a_k,b_k]}) converges jointly in distribution to a vector of Poisson random variables with means lambda_[a_i,b_i] = 8*pi*(b_i^2 - a_i^2) for i = 1, ..., k, but the limiting variables need not be independent; equivalently, only the marginal laws are asymptotically Poisson.
E. The vector (N_{g,[a_1,b_1]}, ..., N_{g,[a_k,b_k]}) converges jointly in distribution to a vector of independent Poisson random variables with means lambda_[a_i,b_i] = 8*pi*(b_i^2 - a_i^2), provided the intervals [a_1,b_1], ..., [a_k,b_k] are pairwise disjoint and satisfy b_i < a_{i+1} with all endpoints bounded away from 0 independently of g.

The stem defines the objects carefully but does not explicitly state the limiting law, independence, or the exact intensity, so the correct answer is not leaked.

The item is close to a theorem-recall question: it essentially asks for the precise asymptotic statement. However, it is not a pure verbatim restatement because the choices force discrimination between nearby variants (joint vs marginal convergence, independence, exact mean, extra hypotheses).

Some reasoning is needed to identify the strongest correct conclusion among plausible alternatives, but the task mainly tests recognition of the exact theorem rather than substantial derivation or synthesis from first principles.

The distractors are mathematically plausible and target realistic failure modes: wrong scaling of the mean, confusing marginal Poisson limits with joint independence, weakening independence, and adding unnecessary technical assumptions.

A solid theorem-discrimination MCQ with good distractors and little answer leakage, but it leans more toward precise recall of a known result than toward genuinely generative mathematical reasoning.