[\cite{KirkpatrickMeckes2013,KirkpatrickNawaz2016,FangKoike2022}]
  \label{Theorem: high temp O(N)}
  Fix $N \geq 2$ and let $W_{n} := \sqrt{N-\beta}\,S_{n}/\sqrt{n}$ and $Z \sim N\left(0, \mathbf{I}\right)$, where $\mathbf{I}\in\mathbb{R}^{d\times d}$ is the
  identity matrix. For any $\beta < N$ there exists $c(N,\beta)<\infty$ such that
    \begin{equation}
         d(\mathcal{L}(W_n),\mathcal{L}(Z)) \leq \frac{c(N,\beta)}{\sqrt{n}}.
    \end{equation}

The $O(N)$ model was introduced in~\cite{Stanley1968} as a natural generalization of the Ising, $XY$ and Heisenberg
models. For integer $N,n\ge1$, let $\SS^{N-1}:=\{x\in\mathbb{R}^N:|x|=1\}$ denote the set of Euclidean unit vectors in $\mathbb{R}^N$, and let $P_{N,n}$
denote the $n$-fold product of uniform measure on $\SS^{N-1}$. The mean-field $O(N)$ model is then defined for $\beta\ge0$ by the probability
measure $\PP_{N,n,\beta}$ whose density with respect to $P_{N,n}$ is proportional to
\begin{equation}
  \exp\left(\frac{\beta}{2n}\sum_{i,j=1}^n \sigma_i\cdot\sigma_j\right), \qquad \sigma\in (\SS^{N-1})^n.
  \label{eq:mean-field O(N) measure}
\end{equation}

It is known that, with respect to the mean-field measure~\eqref{eq:mean-field O(N) measure}, the magnetization
\begin{equation}
  \label{eq:magnetization definition}
S_n(\sigma):=\sum_{i=1}^n\sigma_i
\end{equation}
obeys a law of large numbers iff $\beta\le N$; see~\cite{EllisNewman1978,Ellis1985} for the $N=1$ case, \cite{KirkpatrickMeckes2013} for
the $N=3$ case, and~\cite{KirkpatrickNawaz2016} for the general case of $N\ge2$.
In the subcritical phase, $\beta<N$, it is also known~\cite{EllisNewman1978,Ellis1985,KirkpatrickMeckes2013,KirkpatrickNawaz2016} that
$S_n/\sqrt{n}$ obeys a Gaussian central limit theorem. By contrast, at the critical point, $\beta=N$,
it has long been known that $n^{-3/4}\,S_n$ converges weakly to a random vector in $\mathbb{R}^N$ with Lebesgue density
proportional to $\exp(-a_N |x|^4)$, where $a_N:=N^2/(4N+8)$. This was established for $N=1$ in~\cite{SimonGriffiths1973},
and then generalised to $N\ge2$ in~\cite{DunlopNewman1975}.
Analogous limit theorems have been subsequently established for other mean-field spin models, including the Curie-Weiss-Potts
model~\cite{EllisWang1990}, and Blume-Emery-Griffiths model~\cite{EllisOttoTouchette2005}.

A natural question associated with a given limit theorem is to determine its rate of convergence. It was shown
in~\cite{EichelsbacherLowe2010} via Stein's method, that for the $N=1$ model with $\beta<1$, the Kolmogorov distance
between the distribution of $\sqrt{1-\beta}\, S_n/\sqrt{n}$ and a standard univariate normal is bounded above by
$c(\beta)/\sqrt{n}$. Utilising a multivariate generalisation of Stein's method established
in~\cite{ChatterjeeMeckes2008,ReinertRollin2009,Meckes2009}, analogous results~\cite{KirkpatrickMeckes2013,KirkpatrickNawaz2016} were then
shown to hold in the disordered phase, $\beta<N$, of the $O(N)$ model for any $N\ge2$.

In the critical case, it was shown in~\cite{EichelsbacherLowe2010,ChatterjeeShao2011} using nonnormal extensions of Stein's method,
that for the $N=1$ model with $\beta=1$ the Kolmogorov distance between the distribution of $n^{-3/4}\, S_n$ and the probability
measure with Lebesgue density proportional to $\exp(-x^4/12)$ is again bounded above by $c/\sqrt{n}$. We establish here a natural
multivariate generalisation of this result to all $N\ge 2$.

Our method relies on a general multivariate nonnormal Stein's method framework established
in~\cite{FangShaoXu2019,FangShaoXu2019correction}, which in turn can be viewed as a multivariate generalization
of~\cite{ChatterjeeShao2011}.
As explicitly noted in~\cite{FangShaoXu2019}, however, their underlying assumptions
do not hold for densities of the form $\exp(-a|x|^m)$ with $m>2$, and so their results cannot be applied to the critical mean-field $O(N)$
model. Nevertheless, as shown here, a generalization of~\cite[Theorem 2.5]{FangShaoXu2019} can be established which does indeed cover the
case of the critical $O(N)$ model. We present this result in Theorem~\ref{Theorem: main result} of Section~\ref{sec: Stein's method}.

The distributional bounds we present are in terms of the Wasserstein distance. The Wasserstein distance between the laws $\mathcal{L}(X)$, $\mathcal{L}(Y)$
of two random vectors $X,Y\in\mathbb{R}^d$ is
\begin{equation}
  d(\mathcal{L}(X),\mathcal{L}(Y)):= \inf_{(X,Y)}\mathbb{E}|X-Y| = \sup_{h \in \mathrm{Lip}}\left|\mathbb{E}\, h(X) - \mathbb{E}\, h(Y)\right|
  \label{eq: Wasserstein definition}
\end{equation}
where the infimum is taken over all couplings of $\mathcal{L}(X)$ and $\mathcal{L}(Y)$, and $\mathrm{Lip}$ denotes the set of all functions $h:\mathbb{R}^d\to\mathbb{R}$ such
that $|h(x)-h(y)|\le |x-y|$ for all $x,y\in\mathbb{R}^d$. We note that convergence in Wasserstein distance implies weak convergence; see
e.g.~\cite{GibbsSu2002}.

In the disordered phase, the bounds originally presented in~\cite{KirkpatrickMeckes2013,KirkpatrickNawaz2016} were not in terms of the
Wasserstein distance, but instead in terms of an integral probability metric defined by a smaller, smoother, class of test functions. These
bounds were proved by applying a general multivariate normal approximation theorem presented in~\cite{Meckes2009}. However, recent results
giving multivariate normal approximation in the Wasserstein distance allow the results from~\cite{KirkpatrickNawaz2016} to be immediately
sharpened to the Wasserstein distance. Indeed, substituting results given in~\cite[Lemmas 1 and 2]{KirkpatrickNawaz2016} into~\cite[Theorem
  1.1]{FangKoike2022} yields the following.

The $O(N)$ model was introduced in~\cite{Stanley1968} as a natural generalization of the Ising, $XY$ and Heisenberg
models. For integer $N,n\ge1$, let $\SS^{N-1}:=\{x\in\mathbb{R}^N:|x|=1\}$ denote the set of Euclidean unit vectors in $\mathbb{R}^N$, and let $P_{N,n}$
denote the $n$-fold product of uniform measure on $\SS^{N-1}$. The mean-field $O(N)$ model is then defined for $\beta\ge0$ by the probability
measure $\PP_{N,n,\beta}$ whose density with respect to $P_{N,n}$ is proportional to
\begin{equation}
 \exp\left(\frac{\beta}{2n}\sum_{i,j=1}^n \sigma_i\cdot\sigma_j\right), \qquad \sigma\in (\SS^{N-1})^n.
 \label{eq:mean-field O(N) measure}
\end{equation}

It is known that, with respect to the mean-field measure~\eqref{eq:mean-field O(N) measure}, the magnetization
\begin{equation}
 \label{eq:magnetization definition}
S_n(\sigma):=\sum_{i=1}^n\sigma_i
\end{equation}
obeys a law of large numbers iff $\beta\le N$; see~\cite{EllisNewman1978,Ellis1985} for the $N=1$ case, \cite{KirkpatrickMeckes2013} for
the $N=3$ case, and~\cite{KirkpatrickNawaz2016} for the general case of $N\ge2$.
In the subcritical phase, $\beta<N$, it is also known~\cite{EllisNewman1978,Ellis1985,KirkpatrickMeckes2013,KirkpatrickNawaz2016} that
$S_n/\sqrt{n}$ obeys a Gaussian central limit theorem. By contrast, at the critical point, $\beta=N$,
it has long been known that $n^{-3/4}\,S_n$ converges weakly to a random vector in $\mathbb{R}^N$ with Lebesgue density
proportional to $\exp(-a_N |x|^4)$, where $a_N:=N^2/(4N+8)$. This was established for $N=1$ in~\cite{SimonGriffiths1973},
and then generalised to $N\ge2$ in~\cite{DunlopNewman1975}.
Analogous limit theorems have been subsequently established for other mean-field spin models, including the Curie-Weiss-Potts
model~\cite{EllisWang1990}, and Blume-Emery-Griffiths model~\cite{EllisOttoTouchette2005}.

A natural question associated with a given limit theorem is to determine its rate of convergence. It was shown
in~\cite{EichelsbacherLowe2010} via Stein's method, that for the $N=1$ model with $\beta<1$, the Kolmogorov distance
between the distribution of $\sqrt{1-\beta}\, S_n/\sqrt{n}$ and a standard univariate normal is bounded above by
$c(\beta)/\sqrt{n}$. Utilising a multivariate generalisation of Stein's method established
in~\cite{ChatterjeeMeckes2008,ReinertRollin2009,Meckes2009}, analogous results~\cite{KirkpatrickMeckes2013,KirkpatrickNawaz2016} were then
shown to hold in the disordered phase, $\beta<N$, of the $O(N)$ model for any $N\ge2$.

In the critical case, it was shown in~\cite{EichelsbacherLowe2010,ChatterjeeShao2011} using nonnormal extensions of Stein's method,
that for the $N=1$ model with $\beta=1$ the Kolmogorov distance between the distribution of $n^{-3/4}\, S_n$ and the probability
measure with Lebesgue density proportional to $\exp(-x^4/12)$ is again bounded above by $c/\sqrt{n}$. We establish here a natural
multivariate generalisation of this result to all $N\ge 2$.

The distributional bounds we present are in terms of the Wasserstein distance. The Wasserstein distance between the laws $\mathcal{L}(X)$, $\mathcal{L}(Y)$
of two random vectors $X,Y\in\mathbb{R}^d$ is
\begin{equation}
 d(\mathcal{L}(X),\mathcal{L}(Y)):= \inf_{(X,Y)}\mathbb{E}|X-Y| = \sup_{h \in \mathrm{Lip}}\left|\mathbb{E}\, h(X) - \mathbb{E}\, h(Y)\right|
 \label{eq: Wasserstein definition}
\end{equation}
where the infimum is taken over all couplings of $\mathcal{L}(X)$ and $\mathcal{L}(Y)$, and $\mathrm{Lip}$ denotes the set of all functions $h:\mathbb{R}^d\to\mathbb{R}$ such
that $|h(x)-h(y)|\le |x-y|$ for all $x,y\in\mathbb{R}^d$. We note that convergence in Wasserstein distance implies weak convergence; see
e.g.~\cite{GibbsSu2002}.

In the disordered phase, the bounds originally presented in~\cite{KirkpatrickMeckes2013,KirkpatrickNawaz2016} were not in terms of the
Wasserstein distance, but instead in terms of an integral probability metric defined by a smaller, smoother, class of test functions. These
bounds were proved by applying a general multivariate normal approximation theorem presented in~\cite{Meckes2009}. However, recent results
giving multivariate normal approximation in the Wasserstein distance allow the results from~\cite{KirkpatrickNawaz2016} to be immediately
sharpened to the Wasserstein distance. Indeed, substituting results given in~\cite[Lemmas 1 and 2]{KirkpatrickNawaz2016} into~\cite[Theorem
 1.1]{FangKoike2022} yields the following.

\begin{equation}
  \exp\left(\frac{\beta}{2n}\sum_{i,j=1}^n \sigma_i\cdot\sigma_j\right), \qquad \sigma\in (\sphere)^n.
  \label{eq:mean-field O(N) measure}
\end{equation}

\begin{lemma}\label{lemma: Critical remainders}
 Let $(W, W')$ be as described above. Then
 \begin{equation}\label{eq: W' - W conditioned on sigma}
 \EE[W'-W|\sigma] = \frac{1}{Nn^{3/2}}\left(-\frac{N^{2}}{N+2}|W|^{2}W + R_{1}\right),
 \end{equation}
 where
 \begin{equation}\label{eq: remainder 1}
 \begin{split}
 R_{1} &= \frac{N}{n^{1/4}}\sum_{i=1}^{n}\left[m_{(i)}f_{N}(|m_{(i)}|) - mf_{N}(|m|)\right] \\
 & \quad + Nn^{3/4}\left(f_{N}(|m|) - 1 + \frac{N|m|^{2}}{(N+2)}\right)m.
 \end{split}
 \end{equation}
 and
 \begin{equation}\label{eq: W' - W W' - W T conditioned on sigma}
 \begin{split}
 \EE[(W'-W)(W'-W)^{\transpose}|\sigma] &= 2\frac{1}{Nn^{3/2}}\left(\id + R_{2}\right),
 \end{split}
 \end{equation}
 where
 \begin{equation}\label{eq: remainder 2}
 \begin{split}
 R_{2} &= \left[\frac{N}{2n}\sum_{i=1}^{n}\sigma_{i}\sigma_{i}^{\transpose} - \frac{1}{2}\id\right] -
 \frac{N}{\sqrt{n}}WW^{\transpose} + \frac{N}{n^{2}}\sum_{i=1}^{n}\sigma_{i}\sigma_{i}^{\transpose}\\
 &\quad + \frac{N}{2n}\sum_{i=1}^{n}\,g_{N}(|m_{(i)}|)\,|m_{(i)}|^2\,
 \left[\sigma_{i}m_{(i)}^{\transpose}+ m_{(i)} \sigma_{i}^{\transpose} - \frac{\id}{N}\right]
 + \frac{N}{2n}\sum_{i=1}^{n}g_{N}(|m_{(i)}|)m_{(i)}m_{(i)}^\transpose.
 \end{split}
 \end{equation}
\end{lemma}
\begin{proof}
 We follow a similar strategy to the proof~\cite[Lemma 1]{KirkpatrickNawaz2016} (see also \cite[Lemma 11]{KirkpatrickMeckes2013}).
 The key ingredients are the small $x$ asymptotics for $f_N(x)$ given in~\eqref{eq: fN asymptotics}, combined with the fact
 that, as shown in~\cite[Theorem 3]{KirkpatrickNawaz2016}, the quantity $m$ concentrates at 0 as $n\to \infty$ when $\beta = N$.

Finally, to establish~\eqref{eq: Bound hilbert schmidt norm 2}, we fix $\epsilon>0$ and $y\in\RR^d$ with $|y| \leq 1$ and apply a
 similar decomposition to that used in~\cite{FangShaoXu2019}. Specifically, defining
 \begin{equation}\label{eq: Phi definition}
 \Phi(t) :=\left[D^{2}P_{t}h(x + \epsilon y)[u, v] - D^{2}P_{t}h(x)[u, v]\right],
 \end{equation}
 it follows by~\eqref{eq: second derivative of f} that
 \begin{equation}\label{eq: rewriting f(x + epsilon u) - f(x)}
 D_{v}D_{u}f(x + \epsilon y) - D_{v}D_{u}f(x) = -\int_{0}^{\epsilon^{2}\wedge \epsilon}\Phi(t)\,\dt -
 \int_{\epsilon^{2}\wedge \epsilon}^{\infty}\Phi(t)\,\dt.
 \end{equation}
 Equation~\eqref{eq: Ptphi derivative 2} implies
 \begin{equation}\label{eq: bound int 0 to epsilon squared}
 \begin{split}
 \left|\int_{0}^{\epsilon^{2}\wedge \epsilon}\Phi(t) \, \dt\right| &\leq 2|u||v|\int_{0}^{\epsilon^{2}\wedge \epsilon} S(t)\, \dt,\\
 &\leq 2\,|u| |v|\,(\epsilon\wedge \sqrt{\epsilon})\, \left(s_0( \epsilon\wedge \sqrt{\epsilon}) +2s_{-1/2}\right).
 \end{split}
 \end{equation}
 If $\epsilon\le 1$ then $\epsilon\wedge \sqrt{\epsilon}=\epsilon$, in which case
 \begin{equation}
 (\epsilon\wedge \sqrt{\epsilon})\, \left(s_0( \epsilon\wedge \sqrt{\epsilon}) +2s_{-1/2}\right) =
 \epsilon\, (s_0\epsilon +2s_{-1/2})\,\leq \epsilon (s_0 + 2s_{-1/2}).
 \end{equation}
 While if $\epsilon>1$ then $\epsilon\wedge\sqrt{\epsilon}=\sqrt{\epsilon}$, and so
 \begin{equation}
 (\epsilon\wedge \sqrt{\epsilon})\, \left(s_0( \epsilon\wedge \sqrt{\epsilon}) +2s_{-1/2}\right) =
 \sqrt{\epsilon}\, \left(s_0\sqrt{\epsilon}+2s_{-1/2}\right)\leq
 \epsilon\left(s_{0} +2s_{-1/2}\right).
 \end{equation}
 We therefore conclude from~\eqref{eq: bound int 0 to epsilon squared} that for all $\epsilon>0$
 \begin{equation}\label{eq: int bound 0 to min}
 \left|\int_{0}^{\epsilon^{2}\wedge \epsilon}\Phi(t) \, \dt\right| \leq 2\epsilon (s_0 + 2s_{-1/2})|u| |v|.
 \end{equation}
 Now consider the remaining term in~\eqref{eq: rewriting f(x + epsilon u) - f(x)}. It follows from~\eqref{eq: Ptphi derivative 3} that
 $D^2P_t h(x+\cdot)[u,v]$ is differentiable with bounded derivative, and so the fundamental theorem of calculus implies
 \begin{equation}
 \int_{\epsilon^{2}\wedge \epsilon}^{\infty}\Phi(t)\,\dt
 = \epsilon\int_{\epsilon^{2}\wedge \epsilon}^{\infty}\int_{0}^{1} D^{3}P_{t}h(x + \epsilon r y)[u, v, y]\, \dr\, \dt.
 \end{equation}
 Therefore, using $|y|\le1$, and again applying~\eqref{eq: Ptphi derivative 3}, yields
 \begin{equation}
 \label{eq: phi tail bound}
 \begin{split}
 \left|\int_{\epsilon^{2}\wedge \epsilon}^{\infty}\Phi\,\dt\right|
 &\leq \epsilon |u||v| \int_{\epsilon\wedge\epsilon^{2}}^{\infty}\, Q(t) \,e^{-\theta t/4} \dt,\\
 &\leq \epsilon |u||v|
 \left[\frac{128q_2}{\theta^3}+\frac{16q_1}{\theta^2}+\frac{4q_0}{\theta}+\frac{2\sqrt{\pi}q_{-1/2}}{\sqrt{\theta}}
 + q_{-1} E_1\left(\frac{\theta(\epsilon^{2}\wedge \epsilon)}{4}\right)\right]
 \end{split}
 \end{equation}
 where $E_{1}(\cdot)$ denotes the exponential integral~\cite[Equation 5.1.1]{AbramowitzStegun1972}. Applying~\cite[Equation 5.1.20]{AbramowitzStegun1972}
 yields
 \begin{equation}
 \label{eq: bound on exponential integral}
 \begin{split}
 E_1\left(\frac{\theta(\epsilon^{2}\wedge \epsilon)}{4}\right) &\le 2\left(1\vee \left|\log \frac{\theta(\epsilon^{2}\wedge \epsilon)}{4} \right|\right)\\
 &\le 4\left(1+\left|\log\frac{\theta}{4}\right|\right)\left(1\vee |\log\epsilon|\right).
 \end{split}
 \end{equation}
 Applying~\eqref{eq: bound on exponential integral} to~\eqref{eq: phi tail bound}, and combining the result with~\eqref{eq: int bound 0
 to min} and~\eqref{eq: rewriting f(x + epsilon u) - f(x)} then yields
 \begin{equation}
 \sup_{x,y\in\RR^d : |y|\le 1} |D_{v}D_{u}f(x + \epsilon y) - D_{v}D_{u}f(x)|
 \le
 K_3\,|u|\, |v|\, \epsilon\, (1\vee|\log\epsilon|)
 \label{eq: third main bound}
 \end{equation}
 with
 \begin{equation}
 K_3 :=
 2(s_0 +2s_{-1/2}) +
 \frac{128q_2}{\theta^3}+\frac{16q_1}{\theta^2}+\frac{4q_0}{\theta}+\frac{2\sqrt{\pi}q_{-1/2}}{\sqrt{\theta}}
 + 4\left(1+\left|\log\frac{\theta}{4}\right|\right) q_{-1}.
 \end{equation}

The analogue of Proposition~\ref{prop: mean-square bounds} in~\cite{FangShaoXu2019} is Lemma 5.2. While the proof of the latter is
essentially immediate under their assumption of strict convexity of $V$,
the proof of Proposition~\ref{prop: mean-square bounds} under Assumption~\ref{assumption: assumptions on V} is somewhat more involved. A key
ingredient in its proof is the following lemma, whose proof is deferred to the end of this section.
\begin{lemma}\label{lemma: E(t) bound}
 If $V$ satisfies Assumption~\ref{assumption: assumptions on V}, then for all $x\in\RR^d$ and $0\le s\le t<\infty$
 $$
 \EE \exp\left(-2\int_s^t \rho(X_r^x) \dr \right) \le C_2^2 e^{-2\theta (t-s)}.
 $$
\end{lemma}
In addition, we will also require the following Gr\"{o}nwall-type lemma.
\begin{lemma}\label{lemma: Gronwall type}
 Let $u: \RR \to \RR^{d}$ be such that $|u(t)|^{2} \in C^{1}(\RR, \RR)$. Suppose it satisfies the following differential inequality
 \begin{equation}
 \frac{\diff}{\dt}|u(t)|^{2} \leq 2a(t)|u(t)|^{2} + 2b(t)|u(t)|, \qquad \forall t \in (0, T),
 \end{equation}
 where $T \in (0, \infty)$, $a, b \in L^{1}([0, T])$ and $b \geq 0$. If $|u(0)| = 0$ then
 \begin{equation}
 |u(t)| \leq \int_{0}^{t}b(s)\exp\left(\int_{s}^{t}a(r)\, \dr\right)\, \ds, \qquad \forall t \in [0, T).
 \end{equation}
\end{lemma}
\begin{proof}
 Fix $t \in (0, T)$ and let $\tau := \sup\{0 \leq s \leq t: |u(s)| = 0\}$. If $\tau = t$ then the statement holds trivially since $b
 \geq 0$. If $\tau < t$ then $|u(t)|$ is differentiable on $(\tau, t]$ and gives
 \begin{equation}
 \frac{\diff}{\ds}|u(s)| \leq a(s)|u(s)| + b(s), \qquad s \in (\tau, t].
 \end{equation}
 Since $|u(\tau)| = 0$, by the Generalised Jones Inequality \cite[Theorem 1.2.2]{Qin2016} we have
 \begin{equation}
 |u(t)| \leq \int_{\tau}^{t}b(s)\exp\left(\int_{s}^{t}a(r)\, \dr\right)\, \ds \leq \int_{0}^{t}b(s)\exp\left(\int_{s}^{t}a(r)\, \dr\right)\, \ds,
 \end{equation}
 using $b \geq 0$. This completes the proof.
\end{proof}

Our main result in the current work is the analogous multivariate nonnormal approximation holding at criticality.
\begin{theorem}
  \label{Theorem: Critical O(N)}
  Fix $N \geq 2$ and let $W_{n} := n^{-3/4}\,S_{n}$. If $\beta=N$, then there exists $c(N)<\infty$ such that
    \begin{equation}
        d(\mathcal{L}(W_n),\mu) \leq \frac{c(N)}{\sqrt{n}},
    \end{equation}
  where $\mu$ has Lebesgue density proportional to $\exp\left(-a_N |x|^{4}\right)$ and $a_N:=N^2/(4N+8)$.
\end{theorem}

\begin{remark}
As a corollary of Theorem~\ref{Theorem: Critical O(N)} it follows that for any $N\ge2$, the Wasserstein distance between the distribution of
$n^{-3/4}\,|S_n|$ and the probability measure with Lebesgue density proportional to $r^{N-1}\exp(-a_N r^4)$, is again bounded above by
$c(N)/\sqrt{n}$. This is analogous to the univariate nonnormal approximation presented in~\cite[Theorem 6]{KirkpatrickNawaz2016} for
$|S_n|^2$, however the bound given there is not in terms of Wasserstein distance, but in terms of an integral probability metric defined by
a smaller class of test functions, and contains an additional logarithmic factor.
\end{remark}

The remainder of this paper is organised as follows. Section~\ref{sec: Stein's method} introduces the relevant background on Stein's method, and
provides the statement of our main result on Wasserstein approximation, Theorem~\ref{Theorem: main result}. In Section \ref{sec: O(N) model}
we apply Theorem~\ref{Theorem: main result} to prove Theorem~\ref{Theorem: Critical O(N)}.
Section~\ref{sec: sdes and semigroups} provides preliminary results related to a class of stochastic differential equations, and their
related stochastic semigroups, which provide the framework for the proof of Theorem \ref{Theorem: main result} given in Section \ref{sec: proof of main
  result}. The key result used in this proof is a bound on the derivatives of the relevant stochastic semigroups.
These bounds are proved in Section~\ref{sec: semigroup bounds}, using Elworthy-Li formulae and
bounds on the spatial derivatives of the solutions to the corresponding stochastic differential equation
proved in Section~\ref{sec: variation process bounds}.
Finally, the appendix contains the proof of a proposition stated in Section~\ref{sec: sdes and semigroups}.

\subsection{Notation}
\label{subsec:notation}
We will denote the set of positive integers by $\ZZ_+$, and the set of natural numbers by $\mathbb{N}:=\ZZ_+\cup\{0\}$.
For $d\in\ZZ_+$ and $x,y\in \mathbb{R}^{d}$, we let $\<x,y\>$ denote the Euclidean inner product, and set $|x| = \sqrt{\<x,x\>}$. For two
matrices $A, B \in \mathbb{R}^{d \times d}$ we let $\<A,B\>_{{}} = \sum_{i, j = 1}^{d}A_{i, j}B_{i, j}$ denote the Hilbert-Schmidt inner product,
and denote the corresponding norm by $\|A\|_{{}} = \sqrt{\< A, A \>_{{}}} = \sqrt{\text{Tr}(A^{\mathrm{T}}A)}$. We will denote the operator
norm of $A\in\mathbb{R}^{d\times d}$ by $\|A\|_{\mathrm{op}}:=\sup_{x\in\mathbb{S}^{d-1}}|Ax|$.

For open sets $U_1\subseteq \mathbb{R}^\ell$ and $U_2\subseteq\mathbb{R}^m$ we let $C^{k}(U_1;U_2)$ denote the set of $k$-times continuously differentiable
$f:U_1\to U_2$. If $U_2=\mathbb{R}$ we abbreviate $C^k(U_1):=C^k(U_1,U_2)$, and if additionally $U_1=\mathbb{R}^d$ we write simply $C^k$.
We let $B_b$ denote the
space of $f:\mathbb{R}^d\to\mathbb{R}$ which are bounded and Borel measurable, and let $C_b$ denote the subspace of $B_b$ of uniformly continuous such functions. We will then make
use of the convenient abbreviation $\mathrm{Lip}^{\infty}_b:=\mathrm{Lip}\cap C^{\infty}\cap C_b$.

For $f\in C^1(\mathbb{R}^d;\mathbb{R})$, we let $D_u f(x)$ denote the directional derivative of $f$ in the direction $u\in\mathbb{R}^d$, and we extend the
definition to $f\in C^1(\mathbb{R}^d;\mathbb{R}^m)$, by then defining $D_u f(x)$ entrywise. For given $f\in C^3(\mathbb{R}^d;\mathbb{R}^m)$ and $u,v,w,x\in\mathbb{R}^d$ we
define
\begin{equation}
  \label{eq: D notation}
  \begin{split}
    Df(x)[u]&:= D_u f(x)\\
    D^2f(x)[u,v]&:=D_v D_u f(x)\\
    D^3f(x)[u,v,w]&:=D_w D_v D_u f(x)
  \end{split}
\end{equation}
We note that for each given $x\in\mathbb{R}^d$ and $i=1,2,3$, the map $D^i g(x) : (\mathbb{R}^d)^i\to\mathbb{R}^d$ is multilinear and symmetric.

For $f \in C^{1}(\mathbb{R}^d;\mathbb{R})$, we let $\nabla f$ denote the gradient of $f$, so that
\begin{equation}
  Df(x)[u]=\<\nabla f(x),u\>,
\end{equation}
and set $\|\nabla f\|_\infty:=\sup_{x \in \mathbb{R}^{d}}|\nabla f(x)|$.
Finally, for $f \in C^{2}(\mathbb{R}^{d};\mathbb{R})$, we let $\nabla^2 f$ denote the Hessian of $f$,
so that
\begin{equation}
D^2 f(x)[u,v] =\<u,\nabla^2 f(x) v\>,
\end{equation}
and let $\Delta$ denote the Laplacian.

The introduction does not give a step-by-step proof of Theorem~\ref{Theorem: high temp O(N)}, but it does indicate how the analogous critical result is proved: the paper "introduces the relevant background on Stein's method" and states a main Wasserstein-approximation theorem (Theorem~\ref{Theorem: main result}); then "we apply Theorem~\ref{Theorem: main result} to prove Theorem~\ref{Theorem: Critical O(N)}." The proof framework for Theorem~\ref{Theorem: main result} uses "a class of stochastic differential equations, and their related stochastic semigroups"; "the key result used in this proof is a bound on the derivatives of the relevant stochastic semigroups." These derivative bounds are proved "using Elworthy-Li formulae" and "bounds on the spatial derivatives of the solutions to the corresponding stochastic differential equation" (via "variation process bounds"). No direct proof sketch for Theorem~\ref{Theorem: high temp O(N)} itself appears in the post-theorem introduction.

The introduction does not give a step-by-step proof of the main theorem, but it does indicate how the analogous critical result is proved: the paper introduces the relevant background on Stein's method and states a main Wasserstein-approximation theorem.

\begin{theorem}\label{Theorem: main result}
  Suppose $V$ satisfies Assumption \ref{assumption: assumptions on V}, and let $\mu$ have Lebesgue density proportional to
  $e^{-V}$.
  Let $W,W'$ be identically distributed $\RR^d$-valued random variables, defined on the same probability space, such that if $\delta:=W'-W$ then
  $W$ and $\delta\delta^\transpose$ are both integrable.
  Let $\sF\supseteq\sigma(W)$ be a $\sigma$-algebra, let $\lambda>0$, and define $R_1$ and $R_2$ so that
    \begin{equation}
      \mathbb{E}[\delta|\mathcal{F}] = \lambda(-\grad V(W) + R_{1}),
      \label{eq: Regression Property}
    \end{equation}
    and
    \begin{equation}
      \mathbb{E}\left[\delta\,\delta^{\transpose}\big|\mathcal{F}\right] = 2\lambda(\id + R_{2}).
      \label{eq: Correlation Property}
    \end{equation}
    Then there exists a constant $C\in(0,\infty)$ such that
    \begin{equation}
      d(\sL(W),\mu)
      \leq C\left\lbrace \frac{1}{\lambda}\mathbb{E} [|\delta|^{3}(|\log|\delta|| \vee 1)] + \mathbb{E}|R_{1}|
      + \sqrt{d}\,\mathbb{E}\,\|R_{2}\| \right\rbrace.
    \end{equation}
\end{theorem}

Then this theorem is applied to prove the following critical result.

\begin{theorem}
  \label{Theorem: Critical O(N)}
  Fix $N \geq 2$ and let $W_{n} := n^{-3/4}\,S_{n}$. If $\beta=N$, then there exists $c(N)<\infty$ such that
    \begin{equation}
        \wass(\sL(W_n),\mu) \leq \frac{c(N)}{\sqrt{n}},
    \end{equation}
  where $\mu$ has Lebesgue density proportional to $\exp\left(-a_N |x|^{4}\right)$ and $a_N:=N^2/(4N+8)$.
\end{theorem}

The proof framework for the Wasserstein-approximation theorem above uses a class of stochastic differential equations and their related stochastic semigroups; the key result used in this proof is a bound on the derivatives of the relevant stochastic semigroups. These derivative bounds are proved using Elworthy--Li formulae and bounds on the spatial derivatives of the solutions to the corresponding stochastic differential equation (via variation process bounds). No direct proof sketch for the main theorem itself appears in the post-theorem introduction.

Existential–Universal
Inequality or Bound

Question:
Let $N\ge 2$. In the mean-field $O(N)$ model, the spin configuration $\sigma=(\sigma_1,\dots,\sigma_n)\in (\mathbb S^{N-1})^n$ has law with density proportional to
\[
\exp\!\left(\frac{\beta}{2n}\sum_{i,j=1}^n \sigma_i\cdot \sigma_j\right)
\]
with respect to the $n$-fold product of the uniform measure on the unit sphere $\mathbb S^{N-1}\subset\mathbb R^N$. Let the magnetization be
\[
S_n=\sum_{i=1}^n \sigma_i,
\]
and define
\[
W_n:=\sqrt{N-\beta}\,\frac{S_n}{\sqrt n}.
\]
Let $Z\sim N(0,\mathbf I)$ be the standard Gaussian vector in $\mathbb R^N$, where $\mathbf I$ is the identity matrix, and let the Wasserstein distance between laws be
\[
d(\mathcal L(X),\mathcal L(Y)):=\inf_{(X,Y)}\mathbb E|X-Y|=\sup_{h\in \mathrm{Lip}}\big|\mathbb Eh(X)-\mathbb Eh(Y)\big|,
\]
where the supremum is over all $1$-Lipschitz functions $h$. For a fixed inverse temperature $\beta<N$, which quantitative estimate holds?

There exists a finite constant $c(N,\beta)$, depending only on $N$ and $\beta$, such that for every $n\ge 1$,
\[
d\big(\mathcal L(W_n),\mathcal L(Z)\big)\le \frac{c(N,\beta)}{\sqrt n}.
\]

Choices:
B.
There exists a finite constant $c(N)$, depending only on $N$, such that for every $\beta<N$ and every $n\ge 1$,
\[
d\big(\mathcal L(W_n),\mathcal L(Z)\big)\le \frac{c(N)}{\sqrt n}.
\]
C.
For each fixed $\beta<N$, one has
\[
d\big(\mathcal L(W_n),\mathcal L(Z)\big)\longrightarrow 0\qquad\text{as }n\to\infty.
\]
D.
There exists a finite constant $c(N,\beta)$, depending only on $N$ and $\beta$, such that for every $n\ge 1$,
\[
d\big(\mathcal L(W_n),\mathcal L(Z)\big)\le \frac{c(N,\beta)}{n}.
\]
E.
There exists a finite constant $c(N,\beta)<\infty$, depending only on $N$ and $\beta$, such that for every $n\ge 1$,
\[
d\big(\mathcal L\!\left(n^{-3/4}S_n\right),\mathcal L(Z)\big)\le \frac{c(N,\beta)}{\sqrt n}.
\]

explicit $n^{-1/2}$ Wasserstein rate and quantitative constant

subcritical scaling/normal limit replaced by critical scaling

The stem does not reveal the quantitative conclusion or single out the correct option. It gives the model and asks which theorem-level statement is valid in the regime β<N, without explicit answer leakage.

The item is close to a theorem-recall question: the correct option is essentially the precise theorem statement. However, it is not a pure tautology because the alternatives vary meaningfully in rate, uniformity in β, and inclusion of the critical case.

Some reasoning is needed to distinguish the exact quantitative result from weaker or overstrong variants, especially regarding dependence on β, the exclusion of β=N, and the 1/√n rate. Still, the question mainly tests recognition of the theorem rather than substantial derivation.

The distractors are plausible and mathematically targeted: a weaker true asymptotic statement, an overstrong uniform-in-β claim, an incorrect extension to criticality, and an overly fast rate. These reflect common failure modes in interpreting limit theorems.

A solid MCQ with no real answer leakage and strong distractors, but it remains somewhat theorem-recall based rather than deeply generative.


Question:
Let $N\ge 2$. In the mean-field $O(N)$ model, the spin configuration $\sigma=(\sigma_1,\dots,\sigma_n)\in (\mathbb S^{N-1})^n$ has law with density proportional to
\[
\exp\!\left(\frac{\beta}{2n}\sum_{i,j=1}^n \sigma_i\cdot \sigma_j\right)
\]
with respect to the $n$-fold product of the uniform measure on the unit sphere $\mathbb S^{N-1}\subset\mathbb R^N$. Let the magnetization be
\[
S_n=\sum_{i=1}^n \sigma_i,
\]
and define
\[
W_n:=\sqrt{N-\beta}\,\frac{S_n}{\sqrt n}.
\]
Let $Z\sim N(0,\mathbf I)$ be the standard Gaussian vector in $\mathbb R^N$, where $\mathbf I$ is the identity matrix, and let the Wasserstein distance between laws be
\[
d(\mathcal L(X),\mathcal L(Y)):=\inf_{(X,Y)}\mathbb E|X-Y|=\sup_{h\in \mathrm{Lip}}\big|\mathbb Eh(X)-\mathbb Eh(Y)\big|,
\]
where the supremum is over all $1$-Lipschitz functions $h$. For a fixed inverse temperature $\beta<N$, which quantitative estimate holds?

Choices:
B.
There exists a finite constant $c(N)$, depending only on $N$, such that for every $\beta<N$ and every $n\ge 1$,
\[
d\big(\mathcal L(W_n),\mathcal L(Z)\big)\le \frac{c(N)}{\sqrt n}.
\]
C.
For each fixed $\beta<N$, one has
\[
d\big(\mathcal L(W_n),\mathcal L(Z)\big)\longrightarrow 0\qquad\text{as }n\to\infty.
\]
D.
There exists a finite constant $c(N,\beta)$, depending only on $N$ and $\beta$, such that for every $n\ge 1$,
\[
d\big(\mathcal L(W_n),\mathcal L(Z)\big)\le \frac{c(N,\beta)}{n}.
\]
E.
There exists a finite constant $c(N,\beta)<\infty$, depending only on $N$ and $\beta$, such that for every $n\ge 1$,
\[
d\big(\mathcal L\!\left(n^{-3/4}S_n\right),\mathcal L(Z)\big)\le \frac{c(N,\beta)}{\sqrt n}.
\]

The stem does not state the quantitative conclusion or the sharp rate. It sets up the model and normalization, but the respondent still has to choose among different rates, scalings, and parameter dependences.

The correct option is essentially the theorem statement itself: a subcritical CLT with Wasserstein error of order n^{-1/2} and a constant depending on (N, beta). The item mainly asks for recall of the exact conclusion rather than a substantively reformulated consequence.

Some reasoning is required to reject stronger, weaker, or mis-scaled alternatives, especially the uniform-in-beta, n^{-1}, and critical-scaling distractors. However, the task still mostly tests recognition of the exact theorem-level estimate rather than deeper generative derivation.

The distractors are plausible and mathematically meaningful: uniformity in beta, mere convergence without rate, an overly strong n^{-1} rate, and an incorrect critical scaling. These align well with common mistakes about sharpness, parameter dependence, and phase-dependent normalization.

A mathematically well-constructed item with strong distractors, but it is close to a direct theorem-recall question rather than a high-generative-reasoning MCQ.


\begin{rem}\label{rem:K-mod}
According to the author's knowledge based on \cite{SS17, LX19, ADL21, ACC+, ACD+, CDG+, CT23, DJKHQ24, LZ24, Zha24, CFFK24}, we give a list of families of smooth Fano threefolds with volume $V\geq 11$ whose K-moduli compactifications are currently unknown. We hope Theorem \ref{thm:K-moduli} can be used to study these families in the future. We follow Mori--Mukai's notation for the numbering of these families.
\begin{enumerate}
    \item $V\geq 26$: \textnumero 2.20, 2.21, 3.11;
    \item $22\leq V<26$: \textnumero 1.10, 2.16, 2.17, 3.6, 3.7, 3.8;
    \item $11\leq V<22$: \textnumero 1.6, 1.7, 1.8, 1.9, 2.5, 2.6, 2.7, 2.8, 2.9, 2.10, 2.11, 2.12, 2.13, 2.14, 3.1, 3.2, 3.3, 3.4, 3.5.
\end{enumerate}
\end{rem}

The local volume $\hvol(x,X)$ of a klt singularity $x\in X$, introduced by Chi Li in \cite{Li18}, is a numerical invariant that has played a crucial role in the recent development of a local K-stability theory for singularities; see \cite{LLX18, Zhu25} for recent surveys. It is thus natural to further investigate the distribution of local volumes in a fixed dimension $n$, such as explicit bounds and gaps. In \cite{LX19} it was shown that $\hvol(x,X)\leq n^n$ and equality holds if and only if $x\in X$ is smooth. Then it was conjectured in \cite{SS17}, known as the ODP Gap Conjecture, that the second largest local volume is $2(n-1)^n$ which is obtained by the ordinary double point. This conjecture is confirmed in dimension at most $3$ \cite{LL19, LX19} and for local complete intersection singularities \cite{Liu22}. More recently, it was shown in \cite{XZ24}, as a consequence of their local boundedness result for K-semistable Fano cone singularities, that the local volumes in a fixed dimension are discrete away from zero. See also \cite{HLQ23, Zhu24, LMS23} for previous results in lower dimensions and \cite{HLQ24} for general coefficient sets.

In this article, we establish an optimal upper bound for local volumes of Gorenstein canonical non-hypersurface threefold singularities. Our main result goes as follows.

In this article, we establish an optimal upper bound for local volumes of Gorenstein canonical non-hypersurface threefold singularities. Our main result goes as follows.

Theorem \ref{thm:nv-can} can be thought of as a refinement of the ODP Gap Theorem for threefolds in \cite{LX19} that any non-smooth threefold singularity $x\in X$ satisfies $\hvol(x,X)\leq 16$, and equality holds if and only if $x\in X$ is an $A_1$-singularity.

\begin{thm}\label{thm:nv-3fold}
Let $x\in X$ be a klt threefold singularity. Then $\hvol(x,X)\geq 9$ if and only if $x\in X$ is either a hypersurface singularity of type $cA_{\leq 2}$, or a cyclic quotient singularity of type $\frac{1}{2}(1,1,1)$, $\frac{1}{3}(1,1,0)$, $\frac{1}{3}(1,1,1)$, or $\frac{1}{3}(1,1,2)$.
\end{thm}

\begin{thm}\label{thm:K-moduli}
Let $X$ be a K-semistable $\bQ$-Fano threefold with volume $(-K_X)^3 = V$.
\begin{enumerate}
 \item If $V\geq 26$, then $X$ has only $cA_1$-singularities or cyclic quotient singularities of type $\frac{1}{2}(1,1,1)$.
 \item If $V\geq 22$, then $X$ has only $cA_1$-singularities, isolated $cA_2$-singularities, $D_\infty$-singularities, or cyclic quotient singularities of type $\frac{1}{2}(1,1,1)$.
 \item If $V\geq 11$, then any non-Gorenstein singularity $x\in X$ is a cyclic quotient of a (possibly smooth) hypersurface singularity of type $cA_{\leq 2}$.
\end{enumerate}
If in addition $X$ is $\bQ$-Gorenstein smoothable, then it does not admit cyclic quotient singularities of type $\frac{1}{2}(1,1,1)$.
\end{thm}

\begin{thm}\label{thm:nv-mld}
Let $x\in X$ be a klt threefold singularity. Then
\begin{equation}\label{eq:nv-mld}
 \hvol(x,X) \leq 9 \cdot \mld(x,X).
\end{equation}
Moreover, equality holds if and only if $x\in X$ is a cyclic quotient singularity of type $\frac{1}{r}(1,1,1)$ for some $r\in \bZ_{>0}$.
\end{thm}

\begin{proof}[Proof of Theorem \ref{thm:nv-3fold}]
 The ``if'' part is a direct consequence of Example \ref{expl:quotient} and Proposition \ref{prop:nv-hyp}.
 For the ``only if'' part, let $(\tx\in \tX)\to (x\in X)$ be the index $1$ cover of $K_X$ whose Cartier index is denoted by $d$. Then by Theorem \ref{thm:finite-deg} we have
 \[
 \hvol(\tx, \tX) = d\cdot \hvol(x,X)\geq 9d.
 \]
 If $d=1$, i.e.\ $x\in X$ is Gorenstein, then by Theorem \ref{thm:crex-cDV} we know that $x\in X$ is either a hypersurface singularity or a cyclic quotient singularity of type $\frac{1}{3}(1,1,1)$. The hypersurface singularity case follows from Proposition \ref{prop:nv-hyp}. If $d\geq 2$, then we have $\hvol(\tx, \tX) \geq 18$ which implies that $\tx\in \tX$ is smooth and $d \leq 3$ by Theorem \ref{thm:ODP-gap}. Thus $x\in X$ is a cyclic quotient singularity of order $2$ or $3$. If $d=2$, then $x\in X$ is of type $\frac{1}{2}(1,1,1)$ as the other case $\frac{1}{2}(1,1,0)$ is Gorenstein. If $d=3$, then $x\in X$ is of type $\frac{1}{3}(1,1,0)$ or $\frac{1}{3}(1,1,2)$ as the other two cases $\frac{1}{3}(1,2,0)$ or $\frac{1}{3}(1,1,1)$ are both Gorenstein.
\end{proof}

Since $K_{\tX}$ is Cartier, we know that $\mld(\tx,\tX) \geq 1$ is a positive integer. If $\crex(\tx, \tX) = 0$, then $\mld(\tx, \tX) = 1$ and \eqref{eq:nv-mld-cover} follows by Theorem \ref{thm:nv-can} with equality if and only if $\tx\in \tX$ is a cyclic quotient singularity of type $\frac{1}{3}(1,1,1)$. If $\crex(\tx,\tX) >0$, then by Theorem \ref{thm:crex-cDV} we know that $\tx\in \tX$ is a cDV singularity. Then we have three cases: $\tx\in \tX$ is smooth, non-smooth terminal, or non-terminal. The smooth case is obvious as $\mld(\tx,\tX) = 3$ and $\hvol(\tx,\tX) = 27$. For the terminal case, we have $\mld(\tx,\tX) = 2$ by \cite{Mar96}. Thus \eqref{eq:nv-mld-cover} follows from Theorem \ref{thm:ODP-gap} as $\hvol(\tx, \tX)\leq 16 < 18 = 9\cdot \mld(\tx,\tX)$. The last case is when $\tx\in \tX$ is not terminal. Let $\phi: \tZ\to \tX$ be a terminalization. Then we know that $\phi^{-1}(\tx)$ has dimension $1$ since $\phi$ is exceptional over $\tx$ and there are no crepant exceptional divisors over $\tx\in \tX$ by the assumption that $\crex(\tx, \tX) = 0$. Let $\eta$ be a generic point of $\phi^{-1}(\tx)$. Since $\tZ$ is terminal, we know that $\tZ$ is smooth at $\eta$. Thus we have $\mld(\tx,\tX) \leq \mld(\eta, \tZ) = 2$. Moreover, since $\crex(\tx,\tX) = 0$ we have $\mld(\tx,\tX)>1$. Combining these together, we have $\mld(\tx, \tX) = 2$. Thus \eqref{eq:nv-mld-cover} follows again from Theorem \ref{thm:ODP-gap}.

\begin{que}\label{que:vol-list}
Let $\hVol_3$ denote the set of local volumes of all klt threefold singularities $x\in X$. By \cite{XZ24} (see also \cite{Zhu24, LMS23}) we know that $0$ is the only accumulation point of $\hVol_3$. Moreover, Theorem \ref{thm:nv-3fold} implies that if $\hvol(x,X)\in \hVol_3 \cap [9, 27]$ then $x\in X$ is either a quotient singularity of order at most $3$ or a hypersurface singularity of type $cA_{\leq 2}$.
 From Example \ref{expl:ADE} we have the following containment
\[
 \hVol_3 \cap [9, 27] \supset \left\{9, \tfrac{2048}{225}, \tfrac{250}{27}, \tfrac{343}{36}, 6\sqrt{3}, \tfrac{32}{3}, \tfrac{27}{2}, \tfrac{125}{9}, 16, 27\right\}.
\]
Is this containment an equality?
\end{que}

\begin{expl}\label{expl:ADE}
\begin{enumerate}
    \item If $x\in X$ is an $A_k$-singularity, that is, locally analytically given by $x_1 x_2 + x_3^2 + x_4^{k+1} = 0$, then its local volume is $16$ when $k=1$, $\frac{125}{9}$ when $k=2$, and $\frac{27}{2}$ when $k\geq 3$. See \cite[Section 5]{Li18}, \cite[Example 4.7]{LL19} and \cite[Example 7.1.2]{LX20}.
\item If $x\in X$ is a $cA_1$-singularity but not an $A_k$-singularity, then $x\in X$ has to be a transversal $A_1$-singularity with $\hvol(x,X) = \frac{27}{2}$.
\item If $x\in X$ is defined by $x_1 x_2 + x_3^3 + x_4^k=0$ for $3\leq k\leq 6$, then by \cite{CS15} and \cite{LST25} we have $\hvol(x,X) = \frac{4(3+k)^3}{9k^2}$. Thus we get values $\frac{32}{3}, \frac{343}{36}, \frac{2048}{225}, 9$ for $k = 3,4,5,6$. Note that  $k=3,4,5$ correspond to $D_4$, $E_6$, and $E_8$-singularities.
\item If $x\in X$ is a $D_k$-singularity with $k\geq 5$, then we have local equation $x_1x_2 + x_3^2 x_4 + x_4^{k-1} = 0$. Then by \cite[Appendix, Example 4.1]{LX18} we know that its local volume is computed by the monomial valuation $v_{\bfw}|_{K(X)}$ with weight $\bfw=(1, 1, \sqrt{3}-1, 4-2\sqrt{3})$ whose K-semistable Fano cone degeneration is precisely the $D_\infty$-singularity $x_1x_2 + x_3^2 x_4 = 0$. Thus we have $\hvol(x, X) = 6\sqrt{3}$ which is the same as the local volume of a $D_\infty$-singularity.
\item If $x\in X$ is an $E_7$-singularity, then we have local equation $x_1x_2 + x_3^3 + x_3 x_4^3 =0$. By \cite[Section 8]{Li18} and \cite[Example 7.1.3]{LX20} we know that $\hvol(x, X) = \frac{250}{27}$.
\end{enumerate}

\end{expl}

\begin{thm}\label{thm:nv-3fold}
Let $x\in X$ be a klt threefold singularity. Then $\hvol(x,X)\geq 9$ if and only if  $x\in X$ is either a hypersurface singularity of type $cA_{\leq 2}$, or a cyclic quotient singularity of type $\frac{1}{2}(1,1,1)$, $\frac{1}{3}(1,1,0)$, $\frac{1}{3}(1,1,1)$, or $\frac{1}{3}(1,1,2)$.
\end{thm}

\begin{thm}\label{thm:nv-can}
Let $x\in X$ be a Gorenstein canonical non-hypersurface threefold singularity. Then $\hvol(x,X)\leq 9$, and equality holds if and only if $x\in X$ is a quotient singularity  of type $\frac{1}{3}(1,1,1)$.
\end{thm}

Theorem \ref{thm:nv-can} can be thought of as a refinement of the ODP Gap Theorem for threefolds in \cite{LX19} that any non-smooth threefold singularity $x\in X$ satisfies $\hvol(x,X)\leq 16$, and equality holds if and only if $x\in X$ is an $A_1$-singularity.

As a consequence, we have the following characterization of threefold singularities with local volumes at least $9$. See Example \ref{expl:ADE} for many examples of such singularities and their local volumes, and Question \ref{que:vol-list} for a conjectural list of all possible local volumes that are at least $9$.

\begin{thm}\label{thm:nv-3fold}
Let $x\in X$ be a klt threefold singularity. Then $\hvol(x,X)\geq 9$ if and only if  $x\in X$ is either a hypersurface singularity of type $cA_{\leq 2}$, or a cyclic quotient singularity of type $\frac{1}{2}(1,1,1)$, $\frac{1}{3}(1,1,0)$, $\frac{1}{3}(1,1,1)$, or $\frac{1}{3}(1,1,2)$.
\end{thm}

The above results have some notable applications to the study of K-moduli of Fano threefolds through the local-to-global volume comparison \cite{Liu18}.

\begin{thm}\label{thm:K-moduli}
Let $X$ be a K-semistable $\mathbb{Q}$-Fano threefold with volume $(-K_X)^3 = V$.
\begin{enumerate}
    \item If $V\geq 26$, then $X$ has only $cA_1$-singularities or  cyclic quotient singularities of type $\frac{1}{2}(1,1,1)$.
    \item If $V\geq 22$, then $X$ has only $cA_1$-singularities, isolated $cA_2$-singularities, $D_\infty$-singularities, or cyclic quotient singularities of type $\frac{1}{2}(1,1,1)$.
    \item If $V\geq 11$, then any non-Gorenstein singularity $x\in X$ is a cyclic quotient of a (possibly smooth) hypersurface singularity of type $cA_{\leq 2}$.
\end{enumerate}
If in addition $X$ is $\mathbb{Q}$-Gorenstein smoothable, then it does not admit cyclic quotient singularities of type $\frac{1}{2}(1,1,1)$.
\end{thm}

Theorem \ref{thm:K-moduli} can be viewed as a strengthening of a result proven in \cite{LX19, LZ24} stating that $X$ is Gorenstein canonical if $V\geq 20$. Such  restrictions on singularities played a key role in the description of K-moduli spaces for various families of Fano varieties via the moduli continuity method, see e.g.\ \cite{MM93, OSS16, SS17, LX19, Liu22, ADL21, LZ24, Zha24}. We expect Theorem \ref{thm:K-moduli} to be useful in the future study of K-moduli spaces for Fano threefolds. Specifically, according to \cite{ACC+}, there are precisely $79$ out of $105$ families of smooth Fano threefolds that contain K-semistable members, i.e.\ whose K-moduli spaces are non-empty. Among these $79$ families, Theorem \ref{thm:K-moduli} (1), (2), and (3) can be applied to $35$, $46$, and $68$ families, respectively, where   $3$, $9$, and $28$ families respectively have unknown K-moduli compactifications to the author's knowledge. See Remark \ref{rem:K-mod} for a list of these families.

As another application, we establish a sharp comparison between the local volume and the minimal log discrepancy (mld) for klt threefold singularities, answering  \cite[Question 6.16]{LLX18} affirmatively in dimension $3$.

\begin{thm}\label{thm:nv-mld}
Let $x\in X$ be a klt threefold  singularity. Then
\begin{equation}\label{eq:nv-mld}
    \hvol(x,X) \leq 9 \cdot \mathrm{mld}(x,X).
\end{equation}
Moreover, equality holds if and only if $x\in X$ is a cyclic quotient singularity of type $\frac{1}{r}(1,1,1)$ for some $r\in \bZ_{>0}$.
\end{thm}

Note that a weaker inequality $\hvol(x,X)< n^n\cdot \mathrm{mld}(x,X)$ was proven in any dimension $n$ in \cite[Theorem 6.13]{LLX18}.

Let us briefly explain the strategy of the proof of Theorems \ref{thm:nv-can} and \ref{thm:nv-3fold}, which is similar to \cite{LX19} but technically more involved. For hypersurface singularities, we use local equations and monomial valuations on the ambient space to get upper bounds for its local volumes.  Suppose $x\in X$ is a Gorenstein canonical non-hypersurface threefold singularity. Then the hope is to either find a good divisor with small normalized volume or to bound local volumes on a crepant birational model. This is done by running a suitable MMP on the terminalization to contract crepant exceptional divisors one by one. Compared to \cite{LX19}, the new difficulty lies in the fact that in order to achieve the local volume bound $9$ ($=$ local volume of a transversal $A_2$-singularity), we need to analyze the behavior of contractions of two exceptional divisors, as contracting only one divisor may create a transversal $A_1$-singularity whose local volume is $\frac{27}{2}$, which is not strong enough for our estimate. Then we reduce to the case when there are at most two crepant exceptional divisors over $x\in X$ and apply the detailed analysis  from \cite{Lau77, Rei76, Rei80, Rei94}. In some cases we are only able to get a good crepant birational model after an \'etale base change using Artin approximation theorem (see Section \ref{sec:Artin}), a result that is of independent interest.

\subsection*{Acknowledgements} The author would like to thank Anne-Sophie Kaloghiros, J\'anos Koll\'ar, Andrea Petracci, Jakub Witaszek, Chenyang Xu, and Junyan Zhao for helpful discussions and comments. The author is partially supported by NSF CAREER Grant DMS-2237139 and an AT\&T Research Fellowship from Northwestern University.

“Let us briefly explain the strategy of the proof of Theorems \ref{thm:nv-can} and \ref{thm:nv-3fold}, which is similar to \cite{LX19} but technically more involved.”

- For hypersurface singularities: “we use local equations and monomial valuations on the ambient space to get upper bounds for its local volumes.”

- For a Gorenstein canonical non-hypersurface threefold singularity $x\in X$: the approach is “to either find a good divisor with small normalized volume or to bound local volumes on a crepant birational model.”

- This is achieved “by running a suitable MMP on the terminalization to contract crepant exceptional divisors one by one.”

- New technical issue for the bound $\hvol\le 9$: “in order to achieve the local volume bound $9$ ($=$ local volume of a transversal $A_2$-singularity), we need to analyze the behavior of contractions of two exceptional divisors, as contracting only one divisor may create a transversal $A_1$-singularity whose local volume is $\frac{27}{2}$, which is not strong enough for our estimate.”

- Reduction step: “Then we reduce to the case when there are at most two crepant exceptional divisors over $x\in X$ and apply the detailed analysis from \cite{Lau77, Rei76, Rei80, Rei94}.”

- In some cases: “we are only able to get a good crepant birational model after an \'etale base change using Artin approximation theorem (see Section \ref{sec:Artin}).”

Let us briefly explain the strategy of the proof of \begin{thm}\label{thm:nv-3fold}
Let $x\in X$ be a klt threefold singularity. Then $\hvol(x,X)\geq 9$ if and only if  $x\in X$ is either a hypersurface singularity of type $cA_{\leq 2}$, or a cyclic quotient singularity of type $\frac{1}{2}(1,1,1)$, $\frac{1}{3}(1,1,0)$, $\frac{1}{3}(1,1,1)$, or $\frac{1}{3}(1,1,2)$.
\end{thm}
and, in establishing the main theorem, the proof is similar to \cite{LX19} but technically more involved.

- For hypersurface singularities: we use local equations and monomial valuations on the ambient space to get upper bounds for its local volumes.

- For a Gorenstein canonical non-hypersurface threefold singularity $x\in X$: the approach is to either find a good divisor with small normalized volume or to bound local volumes on a crepant birational model.

- This is achieved by running a suitable MMP on the terminalization to contract crepant exceptional divisors one by one.

- New technical issue for the bound $\hvol\le 9$: in order to achieve the local volume bound $9$ ($=$ local volume of a transversal $A_2$-singularity), we need to analyze the behavior of contractions of two exceptional divisors, as contracting only one divisor may create a transversal $A_1$-singularity whose local volume is $\frac{27}{2}$, which is not strong enough for our estimate.

- Reduction step: then we reduce to the case when there are at most two crepant exceptional divisors over $x\in X$ and apply the detailed analysis from \cite{Lau77, Rei76, Rei80, Rei94}.

- In some cases: we are only able to get a good crepant birational model after an \'etale base change using Artin approximation theorem (see Section \ref{sec:Artin}).

Biconditional or Equivalence
Inequality or Bound

Question: Let $x\in X$ be a Gorenstein canonical non-hypersurface threefold singularity, and let $\hvol(x,X)$ denote its local volume. Here, “non-hypersurface” means that the singularity is not analytically a hypersurface singularity, and a quotient singularity of type $\frac{1}{3}(1,1,1)$ means the cyclic quotient of $\mathbb{A}^3$ by the order-$3$ action with weights $(1,1,1)$. Which quantitative estimate holds for $\hvol(x,X)$ under these assumptions?

$\hvol(x,X)\le 9$, and equality holds if and only if $x\in X$ is a quotient singularity of type $\frac{1}{3}(1,1,1)$.

Choices:
B. $\hvol(x,X)\le \frac{27}{2}$, and equality holds if and only if $x\in X$ is a transversal $A_1$-singularity.
C. $\hvol(x,X)\le 9$.
D. $\hvol(x,X)< 9$, unless $x\in X$ is a quotient singularity of type $\frac{1}{3}(1,1,1)$, in which case $\hvol(x,X)=9$.
E. $\hvol(x,X)\le 9$, and equality holds if and only if $x\in X$ is a cyclic quotient singularity of type $\frac{1}{r}(1,1,1)$ for some $r\in \mathbb{Z}_{>0}$.

two-divisor analysis needed to improve bound from 27/2 down to 9

equality case at the sharp boundary value 9

special equality classification restricted to the single non-hypersurface Gorenstein case r=3

The stem does not explicitly reveal the correct analytic type. It asks for an equivalent characterization of the equality case, without embedding the answer in the wording.

This is essentially a direct restatement of the theorem characterizing when the local volume reaches 9. It tests recall of the exact equality case rather than selection among independently derived conclusions.

Some discrimination is required because the distractors are nearby variants (arbitrary cyclic quotient, weaker quotient condition, extra geometric alternative), but the main task is still theorem recall rather than genuine multi-step reasoning.

The distractors are plausible and mathematically meaningful: one weakens the conclusion, one overgeneralizes the quotient type, one introduces a tempting alternative singularity case, and one offers a technical but incorrect birational criterion.

A mathematically clean MCQ with strong distractors and no answer leakage, but it is largely tautological: it mainly checks recall of the precise theorem statement rather than generative reasoning.