[\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


\label{theorem: bound rango}
For every $k\geq 2$ and every $d_1,\dots,d_k\geq 3$ we have
$$
\rk_{\mathbf{d}}(W_{d_1}\otimes \cdots \otimes W_{d_k})\leq  2^{k-1}\left(d_1+\cdots+d_k-2k+2\right).
$$

\begin{theorem}\label{theorem: bound rango}
For every $k\geq 2$ and every $d_1,\dots,d_k\geq 3$ we have
$$
\rk_{\bfd}(W_{d_1}\otimes \cdots \otimes W_{d_k})\leq  2^{k-1}\left(d_1+\cdots+d_k-2k+2\right).
$$
\end{theorem}

\label{subsection: commenti sul bound}

As already emphasized, the general idea of bounding the rank of tensor product of $W$-states by looking at the containment of $W_{d_1}\otimes \cdots \otimes W_{

\begin{proposition}\label{prop: rango TJ}
Let $k\geq 2$, let $\bfd\in \NN_{\geq 3}^k$ and let $d=d_1+\cdots +d_k$. Let $\mathcal{F}\subseteq \CC[u,v]_d$ be the family of degree $d$-homogenous polynomial defined as
$$
\mathcal{F}= \left\{  F=\sum_{i=0}^{k-2}a_i\binom{d}{i}u^{d-i}v^i+\binom{d}{d-k}a_{d-k}u^{k}v^{d-k}+\sum_{i=d-k+2}^d a_i \binom{d}{i}u^{d-i}v^i, \text{ for }a_i\in \CC \text{ with } a_{d-k}\neq 0 \right\}.
$$
Then
$$
\min_{F\in\mathcal F}\rk F=d-2k+2.
$$
In particular, if  $2k\leq\left\lfloor \frac d2\right\rfloor +1$ the minimum is achieved by any $F\in\mathcal F$ with $a_{k-2}\neq 0$. Otherwise, the minimum is achieved by any $F\in \mathcal{F}$ with $a_{k-2}\neq 0$ and $a_i=0$ for $i\in \{0,\dots,k-3\} \cup \{d-k+2,\dots,d \}$.
\end{proposition}

Computing the rank of a tensor is generally NP-hard, see \cite{Has90, rankNPhard}. For this reason, one often focuses on special families of tensors that serve as meaningful benchmarks either because they originate from applications, or because they exhibit particularly interesting behavior. One such family is the tensor product of generalized $W$-state: $$W_{d_1}\otimes \cdots \otimes W_{d_k}\in \underbrace{\mathbb{C}^2\otimes \cdots \otimes \mathbb{C}^2}_{d_1+\cdots +d_k-\text{times}}
$$
where each $W$-state $W_d$ is defined as
$$
W_d=\sum_{i=1}^d e^{(1)}_{1}\otimes \cdots \otimes e^{(i-1)}_1\otimes e^{(i)}_2 \otimes e^{(i+1)}_1\otimes \cdots \otimes e^{(d)}_1 \in (\mathbb{C}^2)^{\otimes d}, \quad \spann\{e^{(i)}_1,e^{(i)}_2 \}\cong \mathbb{C}^2.
$$
Focusing for a moment on a single copy, the generalized $W$-state itself is a highly interesting tensor. It is a symmetric tensor and one of the simplest examples illustrating the non-semicontinuity of tensor rank. Moreover, the expression of the general element of the tangential variety $\tau(\nu((\mathbb{P}^1)^{\times d}))$ of the Segre image $\nu((\mathbb{P}^1)^{\times d})$ is $W_d$. This is because actually $W_d\in \langle \nu(Z)\rangle$, where $Z\subset (\mathbb{P}^1)^{\times d}$ is a zero-dimensional scheme of length 2 supported at $\otimes_{i=1}^d e^{(i)}_{1} $, called a 2-jet \cite{alexander1997interpolation}, so actually $W_d$ lies on a line that is tangent to $\nu((\mathbb{P}^1)^{\times d})$ at $\otimes_{i=1}^d e^{(i)}_{1} $. The $W$-state is a rank-$d$ tensor and it is an example of tensor for which tensor rank and symmetric tensor rank coincide \cite{rankTangentialBB}. It has infinitely many decompositions computing its rank and in particular, one can choose any elementary tensor except for $\otimes_{i=1}^d e^{(i)}_{1}$ to be part of one of its minimal (symmetric) rank decomposition \cite{carlini2017waring, BOS}.

From the point of view of complexity theory, $W$-states are related to the so-called Coppersmith-Winogard tensors \cite{coppersmith1987matrix}, which are tensors of interest in the study of the matrix multiplication tensor. In particular, $W$-states are the outer structure of many tensors, including Strassen tensor and truncated polynomial multiplication, used in the so-called \emph{laser method}, a strategy introduced by V. Strassen in \cite{Str86} for computing the complexity of matrix multiplication.

In the three-factors case $W_3$ plays a historical central role in quantum information theory, being one of the two fundamental classes of genuine tripartite entanglement \cite{dur2000three}, and this is where the name \emph{$W$-state} comes from. This makes the $W$-state a structurally very rich object at the intersection of geometry, algebraic complexity, and quantum information.

Given how special $W_d$ is, it is natural to expect that the tensor product of several copies, $W_{d_1}\otimes \cdots \otimes W_{d_k} $, is also quite special. This expectation is reinforced for instance by the role that the tensor product of two copies of $W$-states had in the study of the multiplicativity of tensor rank under tensor product: indeed, \cite{CJZ} proved that the rank of $W_3\otimes W_3$ is strictly less than the naive multiplicative guess $3\cdot 3$. This was the first explicit example showing that the rank of the tensor product of two tensors is not the product of the two ranks. Shortly after, \cite{CF18} proved that the rank of $W_3\otimes W_3$ is 8. A systematic approach to the strict submulticativity property can be found in \cite{BBGOV}, while, motivated by applications in complexity theory and quantum information theory, the submultiplicativity of the Kronecker tensor product of many copies of $W$-states has been studied for instance in \cite{chen2010tensor, zuiddam2017note}. Some Kronecker products of $W$-states show also other interesting connections. For instance, $W_3\boxtimes W_3$ is a symmetric tensor corresponding to the cubic hypersurface of $\mathbb{P}^3$ made of a quadric and a tangent hyperplane and, by \cite{Seg42}, this is the only quaternary cubic having maximal rank 7. As a second example, a particular projection of $W_3\boxtimes W_3\boxtimes W_3$ is the so-called Bjorklund-Kaski tensor, which appears in \cite{BK24} and is used for proving that the asymptotic rank conjecture \cite{Str94} and the set cover conjecture \cite{CFKLMMPS15, KT19} cannot be both true.

Hence, determining the rank of $W_{d_1}\otimes \cdots \otimes W_{d_k}$ becomes a particularly meaningful problem. \cite{BBCG} gave the first bounds on the rank of the tensor product of an arbitrary number of $W$-states
 and these remain the best known results to date, with the notable exception of the case $k=2$ where the bound was later improved in \cite[Theorem 1.8(i)]{gal}.

In this paper, we make progress along this direction by providing a new sharp upper bound on the rank of tensor products of generalized $W$-states $W_{d_1}\otimes \cdots \otimes W_{d_k}\in \bigotimes_{i=1}^k(\mathbb{C}^{2})^{\otimes d_i}$. We remark that $W_{d_1}\otimes \cdots \otimes W_{d_k}$ is a partially symmetric tensor and for a partially symmetric tensor $T\in \mathop{\rm Sym}\nolimits^{d_1}\mathbb{C}^{n_1}\otimes \cdots \otimes \mathop{\rm Sym}\nolimits^{d_k}\mathbb{C}^{n_k}$ the partially symmetric rank of $T$, denoted as $\rk_{\mathbf{d}}(T)$, is the minimum integer $r$ such that $T=\sum_{i=1}^r (v^{(1)}_i)^{\otimes d_1}\otimes \cdots \otimes (v^{(k)}_i)^{\otimes d_k}$, where $v^{(j)}_i\in \mathbb{C}^{n_j}$. Our main result is the following.

In this paper, we make progress along this direction by providing a new sharp upper bound on the rank of tensor products of generalized $W$-states $W_{d_1}\otimes \cdots \otimes W_{d_k}\in \bigotimes_{i=1}^k(\CC^{2})^{\otimes d_i}$. We remark that $W_{d_1}\otimes \cdots \otimes W_{d_k}$ is a partially symmetric tensor and for a partially symmetric tensor $T\in \sym^{d_1}\CC^{n_1}\otimes \cdots \otimes \sym^{d_k}\CC^{n_k}$ the partially symmetric rank of $T$, denoted as $\rk_{\bfd}(T)$, is the minimum integer $r$ such that $T=\sum_{i=1}^r (v^{(1)}_i)^{\otimes d_1}\otimes \cdots \otimes (v^{(k)}_i)^{\otimes d_k}$, where $v^{(j)}_i\in \CC^{n_j}$. Our main result is the following.

In addition to providing a theoretical upper bound on the tensor rank, \Cref{theorem: bound rango} is constructive. Indeed, the proof yields an explicit decomposition of length equal to the bound that can be achieved by computing the rank decomposition of essentially one binary form (see \Cref{algo}). This decreases a lot the computational cost for computing a partially symmetric decomposition of $W_{d_1}\otimes\cdots\otimes W_{d_k}$; see the forthcoming \Cref{remark: commento algo}.

As an immediate consequence the same bound holds also for the rank of tensor product of $W$-states. \begin{corollary}\label{corollary: bound rango vero}
Let $k\geq 2$, fix positive integers $d_i\geq 3$ for $i=1,\dots,k$.
The rank of $W_{d_1}\otimes \cdots \otimes W_{d_k}\in \bigotimes_{i=1}^k(\CC^{2})^{\otimes d_i}$ is at most $2^{k-1}(d_1+\dots+d_k-2k+2)$.
\end{corollary}

We have now all the tools to prove our main \Cref{theorem: bound rango}, which we restate below for convenience.
\begin{theorem*}(\Cref{theorem: bound rango}) Let $k\geq 2$, $\bfd\in \NN_{\geq 3}^k$ and $d=d_1+\cdots +d_k$. Then
$$
\rk_{\bfd}(W_{d_1}\otimes \cdots \otimes W_{d_k})\leq 2^{k-1}(d-2k+2).
$$
\end{theorem*}
\begin{proof}
By \Cref{remark: W-state sta nel quadrello} $W_{d_1}\otimes\cdots \otimes W_{d_k}$ belongs to $\spann\left\{ \cup_{J\subseteq [2,k]} \mathcal{C}_{\bfd}^J\right\} $, so we have
$$
\prod_{j=1}^kd_j W_{d_1}\otimes\cdots \otimes W_{d_k}=\sum_{J\subseteq [2,k]}T^J, \,\hbox{ for some }T^J\in\spann\mathcal{C}_{\bfd}^J.
$$
Writing $T^J$ with respect to the basis of \Cref{remark: multi hom binary in forma binaria emplice}, we have
$$T^J=\sum_{s=0}^{d}\alpha_s^J\left(\sum_{\bfi\in\mathcal A_{\bfd,s}}\varepsilon_\bfi^Jx_{1,0}^{i_1}x_{1,1}^{d_1-i_1}\otimes\cdots\otimes x_{k,0}^{i_k}x_{k,1}^{d_k-i_k}\right),$$
where the $\alpha^J_s$'s are a solution of the linear system $\mathcal S$ of \Cref{prop: mortale}.
Using the isomorphism $\varphi^J$ defined in \Cref{remark: multi hom binary in forma binaria emplice}, each $T^J$ can be seen as a binary form of degree $d$ in $\spann\mathcal C_\bfd^J$. More precisely, we have
$$\varphi^J(T^J)=\sum_{s=0}^d\alpha_s^J\binom{d}{s}u^{d-s}v^s.$$
It follows that $\rk_\bfd T^J\leq\rk\varphi^J(T^J)$, where $\rk\varphi^J(T^J)$ is the Waring rank of the binary form $\varphi^J(T^J)$ and, by \Cref{remark: multi hom binary in forma binaria emplice}, it is equal to the $\mathcal C_\bfd^J$-rank of $T^J$.
By \Cref{prop: mortale}, we have $\alpha_s^J=0$ for any $s\in[k-1,d-k-1]\cup\left\{d-k+1\right\}$ and $\alpha^J_{d-k}\neq 0$. Moreover, whether $2k\leq\left\lfloor\frac{d}{2}\right\rfloor+1$ or $2k>\left\lfloor\frac{d}{2}\right\rfloor+1$, by \Cref{prop: mortale} we can choose the $\alpha_s^J$'s such that $\varphi^J(T^J)$ is a minimizer as in \Cref{prop: rango TJ}, so that $\rk (\varphi^J(T^J))=d-2k+2$. Hence, we have
\begin{equation*}
\rk_{\bfd}(W_{d_1}\otimes \cdots \otimes W_{d_k})\leq\sum_{J\subseteq[2,k]}\rk_\bfd T^J\leq \sum_{J\subseteq[2,k]}\rk \varphi^J(T^J)= 2^{k-1}(d-2k+2).\qedhere
\end{equation*}
\end{proof}
Since for a tensor $T\in \sym^{d_1}\CC^2\otimes \cdots \otimes \sym^{d_k}\CC^2$ we have $\rk(T)\leq \rk_{\bf d}(T)$, the previous result gives a bound also on the tensor rank of $W_{d_1}\otimes \cdots \otimes W_{d_k}$.

\Cref{example: step4}, together with \Cref{example: step1}, \Cref{example: step2} and \Cref{example: step3}, concretely shows that, following our procedure, it is possible to construct an explicit decomposition of $W_{d_1}\otimes\cdots\otimes W_{d_k}$ having length as in the bound of \Cref{theorem: bound rango} as we now explain in detail.
\begin{remark}\label{remark: commento algo}
Let $\bfd\in\NN^k_{\geq 3}$. For $s\in[0,d]\setminus\{d-k\}$ we choose one $\bf i \in \A_{\bf d,s }$ and set $\varepsilon^J_{\bf i}=1$ for any $J\subseteq[2,k]$, and for $s=d-k$ and $\bfi=(d_1-1,\dots,d_k-1)$ we set $\varepsilon_\bfi^J=1$ for any $J\subseteq[2,k]$. In order to find a decomposition of $W_{d_1}\otimes\cdots\otimes W_{d_k}$ following our procedure, we have to first find $T^J\in\spann(\mathcal C_{\bfd}^J)$ such that
$$\prod_{j=1}^kd_jW_{d_1}\otimes\cdots\otimes W_{d_k}=\sum_{J\subseteq[2,k]}T^J.$$
The coordinates of the $T^J$'s are of the form
$$T^J=(\varepsilon_{\bfi}^J\alpha_s^J)_{\substack{0\leq s\leq d\\ \bfi\in\A_{\bfd,s}}}$$
and, by \Cref{prop: rango TJ}, it is enough to take $\alpha_{k-2}^J\neq0$ and $\alpha_{d-k}^J\neq 0$ for every $J$ to minimize the length of the decomposition. The easiest possible choice is given by \Cref{prop: mortale}: we take $\alpha_s=0$ for $s\in[0,d]\setminus\{k-2,d-k\}$, $\alpha_{k-2}=(-1)^{|J|}$ and $\alpha_{d-k}=2^{k-1}$. Hence, when we compute $\varphi^J(T^J)$ we only have two possibilities:
$$\varphi^J(T^J)=\begin{cases}
 \binom{d}{k-2}u^{d-k+2}v^{k-2}+2^{k-1}\binom{d}{d-k}u^kv^{d-k},&\text{ if $|J|$ is even}\\
 -\binom{d}{k-2}u^{d-k+2}v^{k-2}+2^{k-1}\binom{d}{d-k}u^kv^{d-k},&\text{ otherwise.}
\end{cases}$$
As a consequence, it is enough to apply the Sylvester's algorithm to only two binary forms. Once a minimal decomposition of the two possible $\varphi^J(T^J)$'s is found it is enough to apply $(\varphi^J)^{-1}$ for all the $J\subseteq[2,k]$ to find a partially symmetric decomposition of the product of $W$-states. Note that this drastically reduce the computational cost. Indeed, the decompositions of the two binary forms and the application of $(\varphi^J)^{-1}$ are computationally cheap. The core of the advantage of this method is that the combinatorics of the $\varepsilon_\bfi^J$'s allows to avoid many computations. We summarized the procedure in \Cref{algo}.
\end{remark}

\begin{theorem}\label{theorem: bound rango bordo}
Let $k\geq 2$, let $\bfd\in \NN_{\geq 3}^k$ and let $d=d_1+\cdots+d_k$. Then
$$
\underline{\rk}_\bfd(W_{d_1}\otimes\cdots\otimes W_{d_k})= 2^{k}.
$$
\end{theorem}
\begin{proof}
We use the catalecticant bound for border rank of \cite[Corollary 5.5]{gal}. In order to do that we have to produce a partially symmetric flattening with rank at least $2^k$. Let us consider the following flattening of $W_{d_1}\otimes \cdots \otimes W_{d_k}$:
\[
F : \sym^2(\mathbb{C}^2) \otimes \dots \otimes \sym^2(\mathbb{C}^2) \rightarrow
\sym^{d_1-2}(\mathbb{C}^2)^* \otimes \dots \otimes \sym^{d_k-2}(\mathbb{C}^2)^*.
\]
Let $\mathbb{C}[\alpha_{1,0},\alpha_{1,1},\dots,\alpha_{k,0},\alpha_{k,1}]$ be the ring of multigraded derivations on $\CC[x_{1,0},x_{1,1},\dots,x_{k,0},x_{k,1}]$ with $\alpha_{i,j}$ defined as the dual of $x_{i,j}$. By apolarity theory, the rank of the linear map $F$ is exactly the dimension of the degree $(2,\dots,2)$ component of the quotient of the multigraded ring $\mathbb{C}[\alpha_{1,0},\alpha_{1,1},\dots,\alpha_{k,0},\alpha_{k,1}]$ by the ideal $I_{d}$ apolar to the W-state, i.e. $I_{d}=(\alpha_{1,0}^2,\dots,\alpha_{k,0}^2,\alpha_{1,1}^{d_1},\dots,\alpha_{k,1}^{d_k})$. For $\varepsilon_i \in \{0,1\}$ the $2^k$ forms
$$F_{\varepsilon_1,\dots,\varepsilon_{k}}=\alpha_{1,0}^{\varepsilon_1}\alpha_{1,1}^{2-\varepsilon_1} \dots \alpha_{k,0}^{\varepsilon_k}\alpha_{k,1}^{2-\varepsilon_k}$$ are linear independent of multidegree $(2,\dots, 2)$, with $F_{\varepsilon_1,\dots,\varepsilon_{k}} \notin I_d$. This concludes the proof.
\end{proof}

\Cref{theorem: bound rango} improves \cite[Theorem 3.6]{BBCG} by $2^k(k-1)$. For $k=2$ it coincides with the bound provided in \cite[Theorem 1.8(i)]{gal} and when all $d_i=3$ it matches the bound of \cite[Theorem 3.3]{BBCG}. The bound of \Cref{theorem: bound rango} is sharp as for $k=2$ and $d_1=d_2=3$ it gives rank 8.

 In addition to providing a theoretical upper bound on the tensor rank, \Cref{theorem: bound rango} is constructive. Indeed, the proof yields an explicit decomposition of length equal to the bound that can be achieved by computing the rank decomposition of essentially one binary form (see \Cref{algo}). This decreases a lot the computational cost for computing a partially symmetric decomposition of $W_{d_1}\otimes\cdots\otimes W_{d_k}$; see the forthcoming \Cref{remark: commento algo}.

 As an immediate consequence the same bound holds also for the rank of tensor product of $W$-states. \begin{corollary}\label{corollary: bound rango vero}
Let $k\geq 2$, fix positive integers $d_i\geq 3$ for $i=1,\dots,k$.
The rank of $W_{d_1}\otimes \cdots \otimes W_{d_k}\in \bigotimes_{i=1}^k(\mathbb{C}^{2})^{\otimes d_i}$ is at most $2^{k-1}(d_1+\dots+d_k-2k+2)$.
\end{corollary}

Our methods to achieve this result are geometric and, as in \cite[Theorem 3.6]{BBCG}, they rely on finding a curve in $\mathcal{C}\subset (\mathbb{P}^1)^{\times k}$ such that the span of the image of $\mathcal{C}$ under the Segre-Veronese embedding contains $W_{d_1}\otimes \cdots \otimes W_{d_k}$.

\subsection*{Outline of the paper}  We set up our notation and give standard preliminaries in \Cref{section: preliminaries}, while \Cref{section: solving the system} contains the main technical results to get our bounds. We start \Cref{section: computing the bounds} by briefly reviewing the theory of catalecticants needed to compute the rank of a family of binary homogeneous polynomials (\Cref{prop: rango TJ}). Then, we prove our main \Cref{theorem: bound rango} and in \Cref{algo} we give a recipe to explicitly compute a partially symmetric decomposition of length given by our bound. \Cref{theorem: bound rango bordo} computes the border partially symmetric rank of $W_{d_1}\otimes \cdots \otimes W_{d_k}$. We conclude with a detailed comment on the method in \Cref{subsection: commenti sul bound}.

\subsection*{Acknowledgements}
The authors are grateful to Fulvio Gesmundo, Alessandro Gimigliano and Joachim Jelisiejew for interesting conversations on the topic.

S. Canino has been funded by the Italian Ministry of
University and Research in the framework of the Call for Proposals for
scrolling of final rankings of the PRIN 2022 call - Protocol no.
2022NBN7TL.
A. Casarotti has been funded by the European Union under the project NextGenerationEU. PRIN 2022, CUP: F53D23002600006.
P. Santarsiero was supported by the European Union under NextGenerationEU. PRIN 2022, Prot. 2022E2Z4AK and PRIN 2022 SC-CUP: I53C24002240006.

The post-theorem text does not give a step-by-step proof sketch for \Cref{theorem: bound rango}, but it indicates the proof’s nature and main idea: the result is obtained by “geometric” methods and “rely on finding a curve in $\mathcal{C}\subset (\mathbb{P}^1)^{\times k}$ such that the span of the image of $\mathcal{C}$ under the Segre-Veronese embedding contains $W_{d_1}\otimes \cdots \otimes W_{d_k}$.” It also states the proof is “constructive,” yielding “an explicit decomposition of length equal to the bound” that “can be achieved by computing the rank decomposition of essentially one binary form (see \Cref{algo}).”

The post-theorem text does not give a step-by-step proof sketch for the main theorem, but it indicates the proof’s nature and main idea: the result is obtained by “geometric” methods and “rely on finding a curve in $\mathcal{C}\subset (\mathbb{P}^1)^{\times k}$ such that the span of the image of $\mathcal{C}$ under the Segre-Veronese embedding contains $W_{d_1}\otimes \cdots \otimes W_{d_k}$.” It also states the proof is “constructive,” yielding “an explicit decomposition of length equal to the bound” that “can be achieved by computing the rank decomposition of essentially one binary form (see
\label{algo}
\end{algorithm}
\subsection{Further remarks on \Cref{theorem: bound rango} }\label{subsection: commenti sul bound}

As already emphasized, the general idea of bounding the rank of tensor).”

Question:
Let \(k\ge 2\) and let \(d_1,\dots,d_k\ge 3\) be integers, and write \(\mathbf d=(d_1,\dots,d_k)\). For each \(d\), define the generalized \(W\)-state
\[
W_d=\sum_{i=1}^d e^{(1)}_1\otimes\cdots\otimes e^{(i-1)}_1\otimes e^{(i)}_2\otimes e^{(i+1)}_1\otimes\cdots\otimes e^{(d)}_1\in (\mathbb C^2)^{\otimes d},
\]
where each \(\operatorname{span}\{e^{(i)}_1,e^{(i)}_2\}\cong \mathbb C^2\). View
\(W_{d_1}\otimes\cdots\otimes W_{d_k}\) as an element of
\(\operatorname{Sym}^{d_1}(\mathbb C^2)\otimes\cdots\otimes \operatorname{Sym}^{d_k}(\mathbb C^2)\), and let \(\operatorname{rk}_{\mathbf d}\) denote its partially symmetric rank, i.e. the minimum number of summands of the form \(\ell_1^{d_1}\otimes\cdots\otimes \ell_k^{d_k}\) needed to express it. Which statement holds for every such choice of \(k,d_1,\dots,d_k\)?

\[\operatorname{rk}_{\mathbf d}(W_{d_1}\otimes\cdots\otimes W_{d_k})\le 2^{k-1}\bigl(d_1+\cdots+d_k-2k+2\bigr).\]

Choices:
B. \[\operatorname{rk}_{\mathbf d}(W_{d_1}\otimes\cdots\otimes W_{d_k})\le 2^{k}\bigl(d_1+\cdots+d_k-2k+2\bigr).\]
C. \[\operatorname{rk}_{\mathbf d}(W_{d_1}\otimes\cdots\otimes W_{d_k})\le 2^{k-1}(d_1+\cdots+d_k).\]
D. \[\operatorname{rk}_{\mathbf d}(W_{d_1}\otimes\cdots\otimes W_{d_k})\le 2^{k-1}\bigl(d_1\cdots d_k-2k+2\bigr).\]
E. \[\operatorname{rk}_{\mathbf d}(W_{d_1}\otimes\cdots\otimes W_{d_k})= 2^{k-1}\bigl(d_1+\cdots+d_k-2k+2\bigr).\]


\begin{theorem}\label{thm:main}
Let $M,N\in\N$, with $\lcm(M,N)$ not
divisible by $2^4$ or the square of an odd prime.
Let $f\in S_k^{\rm new}(\Gamma_0(M))$ and $g\in S_\ell^{\rm new}(\Gamma_0(N))$
be non-zero cusp forms with real Fourier coefficients
$a_f(n),a_g(n)$. Then the following are equivalent:
\begin{enumerate}
\item $a_f(n)a_g(n)\ge0$ for all sufficiently large $n\in\N$;
\item $(M,k)=(N,\ell)$ and $f/g$ is a positive constant.
\end{enumerate}
Thus if $f$ and $g$ are not proportional then
$\{n\in\N:\epsilon a_f(n)a_g(n)>0\}$
is infinite for each $\epsilon\in\{\pm1\}$.
\end{theorem}

Let $f\in S_k(\Gamma_0(M))$ and $g\in S_\ell(\Gamma_0(N))$ be modular
forms with real Fourier coefficients $a_f(n)$ and $a_g(n)$. Various
authors have investigated the extent to which $f$ and $g$ are distinguished
by the signs of $a_f(n)$ and $a_g(n)$; see for instance
\cite{KLSW,Matomaki,GKR} and the references therein.
For Hecke eigenforms,
it follows from the recent proof of joint Sato--Tate by Wong \cite{Wong}
that if $f$ and $g$ are distinct, non-CM, normalized newforms
then each of the sets
$\{p\text{ prime}:\epsilon a_f(p)a_g(p)>0\}$
for $\epsilon\in\{\pm1\}$
has natural density $\frac12$.
In another direction, Gun, Kohnen, and Rath \cite{GKR} considered
forms $f$ and $g$ of distinct weights that are not necessarily
Hecke eigenforms; it follows from their result that if $k\ne\ell$ and
$a_f(n)a_g(n)$ is not identically zero\footnote{One can
find examples of non-zero $f$ and $g$ of distinct weights such that
$a_f(n)a_g(n)=0$ identically, so this hypothesis cannot be removed.}
then each of the sets $\{n\in\mathbb{N}:\epsilon a_f(n)a_g(n)>0\}$ is infinite.

In both of these results the forms in question are orthogonal,
so it is natural to expect sign changes to occur.
Without orthogonality the situation is much less clear, but
one might guess that the signs of $a_f(n)$ determine $f$ up to
scalar multiplication. That is not always the case (see the remarks below),
but we show that it is true in many cases:

Our proof makes use of the joint Sato--Tate equidistribution
of any two non-CM, twist-inequivalent newforms, proven by
Wong \cite{Wong}, based on the spectacular results
of Barnet-Lamb et al.\ \cite{BGG}. (See also Newton and Thorne
\cite{Newton-Thorne} for the recent strengthening
to full automorphy of symmetric powers for Hilbert modular forms
of regular weight, and Thorner \cite{Thorner} for results with an
effective rate of convergence.)
Precisely, let $\varphi\in S_k^{\rm new}(\Gamma_0(M))$,
$\psi\in S_\ell^{\rm new}(\Gamma_0(N))$
be distinct normalized Hecke eigenforms of conductors $M,N$ as in the theorem,
and write their Fourier coefficients at primes $p$ in the form
\[
a_\varphi(p)=2p^{\frac{k-1}{2}}\cos(2\pi\theta_\varphi(p)),
\quad
a_\psi(p)=2p^{\frac{\ell-1}{2}}\cos(2\pi\theta_\psi(p)),
\quad\text{where }
(\theta_\varphi(p),\theta_\psi(p))\in\bigl[0,\tfrac12\bigr]^2.
\]
Then by \cite[Theorem~1.1]{Wong}, for any box
$B=[\alpha,\beta]\times[\gamma,\delta]\subseteq[0,\frac12]^2$, we have
\[
\lim_{x\to\infty}
\frac{\#\{p\le x:(\theta_\varphi(p),\theta_\psi(p))\in B\}}{\pi(x)}
=\int_B\bigl(4\sin(2\pi u)\sin(2\pi v)\bigr)^2\,du\,dv.
\]
As we will see in Lemma~\ref{lem:linindep}, it follows that
$1$, $\theta_\varphi(p)$, and $\theta_\psi(p)$ are linearly independent
over $\Q$ for all $p$ in a set of density $1$.

We note that mutual independence of the angles $\theta_\varphi$ for
more than two newforms is not known, though it would follow from
the functoriality of arbitrary products of symmetric powers.
This presents an obstacle to proving Theorem~\ref{thm:main} using
only information on $a_f(p)a_g(p)$ at primes $p$.
(We will however give such a proof when $f$ and $g$ have different
weights or levels.) Instead we take an approach that is possibly of independent
interest, showing that for almost any tuple of
distinct primes $p_1,\ldots,p_d$, where $d=\dim S_k^{\rm new}(\Gamma_0(M))$,
the non-zero $f\in S_k^{\rm new}(\Gamma_0(M))$ with real coefficients
are determined modulo scalars by the signs of $a_f(n)$
for $n\in\{p_1^{k_1}\cdots p_d^{k_d}:k_1,\ldots,k_d\ge0\}$;
see Proposition~\ref{prop:dense} for
the precise statement, and \cite{Amri,GMP} for some related results.

Let $\{\varphi_1,\ldots,\varphi_d\}$
and $\{\psi_1,\ldots,\psi_{d'}\}$ be normalized Hecke eigenbases
for $S_k^{\rm new}(\Gamma_0(M))$ and $S_\ell^{\rm new}(\Gamma_0(N))$, respectively.
Write
\[
\varphi_i(z)=\sum_{n=1}^\infty
\lambda_{\varphi_i}(n)n^{\frac{k-1}{2}}e(nz),
\quad
\psi_j(z)=\sum_{n=1}^\infty
\lambda_{\psi_j}(n)n^{\frac{\ell-1}{2}}e(nz),
\]
and
\[
\lambda_f(n)\coloneq\frac{a_f(n)}{n^{\frac{k-1}{2}}}
=\sum_{i=1}^d u_i\lambda_{\varphi_i}(n),
\quad
\lambda_g(n)\coloneq\frac{a_g(n)}{n^{\frac{\ell-1}{2}}}
=\sum_{j=1}^{d'} v_j\lambda_{\psi_j}(n).
\]
Our hypothesis on $\lcm(M,N)$ implies that
$\varphi_1,\ldots,\varphi_d,\psi_1,\ldots,\psi_{d'}$ are all
twist minimal without CM, and no two are twist equivalent.
By \cite[\S3, Theorem~M]{Ramakrishnan}, for any pair of indices $(i,j)$,
there is a cuspidal automorphic representation $\pi_{ij}$ of $\GL(4)$
with $p$th Dirichlet coefficient $\lambda_{\pi_{ij}}(p)$ satisfying
$\lambda_{\pi_{ij}}(p)=\lambda_{\varphi_i}(p)\lambda_{\psi_j}(p)$
for primes $p\nmid MN$.
Furthermore, following the proof of
\cite[Lemma~4.5.8]{Ramakrishnan}, one can see that
$\pi_{ij}\cong\pi_{i'j'}$ if and only if $(i,j)=(i',j')$.

\begin{lemma}\label{lem:dense}
For any $p\in S$ and any index $i_1$, there are distinct
non-zero integers $n_1,\ldots,n_d$ such that
\begin{enumerate}
\item $n_{i_1}\nmid n_i$ for $i\ne i_1$;
\item for any $t\in\R$ and any $\varepsilon>0$, there exists
$n\in\N$ such that
\[
\min\{|n\theta_i(p)-n_it-m|:m\in\Z\}<\varepsilon
\quad\text{for }i=1,\ldots,d.
\]
\end{enumerate}
\end{lemma}
\begin{proof}
Fix $p\in S$ and an index $i_1$, and consider the vector space
\[
V=\Q+\Q\theta_1(p)+\cdots+\Q\theta_d(p).
\]
Write $\dim V=m+1\ge3$.
By permuting the indices if necessary, we may assume that
$\{1,\theta_1(p),\ldots,\theta_m(p)\}$ is a basis for $V$, and we may choose
the permutation to map $i_1$ to $1$.
For $i=1,\ldots,m$ we write
\[
\theta_i(p)=\sum_{j=1}^m\alpha_{ij}\theta_j(p) + \beta_i
\]
for some $\alpha_{ij},\beta_i\in\Q$.
By our construction, no two of the vectors
$(\alpha_{ij})_{j=1,\ldots,d}$ are colinear.

\begin{proposition}\label{prop:dense}
Let $p_1,\ldots,p_d\in S$ be distinct primes. Then
\[
\bigl\{
\bigl[a_1\bigl(p_1^{k_1}\cdots p_d^{k_d}\bigr):\cdots:a_d\bigl(p_1^{k_1}\cdots p_d^{k_d}\bigr)\bigl]\in\RP^{d-1}:
k_1,\ldots,k_d\ge0
\bigl\}
\]
is dense in $\RP^{d-1}$.
\end{proposition}
\begin{proof}
Let $\theta_{ij}=\theta_j(p_i)$. For each $i$ we
apply Lemma~\ref{lem:dense} with $p=p_i$ and $i_1=i$,
and we denote the resulting integers $n_{i1},\ldots,n_{id}$.
Let $t=(t_1,\ldots,t_d)$, and
consider the vectors $F(t)=(F_j(t))_{j=1,\ldots,d}$, where
\[
F_j(t)=\prod_{i=1}^d\frac{\sin(2\pi n_{ij}t_i)}{\sin(2\pi\theta_{ij})}
=c_j\prod_{i=1}^d\sin(2\pi n_{ij}t_i),
\]
for some $c_j\in\R_{>0}$.
Then
\[
\frac{\partial F_j}{\partial t_i}
=2\pi c_jn_{ij}\cos(2\pi n_{ij}t_i)
\prod_{\substack{1\le i'\le d\\i'\ne i}}
\sin(2\pi n_{i'j}t_{i'}).
\]
By construction we have $n_{ii}\nmid n_{ij}$ for $j\ne i$,
and it follows that $F$ vanishes at the point
$t=(\frac1{2n_{ii}})_{i=1,\ldots,d}$, and
the Jacobian matrix there is diagonal with non-zero determinant.
By the inverse function theorem, the image of $F$ contains
an open neighborhood of the origin.

To complete the proof of Theorem~\ref{thm:main}, write
$f=\sum_{i=1}^d u_i\varphi_i$ and
$g=\sum_{i=1}^d v_i\varphi_i$
for some non-zero vectors $u=(u_1,\ldots,u_d),v=(v_1,\ldots,v_d)\in\R^d$.
If $f$ and $g$ are not proportional
then we may choose a vector $w$ such that
$w\cdot u=0$ and $w\cdot v\ne0$.
Then for any $\delta\in\R$, we have
\[
(w+\delta u)\cdot u=\delta(u\cdot u)
\quad\text{and}\quad
(w+\delta u)\cdot v=(w\cdot v)+\delta(u\cdot v).
\]
Choosing $\delta$ such that $\delta(w\cdot v)<0$ and
$\delta|u\cdot v|<|w\cdot v|$, we see that
$(w+\delta u)\cdot u$ and $(w+\delta u)\cdot v$ have different signs.
Applying Proposition~\ref{prop:dense} we can find
an arbitrarily large $n$ such that $[a_1(n):\cdots:a_d(n)]$ approximates
the image of $w+\delta u$ in $\RP^{d-1}$
arbitrarily closely, and it follows that
$a_f(n)$ and $a_g(n)$ have different signs.
This is a contradiction, so $f$ and $g$ must be proportional,
meaning that $c=f/g$ is constant.
Since $f$ and $g$ are non-zero and $a_f(n)a_g(n)\ge0$
for sufficiently large $n$, it follows that $c>0$.

\begin{remarks}\
\begin{enumerate}
\item The restriction on $\lcm(M,N)$
is necessary when $(M,k)=(N,\ell)$ and $Mk$ is sufficiently large,
since otherwise one can
choose a fundamental discriminant $\Delta\ne1$ with $\Delta^2\mid M$
and a twist-minimal, non-CM newform $f$ of conductor $M$.
Setting $g=2f+f\times\left(\frac{\Delta}{\cdot}\right)$, we have
$a_f(n)a_g(n)=a_f(n)^2(2+\left(\frac{\Delta}{n}\right))\ge0$.
\item The restriction to the new subspaces of
$S_k(\Gamma_0(M))$ and $S_\ell(\Gamma_0(N))$ is also
necessary in some cases, even under the assumption that $M=N$.
For example, let $f$ be the newform associated to an
elliptic curve $E$ of squarefree conductor $N_E$, and set $g(z)=f(z)+f(pz)$,
where $p>3$ is a supersingular prime for $E$. Then
$f,g\in S_2(\Gamma_0(pN_E))$, and $a_f(n)a_g(n)=a_f(n)^2\ge0$.
\item It will be clear from the proof that condition (1) of the theorem
can be substantially weakened. For instance, it suffices to have
$\liminf_{\substack{n\to\infty\\\gcd(n,q)=1}}\frac{a_f(n)a_g(n)}{n^{\frac{k+\ell}{2}-1}}\ge0$
for some fixed modulus $q$.
\item
However, infinitely many $n$ are required when $(M,k)=(N,\ell)$
and $\dim S_k^{\rm new}(\Gamma_0(M))>1$, i.e.\ the first sign
change of $a_f(n)a_g(n)$ cannot be effectively bounded.
To see this, let $\varphi,\psi\in S_k^{\rm new}(\Gamma_0(M))$ be distinct
normalized newforms, and set $f=\varphi+\pi\psi$,
$g_\varepsilon=f+\varepsilon\varphi$ for some $\varepsilon\ne0$.
Then $\min\{n\in\mathbb{N}:a_f(n)a_{g_\varepsilon}(n)<0\}\to\infty$ as $\varepsilon\to0$.
\end{enumerate}
\end{remarks}

Our proof makes use of the joint Sato--Tate equidistribution
of any two non-CM, twist-inequivalent newforms, proven by
Wong \cite{Wong}, based on the spectacular results
of Barnet-Lamb et al.\ \cite{BGG}. (See also Newton and Thorne
\cite{Newton-Thorne} for the recent strengthening
to full automorphy of symmetric powers for Hilbert modular forms
of regular weight, and Thorner \cite{Thorner} for results with an
effective rate of convergence.)
Precisely, let $\varphi\in S_k^{\rm new}(\Gamma_0(M))$,
$\psi\in S_\ell^{\rm new}(\Gamma_0(N))$
be distinct normalized Hecke eigenforms of conductors $M,N$ as in the theorem,
and write their Fourier coefficients at primes $p$ in the form
\[
a_\varphi(p)=2p^{\frac{k-1}{2}}\cos(2\pi\theta_\varphi(p)),
\quad
a_\psi(p)=2p^{\frac{\ell-1}{2}}\cos(2\pi\theta_\psi(p)),
\quad\text{where }
(\theta_\varphi(p),\theta_\psi(p))\in\bigl[0,\tfrac12\bigr]^2.
\]
Then by \cite[Theorem~1.1]{Wong}, for any box
$B=[\alpha,\beta]\times[\gamma,\delta]\subseteq[0,\frac12]^2$, we have
\[
\lim_{x\to\infty}
\frac{\#\{p\le x:(\theta_\varphi(p),\theta_\psi(p))\in B\}}{\pi(x)}
=\int_B\bigl(4\sin(2\pi u)\sin(2\pi v)\bigr)^2\,du\,dv.
\]
As we will see in Lemma~\ref{lem:linindep}, it follows that
$1$, $\theta_\varphi(p)$, and $\theta_\psi(p)$ are linearly independent
over $\mathbb{Q}$ for all $p$ in a set of density $1$.

We note that mutual independence of the angles $\theta_\varphi$ for
more than two newforms is not known, though it would follow from
the functoriality of arbitrary products of symmetric powers.
This presents an obstacle to proving Theorem~\ref{thm:main} using
only information on $a_f(p)a_g(p)$ at primes $p$.
(We will however give such a proof when $f$ and $g$ have different
weights or levels.) Instead we take an approach that is possibly of independent
interest, showing that for almost any tuple of
distinct primes $p_1,\ldots,p_d$, where $d=\dim S_k^{\rm new}(\Gamma_0(M))$,
the non-zero $f\in S_k^{\rm new}(\Gamma_0(M))$ with real coefficients
are determined modulo scalars by the signs of $a_f(n)$
for $n\in\{p_1^{k_1}\cdots p_d^{k_d}:k_1,\ldots,k_d\ge0\}$;
see Proposition~\ref{prop:dense} for
the precise statement, and \cite{Amri,GMP} for some related results.

\subsection*{Acknowledgements}
I thank Jonathan Bober and Oleksiy Klurman for
many helpful conversations.

Our proof uses the joint Sato--Tate equidistribution of any two non-CM, twist-inequivalent newforms (Wong), based on Barnet-Lamb et al.  Writing for primes $p$
\[
a_\varphi(p)=2p^{\frac{k-1}{2}}\cos(2\pi\theta_\varphi(p)),\qquad a_\psi(p)=2p^{\frac{\ell-1}{2}}\cos(2\pi\theta_\psi(p)),\qquad (\theta_\varphi(p),\theta_\psi(p))\in[0,\tfrac12]^2,
\]
Wong’s theorem gives equidistribution of $(\theta_\varphi(p),\theta_\psi(p))$ in $[0,\tfrac12]^2$ with respect to density $(4\sin(2\pi u)\sin(2\pi v))^2\,du\,dv$.  As noted, “it follows that $1$, $\theta_\varphi(p)$, and $\theta_\psi(p)$ are linearly independent over $\mathbb{Q}$ for all $p$ in a set of density $1$” (via Lemma~\ref{lem:linindep}).

Because “mutual independence of the angles $\theta_\varphi$ for more than two newforms is not known,” this “presents an obstacle to proving Theorem~\ref{thm:main} using only information on $a_f(p)a_g(p)$ at primes $p$” (except that “we will however give such a proof when $f$ and $g$ have different weights or levels”).  Instead, the approach is to show that “for almost any tuple of distinct primes $p_1,\ldots,p_d$, where $d=\dim S_k^{\rm new}(\Gamma_0(M))$, the non-zero $f\in S_k^{\rm new}(\Gamma_0(M))$ with real coefficients are determined modulo scalars by the signs of $a_f(n)$ for $n\in\{p_1^{k_1}\cdots p_d^{k_d}:k_1,\ldots,k_d\ge0\}$,” referring to Proposition~\ref{prop:dense} for the precise statement.

Our proof uses the joint Sato--Tate equidistribution of any two non-CM, twist-inequivalent newforms (Wong), based on Barnet-Lamb et al.  Writing for primes $p$
\[
a_\varphi(p)=2p^{\frac{k-1}{2}}\cos(2\pi\theta_\varphi(p)),\qquad a_\psi(p)=2p^{\frac{\ell-1}{2}}\cos(2\pi\theta_\psi(p)),\qquad (\theta_\varphi(p),\theta_\psi(p))\in[0,\tfrac12]^2,
\]
Wong’s theorem gives equidistribution of $(\theta_\varphi(p),\theta_\psi(p))$ in $[0,\tfrac12]^2$ with respect to density $(4\sin(2\pi u)\sin(2\pi v))^2\,du\,dv$.  As noted, “it follows that $1$, $\theta_\varphi(p)$, and $\theta_\psi(p)$ are linearly independent over $\mathbb{Q}$ for all $p$ in a set of density $1$”, i.e.
\begin{lemma}\label{lem:linindep}
There is a set $S\subseteq\{p\text{ prime}:p\nmid M\}$ of natural density $1$
such that, for any $p\in S$ and any pair $i\ne j$, the numbers
$1$, $\theta_i(p)$, and $\theta_j(p)$ are linearly independent
over $\Q$.
\end{lemma}

Because “mutual independence of the angles $\theta_\varphi$ for more than two newforms is not known,” this “presents an obstacle to proving the main theorem using only information on $a_f(p)a_g(p)$ at primes $p$” (except that “we will however give such a proof when $f$ and $g$ have different weights or levels”).  Instead, the approach is to show that “for almost any tuple of distinct primes $p_1,\ldots,p_d$, where $d=\dim S_k^{\rm new}(\Gamma_0(M))$, the non-zero $f\in S_k^{\rm new}(\Gamma_0(M))$ with real coefficients are determined modulo scalars by the signs of $a_f(n)$ for $n\in\{p_1^{k_1}\cdots p_d^{k_d}:k_1,\ldots,k_d\ge0\}$,” and for the precise statement we use the following proposition.
\begin{proposition}\label{prop:dense}
Let $p_1,\ldots,p_d\in S$ be distinct primes. Then
\[
\bigl\{
\bigl[a_1\bigl(p_1^{k_1}\cdots p_d^{k_d}\bigr):\cdots:a_d\bigl(p_1^{k_1}\cdots p_d^{k_d}\bigr)\bigl]\in\RP^{d-1}:
k_1,\ldots,k_d\ge0
\bigl\}
\]
is dense in $\RP^{d-1}$.
\end{proposition}

\label{thm:main}
Let $M,N\in\mathbb{N}$, with $\lcm(M,N)$ not
divisible by $2^4$ or the square of an odd prime.
Let $f\in S_k^{\rm new}(\Gamma_0(M))$ and $g\in S_\ell^{\rm new}(\Gamma_0(N))$
be non-zero cusp forms with real Fourier coefficients
$a_f(n),a_g(n)$. Then the following are equivalent:
\begin{enumerate}
\item $a_f(n)a_g(n)\ge0$ for all sufficiently large $n\in\mathbb{N}$;
\item $(M,k)=(N,\ell)$ and $f/g$ is a positive constant.
\end{enumerate}
Thus if $f$ and $g$ are not proportional then
$\{n\in\mathbb{N}:\epsilon a_f(n)a_g(n)>0\}$
is infinite for each $\epsilon\in\{\pm1\}$.,

Biconditional or Equivalence
Implication

Question: Let $M,N\in\mathbb N$ be such that $\operatorname{lcm}(M,N)$ is divisible neither by $2^4$ nor by the square of any odd prime. Let $f\in S_k^{\rm new}(\Gamma_0(M))$ and $g\in S_\ell^{\rm new}(\Gamma_0(N))$ be nonzero new cusp forms of weights $k,\ell$ and levels $M,N$, with real Fourier coefficients $a_f(n)$ and $a_g(n)$. Which statement about the signs of the products $a_f(n)a_g(n)$ is valid under these hypotheses?

The condition $a_f(n)a_g(n)\ge 0$ for all sufficiently large $n\in\mathbb N$ is equivalent to saying that $(M,k)=(N,\ell)$ and $f/g$ is a positive constant, i.e. $f=cg$ for some $c>0$. Consequently, if $f$ and $g$ are not proportional, then for each $\epsilon\in\{\pm1\}$ the set $\{n\in\mathbb N: \epsilon\, a_f(n)a_g(n)>0\}$ is infinite.

Choices:
B. The condition $a_f(n)a_g(n)\ge 0$ for all sufficiently large $n\in\mathbb N$ is equivalent to saying that $(M,k)=(N,\ell)$ and $f/g$ is a constant, possibly negative. Consequently, if $f$ and $g$ are not proportional, then there is at least one $\epsilon\in\{\pm1\}$ for which the set $\{n\in\mathbb N: \epsilon\, a_f(n)a_g(n)>0\}$ is infinite.
C. If $(M,k)=(N,\ell)$ and $f/g$ is a positive constant, then $a_f(n)a_g(n)\ge 0$ for all sufficiently large $n\in\mathbb N$. In particular, this condition is sufficient for eventual nonnegativity of the products $a_f(n)a_g(n)$.
D. The condition $a_f(p)a_g(p)\ge 0$ for all sufficiently large primes $p$ is equivalent to saying that $(M,k)=(N,\ell)$ and $f/g$ is a positive constant, i.e. $f=cg$ for some $c>0$. Consequently, if $f$ and $g$ are not proportional, then for each $\epsilon\in\{\pm1\}$ the set $\{p\text{ prime}: \epsilon\, a_f(p)a_g(p)>0\}$ is infinite.
E. If $a_f(n)a_g(n)\ge 0$ for infinitely many $n\in\mathbb N$, then necessarily $(M,k)=(N,\ell)$ and $f/g$ is a positive constant, i.e. $f=cg$ for some $c>0$. Consequently, if $f$ and $g$ are not proportional, then $a_f(n)a_g(n)<0$ for all but finitely many $n$.


\label{sec1}

Let $G$ be a compact Lie group, and $P$ a principal $G$-bundle over the $4$-dimensional sphere $S^4$. The $G$-gauge group over the base space $S^4$ is the topological group of $G$-bundle automorphisms of $P$. Suppose that $G$ is simply-connected and simple. Its classifying space $BG$ is $3$-connected and $\pi_4(BG)\cong \mathbb{Z}$. Let us denote the homotopy class of a map $f$ by the same symbol $f$. Then, the principal $G$-bundle $P$ is classified by a map $k\colon S^4 \to BG$ in $\pi_4(BG)\cong \mathbb{Z}$ and there are infinitely many isomorphism classes of principal $G$-bundles over $S^4$. In \cite{kono-1991}, Kono classified the homotopy types of $SU(2)$-gauge groups over $S^4$ and showed that there are six homotopy types of these groups.

Since then, many classification results on the homotopy types of gauge groups of low-rank simple Lie groups have been obtained. Among compact Lie groups, $SU(2)$ is the rank $1$ simply-connected simple Lie group. There are three simply-connected simple compact Lie groups of rank $2$: $SU(3)$, $Sp(2)$, and $G_2$. For $G=SU(3)$, Hamanaka and Kono \cite{hamanaka-kono-2006} proved that $\mathcal{G}_k\simeq \mathcal{G}_{k'}$ if and only if $(k, 24)=(k', 24)$, where $(m,n)$ is the greatest common divisor of $m$ and $n$. For $G=Sp(2)$, Theriault \cite{theriault-2010} showed $\mathcal{G}_k\simeq_{(p)} \mathcal{G}_{k'}$ at any prime $p$ if and only if $(k, 40)=(k', 40)$, where $\simeq_{(p)}$ means $p$-local homotopy equivalence. Then, for $G=G_2$, Kishimoto, Theriault and Tsutaya \cite{ktt-2017} showed that $\mathcal{G}_k\simeq_{(p)} \mathcal{G}_{k'}$ at any prime $p$ if $(k, 168)=(k', 168)$ and if $\mathcal{G}_k \simeq_{(p)} \mathcal{G}_{k'}$ at any prime $p$, then $(k,84)=(k', 84)$. The discrepancy between $(k, 168)=(k',168)$ and $(k, 84)=(k', 84)$ has remained as an open problem.

In this paper, we complete the classification for rank $2$ simple Lie groups up to $p$-local homotopy equivalence at any prime $p$ by proving the following.

In this paper, we complete the classification for rank $2$ simple Lie groups up to $p$-local homotopy equivalence at any prime $p$ by proving the following.

Theorem~\ref{theorem:1.1} (2) is equivalent to the following local form. \begin{proposition}\label{proposition:1.2}
Let $\mathcal{G}_k$ be the gauge group of a principal $G_2$-bundle over $S^4$ whose classifying map is $k\in \mathbb{Z}\cong \pi_{4}(BG_2)$.
Then, we have
\begin{itemize}
\item[{\rm (1)}] $\mathcal{G}_k\simeq_{(2)} \mathcal{G}_{k'}$ if $(k, 4)=(k', 4)$,
\item[{\rm (2)}]
$\mathcal{G}_k\simeq_{(3)} \mathcal{G}_{k'}$ if $(k, 3)=(k', 3)$,
\item[{\rm (3)}] $\mathcal{G}_k\simeq_{(7)} \mathcal{G}_{k'}$ if $(k, 7)=(k', 7)$
and
\item[{\rm (4)}] $\mathcal{G}_k\simeq_{(p)} \mathcal{G}_{k'} \simeq _{(p)} G_2 \times \Omega_0^4 G_2$ for $p\not=2,3,7$.
\end{itemize}
\end{proposition}

First, we recall the basic method for determining the homotopy types of $G$-gauge groups over $S^4$, which relates their homotopy types to Samelson products. We begin with the following homotopy fiber sequence of mapping spaces.
\[
\Omega {\mathrm{Map}} (S^4, BG)_k \to G \to {\mathrm{Map}}_* (S^4, BG)_k \to {\mathrm{Map}} (S^4, BG)_k \stackrel{\mathrm{ev}}{\longrightarrow} BG,
\]
where $\mathrm{Map} (S^4, BG)_k$ is the connected component of the space of continuous maps containing the map $k\colon S^4\to BG$, $\mathrm{Map}_* (S^4, BG)_k$ is its subspace consisting of base point preserving maps, and $\mathrm{ev}$ is the evaluation map.
There is a homotopy equivalence $\Omega_0^3 G\simeq \mathrm{Map}_{*}(S^4, BG)_0 \simeq \mathrm{Map}_{*}(S^4, BG)_k$ and Gottlieb \cite{gottlieb-1972} showed that the classifying space of the gauge group $B\mathcal{G}_k$ is homotopy equivalent to the mapping space $ \mathop{\mathrm{Map}} (S^4, BG)_k$. Therefore, we have the following fiber sequence.
\[
\mathcal{G}_k \to G \stackrel{\partial_k}{\longrightarrow} \Omega_0^3 G \to B\mathcal{G}_k \to BG.
\]
Thus, the $G$-gauge group $\mathcal{G}_k$ is homotopy equivalent to the homotopy fiber of the map $\partial_k$. Let $i_3\colon S^3 \to G$ be the inclusion map of the bottom cell. Then, Lang \cite{lang-1973} proved that the map $\partial_k$ is the triple adjoint of the Samelson product $\langle k\cdot i_3, 1\rangle$ where $1$ is the identity map of $G$. By the linearity of the Samelson product, we have $\langle k\cdot i_3, 1\rangle\simeq k \cdot \langle i_3, 1\rangle$. Furthermore, in \cite{theriault-2010}, Theriault showed that $\mathcal{G}_k$ is $p$-locally homotopy equivalent to $\mathcal{G}_{k'}$ at all prime $p$ if $(k, m)=(k',m)$ where $m$ is the order of the Samelson product $\langle i_3, 1\rangle$.

\begin{proposition}\label{proposition:2.1}
The following holds.
\begin{itemize}
\item[{\rm (1)}] $\pi_{8}(S^{6})=\mathbb{Z}/2\{ \eta_{6}^{2}\}$, $\pi_{11}(S^{6})=\mathbb{Z}\{ \Delta(\iota_{13})\}$, $\pi_{14}(S^{6})_{{(2)}}=\mathbb{Z}/8\{\bar{\nu}_6 \} \oplus \mathbb{Z}/2\{ \varepsilon_6 \}$,
\item[{\rm (2)}] $\pi_{14}(S^{8})=\mathbb{Z}/2\{ \nu_{8}^{2} \}$, $\pi_{14}(S^{9})=\{0\}$, $\pi_{14}(S^{10})=\{0\}$.
\item[{\rm (3)}] $\eta_{n}\nu_{n+1}\simeq 0$ for $n\geq 5$.
\end{itemize}
\end{proposition}

\begin{proposition}[Mimura] \label{proposition:2.2}
Let $i\colon SU(3)\to G_2$ be the inclusion map and
$p\colon G_2\to S^6=G_2/SU(3)$ the projection map.
By $\langle \alpha \rangle$, we denote an element in $\pi_i(G_2)$ such that $p_*(\langle \alpha\rangle)\simeq \alpha$.
The following holds.
\begin{itemize}
\item[{\rm (1)}] $\pi_{11}(G_{2})=\mathbb{Z}\{ \langle 2 \Delta(\iota_{13})\rangle\} \oplus \mathbb{Z}/2\{ i_{*}([\nu_{5}^{2}])\}$.
\item[{\rm (2)}] $\pi_{14}(G_{2})_{(2)}=\mathbb{Z}/8\{ \langle \bar{\nu}_{6}+\varepsilon_{6}\rangle\} \oplus \mathbb{Z}/2\{ i_{*}([\nu_{5}^{2}])\circ \nu_{11}\}$
\end{itemize}
\end{proposition}

\begin{theorem}[{\oshima} \cite{oshima-2005}*{Theorem 2.1}] \label{theorem:3.1}
There is a map $\gamma \colon S^{11}\to G_2$ such that
\begin{itemize}
\item[{\rm (1)}]
$p\circ \gamma\simeq 2\Delta(\iota_{13})$,
\item[{\rm (2)}] $\langle i_3, \gamma\rangle\in \pi_{14}(G_2)$ has order $21=3\cdot 7$.
\end{itemize}
\end{theorem}

\begin{proposition}\label{proposition:3.2}
Let $i_{8}\colon S^8\to P^9(2)$ be the inclusion map of the bottom cell. Then, the following hold.
\begin{itemize}
\item[{\rm (1)}] ${i_8}_{*}\colon \pi_{14}(S^8)\to \pi_{14}(P^9(2))$ is an isomorphism.
\item[{\rm (2)}] For maps $f\colon S^{14}\to S^8$, and $g\colon S^{8}\to S^{6}$,
$g \circ f \simeq 0$ in $\pi_{14}(S^6)$.
\item[{\rm (3)}] If a map $f\colon S^{14}\to S^{6}$ factors through $P^9(2)$, then $f\simeq 0$ in $\pi_{14}(S^{6})$.
\end{itemize}
\end{proposition}

\begin{proposition}\label{proposition:3.3}
The following holds.
\begin{itemize}
\item[{\rm (1)}] $\pi_{11}(C)\simeq \mathbb{Z}\{ \bar{i}_{6}\circ \Delta(\iota_{13})\} \oplus \mathbb{Z}\{ \bar{s}\}$,
\item[{\rm (2)}] $\mathop{\mathrm{Ker}} \bar{p}_{6\; *} \colon \pi_{11}(C)\to \pi_{11}(S^6)$ is generated by $w\simeq a \bar{i}_{6}\circ \Delta(\iota_{13})+b \bar{s}$ where $a\in \mathbb{Z}$ and $b=1$ or $a$ is odd and $b=2$.
\item[{\rm (3)}] $j\circ q\circ \gamma\simeq cw$ where $c\in \{ \pm 1, \pm 2\}$.
\item[{\rm (4)}] $p_{11}\circ q\circ \gamma \simeq bc (\iota_{11})$ where $bc \in \{ \pm 1, \pm 2, \pm 4\}$.
\end{itemize}
\end{proposition}

\begin{theorem}\label{theorem:1.1}
Let $\mathcal{G}_k$ be the gauge group of a principal $G_2$-bundle over $S^4$ whose classifying map is $k\in \mathbb{Z}\cong \pi_{4}(BG_2)$. The following holds.
\begin{itemize}
\item[{\rm (1)}]
If $\mathcal{G}_k\simeq \mathcal{G}_{k'}$, then $(k, 84)=(k', 84)$ and
\item[{\rm (2)}]  $\mathcal{G}_k\simeq_{(p)} \mathcal{G}_{k'}$ at any prime $p$ if $(k, 84)=(k', 84)$.
\end{itemize}
\end{theorem}

Theorem~\ref{theorem:1.1} (2) is equivalent to the following local form. \begin{proposition}\label{proposition:1.2}
Let $\mathcal{G}_k$ be the gauge group of a principal $G_2$-bundle over $S^4$ whose classifying map is $k\in \mathbb{Z}\cong \pi_{4}(BG_2)$.
Then, we have
\begin{itemize}
\item[{\rm (1)}] $\mathcal{G}_k\simeq_{(2)} \mathcal{G}_{k'}$  if $(k, 4)=(k', 4)$,
\item[{\rm (2)}]
$\mathcal{G}_k\simeq_{(3)} \mathcal{G}_{k'}$  if $(k, 3)=(k', 3)$,
\item[{\rm (3)}] $\mathcal{G}_k\simeq_{(7)} \mathcal{G}_{k'}$  if $(k, 7)=(k', 7)$
and
\item[{\rm (4)}] $\mathcal{G}_k\simeq_{(p)} \mathcal{G}_{k'} \simeq _{(p)} G_2 \times \Omega_0^4 G_2$ for $p\not=2,3,7$.
\end{itemize}
\end{proposition}

Theorem~\ref{theorem:1.1} (1) was established in \cite{ktt-2017}*{Theorem 1.2 (a)} and Proposition~\ref{proposition:1.2} (2), (3), (4) were established in \cite{ktt-2017}*{Propositions 4.6 and 3.2}. In this paper, we prove Proposition~\ref{proposition:1.2} (1) and complete the proof of Theorem~\ref{theorem:1.1} assuming the results of \cite{ktt-2017}.

First, we recall the basic method for determining the homotopy types of $G$-gauge groups over $S^4$, which relates their homotopy types to Samelson products. We begin with the following homotopy fiber sequence of mapping spaces.
\[
\Omega {\mathrm{Map}} (S^4, BG)_k  \to G \to  {\mathrm{Map}}_* (S^4, BG)_k \to {\mathrm{Map}} (S^4, BG)_k \stackrel{\mathrm{ev}}{\longrightarrow} BG,
\]
where $\mathrm{Map} (S^4, BG)_k$ is the connected component of the space of continuous maps containing the map $k\colon S^4\to BG$, $\mathrm{Map}_* (S^4, BG)_k$ is its subspace consisting of base point preserving maps,  and $\mathrm{ev}$ is the evaluation map.
There is a homotopy equivalence  $\Omega_0^3 G\simeq \mathrm{Map}_{*}(S^4, BG)_0 \simeq \mathrm{Map}_{*}(S^4, BG)_k$ and Gottlieb \cite{gottlieb-1972} showed that the classifying space of the gauge group $B\mathcal{G}_k$ is homotopy equivalent to the mapping space $ \mathop{\mathrm{Map}} (S^4, BG)_k$. Therefore, we have the following fiber sequence.
\[
\mathcal{G}_k  \to G \stackrel{\partial_k}{\longrightarrow} \Omega_0^3 G \to B\mathcal{G}_k \to BG.
\]
Thus, the $G$-gauge group $\mathcal{G}_k$ is homotopy equivalent to the homotopy fiber of the map $\partial_k$. Let $i_3\colon S^3 \to G$ be the inclusion map of the bottom cell. Then, Lang \cite{lang-1973} proved that the map $\partial_k$ is the triple adjoint of the Samelson product $\langle k\cdot i_3, 1\rangle$ where $1$ is the identity map of $G$. By the linearity of the Samelson product, we have $\langle k\cdot i_3, 1\rangle\simeq k \cdot \langle i_3, 1\rangle$. Furthermore, in \cite{theriault-2010}, Theriault showed that $\mathcal{G}_k$ is $p$-locally homotopy equivalent to $\mathcal{G}_{k'}$ at all prime $p$ if $(k, m)=(k',m)$ where $m$ is the order of the Samelson product $\langle i_3, 1\rangle$.

For an abelian group $A$, we denote by $A_{(p)}$ its localization at the prime $p$. If $A$ is finite, then $A_{(p)}$ is the $p$-primary subgroup of $A$. By \cite{ktt-2017}*{Corollary 6.6}, the order of $\langle i_3, 1\rangle$ in $[\Sigma^3 G_2, G_2]_{(2)}$ is at least $4$. Therefore, to show Proposition~\ref{proposition:1.2} (1), it suffices to prove the following proposition.

\begin{proposition}\label{proposition:1.3}  The order of the Samelson product $\langle i_3, 1\rangle$ in $[\Sigma^3 G_2, G_2]_{(2)}$ is at most $4$.
\end{proposition}

As for the order of the Samelson product $\langle i_3, 1\rangle$ in $[\Sigma^3 G_2, G_2]$, together with \cite{ktt-2017}*{Lemma 3.1, Proposition 4.5 and Corollary 6.6}, by Proposition~\ref{proposition:1.3}, we have the following theorem.

\begin{theorem}\label{theorem:1.4}
The order of the Samelson product $\langle i_3, 1\rangle$ in $[\Sigma^3 G_2, G_2]$ is $84$.
\end{theorem}

Next, we recall some properties of a space $V:=G_2/SU(2)$, which played an important role in \cite{ktt-2017} and is crucial in this paper. Let $q\colon G_2\to V$ be the obvious projection map. By \cite{ktt-2017}*{Lemmas 7.1 and 7.2}, the Samelson product $3 \cdot \langle i_3, 1\rangle\colon \Sigma^3 G_2\to G_2$ factors through $\Sigma^3 q \colon \Sigma^3 G_2 \to \Sigma^3 V$.
Let us write this factorization as follows.
 \[
3\cdot \langle i_3, 1\rangle\simeq \psi\circ \Sigma^3 q.
\]
Let $P^{n+1}(2)$ be the mapping cone of the degree $2$ map $\times 2 \colon S^n\to S^n$.
The $6$-skeleton of $V=S^5\cup e^6\cup e^{11}$ is $P^6(2)$ and, by \cite{ktt-2017}{Lemma 7.4}, there is a homotopy equivalence $\Sigma^3 V\simeq P^9(2) \vee S^{14}$.  Let \[
p_{11}\colon V \to S^{11}
\]
 be the pinch map to the top cell, collapsing the $6$-skeleton to the base point. We denote by
 \[
 i_6\colon P^6(2) \to V
 \]
  the inclusion map. Then, the above homotopy equivalence provides maps $s'\colon \Sigma^3 V \to P^9(2)$ and $s''\colon S^{14}\to \Sigma^3V$ such that the identity map of $\Sigma^3 V$ is homotopic to
  \[
  \Sigma^3 i_6 \circ s' + s'' \circ \Sigma^3 p_{11}.
  \] The following proposition is what we prove in the rest of this paper.

\begin{proposition} \label{proposition:1.5}
We have  $4(\psi\circ s'')\simeq 0$  in $\pi_{14}(G_2)_{(2)}$. \end{proposition}

\begin{proof}[Proof of Proposition~\ref{proposition:1.3}] We consider the following decomposition in $[\Sigma^3 G_2, G_2]$. \[ 3 \cdot \langle i_3, 1\rangle\simeq \psi \circ \Sigma^3 i_6\circ s' \circ \Sigma^3 q +\psi \circ s'' \circ \Sigma^3 (p_{11}\circ q). \]
By \cite{ktt-2017}*{Lemma 5.2 (a) and (b)}, the order of the identity map of $P^9(2)$ is $4$. Hence, we have
$4 (\Sigma^3 i_6\circ s' )\simeq 0$. Therefore, by Proposition~\ref{proposition:1.5}, the order of $ \langle i_3, 1\rangle$ in $[\Sigma^3 G_2, G_2]_{(2)}$ is at most $4$.  \end{proof}

In what follows, we use the symbol $p$ to express the map $p\colon G_2\to S^6\simeq G_2/SU(3)$ only. When we need to indicate the generator of a cyclic group, we write $\mathbb{Z}/m\{ a\}$ for the cyclic group of order $m$ generated by $a$.

This paper is organized as follows. In Section \ref{sec2}, we collect some facts on homotopy groups of $G_2$ and $S^6$ and reduce the problem on the $2$-primary homotopy group $\pi_{14}(G_2)_{(2)}$ to that of $\pi_{14}(S^6)_{(2)}$. In Section \ref{sec3}, we prove Proposition~\ref{proposition:1.5}.

To complete Theorem~\ref{theorem:1.1} (2) it suffices to prove the local statement Proposition~\ref{proposition:1.2}. The paper reduces this to proving Proposition~\ref{proposition:1.2} (1) (the $2$-local case), since (2),(3),(4) are cited from \cite{ktt-2017}.

The method recalled is: using Gottlieb’s identification $B\mathcal{G}_k\simeq \mathrm{Map}(S^4,BG)_k$ and the resulting fiber sequence
\[
\mathcal{G}_k\to G\xrightarrow{\partial_k} \Omega_0^3 G\to B\mathcal{G}_k\to BG,
\]
so $\mathcal{G}_k$ is the homotopy fiber of $\partial_k$. By Lang, $\partial_k$ is the triple adjoint of the Samelson product $\langle k\cdot i_3,1\rangle$, hence by linearity $\langle k\cdot i_3,1\rangle\simeq k\cdot\langle i_3,1\rangle$. Theriault’s result is then invoked: $\mathcal{G}_k\simeq_{(p)}\mathcal{G}_{k'}$ for all primes $p$ if $(k,m)=(k',m)$ where $m$ is the order of $\langle i_3,1\rangle$.

For $G=G_2$ at $p=2$, \cite{ktt-2017} gives that the order of $\langle i_3,1\rangle$ in $[\Sigma^3G_2,G_2]_{(2)}$ is at least $4$, so “to show Proposition~\ref{proposition:1.2} (1), it suffices” to prove Proposition~\ref{proposition:1.3}: the order is at most $4$.

The proof sketch for Proposition~\ref{proposition:1.3} is given: using that $3\cdot\langle i_3,1\rangle$ factors through $\Sigma^3q\colon \Sigma^3G_2\to \Sigma^3V$ (with $V=G_2/SU(2)$), write
\[
3\cdot\langle i_3,1\rangle\simeq \psi\circ \Sigma^3q.
\]
With the homotopy decomposition $\Sigma^3V\simeq P^9(2)\vee S^{14}$, choose maps $s'\colon \Sigma^3V\to P^9(2)$ and $s''\colon S^{14}\to \Sigma^3V$ so that
\[
\mathrm{id}_{\Sigma^3V}\simeq \Sigma^3i_6\circ s'\; +\; s''\circ \Sigma^3p_{11}.
\]
Then
\[
3\cdot\langle i_3,1\rangle\simeq \psi\circ \Sigma^3i_6\circ s'\circ \Sigma^3q\; +\; \psi\circ s''\circ \Sigma^3(p_{11}\circ q).
\]
Using \cite{ktt-2017} (Lemma 5.2(a),(b)) that “the order of the identity map of $P^9(2)$ is $4$,” one gets $4(\Sigma^3i_6\circ s')\simeq 0$. The remaining summand is controlled by Proposition~\ref{proposition:1.5}, namely $4(\psi\circ s'')\simeq 0$ in $\pi_{14}(G_2)_{(2)}$. Combining these shows “the order of $\langle i_3,1\rangle$ in $[\Sigma^3G_2,G_2]_{(2)}$ is at most $4$,” hence Proposition~\ref{proposition:1.2}(1), and therefore Theorem~\ref{theorem:1.1}(2) via the Samelson-product/Thériault criterion.

(They additionally note that with Proposition~\ref{proposition:1.3} and results of \cite{ktt-2017}, the global order is computed as $84$ (Theorem~\ref{theorem:1.4}).)

To establish the main theorem (2) it suffices to prove the local statement

\begin{proposition}\label{proposition:1.2}
Let $\mathcal{G}_k$ be the gauge group of a principal $G_2$-bundle over $S^4$ whose classifying map is $k\in \mathbb{Z}\cong \pi_{4}(BG_2)$.
Then, we have
\begin{itemize}
\item[{\rm (1)}] $\mathcal{G}_k\simeq_{(2)} \mathcal{G}_{k'}$  if $(k, 4)=(k', 4)$,
\item[{\rm (2)}]
$\mathcal{G}_k\simeq_{(3)} \mathcal{G}_{k'}$  if $(k, 3)=(k', 3)$,
\item[{\rm (3)}] $\mathcal{G}_k\simeq_{(7)} \mathcal{G}_{k'}$  if $(k, 7)=(k', 7)$
and
\item[{\rm (4)}] $\mathcal{G}_k\simeq_{(p)} \mathcal{G}_{k'} \simeq _{(p)} G_2 \times \Omega_0^4 G_2$ for $p\not=2,3,7$.
\end{itemize}
\end{proposition}

The paper reduces this to proving the preceding proposition (1) (the $2$-local case), since (2),(3),(4) are cited from \cite{ktt-2017}.

The method recalled is: using Gottlieb’s identification $B\mathcal{G}_k\simeq \mathrm{Map}(S^4,BG)_k$ and the resulting fiber sequence
\[
\mathcal{G}_k\to G\xrightarrow{\partial_k} \Omega_0^3 G\to B\mathcal{G}_k\to BG,
\]
so $\mathcal{G}_k$ is the homotopy fiber of $\partial_k$. By Lang, $\partial_k$ is the triple adjoint of the Samelson product $\langle k\cdot i_3,1\rangle$, hence by linearity $\langle k\cdot i_3,1\rangle\simeq k\cdot\langle i_3,1\rangle$. Theriault’s result is then invoked: $\mathcal{G}_k\simeq_{(p)}\mathcal{G}_{k'}$ for all primes $p$ if $(k,m)=(k',m)$ where $m$ is the order of $\langle i_3,1\rangle$.

For $G=G_2$ at $p=2$, \cite{ktt-2017} gives that the order of $\langle i_3,1\rangle$ in $[\Sigma^3G_2,G_2]_{(2)}$ is at least $4$, so to show the preceding proposition (1), it suffices to prove the following.

\begin{proposition}\label{proposition:1.3}  The order of the Samelson product $\langle i_3, 1\rangle$ in $[\Sigma^3 G_2, G_2]_{(2)}$ is at most $4$.
\end{proposition}

The proof sketch for the preceding proposition is given: using that $3\cdot\langle i_3,1\rangle$ factors through $\Sigma^3q\colon \Sigma^3G_2\to \Sigma^3V$ (with $V=G_2/SU(2)$), write
\[
3\cdot\langle i_3,1\rangle\simeq \psi\circ \Sigma^3q.
\]
With the homotopy decomposition $\Sigma^3V\simeq P^9(2)\vee S^{14}$, choose maps $s'\colon \Sigma^3V\to P^9(2)$ and $s''\colon S^{14}\to \Sigma^3V$ so that
\[
\mathrm{id}_{\Sigma^3V}\simeq \Sigma^3i_6\circ s'\; +\; s''\circ \Sigma^3p_{11}.
\]
Then
\[
3\cdot\langle i_3,1\rangle\simeq \psi\circ \Sigma^3i_6\circ s'\circ \Sigma^3q\; +\; \psi\circ s''\circ \Sigma^3(p_{11}\circ q).
\]
Using \cite{ktt-2017} (Lemma 5.2(a),(b)) that “the order of the identity map of $P^9(2)$ is $4$,” one gets $4(\Sigma^3i_6\circ s')\simeq 0$. The remaining summand is controlled by

\begin{proposition} \label{proposition:1.5}
We have  $4(\psi\circ s'')\simeq 0$  in $\pi_{14}(G_2)_{(2)}$. \end{proposition}

Combining these shows that the order of $\langle i_3,1\rangle$ in $[\Sigma^3G_2,G_2]_{(2)}$ is at most $4$, hence the preceding proposition (1), and therefore the main theorem (2) via the Samelson-product/Thériault criterion.

(They additionally note that with the proposition above and results of \cite{ktt-2017}, the global order is computed as $84$ (Theorem~\ref{theorem:1.4}).)

Question: Let \(\mathcal{G}_k\) denote the gauge group of a principal \(G_2\)-bundle over \(S^4\) whose classifying map represents \(k\in \pi_4(BG_2)\cong \mathbb Z\). Write \((m,n)\) for the greatest common divisor of integers \(m\) and \(n\), write \(\mathcal{G}_k\simeq \mathcal{G}_{k'}\) for homotopy equivalence, and write \(\mathcal{G}_k\simeq_{(p)} \mathcal{G}_{k'}\) for \(p\)-local homotopy equivalence. Which statement holds for integers \(k\) and \(k'\)?

If \(\mathcal{G}_k\simeq \mathcal{G}_{k'}\), then \((k,84)=(k',84)\). Moreover, if \((k,84)=(k',84)\), then \(\mathcal{G}_k\simeq_{(p)} \mathcal{G}_{k'}\) for every prime \(p\).

Choices:
B. If \(\mathcal{G}_k\simeq \mathcal{G}_{k'}\), then \((k,168)=(k',168)\). Moreover, if \((k,168)=(k',168)\), then \(\mathcal{G}_k\simeq_{(p)} \mathcal{G}_{k'}\) for every prime \(p\).
C. If \((k,84)=(k',84)\), then \(\mathcal{G}_k\simeq_{(2)} \mathcal{G}_{k'}\), \(\mathcal{G}_k\simeq_{(3)} \mathcal{G}_{k'}\), and \(\mathcal{G}_k\simeq_{(7)} \mathcal{G}_{k'}\).
D. If \(\mathcal{G}_k\simeq \mathcal{G}_{k'}\), then \((k,84)=(k',84)\). Moreover, if \((k,84)=(k',84)\), then \(\mathcal{G}_k\simeq \mathcal{G}_{k'}\).
E. If \(\mathcal{G}_k\simeq \mathcal{G}_{k'}\), then \((k,84)=(k',84)\). Moreover, if \((k,4)=(k',4)\), then \(\mathcal{G}_k\simeq_{(p)} \mathcal{G}_{k'}\) for every prime \(p\).