\begin{equation}\label{mH1}
	\mH(R):= \left\{ h\in W^{1,2}(B_R) \,\bigg\vert\, \exists \zeta \in W^{1,2}(\mathbb{S}^{N-1}): \ h(x)=\frac{1}{|x|}\zeta\big( \frac{x}{|x|} \big) \ \text{for $x\in B_R$ a.e.} \right\}.
\end{equation}

\begin{theorem}\label{TH-00}
Let $u$ be a suitable weak solutions. Suppose that
the following two conditions hold:
\begin{equation}
\liminf_{r\to 0}M(r):=\liminf_{r\to 0}\int_{B_r}\!\! \left( \frac{|u|^2}{r^3} + \frac{|\nabla u|^2}{r} \right) <\infty, \quad \liminf_{r\to 0}\frac1{r^2}\int_{B_r}\!\!(|u|^2+2P)u\cdot\frac x{|x|}>0.
\end{equation}
Then  for any $\{r_k\}_{k=1}^\infty, r_k\downarrow 0$ there is a subsequence $r_{k_m}$ such that
the scaled solutions $u_{r_{k_m}}(x)=r_{k_m}u(r_{k_m}x)$, converge to
a homogenous vector field of degree negative one, and hence
$x=0$ is a regular point.
\end{theorem}

\begin{proposition}\label{prop:cubicEst}
	For fixed constants $\delta_{1}, \delta_{2}>0$, there exists $\ep>0$ such that if $(u,p)$ is a Leray-Hopf solution to the Navier-Stokes equations satisfying
	\begin{equation}\label{h-A}
		\frac{1}{R^3} \int_{B_R} \big|u- \mP_R[u] \big|^2 + \frac{1}{R} \int_{ B_R} \big|\nabla u - \nabla \mP_R[u] \big|^2 \le \ep M[u](R)
	\end{equation}
	for some $R>0$, then
	\begin{equation}
		\bigg|\frac{1}{R^2}\int_{B_{R/2}} \big( |u|^2 + 2 p \big) \, u \cdot \frac{x}{|x|} \bigg|  \le \delta_{1} + \delta_{2} \big(M[u](R)\big)^{\frac{3}{2}}.
	\end{equation}
\end{proposition}

In this paper we study the local behavior of the weak solutions of the stationary incompressible
Navier-Stokes equations in five space dimensions
\begin{equation}\label{eq:problem}
\left.\begin{array}{rrr}
u^ju^i_j+P_i=\Delta u^i, \quad i=1, 2, 3, 4, 5,\\
\div u =0,
\end{array}
\right\}
\quad\mbox{in}\ \Omega\subset \mathbb{R}^5
\end{equation}
where $\Omega\subset \mathbb{R}^5$ is a domain .

The existence of weak solutions under various assumptions on the
boundary data and $\Omega$ has been established in \cite{Galdi}, \cite{FR-Pisa}, \cite{Struwe-per}.
Moreover, in  \cite{FR-Pisa}, \cite{FR-arma}, \cite{Struwe-per} the authors constructed
smooth solutions of \eqref{eq:problem-3d}.

The problem \eqref{eq:problem} has a number of similarities with the
dynamic Navier-Stokes system in three space dimensions
\begin{equation}\label{eq:problem-3d}
\left.\begin{array}{rrr}
u_t^i+u^ju^i_j+P_i=\Delta u^i, \quad i=1, 2, 3,\\
\div u =0,
\end{array}
\right\}
\mbox{in}\ \Omega\times(0, T), \Omega\subset \mathbb{R}^3.
\end{equation}
For instance, in both cases $u\in L^{\frac{10}3}_{loc}, P\in L^{\frac 53}_{loc}, \nabla P\in L^{\frac54}_{loc}$, see  \cite{Seregin-book}.
Due to this a number of mathematicians studied the stationary
Navier-Stokes equations in higher dimensions in order to develop
stronger analytical methods which may be applicable to the
dynamic case \eqref{eq:problem-3d}, see \cite{Galdi}.

In this context, of particular interest is the problem of estimating the
dimension of the singular points of suitable weak solutions $u$, i.e. the points where
$u$ is not bounded. Scheffer \cite{Scheffer}
proved such results  for \eqref{eq:problem-3d} and later
Caffarelli, Kohn and Nirenberg \cite{CKN} improved upon it
showing that  the Hausdorff dimension of the singular set in space-time is atmost one.
We note that the latter result can be established by a different method
by looking at the
small perturbations of the Stokes system \cite{Lin}.
For \eqref{eq:problem} the partial regularity is proved in \cite{Struwe}.

At the possible singular point $(x_0, t_0)$ the scale invariance  $u(x, t)\mapsto ru(x_0+rx, t_0+r^2), r>0$ suggests that at the scale $r$, $u$ behaves like $1/r$ near $(x_0, t_0)$.
A natural question that follows from this observation is whether
one can classify the scale invariant solutions.
This has been the main approach towards understanding the structure of possible singularities.
\v{S}ver\'{a}k's classification for the self-similar solutions \cite{Sverak} for Navier-Stokes equations \eqref{eq:problem} shows that
a solution of the form $h(x)=\frac{\zeta(\frac x{|x|})}{|x|}$, with some smooth vectorfield $\zeta$, must be identically zero.

Another questions following from this result is whether the solutions sufficiently close to the self-similar one
are in fact zero. It is easy to see that the self-similar vectorfields $h=\frac{\zeta(\frac x{|x|})}{|x|}$ form a Hilbert subspace $\mathcal{H}(R)$ of
the Sobolev space $\mathcal W^{1, 2}(B_R)$ in $\mathbb{R}^5$ with an appropriately scaled invariant norms,
\begin{equation}\label{mH1}
	\mathcal{H}(R):= \left\{ h\in W^{1,2}(B_R) \,\bigg\vert\, \exists \zeta \in W^{1,2}(\mathbb{S}^{N-1}): \ h(x)=\frac{1}{|x|}\zeta\big( \frac{x}{|x|} \big) \ \text{for $x\in B_R$ a.e.} \right\}.
\end{equation}
Thus the Hilbert projection theorem yields that $\mathcal W^{1,2}(B_R) = \mathcal{H}(R) \oplus \mathcal{H}(R)^{\perp}$ and we can define the the corresponding projection operator as $\mathcal{P}_R[\,\cdot\,]: \mathcal W^{1,2}(B_R)\to \mathcal{H}(R)$. Using this, we can measure the error $u-\mathcal{P}_R[u]$ in terms of the $W^{1,2}$ norm of $u$.

Our work is motivated by the following question.

{\textit{  If a suitable weak solution to the stationary Navier-Stokes system develops a singularity,
 can it asymptotically  become self-similar?}}

Our  main result in this direction can be stated as follows:

In this paper we study the local behavior of the weak solutions of the stationary incompressible
Navier-Stokes equations in five space dimensions
\begin{equation}\label{eq:problem}
\left.\begin{array}{rrr}
u^ju^i_j+P_i=\Delta u^i, \quad i=1, 2, 3, 4, 5,\\
\div u =0,
\end{array}
\right\}
\quad\mbox{in}\ \Omega\subset \mathbb{R}^5
\end{equation}
where $\Omega\subset \mathbb{R}^5$ is a domain .

The problem \eqref{eq:problem} has a number of similarities with the
dynamic Navier-Stokes system in three space dimensions
\begin{equation}\label{eq:problem-3d}
\left.\begin{array}{rrr}
u_t^i+u^ju^i_j+P_i=\Delta u^i, \quad i=1, 2, 3,\\
\div u =0,
\end{array}
\right\}
\mbox{in}\ \Omega\times(0, T), \Omega\subset \mathbb{R}^3.
\end{equation}
For instance, in both cases $u\in L^{\frac{10}3}_{loc}, P\in L^{\frac 53}_{loc}, \nabla P\in L^{\frac54}_{loc}$, see \cite{Seregin-book}.
Due to this a number of mathematicians studied the stationary
Navier-Stokes equations in higher dimensions in order to develop
stronger analytical methods which may be applicable to the
dynamic case \eqref{eq:problem-3d}, see \cite{Galdi}.

In this context, of particular interest is the problem of estimating the
dimension of the singular points of suitable weak solutions $u$, i.e. the points where
$u$ is not bounded. Scheffer \cite{Scheffer}
proved such results for \eqref{eq:problem-3d} and later
Caffarelli, Kohn and Nirenberg \cite{CKN} improved upon it
showing that the Hausdorff dimension of the singular set in space-time is atmost one.
We note that the latter result can be established by a different method
by looking at the
small perturbations of the Stokes system \cite{Lin}.
For \eqref{eq:problem} the partial regularity is proved in \cite{Struwe}.

Another questions following from this result is whether the solutions sufficiently close to the self-similar one
are in fact zero. It is easy to see that the self-similar vectorfields $h=\frac{\zeta(\frac x{|x|})}{|x|}$ form a Hilbert subspace $\mathcal{H}(R)$ of
the Sobolev space $\mathcal W^{1, 2}(B_R)$ in $\mathbb{R}^5$ with an appropriately scaled invariant norms,
\begin{equation}\label{mH1}
 \mathcal{H}(R):= \left\{ h\in W^{1,2}(B_R) \,\bigg\vert\, \exists \zeta \in W^{1,2}(\mathbb{S}^{N-1}): \ h(x)=\frac{1}{|x|}\zeta\big( \frac{x}{|x|} \big) \ \text{for $x\in B_R$ a.e.} \right\}.
\end{equation}
Thus the Hilbert projection theorem yields that $\mathcal W^{1,2}(B_R) = \mathcal{H}(R) \oplus \mathcal{H}(R)^{\perp}$ and we can define the the corresponding projection operator as $\mathcal{P}_R[\,\cdot\,]: \mathcal W^{1,2}(B_R)\to \mathcal{H}(R)$. Using this, we can measure the error $u-\mathcal{P}_R[u]$ in terms of the $W^{1,2}$ norm of $u$.

{\textit{ If a suitable weak solution to the stationary Navier-Stokes system develops a singularity,
 can it asymptotically become self-similar?}}

Our main result in this direction can be stated as follows:

\begin{equation}\label{eq:problem}
\left.\begin{array}{rrr}
u^ju^i_j+P_i=\Delta u^i, \quad i=1, 2, 3, 4, 5,\\
\div u =0,
\end{array}
\right\}
\quad\mbox{in}\ \Omega\subset \R^5
\end{equation}

\begin{equation}\label{eq:problem-3d}
\left.\begin{array}{rrr}
u_t^i+u^ju^i_j+P_i=\Delta u^i, \quad i=1, 2, 3,\\
\div u =0,
\end{array}
\right\}
\mbox{in}\ \Omega\times(0, T), \Omega\subset \R^3.
\end{equation}

{\textit{ If a suitable weak solution to the stationary Navier-Stokes system develops a singularity,
 can it asymptotically become self-similar?}}

If $\liminf_{r\to 0}M(r)<\infty$
then the singularity may occur only if the function
\begin{equation}
\text{\Large$\wp$}(r) =\frac{1}{r^2} \int_{B_r} \Big\{{|u|^2} + 2P \Big\} (u\cdot \frac{x}{|x|}) \, dx .
\end{equation}
takes nonpositive values as $r\to 0$. Moreover,
if $u$ is of the form $\frac{\zeta(\frac x{|x|})}{|x|}$ then one can check that
$\text{\Large$\wp$}(r) =0$. This observation motivates the formulation of
a condition in our next result that allows to control $\text{\Large$\wp$}(r)$.
\begin{theorem}\label{TH}
Let $u$ be a suitable weak solution of \eqref{eq:problem}, $B_1\subset \Omega$, and
\[
m:=\liminf_{R\to 0}M(R)<\infty \qquad \text{where} \quad M(R):=\int_{B_R} \Big( \frac{|u|^2}{R^3} + \frac{|\nabla u|^2}{R} \Big).
\] Let $\mP_R[\,\cdot\,]: \mathcal W^{1,2}(B_R)\to \mH(R)$ be the projection operator for the space \eqref{mH1}. There exists $\ep(m)>0$
such that
if
\begin{equation}\label{eq:13-alt}
\frac{1}{R^3} \int_{B_R} \left| u - \mP_R[u] \right|^2 + \frac{1}{R} \int_{B_R} \left| \nabla u - \nabla \mP_R[u] \right|^2 \le \ep(m) M(R)
\end{equation}
holds
for all $R\in (0, R_0)$ then $u$ is regular at $x=0$.
\end{theorem}

\begin{proposition}\label{prop:dAdr}
 Suppose $N=5$ and $(u,P)$ is a suitable weak solution of \eqref{st-NS-N}. For $r>0$, define
 \begin{equation}
 \left\{
 \begin{aligned}
 &D(r)\vcentcolon= \int_{B_r} \Big\{ \frac{15}{4r^3}|u|^2 + \frac{1}{4r}|\nabla u |^2 + \frac{3}{4r^3}\left| \nabla (|x|u) \right|^2 + \frac{3(r^2-|x|^2)}{4r^3} |\nabla u|^2\Big\}\, dx,\\
 &A(r) \vcentcolon= \frac{1}{r^3}\int_{B_r}(x\cdot \nabla)\frac{|u|^2}{2}\, dx + \frac{9}{4r^3}\int_{ B_r}|u|^2 \, dx - \frac{1}{r^{2}} \int_{B_r} \big\{ \frac{|u|^2}{2} + P \big\} u\cdot \frac{x}{|x|} \, dx .
 \end{aligned}
 \right.
 \end{equation}
 Then the following differential equation holds for $r>0$,
 \begin{equation*}
 \frac{d A}{d r} \ge \frac{1}{r} D(r) + \frac{2}{r^3} \int_{B_r} \Big\{\frac{|u|^2}{2} + P \Big\} (u\cdot \frac{x}{|x|}) \, dx .
 \end{equation*}
\end{proposition}
\begin{proof}
 Let us consider the function
 \begin{equation}
 \phi(x)=
 \left\{
 \begin{array}{lll}
 1 \quad & \mbox{if} \ |x|<r,\\
 \frac{r-|x|}\epsilon \quad &\mbox{if} \ r\le |x|\le r+\epsilon,\\
 0 \quad & \mbox{if}\ |x|>r+\epsilon.\\
 \end{array}
 \right.
 \end{equation}
 We then mollify this function and take $\psi=\phi*\rho_\delta$,
 where $\rho$ is the mollification kernel.
 Note that $\psi\in \mathcal C_c^\infty(\R^N)$.
 We use $\psi$ as a test function in the local
 energy inequality to obtain
 \begin{equation}
 \int |\nabla u|^2 \psi
 \le
 \int \left(-u \nabla u +(|u|^2+2P)u\right)\cdot \nabla \psi.
 \end{equation}

\begin{lemma}\label{lemma:reg}
Suppose $m:=\liminf_{R\to 0}M(R)<\infty$.
 If there exists a sufficiently small $\ep>0$, depending on $m$, such that $(u,P)$ is a suitable weak solution to the Navier-Stokes equations satisfying
 \begin{equation}\label{H-Assump}
 \ep \frac{1}{R^{3}}\int_{B_R} |p| + \frac{1}{R^3} \int_{B_R} \left| u - \mP_R[u] \right|^2 + \frac{1}{R} \int_{B_R} \left| \nabla u - \nabla \mP_R[u] \right|^2 \le \ep M[u](R),
 \end{equation}
 for all $R\in(0,1]$ then $u$ is regular at $x=0$.
\end{lemma}
\begin{proof}
If $m\le 8C_E$ then we can apply Proposition \ref{prop:fM}, and
hence the result follows.
 Now suppose $m\in(8C_E, \infty)$. In light of Lemma \ref{lemma:MBound}, there exists $\ep_1>0$ such that if $(u,P)$ satisfies \eqref{H-Assump} with $\ep\in (0,\ep_1)$ then $M(R)$ is uniformly bounded in $R\in(0,1]$ and we set
 \begin{equation}\label{Mast}
 M_{\ast} \vcentcolon= \sup\limits_{0<R\le 1} M(R) \le 2^5 C_0(m),
 \end{equation}
 where $C_0(m)$ is given by \eqref{eq:Cmm}.
 At this point we can apply Proposition \ref{prop:fM}
 with $C_0=2^5C_0(m)$ by choosing $\ep_0$ sufficiently small, however we can avoid this by using the monotonicity formula.

\begin{lemma}\label{lem:Lin}
If $u$ is a suitable weak solution and
\[
\int_{B_1} |u|^3+|p|^{\frac32}<\ep^*
\]
for some sufficiently small $\ep^*$,
then
\begin{equation}\label{eq:Lin1}
\frac1{\theta^5}\int_{B_\theta}\frac{|u-[u]_\theta|^3}{\theta^{\alpha_0}}+\frac{|p-[p]_\theta|^\frac32}{\theta^{\alpha_0}}\le \frac12 \int_{B_1} |u|^3+|p|^{\frac32},
\end{equation}
for some positive $\theta$ and $\alpha_0\in (0, \frac12)$.
\end{lemma}
\begin{proof}
If \eqref{eq:Lin1} fails, then there would be a sequence of solutions
$(u_i, p_i)$ such that $\int_{B_1}|u_i|^3+|p_i|^{\frac32}:=\epsilon_i\to 0$ but
\eqref{eq:Lin1} is not valid. Introduce
\[
\bar u_i=\frac{u_i}{\epsilon_i}, \quad \bar p_i=\frac{p_i}{\epsilon_i},
\]
then
\begin{equation}\label{eq:Lin4}
\epsilon_i \bar u_i\cdot \nabla \bar u_i+\nabla \bar p_i=\Delta \bar u_i,
\quad
\frac1{\theta^5}\int_{B_\theta}\frac{|\bar u-[\bar u]_\theta|^3}{\theta^{\alpha_0}}
+
\frac{|\bar p-[\bar p]_\theta|^\frac32}{\theta^{\alpha_0}}> \frac12,
\quad \int_{B_1} |\bar u_i|^3+|\bar p_i|^{\frac32}\le 2.
\end{equation}
From the local energy inequality $u\in \mathcal W^{1, 2}_{loc}(B_1).$
Moreover, the following equation is satisfied in distributional sense
\begin{equation}
\Delta \bar p_i=-\epsilon_i \frac{\partial^2 (\bar u^k\bar u^l)}{\partial x_l\partial x_k},
\quad \mbox {in}\ B_1.
\end{equation}
From the Poisson representation theorem we can write $\bar p_i=h_i+g_i$,
where $h_i$ is harmonic in $B_1$, and
\begin{equation}
\left\{
\begin{array}{lll}
\Delta g_i=-\epsilon_i \frac{\partial^2 (\bar u^k\bar u^l)}{\partial x_l\partial x_k}
\quad \mbox {in}\ B_{\frac23},\\
g_i=0 \quad \mbox {on}\ \partial B_{\frac23}.
\end{array}
\right.
\end{equation}
From the Calder\'on-Zygmund estimates
$g_i$ is uniformly bounded in ${L^{5/3}(B_{2/3})}$.
Consequently, $h_i\in L^{3/2}(B_{2/3})$ uniformly, hence
from the local estimates for the harmonic functions
\begin{align}
\int_{B_\theta}|\bar p_i-[\bar p_i]_\theta|^{\frac32}\le
\int_{B_\theta}| h_i-[h_i]_\theta|^{\frac32}
\int_{B_\theta}|g_i-[g_i]_\theta|^{\frac32}\\
\le C_0 \theta^5\theta^{3/2}+C_0\epsilon_i\int_{B_{2/3}}|\bar u_i|^3.
\end{align}
For a suitable subsequence $\bar u_i\to \bar u$ in $\mathcal W^{1, 2}(B_{2/3})$
and $\bar p_i\to \bar p$ strongly in ${L^{3/2}(B_{2/3})}$.
Consequently, for sufficiently large $i$, we have
\begin{equation}\label{eq:Lin2}
\int_{B_\theta}|\bar p_i-[\bar p_i]_\theta|^{\frac32} \le C_0 \theta^5\theta^{3/2}
\end{equation}
Since the limit $\bar u$ solves the Stokes system, then it follows that
$\bar u$ is H\"older continuous with, say, exponent $2\alpha_0$, and therefore
$\int_{B_\theta}|\bar u-[\bar u]_\theta|^{\frac32}\le \frac14 \theta^5\theta^{\alpha_0}$.
From the strong convergence $\bar u_i\to \bar u$ in $L^3(B_{2/3})$, we infer that
\begin{equation}\label{eq:Lin3}
\int_{B_\theta}|\bar u-[\bar u]_\theta|^{\frac32}\le \frac13 \theta^5\theta^{\alpha_0}.
\end{equation}
Combining \eqref{eq:Lin2} and \eqref{eq:Lin2} we get a contradiction with the
second inequality in \eqref{eq:Lin4}.
\end{proof}

The proof of Theorem \ref{TH-00} uses the monotonicity
formula introduced in Proposition \ref{prop:dAdr}, and
a scaling argument. See Lemma \ref{lem:positiveinf} for the proof.
Note that there are no smallness assumptions in the
statement of Theorem \ref{TH-00}.

If $\liminf_{r\to 0}M(r)<\infty$
then the singularity may occur only if the function
\begin{equation}
\text{\Large$\wp$}(r)  =\frac{1}{r^2} \int_{B_r}  \Big\{{|u|^2} + 2P \Big\} (u\cdot \frac{x}{|x|})  \, dx .
\end{equation}
takes nonpositive values as $r\to 0$. Moreover,
if $u$ is of the form $\frac{\zeta(\frac x{|x|})}{|x|}$ then one can check that
$\text{\Large$\wp$}(r) =0$. This observation motivates the formulation of
a condition in our next result that allows to control $\text{\Large$\wp$}(r)$.
\begin{theorem}\label{TH}
Let $u$ be a suitable weak solution of \eqref{eq:problem}, $B_1\subset \Omega$, and
\[
m:=\liminf_{R\to 0}M(R)<\infty \qquad \text{where} \quad M(R):=\int_{B_R} \Big( \frac{|u|^2}{R^3} + \frac{|\nabla u|^2}{R} \Big).
\] Let $\mP_R[\,\cdot\,]: \mathcal W^{1,2}(B_R)\to \mathcal{H}(R)$ be the projection operator for the space \eqref{mH1}. There exists $\varepsilon(m)>0$
such that
if
\begin{equation}\label{eq:13-alt}
\frac{1}{R^3} \int_{B_R} \left| u - \mP_R[u] \right|^2 + \frac{1}{R} \int_{B_R} \left| \nabla u - \nabla \mP_R[u] \right|^2 \le  \varepsilon(m) M(R)
\end{equation}
holds
for all $R\in (0, R_0)$ then $u$ is regular at $x=0$.
\end{theorem}

It is known that if $u\in \mathcal{H}$ then $u=0$ \cite{Sverak}. In this context, Theorem \ref{TH}
states that if $\|u-\mP_R[u]\|_{\mathcal{W}^{1,2}(R)}$ is small compared to $\|u\|_{\mathcal{W}^{1,2}(R)}$ then
$\text{\Large$\wp$}(r)$ is smaller than $M^{\frac32}(r)$, which
after application of Proposition \ref{prop:dAdr} implies that
$u=0$.

As opposed to the main result in \cite{CKN}, we do not assume that $u$ is small in some scale invariant seminorm, reminiscent to
the ``local" Reynolds number $\frac1{r}\int_{B_r}|\nabla u|^2$. This  leads us to the classification of the
self-similar solution of the incompressible Euler equations in $\mathbb{R}^5$.
In fact, we prove that for such solutions
the Bernoulli pressure is zero. This is the first  key point in our proof of the main technical result, Proposition \ref{prop:cubicEst}.

The second key point is the construction of a monotonicity formula for the suitable weak solutions, which follows from
the weak energy inequality.

We compare Theorem \ref{TH} with the well-known
regularity criteria for suitable weak solutions of
\eqref{eq:problem-3d}, which in its most general form, can be stated as follows:
let $Q(R)=B_R\times(-R^2, 0)$ and define the local Reynolds numbers
\[
E(R)=\frac1R\int_{Q(R)}|\nabla u|^2, \quad C(R)=\frac1{R^2}\int_{Q(R)}|u|^3.
\]

Then the following statement holds: for every $M>0$ there is
$\varepsilon(M)>0$ such that
$\limsup_{R\to 0} C(R)<M, \liminf_{R\to 0}E(R)<\epsilon(M)$ imply that the origin is a regular point.

This result can be found in
Seregin's paper \cite{Seregin}, Theorem 1.4. Note that
 $\limsup_{R\to 0} C(R)<M$ implies that there is a constant $C_0(M)$
 depending on $M$ such that $\limsup_{R\to 0} E(R)<C_0(M)$ \cite{Lewis}.

As opposed to this result, we do not impose the finiteness of upper limit
of the Reynolds number. Instead, we assume that the lower limit is finite, i.e.
\[
\liminf_{R\to 0} M(R)<\infty.
\]

This is the replacement of the condition $\limsup_{R\to 0} C(R)<M$ in \cite{Seregin}.

As for the other condition, $\liminf_{R\to 0}E(R)<\epsilon(M)$, it is replaced by closeness assumption: more precisely, we assume that there is a vectorfield $h$
homogeneous of degree negative one such that
$u-h$ has a suitable small norm compared to $M(R)$.
See Section \ref{sec:energy}
for precise definitions.
Hence our conditions are weaker.

The paper is organized as follows:
In Section \ref{sec:mon} we introduce one of our main technical tools, the
monotonicity formula and prove Theorem \ref{TH-00}. In the next section
we classify the self-similar solutions of the incompressible Euler equations in
$\mathbb R^5$, and prove that for such solutions the Bernoulli
pressure  is zero.
In Section \ref{sec:energy} we prove some estimates for the pressure.
Section \ref{sec:technical} contains one of our main estimates of the
cubic term that appears in the local energy inequality.
In order to control the growth of $M(R)$ we prove an iteration result in Section \ref{sec:iteration}, and apply it to obtain a local bound in Section
\ref{sec:sup-est}.
The proof of Theorem \ref{TH} is given in Section.
\ref{sec:proofofmain}.
We also added an appendix at the end of the paper that contains
some estimates and computations used in the proof of Theorem \ref{TH}.

The post-theorem introduction says: “The proof of Theorem \ref{TH-00} uses the monotonicity formula introduced in Proposition \ref{prop:dAdr}, and a scaling argument. See Lemma \ref{lem:positiveinf} for the proof.”

The post-theorem introduction says: “To prove the main theorem, the proof uses the monotonicity formula introduced in the following proposition.

\begin{proposition}\label{prop:dAdr}
	Suppose $N=5$ and $(u,P)$ is a suitable weak solution of \eqref{st-NS-N}. For $r>0$, define
	\begin{equation}
		\left\{
		\begin{aligned}
			&D(r)\vcentcolon=   \int_{B_r} \Big\{ \frac{15}{4r^3}|u|^2 + \frac{1}{4r}|\nabla u |^2 + \frac{3}{4r^3}\left| \nabla (|x|u) \right|^2 + \frac{3(r^2-|x|^2)}{4r^3} |\nabla u|^2\Big\}\, dx,\\
			&A(r) \vcentcolon= \frac{1}{r^3}\int_{B_r}(x\cdot \nabla)\frac{|u|^2}{2}\, dx + \frac{9}{4r^3}\int_{ B_r}|u|^2 \, dx  - \frac{1}{r^{2}} \int_{B_r} \big\{ \frac{|u|^2}{2} + P \big\} u\cdot \frac{x}{|x|}  \, dx .
		\end{aligned}
		\right.
	\end{equation}
	Then the following differential equation holds for $r>0$,
	\begin{equation*}
		\frac{d A}{d r} \ge \frac{1}{r} D(r) + \frac{2}{r^3} \int_{B_r}  \Big\{\frac{|u|^2}{2} + P \Big\} (u\cdot \frac{x}{|x|})  \, dx .
	\end{equation*}
\end{proposition}

It also uses a scaling argument. For the proof, see the following lemma.

\begin{lemma}\label{lem:positiveinf}
Let
\begin{equation}
Q(r):= \int_{B_r} \Big\{ \frac{15}{4r^3}|u|^2
+
\frac{1}{4r}|\nabla u |^2
+
\frac{3(r^2-|x|^2)}{4r^3} |\nabla u|^2\Big\}\, dx,
\end{equation}
and
\begin{equation}
\text{\Large$\wp$}(r)  =\frac{1}{r^2} \int_{B_r}  \Big\{{|u|^2} + 2P \Big\} (u\cdot \frac{x}{|x|})  \, dx .
\end{equation}
If  $\liminf_{r\to 0} M(r)<\infty$ and
\[
\liminf_{r\to0^+}\left[ Q(r)
+\text{\Large$\wp$}(r)
\right]> 0
\]
then $x=0$ is a regular point.
\end{lemma}”,

\label{TH-00}
Let $u$ be a suitable weak solutions. Suppose that
the following two conditions hold:
\begin{equation}
\liminf_{r\to 0}M(r):=\liminf_{r\to 0}\int_{B_r}\!\! \left( \frac{|u|^2}{r^3} + \frac{|\nabla u|^2}{r} \right) <\infty, \quad \liminf_{r\to 0}\frac1{r^2}\int_{B_r}\!\!(|u|^2+2P)u\cdot\frac x{|x|}>0.
\end{equation}
To prove the main theorem, we show that for any $\{r_k\}_{k=1}^\infty, r_k\downarrow 0$ there is a subsequence $r_{k_m}$ such that
the scaled solutions $u_{r_{k_m}}(x)=r_{k_m}u(r_{k_m}x)$, converge to
a homogenous vector field of degree negative one, and hence
$x=0$ is a regular point.,

Question:
Let $(u,P)$ be a suitable weak solution near $x=0$ of the stationary incompressible NavierStokes system in five dimensions,
\[
u^j\partial_j u^i + \partial_i P = \Delta u^i, \qquad \operatorname{div}u=0,
\]
in a domain $\Omega\subset\mathbb{R}^5$. For $r>0$, write
\[
M(r):=\int_{B_r}\left(\frac{|u|^2}{r^3}+\frac{|\nabla u|^2}{r}\right),
\]
where $B_r$ is the ball of radius $r$ centered at $0$. Assume that
\[
\liminf_{r\to 0} M(r)<\infty
\]
and also
\[
\liminf_{r\to 0}\frac1{r^2}\int_{B_r} (|u|^2+2P)\,u\cdot \frac{x}{|x|}>0.
\]
For a scale $r>0$, define the rescaled velocity field by
\[
u_r(x)=r\,u(rx).
\]
Which conclusion holds under these hypotheses?

For every sequence $\{r_k\}_{k=1}^\infty$ with $r_k\downarrow 0$, there exists a subsequence $r_{k_m}$ such that the scaled fields $u_{r_{k_m}}(x)=r_{k_m}u(r_{k_m}x)$ converge to a vector field that is homogeneous of degree $-1$; consequently, $x=0$ is a regular point.

Choices:
B. There exists a sequence $r_k\downarrow 0$ such that the scaled fields $u_{r_k}(x)=r_k u(r_kx)$ converge to a vector field that is homogeneous of degree $-1$; consequently, $x=0$ is a regular point.
C. There exists a sequence $r_k\downarrow 0$ such that, after passing to a subsequence, the scaled fields $u_{r_k}(x)=r_k u(r_kx)$ converge to a vector field that is homogeneous of degree $-1$.
D. For every sequence $\{r_k\}_{k=1}^\infty$ with $r_k\downarrow 0$, the full sequence of scaled fields $u_{r_k}(x)=r_k u(r_kx)$ converges to a unique vector field that is homogeneous of degree $-1$; consequently, $x=0$ is a regular point.
E. For every sequence $\{r_k\}_{k=1}^\infty$ with $r_k\downarrow 0$, there exists a subsequence $r_{k_m}$ such that the scaled fields $u_{r_{k_m}}(x)=r_{k_m}u(r_{k_m}x)$ converge to a vector field that is homogeneous of degree $-1$; moreover, this implies that $x=0$ is a singular point.

dropped regularity conclusion and universal quantifier over sequences

subsequence compactness replaced by full-sequence unique limit

sign of the monotonicity-based regularity consequence

The stem states hypotheses and asks for the resulting conclusion, but it does not explicitly reveal the correct option. The correct answer must be distinguished from nearby variants involving different quantifiers, convergence strength, and regularity consequences.

The item is close to a theorem-recall question: the hypotheses are presented and the conclusion is selected. However, it is not a pure tautology because the choices differ in meaningful ways (every sequence vs. some sequence, subsequence vs. full sequence, regular vs. singular point).

Moderate reasoning is required. A solver must track subtle logical structure and identify the strongest justified conclusion, but the problem mainly tests precise theorem comprehension rather than generating a new mathematical argument.

The distractors are strong and plausible: they reflect common mathematical errors involving quantifier order, unjustified uniqueness/full-sequence convergence, omission of the regularity consequence, and reversal of the conclusion about regularity.

A solid MCQ with no answer leakage and high-quality distractors. Its main weakness is that it functions largely as a theorem-conclusion recognition item, so it tests precision with logical details more than deep generative reasoning.


\begin{equation}
		\label{eq:orth_decomp}
		\{f\in\ell^2(E)\colon \partial f=0\}= \ell_{0,c}^2(E)\oplus_{\ell^2(E)} \{du\mid u\colon G\to \R, \, \Delta_0 u = 0, \, du\in \ell^2(E)\}.
	\end{equation}

\label{sec:Intro}

Let $G=(V,E)$ be countable oriented graph with degrees of vertices bounded from the above. Let $\tilde G$ be the non-oriented graph obtained from $G$ by forgetting the orientation of edges. Pick $D\in\mathbb N$ large enough. Consider all cycles in $\tilde G$ having lengths $\le D$. In $G$, glue each such cycle with a polygon. Choose any orientation of the latter polygons. We arrive to oriented $2$-dimensional complex, denote it by $G^{(2)}$ with implicit dependence on~$D$. Denote by $F$ the set of $2$-dimensional faces in $G^{(2)}$ which are polygons. Sometimes we write $F=FG$ and also $E=EG$ to indicate the dependence of these sets on $G$. Any of sets $V, E, F$ is endowed with counting measure which we denote by $\card$. In graph $G$, we define graph metric $\dist_G$ at $V\cup E$ along edges in $E$ so that any edge has length $1$.

If $\Gamma$ is a finitely generated group, $S$ is any of its generating sets (symmetrized or not) then we may consider $G=\Cay(\Gamma,S)$, Cayley graph of $\Gamma$; then $V=\Gamma$. If $\Gamma$ is also finitely presented, that is, given by a finite number of relations then we assume that $D$ in the definition of $G^{(2)}=\Cay^{(2)}(\Gamma,S)$ is $\ge$ than length of any of the defining relations. For general $G$, we assume that $D$ is such that
\begin{equation*}\mbox{$G^{(2)}$ is simply connected}
\end{equation*}
(and that such $D$ does exist).

A \emph{$k$-cochain}, $k$ is $0$, or $1$, or $2$, is a function from $V$, or $E$, or $F$, respectively, to ${\mathbb R}$. We often understand  cochains as chains.
Discrete differentials (coboundaries)
$$
\{0\mbox{-cochains}\}\xrightarrow{d}\{1\mbox{-cochains}\}\xrightarrow{d}\{2\mbox{-cochains}\}
$$
and boundary operators
$$
\{2\mbox{-cochains}\}\xrightarrow{\partial}\{1\mbox{-cochains}\}\xrightarrow{\partial}\{0\mbox{-cochains}\}
$$
are introduced in the standard way with respect to the orientation of edges and faces.  Since valencies of vertices are bounded, all these operators are also bounded with respect to $\ell^2$-norms on cochains. We have $(d|_{\ell^2(V)})^*=\partial|_{\ell^2(E)}$, $(d|_{\ell^2(E)})^*=\partial|_{\ell^2(F)}$. Indeed, discrete integration by parts is valid for finitely supported cochains and is proved for  $\ell^2$-cochains by $\ell^2$-approximation.
If $\gamma$ is an oriented path in $G$ then we may define $1$-(co)chain $f_\gamma$: for $e\in E$, let $f_\gamma(e)$ be the number of passes of $\gamma$ through $e$ in its direction minus number of passes of $\gamma$ over $e$ in its reversed direction. Then we have $\partial f_\gamma=0$.

Our space of interest is
\begin{equation*}
	\ell_{0,c}^2(E) := \clos_{\ell^2(E)}\{f\colon E\to {\mathbb R}\mid \partial f=0, \, \supp f
	\,\mbox{is finite}\}.\end{equation*}
Any of $1$-cochains at the right-hand side can be (convexly) decomposed into simple finite loops. Thus, $\ell_{0,c}^2(E)$ is $\ell^2$-closed linear span of (co)chains of the form $f_\gamma$ with $\gamma$ a finite loop in $G$.

Laplace operator on $0$-cochains is
$$
-\Delta_0=\partial d\colon \left(\mbox{functions on }V\right)\to \left(\mbox{functions on }V\right).
$$
A discrete integration by parts leads to the following Hodge-type decomposition:

\begin{predl}

	We have
	\begin{equation}
		\label{eq:orth_decomp}
		\{f\in\ell^2(E)\colon \partial f=0\}= \ell_{0,c}^2(E)\oplus_{\ell^2(E)} \{du\mid u\colon G\to {\mathbb R}, \, \Delta_0 u = 0, \, du\in \ell^2(E)\}.
	\end{equation}
\end{predl}

The second summand in the right-hand side of the latter relation is \emph{$\ell^2$-cohomology} of $G$. It is known to be invariant with respect to change of generating system in a group: the factorspace nature of cohomology allows to implement "discrete change of variables" from one to another set of generators. Cohomology is invariant with respect to more general quasiisometries.
Now, we pass to spectral estimates for $1$-cochains. Define non-negative Laplacian operator $\Delta_1:=\partial d+d\partial \colon \ell^2(E)\to \ell^2(E)$. On $\ell^2_{0,c}(E)$, our space of interest, this reduces to $\partial d$.

We have one more Hodge-type decomposition:
$$
\ell^2(E) = \clos_{\ell^2(E)}\{du\mid u \colon V\to {\mathbb R}, \, \supp u \mbox{ is finite}\} \oplus_{\ell^2(E)} \{f\in\ell^2(E)\colon \partial f=0\}.$$
Spectral questions for $\Delta_1$ on the first summand are generally reduced to the same for $\Delta_0$ on $\ell^2(V)$.   What concerns decomposition (\ref{eq:orth_decomp}) for $\{f\in\ell^2(E)\colon \partial f=0\}$, operator $\Delta_1$ vanishes at the second summand of its right-hand side, $\ell^2$-cohomology. Also, by the definition of $\ell^2_{0,c}(E)$ and by $\ell^2$-approximation, we see that $\Delta_1(\ell^2_{0,c}(E))\subset\ell^2_{0,c}(E)$.

\begin{define}
	We say that $\Delta_1$ \emph{has a spectral gap at $\ell^2_{0,c}(E)$} \emph{(}or just that graph $G$ has \emph{{coexact $1$-Laplacian spectral gap}}\emph{)} if $$\spec\left(\Delta_1|_{\ell^2_{0,c}(E)}\right)\cap [0,\varepsilon)=\varnothing$$ for some $\varepsilon>0$ small enough.
\end{define}

Applying discrete integration by parts, we conclude that this is equivalent to the estimate
\begin{equation}
	\label{eq:rot_estim}
\langle f, f \rangle_{\ell^2(E)}	\le 1/\varepsilon\cdot \langle df, df\rangle_{\ell^2(F)}
\end{equation}
for $f\in \ell^2_{0,c}(E)$. It is enough to check the latter only for finitely supported closed $1$-cochains $f$. Also, we conclude that if $1$-Laplacian has a coexact spectral gap then it will be so if we enlarge $D$ in the construction of $G^{(2)}$ or glue some extra faces to $G^{(2)}$ in a locally finite manner.
 We state quasiinvariance result as below, with possibility to add only edges. It seems feasible to preserve spectral gap under more general quasiisometric transformations of a graph, the ones with possibility to add or remove vertices in a locally finite way.
 \begin{predl}
 	\label{predl:general_invariance}
 	Let $G_1=(V,E_1)$ and $G_2=(V,E_2)$ be graphs with the same vertex set $V$ and with $E_1\subset E_2$.  	Assume the following:
 	\begin{enumerate}
 		\item $G_1$ is connected;

 		\item degrees of vertices in $G_2$ are bounded from the above;

 		\item $\sup\limits_{e\in E_2\setminus E_1} \dist_{G_1}(\beg e, \ennd e) < +\infty$ with the obvious notation.
 	\end{enumerate}
	\emph{(}In other words, metrics $\dist_{G_1}$ and $\dist_{G_2}$ on $V$ are bilipshitz equivalent.\emph{)} Then, $G_1$ has a coexact $1$-Laplacian spectral gap  \emph{(}with some $D$ implied at the construction of $G_1^{(2)}$\emph{)}
	if and only if  $G_2$ has such a spectral gap \emph{(}with some $D'$ for $G_2^{(2)}$\emph{)}.
 \end{predl}

 \noindent Proof of this quasiinvariance  is similar to the proof of quasiiinvariance of $\ell^2$-cohomology; both are based on orthogonal projection. We give a detailed argument in the Appendix. Notice also that the proof is constructive: we may estimate $D'$ via $D$ and the supremum from the third assumption of Proposition \ref{predl:general_invariance}, and vice versa.

 \begin{sled}
 	For two Cayley graphs of the same finitely-presented group but with different generating sets, properties of existence of a coexact $1$-Laplacian spectral gap on them  are equivalent.
 \end{sled}

Our main result is

If $\Gamma$ is a finitely generated group, $S$ is any of its generating sets (symmetrized or not) then we may consider $G=\Cay(\Gamma,S)$, Cayley graph of $\Gamma$; then $V=\Gamma$. If $\Gamma$ is also finitely presented, that is, given by a finite number of relations then we assume that $D$ in the definition of $G^{(2)}=\Cay^{(2)}(\Gamma,S)$ is $\ge$ than length of any of the defining relations. For general $G$, we assume that $D$ is such that
\begin{equation*}\mbox{$G^{(2)}$ is simply connected}
\end{equation*}
(and that such $D$ does exist).

A \emph{$k$-cochain}, $k$ is $0$, or $1$, or $2$, is a function from $V$, or $E$, or $F$, respectively, to ${\mathbb R}$. We often understand cochains as chains.
Discrete differentials (coboundaries)
$$
\{0\mbox{-cochains}\}\xrightarrow{d}\{1\mbox{-cochains}\}\xrightarrow{d}\{2\mbox{-cochains}\}
$$
and boundary operators
$$
\{2\mbox{-cochains}\}\xrightarrow{\partial}\{1\mbox{-cochains}\}\xrightarrow{\partial}\{0\mbox{-cochains}\}
$$
are introduced in the standard way with respect to the orientation of edges and faces. Since valencies of vertices are bounded, all these operators are also bounded with respect to $\ell^2$-norms on cochains. We have $(d|_{\ell^2(V)})^*=\partial|_{\ell^2(E)}$, $(d|_{\ell^2(E)})^*=\partial|_{\ell^2(F)}$. Indeed, discrete integration by parts is valid for finitely supported cochains and is proved for $\ell^2$-cochains by $\ell^2$-approximation.
If $\gamma$ is an oriented path in $G$ then we may define $1$-(co)chain $f_\gamma$: for $e\in E$, let $f_\gamma(e)$ be the number of passes of $\gamma$ through $e$ in its direction minus number of passes of $\gamma$ over $e$ in its reversed direction. Then we have $\partial f_\gamma=0$.

\begin{define}
 We say that $\Delta_1$ \emph{has a spectral gap at $\ell^2_{0,c}(E)$} \emph{(}or just that graph $G$ has \emph{{coexact $1$-Laplacian spectral gap}}\emph{)} if $$\spec\left(\Delta_1|_{\ell^2_{0,c}(E)}\right)\cap [0,\varepsilon)=\varnothing$$ for some $\varepsilon>0$ small enough.
\end{define}

Applying discrete integration by parts, we conclude that this is equivalent to the estimate
\begin{equation}
 \label{eq:rot_estim}
\langle f, f \rangle_{\ell^2(E)} \le 1/\varepsilon\cdot \langle df, df\rangle_{\ell^2(F)}
\end{equation}
for $f\in \ell^2_{0,c}(E)$. It is enough to check the latter only for finitely supported closed $1$-cochains $f$. Also, we conclude that if $1$-Laplacian has a coexact spectral gap then it will be so if we enlarge $D$ in the construction of $G^{(2)}$ or glue some extra faces to $G^{(2)}$ in a locally finite manner.
 We state quasiinvariance result as below, with possibility to add only edges. It seems feasible to preserve spectral gap under more general quasiisometric transformations of a graph, the ones with possibility to add or remove vertices in a locally finite way.
 \begin{predl}
 \label{predl:general_invariance}
 Let $G_1=(V,E_1)$ and $G_2=(V,E_2)$ be graphs with the same vertex set $V$ and with $E_1\subset E_2$. Assume the following:
 \begin{enumerate}
 \item $G_1$ is connected;

\begin{sled}
 For two Cayley graphs of the same finitely-presented group but with different generating sets, properties of existence of a coexact $1$-Laplacian spectral gap on them are equivalent.
 \end{sled}

Our main result is

\renewcommand{\abstractname}{}
\begin{abstract}
Let $\Gamma$ be a discrete finitely presented group. Pick any system $S$ of generators in $\Gamma$. In Cayley graph $\Cay(\Gamma)=\Cay(\Gamma, S)$ with edge set~$E$, glue with oriented polygons all the group relations translated to all the points of $\Gamma$; denote the obtained simply connected complex by $\Cay^{(2)}(\Gamma)$.
We study non-negative \emph{Hodge--Laplace operator $\Delta_1$} on edge functions which is defined via complex $\Cay^{(2)}(\Gamma)$; $\Delta_1$ acts on
$$
\ell^2_{0,c}(E):= \clos_{\ell^2(E)}\left\{\mbox{finitely supported closed $1$-(co)chains in }\Cay^{}(\Gamma)\right\}.
$$

We prove the following implication in the spirit of Kesten Theorem: \emph{if $\Delta_1|_{\ell_{0,c}^2(E)}$ has a spectral gap then $\Gamma$ either has exponential growth or is virtually $\mathbb Z$}. \end{abstract}

Our main result is

Let us recall the well-known Kesten Theorem on Laplacian on vertices of a graph:

\begin{theorem}[\cite{K59}] \label{th:Kesten}
 Let $\Gamma$ be a finitely generated group. Then, for $0$-Laplacian in $\Cay(\Gamma)$, $$0\in \spec\left(-\Delta_0|_{\ell^2(\Gamma)}\right)$$ if and only if $\Gamma$ is amenable.
\end{theorem}

If we assume the contrary to Theorem \ref{th:exp_growth}, then, first, $\Gamma$ cannot have two \emph{ends} since in this case $\Gamma$ is virtually cyclic, see, e.g., \cite{Me08}. Second, $\Gamma$ cannot have infinitely many ends because then $\Gamma$ has exponential growth; the latter follows from Stallings Theorem and from results of \cite{HB00} but, of course, can be proved directly. So, by Freudenthal--Hopf Theorem we may assume that $\Gamma$ has one end.

Now, let us briefly recall the proof of Kesten Theorem \ref{th:Kesten} to compare it to our argument. Non-amenability of $\Gamma$ means that
\begin{equation}
\label{eq:isoperimetr}
\|du\|_{\ell^1(E)} \gtrsim \|u\|_{\ell^1(\Gamma)}
\end{equation}
for $u=\mathds 1_E$, $E$ ranges all finite subsets in $\Gamma$. By discrete version of coarea formula, this is equivalent to the same for any finitely supported $u\colon \Gamma\to\R$.
The spectral gap condition $0\notin \spec\left(-\Delta_0|_{\ell^2(\Gamma)}\right)$ means that
$$
\langle u, u\rangle_{\ell^2(\Gamma)} \lesssim \langle du, du\rangle_{\ell^2(E)}, ~~~ u\colon\Gamma\to\R \mbox{ is finitely supported}.
$$
To obtain this from (\ref{eq:isoperimetr}), it remains to insert $u^2$ instead of $u$ to (\ref{eq:isoperimetr}) and apply Cauchy--Bunyakovsky--Schwartz inequality.

This paper is organized as follows. In Section \ref{sec:maueutic}, we explain our interest to the study of $1$-Laplacian spectra. This Section is not used in the proof of Theorem \ref{th:exp_growth}. In Section~\ref{sec:loop}, we prove Lemma \ref{lemma:embed} by dropping lots of geodesic perpendiculars in a branching way. In Section \ref{sec:spectra_homology}, we conclude the proof of Theorem~\ref{th:exp_growth}. This is done by approximating the resolvent $\Delta^{-1}_1$ by polynomials of $\Delta_1$ provided that $0\notin\spec\Delta_1|_{\ell_{0,c}^2(E)}$. Next, we apply this to $1$-cochain given by curve $\gamma$ from Lemma \ref{lemma:embed}. We put some metric control on the approximating process as implemented in $\Cay(\Gamma)$ and also make use of homological nature of $\Delta_1$: this operator is divisible by $\partial\colon \ell^2(F)\to\ell^2(E)$ at $\ell_{0,c}^2(E)$. Finally, in Section \ref{sec:examples} we check the most natural examples of Cayley graphs.

\begin{lemma}[on loop embedding]
\label{lemma:embed}
Suppose that $\Gamma$ has subexponential growth and just one end.

Let  $C>20$, $x\ge 1$. There exist $\mathscr L>2DCx$ and injective naturally parametrized $\gamma\colon \mathbb T_{\mathscr L}\to \Cay(\Gamma)$ such that, for $t_1, t_2\in\mathbb T_{\mathscr L}$ with $\gamma(t_1), \gamma(t_2)\in \Gamma$,
\emph{
\begin{equation}
	\label{eq:x_bilip}
\mbox{if }\dist_{\Cay(\Gamma)}(\gamma(t_1), \gamma(t_2)) \le x
\mbox{ then }
\dist_{\mathbb T_{\mathscr L}}(t_1, t_2) \le Cx.
\end{equation}
}
\end{lemma}

\begin{theorem}[\cite{K59}]	\label{th:Kesten}
	Let $\Gamma$ be a finitely generated group. Then, for $0$-Laplacian in $\Cay(\Gamma)$,  $$0\in \spec\left(-\Delta_0|_{\ell^2(\Gamma)}\right)$$ if and only if $\Gamma$ is amenable.
\end{theorem}

\begin{theorem}
	\label{th:exp_growth}
	Let $\Gamma$ be a countable finitely presented group.
	If $\Delta_1$ has a spectral gap on 	$\ell_{0,c}^2(E)$ 	then either $\Gamma$ has exponential growth, 	 or $\Gamma$ is virtually infinite cyclic. \end{theorem}

Let us recall the well-known Kesten Theorem on Laplacian on vertices of a graph:

\begin{theorem}[\cite{K59}]	\label{th:Kesten}
	Let $\Gamma$ be a finitely generated group. Then, for $0$-Laplacian in $\Cay(\Gamma)$,  $$0\in \spec\left(-\Delta_0|_{\ell^2(\Gamma)}\right)$$ if and only if $\Gamma$ is amenable.
\end{theorem}

Non-amenability of a group, that is, the existence of a spectral gap for $-\Delta_0$, easily implies exponential growth. The reverse is not true, in general. Thus, it is natural to ask, for example, whether Baumslag--Solitar groups $\langle\mathscr a, \mathscr b\mid \mathscr b^{-1}\mathscr a\mathscr b=\mathscr a^n\rangle$, $n\in\mathbb N$, have a spectral gap for $1$-Laplacian. These groups are \emph{non-elementary amenable} but have exponential growth. Such groups are not covered by Theorem \ref{th:exp_growth}, and it is still unclear for the author whether $1$-Laplacian has a spectral gap on them.

If we assume the contrary to Theorem \ref{th:exp_growth}, then, first, $\Gamma$ cannot have two \emph{ends} since in this case $\Gamma$ is virtually cyclic, see, e.g., \cite{Me08}. Second, $\Gamma$ cannot have infinitely many ends because then $\Gamma$ has exponential growth; the latter follows from Stallings Theorem and from results of \cite{HB00} but, of course, can be proved directly. So, by Freudenthal--Hopf Theorem we may assume that $\Gamma$ has one end.

For $\mathscr L>0$, denote by $\mathbb T_{\mathscr{L}}$ a circle of length $\mathscr L$. On $\mathbb T_{\mathscr L}$, one may measure distances along this loop. To prove Theorem \ref{th:exp_growth}, we need the following

\begin{lemma}[on loop embedding]
\label{lemma:embed}
Suppose that $\Gamma$ has subexponential growth and just one end.

Let  $C>20$, $x\ge 1$. There exist $\mathscr L>2DCx$ and injective naturally parametrized $\gamma\colon \mathbb T_{\mathscr L}\to \Cay(\Gamma)$ such that, for $t_1, t_2\in\mathbb T_{\mathscr L}$ with $\gamma(t_1), \gamma(t_2)\in \Gamma$,
\emph{
\begin{equation}
	\label{eq:x_bilip}
\mbox{if }\dist_{\Cay(\Gamma)}(\gamma(t_1), \gamma(t_2)) \le x
\mbox{ then }
\dist_{\mathbb T_{\mathscr L}}(t_1, t_2) \le Cx.
\end{equation}
}
\end{lemma}

In fact, we are able to make $\mathscr L$ arbitrarily large with fixed $x$.

Now, let us briefly recall the proof of Kesten Theorem \ref{th:Kesten} to compare it to our argument. Non-amenability of $\Gamma$ means that
\begin{equation}
\label{eq:isoperimetr}
\|du\|_{\ell^1(E)} \gtrsim \|u\|_{\ell^1(\Gamma)}
\end{equation}
for $u=\mathds 1_E$, $E$ ranges all finite subsets in $\Gamma$. By discrete version of coarea formula, this is equivalent to the same for any finitely supported $u\colon \Gamma\to{\mathbb R}$.
The spectral gap condition $0\notin \spec\left(-\Delta_0|_{\ell^2(\Gamma)}\right)$ means that
$$
\langle u, u\rangle_{\ell^2(\Gamma)} \lesssim \langle du, du\rangle_{\ell^2(E)}, ~~~ u\colon\Gamma\to{\mathbb R} \mbox{ is finitely supported}.
$$
To obtain this from (\ref{eq:isoperimetr}), it remains to insert $u^2$ instead of $u$ to (\ref{eq:isoperimetr}) and apply Cauchy--Bunyakovsky--Schwartz inequality.

In the first step of the latter argument, we assemble a function $u\colon \Gamma\to{\mathbb R}$, say, non-negative one, from its super-level sets $\mathds 1_{\{u\ge \mathscr t\}}$, $\mathscr t$ ranges $[0,+\infty)$; we also assemble $du$ from $d\mathds 1_{\{u\ge \mathscr t\}}$. (Both decompositions are $\ell^1$-convex.) Thus, in Kesten Theorem, we deal with "sets of codimensions $0$ and $1$". At least, we will have such genuine codimensions in the case of a manifold instead of a group, the corresponding result linking spectra and isoperimetry is known as Cheeger--Yau inequality, see \cite{Cheeger}, \cite{Yau}.

Unlike this, in our argument we work with dimension $1$ sets --- loops, in particular, as in Lemma \ref{lemma:embed}. Also, in Section \ref{sec:spectra_homology} we bound $1$-cycles with $2$-dimensional surfaces.

Notice also that an analogue of Cheeger--Yau inequality for $1$-forms was obtained in \cite{BC22} in the case of manifolds. Coexact $1$-Laplacian spectrum is indeed related to appropriate isoperimetric ratio, namely, to $\sup_{\gamma}\inf_h|h|/\length\gamma$ with $\gamma$ ranging homologicaly trivial loops at a manifold $\mathscr M$ and $h$ be a $2$-dimensional chain in $\mathscr M$ bounding $\gamma$; here, $|h|$ is area of $h$. Some Poincar\'e-type estimates for operator $d$ on coclosed $1$-forms are  possible if isoperimetric ratios as above are bounded from the below. But, in \cite{BC22}, authors impose the condition of finite diameter of $\mathscr M$  which is not our case; also, \cite{BC22} does not deal with effects of negative curvature.

What concerns spaces with negative curvature, let us mention recent works \cite{AAGLZ24}, \cite{R23} devoted to $3$-dimensional hyperbolic manifolds. It turns out that, first, $1$-coexact spectral gap is related to exponential growth of torsion $1$-homology of the manifolds; second, there are relations  between the spectral gap and isoperimetric ratios. The latter estimates from \cite{R23} are also volume-dependent, as in \cite{BC22}.

\medskip

This paper is organized as follows. In Section \ref{sec:maueutic}, we explain our interest to the study of $1$-Laplacian spectra. This Section is not used in the proof of Theorem \ref{th:exp_growth}. In Section~\ref{sec:loop}, we prove Lemma \ref{lemma:embed} by dropping lots of geodesic perpendiculars in a branching way. In Section \ref{sec:spectra_homology}, we conclude the proof of Theorem~\ref{th:exp_growth}. This is done by approximating the resolvent $\Delta^{-1}_1$ by polynomials of $\Delta_1$ provided that $0\notin\spec\Delta_1|_{\ell_{0,c}^2(E)}$. Next, we apply this to $1$-cochain given by curve $\gamma$ from Lemma \ref{lemma:embed}. We put some metric control on the approximating process as implemented in $\Cay(\Gamma)$ and also  make use of homological nature of $\Delta_1$: this operator is divisible by $\partial\colon \ell^2(F)\to\ell^2(E)$ at $\ell_{0,c}^2(E)$.  Finally, in Section \ref{sec:examples} we check the most natural examples of Cayley graphs.

\medskip

\noindent {\bf Some notation.} For a set $A$ we denote by $\card A$ the number of its elements.
If $v_1, v_2$ are vertices of some oriented graph then we denote by $\edge(v_1, v_2)$ the oriented edge in the graph under consideration provided that the edge exists. If $e$ is an edge in a oriented graph or $\gamma$ is an oriented path in a metric space then we write $\beg e$, $\beg\gamma$ for their beginnings and $\ennd e$, $\ennd\gamma$ for their endpoints, respectively. The notation $\length\gamma$ is obvious.

We write $\mathcal B_X(x, \rho)$ for the open ball in a metric space $X$ centered in a point $x\in X$ and having radius $\rho\ge 0$.

Assuming the contrary to Theorem~\ref{th:exp_growth}, one first reduces the possibilities for the number of ends: $\Gamma$ "cannot have two \emph{ends} since in this case $\Gamma$ is virtually cyclic"; it also "cannot have infinitely many ends because then $\Gamma$ has exponential growth". Hence, "by Freudenthal--Hopf Theorem we may assume that $\Gamma$ has one end."

Under the hypothesis that $\Gamma$ has subexponential growth and one end, one proves Lemma~\ref{lemma:embed} ("on loop embedding"), producing for given $C>20$, $x\ge1$ an injective loop $\gamma\colon\mathbb T_{\mathscr L}\to\Cay(\Gamma)$ with $\mathscr L>2DCx$ and the metric control (\ref{eq:x_bilip}): if $\dist_{\Cay(\Gamma)}(\gamma(t_1),\gamma(t_2))\le x$ then $\dist_{\mathbb T_{\mathscr L}}(t_1,t_2)\le Cx$ (and "we are able to make $\mathscr L$ arbitrarily large with fixed $x$"). The lemma is proved "by dropping lots of geodesic perpendiculars in a branching way."

The proof of Theorem~\ref{th:exp_growth} is then completed in Section~\ref{sec:spectra_homology}: assuming a spectral gap, i.e. "$0\notin\spec\Delta_1|_{\ell_{0,c}^2(E)}$", one "approximat[es] the resolvent $\Delta^{-1}_1$ by polynomials of $\Delta_1$" and applies this to the "$1$-cochain given by curve $\gamma$ from Lemma~\ref{lemma:embed}$." One also "put[s] some metric control on the approximating process" in $\Cay(\Gamma)$ and uses the "homological nature of $\Delta_1$: this operator is divisible by $\partial\colon \ell^2(F)\to\ell^2(E)$ at $\ell_{0,c}^2(E)$", together with the fact that "we bound $1$-cycles with $2$-dimensional surfaces."

Assuming the contrary to the main theorem, one first reduces the possibilities for the number of ends: $\Gamma$ "cannot have two \emph{ends} since in this case $\Gamma$ is virtually cyclic"; it also "cannot have infinitely many ends because then $\Gamma$ has exponential growth". Hence, "by Freudenthal--Hopf Theorem we may assume that $\Gamma$ has one end."

Under the hypothesis that $\Gamma$ has subexponential growth and one end, one proves the following lemma.

\begin{lemma}[on loop embedding]
\label{lemma:embed}
Suppose that $\Gamma$ has subexponential growth and just one end.

Let  $C>20$, $x\ge 1$. There exist $\mathscr L>2DCx$ and injective naturally parametrized $\gamma\colon \mathbb T_{\mathscr L}\to \Cay(\Gamma)$ such that, for $t_1, t_2\in\mathbb T_{\mathscr L}$ with $\gamma(t_1), \gamma(t_2)\in \Gamma$,
\emph{
\begin{equation}
	\label{eq:x_bilip}
\mbox{if }\dist_{\Cay(\Gamma)}(\gamma(t_1), \gamma(t_2)) \le x
\mbox{ then }
\dist_{\mathbb T_{\mathscr L}}(t_1, t_2) \le Cx.
\end{equation}
}
\end{lemma}

This produces for given $C>20$, $x\ge1$ an injective loop $\gamma\colon\mathbb T_{\mathscr L}\to\Cay(\Gamma)$ with $\mathscr L>2DCx$ and the metric control given by the displayed equation above (and "we are able to make $\mathscr L$ arbitrarily large with fixed $x$"). The lemma is proved "by dropping lots of geodesic perpendiculars in a branching way."

In completing the proof of the main theorem, assuming a spectral gap, i.e. "$0\notin\spec\Delta_1|_{\ell_{0,c}^2(E)}$", one "approximat[es] the resolvent $\Delta^{-1}_1$ by polynomials of $\Delta_1$" and applies this to the "$1$-cochain given by curve $\gamma$ from the lemma above." One also "put[s] some metric control on the approximating process" in $\Cay(\Gamma)$ and uses the "homological nature of $\Delta_1$: this operator is divisible by $\partial\colon \ell^2(F)\to\ell^2(E)$ at $\ell_{0,c}^2(E)$", together with the fact that "we bound $1$-cycles with $2$-dimensional surfaces."

\label{th:exp_growth}
	Let $\Gamma$ be a countable finitely presented group.
	If $\Delta_1$ has a spectral gap on 	$\ell_{0,c}^2(E)$ 	then either $\Gamma$ has exponential growth,	 or $\Gamma$ is virtually infinite cyclic.,

Question:
Let \(\Gamma\) be a countable finitely presented group. Choose a finite generating set \(S\), let \(\operatorname{Cay}(\Gamma,S)\) be its Cayley graph with edge set \(E\), and form the simply connected 2-complex \(\operatorname{Cay}^{(2)}(\Gamma)\) by gluing oriented polygons corresponding to all translates of the defining relations. Let \(\Delta_1\) be the nonnegative Hodge--Laplace operator on edge functions determined by this 2-complex, and let
\[
\ell^2_{0,c}(E):=\overline{\{\text{finitely supported closed 1-cochains on }\operatorname{Cay}(\Gamma,S)\}}^{\,\ell^2(E)}.
\]
Assume that \(\Delta_1\) has a spectral gap on \(\ell^2_{0,c}(E)\), meaning that for some \(\varepsilon>0\),
\[
\operatorname{spec}\bigl(\Delta_1\big|_{\ell^2_{0,c}(E)}\bigr)\cap[0,\varepsilon)=\varnothing.
\]
Which conclusion about \(\Gamma\) holds?

Either \(\Gamma\) has exponential growth, or \(\Gamma\) is virtually infinite cyclic (that is, it has a finite-index subgroup isomorphic to \(\mathbb Z\)).

Choices:
B. Either \(\Gamma\) is nonamenable, or \(\Gamma\) is virtually infinite cyclic.
C. \(\Gamma\) is either of exponential growth or virtually cyclic.
D. If \(\Delta_1\) has a spectral gap on \(\ell^2_{0,c}(E)\), then \(\Gamma\) has exponential growth unless it is finite.
E. Either \(\Gamma\) has exponential growth, or \(\Gamma\) has exactly one end.

replacing growth conclusion by amenability-type alternative

replace virtually infinite cyclic by weaker virtually cyclic

exceptional virtually-Z case replaced by finite-case exception

one-end reduction under contradiction promoted to final alternative

The stem gives only the hypothesis and setup; it does not explicitly state or strongly hint at the specific conclusion "exponential growth or virtually infinite cyclic."

This is essentially a theorem-recall/application item: given a highly specific hypothesis, the student is asked for the resulting conclusion. The options do create some contrast, but the item remains close to a direct theorem restatement.

There is some reasoning pressure because one must distinguish the strongest correct conclusion from nearby alternatives, especially the weaker true option "virtually cyclic." However, for a prepared student the item is mostly recognition/recall rather than deep derivation.

The distractors are mathematically plausible and target realistic confusions: swapping in amenability/nonamenability, weakening "virtually infinite cyclic" to "virtually cyclic," mishandling the exceptional case, or replacing the conclusion with an ends statement.

A solid but theorem-centric MCQ: it avoids answer leakage and has strong distractors, but it mainly tests recognition of a specific result rather than substantial generative reasoning.


The fourth stable homotopy group of spheres is trivial, meaning that $\pi_{n+4}(S^n)=0$ for all $n>5$. In contrast to this, it is a theorem that $S^m$ has no trivial higher homotopy groups when $m\in\{2,3,4,5\}$, as we will briefly explain.

Curtis proved that $\pi_n(S^4)\neq 0$ for all $n\geq 4$ \cite{Cur69}. Curtis also proved that $\pi_n(S^2)\neq 0$ for all $n\not\equiv 1\mod 8$. These same results were obtained (via different methods) by Mimura, Mori, and Oda \cite{MMO75}. The proof that $\pi_n(S^5)\neq 0$ for all $n\geq 5$ was given by Mori \cite{Mor75} and Mahowald \cite{Mah75,Mah82}.

Since $\pi_n(S^2)\cong\pi_n(S^3)$ for all $n\geq 3$, the remaining case was $\pi_n(S^3)$ with $n\equiv 1\mod 8$. This last case was proved by Gray \cite{Gra84}, and later by Ivanov, Mikhailov, and Wu \cite{IMW16} using different methods. In \emph{op.~cit.}, the authors note that \cite[Theorem 5.2(ii)]{Tod66} implies that
\[\mathbb{Z}/p\subseteq\pi_{2(p-1)k+1}(S^3)\]
whenever $\gcd(p,k)=1$ \cite[p.~342, Equation (B)]{IMW16}. It follows that if every positive integer $n$ admits an odd prime $p$ and an integer $k$ such that $\gcd(p,k)=1$ and $4n=(p-1)k$, then $\pi_n(S^3)\neq 0$ for all $n\equiv 1\mod 8$. This leads one to the following definition.

\begin{defn}
    Let $n$ be an integer. A \emph{Toda prime} of $n$ is an odd prime $p$ such that $p-1\mid 4n$ and $\gcd(p,\frac{4n}{p-1})=1$. Denote the set of Toda primes of $n$ by $T(n)$ (see \cref{table:toda sets}), and let $t(n):=|T(n)|$ (see \cref{fig:tn}).
\end{defn}

\begin{table}[t]
\centering
\caption{Sets of Toda primes}\label{table:toda sets}
\begin{tabular}{|r|l|}
\hline
$n$ & $T(n)$\\
\hline
1 & $3,5$\\
2 & $3,5$\\
3 & $5,7,13$\\
4 & $3,5,17$\\
5 & $3,11$\\
6 & $5,7,13$\\
7 & $3,5,29$\\
8 & $3,5,17$\\
9 & $5,7,13,19,37$\\
10 & $3,11,41$\\
\hline
\end{tabular}
\quad
\begin{tabular}{|r|l|}
\hline
$n$ & $T(n)$\\
\hline
11 & $3,5,23$\\
12 & $5,7,13,17$\\
13 & $3,5,53$\\
14 & $3,5,29$\\
15 & $7,11,13,31,61$\\
16 & $3,5,17$\\
17 & $3,5$\\
18 & $5,7,13,19,37,73$\\
19 & $3,5$\\
20 & $3,11,17,41$\\
\hline
\end{tabular}
\quad
\begin{tabular}{|r|l|}
\hline
$n$ & $T(n)$\\
\hline
21 & $5,13,29,43$\\
22 & $3,5,23,89$\\
23 & $3,5,47$\\
24 & $5,7,13,17,97$\\
25 & $3,11,101$\\
26 & $3,5,53$\\
27 & $5,7,13,19,37,109$\\
28 & $3,5,17,29,113$\\
29 & $3,5,59$\\
30 & $7,11,13,31,41,61$\\
\hline
\end{tabular}
\end{table}

If every positive integer has a Toda prime, then one can greatly simplify the proof that $\pi_n(S^3)\neq 0$ for $n\geq 3$. This was asked on MathOverflow \cite{MO} (and attributed to Roman Mikhailov) several years ago.

\begin{ques}[Mikhailov]\label{ques:at least one}
    Does every positive integer have a Toda prime?
\end{ques}

In fact, it appears that every positive integer has at least two Toda primes.

\begin{conj}\label{conj:at least two}
    If $n$ is a positive integer, then $t(n)\geq 2$.
\end{conj}

We tried to answer \cref{ques:at least one} in the affirmative, but our approach hits a snag. To turn our failed attempt into a theorem, we adopt the time-tested tradition of stating our snag as a conjecture (\cref{conj:main}). We will give some heuristic evidence for this conjecture in \cref{sec:heuristic}.

\begin{conj}\label{conj:main}
    Let $n$ be an odd, square-free multiple of 3. Assume that there exists $p\in\{5,7,13\}$ such that $p\mid n$, and that $r\nmid n$ for $r\in\{5,7,13\}-\{p\}$. Finally, assume there exists $q\in T(3p)-\{5,7,13\}$ such that $q\nmid n$. Then $t(n)\geq 4$.
\end{conj}

Since $\pi_n(S^2)\cong\pi_n(S^3)$ for all $n\geq 3$, the remaining case was $\pi_n(S^3)$ with $n\equiv 1\mod 8$. This last case was proved by Gray \cite{Gra84}, and later by Ivanov, Mikhailov, and Wu \cite{IMW16} using different methods. In \emph{op.~cit.}, the authors note that \cite[Theorem 5.2(ii)]{Tod66} implies that
\[\mathbb{Z}/p\subseteq\pi_{2(p-1)k+1}(S^3)\]
whenever $\gcd(p,k)=1$ \cite[p.~342, Equation (B)]{IMW16}. It follows that if every positive integer $n$ admits an odd prime $p$ and an integer $k$ such that $\gcd(p,k)=1$ and $4n=(p-1)k$, then $\pi_n(S^3)\neq 0$ for all $n\equiv 1\mod 8$. This leads one to the following definition.

\begin{defn}
 Let $n$ be an integer. A \emph{Toda prime} of $n$ is an odd prime $p$ such that $p-1\mid 4n$ and $\gcd(p,\frac{4n}{p-1})=1$. Denote the set of Toda primes of $n$ by $T(n)$ (see \cref{table:toda sets}), and let $t(n):=|T(n)|$ (see \cref{fig:tn}).
\end{defn}

If every positive integer has a Toda prime, then one can greatly simplify the proof that $\pi_n(S^3)\neq 0$ for $n\geq 3$. This was asked on MathOverflow \cite{MO} (and attributed to Roman Mikhailov) several years ago.

\begin{conj}\label{conj:at least two}
 If $n$ is a positive integer, then $t(n)\geq 2$.
\end{conj}

We tried to answer \cref{ques:at least one} in the affirmative, but our approach hits a snag. To turn our failed attempt into a theorem, we adopt the time-tested tradition of stating our snag as a conjecture (\cref{conj:main}). We will give some heuristic evidence for this conjecture in \cref{sec:heuristic}.

\begin{conj}\label{conj:main}
 Let $n$ be an odd, square-free multiple of 3. Assume that there exists $p\in\{5,7,13\}$ such that $p\mid n$, and that $r\nmid n$ for $r\in\{5,7,13\}-\{p\}$. Finally, assume there exists $q\in T(3p)-\{5,7,13\}$ such that $q\nmid n$. Then $t(n)\geq 4$.
\end{conj}

\begin{conj}\label{conj:main}
    Let $n$ be an odd, square-free multiple of 3. Assume that there exists $p\in\{5,7,13\}$ such that $p\mid n$, and that $r\nmid n$ for $r\in\{5,7,13\}-\{p\}$. Finally, assume there exists $q\in T(3p)-\{5,7,13\}$ such that $q\nmid n$. Then $t(n)\geq 4$.
\end{conj}

\begin{figure}[p]
    \includegraphics[width=.75\linewidth]{tn.png}
    \caption{$t(n)$ for $n\leq 100000$}\label{fig:tn}
\end{figure}

\begin{ques}[Mikhailov]\label{ques:at least one}
    Does every positive integer have a Toda prime?
\end{ques}

\begin{table}[t]
\centering
\caption{Sets of Toda primes}\label{table:toda sets}
\begin{tabular}{|r|l|}
\hline
$n$ & $T(n)$\\
\hline
1 & $3,5$\\
2 & $3,5$\\
3 & $5,7,13$\\
4 & $3,5,17$\\
5 & $3,11$\\
6 & $5,7,13$\\
7 & $3,5,29$\\
8 & $3,5,17$\\
9 & $5,7,13,19,37$\\
10 & $3,11,41$\\
\hline
\end{tabular}
\quad
\begin{tabular}{|r|l|}
\hline
$n$ & $T(n)$\\
\hline
11 & $3,5,23$\\
12 & $5,7,13,17$\\
13 & $3,5,53$\\
14 & $3,5,29$\\
15 & $7,11,13,31,61$\\
16 & $3,5,17$\\
17 & $3,5$\\
18 & $5,7,13,19,37,73$\\
19 & $3,5$\\
20 & $3,11,17,41$\\
\hline
\end{tabular}
\quad
\begin{tabular}{|r|l|}
\hline
$n$ & $T(n)$\\
\hline
21 & $5,13,29,43$\\
22 & $3,5,23,89$\\
23 & $3,5,47$\\
24 & $5,7,13,17,97$\\
25 & $3,11,101$\\
26 & $3,5,53$\\
27 & $5,7,13,19,37,109$\\
28 & $3,5,17,29,113$\\
29 & $3,5,59$\\
30 & $7,11,13,31,41,61$\\
\hline
\end{tabular}
\end{table}

\begin{conj}\label{conj:main}
 Let $n$ be an odd, square-free multiple of 3. Assume that there exists $p\in\{5,7,13\}$ such that $p\mid n$, and that $r\nmid n$ for $r\in\{5,7,13\}-\{p\}$. Finally, assume there exists $q\in T(3p)-\{5,7,13\}$ such that $q\nmid n$. Then $t(n)\geq 4$.
\end{conj}

In \cref{sec:lemmas}, we state and prove a few simple lemmas. We prove \cref{thm:main} in \cref{sec:proof} by inducting on the number of odd prime factors of $n$. Essentially all of the real work happens in \cref{lem:t(n)=3}. We conclude with \cref{sec:observations}, where we pose a couple questions that arose while working on this project.

\begin{lem}\label{lem:not 5 or not 3 and 5}
 Let $p\in\{3,5\}$. If $p\nmid n$, then $p\in T(n)$. In particular:
 \begin{itemize}
 \item If $5\nmid n$, then $t(n)\geq 1$.
 \item If $3,5\nmid n$, then $t(n)\geq 2$.
 \end{itemize}
\end{lem}
\begin{proof}
 Let $p\in\{3,5\}$. Then $p-1\mid 4n$, and $\gcd(4n,p)=1$ by assumption. Thus $\gcd(\frac{4n}{p-1},p)=1$, so $p\in T(n)$.
\end{proof}

\begin{lem}\label{lem:t(n)=3}
 Assume \cref{conj:main}. Let $n$ be an odd, square-free multiple of 3. Then $t(n)\geq 3$. Moreover, $t(n)=3$ if and only if $T(n)=\{5,7,13\}$.
\end{lem}
\begin{proof}
 We will induct on $\omega(n)$. Our base cases will consist of $\omega(n)\leq 4$. Note that if $p\nmid n$ for each $p\in\{5,7,13\}$, then $T(n)\supseteq\{5,7,13\}$. In particular, we may restrict our attention to multiples of these three primes. Moreover, $t(ap)\geq t(a)$ for any prime $p\not\in T(a)$ by \cref{cor:multiply by prime}, so we may assume that every prime factor $p\mid n$ is a Toda prime of some divisor of $n$.
 \begin{itemize}
 \item The case of $\omega(n)=1$ is just the calculation $T(3)=\{5,7,13\}$.
 \item For $\omega(n)=2$, we just need to compute $t(15)=t(39)=5$ and $t(21)=4$.
 \item For $\omega(n)=3$, we first compute $t(3\cdot 5\cdot 7)=9$ and $t(3\cdot 5\cdot 13)=t(3\cdot 7\cdot 13)=8$. It remains to compute, for each $p\in\{5,7,13\}$, the Toda primes of $3pq$ for each $q\in T(3p)$. Using the code provided in \cref{sec:code}, we find that $t(3pq)\geq 4$ for all such $p,q$.
 \item For $\omega(n)=4$, we first compute $t(3\cdot 5\cdot 7\cdot 13)=16$. For the remaining computations in this case, we use the code in \cref{sec:code}.
 \begin{itemize}
 \item If $\{p,q\}\subseteq\{5,7,13\}$ and $r\in T(3pq)$, then $t(3pqr)\geq 9$.
 \item If $p\in\{5,7,13\}$ and $\{q,r\}\subseteq T(3p)$, then $t(3pqr)\geq 7$.
 \item If $p\in\{5,7,13\}$, $q\in T(3p)$, and $r\in T(3pq)$, then $t(3pqr)\geq 5$.
 \end{itemize}
 \end{itemize}

\begin{cor}\label{prop:divisible by 3}
 Assume \cref{conj:main}. If $3\mid n$, then $t(n)\geq 3$.
\end{cor}
\begin{proof}
 By \cref{cor:multiply by divisor}, we may assume that $n$ is square-free. By \cref{cor:odd}, we may further assume that $n$ is odd. The result now follows from \cref{lem:t(n)=3}.
\end{proof}

Firstly, if there exists $q'\in T(3p)-\{5,7,13,q\}$ such that $q'\nmid n$, then $\{5,7,13,q,q'\}-\{p\}\subseteq T(n)$, and we are done. In fact, the Toda primes of $n$ are precisely those primes among $\{2d+1:d\mid 2n\}$ that are not factors of $n$. Thus if
\begin{equation}\label{eq:heuristic}
\{2d+1\text{ prime}:d\mid 2n\}-(\Omega(n)\cup\{5,7,13,q\})
\end{equation}
is non-empty, then $t(n)\geq 4$. Our heuristic for \cref{conj:main} is that the set $\{2d+1:d\mid 2n\}$ consists of $2^{\omega(n)+1}$ elements, while $\Omega(n)\cup\{5,7,13,q\}$ consists of $\omega(n)+3$ elements.

\begin{prop}\label{prop:t(p)}
Assume $p\geq 7$ is a prime. Let $\vphi$ denote the totient function. If $\vphi(x)=4p$ for some integer $x$, then $T(p)=\{3,5,2p+1\}$ or $\{3,5,4p+1\}$. Otherwise, $T(p)=\{3,5\}$.
\end{prop}
\begin{proof}
 One can directly check that $3,5\in T(p)$ for all primes greater than 5. Now by Euler's product formula, we have $\vphi(x)=p_1^{e_1-1}(p_1-1)\cdots p_m^{e_m-1}(p_m-1)$, where $x=\prod_{i=1}^m p_i^{e_i}$ is the prime factorization of $x$. It follows that there exists $x$ such that $\vphi(x)=4p$ if and only if one of the following cases holds:
 \begin{enumerate}[(i)]
 \item $x=2^2\cdot q$, where $q$ is an odd prime such that $q-1=2p$. In this case, $q$ is a Toda prime of $p$ with $\frac{4p}{q-1}=2$.
 \item $x=2^r\cdot 3\cdot q$, where $r\in\{0,1\}$ and $q$ is an odd prime such that $q-1=2p$. In this case, $q$ is a Toda prime of $p$ with $\frac{4p}{q-1}=2$.
 \item $x=2^r\cdot q$, where $r\in\{0,1\}$ and $q$ is an odd prime such that $q-1=4p$. In this case, $q$ is a Toda prime of $p$ with $\frac{4p}{q-1}=1$.
 \item $x=2^r\cdot 5^2$, where $r\in\{0,1\}$ (in which case $p=5$). This case is not relevant for this lemma, as we have assumed $p\geq 7$.
 \end{enumerate}
 It remains to show that no other primes can be the Toda prime of $p$. To this end, let $q>5$ be a Toda prime of $p$. Then $q-1\mid 4p$, so we either have $q-1=4p$ or $q-1=2p$ (as $q-1$ is even and $p$ is odd). The existence of such a $q$ gives us a solution to $\vphi(x)=4p$ as outlined in cases (i), (ii), and (iii).

\begin{lem}\label{lem:strategy for denoms}
 Let $d$ be a Bernoulli denominator with $F(d)=4a$ for some integer $a$. If $\{2pi+1:i\mid 2a\}$ contains a prime number for each $p\in T(a)$, then \cref{conj:general bernoulli} \eqref{conj:toda for bernoulli} holds for this Bernoulli denominator.
\end{lem}
\begin{proof}
 We know that $p-1\mid 4a$ with $\gcd(\frac{4a}{p-1},p)=1$ for all $p\in T(a)$. Thus $p-1\mid 4am$, and we have $\gcd(\frac{4am}{p-1},p)=1$ if and only if $p\nmid m$. It therefore suffices to show that if $p\mid m$ for some $p\in T(a)$, then $D_{4am}>D_{4a}$.

\begin{conj}\label{conj:main}
    Let $n$ be an odd, square-free multiple of 3. Assume that there exists $p\in\{5,7,13\}$ such that $p\mid n$, and that $r\nmid n$ for $r\in\{5,7,13\}-\{p\}$. Finally, assume there exists $q\in T(3p)-\{5,7,13\}$ such that $q\nmid n$. Then $t(n)\geq 4$.
\end{conj}

\begin{lem}\label{lem:t(n)=3}
    Assume \cref{conj:main}. Let $n$ be an odd, square-free multiple of 3. Then $t(n)\geq 3$. Moreover, $t(n)=3$ if and only if $T(n)=\{5,7,13\}$.
\end{lem}

\begin{thm}\label{thm:main}
    Assume \cref{conj:main}. If $n$ is a positive integer, then $t(n)\geq 1$. If $5\nmid n$, then $t(n)\geq 2$. If $3\mid n$, then $t(n)\geq 3$.
\end{thm}

In \cref{sec:lemmas}, we state and prove a few simple lemmas. We prove \cref{thm:main} in \cref{sec:proof} by inducting on the number of odd prime factors of $n$. Essentially all of the real work happens in \cref{lem:t(n)=3}. We conclude with \cref{sec:observations}, where we pose a couple questions that arose while working on this project.

\subsection*{Acknowledgements}
We thank Nick Andersen, Pace Nielsen, and Kyle Pratt for helpful conversations. The author was partially supported by the NSF (DMS-2502365) and the Simons Foundation.

\label{thm:main}
Assume
\begin{conj}\label{conj:main}
    Let $n$ be an odd, square-free multiple of 3. Assume that there exists $p\in\{5,7,13\}$ such that $p\mid n$, and that $r\nmid n$ for $r\in\{5,7,13\}-\{p\}$. Finally, assume there exists $q\in T(3p)-\{5,7,13\}$ such that $q\nmid n$. Then $t(n)\geq 4$.
\end{conj}
If $n$ is a positive integer, then $t(n)\geq 1$. If $5\nmid n$, then $t(n)\geq 2$. If $3\mid n$, then $t(n)\geq 3$.,

Question: For a positive integer n, call an odd prime p a Toda prime of n if p-1 divides 4n and \(\gcd\!\left(p,\frac{4n}{p-1}\right)=1\). Let \(T(n)\) be the set of Toda primes of n, and let \(t(n)=|T(n)|\). Assume the following hypothesis holds universally: whenever m is an odd, square-free multiple of 3, there exists \(p\in\{5,7,13\}\) such that \(p\mid m\) and \(r\nmid m\) for every \(r\in\{5,7,13\}\setminus\{p\}\), and there exists \(q\in T(3p)\setminus\{5,7,13\}\) with \(q\nmid m\), then \(t(m)\ge 4\). Under this assumption, which quantitative estimate holds for Toda primes of a positive integer n?

For every positive integer n, \(t(n)\ge 1\); moreover, if \(5\nmid n\), then \(t(n)\ge 2\); and if \(3\mid n\), then \(t(n)\ge 3\).

Choices:
B. For every positive integer \(n\), \(t(n)\ge 1\); moreover, if \(3\nmid n\), then \(t(n)\ge 2\); and if \(5\mid n\), then \(t(n)\ge 3\).
C. For every positive integer \(n\), \(t(n)\ge 1\); moreover, if \(5\nmid n\), then \(t(n)\ge 2\).
D. For every positive integer \(n\), \(t(n)\ge 2\); moreover, if \(5\nmid n\), then \(t(n)\ge 3\); and if \(3\mid n\), then \(t(n)\ge 4\).
E. For every positive integer \(n\), \(t(n)\ge 1\); moreover, if \(5\nmid n\), then \(t(n)\ge 2\); and if \(n\) is an odd, square-free multiple of \(3\), then \(t(n)\ge 4\).

special-role of the primes 3 and 5 in the baseline bounds

dropped the final clause requiring \(3\mid n\Rightarrow t(n)\ge 3\)

numerical lower bounds 1/2/3 replaced by stronger 2/3/4

scope of the \(t(m)\ge 4\) hypothesis enlarged by dropping the auxiliary existence conditions on \(p\) and \(q\)

The stem does not explicitly state the target estimate, and there is no direct textual giveaway of choice A. The hypothesis is technical, but it does not itself reveal the final 1/2/3 lower-bound pattern.

The item appears to ask for the theorem-level conclusion under a stated hypothesis, so it is close to theorem recall rather than a fully independent problem. Still, the answer choices differ by quantifiers, exceptional primes, and bound strength, so it is not a pure verbatim restatement.

There is some reasoning pressure in comparing nearby quantitative claims and tracking how the special roles of 3 and 5 affect the bounds. However, the stem is too specialized to support much derivation from first principles, so the task leans more toward recognizing the intended conclusion than generating it.

The distractors are plausible and well targeted: one swaps the roles of 3 and 5, one gives a weaker true-looking statement, one overstates the bounds, and one improperly broadens the hypothesis. These reflect realistic mathematical failure modes involving quantifiers and strength of conclusions.

A reasonably strong MCQ with little answer leakage and high-quality distractors, but it is somewhat theorem-recall driven and only moderately tests genuine generative reasoning.