\begin{equation}
    \label{eq:lawbicolour}
    \mathbb{P} ( (T, L) = (\mathfrak{t}, \boldsymbol{\ell})) \; \propto \;  x_1^{\#\mathfrak{f}_1} x_2^{\#\mathfrak{f}_2} n^{\#\boldsymbol{\ell}},
\end{equation}

\begin{Thm}[Expression and asymptotics for the partition function]
\label{thm:F_ell_main}
We have the exact expression
	\[
	F_\ell
	=
	\int_{\gamma_-}^{\gamma_+}
	\rho(y) y^{\ell} \mathrm{d}y,
    \quad \ell\geq 1,
	\]
where
    \[
    \rho(y) =
    \frac{2^{-\theta - 1/2}}{\pi\theta(\gamma_+\!-\gamma_-)}  (\gamma_+-y)^{1-\theta} \bigg( \Big( \sqrt{2\gamma_+ - \gamma_- -y} + \sqrt{y-\gamma_-}\Big)^{2\theta} - \Big( \sqrt{2\gamma_+ - \gamma_- -y} - \sqrt{y-\gamma_-}\Big)^{2\theta}\bigg),
    \]
    and
    \[
\gamma_+ = 2^{3/2} \cos\left( \frac{\pi\theta}{2}\right) \quad \text{and} \quad \gamma_+-\gamma_- = \frac{2^{3/2}}{\theta} \sin\left(\frac{\pi\theta}{2}\right).
\]
\end{Thm}

\begin{equation}\label{eq:link_FK_Z}
\Big(\frac{1-p}{16}\Big)^{\frac{\#F(\tfrak)+\ell-1}{2}}
\Big(\frac{2p}{1-p}\Big)^{\#\boldsymbol{\ell}+1}
=
n x_c^{\ell+1} x_c^{\#F(\tfrak)-1} n^{\#\boldsymbol{\ell}}
=
n x_c^{\ell+1} Z(\tfrak,\boldsymbol{\ell},x_c,n),
\end{equation}
where $n$ and $x_c$ are as in~\eqref{eq:x_c_formula}.

\subsubsection{The gasket decomposition}
\label{sss:gasket_dec}
The gasket decomposition, introduced in \cite{BBG12a} and further developed in \cite{BBG12}, provides a remarkably effective framework for analysing a broad class of loop models.
In what follows we adapt the presentation of \cite{BBG12} to the particular fully packed setting determined by \eqref{eq: def weight triangulation} and \eqref{eq: def partition triangulation}.

Consider a configuration $(\tfrak,\boldsymbol{\ell})\in \mathbb{T}_\ell$.
To construct its gasket, we first look at the edges of $\tfrak$ that are reachable from the boundary without crossing any loop.
Because the loops are fully packed, these accessible edges divide the map into exactly $\#\boldsymbol{\ell}+1$ faces: the external face, with boundary length~$\ell$, and an additional face of degree~$k$ for each loop of (outer) perimeter~$k$.
The resulting object is by definition the \textbf{gasket}.
Now examine one of the internal faces of degree~$k$ created by the gasket.

\begin{figure}[b!]
  \bigskip
  \begin{center}
    \includegraphics[width=0.5\textwidth]{Ring-partition-function.pdf}
  \end{center}
  \caption{The ring partition function $A^{(1\to 2)}_{k,k'}$ from colour 1 (blue) to colour 2 (red). It accounts for the weight of all the triangles crossed by the purple loop.}
  \label{fig:ring_part}
\end{figure}

If we reinsert the triangles traversed by the corresponding loop, we obtain a ring of triangles whose outer boundary length is~$k$ and whose inner boundary has some length~$k'$.
The ring consists of $k+k'$ triangles in total.
Moreover, the loop-decorated triangulation originally contained inside this loop -- call it $(\tfrak',\boldsymbol{\ell}')\in\mathbb{T}_{k'}$ -- is precisely what was taken out from $(\tfrak,\boldsymbol{\ell})$ to form the corresponding face of the gasket.
In this way, $(\tfrak,\boldsymbol{\ell})$ decomposes into:
(i) the gasket;
(ii) one ring of triangles for each internal gasket face; and
(iii) a fully packed loop-decorated triangulation attached along the inner boundary of each such ring.
The contribution of such a component to the weight $Z(\tfrak',\boldsymbol{\ell}',x,n)$ in \eqref{eq: def weight triangulation} is $n x^{k + k'}Z(\tfrak, \boldsymbol{\ell}, x, n)$,
where $k$ is the outer and $k'$ the inner boundary length of the ring.
This motivates the definition
\begin{equation}\label{eq: gasket decomposition}
g_k := n\sum_{k'=0}^{\infty} A_{k\rightarrow k'} x^{k+k'} F_{k'},
\end{equation}

\begin{Prop}[From hitting times to the partition function]\label{prop:prob_tau}
	There exists a normalising constant $C>0$ such that, for all $\ell\geq 0$,
	\[
	\mathbb{P}(\tau^{\rh} = \ell+1) = C (2x_c)^{\ell+1}F_{\ell}.
	\]
\end{Prop}

\label{sec:intro}

Planar maps (i.e., proper embeddings of graphs in the $2$-sphere, considered up to orientation-preserving homeomorphisms) are a central topic not only in combinatorics but also in probability and mathematical physics due to their conjectured links with Liouville quantum gravity (LQG). In that context, planar maps can be thought of as canonical discretisations of the ``random surfaces'' which are at the core of Polyakov's original formulation of LQG \cite{polyakov1981quantum}. Indeed, in order to describe the gravitational action when gravity is coupled to a matter field, the so-called DDK Ansatz (\cite{David, DistlerKawai}) implies that this can be equivalently described by considering the scaling limit of random planar maps decorated by models of statistical mechanics at their critical point.

In this paper we will consider two such models, which turn out to be bijectively related. We start with the Fortuin--Kasteleyn percolation model with parameter $q\in (0,4)$.
In this model, we sample a pair $(M, \Omega)$ where $M$ is a (rooted) planar map with a fixed number of edges (say $k$ edges), and $\Omega$ is a subset of edges of $M$. In general, the probability of sampling a given decorated map $(\mathfrak m, \omega)$
 depends on an extra percolation parameter $p_0 \in (0,1)$ and a volume weight $s$ and is proportional to
\begin{equation}\label{eq:lawgen}
\bb P ((M, \Omega) = (\mathfrak m, \omega)) \,  \propto  \,
\left(\frac{p_0}{1-p_0}\right)^{\r o(\omega)} q^{\r{cc}(\omega)} s^{\r v(\mathfrak m)},
\end{equation}
where $\r o(\omega)$ and $\r{cc}(\omega)$ are the number of edges and connected components of $\omega$, and $\mathsf{v} (\mathfrak m)$ is the number of vertices of $\mathfrak{m}$.
In particular, conditional on the planar map $M = \mathfrak{m}$, the edge configuration $\Omega$ is sampled as an  FK$(q)$-percolation configuration on $\mathfrak{m}$ (see, e.g., \cite{duminil2013parafermionic} for a general introduction to this model). On such a decorated planar map $( \mathfrak m, \omega)$ there is a natural duality operation $( \mathfrak m, \omega) \to ( \mathfrak m^\dagger, \omega^\dagger)$. Requiring that the law \eqref{eq:lawgen} is invariant under this duality imposes that
 $p_0 = \sqrt{q}/(1+\sqrt{q})$ and $s=1/\sqrt{q}$, as can be checked using Euler's formula.
In that case the law \eqref{eq:lawgen} reduces to
\begin{equation}\label{eq:lawSD}
\bb P ((M, \Omega) = (\mathfrak m, \omega)) \,  \propto  \,
\sqrt{q}^{\# \mathsf{loops}(\mathfrak m, \omega)},
\end{equation}
where $\mathsf{loops}(\mathfrak m, \omega)$ is the set of loops separating $\omega$ and its dual in $\mathfrak m$. This is called the \textbf{self-dual} FK$(q)$-weighted planar map model.
{
In the self-dual case, one can also let the number of edges $k \to \infty$ to obtain, as a local limit of the above, an infinite FK$(q)$-weighted planar map model~\cite{sheffield2016quantum,chen2017basic}.
}

The second model we will discuss is that of the fully packed loop $O(n)$ model on planar triangulations.
In this model we sample a triangulation $T$, together with a fully packed  configuration $L$ of loops, with probability
\begin{equation}
    \label{eq:lawbicolour_sd}
    \mathbb{P} ((T, L) = ( \mathfrak{t}, \boldsymbol{\ell}) ) \; \propto \; x^{\# \mathsf{faces} (\mathfrak{t})} n^{\# \boldsymbol{\ell}}.
\end{equation}
Up to considering the dual map (one may view the loops in $\boldsymbol{\ell}$ as crossing the triangles of $\mathfrak{t}$ or as a subset of edges of the dual map $\mathfrak{t}^\dagger$ of degree two), this is the classical loop $O(n)$ model on planar triangulations, introduced in \cite{DMNS81}. In fact, it is convenient to view the model \eqref{eq:lawbicolour_sd} as the symmetric (i.e., ``self-dual'') specialisation of a more general model where loops are assigned one of two possible colours, and there is a weight $x_1, x_2$ for each face crossed by a loop of colour $i = 1,2$. When $x_1 = x_2$ (the ``self-dual'' case) then this reduces to \eqref{eq:lawbicolour_sd}. This model, introduced in \cite{BBG12a}, is called the twofold loop $O(n)$ model or bicoloured loop $O(n)$ model; see \eqref{eq:lawbicolour} for its precise definition and Section \ref{sec:O(n)_model} for more explanations.

As is well known, and as will be recalled in Section \ref{sec:O(n)_model}, the two models \eqref{eq:lawgen} and \eqref{eq:lawbicolour} (and their self-dual versions, \eqref{eq:lawSD} and \eqref{eq:lawbicolour_sd}) are in measure-preserving bijection with one another, provided that
\begin{equation}\label{eq:xc}
n = \sqrt{q} \quad ; \quad x = x_c=\frac{1}{ \sqrt{8(n+2)}} = \frac{1}{\sqrt{8( \sqrt{q}+2)}}.
\end{equation}

\medskip In many models of planar statistical mechanics, it is widely expected that the self-dual point coincides with its critical point (see for instance \cite{grimmett2018probability} for some background discussion).
In particular, such a statement is the content of a celebrated result of Beffara and Duminil-Copin \cite{beffara2012self} for the random cluster model (equivalently the Fortuin--Kasteleyn percolation model) on the square lattice and for $q\ge 1$. For the related loop $O(n)$ model on the hexagonal lattice, the equality between critical and self-dual point led Nienhuis to predict (\cite{Nienhuis82, Nienhuis84}) that the critical point occurs at $x = x_c = 1/ \sqrt{2 + \sqrt{2-n}}$ for $n \in (0,2]$, and in fact $n \in [-2, 2]$; this remains a famous open problem in this field (see \cite{DCensaios} for a thorough survey).  This should also apply to the two self-dual models planar maps discussed above, \eqref{eq:lawSD} and \eqref{eq:lawbicolour_sd}. This suggests that this model is also \textbf{critical}, although in a sense that needs to be carefully specified.

Furthermore, one of the principal conjectures in the field -- closely related to the DDK ansatz mentioned above -- states that, when suitably renormalised and conformally embedded into the Riemann sphere, self-dual FK$(q)$-weighted planar maps converge to a $\gamma$-LQG surface, where
\[
q = 2 + 2\cos\left(\frac{\pi \gamma^2}{2}\right)
\quad
\text{and}
\quad
\gamma\in(\sqrt{2},2).
\]
In addition, the loops separating primal and dual clusters of $\Omega$ are conjectured to converge jointly with the map to an independent Conformal Loop Ensemble (CLE) with parameter $\kappa' = 16/\gamma^2$.

This is supported by a considerable body of evidence. On the one hand, in the breakthrough paper \cite{sheffield2016quantum}, Sheffield provided a measure-preserving bijection between self-dual FK-weighted planar maps \eqref{eq:lawSD} and inventory accumulations, which in turn correspond to a pair of (non-Markovian) walks, and showed that in the scaling limit this pair of walks converges to a pair of correlated Brownian motions. In combination with the so-called Mating of Trees framework developed by Duplantier, Miller and Sheffield \cite{duplantier2021liouville} this result can be re-interpreted as a form of convergence towards a $\gamma$-LQG surface decorated with an independent space-filling SLE$_{\kappa'}$ curve, albeit for a relatively weak topology (the so-called ``peanosphere'' topology). Building on this foundational work, a number of observables (such as sizes of clusters and their boundaries) have been analysed and the associated exponents computed (see \cite{gwynne2019scaling,gwynne2017scaling,gwynne2015scaling} and \cite{berestycki2017critical}). These values are consistent with those that can be predicted by combining known results on the dimension of SLE$_\kappa$ (\cite{beffara2008dimension}) with the Knizhnik--Polyakov--Zamolodchikov (KPZ) identity, cf.\ \cite{duplantier2021liouville, HKPZ}. See, e.g., \cite[Chapter 4]{BP} for a discussion of these results and additional perspective.

On the other hand,
a classical approach to the loop $O(n)$ model is to make use of the spatial Markov property of the model. This results in the so-called \emph{gasket decomposition} which was in particular used by Borot, Bouttier and Guitter \cite{BBG12, BBG12a, BBG12b} {to establish the phase diagram of the loop-$O(n)$ model}. Building on powerful tools of analytic combinatorics they derived and solved (assuming a certain natural ansatz) an equation for the resolvent of the partition function of the model, which yields fine asymptotics. Such an approach has been made fully rigorous (including the proof of the above ansatz) by works of Budd and Chen \cite{budd2019peeling}, in the case of the so-called \emph{rigid} model on quadrangulations (where loops are constrained to enter and leave a given quadrangle through opposite edges). From this, Chen, Curien and Maillard \cite{ChenCurienMaillard} deduced some scaling limit results for the perimeter cascade of $O(n)$ loops, and Aïdékon, Da Silva and Hu \cite{aidekon2024scaling} proved the scaling limit for the volume of such {rigid} loop-$O(n)$ quadrangulations. A similar approach was also used by Borot, Duplantier and Guitter \cite{BBD16} to determine the nesting statistics in the \emph{bending energy} variant of the $O(n)$ model, {where loops can bend inside a quadrangle at a given energetic cost.}

{However, we emphasise that the aforementioned ansatz (and thus, the phase diagram of the model) has so far only been rigorously established in the rigid case \cite{budd2019peeling}. The proof relies on special symmetries of bipartite maps and cannot be directly extended to the case of triangulations. We comment on the exact nature of the missing step in \cref{sss:gasket_dec}.}

\medskip One of the goals of this paper is to combine these two approaches. For instance, we obtain an exact relation between quantities naturally arising in Sheffield's bijection on the one hand, and partition functions for the loop-$O(n)$ model on the other (see  \cref{prop:prob_tau} for a precise statement). This works as a ``dictionary'' which allows us to translate results between the different points of view. As we will now see, this enables us to establish a number of consequences for both models. Our first main consequence is the complete proof ({including a proof of the above ansatz}) of an exact formula for the partition function $F_\ell$ of the fully packed loop-$O(n)$ model for a given boundary length $\ell \ge 1$, {defined as the sum of the loop-$O(n)$ weights over all triangulations with a boundary of length $\ell$.}
Here and in the rest of the paper, the expression $a_n \sim b_n$ means that  $a_n /b_n\to 1$ as $n\to \infty$.
Set
\begin{equation}
    \theta = \frac{1}{\pi}\arccos\Big(\frac{n}{2}\Big) = \frac1\pi \arccos\Big(\frac{\sqrt{q}}{2}\Big).
\end{equation}

In this paper we will consider two such models, which turn out to be bijectively related. We start with the Fortuin--Kasteleyn percolation model with parameter $q\in (0,4)$.
In this model, we sample a pair $(M, \Omega)$ where $M$ is a (rooted) planar map with a fixed number of edges (say $k$ edges), and $\Omega$ is a subset of edges of $M$. In general, the probability of sampling a given decorated map $(\mathfrak m, \omega)$
 depends on an extra percolation parameter $p_0 \in (0,1)$ and a volume weight $s$ and is proportional to
\begin{equation}\label{eq:lawgen}
\bb P ((M, \Omega) = (\mathfrak m, \omega)) \, \propto \,
\left(\frac{p_0}{1-p_0}\right)^{\r o(\omega)} q^{\r{cc}(\omega)} s^{\r v(\mathfrak m)},
\end{equation}
where $\r o(\omega)$ and $\r{cc}(\omega)$ are the number of edges and connected components of $\omega$, and $\mathsf{v} (\mathfrak m)$ is the number of vertices of $\mathfrak{m}$.
In particular, conditional on the planar map $M = \mathfrak{m}$, the edge configuration $\Omega$ is sampled as an FK$(q)$-percolation configuration on $\mathfrak{m}$ (see, e.g., \cite{duminil2013parafermionic} for a general introduction to this model). On such a decorated planar map $( \mathfrak m, \omega)$ there is a natural duality operation $( \mathfrak m, \omega) \to ( \mathfrak m^\dagger, \omega^\dagger)$. Requiring that the law \eqref{eq:lawgen} is invariant under this duality imposes that
 $p_0 = \sqrt{q}/(1+\sqrt{q})$ and $s=1/\sqrt{q}$, as can be checked using Euler's formula.
In that case the law \eqref{eq:lawgen} reduces to
\begin{equation}\label{eq:lawSD}
\bb P ((M, \Omega) = (\mathfrak m, \omega)) \, \propto \,
\sqrt{q}^{\# \mathsf{loops}(\mathfrak m, \omega)},
\end{equation}
where $\mathsf{loops}(\mathfrak m, \omega)$ is the set of loops separating $\omega$ and its dual in $\mathfrak m$. This is called the \textbf{self-dual} FK$(q)$-weighted planar map model.
{
In the self-dual case, one can also let the number of edges $k \to \infty$ to obtain, as a local limit of the above, an infinite FK$(q)$-weighted planar map model~\cite{sheffield2016quantum,chen2017basic}.
}

The second model we will discuss is that of the fully packed loop $O(n)$ model on planar triangulations.
In this model we sample a triangulation $T$, together with a fully packed configuration $L$ of loops, with probability
\begin{equation}
 \label{eq:lawbicolour_sd}
 \mathbb{P} ((T, L) = ( \mathfrak{t}, \boldsymbol{\ell}) ) \; \propto \; x^{\# \mathsf{faces} (\mathfrak{t})} n^{\# \boldsymbol{\ell}}.
\end{equation}
Up to considering the dual map (one may view the loops in $\boldsymbol{\ell}$ as crossing the triangles of $\mathfrak{t}$ or as a subset of edges of the dual map $\mathfrak{t}^\dagger$ of degree two), this is the classical loop $O(n)$ model on planar triangulations, introduced in \cite{DMNS81}. In fact, it is convenient to view the model \eqref{eq:lawbicolour_sd} as the symmetric (i.e., ``self-dual'') specialisation of a more general model where loops are assigned one of two possible colours, and there is a weight $x_1, x_2$ for each face crossed by a loop of colour $i = 1,2$. When $x_1 = x_2$ (the ``self-dual'' case) then this reduces to \eqref{eq:lawbicolour_sd}. This model, introduced in \cite{BBG12a}, is called the twofold loop $O(n)$ model or bicoloured loop $O(n)$ model; see \eqref{eq:lawbicolour} for its precise definition and Section \ref{sec:O(n)_model} for more explanations.

As is well known, and as will be recalled in Section \ref{sec:O(n)_model}, the two models \eqref{eq:lawgen} and \eqref{eq:lawbicolour} (and their self-dual versions, \eqref{eq:lawSD} and \eqref{eq:lawbicolour_sd}) are in measure-preserving bijection with one another, provided that
\begin{equation}\label{eq:xc}
n = \sqrt{q} \quad ; \quad x = x_c=\frac{1}{ \sqrt{8(n+2)}} = \frac{1}{\sqrt{8( \sqrt{q}+2)}}.
\end{equation}

Furthermore, one of the principal conjectures in the field -- closely related to the DDK ansatz mentioned above -- states that, when suitably renormalised and conformally embedded into the Riemann sphere, self-dual FK$(q)$-weighted planar maps converge to a $\gamma$-LQG surface, where
\[
q = 2 + 2\cos\left(\frac{\pi \gamma^2}{2}\right)
\quad
\text{and}
\quad
\gamma\in(\sqrt{2},2).
\]
In addition, the loops separating primal and dual clusters of $\Omega$ are conjectured to converge jointly with the map to an independent Conformal Loop Ensemble (CLE) with parameter $\kappa' = 16/\gamma^2$.

{However, we emphasise that the aforementioned ansatz (and thus, the phase diagram of the model) has so far only been rigorously established in the rigid case \cite{budd2019peeling}. The proof relies on special symmetries of bipartite maps and cannot be directly extended to the case of triangulations. We comment on the exact nature of the missing step in \cref{sss:gasket_dec}.}

\medskip One of the goals of this paper is to combine these two approaches. For instance, we obtain an exact relation between quantities naturally arising in Sheffield's bijection on the one hand, and partition functions for the loop-$O(n)$ model on the other (see \cref{prop:prob_tau} for a precise statement). This works as a ``dictionary'' which allows us to translate results between the different points of view. As we will now see, this enables us to establish a number of consequences for both models. Our first main consequence is the complete proof ({including a proof of the above ansatz}) of an exact formula for the partition function $F_\ell$ of the fully packed loop-$O(n)$ model for a given boundary length $\ell \ge 1$, {defined as the sum of the loop-$O(n)$ weights over all triangulations with a boundary of length $\ell$.}
Here and in the rest of the paper, the expression $a_n \sim b_n$ means that $a_n /b_n\to 1$ as $n\to \infty$.
Set
\begin{equation}
 \theta = \frac{1}{\pi}\arccos\Big(\frac{n}{2}\Big) = \frac1\pi \arccos\Big(\frac{\sqrt{q}}{2}\Big).
\end{equation}

In particular, we deduce the following asymptotics.
\begin{Cor} \label{cor:asymp_F_ell_main}
 The partition function satisfies:
 \begin{equation} \label{eq:asymp_F_ell_main}
 F_\ell
 \sim
 c \frac{\gamma_+^{\ell}}{\ell^{2-\theta}} \quad \text{as } \ell\to\infty,
 \end{equation}
 where $c =\frac{2^{\theta-1/2}}{\pi \theta} (\gamma_+-\gamma_-)^{\theta-1} \gamma_+^{1-\theta} \Gamma(2-\theta)$, and $\gamma_-$ and $\gamma_+$ are as above.
\end{Cor}

\noindent With \cref{lem:WH_K}, equation \eqref{eq: sd resolvent fourier} implies
\[
2\pi^2 \frac{R_+(\omega)}{K_+(\omega)} = \frac{F_+(\omega)}{K_-(\omega)} + \frac{F_-(\omega)}{K_-(\omega)} \quad \text{for } \omega\in\cal S \text{ such that } \Gamma\left( \frac{1-2\theta+i\omega}{4} \right) \neq \infty.
\]
We now multiply the latter display to remove the pole singularities of $F_+$: for $\omega\in\cal S$ such that $\Gamma\left( \frac{1-2\theta+i\omega}{4} \right) \neq \infty$,
\[
2\pi^2 \frac{R_+(\omega)}{K_+(\omega)} (\omega+i)(\omega+3i)= \left(\frac{F_+(\omega)}{K_-(\omega)} + \frac{F_-(\omega)}{K_-(\omega)}\right)(\omega+i)(\omega+3i).
\]
Now remark that the left-hand side is actually holomorphic on $\cal H_+$, whereas the right-hand side is holomorphic on $\{z\in\bb C, \Im(z)<1-2\theta\}$. We stress that, since $0<n<2$, we have $1-2\theta\in (0, 1)$. Therefore, both sides can be extended to an entire function $S$ in the complex plane, whence
\begin{equation} \label{eq: WH final 1}
R_+(\omega)
=
\frac{1}{2\pi^2} K_+(\omega) \frac{S(\omega)}{(\omega+i)(\omega+3i)}
\quad \text{for } \omega\in \cal H_+,
\end{equation}
and
\begin{equation} \label{eq: WH final 2}
F_+(\omega)+F_-(\omega)
=
K_-(\omega) \frac{S(\omega)}{(\omega+i)(\omega+3i)} \quad \text{for } \omega \text{ such that } \Im(\omega)<1-2\theta.
\end{equation}
We now analyse the behaviours at infinity in \eqref{eq: WH final 1} and \eqref{eq: WH final 2}.
{Since $r_+$ is integrable by Lemma \ref{lem:r(v)/v_int}, we have that $R_+(\omega) = O(1)$ as $|\omega| \to \infty$ with $\omega\in \mathcal{H}_+$. Likewise, writing $ \omega = x + {i} y$ with $y < 1$, we have by Lemma \ref{lem:r(v)/v_int},
\[
|F_-(\omega)| = \Big|\int_{-\infty}^0 f_-(u) \mathrm{e}^{ i \omega u} \mathrm{d} u \Big|\le \int_{-\infty}^0 O(\mathrm{e}^u) \mathrm{e}^{-yu}\mathrm{d} u.
\]
Thus we deduce that $F_- (\omega) = O(1)$ as $|\omega| \to \infty$ with $\Im(\omega)<1-2\theta$.
Combining this with the exact expressions \eqref{eq: expressions K_+ and K_-}, we see that
$S(\omega) = O(\omega)$ as $|\omega|\to\infty$. By Liouville's theorem, $S$ is a degree $1$ polynomial.
We can therefore write it as $S(\omega) = A\omega + B$ for some $A, B \in \mathbb{C}$. We will in fact show it is a constant: indeed, we can read the value of $A$ by taking $\omega\to +\infty$ along the real line in \eqref{eq: WH final 2}. By the Riemann--Lebesgue lemma, we have $F_-(\omega) \to 0$ and this forces $A=0$ (as $F_+ ( \omega) = O ( 1/|\omega|) \to 0$ as $\omega \to \infty$). We conclude that $S$ is actually a constant,
the value of which is determined by \eqref{eq: sd normalisation fourier}.} We get (using the well known identities for the $\Gamma$ function that $\Gamma (1 + x) = x \Gamma( x)$ for at least $x >0$ and $\Gamma(x) \Gamma(1-x) = \pi / \sin(\pi x)$ whenever $x \notin \mathbb{Z}$),
\begin{equation} \label{eq:S=cst}
S = -\frac{8\pi\sqrt{2}}{\theta} \sin\left( \frac{\pi\theta}{2}\right).
\end{equation}

\paragraph{Conclusion.}
We finally arrived at the following explicit expression for $R_+$: by \eqref{eq: WH final 1} and \eqref{eq:S=cst},
\begin{equation}\label{eq: R_+ explicit}
 R_+(\omega) =
 - \frac{4\sqrt{2}}{\pi \theta} \sin\left( \frac{\pi\theta}{2}\right) \frac{K_+(\omega)}{(\omega+i)(\omega+3i)}
 =
 - \frac{4\sqrt{2}}{\pi \theta} \sin\left( \frac{\pi\theta}{2}\right)
 \frac{\Gamma\left( \frac{3+2\theta-i\omega}{4} \right) \Gamma\left( \frac{3-2\theta-i\omega}{4} \right)}{2^{i\omega/2} \Gamma\left(\frac{1-i\omega}{2}\right) (\omega+i)(\omega+3i)}.
\end{equation}
On the other hand, one may now use \eqref{eq: WH final 2} to get the value of $(\gamma_-,\gamma_+)$. Indeed, $F_-$ is holomorphic on $\cal H_-$, while $F_+$ has poles at $-i$ and $-3i$. Thus, we can read off the coefficients $c_0$ and $c_1$ of \eqref{eq: expr_F(omega)} from the right-hand side of \eqref{eq: WH final 2}. We arrive at $c_0 = \frac{4}{\theta} \cos\left(\frac{\pi\theta}{2}\right)\sin\left(\frac{\pi\theta}{2}\right)$ and $c_1 = -\frac{4}{\theta} \sin^2\left(\frac{\pi \theta}{2}\right)$. Recalling the expression of $c_0, c_1$ in \cref{lem:fourier_WH}, this in turn implies that
\begin{equation}\label{eq:gamma+}
\gamma_+ = 2^{3/2} \cos\left( \frac{\pi\theta}{2}\right) \quad \text{and} \quad \gamma_+-\gamma_- = \frac{2^{3/2}}{\theta} \sin\left(\frac{\pi\theta}{2}\right).
\end{equation}
One then recovers $r_+$ as the inverse Fourier transform of the above \eqref{eq: R_+ explicit}:
\begin{equation}\label{eq:r+explicit}
 r_+(v)
 =
 \frac{\gamma_+-\gamma_-}{\sqrt{2}\pi\theta} \mathrm{e}^{-3v} \left( (\mathrm{e}^{2v}+\sqrt{\mathrm{e}^{4v}-1})^{\theta} - (\mathrm{e}^{2v}-\sqrt{\mathrm{e}^{4v}-1})^{\theta} \right).
\end{equation}
We will prove this identity in Appendix \ref{sec:appendix} (note that this differs from the analogous identity in \cite[(38)]{gaudin1989n}, though their final answer appears to be correct).
Undoing the change of variables, we conclude that for $y\in(\gamma_-,\gamma_+)$,
\begin{multline}
 \label{eq:rho_final_expr}
 \rho(y) = \\
 \frac{2^{-\theta - 1/2}}{\pi\theta(\gamma_+-\gamma_-)} (\gamma_+-y)^{1-\theta} \bigg( \Big( \sqrt{2\gamma_+ - \gamma_- -y} + \sqrt{y-\gamma_-}\Big)^{2\theta} - \Big( \sqrt{2\gamma_+ - \gamma_- -y} - \sqrt{y-\gamma_-}\Big)^{2\theta}\bigg).
\end{multline}
Note that, as expected,
 \[
 \rho(y)
 \sim
 c (\gamma_+ - y)^{1-\theta}, \quad \text{as } y\to (\gamma_+)^-.
 \]

Here we verify that the formula computing the inverse Fourier transform of $R_+$ in Section \ref{sec:resolvents} (crucial in the proof of Theorem \ref{thm:F_ell_main}) is as claimed in \eqref{eq:r+explicit}. Recall that
we know from the Wiener--Hopf argument that the Fourier transform $R_+ (\omega) = \textstyle\int_{\mathbb{R}} \mathrm{e}^{ i \omega v} r_+ (v) \mathrm{d} v$ satisfies
\begin{align*}
R_+(\omega) &=
 - \frac{4\sqrt{2}}{\pi \theta} \sin\left( \frac{\pi\theta}{2}\right)
 \frac{\Gamma\left( \frac{3+2\theta-i\omega}{4} \right) \Gamma\left( \frac{3-2\theta-i\omega}{4} \right)}{2^{i\omega/2} \Gamma\left(\frac{1-i\omega}{2}\right) (\omega+i)(\omega+3i)}\\
 & = \frac{\sqrt{2}}{\pi \theta} \sin\left( \frac{\pi\theta}{2}\right)
 \frac{\Gamma\left( \frac{3+2\theta-i\omega}{4} \right) \Gamma\left( \frac{3-2\theta-i\omega}{4} \right)}{2^{i\omega/2}\Gamma ( \tfrac{5 - i \omega}{2})}.
\end{align*}
where we used properties of the Gamma function to simplify the denominator of the fraction.

It is easier to start from the answer, i.e., to consider the function
$$
s_+ (v) : = \frac{(\gamma_+-\gamma_-)}{\sqrt{2} \pi\theta} \mathrm{e}^{-3v} \left( (\mathrm{e}^{2v}+\sqrt{\mathrm{e}^{4v}-1})^{\theta} - (\mathrm{e}^{2v}-\sqrt{\mathrm{e}^{4v}-1})^{\theta} \right), \quad v >0,
$$
compute its Fourier transform $S_+(\omega) = \int_{0}^\infty \mathrm{e}^{ i \omega v} s_+ (v) \mathrm{d} v$ and verify that $S_+(\omega) = R_+(\omega)$. (We will verify this for $\omega \in \mathcal{H}_+$, i.e., where $R_+$ is well defined). Let us write $c =\tfrac{\gamma_+-\gamma_-}{\sqrt{2} \pi\theta}$.

In particular, we deduce the following asymptotics.
\begin{Cor} \label{cor:asymp_F_ell_main}
    The partition function satisfies:
    \begin{equation} \label{eq:asymp_F_ell_main}
	F_\ell
	\sim
	c \frac{\gamma_+^{\ell}}{\ell^{2-\theta}} \quad \text{as } \ell\to\infty,
	\end{equation}
	where $c =\frac{2^{\theta-1/2}}{\pi \theta} (\gamma_+-\gamma_-)^{\theta-1} \gamma_+^{1-\theta} \Gamma(2-\theta)$, and $\gamma_-$ and $\gamma_+$ are as above.
\end{Cor}

\noindent Observe that the form of the asymptotic \eqref{eq:asymp_F_ell_main} is characteristic of the critical (in fact, more precisely, \textbf{non-generic critical} according to the terminology of \cite{BBG12}) phase of the loop-$O(n)$ model, with the exponent $2-\theta$ interpolating between $3/2$ and $2$ (see e.g.\ \cite[Section 5]{BBG12}). More precisely, it lies in the so-called \emph{dense} phase of the model, where loops are conjectured to be non-simple and to touch each other in the scaling limit.

Our second main result determines the exact tail exponents for the size of typical loops and clusters in the original FK$(q)$-weighted planar map model. This sharpens the main theorem of \cite{berestycki2017critical} by identifying the previously implicit $\ell^{o(1)}$ factor as a true constant {(a similar result can also be deduced from  \cite{gwynne2019scaling}, with the $\ell^{o(1)}$ term identified as a slowly varying function)}. We denote by $\mathfrak{L}$ and $\mathfrak{K}$ the typical loop and (filled-in) cluster in the \emph{infinite} FK$(q)$ planar map. These correspond to the local limit of a uniformly chosen loop or (filled-in) cluster in a finite FK$(q)$ map of size $k$, and then sending $k$ to infinity, see \cite[Theorem 1.1]{berestycki2017critical}.
We define the perimeter $|\mathfrak{L}|$ of the loop $\mathfrak{L}$ as the number of triangles it crosses, and the perimeter $|\partial \mathfrak{K}|$ of the cluster as the degree of its external face, see \cref{sec:typ_cluster} for more precise definitions.

\begin{Thm}[Exponents for loops and clusters]
\label{thm:exp_main}
    We have the following tail asymptotics: as $\ell\to\infty$,
    \[
    \mathbb{P}(|\partial \mathfrak{K}| = \ell) \sim \frac{C}{\ell^{3-2\theta}}
    \quad \text{and} \quad
    \mathbb{P}(|\mathfrak{L}| = \ell) \sim \frac{C'}{\ell^{3-2\theta}},
    \]
    where $C$ and $C'$ are positive constants.
\end{Thm}

Our strategy to prove the above two theorems is the following. First, we make some connections between the gasket decomposition approach and Sheffield's hamburger-cheeseburger bijection. This allows us to express the partition function (up to an explicit factor) as a hitting time probability for the burger walk. We use this information in both directions, as it will enable us to: (a) rigorously justify the missing ansatz in the gasket decomposition approach, and then deduce the exact expression of the partition from there; (b) feed this information back into the burger walks to derive exact asymptotics for the hitting times, and thus for loops and clusters.

The paper is organised as follows. \cref{sec:prelim} is dedicated to preliminaries on self-dual FK planar maps, the $O(n)$ model and the Mullin--Bernardi--Sheffield bijection. In \cref{sec:self_dual_critical}, we make some connections between Sheffield's hamburger-cheeseburger walk and the gasket decomposition of Borot, Bouttier and Guitter. In particular, this allows us to prove an ansatz on the \emph{cut} of the associated resolvent function using solely hamburger-cheeseburger arguments: we will derive an explicit formula for the right endpoint $\gamma_+$ of the cut.  We will then use this information in \cref{sec:resolvents}, where we solve the resolvent equation for the fully packed $O(n)$ model on triangulations, thus proving \cref{thm:F_ell_main}. Finally, in \cref{sec:loop-clusters}, we come back to the hamburger-cheeseburger model and deduce \cref{thm:exp_main} by plugging back the information on the partition function.

We stress that this dictionary has been used in \cite{da2025scaling}, which concerns the critical case $q=4$ (equivalently, $n=2$). In that case, the authors prove the analogue of \cref{thm:F_ell_main} by directing solving the gasket decomposition equation (without any extra input) and then deduce the scaling limit of FK$(4)$-weighted planar maps in the peanosphere sense, resolving the remaining open regime in Sheffield's work \cite{sheffield2016quantum}.

\paragraph{Acknowledgments.}
We thank Sasha Glazman, Xingjian Hu, Ellen Powell, Xin Sun, Joonas Turunen and Mo Dick Wong for insightful discussions.
N.B.  acknowledges the support from the Austrian Science Fund (FWF) grants 10.55776/F1002 on ``Discrete random structures: enumeration and scaling limits" and 10.55776/PAT1878824 on ``Random Conformal Fields''.
W.D.S.\ is supported by the Austrian Science Fund (FWF) grant on “Emergent branching structures in random geometry” (DOI: 10.55776/ESP534).

“Our strategy to prove the above two theorems is the following. First, we make some connections between the gasket decomposition approach and Sheffield's hamburger-cheeseburger bijection. This allows us to express the partition function (up to an explicit factor) as a hitting time probability for the burger walk. We use this information in both directions, as it will enable us to: (a) rigorously justify the missing ansatz in the gasket decomposition approach, and then deduce the exact expression of the partition from there; (b) feed this information back into the burger walks to derive exact asymptotics for the hitting times, and thus for loops and clusters.”

More specifically for Theorem~\ref{thm:F_ell_main}: in \cref{sec:self_dual_critical} the authors “prove an ansatz on the \emph{cut} of the associated resolvent function using solely hamburger-cheeseburger arguments” and “derive an explicit formula for the right endpoint $\gamma_+$ of the cut.” Then “in \cref{sec:resolvents}, we solve the resolvent equation for the fully packed $O(n)$ model on triangulations, thus proving \cref{thm:F_ell_main}.”

“Our strategy to prove the above two theorems is the following. First, we make some connections between the gasket decomposition approach and Sheffield's hamburger-cheeseburger bijection. This allows us to express the partition function (up to an explicit factor) as a hitting time probability for the burger walk. We use this information in both directions, as it will enable us to: (a) rigorously justify the missing ansatz in the gasket decomposition approach, and then deduce the exact expression of the partition from there; (b) feed this information back into the burger walks to derive exact asymptotics for the hitting times, and thus for loops and clusters.”

More specifically in establishing the main theorem: next the authors “prove an ansatz on the \emph{cut} of the associated resolvent function using solely hamburger-cheeseburger arguments” and “derive an explicit formula for the right endpoint $\gamma_+$ of the cut.” Then later they “solve the resolvent equation for the fully packed $O(n)$ model on triangulations, thus proving the main theorem.”

Question:
Let \((F_\ell)_{\ell\ge 1}\) be the partition-function sequence in the setting under consideration, and let the parameter \(\theta\) determine constants \(\gamma_+\) and \(\gamma_-\) by
\[
\gamma_+ = 2^{3/2}\cos\!\left(\frac{\pi\theta}{2}\right),
\qquad
\gamma_+ - \gamma_- = \frac{2^{3/2}}{\theta}\sin\!\left(\frac{\pi\theta}{2}\right).
\]
Which explicit formula holds for every integer \(\ell\ge 1\)?

\[
F_\ell = \int_{\gamma_-}^{\gamma_+} \rho(y)\, y^{\ell}\, \mathrm{d}y,
\qquad \ell\ge 1,
\]
where
\[
\rho(y)=\frac{2^{-\theta-1/2}}{\pi\theta(\gamma_+-\gamma_-)}(\gamma_+-y)^{1-\theta}
\bigg(\Big(\sqrt{2\gamma_+-\gamma_- - y}+\sqrt{y-\gamma_-}\Big)^{2\theta}
-\Big(\sqrt{2\gamma_+-\gamma_- - y}-\sqrt{y-\gamma_-}\Big)^{2\theta}\bigg).
\]

Choices:
B.
\[
F_\ell = \int_{\gamma_-}^{\gamma_+} \rho(y)\, y^{\ell}\, \mathrm{d}y,
\qquad \ell\ge 1,
\]
where
\[
\rho(y)=\frac{2^{-\theta-1/2}}{\pi\theta(\gamma_+-\gamma_-)}(y-\gamma_-)^{1-\theta}
\bigg(\Big(\sqrt{2\gamma_+-\gamma_- - y}+\sqrt{y-\gamma_-}\Big)^{2\theta}
-\Big(\sqrt{2\gamma_+-\gamma_- - y}-\sqrt{y-\gamma_-}\Big)^{2\theta}\bigg).
\]
C.
\[
F_\ell = \int_{\gamma_-}^{\gamma_+} \rho(y)\, y^{\ell}\, \mathrm{d}y,
\qquad \ell\ge 1,
\]
for a density \(\rho\) supported on \([\gamma_-,\gamma_+]\).
D.
\[
F_\ell = \int_{\gamma_-}^{\gamma_+} \rho(y)\, y^{\ell}\, \mathrm{d}y,
\qquad \ell\ge 1,
\]
where
\[
\rho(y)=\frac{2^{-\theta-1/2}}{\pi\theta(\gamma_+-\gamma_-)}(\gamma_+-y)^{1-\theta}
\bigg(\Big(\sqrt{2\gamma_+ - \gamma_- - y}+\sqrt{y-\gamma_-}\Big)^{2\theta}
+\Big(\sqrt{2\gamma_+ - \gamma_- - y}-\sqrt{y-\gamma_-}\Big)^{2\theta}\bigg).
\]
E.
\[
F_\ell = \int_{\gamma_-}^{\gamma_+} \rho(y)\, y^{\ell-1}\, \mathrm{d}y,
\qquad \ell\ge 1,
\]
where
\[
\rho(y)=\frac{2^{-\theta-1/2}}{\pi\theta(\gamma_+-\gamma_-)}(\gamma_+-y)^{1-\theta}
\bigg(\Big(\sqrt{2\gamma_+-\gamma_- - y}+\sqrt{y-\gamma_-}\Big)^{2\theta}
-\Big(\sqrt{2\gamma_+-\gamma_- - y}-\sqrt{y-\gamma_-}\Big)^{2\theta}\bigg).
\]

dropped explicit closed formula for the density

moment exponent in the integral representation

The stem does not reveal the correct formula or uniquely signal choice A. It only asks for the explicit integral representation, without giving away the decisive exponent, sign pattern, or endpoint formulas.

This is essentially a direct theorem-recall item: the correct option is the full explicit representation itself. The task is mainly to recognize the exact stated formula rather than infer a new conclusion from premises.

There is some pressure to compare closely related formulas and reject a weaker true statement, but the item mostly tests exact memory/recognition of a known formula rather than genuine generative mathematical reasoning.

The distractors are plausible and mathematically targeted: one alters the boundary exponent, one changes the resolvent-sign structure, one perturbs an endpoint formula, and one offers a weaker-but-true statement. These reflect realistic failure modes.

No major answer leakage and strong distractors, but the question is largely a theorem restatement that emphasizes precise recall over generative reasoning.


Question:
Let \((F_\ell)_{\ell\ge 1}\) be the partition-function sequence in the setting under consideration, and let the parameter \(\theta\) determine constants \(\gamma_+\) and \(\gamma_-\) by
\[
\gamma_+ = 2^{3/2}\cos\!\left(\frac{\pi\theta}{2}\right),
\qquad
\gamma_+ - \gamma_- = \frac{2^{3/2}}{\theta}\sin\!\left(\frac{\pi\theta}{2}\right).
\]
Which explicit formula holds for every integer \(\ell\ge 1\)?

Choices:
B.
\[
F_\ell = \int_{\gamma_-}^{\gamma_+} \rho(y)\, y^{\ell}\, \mathrm{d}y,
\qquad \ell\ge 1,
\]
where
\[
\rho(y)=\frac{2^{-\theta-1/2}}{\pi\theta(\gamma_+-\gamma_-)}(y-\gamma_-)^{1-\theta}
\bigg(\Big(\sqrt{2\gamma_+-\gamma_- - y}+\sqrt{y-\gamma_-}\Big)^{2\theta}
-\Big(\sqrt{2\gamma_+-\gamma_- - y}-\sqrt{y-\gamma_-}\Big)^{2\theta}\bigg).
\]
C.
\[
F_\ell = \int_{\gamma_-}^{\gamma_+} \rho(y)\, y^{\ell}\, \mathrm{d}y,
\qquad \ell\ge 1,
\]
for a density \(\rho\) supported on \([\gamma_-,\gamma_+]\).
D.
\[
F_\ell = \int_{\gamma_-}^{\gamma_+} \rho(y)\, y^{\ell}\, \mathrm{d}y,
\qquad \ell\ge 1,
\]
where
\[
\rho(y)=\frac{2^{-\theta-1/2}}{\pi\theta(\gamma_+-\gamma_-)}(\gamma_+-y)^{1-\theta}
\bigg(\Big(\sqrt{2\gamma_+ - \gamma_- - y}+\sqrt{y-\gamma_-}\Big)^{2\theta}
+\Big(\sqrt{2\gamma_+ - \gamma_- - y}-\sqrt{y-\gamma_-}\Big)^{2\theta}\bigg).
\]
E.
\[
F_\ell = \int_{\gamma_-}^{\gamma_+} \rho(y)\, y^{\ell-1}\, \mathrm{d}y,
\qquad \ell\ge 1,
\]
where
\[
\rho(y)=\frac{2^{-\theta-1/2}}{\pi\theta(\gamma_+-\gamma_-)}(\gamma_+-y)^{1-\theta}
\bigg(\Big(\sqrt{2\gamma_+-\gamma_- - y}+\sqrt{y-\gamma_-}\Big)^{2\theta}
-\Big(\sqrt{2\gamma_+-\gamma_- - y}-\sqrt{y-\gamma_-}\Big)^{2\theta}\bigg).
\]

The stem does not state the explicit density formula or uniquely reveal the correct option. It only provides the definitions of \(\gamma_\pm\), so there is no direct answer leakage.

This is essentially a theorem-recall item asking for the exact explicit formula that holds. It functions as a near-direct restatement of a known result rather than asking for an independently derived conclusion.

There is some reasoning pressure in distinguishing subtle formula variants (endpoint factor, sign choice, moment exponent, weaker-vs-exact statement), but the task mainly tests precise recall/pattern matching rather than genuine generative mathematical reasoning.

The distractors are plausible and mathematically targeted: one swaps the boundary singular factor, one alters the sign structure, one weakens the claim, and one shifts the moment exponent. These reflect realistic failure modes.

A technically well-constructed recall-style MCQ with strong distractors and little answer leakage, but it is mostly tautological and only moderately tests reasoning.


\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 $n$.

The stem does not reveal the conclusion explicitly or by an obvious cue. It states the hypotheses and asks for an equivalent characterization; the correct option is not directly encoded in the wording.

The item is fairly close to a theorem-recall prompt: it asks which statement is equivalent to an eventual sign condition under technical hypotheses. However, it is not a pure verbatim restatement, since the options force the test-taker to distinguish among nearby formulations.

Some reasoning is required to reject subtle variants such as allowing a negative scalar, allowing c = 0, replacing all large n by large primes, or replacing equality of forms by eventual coefficient agreement. Still, the question mainly tests precise theorem recognition rather than deeper derivation.

The distractors are mathematically plausible and target natural failure modes: sign of the proportionality constant, inclusion of the zero scalar, weakening from all n to primes, and confusing eventual coefficient agreement with form equality. They are distinct and well aligned with likely misconceptions.

A solid MCQ with strong distractors and little answer leakage, but it leans more toward theorem recall/recognition than fully generative reasoning.


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?

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 $n$.

The stem does not explicitly reveal the correct choice. It states the hypotheses and asks for the valid conclusion, without giving away the equivalence or the proportionality conclusion.

The correct option is very close to a direct theorem statement under the same hypotheses, so the item leans heavily on theorem recall. However, the presence of nearby variants means it is not a pure verbatim restatement.

There is some reasoning pressure in distinguishing eventual nonnegativity from infinitely many sign occurrences, all integers versus primes, and positive versus arbitrary scalar multiples. But the question is weakened because choice C is also true, so it does not cleanly force identification of a unique strongest conclusion.

Several distractors are plausible and reflect common mistakes with quantifiers and strength of conclusions. However, choice C is a genuinely true weaker consequence of A, so the distractor set is not fully well-formed for a single-answer MCQ.

Moderately good theorem-discrimination item with no answer leakage and mostly plausible distractors, but it is flawed as a single-correct MCQ because a weaker true statement is included among the distractors.


An \emph{$m$-component 2-link} is a smooth submanifold of $S^4$ homeomorphic to a disjoint union $\sqcup^m S^2$.
Given an $m$-component 2-link $L$, a smoothly embedded collection of 3-discs $D_m \subseteq S^4$, with $D_m \cong \sqcup^m D^3$ and $\partial D_m = L$, is called a \emph{spanning 3-disc collection} for $L$.
If $L$ admits a spanning 3-disc collection, I say that $L$ is \emph{trivial}. Fix a standard trivial $m$-component 2-link $U_m \subseteq S^4$.

\begin{document}
\title[Spanning 3-discs for the trivial 2-link]{Spanning 3-discs in the 4-sphere pushed into the 5-disc}

\begin{abstract}
I prove that any two smooth collections of spanning 3-discs for the trivial 2-link in $S^4$ become smoothly isotopic rel.\ boundary after pushing them into $D^5$.
\end{abstract}
\maketitle

An \emph{$m$-component 2-link} is a smooth submanifold of $S^4$ homeomorphic to a disjoint union $\sqcup^m S^2$.
Given an $m$-component 2-link $L$, a smoothly embedded collection of 3-discs $D_m \subseteq S^4$, with $D_m \cong \sqcup^m D^3$ and $\partial D_m = L$, is called a \emph{spanning 3-disc collection} for $L$.
If $L$ admits a spanning 3-disc collection, I say that $L$ is \emph{trivial}. Fix a standard trivial $m$-component 2-link $U_m \subseteq S^4$.

\begin{remark}
 In the case that $m=1$, i.e.\ spanning 3-discs for the trivial 2-knot, this was proven by Hartman in \cite{H}. An alternative argument was given in the introduction to Hughes-Kim-Miller~\cite{HKM}. The $m=1$ case of the proof of \cref{thm} gives another proof that is more direct. The case of multiple connected components is new. I deduce it from an application of surgery theory.
\end{remark}

Since the surgery methods used in the proof have been available for some time, let me mention how this question arose in the modern context.
Budney--Gabai~\cite{BG} showed that there are spanning 3-discs for $U_1$ that are not isotopic rel.\ boundary in $S^4$, but which become isotopic in $D^5$.
The $m=1$ case of \cref{thm} showed that the Budney--Gabai examples are optimal in the sense that it is not possible to construct examples that remain distinct in $D^5$.

Alison Tatsuoka informs me that work in progress with Seungwon Kim and Gheehyun Nahm will produce spanning 3-disc collections for $U_m$, $m \geq 2$, that are not isotopic rel.\ boundary. Their collections are Brunnian, in the sense that every $(m-1)$-component sub-collection is isotopically standard.
In his PhD thesis, Weizhe Niu~\cite{Niu} independently developed alternative methods, extending the computations from~\cite{BG}, that can likely be applied to prove the same result; this is also work in progress of Niu.
These new collections can be seen directly to be standard in $D^5$; \cref{thm} shows that this must always be the case.

If I allow trivial surface links in $S^4$ with positive genus, there is no analogous result. Hughes--Kim--Miller~\cite{HKM} showed that there are pairs of genus $g \geq 2$ handlebodies in $S^4$, with the same surface boundary, that are not isotopic rel.\ boundary, and remain non-isotopic rel.\ boundary after they are pushed into $D^5$. The genus one case is open at the time of writing, but Kim--Nahm--Tatsuoka have announced similar examples in this case too.
Assuming their veracity, it follows that for every trivial surface link in $S^4$ that contains a surface of positive genus, there exist distinct pairs of spanning handlebodies that remain distinct rel.\ boundary in $D^5$.

I can now consider $f_i \colon W_i \to \natural^m S^1 \times D^4$ as an element of the
Browder-Novikov-Sullivan-Wall relative surgery structure set~\cite{W}, denoted $\mathcal{S}(\natural^m S^1 \times D^4, \#^m S^1 \times S^3)$, which sits in the exact sequence (of abelian groups, in this case, since $\natural^m S^1 \times D^4 \cong D^2 \times \natural^m S^1 \times D^2$):
\begin{equation}\label{eqn:ses}
L_6(\Z[F_m]) \to \mathcal{S}(\natural^m S^1 \times D^4, \#^m S^1 \times S^3) \to \mathcal{N}(\natural^m S^1 \times D^4, \#^m S^1 \times S^3) \to L_5(\Z[F_m]).
\end{equation}
Note that since the Whitehead group of $F_m$ is trivial~\cite{St1}, I can ignore $h$ and $s$ decorations, and every $h$-cobordism is an $s$-cobordism. Let $\wt{L}_n(\Z[\pi])$ denote the reduced $L$-theory of $\Z[\pi]$, such that $L_n(\Z[\pi]) \cong L_n(\Z) \oplus \wt{L}_n(\Z[\pi])$.
Recall that $L_n(\Z)=0$ for $n$ odd, and for $k \geq 0$ one has $L_{4k}(\Z) \cong \Z$ and $L_{4k+2}(\Z) \cong \Z/2$.
I will use Cappell~\cite[Corollary~6]{Ca1,Ca2} and Shaneson's~\cite{Sh} calculations
\begin{align*}
 L_5(\Z[F_m]) &\cong L_5(\Z) \oplus \bigoplus^m \wt{L}_5(\Z[\Z]) \cong \oplus^m L_4(\Z) \cong \oplus^m \Z; \\
L_6(\Z[F_m]) &\cong L_6(\Z) \oplus \bigoplus^m \wt{L}_6(\Z[\Z]) \cong L_6(\Z) \oplus \bigoplus^m L_5(\Z) \cong \Z/2.
\end{align*}

\begin{thmx}\label{thm}
  Let $D_m^0$ and $D_m^1$ be spanning 3-disc collections for the trivial 2-link $U_m$ in $S^4$. Then including $S^4 \subseteq D^5$, $D^0_m$ and $D^1_m$ become smoothly isotopic in $D^5$, relative to $U_m$.
\end{thmx}

\begin{remark}
   In the case that $m=1$, i.e.\ spanning 3-discs for the trivial 2-knot, this was proven by Hartman in \cite{H}. An alternative argument was given in the introduction to Hughes-Kim-Miller~\cite{HKM}.  The $m=1$ case of the proof of \cref{thm} gives another proof that is more direct.  The case of multiple connected components is new. I deduce it from an application of surgery theory.
\end{remark}

Since the surgery methods used in the proof have been available for some time, let me mention how this question arose in the modern context.
Budney--Gabai~\cite{BG} showed that there are spanning 3-discs for $U_1$ that are not isotopic rel.\ boundary in $S^4$, but which become isotopic in $D^5$.
The $m=1$ case of \cref{thm} showed that the Budney--Gabai examples are optimal in the sense that it is not possible to construct examples that remain distinct in $D^5$.

Alison Tatsuoka informs me that work in progress with Seungwon Kim and Gheehyun Nahm will produce spanning 3-disc collections for $U_m$, $m \geq 2$, that are not isotopic rel.\ boundary. Their collections are Brunnian, in the sense that every $(m-1)$-component sub-collection is isotopically standard.
In his PhD thesis, Weizhe Niu~\cite{Niu} independently developed alternative methods, extending the computations from~\cite{BG}, that can likely be applied to prove the same result; this is also work in progress of Niu.
These new collections can be seen directly to be standard in $D^5$; \cref{thm} shows that this must always be the case.

If I allow trivial surface links in $S^4$ with positive genus, there is no analogous result.  Hughes--Kim--Miller~\cite{HKM} showed that there are pairs of genus $g \geq 2$ handlebodies in $S^4$, with the same surface boundary, that are not isotopic rel.\ boundary, and remain non-isotopic rel.\ boundary after they are pushed into $D^5$.  The genus one case is open at the time of writing, but Kim--Nahm--Tatsuoka have announced similar examples in this case too.
Assuming their veracity, it follows that for every trivial surface link in $S^4$ that contains a surface of positive genus, there exist distinct pairs of spanning handlebodies that remain distinct rel.\ boundary in $D^5$.

Thus, Theorem~\ref{thm} is optimal: for the isotopy uniqueness result it espouses to hold, one must allow pushing the spanning 3-discs into the 5-disc, and one cannot increase the topological complexity beyond spherical 2-links.

\begin{remark}
  The proof I give of \cref{thm} is analogous to the proof that slice discs $D^2 \subseteq D^4$ for Alexander polynomial one knots are unique up to topological isotopy rel.\ boundary, which was proven in my work with Conway~\cite{CP}. Since \cref{thm} concerns one dimension up, the result holds in the smooth category. In addition, I can work here with free groups of arbitrary rank, whereas in the 4-dimensional case we needed to work in a setting where the fundamental group was good.
\end{remark}

\subsubsection*{Acknowledgements}

Thanks to Seungwon Kim and Maggie Miller for suggesting this question, and to both Anthony Conway and the anonymous referee for helpful comments on previous drafts.  I am very grateful to Sarah Blackwell and Alison Tatsuoka for informing me about potential applications and comparisons.
I was partially supported by EPSRC grants EP/T028335/1 and  EP/V04821X/1.

The post-theorem introduction does not give a step-by-step outline, but it does state the method and analogies: the author says the multi-component case is "deduce[d] ... from an application of surgery theory," and later that "The proof I give of \cref{thm} is analogous to the proof that slice discs $D^2 \subseteq D^4$ for Alexander polynomial one knots are unique up to topological isotopy rel.\ boundary" from \cite{CP}. It is also noted that "Since \cref{thm} concerns one dimension up, the result holds in the smooth category," and that the argument can be carried out "with free groups of arbitrary rank" (contrasting with the 4-dimensional case where the fundamental group needed to be good).

The post-theorem introduction does not give a step-by-step outline, but it does state the method and analogies: the author says the multi-component case is "deduce[d] ... from an application of surgery theory," and later that "The proof I give of To prove the main theorem, … is analogous to the proof that slice discs $D^2 \subseteq D^4$ for Alexander polynomial one knots are unique up to topological isotopy rel.\ boundary" from \cite{CP}. It is also noted that "Since To prove the main theorem, … concerns one dimension up, the result holds in the smooth category," and that the argument can be carried out "with free groups of arbitrary rank" (contrasting with the 4-dimensional case where the fundamental group needed to be good).,

Question: An $m$-component 2-link is a smooth submanifold of $S^4$ homeomorphic to a disjoint union $\bigsqcup^m S^2$. For an $m$-component 2-link $L$, a spanning 3-disc collection for $L$ is a smoothly embedded submanifold $D_m\subseteq S^4$ with $D_m\cong \bigsqcup^m D^3$ and $\partial D_m=L$. Fix a standard trivial $m$-component 2-link $U_m\subseteq S^4$, where “trivial” means that such a spanning 3-disc collection exists. If $D_m^0$ and $D_m^1$ are any two spanning 3-disc collections for $U_m$ in $S^4$, and we view them inside $D^5$ via the inclusion $S^4\subseteq D^5$, which statement holds for every such pair?

The two collections $D_m^0$ and $D_m^1$ become smoothly isotopic in $D^5$ relative to $U_m$; that is, there is a smooth isotopy in $D^5$ from $D_m^0$ to $D_m^1$ that keeps the common boundary $U_m$ fixed.

Choices:
B. The two collections $D_m^0$ and $D_m^1$ become topologically isotopic in $D^5$ relative to $U_m$, but one cannot in general conclude the existence of a smooth isotopy in $D^5$ fixing the boundary.
C. The two collections $D_m^0$ and $D_m^1$ become smoothly ambiently homeomorphic in $D^5$ by a self-diffeomorphism of $D^5$ that restricts to the identity on $U_m$.
D.
If $m=1$, then $D_m^0$ and $D_m^1$ become smoothly isotopic in $D^5$ relative to $U_m$; for $m
eq 1$, this need not hold for arbitrary spanning 3-disc collections of $U_m$.
E. There exists a smooth isotopy in $D^5$ from $D_m^0$ to $D_m^1$ after possibly precomposing one collection with a diffeomorphism of $U_m$ that permutes the components, but not necessarily relative to the fixed boundary $U_m$ componentwise.

smooth-vs-topological conclusion in one-higher dimension

isotopy rel. boundary weakened to boundary-fixing ambient equivalence

The stem does not state the answer outright, but it strongly cues that the relevant conclusion will involve passing from $S^4$ into $D^5$, which narrows the field toward A/C/E.

This is largely a theorem-identification item: the correct option is essentially the precise statement to be recognized, though the alternatives do introduce meaningful nearby variants rather than mere paraphrases.

Some reasoning is needed to distinguish subtle variants—$S^4$ vs. $D^5$, relative-boundary vs. ambient isotopy, smooth vs. topological, and all $m$ vs. only $m=1$—but it mainly tests recognition of the right formulation rather than substantial derivation.

The distractors are strong: they are mathematically close to the target statement, reflect common confusions about dimension shift, weakening/strengthening conclusions, and smooth vs. topological isotopy, and are clearly distinct from one another.

A reasonably well-designed theorem-discrimination MCQ with strong distractors, but it is still somewhat close to a restatement/recognition task rather than a deeply generative reasoning question.


Question: An $m$-component 2-link is a smooth submanifold of $S^4$ homeomorphic to a disjoint union $\bigsqcup^m S^2$. For an $m$-component 2-link $L$, a spanning 3-disc collection for $L$ is a smoothly embedded submanifold $D_m\subseteq S^4$ with $D_m\cong \bigsqcup^m D^3$ and $\partial D_m=L$. Fix a standard trivial $m$-component 2-link $U_m\subseteq S^4$, where “trivial” means that such a spanning 3-disc collection exists. If $D_m^0$ and $D_m^1$ are any two spanning 3-disc collections for $U_m$ in $S^4$, and we view them inside $D^5$ via the inclusion $S^4\subseteq D^5$, which statement holds for every such pair?

Choices:
B. The two collections $D_m^0$ and $D_m^1$ become topologically isotopic in $D^5$ relative to $U_m$, but one cannot in general conclude the existence of a smooth isotopy in $D^5$ fixing the boundary.
C. The two collections $D_m^0$ and $D_m^1$ become smoothly ambiently homeomorphic in $D^5$ by a self-diffeomorphism of $D^5$ that restricts to the identity on $U_m$.
D.
If $m=1$, then $D_m^0$ and $D_m^1$ become smoothly isotopic in $D^5$ relative to $U_m$; for $m
eq 1$, this need not hold for arbitrary spanning 3-disc collections of $U_m$.
E. There exists a smooth isotopy in $D^5$ from $D_m^0$ to $D_m^1$ after possibly precomposing one collection with a diffeomorphism of $U_m$ that permutes the components, but not necessarily relative to the fixed boundary $U_m$ componentwise.

The stem gives definitions and hypotheses but does not explicitly reveal that the spanning 3-disc collections are smoothly isotopic in D^5 relative to the boundary. There is no direct textual leakage of choice A.

This is essentially a direct theorem-recall item: the correct option states the target conclusion in nearly theorem form, rather than asking the student to derive a less explicitly packaged consequence.

Some reasoning is needed to distinguish the strongest smooth relative-isotopy statement from weaker or altered variants, but the item primarily tests recognition/recall of a specific result rather than substantial generative mathematical reasoning.

The distractors are mathematically plausible and distinct: they vary smooth vs. topological, isotopy vs. ambient equivalence, all m vs. only m=1, and relative boundary fixation vs. permutation ambiguity. These reflect realistic confusion points.

A solid theorem-discrimination MCQ with good distractors and no obvious answer leakage, but it is close to a direct restatement of a specific theorem and thus only moderately tests generative reasoning.