\begin{equation}\label{eq: cp_zero}
c'(0)=-\f{\phi_f'(c_f)}\left\lbrace 2\theta-\f{c_f}\integ{x}{-\infty}[0]<|x|\left(e^{\f{x}{c_f}}+1\right)K(x)>\right\rbrace,
\end{equation}
where $\phi_f$ is the speed index function for the front that satisfies $\phi_f'(c_f)<0$.
\end{corollary}
\subsection{Hypotheses} \label{subsec: hyp}
We state our main hypotheses and then offer some remarks as to why they are necessary.
\begin{itemize}
\item[(H1)] The parameters satisfy $0<\theta<\min\{\f{2},\int_{-\infty}^0 K\}$, \, $\f{\gamma}{1+\gamma}<\theta,\,$ and $\epsilon \ll 1$.
\item[(H2)] When $\epsilon=0$, there exists a traveling front solution $u(x,t)=U_f(x+c_ft)$ to \eqref{eq: intro_scalar} with $U_f(\cdot)<\theta$ on $(-\infty,0)$, $U_f(\cdot)>\theta$ on $(0,\infty)$, $U_f(-\infty)=0$, and $U_f(\infty)=1$. The solution is translation invariant, but we will assume $U_f(0)=\theta$ unless stated otherwise.
\item[(H3)] There exists $\alpha>0$ and $\rho>0$ such that $|K(x)|\leq \alpha e^{-\rho|x|}$. Moreover, assume $K$ is continuous and $\int_\R K = 1$.
\end{itemize}
We require $0<\theta<\int_{-\infty}^0 K$ because it is necessary in order for the front to exist, while $\theta <\f{2}$ is required for the pulse. Often $K$ is symmetrical so these conditions are equivalent. To conclude remarks on (H1), we note that the assumption $\frac{\gamma}{1+\gamma}<\theta$ ensures that the line $Q=\frac{U}{\gamma}$ does not intersect $Q=-U+H(U-\theta)$ when $U>\theta$, which would lead to a second fixed point, and the homoclinic orbit would be replaced by a heteroclinic orbit. Hypothesis (H2) ensures we have a leading order inner solution. Finally, hypothesis (H3) is a common assumption in nonlocal problems. Biologically, long-range synaptic interaction strengths decay sharply, while mathematically, exponential decay ensures that terms of order $O\left(\epsilon^{-k}e^{-\f{|C|}{\epsilon}}\right)$ vanish.
\section{Singular Solution,  Previous Results, and Outline}
We review the Heaviside version of the $\epsilon=0$ singular homoclinic orbit construction from \cite{PintoandErmentrout-SpatiallyStructuredActivityinSynapticallyCoupledI.TravelingWaves}, which proves the leading orders of inner and outer solutions can be matched at boundary points. We then highlight how our result advances nonlocal geometric singular perturbation theory approaches.
\subsection{Construction of Singular Homoclinic Orbit}
\subsubsection*{Fast Inner Solutions at $\epsilon=0$}
Writing \eqref{eq: intro_system_1}--\eqref{eq: intro_system_2}, in terms of the traveling coordinate, we obtain the fast inner system
\begin{align}
c\begin{pmatrix} U' \\ Q' \end{pmatrix}
+\begin{pmatrix}
1 & 1\\
-\epsilon & \epsilon\gamma
\end{pmatrix}
\begin{pmatrix} U \\ Q \end{pmatrix}
= \begin{pmatrix} \integ{y}{\R}[]<K(z-y)H(U(y)-\theta)> \\ 0 \end{pmatrix}. \label{eq: fast_null}
\end{align}
When $\epsilon=0$, from the second equation, we have $Q'=0$ so $Q\equiv Q_0$ for some constant $Q_0$. Plugging into the first equation, we recall from \cite{ExistenceandUniqueness-ErmMcLeod} that there may (guaranteed when $K\geq 0$) exist a front with wave speed $c_f>0$ that solves
\begin{equation}
c_f U_f' + U_f=\integ{y}{\R}[]<K(z-y)H(U_f(y)-\theta)>.
\label{eq: front_ODE}
\end{equation}

\begin{theorem}[Existence and Uniqueness of Fast Pulses]\label{thm: E}
When $0<\epsilon \ll 1,$ there exists a unique $C^2$ (modulo translation) fast traveling pulse solution $(U_\epsilon,Q_\epsilon)$ to system \eqref{eq: intro_system_1}-\eqref{eq: intro_system_2}. In particular, with $z=x+ct$, there exists unique $C^1$ curves $c_\epsilon=c(\epsilon)$ and $a_\epsilon=a(\epsilon)$ such that $U_\epsilon(z)<\theta$ on $(-\infty,0)\cup(a_\epsilon,\infty)$ and $U_\epsilon(z)>\theta$ on $(0,a_\epsilon)$. The parameters satisfy $c_\epsilon\to c_f$ and $a_\epsilon\to +\infty$ as $\epsilon \to 0$, where $c_f>0$ is the unique speed of the front solution to \eqref{eq: intro_scalar}. As a homoclinic orbit, $\mathcal{S}_\epsilon=\Bigl\{(U_\epsilon(z),Q_\epsilon(z)): z\in \R \Bigr\}$ satisfies $d_H(\mathcal{S}_\epsilon,\mathcal{S}_0)\to 0$ as $\epsilon\to 0$. Here,
$$
d_H(\mathcal{S}_\epsilon,\mathcal{S}_0)=\max\left\lbrace \sup_{(U,Q)\in \mathcal{S}_0}d\Bigl(\mathcal{S}_\epsilon,(U,Q)\Bigr),\sup_{(U_\epsilon,Q_\epsilon)\in \mathcal{S}_\epsilon}d\Bigl((U_\epsilon,Q_\epsilon),\mathcal{S}_0\Bigr) \right\rbrace
$$
is the Hausdorff distance between sets in $(U,Q)$ space with respect to the standard Euclidean metric, and $\mathcal{S}_0$ is the singular homoclinic orbit at $\epsilon=0$.
\end{theorem}

\label{Section: Intro}
In mammals, cortical waves are an intriguing and observable pattern, both in normal function and pathology. For example, wave propagation is a prominent phenomena seen in higher order sensory and motor cortical processing, believed to encode plasticity and other hierarchical sequencing \cite{sato2022cortical}. This occurs in the visual cortex via feedforward and feedback traveling waves \cite{aggarwal2022v}. By combining high-resolution imaging techniques like fMRI and MEG-EEG, a synergistic approach can provide insight into the connection between retinotopic directional mappings, hemodynamics, and vision-related motor tasks \cite{zanos2015sensorimotor,aquino2012hemodynamic,grabot2025traveling}.
Traveling waves have also been proven to play a pivotal role in cortical organization such as in working memory \cite{luo2025traveling}. In vitro, neural slices have also shown evidence of recorded propagation, often with techniques to block inhibition \cite{avoli2016models,pavan2022vitro}. Sensory processing pathologies such as migraines \cite{o2021migraine} and epilepsy \cite{martinet2017human} are also known to be related to pulse propagation.

Due to a tractability challenge in neuroscience, neural fields are a well-studied macroscopic model that portray the spatial and temporal interactions between patches of neurons as nonlocal integro-differential equations. Originally, the neural field equations were modeled by Wilson and Cowan \cite{WilsonCowan1972} with separate excitatory and inhibitory network populations. Amari \cite{Amari1977} justified a scalar equation of mixed populations with firing rates encoded within integrals of synaptic interactions. Models with one spatial dimension are justified, based on the layering structure of the cortex and practical recording methods. Voltage activity is then measured as spatial and temporal averages with spike times neglected. Efforts have been made to fit Wilson-Cowan and Amari models to data \cite{freestone2011data,rule2019neural,kofinas2023latent}.

With further simplification, Amari introduced the so-called ``Heaviside" activation approach that supplements the usual sigmoidal models with a more tractable alternative. By exploiting the Heaviside structure, Amari showed bumps, oscillations, and traveling waves exist when synaptic coupling kernels take on Mexican hat shapes. Since then, there have been many follow-ups based on Amari's Heaviside assumption.

As a scalar equation, the model takes on the form \cite{Amari1977,ExistenceandUniqueness-ErmMcLeod}
\begin{equation} \label{eq:	 intro_scalar}
u_t=-u+\integ{y}{\R}[]<K(x-y)H(u(y,t)-\theta)>.
\end{equation}
In \eqref{eq: intro_scalar}, $u=u(x,t)$ represents average voltage at position $x$ and time $t$; $K$ is a synaptic coupling kernel, representing homogeneous spatial connectivity strength between presynaptic neural patches at position $y$ with postsynaptic neural patches at position $x$. The parameter $\theta>0$ is the threshold for neurons at $y$ to fire. We ignore previously studied spatial and feedback delays \cite{HuttZhang-TravelingWave,LijZhangSolo,Atay2004} for technical reasons, though there's no obvious reason why they would alter our approach. There are a variety of other modifications of \eqref{eq: intro_scalar} in the literature, but \eqref{eq: intro_scalar} can be categorized as the standard nonlocal Heaviside model without external currents, synaptic depression, stochastic noise, or other nonlinearities.

Due to a Heaviside firing rate, neural patches communicate in an all-or-none manner, justified statistically (and mathematically \cite{Burlakov_etal2025}) as an average of sigmoidals or the limit of steep sigmoidals. While sigmoidal firing rates are more physically appropriate, the Heaviside function provides a favorable mathematical trade-off.

Incorporating more realistic linear feedback such as spike frequency adaptation, Pinto and Ermentrout also studied traveling and standing waves in the singularly perturbed system \cite{pinto2001spatially,PintoandErmentrout-SpatiallyStructuredActivityinSynapticallyCoupledI.TravelingWaves}
\begin{align}
u_t&=-u-q+ \integ{y}{\R}[]<K(x-y)H(u(y,t)-\theta)>, \label{eq: intro_system_1}\\
q_t &=\epsilon (u-\gamma q). \label{eq: intro_system_2}
\end{align}
Here, $0<\epsilon \ll 1$, the function $q$ is a slow linear term, and $\gamma>0$ is a decay constant. Equation \eqref{eq: intro_scalar}, system \eqref{eq: intro_system_1}--\eqref{eq: intro_system_2}, and variations have been studied extensively, leading to vast dynamical systems findings. See \cite{cowan2014personal,ermentrout2010mathematical, bressloff2014waves_wholebook,coombes2014neural,cook2022neural} for background and more historical context.

With traveling waves as our focus, we introduce the traveling coordinate $z=x+c t$ and look for solutions to \eqref{eq: intro_system_1}--\eqref{eq: intro_system_2} of the form $\allowbreak (u(x,t),q(x,t))=(U(z),Q(z))$. Moreover, this solution is defined as a fast traveling pulse if there exists $a=O\left(\f{\epsilon}\right)$ such that $U(z)<\theta$ on $(-\infty,0)\cup (a,\infty)$ and $U(z)>\theta$ on $(0,a)$. Plugging in this ansatz into \eqref{eq: intro_system_1}--\eqref{eq: intro_system_2}, we arrive at the system
\begin{align}
c\begin{pmatrix} U' \\ Q' \end{pmatrix}
+A
\begin{pmatrix} U \\ Q \end{pmatrix}
= \begin{pmatrix} \integ{x}{z-a}[z]<K(x)> \\ 0 \end{pmatrix}, \label{eq: ansatz}
\end{align}
where $A:=\begin{pmatrix}
1 & 1\\
-\epsilon & \epsilon\gamma
\end{pmatrix}$. The matrix $A$ has eigenvalues and eigenvectors given by \begin{align}
\omega_1(\epsilon)&=\frac{1+\gamma\epsilon+\sqrt{(1-\gamma\epsilon)^2-4\epsilon}}{2},\qquad v_1=\begin{pmatrix}1\\\omega_1-1\end{pmatrix}, \\
\omega_2(\epsilon)&=\frac{1+\gamma\epsilon-\sqrt{(1-\gamma\epsilon)^2-4\epsilon}}{2},\qquad v_2=\begin{pmatrix}1\\\omega_2-1\end{pmatrix}.
\end{align}
Under the assumption of fixed $\gamma>0$ and $\epsilon \ll 1$, both $\omega_1$ and $\omega_2$ are positive with
\begin{equation}\label{eq: limits_eig}
\omega_1(0)=1,\qquad \omega_2(0)=0,\qquad \omega_1'(0)=-1, \qquad \omega_2'(0)=1+\gamma.
\end{equation}
Note that
\begin{equation}\label{eq: omega_calcs}
\omega_1 \omega_2 =\det(A)=\epsilon(1 + \gamma), \qquad \omega_1 + \omega_2 = tr(A) = 1+\epsilon \gamma,
\end{equation}
and from \eqref{eq: omega_calcs}, we see that
\begin{equation}
\f{1-\omega_1}{\omega_2}=\f{\omega_2-\epsilon\gamma}{\omega_2}=1-\f{\epsilon\gamma}{\omega_2}=1-\f{\omega_1\gamma}{1+\gamma}. \label{eq: OneMinusOmega1_Omega2}
\end{equation}
Furthermore, by differentiating the equations in \eqref{eq: omega_calcs} twice, we find $\omega_1''\omega_2 +2\omega_1'\omega_2' +\omega_1\omega_2''=0$ and $\omega_1''+\omega_2''=0$ so
\begin{equation}
\omega_1''(0)=-2(1+\gamma),\qquad \omega_2''(0)=2(1+\gamma). \label{eq: omega_secder}
\end{equation}
The formal solution to \eqref{eq: ansatz} is given by \cite{PintoandErmentrout-SpatiallyStructuredActivityinSynapticallyCoupledI.TravelingWaves,Pinto2005}
\begin{align}
U(z)&=\integ{x}{-\infty}[z]<C_x(x-z,c,\epsilon)\left(\integ{y}{x-a}[x]<K(y)>\right)>,\label{eq: U_formal} \\
Q(z)&=\integ{x}{-\infty}[z]<D_x(x-z,c,\epsilon)\left(\integ{y}{x-a}[x]<K(y)>\right)>, \label{eq: Q_formal}\\
U'(z)&=\integ{x}{-\infty}[z]<C_x(x-z,c,\epsilon)\left(K(x)-K(x-a)\right)>, \label{eq: Up_formal}\\
Q'(z)&=\integ{x}{-\infty}[z]<D_x(x-z,c,\epsilon)\left(K(x)-K(x-a)\right)>,\label{eq: Qp_formal}
\end{align}
where
\begin{align}
C(x,c,\epsilon)&=\frac{1}{\omega_1-\omega_2}\left[\f{1-\omega_2}{\omega_1}e^{\frac{\omega_1 x}{c}}-\f{1-\omega_1}{\omega_2}e^{\frac{\omega_2 x}{c}}\right], \label{eq: C}\\
C_x(x,c,\epsilon)&=\frac{1}{c(\omega_1-\omega_2)}\left[(1-\omega_2)e^{\frac{\omega_1 x}{c}}-(1-\omega_1)e^{\frac{\omega_2 x}{c}}\right], \label{eq: Cx}\\
D(x,c,\epsilon)&=\frac{\epsilon}{\omega_1-\omega_2}\left[-\f{\omega_1}e^{\frac{\omega_1 x}{c}}+\f{\omega_2}e^{\frac{\omega_2 x}{c}}\right], \label{eq: D} \\
D_x(x,c,\epsilon)&=\frac{\epsilon}{c(\omega_1-\omega_2)}\left[-e^{\frac{\omega_1 x}{c}}+e^{\frac{\omega_2 x}{c}}\right]. \label{eq: Dx}
\end{align}
Using the properties of $\omega_1$ and $\omega_2$ from \eqref{eq: limits_eig}--\eqref{eq: omega_calcs}, we see that
\begin{align*}
C(0,c,\epsilon)&=\f{\omega_2(1-\omega_2)-\omega_1(1-\omega_1)}{\omega_1\omega_2(\omega_1-\omega_2)}=\f{-1+\omega_1+\omega_2}{\omega_1\omega_2}=\f{\gamma}{1+\gamma}, \numberthis \label{eq: C0}\\
D(0,c,\epsilon)&=\f{\epsilon}{\omega_1\omega_2}=\f{1+\gamma}. \numberthis \label{eq: D0}
\end{align*}

Clearly, $(U(-\infty),Q(-\infty))=(0,0)$, and by splitting the integrals by $\int_{-\infty}^z=\int_{-\infty}^a+\int_a^z$, an easy application of the dominated convergence theorem shows $(U(+\infty),Q(+\infty))=(0,0)$ as well. Hence, even for arbitrary $a,c>0$, the formal solution can be understood as a homoclinic orbit connecting the point $(0,0)$ to itself.

This brings us to the main difficulty and purpose of this paper. When is a solution to \eqref{eq: ansatz} really a solution to \eqref{eq: intro_system_1}--\eqref{eq: intro_system_2}? Assuming only the existence of a front (that we show is always unique), how do we prove the existence of unique parameters $(a(\epsilon),c(\epsilon))$ that solve the compatibility equations $U(0)=U(a)=\theta$? How do we demonstrate that the sub and super threshold regions truly correspond and that the pulse is close to the singular homoclinic orbit? Can we compare the difference between the pulse and front speed up to first order in $\epsilon$? We answer all of these questions.

\subsection{Main Results}\label{subsec: main_goal}
Our main goal is to rigorously prove the existence and uniqueness of fast traveling pulses to system \eqref{eq: intro_system_1}-\eqref{eq: intro_system_2} under minimal assumptions. In particular, we prove the following theorems, assuming hypotheses (H1)--(H3) discussed in \cref{subsec: hyp}.

Incorporating more realistic linear feedback such as spike frequency adaptation, Pinto and Ermentrout also studied traveling and standing waves in the singularly perturbed system \cite{pinto2001spatially,PintoandErmentrout-SpatiallyStructuredActivityinSynapticallyCoupledI.TravelingWaves}
\begin{align}
u_t&=-u-q+ \integ{y}{\R}[]<K(x-y)H(u(y,t)-\theta)>, \label{eq: intro_system_1}\\
q_t &=\epsilon (u-\gamma q). \label{eq: intro_system_2}
\end{align}
Here, $0<\epsilon \ll 1$, the function $q$ is a slow linear term, and $\gamma>0$ is a decay constant. Equation \eqref{eq: intro_scalar}, system \eqref{eq: intro_system_1}--\eqref{eq: intro_system_2}, and variations have been studied extensively, leading to vast dynamical systems findings. See \cite{cowan2014personal,ermentrout2010mathematical, bressloff2014waves_wholebook,coombes2014neural,cook2022neural} for background and more historical context.

With traveling waves as our focus, we introduce the traveling coordinate $z=x+c t$ and look for solutions to \eqref{eq: intro_system_1}--\eqref{eq: intro_system_2} of the form $\allowbreak (u(x,t),q(x,t))=(U(z),Q(z))$. Moreover, this solution is defined as a fast traveling pulse if there exists $a=O\left(\f{\epsilon}\right)$ such that $U(z)<\theta$ on $(-\infty,0)\cup (a,\infty)$ and $U(z)>\theta$ on $(0,a)$. Plugging in this ansatz into \eqref{eq: intro_system_1}--\eqref{eq: intro_system_2}, we arrive at the system
\begin{align}
c\begin{pmatrix} U' \\ Q' \end{pmatrix}
+A
\begin{pmatrix} U \\ Q \end{pmatrix}
= \begin{pmatrix} \integ{x}{z-a}[z]<K(x)> \\ 0 \end{pmatrix}, \label{eq: ansatz}
\end{align}
where $A:=\begin{pmatrix}
1 & 1\\
-\epsilon & \epsilon\gamma
\end{pmatrix}$. The matrix $A$ has eigenvalues and eigenvectors given by \begin{align}
\omega_1(\epsilon)&=\frac{1+\gamma\epsilon+\sqrt{(1-\gamma\epsilon)^2-4\epsilon}}{2},\qquad v_1=\begin{pmatrix}1\\\omega_1-1\end{pmatrix}, \\
\omega_2(\epsilon)&=\frac{1+\gamma\epsilon-\sqrt{(1-\gamma\epsilon)^2-4\epsilon}}{2},\qquad v_2=\begin{pmatrix}1\\\omega_2-1\end{pmatrix}.
\end{align}
Under the assumption of fixed $\gamma>0$ and $\epsilon \ll 1$, both $\omega_1$ and $\omega_2$ are positive with
\begin{equation}\label{eq: limits_eig}
\omega_1(0)=1,\qquad \omega_2(0)=0,\qquad \omega_1'(0)=-1, \qquad \omega_2'(0)=1+\gamma.
\end{equation}
Note that
\begin{equation}\label{eq: omega_calcs}
\omega_1 \omega_2 =\det(A)=\epsilon(1 + \gamma), \qquad \omega_1 + \omega_2 = tr(A) = 1+\epsilon \gamma,
\end{equation}
and from \eqref{eq: omega_calcs}, we see that
\begin{equation}
\f{1-\omega_1}{\omega_2}=\f{\omega_2-\epsilon\gamma}{\omega_2}=1-\f{\epsilon\gamma}{\omega_2}=1-\f{\omega_1\gamma}{1+\gamma}. \label{eq: OneMinusOmega1_Omega2}
\end{equation}
Furthermore, by differentiating the equations in \eqref{eq: omega_calcs} twice, we find $\omega_1''\omega_2 +2\omega_1'\omega_2' +\omega_1\omega_2''=0$ and $\omega_1''+\omega_2''=0$ so
\begin{equation}
\omega_1''(0)=-2(1+\gamma),\qquad \omega_2''(0)=2(1+\gamma). \label{eq: omega_secder}
\end{equation}
The formal solution to \eqref{eq: ansatz} is given by \cite{PintoandErmentrout-SpatiallyStructuredActivityinSynapticallyCoupledI.TravelingWaves,Pinto2005}
\begin{align}
U(z)&=\integ{x}{-\infty}[z]<C_x(x-z,c,\epsilon)\left(\integ{y}{x-a}[x]<K(y)>\right)>,\label{eq: U_formal} \\
Q(z)&=\integ{x}{-\infty}[z]<D_x(x-z,c,\epsilon)\left(\integ{y}{x-a}[x]<K(y)>\right)>, \label{eq: Q_formal}\\
U'(z)&=\integ{x}{-\infty}[z]<C_x(x-z,c,\epsilon)\left(K(x)-K(x-a)\right)>, \label{eq: Up_formal}\\
Q'(z)&=\integ{x}{-\infty}[z]<D_x(x-z,c,\epsilon)\left(K(x)-K(x-a)\right)>,\label{eq: Qp_formal}
\end{align}
where
\begin{align}
C(x,c,\epsilon)&=\frac{1}{\omega_1-\omega_2}\left[\f{1-\omega_2}{\omega_1}e^{\frac{\omega_1 x}{c}}-\f{1-\omega_1}{\omega_2}e^{\frac{\omega_2 x}{c}}\right], \label{eq: C}\\
C_x(x,c,\epsilon)&=\frac{1}{c(\omega_1-\omega_2)}\left[(1-\omega_2)e^{\frac{\omega_1 x}{c}}-(1-\omega_1)e^{\frac{\omega_2 x}{c}}\right], \label{eq: Cx}\\
D(x,c,\epsilon)&=\frac{\epsilon}{\omega_1-\omega_2}\left[-\f{\omega_1}e^{\frac{\omega_1 x}{c}}+\f{\omega_2}e^{\frac{\omega_2 x}{c}}\right], \label{eq: D} \\
D_x(x,c,\epsilon)&=\frac{\epsilon}{c(\omega_1-\omega_2)}\left[-e^{\frac{\omega_1 x}{c}}+e^{\frac{\omega_2 x}{c}}\right]. \label{eq: Dx}
\end{align}
Using the properties of $\omega_1$ and $\omega_2$ from \eqref{eq: limits_eig}--\eqref{eq: omega_calcs}, we see that
\begin{align*}
C(0,c,\epsilon)&=\f{\omega_2(1-\omega_2)-\omega_1(1-\omega_1)}{\omega_1\omega_2(\omega_1-\omega_2)}=\f{-1+\omega_1+\omega_2}{\omega_1\omega_2}=\f{\gamma}{1+\gamma}, \numberthis \label{eq: C0}\\
D(0,c,\epsilon)&=\f{\epsilon}{\omega_1\omega_2}=\f{1+\gamma}. \numberthis \label{eq: D0}
\end{align*}

This brings us to the main difficulty and purpose of this paper. When is a solution to \eqref{eq: ansatz} really a solution to \eqref{eq: intro_system_1}--\eqref{eq: intro_system_2}? Assuming only the existence of a front (that we show is always unique), how do we prove the existence of unique parameters $(a(\epsilon),c(\epsilon))$ that solve the compatibility equations $U(0)=U(a)=\theta$? How do we demonstrate that the sub and super threshold regions truly correspond and that the pulse is close to the singular homoclinic orbit? Can we compare the difference between the pulse and front speed up to first order in $\epsilon$? We answer all of these questions.

\subsection{Main Results}\label{subsec: main_goal}
Our main goal is to rigorously prove the existence and uniqueness of fast traveling pulses to system \eqref{eq: intro_system_1}-\eqref{eq: intro_system_2} under minimal assumptions. In particular, we prove the following theorems, assuming hypotheses (H1)--(H3) discussed in \cref{subsec: hyp}.

In particular, to the author's knowledge, there have been no other results that rigorously study the fast pulse solutions in the Heaviside model when they are broken down by fast and slow representations in inner and outer regions for $\epsilon \ll 1$. Unlike the singular homoclinic orbit, when $\epsilon>0$, the fast and slow dynamics need to be analyzed when they overlap. When $K$ is chosen as the exponential type. we have PDE reductions and applications of classic geometric singular perturbation theory can be applied. We avoid requiring this reduction.

Finally, for $\Gamma_4$, with $z\in R_3$, we have
\begin{align*}
|\Gamma_4(z,\epsilon)|&\leq \f{\alpha}{c_\epsilon(\omega_1-\omega_2)}\integ{x}{0}[z]<\left[(1-\omega_2)e^{\f{\omega_1(x-z)}{c_\epsilon}}+(1-\omega_1)e^{\f{\omega_2(x-z)}{c_\epsilon}}\right]\left(\integ{y}{x}[\infty]<e^{-\rho y}>\right)> \numberthis \label{eq: Gamma4_inequal}\\
&=\f{\alpha}{\rho(\omega_1-\omega_2)}\left[\f{1-\omega_2}{\omega_1-\rho c_\epsilon}\left(e^{-\rho z}-e^{-\f{\omega_1 z}{c_\epsilon}}\right)+\f{1-\omega_1}{\omega_2-\rho c_\epsilon}\left(e^{-\rho z}-e^{-\f{\omega_2 z}{c_\epsilon}}\right)\right] \\
&\leq \f{\alpha}{\rho(\omega_1-\omega_2)}\left[\f{1-\omega_2}{|\omega_1-\rho c_\epsilon|}\left(e^{-\rho (a_\epsilon-z_0)}+e^{-\f{\omega_1 (a_\epsilon-z_0)}{c_\epsilon}}\right)+\f{2(1-\omega_1)}{|\omega_2-\rho c_\epsilon|}\right] \longrightarrow 0
\end{align*}
\end{proof}
\begin{lemma}\label{lemma: unif_aminb_a}
There exists $\epsilon_3\leq \epsilon_2$ such that for all $\epsilon<\epsilon_3$, the uniform estimates on $R_3$ of $U_\epsilon(z)\approx U_b(z-a_\epsilon)$, $U_\epsilon'(z)\approx U_b'(z-a_\epsilon)$, and $Q_\epsilon(z)\approx 1-2\theta$ hold. The estimates are as close as we desire as $\epsilon\to 0$.
\end{lemma}
\begin{proof}
The uniform limit $(U_\epsilon(z)-U_b(z-a_\epsilon)\chi_{R_3}\to 0$ follows directly from \cref{prop: Gamma_errors}. For $Q_\epsilon$, after writing it in a similar way to that of $U_\epsilon$ in \eqref{eq: U_R3}. A tedious adjustment of \cref{prop: Gamma_errors} results in $(Q_\epsilon(z)-\underbrace{Q_b(z-a_\epsilon)}_{=1-2\theta})\chi_{R_3} \to 0$ uniformly. Finally, the back satisfies
\begin{equation} \label{eq: Uback_diff}
c_f U_b'(z-a_\epsilon)+U_b(z-a_\epsilon)+(1-2\theta)=\integ{x}{z-a_\epsilon}[\infty]<K(x)>=1-\integ{x}{-\infty}[z-a_\epsilon]<K(x)>,
\end{equation}
while $U_\epsilon$ satisfies
\begin{equation}\label{eq: U_R3_diff}
c_\epsilon U_\epsilon'(z)+U_\epsilon(z)+Q_\epsilon(z)=1-\left(\int_z^\infty+\int_{-\infty}^{z-a_\epsilon}\right)K(x)\,\mathrm{d}x.
\end{equation}
Subtracting \eqref{eq: Uback_diff} from \eqref{eq: U_R3_diff} leads to the inequality
\begin{align*}
|c_\epsilon U_\epsilon'(z)-c_f U_b'(z-a_\epsilon)| &\leq |U_\epsilon(z)-U_b(z-a_\epsilon)|+|Q_\epsilon(z)-(1-2\theta)| \\
&\phantom{\leq}+\integ{x}{a_\epsilon-z_0}[\infty]<|K(x)|>\longrightarrow 0
\end{align*}
so $(U_\epsilon'(z)-U_b'(z-a_\epsilon))\chi_{R_3}\to 0$ uniformly.
\end{proof}
In conclusion, for $\epsilon<\epsilon_3$, the curve $(U_\epsilon,Q_\epsilon)$ is uniformly close to the (fast time) back in the singular homoclinic orbit, and because $U_\epsilon'(z)\approx U_b'(z-a_\epsilon)$ and $Q_\epsilon'(z)\approx Q_b'(z-a_\epsilon)=0$ as well (where $U_b'(0)<0$), the solution successfully crosses the threshold from above one time at $z=a_\epsilon$. We conclude this subsection with a statement about the Hausdorff distance which is similar to \cref{prop: R1_Hausdorff}. If necessary, ensure $d(s_\epsilon(z),x_b(z-a_\epsilon))\chi_{R_3}<\f{\delta}{2}$ for all $z\in\R$.
\begin{proposition}\label{prop: R3_Hausdorff}
The estimate $d_H(\mathcal{S}_\epsilon^3,\mathcal{B}_0)< \delta$ holds for all $\epsilon<\epsilon_3$. Moreover, $s_\epsilon(a_\epsilon+z_0)\in B_d(\mathcal{M}_0^L,\delta)$.
\end{proposition}
\begin{proof}
Firstly,
\begin{equation}\label{eq: Haus_R3_first}
\sup_{z\in R_3}d(s_\epsilon(z),\mathcal{B}_0)\leq \sup_{z\in R_3}d(s_\epsilon(z),x_b(z-a_\epsilon)) <\f{\delta}{2}.
\end{equation}
On the other hand, let $x_b(z) \in \mathcal{B}_0$. If $|z-a_\epsilon |\leq z_0$, then
$$
d(\mathcal{S}_\epsilon^3,x_b(z-a_\epsilon))\leq d(s_\epsilon(z),x_b(z-a_\epsilon)) <\f{\delta}{2}.
$$
If $z-a_\epsilon>z_0$, then
$
d(\mathcal{S}_\epsilon^3,x_b(z-a_\epsilon))\leq d(s_\epsilon(a_\epsilon+z_0),x_b(z-a_\epsilon))$, which is bounded by
$$
d(x_b(z-a_\epsilon),m_L(1-2\theta))+ d(m_L(1-2\theta),x_b(z_0))+d(x_b(z_0),s_\epsilon(a_\epsilon +z_0))<\delta
$$
using \eqref{eq: z0_second}. The $z-a_\epsilon<-z_0$ case is similar. It is clear that $d_H(\mathcal{S}_\epsilon^3,\mathcal{B}_0)<\delta$.

Finally, for $L_5$, since $e^{-\f{\omega_i z}{c_\epsilon}}<e^{-\f{\omega_i (z-a_\epsilon)}{c_\epsilon}}$, we have
\begin{align*}
|L_5(z,\epsilon)|&< \f{2}{\omega_1-\omega_2}\left[\left(\f{1-\omega_2-\epsilon}{\omega_1}\right) e^{-\f{\omega_1 (z-a_\epsilon)}{c_\epsilon}}+\left|\f{1-\omega_1-\epsilon}{\omega_2}\right| e^{-\f{\omega_2 (z-a_\epsilon)}{c_\epsilon}}\right]\\
&\leq \f{2}{\omega_1-\omega_2}\left[\left(\f{1-\omega_2-\epsilon}{\omega_1}\right) e^{-\f{\omega_1 (\tau_1+\epsilon z_0)}{\epsilon c_\epsilon}}+\left|\f{1-\omega_1-\epsilon}{\omega_2}\right|\right] \longrightarrow 0.
\end{align*}
In conclusion, $W_\epsilon(z)\chi_{R_4^2}\to 0$ uniformly. For the derivative, by adding the $U_\epsilon$ and $Q_\epsilon$ equations, we find
\begin{align*}
c_\epsilon |W_\epsilon'(z)|\leq |W_\epsilon(z)|+\epsilon |U_\epsilon(z)-\gamma Q_\epsilon(z)|+\integ{x}{\f{\tau_1}{\epsilon}}[\infty]<|K(x)|> \longrightarrow 0.
\end{align*}
This concludes the proof.
\end{proof}
After $\epsilon_4\leq \epsilon_3$ is chosen so that $d(s_\epsilon(z),\mathcal{M}_0^L)<\delta$ for all $z\in R_4^2$, we have enough information to complete the proof of \cref{thm: E}.
\begin{proposition}\label{prop: R4_Hausdorff}
The limit $d_H(\mathcal{S}_\epsilon,\mathcal{S}_0)\to 0$ holds.
\end{proposition}
\begin{proof}
Since $s_\epsilon(a_\epsilon+z_0)\in B_d(\mathcal{M}_0^L,\delta)$, we may combine \cref{prop: R4_nearline,lemma: UplusQ_R4_errors} to show $d(\mathcal{S}_\epsilon^4,\mathcal{M}_0^L)<3\delta$ by a similar proof seen in \cref{prop: R2_Haus_proof} on $R_2$. It follows that for $\epsilon<\epsilon_4$, we have
$$
\max \Bigl\{d(\mathcal{S}_\epsilon^1,\mathcal{F}_0),d(\mathcal{S}_\epsilon^2,\mathcal{M}_0^R),d(\mathcal{S}_\epsilon^3,\mathcal{B}_0),d(\mathcal{S}_\epsilon^4,\mathcal{M}_0^L) \Bigr\}<3\delta.
$$
Hence, by \cref{prop: dH_regions}, it follows that $d_H(\mathcal{S}_\epsilon,\mathcal{S}_0)<3\delta$.
\end{proof}
In conclusion, we proved in this section that $U_\epsilon$ crosses the threshold only at $z=0$ and $z=a_\epsilon$. Therefore, the formal homoclinic orbit connecting $(u,q)=(0,0)$ to itself is a true solution to the original system with Heaviside nonlinearity structure. Moreover, the homoclinic orbit is as close to the singular homoclinic orbit as we desire. This completes the proof of \cref{thm: E}.
\section*{Discussion}
In the present study, we successfully completed a rigorous proof of the existence and uniqueness (modulo translation) of fast traveling pulse solutions to the standard nonlocal, singularly perturbed neural field system with Heaviside activation functions and linear feedback. In \cref{sec: calc}, we changed the time scale in the two speed index functions due to the $O\left(\f{\epsilon}\right)$ second threshold crossing point. This allowed us to proceed with first-order approximations of the differences between pulse and front speed, not seen in previous literature. We also showed that, provided they exist, traveling fronts with single threshold crossing points have speed index functions with unique positive roots. Therefore, the corresponding Evans function cannot have positive zeros and $\lambda=0$ is always simple.

As a result of the original work of Pinto and Ermentrout \cite{PintoandErmentrout-SpatiallyStructuredActivityinSynapticallyCoupledI.TravelingWaves}, as well as follow-up mathematical analysis (such as  by Pinto et al. \cite{Pinto2005}), there is a heuristically and computationally justified acceptance of much of \cref{thm: E} (especially for nonnegative kernels), but a complete proof of existence and uniqueness is lacking prior to this paper.

In particular, to the author's knowledge, there have been no other results that rigorously study the fast pulse solutions in the Heaviside model when they are broken down by fast and slow representations in inner and outer regions for $\epsilon \ll 1$. Unlike the singular homoclinic orbit, when $\epsilon>0$, the fast and slow dynamics need to be analyzed when they overlap. When $K$ is chosen as the exponential type. we have PDE reductions and applications of classic geometric singular perturbation theory can be applied. We avoid requiring this reduction.

In order to prove \cref{thm: E}, we derive a number of new estimates of the solutions and their wave speeds and widths. The following corollary summarizes first-order approximations of the parameters. Denote $\tau(\epsilon):=\epsilon a(\epsilon)$, a natural change of variable since $a(\epsilon)=O\left(\f{\epsilon}\right)$ is known based on the fast and slow time scales. The functions $f$ and $g$ are discussed at length in \cref{sec: calc}.
\begin{corollary}[First-Order Approximations of $\tau(\epsilon)$ and $c(\epsilon)$] \label{thm: first_order}Consider the unique pair of $C^1$ curves $\tau_\epsilon=\tau(\epsilon)$ and $c_\epsilon=c(\epsilon)$ from \cref{thm: E}. The limits $\tau(\epsilon)\to \tau_0$ and $c(\epsilon)\to c_f$ hold, where
$$
\tau_0=-\f{c_f}{1+\gamma}\ln(2\theta(1+\gamma)-\gamma).
$$
The derivative of the vector $(\tau(\epsilon),c(\epsilon))^T$ satisfies
\begin{equation}
 \begin{pmatrix}
\tau'(0) \\ c'(0)
\end{pmatrix}
=-J^{-1}(\tau_0,c_f,0)
\begin{pmatrix}
f_\epsilon(\tau_0,c_f,0) \\ g_\epsilon(\tau_0,c_f,0)
\end{pmatrix},
\end{equation}
where $f(\tau(\epsilon),c(\epsilon),\epsilon)=U(0)=\theta$ and $g(\tau(\epsilon),c(\epsilon),\epsilon)=U(a(\epsilon))=\theta$ are the speed index equations for the pulse. Hence, $\tau(\epsilon)\approx \tau_0 +\tau'(0)\epsilon$ and $c(\epsilon)\approx c_f +c'(0)\epsilon$ are first-order approximations of $\tau(\epsilon)$ and $c(\epsilon)$, respectively. In particular,
\begin{equation}\label{eq: cp_zero}
c'(0)=-\f{\phi_f'(c_f)}\left\lbrace 2\theta-\f{c_f}\integ{x}{-\infty}[0]<|x|\left(e^{\f{x}{c_f}}+1\right)K(x)>\right\rbrace,
\end{equation}
where $\phi_f$ is the speed index function for the front that satisfies $\phi_f'(c_f)<0$.
\end{corollary}
\subsection{Hypotheses} \label{subsec: hyp}
We state our main hypotheses and then offer some remarks as to why they are necessary.
\begin{itemize}
\item[(H1)] The parameters satisfy $0<\theta<\min\{\f{2},\int_{-\infty}^0 K\}$, \, $\f{\gamma}{1+\gamma}<\theta,\,$ and $\epsilon \ll 1$.
\item[(H2)] When $\epsilon=0$, there exists a traveling front solution $u(x,t)=U_f(x+c_ft)$ to \eqref{eq: intro_scalar} with $U_f(\cdot)<\theta$ on $(-\infty,0)$, $U_f(\cdot)>\theta$ on $(0,\infty)$, $U_f(-\infty)=0$, and $U_f(\infty)=1$. The solution is translation invariant, but we will assume $U_f(0)=\theta$ unless stated otherwise.
\item[(H3)] There exists $\alpha>0$ and $\rho>0$ such that $|K(x)|\leq \alpha e^{-\rho|x|}$. Moreover, assume $K$ is continuous and $\int_\R K = 1$.
\end{itemize}
We require $0<\theta<\int_{-\infty}^0 K$ because it is necessary in order for the front to exist, while $\theta <\f{2}$ is required for the pulse. Often $K$ is symmetrical so these conditions are equivalent. To conclude remarks on (H1), we note that the assumption $\frac{\gamma}{1+\gamma}<\theta$ ensures that the line $Q=\frac{U}{\gamma}$ does not intersect $Q=-U+H(U-\theta)$ when $U>\theta$, which would lead to a second fixed point, and the homoclinic orbit would be replaced by a heteroclinic orbit. Hypothesis (H2) ensures we have a leading order inner solution. Finally, hypothesis (H3) is a common assumption in nonlocal problems. Biologically, long-range synaptic interaction strengths decay sharply, while mathematically, exponential decay ensures that terms of order $O\left(\epsilon^{-k}e^{-\f{|C|}{\epsilon}}\right)$ vanish.

In order to prove \cref{thm: E}, we derive a number of new estimates of the solutions and their wave speeds and widths. Unlike the singular homoclinic orbit, when $\epsilon>0$, the fast and slow dynamics need to be analyzed when they overlap. When $K$ is chosen as the exponential type. we have PDE reductions and applications of classic geometric singular perturbation theory can be applied. We avoid requiring this reduction.

A key part of the argument is summarized via the “speed index equations for the pulse”: define $\tau(\epsilon):=\epsilon a(\epsilon)$ and use
\[
 f(\tau(\epsilon),c(\epsilon),\epsilon)=U(0)=\theta,\qquad g(\tau(\epsilon),c(\epsilon),\epsilon)=U(a(\epsilon))=\theta.
\]
Then $\tau(\epsilon)\to \tau_0$ and $c(\epsilon)\to c_f$, and the derivative of $(\tau(\epsilon),c(\epsilon))^T$ at $\epsilon=0$ is obtained from
\[
 \begin{pmatrix}
\tau'(0) \\ c'(0)
\end{pmatrix}
=-J^{-1}(\tau_0,c_f,0)
\begin{pmatrix}
 f_\epsilon(\tau_0,c_f,0) \\ g_\epsilon(\tau_0,c_f,0)
\end{pmatrix},
\]
which yields first-order approximations $\tau(\epsilon)\approx \tau_0+\tau'(0)\epsilon$ and $c(\epsilon)\approx c_f+c'(0)\epsilon$ (with $c'(0)$ given explicitly in \eqref{eq: cp_zero}).

In establishing the main theorem, we derive a number of new estimates of the solutions and their wave speeds and widths. Unlike the singular homoclinic orbit, when $\epsilon>0$, the fast and slow dynamics need to be analyzed when they overlap. When $K$ is chosen as the exponential type. we have PDE reductions and applications of classic geometric singular perturbation theory can be applied. We avoid requiring this reduction.

A key part of the argument is summarized via the “speed index equations for the pulse”: define $\tau(\epsilon):=\epsilon a(\epsilon)$ and use
\[
 f(\tau(\epsilon),c(\epsilon),\epsilon)=U(0)=\theta,\qquad g(\tau(\epsilon),c(\epsilon),\epsilon)=U(a(\epsilon))=\theta.
\]
Then $\tau(\epsilon)\to \tau_0$ and $c(\epsilon)\to c_f$, and the derivative of $(\tau(\epsilon),c(\epsilon))^T$ at $\epsilon=0$ is obtained from
\[
 \begin{pmatrix}
\tau'(0) \\ c'(0)
\end{pmatrix}
=-J^{-1}(\tau_0,c_f,0)
\begin{pmatrix}
 f_\epsilon(\tau_0,c_f,0) \\ g_\epsilon(\tau_0,c_f,0)
\end{pmatrix},
\]
which yields first-order approximations $\tau(\epsilon)\approx \tau_0+\tau'(0)\epsilon$ and $c(\epsilon)\approx c_f+c'(0)\epsilon$ (with $c'(0)$ given explicitly in \eqref{eq: cp_zero}).,

[Existence and Uniqueness of Fast Pulses]\label{thm: E}
When $0<\epsilon \ll 1,$ there exists a unique $C^2$ (modulo translation) fast traveling pulse solution $(U_\epsilon,Q_\epsilon)$ to the system
\begin{align}\nu_t&=-u-q+ \\integ{y}{\R}[]<K(x-y)H(u(y,t)-\theta)>, \\label{eq: intro_system_1}\\
q_t &=\epsilon (u-\gamma q). \\label{eq: intro_system_2}
\end{align}
In particular, with $z=x+ct$, there exists unique $C^1$ curves $c_\epsilon=c(\epsilon)$ and $a_\epsilon=a(\epsilon)$ such that $U_\epsilon(z)<\theta$ on $(-\infty,0)\cup(a_\epsilon,\infty)$ and $U_\epsilon(z)>\theta$ on $(0,a_\epsilon)$. The parameters satisfy $c_\epsilon\to c_f$ and $a_\epsilon\to +\infty$ as $\epsilon \to 0$, where $c_f>0$ is the unique speed of the front solution to \eqref{eq: intro_scalar}. As a homoclinic orbit, $\mathcal{S}_\epsilon=\Bigl\{(U_\epsilon(z),Q_\epsilon(z)): z\in \R \Bigr\}$ satisfies $d_H(\mathcal{S}_\epsilon,\mathcal{S}_0)\to 0$ as $\epsilon\to 0$. Here,
$$
d_H(\mathcal{S}_\epsilon,\mathcal{S}_0)=\max\left\lbrace \sup_{(U,Q)\in \mathcal{S}_0}d\Bigl(\mathcal{S}_\epsilon,(U,Q)\Bigr),\sup_{(U_\epsilon,Q_\epsilon)\in \mathcal{S}_\epsilon}d\Bigl((U_\epsilon,Q_\epsilon),\mathcal{S}_0\Bigr) \right\rbrace
$$
is the Hausdorff distance between sets in $(U,Q)$ space with respect to the standard Euclidean metric, and $\mathcal{S}_0$ is the singular homoclinic orbit at $\epsilon=0$.

Question:
Consider the nonlocal neural field system
\[
 u_t=-u-q+\int_{\mathbb R} K(x-y)\,H(u(y,t)-\theta)\,dy,
 \qquad
 q_t=\epsilon\,(u-\gamma q),
\]
where $H$ is the Heaviside function, $\gamma>0$ is fixed, and $0<\epsilon\ll 1$. Suppose one looks for traveling-wave solutions of the form $(u(x,t),q(x,t))=(U(z),Q(z))$ with $z=x+ct$. Let
\[
\mathcal S_\epsilon:=\{(U_\epsilon(z),Q_\epsilon(z)):z\in\mathbb R\},
\]
and for sets $A,B\subset \mathbb R^2$ define the Hausdorff distance by
\[
d_H(A,B)=\max\Bigl\{\sup_{p\in B}\inf_{a\in A}\|a-p|,\;\sup_{q\in A}\inf_{b\in B}\,|q-b|\Bigr\},
\]
with $|\cdot|$ the standard Euclidean norm. Let $\mathcal S_0$ denote the singular homoclinic orbit at $\epsilon=0$, and let $c_f>0$ be the unique wave speed of the front for the associated scalar equation
\[
u_t=-u+\int_{\mathbb R} K(x-y)\,H(u(y,t)-\theta)\,dy.
\]
Which statement holds?

For all sufficiently small $\epsilon>0$, there exists a unique $C^2$ fast traveling pulse solution $(U_\epsilon,Q_\epsilon)$ of the system, unique modulo translation in $z$. In particular, there exist unique $C^1$ functions $c_\epsilon=c(\epsilon)$ and $a_\epsilon=a(\epsilon)$ such that
\[
U_\epsilon(z)<\theta \text{ on }(-\infty,0)\cup(a_\epsilon,\infty),
\qquad
U_\epsilon(z)>\theta \text{ on }(0,a_\epsilon).
\]
Moreover,
\[
c_\epsilon\to c_f,
\qquad
 a_\epsilon\to +\infty
\quad\text{as }\epsilon\to 0,
\]
and, viewing $\mathcal S_\epsilon$ as a homoclinic orbit,
\[
d_H(\mathcal S_\epsilon,\mathcal S_0)\to 0
\quad\text{as }\epsilon\to 0.
\]

Choices:
B.
For all sufficiently small $\epsilon>0$, there exists a unique $C^2$ fast traveling pulse solution $(U_\epsilon,Q_\epsilon)$ of the system, unique modulo translation in $z$. In particular, there exist unique $C^1$ functions $c_\epsilon=c(\epsilon)$ and $a_\epsilon=a(\epsilon)$ such that
\[
U_\epsilon(z)<\theta \text{ on }(-\infty,0)\cup(a_\epsilon,\infty),
\qquad
U_\epsilon(z)>\theta \text{ on }(0,a_\epsilon).
\]
Moreover,
\[
c_\epsilon\to c_f,
\qquad
 a_\epsilon\to a_0<\infty
\quad\text{as }\epsilon\to 0,
\]
and, viewing $\mathcal S_\epsilon$ as a homoclinic orbit,
\[
d_H(\mathcal S_\epsilon,\mathcal S_0)\to 0
\quad\text{as }\epsilon\to 0.
\]
C.
For all sufficiently small $\epsilon>0$, there exists a $C^2$ fast traveling pulse solution $(U_\epsilon,Q_\epsilon)$ of the system, unique modulo translation in $z$, with associated $c_\epsilon$ and $a_\epsilon$ such that
\[
U_\epsilon(z)<\theta \text{ on }(-\infty,0)\cup(a_\epsilon,\infty),
\qquad
U_\epsilon(z)>\theta \text{ on }(0,a_\epsilon),
\]
and
\[
c_\epsilon\to c_f,
\qquad
 a_\epsilon\to +\infty
\quad\text{as }\epsilon\to 0.
\]
D.
For all sufficiently small $\epsilon>0$, there exists a unique $C^2$ fast traveling pulse solution $(U_\epsilon,Q_\epsilon)$ of the system, unique modulo translation in $z$. In particular, there exist $C^1$ functions $c_\epsilon=c(\epsilon)$ and $a_\epsilon=a(\epsilon)$ such that
\[
U_\epsilon(z)<\theta \text{ on }(-\infty,0)\cup(a_\epsilon,\infty),
\qquad
U_\epsilon(z)>\theta \text{ on }(0,a_\epsilon).
\]
Moreover, there is a constant $M>0$ independent of $\epsilon$ such that $a_\epsilon\le M$ for all sufficiently small $\epsilon$, and
\[
c_\epsilon\to c_f
\quad\text{as }\epsilon\to 0.
\]
Finally,
\[
d_H(\mathcal S_\epsilon,\mathcal S_0)\to 0
\quad\text{as }\epsilon\to 0.
\]
E.
For all sufficiently small $\epsilon>0$, there exists a unique $C^2$ fast traveling pulse solution $(U_\epsilon,Q_\epsilon)$ of the system, unique modulo translation in $z$. In particular, there exist unique $C^1$ functions $c_\epsilon=c(\epsilon)$ and $a_\epsilon=a(\epsilon)$ such that
\[
U_\epsilon(z)<\theta \text{ on }(-\infty,0)\cup(a_\epsilon,\infty),
\qquad
U_\epsilon(z)>\theta \text{ on }(0,a_\epsilon).
\]
Moreover,
\[
c_\epsilon\to c_f,
\qquad
 a_\epsilon\to +\infty
\quad\text{as }\epsilon\to 0,
\]
and, viewing $\mathcal S_\epsilon$ as a homoclinic orbit,
\[
d_H(\mathcal S_\epsilon,\mathcal S_0)=0
\quad\text{for all sufficiently small }\epsilon>0.
\]

The stem does not state the correct conclusion directly and does not single out choice A. It only sets up the system, notation, and asks which theorem-like statement is valid.

This is largely a theorem-recall item: the answer choices are near-verbatim variants of a precise existence/convergence statement. There is some comparison among competing conclusions, but the task is still close to recognizing the exact theorem formulation.

Moderate reasoning is needed to distinguish asymptotic behaviors such as $a_\epsilon\to+\infty$ versus bounded limits, and convergence versus exact equality. However, it mainly tests precision/recall of a known result rather than substantial derivation or synthesis from the stem.

Several distractors are plausible and target common errors: replacing asymptotic convergence by exact equality, forcing bounded pulse width, or altering the limit of $a_\epsilon$. But choice C is a weaker statement that is also true, which weakens the single-correct-answer structure.

A technically precise MCQ with little answer leakage and reasonably plausible distractors, but it is heavily theorem-recall based and is weakened by the presence of a true-but-weaker option, which reduces answer uniqueness and generative depth.


\begin{Definition}\label{def:phi homogenization}
Fix $\lambda = \lambda(n) > 0$ sufficiently small such that $\sigma = C_1(n)\lambda$ satisfies $4^{n+1}\sigma < c(n)$, where $C_1(n)$ is the one in Lemma \ref{lem:covering Lemma} and $c(n)$ is the one in Proposition \ref{prop:basic}. For $\Lambda\ge \max\{\Lambda_0,\Lambda_1,\Lambda_2\}$, and any nonnegative functions $v \in C(\Omega)$, we define its $\phi$-homogenization $M_{\phi,\Lambda}v(x):=M_{\phi,\lambda,\Lambda}v(x)$ in $\frac{7}{8}\Omega$ through the following maximal averaging type operator:
\begin{equation}\label{eq:phi homogenization}
\begin{split}
M_{\phi,\Lambda}v(x)=  \sup\left\{ t\in\R:\; \exists \ \check{h}(x)<h<  \hat{h}(x),\ s.t., \
\frac{\mu ( \{v> t\} \cap S_h (x)  )}{\mu(S_h (x))} > \lambda \right\}.
\end{split}
\end{equation}
\end{Definition}

\begin{Theorem}\label{thm:uniqueness within periodic}
Let $\mu \not\equiv 0$ be a nonnegative locally finite Borel measure satisfying \eqref{eq:periodic f}, and let $P(x)=\frac{1}{2}x^{\top}Ax+b\cdot x+c$ be a quadratic function with $A\in \mathcal{S}_+^{n\times n}$ satisfying \eqref{eq:A compatible}. Then there exists a unique periodic solution (up to addition of constants) to \eqref{eq:periodic equation ma} with quadratic component $P$.
\end{Theorem}

In this paper, we study convex solutions to the Monge-Amp\`ere equation
\begin{equation}\label{eq:periodic equation ma}
\det D^2 u = \mu \quad \text{in } \mathbb{R}^n,
\end{equation}
where $\mu\not\equiv 0$ is a nonnegative locally finite Borel measure on $\mathbb{R}^n$ and is periodic in $n$ linearly independent directions. By exploiting the affine invariance of the equation, we may, without loss of generality, restrict our analysis to the case where $\mu $ is periodic with respect to the integer lattice $\mathbb{Z}^n := \{(k_1,\dots,k_n) \mid k_i \in \mathbb{Z}\} \subset \mathbb{R}^n$, i.e., for any Borel set $E \subset \mathbb{R}^n$,
\begin{equation}\label{eq:periodic f}
\mu (E+z)=\mu (E),\quad \forall \ z \in \mathbb{Z}^n.
\end{equation}

Let $\mathcal{S}_+^{n\times n}$ denote the set of positive definite symmetric $n\times n$ matrices.
\begin{Definition}[Periodic solution]\label{def:periodic solution}
A \emph{periodic solution} to \eqref{eq:periodic equation ma} is a convex function $u \in C(\mathbb{R}^n)$ satisfying \eqref{eq:periodic equation ma} in the Alexandrov sense, and can be decomposed as
\begin{equation}\label{eq:periodic solution}
u(x) = v(x) +P(x),
\end{equation}
where $v$ is periodic with respect to $\mathbb{Z}^n$ and $P(x)=\frac{1}{2}x^{\top}Ax+b\cdot x+c$ is a quadratic function with $A\in \mathcal{S}_+^{n\times n}$, $b\in\mathbb{R}^n$ and $c\in\mathbb{R}$.
\end{Definition}

Note that if $u$ is a periodic solution to \eqref{eq:periodic equation ma} and is expressed as \eqref{eq:periodic solution}, then the quadratic component $P(x)$ always satisfies the compatibility condition
\begin{equation}\label{eq:A compatible}
 \det A =\mu (\mathbb{T}^n),
\end{equation}
where $\mathbb{T}^n=\mathbb{R}^n/\mathbb{Z}^n$.

Our first result is the uniqueness of periodic solutions.

\begin{equation}\label{eq:periodic equation ma}
\det D^2 u = \mu \quad \text{in } \R^n,
\end{equation}

Let $\mu \not\equiv 0$ be a nonnegative locally finite periodic Borel measure on $\R^n$. We show that any convex solution to the Monge-Amp\`ere equation
\[
\det D^2 u = \mu \quad \text{in } \R^n
\]
admits a unique decomposition (up to addition of constants) as the sum of a quadratic polynomial and a periodic function. This result extends,
in full generality, the earlier works for the case $\mu=f(x)\,\ud x$: when $\log f \in C^\alpha$, it was established by Caffarelli and Li; and
when $\log f$ is merely bounded, it was proved by Li and Lu. Our result thus answers a question raised by Li and Lu. A key ingredient in the proof is a new dichotomous Harnack-type inequality for linearized Monge-Amp\`ere equations with nonnegative periodic measures.

Let $\mathcal{S}_+^{n\times n}$ denote the set of positive definite symmetric $n\times n$ matrices.
\begin{Definition}[Periodic solution]\label{def:periodic solution}
A \emph{periodic solution} to \eqref{eq:periodic equation ma} is a convex function $u \in C(\mathbb{R}^n)$ satisfying \eqref{eq:periodic equation ma} in the Alexandrov sense, and can be decomposed as
\begin{equation}\label{eq:periodic solution}
u(x) = v(x) +P(x),
\end{equation}
where $v$ is periodic with respect to $\mathbb{Z}^n$ and $P(x)=\frac{1}{2}x^{\top}Ax+b\cdot x+c$ is a quadratic function with $A\in \mathcal{S}_+^{n\times n}$, $b\in\R^n$ and $c\in\R$.
\end{Definition}

Our first result is the uniqueness of periodic solutions.

The existence of periodic solutions follows easily from Theorem 2.1 in Li \cite{li1990existence} and its proof. For the uniqueness, when $\mu=f(x)\,\ud x$ is a positive measure, the case $\log f \in C^\alpha$ was proved by Li \cite{li1990existence}, and the case $\log f\in L^\infty$ was shown by Li-Lu \cite{li2019monge}. Our theorem extends these results to general periodic Borel measures.

\begin{Theorem}\label{thm:solutions are periodic}
Let $\mu \not\equiv 0$ be a nonnegative locally finite Borel measure satisfying \eqref{eq:periodic f}, and let $u$ be a convex solution of \eqref{eq:periodic equation ma}. Then $u$ is a periodic solution in the sense of Definition \ref{def:periodic solution}.
\end{Theorem}

\begin{Proposition}\label{prop:semiconcave}
Let $u$ be a convex solution to \eqref{eq:periodic equation ma} with $\mu$ being a $\mathbb{Z}^n/8n$-periodic measure (where $\mathbb{Z}^n := \Gamma_{\{e_1,\cdots,e_n\}}$) satisfying $\mu(\mathbb{Z}^n) = 1$.
Under the nondegeneracy condition
\begin{equation}\label{eq:nondegenerate}
\Delta^2_{i} u(0):=\Delta^2_{e_i} u(0) > 0 \quad \text{for all } 1 \leq i \leq n,
\end{equation}
there exists a unique compatible matrix $A$ satisfying:
\[
\sup_{ \R^n}\Delta_z^2 u=z^{\top}Az ,\quad \forall z \in \mathbb{Z}^n.
\]
\end{Proposition}
\begin{proof}
Without loss of generality, we may assume $u(0) = 0$ and $u \geq 0$.
Let us define $S_h := \{x \in \mathbb{R}^n : u(x) \leq h\}$, and select $\Lambda_u>0$ such that the section $S_{\Lambda_u}$ satisfies $\kappa_{\mathbb{Z}^n}(S_{\Lambda_u}) \geq \Lambda_0^2$.
Let us now normalize $S_h$ such that $B_1(0) \subset T_h^{-1} S_h \subset B_n(0)$ and consider the normalized function
\[
u_{h}(x) =\frac{u(T_h x)}{(\det T_h)^{\frac{2}{n}}} \quad \text{with } \det D^2 u_h= \mu_h=\mu \circ T_h .
\]
Due to the periodicity of $\mu$ and large $h$, we know that
\[
c(n)h^{\frac{n}{2}} \leq \det T_h \leq C(n)h^{\frac{n}{2}},
\]
and thus, $c(n)h^{\frac{n}{2}} \leq |S_h| \leq C(n)h^{\frac{n}{2}}$. Applying Proposition \ref{prop:basic} to $\phi=u_h-h(\det T_h)^{-\frac{2}{n}}$, we obtain that for sufficiently large $h$,
\[
c(n) (h/\Lambda_u)^{\frac{1}{3}} S_{\Lambda_u} \subset S_h \subset C(n) (h/\Lambda_u)^{\frac{2}{3}} S_{\Lambda_u},
\]
and consequently, $B_{c(n) h^{\frac{1}{4}}}\subset S_h\subset B_{C(n) h^{\frac{3}{4}}}$ for all large $h$. That is, for sufficiently large $h$, we now have
\[
\quad c(n)h^{\frac{1}{4}} I\leq T_h \leq C(n)h^{\frac{3}{4}} I.
\]
The transformation law \eqref{eq:transformation law} then yields the following estimates at corresponding points
\begin{equation}\label{eq:transformh}
c(n)h^{-\frac{1}{2}}\Delta^2_{e_i} u \leq \Delta^2_{T_h^{-1} e_i} u_h \leq C(n)h^{\frac{1}{2}}\Delta^2_{e_i} u.
\end{equation}

\textbf{Step 4.} Suppose for some sequence $h \to \infty$, the rescaled functions $\tilde{u}_h$ locally converge to a quadratic form $P_A(x) = \frac{1}{2}x^{\top}A x$ for $A \in \mathcal{S}_+^{n\times n}$ with $\det A = 1$.
For each fixed direction $i$, we denote
\[
\alpha=\sup_{\R^n}\Delta^2_{e_i} u,\quad \beta= e_i^{\top} A e_i.
\]
Note that the estimate \eqref{eq:w21 estimate} continues to hold for $\tilde{u}_{h}$, from which we derive $\alpha \geq \beta$.
We claim
\begin{equation}\label{eq:w2infty sup}
\alpha=\beta.
\end{equation}
Assume by contradiction that $\alpha = \beta + 4s$ for some $s > 0$.
Through rescaling from infinity, we may assume without loss of generality that there exist points $x_h \in B_{1/8}$ satisfying
\[
\alpha- \Delta^2_{h^{\frac{1}{2}}e_i} \tilde{u}_{h}(x_h)= \inf_{B_{1/8}} (\alpha- \Delta^2_{h^{\frac{1}{2}} e_i} \tilde{u}_{h} ) \leq a_h,
\]
where $a_h \to 0 $ as $h \to \infty$. Define the normalized function $v = (\alpha - \Delta^2_{h^{\frac{1}{2}}e_i} \tilde{u}_{h})/a_h$ and set $\delta = s/[C(\beta + 2s)]$. Applying Lemma \ref{lem:Theorem 7.3.1 modify} yields with $\phi=\tilde{u}_h-1$ and $\Omega=S_{1,h}=\{\tilde{u}_h \leq 1\} $, we obtain that
\[
\tilde{\mu}_h ( \{v\leq M_1(n,\delta) \} \cap S_{1,h} ) \geq (1-\delta) \tilde{\mu}_h(S_{1,h} ),
\]
where $\tilde{\mu}_h=\tilde{f}_{h}\,\ud x$. For sufficiently small $a_h$, we consequently obtain
\[
\tilde{\mu}_h\left(\left\{\Delta^2_{h^{\frac{1}{2}} e_i} \tilde{u}_h>\beta +2 s\right\} \cap S_{1,h}\right)= \tilde{\mu}_h\left(\left\{v <\frac{2s}{a_h}\right\} \cap S_{1,h}\right)\geq (1-\delta) \tilde{\mu}_h (S_{1,h}).
\]
This leads to the integral lower bound
\[
\int_{S_{1,h}} \Delta^2_{h^{\frac{1}{2}} e_i} \tilde{u}_h \ud \tilde{\mu}_h \geq (1-\delta) (\beta+2s) \tilde{\mu}_h (S_{1,h})\geq
(\beta+s) \tilde{\mu}_h (S_{1,h}).
\]
This contradicts the uniform estimate \eqref{eq:w21 estimate} after performing a suitable rescaling.

\begin{proof}[Proof of Theorems \ref{thm:uniqueness within periodic} and \ref{thm:solutions are periodic}]
Let $u$ be a global solution of \eqref{eq:periodic equation ma}.
Clearly, $u$ cannot be linear on any ray segment. Indeed, if it were linear on a ray, after subtracting its supporting hyperplane on this ray and by using the convexity of $u$, $u$ would be bounded within any cylinder having that ray as its axis. This would imply that the associated measure $\mu$ is finite on any strictly smaller cylinder, contradicting the assumptions that $\mu$ is periodic and $\mu \not\equiv 0$. Therefore, there exists a sufficiently large integer $R$ such that $S_1(0) \subset B_R(0)$.
We now rescale $u$ by defining $u(8nRx)/64n^2R^2$. For simplicity, we continue to denote this rescaled function as $u$. Under this scaling, one can verify that condition \eqref{eq:nondegenerate} in Proposition \ref{prop:semiconcave} is satisfied. Hence, by Proposition \ref{prop:semiconcave}, there exists a compatible matrix $A \in \mathcal{S}_+^{n \times n}$ such that
\[
\sup_{ \R^n}\Delta_z^2 u=z^{\top}Az,\quad \forall z \in \mathbb{Z}^n.
\]
Let $w$ be a periodic solution to \eqref{eq:periodic equation ma} in the sense of Definition \ref{def:periodic solution} with quadratic part $\frac{1}{2}z^{\top} A z $. Then $\Delta_z^2 w=z^{\top}Az$. Let
\[
v(x)=u(x)-w(x).
\]
By subtracting a suitable linear function, we now assume that $v(0)=0$ and
\[
v(e_i)=v(-e_i),\quad 1\leq i\leq n.
\]
Noting that $\sup_{\R^n} \Delta_z^2 v = \sup_{\R^n} \Delta_z^2 (u - w) = 0$, we then find that
\[
v(\pm ke_i)\leq 0 \quad \text{for all } k\in \mathbb{Z},\ 1\leq i\leq n.
\]

The existence of periodic solutions follows easily from Theorem 2.1 in Li \cite{li1990existence} and its proof.  For the uniqueness, when $\mu=f(x)\,\mathrm{d} x$ is a positive measure, the case $\log f \in C^\alpha$ was proved by Li \cite{li1990existence}, and the case  $\log f\in L^\infty$ was shown by Li-Lu \cite{li2019monge}. Our theorem extends these results to general periodic Borel measures.

Our second result is the classification below, which extends the main results  of Caffarelli-Li \cite{caffarelli2004liouville} and Li-Lu \cite{li2019monge} to the full generality. In particular, it provides a complete answer to Question 1.1 in \cite{li2019monge}.

\begin{Theorem}\label{thm:solutions are periodic}
Let $\mu \not\equiv 0$ be a nonnegative locally finite Borel measure satisfying \eqref{eq:periodic f}, and let $u$ be a convex solution of \eqref{eq:periodic equation ma}. Then $u$ is a periodic solution in the sense of Definition \ref{def:periodic solution}.
\end{Theorem}

Caffarelli-Li \cite{caffarelli2004liouville} first proved Theorem \ref{thm:solutions are periodic} for positive measures $\mu=f(x)\,\mathrm{d} x$ with $\log f \in C^\alpha$, and conjectured the extension to merely bounded $\log f$. This conjecture was resolved by Li-Lu \cite{li2019monge}, who further asked whether the result holds for the nonnegative case $f \geq 0$ with $f \not\equiv 0$. Theorem \ref{thm:solutions are periodic} affirmatively answers this question -- and indeed establishes it in full generality.

\begin{Remark}
When $\mu \equiv 0$, equation \eqref{eq:periodic equation ma} reduces to the homogeneous Monge-Amp\`ere equation $\det D^2u = 0$. The solutions are precisely those functions that are linear along certain directions (see, e.g., Caffarelli-Nirenberg-Spruck \cite{caffarelli1986dirichlet}).
In particular, writing $x = (x',x_n) \in \mathbb{R}^{n-1} \times \mathbb{R}$, any convex function of the form $u(x) = w(x')$ is a solution, and such solutions are not necessarily periodic.
\end{Remark}

Since $\mu$ is assumed to be nonnegative,  equation \eqref{eq:periodic equation ma} represents a possibly degenerate Monge-Amp\`ere equation, for which many arguments in  \cite{caffarelli2004liouville,li2019monge} do not apply.  The principal challenge stems from the behavior of $u$ on small sections: the associated measure $\mu$ may not satisfy the doubling condition. This failure implies that sections lose the engulfing property, consequently invalidating the Vitali or  Besicovitch covering lemma which is essential for the Calder\'on-Zygmund decomposition on sections.
Therefore, we cannot employ directly the weak Harnack inequalities of Caffarelli-Guti\'errez \cite{caffarelli1997properties} for subsolutions or supersolutions of linearized Monge-Amp\`ere equations.

Nonetheless, by exploiting the equation's periodic structure, we derive truncated weak Harnack inequalities for both supersolutions and subsolutions via a maximal-type operator (see Definition \ref{def:phi homogenization}). This constitutes one of the key innovations of the present work. For supersolutions, we address the challenge by excluding contributions from small sections. Theorem \ref{thm:harnack super} establishes decay estimates for homogenized level-sets of supersolutions within our framework. For subsolutions, for which standard $L^{\infty}$ bounds fail, Theorem \ref{thm:harnack sub} establishes an alternative dichotomy regarding their growth behavior: either the $L^{\infty}$ bound holds or the $L^{\infty}$ norm grows exponentially. Finally, we obtain  in Theorem \ref{thm:harnack} a Harnack inequality with a quantitatively controlled error term.

After overcoming this critical difficulty arising from the degeneracy of equation \eqref{eq:periodic equation ma}, the rest proof focuses on analyzing the second-order difference quotient $\Delta^2_e u$ and the deviation between $u$ and its periodic counterpart. As observed in \cite{caffarelli2004liouville}, these quantities naturally arise as subsolutions and supersolutions to carefully selected linearized Monge-Amp\`ere equations. This is another point where the periodicity of the measure plays a crucial role. We then establish the quadratic behavior of solutions near infinity, extending the arguments in \cite{caffarelli2004liouville,li2019monge} for positive measures $\mu=f(x)\mathrm{d} x$ to  the degenerate case considered here.  In this step, we must also overcome additional difficulties due to the degeneracy of
 \eqref{eq:periodic equation ma}. Consequently, when Theorem \ref{thm:harnack} is applied to the difference of two solutions, exponential growth is precluded. Hence, the classical Harnack inequality holds for this difference, and our theorems would follow.

This paper is organized as follows. In Section \ref{sec:harnack}, we study convex functions whose Monge-Amp\`ere measure has small period, establishing a dichotomous Harnack type  inequality for linearized Monge-Amp\`ere equations. Section \ref{sec:semiconvavity} investigates  the uniqueness of compatible quadratic component of each solution in the sense of Proposition \ref{prop:semiconcave}. Finally, in Section \ref{sec:main theorem}, we prove  Theorems \ref{thm:uniqueness within periodic} and \ref{thm:solutions are periodic}.

The post-theorem introduction says: existence of periodic solutions “follows easily from Theorem 2.1 in Li \cite{li1990existence} and its proof,” while Theorem~\ref{thm:uniqueness within periodic} “extends these results to general periodic Borel measures.” It then explains the main difficulty in proving the results (including Theorem~\ref{thm:uniqueness within periodic}) in the nonnegative/degenerate setting: “the associated measure $\mu$ may not satisfy the doubling condition,” so “sections lose the engulfing property,” which “invalidat[es] the Vitali or Besicovitch covering lemma … essential for the Calder\'on-Zygmund decomposition on sections,” and therefore one “cannot employ directly the weak Harnack inequalities of Caffarelli-Guti\'errez.”

To overcome this, “by exploiting the equation's periodic structure,” the authors “derive truncated weak Harnack inequalities for both supersolutions and subsolutions via a maximal-type operator.” For supersolutions they do so “by excluding contributions from small sections,” obtaining “decay estimates for homogenized level-sets” (Theorem~\ref{thm:harnack super}); for subsolutions (where “standard $L^{\infty}$ bounds fail”), they prove a “dichotomy” (Theorem~\ref{thm:harnack sub}): “either the $L^{\infty}$ bound holds or the $L^{\infty}$ norm grows exponentially”; combining these yields “a Harnack inequality with a quantitatively controlled error term” (Theorem~\ref{thm:harnack}).

“After overcoming this critical difficulty,” the remaining proof “focuses on analyzing the second-order difference quotient $\Delta^2_e u$ and the deviation between $u$ and its periodic counterpart,” which “arise as subsolutions and supersolutions to carefully selected linearized Monge-Amp\`ere equations,” with periodicity “play[ing] a crucial role.” They then “establish the quadratic behavior of solutions near infinity,” extending arguments from \cite{caffarelli2004liouville,li2019monge} to the degenerate case. Finally, applying Theorem~\ref{thm:harnack} “to the difference of two solutions,” “exponential growth is precluded,” so “the classical Harnack inequality holds for this difference,” and “our theorems would follow.”

The post-theorem introduction says: existence of periodic solutions “follows easily from Theorem 2.1 in Li \cite{li1990existence} and its proof,” while in establishing the main theorem it “extends these results to general periodic Borel measures.” It then explains the main difficulty in proving the results (including the main theorem) in the nonnegative/degenerate setting: “the associated measure $\mu$ may not satisfy the doubling condition,” so “sections lose the engulfing property,” which “invalidat[es] the Vitali or Besicovitch covering lemma … essential for the Calder\'on-Zygmund decomposition on sections,” and therefore one “cannot employ directly the weak Harnack inequalities of Caffarelli-Guti\'errez.”

To overcome this, “by exploiting the equation's periodic structure,” the authors “derive truncated weak Harnack inequalities for both supersolutions and subsolutions via a maximal-type operator.” We first use the following theorem for supersolutions:
\begin{Theorem}\label{thm:harnack super}
Assume that $v \in C^2(\Omega)$ is a nonnegative supersolution of \eqref{eq:linearized ma phi} in $\Omega$.
Then there exist positive constants  $\Lambda_6(n)$, $\epsilon(n) $ and $C(n) $ such that if
\[
\inf_{B_{1/4}} v \leq 1,
\]
then for $\Lambda\ge \Lambda_6(n)$, we have
\begin{equation}\label{eq:harnack super}
\mu(\{M_{\phi,\Lambda}v >  t\} \cap B_{1/2})< Ct^{-\epsilon} .
\end{equation}
\end{Theorem}
For subsolutions (where “standard $L^{\infty}$ bounds fail”), they prove the following “dichotomy”:
\begin{Theorem}\label{thm:harnack sub}
Assume that $v \in C^2(\Omega)$ with $v \geq 0$ is a subsolution of \eqref{eq:linearized ma phi} in $\Omega$.
Then, there exist positive constants $\Lambda_7(n)$ and $\beta(n)$ such that for any $\epsilon > 0$, there is a constant $C_2(n, \epsilon)$ with the following property: If $\Lambda\ge \Lambda_7(n)$ and
\[
\min\{\mu(\{v >  t\}  \cap B_{1/2}), \mu(\{M_{\phi,\Lambda}v >  t\} \cap B_{1/2})  \} \leq t^{-\epsilon}   \quad  \forall t>0,
\]
then either
\[
\sup_{B_{1/4} } v  \leq C_2(n,\epsilon),
\]
or for $\kappa =\kappa(B_1)\geq \Lambda^2$ that
\[
\sup_{B_{1/2} } v  \geq e^{\beta \kappa^{\frac{1}{3}} } \sup_{B_{1/4} }v.
\]
\end{Theorem}
Combining these yields “a Harnack inequality with a quantitatively controlled error term,” namely the following theorem:
\begin{Theorem}\label{thm:harnack}
Let $\Omega$ and $\widetilde{\Omega}$ be open convex subsets of $\mathbb{R}^n$ satisfying $B_1\subset \Omega \subset B_n, B_1\subset \widetilde{\Omega} \subset B_n$, $n\geq 2$, and let $\phi\in C^2(\overline{\Omega})$ and $\tilde{\phi}\in C^2(\overline{\widetilde{\Omega}})$ be convex functions satisfying,
\[
\mu_\phi =\mu \quad \text{in } \Omega, \quad \phi=0 \quad \text{on } \partial \Omega,
\]
\[
\mu_{\tilde{\phi}} =\mu \quad \text{in }  \widetilde{\Omega}, \quad \tilde{\phi}=0 \quad \text{on } \partial \widetilde{\Omega},
\]
where $\mu$ is a periodic Borel measure defined on $\R^n$ satisfying \eqref{eq:period gamma},  \eqref{eq:normalized assumption}, and \eqref{eq:small period assumption}. Assume that $v\in C^2(\Omega) \cap C^2(\widetilde{\Omega})$ with $v\geq 0$ satisfies
\[
L_{\phi}v \leq 0   \quad  \text{in } \Omega, \quad L_{\tilde{\phi}}v \geq 0   \quad  \text{in }\widetilde{\Omega}.
\]

Then there exist positive constants $\beta(n)$, $\Lambda_9(n)$,  and $C(n)$ such that if $\Lambda>\Lambda_9(n)$, then either
\[
\sup_{B_{1/4}}v\leq C\inf_{B_{1/4}}v,
\]
or for $\kappa =\kappa(B_1)\geq \Lambda^2$ that
\[
\sup_{B_{1/2}} v  \geq e^{\beta \kappa^{\frac{1}{6}}}\sup_{B_{1/4}}v.
\]
Consequently, we have
\[
\sup_{B_{1/4}}v \leq C\inf_{B_{1/4}}v+e^{-\beta \kappa^{\frac{1}{6}}}\sup_{B_{1/2}} v.
\]
\end{Theorem}

“After overcoming this critical difficulty,” the remaining proof “focuses on analyzing the second-order difference quotient $\Delta^2_e u$ and the deviation between $u$ and its periodic counterpart,” which “arise as subsolutions and supersolutions to carefully selected linearized Monge-Amp\`ere equations,” with periodicity “play[ing] a crucial role.” They then “establish the quadratic behavior of solutions near infinity,” extending arguments from \cite{caffarelli2004liouville,li2019monge} to the degenerate case. Finally, applying the preceding theorem (the Harnack inequality with controlled error term) “to the difference of two solutions,” “exponential growth is precluded,” so “the classical Harnack inequality holds for this difference,” and this completes the proof strategy for the main theorem.

\label{thm:uniqueness within periodic}
Let $\mu \not\equiv 0$ be a nonnegative locally finite Borel measure satisfying
\begin{equation}\label{eq:periodic f}
\mu (E+z)=\mu (E),\quad \forall \ z \in \mathbb{Z}^n.
\end{equation}
and let $P(x)=\frac{1}{2}x^{\top}Ax+b\cdot x+c$ be a quadratic function with $A\in \mathcal{S}_+^{n\times n}$ satisfying
\begin{equation}\label{eq:A compatible}
 \det A =\mu (\mathbb{T}^n),
\end{equation}
Then, to prove the main theorem, we show that there exists a unique periodic solution (up to addition of constants) to
\begin{equation}\label{eq:periodic equation ma}
\det D^2 u = \mu \quad \text{in } \R^n,
\end{equation}
with quadratic component $P$.

Question: Let \(\mu \not\equiv 0\) be a nonnegative locally finite Borel measure on \(\mathbb{R}^n\) that is periodic with respect to \(\mathbb{Z}^n\), meaning that \(\mu(E+z)=\mu(E)\) for every Borel set \(E\subset \mathbb{R}^n\) and every \(z\in \mathbb{Z}^n\). Let \(P(x)=\frac12 x^{\top}Ax+b\cdot x+c\), where \(A\in \mathcal{S}_+^{n\times n}\) is positive definite symmetric, \(b\in\mathbb{R}^n\), \(c\in\mathbb{R}\), and \(\det A=\mu(\mathbb{T}^n)\) with \(\mathbb{T}^n=\mathbb{R}^n/\mathbb{Z}^n\). A periodic solution with quadratic component \(P\) means a convex function \(u\in C(\mathbb{R}^n)\) satisfying \(\det D^2u=\mu\) in \(\mathbb{R}^n\) in the Alexandrov sense and admitting a decomposition \(u(x)=v(x)+P(x)\), where \(v(x+z)=v(x)\) for all \(z\in\mathbb{Z}^n\). Which statement holds?

There exists a periodic solution with quadratic component \(P\), and it is unique up to addition of constants. Equivalently, there exists a convex \(u\in C(\mathbb{R}^n)\) solving \(\det D^2u=\mu\) in the Alexandrov sense such that \(u(x)=v(x)+P(x)\) for some \(\mathbb{Z}^n\)-periodic function \(v\), and whenever \(u_1=u_1^{\mathrm{per}}+P\) and \(u_2=u_2^{\mathrm{per}}+P\) are two such periodic solutions, \(u_1-u_2\) is constant on \(\mathbb{R}^n\).

Choices:
B. There exists a periodic solution with quadratic component \(P\), and it is unique in the strict sense. Equivalently, there exists a convex \(u\in C(\mathbb{R}^n)\) solving \(\det D^2u=\mu\) in the Alexandrov sense such that \(u(x)=v(x)+P(x)\) for some \(\mathbb{Z}^n\)-periodic function \(v\), and whenever \(u_1=u_1^{\mathrm{per}}+P\) and \(u_2=u_2^{\mathrm{per}}+P\) are two such periodic solutions, one has \(u_1=u_2\) on \(\mathbb{R}^n\).
C. There exists at most one periodic solution with quadratic component \(P\) up to addition of constants. Equivalently, whenever \(u_1=u_1^{\mathrm{per}}+P\) and \(u_2=u_2^{\mathrm{per}}+P\) are two such periodic solutions of \(\det D^2u=\mu\) in the Alexandrov sense, \(u_1-u_2\) is constant on \(\mathbb{R}^n\).
D. For every quadratic function \(P(x)=\frac12 x^{\top}Ax+b\cdot x+c\) with \(A\in \mathcal{S}_+^{n\times n}\), there exists a periodic solution with quadratic component \(P\), and it is unique up to addition of constants; in particular, the compatibility condition \(\det A=\mu(\mathbb{T}^n)\) is not needed for existence or uniqueness.
E. There exists a periodic solution with quadratic component \(P\), and any two such periodic solutions differ by an affine function on \(\mathbb{R}^n\). Equivalently, there exists a convex \(u\in C(\mathbb{R}^n)\) solving \(\det D^2u=\mu\) in the Alexandrov sense such that \(u(x)=v(x)+P(x)\) for some \(\mathbb{Z}^n\)-periodic function \(v\), and whenever \(u_1=u_1^{\mathrm{per}}+P\) and \(u_2=u_2^{\mathrm{per}}+P\) are two such periodic solutions, \(u_1-u_2=\ell\) for some affine function \(\ell\).

The stem gives hypotheses and definitions but does not explicitly state the conclusion. It does not directly reveal that existence plus uniqueness up to constants is the correct option.

The item is close to a theorem-recall question: under the stated assumptions, the student is asked to identify the correct conclusion. However, the alternatives are meaningfully varied, so it is not a pure verbatim restatement.

Some reasoning is needed to distinguish strict uniqueness, uniqueness up to constants, mere uniqueness without existence, and incorrect affine ambiguity. But the task is still mainly recognition of the theorem rather than substantial derivation.

The distractors are strong and mathematically plausible: they test common overstatements and confusions about compatibility conditions and uniqueness classes. They are distinct and well-targeted.

A solid theorem-identification MCQ with strong distractors and little answer leakage, though it leans more toward recall of a known result than deep generative reasoning.