\label{th:main}
Fix a truncation parameter $m_0\ge1$ and
assume that:
\begin{enumerate}[---]
\item the space dimension is sufficiently large depending on $m_0$, in the sense that
\begin{equation}\label{eq:def de d_0}
d\ge d_0=\left\{\begin{array}{cl}
2,&\text{if $m_0=1$},\\
8,&\text{if $m_0=2$},\\
28m_0+70,&\text{if $m_0\ge3$},
\end{array}\right.\end{equation}
\item the diffusion constant $\kappa$ is sufficiently large, in the sense that $\kappa\ge C_{m_0}\int_{\mathbb R^d}|k|\V(k)\,dk$ for some constant~$C_{m_0}>0$ depending only on $m_0$. In case $m_0=1$, this condition can be dropped.
\end{enumerate}
Then, for any initial condition $\mathfrak G\in C_c^\infty(\mathbb R^d)$, the solution $(G_m^{N,m_0})_{0\le m\le m_0}$ of the simplified hierarchy~\eqref{eq:hierarchy-simpl}--\eqref{eq:hierarchy-simpl-ci} satisfies that the time-rescaled tagged-particle density
\[(\tau,v)\mapsto G_0^{N,m_0}(N\tau,v)\]
converges strongly in $L^2_\loc(\mathbb R^+;L^2(M\, dv))$ to the solution $G_0$ of the Fokker-Planck equation~\eqref{eq:FP-exp-re}.
More precisely, we have
\begin{equation*}
\Big(\int_0^\infty e^{-2\tau}\|G_0^{N,m_0}(N\tau)-G_0(\tau)\|_{L^2(M\,dv)}^2\,d\tau\Big)^\frac12 \,\lesssim\,N^{-\frac1{12}}\|\<(\nabla_{v_0},v_0)\>^{4m_0+10}\mathfrak G\|_{L^2(M\, dv)},
\end{equation*}
and this bound can be improved to $N^{-1}\|\<(\nabla_{v_0},v_0)\>^{4m_0+21}\mathfrak G\|_{L^2(M\,dv)}$ if $d\ge 28m_0+147$.

\begin{eqnarray}
A_0(v)&:=&(B_0\ast M)(v),\label{eq:coeff-LB-re}\\
B_0(v)&:=&\frac{\Lambda_\Vc}{|v|}\Big(\Id-\frac{v\otimes v}{|v|^2}\Big)=\Lambda_\Vc\nabla^2|v|,\nonumber
\end{eqnarray}

\begin{equation}\label{eq:FN-diag-rem}
F_N\,:=\,
N\big(\tfrac{i}{\sqrt N}\big)^{2\ell+m+2}~{\tiny\begin{tikzpicture}[baseline={([yshift=-3ex]current bounding box.center)},scale=0.8]
\begin{scope}[every node/.style={circle,draw,fill=white,inner sep=0pt,minimum size=3pt}]
\node (0) at (-0.5,0) {};
\node (1) at (0.5,0) {};
\node (2) at (0,0.6) {};
\node (3) at (2,0) {};
\node (4) at (1.5,0.6) {};
\node (5) at (2.5,0) {};
\node (6) at (-1,-0.5) {};
\node (7) at (-3,-1.3) {};
\end{scope}
\draw[decorate,decoration={snake,amplitude=1pt,segment length=3pt}] (0) -- (1);
\draw[decorate,decoration={snake,amplitude=1pt,segment length=3pt}] (1) -- (3);
\draw[decorate,decoration={snake,amplitude=1pt,segment length=3pt}] (3) -- (5);
\draw (0) -- (-0.5,0.6);
\draw[decorate,decoration={snake,amplitude=1pt,segment length=3pt}] (-0.5,0.6) -- (2);
\draw (2) -- (0,0.3);
\draw[decorate,decoration={snake,amplitude=1pt,segment length=3pt}] (0,0.3) -- (0.5,0.3);
\draw (0.5,0.3) -- (1);
\draw (4) -- (1.5,0.3);
\draw[decorate,decoration={snake,amplitude=1pt,segment length=3pt}] (1.5,0.3) -- (2,0.3);
\draw (2,0.3) -- (3);
\draw[decorate,decoration={snake,amplitude=1pt,segment length=3pt}] (2) -- (4);
\draw[decorate,decoration={snake,amplitude=1pt,segment length=3pt}] (4) -- (2.5,0.6);
\draw (2.5,0.6) -- (5);
\draw[decorate,decoration={snake,amplitude=1pt,segment length=3pt}] (0) -- (-1,0);
\draw (-1,0) -- (6);
\draw[decorate,decoration={snake,amplitude=1pt,segment length=3pt}] (2.5,-0.5) -- (6);
\draw[decorate,decoration={snake,amplitude=1pt,segment length=3pt}] (6) -- (-1.5,-0.5);
\draw (-1.5,-0.5) -- (-1.5,-0.65);
\draw[dotted] (-1.5,-0.5) -- (-1.5,-1);
\draw (7) -- (-3,-1.1);
\draw[dotted] (-3,-1.1) -- (-3,-0.75);
\draw[decorate,decoration={snake,amplitude=1pt,segment length=3pt}] (7) -- (2.5,-1.3);
\node at (1,0.3) {\ldots};
\node at (1,0.8) {($\ell$ times)};
\node at (-2.3,-0.5) {($m$ times)};
\end{tikzpicture}}~\Lc g^{N,m_0}_m.
\end{equation}

\begin{equation}\label{eq:coeff-LB-re-LB!}
B(v,w)=\int_{\R^d}(k\otimes k)\,\pi\V(k)^2\frac{\delta(k\cdot w)}{|\e(k,k\cdot v)|^2}\,\frac{dk}{(2\pi)^{d}}.
\end{equation}

\begin{equation}\label{eq:orthogonality0}
\int_{\T^d\times\R^d}G_{N,m}(z_0,\ldots,z_m)\,M(z_j)\,dz_j=0,\qquad\text{for all $1\le j\le m$},
\end{equation}

\begin{eqnarray}
A(v)&:=&\int_{\R^d}B(v,v-v_*)M(v_*)\,dv_*,\label{eq:coeff-LB}\\
B(v,w)&:=&\sum_{k\in2\pi\Z^d}(k\otimes k)\,\pi\V(k)^2\frac{\delta(k\cdot w)}{|\e(k,k\cdot v)|^2},\nonumber\\
\e(k,k\cdot v)&:=&1+\V(k)\int_{\R^d}\frac{k\cdot \nabla M(v_*)}{k\cdot(v-v_*)-i0}\,dv_*.\nonumber
\end{eqnarray}

\subsection{General overview}
Consider the dynamics of a tagged particle (labeled `$0$') in a system of $N+1$ interacting particles in the $d$-dimensional box $\mathbb T^d=[-\frac12,\frac12]^d$ with periodic boundary conditions, as described by Newton's equations
\begin{equation*}
\left\{\begin{array}{l}
\tfrac{d}{dt}X_{j}=V_j,\\[2mm]
\tfrac{d}{dt}V_{j}=-\frac1N\sum_{0\le l\le N}\nabla\Vc(X_j-X_l),\qquad0\le j\le N,\\[2mm]
(X_{j},V_j)|_{t=0}=(X_j^\circ,V_j^\circ),
\end{array}\right.
\end{equation*}
where $\{Z_j:=(X_j,V_j)\}_{0\le j\le N}$ is the set of positions and velocities of the particles in the phase space $\mathbb T^d\times\mathbb R^d$, where $\Vc:\mathbb T^d\to\mathbb R$ is a (long-range) interaction potential, and where the mean-field scaling is considered. For simplicity, we focus throughout this work on the case of a smooth potential~$\Vc\in C^\infty_c$.
In terms of a probability density $F_N$ on the $N$-particle phase space $(\mathbb T^d\times\mathbb R^d)^N$, this system of ODEs for trajectories leads formally to the Liouville equation
\begin{equation}\label{eq:Liouville}
\left\{\begin{array}{l}
\partial_tF_N+\sum_{0\le j\le N}v_j\cdot\nabla_{x_j}F_N\,=\,\frac1N\sum_{0\le j, l\le N}\nabla\Vc(x_j-x_l)\cdot\nabla_{v_j}F_N,\\[1mm]
F_N|_{t=0}\,=\,F_N^\circ.
\end{array}\right.
\end{equation}
For this system, we aim to study one of the predictions of the Lenard-Balescu theory, that is, the slow thermalization of the tagged particle `$0$' when the background particles are initially at thermal equilibrium. For simplicity, we assume that the tagged particle has initially spatially-homogeneous distribution: this ensures that the mean-field force on the tagged particle vanishes, so that thermalization becomes the leading effect. In other words, we choose initially
\[F_N^\circ(z_0,\ldots,z_N)\,=\,f^\circ(v_0)M_{N}(z_1,\ldots,z_N),\qquad z_j=(x_j,v_j),\]
where $f^\circ\in \mathcal{P}(\mathbb R^d)$ is the initial velocity density of the tagged particle
and where $M_{N}$ stands for the Gibbs equilibrium
\[M_{N}(z_1,\ldots,z_N)\,:=\,Z_{N}^{-1}\exp\bigg(-\frac\beta2\sum_{1\le j\le N}|v_j|^2-\frac\beta{2N}\sum_{1\le j,l\le N\atop j\ne l}\Vc(x_j-x_l)\bigg),\]
for some fixed inverse temperature $\beta\in(0,\infty)$ and normalization factor $Z_{N}$.
We emphasize that background particles are exchangeable
and that the system is invariant under spatial translations.
In this setting, due to interactions with the background, the Lenard-Balescu theory predicts that the tagged particle should slowly thermalize on timescales $t\gg N$ and progressively acquire a Maxwellian velocity distribution as the background itself,
\[M(v)\,:=\,(\tfrac\beta{2\pi})^\frac d2e^{-\frac\beta2|v|^2}.\]
More precisely, focussing on the velocity distribution of the tagged particle
\[f_{N,0}(t,v)\,:=\,\int_{\mathbb T^d\times(\mathbb T^d\times\mathbb R^d)^N}F_{N}(t,x,v,z_1,\ldots,z_N)\,dxdz_1\ldots dz_N,\]
it is predicted in~\cite{Piasecki-81,PS-87,DSR-21} (see also~\cite{RR-60,TGM-64} for the corresponding test particle problem without back-reaction) that the time-rescaled density $f_{N,0}(N\tau,v)$ should converge as $N\uparrow\infty$ to the solution~$f(\tau,v)$ of the linearized Lenard-Balescu equation at Maxwellian equilibrium (without loss term), which takes form of the following Fokker-Planck equation,
\begin{equation}\label{eq:FP-exp}
\left\{\begin{array}{l}
\partial_\tau f=\Div_v(A (\nabla+\beta v)f),\\
f|_{\tau=0}=f^\circ,
\end{array}\right.
\end{equation}
with diffusion tensor field $A$ given by the (periodic) Lenard-Balescu formula
\begin{eqnarray}
A(v)&:=&\int_{\mathbb R^d}B(v,v-v_*)M(v_*)\,dv_*,\label{eq:coeff-LB}\\
B(v,w)&:=&\sum_{k\in2\pi\mathbb Z^d}(k\otimes k)\,\pi\V(k)^2\frac{\delta(k\cdot w)}{|\varepsilon(k,k\cdot v)|^2},\nonumber\\
\varepsilon(k,k\cdot v)&:=&1+\V(k)\int_{\mathbb R^d}\frac{k\cdot \nabla M(v_*)}{k\cdot(v-v_*)-i0}\,dv_*.\nonumber
\end{eqnarray}
In this formula, the dispersion function $\varepsilon$ modulating the collision kernel $B$ accounts for collective screening effects at equilibrium.
This Fokker-Planck equation~\eqref{eq:FP-exp} quantifies the thermalization
\[f(\tau,v)\to M(v)\qquad\text{as $\tau=N^{-1}t\uparrow\infty$.}\]
A rigorous justification is still beyond reach at the moment, and we refer to~\cite{VW-18,Winter-21,DSR-21,MD-21} for some partial results on the topic. To date, the best result is the consistency obtained in~\cite{DSR-21}, which only holds at best for relatively short times $\tau=N^{-1}t\ll N^{-3/4}$, thus missing the thermalization timescale $\tau\sim1$. We emphasize that the difficulty includes understanding the emergence of irreversibility, which here would only occur on long times as a fluctuation around the (trivial) mean-field behavior. In the present work, we introduce a simplified, hierarchical toy model for which a full derivation can be achieved, which will shed some new light on the problem.

\subsection{BBGKY approach}
To motivate our simplified model, we start by briefly recalling the standard BBGKY framework for addressing the above thermalization problem.
Intuitively, the slow relaxation of the tagged particle arises from the nontrivial correlations it develops with the background. To capture these effects, we introduce a suitable notion of correlation functions.
For $0\le m\le N$, we first define the joint density of the tagged particle with $m$ background particles as the following marginal of the $N$-particle density $F_N$,
\[F_{N,m}(z_0,\ldots,z_m)\,:=\,\int_{(\mathbb T^d\times\mathbb R^d)^{N-m}}F_N(z_0,\ldots ,z_N)\,dz_{m+1}\ldots dz_N,\]
which is symmetric in its last $m$ variables $\{z_j\}_{1\le j\le m}$. Next, for $0\le m\le N$, we define the $m$-th order correlation function (or cumulant) by
\begin{equation}\label{eq:def-CNm}
G_{N,m}(z_0,\ldots,z_m)\,:=\,\sum_{j=0}^m(-1)^{m-j}\sum_{\sigma\in \mathcal S_j^m}\frac{F_{N,j}}{M^{\otimes j+1}}(z_0,z_\sigma),
\end{equation}
where $\mathcal S_j^m$ stands for the set of all subsets of $\{1,\ldots,m\}$ with $j$ elements and where we use the short-hand notation $z_\sigma=(z_{i_1},\ldots,z_{i_j})$ for $\sigma=\{i_1,\ldots,i_j\}$. This definition ensures the orthogonality property
\begin{equation}\label{eq:orthogonality0}
\int_{\mathbb T^d\times\mathbb R^d}G_{N,m}(z_0,\ldots,z_m)\,M(z_j)\,dz_j=0,\qquad\text{for all $1\le j\le m$},
\end{equation}
and also ensures that marginals can be recovered by means of a (linear) cluster expansion,
\begin{equation}\label{eq:cluster}
F_{N,m}(z_0,\ldots,z_m)\,=\,M^{\otimes m+1}(z_0,\ldots,z_m)\sum_{j=0}^m\sum_{\sigma\in\mathcal S_j^m}G_{N,j}(z_0,z_\sigma),\qquad\text{for all $0\le m\le N$}.
\end{equation}
By orthogonality~\eqref{eq:orthogonality0}, we deduce that correlation functions satisfy
\begin{equation}\label{eq:unitarity0}
\sum_{m=0}^N\binom{N}{m}\int_{(\mathbb T^d\times\mathbb R^d)^{m+1}}|G_{N,m}|^2M^{\otimes m+1}\,=\,\int_{(\mathbb T^d\times\mathbb R^d)^{N+1}}\frac{|F_N|^2}{M^{\otimes N+1}}.
\end{equation}
The right-hand side in this identity would be a conserved quantity if $M^{\otimes N+1}$ were replaced by the Gibbs ensemble~$M_{N+1}$. Still, it can be checked to be uniformly controlled in time: as shown in~\cite[Lemma~2.2]{DSR-21}, in the spirit of~\cite{BGSR-17}, we can deduce for all $0\le m\le N$ and $t\ge0$,
\begin{equation}\label{eq:apriori-CNm}
\Big(\int_{(\mathbb T^d\times\mathbb R^d)^{m+1}}|G_{N,m}(t)|^2M^{\otimes m+1}\Big)^\frac12\,\lesssim\,\binom{N}{m}^{-\frac12}\,\lesssim\,m^{\frac m2}N^{-\frac m2}.
\end{equation}
This can be viewed as some form of chaos estimates, showing that higher-order correlations are tinier in the limit $N\uparrow\infty$. Yet, the scaling in $N$ is not expected to be optimal: in particular, $G_{N,1}$ is expected to be only $O(N^{-1})$ instead of $O(N^{-1/2})$, cf.~\cite{DSR-21}.

To get more precise estimates on correlations, we need to go back to the Liouville equation~\eqref{eq:Liouville} for~$F_N$. Inserting the cluster expansion~\eqref{eq:cluster}, we check that correlations satisfy a hierarchy of equations of the following form, for $0\le m\le N$,
\begin{equation}\label{eq:hierarchy-true}
\partial_tG_{N,m}+iL_{N,m}G_{N,m}=iS^+_{N,m}G_{N,m+1}+\tfrac1N\Big(iS^\circ_{N,m}G_{N,m}+iS^-_{N,m}G_{N,m-1}+iS^=_{N,m}G_{N,m-2}\Big),
\end{equation}
where for notational convenience we set $G_{N,m}\equiv0$ for $m>N$ or $m<0$, where the operators~$S^\square_{N,m}$
are viewed as some `creation/annihilation' operators on the space of correlations,
\begin{equation*}
\begin{array}{rllll}
S_{N,m}^+&:&L^2(M)^{\otimes m+1}&\to&L^2(M)^{\otimes m}\\
S_{N,m}^\circ&:&L^2(M)^{\otimes m}&\to&L^2(M)^{\otimes m}\\
S_{N,m}^-&:&L^2(M)^{\otimes m-1}&\to&L^2(M)^{\otimes m},\\
S_{N,m}^=&:&L^2(M)^{\otimes m-2}&\to&L^2(M)^{\otimes m},
\end{array}
\end{equation*}
and where~$iL_{N,m}$ stands for the $m$-particle linearized Vlasov operator
\begin{multline}\label{eq:lin-mf0}
iL_{N,m}G_{N,m}\,=\,\sum_{j=0}^mv_j\cdot\nabla_{x_j}G_{N,m}\\[-3.5mm]
+\tfrac{N+1-m}N\sum_{j=1}^m\beta v_j\cdot\int_{\mathbb T^d\times\mathbb R^d}\nabla \Vc(x_j-x_{*})G_{N,m}(z_{[0,m]\setminus \{j\}},z_*)\,M(v_{*})\,dz_{*}.
\end{multline}
We refer to~\cite[Lemma~2.4]{DSR-21} for a detailed formulation of this hierarchy.

If the a priori correlation estimates~\eqref{eq:apriori-CNm} were known to hold in a stronger, smooth topology, a direct analysis of the above hierarchy would readily yield the expected kinetic limit $f_{N,0}(N\tau,v)\to f(\tau,v)$; see~\cite{DSR-21}.
However, due to filamentation effects in phase space, correlations become increasingly oscillatory over time and are only controlled a priori in $L^2$. This leads to possible resonances that may, in principle, prevent convergence to the kinetic limit. Consequently, in~\cite{DSR-21}, we only managed to establish a partial consistency result, valid on some intermediate timescale~$t\sim N^r$ with $r<\frac1{18}$ (which could be improved at best to~$r<\frac14$).

This motivates a more refined analysis of the hierarchy~\eqref{eq:hierarchy-true}, with the goal of tracking the oscillatory structure of correlations in greater detail and demonstrating that resonances cannot occur. An important observation --- crucial to the present work but not exploited in earlier studies --- is that the operator~$S_{N,m}^+$ involves a velocity average: hence, while filamentation is a priori viewed as a problem, phase mixing may in fact be leveraged beneficially in some terms.

\subsection{A simplified hierarchy}\label{sec:simpl}
As a first step toward analyzing the exact hierarchy~\eqref{eq:hierarchy-true}, we introduce a simplified setting that preserves the essential features of the original system, but for which we are able to rigorously justify the expected thermalization.
We start with the following two convenient simplifications, which do not affect the physical relevance of the model:
\begin{enumerate}[(H1)]
\item\label{H1} To avoid specific resonance issues on the torus, we consider the problem on the whole space~$\mathbb R^d$ instead of the periodic box $\mathbb T^d$. Physically, this amounts to considering a system of $NL^d$ background particles in a rescaled box $[-\frac L2,\frac L2]^d$ with periodic boundary conditions, and taking the large-box limit $L\uparrow\infty$ as $N\uparrow\infty$. This does not change much in the system physically, but has a very important simplifying effect as Fourier variables become continuous.
\smallskip\item\label{H2} We include a small $O(\frac1N)$ diffusion in velocity in the particle system. As this only acts on the slow relaxation timescale $t\sim N$, we do not expect it to change much to the problem, but it is useful to simplify the analysis.
\end{enumerate}
We introduce two additional simplifications, which we believe are not essential to our arguments but are highly convenient for streamlining the computations:
\begin{enumerate}[(H3)]
\item[(H3)]\label{H1} The linearized mean-field operator $iL_{N,m}$, cf.~\eqref{eq:lin-mf0}, is replaced by pure transport: this allows to avoid many technicalities and in particular to perform direct computations in Fourier space instead of appealing to linear Landau damping. Physically, this amounts to neglecting collective screening effects.
\smallskip\item[(H4)]\label{H2} In the exact hierarchy~\eqref{eq:hierarchy-true}, the creation and annihilation operators $S_{N,m}^\pm$ are not exact adjoints: they only become approximately so in the limit $N\uparrow\infty$. For finite $N$, the additional operators~$S_{N,m}^\circ$ and~$S_{N,m}^=$ correct this lack of adjointness, so as to ensure the approximate unitarity~\eqref{eq:unitarity0}. This reflects the fact that the Gibbs equilibrium $M_{N+1}$ differs from the tensorized mean-field ensemble~$M^{\otimes N+1}$ around which correlations are defined, cf.~\eqref{eq:def-CNm}. To simplify the structure, we replace $S_{N,m}^{\pm}$ by operators $S_m^\pm$ that are truly adjoint and are independent of $N$. The additional operators $S_{N,m}^\circ$ and~$S_{N,m}^=$ are then set to $0$. This leads to a hierarchy with a neater unitarity structure, in particular making~\eqref{eq:apriori-CNm} trivial.
\end{enumerate}
Finally, we introduce a last simplification, which plays a crucial role in our analysis and constitutes the main restriction of the present work:
\begin{enumerate}[(H5)]
\smallskip\item[(H5)]\label{H5} We truncate the hierarchy at a fixed order $m_0\ge1$ independently of $N$, thus setting $G_{N,m}\equiv0$ in~\eqref{eq:hierarchy-true} for all~$m>m_0$. This means that correlations of the tagged particle are restricted to involve at most $m_0$ background particles at once.
\end{enumerate}
As we shall see, truncating the hierarchy enables us to control the complexity of possible Feynman diagrams in the perturbative expansion. Nonetheless, the model remains highly nontrivial as it still leads to an infinite Dyson series; see Section~\ref{sec:diag}.

Let us introduce more precisely the simplified hierarchy that we will study. In view of~(H1), with the large-box limit, the phase space for the particles is now
\[\mathbb D:=\mathbb R^d\times\mathbb R^d\,\ni\,z=(x,v).\]
Recalling the symmetry, invariance, and orthogonality properties of correlation functions~\eqref{eq:def-CNm}, the state space for simplified correlations is similarly chosen as the direct sum
\[\mathcal H\,:=\,\bigoplus_{m=0}^\infty\Hc_m,\]
where $\Hc_m$ is the set of functions $G_m:\mathbb D^{m+1}\to\mathbb R:(z_0,\ldots,z_m)\mapsto G_m(z_0,\ldots,z_m)$ that are symmetric in their last $m$ variables $z_1,\ldots,z_m$, that are invariant under spatial translations $(z_0,\ldots,z_m)\mapsto(x_0+x,v_0,\ldots,x_m+x,v_m)$, $x\in\mathbb R^d$, and such that
\begin{gather*}
\int_{\mathbb D^{m+1}} \delta(x_0)\,|G_m(z_{[m]})|^2\,M^{\otimes m+1}(v_{[m]})\,dz_{[m]}<\infty,\\
\int_\mathbb D G_m(z_{[m]})\,M(v_j)\,dz_j=0,\quad\text{for all $1\le j\le m$,}
\end{gather*}
where we set for abbreviation $[m]:=\{0,\ldots,m\}$.
Recalling the unitarity structure~\eqref{eq:unitarity0} with $\binom{N}{m}\sim \frac{N^m}{m!}$ as $N\uparrow\infty$ for fixed $m$, we endow the space~$\Hc_m$ with the Hilbert norm
\begin{equation}\label{eq:defHm}
\|G_m\|_{\Hc_m}^2\,:=\,\langle G_m,G_m\rangle_{\Hc_m}\,:=\,\frac1{m!}\int_{\mathbb D^{m+1}}\delta(x_0)\,|G_m(z_{[m]})|^2\,M^{\otimes m+1}(v_{[m]})\,dz_{[m]},
\end{equation}
thus leading to the following norm on the direct sum~$\mathcal H$,
\[\|(G_m)_m\|_\mathcal H^2\,:=\,\sum_{m=0}^\infty\|G_m\|_{\Hc_m}^2.\]
By spatial homogeneity, setting $G_0(v_0)\equiv G_0(z_0)$, we emphasize that $\Hc_0$ is identified with the weighted space~$L^2(M\,dv)$.
In this setting, for all $m\ge0$, we consider the skew-adjoint transport operator $iL_m$ and the self-adjoint velocity-diffusion operator $D_m$ on $\Hc_m$,
\begin{equation}\label{eq:mainop}
iL_m:=\sum_{0\le j\le m}v_j\cdot\nabla_{x_j},\qquad
D_m:=-\sum_{0\le j\le m}(\nabla_{v_j}-\tfrac\beta2 v_j)^2.
\end{equation}
Next, only keeping the leading contributions in the actual BBGKY creation and annihilation operators described e.g.\@ in~\cite{DSR-21}, we define
\begin{eqnarray}
iS_m^-G_{m-1}(z_{[m]})&:=&\sum_{0\le j\le m}\sum_{1\le l\le m\atop l\ne j}\nabla\Vc(x_j-x_l)\cdot(\nabla_{v_j}-\tfrac\beta2 v_j)G_{m-1}(z_{[m]\setminus\{l\}}),\label{eq:def-Sm+-}\\
iS_{m-1}^+G_{m}(z_{[m-1]})&:=&\sum_{0\le j\le m-1}\int_\mathbb D\nabla\Vc(x_j-x_{m})\cdot(\nabla_{v_j}-\tfrac\beta2 v_j)G_{m}(z_{[m]})M(v_{m})\,dz_{m},\nonumber
\end{eqnarray}
in terms of the interaction potential $\Vc\in C^\infty_c(\mathbb R^d)$.
These simplified operators have the advantage of satisfying the exact adjointness relation
\begin{equation}\label{eq:adjS}
\langle H_{m-1},S_{m-1}^+G_{m}\rangle_{\Hc_{m-1}}=\langle S_{m}^-H_{m-1}, G_{m}\rangle_{\Hc_m},\qquad m\ge1.
\end{equation}
Instead of~\eqref{eq:hierarchy-true},
letting $m_0$ denote the maximal order of correlations, cf.~(H5), and including an $O(\frac1N)$ diffusion in velocity, cf.~(H2), we consider the following simplified hierarchy of equations on $\mathcal H$,
\begin{equation}\label{eq:hierarchy-simpl}
\fbox{\text{$\partial_tG_m^{N,m_0}+(iL_m+\tfrac\kappa ND_m)G_m^{N,m_0}=iS_m^+G_{m+1}^{N,m_0}+\tfrac1N iS_m^-G_{m-1}^{N,m_0},\qquad 0\le m\le m_0,$}}
\end{equation}
where we set $G_{m}^{N,m_0}\equiv0$ for $m>m_0$ or $m<0$, and where we let $\kappa>0$ be some diffusion constant. Regarding initial data, we set
\begin{equation}\label{eq:hierarchy-simpl-ci}
G_m^{N,m_0}|_{t=0}\,=\,\left\{\begin{array}{lll}
\mathfrak G&:&m=0,\\
0&:&m\ne0,
\end{array}\right.
\end{equation}
for some initial velocity density $\mathfrak G\in\Hc_0$.
While in the original hierarchy~\eqref{eq:hierarchy-true} we have $m_0=N\uparrow\infty$, we consider here a fixed truncation parameter $m_0$ independently of $N\uparrow\infty$.

By the exact adjointness relation~\eqref{eq:adjS} for $S_m^\pm$,
recalling that the transport operator $iL_m$ is skew-adjoint and that $D_m$ is nonnegative, we directly find
\[\frac{d}{dt}\sum_{m=0}^{m_0}N^{m}\|G_m^{N,m_0}\|_{\Hc_m}^2\,=\,-\frac{2\kappa}N\sum_{m=0}^{m_0}N^{m}\sum_{j=1}^m\|(\nabla_{v_j} -\tfrac\beta2v_j)\,G_m^{N,m_0}\|_{\Hc_m}^2\,\le\,0.\]
Hence, for all $0\le m\le m_0$ and $t\ge0$, we deduce
\begin{equation}\label{eq:apriori-GNm-mod}
\|G_m^{N,m_0}(t)\|_{\Hc_m}\,\le\,N^{-\frac m2}\|\mathfrak G\|_{\Hc_0},
\end{equation}
which is the analogue of~\eqref{eq:apriori-CNm} in the present simplified setting.

Similarly as for the original hierarchy~\eqref{eq:hierarchy-true}, for fixed $m_0$, the time-rescaled tagged particle density~$G_0^{N,m_0}(N\tau,v)$ is now expected to converge weakly as $N\uparrow\infty$ to the solution $G_0(\tau,v)$ of the following Fokker-Planck equation,
\begin{equation}\label{eq:FP-exp-re}
\left\{\begin{array}{l}
\partial_\tau G_0=(\nabla_v-\tfrac\beta 2v)\cdot (\kappa\operatorname{Id}+A_0)(\nabla_{v}-\tfrac\beta 2v)G_0,\\
G_0|_{\tau=0}=\mathfrak G,
\end{array}\right.
\end{equation}
where the diffusion tensor field $A_0\ge0$ is given by the Landau formula
\begin{eqnarray}
A_0(v)&:=&(B_0\ast M)(v),\label{eq:coeff-LB-re}\\
B_0(v)&:=&\frac{\Lambda_\Vc}{|v|}\Big(\operatorname{Id}-\frac{v\otimes v}{|v|^2}\Big)=\Lambda_\Vc\nabla^2|v|,\nonumber
\end{eqnarray}
with the explicit prefactor
$\Lambda_\Vc:=\frac{\omega_{d-1}}{d\omega_d}\int_{\mathbb R^d}|k|\pi\V(k)^2\frac{dk}{(2\pi)^d}$,
where $\omega_n$ stands for the volume of the $n$-dimensional unit ball.

\begin{rem}[From Lenard-Balescu to Landau kernel]
Formula~\eqref{eq:coeff-LB-re} for the diffusion tensor can be directly compared with the original Lenard-Balescu expression~\eqref{eq:coeff-LB}, once the different simplifying assumptions of the model are taken into account:
\begin{enumerate}[---]
\item As we now consider a large-box limit, cf.~(H1), Fourier variables become continuous and the collision kernel~$B$ in~\eqref{eq:coeff-LB} is replaced by its continuum version
\begin{equation}\label{eq:coeff-LB-re-LB!}
B(v,w)=\int_{\mathbb R^d}(k\otimes k)\,\pi\V(k)^2\frac{\delta(k\cdot w)}{|\varepsilon(k,k\cdot v)|^2}\,\frac{dk}{(2\pi)^{d}}.
\end{equation}
\item As we neglect collective screening, cf.~(H3), we replace the dispersion function $\varepsilon$ by the constant~$1$. A direct computation then shows that the collision kernel reduces precisely to the above Landau kernel~$B_0$,
\begin{equation}\label{eq:comput-Landau}
B(v,w)\leadsto \int_{\mathbb R^d}(k\otimes k)\,\pi\V(k)^2\delta(k\cdot w)\,\frac{dk}{(2\pi)^{d}}\,=\,\frac{\Lambda_\Vc}{|w|}\Big(\operatorname{Id}-\frac{w\otimes w}{|w|^2}\Big).
\end{equation}
\item As we have introduced $O(\frac1N)$ diffusion in velocity, cf.~(H2), we add identity to the obtained diffusion tensor in~\eqref{eq:FP-exp-re}.
\end{enumerate}
\end{rem}
\begin{rem}[Modified equilibrium structure]\label{rem:mod-equ}
Beyond the value of the diffusion tensor, we note that the equilibrium structure in the Fokker-Planck equation~\eqref{eq:FP-exp-re} also differs slightly from that in~\eqref{eq:FP-exp}. More precisely, as the velocity density is recovered via $F_0=MG_0$, we find that~\eqref{eq:FP-exp-re} describes relaxation to~$\sqrt M$ instead of~$M$. This discrepancy arises from our symmetric choice of the zeroth-order terms in the definition of the creation and annihilation operators~\eqref{eq:def-Sm+-} (and accordingly for the velocity-diffusion operator~\eqref{eq:mainop}). While this choice is made primarily for computational convenience, it has no essential impact on the results.
\end{rem}

\subsection{Main result}
In the framework of the simplified hierarchy~\eqref{eq:hierarchy-simpl}--\eqref{eq:hierarchy-simpl-ci}, for a fixed truncation parameter $m_0\ge1$, we rigorously prove thermalization and derive the expected Fokker-Planck equation~\eqref{eq:FP-exp-re} for the tagged particle density.

Similarly as for the original hierarchy~\eqref{eq:hierarchy-true}, for fixed $m_0$, the time-rescaled tagged particle density~$G_0^{N,m_0}(N\tau,v)$ is now expected to converge weakly as $N\uparrow\infty$ to the solution $G_0(\tau,v)$ of the following Fokker-Planck equation,
\begin{equation}\label{eq:FP-exp-re}
\left\{\begin{array}{l}
\partial_\tau G_0=(\nabla_v-\tfrac\beta 2v)\cdot (\kappa\operatorname{Id}+A_0)(\nabla_{v}-\tfrac\beta 2v)G_0,\\
G_0|_{\tau=0}=\mathfrak G,
\end{array}\right.
\end{equation}
where the diffusion tensor field $A_0\ge0$ is given by the Landau formula
\begin{eqnarray}
A_0(v)&:=&(B_0\ast M)(v),\label{eq:coeff-LB-re}\\
B_0(v)&:=&\frac{\Lambda_\Vc}{|v|}\Big(\operatorname{Id}-\frac{v\otimes v}{|v|^2}\Big)=\Lambda_\Vc\nabla^2|v|,\nonumber
\end{eqnarray}
with the explicit prefactor
$\Lambda_\Vc:=\frac{\omega_{d-1}}{d\omega_d}\int_{\mathbb R^d}|k|\pi\V(k)^2\frac{dk}{(2\pi)^d}$,
where $\omega_n$ stands for the volume of the $n$-dimensional unit ball.

\begin{rem}[From Lenard-Balescu to Landau kernel]
Formula~\eqref{eq:coeff-LB-re} for the diffusion tensor can be directly compared with the original Lenard-Balescu expression~\eqref{eq:coeff-LB}, once the different simplifying assumptions of the model are taken into account:
\begin{enumerate}[---]
\item As we now consider a large-box limit, cf.~(H1), Fourier variables become continuous and the collision kernel~$B$ in~\eqref{eq:coeff-LB} is replaced by its continuum version
\begin{equation}\label{eq:coeff-LB-re-LB!}
B(v,w)=\int_{\mathbb R^d}(k\otimes k)\,\pi\V(k)^2\frac{\delta(k\cdot w)}{|\varepsilon(k,k\cdot v)|^2}\,\frac{dk}{(2\pi)^{d}}.
\end{equation}
\item As we neglect collective screening, cf.~(H3), we replace the dispersion function $\varepsilon$ by the constant~$1$. A direct computation then shows that the collision kernel reduces precisely to the above Landau kernel~$B_0$,
\begin{equation}\label{eq:comput-Landau}
B(v,w)\leadsto \int_{\mathbb R^d}(k\otimes k)\,\pi\V(k)^2\delta(k\cdot w)\,\frac{dk}{(2\pi)^{d}}\,=\,\frac{\Lambda_\Vc}{|w|}\Big(\operatorname{Id}-\frac{w\otimes w}{|w|^2}\Big).
\end{equation}
\item As we have introduced $O(\frac1N)$ diffusion in velocity, cf.~(H2), we add identity to the obtained diffusion tensor in~\eqref{eq:FP-exp-re}.
\end{enumerate}
\end{rem}
\begin{rem}[Modified equilibrium structure]\label{rem:mod-equ}
Beyond the value of the diffusion tensor, we note that the equilibrium structure in the Fokker-Planck equation~\eqref{eq:FP-exp-re} also differs slightly from that in~\eqref{eq:FP-exp}. More precisely, as the velocity density is recovered via $F_0=MG_0$, we find that~\eqref{eq:FP-exp-re} describes relaxation to~$\sqrt M$ instead of~$M$. This discrepancy arises from our symmetric choice of the zeroth-order terms in the definition of the creation and annihilation operators~\eqref{eq:def-Sm+-} (and accordingly for the velocity-diffusion operator~\eqref{eq:mainop}). While this choice is made primarily for computational convenience, it has no essential impact on the results.
\end{rem}

\subsection{Main result}
In the framework of the simplified hierarchy~\eqref{eq:hierarchy-simpl}--\eqref{eq:hierarchy-simpl-ci}, for a fixed truncation parameter $m_0\ge1$, we rigorously prove thermalization and derive the expected Fokker-Planck equation~\eqref{eq:FP-exp-re} for the tagged particle density.

\subsection{Main result}
In the framework of the simplified hierarchy~\eqref{eq:hierarchy-simpl}--\eqref{eq:hierarchy-simpl-ci}, for a fixed truncation parameter $m_0\ge1$, we rigorously prove thermalization and derive the expected Fokker-Planck equation~\eqref{eq:FP-exp-re} for the tagged particle density.

\subsection{Dyson series expansion}\label{sec:diag}
Applying Laplace transform in the form~\eqref{eq:Lap}, the simplified hierarchy~\eqref{eq:hierarchy} reads as follows,
\begin{equation}\label{eq:Duhamel}
\left\{\begin{array}{l}
\big(1+i\alpha+\kappa\tfrac{t_N}{N}\hat D_0\big)\Lc g_0^{N,m_0}=\mathfrak g+\tfrac{t_N}{\sqrt N}i\hat S_0^+\Lc g_{1}^{N,m_0},\\[2mm]
\big(\tfrac{1+i\alpha}{t_N}+i\hat L_m+\tfrac\kappa N\hat D_m\big)\Lc g_m^{N,m_0}=\tfrac1{\sqrt N}\Big(i\hat S_m^+\Lc g_{m+1}^{N,m_0}+i\hat S_m^-\Lc g_{m-1}^{N,m_0}\Big),\qquad 1\le m\le m_0.
\end{array}\right.
\end{equation}
Iteratively solving this hierarchy, we obtain an infinite Dyson series for each correlation function $g_m^{N,m_0}$. Each term in this expansion consists of a sequence of resolvents
\[(\tfrac{1+i\alpha}{t_N}+i\hat L_m+\tfrac\kappa N\hat D_m)^{-1},\]
interlaced with creation or annihilation operators $\hat S_m^\pm$, acting on the initial data $\mathfrak g$.
To organize these contributions, we introduce a representation using Feynman-type diagrams:
\begin{enumerate}[---]
\item Each connected horizontal line corresponds to a different particle, with the lower line corresponding to the tagged particle `$0$'.
\smallskip\item Parallel horizontal segments stand for free propagators,
\[\big(\tfrac{1+i\alpha}{t_N}+i\hat L_1+\tfrac\kappa N\hat D_1\big)^{-1}=
{\tiny\begin{tikzpicture}[baseline={([yshift=-1ex]current bounding box.center)},scale=0.8]
\draw (0,0) -- (1,0);
\draw (0,0.3) -- (1,0.3);
\end{tikzpicture}}\qquad
\big(\tfrac{1+i\alpha}{t_N}+i\hat L_2+\tfrac\kappa N\hat D_2\big)^{-1}=
{\tiny\begin{tikzpicture}[baseline={([yshift=-1ex]current bounding box.center)},scale=0.8]
\draw (0,0) -- (1,0);
\draw (0,0.3) -- (1,0.3);
\draw (0,0.6) -- (1,0.6);
\end{tikzpicture}}\qquad\text{etc.}\]
\item Vertical segments merging two horizontal levels are viewed as ``collisions'' and correspond to applying creation or annihilation operators,
\[\hat S^{m,+}_{j,l}
={\tiny\begin{tikzpicture}[baseline={([yshift=-1ex]current bounding box.center)},scale=0.8]
\begin{scope}[every node/.style={circle,draw,fill=white,inner sep=0pt,minimum size=3pt}]
\node (1) at (0,0) {};
\end{scope}
\draw (1) -- (0,0.5);
\draw[dotted] (-0.4,0) -- (1) -- (0.4,0);
\draw[dotted] (0,0.5) -- (0.4,0.5);
\node at (1.5,0.5) {\tiny (particle $l$)};
\node at (1.5,0) {\tiny (particle $j$)};
\end{tikzpicture}}\qquad
\hat S^{m,-}_{j,l}
={\tiny\begin{tikzpicture}[baseline={([yshift=-1ex]current bounding box.center)},scale=0.8]
\begin{scope}[every node/.style={circle,draw,fill=white,inner sep=0pt,minimum size=3pt}]
\node (1) at (0,0) {};
\end{scope}
\draw (1) -- (0,0.5);
\draw[dotted] (-0.4,0) -- (1) -- (0.4,0);
\draw[dotted] (0,0.5) -- (-0.4,0.5);
\node at (1.5,0.5) {\tiny (particle $l$)};
\node at (1.5,0) {\tiny (particle $j$)};
\end{tikzpicture}}\]
We emphasize that for~$\hat S_{j,l}^{m,-}$ the annihilated index $l$ only runs over $1\le l\le m$, meaning that the tagged particle `$0$' is never annihilated.
\smallskip\item Diagrams are arbitrary compositions of the above two ingredients, alternating between free propagators and collisions.
The \emph{complexity of a diagram} is the maximal number of horizontal levels that appear at once on top of the lower level: by assumption, it is always bounded by the truncation parameter~$m_0$. Viewing time as flowing from right to left, we call \emph{input} (resp.\@ \emph{output}) variables the list of particle indices that are present at the right (resp.\@ left) of the diagram.
\end{enumerate}
To give a concrete example,
\[{\tiny\begin{tikzpicture}[baseline={([yshift=-1ex]current bounding box.center)},scale=0.8]
\begin{scope}[every node/.style={circle,draw,fill=white,inner sep=0pt,minimum size=3pt}]
\node (1) at (0,0) {};
\node (2) at (0.5,0) {};
\node (3) at (1,0.5) {};
\node (4) at (1.5,0) {};
\end{scope}
\draw (1) -- (2) -- (4) -- (1.5,0.5) -- (3) -- (0,0.5) -- (1);
\draw (2) -- (0.5,0.25) -- (1,0.25) -- (3);
\end{tikzpicture}}
\,=\, S_{0,1}^{0,+}\big(\tfrac{1+i\alpha}{t_N}+i\hat L_1+\tfrac\kappa N\hat D_1\big)^{-1}S_{0,2}^{1,+}\big(\tfrac{1+i\alpha}{t_N}+i\hat L_2+\tfrac\kappa N\hat D_2\big)^{-1}S_{1,2}^{2,-}\big(\tfrac{1+i\alpha}{t_N}+i\hat L_1+\tfrac\kappa N\hat D_1\big)^{-1}S_{0,1}^{1,-},\]
and thus, inserting the definition of the operators and carefully tracking the Fourier variables,
\begin{multline*}
{\tiny\begin{tikzpicture}[baseline={([yshift=-1ex]current bounding box.center)},scale=0.8]
\begin{scope}[every node/.style={circle,draw,fill=white,inner sep=0pt,minimum size=3pt}]
\node (1) at (0,0) {};
\node (2) at (0.5,0) {};
\node (3) at (1,0.5) {};
\node (4) at (1.5,0) {};
\end{scope}
\draw (1) -- (2) -- (4) -- (1.5,0.5) -- (3) -- (0,0.5) -- (1);
\draw (2) -- (0.5,0.25) -- (1,0.25) -- (3);
\end{tikzpicture}}\,\mathfrak g(\alpha,v_0)
\,=\, \int_{\Dd^2}
\sqrt M(v_1)\,k_1\V(k_1)\cdot\nabla_{v_0}\Big(\tfrac{1+i\alpha}{t_N}+ik_1\cdot(v_1-v_0)-\tfrac\kappa N\triangle_{v_{[1]}}\Big)^{-1}\\
\times \sqrt M(v_2)\,k_2\V(k_2)\cdot\nabla_{v_0}\Big(\tfrac{1+i\alpha}{t_N}+ik_1\cdot(v_1-v_0)+ik_2\cdot(v_2-v_0)-\tfrac\kappa N\triangle_{v_{[2]}}\Big)^{-1}\\
\times \sqrt M(v_2)\,k_2\V(k_2)\cdot\nabla_{v_0}\Big(\tfrac{1+i\alpha}{t_N}+i(k_1+k_2)\cdot(v_1-v_0)-\tfrac\kappa N\triangle_{v_{[1]}}\Big)^{-1}\\
\times\sqrt M(v_1)\,(k_1+k_2)\V(k_1+k_2)\cdot\nabla_{v_0}\mathfrak g(v_0)\,d^*\hat z_1d^*\hat z_2,
\end{multline*}
where for an index set $A$ we let $\triangle_{v_A}=\sum_{j\in A}\triangle_{v_{j}}$.
Occasionally, we shall indicate the associated Fourier momentum variables above each horizontal segment, thus representing the above integral as
\[=\,\int_{(\R^d)^2}
{\tiny\begin{tikzpicture}[baseline={([yshift=-2ex]current bounding box.center)},scale=0.8]
\begin{scope}[every node/.style={circle,draw,fill=white,inner sep=0pt,minimum size=3pt}]
\node (1) at (0,0) {};
\node (2) at (1.5,0) {};
\node (3) at (3,0.8) {};
\node (4) at (4.5,0) {};
\end{scope}
\draw (1) -- (2) -- (4) -- (4.5,0.8) -- (3) -- (0,0.8) -- (1);
\draw (2) -- (1.5,0.4) -- (3,0.4) -- (3);
\node at (0.75,0.17) {$-k_1$};
\node at (1.5,0.97) {$k_1$};
\node at (3,0.17) {$-k_1-k_2$};
\node at (2.25,0.57) {$k_2$};
\node at (3.75,0.97) {$k_1+k_2$};
\end{tikzpicture}}
\,\mathfrak g(v_0)\,d^*k_1d^*k_2.\]
Note that the definition of the creation and annihilation operators ensures the conservation of the sum of Fourier variables at each collision.

As detailed in Section~\ref{sec:strategy}, the proof relies on a careful analysis of the Dyson expansion of the hierarchy in terms of Feynman diagrams and builds on three key new ingredients:
\begin{enumerate}[---]
\item {\it Renormalization:} The Dyson series must be renormalized to eliminate the leading recollisions, which amounts to factoring out part of the expansion. Concretely, the free transport propagators around which the expansion is performed are replaced by renormalized propagators, obtained by adding to the free transport an approximate (non-Markovian) version of the limiting Fokker-Planck operator (so-called ``hat operator'' below).
\smallskip\item {\it Phase mixing:} Since annihilation operators involve velocity averages, cf.~\eqref{eq:def-Sm+-}, phase mixing can be effectively exploited in several terms of the expansion. In our approach, this is achieved by systematically applying contour deformations in all the integrals over free velocity variables.
\smallskip\item {\it Hypoelliptic estimates:} As explained, the renormalization effectively augments the free transport with a (non-Markovian) Fokker-Planck-type operator carrying a prefactor of order $O(\frac1N)$. By treating this, heuristically, as a genuine Fokker-Planck operator, filamentation effects in phase space are mitigated through the induced $O(\frac1N)$ velocity diffusion. This allows to replace naive resolvent estimates with hypoelliptic ones, which substantially improve the scaling in $N$. Since such estimates for the actual renormalized propagators are not directly accessible due to non-Markovian effects, we exploit the additional $O(\frac1N)$ velocity diffusion included for simplicity in the system to justify these improved bounds.
\end{enumerate}
As discussed heuristically in Section~\ref{sec:limitation} below, the restriction~\eqref{eq:def de d_0} on the space dimension relative to the truncation parameter arises from geometric constraints in the complex deformations used for phase-mixing arguments and does not appear to be avoidable at present. Several refinements of the method could improve the value of $d_0$ in~\eqref{eq:def de d_0}, but none seem enough to remove the linear dependence on the truncation parameter.

\begin{rem}[Scope and limitations of the model]
Several of the simplifying assumptions in our model could be relaxed without substantial modifications. In particular, the free transport operator could be replaced by the linearized Vlasov operator appearing in the original hierarchy, cf.~\eqref{eq:lin-mf0}, in which case the Landau kernel in the formula~\eqref{eq:coeff-LB-re} for the diffusion tensor would be replaced by the Lenard-Balescu kernel~\eqref{eq:coeff-LB-re-LB!}. Likewise, the modification of the equilibrium structure noted in Remark~\ref{rem:mod-equ} could be removed with only minor adjustments.
A more consequential simplification of the model concerns the choice of creation and annihilation operators as exact adjoints; we expect that the approximate unitarity of the original hierarchy~\eqref{eq:hierarchy-true} would still suffice for the analysis, but this remains to be verified. Another technical simplification is the inclusion of an $O(\frac1N)$ velocity diffusion in the system, which we cannot yet dispense with. Finally, the most restrictive assumption is, of course, the truncation of the hierarchy itself, which currently appears unavoidable.
\end{rem}

\begin{rem}[Related work]
While completing this work, we became aware of an ongoing parallel investigation by Bodineau and Le Bris~\cite{BLB}, which addresses another simplified model in the direction of a rigorous justification of Lenard-Balescu thermalization. Specifically, they derive a similar linearized Landau equation for a tagged particle interacting via mean-field forces within an ideal Rayleigh gas --- namely, a background of particles that do not interact with each other, only indirectly through the tagged particle.
Their analysis follows a trajectorial approach, but our hierarchical framework could in principle be applied to their model as well. In that case, the diagrammatic structure is significantly simpler: since background particles do not interact, the creation and annihilation operators always correspond to collisions involving the tagged particle, and the pathological diagrams~\eqref{eq:err-ex} and~\eqref{eq:FN-diag-rem} that constitute the main difficulty in our analysis would be absent. In this simplified setting, we expect that the truncation of the hierarchy could be avoided, leading to a hierarchical proof of their result. We note, however, that our approach would still require the inclusion of an $O(\frac1N)$ velocity diffusion in the system, which is not needed in~\cite{BLB}.
\end{rem}

As detailed in Section~\ref{sec:strategy}, the proof of Theorem~\ref{th:main} relies on a careful analysis of the Dyson expansion of the hierarchy in terms of Feynman diagrams and builds on three key new ingredients:
\begin{enumerate}[---]
\item \textit{Renormalization:} The Dyson series is renormalized to eliminate the leading recollisions, by \emph{factoring out part of the expansion}. Concretely, the free transport propagators are replaced by \emph{renormalized propagators}, obtained by adding to free transport an \emph{approximate (non-Markovian) version of the limiting Fokker-Planck operator} (the ``hat operator'').
\item \textit{Phase mixing:} Because annihilation operators involve velocity averages (cf.~\eqref{eq:def-Sm+-}), phase mixing is exploited in several terms of the expansion by \emph{systematically applying contour deformations} in the integrals over free velocity variables.
\item \textit{Hypoelliptic estimates:} Renormalization augments free transport with a (non-Markovian) Fokker-Planck-type operator with prefactor $O(\tfrac1N)$. Treating this heuristically as a genuine Fokker-Planck operator, filamentation effects are mitigated via the induced $O(\tfrac1N)$ velocity diffusion, allowing one to \emph{replace naive resolvent estimates with hypoelliptic ones} to improve the scaling in $N$. Since such estimates for the actual renormalized propagators are not directly accessible due to non-Markovian effects, the argument \emph{exploits the additional $O(\tfrac1N)$ velocity diffusion included for simplicity in the system} to justify these improved bounds.
\end{enumerate}
It is also noted that the dimension restriction \eqref{eq:def de d_0} (relative to the truncation parameter) arises from \emph{geometric constraints in the complex deformations used for phase-mixing arguments} and "does not appear to be avoidable at present."

As detailed in Section~\ref{sec:strategy}, the proof of the main theorem relies on a careful analysis of the Dyson expansion of the hierarchy in terms of Feynman diagrams and builds on three key new ingredients:
\begin{enumerate}[---]
\item \textit{Renormalization:} The Dyson series is renormalized to eliminate the leading recollisions, by \emph{factoring out part of the expansion}. Concretely, the free transport propagators are replaced by \emph{renormalized propagators}, obtained by adding to free transport an \emph{approximate (non-Markovian) version of the limiting Fokker-Planck operator} (the ``hat operator'').
\item \textit{Phase mixing:} Because annihilation operators involve velocity averages (cf.
\begin{eqnarray}
iS_m^-G_{m-1}(z_{[m]})&:=&\sum_{0\le j\le m}\sum_{1\le l\le m\atop l\ne j}\nabla\Vc(x_j-x_l)\cdot(\nabla_{v_j}-\tfrac\beta2 v_j)G_{m-1}(z_{[m]\setminus\{l\}}),\label{eq:def-Sm+-}\\
iS_{m-1}^+G_{m}(z_{[m-1]})&:=&\sum_{0\le j\le m-1}\int_\Dd\nabla\Vc(x_j-x_{m})\cdot(\nabla_{v_j}-\tfrac\beta2 v_j)G_{m}(z_{[m]})M(v_{m})\,dz_{m},\nonumber
\end{eqnarray}
), phase mixing is exploited in several terms of the expansion by \emph{systematically applying contour deformations} in the integrals over free velocity variables.
\item \textit{Hypoelliptic estimates:} Renormalization augments free transport with a (non-Markovian) Fokker-Planck-type operator with prefactor $O(\tfrac1N)$. Treating this heuristically as a genuine Fokker-Planck operator, filamentation effects are mitigated via the induced $O(\tfrac1N)$ velocity diffusion, allowing one to \emph{replace naive resolvent estimates with hypoelliptic ones} to improve the scaling in $N$. Since such estimates for the actual renormalized propagators are not directly accessible due to non-Markovian effects, the argument \emph{exploits the additional $O(\tfrac1N)$ velocity diffusion included for simplicity in the system} to justify these improved bounds.
\end{enumerate}
It is also noted that the dimension restriction
\begin{equation}\label{eq:def de d_0}
d\ge d_0=\left\{\begin{array}{cl}
2,&\text{if $m_0=1$},\\
8,&\text{if $m_0=2$},\\
28m_0+70,&\text{if $m_0\ge3$},
\end{array}\right.\end{equation}
(relative to the truncation parameter) arises from \emph{geometric constraints in the complex deformations used for phase-mixing arguments} and "does not appear to be avoidable at present."

\label{th:main}
Fix a truncation parameter $m_0\ge1$ and
assume that:
\begin{enumerate}[---]
\item the space dimension is sufficiently large depending on $m_0$, in the sense that
\begin{equation}\label{eq:def de d_0}
d\ge d_0=\left\{\begin{array}{cl}
2,&\text{if $m_0=1$},\\
8,&\text{if $m_0=2$},\\
28m_0+70,&\text{if $m_0\ge3$},
\end{array}\right.\end{equation}
\item the diffusion constant $\kappa$ is sufficiently large, in the sense that $\kappa\ge C_{m_0}\int_{\mathbb R^d}|k|\V(k)\,dk$ for some constant~$C_{m_0}>0$ depending only on $m_0$. In case $m_0=1$, this condition can be dropped.
\end{enumerate}
Then, for any initial condition $\mathfrak G\in C_c^\infty(\mathbb R^d)$, the solution $(G_m^{N,m_0})_{0\le m\le m_0}$ of the simplified hierarchy
\begin{equation}\label{eq:hierarchy-simpl}
\fbox{\text{$\partial_tG_m^{N,m_0}+(iL_m+\tfrac\kappa ND_m)G_m^{N,m_0}=iS_m^+G_{m+1}^{N,m_0}+\tfrac1N iS_m^-G_{m-1}^{N,m_0},\qquad 0\le m\le m_0,$}}
\end{equation}
with initial condition
\begin{equation}\label{eq:hierarchy-simpl-ci}
G_m^{N,m_0}|_{t=0}\,=\,\left\{\begin{array}{lll}
\mathfrak G&:&m=0,\\
0&:&m\ne0,
\end{array}\right.
\end{equation}
satisfies that the time-rescaled tagged-particle density
\[(\tau,v)\mapsto G_0^{N,m_0}(N\tau,v)\]
converges strongly in $L^2_\loc(\mathbb R^+;L^2(M\, dv))$ to the solution $G_0$ of the Fokker-Planck equation
\begin{equation}\label{eq:FP-exp-re}
\left\{\begin{array}{l}
\partial_\tau G_0=(\nabla_v-\tfrac\beta 2v)\cdot (\kappa\Id+A_0)(\nabla_{v}-\tfrac\beta 2v)G_0,\\
G_0|_{\tau=0}=\mathfrak G,
\end{array}\right.
\end{equation}
More precisely, we have
\begin{equation*}
\Big(\int_0^\infty e^{-2\tau}\|G_0^{N,m_0}(N\tau)-G_0(\tau)\|_{L^2(M\,dv)}^2\,d\tau\Big)^\frac12 \,\lesssim\,N^{-\frac1{12}}\|\<(\nabla_{v_0},v_0)\>^{4m_0+10}\mathfrak G\|_{L^2(M\, dv)},
\end{equation*}
and this bound can be improved to $N^{-1}\|\<(\nabla_{v_0},v_0)\>^{4m_0+21}\mathfrak G\|_{L^2(M\,dv)}$ if $d\ge 28m_0+147$.

Question:
Let $m_0\ge 1$. Assume the space dimension $d$ satisfies
\[
d\ge d_0:=\begin{cases}
2,& m_0=1,\\
8,& m_0=2,\\
28m_0+70,& m_0\ge 3,
\end{cases}
\]
and assume the diffusion constant $\kappa$ is large enough in the sense that
\[
\kappa\ge C_{m_0}\int_{\mathbb R^d}|k|\,\mathcal V(k)\,dk
\]
for some constant $C_{m_0}>0$ depending only on $m_0$ (this assumption is not needed when $m_0=1$). For any initial datum $\mathfrak G\in C_c^\infty(\mathbb R^d)$, let $(G_m^{N,m_0})_{0\le m\le m_0}$ solve
\[
\partial_t G_m^{N,m_0}+\Big(iL_m+\frac\kappa N D_m\Big)G_m^{N,m_0}
= iS_m^+G_{m+1}^{N,m_0}+\frac{i}{N}S_m^-G_{m-1}^{N,m_0},\qquad 0\le m\le m_0,
\]
with initial data
\[
G_m^{N,m_0}\big|_{t=0}=\begin{cases}
\mathfrak G,& m=0,\\
0,& m\ne 0.
\end{cases}
\]
Define the diffusion tensor
\[
A_0(v)=(B_0*M)(v),\qquad
B_0(v)=\frac{\Lambda_{\mathcal V}}{|v|}\Big(\mathrm{Id}-\frac{v\otimes v}{|v|^2}\Big)=\Lambda_{\mathcal V}\nabla^2|v|,
\]
where
\[
\Lambda_{\mathcal V}:=\frac{\omega_{d-1}}{d\omega_d}\int_{\mathbb R^d}|k|\,\pi\,\mathcal V(k)^2\frac{dk}{(2\pi)^d},
\]
and let $G_0(\tau,v)$ solve the Fokker--Planck equation
\[
\partial_\tau G_0=(\nabla_v-\tfrac\beta 2 v)\cdot (\kappa\mathrm{Id}+A_0)(\nabla_v-\tfrac\beta 2 v)G_0,
\qquad G_0\big|_{\tau=0}=\mathfrak G.
\]
Here $L^2(M\,dv)$ denotes the weighted space with measure $M(v)\,dv$. Which quantitative estimate holds for the rescaled tagged-particle density $(\tau,v)\mapsto G_0^{N,m_0}(N\tau,v)$?

The rescaled density $G_0^{N,m_0}(N\tau,v)$ converges strongly to $G_0(\tau,v)$ in $L^2_{\mathrm{loc}}(\mathbb R^+;L^2(M\,dv))$, and more precisely
\[
\left(\int_0^\infty e^{-2\tau}\,\big\|G_0^{N,m_0}(N\tau)-G_0(\tau)\big\|_{L^2(M\,dv)}^2\,d\tau\right)^{1/2}
\lesssim N^{-1/12}\,\big\|\langle(\nabla_{v_0},v_0)\rangle^{4m_0+10}\mathfrak G\big\|_{L^2(M\,dv)}.
\]
Moreover, if in addition $d\ge 28m_0+147$, then this improves to
\[
\left(\int_0^\infty e^{-2\tau}\,\big\|G_0^{N,m_0}(N\tau)-G_0(\tau)\big\|_{L^2(M\,dv)}^2\,d\tau\right)^{1/2}
\lesssim N^{-1}\,\big\|\langle(\nabla_{v_0},v_0)\rangle^{4m_0+21}\mathfrak G\big\|_{L^2(M\,dv)}.
\]

Choices:
B.
The rescaled density $G_0^{N,m_0}(N\tau,v)$ converges strongly to $G_0(\tau,v)$ in $L^2_{\mathrm{loc}}(\mathbb R^+;L^2(M\,dv))$, and more precisely
\[
\left(\int_0^\infty e^{-2\tau}\,\big\|G_0^{N,m_0}(N\tau)-G_0(\tau)\big\|_{L^2(M\,dv)}^2\,d\tau\right)^{1/2}
\lesssim N^{-1/12}\,\big\|\langle(\nabla_{v_0},v_0)\rangle^{4m_0+10}\mathfrak G\big\|_{L^2(M\,dv)}.
\]
Moreover, if in addition $d\ge 28m_0+70$, then this improves to
\[
\left(\int_0^\infty e^{-2\tau}\,\big\|G_0^{N,m_0}(N\tau)-G_0(\tau)\big\|_{L^2(M\,dv)}^2\,d\tau\right)^{1/2}
\lesssim N^{-1}\,\big\|\langle(\nabla_{v_0},v_0)\rangle^{4m_0+21}\mathfrak G\big\|_{L^2(M\,dv)}.
\]
C.
The rescaled density $G_0^{N,m_0}(N\tau,v)$ converges strongly to $G_0(\tau,v)$ in $L^2_{\mathrm{loc}}(\mathbb R^+;L^2(M\,dv))$, and more precisely
\[
\left(\int_0^\infty e^{-2\tau}\,\big\|G_0^{N,m_0}(N\tau)-G_0(\tau)\big\|_{L^2(M\,dv)}^2\,d\tau\right)^{1/2}
\lesssim N^{-1/12}\,\big\|\langle(\nabla_{v_0},v_0)\rangle^{4m_0+21}\mathfrak G\big\|_{L^2(M\,dv)}.
\]
D.
The rescaled density $G_0^{N,m_0}(N\tau,v)$ converges strongly to $G_0(\tau,v)$ in $L^2_{\mathrm{loc}}(\mathbb R^+;L^2(M\,dv))$, and more precisely
\[
\left(\int_0^\infty e^{-2\tau}\,\big\|G_0^{N,m_0}(N\tau)-G_0(\tau)\big\|_{L^2(M\,dv)}^2\,d\tau\right)^{1/2}
\lesssim N^{-1/12}\,\big\|\langle(\nabla_{v_0},v_0)\rangle^{4m_0+10}\mathfrak G\big\|_{L^2(M\,dv)}.
\]
Moreover, there exists a constant $C>0$ independent of $m_0$ such that, whenever
$d\ge 28m_0+147$ and $\kappa\ge C\int_{\mathbb R^d}|k|\,\mathcal V(k)\,dk$, the estimate improves to
\[
\left(\int_0^\infty e^{-2\tau}\,\big\|G_0^{N,m_0}(N\tau)-G_0(\tau)\big\|_{L^2(M\,dv)}^2\,d\tau\right)^{1/2}
\lesssim N^{-1}\,\big\|\langle(\nabla_{v_0},v_0)\rangle^{4m_0+21}\mathfrak G\big\|_{L^2(M\,dv)}.
\]
E.
The rescaled density $G_0^{N,m_0}(N\tau,v)$ converges strongly to $G_0(\tau,v)$ in $L^2_{\mathrm{loc}}(\mathbb R^+;L^2(M\,dv))$, and more precisely
\[
\left(\int_0^\infty e^{-2\tau}\,\big\|G_0^{N,m_0}(N\tau)-G_0(\tau)\big\|_{L^2(M\,dv)}^2\,d\tau\right)^{1/2}
\lesssim N^{-1}\,\big\|\langle(\nabla_{v_0},v_0)\rangle^{4m_0+10}\mathfrak G\big\|_{L^2(M\,dv)}.
\]
Moreover, if in addition $d\ge 28m_0+147$, then one has the same bound with the stronger norm $\big\|\langle(\nabla_{v_0},v_0)\rangle^{4m_0+21}\mathfrak G\big\|_{L^2(M\,dv)}$ on the right-hand side.

extra-dimension threshold for the $N^{-1}$ improvement

dropped the improved $N^{-1}$ conclusion under the extra dimension hypothesis

dependence of the largeness constant on $m_0$

baseline scaling improved from $N^{-1/12}$ to $N^{-1}$ without the extra high-dimensional hypothesis

The stem gives the hypotheses and asks for the resulting estimate, but it does not explicitly reveal the correct conclusion. There is no direct answer leakage beyond the fact that the question is about a specific theorem statement.

This is essentially a direct theorem-recall item: the stem presents the full setup and asks which conclusion holds. The correct option is basically the theorem’s conclusion restated verbatim.

Selecting the correct option requires some discrimination among close variants (dimension threshold, rate, norm strength, parameter dependence), but it does not demand substantial derivation or generative mathematical reasoning.

The distractors are plausible and well targeted: they vary sharp thresholds, weaken the true statement, overstrengthen the rate, or alter quantifier dependence. These reflect realistic mistakes in reading or recalling a technical result.

A solid theorem-recall MCQ with strong distractors, but it is highly tautological and only weakly tests genuine generative reasoning.


\label{theorem: radius of essential spectrum}
    On the half-plane $\Re(\beta) >(1-\alpha)/2$, the radius $\rho_e(L_\beta):= \sup_{\lambda\in \sigma_\varepsilon(L_\beta)}\left\lvert\lambda\right\rvert$ of the essential spectrum satisfies
    \[\rho_e(L_\beta)\leq \rho(\mathcal{L}_{\Re(\beta) +\alpha}):=\sup_{\lambda\in \sigma(\mathcal{L}_{\beta+\alpha})}\left\lvert\lambda\right\rvert .\]
    Furthermore, the generalised eigenspaces belonging to eigenvalues $\lambda$ with $\left\lvert\lambda\right\rvert>\rho_e(L_\beta)$ consist of analytic functions which extend to generalised eigenfunctions of $\mathcal{L}_\beta$.

\begin{theorem}\label{theorem: radius of essential spectrum}
    On the half-plane $\Re(\beta) >(1-\alpha)/2$, the radius $\rho_e(L_\beta):= \sup_{\lambda\in \sigma_\varepsilon(L_\beta)}\abs{\lambda}$ of the essential spectrum satisfies
    \[\rho_e(L_\beta)\leq \rho(\mathcal{L}_{\Re(\beta) +\alpha}):=\sup_{\lambda\in \sigma(\mathcal{L}_{\beta+\alpha})}\abs{\lambda} .\]
    Furthermore, the generalised eigenspaces belonging to eigenvalues $\lambda$ with $\abs{\lambda}>\rho_e(L_\beta)$ consist of analytic functions which extend to generalised eigenfunctions of $\mathcal{L}_\beta$.
\end{theorem}

\begin{remark}\label{remark: abstract}
    The zeroes of $Z(s)$ on the line $s=1/2+it$ are those for which there are eigenvalues of $\Delta$ called \enquote{Maass cusp forms} with eigenvalue $s(1-s)$. Furthermore the multiplicity of the zeroes and the eigenspaces of $\Delta$ match. The other zeroes of $Z(s)$ are a simple zero at $s=1$ and at the values of $s$ for which $2s$ is a nontrivial zero of the Riemann zeta function $\zeta$. Since $\rho(\mathcal{L}_a)<1$ when $a>1$, we see that the eigenspaces of $\mathcal{L}_\beta$ and $L_\beta$ for the eigenvalues $\lambda=\pm 1$ are identical when $1>\alpha>1-\Re(\beta)$. In particular, the eigenspaces associated to the Maass cusp forms are preserved when $\alpha >1/2$, and the eigenspaces associated to the nontrivial zeroes of the Riemann zeta function\footnote{In fact, the zeroes of $Z(s)$ associated with the nontrivial zeroes of $\zeta$ always occur due to $1$ being an eigenvalue of $\mathcal{L}_\beta$.} to the right of the critical line are preserved when $\alpha=3/4$.
\end{remark}

Let $G:(0,1)\to [0,1)$ be the Gauss map
\begin{equation}\label{equation: gauss map}
 G(x) = \frac{1}{z}-\left\lfloor\frac{1}{z}\right\rfloor.
\end{equation}
We associate to $G$ the following family of transfer operators which we formally define as acting on some space of functions $f:[0,1]\to \mathbb{C}$:
\begin{equation}\label{equation: Mayer transfer operator}
 \mathcal{L}_{\beta}(f)(z)=\sum_{n=1}^\infty \frac{1}{(n+z)^{2\beta}}f\left(\frac{1}{n+z}\right),
\end{equation}
where $\beta \in \{z\in \mathbb{C}: \Re(\beta)>1/2\}$. When $\beta=1$, this is the familiar Ruelle-Perron-Frobenius operator for $G$ and can formally be interpreted as the right-adjoint of the Koopman operator $f\mapsto f\circ G$. The operators in the case $\beta \neq 1$ were first introduced by Mayer \cite{Mayer90}, who considered them acting on the Banach space of functions $A_\infty(D)$ which are holomorphic on $D:=\{z\in\mathbb{C}: \left\lvertz-1\right\rvert< 3/2\}$ with continuous extension to $\overline{D}$.

For this choice of Banach space, Mayer also showed that the operator-valued function $\beta\mapsto \mathcal{L}_{\beta}$ admits a meromorphic continuation on $\mathbb{C}$, with simple poles at $\beta=k/2$ for $k\in\{1,0,-1,-2,\ldots\}$. The continuation for $\Re(\beta) >0$ can be obtained by rewriting $f=f(0)+f^\star$ with $f^\star:=f-f(0)$ and noting that
\begin{equation}\label{equation: Mayer transfer extended}
 \mathcal{L}_{\beta}(f)(z)=f(0)\zeta_H(2\beta,z+1)+\sum_{n=1}^\infty \frac{1}{(n+z)^{2\beta}}(f^\star)\left(\frac{1}{n+z}\right),
\end{equation}
where $\zeta_H(s,z)=\sum_{n=0}^\infty (n+z)^{-s}$ is the Hurwitz zeta function, which has an analytic continuation for all $s\neq 1$ and a simple pole of residue $1$ at $s=1$. In fact, it can further be analytically continued to $\beta \geq -k/2$ for any $k\in\mathbb{N}$ by noticing that
\begin{equation}\label{equation: Mayer transfer extended2}
 \mathcal{L}_{\beta}(f)(z)=\sum_{n=0}^k\frac{d^nf}{dz^n}(0)\frac{\zeta_H(n+2\beta,z+1)}{n!}+\sum_{n=1}^\infty \frac{1}{(n+z)^{2\beta}}(f^\star)\left(\frac{1}{n+z}\right),
\end{equation}
where $f^\star(z)=f(z)- \sum_{n=0}^k\frac{d^nf}{d^nz}(0)z^n/n!$.
Furthermore $\mathcal{L}_{\beta}$ is a nuclear operator on $A_\infty(D)$ of order $0$ for all $\beta$.

The Mayer transfer operator encodes geometric information about the modular surface, which is the quotient orbifold $\mathbf{M}$ obtained from considering the Poincaré upper half plane $\mathbb{H}:\{x+iy:x,y\in \mathbb{R}, y>0\}$ equipped with the usual Riemannian metric $ds^2=(dx^2+dy^2)/y^2$ and identifying points via equivalence relation $\mathbb{H}\ni z\sim (az+b)/(cz+d)$ for all $a,b,c,d\in\mathbb{Z}$ with $ad-bc=1$. Since $\mathcal{L}_{\beta}$ is a nuclear operator, the Fredholm determinants $\det(1-\mathcal{L}_{\beta})$ and $\det(1+\mathcal{L}_{\beta})$ are well-defined. These relate to Selberg Zeta function $Z(s)$ by the formula
\begin{equation}\label{equation: zeta functions}
 Z(s)= \det(1-\mathcal{L}_s)(1+\mathcal{L}_s),
\end{equation}
see \cite{Mayer91}.
The Selberg zeta function is closely connected with the spectral theory of the Laplace-Beltrami operator $\Delta:=-y^2(d^2/dx^2+d^2/dy^2)$ \cite{Iwaniec}. Equation \eqref{equation: Mayer transfer extended} is thus a way of connecting the spectral theory of $\Delta$ with that of $\mathcal{L}_\beta$. This connection has been elaborated upon by Lewis, Zagier, Mayer and Chang \cite{Lewis97,ZagierLewis,MayerChang96,MayerChang98}.

From a dynamical perspective, it is interesting to see to what extent the aforementioned spectral properties of $\mathcal{L}_s$ survive when we consider e.g. Hölder continuous functions instead. For $\alpha\in (0,1)$, let $C^\alpha([0,1])$ denote the space of $\alpha$-Hölder functions on $[0,1]$. If $\alpha = k+\eta$ with $k\in\mathbb{N}$ and $\eta\in [0,1)$, let $C^\alpha([0,1])$ be the set of functions for which the $k$-th derivative exists and is continuous or $\eta$-Hölder if $\eta=0$ or $\eta\in (0,1)$ respectively. Denote by $L_\beta$ the operator \eqref{equation: Mayer transfer extended2} acting on $C^\alpha([0,1])$, which we shall show to be well-defined and continuous for large enough $\alpha$. These operators are not compact, but we can decompose the spectrum $\sigma(L_\beta)$ into the essential spectrum $\sigma_e(L_\beta)$ and the discrete spectrum $\sigma_{\mathrm{disc}}(L_\beta)$ which consists of isolated eigenvalues of finite algebraic multiplicity. We prove the following.

\begin{equation}\label{equation: Mayer transfer extended}
    \mathcal{L}_{\beta}(f)(z)=f(0)\zeta_H(2\beta,z+1)+\sum_{n=1}^\infty \frac{1}{(n+z)^{2\beta}}(f^\star)\left(\frac{1}{n+z}\right),
\end{equation}

\begin{equation}\label{equation: Mayer transfer extended2}
    \mathcal{L}_{\beta}(f)(z)=\sum_{n=0}^k\frac{d^nf}{dz^n}(0)\frac{\zeta_H(n+2\beta,z+1)}{n!}+\sum_{n=1}^\infty \frac{1}{(n+z)^{2\beta}}(f^\star)\left(\frac{1}{n+z}\right),
\end{equation}

Many of the usual theorems in complex analysis continue to hold for Banach or topological vector space-valued functions. In particular, holomorphic implies infinitely differentiable with a locally converging Taylor expansion. The unique extension theorem also continues to hold.
Finally, note that it is straightforward to show that $\beta\to L_\beta$ is a meromorphic function on $\Re(\beta)>(1-\alpha)/2$ for any $\alpha\in\mathbb{R}_{>0}$ with derivative
\[ \left(\frac{d L_\beta}{d\beta}\right)(f)(z)= -\sum_{n=0}^k(n+2\beta)\frac{d^nf}{dz^n}(0)\frac{\zeta_H(n+2\beta+1,z+1)}{n!} -2 \sum_{n=1}^\infty \frac{\ln(n+z)}{(n+z)^{2\beta}}(f^\star)\left(\frac{1}{n+z}\right).\]
\section{Uniformly Bounded Essential Spectrum}\label{section: uniformly bounded essential spectrum}
By \eqref{equation: Nussbaum}, we can estimate the essential spectral radius by finding for each $l\in \mathbb{N}$ a compact operator $T'$ for which $\|L_\beta^l - T' \|$ is small. A first naive approach would be to consider the operator $P$ which sends $f$ to a piecwise linear approximation that interpolates $f$ on the points $0,\frac{1}{N},\frac{2}{N},\cdots,1$ and considering $\|L_\beta^l-L_\beta^l\circ P\|$ as $N\to \infty$. For $\Re(\beta)>0$, we indeed have by Lemma \ref{lemma: mayer operator inequalities} that for any $\varepsilon>0$ there exists some $N$ for which $\lvert (L_\beta-L_\beta\circ P) (f)\rvert_\alpha(x)\leq \mathfrak{L}_{\Re(\beta)+\alpha}((1+\varepsilon)|f-P(f)|_\alpha)(x)+\varepsilon$. However, we do not have sufficient control over the $\|\cdot\|_\infty$-norm. We remedy this by considering a \textit{countable} interpolation which ensures $(L_\beta^l\circ P)f$ remains a (nonlinear) interpolation of
$L_\beta^l f$.

Let us now prove most of Theorem \ref{theorem: radius of essential spectrum}.
\begin{theorem}\label{theorem: spectral theorem}
Let $0<\alpha<1 $. Then for all $\beta$ with $\Re(\beta)>(1-\alpha)/2$ the transfer operator $L_\beta$ satisfies
\[\rho_e(L_\beta) \leq \rho(\mathfrak{L}_{\Re(\beta)+\alpha}).\]
\end{theorem}
\begin{proof}
 By Nussbaum's formula and Proposition \ref{proposition: compact operator}, it suffices to prove that there is some constant $C''>0$ for which \[\limsup_{N\to\infty} \|L_{\beta}^l -L_{\beta}^l \circ P_{l,N} \| \leq C''\left\|\mathfrak{L}_{\Re(\beta)+\alpha}^l\right\| \text{ for all } l.\]

\end{proof}
\begin{remark}\label{remark: spectral radius}
By using the spectral radius formula $\rho(\mathfrak{L}_{\Re(\beta)+\alpha})=\limsup_{l\to\infty}\|\mathfrak{L}_{\Re(\beta)+\alpha}^l\|^{1/l}$ and bounding every function $f\in L^\infty([0,1])$ by a constant, we obtain that $\rho(\mathfrak{L}_{\Re(\beta)+\alpha})\leq \rho(\mathcal{L}_{\Re(\beta)+\alpha})$ and hence that
\[\rho(\mathfrak{L}_{\Re(\beta)+\alpha}) = \lambda_1(\Re(\beta)+\alpha),\]
where $\lambda_1(t)$ for $t>1/2$ is the eigenvalue of maximum modulus of $\mathcal{L}_{t}$. The eigenvalue is guaranteed to be simple and positive by the Perron-Frobenius theorem. In fact $\lambda_1(t)$ is a real-analytic and decreasing function of $t$ with $\lambda(1)=1$, see \cite{Mayer90,Vallee98}.
\end{remark}
Theorem \ref{theorem: radius of essential spectrum} for $\alpha \in (0,1)$ then follows from Remark \ref{remark: spectral radius} and from the following corollary of Theorem \ref{theorem: spectral theorem}.
\begin{corollary}\label{Corollary: Ruelle}
 Let $\lambda \in \mathbb{C}$ satisfy $\abs{\lambda}>\rho(\mathfrak{L}_{\Re(\beta)+\alpha})$. Then $\lambda$ is an eigenvalue of $L_\beta$ if and only if it is an eigenvalue of $\mathcal{L}_\beta$. In this case the respective eigenspaces $E_\lambda^\alpha$ and $E_\lambda^\omega$ are equal.
\end{corollary}
This is an example of a more general principle for operators on Banach spaces, see e.g. \cite{Grabiner}. For our purposes, we may directly adapt a proof due to Ruelle (Corollary 3.3 in \cite{Ruelle89}), which we provide in the appendix.

\end{proof}
We see by Proposition $4.1$ that a generic solution of \eqref{equation: three term functional} in $C^\alpha((-1,\infty))$ satisfies $f= O(z^{1-2\Re(\beta)})$ if $\Re(\beta)>(1-\alpha)/2$. On the other hand, we see that the condition $Q=0$ in \eqref{equation: Qinfinity} or in \eqref{equation: Q0} is equivalent to the condition $L_\beta f= \lambda f$, which by Theorem \ref{theorem: radius of essential spectrum} implies $f$ extends to a holomorphic function in $\mathbb{C}\setminus (-\infty,-1]$ if $\abs{\lambda}\leq \rho(\mathcal{L}_{\Re(\beta)+\alpha})$. For example, we have the following corollary of Proposition \ref{equation: proposition: }, which we can view as a stronger version of the Bootstrapping procedure by Lewis and Zagier in \cite{ZagierLewis}.
\begin{corollary}
 Let $\Re(\beta)=1/2$. Suppose $f:(-1,\infty)\to \mathbb{C}$ is some solution of \eqref{equation: three term functional} for $\abs{\lambda} = 1$. Assume furthermore it satisfies the mild growth condition $f(z)= C_0x^{1-2\beta}+o(1)$ as $z\to \infty$.
 Then if $f$ is $1/2+\varepsilon$-Hölder for some $\varepsilon>0$, it is automatically holomorphically extendable to the cut plane $\mathbb{C}\setminus (-\infty,-1]$.
\end{corollary}
This corollary immediately follows from Theorem \ref{theorem: radius of essential spectrum} and the fact that $\rho(\mathcal{L}_a)<1$ if $a>1$. We remark that when $\Re(\beta)=1/2$ and $\lambda =1$ or $-1$, the space of eigenvalues of $\mathcal{L}_{\beta}$ is in bijection with the space of even, respectively uneven Maass cusp forms on the modular surface with eigenvalue $\beta(1-\beta)$ and we have that $C_0=0$, i.e. $f(z)= o(1)$ as $z\to \infty$, see \cite{Lewis97, ZagierLewis}.

\begin{proposition}\label{Proposition: Constructing Q}
 Let $\alpha>0$, $\lambda\in\mathbb{C}\setminus\{0\}$ and let $Q\in C^\alpha(\mathbb{R}/\mathbb{Z})$.
 For all $\beta$ for which $\Re(\beta)> (1-\alpha)/2$ and \[r(\mathcal{L}_{\Re(\beta)+\alpha})< \abs{\lambda},\]
 the function
 \[f=(1 - \lambda^{-1}L_\beta)^{-1}Q,\]
 is a meromorphic expression over all $\beta$ satisfying the conditions in this proposition. Furthermore $f$ satisfies \eqref{equation: three term functional}, \eqref{equation: Qinfinity} and \eqref{equation: Q0}.
\end{proposition}
\begin{proof}
 We interpret $(\lambda - L_\beta)^{-1}$ in the sense of Remark \ref{remark: extending L_beta}. More specifically, for $L_\beta$ acting on $ C^r([a,(1-a)^{-1}])$, the expression $(1 - \lambda^{-1}L_\beta)^{-1}\left(\rstr{Q}{[a,(1-a)^{-1}]}\right)$ is meromorphic and satisfies
\begin{equation}\label{equation: expansion for resolvent}
 \begin{split}
 &(1 - \lambda^{-1}L_\beta)^{-1}\left(\rstr{Q}{[a,(1-a)^{-1}]}\right)(z)=\sum_{n=0}^\infty \lambda^{-n}L_\beta^n\left(\rstr{Q}{[a,(1-a)^{-1}]}\right)(z).\\
 &= Q(z)+\sum_{n=1}^\infty \lambda^{-n}\sum_{a_1,\ldots,a_n\in\mathbb{N}} Q([a_1,\ldots,a_n+z])\Pi_{k=1}^n[a_k,\ldots,a_n+z]^{2\beta}
 \end{split}
\end{equation}
if $\Re(\beta)$ is large enough. By Theorem \ref{theorem: radius of essential spectrum}, we see that $(1 - \lambda^{-1}L_\beta)^{-1}$ is a meromorphic operator-valued function for all $a\leq1$. Hence the functions $(1 - \lambda^{-1}L_\beta)^{-1}\left(\rstr{Q}{[a,(1-a)^{-1}]}\right)$ are
meromorphic functions over the half-plane consisting of all $\beta$ satisfying the conditions in the proposition. We see by the second line of \eqref{equation: expansion for resolvent} and uniqueness of analytic continuation that these functions agree for different $a$ on the intersection of their domains. Hence we may define $(1 - \lambda^{-1}L_\beta)^{-1}Q(z)$ for any $z\in (-1,\infty)$ by choosing $a$ in \eqref{equation: expansion for resolvent} small enough.

The statement in the abstract concerning Maass cusp forms and nontrivial zeroes of the Riemann zeta function, which we make more precise in Remark \ref{remark: abstract}, turns out to follow immediately from this theorem.

Comparing the behaviour of transfer operators for different levels of regularity is a somewhat common theme in dynamics. One typically expects for a transfer operator associated to a sufficiently \enquote{nice} expanding map that the essential spectral radius vanishes for the operator acting on spaces of increasingly smooth functions, see e.g. \cite{Butterley}. In a sense the above theorem is therefore not too suprising. Indeed, we shall show that if $\alpha$ is a natural number, the proof of Theorem \ref{theorem: radius of essential spectrum} is a mundane application of standard techniques.

 However, the proof of the case $1/2<\alpha < 1$ is perhaps more interesting. In that case, we still obtain uniformly bounded estimates for $\rho_e(L_\beta)$ on vertical lines $\Re(\beta)=\sigma$ even when $\sigma < 1/2$, where the function $\beta\mapsto L_\beta$ is unbounded. However, by \eqref{equation: Mayer transfer operator}, the unboundedness in $\beta$ is due to the contribution rank one operator $f\mapsto f(0)\zeta_H(2\beta,z+1)$, which does not contribute to the essential spectral radius. Nevertheless, we do not obtain this result using a Doeblin-Fortet-Lasota-Yorke inequality, so our proof is substantially different from the standard approaches to these type of problems.

In Section \ref{section: three-term functional equation}, we interpret this result in terms of Lewis' three-term functional equation. In particular, we can interpret this as an extension of the \enquote{bootstrapping} result of Lewis and Zagier, who proved in \cite{ZagierLewis} that real-analytic solutions of this equation on $(1,\infty)$ satisfying certain growth conditions automatically extend to holomorphic functions on the cut plane $\mathbb{C}\backslash(-\infty,-1]$. We show that for $\Re(\beta)>0$, we may even replace real-analyticity with a \textit{Hölder condition} and obtain that it is still the restriction of a holomorphic function on $\mathbb{C}\backslash(-\infty,-1]$.

If $\alpha$ is a natural number, “the proof of Theorem \ref{theorem: radius of essential spectrum} is a mundane application of standard techniques.”

For $1/2<\alpha<1$, the proof is “substantially different from the standard approaches”: one “still obtain[s] uniformly bounded estimates for $\rho_e(L_\beta)$ on vertical lines $\Re(\beta)=\sigma$ even when $\sigma<1/2$, where the function $\beta\mapsto L_\beta$ is unbounded,” and this is because “the unboundedness in $\beta$ is due to the contribution rank one operator $f\mapsto f(0)\zeta_H(2\beta,z+1)$, which does not contribute to the essential spectral radius.” The result is “not obtain[ed]… using a Doeblin-Fortet-Lasota-Yorke inequality.”

If $\alpha$ is a natural number, the proof of the main theorem is a mundane application of standard techniques.

For $1/2<\alpha<1$, the proof is substantially different from the standard approaches: one still obtains uniformly bounded estimates for $\rho_e(L_\beta)$ on vertical lines $\Re(\beta)=\sigma$ even when $\sigma<1/2$, where the function $\beta\mapsto L_\beta$ is unbounded, and this is because the unboundedness in $\beta$ is due to the contribution rank one operator $f\mapsto f(0)\zeta_H(2\beta,z+1)$, which does not contribute to the essential spectral radius. The result is not obtained using a Doeblin-Fortet-Lasota-Yorke inequality.

Question:
Let lpha>0, write lpha=k+b7 with k\in\mathbb N and \eta\in[0,1), and let C^\alpha([0,1]) be the usual Hf6lder space. For \beta\in\mathbb C, define the analytically continued Mayer transfer operator L_\beta on C^\alpha([0,1]) by
\[
L_{\beta}(f)(z)=\sum_{n=0}^k\frac{f^{(n)}(0)}{n!}\,\zeta_H(n+2\beta,z+1)+\sum_{m=1}^\infty \frac{1}{(m+z)^{2\beta}}\,f^*\!\left(\frac{1}{m+z}\right),
\]
where
\[
f^*(z)=f(z)-\sum_{n=0}^k\frac{f^{(n)}(0)}{n!}z^n,
\]
and \zeta_H(s,z)=\sum_{m=0}^\infty (m+z)^{-s} is the Hurwitz zeta function. Let \mathcal L_\gamma denote the corresponding holomorphic Mayer transfer operator
\[
\mathcal L_\gamma(g)(z)=\sum_{m=1}^\infty \frac{1}{(m+z)^{2\gamma}}\,g\!\left(\frac{1}{m+z}\right).
\]
If \sigma_e(L_\beta) denotes the essential spectrum of L_\beta and
\[
\rho_e(L_\beta):=\sup_{\lambda\in\sigma_e(L_\beta)}|\lambda|,
\qquad
\rho(\mathcal L_{\Re(\beta)+\alpha}):=\sup_{\mu\in\sigma(\mathcal L_{\beta+\alpha})}|\mu|,
\]
which statement holds for all \beta with \Re(\beta)>(1-\alpha)/2?

For every \beta with \Re(\beta)>(1-\alpha)/2, the essential spectral radius satisfies
\[
\rho_e(L_\beta)\le \rho(\mathcal L_{\Re(\beta)+\alpha}).
\]
Moreover, for any eigenvalue \lambda of L_\beta with |\lambda|>\rho_e(L_\beta), the corresponding generalized eigenspace consists of analytic functions, and those functions extend to generalized eigenfunctions of \mathcal L_\beta.

Choices:
B.
For every \beta with \Re(\beta)\ge (1-\alpha)/2, the essential spectral radius satisfies
\[
\rho_e(L_\beta)\le \rho(\mathcal L_{\beta+\alpha}).
\]
Moreover, for any eigenvalue \lambda of L_\beta with |\lambda|\ge \rho_e(L_\beta), the corresponding generalized eigenspace consists of analytic functions, and those functions extend to generalized eigenfunctions of \mathcal L_\beta.
C.
For every \beta with \Re(\beta)>(1-\alpha)/2, the essential spectral radius satisfies
\[
\rho_e(L_\beta)\le \rho(\mathcal L_{\Re(\beta)+\alpha}).
\]
D.
For every \beta with \Re(\beta)>(1-\alpha)/2, the essential spectral radius satisfies
\[
\rho_e(L_\beta)< \rho(\mathcal L_{\Re(\beta)+\alpha}).
\]
Moreover, for any eigenvalue \lambda of L_\beta with |\lambda|>\rho_e(L_\beta), the corresponding generalized eigenspace consists of holomorphic functions on a neighborhood of [0,1], and every such generalized eigenfunction extends uniquely to an eigenfunction of \mathcal L_\beta with the same eigenvalue.
E.
For every \beta with \Re(\beta)>(1-\alpha)/2, the essential spectral radius satisfies
\[
\rho_e(L_\beta)\le \rho(\mathcal L_{\Re(\beta)}).
\]
Moreover, for any eigenvalue \lambda of L_\beta with |\lambda|>\rho_e(L_\beta), the corresponding generalized eigenspace consists of analytic functions, and those functions extend to generalized eigenfunctions of \mathcal L_{\Re(\beta)}.

strict half-plane condition and dependence on \Re(\beta) rather than full \beta

analyticity and extension of generalized eigenspaces above the essential spectral radius

generalized eigenfunctions vs eigenfunctions; non-strict radius inequality strengthened to strict and extension target sharpened

loss of the +\alpha shift coming from Hölder regularity in the comparison operator

The stem gives definitions and asks for the valid conclusion, but it does not explicitly state the theorem’s conclusion or otherwise reveal the correct option. The answer must be identified from the choices.

The item is close to a theorem-restatement question: the correct option essentially reproduces the full theorem, while distractors are perturbed variants. It is not fully tautological because the choices encode competing conclusions, but it still mainly tests exact theorem recall.

Some reasoning is needed to distinguish subtle mathematical differences such as strict vs. non-strict inequalities, dependence on Re(β) versus β, the +α shift, and generalized eigenspaces versus eigenfunctions. However, the task is still mostly recognition of the precise theorem statement rather than substantial derivation.

The distractors are strong: they are mathematically close to the correct statement, vary in realistic ways, and reflect common failure modes such as boundary-condition weakening, unjustified strengthening, dropping the regularity shift, or omitting part of the conclusion.

A solid theorem-identification MCQ with good distractors and little answer leakage, but it leans more toward precise recall of a known statement than genuine generative mathematical reasoning.


\label{thm:gawron-generalization}
	If $z\in \mathcal S$, then $|z| > \tfrac 14$.

\begin{thm}\label{thm:MO-ineq}
	For all real numbers $t$ and all positive integers $n$,
	\begin{equation}\label{eqn:tao-ineq}
		\abs{\left(1+\frac{it}n\right)^n - 1} \geq \big|e^{it} - 1\big| = 2\sin\frac t2.
	\end{equation}
\end{thm}

\begin{thm}\label{thm:roots-disk}
	All roots of $\stern n\lm$ lie outside the disk $\{\abs{z-2}\leq 1\}\subseteq\C$.
\end{thm}

\begin{thm}\label{thm:gawron-generalization}
	If $z\in \calS$, then $|z| > \tfrac 14$.
\end{thm}

\begin{conj}\label{conj:location-roots}
	All elements of $\calS$ lie in the half-plane $\{\Re w < 1\}$.
\end{conj}

\begin{figure}[ht]
	\centering
	\captionsetup{width=0.75\linewidth}
	\includegraphics[scale=0.75]{sternpoly-roots-deg-21-BW}
	\caption{\small The zeros of $\stern n\lm$ in the range $\{a+bi:-4\leq a\leq 1, |b|\leq 3\}$, where $1\leq n<2^{21}$ is odd.}
	\label{fig:stern-roots}
\end{figure}

\begin{corr}\label{thm:prime-poly-irred}
	For each prime number $p$, the Stern polynomial $\stern p \lm$ is irreducible in $\Q$.
\end{corr}

The Stern sequence $(s_n)_{n\geq 0}$ of positive integers, named after Moritz Abraham Stern, is given by $s_0 = 0$, $s_1 = 1$, and
\begin{equation}\label{eqn:stern-seq}
 s_{2n} = s_n,\quad s_{2n+1} = s_n + s_{n+1}.
\end{equation}
There is a vast body of literature (e.g. \cite{Coons2014,Northshield2010,Reznick2008,Stanley2019} and the references therein) about the Stern sequence. Perhaps the most striking property of the Stern sequence is that every positive rational number appears exactly once in the sequence $(s_{n+1}/s_n)_{n\geq 1}$, giving an explicit enumeration of $\mathbb Q^+$. Stern proved this fact 15 years before Cantor introduced the notion of a countable set!

\par In \cite{Klavzar2007}, Klav\v{z}ar, Milutinovi\'{c}, and Petr define a polynomial analogue $\stern n\lambda$ of the Stern sequence via $\stern 0\lambda = 0$, $\stern 1\lambda = 1$, and
\begin{equation}\label{eqn:stern-def}
 \stern{2n}\lambda = \lambda \stern n\lambda,\quad \stern{2n+1}\lambda = \stern n\lambda + \stern{n+1}\lambda.
\end{equation}
The first $16$ terms of the sequence $\stern n\lambda$ are given in Table \ref{table:stern-16}. This table suggests many patterns, several of which are recorded in Proposition \ref{prop:basic-patterns}.

\begin{table}[ht]
 \input{table-stern-16}
 \caption{\small The first $16$ terms of the sequence $\stern n\lambda$.}
 \label{table:stern-16}
\end{table}

While there are several papers which analyze the sequence $\stern n\lambda$ (see \cite{Dilcher2017,Dilcher2018,Gawron2014,Ulas2011} and the references therein), not much is known about the zeros of Stern polynomials. Because $\stern{2n}\lambda = \lambda \stern n \lambda$ for all $n\geq 1$, it suffices to examine the roots of $\stern n\lambda$ only when $n$ is odd. Define
\[
\mathcal S\coloneqq \{z\in\mathbb C: \stern n z = 0\text{ for some odd }n\geq 1\}.
\]
In \cite[Theorem 2.1]{Gawron2014}, Gawron proves that $0$, ${-}1$, ${-}\tfrac12$, and $-\tfrac13$ are the only rational zeros of any Stern polynomial by showing, more generally, that any real number in the closed interval $[-\tfrac14,\tfrac14]$ cannot be in $\mathcal S$. His methods extend easily to show that any complex number $z$ with $|z| \leq \tfrac14$ cannot be in $\mathcal S$. For completeness, we record the proof here.

\begin{prop}\label{prop:basic-patterns}
	Let $n$ be a nonnegative integer.

	\vspace{-3pt}

	\begin{enumerate}
		\item $\stern{n}{0} = 0$ when $n$ is even, and $\stern n 0 = 1$ when $n$ is odd.
		\item $\stern{n}{1} = s_n$, where $s_n$ is the Stern sequence \eqref{eqn:stern-seq}.
		\item $\stern{n}{2} = n$.
		\item When $n = 2^r$ for a nonnegative integer $r$, $\stern n \lm = \lm^r$.
		\item When $n = 2^r - 1$ for a nonnegative integer $r$, $\stern n\lm = (\lm)_r$.
	\end{enumerate}
\end{prop}

\begin{table}[ht]
	\input{table-stern-16}
	\caption{\small The first $16$ terms of the sequence $\stern n\lm$.}
	\label{table:stern-16}
\end{table}

While there are several papers which analyze the sequence $\stern n\lm$ (see \cite{Dilcher2017,Dilcher2018,Gawron2014,Ulas2011} and the references therein), not much is known about the zeros of Stern polynomials. Because $\stern{2n}\lm = \lm \stern n \lm$ for all $n\geq 1$, it suffices to examine the roots of $\stern n\lm$ only when $n$ is odd. Define
\[
\calS\coloneqq \{z\in\C: \stern n z = 0\text{ for some odd }n\geq 1\}.
\]
In \cite[Theorem 2.1]{Gawron2014}, Gawron proves that $0$, ${-}1$, ${-}\tfrac12$, and $-\tfrac13$ are the only rational zeros of any Stern polynomial by showing, more generally, that any real number in the closed interval $[-\tfrac14,\tfrac14]$ cannot be in $\calS$. His methods extend easily to show that any complex number $z$ with $|z| \leq \tfrac14$ cannot be in $\calS$. For completeness, we record the proof here.

\begin{proof}
 Let $z$ be any complex number with $|z| \leq \tfrac 14$. Let $b_n\coloneqq |\stern n z|$ for each positive integer $n$. We show more strongly that
 \begin{equation}\label{eqn:stronger-statement}
 b_{2n+1} > \tfrac12\max\{b_n,b_{n+1}\} > 0
 \end{equation}
 for all $n\geq 1$.

\begin{thm}[Parabola Theorem, {\cite[Theorem 3.43]{Lorentzen2008}}]\label{thm:parabola}
 For fixed $|\alpha| < \tfrac\pi 2$, let
 \[
 V_\alpha\coloneqq -\tfrac12 + e^{i\alpha}\overline{\mathbb H} = \{w\in\C: \Re(we^{- i\alpha})\geq -\tfrac12\cos\alpha\}
 \]
 and
 \[
 E_\alpha \coloneqq \{a\in\C: |a| - \Re(ae^{-2i\alpha})\leq \tfrac12\cos^2\alpha\}.
 \]
 Then $E_\alpha$ is the element set for continued fractions $\boldsymbol{K}(a_n | 1)$ corresponding to the value set $V_\alpha$.
\end{thm}
The region $V_\alpha$ is a half-plane whose boundary is a line intersecting the real axis at $z = -\tfrac12$. In particular, because $|\alpha| < \tfrac\pi 2$, $V_\alpha$ contains the half-line $[-\tfrac12,\infty)$. The boundary of $E_\alpha$ is a parabola with focus at the origin and vertex at $-\tfrac14 e^{2i\alpha}\cos^2\alpha$. This parabola intersects the real axis at $z = -\tfrac14$.

We now tie the continued fraction theory back to Stern polynomials.
Recall that Corollary \ref{thm:stern-cont-frac} expresses the ratio
of two Stern polynomials as a continued fraction with elements in $\C$.
This form on its own is difficult to work with. We instead note (e.g. \cite[Corollary 2.15]{Lorentzen2008}) that any continued fraction $\boldsymbol{K}(a_n|b_n)$ with $b_j\neq 0$ is equivalent to the continued fraction $\boldsymbol{K}(c_n|1)$, where $c_j = \tfrac{a_j}{b_jb_{j+1}}$. As an explicit example,
\begin{equation}\label{eqn:cfrac-example}
 \frac{\stern{[[2,3,5]]}\lm}{\stern{[[3,5]]}\lm} = (\lm)_2 + \cfrac{\lm^2}{(\lm)_3 + \cfrac{\lm^3}{(\lm)_5}}
 = (\lm)_2 \left[1 + \cfrac{\dfrac{\lm^2}{(\lm)_2(\lm)_3}}{1 + \cfrac{\lm^3}{(\lm)_3(\lm)_5}}\right].
\end{equation}
\begin{remark}
 While continued fractions of the form $\boldsymbol{K}(1|d_n)$ are more common, the coefficients $d_n$ are significantly messier than the coefficients $c_n$ (see \cite[Corollary 2.15]{Lorentzen2008}). We opt to use the less-common form $\boldsymbol{K}(c_n|1)$ to simplify our analysis.
\end{remark}
Equation \eqref{eqn:cfrac-example} suggests that the ratios $\tfrac{z^a}{(z)_a(z)_b}$, where $a$ and $b$ are any positive integers, may play some importance.
For this reason, we define $z_{a,b}\coloneqq \tfrac{z^a}{(z)_a(z)_b}$ and
\[
\calA_z\coloneqq \left\{z_{a,b}:(a,b)\in\N^2\right\}.
\]
Suppose $\calA_z\subseteq E_\alpha$ for some angle $\alpha$ with $|\alpha| < \tfrac{\pi}2$.
Then $V_\alpha$ is a value set for the continued fraction \eqref{eqn:stern-cont-frac} at $\lm = z$. In particular, $\infty$ is not a possible value for this fraction, so $\stern n z \neq 0$ for any $z$.

\begin{itemize}
 \item First suppose $x\in[0,\pi]$. Because both $\sin x$ and $x - \tfrac{x^2}{\pi}$ are symmetric about the axis $x = \tfrac\pi 2$, it suffices to prove the inequality for $x\in[0,\tfrac\pi 2]$. Within this smaller interval, cosine is concave down with $\cos 0 = 1$ and $\cos\tfrac \pi 2 = 0$, so
 \begin{equation}\label{eqn:cos-ineq}
 \cos x \geq 1 - \frac{2x}{\pi}.
 \end{equation}
 Integrating both sides of \eqref{eqn:cos-ineq} yields \eqref{ineq-sinc} in this case.
 \item Now suppose $x\notin[0,\pi]$. By symmetry, it suffices to prove the inequality for $x\geq \pi$. In this interval, $1 - \tfrac{2x}{\pi} \leq -1 \leq \cos x$. It follows that \eqref{eqn:cos-ineq}, and thus \eqref{ineq-sinc}, holds as well.
 \end{itemize}
\end{proof}

\begin{thm}\label{thm:min-geom-series}
 Let $n$ be a positive integer, and suppose $z\in\calB$. Then $|(z)_n| \geq \min(n,\tfrac{11}2)$.
\end{thm}

Now let
 \[
 M\coloneqq 2\pi\left(1 - \frac{5.5}{n}\right) = \pi\left(2 - \frac{11}n\right).
 \]
 There are two cases to consider. First, suppose $t\leq M$. Then
 \[
 n\left(1 - \frac{t}{2\pi}\right) \geq n\left(1 - \frac{M}{2\pi}\right) = \frac{11}2.
 \]
 Now suppose $t\geq M$. Because $n\geq 15$, the coefficient $\tfrac{1}{2n} - \tfrac{11}{8n^2}$ is positive, so
 \begin{equation}\label{eqn:asymp-w-tn}
 t + t^3\left(\frac{1}{2n} - \frac{11}{8n^2}\right) \geq \pi\left(2 - \frac{11}n\right) + \pi^3\left(2 - \frac{11}n\right)^3\left(\frac{1}{2n} - \frac{11}{8n^2}\right).
 \end{equation}
 To estimate \eqref{eqn:asymp-w-tn}, let $x = \tfrac 1n$, so it suffices to analyze the polynomial
 \[
 P(x) := \pi(2-11x) + \pi^3(2-11x)^3(\tfrac 12x - \tfrac{11}{8}x^2).
 \]
 The polynomial $P$ has exactly one critical point in the interval $[0,\tfrac 1{15}]$, occurring at $x = x_0 \approx 0.024$. In particular, $P$ is increasing on $[0,x_0]$ and decreasing on $[x_0,\tfrac1{15}]$. Compute
 \[
 P(0) = 2\pi\quad\text{and}\quad P\left(\frac 1{15}\right) = \frac{19 \pi }{15}+\frac{336091 \pi ^3}{6075000} \approx 5.695 > \frac{11}2.
 \]
 It follows that $P(x) > \tfrac{11}2$ for all $x\in[0,\tfrac{1}{15}]$, and thus $|w_{n,t}^n - 1| \geq M|w_{n,t}-1|$ for all $t\in[0,2\pi]$ and $n\geq 15$.
\end{proof}

\begin{prop}\label{prop:silli}
 Let $z\in\calB^+$. Then $0\leq \arg z \leq \tfrac\pi 6$ and
 \begin{equation}\label{eqn:re-ineq}
 |\Re z^{-4}|\leq |\Re z^{-2}|.
 \end{equation}
\end{prop}

\begin{corr}[{\cite[Theorem 1]{Schinzel2014}}]\label{thm:stern-cont-frac}
	Let $a_1,\ldots, a_t$ be positive integers, where $t\geq 2$.  Then
	\begin{equation}\label{eqn:stern-cont-frac}
		\frac{\stern{[[a_1,\ldots, a_t]]}{\lm}}{\stern{[[a_2,\ldots, a_t]]}{\lm}}
		=
		(\lm)_{a_1} + \cfrac{\lm^{a_1}}{(\lm)_{a_2} + \cfrac{\lm^{a_2}}{\cdots + \cfrac{\cdots}{(\lm)_{a_{t-1}} + \cfrac{\lm^{a_{t-1}}}{(\lm)_{a_t}}}}}\,.
	\end{equation}
\end{corr}

\begin{proof}
	Let $z$ be any complex number with $|z| \leq \tfrac 14$.  Let $b_n\coloneqq |\stern n z|$ for each positive integer $n$.  We show more strongly that
	\begin{equation}\label{eqn:stronger-statement}
		b_{2n+1} > \tfrac12\max\{b_n,b_{n+1}\} > 0
	\end{equation}
	for all $n\geq 1$.

	\par The proof of \eqref{eqn:stronger-statement} proceeds by induction on $n$.  The base case, $n = 1$, follows because
	\[
	b_3 = |z + 1| \geq \tfrac 34 > \tfrac 12
	= \tfrac 12\max\{1, |z|\} = \tfrac 12\max\{b_1,b_2\}.
	\]

	\par There are two cases to consider.  First assume $n=2k$ is even.  Then
	\begin{align*}
		b_{4k+1} &= |\stern{4k+1}z| = |\stern{2k+1} z + \stern{2k} z|\\
		&= |z \stern kz + \stern{2k+1} z| \geq b_{2k+1} - \tfrac14 b_k\\
		&\geq b_{2k+1} - \tfrac12 b_{2k+1} = \tfrac12 b_{2k+1}.
	\end{align*}
	Moreover, $b_{2k+1} \geq \frac12 b_k$, and thus $\max\{b_{2k},b_{2k+1}\} = b_{2k+1} > 0$.  In this case, our inequality is proved.

	\par Now assume $n = 2k + 1$ is odd.  Then
	\begin{align*}
		b_{4k+3} &= |\stern{4k+3}z| = |\stern{2k+1} z + \stern{2k+2} z|\\
		&= |\stern{2k+1} z + z\stern{k+1} z| \geq b_{2k+1} - \frac 14 b_{k+1}\\
		&\geq b_{2k+1} - \tfrac12 b_{2k+1} = \tfrac12 b_{2k+1}.
	\end{align*}
	Moreover, $b_{2k+1} \geq \frac12 b_{k+1}$, and so in this case we also have $\max\{b_{2k},b_{2k+1}\} = b_{2k+1} > 0$.  We have exhausted both cases, completing the proof of Theorem \ref{thm:gawron-generalization}.
\end{proof}

In \cite{Dilcher2017}, Dilcher et. al. focus more specifically on the complex roots of $\stern n\lambda$.  Their paper makes the following conjecture.

\begin{conj}\label{conj:location-roots}
	All elements of $\mathcal S$ lie in the half-plane $\{\operatorname{Re} w < 1\}$.
\end{conj}

By generalizing the Enestrom-Kakeya theorem, they prove Conjecture \ref{conj:location-roots} for several classes of positive integers $n$ taking the form $2^n\pm k$, where $k$ is fixed and $2^n \geq k$.  These are the only two papers the author could find which discuss the complex zeros of $\stern n\lambda$.

\par Figure \ref{fig:stern-roots} shows a snapshot of $\mathcal S$.  One striking feature of this figure is the contrasting behavior of these roots within the half-planes $\{\operatorname{Re} w\geq 0\}$ and $\{\operatorname{Re} w < 0\}$.  These differences present difficulties in fully characterizing the geometry of $\mathcal S$.

\begin{figure}[ht]
	\centering
	\captionsetup{width=0.75\linewidth}
	\includegraphics[scale=0.75]{sternpoly-roots-deg-21-BW}
	\caption{\small The zeros of $\stern n\lambda$ in the range $\{a+bi:-4\leq a\leq 1, |b|\leq 3\}$, where $1\leq n<2^{21}$ is odd.}
	\label{fig:stern-roots}
\end{figure}

In this paper, we partially resolve Conjecture \ref{conj:location-roots} by establishing the following result.

\begin{thm}\label{thm:roots-disk}
	All roots of $\stern n\lambda$ lie outside the disk $\{\left|z-2\right|\leq 1\}\subseteq\mathbb C$.
\end{thm}

To prove Theorem \ref{thm:roots-disk}, we use a continued fraction representation for a ratio of Stern polynomials (Theorem \ref{thm:stern-cont-frac}) independently discovered by Reznick \cite{Reznick2008} and Schinzel \cite{Schinzel2014}.  This allows us to use the Parabola Theorem (Theorem \ref{thm:parabola}) to show that, for certain values of $z\in\mathbb C$, the denominators of these continued fractions can never be zero.  Along the way, we establish inequalities in $\mathbb C$ relating to the sums $1 + z + \cdots + z^{n-1}$ which may be of independent interest.  The most notable of these inequalities is Theorems \ref{thm:MO-ineq}, which proves a lower bound for this geometric series whenever $\operatorname{Re} z \geq 1$.

As a corollary, we obtain the following surprising fact, resolving a conjecture of Ulas and Ulas (\cite{Ulas2011}).

\begin{corr}\label{thm:prime-poly-irred}
	For each prime number $p$, the Stern polynomial $\stern p \lambda$ is irreducible in $\mathbb Q$.
\end{corr}
Ulas and Ulas had verified this conjecture computationally for the first million primes $p$.  Additionally, Schinzel in \cite{Schinzel2011} proved Corollary \ref{thm:prime-poly-irred} for all primes $p < 2017$ by using finite differences to bound the leading coefficient of any proper divisor of $\stern n\lambda$.  However, these previous attempts to prove the conjecture were algebraic in nature, whereas our proof depends on the analytic properties of $\mathcal S$.

\paragraph{Acknowledgments.} This paper is adapted from the author's PhD dissertation \cite{Altizio2025}.  The author thanks his advisor, Dr. Bruce Reznick, for helpful correspondence.

Let $z$ be any complex number with $|z|\leq \tfrac14$ and set $b_n\coloneqq |\stern n z|$. The proof shows the stronger claim
\begin{equation}\label{eqn:stronger-statement}
 b_{2n+1} > \tfrac12\max\{b_n,b_{n+1}\} > 0
\end{equation}
for all $n\ge1$, and proceeds by induction on $n$.

Base case $n=1$:
\[
 b_3=|z+1|\ge \tfrac34>\tfrac12=\tfrac12\max\{1,|z|\}=\tfrac12\max\{b_1,b_2\}.
\]

Inductive step splits into two cases.

(i) If $n=2k$ is even, then
\[
 b_{4k+1}=|\stern{4k+1}z|=|\stern{2k+1}z+\stern{2k}z|=|z\,\stern k z+\stern{2k+1}z|
 \ge b_{2k+1}-\tfrac14 b_k\ge b_{2k+1}-\tfrac12 b_{2k+1}=\tfrac12 b_{2k+1}.
\]
Moreover, $b_{2k+1}\ge \tfrac12 b_k$, hence $\max\{b_{2k},b_{2k+1}\}=b_{2k+1}>0$, giving \eqref{eqn:stronger-statement}.

(ii) If $n=2k+1$ is odd, then
\[
 b_{4k+3}=|\stern{4k+3}z|=|\stern{2k+1}z+\stern{2k+2}z|=|\stern{2k+1}z+z\,\stern{k+1}z|
 \ge b_{2k+1}-\tfrac14 b_{k+1}\ge b_{2k+1}-\tfrac12 b_{2k+1}=\tfrac12 b_{2k+1}.
\]
Moreover, $b_{2k+1}\ge \tfrac12 b_{k+1}$, so again $\max\{b_{2k},b_{2k+1}\}=b_{2k+1}>0$. These two cases exhaust the induction, completing the proof of Theorem~\ref{thm:gawron-generalization}.

Question:
For the Stern polynomials \(\operatorname{St}_n(\lambda)\) defined by \(\operatorname{St}_0(\lambda)=0\), \(\operatorname{St}_1(\lambda)=1\), and for all integers \(n\ge 0\),
\[
\operatorname{St}_{2n}(\lambda)=\lambda\,\operatorname{St}_n(\lambda),\qquad
\operatorname{St}_{2n+1}(\lambda)=\operatorname{St}_n(\lambda)+\operatorname{St}_{n+1}(\lambda),
\]
define
\[
\mathcal S:=\{z\in\mathbb C: \operatorname{St}_n(z)=0\text{ for some odd }n\ge 1\}.
\]
Which quantitative estimate holds for every \(z\in\mathcal S\)?

Every \(z\in\mathcal S\) satisfies \(|z|>\tfrac14\).

Choices:
B. Every \(z\in\mathcal S\) satisfies \(|z|\ge \tfrac14\).
C. Every \(z\in\mathcal S\) satisfies \(|z|>0\).
D. Every \(z\in\mathcal S\) satisfies \(\Re(z)>-\tfrac14\).
E. Every \(z\in\mathcal S\) satisfies \(|z|>\tfrac13\).

The stem defines the Stern polynomials and the set \(\mathcal S\), but it does not explicitly state or strongly hint at the bound \(|z|>\tfrac14\). The correct estimate is not leaked by wording alone.

This is close to a theorem-recall item: after defining \(\mathcal S\), it asks directly which universal estimate holds. The multiple choices introduce some comparison, but the task is still largely a reformulation of the target conclusion rather than a genuinely new inference problem.

There is some reasoning pressure because the student must distinguish between a sharp strict bound, a weak true statement, and stronger or unrelated false variants. However, it mainly tests recognition of the known result rather than substantial derivation or synthesis.

The distractors are well designed: \(\ge \tfrac14\) probes strict vs. non-strict misunderstanding, \(|z|>0\) is a weaker true statement, \(|z|>\tfrac13\) is an overstrengthening trap, and \(\Re(z)>-\tfrac14\) is a distinct but irrelevant-looking alternative. They are plausible and target common mathematical failure modes.

A solid MCQ with strong distractors and no answer leakage, but it is somewhat theorem-recall based and only moderately tests generative reasoning.