[Structural output from primitive cycles]
\label{thm:structural}
Assume the standing assumptions of Remark~\ref{rem:standing-assumptions},
so that the scaling-limit kernel $K_L$ exists.
Then there exists a nonnegative kernel $\Phi\in L^1(\mathbb{R})$ such that:
\begin{enumerate}[label=(\roman*)]
\item $\Phi\in\mathrm{PF}_\infty$;
\item its bilateral Laplace transform $\mathcal B\Phi$ satisfies the reflection law
$\mathcal B\Phi(s)=\mathcal B\Phi(\tfrac12-s)$ on the common domain of absolute convergence;
\item $\mathcal B\Phi$ admits a canonical Schoenberg--Edrei--Karlin representation
\[
\mathcal B\Phi(s)=\frac{E(s)}{\Psi(s)},
\]
where $\Psi$ is Laguerre--P\'olya and hence has only real zeros.
\end{enumerate}

\begin{remark}[Translation-invariant specialization]
\label{rem:translation-invariant}
For the purposes of this paper we restrict to the translation-invariant case
\(
a_j\equiv a>0
\)
(independent of $j$), so that \eqref{eq:generator} is the usual continuous-time simple
random walk on $\Z/N\Z$ with jump rate $a$. In this case the diffusive scaling limit is
governed by the one-dimensional heat equation with diffusion constant \(D=a\), and the
Fourier analysis on $\Z/N\Z$ leading to the theta-series form is completely explicit.
Extensions to uniformly elliptic periodic environments, or to quantitative homogenization
regimes, can also yield a uniform local CLT and the same scaling limit; we do not pursue
those generalizations here.
\end{remark}

\begin{definition}[Archimedean completion operator]
\label{def:arch-operator}
For a sufficiently smooth function $f:(0,\infty)\to\R$, define
\begin{equation}\label{eq:arch-operator}
(\mathcal A f)(t):=\frac{d}{dt}\Bigl(t^{3/2}\frac{d}{dt}f(t)\Bigr),\qquad t>0.
\end{equation}
\end{definition}

\begin{remark}[What this proof does and does not use]
We do \emph{not} use any Mellin transform, any identification with $\Xi$, nor any
analytic continuation beyond the natural half-planes of convergence. The only external
inputs are the product formula \eqref{eq:invGamma-product} and the standard
Schoenberg--Edrei--Karlin criterion linking Laguerre--P\'olya denominators to
$\mathrm{PF}_\infty$ kernels.
\end{remark}

\section{The bridge problem (open)}
\label{sec:future-directions}

The constructions developed in this paper naturally suggest a further analytic
problem that we do not address here. On the one hand, the primitive finite dynamics
considered above give rise, in a canonical and entirely Archimedean manner, to a
nonnegative integrable kernel $\Phi$ whose bilateral Laplace transform admits a
Schoenberg--Edrei--Karlin factorization $\mathcal B\Phi=E/\Psi$, with $\Psi$ belonging
to the Laguerre--P\'olya class\cite{Levin1964,PolyaSchur1914}. On the other hand, the Archimedean completion procedure
introduced here produces, at a distinguished self-dual normalization, a Mellin
transform coinciding with the classical function $\Xi(2z)$.

\smallskip
\noindent\textbf{Sanity checks for any future identification.}
Any proposed identification should, at minimum, respect (i) the functional symmetries arising from the canonical centering and self-dual normalization, and (ii) the intrinsic total-positivity/Laguerre--P\'olya structure on the $\Psi$-side\cite{Schoenberg1951,Edrei1952,karlin1968total,Levin1964}.

A natural question is whether these two analytic structures can be directly related.
More precisely, one may ask whether the Laguerre--P\'olya datum $\Psi$ arising from the
total-positivity side can be identified with $\Xi(2z)$ up to multiplication by a
zero-free entire factor, or whether a suitable rigidity principle forces such an
identification once both objects are placed in a common analytic framework. Any such
identification would amount to a comparison of the zero divisors of two functions that
originate from a priori unrelated constructions.

We view this identification problem as a promising direction for future work. Its
resolution would require additional analytic input beyond the scope of the present
paper, and we therefore leave it open.

\bibliographystyle{plain}
\bibliography{citations}

\appendix
\section{Technical analytic justifications}
\label{app:technical}

This appendix collects several analytic estimates and justifications that are
used implicitly or explicitly in the main text. None of the results here
introduce new objects or alter the logical structure of the argument; they serve
only to justify standard limiting procedures, termwise operations, and boundary
manipulations already invoked.

\subsection{Uniform local central limit bounds}
\label{app:ULCLT}

In Section~\ref{sec:completion-selfdual} we appealed to a uniform local central limit theorem (ULCLT) to pass
from the discrete primitive dynamics to the scaling-limit kernel $K_L$. We record
here a representative bound sufficient for all uses in the paper.

\begin{lemma}[Uniform local CLT estimate]
\label{lem:ULCLT}
Let $(X_t^{(N)})_{t\ge0}$ be the rescaled primitive process on the $N$-cycle with
effective diffusion constant $D>0$, and let $p_t^{(N)}(j)$ denote its transition
kernel. Then for every compact interval $[t_0,t_1]\subset(0,\infty)$ there exists
a constant $C=C(t_0,t_1)$ such that
\[
\sup_{t\in[t_0,t_1]}\sup_{j\in\mathbb Z}
\Bigl|
p_t^{(N)}(j)
-
\frac{1}{\sqrt{4\pi Dt}}\exp\!\Bigl(-\frac{j^2}{4Dt}\Bigr)
\Bigr|
\le
\frac{C}{N},
\]
for all sufficiently large $N$.
\end{lemma}

\begin{proof}[Sketch]
This is a standard consequence of Fourier analysis on the discrete torus together
with Taylor expansion of the dispersion relation near the origin. Uniformity in
$t\in[t_0,t_1]$ follows from compactness. Precise proofs may be found, for example,
in classical treatments of random walks on finite groups.
\end{proof}

\begin{remark}
Lemma~\ref{lem:ULCLT} is used only to justify dominated convergence and termwise
limits when passing to the theta-series representation of the completed kernel.
No rate sharper than $O(N^{-1})$ is required.
\end{remark}

\begin{theorem}[Archimedean Mellin identification]
\label{thm:arch-mellin-identification}
Assume the self-dual scale $L^2=4\pi D$, so that $\widetilde K_{\mathrm{arch}}=\Theta$
by Lemma~\ref{lem:arch-equals-Theta-new}.
Define, for $z\in\mathbb C$,
\begin{equation}
\label{eq:Farch-def-new}
F_{\mathrm{arch}}(z)
:=
\int_0^\infty \widetilde K_{\mathrm{arch}}(t)\,t^{\frac34+iz}\,\frac{dt}{t}.
\end{equation}
Then for all $z\in\mathbb C$,
\begin{equation}
\label{eq:Farch-equals-Xi}
F_{\mathrm{arch}}(z)=\Xi(2z).
\end{equation}
\end{theorem}

\begin{equation}
\label{eq:Phi-def}
\Phi(x)
:=
e^{x/4}\,\widetilde K_{\mathrm{sym}}(e^{x})
=
e^{-x/4}\,\widetilde K(e^{x}),
\qquad x\in\mathbb R.
\end{equation}

\label{tab:notation}
\end{table}

\section{Acknowledgements}
The author would like to thank Krishnaswami Alladi, Tiziano Valentinuzzi and Kenneth Valpey for helpful discussions.

\section{Self-dual Ar

\begin{tabular}{|p{0.10\linewidth}|p{0.52\linewidth}|p{0.30\linewidth}|}
\hline
\textbf{Step} & \textbf{Input $\to$ Output} & \textbf{Main tool} \\
\hline
1 &
local dynamics $\to$ theta-series kernel $\widetilde K$ &
lift/periodize + ULCLT$^*$
+ dominated convergence
\\ \hline
2 &
$\widetilde K \to$ self-dual normalization and Archimedean completion $\mathcal A$ &
Jacobi inversion + completion operator
\\ \hline
3 &
$\widetilde K \to \Phi \to (\Psi,\ \mathcal B\Phi=E/\Psi)$ &
complete monotonicity $\Rightarrow \mathrm{PF}_\infty$ +
Schoenberg--Edrei--Karlin
\\ \hline
4 &
$\mathcal A(K_L-1)\to \Theta \to \Xi$ (anchor) and open bridge problem &
Archimedean completion + Mellin;
Comparison problem (open)
\\ \hline
\end{tabular}

\caption{Proof map (Steps~1--4). $^*$ULCLT = uniform local central limit theorem.}
\label{tab:proof-map}
\end{table}

The proof relies on a small number of canonical kernels and transforms; these are summarized in Table~\ref{tab:notation} with precise definitions and references.

\begin{table}[h!]
\centering
\setlength{\arrayrulewidth}{0.8pt}
\renewcommand{\arraystretch}{1.25}

\begin{tabular}{|p{0.22\linewidth}|p{0.70\linewidth}|}
\hline
\textbf{Symbol} & \textbf{Meaning} \\
\hline
$K_L(t)$ &
Scaling-limit trace
(Definition~\ref{def:scaling-limit}).
\\ \hline
$\widetilde K(t)$ &
Renormalized trace kernel
(Theorem~\ref{thm:theta-series-form-recall}).
\\ \hline
$\widetilde K_{\mathrm{sym}}(t)=t^{-1/2}\widetilde K(t)$ &
Symmetric half-density kernel
(Definition~\ref{def:half-density}).
\\ \hline
$\Phi(x)$ &
Logarithmic kernel
$\Phi(x)=e^{x/4}\,\widetilde K_{\mathrm{sym}}(e^x)$
(equation~\eqref{eq:Phi-def}).
\\ \hline
$\mathcal B\Phi(s)$ &
Bilateral Laplace transform
$\displaystyle \mathcal B\Phi(s)=\int_{\R}\Phi(x)e^{-s x}\,dx$
(on $\Re(s)>-\tfrac14$).
\\ \hline
$\Psi(s)$ &
Laguerre--P\'olya entire function with only real zeros satisfying
$\mathcal B\Phi(s)=\frac{E(s)}{\Psi(s)}$ on a strip
(Theorem~\ref{thm:SEK}).
\\ \hline
$F(z)$ &
Fourier transform of $\Phi$,
$\displaystyle F(z)=\int_{\R}\Phi(x)e^{izx}\,dx$;
boundary value $F(z)=\mathcal B\Phi(-iz)$ where both sides converge.
\\ \hline
$\mathcal A$ &
Archimedean completion operator
(Definition~\ref{def:arch-operator}).
\\ \hline
$\widetilde K_{\mathrm{arch}}$ &
Completed Archimedean kernel
$\widetilde K_{\mathrm{arch}}:=\mathcal A(K_L-1)$
(Definition~\ref{def:arch-theta-kernel}).
\\ \hline
$F_{\mathrm{arch}}(z)$ &
Mellin transform
$\displaystyle F_{\mathrm{arch}}(z)=\int_0^{\infty}
\widetilde K_{\mathrm{arch}}(t)\,
t^{\frac14+iz}\frac{dt}{t}$;
at the self-dual scale,
$F_{\mathrm{arch}}(z)=\Xi(2z)$
(Theorem~\ref{thm:arch-mellin-identification}).
\\ \hline
$\Psi_c(z)$ &
Centered Laguerre--P\'olya function
$\Psi_c(z):=\Psi(\tfrac14+iz)$. A central open problem is to compare $\Psi_c$ with the Mellin-side
function $\Xi(2z)$ up to multiplication by a zero-free entire factor (Section~\ref{sec:future-directions}).
\\ \hline
\end{tabular}

\begin{definition}[Archimedean-completed kernel]
\label{def:arch-theta-kernel}
Let $K_L(t)$ be the scaling-limit trace. Define the Archimedean-completed kernel
\[
\widetilde K_{\mathrm{arch}}(t) :=\bigl(\mathcal{A}(K_L-1)\bigr)(t),\qquad t>0.
\]
\end{definition}

\begin{theorem}[Theta-series form of the scaling-limit trace]
\label{thm:theta-series-form-recall}
For every $L>0$ and $t>0$,
\begin{equation}\label{eq:KL-theta-recall}
K_L(t)=\sum_{n\in\Z}\exp\!\Bigl(-\frac{4\pi^2D}{L^2}\,n^2\,t\Bigr).
\end{equation}
\end{theorem}

\label{sec:background}

Across several foundational currents in mathematics, a shared goal is to
recover high-level objects from minimal local principles and universal symmetry
requirements, rather than from distinguished presentations. In the present setting, we ask
whether analytic structures that are usually introduced through arithmetic data can instead
be forced by locality, reversibility, symmetry, and scaling alone. We use the term
``primitive'' in this sense: the input is only finite local dynamics, while the outputs are
determined by the invariances that survive in the scaling limit. More specifically, the aim herein is to show that two classical analytic objects can arise canonically
from primitive input: finite, local, reversible Markov dynamics on discrete cycles, introduced
with no reference to primes, Euler products, or spectral/arithmetic data.

The first output is a \emph{total-positivity object}: a logarithmic kernel $\Phi$ whose
bilateral Laplace transform admits a canonical Schoenberg--Edrei--Karlin factorization~\cite{Schoenberg1951,Edrei1952,karlin1968total}.
Equivalently, the reciprocal transform is a Laguerre--P\'olya entire function $\Psi$ and
therefore has only real zeros. The second output is an \emph{Archimedean completion object}:
an explicit completion operator $\mathcal A$ that restores Mellin self-duality at a
canonical normalization, producing a completed kernel whose Mellin transform coincides
(within a natural domain) with the classical Riemann $\Xi$-function.

These constructions are logically independent but share the same primitive source, and they
motivate a natural identification question left open here (see
Section~\ref{sec:future-directions}): to what extent can the Laguerre--P\'olya datum $\Psi$
be identified with the $\Xi$-function after the normalizations are fixed?

Several classical approaches emphasize representations of $\Xi$ as an integral transform. One may express $\Xi$ through Mellin transforms of theta functions, through Fourier transforms of rapidly decaying kernels, or through trace formulas that relate zeros to spectral data\cite{Selberg1956,BerryKeating1999}. These representations highlight a recurring theme: analytic properties of $\Xi$ often reflect structural features of an underlying kernel. In particular, positivity and symmetry at the level of kernels tend to propagate to strong constraints on the associated transforms.

Heat kernels and theta series provide a natural language\cite{Chandrasekharan1985,SteinShakarchi2003} for these ideas. The fundamental solution of the heat equation on the real line is Gaussian, and its periodization produces classical theta functions. Poisson summation expresses a precise duality between spatial periodization and frequency localization. These mechanisms are robust and do not depend on arithmetic input. They arise whenever one studies diffusion on spaces with discrete symmetries. In analytic number theory, theta series appear as generating functions encoding spectral or lattice data, and they often serve as intermediaries between local and global descriptions.

In this work, heat kernels play a central conceptual role. Rather than appearing as analytic tools applied to $\zeta(s)$, they arise from elementary local dynamics. The guiding idea is that long-time diffusion captures universal behavior, while finite-volume effects introduce discrete structure. When these effects are combined through exact lift and periodization identities, theta series emerge canonically. This viewpoint places theta functions in a broader probabilistic and geometric setting, where they reflect general principles rather than special constructions.

A second strand of background comes from the theory of total positivity\cite{Pinkus2010,karlin1968total,GantmacherKrein2002}. Total positivity originated in work of Schoenberg and was developed extensively by Karlin and others. It concerns kernels whose minors of all orders are nonnegative, a property that implies strong variation-diminishing and rigidity phenomena. Within this theory, a distinguished role is played by P\'olya frequency functions of infinite order. In the form used here (Schoenberg--Edrei--Karlin), the bilateral Laplace transform of a $\mathrm{PF}_\infty$ function extends meromorphically and its reciprocal is a Laguerre--P\'olya entire function. Entire functions in the Laguerre--P\'olya class\cite{Levin1964,PolyaSchur1914} are limits of polynomials with only real zeros, and hence have only real zeros.

Closely related to this viewpoint is the Hermite--Biehler theory\cite{Krein1947,deBranges1968} of entire functions and its modern formulation within de Branges spaces\cite{deBranges1968}. Hermite--Biehler theory provides criteria ensuring that an entire function has all its zeros on a line, typically the real axis, based on positivity properties of associated kernels or on analytic inequalities in the upper half-plane. De Branges' work showed that these ideas can be organized into a Hilbert space framework with powerful classification theorems. Although the present paper does not rely on the full machinery of de Branges spaces\cite{deBranges1968}, the Hermite--Biehler perspective clarifies why positivity at the kernel level leads naturally to real-zero conclusions.

We now introduce the central object of the paper and state the main results.

\subsection{Primitive dynamical model}
\label{sec:model}

For each integer $N\ge 1$ we consider a continuous-time, reversible, nearest-neighbor Markov process on the discrete cycle $\mathbb{Z}/N\mathbb{Z}$. The dynamics are specified by a collection of strictly positive conductances $\{a_j\}_{j\in\mathbb{Z}/N\mathbb{Z}}$, where $a_j$ is associated with the undirected edge between $j$ and $j+1$ (indices taken modulo $N$). The generator $\mathcal{L}_N$ acts on functions $f:\mathbb{Z}/N\mathbb{Z}\to\mathbb{R}$ by
\begin{equation}
\label{eq:generator}
(\mathcal{L}_N f)(j)
=
a_j\,\bigl(f(j+1)-f(j)\bigr)
\;+
a_{j-1}\,\bigl(f(j-1)-f(j)\bigr).
\end{equation}

\begin{remark}[Translation-invariant specialization]
\label{rem:translation-invariant}
For the purposes of this paper we restrict to the translation-invariant case
\(
a_j\equiv a>0
\)
(independent of $j$), so that \eqref{eq:generator} is the usual continuous-time simple
random walk on $\mathbb{Z}/N\mathbb{Z}$ with jump rate $a$. In this case the diffusive scaling limit is
governed by the one-dimensional heat equation with diffusion constant \(D=a\), and the
Fourier analysis on $\mathbb{Z}/N\mathbb{Z}$ leading to the theta-series form is completely explicit.
Extensions to uniformly elliptic periodic environments, or to quantitative homogenization
regimes, can also yield a uniform local CLT and the same scaling limit; we do not pursue
those generalizations here.
\end{remark}

We write $p^{\mathrm{cyc}}_t(j,k)$ for the associated heat kernel,
\(p^{\mathrm{cyc}}_t(j,k)=(e^{t\mathcal{L}_N}\mathbf{1}_{\{k\}})(j)\), and we denote by $D>0$ the macroscopic diffusion constant appearing in the Gaussian scaling limit established later.

As described above, the trace of the heat kernel associated with the finite dynamics captures global information about the system but contains a universal singular contribution reflecting diffusive behavior. This singular term is independent of the fine structure of the dynamics and depends only on the macroscopic scaling. To isolate the genuinely structural content, we subtract this term in a canonical way. The resulting object is the completed trace kernel.

\begin{definition}[Scaling--limit completed trace kernel]
\label{def:scaling-limit}
Fix a macroscopic length $L>0$ and choose a scaling parameter $s\to\infty$ with $N=N(s)$ such that $N/s\to L$.
Define the scaling--limit trace
\[
K_L(t):=\lim_{s\to\infty} N(s)\,p^{\mathrm{cyc}}_{s^2 t}(0,0),\qquad t>0,
\]
and set the (scaling--limit) completed trace kernel
\[
\widetilde K_L(t):=K_L(t)-\frac{L}{\sqrt{4\pi D t}}.
\]
When $L$ is fixed we suppress the subscript and write $\widetilde K$.
\end{definition}

\begin{definition}[Archimedean (half--density) completion]
\label{def:half-density}
Define the \emph{Archimedean-normalized} (``half--density'') kernel by
\[
\widetilde K_{\mathrm{sym}}(t):=t^{-1/2}\,\widetilde K(t),\qquad t>0.
\]
\end{definition}

The existence of the scaling limit in Definition~\ref{def:scaling-limit}, together with the basic domination estimates needed to justify termwise limits, follows from the standing assumptions recorded below; for convenience we collect the relevant analytic justifications in Appendix~\ref{app:technical}.

\begin{remark}[Standing assumptions and existence of the scaling limit]
\label{rem:standing-assumptions}
We work throughout in a regime where the scaling-limit trace $K_L(t)$ exists for every $L>0$ and $t>0$, with convergence locally uniform in $t$ on compact subsets of $(0,\infty)$. In particular, under the translation-invariant specialization of Remark~\ref{rem:translation-invariant} this limit can be obtained explicitly by Fourier analysis on $\mathbb{Z}/N\mathbb{Z}$. Under these assumptions the limit admits the explicit theta-series form stated in Theorem~\ref{thm:theta-series-form-recall}. Analytic interchanges (limits, sums, differentiation, and Mellin/Laplace integrals) are justified by the domination estimates recorded in Appendix~\ref{app:technical}.
\end{remark}

\medskip

\noindent
In addition to the total-positivity output (producing a canonical Laguerre--P\'olya function) and
the Archimedean completion output (producing the classical theta/Mellin representation of $\Xi$),
we isolate a single remaining analytic identification problem: relate the Laguerre--P\'olya datum
$\Psi$ to the Mellin-side function $\Xi(2\cdot)$ up to multiplication by a zero-free entire factor.
We treat this as an open ``bridge'' problem and discuss it in Section~\ref{sec:future-directions}.

The first output is a \emph{total-positivity object}: a logarithmic kernel $\Phi$ whose
bilateral Laplace transform admits a canonical Schoenberg--Edrei--Karlin factorization~\cite{Schoenberg1951,Edrei1952,karlin1968total}.
Equivalently, the reciprocal transform is a Laguerre--P\'olya entire function $\Psi$ and
therefore has only real zeros. The second output is an \emph{Archimedean completion object}:
an explicit completion operator $\mathcal A$ that restores Mellin self-duality at a
canonical normalization, producing a completed kernel whose Mellin transform coincides
(within a natural domain) with the classical Riemann $\Xi$-function.

A second strand of background comes from the theory of total positivity\cite{Pinkus2010,karlin1968total,GantmacherKrein2002}. Total positivity originated in work of Schoenberg and was developed extensively by Karlin and others. It concerns kernels whose minors of all orders are nonnegative, a property that implies strong variation-diminishing and rigidity phenomena. Within this theory, a distinguished role is played by P\'olya frequency functions of infinite order. In the form used here (Schoenberg--Edrei--Karlin), the bilateral Laplace transform of a $\mathrm{PF}_\infty$ function extends meromorphically and its reciprocal is a Laguerre--P\'olya entire function. Entire functions in the Laguerre--P\'olya class\cite{Levin1964,PolyaSchur1914} are limits of polynomials with only real zeros, and hence have only real zeros.

\begin{definition}[Scaling--limit completed trace kernel]
\label{def:scaling-limit}
Fix a macroscopic length $L>0$ and choose a scaling parameter $s\to\infty$ with $N=N(s)$ such that $N/s\to L$.
Define the scaling--limit trace
\[
K_L(t):=\lim_{s\to\infty} N(s)\,p^{\mathrm{cyc}}_{s^2 t}(0,0),\qquad t>0,
\]
and set the (scaling--limit) completed trace kernel
\[
\widetilde K_L(t):=K_L(t)-\frac{L}{\sqrt{4\pi D t}}.
\]
When $L$ is fixed we suppress the subscript and write $\widetilde K$.
\end{definition}

\begin{remark}[Standing assumptions and existence of the scaling limit]
\label{rem:standing-assumptions}
We work throughout in a regime where the scaling-limit trace $K_L(t)$ exists for every $L>0$ and $t>0$, with convergence locally uniform in $t$ on compact subsets of $(0,\infty)$. In particular, under the translation-invariant specialization of Remark~\ref{rem:translation-invariant} this limit can be obtained explicitly by Fourier analysis on $\mathbb{Z}/N\mathbb{Z}$. Under these assumptions the limit admits the explicit theta-series form stated in Theorem~\ref{thm:theta-series-form-recall}. Analytic interchanges (limits, sums, differentiation, and Mellin/Laplace integrals) are justified by the domination estimates recorded in Appendix~\ref{app:technical}.
\end{remark}

\medskip

\noindent
In addition to the total-positivity output (producing a canonical Laguerre--P\'olya function) and
the Archimedean completion output (producing the classical theta/Mellin representation of $\Xi$),
we isolate a single remaining analytic identification problem: relate the Laguerre--P\'olya datum
$\Psi$ to the Mellin-side function $\Xi(2\cdot)$ up to multiplication by a zero-free entire factor.
We treat this as an open ``bridge'' problem and discuss it in Section~\ref{sec:future-directions}.

\begin{abstract}
Starting from finite, local, reversible Markov dynamics on discrete cycles, we construct a
scaling-limit renormalized trace kernel admitting an exact theta-series representation.
The construction is entirely Archimedean and uses no Euler products, primes, or
arithmetic spectral input.
From this limit we define a logarithmic kernel $\Phi$ and prove that it lies in the
P\'olya frequency class $\mathrm{PF}_\infty$, yielding via the Schoenberg--Edrei--Karlin
classification a canonical Laguerre--P\'olya function $\Psi$.
Independently, we introduce an Archimedean completion operator and show that, at a
self-dual normalization, the completed kernel coincides with the classical theta kernel,
whose Mellin transform is the Riemann $\Xi$-function.
We isolate a single remaining analytic problem relating $\Psi$ to $\Xi(2\cdot)$.
\end{abstract}

\noindent
In addition to the total-positivity output (producing a canonical Laguerre--P\'olya function) and
the Archimedean completion output (producing the classical theta/Mellin representation of $\Xi$),
we isolate a single remaining analytic identification problem: relate the Laguerre--P\'olya datum
$\Psi$ to the Mellin-side function $\Xi(2\cdot)$ up to multiplication by a zero-free entire factor.
We treat this as an open ``bridge'' problem and discuss it in Section~\ref{sec:future-directions}.

\begin{theorem}[Archimedean Mellin identification]
\label{thm:arch-overview}
At the self-dual scale fixed in Section~\ref{sec:completion-selfdual}, the Archimedean-completed
kernel $\widetilde K_{\mathrm{arch}}:=\mathcal A(K_L-1)$ has Mellin transform
\[
F_{\mathrm{arch}}(z)=\Xi(2z)
\]
as in Theorem~\ref{thm:arch-mellin-identification}.
\end{theorem}

\begin{tabular}{|p{0.22\linewidth}|p{0.70\linewidth}|}
\hline
\textbf{Symbol} & \textbf{Meaning} \\
\hline
$K_L(t)$ &
Scaling-limit trace
(Definition~\ref{def:scaling-limit}).
\\ \hline
$\widetilde K(t)$ &
Renormalized trace kernel
(Theorem~\ref{thm:theta-series-form-recall}).
\\ \hline
$\widetilde K_{\mathrm{sym}}(t)=t^{-1/2}\widetilde K(t)$ &
Symmetric half-density kernel
(Definition~\ref{def:half-density}).
\\ \hline
$\Phi(x)$ &
Logarithmic kernel
$\Phi(x)=e^{x/4}\,\widetilde K_{\mathrm{sym}}(e^x)$
(equation~\eqref{eq:Phi-def}).
\\ \hline
$\mathcal B\Phi(s)$ &
Bilateral Laplace transform
$\displaystyle \mathcal B\Phi(s)=\int_{\R}\Phi(x)e^{-s x}\,dx$
(on $\Re(s)>-\tfrac14$).
\\ \hline
$\Psi(s)$ &
Laguerre--P\'olya entire function with only real zeros satisfying
$\mathcal B\Phi(s)=\frac{E(s)}{\Psi(s)}$ on a strip
(Theorem~\ref{thm:SEK}).
\\ \hline
$F(z)$ &
Fourier transform of $\Phi$,
$\displaystyle F(z)=\int_{\R}\Phi(x)e^{izx}\,dx$;
boundary value $F(z)=\mathcal B\Phi(-iz)$ where both sides converge.
\\ \hline
$\mathcal A$ &
Archimedean completion operator
(Definition~\ref{def:arch-operator}).
\\ \hline
$\widetilde K_{\mathrm{arch}}$ &
Completed Archimedean kernel
$\widetilde K_{\mathrm{arch}}:=\mathcal A(K_L-1)$
(Definition~\ref{def:arch-theta-kernel}).
\\ \hline
$F_{\mathrm{arch}}(z)$ &
Mellin transform
$\displaystyle F_{\mathrm{arch}}(z)=\int_0^{\infty}
\widetilde K_{\mathrm{arch}}(t)\,
t^{\frac14+iz}\frac{dt}{t}$;
at the self-dual scale,
$F_{\mathrm{arch}}(z)=\Xi(2z)$
(Theorem~\ref{thm:arch-mellin-identification}).
\\ \hline
$\Psi_c(z)$ &
Centered Laguerre--P\'olya function
$\Psi_c(z):=\Psi(\tfrac14+iz)$. A central open problem is to compare $\Psi_c$ with the Mellin-side
function $\Xi(2z)$ up to multiplication by a zero-free entire factor (Section~\ref{sec:future-directions}).
\\ \hline
\end{tabular}

\begin{proposition}[Reflection law for the Step-2 transform]
\label{prop:bilaplace-reflection}
Assume \eqref{eq:Ksym-selfdual}. Then for every $s\in\mathbb C$ with
\[
-\tfrac14<\Re(s)<\tfrac34
\]
(one precisely needs absolute convergence of both sides), one has the exact identity
\begin{equation}
\label{eq:bilaplace-reflection}
\mathcal B\Phi(s)=\mathcal B\Phi\!\left(\tfrac12-s\right).
\end{equation}
Consequently, for all $t\in\mathbb R$ one has
\begin{equation}
\label{eq:bilaplace-real-on-centerline}
\mathcal B\Phi\!\left(\tfrac14+it\right)\in\mathbb R.
\end{equation}
\end{proposition}

\begin{theorem}[Schoenberg--Edrei--Karlin]
\label{thm:SEK}
Let $\Phi\in L^1(\mathbb R)\cap \mathrm{PF}_\infty$ be nonnegative.
Then its bilateral Laplace transform admits the representation
\begin{equation}
\label{eq:SEK-form}
\mathcal B\Phi(s)
=
\frac{E(s)}{\Psi(s)},
\end{equation}
where:
\begin{enumerate}[label=(\roman*)]
\item $E(s)$ is an entire function with no zeros,
\item $\Psi(s)$ is an entire function of Laguerre--Pólya type,
\item all zeros of $\Psi$ are real,
\item the representation is unique up to multiplication of $E$ and $\Psi$ by the
same nonzero constant.
\end{enumerate}
\end{theorem}

\begin{theorem}[Main theorem (assembled)]
\label{thm:main-assembled}
Assume the strong regime hypotheses of Section~\ref{sec:completion-selfdual}, so that the scaling-limit kernel
$K_L$ exists and the self-dual normalization can be fixed. Then:
\begin{enumerate}[label=(\roman*)]
\item (\emph{Unconditional structural output.})
There exists a nonnegative kernel $\Phi\in L^1(\mathbb R)$ satisfying:
\begin{itemize}
\item $\Phi\in\mathrm{PF}_\infty$;
\item its bilateral Laplace transform $\mathcal B\Phi$ satisfies the reflection law
$\mathcal B\Phi(s)=\mathcal B\Phi(\tfrac12-s)$ on the common domain of absolute convergence;
\item $\mathcal B\Phi$ admits a canonical Schoenberg--Edrei--Karlin representation
$\mathcal B\Phi(s)=E(s)/\Psi(s)$ with $\Psi$ Laguerre--P\'olya and hence having only real zeros.
\end{itemize}
\item (\emph{Unconditional Archimedean Mellin identification.})
The Archimedean-completed kernel $\widetilde K_{\mathrm{arch}}$ has Mellin transform
$F_{\mathrm{arch}}(z)=\Xi(2z)$ as in Theorem~\ref{thm:arch-mellin-identification}.
\end{enumerate}
\end{theorem}

\begin{proof}
We compute the bilateral Laplace transform on the half-plane $\Re(s)>0$:
\[
\mathcal B\phi_u(s)
=
\int_{\mathbb R} e^{-u e^{-x}}e^{-s x}\,dx.
\]
With the change of variables $y=e^{-x}$ (so $y\in(0,\infty)$ and $dx=-dy/y$), this becomes
\[
\mathcal B\phi_u(s)
=
\int_0^\infty e^{-u y} y^{s-1}\,dy
=
u^{-s}\Gamma(s),
\qquad \Re(s)>0.
\]
Now recall the classical Weierstrass product for the reciprocal Gamma function:
\begin{equation}
\label{eq:invGamma-product}
\frac{1}{\Gamma(s)}
=
s\,e^{\gamma s}\prod_{n=1}^\infty\Bigl(1+\frac{s}{n}\Bigr)e^{-s/n},
\end{equation}
where $\gamma$ is Euler's constant. The right-hand side is an entire function whose
zeros are exactly the nonpositive integers, all real and simple, and whose product
representation is of Laguerre--P\'olya type. In particular,
\[
\Psi(s):=\frac{1}{\Gamma(s)}
\quad\text{is Laguerre--P\'olya,}
\qquad
E(s):=u^{-s}=e^{-s\log u}
\quad\text{is entire and zero-free.}
\]
Thus on $\Re(s)>0$ we have the representation
\[
\mathcal B\phi_u(s)=\frac{E(s)}{\Psi(s)}.
\]

\begin{theorem}[Archimedean Mellin identification]
\label{thm:arch-overview}
At the self-dual scale fixed in Section~\ref{sec:completion-selfdual}, the Archimedean-completed
kernel $\widetilde K_{\mathrm{arch}}:=\mathcal A(K_L-1)$ has Mellin transform
\[
F_{\mathrm{arch}}(z)=\Xi(2z)
\]
as in Theorem~\ref{thm:arch-mellin-identification}.
\end{theorem}

The remainder of the paper constructs the kernel $\Phi$, proves Theorem~\ref{thm:main-assembled}, and isolates each logical seam explicitly.

\medskip

The paper is organized as follows. Section~\ref{sec:completion-selfdual} fixes the self-dual Archimedean normalization and records the resulting theta-series form of the scaling-limit trace. Section~\ref{sec:log-reflection} constructs the logarithmic kernel $\Phi$ and establishes the exact reflection symmetry for its bilateral Laplace transform. Section~\ref{sec:total-positivity} proves total positivity and derives the Schoenberg--Edrei--Karlin factorization $\mathcal B\Phi=E/\Psi$. Section~\ref{sec:arch-mellin-xi} identifies the Archimedean-completed Mellin transform with $\Xi(2\cdot)$ and records the Archimedean Mellin identification with $\Xi(2\cdot)$, while isolating the remaining identification problem (Section~\ref{sec:future-directions}). Section~\ref{sec:completion-ledger} assembles the argument and isolates the remaining seam. Section~\ref{sec:tp-details} provides a detailed proof of total positivity. Appendix~\ref{app:technical} collects analytic justifications for termwise operations and boundary terms.

\subsection{Map of the proof}\label{subsec:map-proof}
The argument is organized into four steps, each producing a new object from the previous one.
For quick reference we summarize the pipeline in Table~\ref{tab:proof-map}.

\begin{table}[h!]
\centering
\setlength{\arrayrulewidth}{0.8pt}
\renewcommand{\arraystretch}{1.25}

\begin{tabular}{|p{0.10\linewidth}|p{0.52\linewidth}|p{0.30\linewidth}|}
\hline
\textbf{Step} & \textbf{Input $\to$ Output} & \textbf{Main tool} \\
\hline
1 &
local dynamics $\to$ theta-series kernel $\widetilde K$ &
lift/periodize + ULCLT$^*$
+ dominated convergence
\\ \hline
2 &
$\widetilde K \to$ self-dual normalization and Archimedean completion $\mathcal A$ &
Jacobi inversion + completion operator
\\ \hline
3 &
$\widetilde K \to \Phi \to (\Psi,\ \mathcal B\Phi=E/\Psi)$ &
complete monotonicity $\Rightarrow \mathrm{PF}_\infty$ +
Schoenberg--Edrei--Karlin
\\ \hline
4 &
$\mathcal A(K_L-1)\to \Theta \to \Xi$ (anchor) and open bridge problem &
Archimedean completion + Mellin;
Comparison problem (open)
\\ \hline
\end{tabular}

\caption{Proof map (Steps~1--4). $^*$ULCLT = uniform local central limit theorem.}
\label{tab:proof-map}
\end{table}

The proof relies on a small number of canonical kernels and transforms; these are summarized in Table~\ref{tab:notation} with precise definitions and references.

\begin{table}[h!]
\centering
\setlength{\arrayrulewidth}{0.8pt}
\renewcommand{\arraystretch}{1.25}

\begin{tabular}{|p{0.22\linewidth}|p{0.70\linewidth}|}
\hline
\textbf{Symbol} & \textbf{Meaning} \\
\hline
$K_L(t)$ &
Scaling-limit trace
(Definition~\ref{def:scaling-limit}).
\\ \hline
$\widetilde K(t)$ &
Renormalized trace kernel
(Theorem~\ref{thm:theta-series-form-recall}).
\\ \hline
$\widetilde K_{\mathrm{sym}}(t)=t^{-1/2}\widetilde K(t)$ &
Symmetric half-density kernel
(Definition~\ref{def:half-density}).
\\ \hline
$\Phi(x)$ &
Logarithmic kernel
$\Phi(x)=e^{x/4}\,\widetilde K_{\mathrm{sym}}(e^x)$
(equation~\eqref{eq:Phi-def}).
\\ \hline
$\mathcal B\Phi(s)$ &
Bilateral Laplace transform
$\displaystyle \mathcal B\Phi(s)=\int_{\mathbb{R}}\Phi(x)e^{-s x}\,dx$
(on $\Re(s)>-\tfrac14$).
\\ \hline
$\Psi(s)$ &
Laguerre--P\'olya entire function with only real zeros satisfying
$\mathcal B\Phi(s)=\frac{E(s)}{\Psi(s)}$ on a strip
(Theorem~\ref{thm:SEK}).
\\ \hline
$F(z)$ &
Fourier transform of $\Phi$,
$\displaystyle F(z)=\int_{\mathbb{R}}\Phi(x)e^{izx}\,dx$;
boundary value $F(z)=\mathcal B\Phi(-iz)$ where both sides converge.
\\ \hline
$\mathcal A$ &
Archimedean completion operator
(Definition~\ref{def:arch-operator}).
\\ \hline
$\widetilde K_{\mathrm{arch}}$ &
Completed Archimedean kernel
$\widetilde K_{\mathrm{arch}}:=\mathcal A(K_L-1)$
(Definition~\ref{def:arch-theta-kernel}).
\\ \hline
$F_{\mathrm{arch}}(z)$ &
Mellin transform
$\displaystyle F_{\mathrm{arch}}(z)=\int_0^{\infty}
\widetilde K_{\mathrm{arch}}(t)\,
t^{\frac14+iz}\frac{dt}{t}$;
at the self-dual scale,
$F_{\mathrm{arch}}(z)=\Xi(2z)$
(Theorem~\ref{thm:arch-mellin-identification}).
\\ \hline
$\Psi_c(z)$ &
Centered Laguerre--P\'olya function
$\Psi_c(z):=\Psi(\tfrac14+iz)$. A central open problem is to compare $\Psi_c$ with the Mellin-side
function $\Xi(2z)$ up to multiplication by a zero-free entire factor (Section~\ref{sec:future-directions}).
\\ \hline
\end{tabular}

\caption{Notation used throughout the proof.}
\label{tab:notation}
\end{table}

The introduction gives a proof map rather than a detailed argument. It says the paper will “constructs the kernel $\Phi$” and establishes the three outputs of Theorem~\ref{thm:structural} by isolating “each logical seam explicitly,” with the following pipeline:

1. **Local dynamics $\to$ theta-series kernel $\widetilde K$** using “lift/periodize + ULCLT$^*$ + dominated convergence.”

2. **$\widetilde K \to$ self-dual normalization and Archimedean completion $\mathcal A$** via “Jacobi inversion + completion operator.”

3. **$\widetilde K \to \Phi \to (\Psi,\ \mathcal B\Phi=E/\Psi)$**: construct the “logarithmic kernel”
\[\Phi(x)=e^{x/4}\,\widetilde K_{\mathrm{sym}}(e^x),\]
then prove “complete monotonicity $\Rightarrow \mathrm{PF}_\infty$ + Schoenberg--Edrei--Karlin,” yielding total positivity and the “Schoenberg--Edrei--Karlin factorization $\mathcal B\Phi=E/\Psi$,” with $\Psi$ in Laguerre--P\'olya.

Additionally, it states that Section~\ref{sec:log-reflection} “establishes the exact reflection symmetry for its bilateral Laplace transform,” i.e. the reflection law for $\mathcal B\Phi$.

4. **Archimedean Mellin anchor**: “$\mathcal A(K_L-1)\to \Theta \to \Xi$ (anchor) and open bridge problem” via “Archimedean completion + Mellin; Comparison problem (open).”

The introduction gives a proof map rather than a detailed argument. It says the paper will “constructs the kernel $\Phi$” and establishes the three outputs of the main theorem by isolating “each logical seam explicitly,” with the following pipeline:

1. **Local dynamics $\to$ theta-series kernel $\widetilde K$** using “lift/periodize + ULCLT$^*$ + dominated convergence.”

2. **$\widetilde K \to$ self-dual normalization and Archimedean completion $\mathcal A$** via “Jacobi inversion + completion operator.”

3. **$\widetilde K \to \Phi \to (\Psi,\ \mathcal B\Phi=E/\Psi)$**: construct the “logarithmic kernel”
\[\Phi(x)=e^{x/4}\,\widetilde K_{\mathrm{sym}}(e^x),\]
then prove “complete monotonicity $\Rightarrow \mathrm{PF}_\infty$ + Schoenberg--Edrei--Karlin,” yielding total positivity and the “Schoenberg--Edrei--Karlin factorization $\mathcal B\Phi=E/\Psi$,” with $\Psi$ in Laguerre--P\'olya.

Additionally, it states that later the paper “establishes the exact reflection symmetry for its bilateral Laplace transform,” i.e. the reflection law for $\mathcal B\Phi$.

4. **Archimedean Mellin anchor**: “$\mathcal A(K_L-1)\to \Theta \to \Xi$ (anchor) and open bridge problem” via “Archimedean completion + Mellin; Comparison problem (open).”

[Structural output from primitive cycles]
\label{thm:structural}
\begin{remark}[Standing assumptions and existence of the scaling limit]
\label{rem:standing-assumptions}
We work throughout in a regime where the scaling-limit trace $K_L(t)$ exists for every $L>0$ and $t>0$, with convergence locally uniform in $t$ on compact subsets of $(0,\infty)$. In particular, under the translation-invariant specialization of Remark~\ref{rem:translation-invariant} this limit can be obtained explicitly by Fourier analysis on $\Z/N\Z$. Under these assumptions the limit admits the explicit theta-series form stated in Theorem~\ref{thm:theta-series-form-recall}. Analytic interchanges (limits, sums, differentiation, and Mellin/Laplace integrals) are justified by the domination estimates recorded in Appendix~\ref{app:technical}.
\end{remark}
Assume the standing assumptions above, so that the scaling-limit kernel $K_L$ exists.
Then there exists a nonnegative kernel $\Phi\in L^1(\mathbb{R})$ such that:
\begin{enumerate}[label=(\roman*)]
\item $\Phi\in\mathrm{PF}_\infty$;
\item its bilateral Laplace transform $\mathcal B\Phi$ satisfies the reflection law
$\mathcal B\Phi(s)=\mathcal B\Phi(\tfrac12-s)$ on the common domain of absolute convergence;
\item $\mathcal B\Phi$ admits a canonical Schoenberg--Edrei--Karlin representation
\[
\mathcal B\Phi(s)=\frac{E(s)}{\Psi(s)},
\]
where $\Psi$ is Laguerre--P\'olya and hence has only real zeros.
\end{enumerate}

Question:
Fix a macroscopic length $L>0$ and define the scaling-limit trace by
\[
K_L(t):=\lim_{s\to\infty} N(s)\,p^{\mathrm{cyc}}_{s^2 t}(0,0),\qquad t>0,
\]
where $N=N(s)$ satisfies $N/s\to L$. Assume that for every $L>0$ and $t>0$ this limit exists, with convergence locally uniform in $t$ on compact subsets of $(0,\infty)$, so that the scaling-limit kernel $K_L$ exists. Let $\mathcal B\Phi(s):=\int_{\mathbb R} e^{-sx}\Phi(x)\,dx$ denote the bilateral Laplace transform on its domain of absolute convergence, and let $\mathrm{PF}_\infty$ denote the Pólya frequency class of infinite order. Which statement holds under these assumptions?

There exists a nonnegative kernel $\Phi\in L^1(\mathbb R)$ such that $\Phi\in\mathrm{PF}_\infty$, its bilateral Laplace transform satisfies the reflection law $\mathcal B\Phi(s)=\mathcal B\Phi(\tfrac12-s)$ on the common domain of absolute convergence, and $\mathcal B\Phi$ admits a canonical Schoenberg--Edrei--Karlin representation
\[
\mathcal B\Phi(s)=\frac{E(s)}{\Psi(s)},
\]
where $\Psi$ is a Laguerre--Pólya entire function and hence has only real zeros.

Choices:
B.
There exists a nonnegative kernel $\Phi\in L^1(\mathbb R)$ such that $\Phi\in\mathrm{PF}_\infty$, its bilateral Laplace transform satisfies the reflection law $\mathcal B\Phi(s)=\mathcal B\Phi(-s)$ on the common domain of absolute convergence, and $\mathcal B\Phi$ admits a canonical Schoenberg--Edrei--Karlin representation
\[
\mathcal B\Phi(s)=\frac{E(s)}{\Psi(s)},
\]
where $\Psi$ is a Laguerre--Pólya entire function and hence has only real zeros.
C.
There exists a nonnegative kernel $\Phi\in L^1(\mathbb R)$ such that $\Phi\in\mathrm{PF}_\infty$ and $\mathcal B\Phi$ admits a canonical Schoenberg--Edrei--Karlin representation
\[
\mathcal B\Phi(s)=\frac{E(s)}{\Psi(s)},
\]
where $\Psi$ is a Laguerre--Pólya entire function and hence has only real zeros.
D.
There exists a nonnegative kernel $\Phi\in L^1(\mathbb R)$ such that $\Phi\in\mathrm{PF}_\infty$, its bilateral Laplace transform satisfies the reflection law $\mathcal B\Phi(s)=\mathcal B\Phi(\tfrac12-s)$ for all $s\in\mathbb C$, and $\mathcal B\Phi$ admits a canonical Schoenberg--Edrei--Karlin representation
\[
\mathcal B\Phi(s)=\frac{E(s)}{\Psi(s)},
\]
where $\Psi$ is a Laguerre--Pólya entire function and hence has only real zeros.
E.
There exists a nonnegative kernel $\Phi\in L^1(\mathbb R)$ such that its bilateral Laplace transform satisfies the reflection law $\mathcal B\Phi(s)=\mathcal B\Phi(\tfrac12-s)$ on the common domain of absolute convergence, and $\mathcal B\Phi$ admits a canonical Schoenberg--Edrei--Karlin representation
\[
\mathcal B\Phi(s)=\frac{E(s)}{\Psi(s)},
\]
where $\Psi$ is a Laguerre--Pólya entire function and hence has only real zeros, with $\Phi$ therefore necessarily in \mathrm{PF}_\infty.

direction of SEK implication versus PF_infty hypothesis


Question:
Suppose one works in a regime where, for every macroscopic length $L>0$ and every $t>0$, the scaling-limit trace
\[
K_L(t):=\lim_{s\to\infty} N(s)\,p^{\mathrm{cyc}}_{s^2 t}(0,0)
\]
exists, with convergence locally uniform in $t$ on compact subsets of $(0,\infty)$. Here $\mathrm{PF}_\infty$ denotes the class of P\'olya frequency kernels of infinite order, $\mathcal B\Phi(s)=\int_{\mathbb R} e^{-sx}\Phi(x)\,dx$ is the bilateral Laplace transform, and a Schoenberg--Edrei--Karlin factorization means a representation $\mathcal B\Phi(s)=E(s)/\Psi(s)$ with $E$ entire and zero-free and $\Psi$ an entire Laguerre--P\'olya function (so all zeros of $\Psi$ are real). Which existence statement holds under these assumptions?

There exists a nonnegative kernel $\Phi\in L^1(\mathbb{R})$ such that $\Phi\in\mathrm{PF}_\infty$, its bilateral Laplace transform satisfies the reflection law $\mathcal B\Phi(s)=\mathcal B\Phi(\tfrac12-s)$ on the common domain of absolute convergence, and $\mathcal B\Phi$ admits a Schoenberg--Edrei--Karlin factorization
\[
\mathcal B\Phi(s)=\frac{E(s)}{\Psi(s)},
\]
where $\Psi$ is Laguerre--P\'olya and hence has only real zeros.

Choices:
B.
There exists a nonnegative kernel $\Phi\in L^1(\mathbb{R})$ such that $\Phi\in\mathrm{PF}_\infty$, its bilateral Laplace transform extends to an entire function satisfying the reflection law $\mathcal B\Phi(s)=\mathcal B\Phi(\tfrac12-s)$ for all $s\in\mathbb C$, and $\mathcal B\Phi$ admits a Schoenberg--Edrei--Karlin factorization
\[
\mathcal B\Phi(s)=\frac{E(s)}{\Psi(s)},
\]
where $\Psi$ is Laguerre--P\'olya and hence has only real zeros.
C.
There exists a nonnegative kernel $\Phi\in L^1(\mathbb{R})$ such that $\Phi\in\mathrm{PF}_\infty$ and $\mathcal B\Phi$ admits a Schoenberg--Edrei--Karlin factorization
\[
\mathcal B\Phi(s)=\frac{E(s)}{\Psi(s)},
\]
where $\Psi$ is Laguerre--P\'olya and hence has only real zeros.
D.
For every nonnegative kernel $\Phi\in L^1(\mathbb{R})$ with $\Phi\in\mathrm{PF}_\infty$, its bilateral Laplace transform satisfies the reflection law $\mathcal B\Phi(s)=\mathcal B\Phi(\tfrac12-s)$ on the common domain of absolute convergence, and $\mathcal B\Phi$ admits a Schoenberg--Edrei--Karlin factorization
\[
\mathcal B\Phi(s)=\frac{E(s)}{\Psi(s)},
\]
where $\Psi$ is Laguerre--P\'olya and hence has only real zeros.
E.
There exists a nonnegative kernel $\Phi\in L^1(\mathbb{R})$ such that its bilateral Laplace transform satisfies the reflection law $\mathcal B\Phi(s)=\mathcal B\Phi(\tfrac12-s)$ on the common domain of absolute convergence, and $\mathcal B\Phi$ admits a Schoenberg--Edrei--Karlin factorization
\[
\mathcal B\Phi(s)=\frac{E(s)}{\Psi(s)},
\]
where $\Psi$ is Laguerre--P\'olya and hence has only real zeros.


\label{newlemma}
Let $D$ be a square-free integer such that $\operatorname{rank}_{\mathbb Z}E_0^{D}(\mathbb Q)=0$. Then $E_0(\mathbb Q(\sqrt{D}))=E_0(\mathbb Q)$, except when $D=\pm 2$.

\begin{conj}\label{Conj}
There are infinitely many non-constant arithmetic progressions of five squares pro\-perly defined over a quadratic extension $K_1$ (resp. $K_2$) of $\Q(\sqrt{-2})$ (resp. of $\Q(\sqrt{2}))$.
\begin{itemize}
\item[(i)] In the case $\Q(\sqrt{-2})$, any of them are, up to equivalence, of the form $\left(a^2,b^2,-2c^2,-m d^2,-2e^2\right)$, where $a,b,c,d,e,m\in\Z$ and $m>0$ is square-free. In particular, $K_1=\Q(\sqrt{-2},\sqrt{-m})$.
\item[(ii)] In the case $\Q(\sqrt{2})$, any of them are, up to equivalence, of the form $\left(2 a^2,b^2,2c^2,m,e^2\right)$, where $a,b,c,e,m\in\Z$ and $m$ is square-free. In particular, $K_2=\Q(\sqrt{2},\sqrt{m})$.
\end{itemize}
\end{conj}

Fermat stated that there are no non-constant arithmetic progressions of four squares over $\mathbb{Q}$ in 1640 and Euler proved this in 1780. It is natural to ask for what would happen in number fields. Over quadratic fields, Xarles \cite{art17} showed that there are no non-constant arithmetic progressions of six squares and Gonz\'alez-Jim\'enez and Xarles \cite{GJX} characterized non-constant arithmetic progressions of five squares. Over cubic fields, Bremner and Siksek \cite{art5} showed that there are no non-constant arithmetic progressions of five squares.

In this paper, we extend the results of  Xarles \cite{art17} and of Gonz\'alez-Jim\'enez and Xarles \cite{GJX}  to quadratic extensions of $\mathbb Q(\sqrt{D})$, where $D$ is a square-free integer satisfying certain conditions on the $D$-quadratic twist of the elliptic curves
$$
E_0\,:\, y^2 = x^3 - x^2 - 9x + 9 \qquad \mbox{and}\qquad E_1\,:\, y^2 = x^3 - x^2 + x.
$$
For any elliptic curve $E$ and integer $D$, we denote by $E^D$ the $D$-quadratic twist of $E$.

{
Let $n$ be a positive integer and $K$ be a field. Let $a_1,\dots,a_n\in K$ such that $(a_1^2,a_2^2,\dots,a_n^2)$ is an arithmetic progression of length $n$. We say that this arithmetic progression is equivalent to $(s^2 a_1^2,s^2 a_2^2,\dots,s^2a_n^2)$ for any $s\in K^*$ and to the arithmetic progression $(a_n^2,\dots,a_2^2,a_1^2)$. We say that $(a_1^2,a_2^2,\dots,a_n^2)$ is properly defined over a number field $K$ if $a_1,\dots,a_n\in K$  and  $\{a_1,\dots,a_n\}\not\subset F$ for any proper subfield $F$ of $K$.} Let $i=\sqrt{-1}$.

\

The main theorems of the paper are the following.

\begin{thm}\label{main}
Let $D$ be a square-free integer, $D\ne -1,\pm 2,3$, such that $\operatorname{rank}_{\mathbb Z}E_1^{\pm D}(\mathbb Q)= 0$ and $K$ a quadratic extension of $\Q(\sqrt{D})$. Then
\begin{itemize}
\item[(A)] If $\operatorname{rank}_{\mathbb Z}E_0^{D}(\mathbb Q)=0$, then there does not exist any non-constant arithmetic progression of five squares properly defined over $K$.

\item[(B)] If $\operatorname{rank}_{\mathbb Z}E_0^{D}(\mathbb Q)\ne 0$ and the class number of $\Q(\sqrt{D})$ is $1$, then a non-constant arithmetic progression of five squares properly defined over $K$ is, up to equivalence, of the form $\left(a^2,b^2,c^2,\alpha \, d^2,e^2\right)$, where $a,b,c,d,e,\alpha \in\mathbb Q(\sqrt{D})$ and $\alpha$ is non-square.
\end{itemize}
\end{thm}

\begin{thm}\label{maincor}
There exists a non-constant arithmetic progression of five squares properly defined over a quadratic extension $K$ of
\begin{itemize}
    \item $\mathbb Q(i)$ if and only if $K=\mathbb Q(i,\sqrt{2})$. In this case, $(-2,-1, 0,1,2)$ is the unique, up to equivalence, non-constant arithmetic progression of five squares properly defined over $K$.
     \item $\mathbb Q(\sqrt{3})$ if and only if $K=\mathbb Q(\sqrt{3},\sqrt{2})$. In this case, $(0,1,2,3,4)$ is the unique, up to equivalence, non-constant arithmetic progression of five squares properly defined over $K$.
     \item $\mathbb Q(\sqrt{2})$ if and only if $K=\mathbb Q(\sqrt{2},\sqrt{3})$, $K=\mathbb Q(\sqrt{2},i)$ or $K=\mathbb Q(\sqrt{2},\sqrt{\alpha})$, for some squarefree $\alpha \in\mathbb Q(\sqrt{2})$. In the first case, $(0,1,2,3,4)$ is the unique, up to equivalence, non-constant arithmetic progression of five squares properly defined over $\mathbb Q(\sqrt{2},\sqrt{3})$; and in the second case, $(-2,-1, 0,1,2)$ is the unique, up to equivalence, non-constant arithmetic progression of five squares properly defined over $\mathbb Q(\sqrt{2},i)$. In the last case, a non-constant arithmetic progression of five squares properly defined over $K$ is, up to equivalence, of the form $\left(a^2,b^2,c^2,\alpha \, d^2,e^2\right)$ where $a,b,c,d,e \in\mathbb Q(\sqrt{2})$.
     \item $\mathbb Q(\sqrt{-2})$ if and only if $K=\mathbb Q(\sqrt{-2},i)$ or $K=\mathbb Q(\sqrt{-2},\sqrt{\alpha})$, for some squarefree $\alpha \in\mathbb Q(\sqrt{-2})$. In the first case, $(-2,-1, 0,1,2)$ is the unique, up to equivalence, non-constant arithmetic progression of five squares properly defined over $\mathbb Q(\sqrt{-2},i)$. In the second case, a non-constant arithmetic progression of five squares properly defined over $K$ is, up to equivalence, of the form $\left(a^2,b^2,c^2,\alpha \, d^2,e^2\right)$ where $a,b,c,d,e \in\mathbb Q(\sqrt{-2})$.
\end{itemize}
\end{thm}
\begin{remark}
Note that $\operatorname{rank}_{\mathbb Z}E_1^{\pm D}(\mathbb Q)= 0$ and $\operatorname{rank}_{\mathbb Z}E_0^{D}(\mathbb Q)= 0$ if $D\in\{-1,\pm 2,3\}$.
\end{remark}

\begin{thm}\label{six}
Let $D$ be a square-free integer such that
$$
\operatorname{rank}_{\mathbb Z}E_1^{\pm D}(\mathbb Q)= 0\,,
$$
and $\operatorname{rank}_{\mathbb Z}E_0^{D}(\mathbb Q)=0$ or if $\operatorname{rank}_{\mathbb Z}E_0^{D}(\mathbb Q)\ne 0$  the class number of $\Q(\sqrt{D})$ is $1$. Then there does not exist any non-constant arithmetic progression of six squares properly defined over any quadratic extension of $\Q(\sqrt{D})$.
\end{thm}

\begin{remark}
The following list shows all the square-free integers $D$, $|D|<200$, satisfying $\operatorname{rank}_{\mathbb Z}E_1^{\pm D}(\mathbb Q)= 0$ and  $\operatorname{rank}_{\mathbb Z}E_0^{D}(\mathbb Q)= 0$\,:
$$
-158, -123, -62, -51, -3, -2, -1, 2, 3, 7, 31, 51, 62, 79, 103, 110, 123, 127,
151, 158, 194, 195, 199.
$$
If we allow $\operatorname{rank}_{\mathbb Z}E_0^{D}(\mathbb Q)\ne 0$ and the class number of $\Q(\sqrt{D})$ is $1$ then $D\in\{-7, 38, 86\}$.
\end{remark}

\begin{remark}
There exist square-free integers $D$ and non-constant arithmetic progressions of six squares properly defined over a quadratic extension of $\Q(\sqrt{D})$. For example, when $D=409$ then $$(7^2, 13^2, 17^2, (\sqrt{409})^2, 23^2, (\sqrt{649})^2)$$ is an arithmetic progression properly defined over $\mathbb Q(\sqrt{409},\sqrt{649})$.
\end{remark}

\begin{conj}\label{Conj}
There are infinitely many non-constant arithmetic progressions of five squares pro\-perly defined over a quadratic extension $K_1$ (resp. $K_2$) of $\mathbb Q(\sqrt{-2})$ (resp. of $\mathbb Q(\sqrt{2}))$.
\begin{itemize}
\item[(i)] In the case $\mathbb Q(\sqrt{-2})$, any of them are, up to equivalence, of the form $\left(a^2,b^2,-2c^2,-m d^2,-2e^2\right)$, where $a,b,c,d,e,m\in\mathbb Z$ and $m>0$ is square-free. In particular, $K_1=\mathbb Q(\sqrt{-2},\sqrt{-m})$.
\item[(ii)] In the case $\mathbb Q(\sqrt{2})$, any of them are, up to equivalence, of the form $\left(2 a^2,b^2,2c^2,m,e^2\right)$, where $a,b,c,e,m\in\mathbb Z$ and $m$ is square-free. In particular, $K_2=\mathbb Q(\sqrt{2},\sqrt{m})$.
\end{itemize}
\end{conj}

Conjecture \ref{Conj} is supported by the calculation in Section \ref{sect_para}. For the cases $D\in\{-7, 38, 86\}$, there is not a similar conjecture.

Our approach is different from the approach in \cite{GJX} and \cite{art17}. We follow the approach in Mordell's paper \cite{art12}, where he gave alternative proofs to the results of Aigner \cite{art1} and Faddeev \cite{art7} on the solution of the equation $x^4+y^4=1$ in quadratic and cubic number fields. Mordell's approach has the advantage that it is very concrete in calculation.

Another important ingredient  is the theory concerning the growth of the torsion subgroup of elliptic curves under base change. Upon assuming $\operatorname{rank}_{\mathbb Z}E_1^{\pm D}(\mathbb Q)= 0$, it becomes imperative to determine the torsion subgroup of certain elliptic curves defined over $\mathbb Q$ over any quadratic field. The answer to this inquiry lies within the LMFDB online database  \cite{lmfdb}. The theory of torsion growth has undergone extensive scrutiny in recent years. Particularly noteworthy references are \cite{GJN1,GJN2} and \cite{GJT1,GJT2,N} (for quadratic fields).

In the following subsection we include all the elliptic curves that are relevant to this article, together with the necessary information concerning rational points over quadratic fields. This material will be used in the proofs of the previous theorems.

\subsection*{Data of some elliptic curves}
Table \ref{TableC} shows the elliptic curves used in this article. For each curve \( C_k \), with \( k = 0,1,\dots,6 \), the third column gives the corresponding label in the LMFDB database \cite{lmfdb}, while the fourth column lists the corresponding \(\mathbb{Q}\)-isomorphic elliptic curve.

\begin{center}
\renewcommand{\arraystretch}{1.3}
\begin{longtable}{|l|l|l|l|}
\caption{Curves} \label{TableC} \\
\hline
$C_0$ & $y^2 = x^4-2x^3+2x^2+2x+1$  &  \href{https://www.lmfdb.org/EllipticCurve/Q/192a2/}{\texttt{192.a2}} &  $E_0$\\
\hline
$C_1$& $y^2 = (x+4)(x^2+4x+16)$  & \href{https://www.lmfdb.org/EllipticCurve/Q/24a4/}{\texttt{24.a5}} & $E_1$\\
\hline
$C_2$& $y^2 = x(x^2+4x+16)$  & \href{https://www.lmfdb.org/EllipticCurve/Q/48a4/}{\texttt{48.a5}}&  $E^{-1}_1$\\
\hline
$C_3$& $y^2 = x(x+4)(x^2+4x+16)$ &  \href{https://www.lmfdb.org/EllipticCurve/Q/24a4/}{\texttt{24.a5}}&  $E_1$\\
\hline
$C_4$& $y^2 = x(x^2+14x+1)$ &  \href{https://www.lmfdb.org/EllipticCurve/Q/24a3/}{\texttt{24.a2}}&  $E_4$\\
\hline
$C_5$ & $y^2 = x^4+4x^2+16$ &  \href{https://www.lmfdb.org/EllipticCurve/Q/48a1/}{\texttt{48.a4}} &  $E^{-1}_6$\\
\hline
$C_6$ & $y^2 = x^4+14x^2+1$  &  \href{https://www.lmfdb.org/EllipticCurve/Q/24a1/}{\texttt{24.a4}} &  $E_6$\\
\hline
\end{longtable}
\end{center}
Note that the elliptic curves $E_1,E_4$ and $E_6$ belongs to the same $\mathbb Q$-isogeny class.  In particular, for any integer $D$, the ranks of any $D$-quadratic twist coincide. In particular, we have $\operatorname{rank}_{\mathbb Z}C_k(\mathbb Q)=0$ for $k=1,\dots,6$. Let $E$ be an elliptic curve defined over $\mathbb Q$. Using the known formula (see \cite{Kra})
$$
\operatorname{rank}_{\mathbb Z}E(\Q(\sqrt{D}))=\operatorname{rank}_{\mathbb Z}E(\mathbb Q)+
\operatorname{rank}_{\mathbb Z}E^D(\mathbb Q),
$$
we obtain that $\operatorname{rank}_{\mathbb Z}E_1^{\pm D}(\mathbb Q)= 0$ is equivalent to  $\operatorname{rank}_{\mathbb Z}C_k(\Q(\sqrt{D}))=0$ for $k=1,\dots,6$. Assuming $\operatorname{rank}_{\mathbb Z}E_1^{\pm D}(\mathbb Q)= 0$ for the square-free integer $D$, to compute $C_k(\Q(\sqrt{D}))$, for $k=1,\dots,6$, we need to compute the torsion subgroup over $\Q(\sqrt{D})$. In our particular case, given an elliptic curve $E$ defined over $\mathbb Q$, there are only a finite number of square-free integers $D$ such that $E(\mathbb Q)_{\operatorname{tors}}\ne E(\Q(\sqrt{D}))_{\operatorname{tors}}$. If the conductor of $E$ is less than $400000$, these square-free integers appear in the LMFDB database \cite{lmfdb} and are listed in Table \ref{growth}.
\begin{center}
\begin{longtable}{|c|c|}
\caption{Torsion growth over quadratic fields}\label{growth}\\
\hline
$E$ & \mbox{$D$ such that $E(\mathbb Q)_{\operatorname{tors}}\ne
E(\Q(\sqrt{D}))_{\operatorname{tors}}$}\\
\hline

$E_0$ & $-2$ \\
\hline
$E_1$   &  $-1$, $\pm 3$ \\
\hline
$E^{-1}_1$     &  $-1$,$\pm 3$\\
\hline
$E_4$ & 2,3,6\\
\hline
$E_6$ & $\emptyset$ \\
\hline
$E_6^{-1}$ & $-1$,$\pm 3$\\
\hline
\end{longtable}
\end{center}
Note that for those elliptic curves and values of $D$, we have $\operatorname{rank}_{\mathbb Z}E^D(\mathbb Q)=0$ except in the case $E^6_4$. Finally, for any square-free integer $D$, Tables \ref{points} and \ref{pointsC4}  show  $C_k(\Q(\sqrt{D}))$ for $k=1,\dots,6$. When $D=-1,2,\pm 3$, the points at the second column correspond to $C_k(\mathbb Q)$ and they must be added to the corresponding set $C_k(\Q(\sqrt{D}))$.

\renewcommand{\arraystretch}{1.3}
\begin{table}[htbp]
\begin{center} \caption{$C_k(\Q(\sqrt{D}))$ for $k=1,2,3,5,6$.}\label{points}
\begin{tabular}{|c|c|c|c|c|}
\cline{2-5}
 \multicolumn{1}{c|}{}
 & $D\ne -1,\pm 3$ & $D=-1$ & $D=-3$ & $D=3$  \\
\hline
$C_1$ & $\begin{array}{c}(-4,0)\\ (0,\pm 8) \end{array}$& $(-4\pm 4i,\pm8)$ & $\begin{array}{c}(-8,\pm8\sqrt{-3})\\(-2\pm 2 \sqrt{-3},0)\end{array}$ & $\begin{array}{c}
(4-4\sqrt{3},\pm(24-16\sqrt{3}))\\
(4+4 \sqrt{3},\pm(24+16\sqrt{3}))
\end{array}$  \\
\hline
$C_2$ & $(0,0)$ & $\begin{array}{c}(\pm 4i,\pm 8i)\\(-4,\pm 8i) \end{array}$& $(-2\pm 2\sqrt{-3},0)$ & $(4,\pm 8\sqrt{3}) $ \\
\hline
$C_3$ & $\begin{array}{c} (0,0)\\(-4,0) \end{array}$& $(-2\pm 2 i,\pm 8i)$ & $\begin{array}{c}(-2\pm 2\sqrt{-3},0)\\(-2,\pm 4 \sqrt{-3})\end{array}$ & $(-2\pm 2 \sqrt{3},\pm 8\sqrt{3})$  \\
\hline
$C_5$ & $(0,\pm 4)$ & $(\pm 2i,\pm 4)$ & $(\pm1\pm\sqrt{-3},0)$ & $(\pm 2,\pm 4\sqrt{3})$  \\
\hline
$C_6$ & $\begin{array}{c} (0,\pm 1)\\ (\pm 1,\pm 4) \end{array}$ & $-$ & $-$ & $- $ \\
\hline
\end{tabular}
\end{center}
\end{table}

\renewcommand{\arraystretch}{1.3}
\begin{table}[htbp]
\begin{center} \caption{$C_4(\Q(\sqrt{D}))$}\label{pointsC4}
\begin{tabular}{|c|c|c|c|}
\cline{2-4}
 \multicolumn{1}{c|}{} & $D\ne 2,3,6$ & $D=3$ & $D=2$  \\
\hline
$C_4$ & $\begin{array}{c} (0,0)\\ (1,\pm 4)  \end{array}$ & $\begin{array}{c}(-1,-2\sqrt{3})\\ (7\pm 4\sqrt{3},0)\end{array}$ & $\begin{array}{c}
(-3-2\sqrt{2},\pm(8+6\sqrt{2}))\\
(-3+2\sqrt{2},\pm(8-6\sqrt{2}))\\
\end{array}$\\
   \hline
\end{tabular}
\end{center}
\end{table}

\newpage

\begin{thm}\label{main}
Let $D$ be a square-free integer, $D\ne -1,\pm 2,3$, such that $\operatorname{rank}_{\mathbb Z}E_1^{\pm D}(\mathbb Q)= 0$ and $K$ a quadratic extension of $\Q(\sqrt{D})$. Then
\begin{itemize}
\item[(A)] If $\operatorname{rank}_{\mathbb Z}E_0^{D}(\mathbb Q)=0$, then there does not exist any non-constant arithmetic progression of five squares properly defined over $K$.

\begin{thm}\label{six}
Let $D$ be a square-free integer such that
$$
\operatorname{rank}_{\mathbb Z}E_1^{\pm D}(\mathbb Q)= 0\,,
$$
and $\operatorname{rank}_{\mathbb Z}E_0^{D}(\mathbb Q)=0$ or if $\operatorname{rank}_{\mathbb Z}E_0^{D}(\mathbb Q)\ne 0$ the class number of $\Q(\sqrt{D})$ is $1$. Then there does not exist any non-constant arithmetic progression of six squares properly defined over any quadratic extension of $\Q(\sqrt{D})$.
\end{thm}

\begin{center}
\renewcommand{\arraystretch}{1.3}
\begin{longtable}{|l|l|l|l|}
\caption{Curves} \label{TableC} \\
\hline
$C_0$ & $y^2 = x^4-2x^3+2x^2+2x+1$ & \href{https://www.lmfdb.org/EllipticCurve/Q/192a2/}{\texttt{192.a2}} & $E_0$\\
\hline
$C_1$& $y^2 = (x+4)(x^2+4x+16)$ & \href{https://www.lmfdb.org/EllipticCurve/Q/24a4/}{\texttt{24.a5}} & $E_1$\\
\hline
$C_2$& $y^2 = x(x^2+4x+16)$ & \href{https://www.lmfdb.org/EllipticCurve/Q/48a4/}{\texttt{48.a5}}& $E^{-1}_1$\\
\hline
$C_3$& $y^2 = x(x+4)(x^2+4x+16)$ & \href{https://www.lmfdb.org/EllipticCurve/Q/24a4/}{\texttt{24.a5}}& $E_1$\\
\hline
$C_4$& $y^2 = x(x^2+14x+1)$ & \href{https://www.lmfdb.org/EllipticCurve/Q/24a3/}{\texttt{24.a2}}& $E_4$\\
\hline
$C_5$ & $y^2 = x^4+4x^2+16$ & \href{https://www.lmfdb.org/EllipticCurve/Q/48a1/}{\texttt{48.a4}} & $E^{-1}_6$\\
\hline
$C_6$ & $y^2 = x^4+14x^2+1$ & \href{https://www.lmfdb.org/EllipticCurve/Q/24a1/}{\texttt{24.a4}} & $E_6$\\
\hline
\end{longtable}
\end{center}
Note that the elliptic curves $E_1,E_4$ and $E_6$ belongs to the same $\mathbb Q$-isogeny class. In particular, for any integer $D$, the ranks of any $D$-quadratic twist coincide. In particular, we have $\operatorname{rank}_{\mathbb Z}C_k(\mathbb Q)=0$ for $k=1,\dots,6$. Let $E$ be an elliptic curve defined over $\mathbb Q$. Using the known formula (see \cite{Kra})
$$
\operatorname{rank}_{\mathbb Z}E(\Q(\sqrt{D}))=\operatorname{rank}_{\mathbb Z}E(\mathbb Q)+
\operatorname{rank}_{\mathbb Z}E^D(\mathbb Q),
$$
we obtain that $\operatorname{rank}_{\mathbb Z}E_1^{\pm D}(\mathbb Q)= 0$ is equivalent to $\operatorname{rank}_{\mathbb Z}C_k(\Q(\sqrt{D}))=0$ for $k=1,\dots,6$. Assuming $\operatorname{rank}_{\mathbb Z}E_1^{\pm D}(\mathbb Q)= 0$ for the square-free integer $D$, to compute $C_k(\Q(\sqrt{D}))$, for $k=1,\dots,6$, we need to compute the torsion subgroup over $\Q(\sqrt{D})$. In our particular case, given an elliptic curve $E$ defined over $\mathbb Q$, there are only a finite number of square-free integers $D$ such that $E(\mathbb Q)_{\operatorname{tors}}\ne E(\Q(\sqrt{D}))_{\operatorname{tors}}$. If the conductor of $E$ is less than $400000$, these square-free integers appear in the LMFDB database \cite{lmfdb} and are listed in Table \ref{growth}.
\begin{center}
\begin{longtable}{|c|c|}
\caption{Torsion growth over quadratic fields}\label{growth}\\
\hline
$E$ & \mbox{$D$ such that $E(\mathbb Q)_{\operatorname{tors}}\ne
E(\Q(\sqrt{D}))_{\operatorname{tors}}$}\\
\hline

$E_0$ & $-2$ \\
\hline
$E_1$ & $-1$, $\pm 3$ \\
\hline
$E^{-1}_1$ & $-1$,$\pm 3$\\
\hline
$E_4$ & 2,3,6\\
\hline
$E_6$ & $\emptyset$ \\
\hline
$E_6^{-1}$ & $-1$,$\pm 3$\\
\hline
\end{longtable}
\end{center}
Note that for those elliptic curves and values of $D$, we have $\operatorname{rank}_{\mathbb Z}E^D(\mathbb Q)=0$ except in the case $E^6_4$. Finally, for any square-free integer $D$, Tables \ref{points} and \ref{pointsC4} show $C_k(\Q(\sqrt{D}))$ for $k=1,\dots,6$. When $D=-1,2,\pm 3$, the points at the second column correspond to $C_k(\mathbb Q)$ and they must be added to the corresponding set $C_k(\Q(\sqrt{D}))$.

\renewcommand{\arraystretch}{1.3}
\begin{table}[htbp]
\begin{center} \caption{$C_4(\Q(\sqrt{D}))$}\label{pointsC4}
\begin{tabular}{|c|c|c|c|}
\cline{2-4}
 \multicolumn{1}{c|}{} & $D\ne 2,3,6$ & $D=3$ & $D=2$ \\
\hline
$C_4$ & $\begin{array}{c} (0,0)\\ (1,\pm 4) \end{array}$ & $\begin{array}{c}(-1,-2\sqrt{3})\\ (7\pm 4\sqrt{3},0)\end{array}$ & $\begin{array}{c}
(-3-2\sqrt{2},\pm(8+6\sqrt{2}))\\
(-3+2\sqrt{2},\pm(8-6\sqrt{2}))\\
\end{array}$\\
 \hline
\end{tabular}
\end{center}
\end{table}

\newpage

\newpage

\begin{proof}
Table~\ref{growth} shows that the torsion subgroup grows only when $D=-2$. Thus, for $D\neq -2$, we have $E_0(\Q(\sqrt{D}))_{\operatorname{tors}}=E_0(\Q)_{\operatorname{tors}}$. Therefore, if $D\neq -2$ and $\operatorname{rank}_{\Z}E_0^{D}(\Q)=0$, we deduce that $E_0(\Q(\sqrt{D}))$ is isomorphic to $E_0(\Q)$. Let us prove that the only case where $E_0(\Q(\sqrt{D}))\neq E_0(\Q)$ under these assumptions occurs when $D=2$. Assume that there exists $P\in E_0(\Q)$ and $R\in E_0(\Q(\sqrt{D}))\setminus E_0(\Q)$ such that $P=nR$ for some $n\in \Z$. Let $\sigma\in \operatorname{Gal}(\Q(\sqrt{D})/\Q)$ be the nontrivial automorphism. Then $P=n\,\sigma(R)$, and hence $n(R-\sigma(R))=\mathcal{O}$. This implies that $R-\sigma(R)\in E_0(\Q)[n]$. Since $E_0(\Q)_{\operatorname{tors}}=E_0[2]$, we conclude that $n=2$.

\section{Setting}
Let $D$ be a square-free integer and $K$ be a quadratic extension of $\mathbb{Q}(\sqrt{D})$. If $S\subseteq K$, we denote by $S^2$ to the set of squares in $S$. Let $a, b, c, d, e \in K$ be such that $\left(a^{2}, b^{2}, c^{2}, d^{2}, e^{2}\right)$ is an arithmetic progression. Then $[a: b: c: d: e]\in V(K) \subset \mathbb{P}^4(K)$, where $V$ is the genus 5 curve defined by the system of equations
$$
V\,\,:\,\, \{a^{2}+e^{2}=2 c^{2}\,,\,\, a^{2}+c^{2}=2 b^{2}\,,\,\,c^{2}+e^{2}=2 d^{2} \}.
$$
Equation $a^2+e^2=2c^2$ has the parametrization
\begin{equation}\label{eq2}
[a: c: e]=\left[t^{2}-2 t-1: t^{2}+1: t^{2}+2 t-1\right]
\end{equation}
with $t \in K$. Applying \eqref{eq2} in the equations of $V$ gives, up to sign,
\begin{equation}\label{eq3}
\left\{
\begin{array}{l}
a = t^{2}-2 t-1, \\
b= \sqrt{t^{4}-2 t^{3}+2 t^{2}+2 t+1}, \\
c= t^{2}+1, \\
d= \sqrt{t^{4}+2 t^{3}+2 t^{2}-2 t+1}, \\
e= t^{2}+2 t-1.
\end{array}
\right.
\end{equation}
Replacing $t$ by $-t$ in \eqref{eq3} gives the arithmetic progression $(e^2,d^2,c^2,b^2,a^2)$, equivalent to $(a^2,b^2,c^2,d^2,e^2)$, and the constant arithmetic progressions correspond to the cases $t=0,\pm 1$.

\begin{prop}\label{prop}
Let $D$ be a square-free integer and $K$ be a quadratic extension of $\mathbb{Q}(\sqrt{D})$. Let $a, b, c, d, e \in K$ be such that $\left(a^{2}, b^{2}, c^{2}, d^{2}, e^{2}\right)$ is a non elementary arithmetic progression given by \eqref{eq3}. If $\operatorname{rank}_{\Z}E_1^{\pm D}(\Q)= 0\,,$ then
\begin{enumerate}
\item\label{prop1} $r \in \QD$,
\item\label{prop2} $s \in \QD$,
\item\label{prop3} $t\in\QD$,
\item\label{prop4} $t\in\Q$, if $D\ne \pm 2$ and $\operatorname{rank}_{\Z}E_0^{D}(\Q)=0$.
\end{enumerate}
\end{prop}

\section{Proof of Proposition \ref{prop}\,\eqref{prop4}}\label{tinQ}
Let {$D\ne \pm 2$} be a square-free integer satisfying $\operatorname{rank}_{\Z}E_1^{\pm D}(\Q)= 0$ and $\operatorname{rank}_{\Z}E_0^{D}(\Q)=0$. Let \( t \in K \setminus \{0, \pm 1, \pm i, \pm 1 \pm \sqrt{2}\} \) corresponding to a non-elementary arithmetic progression, as described in ~\eqref{eq3}. We prove that $t \in \Q$. Since $[K:\QD]=2$, there exists $\alpha \in K$ such that $\alpha^2 \in \QD$ and $K=\Q(\sqrt{D},\alpha)$. By \eqref{eq6}, there exist $\gamma_1,\delta_1,\gamma_2,\delta_2\in \QD$ such that
\begin{equation}\label{Section6.E1}
\begin{cases}
 t^{4}-2 t^{3}+2 t^{2}+2 t+1&\!\!\!\!=\left(\delta_{1}+\gamma_{1}\alpha\right)^{2}, \\
 t^{4}+2 t^{3}+2 t^{2}-2 t+1&\!\!\!\!=\left(\delta_{2}+\gamma_{2} \alpha\right)^{2}.
\end{cases}
\end{equation}
Since $t \in \Q(\sqrt{D})$ and $\alpha \notin \QD$, we have $\gamma_{1} \delta_{1}=\gamma_{2} \delta_{2}=0$. There are four cases:

Let us prove Theorem \ref{main}:
\begin{itemize}
\item[(A)] Assume $\operatorname{rank}_{\Z}E_0^{D}(\Q)=0$. Since $D\ne \pm 2$. By Proposition \ref{prop} \eqref{prop4}, we have $t\in \Q$. In this case we have \( C_0(\mathbb{Q}(\sqrt{D})) = C_0(\mathbb{Q}) \). Hence, if the value of \( G(t) \) is a square over \( \mathbb{Q}(\sqrt{D}) \), it is already a square over \( \mathbb{Q} \). Consequently, four out of the five terms of the arithmetic progression are defined over \( \mathbb{Q} \), and therefore the full arithmetic progression of five terms is properly defined over a quadratic extension of \( \mathbb{Q} \).
\item[(B)] Assume $\operatorname{rank}_{\Z}E_0^{D}(\Q)\ne 0$ and that the class number of $\QD$ is $1$. By Proposition \ref{prop} \eqref{prop3}, we have $t\in \QD$. Since the ring of integers of $\QD$ is a unique factorization domain, there exist $\alpha,\delta\in \Q(\sqrt{D})$ such that $G(-t)=\alpha\,\delta^2$. We can conclude that \(\alpha\) is not a square in \(\mathbb{Q}(\sqrt{D})\), since otherwise there would exist a nontrivial arithmetic progression of five squares in \(\mathbb{Q}(\sqrt{D})\). However, this is impossible because \cite[Proposition~5.2]{GJX} states that a necessary condition is that \(\operatorname{rank} E_1^{\pm D}(\mathbb{Q}) \geq 2\). Then the arithmetic progression $(a^2,b^2,c^2,d^2,e^2)$ is properly defined over $K=\Q(\sqrt{D},\omega)$, where $\omega^2=\alpha$.
\end{itemize}
Now, the proof of Theorem \ref{maincor} assuming that $(a^2,b^2,c^2,d^2,e^2)$ is not an elementary arithmetic progression:
\begin{itemize}
\item $D\in\{-1,3\}$: The proof is analogous to the case $(A)$ since $\operatorname{rank}_{\Z}E_0^{D}(\Q)=0$ and {$D\ne \pm 2$}.
\item $D= \pm 2$: In this case $\operatorname{rank}_{\Z}E_0^{D}(\Q)=0$. By Proposition \ref{prop} \eqref{prop3} we have $t\in \Q(\sqrt{D})$. The proof is analogous to the case $(B)$ since the class number of $\Q(\sqrt{D})$ is $1$.
\end{itemize}

\section{Parametrization in the cases $\Q(\sqrt{-2})$ and $\Q(\sqrt{2})$}\label{sect_para}
Let \(D\) be a square-free integer such that \(\operatorname{rank}_{\mathbb{Z}} E_1^{\pm D}(\mathbb{Q}) = 0\). We have proved that any non-elementary arithmetic progression of five squares over a quadratic extension of \(\QD\) is equivalent to
\[
\bigl((t^{2}-2t-1)^2,\, \beta^2,\, (t^{2}+1)^2,\, \alpha\delta^2,\, (t^{2}+2t-1)^2 \bigr),
\]
where \(t, \alpha, \beta, \delta \in \mathbb{Q}(\sqrt{D})\) satisfy \(\beta^2 = G(t)\) and \(\alpha\delta^2 = G(-t)\). In particular, for any non-elementary arithmetic progression of five squares, there exists a point \(R \in E_0(\mathbb{Q}(\sqrt{D}))\), and conversely. Note that this correspondence is not one-to-one. This construction recovers the one developed in \cite{GJX} for the special case $D=1$. That is, for any \(R \in E_0(\mathbb{Q})\) we obtain an arithmetic progression of five squares over a quadratic field, except when \(R \in E_0(\mathbb{Q})_{\operatorname{tors}}\), in which case the progression is trivial. Note that \(\operatorname{rank}_{\mathbb{Z}} E_1^{\pm 1}(\mathbb{Q}) = 0\). Let $T_1=(-3,0)$, $T_2=(1,0)$ and $P=(0,3)$, then
$$
E_0(\Q)=\langle T_1 \rangle \oplus \langle T_2 \rangle \oplus \langle P \rangle.
$$
If $n\in\Z$, $n\ge 0$, the set of points $S_n=\{n_1T_1+n_2T_2+m P \,:\, n_1,n_2\in\{0,1\} \,,\,m\in\{n,-n-1\}\}$ corresponds to the same arithmetic progression of five squares, up to equivalence. For example, if $n=1$ the set $S_1$ corresponds to the arithmetic progression $(7^2, 13^2, 17^2, (\sqrt{409})^2, 23^2)$.

\begin{proof}
Table~\ref{growth} shows that the torsion subgroup grows only when $D=-2$. Thus, for $D\neq -2$, we have $E_0(\mathbb Q(\sqrt{D}))_{\operatorname{tors}}=E_0(\mathbb Q)_{\operatorname{tors}}$. Therefore, if $D\neq -2$ and $\operatorname{rank}_{\mathbb Z}E_0^{D}(\mathbb Q)=0$, we deduce that $E_0(\mathbb Q(\sqrt{D}))$ is isomorphic to $E_0(\mathbb Q)$. Let us prove that the only case where $E_0(\mathbb Q(\sqrt{D}))\neq E_0(\mathbb Q)$ under these assumptions occurs when $D=2$. Assume that there exists $P\in E_0(\mathbb Q)$ and $R\in E_0(\mathbb Q(\sqrt{D}))\setminus E_0(\mathbb Q)$ such that $P=nR$ for some $n\in \mathbb Z$. Let $\sigma\in \operatorname{Gal}(\mathbb Q(\sqrt{D})/\mathbb Q)$ be the nontrivial automorphism. Then $P=n\,\sigma(R)$, and hence $n(R-\sigma(R))=\mathcal{O}$. This implies that $R-\sigma(R)\in E_0(\mathbb Q)[n]$. Since $E_0(\mathbb Q)_{\operatorname{tors}}=E_0[2]$, we conclude that $n=2$.

Now recall that $E_0(\mathbb Q)=\langle T_1 \rangle \oplus \langle T_2 \rangle \oplus \langle P_0 \rangle$, where $T_1=(-3,0)$, $T_2=(1,0)$, and $P_0=(5,8)$. Thus any $P\in E_0(\mathbb Q)$ can be written as $ P=n_1T_1+n_2T_2+mP_0$ for some $n_1,n_2\in \{0,1\}$ and $m\in \mathbb Z$. It is therefore enough to check the existence of such $R$ when $P=n_1T_1+n_2T_2+P_0$ for some $n_1,n_2\in \{0,1\}$. Let us prove the above fact: suppose first that $m=2n$ for some $n\neq 0$. Then we would obtain $2(R+nP_0)=n_1T_1+n_2T_2$,  which is impossible since the left-hand side has infinite order while the right-hand side has order  { dividing} $2$. Suppose instead that $m=2n+1$ for some $n\neq 0$. Then we obtain $2(R+nP_0)=n_1T_1+n_2T_2+P_0$, which gives the desired relation.

{ Finally, one computes that the only solutions to $2 R=n_1T_1+n_2T_2+P_0$, $n_1,n_2\in \{0,1\}$, are given by $n_1=n_2=0$ and $R=(1+2\sqrt{2},-4)+T$ with $T\in E[2]$.}  This completes the proof.
\end{proof}

{\bf Acknowledgments.} Nguyen Xuan Tho is also supported by Vietnam Institute for Advanced Study in Mathematics (VIASM) from April 2025 to May 2025. The author really appreciates the Institute for their help and funding. We would like to thank Xavier Xarles for pointing out that the elliptic curves $E_1,E_4$ and $E_6$ belongs to the same $\mathbb Q$-isogeny class. The authors would like to thank the referees for their careful reading of the manuscript and for their valuable suggestions. In particular, we are grateful to the last referee, whose extensive and detailed comments greatly enhanced the overall quality of the paper. We are sincerely thankful for their contributions.

\

All computations in this paper are done in \texttt{Magma} \cite{Magma} and \texttt{Mathematica} \cite{mathematica}.

Table~\ref{growth} is used to note that the torsion subgroup grows only when $D=-2$; hence for $D\neq -2$ one has $E_0(\mathbb Q(\sqrt D))_{\operatorname{tors}}=E_0(\mathbb Q)_{\operatorname{tors}}$, and with $\operatorname{rank}_{\mathbb Z}E_0^{D}(\mathbb Q)=0$ this yields that $E_0(\mathbb Q(\sqrt D))$ is isomorphic to $E_0(\mathbb Q)$.

To see when equality can fail under these assumptions, the argument assumes there exist $P\in E_0(\mathbb Q)$ and $R\in E_0(\mathbb Q(\sqrt D))\setminus E_0(\mathbb Q)$ with $P=nR$. Let $\sigma\in \operatorname{Gal}(\mathbb Q(\sqrt D)/\mathbb Q)$ be nontrivial. Then $P=n\,\sigma(R)$, so $n(R-\sigma(R))=\mathcal O$, hence $R-\sigma(R)\in E_0(\mathbb Q)[n]$. Since $E_0(\mathbb Q)_{\operatorname{tors}}=E_0[2]$, it follows that $n=2$.

Using $E_0(\mathbb Q)=\langle T_1\rangle\oplus\langle T_2\rangle\oplus\langle P_0\rangle$ with $T_1=(-3,0)$, $T_2=(1,0)$, $P_0=(5,8)$, write $P=n_1T_1+n_2T_2+mP_0$ ($n_1,n_2\in\{0,1\}$, $m\in\mathbb Z$). One reduces to checking the existence of $R$ only when $P=n_1T_1+n_2T_2+P_0$. Indeed, if $m=2n\neq 0$ then $2(R+nP_0)=n_1T_1+n_2T_2$, impossible because the left side has infinite order while the right side has order dividing $2$; if $m=2n+1$ with $n\neq 0$ then $2(R+nP_0)=n_1T_1+n_2T_2+P_0$, giving the desired relation.

Finally, it is computed that the only solutions to $2R=n_1T_1+n_2T_2+P_0$ with $n_1,n_2\in\{0,1\}$ occur for $n_1=n_2=0$ and $R=(1+2\sqrt 2,-4)+T$ with $T\in E[2]$, so the exceptional case is $D=2$ (together with the earlier torsion-growth exception $D=-2$).

\begin{longtable}{|c|c|}
\caption{Torsion growth over quadratic fields}\label{growth}\\
\hline
$E$ & \mbox{$D$ such that $E(\Q)_{\operatorname{tors}}\ne
E(\QD)_{\operatorname{tors}}$}\\
\hline

$E_0$ & $-2$ \\
\hline
$E_1$   &  $-1$, $\pm 3$ \\
\hline
$E^{-1}_1$     &  $-1$,$\pm 3$\\
\hline
$E_4$ & 2,3,6\\
\hline
$E_6$ & $\emptyset$ \\
\hline
$E_6^{-1}$ & $-1$,$\pm 3$\\
\hline
\end{longtable}

This table is used to note that for $E_0$ the torsion subgroup grows only when $D=-2$; hence for $D\neq -2$ one has $E_0(\mathbb Q(\sqrt D))_{\operatorname{tors}}=E_0(\mathbb Q)_{\operatorname{tors}}$, and with $\operatorname{rank}_{\mathbb Z}E_0^{D}(\mathbb Q)=0$ this yields that $E_0(\mathbb Q(\sqrt D))$ is isomorphic to $E_0(\mathbb Q)$.

To see when equality can fail under these assumptions, the argument assumes there exist $P\in E_0(\mathbb Q)$ and $R\in E_0(\mathbb Q(\sqrt D))\setminus E_0(\mathbb Q)$ with $P=nR$. Let $\sigma\in \operatorname{Gal}(\mathbb Q(\sqrt D)/\mathbb Q)$ be nontrivial. Then $P=n\,\sigma(R)$, so $n(R-\sigma(R))=\mathcal O$, hence $R-\sigma(R)\in E_0(\mathbb Q)[n]$. Since $E_0(\mathbb Q)_{\operatorname{tors}}=E_0[2]$, it follows that $n=2$.

Using $E_0(\mathbb Q)=\langle T_1\rangle\oplus\langle T_2\rangle\oplus\langle P_0\rangle$ with $T_1=(-3,0)$, $T_2=(1,0)$, $P_0=(5,8)$, write $P=n_1T_1+n_2T_2+mP_0$ ($n_1,n_2\in\{0,1\}$, $m\in\mathbb Z$). One reduces to checking the existence of $R$ only when $P=n_1T_1+n_2T_2+P_0$. Indeed, if $m=2n\neq 0$ then $2(R+nP_0)=n_1T_1+n_2T_2$, impossible because the left side has infinite order while the right side has order dividing $2$; if $m=2n+1$ with $n\neq 0$ then $2(R+nP_0)=n_1T_1+n_2T_2+P_0$, giving the desired relation.

Finally, it is computed that the only solutions to $2R=n_1T_1+n_2T_2+P_0$ with $n_1,n_2\in\{0,1\}$ occur for $n_1=n_2=0$ and $R=(1+2\sqrt 2,-4)+T$ with $T\in E[2]$, so the exceptional case is $D=2$ (together with the earlier torsion-growth exception $D=-2$).

Question: Let \(E_0\) denote the elliptic curve labeled \(192.a2\) in the context, and for a square-free integer \(D\), let \(E_0^D\) be the \(D\)-quadratic twist of \(E_0\). Which statement holds for every square-free integer \(D\) satisfying \(\operatorname{rank}_{\mathbb Z} E_0^D(\mathbb Q)=0\)?

If \(D\neq \pm 2\), then the group of \(\mathbb Q(\sqrt{D})\)-rational points on \(E_0\) equals the group of \(\mathbb Q\)-rational points: \(E_0(\mathbb Q(\sqrt{D}))=E_0(\mathbb Q)\).

Choices:
B. If \(D\neq -2\), then the group of \(\mathbb Q(\sqrt{D})\)-rational points on \(E_0\) equals the group of \(\mathbb Q\)-rational points: \(E_0(\mathbb Q(\sqrt{D}))=E_0(\mathbb Q)\).
C. If \(D\neq \pm 2\), then every \(\mathbb Q(\sqrt{D})\)-rational point on \(E_0\) is already \(\mathbb Q\)-rational up to torsion; equivalently, the quotient group \(E_0(\mathbb Q(\sqrt{D}))/E_0(\mathbb Q)\) is torsion.
D. There exists a finite set \(S\subset \mathbb Z\), independent of \(D\), such that whenever \(D\notin S\) and \(\operatorname{rank}_{\mathbb Z} E_0^D(\mathbb Q)=0\), one has \(E_0(\mathbb Q(\sqrt{D}))_{\operatorname{tors}}=E_0(\mathbb Q)_{\operatorname{tors}}\), and hence \(E_0(\mathbb Q(\sqrt{D}))=E_0(\mathbb Q)\).
E. If \(\operatorname{rank}_{\mathbb Z} E_0^D(\mathbb Q)=0\), then every point of \(E_0(\mathbb Q(\sqrt{D}))\) is twice a point of \(E_0(\mathbb Q)\); in particular, \(E_0(\mathbb Q(\sqrt{D}))=E_0(\mathbb Q)\) unless \(D=\pm 2\).