\begin{theorem}\label{thm:MainIntro}
	Fix $d\geq 2$, $\gamma>0$ and $0\leq\lambda\leq 1$. Consider the visibility region $Z_{\gamma,\lambda,d}$ of $o$ in a Poisson process of $\lambda$-geodesic hyperplanes in $\mathbb{H}^d$ with intensity measure $\gamma\nu_\lambda^o$.
	\begin{itemize}
	\item[(a)] Let $\gamma_{\mathrm{crit}}$ be as in \eqref{eq:GammaCrit}. If $\gamma<\gamma_{\mathrm{crit}}$, then $Z_{\gamma,\lambda,d}$ is unbounded with strictly positive probability, whereas if $\gamma>\gamma_{\mathrm{crit}}$, then $Z_{\gamma,\lambda,d}$ is almost surely bounded. Moreover, if $d=2$, then $Z_{\gamma_{\mathrm{crit}},\lambda,2}$ is almost surely bounded.

	\item[(b)] If $\gamma>\gamma_{\mathrm{crit}}$ then
	\begin{equation}\label{eq:MeanVolume}
		\mathbb{E}{\rm vol}_d(Z_{\gamma,\lambda,d})=\mathbb{E}{\rm vol}_d(Z_{\gamma,0,d}) = \pi^{\frac{d-1}{2}} \Gamma\left(\frac{d+1}{2}\right){\Gamma\big({\gamma_d^\ast-d+1\over 2}\big)\over\Gamma\big({\gamma_d^\ast+d+1\over 2}\big)}
	\end{equation}
	with $\gamma_d^\ast := \gamma\,{\Gamma({d\over 2})\over 2\sqrt{\pi}\Gamma({d+1\over 2})}$. On the other hand, $\mathbb{E}{\rm vol}_d(Z_{\gamma,\lambda,d})=+\infty$ for all $\gamma\leq\gamma_{\mathrm{crit}}$.
	\end{itemize}
	\end{theorem}

Random tessellations generated by Poisson hyperplane processes in Euclidean space form a classical and well-studied topic in stochastic geometry.  In $\mathbb{R}^d$, a stationary Poisson hyperplane process gives rise to a mosaic of convex cells, and many aspects of its geometry have been analyzed in detail, including the cell containing the origin, see \cite{HugSchneiderBook} and Chapter 10 of \cite{SW08}.

	When passing from Euclidean space to hyperbolic space $\mathbb{H}^d$ of constant negative curvature  $-1$, the situation becomes richer, since hyperbolic geometry admits several natural analogues of Euclidean hyperplanes. Besides totally geodesic hyperplanes, which are isometric copies of $\mathbb{H}^{d-1}$, one encounters equidistant hypersurfaces, which are defined as the set of points at fixed distance from a given geodesic hyperplane. A further limiting case is provided by horospheres, which arise as limits of equidistant hypersurfaces whose base hyperplane recedes towards the ideal boundary of $\mathbb{H}^d$. These three families can be unified by the notion of
	\emph{$\lambda$--geodesic hyperplanes} as introduced in \cite{Solanes}. For $\lambda \in [0,1]$, a
	$\lambda$--geodesic hyperplane is a complete totally umbilical hypersurface in $\mathbb{H}^d$ with normal curvature $\lambda $. Thus, $\lambda = 0$ corresponds to totally geodesic hyperplanes, which have vanishing intrinsic curvature, $0 < \lambda < 1$ yields equidistant hypersurfaces, whose curvature depends continuously on~$\lambda$, and $\lambda = 1$ gives horospheres.

	Every $\lambda$--geodesic hyperplane separates $\mathbb{H}^d$ into two connected components.  As we are not interested in all such hypersurfaces, we introduce the space $\mathrm{Hyp}_\lambda^o$ of all $\lambda$--geodesic hyperplanes for which the distinguished point
	$o \in \mathbb{H}^d$, which we refer to as the origin, does not lie on the convex side.  Let $\nu_\lambda$ denote the natural isometry-invariant measure on the space of all $\lambda$-geodesic hyperplanes, as introduced in Section \ref{sec:lambda-geodesic_hyperplanes} below,  and $\nu_\lambda^o$ its restriction to $\mathrm{Hyp}_\lambda^o$. For an intensity parameter $\gamma > 0$, we define $\eta_{\gamma,\lambda}$
	to be a Poisson point process on $\mathrm{Hyp}_\lambda^o$ with intensity measure $\gamma \nu_\lambda^o$.  Each hyperplane in $\eta_{\gamma,\lambda}$ acts as a random obstacle to visibility from the reference point~$o$. For a point $x \in \mathbb{H}^d$, we say that $x$ is visible from~$o$ if the geodesic segment $[o,x]$ connecting $o$ with $x$ does not intersect any $H \in \eta_{\gamma,\lambda}$.
	This leads to the random closed set
	\begin{equation}\label{eq:DefVisibilitySet}
	Z_{\gamma,\lambda,d}
	:= \{\, x \in \mathbb{H}^d :
	[o,x] \cap H = \varnothing
	\text{ for all } H \in \eta_{\gamma,\lambda}
	\,\},
	\end{equation}
	which we call the \emph{visibility region} of~$o$. Equivalently, for $\lambda=0$, $Z_{\gamma,0,d}$ is the connected component of~$o$ in the complement of the union of all hyperplanes in $\eta_{\gamma,0}$. Simulations of Poisson processes of $\lambda$-geodesic hyperplanes for different values of $\lambda$ together with the corresponding visibility regions are shown in Figure \ref{fig: simulation}.

		\begin{figure}
		\centering
		\begin{minipage}{.3\linewidth}
			\hspace*{-0.4cm}\includegraphics[scale=0.09]{l=0_g=2_c.pdf}
			\subcaption{$\lambda=0, \gamma=2<\gamma_{\mathrm{crit}}$}
		\end{minipage}\quad
		\begin{minipage}{.3\linewidth}
			\hspace*{-0.4cm}	\includegraphics[scale=0.09]{l=05_g=2_c.pdf}
			\subcaption{$\lambda=0.5, \gamma=2<\gamma_{\mathrm{crit}}$}
		\end{minipage}\quad
		\begin{minipage}{.3\linewidth}
			\hspace*{-0.4cm}\includegraphics[scale=0.09]{l=1_g=2_c.pdf}
			\subcaption{$\lambda=1, \gamma=2<\gamma_{\mathrm{crit}}$}
		\end{minipage}
		\begin{minipage}{.3\linewidth}
			\hspace*{-0.4cm}\includegraphics[scale=0.09]{l=0_g=pi_c.pdf}
			\subcaption{$\lambda=0, \gamma=\pi=\gamma_{\mathrm{crit}}$}
		\end{minipage}\quad
		\begin{minipage}{.3\linewidth}
			\hspace*{-0.4cm}	\includegraphics[scale=0.09]{l=05_g=pi_c.pdf}
			\subcaption{$\lambda=0.5, \gamma=\pi=\gamma_{\mathrm{crit}}$}
		\end{minipage}\quad
		\begin{minipage}{.3\linewidth}
			\hspace*{-0.4cm}\includegraphics[scale=0.09]{l=1_g=pi_c.pdf}
			\subcaption{$\lambda=1,\gamma=\pi=\gamma_{\mathrm{crit}}$}
		\end{minipage}
		\begin{minipage}{.3\linewidth}
			\hspace*{-0.4cm}\includegraphics[scale=0.09]{l=0_g=7_c.pdf}
			\subcaption{$\lambda=0, \gamma=7>\gamma_{\mathrm{crit}}$}
		\end{minipage}\quad
		\begin{minipage}{.3\linewidth}
			\hspace*{-0.4cm}	\includegraphics[scale=0.09]{l=05_g=7_c.pdf}
			\subcaption{$\lambda=0.5, \gamma=7>\gamma_{\mathrm{crit}}$}
		\end{minipage}\quad
		\begin{minipage}{.3\linewidth}
			\hspace*{-0.4cm}\includegraphics[scale=0.09]{l=1_g=7_c.pdf}
			\subcaption{$\lambda=1, \gamma=7>\gamma_{\mathrm{crit}}$}
		\end{minipage}
		\caption{Simulation of a Poisson process of $\lambda$-geodesic hyperplanes in $\mathrm{Hyp}_\lambda^o$ with the corresponding visibility set in red in the Poincar\'e ball model. Note that for $d=2$, $\gamma_{\mathrm{crit}}=\pi$.}
		\label{fig: simulation}
	\end{figure}

	The case $\lambda = 0$, corresponding to totally geodesic hyperplanes, has been studied in detail in previous works \cite{BJST,BuehlerGusakovaRecke,BuehlerHugThaele,GKT22,PorretBlanc}.  These papers show that the visibility region $Z_{\gamma,0,d}$ may be unbounded, and that its behaviour undergoes a sharp phase transition as the intensity~$\gamma$ varies.  More precisely, there exists the critical value
	\begin{equation}\label{eq:GammaCrit}
	\gamma_{\mathrm{crit}}:= \sqrt{\pi}(d-1)^2{\Gamma({d-1\over 2})\over\Gamma({d\over 2})}
	\end{equation}
	such that
	\begin{itemize}
		\item if $\gamma < \gamma_{\mathrm{crit}}$, then
		$Z_{\gamma,0,d}$ is unbounded with strictly positive probability,
		\item if $\gamma \geq \gamma_{\mathrm{crit}}$, then
		$Z_{\gamma,0,d}$ is almost surely bounded.
	\end{itemize}
	The critical case $\gamma = \gamma_{\mathrm{crit}}$ was treated in \cite{GKT22,PorretBlanc} for $d=2$ and in \cite{BuehlerGusakovaRecke} for general dimensions. Thus, for small intensities, there remains a positive chance of seeing arbitrarily far from the origin, whereas for large intensities the random hypersurfaces almost surely form a finite ``cocoon'' around~$o$.  In the latter regime, the expected hyperbolic volume of the bounded visibility region has been computed in \cite[Theorem 7.1]{BuehlerHugThaele}, yielding an explicit expression in terms of the dimension~$d$ and the intensity parameter~$\gamma$.

	In the present work we investigate the same questions for general
	$\lambda$-geodesic hyperplanes.  One might expect the behaviour of the visibility region to depend sensitively on the parameter $\lambda$, since varying $\lambda$ interpolates between different geometric objects. Surprisingly, our results show that the situation is completely rigid with respect to~$\lambda$.  The critical intensity at which the transition from unbounded to bounded
	visibility occurs is \emph{identical} for all $\lambda \in [0,1]$, and in
	the bounded regime the expected volume of the visibility region
	$Z_{\gamma,\lambda,d}$ agrees exactly with the expression known from the case $\lambda = 0$.  In other words, neither the existence of infinite sightliness nor the size of the bounded visibility region depends in any way on the geometric nature of the obstructing hypersurfaces.  Our main findings are summarized in the following result.

Every $\lambda$--geodesic hyperplane separates $\mathbb{H}^d$ into two connected components. As we are not interested in all such hypersurfaces, we introduce the space $\mathrm{Hyp}_\lambda^o$ of all $\lambda$--geodesic hyperplanes for which the distinguished point
 $o \in \mathbb{H}^d$, which we refer to as the origin, does not lie on the convex side. Let $\nu_\lambda$ denote the natural isometry-invariant measure on the space of all $\lambda$-geodesic hyperplanes, as introduced in Section \ref{sec:lambda-geodesic_hyperplanes} below, and $\nu_\lambda^o$ its restriction to $\mathrm{Hyp}_\lambda^o$. For an intensity parameter $\gamma > 0$, we define $\eta_{\gamma,\lambda}$
 to be a Poisson point process on $\mathrm{Hyp}_\lambda^o$ with intensity measure $\gamma \nu_\lambda^o$. Each hyperplane in $\eta_{\gamma,\lambda}$ acts as a random obstacle to visibility from the reference point~$o$. For a point $x \in \mathbb{H}^d$, we say that $x$ is visible from~$o$ if the geodesic segment $[o,x]$ connecting $o$ with $x$ does not intersect any $H \in \eta_{\gamma,\lambda}$.
 This leads to the random closed set
 \begin{equation}\label{eq:DefVisibilitySet}
 Z_{\gamma,\lambda,d}
 := \{\, x \in \mathbb{H}^d :
 [o,x] \cap H = \varnothing
 \text{ for all } H \in \eta_{\gamma,\lambda}
 \,\},
 \end{equation}
 which we call the \emph{visibility region} of~$o$. Equivalently, for $\lambda=0$, $Z_{\gamma,0,d}$ is the connected component of~$o$ in the complement of the union of all hyperplanes in $\eta_{\gamma,0}$. Simulations of Poisson processes of $\lambda$-geodesic hyperplanes for different values of $\lambda$ together with the corresponding visibility regions are shown in Figure \ref{fig: simulation}.

\begin{figure}
 \centering
 \begin{minipage}{.3\linewidth}
 \hspace*{-0.4cm}\includegraphics[scale=0.09]{l=0_g=2_c.pdf}
 \subcaption{$\lambda=0, \gamma=2<\gamma_{\mathrm{crit}}$}
 \end{minipage}\quad
 \begin{minipage}{.3\linewidth}
 \hspace*{-0.4cm} \includegraphics[scale=0.09]{l=05_g=2_c.pdf}
 \subcaption{$\lambda=0.5, \gamma=2<\gamma_{\mathrm{crit}}$}
 \end{minipage}\quad
 \begin{minipage}{.3\linewidth}
 \hspace*{-0.4cm}\includegraphics[scale=0.09]{l=1_g=2_c.pdf}
 \subcaption{$\lambda=1, \gamma=2<\gamma_{\mathrm{crit}}$}
 \end{minipage}
 \begin{minipage}{.3\linewidth}
 \hspace*{-0.4cm}\includegraphics[scale=0.09]{l=0_g=pi_c.pdf}
 \subcaption{$\lambda=0, \gamma=\pi=\gamma_{\mathrm{crit}}$}
 \end{minipage}\quad
 \begin{minipage}{.3\linewidth}
 \hspace*{-0.4cm} \includegraphics[scale=0.09]{l=05_g=pi_c.pdf}
 \subcaption{$\lambda=0.5, \gamma=\pi=\gamma_{\mathrm{crit}}$}
 \end{minipage}\quad
 \begin{minipage}{.3\linewidth}
 \hspace*{-0.4cm}\includegraphics[scale=0.09]{l=1_g=pi_c.pdf}
 \subcaption{$\lambda=1,\gamma=\pi=\gamma_{\mathrm{crit}}$}
 \end{minipage}
 \begin{minipage}{.3\linewidth}
 \hspace*{-0.4cm}\includegraphics[scale=0.09]{l=0_g=7_c.pdf}
 \subcaption{$\lambda=0, \gamma=7>\gamma_{\mathrm{crit}}$}
 \end{minipage}\quad
 \begin{minipage}{.3\linewidth}
 \hspace*{-0.4cm} \includegraphics[scale=0.09]{l=05_g=7_c.pdf}
 \subcaption{$\lambda=0.5, \gamma=7>\gamma_{\mathrm{crit}}$}
 \end{minipage}\quad
 \begin{minipage}{.3\linewidth}
 \hspace*{-0.4cm}\includegraphics[scale=0.09]{l=1_g=7_c.pdf}
 \subcaption{$\lambda=1, \gamma=7>\gamma_{\mathrm{crit}}$}
 \end{minipage}
 \caption{Simulation of a Poisson process of $\lambda$-geodesic hyperplanes in $\mathrm{Hyp}_\lambda^o$ with the corresponding visibility set in red in the Poincar\'e ball model. Note that for $d=2$, $\gamma_{\mathrm{crit}}=\pi$.}
 \label{fig: simulation}
 \end{figure}

The case $\lambda = 0$, corresponding to totally geodesic hyperplanes, has been studied in detail in previous works \cite{BJST,BuehlerGusakovaRecke,BuehlerHugThaele,GKT22,PorretBlanc}. These papers show that the visibility region $Z_{\gamma,0,d}$ may be unbounded, and that its behaviour undergoes a sharp phase transition as the intensity~$\gamma$ varies. More precisely, there exists the critical value
 \begin{equation}\label{eq:GammaCrit}
 \gamma_{\mathrm{crit}}:= \sqrt{\pi}(d-1)^2{\Gamma({d-1\over 2})\over\Gamma({d\over 2})}
 \end{equation}
 such that
 \begin{itemize}
 \item if $\gamma < \gamma_{\mathrm{crit}}$, then
 $Z_{\gamma,0,d}$ is unbounded with strictly positive probability,
 \item if $\gamma \geq \gamma_{\mathrm{crit}}$, then
 $Z_{\gamma,0,d}$ is almost surely bounded.
 \end{itemize}
 The critical case $\gamma = \gamma_{\mathrm{crit}}$ was treated in \cite{GKT22,PorretBlanc} for $d=2$ and in \cite{BuehlerGusakovaRecke} for general dimensions. Thus, for small intensities, there remains a positive chance of seeing arbitrarily far from the origin, whereas for large intensities the random hypersurfaces almost surely form a finite ``cocoon'' around~$o$. In the latter regime, the expected hyperbolic volume of the bounded visibility region has been computed in \cite[Theorem 7.1]{BuehlerHugThaele}, yielding an explicit expression in terms of the dimension~$d$ and the intensity parameter~$\gamma$.

In the present work we investigate the same questions for general
 $\lambda$-geodesic hyperplanes. One might expect the behaviour of the visibility region to depend sensitively on the parameter $\lambda$, since varying $\lambda$ interpolates between different geometric objects. Surprisingly, our results show that the situation is completely rigid with respect to~$\lambda$. The critical intensity at which the transition from unbounded to bounded
 visibility occurs is \emph{identical} for all $\lambda \in [0,1]$, and in
 the bounded regime the expected volume of the visibility region
 $Z_{\gamma,\lambda,d}$ agrees exactly with the expression known from the case $\lambda = 0$. In other words, neither the existence of infinite sightliness nor the size of the bounded visibility region depends in any way on the geometric nature of the obstructing hypersurfaces. Our main findings are summarized in the following result.

\begin{prop}\label{prop:gamma=gamma_crit}
 We assume the same set-up as in Theorem \ref{thm:CoveringGenerald}. For $d=2$ and $\gamma=\gamma_{\mathrm{crit}}$ it holds
 $\displaystyle\mathbb{P}[\limsup_{n\to\infty} \mathcal{S}^\circ(u_n, \varphi(r_n)) = \mathbb{S}^{1}] = 1$. Thus, the visibility set $Z_{\gamma_\mathrm{crit},\lambda,2}$ in \eqref{eq:DefVisibilitySet} is almost surely bounded.
 \end{prop}
 \begin{proof}
 As before, let $0<r_1<r_2<\dots$ be such that they form an inhomogeneous Poisson point process on $(0,1)$ with intensity function $f$ defined in \eqref{eq:intensity_r}. Then, in the Poincar\'e disc model a $\lambda$-geodesic hyperplane $H(r_n,u)$ of distance $r_n$ to $o$ in direction $u\in\mathbb{S}^1$ covers an arc of length $2\arccos(\varphi(r_n))$ for $n\in\mathbb{N}$ (see Figure \ref{fig: configuration_caps}). Now, by \cite[Equation (1)]{Shepp}, Proposition \ref{prop:gamma=gamma_crit} follows once we have shown that
 $$\sum_{n=1}^{\infty}\frac{1}{n^2}e^{\ell_1+\ldots+\ell_n}=\infty,$$
 where $2\pi\ell_i$ is the arc length covered by $r_i$, i.e.\ $\ell_i=\frac{1}{\pi}\arccos(\varphi(r_i))$ for $i\in\mathbb{N}$. Combining \eqref{eq: 1-phi(rn)_asymptotics}, $\gamma_{\mathrm{crit}}=\pi$ and the refined $d=2$ asymptotics from Lemma~\ref{lemma:asymptotics_rn} we can sharpen the estimate for the arc lengths $\ell_i$.
 Indeed, for $d=2$ and $\gamma=\gamma_{\mathrm{crit}}=\pi$, Lemma~\ref{lemma:asymptotics_rn} yields almost surely
 \begin{equation*}
 1-r_i=\frac{\pi(1+\lambda)}{i} +O\left(\sqrt{\frac{\log\log(i)}{i^3}}\right),\qquad i\to\infty.
 \end{equation*}
 Together with Lemma~\ref{lemma:asymptotics_varphi} this implies almost surely
 \begin{equation*}
 \delta_i:=1-\varphi(r_i)
 =\frac{(1-r_i)^2}{2(1+\lambda)^2}+O\big((1-r_i)^3\big)
 =\frac{\pi^2}{2i^2}+O\left(\sqrt{\frac{\log\log(i)}{i^5}}\right),
 \qquad i\to\infty.
 \end{equation*}
 Using the Taylor expansions $\arccos(1-x)=\sqrt{2x}+O(x^{3/2})$ as $x\downarrow 0$ and $\sqrt{1+x}=1+\frac{x}{2}+O(x^2)$ as $x\downarrow 0$, we obtain almost surely
 \[
 \arccos(\varphi(r_i))
 =\arccos(1-\delta_i)
 =\sqrt{2\delta_i}+O(\delta_i^{3/2})
 =\frac{\pi}{i}+O\left(\sqrt{\frac{\log\log(i)}{i^3}}\right),
 \qquad i\to\infty.
 \]
 Consequently,
 \begin{equation*}
 \ell_i=\frac{1}{\pi}\arccos(\varphi(r_i))
 =\frac{1}{i}+O\left(\sqrt{\frac{\log\log(i)}{i^3}}\right),
 \qquad i\to\infty,
 \end{equation*}
 almost surely. Hence, as $\sum_{k=1}^\infty \sqrt{\frac{\log\log(k)}{k^3}}<\infty,$ this
 implies that there exists $c\in\mathbb{R}$ such that
 \[
 \sum_{k=1}^{n}\ell_k
 \ge \sum_{k=1}^{n}\frac{1}{k}+c
 \ge \ln(n)+c,
 \]
 where we used $\sum_{k=1}^n \frac1k \ge \ln(n)$.
 Thus,
 \begin{align*}
 \sum_{n=1}^{\infty}\frac{1}{n^2}e^{\ell_1+\ldots+\ell_n}
 &\ge \sum_{n=1}^{\infty}\frac{1}{n^2}e^{\ln(n)+c}
 = \sum_{n=1}^\infty \frac{e^{c}}{n}
 =\infty,
 \end{align*}
 which completes the proof.
 \end{proof}

\begin{proof}[Proof of Theorem \ref{thm:MainIntro} (b)]
 Fix $u\in\mathbb{S}^{d-1}$ and let $\ell(h)$ be a geodesic segment of length $h>0$ starting at $o$ in an arbitrary direction. Then by Corollary \ref{coro:mu(F_[o,l])} we have
 \begin{align*}
 \p(s_u(\eta_{\gamma,\lambda})>h)&=\p(\eta_{\gamma,\lambda}([\ell(h)]_\lambda)=0)
 =\exp\big({-\mu_{\gamma,\lambda}([\ell(h)]_\lambda)}\big)=\exp\Big({-\gamma\,\frac{\Gamma(\frac{d}{2})}{2\sqrt{\pi}\,\Gamma(\frac{d+1}{2})}\,h}\Big),
 \end{align*}
 independently of $\lambda$.
 Hence, the random variable $s_u(\eta_{\gamma,\lambda})$ is exponentially distributed with parameter $\gamma\,\frac{\Gamma(\frac{d}{2})}{2\sqrt{\pi}\,\Gamma(\frac{d+1}{2})}$ and by the polar decomposition of hyperbolic space we have
 \begin{align*}
 \mathbb{E}{\rm vol}_d(Z_{\gamma,\lambda,d})
 &={2\pi^{d/2}\over\Gamma({d\over 2})}\E\Bigg[\int_{\mathbb{S}^{d-1}}\int_0^{s_u(\eta_{\gamma,\lambda})}\sinh^{d-1}(s)\;\mathrm{d}s\sigma_d(\mathrm{d}u)\Bigg]\\
 &={2\pi^{d/2}\over\Gamma({d\over 2})}\int_{\mathbb{S}^{d-1}}\int_0^{\infty}\p(s_u(\eta_{\gamma,\lambda})\geq s)\sinh^{d-1}(s)\;\mathrm{d}s\sigma_d(\mathrm{d}u)\Bigg]\\
 &={2\pi^{d/2}\over\Gamma({d\over 2})}\int_0^\infty \exp\Big({-\gamma\,\frac{\Gamma(\frac{d}{2})}{2\sqrt{\pi}\,\Gamma(\frac{d+1}{2})}\,s}\Big)\sinh^{d-1}(s)\;\mathrm{d}s,
 \end{align*}
 where $\sigma_d$ denotes the normalized spherical Lebesgue measure on $\mathbb{S}^{d-1}$.
 According to Identity 3.541.1 in \cite{GradRysz} (see also \cite[Equation (6.9)]{BuehlerHugThaele}) it holds that
 \begin{equation}\label{eq:SinhExpIntegral}
 \int_0^\infty\sinh^{d-1}(s)e^{-as}\,\mathrm{d}s=\frac{(d-1)!}{2^d}\frac{\Gamma(\frac{a-d+1}{2})}{\Gamma(\frac{a+d+1}{2})},
 \end{equation}
 whenever $a>d-1$. Thus, the expected volume is finite if and only if $\gamma\,\frac{\Gamma(\frac{d}{2})}{2\sqrt{\pi}\,\Gamma(\frac{d+1}{2})}>d-1$ or $\gamma>\frac{2(d-1)\sqrt{\pi}\Gamma(\frac{d+1}{2})}{\Gamma(\frac{d}{2})}=(d-1)^2\sqrt{\pi}\frac{\Gamma(\frac{d-1}{2})}{\Gamma(\frac{d}{2})}=\gamma_{\mathrm{crit}}$ and, in this case,
 $$
 \mathbb{E}{\rm vol}_d(Z_{\gamma,\lambda,d})= \mathbb{E}{\rm vol}_d(Z_{\gamma,0,d}) = {2^{1-d}\pi^{d/2}(d-1)!\over \Gamma({d\over 2})}{\Gamma\big({\gamma_d^\ast-d+1\over 2}\big)\over\Gamma\big({\gamma_d^\ast+d+1\over 2}\big)}\quad\text{with}\quad \gamma_d^\ast := \gamma\,{\Gamma({d\over 2})\over2\sqrt{\pi}\Gamma({d+1\over 2})}.
 $$
 To arrive at the desired expression, we simplify the factor ${2^{1-d}\pi^{d/2}(d-1)!\over \Gamma({d\over 2})}$ using the Legendre duplication formula.
 It follows that
 \begin{align*}
 {2^{1-d}\pi^{d/2}(d-1)!\over \Gamma({d\over 2})}&=2^{1-d}\pi^{d/2}(d-1)! \frac{\Gamma(\frac{d+1}{2})}{2^{1-d}\sqrt{\pi}\Gamma(d)}= \pi^{\frac{d-1}{2}} \Gamma\left(\frac{d+1}{2}\right).
 \end{align*}
 Plugging this into the expression for $\mathbb{E}{\rm vol}_d(Z_{\gamma,\lambda,d})$ completes the proof.
\end{proof}

\begin{rema}\rm
 The notion of $\lambda$-geodesic hyperplanes as totally umbilical hypersurfaces of $\mathbb{H}^d$ can be extended to the case $\lambda>1$. In that situation, a $\lambda$-geodesic 'hyperplane' is a hyperbolic sphere of radius
 \[
 R_\lambda : = \operatorname{artanh}\Big(\frac{1}{\lambda}\Big).
 \]
 Hence, in this regime the visibility question in presence of a Poisson process of such spheres reduces to that in a Boolean model of balls with fixed radius $R_\lambda$, which in turn has been studied in \cite{L96}. As for $\lambda\in[0,1]$, there exists a critical intensity $\gamma_{B,\mathrm{crit}}>0$ such that the visibility region is unbounded with positive probability if $\gamma<\gamma_{B,\mathrm{crit}}$ and almost surely bounded for $\gamma>\gamma_{B,\mathrm{crit}}$. However, a key difference to the case $\lambda\in[0,1]$ is that, for $\lambda>1$, this critical intensity depends on $\lambda$. By \cite[p.\ 447]{L96} and \cite[Remark 6.5]{BuehlerHugThaele} it is given by $$\gamma_{B,\mathrm{crit}}=\frac{(d-1)\Gamma(\frac{d+1}{2})}{\pi^{(d-1)/2}\sinh(R_\lambda)}=\frac{(d-1)\Gamma(\frac{d+1}{2})}{\pi^{(d-1)/2}(\lambda^2-1)^{(d-1)/2}}.$$ In the bounded phase, the expected visible volume is derived in \cite[Theorem 6.2]{BuehlerHugThaele}. It can be expressed in the same way as the mean visible volume in \eqref{eq:MeanVolume} by replacing $\gamma_d^*$ by $v_d^*=\gamma\frac{\pi^{(d-1)/2}}{\Gamma(\frac{d+1}{2})}\sinh(R_\lambda)=\gamma\frac{\pi^{(d-1)/2}}{\Gamma(\frac{d+1}{2})(\lambda^2-1)^{(d-1)/2}}$.
\end{rema}

The proofs of the results in (a) and (b) rely on two complementary ingredients.
	For the existence of a phase transition, we adapt a covering criterion due to Hoffmann--J\o rgensen \cite{H73}. This method parallels the approach in \cite{GKT22} used in the case $\lambda = 0$, and it  shows that the critical threshold does not depend on the geometric nature of the
	hyperplanes, i.e., on $\lambda$. The concrete computations rely on a subtle cancellation, which eliminates the $\lambda$-dependence from the relevant density terms.
	The computation of the expected volume in the bounded phase reduces to an integral-geometric question: one must determine the measure $\nu_\lambda^o$ of $\lambda$--geodesic hyperplanes that intersect a fixed geodesic segment of length~$h>0$.
	For $\lambda = 0$, this quantity can be determined using the classical Crofton formula in hyperbolic space. In fact, if $x_h$ is a point with hyperbolic distance $h$ from $o$ we have
	\begin{equation}\label{eq:IndicatorToEulerChar}
	\int_{{\rm Hyp}_0^o}{\bf 1}\{H\cap[o,x_h]\neq\varnothing\}\,\nu_0^o(\mathrm{d} H) = \int_{{\rm Hyp}_0^o}\chi(H\cap[o,x_h])\,\nu_0^o(\mathrm{d} H) = {\Gamma({d\over 2})\over 2\sqrt{\pi}\Gamma({d+1\over 2})}\,h,
	\end{equation}
	where $\chi$ denotes the Euler characteristic, see \cite[Section 7]{BuehlerHugThaele}.  However, the Crofton argument breaks down for $\lambda>0$, since a
	$\lambda$--geodesic hyperplane may intersect a geodesic segment either once or twice. This implies that for all such $\lambda$ the indicator function cannot be replaced by the Euler characteristic and it seems that no direct integral-geometric relation is available.  We overcome this  by introducing an explicit parametrization of $\lambda$--geodesic hyperplanes and carrying out the computation directly.  The resulting expression turns out to be independent of~$\lambda$ and again proportional to the length~$h$. This invariance is a surprising integral-geometric fact of independent interest and is ultimately responsible for the $\lambda$--independence of the expected volume as described in Theorem \ref{thm:MainIntro}.

	Our findings are placed within the broader context of stochastic geometry in hyperbolic space, a field that has seen significant recent activity. This area encompasses a rich variety of random structures adapted to negative curvature. A selection of recent results includes the asymptotic geometry of random polytopes \cite{BesauThaele,FodorGruenfelder}, the investigation of the Boolean model \cite{BJST,BuehlerHugThaele,HugLastSchulte,Tykesson} and the study of hyperbolic random graphs, which connects geometric probability with network science \cite{BlaesiusETal,FountulakisEtal,KrioukovEtal}. The investigation of Poisson point processes and their associated Voronoi tessellations remained a central theme, with foundational works including \cite{Isokawa}. A particular focus has recently developed around ideal Poisson--Voronoi tessellations and their applications, where the generating points reside on the boundary at infinity \cite{Achille,AchilleEtal,BudzinskiEtal,Fraczyk}. Our work complements this diverse landscape by focusing on the geometry induced by Poisson processes of $\lambda$-geodesic hyperplanes, uncovering a surprising universality in visibility properties.

	\medspace

	The paper is organized as follows. Section \ref{sec:Background} collects the necessary background on hyperbolic geometry to make the paper self-contained. In Section \ref{sec:Visibility}, we establish the visibility phase transition, proving Theorem \ref{thm:MainIntro}(a). Section \ref{sec:IntegralGeometrySegment} is dedicated to the core integral-geometric mechanism: we prove that the Crofton-type identity \eqref{eq:IndicatorToEulerChar} holds universally for all $\lambda\in[0,1]$. Finally, we apply this result in Section \ref{sec:ExpectedVolume} to derive the mean visible volume formula in Theorem \ref{thm:MainIntro}(b).

The introduction states that the proofs of Theorem~\ref{thm:MainIntro}(a) and (b) “rely on two complementary ingredients.”

1. **Phase transition / (a):** “For the existence of a phase transition, we adapt a covering criterion due to Hoffmann--J\o rgensen \cite{H73}.” This “parallels the approach in \cite{GKT22} used in the case $\lambda=0$,” and “shows that the critical threshold does not depend on the geometric nature of the hyperplanes, i.e., on $\lambda$.” The needed computations “rely on a subtle cancellation, which eliminates the $\lambda$-dependence from the relevant density terms.”

2. **Expected volume / (b):** “The computation of the expected volume in the bounded phase reduces to an integral-geometric question: one must determine the measure $\nu_\lambda^o$ of $\lambda$--geodesic hyperplanes that intersect a fixed geodesic segment of length~$h>0$.” For $\lambda=0$ this is obtained via “the classical Crofton formula,” yielding \eqref{eq:IndicatorToEulerChar}. For $\lambda>0$ “the Crofton argument breaks down … since a $\lambda$--geodesic hyperplane may intersect a geodesic segment either once or twice,” so “the indicator function cannot be replaced by the Euler characteristic” and “no direct integral-geometric relation is available.” They “overcome this by introducing an explicit parametrization of $\lambda$--geodesic hyperplanes and carrying out the computation directly.” The “resulting expression turns out to be independent of~$\lambda$ and again proportional to the length~$h$,” and “is ultimately responsible for the $\lambda$--independence of the expected volume as described in Theorem~\ref{thm:MainIntro}.”

Finally, they indicate the proof structure: Section~\ref{sec:Visibility} proves Theorem~\ref{thm:MainIntro}(a) via the phase transition argument; Section~\ref{sec:IntegralGeometrySegment} proves the universal Crofton-type identity; Section~\ref{sec:ExpectedVolume} applies it to derive Theorem~\ref{thm:MainIntro}(b).

The introduction states that the proofs establishing the main theorem (parts (a) and (b)) “rely on two complementary ingredients.”

1. **Phase transition / (a):** “For the existence of a phase transition, we adapt a covering criterion due to Hoffmann--J\o rgensen \cite{H73}.” This “parallels the approach in \cite{GKT22} used in the case $\lambda=0$,” and “shows that the critical threshold does not depend on the geometric nature of the hyperplanes, i.e., on $\lambda$.” The needed computations “rely on a subtle cancellation, which eliminates the $\lambda$-dependence from the relevant density terms.”

2. **Expected volume / (b):** “The computation of the expected volume in the bounded phase reduces to an integral-geometric question: one must determine the measure $\nu_\lambda^o$ of $\lambda$--geodesic hyperplanes that intersect a fixed geodesic segment of length~$h>0$.” For $\lambda=0$ this is obtained via “the classical Crofton formula,” yielding
\begin{equation}\label{eq:IndicatorToEulerChar}
	\int_{{\rm Hyp}_0^o}{\bf 1}\{H\cap[o,x_h]\neq\varnothing\}\,\nu_0^o(\dint H) = \int_{{\rm Hyp}_0^o}\chi(H\cap[o,x_h])\,\nu_0^o(\dint H) = {\Gamma({d\over 2})\over 2\sqrt{\pi}\Gamma({d+1\over 2})}\,h,
	\end{equation}
For $\lambda>0$ “the Crofton argument breaks down … since a $\lambda$--geodesic hyperplane may intersect a geodesic segment either once or twice,” so “the indicator function cannot be replaced by the Euler characteristic” and “no direct integral-geometric relation is available.” They “overcome this by introducing an explicit parametrization of $\lambda$--geodesic hyperplanes and carrying out the computation directly.” The “resulting expression turns out to be independent of~$\lambda$ and again proportional to the length~$h$,” and “is ultimately responsible for the $\lambda$--independence of the expected volume as described in establishing the main theorem.”

Finally, they indicate the proof structure: Section~\ref{sec:Visibility} proves part (a) of the main theorem via the phase transition argument; Section~\ref{sec:IntegralGeometrySegment} proves the universal Crofton-type identity; Section~\ref{sec:ExpectedVolume} applies it to derive part (b) of the main theorem.

\label{thm:MainIntro}
	Fix $d\geq 2$, $\gamma>0$ and $0\leq\lambda\leq 1$. Consider the visibility region $Z_{\gamma,\lambda,d}$ of $o$ in a Poisson process of $\lambda$-geodesic hyperplanes in $\mathbb{H}^d$ with intensity measure $\gamma\nu_\lambda^o$.
	\begin{itemize}
	\item[(a)] Let $\gamma_{\mathrm{crit}}$ be as in
	\begin{equation}\label{eq:GammaCrit}
	\gamma_{\mathrm{crit}}:= \sqrt{\pi}(d-1)^2{\Gamma({d-1\over 2})\over\Gamma({d\over 2})}
	\end{equation}
	If $\gamma<\gamma_{\mathrm{crit}}$, then $Z_{\gamma,\lambda,d}$ is unbounded with strictly positive probability, whereas if $\gamma>\gamma_{\mathrm{crit}}$, then $Z_{\gamma,\lambda,d}$ is almost surely bounded. Moreover, if $d=2$, then $Z_{\gamma_{\mathrm{crit}},\lambda,2}$ is almost surely bounded.

	\item[(b)] If $\gamma>\gamma_{\mathrm{crit}}$ then
	\begin{equation}\label{eq:MeanVolume}
		\mathbb{E}{\rm vol}_d(Z_{\gamma,\lambda,d})=\mathbb{E}{\rm vol}_d(Z_{\gamma,0,d}) = \pi^{\frac{d-1}{2}} \Gamma\left(\frac{d+1}{2}\right){\Gamma\big({\gamma_d^\ast-d+1\over 2}\big)\over\Gamma\big({\gamma_d^\ast+d+1\over 2}\big)}
	\end{equation}
	with $\gamma_d^\ast := \gamma\,{\Gamma({d\over 2})\over 2\sqrt{\pi}\Gamma({d+1\over 2})}$. On the other hand, $\mathbb{E}{\rm vol}_d(Z_{\gamma,\lambda,d})=+\infty$ for all $\gamma\leq\gamma_{\mathrm{crit}}$.
	\end{itemize},

Question:
Let $d\ge 2$, $\gamma>0$, and $0\le \lambda\le 1$. In hyperbolic space $\mathbb H^d$, fix an origin $o$, let $\eta_{\gamma,\lambda}$ be a Poisson point process on the space of $\lambda$-geodesic hyperplanes with intensity measure $\gamma\nu_\lambda^o$, and define the visibility region of $o$ by
\[
Z_{\gamma,\lambda,d}:=\{x\in \mathbb H^d:[o,x]\cap H=\varnothing\text{ for all }H\in \eta_{\gamma,\lambda}\},
\]
where $[o,x]$ is the geodesic segment from $o$ to $x$. Which statement about the boundedness and expected hyperbolic volume ${\rm vol}_d(Z_{\gamma,\lambda,d})$ holds?

If
\[
\gamma_{\mathrm{crit}}:=\sqrt{\pi}(d-1)^2\frac{\Gamma\big(\frac{d-1}{2}\big)}{\Gamma\big(\frac d2\big)},
\]
then: (i) if $\gamma<\gamma_{\mathrm{crit}}$, the visibility region $Z_{\gamma,\lambda,d}$ is unbounded with strictly positive probability; if $\gamma>\gamma_{\mathrm{crit}}$, then $Z_{\gamma,\lambda,d}$ is almost surely bounded; and in the critical case, when $d=2$, $Z_{\gamma_{\mathrm{crit}},\lambda,2}$ is almost surely bounded. (ii) If $\gamma>\gamma_{\mathrm{crit}}$, then
\[
\mathbb E\,\mathrm{vol}_d(Z_{\gamma,\lambda,d})=\mathbb E\,\mathrm{vol}_d(Z_{\gamma,0,d})
=\pi^{\frac{d-1}{2}}\Gamma\!\left(\frac{d+1}{2}\right)
\frac{\Gamma\!\left(\frac{\gamma_d^\ast-d+1}{2}\right)}{\Gamma\!\left(\frac{\gamma_d^\ast+d+1}{2}\right)},
\]
where
\[
\gamma_d^\ast:=\gamma\,\frac{\Gamma\big(\frac d2\big)}{2\sqrt{\pi}\,\Gamma\big(\frac{d+1}{2}\big)}.
\]
On the other hand, $\mathbb E\,\mathrm{vol}_d(Z_{\gamma,\lambda,d})=+\infty$ for all $\gamma\le \gamma_{\mathrm{crit}}$.

Choices:
B.
If
\[
\gamma_{\mathrm{crit}}:=\sqrt{\pi}(d-1)^2\frac{\Gamma\big(\frac{d-1}{2}\big)}{\Gamma\big(\frac d2\big)},
\]
then: (i) if $\gamma<\gamma_{\mathrm{crit}}$, the visibility region $Z_{\gamma,\lambda,d}$ is unbounded with strictly positive probability; if $\gamma\ge \gamma_{\mathrm{crit}}$, then $Z_{\gamma,\lambda,d}$ is almost surely bounded. (ii) If $\gamma\ge \gamma_{\mathrm{crit}}$, then
\[
\mathbb E\,\mathrm{vol}_d(Z_{\gamma,\lambda,d})=\mathbb E\,\mathrm{vol}_d(Z_{\gamma,0,d})
=\pi^{\frac{d-1}{2}}\Gamma\!\left(\frac{d+1}{2}\right)
\frac{\Gamma\!\left(\frac{\gamma_d^\ast-d+1}{2}\right)}{\Gamma\!\left(\frac{\gamma_d^\ast+d+1}{2}\right)},
\]
where
\[
\gamma_d^\ast:=\gamma\,\frac{\Gamma\big(\frac d2\big)}{2\sqrt{\pi}\,\Gamma\big(\frac{d+1}{2}\big)}.
\]
On the other hand, $\mathbb E\,\mathrm{vol}_d(Z_{\gamma,\lambda,d})=+\infty$ for all $\gamma< \gamma_{\mathrm{crit}}$.
C.
If
\[
\gamma_{\mathrm{crit}}:=\sqrt{\pi}(d-1)^2\frac{\Gamma\big(\frac{d-1}{2}\big)}{\Gamma\big(\frac d2\big)},
\]
then: (i) if $\gamma<\gamma_{\mathrm{crit}}$, the visibility region $Z_{\gamma,\lambda,d}$ is unbounded with strictly positive probability, while if $\gamma>\gamma_{\mathrm{crit}}$, then $Z_{\gamma,\lambda,d}$ is almost surely bounded. (ii) If $\gamma>\gamma_{\mathrm{crit}}$, then $\mathbb E\,\mathrm{vol}_d(Z_{\gamma,\lambda,d})<\infty$, whereas $\mathbb E\,\mathrm{vol}_d(Z_{\gamma,\lambda,d})=+\infty$ for all $\gamma\le \gamma_{\mathrm{crit}}$.
D.
If
\[
\gamma_{\mathrm{crit}}(\lambda):=\sqrt{\pi}(d-1)^2\frac{\Gamma\big(\frac{d-1}{2}\big)}{\Gamma\big(\frac d2\big)}(1+\lambda),
\]
then: (i) if $\gamma<\gamma_{\mathrm{crit}}(\lambda)$, the visibility region $Z_{\gamma,\lambda,d}$ is unbounded with strictly positive probability; if $\gamma>\gamma_{\mathrm{crit}}(\lambda)$, then $Z_{\gamma,\lambda,d}$ is almost surely bounded; and in the critical case, when $d=2$, $Z_{\gamma_{\mathrm{crit}}(\lambda),\lambda,2}$ is almost surely bounded. (ii) If $\gamma>\gamma_{\mathrm{crit}}(\lambda)$, then
\[
\mathbb E\,\mathrm{vol}_d(Z_{\gamma,\lambda,d})
=\pi^{\frac{d-1}{2}}\Gamma\!\left(\frac{d+1}{2}\right)
\frac{\Gamma\!\left(\frac{\widetilde\gamma_{d,\lambda}-d+1}{2}\right)}{\Gamma\!\left(\frac{\widetilde\gamma_{d,\lambda}+d+1}{2}\right)},
\]
where
\[
\widetilde\gamma_{d,\lambda}:=\gamma\,\frac{\Gamma\big(\frac d2\big)}{2\sqrt{\pi}\,\Gamma\big(\frac{d+1}{2}\big)}(1+\lambda).
\]
On the other hand, $\mathbb E\,\mathrm{vol}_d(Z_{\gamma,\lambda,d})=+\infty$ for all $\gamma\le \gamma_{\mathrm{crit}}(\lambda)$.
E.
If
\[
\gamma_{\mathrm{crit}}:=\sqrt{\pi}(d-1)^2\frac{\Gamma\big(\frac{d-1}{2}\big)}{\Gamma\big(\frac d2\big)},
\]
then: (i) if $\gamma<\gamma_{\mathrm{crit}}$, the visibility region $Z_{\gamma,\lambda,d}$ is unbounded with strictly positive probability; if $\gamma>\gamma_{\mathrm{crit}}$, then $Z_{\gamma,\lambda,d}$ is almost surely bounded; and in the critical case, when $d=2$, $Z_{\gamma_{\mathrm{crit}},\lambda,2}$ is almost surely bounded. (ii) If $\gamma>\gamma_{\mathrm{crit}}$, then
\[
\mathbb E\,\mathrm{vol}_d(Z_{\gamma,\lambda,d})=\mathbb E\,\mathrm{vol}_d(Z_{\gamma,0,d})
=\pi^{\frac{d-1}{2}}\Gamma\!\left(\frac{d+1}{2}\right)
\frac{\Gamma\!\left(\frac{\gamma_d^\ast-d+1}{2}\right)}{\Gamma\!\left(\frac{\gamma_d^\ast+d+1}{2}\right)},
\]
where
\[
\gamma_d^\ast:=\gamma\,\frac{\Gamma\big(\frac d2\big)}{2\sqrt{\pi}\,\Gamma\big(\frac{d+1}{2}\big)}.
\]
Moreover, for every fixed $R>0$ there is a deterministic constant $c_R>0$, independent of $\lambda\in[0,1]$, such that whenever $\gamma>\gamma_{\mathrm{crit}}$,
\[
\mathbb P\big({\rm vol}_d(Z_{\gamma,\lambda,d})\le R\big)\ge 1-e^{-c_R\gamma}.
\]
On the other hand, $\mathbb E\,\mathrm{vol}_d(Z_{\gamma,\lambda,d})=+\infty$ for all $\gamma\le \gamma_{\mathrm{crit}}$.

strictness at the critical threshold for expected volume

dropped the explicit universal gamma-function formula and the equality with the case lambda=0

lambda-independence of the segment-intersection measure and critical threshold

added a uniform quantitative tail bound not provided by the theorem or sketch


Consider the following problem with nonlinear Neumann boundary condition
\begin{equation}\label{con}
\begin{cases}
u_t - \Delta u = 0, & (x,t)\in \Omega\times\mathbb{R}^+, \\
\partial_{\nu} u = \lambda g(u), & (x,t)\in \partial\Omega\times\mathbb{R}^+, \\
u(x,0)=u_0(x), & x\in \Omega,
\end{cases}
\end{equation}
where $\Omega \subset \mathbb{R}^2$ is a smooth, bounded, and convex domain,
$\nu$ denotes the outward unit normal vector to $\partial\Omega$,
$\lambda$ is a positive parameter, and $g$ is a bistable nonlinearity,
here taken in its prototypical form $g(u)=u-u^3$.

A non-constant stationary solution of \eqref{con} that is linearly stable will be called a {\it pattern}.

In order to state our main result, we recall that the \emph{inradius} of a set
$\Omega \subset \mathbb{R}^2$ is defined by
\[
\mathrm{inr}(\Omega) := \sup \left\{ r > 0 \; ; \; \exists\, x \in \Omega
\ \text{such that} \ B_r(x) \subset \Omega \right\},
\]
where $B_r(x)$ denotes the open ball of radius $r$ centered at $x$.

The \emph{diameter} of $\Omega$ is defined by
\[
D(\Omega) := \sup \left\{ |x-y| \; ; \; x,y \in \Omega \right\}.
\]

The main result of this work is stated as follows.

\abstract{We consider the heat equation in a smooth bounded convex domain
$\Omega \subset \mathbb{R}^2$ with nonlinear Neumann boundary condition
$\partial_\nu u = \lambda (u - u^3)$.
Stable non-constant stationary solutions do not exist when $\Omega$ is a ball.
We show that this behavior is not a consequence of convexity alone.
More precisely, if the inradius of $\Omega$ is fixed and its diameter is sufficiently large, then there exists $\lambda>0$ for which the problem admits such a solution.
The result reveals a geometric mechanism for the emergence of stable non-constant stationary solutions in elongated convex domains.}

Consider the following problem with nonlinear Neumann boundary condition
\begin{equation}\label{con}
\begin{cases}
u_t - \Delta u = 0, & (x,t)\in \Omega\times\mathbb{R}^+, \\
\partial_{\nu} u = \lambda g(u), & (x,t)\in \partial\Omega\times\mathbb{R}^+, \\
u(x,0)=u_0(x), & x\in \Omega,
\end{cases}
\end{equation}
where $\Omega \subset \mathbb{R}^2$ is a smooth, bounded, and convex domain,
$\nu$ denotes the outward unit normal vector to $\partial\Omega$,
$\lambda$ is a positive parameter, and $g$ is a bistable nonlinearity,
here taken in its prototypical form $g(u)=u-u^3$.

The main result of this work is stated as follows.

Theorem \ref{t1} reveals a purely geometric mechanism for pattern
formation. While the inradius remains fixed, increasing the diameter
forces the emergence of patterns for appropriate values
of $\lambda$. In other words, sufficiently elongated convex domains (see Figure \ref{fig1})
necessarily admit patterns. This phenomenon is independent of symmetry assumptions and depends
only on the convexity and aspect ratio of the domain.

\begin{lemma}\label{LFU}
Let $\Omega \subset \mathbb{R}^n$ be a bounded convex domain,
and denote $D=D(\Omega)$.
Then there exists a constant $C=C(n)>0$ such that,
for every $u\in H^1(\Omega)$,
\[
\|u-u_{\partial\Omega}\|_{L^2(\partial\Omega)}
\le
C D^{1/2}\|\nabla u\|_{L^2(\Omega)},
\]
where
\[
u_{\partial\Omega}
=
\frac{1}{|\partial\Omega|}
\int_{\partial\Omega} u\ d\sigma.
\]
\end{lemma}

\begin{theorem}\label{tmi}
Let $\Omega \subset \mathbb{R}^2$ be a bounded convex domain and again let
$
D := D(\Omega).
$
Then there exists a constant $c>0$ depending only on the dimension such that
\[
\mu_1(\Omega) \ge \frac{c}{D}.
\]
\end{theorem}

\begin{lemma}\label{le1}
Let $\Omega \subset \mathbb{R}^n$ ($n \ge 2$) be a smooth bounded domain,
$\Omega_l$ and $\Omega_r$ two disjoint subdomains of $\Omega$
with smooth boundaries, and
$S_j = \partial\Omega \cap \partial\Omega_j$,
$\mathcal{H}^{n-1}(S_j) > 0$ $(j=l,r)$.
For $p > n$, we define the set
\begin{equation}\label{L}
\Lambda(\Omega)
=
\left\{\begin{aligned}
v \in W^{1,p}(\Omega) :\quad
&-1 \le v(x) \le 1,\; x \in \overline{\Omega},\\
&\int_{S_l} v \, d\sigma < 0,\qquad
\int_{S_r} v \, d\sigma > 0,\\
&E(v) < \varepsilon_0 - G(1)\mathcal{H}^{n-1}(\partial\Omega)
\end{aligned}\right\},
\end{equation}
where
\[
\varepsilon_0
=
G(1)
\min
\left\{\begin{aligned}
&\mathcal{H}^{n-1}(S_l)\min\{\lambda,\,\mu_1(\Omega_l)\},\\
&\mathcal{H}^{n-1}(S_r)\min\{\lambda,\,\mu_1(\Omega_r)\}
\end{aligned}\right\},
\]
and $\mu_1(\Omega_j)$ is the first positive eigenvalue of the Steklov
problem \ref{ST} defined in $\Omega_j$ $(j=l,r)$.

For convenience, in this section we denote a point in $\mathbb{R}^2$ by $(x,y)$. Our objective is to prove that if $\Omega \subset \mathbb{R}^2$ is smooth bounded convex domain
 with inradius $r$ and diameter $D$ (see Figure \ref{fig1}) large enough, then one can find $\lambda > 0$ such that
$
\Lambda(\Omega) \neq \emptyset.
$, where $\Lambda(\Omega)$ is the set defined in \eqref{L}.

\begin{figure}[ht]
\centering
\def\svgwidth{1\linewidth}
\input{fig3.pdf_tex}
\caption{Convex domain
$\Omega \subset \mathbb{R}^2$ with inradius $r$ and diameter $D$.}
\label{fig1}
\end{figure}

\begin{theorem}\label{t1} Let $\Omega \subset \mathbb{R}^2$ be a smooth bounded convex domain such that $\mathrm{inr}(\Omega)=r$ and $D(\Omega)=D$. Then if $D$ is  large enough, there exists $\lambda>0$ for which problem \eqref{con} admits a pattern. \end{theorem}

Theorem \ref{t1} reveals a purely geometric mechanism for pattern
formation. While the inradius remains fixed, increasing the diameter
forces the emergence of patterns for appropriate values
of $\lambda$. In other words, sufficiently elongated convex domains (see Figure \ref{fig1})
necessarily admit patterns. This phenomenon is independent of symmetry assumptions and depends
only on the convexity and aspect ratio of the domain.

From a biological perspective, Theorem~\ref{t1} suggests that
geometric elongation of the domain can counterbalance
the homogenizing action of diffusion in the interior.
When the diameter becomes sufficiently large compared to the inradius,
the global geometric structure weakens diffusive stabilization
and allows patterns to arise.

The relation between convexity and stability of non-constant stationary solutions
has been a central theme in the study of semilinear parabolic equations.
In the classical works of Casten--Holland \cite{CH} and Matano \cite{matano}, the nonlinearity
acts in the interior of the domain and homogeneous Neumann boundary
conditions are imposed. In that setting, it was shown that in convex
domains patterns cannot exist under very general assumptions on the
nonlinearity. These results reveal a strong rigidity
phenomenon induced by convexity.

In contrast, in the present work the nonlinearity acts through the
boundary condition. Theorem~\ref{t1} shows that, in this setting,
convexity by itself does not constitute a structural obstruction
to pattern formation.
Although it is known that no pattern can exist when $\Omega$ is a ball
(see \cite{NLM}), this reflects the special symmetry of the ball rather
than convexity per se.

To the best of our knowledge, no previous result establishes a direct
criterion for the existence or absence of patterns in terms of global
geometric quantities such as the inradius and the diameter. Conversely,
much of the existing literature connects the existence of patterns to
curvature effects or other local geometric properties of the boundary, for instance, see \cite{CH,SAE,FA,SN} and references therein.

It is worth mentioning that in \cite{CJ}, through a computer-assisted analysis
combined with bifurcation techniques, strong evidence was provided for
the existence of patterns for problem \eqref{con} in the case
$g(u)=u-u^3$, $\lambda>2,84083164$, and $\Omega$ equal to the unit square,
which is a convex planar domain. The authors further conjectured that the same phenomenon should persist
for smooth convex domains obtained by suitably rounding the corners of the square.

In addition to \cite{CJ}, a proof in the convex case was mentioned in \cite{CS,DP}
as part of work in preparation. To the best of our knowledge, a
complete proof has not yet appeared in the literature.

In the same framework of boundary reactions in a bidimensional domain, Del Pino {\it et al.} \cite{DP}
constructed solutions with multiple boundary transitions, addressing existence and qualitative properties, but not stability. The existence of patterns has been obtained under stronger geometric or structural assumptions.
In \cite{AG,C2,C3}, existence results were proved   in special classes of non-convex bounded domains, such as dumbbell-type and perfored domains.
Moreover, when the boundary nonlinearity depends explicitly on the spatial variable, namely $g=g(x,u)$, existence of patterns has been established in \cite{ARRIETA} and for problems with variable diffusivity, see \cite{ACM}.

The proof of Theorem \ref{t1} is based on the construction of a specific set, denoted by $\Lambda(\Omega)$, associated with the geometry of $\Omega$ and the positive flow generated by \eqref{con}. The underlying strategy ensures that, whenever $\Lambda(\Omega)$ is nonempty, it contains the desired pattern. Thus, the core of the argument consists in proving that $\Lambda(\Omega)\neq\emptyset$. This approach appears to have been introduced in \cite{matano} and has since been employed in different contexts, for instance in \cite{ar1, C2}, as well as in some subsequent works \cite{ACM}.

The present work is organized as follows. Section 2 is devoted to the preliminary results needed for the proof of Theorem \ref{t1}. In particular, in Theorem \ref{tmi} we establish a lower bound for the first positive Steklov eigenvalue, expressed in terms of the diameter of the domain and a constant depending only on the dimension. This estimate, which may be of independent interest in its own right, plays a key role in the proof of Theorem \ref{t1}, carried out in Section 3. Section 4 concludes the paper with some final remarks.

The proof of Theorem \ref{t1} is based on “the construction of a specific set, denoted by $\Lambda(\Omega)$, associated with the geometry of $\Omega$ and the positive flow generated by \eqref{con}.” The strategy is that “whenever $\Lambda(\Omega)$ is nonempty, it contains the desired pattern,” so “the core of the argument consists in proving that $\Lambda(\Omega)\neq\emptyset$.”

A key ingredient used in this program is that “in Theorem \ref{tmi} we establish a lower bound for the first positive Steklov eigenvalue, expressed in terms of the diameter of the domain and a constant depending only on the dimension,” and “this estimate … plays a key role in the proof of Theorem \ref{t1}.”

In establishing the main theorem, the proof is based on “the construction of a specific set, denoted by $\Lambda(\Omega)$, associated with the geometry of $\Omega$ and the positive flow generated by
\begin{equation}\label{con}
\begin{cases}\nu_t - \Delta u = 0, & (x,t)\in \Omega\times\mathbb{R}^+, \\
\partial_{\nu} u = \lambda g(u), & (x,t)\in \partial\Omega\times\mathbb{R}^+, \\
u(x,0)=u_0(x), & x\in \Omega,
\end{cases}
\end{equation}
.” The strategy is that “whenever $\Lambda(\Omega)$ is nonempty, it contains the desired pattern,” so “the core of the argument consists in proving that $\Lambda(\Omega)\neq\emptyset$.”

We first prove the following theorem.
\begin{theorem}\label{tmi}
Let $\Omega \subset \mathbb{R}^2$ be a bounded convex domain and again let
$
D := D(\Omega).
$
Then there exists a constant $c>0$ depending only on the dimension such that
\[
\mu_1(\Omega) \ge \frac{c}{D}.
\]
\end{theorem}
This estimate plays a key role in the proof of the main theorem.

\label{t1} Let $\Omega \subset \mathbb{R}^2$ be a smooth bounded convex domain such that $\mathrm{inr}(\Omega)=r$ and $D(\Omega)=D$. Then if $D$ is  large enough, there exists $\lambda>0$ for which the problem
\begin{equation}\label{con}
\begin{cases}\nu_t - \Delta u = 0, & (x,t)\in \Omega\times\mathbb{R}^+, \\
\partial_{\nu} u = \lambda g(u), & (x,t)\in \partial\Omega\times\mathbb{R}^+, \\
u(x,0)=u_0(x), & x\in \Omega,
\end{cases}
\end{equation}
admits a pattern.

Question:
Let \(\Omega \subset \mathbb{R}^2\) be a smooth bounded convex domain with inradius
\[
\mathrm{inr}(\Omega):=\sup\{r'>0:\exists x\in\Omega\text{ such that }B_{r'}(x)\subset \Omega\}=r
\]
and diameter
\[
D(\Omega):=\sup\{|x-y|:x,y\in\Omega\}=D.
\]
Consider the heat equation with nonlinear Neumann boundary condition
\[
\begin{cases}
u_t-\Delta u=0, & (x,t)\in \Omega\times\mathbb{R}^+,\\
\partial_\nu u=\lambda g(u), & (x,t)\in \partial\Omega\times\mathbb{R}^+,\\
u(x,0)=u_0(x), & x\in \Omega,
\end{cases}
\]
where \(\nu\) is the outward unit normal and \(g(u)=u-u^3\). A pattern means a non-constant stationary solution that is linearly stable. If \(D\) is sufficiently large, which conclusion about this problem is valid?

There exists \(\lambda>0\) such that the problem admits a pattern; equivalently, for some positive \(\lambda\), it has a linearly stable non-constant stationary solution.

Choices:
B. There exists a constant \(\lambda_0>0\), depending only on the dimension, such that for every \(\lambda\ge \lambda_0\) the problem admits a pattern whenever \(D\) is sufficiently large.
C. There exists \(\lambda>0\) such that the problem admits a stationary solution that is non-constant.
D. For every sufficiently large diameter \(D\), and for every smooth bounded convex domain \(\Omega\subset\mathbb{R}^2\) with \(\mathrm{inr}(\Omega)=r\) and \(D(\Omega)=D\), the problem admits a pattern for every \(\lambda>0\).
E. If \(D\) is sufficiently large, then for every \(\lambda>0\) there exists a linearly stable non-constant stationary solution of the problem.

existential dependence of the admissible parameter \(\lambda\)

linear stability requirement in the definition of pattern

quantifier order on \(\lambda\): existence replaced by universality

parameter-selection mechanism via nonemptiness of \(\Lambda(\Omega)\) for some suitable \(\lambda\)


Question:
Consider the heat equation with nonlinear Neumann boundary condition
\[
\begin{cases}
u_t-\Delta u=0, & (x,t)\in \Omega\times \mathbb{R}^+,\\
\partial_\nu u=\lambda g(u), & (x,t)\in \partial\Omega\times \mathbb{R}^+,\\
u(x,0)=u_0(x), & x\in \Omega,
\end{cases}
\]
on a smooth bounded convex domain \(\Omega\subset \mathbb{R}^2\), where \(\nu\) is the outward unit normal, \(\lambda>0\), and \(g(u)=u-u^3\). A \emph{pattern} means a non-constant stationary solution of this problem that is linearly stable. The inradius of \(\Omega\) is
\[
\mathrm{inr}(\Omega):=\sup\{r>0:\exists x\in\Omega\text{ such that }B_r(x)\subset \Omega\},
\]
and the diameter is
\[
D(\Omega):=\sup\{|x-y|:x,y\in\Omega\}.
\]
Let \(\Omega\subset\mathbb{R}^2\) be a smooth bounded convex domain with \(\mathrm{inr}(\Omega)=r\) and \(D(\Omega)=D\). If \(D\) is sufficiently large, which existence statement holds?

Choices:
B. For every \(\lambda>0\), if \(D\) is sufficiently large then the problem admits a pattern; equivalently, every positive choice of \(\lambda\) yields a linearly stable non-constant stationary solution on such elongated convex domains.
C. There exists \(\lambda>0\) such that the problem admits a non-constant stationary solution.
D. There exists \(\lambda_0>0\), depending only on the dimension and not on the particular domain \(\Omega\), such that whenever \(D\) is sufficiently large the problem admits a pattern for every \(\lambda\ge \lambda_0\).
E. If \(D\) is sufficiently large, then for every smooth bounded convex domain \(\Omega\subset\mathbb{R}^2\) with \(\mathrm{inr}(\Omega)=r\) and \(D(\Omega)=D\), the problem admits a unique pattern for some \(\lambda>0\).

domain-dependent parameter selection for \(\lambda\)

mere existence of a pattern upgraded to uniqueness


[Half-lift principle for sums of squares]\label{thm:half-lift-principle}
Let $d\ge 1$ and set
\[
  Q_d(x_1,\dots,x_d):=x_1^2+\cdots+x_d^2.
\]
For $n\ge 1$ and $a\in\mathbb{Z}$ let
\[
  N_{d,n}(a):=\#\{x\in (\mathbb{Z}/2^n\mathbb{Z})^d: Q_d(x)\equiv a\pmod{2^n}\}.
\]
Assume $n\ge 3$ and $4\nmid a$.
Then
\[
  N_{d,n+1}(a)=2^{d-1}\,N_{d,n}(a).
\]
Equivalently, among the residue classes modulo $2^n$ satisfying $Q_d\equiv a$, exactly half admit lifts to level $2^{n+1}$; moreover, whenever a class lifts, \emph{all} $2^d$ lifts in its fibre are solutions (cf.\ Lemma~\ref{lem:fibre-invariance}).

\begin{remark}\label{rem:half-lift-hensel}
For $d=3$ one can improve Step~2 by choosing a single involution direction on each parity class (as in the original ``three-squares'' half-lift lemma).
The minimal-index partition above is slightly less symmetric but works uniformly for all $d\ge 1$.
In particular, the growth factor $2^{d-1}$ coincides with the ``smooth'' Hensel factor $p^{d-1}$ specialised to $p=2$, even though the Jacobian of the bilinear quadratic form vanishes modulo $2$.
\end{remark}

\label{tab:block-densities}
\end{table}

\begin{corollary}[Dyadic vanishing thresholds in the $Q$- and $q$-normalisations]
\label{cor:dyadic-vanishing-thresholds}
Let $p=2$.
Let $L$ be an \emph{even}

\begin{equation}\label{eq:2xy}
  2xy\equiv s\pmod{p^m}.
\end{equation}

\begin{theorem}[Prime-uniform density for the hyperbolic plane]
\label{thm:prime-uniform-h0}
Let $p$ be a prime and let $e\ge 0$.
Let $H_0$ denote the rank-$2$ hyperbolic plane over $\Zp$ with bilinear form
\(\langle (x,y),(x',y')\rangle = xy'+x'y\).
On the scaled lattice $p^eH_0$ consider the quadratic form
\[
  q_e(v):=\tfrac12\langle v,v\rangle = p^e xy\in\Zp.
\]
For $m\ge 1$ and $t\in\Zp$ set
\[
  r^{(q)}_{m}(t;p^eH_0)
  :=\#\{(x,y)\in(\Z/p^m\Z)^2 : p^e xy\equiv t\!\!\pmod{p^m}\},
\]
and define the associated density (when the limit exists) by
\[
  \alpha_p^{(q)}(t;p^eH_0)
  :=\lim_{m\to\infty}p^{-m}\,r^{(q)}_{m}(t;p^eH_0).
\]
Let $t\in\Zp\setminus\{0\}$ and write $v:=v_p(t)$.
Then
\[
  \alpha_p^{(q)}(t;p^eH_0)=
  \begin{cases}
    0, & v<e,\\[2pt]
    (v-e+1)(p-1)p^{e-1}, & v\ge e.
  \end{cases}
\]
\smallskip
\noindent In particular, under the $q=\langle\cdot,\cdot\rangle/2$ normalisation, both the vanishing threshold and the density formula for the scaled hyperbolic plane are uniform for all primes $p$, including $p=2$.
\end{theorem}

\begin{lemma}[Counts for the anisotropic even plane]\label{lem:H1}
Let $m\ge 1$ and $t\in\Ztwo$.
Write $v:=v_2(t)$ with the convention $v=\infty$ if $t=0$.
Then:
\begin{altenumeratealpha}
\item If $t\ne 0$ and $v$ is even, then $A_m(t)=0$ for all $m>v$.
\item If $t\ne 0$ and $v$ is odd, then $A_m(t)=3\cdot 2^m$ for all $m>v$.
\item If $t\equiv 0\pmod{2^m}$, then $A_m(t)=4^{\lceil m/2\rceil}$.
\end{altenumeratealpha}
\end{lemma}

\begin{corollary}[Scaled hyperbolic block]\label{cor:density-hyp-scaled}
Let $e\ge 0$ and let $L=2^eH_0$, so that $Q(x,y)=2^{e+1}xy$.
Let $t\in\Ztwo\setminus\{0\}$ and set $v:=v_2(t)$.
If $v<e+1$ then $\alpha_2(t;L)=0$.
If $v\ge e+1$ then
\[
  \alpha_2(t;L)=2^e\,(v-e).
\]
\end{corollary}

\begin{definition}[Basic dyadic blocks]\label{def:dyadic-base-blocks}
For the explicit computations in this paper we single out the following quadratic $\Ztwo$-lattices; we refer to them as the \emph{basic dyadic blocks}:
\begin{altenumerate}
\item \textbf{Type~I (rank $1$).} The lattice $\langle 2^a u\rangle$ with Gram matrix $[2^a u]$, where $a\ge 0$ and $u\in\Ztwo^\times$.
\item \textbf{Type~II (rank $2$, even).} The lattice $2^a H_\varepsilon$ with Gram matrix $2^a H_\varepsilon$, where $a\ge 0$ and
\[
  H_0:=\begin{pmatrix}0&1\\1&0\end{pmatrix}\qquad\text{and}\qquad
  H_1:=\begin{pmatrix}2&1\\1&2\end{pmatrix}.
\]
\end{altenumerate}
Only the square class of the unit $u$ (equivalently $u\bmod 8$) will matter for solvability in the rank-one case.
\end{definition}

\begin{lemma}[Counting $2xy\equiv s\pmod{2^m}$ at $p=2$]\label{lem:2xy-2}
Let $m\ge 1$ and $s\in\Z/2^m\Z$.
Write $v:=v_2(s)$ with the convention $v=m$ if $s\equiv 0\pmod{2^m}$.
Then:
\begin{altenumerate}
\item If $v=0$ (i.e.\ $s$ is odd), then $M_m(s)=0$.
\item If $1\le v<m$, then $M_m(s)=v\cdot 2^m$.
\item If $v=m$ (i.e.\ $s\equiv 0\pmod{2^m}$), then $M_m(0)=(m+1)2^m$.
\end{altenumerate}
\end{lemma}

\begin{corollary}[Scaled hyperbolic plane at odd primes]\label{cor:density-hyp-scaled-odd}
Let $p$ be an odd prime and let $e\ge 0$.
Let $L=p^eH_0$, so that $Q(x,y)=2p^e xy$.
Let $t\in\Zp\setminus\{0\}$ and set $v:=v_p(t)$.
If $v<e$ then $\alpha_p(t;L)=0$.
If $v\ge e$, then
\[
  \alpha_p(t;L)=(v-e+1)(p-1)p^{e-1}.
\]
\end{corollary}

\begin{proposition}[Jordan decomposition]\label{prop:jordan}
Let $L$ be a nondegenerate quadratic lattice over $\Ztwo$.
Then there exists an orthogonal decomposition
\[
  L \;\cong\; \perp_{a\ge 0} 2^a L_a,
\]
with each $L_a$ a unimodular quadratic $\Ztwo$-lattice (possibly $0$).
Moreover, the multiset of integers $a$ together with the multiplicities $\mathrm{rank}(L_a)$ (equivalently, the Jordan \emph{scales} and \emph{ranks}) is uniquely determined by $L$.
\end{proposition}

\begin{lemma}[Counting $2xy\equiv s\pmod{p^m}$ at odd primes]\label{lem:2xy-odd}
Let $p$ be an odd prime, $m\ge 1$, and $s\in\Z/p^m\Z$.
Write $v:=v_p(s)$ with the convention $v=m$ if $s\equiv 0\pmod{p^m}$.
Then:
\begin{altenumerate}
\item If $0\le v<m$, then $M_{p,m}(s)=(v+1)(p-1)p^{m-1}$.
\item If $v=m$ (i.e.\ $s\equiv 0\pmod{p^m}$), then $M_{p,m}(0)=(m(p-1)+p)p^{m-1}$.
\end{altenumerate}
\end{lemma}

\begin{proposition}[Orthogonal-sum convolution]\label{prop:orthogonal-convolution}
Let $L$ and $M$ be quadratic $\Zp$-lattices, and equip $L\oplus M$ with the orthogonal direct-sum quadratic form $Q_{L\oplus M}(v,w):=Q_L(v)+Q_M(w)$.
Then for every $m\ge 1$ and $t\in\Zp$ one has the finite-level identity
\[
  r_m(t;L\oplus M)
  \,=\,
  \sum_{s\in \Z/p^m\Z} r_m(s;L)\,r_m(t-s;M).
\]
\end{proposition}

\label{sec:intro}

Local representation densities of quadratic lattices are the local factors in many global counting problems.
They appear in the Siegel series and Siegel's mass formula, and they control Fourier coefficients of theta series and Eisenstein series.
In concrete arithmetic applications one is often forced to evaluate these local factors for specific lattices, and the dyadic prime is typically the main obstacle.

At odd primes, once the relevant solution locus is smooth modulo $p$, Hensel lifting implies that reduction maps between solution sets have constant fibre size $p^{n-1}$, and the density stabilises quickly.
At the dyadic prime this mechanism can break down even for very simple lattices.
In the bilinear-lattice convention $Q(v)=\langle v,v\rangle$, the linearisation of $Q-t$ at a solution $\bar v$ is the functional $h\mapsto 2\langle \bar v,h\rangle$, which vanishes modulo~$2$.
Thus the standard Jacobian criterion yields no lifting information.

Every nondegenerate quadratic lattice over $\Z_2$ admits a Jordan decomposition into scaled unimodular constituents (Proposition~\ref{prop:jordan}).
Accordingly, explicit formulas for the basic dyadic blocks are a natural starting point for dyadic computations.
In this paper we determine the finite-level representation counts and local densities for the basic dyadic blocks of Definition~\ref{def:dyadic-base-blocks} (rank-$1$ Type~I blocks and the even planes $2^aH_\varepsilon$), together with the anisotropic ternary lattice $L_3$.

The main new input is a very simple involution argument, the \emph{half-lift}.
For diagonal sums of squares $Q_d=x_1^2+\cdots+x_d^2$ and a primitive target $a$ with $4\nmid a$, translation by $2^{n-1}u$ in a primitive direction produces a free involution on the solution set modulo $2^n$ whose orbits control lifts to level $2^{n+1}$.
In each orbit exactly one residue class lifts, and whenever a class lifts, all lifts in its fibre are solutions.
This yields the stable recursion $N_{d,n+1}(a)=2^{d-1}N_{d,n}(a)$ for $n\ge 3$ (Theorem~\ref{thm:half-lift-principle}).
For $d=3$ it leads to closed forms for the three-squares congruence counts and hence an explicit formula for $\alpha_2(t;L_3)$.

\smallskip
We fix the conventions for the finite-level representation counts $r_m(t;L)$ and the local density $\alpha_p(t;L)$ in \S\ref{sec:prelim}.
At $p=2$ and for even lattices there is a second common normalisation $q:=Q/2$; we work with $Q$ unless explicitly indicated, and Lemma~\ref{lem:q-dictionary} gives the precise conversion.

General formulas for the dyadic Siegel series are known, expressed for instance in terms of Gross--Keating invariants or group-scheme methods (see for example
\cite{grossKeating1993,choYamauchi2020,ganYu2000,ikedaKatsurada2018,kitaokaBook}).
The densities computed here agree with these formulas (cf.\ Yang's explicit expressions~\cite{yang1998}), but this serves only as an \emph{external consistency check} and is not used as an input to any argument.

At odd primes, the hyperbolic-plane computations of \S\ref{sec:hyperbolic} are classical; we include a self-contained counting argument to keep the exposition uniform and to make the prime-uniform statement of Theorem~\ref{thm:prime-uniform-h0} transparent.
The genuinely new feature of this paper is dyadic: the half-lift involution of Theorem~\ref{thm:half-lift-principle} and the resulting elementary derivation of the dyadic block densities without appealing to the full Siegel-series machinery.

All computations in the body of the paper are organised around the same elementary theme:
describe the fibres of the reduction maps $L/2^{m+1}L\to L/2^mL$ on the relevant solution sets.
Concretely,
\begin{altenumerate}
\item for the split even plane $H_0$ this reduces to valuation-stratified counting for the bilinear congruence $2xy\equiv t$ (see Lemmas~\ref{lem:2xy-odd} and~\ref{lem:2xy-2});
\item for the anisotropic plane $H_1$ it becomes a lifting recursion for a norm congruence in the unramified quadratic extension of~$\Q_2$ (see Lemma~\ref{lem:H1});
\item for the anisotropic ternary block $L_3$ one needs an additional involution (the half-lift) because the naive dyadic linearisation carries no information (see Theorem~\ref{thm:half-lift-principle}).
\end{altenumerate}

This paper focuses on the basic dyadic blocks (Definition~\ref{def:dyadic-base-blocks}) and the anisotropic ternary lattice $L_3$.
Once the basic block counts are known, representation counts for orthogonal sums reduce to finite convolutions (Proposition~\ref{prop:orthogonal-convolution}).
We do not attempt closed forms for general orthogonal sums.

\smallskip
\noindent
For quick reference, Table~\ref{tab:block-densities} in \S\ref{sec:densities} lists the final density values.
Readers primarily interested in the dyadic novelty may start with \S\ref{sec:half-lift}.
\subsection{Main results}

Let $p$ be a prime.
Section~\ref{sec:hyperbolic} treats the hyperbolic plane $H_0$ over $\Z_p$ for every $p$ and gives explicit formulas for the congruence counts $M_{p,m}(s)$ in~\eqref{eq:2xy}, together with their generating series and stable-range behaviour.
Section~\ref{sec:densities} records the resulting density formulas for $\alpha_p(t;H_0)$ and the basic dyadic blocks.
In particular, after passing to the standard normalisation $q=\langle\cdot,\cdot\rangle/2$ at $p=2$ (Lemma~\ref{lem:q-dictionary}), the density of $H_0$ and its scalings admits a single prime-uniform expression (Theorem~\ref{thm:prime-uniform-h0}).
The remaining sections specialise to $p=2$ and evaluate the basic dyadic blocks.

We therefore recall the dyadic Jordan decomposition (Proposition~\ref{prop:jordan}) and the basic dyadic blocks (Definition~\ref{def:dyadic-base-blocks}) that will be treated explicitly.
In addition to the rank-$2$ even blocks, the rank-$3$ lattice
\begin{equation}\label{eq:L3}
  L_3 := \langle 2\rangle^{\oplus 3}\qquad\text{with }Q(x)=2(x_1^2+x_2^2+x_3^2)
\end{equation}
plays a distinguished role: it is the simplest anisotropic even ternary lattice over $\Z_2$, and its representation counts amount to the classical three-squares congruence problem.
Indeed, if $x_1^2+x_2^2+x_3^2=0$ in $\Z_2$, then reducing modulo $8$ forces $x_1,x_2,x_3$ to be even, and iterating gives $x_1=x_2=x_3=0$; hence $L_3$ is anisotropic over $\Q_2$.

Our first main theorem is a lifting principle for diagonal sums of squares in arbitrary dimension, under the mild primitivity hypothesis $4\nmid a$.

The main new input is a very simple involution argument, the \emph{half-lift}.
For diagonal sums of squares $Q_d=x_1^2+\cdots+x_d^2$ and a primitive target $a$ with $4\nmid a$, translation by $2^{n-1}u$ in a primitive direction produces a free involution on the solution set modulo $2^n$ whose orbits control lifts to level $2^{n+1}$.
In each orbit exactly one residue class lifts, and whenever a class lifts, all lifts in its fibre are solutions.
This yields the stable recursion $N_{d,n+1}(a)=2^{d-1}N_{d,n}(a)$ for $n\ge 3$ (Theorem~\ref{thm:half-lift-principle}).
For $d=3$ it leads to closed forms for the three-squares congruence counts and hence an explicit formula for $\alpha_2(t;L_3)$.

\smallskip
We fix the conventions for the finite-level representation counts $r_m(t;L)$ and the local density $\alpha_p(t;L)$ in \S\ref{sec:prelim}.
At $p=2$ and for even lattices there is a second common normalisation $q:=Q/2$; we work with $Q$ unless explicitly indicated, and Lemma~\ref{lem:q-dictionary} gives the precise conversion.

All computations in the body of the paper are organised around the same elementary theme:
describe the fibres of the reduction maps $L/2^{m+1}L\to L/2^mL$ on the relevant solution sets.
Concretely,
\begin{altenumerate}
\item for the split even plane $H_0$ this reduces to valuation-stratified counting for the bilinear congruence $2xy\equiv t$ (see Lemmas~\ref{lem:2xy-odd} and~\ref{lem:2xy-2});
\item for the anisotropic plane $H_1$ it becomes a lifting recursion for a norm congruence in the unramified quadratic extension of~$\Q_2$ (see Lemma~\ref{lem:H1});
\item for the anisotropic ternary block $L_3$ one needs an additional involution (the half-lift) because the naive dyadic linearisation carries no information (see Theorem~\ref{thm:half-lift-principle}).
\end{altenumerate}

Let $p$ be a prime.
Section~\ref{sec:hyperbolic} treats the hyperbolic plane $H_0$ over $\Z_p$ for every $p$ and gives explicit formulas for the congruence counts $M_{p,m}(s)$ in~\eqref{eq:2xy}, together with their generating series and stable-range behaviour.
Section~\ref{sec:densities} records the resulting density formulas for $\alpha_p(t;H_0)$ and the basic dyadic blocks.
In particular, after passing to the standard normalisation $q=\langle\cdot,\cdot\rangle/2$ at $p=2$ (Lemma~\ref{lem:q-dictionary}), the density of $H_0$ and its scalings admits a single prime-uniform expression (Theorem~\ref{thm:prime-uniform-h0}).
The remaining sections specialise to $p=2$ and evaluate the basic dyadic blocks.

We therefore recall the dyadic Jordan decomposition (Proposition~\ref{prop:jordan}) and the basic dyadic blocks (Definition~\ref{def:dyadic-base-blocks}) that will be treated explicitly.
In addition to the rank-$2$ even blocks, the rank-$3$ lattice
\begin{equation}\label{eq:L3}
 L_3 := \langle 2\rangle^{\oplus 3}\qquad\text{with }Q(x)=2(x_1^2+x_2^2+x_3^2)
\end{equation}
plays a distinguished role: it is the simplest anisotropic even ternary lattice over $\Z_2$, and its representation counts amount to the classical three-squares congruence problem.
Indeed, if $x_1^2+x_2^2+x_3^2=0$ in $\Z_2$, then reducing modulo $8$ forces $x_1,x_2,x_3$ to be even, and iterating gives $x_1=x_2=x_3=0$; hence $L_3$ is anisotropic over $\Q_2$.

Our first main theorem is a lifting principle for diagonal sums of squares in arbitrary dimension, under the mild primitivity hypothesis $4\nmid a$.

The main new input is a very simple involution argument, the \emph{half-lift}.
For diagonal sums of squares $Q_d=x_1^2+\cdots+x_d^2$ and a primitive target $a$ with $4\nmid a$, translation by $2^{n-1}u$ in a primitive direction produces a free involution on the solution set modulo $2^n$ whose orbits control lifts to level $2^{n+1}$.
In each orbit exactly one residue class lifts, and whenever a class lifts, all lifts in its fibre are solutions.
This yields the stable recursion $N_{d,n+1}(a)=2^{d-1}N_{d,n}(a)$ for $n\ge 3$ (Theorem~\ref{thm:half-lift-principle}).
For $d=3$ it leads to closed forms for the three-squares congruence counts and hence an explicit formula for $\alpha_2(t;L_3)$.

Our first main theorem is a lifting principle for diagonal sums of squares in arbitrary dimension, under the mild primitivity hypothesis $4\nmid a$.

\begin{remark}[Scope of the half-lift involution]\label{rem:half-lift-scope}
The involution argument proving Theorem~\ref{thm:half-lift-principle} uses the specific diagonal form
$Q_d=x_1^2+\cdots+x_d^2$ and the primitivity hypothesis $4\nmid a$.
The primitivity is essential in general: for example, in dimension $d=4$ one checks directly that
\(N_{4,4}(8)=1536\) while \(N_{4,3}(8)=128\), so the ratio is $12\ne 2^{d-1}=8$.
On a conceptual level, two features of the proof break down once these hypotheses are relaxed:
if $4\mid a$ then solutions modulo $2^n$ may have all coordinates even, so the first step (which extracts an odd coordinate to choose an involution direction) can fail; and for $n=2$ the quadratic error term in the expansion \eqref{eq:expansion-d} no longer vanishes modulo $2^{n+1}$, so the translation $x\mapsto x+2^{n-1}u$ need not toggle $Q_d(x)$ by a clean additive $2^n$.
We do not claim a dimension-free half-lift recursion for arbitrary diagonal unit coefficients (or for general dyadic lattices) without further reduction.
In the dyadic setting, the half-lift mechanism is used here only to analyse the even ternary lattice $L_3$ in \eqref{eq:L3};
rank-$1$ Type~I blocks are handled separately via explicit square-root counts (Appendix~\ref{app:typeI}).
\end{remark}

\begin{lemma}[Dictionary with the $q=\langle\cdot,\cdot\rangle/2$ convention]\label{lem:q-dictionary}
Assume $L$ is \emph{even}, i.e.\ $Q(L)\subset 2\Ztwo$, and set $q(v):=Q(v)/2\in\Ztwo$.
For $m\ge 1$ define
\[
 r_m^{(q)}(t;L):=\#\{v\in L/2^mL : q(v)\equiv t\!\!\pmod{2^m}\}.
\]
Then for $t\in\Ztwo$ and $n:=\mathrm{rank}_{\Ztwo}(L)$:
\begin{altenumerate}
\item If $t$ is odd, then $r_m(t;L)=0$ for all $m\ge 1$.
\item If $t=2t'$ is even, then for every $m\ge 2$ one has
\[
 r_m(t;L)=2^{n}\,r_{m-1}^{(q)}(t';L).
\]
In particular, $r_1(2t';L)=2^n$.
\end{altenumerate}
Consequently, whenever the limits exist,
\[\alpha_2(2t';L)=2\,\alpha_2^{(q)}(t';L),\qquad
 \alpha_2^{(q)}(t';L):=\lim_{m\to\infty}2^{-m(n-1)}\,r_m^{(q)}(t';L).\]
\end{lemma}

\smallskip
\noindent\emph{Case $0\le v<m$.}
Stratify by $i:=v_p(x)$.
For each $0\le i\le v$, the number of residue classes $x\bmod p^m$ with $v_p(x)=i$ is $(p-1)p^{m-i-1}$.
Write $x=p^i x'$ with $x'\in(\Z/p^{m-i}\Z)^\times$.
Then $xy\equiv t\pmod{p^m}$ becomes
\[
 x'\,y\equiv p^{v-i}t'\pmod{p^{m-i}},\qquad t':=t/p^v\in(\Z/p^{m-v}\Z)^\times.
\]
As $x'$ is a unit modulo $p^{m-i}$, there is a unique solution $y_0\bmod p^{m-i}$.
It has exactly $p^i$ lifts modulo $p^m$, so for each fixed $i$ the number of pairs with $v_p(x)=i$ is
\[(p-1)p^{m-i-1}\cdot p^i=(p-1)p^{m-1}.
\]
Summing over $i=0,1,\dots,v$ yields $M_{p,m}(s)=(v+1)(p-1)p^{m-1}$.

\begin{lemma}[Fibre invariance for sums of squares]\label{lem:fibre-invariance}
Let $d\ge 1$ and $n\ge 1$.
For any $x\in(\Z/2^n\Z)^d$ and any $z\in(\Z/2\Z)^d$ one has
\[
 Q_d(x+2^n z)\equiv Q_d(x)\pmod{2^{n+1}}.
\]
Consequently, for any $a\in\Z$ and any residue class $\bar x\in(\Z/2^n\Z)^d$, the number of lifts $\tilde x\in(\Z/2^{n+1}\Z)^d$ of $\bar x$ satisfying
$Q_d(\tilde x)\equiv a\pmod{2^{n+1}}$
is either $0$ or the full fibre size $2^d$.
\end{lemma}

Let
$S_n(a):=\{x\in(\Z/2^n\Z)^d: f(x)\equiv 0\pmod{2^n}\}$,
so that $\#S_n(a)=N_{d,n}(a)$.
By Lemma~\ref{lem:fibre-invariance}, for each $\bar x\in(\Z/2^n\Z)^d$ the congruence condition modulo $2^{n+1}$ is constant across the entire fibre of its lifts.
In particular, if we view $f(\bar x)\bmod 2^{n+1}$ as the residue obtained by evaluating $f$ on any lift of $\bar x$ to $(\Z/2^{n+1}\Z)^d$ (well-defined by the lemma), then
\[
 N_{d,n+1}(a)=2^d\cdot\#\{\bar x\in S_n(a): f(\bar x)\equiv 0\pmod{2^{n+1}}\}.
\]
We show the subset on the right has size exactly $\tfrac12\#S_n(a)$.

\smallskip
\noindent\emph{(ii)} Assume $b\not\equiv 0\pmod{2^m}$ and write $b=2^{2j}c$ with $j\ge 0$, $2j<m$, and $c$ odd.
If $x^2\equiv 2^{2j}c\pmod{2^m}$, then necessarily $2^j\mid x$.
Write $x=2^j y$.
Dividing by $2^{2j}$ reduces the congruence to
\begin{equation}\label{eq:sqroot-reduced}
 y^2\equiv c\pmod{2^{k}},\qquad k:=m-2j\ge 1.
\end{equation}
The condition \eqref{eq:sqroot-reduced} depends only on the class of $y$ modulo $2^k$.
Conversely, each residue class $\bar y\in\Z/2^k\Z$ has exactly $2^j$ lifts to a class modulo $2^{m-j}$.
Hence the number of solutions $x\bmod 2^m$ equals
\[
 2^j\cdot\#\{y\bmod 2^k: y^2\equiv c\pmod{2^k}\}.
\]

\label{app:typeI}

We record the square-root counts modulo $2^m$ that enter the rank-$1$ dyadic blocks.
These counts are classical, but we include them so that the evaluation of the base blocks can be

\begin{equation}\label{eq:L3}
  L_3 := \langle 2\rangle^{\oplus 3}\qquad\text{with }Q(x)=2(x_1^2+x_2^2+x_3^2)
\end{equation}

\begin{equation}\label{eq:expansion-d}
  f(\tau_u(x))\equiv f(x)\pmod{2^n},\qquad
  f(\tau_u(x))\equiv f(x)+2^n(x\cdot u)\pmod{2^{n+1}},
\end{equation}

\begin{lemma}[Fibre invariance for sums of squares]\label{lem:fibre-invariance}
Let $d\ge 1$ and $n\ge 1$.
For any $x\in(\Z/2^n\Z)^d$ and any $z\in(\Z/2\Z)^d$ one has
\[
  Q_d(x+2^n z)\equiv Q_d(x)\pmod{2^{n+1}}.
\]
Consequently, for any $a\in\Z$ and any residue class $\bar x\in(\Z/2^n\Z)^d$, the number of lifts $\tilde x\in(\Z/2^{n+1}\Z)^d$ of $\bar x$ satisfying
$Q_d(\tilde x)\equiv a\pmod{2^{n+1}}$
is either $0$ or the full fibre size $2^d$.
\end{lemma}

\begin{theorem}[Half-lift principle for sums of squares]\label{thm:half-lift-principle}
Let $d\ge 1$ and set
\[
  Q_d(x_1,\dots,x_d):=x_1^2+\cdots+x_d^2.
\]
For $n\ge 1$ and $a\in\Z$ let
\[
  N_{d,n}(a):=\#\{x\in (\Z/2^n\Z)^d: Q_d(x)\equiv a\pmod{2^n}\}.
\]
Assume $n\ge 3$ and $4\nmid a$.
Then
\[
  N_{d,n+1}(a)=2^{d-1}\,N_{d,n}(a).
\]
Equivalently, among the residue classes modulo $2^n$ satisfying $Q_d\equiv a$, exactly half admit lifts to level $2^{n+1}$; moreover, whenever a class lifts, \emph{all} $2^d$ lifts in its fibre are solutions (cf.\ Lemma~\ref{lem:fibre-invariance}).
\end{theorem}

\begin{remark}[Scope of the half-lift involution]\label{rem:half-lift-scope}
The involution argument proving Theorem~\ref{thm:half-lift-principle} uses the specific diagonal form
$Q_d=x_1^2+\cdots+x_d^2$ and the primitivity hypothesis $4\nmid a$.
The primitivity is essential in general: for example, in dimension $d=4$ one checks directly that
\(N_{4,4}(8)=1536\) while \(N_{4,3}(8)=128\), so the ratio is $12\ne 2^{d-1}=8$.
On a conceptual level, two features of the proof break down once these hypotheses are relaxed:
if $4\mid a$ then solutions modulo $2^n$ may have all coordinates even, so the first step (which extracts an odd coordinate to choose an involution direction) can fail; and for $n=2$ the quadratic error term in the expansion \eqref{eq:expansion-d} no longer vanishes modulo $2^{n+1}$, so the translation $x\mapsto x+2^{n-1}u$ need not toggle $Q_d(x)$ by a clean additive $2^n$.
We do not claim a dimension-free half-lift recursion for arbitrary diagonal unit coefficients (or for general dyadic lattices) without further reduction.
In the dyadic setting, the half-lift mechanism is used here only to analyse the even ternary lattice $L_3$ in \eqref{eq:L3};
rank-$1$ Type~I blocks are handled separately via explicit square-root counts (Appendix~\ref{app:typeI}).
\end{remark}

For $d=3$, Theorem~\ref{thm:half-lift-principle} combines with a simple $4$-adic descent and a base computation modulo $8$ to yield closed forms for the three-squares counts once one reaches the stable range $n\ge 2k+3$ for $a=4^k a_0$ with $4\nmid a_0$ (and hence an explicit formula for $\alpha_2(t;L_3)$).
More generally, in the stable range the half-lift recursion has growth factor $2^{d-1}$, which is exactly the ``smooth'' Hensel factor $p^{d-1}$ specialised to $p=2$ (Remark~\ref{rem:half-lift-hensel}).
We also record explicit formulas for the even rank-$2$ Jordan blocks (hyperbolic and anisotropic).
As a consequence we obtain explicit densities $\alpha_2(t;B)$ for each basic dyadic block $B$ of Definition~\ref{def:dyadic-base-blocks}.

\begin{theorem}[Prime-uniform density for the hyperbolic plane]
\label{thm:prime-uniform-h0}
Let $p$ be a prime and let $e\ge 0$.
Let $H_0$ denote the rank-$2$ hyperbolic plane over $\Z_p$ with bilinear form
\(\langle (x,y),(x',y')\rangle = xy'+x'y\).
On the scaled lattice $p^eH_0$ consider the quadratic form
\[
  q_e(v):=\tfrac12\langle v,v\rangle = p^e xy\in\Z_p.
\]
For $m\ge 1$ and $t\in\Z_p$ set
\[
  r^{(q)}_{m}(t;p^eH_0)
  :=\#\{(x,y)\in(\mathbb{Z}/p^m\mathbb{Z})^2 : p^e xy\equiv t\!\!\pmod{p^m}\},
\]
and define the associated density (when the limit exists) by
\[
  \alpha_p^{(q)}(t;p^eH_0)
  :=\lim_{m\to\infty}p^{-m}\,r^{(q)}_{m}(t;p^eH_0).
\]
Let $t\in\Z_p\setminus\{0\}$ and write $v:=v_p(t)$.
Then
\[
  \alpha_p^{(q)}(t;p^eH_0)=
  \begin{cases}
    0, & v<e,\\[2pt]
    (v-e+1)(p-1)p^{e-1}, & v\ge e.
  \end{cases}
\]
\smallskip
\noindent In particular, under the $q=\langle\cdot,\cdot\rangle/2$ normalisation, both the vanishing threshold and the density formula for the scaled hyperbolic plane are uniform for all primes $p$, including $p=2$.
\end{theorem}

Theorem~\ref{thm:prime-uniform-h0} is a formal consequence of the $Q$-normalised density formulas for the hyperbolic plane (Corollaries~\ref{cor:density-hyp-scaled-odd} and~\ref{cor:density-hyp-scaled}) together with the conversion Lemma~\ref{lem:q-dictionary}; we give the short proof in \S\ref{sec:densities}.

\subsection*{Organisation}
Section~\ref{sec:prelim} fixes notation and recalls the dyadic Jordan decomposition and the basic dyadic blocks.
Section~\ref{sec:hyperbolic} treats the hyperbolic plane $H_0$ uniformly for all primes $p$ by explicit congruence counting, including closed forms for its stable-range counts and generating series.
Section~\ref{sec:anisotropic} treats the anisotropic even plane; here one can recast the counting problem as a norm-congruence in an unramified quadratic extension, where the unit locus admits a clean lifting recursion.
Section~\ref{sec:half-lift} proves the half-lift principle and derives closed forms for the anisotropic ternary lattice $L_3$.
Section~\ref{sec:densities} summarises the resulting density formulas.
Appendix~\ref{app:typeI} records the rank-$1$ (Type~I) dyadic block counts.

\subsection*{Computational checks}

The arguments in this paper are self-contained and elementary, and they do not rely on computation.
As an external consistency check, the arXiv source bundle includes ancillary files containing finite-level verifiers for the congruence counts and a Lean~4 formalisation of the half-lift involution (see \texttt{anc/verify.py} and the directory \texttt{anc/lean/}; build instructions are in \texttt{anc/README\_CODE.txt}).

The post-theorem introduction does not give a full step-by-step proof sketch, but it does describe the key mechanism and where it can fail. It says that the proof of Theorem~\ref{thm:half-lift-principle} is an \emph{“involution argument”} using the \emph{“specific diagonal form $Q_d=x_1^2+\cdots+x_d^2$”} together with the \emph{“primitivity hypothesis $4\nmid a$.”} Conceptually, it highlights two main ingredients/steps behind the involution:

1. A first step that \emph{“extracts an odd coordinate to choose an involution direction”}; this can fail when $4\mid a$ because \emph{“solutions modulo $2^n$ may have all coordinates even.”}

2. A translation step \emph{“$x\mapsto x+2^{n-1}u$”} that should \emph{“toggle $Q_d(x)$ by a clean additive $2^n$.”} This relies on the fact that for $n\ge 3$ \emph{“the quadratic error term in the expansion \eqref{eq:expansion-d} … vanishes modulo $2^{n+1}$,”} whereas for $n=2$ it \emph{“no longer vanishes modulo $2^{n+1}$,”} so the toggling may fail.

The post-theorem introduction does not give a full step-by-step proof sketch, but it does describe the key mechanism and where it can fail. It says that, in establishing the main theorem, the argument is an \emph{“involution argument”} using the \emph{“specific diagonal form $Q_d=x_1^2+\cdots+x_d^2$”} together with the \emph{“primitivity hypothesis $4\nmid a$.”} Conceptually, it highlights two main ingredients/steps behind the involution:

1. A first step that \emph{“extracts an odd coordinate to choose an involution direction”}; this can fail when $4\mid a$ because \emph{“solutions modulo $2^n$ may have all coordinates even.”}

2. A translation step \emph{“$x\mapsto x+2^{n-1}u$”} that should \emph{“toggle $Q_d(x)$ by a clean additive $2^n$.”} This relies on the fact that for $n\ge 3$ \emph{“the quadratic error term in the expansion
\begin{equation}\label{eq:expansion-d}
  f(\tau_u(x))\equiv f(x)\pmod{2^n},\qquad
  f(\tau_u(x))\equiv f(x)+2^n(x\cdot u)\pmod{2^{n+1}},
\end{equation}
… vanishes modulo $2^{n+1}$,”} whereas for $n=2$ it \emph{“no longer vanishes modulo $2^{n+1}$,”} so the toggling may fail.

[Half-lift principle for sums of squares]\label{thm:half-lift-principle}
Let $d\ge 1$ and set
\[
  Q_d(x_1,\dots,x_d):=x_1^2+\cdots+x_d^2.
\]
For $n\ge 1$ and $a\in\mathbb{Z}$ let
\[
  N_{d,n}(a):=\#\{x\in (\mathbb{Z}/2^n\mathbb{Z})^d: Q_d(x)\equiv a\pmod{2^n}\}.
\]
Assume $n\ge 3$ and $4\nmid a$.
Then
\[
  N_{d,n+1}(a)=2^{d-1}\,N_{d,n}(a).
\]
Equivalently, among the residue classes modulo $2^n$ satisfying $Q_d\equiv a$, exactly half admit lifts to level $2^{n+1}$; moreover, whenever a class lifts, \emph{all} $2^d$ lifts in its fibre are solutions, by the following lemma.

\begin{lemma}[Fibre invariance for sums of squares]\label{lem:fibre-invariance}
Let $d\ge 1$ and $n\ge 1$.
For any $x\in(\Z/2^n\Z)^d$ and any $z\in(\Z/2\Z)^d$ one has
\[
  Q_d(x+2^n z)\equiv Q_d(x)\pmod{2^{n+1}}.
\]
Consequently, for any $a\in\Z$ and any residue class $\bar x\in(\Z/2^n\Z)^d$, the number of lifts $\tilde x\in(\Z/2^{n+1}\Z)^d$ of $\bar x$ satisfying
$Q_d(\tilde x)\equiv a\pmod{2^{n+1}}$
is either $0$ or the full fibre size $2^d$.
\end{lemma}

Question:
Let \(d\ge 1\) and define the diagonal quadratic form
\[
Q_d(x_1,\dots,x_d)=x_1^2+\cdots+x_d^2.
\]
For \(n\ge 1\) and \(a\in\mathbb Z\), let
\[
N_{d,n}(a):=\#\{x\in(\mathbb Z/2^n\mathbb Z)^d: Q_d(x)\equiv a\pmod{2^n}\}.
\]
Assume \(n\ge 3\) and \(4\nmid a\). For a residue class \(\bar x\in(\mathbb Z/2^n\mathbb Z)^d\), its fibre at level \(2^{n+1}\) means the set of lifts \(\tilde x\in(\mathbb Z/2^{n+1}\mathbb Z)^d\) reducing to \(\bar x\), which has size \(2^d\). Which quantitative statement holds?

One has
\[
N_{d,n+1}(a)=2^{d-1}N_{d,n}(a).
\]
Equivalently, among the classes \(\bar x\in(\mathbb Z/2^n\mathbb Z)^d\) with \(Q_d(\bar x)\equiv a\pmod{2^n}\), exactly half admit lifts to level \(2^{n+1}\); and whenever such a class lifts, all \(2^d\) elements in its fibre are solutions modulo \(2^{n+1}\).

Choices:
B.
One has
\[
N_{d,n+1}(a)=2^{d-1}N_{d,n}(a)
\]
for every \(n\ge 2\) and every \(a\in\mathbb Z\) with \(2\nmid a\). Equivalently, among the classes \(\bar x\in(\mathbb Z/2^n\mathbb Z)^d\) with \(Q_d(\bar x)\equiv a\pmod{2^n}\), exactly half admit lifts to level \(2^{n+1}\); and whenever such a class lifts, all \(2^d\) elements in its fibre are solutions modulo \(2^{n+1}\).
C. Whenever a class \(\bar x\in(\mathbb Z/2^n\mathbb Z)^d\) with \(Q_d(\bar x)\equiv a\pmod{2^n}\) admits one lift to level \(2^{n+1}\), in fact all \(2^d\) elements in its fibre are solutions modulo \(2^{n+1}\).
D.
One has
\[
N_{d,n+1}(a)=2^{d-1}N_{d,n}(a)
\]
for every \(n\ge 3\) and every primitive target \(a\in\mathbb Z\). Equivalently, among the classes \(\bar x\in(\mathbb Z/2^n\mathbb Z)^d\) with \(Q_d(\bar x)\equiv a\pmod{2^n}\), exactly half admit lifts to level \(2^{n+1}\); and whenever such a class lifts, all \(2^d\) elements in its fibre are solutions modulo \(2^{n+1}\).
E.
One has
\[
N_{d,n+1}(a)=2^{d}N_{d,n}(a).
\]
Equivalently, every class \(\bar x\in(\mathbb Z/2^n\mathbb Z)^d\) with \(Q_d(\bar x)\equiv a\pmod{2^n}\) admits lifts to level \(2^{n+1}\), and whenever such a class lifts, all \(2^d\) elements in its fibre are solutions modulo \(2^{n+1}\).

dropped the exact-half orbit-counting conclusion and recursion, retaining only fibre invariance

hypothesis 4\nmid a weakened to mere primitivity of a

replaced half of the classes lifting by all classes lifting


Question:
Let \(d\ge 1\) and define the diagonal quadratic form
\[
Q_d(x_1,\dots,x_d)=x_1^2+\cdots+x_d^2.
\]
For \(n\ge 1\) and \(a\in\mathbb Z\), let
\[
N_{d,n}(a):=\#\{x\in(\mathbb Z/2^n\mathbb Z)^d: Q_d(x)\equiv a\pmod{2^n}\}.
\]
Assume \(n\ge 3\) and \(4\nmid a\). For a residue class \(\bar x\in(\mathbb Z/2^n\mathbb Z)^d\), its fibre of lifts to level \(2^{n+1}\) means the set of all \(\tilde x\in(\mathbb Z/2^{n+1}\mathbb Z)^d\) reducing to \(\bar x\), which has size \(2^d\). Which conclusion about the counts \(N_{d,n}(a)\) and the lifting behavior from modulus \(2^n\) to modulus \(2^{n+1}\) holds?

One has
\[
N_{d,n+1}(a)=2^{d-1}N_{d,n}(a).
\]
Equivalently, among the residue classes modulo \(2^n\) satisfying \(Q_d\equiv a\pmod{2^n}\), exactly half admit lifts to level \(2^{n+1}\); and whenever a class admits such a lift, all \(2^d\) elements in its fibre of lifts are solutions modulo \(2^{n+1}\).

Choices:
B.
One has
\[
N_{d,n+1}(a)=2^{d-1}N_{d,n}(a)
\]
for every \(n\ge 2\) and every \(a\in\mathbb Z\) with \(2\nmid a\). Equivalently, among the residue classes modulo \(2^n\) satisfying \(Q_d\equiv a\pmod{2^n}\), exactly half admit lifts to level \(2^{n+1}\); and whenever a class admits such a lift, all \(2^d\) elements in its fibre of lifts are solutions modulo \(2^{n+1}\).
C.
Whenever a residue class modulo \(2^n\) satisfying \(Q_d\equiv a\pmod{2^n}\) admits a lift to level \(2^{n+1}\), all \(2^d\) elements in its fibre of lifts are solutions modulo \(2^{n+1}\). Consequently,
\[
N_{d,n+1}(a)\le 2^d N_{d,n}(a).
\]
D.
There exists a subset \(T\subseteq (\mathbb Z/2^n\mathbb Z)^d\) of residue classes with \(Q_d\equiv a\pmod{2^n}\), of cardinality \(\tfrac12 N_{d,n}(a)\), such that every class in \(T\) admits exactly one lift to a solution modulo \(2^{n+1}\), while the remaining classes admit none. In particular,
\[
N_{d,n+1}(a)=2^{d-1}N_{d,n}(a).
\]
E.
One has
\[
N_{d,n+1}(a)=2^d N_{d,n}(a).
\]
Equivalently, every residue class modulo \(2^n\) satisfying \(Q_d\equiv a\pmod{2^n}\) admits lifts to level \(2^{n+1}\), and whenever a class admits a lift, all \(2^d\) elements in its fibre of lifts are solutions modulo \(2^{n+1}\).

dropped the exact half-of-classes lifting statement and the exact recursion

fibre invariance changed from all 2^d lifts in a lifting fibre to exactly one lift per lifting class

half-lift involution replaced by universal lifting of every class