\section{Two-branch states, and why many branches have no critical distance}
\label{sec:twobranch}

\subsection{The published distances as special cases}

With two branches per body each space of patterns is a single dipole, and the canonical correlation is the
cosine between the two dipoles. For $S_A=\{a_1,a_2\}$, $S_B=\{b_1,b_2\}$, Eq.~\eqref{eq:rhomax} is
\begin{equation}
  s=\frac{|f(a_1b_1)+f(a_2b_2)-f(a_1b_2)-f(a_2b_1)|}{2\sqrt{D(L_A)D(L_B)}},\qquad D(L)=f_0-f(L),
  \label{eq:twobranch}
\end{equation}
with $L_A=|a_1-a_2|$, $L_B=|b_1-b_2|$ and $f(xy)=f(|x-y|)$ (point masses). The cosine depends on how the
dipoles are placed, so a critical distance belongs to a geometry and not to the model. Four limiting
families have closed forms (Table~\ref{tab:twobranch}, Fig.~\ref{fig:twobranch}): superpositions of length
$L\to0$ along the line joining the bodies (collinear); superpositions perpendicular to that line
(transversal), for $L\to0$ and for $L\to\infty$; and a vanishing superposition of $A$ midway between two
branches of $B$, each at distance $d$ (straddle). In each family $s$ increases monotonically towards its
limit, which is not attained. Setting the limits equal to $\sth$ reproduces the two distances
of~\cite{TrilloNavascues2025} and adds two. For every $C\geq1$ the straddle family has the largest critical
distance of the four (Table~\ref{tab:Cdep} gives the dependence on $C$).

\begin{table}[h]
\centering
\caption{The four limiting two-branch families: the limit of the correlation $s$, with
$\kappa_0=-f''(0)=1/(6\sqrt{\pi}\,\sigma^3)$ and $f(d)/f_0=\sqrt{\pi}\,\sigma\operatorname{erf}(d/2\sigma)/d$,
and the critical distance at $C=1$, where the limit equals $\sth(1)$; $d$ is the closest approach.}
\label{tab:twobranch}
\begin{tabular}{llc}
\toprule
family & limit of $s$ & critical $d/\sigma$\\
\midrule
collinear, $L\to0$~\cite{TrilloNavascues2025} & $|f''(d)|/\kappa_0$ & 0.850872\\
transversal, $L\to0$ & $|f'(d)/d|/\kappa_0$ & 1.547134\\
transversal, $L\to\infty$~\cite{TrilloNavascues2025} & $f(d)/f_0$ & 2.210928\\
straddle & $\sqrt2|f'(d)|/\sqrt{\kappa_0(f_0-f(2d))}$ & 2.738073\\
\bottomrule
\end{tabular}
\end{table}

\begin{figure}[t]
\centering
\includegraphics[width=0.64\textwidth]{fig-twobranch.pdf}
\caption{Canonical correlation of the four limiting two-branch families against the separation, with the
thresholds $\sth(1)$ and $\sth(2)$; dots mark the critical distances at $C=1$. In the shaded region
($d\geq2.75\,\sigma$ at slab separation) no two-branch state of any geometry is entangled at $C\geq1$
(Proposition~\ref{prop:ceilings}).}
\label{fig:twobranch}
\end{figure}

Most designs give both masses the same superposition, translated; the design of~\cite{Bose2017} is one.
For this parallel class the transversal limit is the supremum.

\begin{theorem}[Parallel superpositions]
\label{thm:parallel}
Let $S_A=\{a,a+w\}$ and $S_B=\{a+r,a+r+w\}$ for arbitrary vectors $w$, $r$, and let $d$ be the closest
approach. Then
\begin{equation}
  s\leq\max\big(f(d)/f_0,\;c_*\big),\qquad c_*=\Lambda_+L_{1/2}^2/f_0\approx0.635,
  \label{eq:parallel}
\end{equation}
where $\Lambda_+=\max f''$ and $f(L_{1/2})=f_0/2$ (values in Table~\ref{tab:constants}). For every $C$ with
$\sth(C)\geq c_*$, in particular every $C\geq1$, no parallel two-branch state with closest approach
$d\geq d_*(C)$ is entangled, where $f(d_*)/f_0=\sth(C)$ ($d_*=2.21\,\sigma$ at $C=1$); for every
$d<d_*(C)$ some are.
\end{theorem}

Within the parallel class, then, the model entangles at $C=1$ only below $2.21\,\sigma$, a supremum that
long transversal superpositions approach and no state attains. The proof
(Appendix~\ref{app:bounds}) uses that $h(x)=f(\sqrt x)$ is a mixture of decaying exponentials, hence convex
and log-convex. For arbitrary two-branch geometries we have only computer-assisted ceilings.

\begin{proposition}[Two-branch ceilings]
\label{prop:ceilings}
For every $C\geq1$ no two-branch state is entangled by the model at slab separation $d\geq2.75\,\sigma$
(the branches of $A$ and $B$ on opposite sides of a plane, at normal distance $\geq d$), nor at closest
approach $d\geq3.78\,\sigma$.
\end{proposition}

The inequalities that bound $s$ are proven; the final maximisation over the two splittings is a
floating-point evaluation without interval arithmetic, with a margin of $0.36\%$ below $1/\sqrt2$
(Appendix~\ref{app:bounds}). A numerical search over all two-branch geometries found nothing above the
closed forms of Table~\ref{tab:twobranch}; we have not proven these to be the suprema. So at $C=1$ the
largest two-branch distance lies between $2.21\,\sigma$ and $2.75\,\sigma$ at slab separation, between
$2.74\,\sigma$ and $3.78\,\sigma$ at closest approach, and equals $2.21\,\sigma$ within the parallel class.

\subsection{Many branches: no critical distance}

With many branches the condenser argument of Section~\ref{sec:idea} becomes a theorem: sheets of branches
that are wide compared with the gap exceed any threshold below one.

\begin{theorem}[No critical distance]
\label{thm:nodistance}
For every $d>0$ and every $C<\infty$ there are finite branch sets at slab separation $d$ with
$\rmax>\sth$. For uniform coaxial discs of radius $a$ at gap $d$ (lattices of point branches), each body
carrying in addition a far copy of its disc,
\begin{equation}
  \rmax\geq\operatorname{erf}(d/2\sigma)\Big(1-\frac{3\pi d}{8a}\Big),
  \label{eq:discbound}
\end{equation}
so the model entangles such states once $a/d>(3\pi/8)/\big(1-\sth/\operatorname{erf}(d/2\sigma)\big)$,
which is $4.02$ at $C=1$ and $d\gg\sigma$.
\end{theorem}

The condition is on the extent of the coherent superposition in units of the gap, not on $d/\sigma$. The
far copy is a second disc of branches, far from both near discs, on which the pattern has the opposite
sign: it makes the pattern neutral without changing the correlation, and it means that the coherent extent
of the state is really the distance to the copy (Section~\ref{sec:extent} gives the sharp extent law, also
for a single sheet). Since $\rmax=1$ for every pair of open supports
(Remark~\ref{rem:ideal}), the absence of a critical distance is not by itself a physical prediction. What
matters is how much entanglement the model can produce, and how fast.