\section{Numerical values}
\label{app:values}

This appendix collects the precise values behind the rounded numbers of the main text. Lengths are in units
of $\sigma$ unless stated, and point masses are assumed except where $\seff$ appears. Every entry is a root,
a maximum or a top eigenvalue of a formula stated in the text.

\begin{table}[h]
\centering
\caption{Constants that enter the theorems, the rate bounds and the estimates of the main text, with the
place where each is used. Values are at $C=1$ unless stated.}
\label{tab:constants}
\small
\begin{tabular}{lll}
\toprule
quantity & where & value\\
\midrule
$\Lambda_+=\max f''$ & Theorem~\ref{thm:parallel} & $0.311326\,\kappa_0$, at $r=3.2273\,\sigma$\\
$L_{1/2}$, $f(L_{1/2})=f_0/2$ & Theorem~\ref{thm:parallel} & $3.497418\,\sigma$\\
$c_*=\Lambda_+L_{1/2}^2/f_0$ & Theorem~\ref{thm:parallel} & $0.634687$; $\sth(C)\geq c_*$ for $C\geq0.8213$\\
straddle above transversal & Table~\ref{tab:twobranch} & for $d<3.699\,\sigma$, i.e.\ every $C>0.5396$\\
sufficient extent, uniform discs & Theorem~\ref{thm:nodistance} & $a/d>4.022$ ($d\gg\sigma$)\\
$\kappa$, any product state & Theorem~\ref{thm:ratebounds}(i) & $\leq\sqrt2-1=0.4142$\\
slab ceiling $B(1)$ & Eq.~\eqref{eq:BC} & $0.067640$\\
planar ceiling at $d=3\,\sigma$ (finite $\sigma$) & Appendix~\ref{app:bounds} & $0.104$\\
Gaussian sheets $\kappa_G$ & Eq.~\eqref{eq:kappaG} & $0.027108$ at $b=3.15\,d$; zero for $b/d\leq1.465$\\
uniform disc, uniform witness & Fig.~\ref{fig:rate} & $\kappa=0.0264$ at $a=6.12\,d$\\
$1/\Rr$, slab ceiling; Gaussian sheets & Eq.~\eqref{eq:ratio} & $8.3\,d/\seff$; $20.8\,d/\seff$\\
negativity bound, $\ell,d\gg\sigma$ & Theorem~\ref{thm:negativity} & $0.191\,\sigma/d$; $0.069\,\sigma/d$ at $C=2$\\
peak by the law, Gaussian sheets & Eq.~\eqref{eq:law} & $8.8\times10^{-3}\,\seff/d$\\
peak by the law, slab ceiling & Eq.~\eqref{eq:law} & $2.2\times10^{-2}\,\seff/d$\\
\bottomrule
\end{tabular}
\end{table}

\begin{table}[h]
\centering
\caption{Dependence on the noise strength $C$. Two-branch critical distances (in units of $\sigma$) of the
transversal ($L\to\infty$) and straddle families; the threshold on $\sigma$ above which the design
of~\cite{Bose2017} is entangled; and the smallest $a/d$ for which discs of radius $a$ at gap $d$ (each with
a far copy, $\sigma\ll d$) are entangled, for the uniform pattern, Eq.~\eqref{eq:Phi}, and the optimal
axisymmetric pattern, Eq.~\eqref{eq:love}.}
\label{tab:Cdep}
\small
\begin{tabular}{cccccc}
\toprule
$C$ & transversal & straddle & design, $\sigma>$ & uniform pattern & optimal pattern\\
\midrule
1 & 2.2109 & 2.7381 & $524.494~\mu$m & 2.8316 & 2.7706\\
1.25 & 1.8203 & 2.4425 & $617~\mu$m & 4.1316 & 4.0642\\
2 & 1.1845 & 1.9106 & $910~\mu$m & 9.7539 & 9.6453\\
5 & 0.4870 & 1.1832 & $2158~\mu$m & 58.810 & 58.041\\
\bottomrule
\end{tabular}
\end{table}

\begin{table}[h]
\centering
\caption{The ratio $\Rr=\Gent/\Gsat$ in units of $\seff/d$. The first column is the Gaussian sheet of
Eq.~\eqref{eq:kappaG} ($\sigma\to0$, zero thickness); the second the slab bound~\eqref{eq:BC} ($\sigma\ll d$);
the third Theorem~\ref{thm:ratebounds}(i), any product state, any $\sigma$.}
\label{tab:ratio}
\small
\begin{tabular}{cccc}
\toprule
$C$ & Gaussian sheets & slab-separated, ceiling & any product state, ceiling\\
\midrule
1 & 0.0481 & 0.120 & 0.734\\
1.25 & 0.0235 & 0.0590 & 0.497\\
2 & 0.00453 & 0.0115 & 0.209\\
5 & 0.000135 & 0.00034 & 0.035\\
\bottomrule
\end{tabular}
\end{table}

\paragraph{Extent.} At $C=1$ the optimal pattern needs a coherent diameter of $5.54$ gaps, the uniform one
$5.66$. At large $C$, $1-\Phi\simeq(3\pi/8)\,d/a$ gives $a/d\simeq(3\pi/4)C^2=2.356\,C^2$ for the uniform
pattern. Without the far copy, the best neutral pattern found on a single disc is dipolar, the odd block of
Love's operator: it needs $a/d>7.390$, a coherent diameter of $14.8$ gaps, against $a/d>11.34$ ($22.7$
gaps) for the best axisymmetric neutral pattern. For hexagonal lattice patches with a far copy at $C=1$, the
smallest number of point branches per mass that reaches the threshold is $15.2\,d/\sigma$ at
$d=12\,\sigma$ and $15.6\,d/\sigma$ at $d=24\,\sigma$.

\paragraph{The leading-order law at finite $\xi$.} From
$n\lambda_{\min}\simeq e^{-2\tau}[e^\tau-1-\xi\tau]$, $n$ times the peak is $\xi/2e-0.239+O(1/\xi)$, and
it vanishes at $\xi=1$.

\paragraph{An explicit state.} Take for each mass a hexagonal patch of 19 point branches with spacing
$8\,\sigma$, plus its far copy (38 branches per mass), the two patches facing each other at slab separation
$d=3.52\,\sigma$, and $C=1$. Then $\rmax=0.813>\sth$, $\kappa=0.0258$, $\Rr=0.0130$ and
$\Rr/2e=2.4\times10^{-3}$. Exact diagonalisation of the $1444$-dimensional partial transpose gives a
smallest eigenvalue of $-1.6\times10^{-3}$ at $t'=0.5$, when each body's coherence is still $0.75$; the
peak lies below the large-$\xi$ value because this state is sparse. Unitary Newtonian gravity ($1/r$
coupling, no noise) on the same state reaches $-0.41$.

\paragraph{The design of~\cite{Bose2017}.} At $C=1$ the two-branch threshold is $\sigma>524.494~\mu$m. No
two-branch geometry with closest approach $200~\mu$m is entangled unless $\sigma>73.0~\mu$m (straddle) or
$90.5~\mu$m (slab), according to the numerical suprema of Section~\ref{sec:twobranch};
Proposition~\ref{prop:ceilings} guarantees $\sigma>52.9~\mu$m (any geometry) or $72.7~\mu$m (slab). For
$\sigma\ll R$ and $C=1$ the coherence time is $1/\Gsat=11.6$~ms, and over the $2.5$~s of the proposed
experiment the dephasing exponent is $216$; over the same time unitary Newtonian gravity gives the
two-branch design an entangling phase of $0.314$~rad. The best Gaussian sheet at $d=200~\mu$m has
$b\approx630~\mu$m, $\Rr=9.9\times10^{-5}$, and by Eq.~\eqref{eq:law} a peak negativity of
$1.8\times10^{-5}$ at $5.8$~ms.