\section{The Bose \emph{et al.}\ design and the inference from an observation}
\label{sec:experiments}

\subsection{What the model predicts for the design}

In the design of~\cite{Bose2017} the branches of $A$ sit at $0$ and $250~\mu$m, those of $B$ at $450$ and
$700~\mu$m, collinear, with closest approach $d=200~\mu$m and $m=10^{-14}$~kg (a diamond sphere of radius
$R=0.879~\mu$m). This is a parallel two-branch configuration, so by Theorem~\ref{thm:parallel} the model
can entangle it only if $\sigma$ is comparable to the separation. Eq.~\eqref{eq:twobranch} gives the exact
condition: $\sigma>524~\mu$m at $C=1$, and more at larger $C$ (Table~\ref{tab:Cdep}). With such a $\sigma$
the model's force is no longer Newtonian at laboratory distances: at $\sigma=524~\mu$m the smeared force
between two point masses $1$~mm apart is $0.39$ of its Newtonian value, and torsion-balance tests of the
inverse-square law at such separations show no deviation at that level~\cite{Lee2020}. Other two-branch
geometries at the same closest approach lower the threshold on $\sigma$ only to some tens of micrometres
(Appendix~\ref{app:values}).

Many-branch states at the same separation can be entangled for any $\sigma$, but only at the rates of
Section~\ref{sec:rate}. For $\sigma\ll R$ and $C=1$ each body dephases at $\Gsat=86~\mathrm{s}^{-1}$, while
the best sheet state known entangles at $\Gent=8.6\times10^{-3}~\mathrm{s}^{-1}$: the superposition is lost
about ten thousand times faster than entanglement builds up. By the law~\eqref{eq:law} the negativity then
peaks at order $10^{-5}$, half a dephasing time after preparation (Appendix~\ref{app:values}).

\subsection{What an observed entanglement excludes}

An observed entanglement can still exclude the model, through the geometry of the state or the size and
time course of the signal, but not through the separation alone.

\emph{Remains true.} (i) Two-branch states with ideal branches are not entangled by the model at any time
unless the separation is comparable to $\sigma$: at $C=1$, below $2.21\,\sigma$ for parallel superpositions
(Theorem~\ref{thm:parallel}) and below $2.75\,\sigma$ at slab separation for any two-branch geometry
(Proposition~\ref{prop:ceilings}, computer-assisted). For the design above, with ideal branches, a positive
result of the expected size excludes the model for every $\sigma$ below $524~\mu$m. (ii) For any
state of many branches that are more than a few $\sigma$ apart, the negativity stays below $0.19\,\seff/d$
at $C=1$ (Theorem~\ref{thm:negativity}), against about $0.4$ for unitary gravity. An entanglement signal of
the size the quantum prediction gives, or one that oscillates or persists beyond a few dephasing times
$1/\Gsat$, is incompatible with the model. (iii) Before entanglement is at issue, the survival of each
body's superposition over the experiment already tests the model.

\emph{No longer true.} (i) ``The Di\'{o}si--Penrose dynamics cannot entangle masses further apart than
$0.85\,\sigma$ (or $2.21\,\sigma$)'', read as a statement about the model. It is a statement about
particular two-branch geometries. The same applies to ``the dynamics is separability-preserving for
$d\gg\sigma$''.
(ii) The geometry-free inference ``any entanglement detected beyond $2.2\,\sigma$ falsifies the model,
whatever the state and whatever its size''.

\emph{Not decided.} For wavepackets of finite width the criterion is degenerate (Remark~\ref{rem:ideal}):
every such state is entangled at once, and we give no bound on how much. The design therefore excludes the
model for a result of the expected size, not for any positive result.