\section{Introduction}
\label{sec:intro}

\subsection*{The question}

If gravity is a quantum channel, two masses that are each in a superposition of positions become entangled
through their gravitational interaction, and the detection of that entanglement has been proposed as a
witness of the quantum nature of gravity~\cite{Bose2017,Marletto2017}. The witness needs a classical
benchmark. The Di\'{o}si--Penrose model~\cite{Diosi1987,Diosi1989,Penrose1996}, in the form of Tilloy and
Di\'{o}si~\cite{TilloyDiosi2016,TilloyDiosi2017}, is one. Its Newtonian field is sourced by continuously
monitored mass densities, smeared over a length $\sigma$, and the monitoring dephases superpositions of
position, a decoherence that the positivity of the dynamics forces~\cite{KafriTaylorMilburn2014}. Trillo
and Navascu\'{e}s showed that this model nevertheless entangles two masses, in the same geometries in which
the quantum prediction is made~\cite{TrilloNavascues2025}. For two branches per mass they found critical
distances $d_c$, about $0.85\,\sigma$ for collinear superpositions of vanishing size and $2.21\,\sigma$ for
transversal superpositions of diverging size, and asked: ``Is there a way to observe GIE [gravitationally
induced entanglement] beyond $d_c$?'' We answer in the form: for which states does the model entangle two
masses, and how much?

\subsection*{Why it was open}

Later work quotes $d_c\approx0.85\,\sigma$ as the separation below which the model
entangles~\cite{AngeliCarlesso2025,DiBiagio2025}. That is correct for small packets with two branches;
nothing was known for more. Wang, Pedernales and Plenio, in a study of local error correction, gave a
separability lemma for one pair of coupled operators, found the Di\'{o}si--Penrose dynamics of their
two-level codes separable outside a small region of $d/\sigma$, and left the exact classification as
``beyond our scope''~\cite{WangPedernalesPlenio2026}. With more than two branches per mass several
operators are coupled at once, and a condition on each pair does not in general give a condition on the
whole. An exact answer needs a structure that decouples them.

\subsection*{The answer}

The structure exists. The model uses one kernel, the Coulomb energy between smeared mass distributions,
three times: for the Newtonian coupling of the two bodies, for the noise that is correlated between them,
and for the dephasing of each body by itself. With no kinetic term, two rigid bodies in a product state on
finite sets of position branches therefore become entangled if and only if
\begin{equation}
  \rmax(S_A,S_B)\;>\;\frac{C}{\sqrt{1+C^2}},
  \label{eq:headline-criterion}
\end{equation}
where $\rmax$ is the largest canonical correlation~\cite{Hotelling1936} between neutral charge patterns on
the two branch sets, the cosine of the smallest angle between them with the Coulomb energy as inner
product, and $C\geq1$ is the noise strength ($C=1$ in~\cite{TrilloNavascues2025}). Below the threshold the
evolution is a separable operation for all time (Theorem~\ref{thm:criterion}).

Two branches per body give one pattern each, and the published critical distances follow. Many branches
give each body a space of patterns, and two such spaces can be aligned at any separation: the model has no
critical distance. What it has is a limit on the size of the entanglement, which is produced at a rate
$\Gent$ that is a small fraction of the rate $\Gsat$ at which the same dynamics dephases each body,
\begin{equation}
  \frac{\Gent}{\Gsat}\;=\;\frac{\sqrt{\pi}\,\kappa}{C}\,\frac{\seff}{d}
  \;\leq\;0.12\,\frac{\seff}{d}\qquad(C=1,\ \text{slab-separated supports},\ \sigma\ll d).
  \label{eq:headline-ratio}
\end{equation}
Here $\kappa$ is a dimensionless rate constant, $d$ the width of a slab that separates the branches of the
two bodies, $\Gsat = C\,Gm^2/(\sqrt{\pi}\,\hbar\,\seff)$ the saturated Di\'{o}si--Penrose dephasing rate,
and $\seff$ the effective size of a smeared body ($\seff=\sigma$ for point masses). The entanglement
therefore peaks at about $\Gent/(2e\Gsat)$, half a dephasing time after preparation, and then decays. For
the design of Bose \emph{et al.}~\cite{Bose2017} at $C=1$ the two-branch state is entangled only if
$\sigma>524~\mu$m, and for $\sigma$ below the radius of the masses the best many-branch state known
entangles at $8.6\times10^{-3}~\mathrm{s}^{-1}$ while each mass dephases at $86~\mathrm{s}^{-1}$.

\subsection*{What is not shown}

The ``if and only if'' is a statement about position-eigenstate branches. For wavefunctions whose support
has interior the canonical correlation is one, and we give no bound on what wavepackets of finite width
produce. Beyond the initial rate, all results are for a vanishing kinetic term. The preparation of a
many-branch state, during which the same dephasing acts, is not modelled. The law for the size of the
entanglement is derived at leading order without an error bound; the proven bound needs many branches,
more than a few $\sigma$ apart. The two-branch optimum over all geometries rests on a numerical search and
a computer-assisted ceiling, and the best rate constant is bracketed, not determined.
Section~\ref{sec:discussion} gives the full scope.