\section{Scope and discussion}
\label{sec:discussion}

\emph{Position-eigenstate branches.} Theorem~\ref{thm:criterion} is exact for branches that are position
eigenstates. For wavefunctions whose support has interior the yes/no question has no content
(Remark~\ref{rem:ideal}); Theorem~\ref{thm:continuum} gives the entangling direction, with the
rate~\eqref{eq:rate} in integral form. The negativity bound~\eqref{eq:negbound} is proven for finite branch
sets only; we give none in terms of the packet width.

\emph{The kinetic term.} All results are for $H=0$. The initial rate~\eqref{eq:rate} and the bounds of
Theorem~\ref{thm:ratebounds} are unchanged by any local Hamiltonian defined on the initial state, and local
Hamiltonians acting within a finite set of sites cannot make a sub-threshold region entangling
(Theorem~\ref{thm:criterion}(b)). Beyond that we control nothing: free packets acquire full support at once,
and the negativity bound has not been extended to them. For $m=10^{-14}$~kg the relevant experiments sit
deep in the static regime, which is an argument, not a bound.

\emph{Preparation.} Every result assumes the product state exists at $t=0$. The same dephasing acts while
it is made: a superposition over at least about $15\,d/\sigma$ positions, more than five gaps across, must
be made in a time short compared with $1/\Gsat$ ($11.6$~ms for the mass above).

\emph{What is numerical.} The critical distances, the constants $\Lambda_+$, $L_{1/2}$, the extent
thresholds and the rate constants are roots and maxima of explicit one-dimensional functions, or top
eigenvalues of converged discretisations of Love's operator (Appendix~\ref{app:values}).
Proposition~\ref{prop:ceilings} is computer-assisted, in floating point, without interval arithmetic. That
the closed forms of Table~\ref{tab:twobranch} are the two-branch suprema over all geometries rests on
numerical search. On discs, the optimality of the axisymmetric sector and of the dipolar pattern on a
single disc is numerical. The leading-order law~\eqref{eq:law} is derived without an error bound and was
compared with exact partial transposes of lattice sheets only.

\emph{Also open.} Separability after the negative eigenvalue has returned to zero; the best rate constant,
between $0.027$ and $0.068$ at $C=1$; non-planar supports; noise kernels other than the Newtonian
one; $C<1$, which is not a classical model; the unsmeared-force variant, which is not a
measurement-and-feedback model and to which the pair decomposition does not apply.

\emph{Discussion.} The Di\'{o}si--Penrose model is a classical channel with a definite structure: one kernel
in force and noise, and a local dephasing equal to the Gram form of that kernel. That structure gives an
exact criterion, a canonical correlation against $C/\sqrt{1+C^2}$, which answers the question
of~\cite{TrilloNavascues2025}. The model can in principle produce entanglement at any separation, with
many branches spread coherently over several times the separation. Its generator is not
separability-preserving at any distance, so it is not a local-operations-and-classical-communication
channel, and a critical distance is a property of a two-branch geometry, not of the model. But it produces
entanglement at a rate of order $\seff/d$ times the rate at which it destroys the superposition itself,
and for branches more than a few $\sigma$ apart the entanglement never exceeds a fraction of order
$\seff/d$ of what unitary gravity gives. For entanglement-witness experiments the discriminants are the
size and time course of the signal and, before them, the survival of the single-body superposition. The
separation alone discriminates only for two-branch states.