\section{The idea: one kernel, one comparison}
\label{sec:idea}

In the Di\'{o}si--Penrose model the pull between two bodies and the noise that decoheres them are one
quantity, the Coulomb energy $\ip{u}{v}=\int\!\!\int u(x)v(y)/|x-y|$ between smeared mass distributions.
Entanglement then becomes a matter of one angle against one number.

\emph{One kernel.} In the master equation, Eq.~\eqref{eq:master}, the Newtonian interaction and the
monitoring noise integrate the same smeared mass density against the same kernel $1/|x-y|$. On position
branches a coherence sees only the difference of two mass distributions. For one body with branches $a$,
$a'$ this is a dipole: the smeared body at $a$, minus the smeared body at $a'$. Let $u$ and $v$ be such
dipoles of bodies $A$ and $B$, weighted by the masses. With no kinetic term, the coherence between the
configurations $ab$ and $a'b'$ acquires the Newtonian phase and decays at a rate proportional to
$C\,\|u+v\|^2=C\,(\|u\|^2+\|v\|^2+2\ip{u}{v})$, with $\|u\|^2=\ip{u}{u}$ (Eq.~\eqref{eq:Lambda}). The
norms dephase each body by itself, at the Di\'{o}si--Penrose rate; the cross term is noise correlated
between the bodies; the Newtonian phase is the same Coulomb energy, evaluated between them. A classical
field that pushes two bodies must also decohere them~\cite{KafriTaylorMilburn2014}; here push and
decoherence are one bilinear form.

\emph{One comparison.} Entanglement first appears as a negative eigenvalue of the partial transpose. Its
witness reads a neutral pattern $q_1$ on the branches of $A$, a combination of such dipoles, and a pattern
$q_2$ on those of $B$. At first order in time the eigenvalue has the sign of
\begin{equation}
  C\,\|q_1\|^2+C\,\|q_2\|^2-2\,\mathrm{Re}\big[(C+i)\ip{q_1}{q_2}\big].
  \label{eq:idea-form}
\end{equation}
The first two terms are the cost, the dephasing of the collective coherences that the witness reads. The
last is the gain: the force supplies the $i$, the correlated noise the $C$, and with the best relative
phase it is $2\sqrt{1+C^2}\,|\ip{q_1}{q_2}|$. Cost and gain are measured in one metric, so their comparison
is an angle: with the best relative scale the form is negative if and only if the cosine
$|\ip{q_1}{q_2}|/(\|q_1\|\|q_2\|)$ exceeds $C/\sqrt{1+C^2}$. Masses, amplitudes and time do not enter.
Noise alone would put the threshold at one, which no cosine exceeds; the force brings it down, to
$1/\sqrt2$ at $C=1$. The patterns always split into pairs of directions, one on each body, that couple to
no other pair; below the threshold the evolution on each pair is a mixture of local operations, and the
state stays separable for all time.

\emph{Two branches: one dipole each.} With two branches a body has a single pattern, so the geometry fixes
the angle. For two long dipoles side by side at distance $d$, each end faces the like end of the other: the
mutual energy is that of two smeared points at distance $d$, the self-energy that of one, and the cosine is
their ratio, $\sqrt\pi\,\sigma\operatorname{erf}(d/2\sigma)/d$, which falls below $1/\sqrt2$ at
$d=2.21\,\sigma$. Other geometries change the number, not the conclusion: one fixed direction per body is
aligned with the other only within a few $\sigma$. These are the critical distances
of~\cite{TrilloNavascues2025}.

\emph{Many branches: a condenser.} With many branches each body has a space of patterns, and two spaces can
contain nearly parallel vectors where two given vectors do not. Spread the branches of $A$ over a disc of
radius $a$, those of $B$ over the same disc displaced by the gap $d$, and give both the same uniform
charge. Their difference is a charged parallel-plate condenser, whose field is confined to the gap: its
energy is smaller than the self-energy of one plate by a factor of order $d/a$, and the cosine is one minus
a term of that order. Discs several gaps wide carry nearly parallel patterns, however large $d/\sigma$ is.
(A distant copy of each disc with the opposite charge makes the patterns neutral; the self-energies of the
single smeared points enter divided by the number of branches.) There is no critical distance.

\emph{Why the entanglement stays small.} The form~\eqref{eq:idea-form} fixes the size as well as the sign.
Its gain is a Newtonian energy at distance $d$, so the negative eigenvalue grows at a rate $\Gent$ of order
$Gm^2/(\hbar d)$ at most. Meanwhile each pair of branches of one body dephases at a rate set by the body's
own self-energy, which saturates for a rigid body: two positions further apart than the effective size
$\seff$ of the smeared body dephase at $\Gsat=CGm^2/(\sqrt\pi\,\hbar\,\seff)$, however far apart they are.
Many branches dilute the self-energy of the pattern that the witness reads, which is how they make the sign
negative; they do not dilute the dephasing of the state that carries it. At leading order, for many
branches, $\lambda_{\min}(t)\simeq-\Gent\,t\,e^{-2\Gsat t}$, with one factor $e^{-\Gsat t}$ for the
coherence on each body and $\Gent/\Gsat=O(\seff/d)$. The entanglement grows for half a dephasing time,
peaks at $\Gent/(2e\Gsat)$, and decays.