\pdfoutput=1
\documentclass[11pt,a4paper]{article}
\usepackage{qgc}
\usepackage{qgc-notation}
\hypersetup{
  pdftitle={Entanglement from the Diosi-Penrose Model: An Exact Criterion, and Why It Stays Small at Any Distance},
  pdfsubject={Exact entanglement criterion for the Tilloy-Diosi / Diosi-Penrose master equation on position branches; entangling rate against the model's own dephasing rate},
  pdfkeywords={Diosi-Penrose model, Tilloy-Diosi model, gravitationally induced entanglement, classical gravity, separability, canonical correlation, negativity}
}
\newcommand{\rmax}{\rho_{\max}}
\newcommand{\sth}{s_*}
\newcommand{\Gsat}{\Gamma_{\rm sat}}
\newcommand{\Gent}{\Gamma_{\rm ent}}
\newcommand{\seff}{\sigma_{\rm eff}}
\newcommand{\Rr}{\mathcal{R}}
\newcommand{\ip}[2]{\langle #1,#2\rangle}
\makeatletter
\newcommand{\compacttoc}{\begingroup\small\setcounter{tocdepth}{1}%
  \renewcommand*\l@section[2]{\@dottedtocline{1}{0em}{1.6em}{##1}{##2}}%
  \tableofcontents\endgroup}
\makeatother
\title{Entanglement from the Di\'{o}si--Penrose Model:\\An Exact Criterion, and Why It Stays Small at Any Distance}
\author{\qgcauthor}
\date{November 11, 2026}

\begin{document}
\maketitle
\begin{abstract}
Trillo and Navascu\'{e}s showed that the Di\'{o}si--Penrose model, in the Tilloy--Di\'{o}si form with
the same smearing length $\sigma$ in force and noise, entangles two masses in superposition when they
are closer than about $2.2\,\sigma$, and asked whether gravitationally induced entanglement can be observed
beyond that distance. We answer for two rigid bodies whose branches are position eigenstates on finite
sets, without kinetic term, for noise strength $C$. One kernel, the Coulomb energy of the smeared bodies,
sets both the Newtonian coupling and the correlated noise, and the question becomes a single geometric
comparison: a product state becomes entangled if and only if the largest canonical correlation between
neutral charge patterns on the two branch sets exceeds $C/\sqrt{1+C^2}$. Below this threshold the
evolution is a separable operation at all times. The published critical distances are two-branch cases of
the criterion: for parallel two-branch superpositions the supremum is $2.21\,\sigma$ at $C=1$, and a
straddling geometry reaches $2.74\,\sigma$. With many branches there is no critical distance. What limits
the effect is the model's own dephasing, which sits in the same quadratic form. For bodies of effective
size $\seff$ ($\sigma$ for point masses) separated by a slab of width $d\gg\sigma$, the entangling rate at
$C=1$ is at most $0.12\,\seff/d$ times the single-body dephasing rate. The entanglement is therefore
transient and, for well-separated branches, a fraction of order $\seff/d$ of what unitary Newtonian gravity
gives. Entanglement of the size unitary gravity predicts still excludes the model; the inference from any
detected entanglement beyond a fixed multiple of $\sigma$ does not hold.
\end{abstract}
\compacttoc
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\clearpage
\input{sections/introduction}
\input{sections/idea}
\input{sections/criterion}
\input{sections/twobranch}
\input{sections/rate}
\input{sections/experiments}
\input{sections/discussion}
\appendix
\input{appendices/proofs}
\input{appendices/bounds}
\input{appendices/values}
\input{backmatter}
\end{document}