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\title{Spacetime from Self-Modeling: Einstein's Equations\\from a Self-Modeling Lattice via Jacobson's Thermodynamic Argument}

\author{Bryan Ehrlich}
\affiliation{Independent researcher}

\date{\today}

\begin{abstract}
We show that the locality of self-modeling---the principle that a physical
system contains a faithful model of itself, updated through its
boundary---yields Einstein's field equations as the leading-order effective
description at long wavelengths, given lattice topology and
antiferromagnetic coupling as inputs. Self-modeling forces
$M_n(\mathbb{C})^{sa}$ quantum mechanics~\cite{Paper5}. The self-modeling
interaction Hamiltonian $H = \sum_{\langle x,y\rangle} J\, \swapop$,
forced by diagonal $U(n)$ covariance, produces area-law entanglement
entropy $\ent = \eta\,\mathcal{A}$. Applying Jacobson's thermodynamic
argument~\cite{Jacobson1995}---the Clausius relation
$\delta Q = T\,\delta\ent$ on local Rindler horizons, with the Unruh
temperature and the Raychaudhuri equation---yields
\begin{equation*}
  \Gab + \Lambda\, \gab = 8\pi \Gnewton\, \Tab
\end{equation*}
with $\Gnewton = 1/(4\eta)$, in all $d + 1 \geq 3$ spacetime dimensions.
We present numerical verification on small lattices ($N = 8$--$20$) and
precisely identify remaining gaps: emergent Lorentz invariance, the
lattice Bisognano--Wichmann property, and the local equilibrium condition.
\end{abstract}

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\input{sections/introduction}
\input{sections/lattice}
\input{sections/arealaw}
\input{sections/einstein}
\input{sections/numerical}
\input{sections/discussion}

\begin{acknowledgments}
[Acknowledgments placeholder.]
\end{acknowledgments}

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