Breaking the Entanglement-Structure Trade-off: Many-Body Localization Protects Emergent Holographic Geometry in Random Tensor Networks

We present a systematic numerical investigation of the "entanglement geometry gravity" chain in random tensor networks (RTN) established by the ER EPR conjecture and Jacobson's thermodynamic derivation. First, we verify the kinematic foundation: the entanglement first law δlangle Krangle=δS slope=1.000, the encoding of geometry by mutual information correlation=0.92, and the locality of holographic perturbations (3.3x). We also confirm that gravitational dynamics (JT gravity) does not emerge, identifying a sharp kinematics-dynamics boundary. Second, and more importantly, we discover that many-body localization (MBL) is the mechanism that protects emergent holographic geometry from thermalization. Replacing Haar-random evolution geometry lifetime $tsim6$ with an XXZ Hamiltonian plus on-site disorder, we observe a finite-size crossover at disorder strength W_capprox10-12 above which mutual-information-lattice correlations persist indefinitely (r>0.5 for t>50). We map the full parameter space: the optimal regime is a near-Ising anisotropy Δapprox50 with W=30 yielding r=0.779pm0.002 confirmed by a fine scan over $Δin[30,70]$; only holographic (RTN) initial states sustain geometry, while product, Néel, and Bell-pair states do not. MBL preserves the spatial structure of entanglement (adjacent/non-adjacent MI ratio 2.6-4.2x vs. 1.0x in the thermal phase), rather than its total amount. A comparison with classical cellular automata reveals that MBL uniquely breaks the entanglement-structure trade-off imposed by quantum monogamy: classical systems achieve spatial structure only at the cost of negligible mutual information, while MBL sustains both.