Emergent Hyperbolic Geometry from Informational Renormalization Dynamics A Bridge Between IVES and Tensor Network Models

We study the spectral properties of the linearised renormalisation-group (RG) operator arising from MERA-like tensor-network circuits. Using a finite-dimensional approximation of the coarse-graining map, we construct the Jacobian of the RG transformation via finite differences and analyse its spectrum across ensembles of unitary disentanglers. We perform both Haar-random sampling and structured two-parameter scans of the circuit space, together with numerical RG flow simulations. Our results show that a significant fraction of generic circuits exhibit contractive behaviour, while a connected region in parameter space associated with causal light-conepreserving disentanglers forms a stable basin under RG flow. Monte Carlo statistics and phase diagrams indicate non-trivial structure in the universality classes of entanglement renormalization dynamics. We emphasise that this work is purely numerical and concerns the stability properties of quantum information RG maps. Any connection to spacetime physics is purely interpretational and not part of the formal results.