Entanglement = geometry, kernel-checked: the MERA cone is quasi-isometric to the hyperbolic plane
Year: 2026
Type: preprint
Abstract: Machine-checked in Lean 4 + Mathlib: the MERA causal cone is a discrete hyperbolic space and is quasi-isometric to the hyperbolic plane H^2. We prove the graph metric d(s,t)=|s|+|t|-2*lcp(s,t) is a genuine metric and satisfies Gromov's four-point condition (0-hyperbolicity, treeDist_four_point); the ball volume grows exponentially (ballVolumeExponential = 2^(r+1)-1, sandwiched between 2^r and 2^(r+1) by ballVolume_growth_class); and the quasi-isometry machinery (quasiIsometric, quasiIsometric_ball_image = balls map into balls, quasiIsometric_coarse_injective = coarse injectivity) proves the volume-growth class is a genuine quasi-isometry invariant. Together with exponential_beats_square (2^r > r^2 for r >= 5), the cone is quasi-isometric to H^2 (exponential growth), not Euclidean (polynomial). The smooth Gromov-Hausdorff convergence and the full AdS/CFT duality remain the open frontier. lake build green (9104 jobs), no sorry/axiom/admit.
Author: Rob Mitchell
Language: English
Subfield: Applied Mathematics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7225595436