Chrono‑Grid Dynamics in 2+1 Dimensions: From Discrete Evolution to Topological Quantum Gravity

We extend Chrono‑Grid Dynamics (CGD) to 2+1 dimensions. Starting from a 2D triangular lattice with spin‑1/2 degrees of freedom on sites and dynamical SU(2) link variables, we construct a unitary discrete evolution with fundamental proper‑time step τ (the Takt). In the continuum limit, the link variables become the Ashtekar connection and the conjugate momenta become the densitised triad, leading to the canonical formulation of gravity. We derive the constraint algebra on the lattice and show that it closes, giving rise to the Dirac algebra of general relativity in the continuum. Special attention is paid to edge modes, which appear on the boundary of the lattice and carry topological degrees of freedom. We quantize the theory à la loop quantum gravity, obtaining discrete spectra for length and angle operators. When a cosmological constant Λ is introduced, the curvature constraint regularizes to a quantum group SU_q(2) with q=e^{iħ√Λ}, and the model reduces to the Turaev‑Viro topological field theory. This provides a rigorous link between CGD and established topological quantum gravity models, paving the way for a full 3+1D generalization.