Quantum Geometry of a Tetrahedron in Chrono-Grid Dynamics: A Numerical Study

We present a detailed numerical study of the quantum geometry of a single tetrahedron within the framework of Chrono-Grid Dynamics (CGD). Starting from spin-network states on a tetrahedral lattice, we construct the kinematical Hilbert space spanned by admissible spin configurations satisfying triangular inequalities on every vertex. Using the 6j symbols of SU(2), we define a Hamiltonian operator whose eigenstates are candidates for physical states satisfying the Hamiltonian constraint. The Hamiltonian matrix is built in a sparse format and diagonalized using iterative methods, allowing us to reach truncations up to spin jmax = 2.0 (2371 states) and a sampled estimate for jmax = 2.5 (~7000 states). We analyze the ground state energy, the expectation value of the volume, the inverse participation ratio (IPR), the Shannon entropy, and the full energy spectrum. The ground state energy converges to zero within numerical precision, confirming that the Hamiltonian constraint can be satisfied. The ground state is highly delocalized, with IPR indicating participation of more than half of the Hilbert space. The quantum expectation value of the volume is significantly larger than the classical volume of the most probable configuration, revealing strong quantum fluctuations and the non-classical nature of the emergent geometry. These results provide the first numerical evidence that CGD yields a consistent quantized geometry and pave the way for studies of larger lattice systems.