Emergent Einstein Gravity from Discrete-Time Quantum Walks with SU(4) Coin Operators

We show that linearized Einstein gravity emerges from discrete-time quantum walks (DTQWs) with SU(4) directional coins on a lattice, with no metric or Einstein–Hilbert action imposed. The 15 generators of su(4), realized in a Dirac gamma-matrix basis of Cl(3,1), naturally encode the connection (6), vierbein (4), torsion (4), and cosmological (1) sectors. The Wilson plaquette action reproduces the MacDowell–Mansouri gravitational action. Palatini reduction of the full 30×30 Hessian yields massless spin-2 dispersion, Newtonian potentials Φ/Ψ ≈ 1, and 2 candidate propagating modes. A 4D one-loop fermion determinant dynamically generates a torsion mass gap consistent with a physical continuum-limit phenomenon. Code and data for full reproducibility are included.