The Angular Tetravalent Lattice: A Classical Geometric Model with a Non-Unitary Quantum-Walk Extension and Chiral Edge Modes

We introduce the discrete angular tetravalent lattice as a minimal classical geometric model. The system consists of a four-component density evolving under strict +90° cyclic permutations and conditional spatial shifts. We construct a minimal non-unitary quantum walk with complex amplitudes and prove that, under a full quantum dephasing channel, its intensity evolution exactly recovers the classical tetravalent process. By analysing the continuum limit of this decohered process we demonstrate a duality: in the passive regime the system yields a dissipative complex wave equation, while in the active regime the continuum dynamics acquire the algebraic skeleton of the two-dimensional Weyl operator ∂x + i∂y. Quantitative scaling tests confirm a wave-speed duality: the absolute information edge propagates at the hyperbolic light-cone speed c = 1, while the macroscopic density ridge propagates at c = 1/2. Furthermore, a spatial domain wall between opposite chiral cycles (A and Aᵀ) traps a robust one-dimensional chiral edge current driven by classical skipping orbits; the current remains stable under strong spatial disorder. We discuss the scope of this construction relative to phenomenological active quantum walks, clarifying the boundary between classical geometric mechanisms and fully coherent multi-sector quantum dynamics.