Emergent Minimal Gauge Coupling from C4v Symmetry Reduction on the 4.8.8 Lattice

We analyse the symmetry structure of the continuous-time quantum walk on the 4.8.8 (truncated square) Archimedean tiling, whose bifurcated band structure was recently char- acterised via OTOC echo protocols [1]. Decomposing the 8 ×8 Bloch Hamiltonian into irreducible representations of the C4v point group at the Γ point, we show that the gapped slow branch transforms as the fully symmetric scalar representation A1, while the gapless fast branches transform as the two-dimensional vector representation E. At Γ, the Wigner-Eckart theorem enforces strict orthogonality between these sectors: ⟨A1|H|E⟩= 0. Away from Γ, symmetry reduction C4v →Cs breaks this selection rule. We compute the inter-branch matrix element exactly and find M=− i 2 sin kx for the Ex component, yielding a momentum- linear vertex M∝i(k·E) that is structurally identical to the p·A minimal coupling of quantum electrodynamics. We then promote the tight-binding hopping amplitudes to dy- namical U(1) link variables in a Wilson lattice gauge theory, demonstrate via a weak-field expansion that the gauge-covariant Hamiltonian reproduces the same vertex, and identify a two-coupling structure (β4,β8) rigorously locked by the tiling geometry. We formulate the velocity-unification conjecture—that renormalisation group flow drives the distinct bare group velocities toward a common infrared fixed point—as a concrete numerical programme for lattice Monte Carlo simulation.