Cubic Anisotropic Lorentz Violation: A Distinctive Falsifiable Signature of the Discrete Cost Metric

The minimal discrete épure carries an L¹ (rhombic) cost metric, not an isotropic continuum. We show this implies a distinctive form of Lorentz violation: anisotropic and of cubic symmetry. Expanding the lattice propagator, the leading correction scales as Σᵢkᵢ⁴ (sum of fourth powers per axis), not as (Σkᵢ²)² (isotropic); the difference is the anisotropy. For a line of sight n, the high-energy photon speed acquires a directional modulation v(E,n)=c[1−ξ(E/E_Pl)²A(n)] with A(n)=Σnᵢ⁴−3/5, ranging from −0.27 (diagonal) to +0.40 (axis), the coefficient tied to the previously derived ξ₂=37/1296. The observable consequence is a cubic sky pattern in the energy-dependent time delays of gamma-ray bursts, with maxima along three orthogonal axes. This is sharply discriminating: general relativity predicts no Lorentz violation; standard and loop-quantum-gravity Lorentz violation is isotropic (a function of energy alone); the discrete cost metric alone predicts cubic anisotropy, because its (1,2) structure breaks isotropy where an isotropic spin network does not. We state the reservation honestly: the pattern presupposes a preferred orientation of the cosmic lattice; if averaged isotropically on large scales, the amplitude is washed out — but the presence or absence of the cubic pattern is falsifiable. The form is derived; the amplitude is conditional. Test with Fermi-LAT and CTA.