The Lattice Graviton Moves on the Lattice Laplacian's Cone: Exact dispersion of gravitational waves in Regge calculus on the Kuhn triangulation, its direction dependence, and the cosmic-ray bound on the lattice spacing

Linearize the Regge action about flat space on the Kuhn triangulation of the hypercubic lattice and ask how a gravitational wave moves. The answer is exact. The two transverse-traceless polarizations obey ∑μ 4 sin²(kμ a/2) = 0, the dispersion relation of the nearest-neighbor scalar Laplacian, to all orders in the spacing a, with no birefringence, and no other wave vector in the complex Brillouin zone carries a propagating mode. The proof rests on the factorization of the lattice weak-field action found by Roček and Williams; a forty-digit computation in the raw edge variables confirms it, and a fifth exact zero mode of the Hessian, the strain of the hypercube diagonal, is explained by the right-angled geometry of the triangulation. With a lattice axis as time the phase velocity is 1 − (1 + ∑j nⱼ⁴) k² a²/24, subluminal in every direction; the coefficient lies between 1/18 and 1/12, and of the four lattice directions the axis is the only one whose continuation gives a real, subluminal cone. This is a dimension-six Lorentz-violating graviton with coefficient s̄⁽⁶⁾(n̂) = (1 + ∑j nⱼ⁴) a²/12. If matter moves on the light cone, the survival of the highest-energy cosmic rays against gravitational Cherenkov radiation bounds a < 1.4 × 10⁻³¹ m, about 9 × 10³ Planck lengths, under the conservative reading of the data (iron primaries, a tenth of the energy in the emitting parton, sources at 10 Mpc); a Planck-scale spacing sits a factor 5 × 10⁷ inside the bound and loses partons to gravitons only above 9 × 10¹⁹ eV.