On the Kuhn Lattice, the Long-Wavelength Vertices of Regge's Action Are One Half of Einstein's Through Quartic Order

On the Kuhn lattice, the long-wavelength (two-derivative) quadratic, cubic and quartic vertices of Regge's action around a flat background are one half of those of the Einstein–Hilbert density √|g| R, for all polarizations and conserved momenta, in the metric sector, on the Euclidean lattice and on the Lorentzian one with flat metric I−11ᵀ. The ½ is one normalization at all three orders, not a changed coupling. On the Euclidean lattice these identities also follow from Cheeger, Müller and Schrader's convergence theorem; here they come with explicit tables and a finite, exact proof. The quartic table, rational in the background parameter κ, is Regge's own fourth derivative at every Euclidean κ; its contact vertex is half Einstein's at every κ off three poles. For every κ > ½, beyond that theorem's reach, the Lorentzian action's derivatives are √(4κ−1) times the same tables. At κ = 1 the five non-metric edge modes decouple from these vertices at tree level; at κ = 0 the body diagonal is a non-gauge zero mode, and the results describe the metric truncation. The tree four-graviton amplitude from the tables is gauge invariant, with general relativity's helicity selection and MHV amplitude. No hinge shape cost, linear in the deficit with a coefficient set by the hinge's class and three squared lengths, completes a finite change of the coupling at quartic order (metric sector; κ = 0, 1/10, 1; one stated inference). Every theorem is proved in the paper or by exact computation in its supplement.