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. The supplement (version 1.0, archived with the paper) holds the tables T₂, T₃ and T₄ as rational functions of κ and short scripts that recheck in exact arithmetic every identity and certificate the proofs use, in eight parts, S1 to S8, each with its own README: the tables (S1); the derivatives of the Euclidean and Lorentzian simplex functions against the tables, the sixty shape-cost tables and the lower jets of the shape costs (S2); the quadratic and cubic vertex identities, the lower orders and the paper's two examples (S3); the quartic vertex identity (S4); the total-derivative step to √|g| R (S5); the fifteen-class quadratic form, the body diagonal, the decoupling of the non-metric modes and the quartic obstruction with them (S6); the Noether identities, gauge invariance, the gauge-parameter lemma, helicity selection and the MHV value at κ = 0 and κ = 1 (S7); and the certificates of the quartic obstruction in the metric sector (S8). Its index maps each result of the paper to its checks, and its file SHA256SUMS lists the SHA-256 hash of every file; the paper prints the hash of SHA256SUMS.