Regge's Slope and the Long-Wave Einstein–Hilbert Vertices from Flat-Space Stationarity on the Recognition Lattice

Take a lattice action of Regge-calculus form. On the periodic Kuhn triangulation of four-dimensional record space, each hinge contributes a smooth charge C(A, δ) of its area A and deficit angle δ, and the charge vanishes at zero deficit. Ask that flat space be a solution in every frame in which the lattice's time diagonal is uniformly stretched beyond the value κ = 1. Then the charge's slope ∂δC(A, 0) must equal Regge's, σ(1) A, at every area A ≥ 1/2. The proof turns the vacuum condition into two functional equations whose only continuous solutions are linear. With that slope, the long-wave quadratic action on plane waves around the flat κ = 1 lattice is exactly σ(1)/ħ times the Fierz–Pauli action, and the cubic vertex is the same multiple of the Einstein–Hilbert cubic vertex. Both come from one common ½∫√|g| R in lattice units, whatever the charge does at higher order in δ. Newton's constant stays free. Two cautions belong with the result. First, the vacuum condition is equivalent to Regge's slope, so the theorem recasts Regge's slope as the statement that flat space stays a vacuum at every time-stretch; it does not derive Einstein's equations from nothing. Second, it covers the long-wave plane-wave vertices at second and third order around flat space, not the full nonlinear theory. The condition is not a formality: a smooth charge can meet every classical long-wave clause and still fail it, and stationarity beyond any threshold above κ = 1 is not enough. Extensivity of the charge over the pieces of a hinge gives the same conclusions and is strictly stronger. For the Regge action itself, the lattice's own long-wave solutions at κ = 1 with point-mass jets have post-Newtonian parameters γ = β = 1, and the isotropic Schwarzschild expansion is such a solution near a point, through second order. Every theorem is formally verified. Preliminary preprint; not peer reviewed.