Gravity from the Recognition Ledger: A Finite Sum over Geometries with Einstein's Equations at Every Order

Events are records on a ledger: four counters advance one step, a posting, at a time, and the records form a lattice. Quantum gravity on that lattice is a sum over geometries, each carrying a phase set by an action. Each triangle of the lattice, a hinge, adds to the action a hinge charge that depends on its area and deficit angle, the curvature it carries; Regge's charge is their product times an action scale. This paper asks why that charge. The records form the Kuhn lattice, with flat metric I − κ11ᵀ, and the paper identifies κ = 1 on the transfer reading of light's lattice step. At that value the Lorentzian sum over Regge geometries, with a model measure, is finite on every finite window of squared edge lengths. At leading order in the lattice spacing, the metric part of the field equation is Einstein's at every order in the field amplitude. Both are published, for Regge's action. The new results, proved in full, answer by two routes. Gauge consistency asks that the lattice's version of a change of coordinates remain a symmetry where Einstein's theory needs one. Take a charge that is the deficit times a rate depending on the area alone. Near flat space and through cubic order, the gauge tests use only that rate's value, slope and curvature at each flat area. On the Euclidean lattice, κ = 0, they allow exactly Regge's values, up to a common scale, so gauge consistency fixes the action through cubic order. The record clause says that one posting on a hinge carries an action proportional to the hinge's area. Unlike the tests, it constrains the whole function: with it, such a charge, or one depending only on the product Aδ, is exactly Regge's, scale included. At κ = 1 the lattice's cubic vertex is Einstein's, with the normalization of (1/2)∫√|g| R, so s Σ_h A_h δ_h is the Einstein–Hilbert action at G = 1/(8πs) through cubic order at long wavelength. It claims no unitarity, black holes or massless graviton, and names every input it uses. At one loop the computed zero-momentum term is nonzero, with or without the derived weight (numerical). Under the paper's premises on the action, whether that excludes a massless graviton rests on three open inputs: a positive lower bound on that term's imaginary part, a complete-gauge bound on how the self-energy changes near zero momentum, and a proof that its real part is not zero.