The Long-Wave Regge Graviton on the κ = 1 Kuhn Lattice and the Recognition Cost's Quadratic Form

On the Kuhn lattice, a triangulated four-torus with flat Lorentzian metric I − 𝟏𝟏ᵀ and time along 𝟏 = (1, 1, 1, 1) (κ = 1), we take Regge's action as the gravitational action. On a torus of side N, N⁻² times its flat second variation on a plane wave of fixed integer wave vector tends to −√3π²/2 times the Fierz–Pauli form: a preprint result, confirmed here by an independent exact computation over ℚ(√3), Schläfli's identity included (inputs printed, script deposited). On the graviton's two polarizations, transverse-traceless strains H orthogonal to 𝟏, Fierz–Pauli is the squared wave covector times tr H², twice the quadratic part of the recognition cost J(x) = ½(x + x⁻¹) − 1, an identity that uses J only through J″(1) = 1. In the Euclidean hypercubic model, not on the lattice, cube symmetry and gauge invariance fix Fierz–Pauli up to one factor. The ledger's charge, a quadratic form on spatial strains fixed by premises on how strained lattice steps are recorded, is, given the lattice's symmetries, ½ tr H² on every polarization exactly when it is so on the plus and cross strains of the face wave (1, 1, −1, −1) along the face axes (1, −1, 1, −1) and (1, −1, −1, 1); at a three-fold-axis wave, or for a rotated pair, two such numbers do not suffice. Three premise routes, none derived, give both numbers; two writing rules that the premises exclude fail only by birefringence, the premises' sensitive test inside the paper. The thesis is stated, not tested: The graviton is not a field added to the recognition ledger. It is the cost the ledger charges for comparing a geometry with flat space. These premises make it precise on the polarizations; nothing here about the graviton or the ledger's charge uses more of J than J″(1) = 1 and J > 0 away from 1. The scope is linear gravity around flat space.