On Its Two Polarizations the Graviton's Strain Factor Is Twice the Recognition Cost's Long-Wave Form in the Fierz–Pauli Normalization

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. Version 1.1. Version 1.1 answers the review of version 1.0. Its central correction is the statement of the thesis's precise form: the two numbers that fix a lattice-symmetric charge of spatial strains on every polarization are now named where they hold, on the plus and cross strains of the face wave (1, 1, −1, −1), diagonal and off-diagonal in the frame of the face axes, and a new paragraph proves that two numbers at a wave along a three-fold axis, or for a rotated pair, do not suffice. The paper is retitled "The Long-Wave Regge Graviton on the κ = 1 Kuhn Lattice and the Recognition Cost's Quadratic Form" and now leads with what it proves: the long-wave limit of Regge's action on the κ = 1 Kuhn lattice, −√3π²/2 times Fierz–Pauli, confirmed independently; Fierz–Pauli on the two polarizations as twice the recognition cost's quadratic form, an identity that uses the cost only through J″(1) = 1; the cube-group lemma, in the Euclidean hypercubic model; and the three premise routes for the ledger's charge, with birefringence as their sensitive test, while the thesis keeps the author's words and is labelled as stated, not tested. The exact computation of Section 6.3 is now reproducible: Appendix A prints the flat derivatives of one simplex, Schläfli's identity and its first derivative are checked exactly at the flat lattice, and the script, in exact arithmetic over ℚ(√3), is released with its output tables in the supplementary files of this record. All the other referee items are addressed, among them the definitions of the strain factor and the ledger's charge, the long-wave qualifier with the finite-N polarization splitting (about 2.78/N²), the uniqueness of the cost from d'Alembert's equation, the comparison cost read as the Jeffreys divergence with its Fisher–Rao and DeWitt quadratic part, the physical content of each identification, a shorter abstract, the literature on linearized Regge calculus, and the known cost of κ = 1, its massless corner doublers. Supplementary files: regge_exact.py (the Section 6.3 exact computation in exact arithmetic over ℚ(√3)), its output regge_exact_output.txt, and README.md.