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

The recognition cost J(x) = ½(x + 1/x) − 1 is the one continuous cost, zero at x = 1, that obeys its composition law and the calibration J″(1) = 1. This paper's thesis, an identification and not a theorem, is: 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. A geometry here is a metric matrix in the counter coordinates, and the thesis is read on transverse-traceless strains in the gauge H·1 = 0. Its precise form is Proposition 6.2: if the ledger's charge Q of spatial strains has the lattice's symmetries, charges one polarization of one wave as J does and charges the two alike, then Q is J's long-wave form J_lw(X) = ½ tr X² on every transverse-traceless strain. These properties follow, at a record charge set by J, an identification, from either of two named premise sets, neither derived: the ledger writes each strained step by its least-charged record, or by one comparison with local shares. Take Regge's action on the κ = 1 Kuhn lattice as the gravitational action, a model, with κ = 1 an identification. For every integer wave m that is not a multiple of the diagonal and every such strain H, lim N⁻² H_N(um,H, um,H) = −(√3π²/2) η(m) · 2Q(H) as N → ∞, with H_N the flat Hessian on the torus of side N and η(m) the squared covector. The identity behind this, Fierz–Pauli = η(m) · 2J_lw on these strains, uses J only through J″(1) = 1, so by itself it does not test the thesis. Regge's action fixes the whole transverse-traceless form; J's calibration only sets its unit: the coefficient is one in the Fierz–Pauli normalization. Locality, in the Euclidean model, fixes the trace part. None of them fixes the strength. At null waves both sides vanish: the identity concerns the off-shell kinetic operator on the helicity-±2 strains in the lattice's rest frame. New here are lattice results: two numbers at one wave fix a lattice-symmetric charge on every polarization, equal step weights are birefringent, and cube symmetry with gauge invariance fixes Fierz–Pauli up to one factor. Through the posting identification, Għ = λ_rec²/(4 ln 2): this fixes the lattice length λ_rec and does not predict G. The scope is linear gravity around flat space.