Nonlinear Optics without Constitutive Postulates: The Charlton Geometry Route to χ^(3)

We derive linear refraction (Snell and prism optics) in Charlton Geometry as a transfer identity of geometric optics across an interface, with the refractive index encoded as a spatial conformal factor in the optical metric. We then prove that a Kerr-type cubic nonlinearity is unavoidable in CG because the constitutive map F ↦ G := *{η[ω]} F depends on ω, so wave-induced perturbations of ω feed back into *{η[ω]} and generate cubic response in the excitation. We present an explicit susceptibility tensor χ^(3) as a contraction of the first metric-variation of the Hodge star with the congruence-kinematic decomposition of ∇u, and we reduce it to an isotropic scalar law χ^(3) ∼ c₁ θ + c₂ ‖ϖ‖² under a standard isotropy hypothesis. Two calibrations are left explicit as obligations: the mapping from CG-normalized fields to SI units (vacuum impedance) and the resulting numerical scale of n₂.