Newton's Constant from the UV Fixed Point: Gravitational Damping of the Skyrme Coupling and the Structural Prediction G·M_Sk² = Kħc

Paper 7 of the Field of Resonance (FoR) series, version 2. This version substantially extends v1a (DOI: 10.5281/zenodo.19246232) with three new results and a new section situating the work within the Asymptotic Safety programme. The gravitational damping coefficient c_lin is derived from first principles via the Wetterich functional renormalisation group trace for the dimension-6 Skyrme operator in the curved SU(2) TEGR background. Using the Barvinsky-Vilkovisky heat kernel ratio a₄(□²)/a₄(□) = 1/2 for biharmonic operators, the result is c_lin(g̃,λ̃) = 5g̃/[12π(1−2λ̃)²], replacing the placeholder estimate of v1a. The three-coupling stability matrix of the fully coupled gravity-Skyrme system yields a new critical exponent θ_e ≈ −1.74. Since θ_e < 0, the Skyrme direction is UV-irrelevant: e* is uniquely fixed by the gravitational fixed point with no additional free parameter. The theory has exactly two UV-relevant directions, as required by Asymptotic Safety predictivity. The structural prediction G·M_Sk² = K·ħc is expressed in terms of the scheme-independent critical exponents θ₁ and θ₂ as K ≈ 0.054·θ₁³·⁹³·θ₂⁻⁵·²⁹. Using the Padé-resummed result θ₁ = 2.49 ± 0.03, θ₂ = 1.54 ± 0.05 (Falls-Litim-Platania 2020), this gives K ≈ 0.20–0.36 and M_Sk ≈ 0.45–0.60 × M_Pl — the preon Skyrmion mass is predicted to be slightly sub-Planckian. This is a factor of ten improvement in predictive range over v1a (which had K ∈ 28–800).