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. This paper addresses Open Problem 1 (OP1) of the FoR programme — the derivation of Newton's constant G without G as prior input. The one-loop beta function for the Skyrme coupling e is derived from chiral perturbation theory (Gasser-Leutwyler): β(e) = +b_Sk·e³ with b_Sk = 3/(256π²). The positive sign means e grows toward the ultraviolet with no non-Gaussian fixed point in isolation. When the coupled gravity-Skyrme system is considered, gravitational damping of the Skyrme running introduces a term −c_lin·g̃·e/(1−2λ̃)² into β(e). At the Asymptotic Safety gravitational UV fixed point (g̃*, λ̃*), the balance β(e*)=0 uniquely determines a non-Gaussian Skyrme fixed point: (e*)² = c_lin·g̃*/[b_Sk·(1−2λ̃*)²], with e* ≈ 2.3–7.0 depending on AS scheme. The core structural prediction follows: G·M_Sk² = K·ħc, where K = g̃*·C_M²·C_a²/(2π²·(e*)⁴), C_a = √2·π (exact, FCC X-point geometry), and C_M = 73 (Adkins-Nappi-Witten Skyrmion profile). Every factor in K is determined by the UV fixed point without G as input. Numerically K ≈ 28–800 (scheme-dependent), giving M_Sk ≈ 5–28 × M_Pl. Newton's constant is not independently fundamental: G = K·ħc/M_Sk² is the gravitational coupling set by the mass of the fundamental topological object (the preon Skyrmion). The one remaining open step is the non-perturbative computation of c_lin from the Seeley-DeWitt heat kernel of the Skyrme operator in the curved SU(2) TEGR background, which would fix K precisely and make G a pure prediction in SI units. This paper is part of the Field of Resonance series. Related works: Paper 1 v9c (DOI: 10.5281/zenodo.19221656), Paper 6 v2b (DOI: 10.5281/zenodo.19244557).