Theory of the Gravitational Constant. Part VI. Non-Linear Gravitation. Formulation of the Newton 2.0 Law

This paper represents Part VI of a comprehensive investigation into effective gravitational coupling and presents the mathematical derivation and verification of the vector-tensor law of non-linear gravitation ("NEWTON 2.0") across galactic and local scales. Based on the rigorous integration of wave solutions of the Klein–Gordon–Yukawa equation for the massless graviton regime (α = 0) using the Dirac delta function formalism, a linear variation law for the effective gravitational constant is derived and generalized via Maclaurin series summation into the non-linear exponential function: Geff(R) = GN · exp(√(aN / a(R))). For the first time, the reference baseline acceleration constant aN = 6.6743 × 10⁻¹¹ m/s² (numerically equal to the Newtonian gravitational constant GN) is introduced. It is mathematically proven that the force dynamics equation incorporates negative feedback that prevents infinite potential growth and ensures the stability of dynamical systems. Crucially, without introducing artificial interpolation functions or a fourth-root mass dependence (V ∝ M¹/⁴) a priori, the intrinsic dynamics of this exponential series naturally converges on galactic scales into a flat velocity plateau, providing the first rigorous mathematical foundation for Modified Newtonian Dynamics (MOND) and the empirical Tully–Fisher relation. Within the developed "dominant field principle" and vacuum vector anisotropy, the 1798 Cavendish experiment is mathematically validated, demonstrating that Cavendish's measured value GN = 6.754 × 10⁻¹¹ m³·kg⁻¹·s⁻² (Δ = 1.1933%) is a direct physical consequence of non-linear gravitation in weak horizontal fields rather than instrumental error. Furthermore, the NEWTON 2.0 model explains outer planetary anomalies and the Pioneer anomaly, and precisely reproduces the stellar rotation curves of the Milky Way, Andromeda (M31), and the giant elliptical galaxy IC 1101 using pure baryonic mass without invoking non-baryonic Dark Matter.