Paper VI: The Resolution of the Dark Matter Problem from Quantum Geometrodynamics

We demonstrate that the "dark matter" phenomenology observed at galactic, cluster, and cosmological scales emerges entirely from the nonlinear self-interaction of the gravitational field within the Quantum Geometrodynamics (QGD) framework, with no exotic particles and no free parameters. The argument proceeds in three stages. Galactic scale. The universal acceleration a₀ = c²/(2πRU) is derived from two independent routes — the Planck force distributed over the cosmic mass, and the intrinsic centripetal scale of the S³ geometry. The coupled Tikbon–Einstein field equations in static spherical symmetry reduce to a nonlinear master ODE whose unique power-law vacuum solution has exponent n = 1 (force ∝ 1/r). Matching to the Newtonian interior at aN = a₀ yields the MOND force law aobs = √(aN · a₀) and the baryonic Tully–Fisher relation vflat⁴ = GMb a₀. The predicted Milky Way rotation velocity (219 km/s) agrees with observation (220–230 km/s) to within ~1%. Cluster scale. Standard MOND underestimates cluster masses by a factor of ~2. We show that the nonlinear cross-terms in the Tikbon master equation, activated when two sub-clusters' MOND regions overlap, produce an interference amplification that raises the Bullet Cluster mass ratio from KMOND = 2.42 to KTikbon = 4.38, within 10% of the observed Kobs = 4.84. The solitonic character of the Tikbon field explains the spatial offset between gas and lensing mass. An independent holographic derivation via the CFT Ward identity confirms the interference pattern from the boundary theory. A universal mass ceiling Mmax(R) = a₀R²/G is derived from Gauss's law; a survey of 10 galaxy clusters confirms that no system violates this bound. Cosmological scale. From Gauss's law applied to the S³ volume, the ratio of the effective gravitational mass (induced by a₀) to the critical mass yields: ΩDM = 1/π ≈ 0.3183 in 1% agreement with the Planck 2018 value Ωm = 0.315 ± 0.007. This result contains zero adjustable parameters. All predictions follow from a single cosmological input — a₀ = c²/(2πRU) — which is itself derived from the QGD axioms. Dark matter is not a substance but the self-energy of the gravitational field on S³. Keywords: dark matter, MOND, Tully–Fisher relation, galaxy rotation curves, Bullet Cluster, Tikbon field, gravitational self-energy, S³ topology, cosmological density Related papers: Foundational axioms and dark sector in Paper I. Tikbon field theory in Paper V. Tri-Unity of forces in Paper IV. Dimensional analysis in Paper 0.