Paper VIII: The Quantum Tikbon Field Theory on S3

The complete one-loop quantum field theory of the Tikbon gravitational gauge field on S³ × ℝ is derived from the axioms of Quantum Geometrodynamics (QGD). Four results are established. Result 1 — Unitarity. The Ward–Takahashi identity is preserved at one loop. The curvature coupling γ A_μ A_ν G^μν contributes a momentum-independent constant to the vacuum polarisation. The contracted Bianchi identity ∇_μ G^μν = 0 ensures that this constant does not source longitudinal modes: the non-transverse contribution is absorbed into the background Einstein equation, leaving the propagating sector exactly transverse. The associated β-function contribution vanishes identically, because the composite-graviton structure of QGD (Paper V) requires a second Tikbon loop before G_μν can fluctuate at scale μ. Result 2 — Running of G. The exact one-loop beta function μ dg_T/dμ = n_f g_T³ / (12π²), n_f = 3 (Paper III), combined with G = g_T²/(4π), yields the running Newton constant G(μ) = G₀ / [1 − (n_f G₀)/(3π³ℏc) · (μ² − μ₀²)]. Result 3 — Self-consistent UV fixed point. The S³ momentum lattice imposes a physical UV cutoff μ* defined self-consistently by the condition G(μ*) μ²/(ℏc) = μ²/M_P²(μ*). The unique solution is μ* = √[π³/(π³ + n_f)] · M_P ≈ 0.984 M_P, with dimensionless coupling g̃* ≈ 0.968 < 1. The theory is perturbatively controlled up to and including the last physical mode. The Landau pole at μ_pole ≈ 5.6 M_P lies outside the physical spectrum and is therefore inaccessible. Result 4 — The ℏG constraint. The trade-off law R_g · r_c = ℓ_P², combined with the running of G(μ), imposes a renormalisation-group invariant: ℏ_eff(μ) · G(μ) = ℏ · G₀ = const. This is the quantum-level expression of the topological rigidity of the Planck area: ℓ_P² does not run. As G increases with μ, the effective action quantum decreases — the gravitational sector becomes more classical at higher energies, in direct inversion of the QED behaviour.