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 dgT/dμ = nf gT³ / (12π²), nf = 3 (Paper III), combined with G = gT²/ (4π), yields the running Newton constant G (μ) = G₀ / 1 − (nf 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/MP² (μ*). The unique solution is μ* = √π³/ (π³ + nf) · MP ≈ 0. 984 MP, with dimensionless coupling g̃* ≈ 0. 968 < 1. The theory is perturbatively controlled up to and including the last physical mode. The Landau pole at μₚole ≈ 5. 6 MP lies outside the physical spectrum and is therefore inaccessible. Result 4 — The ℏG constraint. The trade-off law Rg · rc = ℓP², combined with the running of G (μ), imposes a renormalisation-group invariant: ℏₑff (μ) · G (μ) = ℏ · G₀ = const. This is the quantum-level expression of the topological rigidity of the Planck area: ℓP² 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.