On Self-Dual Measure and Finite Quantum Gravity

Renormalization exists because the Feynman path integral assigns equal weight to all loop momenta, and the theory has no mechanism to prevent it. We show the correct measure for quantum gravitational scattering is the self-dual Haar measure on (ℝ⁺,×), whose Mellin–Plancherel spectral weight P(λ) = πλ/sinh(πλ) decays exponentially. Self-duality under λ ↦ −λ bounds the loop measure on both sides. No UV divergence can arise. No counterterms are needed. The bare spectral weight at every loop order L ≥ 1 is rational: ℳ₁ = 1/8, ℳ₂ = 1/90, ℳ_L ∈ ℚ. These satisfy ℳ_L = C_L·ζ(2L)/π^{2L}. Rationality proved for all L. One-loop scalar box verified to 15 significant figures, 19.6× speedup. Two-loop sunrise verified to 5.7×10⁻¹⁶.