On Self-Dual Measure and Finite Quantum Gravity

The Feynman prescription weights every loop energy equally and diverges. We replace the flat spectral weight with the Plancherel measure of SL (2, ℂ): P (λ) = πλ/sinh (πλ), the unique weight forced by locality, unitarity, and dimensional reduction in the shadow spectral representation of celestial holography. Stripped of kinematics, the bare L-loop measure obeys the unconditional bound: ℳL ≤ (1/8) L, so the loop expansion is ultraviolet finite at every order with no regularization and no counterterms. The measure separates cleanly from the kinematics: all transcendental content of an amplitude resides in the conformal block KL (s, t), never in ℳL. At one and two loops the measure is exactly rational: ℳ₁ = 1/8 and ℳ₂ = 1/90, the Bernoulli fractions of ζ (2) and ζ (4) ; we show that this happens precisely because the ladder's path graph coincides with the complete graph only at L ≤ 2, and we show by an elementary wall-count invariant that the coincidence cannot recur at any higher order, with the certified 44-digit value of ℳ₃ failing every integer-relation test for rationality. Loop integrands arise as shadow discontinuities of tree amplitudes, extending the method first presented in Loop Amplitudes from Shadow Discontinuities; the one-loop box is verified to fifteen significant figures with a 19. 6-fold speedup over the Feynman parametrization.