On Self-Dual Measure and Finite Quantum Gravity

Renormalization exists because the Feynman path integral assigns equal weight to all loop momenta, including arbitrarily large ones, and the theory has no mechanism to prevent it. We show that the correct measure for quantum gravitational scattering is not the flat measure d4ℓ/(2π)4 but the self-dual Haar measure on the multiplicative group (ℝ+,×), whose Mellin–Plancherel spectral weight P(λ) = πλ/sinh(πλ) decays exponentially as λ → ∞. Because this weight is self-dual under λ ↦ −λ, the spectral signature of the orientation exchange across null infinity, the loop measure is bounded on both sides simultaneously. No ultraviolet divergence can arise. No counterterms are needed. The renormalization program, with its regularization schemes, renormalization group flows, and order-by-order subtraction of infinities, is not incorrect; it is unnecessary given the right measure. The evidence is direct. The bare spectral weight at every loop order L ≥ 1, stripped of all kinematic data, is a positive rational number: ℳ1 = 1/8, ℳ2 = 1/90, ℳL ∈ ℚ>0 for all L. The transcendental content of loop amplitudes, factors of π, values of ζ(n), polylogarithms, enters exclusively through the kinematic conformal block KL(s,t), not through the measure. This separation, invisible in the Feynman parametrization, is manifest in the shadow spectral representation and follows from a single algebraic fact: Euler's formula ζ(2m) = (2π)2m|B2m|/(2·(2m)!) cancels every power of π introduced by the L-fold convolution of P, leaving Bernoulli arithmetic. The rationality is proved as a closed-form theorem for all L. The one-loop scalar box integral is verified against the standard result to 15 significant figures with no regularization. The 19.6-fold improvement in computational efficiency over Feynman parametrization at one loop is a structural consequence of the dimensional reduction: the shadow spectral representation converts a 3L-dimensional Feynman integral into an L-dimensional spectral integral over the principal series Δ ∈ 1+iℝ. The framework is the shadow discontinuity method in celestial holography: one-loop integrands arise as discontinuities of six-point tree amplitudes across shadow poles Δi+Δj=2; L-loop integrands as L-fold discontinuities of (4+2L)-point tree amplitudes. The loop expansion is not an independent quantization procedure added to the classical theory. It is an analytic unfolding of tree-level data already present in the celestial operator algebra, controlled at every order by the same measure whose self-duality encodes the orientation exchange at the boundary.