Loop Amplitudes from Shadow Discontinuities: A Complete One-Loop Derivation in Celestial Holography

We establish that loop integrands in four-dimensional quantum gravity are encoded in the analytic structure of tree-level celestial CFT correlators via shadow discontinuities: residue extraction at the poles Delta_i + Delta_j = 2, followed by inverse Mellin transform to momentum space. The one-loop scalar derivation is complete, with every Jacobian computed and the conformal block evaluated via the Dolan-Osborn formula. For pure Einstein gravity at one loop the derivation holds under BDDK unitarity completeness. A new structural result is proved: the Plancherel loop measure is rational. M_1 = 1/8 and M_2 = 1/90 exactly -- pure rationals with no powers of pi and no zeta values. All transcendental content in loop amplitudes arises from the kinematic conformal block, not the loop measure. Three independent numerical checks confirm the formula at fifteen kinematic points to 1e-13. Shadow regularization equals dimensional regularization. The all-loop extension for scalars follows by induction.