Metric Source Completions and Integrability in Celestial Gravity

RESUMO

We study metric-source responses of celestial Einstein amplitudes while retaining their momentum-conservation distribution. The consecutive opposite-helicity response is related to the simultaneous gravitational contact at arbitrary hard multiplicity. Their difference contains a Lorentz transformation that annihilates the conservation delta function, determining the normal recoil contribution and its relation to the undifferentiated coefficient. A six-graviton tree calculation evaluates both coefficients on a real scattering channel. For the summed homogeneous contact, momentum conservation cancels the leading angular singularities. Uniform bounds at coincident soft directions and hard punctures yield integrated endpoint estimates that determine the ordered Mellin residues after smooth source smearing. The mixed Mellin divisor also retains logarithms of soft--hard invariant ratios at generic conformal weight. In Weyl--Beltrami variables, the quadratic reference Ward response combines soft-leg, shadow-map and spin-frame terms into the transported metric Hessian, with a nonzero symmetric mixed susceptibility. Relative shadow primitives require boundary currents when represented across logarithmic cuts; their source variation fixes the accompanying contour contribution. Joint factorization and angular expansions organize the additional NMHV relative-contour terms into residue coefficients that distinguish metric integrability from matching to a specified source functional. An independent vacuum completion supplies the Euler response, which fixed-area harmonic combinations separate from local curvature terms at bounded derivative order.