Bandlimited Green's Functions from Minkowski Isometry: A Geometric Derivation of Celestial Conformal Weights

We present a complete, parameter-free derivation of the bandlimited Green's function G = \sin (\sqrt{- \sigma^2}) and the celestial conformal weights \Delta_l = l + 1 from a single input: four-dimensional Minkowski spacetime (M,\eta_{\mu \nu}). The derivation proceeds in three steps. First, the exactness of the exponential map \exp_p: T_p M \to M in flat spacetime, combined with the requirement that discrete sampling be isometrically equivalent to the continuous field, uniquely determines—via the Whittaker interpolation theorem—the reproducing kernel G = \sin (\Omega \sqrt{- \sigma^2}). Second, the null geodesic locus \sigma^2 = 0 emerges as the natural boundary through the reproducing kernel normalisation condition K(x,x) = 1; restriction to this null hypersurface induces a signature flip from Lorentzian (- , + , + , +) to Euclidean (+, +) on the transverse S^2. Third, the SL(2,\mathbb{C}) principal-series representation on the Euclidean celestial sphere, combined with the spherical Bessel decomposition of G, yields \Delta_l = l + 1 as a pure spectral theorem with no free parameters. The result is cross-validated by five independent routes: Kempf's operator-theoretic reconstruction, the present geometric construction, a boundary RKHS derivation, Pasterski-Shao-Strominger from scattering amplitudes, and Gover-Shaukat-Waldron tractor calculus providing the SO(4,2) group-theoretic skeleton explaining why all five routes converge. The scale \Omega is structurally irrelevant: all physical conclusions depend only on the Minkowski metric. We identify the null-geodesic data set as a natural basis for geometric consistency checks, and note that if the universe is a quantum state, the multi-path convergence in principle circumvents the classical cosmic variance bound.