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Abstract Analytic continuation from Minkowski space to (2, 2) split signature spacetime has proven to be a powerful tool for the study of scattering amplitudes. Here we show that, under this continuation, null infinity becomes the product of a null interval with a celestial torus (replacing the celestial sphere) and has only one connected component. Spacelike and timelike infinity are time-periodic quotients of AdS3. These three components of infinity combine to an S 3 represented as a toric fibration over the interval. Privileged scattering states of scalars organize into SL(2, ℝ) L ×SL(2, ℝ) R conformal primary wave functions and their descendants with real integral or half-integral conformal weights, giving the normally continuous scattering problem a discrete character.