  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 AdS$_3$. 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 $SL(2,\mathbb{R})_L
\times SL(2,\mathbb{R})_R$ conformal primary wave functions and their
descendants with real integral or half-integral conformal weights, giving the
normally continuous scattering problem a discrete character.
