  Celestial holography provides a reformulation of scattering amplitudes in
four dimensional asymptotically flat spacetimes in terms of conformal
correlators of operators on the two dimensional celestial sphere in a basis of
boost eigenstates. A basis of {massless particle} states has previously been
identified in terms of conformal primary wavefunctions labeled by a boost
weight $\Delta = 1 + i\lambda$ with $\lambda \in \mathbb{R}$. Here we show that
a {\it discrete} orthogonal and complete basis exists for $\Delta \in
\mathbb{Z}$. This new basis consists of a tower of discrete memory and
Goldstone observables, which are conjugate to each other and allow to
reconstruct gravitational signals belonging to the Schwartz space. We show how
generalized dressed states involving the whole tower of Goldstone operators can
be constructed and evaluate the higher spin Goldstone 2-point functions.
Finally, we recast the tower of higher spin charges providing a representation
of the $w_{1+\infty}$ loop algebra (in the same helicity sector) in terms of
the new discrete basis.
