  Black holes have an enormous underlying space of microstates, but universal
macroscopic physics characterized by mass, charge and angular momentum as well
as a causally disconnected interior. This leads two related puzzles: (1) How
does the effective factorization of interior and exterior degrees of freedom
emerge in gravity?, and (2) How does the underlying degeneracy of states wind
up having a geometric realization in the horizon area and in properties of the
singularity? We explore these puzzles in the context of an incipient black hole
in the AdS/CFT correspondence, the microstates of which are dual to half-BPS
states of the $\mathcal{N}=4$ super-Yang-Mills theory. First, we construct a
code subspace for this black hole and show how to organize it as a tensor
product of a universal macroscopic piece (describing the exterior), and a
factor corresponding to the microscopic degrees of freedom (describing the
interior). We then study the classical phase space and symplectic form for
low-energy excitations around the black hole. On the AdS side, we find that the
symplectic form has a new physical degree of freedom at the stretched horizon
of the black hole, reminiscent of soft hair, which is absent in the
microstates. We explicitly show how such a soft mode emerges from the
microscopic phase space in the dual CFT via a canonical transformation and how
it encodes partial information about the microscopic degrees of freedom of the
black hole.
