We propose that every supersymmetric four dimensional black hole of finite area can be split up into microstates made up of primitive half-BPS atoms. The mutual non-locality of the charges of these atoms binds the state together. In support of this proposal, we display a class of smooth, horizon-free, four dimensional supergravity solutions carrying the charges of black holes, with multiple centers each carrying the charge of a half-BPS state. At vanishing string coupling the solutions collapse to a bound system of intersecting D-branes. At weak coupling the system expands into the non-compact directions forming a topologically complex geometry. At strong coupling, a new dimension opens up, and the solutions form a foam of spheres threaded by flux in M-theory. We propose that this transverse growth of the underlying bound state of constitutent branes is responsible for the emergence of black hole horizons for coarse-grained observables. As such, it suggests the link between the D-brane and spacetime foam approaches to black hole entropy.
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