  We construct examples of black hole formation from regular, one-ended
asymptotically flat Cauchy data for the Einstein-Maxwell-charged scalar field
system in spherical symmetry which are exactly isometric to extremal
Reissner-Nordstr\"om after a finite advanced time along the event horizon.
Moreover, in each of these examples the apparent horizon of the black hole
coincides with that of a Schwarzschild solution at earlier advanced times. In
particular, our result can be viewed as a definitive disproof of the "third law
of black hole thermodynamics."
  The main step in the construction is a novel $C^k$ characteristic gluing
procedure, which interpolates between a light cone in Minkowski space and a
Reissner-Nordstr\"om event horizon with specified charge to mass ratio $e/M$.
Our setup is inspired by the recent work of Aretakis-Czimek-Rodnianski on
perturbative characteristic gluing for the Einstein vacuum equations. However,
our construction is fundamentally nonperturbative and is based on a finite
collection of scalar field pulses which are modulated by the Borsuk-Ulam
theorem.
