  This is the first part of the four-paper sequence, which establishes the
Threshold Conjecture and the Soliton Bubbling vs.~Scattering Dichotomy for the
energy critical hyperbolic Yang--Mills equation in the (4 + 1)-dimensional
Minkowski space-time.
  The primary subject of this paper, however, is another PDE, namely the energy
critical Yang--Mills heat flow on the 4-dimensional Euclidean space. Our first
goal is to establish sharp criteria for global existence and asymptotic
convergence to a flat connection for this system in $\dot{H}^{1}$, including
the Dichotomy Theorem (i.e., either the above properties hold or a harmonic
Yang--Mills connection bubbles off) and the Threshold Theorem (i.e., if the
initial energy is less than twice that of the ground state, then the above
properties hold). Our second goal is to use the Yang--Mills heat flow in order
to define the caloric gauge, which will play a major role in the analysis of
the hyperbolic Yang--Mills equation in the subsequent papers.
