  This is the second part in a four-paper sequence, which establishes the
Threshold Conjecture and the Soliton Bubbling vs.~Scattering Dichotomy for the
hyperbolic Yang--Mills equation in the $(4+1)$-dimensional space-time. This
paper provides the key gauge-dependent analysis of the hyperbolic Yang--Mills
equation.
  We consider topologically trivial solutions in the caloric gauge, which was
defined in the first paper arXiv:1709.08599 using the Yang--Mills heat flow. In
this gauge, we establish a strong form of local well-posedness, where the time
of existence is bounded from below by the energy concentration scale. Moreover,
we show that regularity and dispersive behavior of the solution persists as
long as energy dispersion is small. We also observe that fixed-time regularity
(but not dispersive) properties in the caloric gauge may be transferred to the
temporal gauge without any loss, proving as a consequence small data global
well-posedness in the temporal gauge.
  The results in this paper are used in the subsequent papers arXiv:1709.08604,
arXiv:1709.08606 to prove the sharp Threshold Theorem in caloric gauge in the
trivial topological class, and the dichotomy theorem in arbitrary topological
classes.
