  Heat flows in 1+1 dimensional stochastic environment converge after scaling
to the random geometry described by the directed landscape. In this first part,
we show that the O'Connell-Yor polymer and the KPZ equation converge to the KPZ
fixed point.
  The key is that one-dimensional Baik-Ben Arous-Peche statistics characterize
the KPZ fixed point. This yields a general and elementary method that shows
convergence based on previously established limit theorems.
  Independently, at the same time and place Quastel and Sharkar gave an
unrelated proof of KPZ convergence. The methods invite extensions in different
directions: ours to polymer models, and theirs to interacting particle systems.
