  Massive galaxy-scale outflows of gas are one of the most commonly-invoked
mechanisms to regulate the growth and evolution of galaxies throughout the
universe. While the gas in outflows spans a large range of temperatures and
densities, the cold molecular phase is of particular interest because molecular
outflows may be capable of suppressing star formation in galaxies by removing
the star-forming gas. We have conducted the first survey of molecular outflows
at z > 4, targeting 11 strongly-lensed dusty, star-forming galaxies (DSFGs)
with high-resolution Atacama Large Millimeter Array (ALMA) observations of OH
119um absorption as an outflow tracer. In this first paper, we give an overview
of the survey, focusing on the detection rate and structure of molecular
outflows. We find unambiguous evidence for outflows in 8/11 (73%) galaxies,
more than tripling the number known at z > 4. This implies that molecular winds
in z > 4 DSFGs must have both a near-unity occurrence rate and large opening
angles to be detectable in absorption. Lensing reconstructions reveal that
500pc-scale clumpy structures in the outflows are common. The individual clumps
are not directly resolved, but from optical depth arguments we expect that
future observations will require 50-200pc spatial resolution to do so. We do
not detect high-velocity [CII] wings in any of the sources with clear OH
outflows, indicating that [CII] is not a reliable tracer of molecular outflows.
Our results represent a first step toward characterizing molecular outflows at
z > 4 at the population level, demonstrating that large-scale outflows are
ubiquitous among early massive, dusty galaxies.
