  Final measurements and analysis are reported from the first-generation
Holometer, the first instrument capable of measuring correlated variations in
space-time position at strain noise power spectral densities smaller than a
Planck time. The apparatus consists of two co-located, but independent and
isolated, 40 m power-recycled Michelson interferometers, whose outputs are
cross-correlated to 25 MHz. The data are sensitive to correlations of
differential position across the apparatus over a broad band of frequencies up
to and exceeding the inverse light crossing time, 7.6 MHz. By measuring with
Planck precision the correlation of position variations at spacelike
separations, the Holometer searches for faint, irreducible correlated position
noise backgrounds predicted by some models of quantum space-time geometry. The
first-generation optical layout is sensitive to quantum geometrical noise
correlations with shear symmetry---those that can be interpreted as a
fundamental noncommutativity of space-time position in orthogonal directions.
General experimental constraints are placed on parameters of a set of models of
spatial shear noise correlations, with a sensitivity that exceeds the
Planck-scale holographic information bound on position states by a large
factor. This result significantly extends the upper limits placed on models of
directional noncommutativity by currently operating gravitational wave
observatories.
