  Central to the enigma of the cuprates is ubiquitous electronic inhomogeneity
arising from a variety of electronic orders that coexist with
superconductivity, the individual signatures of which have been impossible to
disentangle despite four decades of intense research. This strong nanoscale
inhomogeneity complicates interpretation of measurements both by probes which
average over this inhomogeneity and those, like scanning tunneling microscopy
(STM), which should be able to spatially resolve variations driven by both
order and inhomogeneity. Here, we develop a novel technique that directly
acknowledges this electronic inhomogeneity and extracts statistically
significant features from scanning tunneling spectroscopic data. Applying our
novel technique to single and bilayer Bi-based cuprates spanning a large doping
range, we peer through local inhomogeneities and find that the gap breaks
translational and rotational symmetries and varies periodically in a four-fold
pattern. Our direct observation of a symmetry breaking gap in the single
particle tunneling spectra adds strong credence to the pair density wave
hypotheses supposed to exist in these materials. We also discuss various
implications of our observations and, in particular, how they can explain the
origin of the low energy checkerboard pattern.
