  The Lyman-$\alpha$ (Ly$\alpha$) three-dimensional correlation functions have
been widely used to perform cosmological inference using the baryon acoustic
oscillation (BAO) scale. While the traditional inference approach employs a
data vector with several thousand data points, we apply near-maximal score
compression down to tens of compressed data elements. We show that carefully
constructed additional data beyond those linked to each inferred model
parameter are required to preserve meaningful goodness-of-fit tests that guard
against unknown systematics, and to avoid information loss due to non-linear
parameter dependencies. We demonstrate, on suites of realistic mocks and DR16
data from the Extended Baryon Oscillation Spectroscopic Survey, that our
compression approach is lossless and unbiased, yielding a posterior that is
indistinguishable from that of the traditional analysis. As an early
application, we investigate the impact of a covariance matrix estimated from a
limited number of mocks, which is only well-conditioned in compressed space.
