  Leveraging the Baryon Acoustic Oscillations (BAO) feature present in
clustering 2-point statistics, we aim to measure cosmological distances
independently of the underlying background cosmological model. However this
inference is complicated by late-time non-linearities that introduce model and
tracer dependencies in the clustering correlation function and power spectrum,
which must be properly accounted for. With this in mind, we introduce the
"Purely-Geometric-BAO," which provides a rigorous tool to measure cosmological
distances without assuming a specific background cosmology. We focus on the
2-point clustering correlation function monopole, and show how to implement
such an inference scheme employing two different methodologies: the Linear
Point standard ruler (LP) and correlation-function model-fitting (CF-MF). For
the first time we demonstrate how, by means of the CF-MF, we can measure very
precisely the sound-horizon/isotropic-volume-distance ratio,
$r_{d}/D_{V}(\bar{z})$, while correctly propagating all the uncertainties.
Using synthetic data, we compare the outcomes of the two methodologies, and
find that the LP provides up to $50\%$ more precise measurements than the
CF-MF. Finally, we test a procedure widely employed in BAO analyses: fitting
the 2-point function while fixing the cosmological and the non-linear-damping
parameters at fiducial values. We find that this underestimates the distance
errors by nearly a factor of $2$. We thus recommend that this practice be
reconsidered, whether for parameter determination or model selection.
