The cosmic tetrarchy: four estimators breaking the assumption degeneracy in cosmological distance tensions

The origin of cosmological distance tensions remains a central open question in precision cosmology, complicated by the fact that most consistency tests between datasets cannot isolate which physical assumption is responsible for an observed discrepancy. We address this by reformulating the standard cosmological framework as a single null test: the requirement that the dimensionless sound-horizon ratio $r_{\rm d}/r_{\rm d}^{\rm fid}$ be one redshift-independent number. We show that this test admits four complementary measurements, obtained by combining Baryon Acoustic Oscillation (BAO) data with either Type Ia supernovae (SNIa) or cosmic chronometers (CC), in either the transverse or the radial direction. The four channels rely on distinct subsets of physical assumptions -- distance-ladder calibration, the distance duality relation, spatial flatness, and the standard-ruler picture -- and one of them, the radial CC-anchored channel, requires none and serves as the natural reference of the framework. The pattern of agreement or disagreement among the four therefore localises the assumption responsible for any observed tension. We refer to this fourfold decomposition as the \emph{cosmic tetrarchy} and evaluate it on DESI DR2 BAO data combined with Pantheon+ and cosmic chronometers, using both a binned analysis with full analytic covariance propagation and a non-parametric Gaussian Process reconstruction. We find that current data are compatible with a single, redshift-independent sound-horizon scale; when comparing the different estimators, we find hints of discrepancies between those based on SNIa and those relying on CC, although with a lack of statistical significance, which might hint to a manifestation of the Hubble tension or to the breaking of assumptions such as the distance duality relation.