Testing Schouten--Codazzi Gravity: Cosmological Constraints and Deviations from $Λ$CDM
Schouten--Codazzi gravity (SCG) admits a cosmological background that extends flat $Λ$CDM by one integration constant, $γ_0$. Its contribution to the Friedmann equation scales as $(1+z)$ and corresponds to an effective component with equation of state $w_{γ_0}=-2/3$. We constrain this extension using 15 combinations of Type~Ia supernovae, DESI~DR2 baryon acoustic oscillations, and compressed cosmic microwave background information, supplemented by a big-bang nucleosynthesis prior as specified in the analysis. Supernovae paired with either BAO or CMB favor positive $Ω_{γ_0}\equivγ_0/H_0^2$, with approximate marginal departures from zero of $1.7$--$3.2σ$. In contrast, all three supernovae$+$BAO$+$CMB combinations are consistent with $Ω_{γ_0}=0$ within their 68\% credible intervals. Their central log Bayes factors favor $Λ$CDM, with $Δ\ln\mathcal{Z}$ between $-2.54$ and $-1.13$, and the inferred Hubble constants remain close to those of the corresponding $Λ$CDM fits. These results provide a background-level test of SCG conditional on the adopted priors and likelihood approximations, particularly the model assumptions entering the CMB compression.