Mapping the redshift drift at various redshifts through cosmography

The redshift drift provides a kinematic test of the cosmic expansion history through the slow time variation of the redshift of comoving sources. Motivated by the expected Sandage-Loeb measurements from future facilities, we investigate the drift within a cosmographic framework, modeling the Hubble rate through both a second-order Taylor expansion and a $(2,1)$ Pad\'e approximant. We constrain the cosmographic parameters $(H_0,q_0,j_0)$ by combining Pantheon+ and SH0ES type Ia supernovae with gamma-ray bursts and then examine the impact of adding baryon acoustic oscillation measurements from the second DESI data release. The resulting constraints are used to construct a mock Sandage-Loeb catalog, after which the analyses are repeated including the simulated drift data. In this way, we assess the internal consistency of the reconstructed background rather than perform an independent forecast. Accordingly, we find that, for the SNeIa+GRB analysis, the Taylor reconstruction is compatible at the $1\sigma$ level with the $\omega_0\omega_1$CDM scenario, whereas the Pad\'e parameterization improves the agreement of $q_0$ with the $\Lambda$CDM paradigm. Once DESI BAO data are included, the agreement with the reference background models weakens to the $2\sigma$ level. The addition of the mock Sandage-Loeb sample mainly tightens the bounds on $q_0$ and $j_0$, with moderate shifts in the central values. We finally compare the reconstructed redshift drift with the corresponding behavior predicted by the $\Lambda$CDM and $\omega_0\omega_1$CDM scenarios.