C. GordonAndrei CuceuAndreu Font-RiberaH. K. Herrera-AlcantarJessica Nicole Aguilar Steven AhlenD. BianchiDavid H. BrooksT. ClaybaughShaun ColeAxel de la Macorra2026-03-222026-03-22202510.1093/mnras/staf2035https://doi.org/10.1093/mnras/staf2035https://andeanlibrary.org/handle/123456789/78464ABSTRACT In preparation for the first cosmological measurements from the full shape of the Lyman-$\alpha$ (Ly $\alpha$) forest from Dark Energy Spectroscopic Instrument (DESI), we must carefully model all relevant systematics that might bias our analysis. It has been shown that random quasar redshift errors produce a smoothing effect on the mean quasar continuum in the Ly $\alpha$ forest region. This, in turn, gives rise to spurious features in the Ly $\alpha$ autocorrelation and its cross-correlation with quasars. Using synthetic data sets based on the DESI survey, we confirm that the impact on baryon acoustic oscillation measurements is small, but that a bias is introduced to parameters which depend on the full shape of our correlations. We combine a model of this contamination in the cross-correlation with a new model we introduce here for the autocorrelation. These are parametrized by three parameters, which, when included in a joint fit to both correlation functions, successfully eliminate any impact of redshift errors on our full-shape constraints. We also present a strategy for removing this contamination from real data, by removing $\sim$0.3 per cent of correlating pairs.enPhysicsQuasarRedshiftSpurious relationshipAutocorrelationDark energyAstrophysicsSmoothingBaryonCosmologyarticle