Validation of the DESI DR2 Ly$α$ forest full-shape analysis

We present the validation of the Dark Energy Spectroscopic Instrument (DESI) Data Release 2 (DR2) Lyman-$\alpha$ (Ly$\alpha$) forest full-shape analysis. This analysis combines three-dimensional Ly$\alpha$ forest auto-correlations and cross-correlations with quasars to extract information from both the baryon acoustic oscillation (BAO) feature and the broadband clustering signal, with primary emphasis on the Alcock-Paczynski (AP) measurement. Compared to the DESI DR1 analysis, the DR2 validation uses substantially larger and more realistic mock datasets, including CoLoRe 2LPT and AbacusSummit Ly$\alpha$ forest simulations. The modeling framework is also improved through analytic marginalization over small scales ($<10$ $h^{-1}$Mpc) and the impact of ultraviolet background fluctuations. The validation program was completed prior to unblinding and defines quantitative requirements for the cosmological parameters of interest, which are evaluated using hundreds of mock realizations. We further test the analysis through independent fits to the auto- and cross-correlations, multiple catalog splits, and a broad suite of analysis and modeling variations applied to both mocks and blinded observational data. We find that the BAO and AP parameters satisfy all validation requirements and remain stable across all tests. In contrast, mock studies reveal a significant bias in the inferred growth-rate parameter $f\sigma_8$, leading us to exclude this measurement from the final analysis. The consistency across mocks, data splits, and robustness tests demonstrates that the DR2 Ly$\alpha$ full-shape analysis provides a reliable and substantially improved broadband AP measurement over previous Ly$\alpha$ forest studies.