THE MODULARITY TRAP IN PRIVATE INFERENCE

Orthogonal scores remove the first-order effect of nuisance estimation; differential privacy adds a second operation, bounding the contribution of each record. We show that where the bound is placed matters: clipping the completed orthogonal score can change the nuisance rates required for inference, even when the clip keeps the null mean exactly zero. No bounded nonlinear scalar map escapes the effect, since preserving every centered nuisance tangent forces affinity. An exact residual-product construction makes the consequence sharp. With nuisance errors of order $m^{-\gamma }$ for $1/4<\gamma <1/2$, the inherited zero-centered test on the clipped product rejects the null with probability tending to one, while the Prediction-Bounded Orthogonal Score (PBOS), bounding a centered residual multiplier instead of the product, keeps size $\alpha$ under the same rates and admits finite-sample calibration with a lattice release. The rate gap resurfaces as a threshold gap once each method is recalibrated, and the threshold gap as a power gap: in text-representation audits under strong confounding, both placements retain nearly identical standardized signal, yet PBOS attains $92.27\%$ power against $73.93\%$ for product clipping.