Debiased and Simultaneous Inference for Heterogeneous Factorial Effect Modifiers via Residualized Walsh-Hadamard Scores

In factorial randomized experiments with $K$ binary treatment components, we study which baseline covariates modify component main effects and interactions. We define effect-modifier cells $(j,S)$, $j\in [p]$ and $\mathrm{\varnothing }\mathrm{\ne }S\subseteq [K]$, through the coefficients ${\theta }_{jS}^{\ast }$ of population linear projections of conditional Walsh–Hadamard contrasts ${\tau }_S$ onto standardized covariates. Using baseline-robust direct scores, we develop frequentist inference for all $p(2^K-1)$ cells. First, we prove a debiased central limit theorem with global insensitivity: the residualization baseline enters the cross-fitted score through an exactly conditionally mean-zero perturbation. The influence-function remainder therefore requires only foldwise weighted $L_2$ consistency of the baseline estimator, with no prescribed polynomial rate or nuisance product-rate condition. A separate, explicit penalty-scale condition controls the baseline’s effect on the score Lasso. Second, under explicitly stated uniform nuisance, baseline, studentization, and influence-array conditions derived from bounded sparse-regression primitives, we establish a high-dimensional Gaussian approximation over the full grid, calibrated by a Walsh-spectral multiplier bootstrap. This yields simultaneous confidence bands and Romano–Wolf step-down selection of modified cells with strong familywise error control, without effect-heredity assumptions. The factorial design is fixed in the asymptotic analysis, while the covariate dimension may grow; all $2^K-1$ contrasts are covered jointly.