Reliable high-dimensional variable selection requires scalable error-controlling methods. The Terminating-Random Experiments (T-Rex) selector estimates the false discovery rate (FDR) by aggregating early-terminated forward-selection paths in which predictors compete with synthetic dummies. We address two remaining challenges: i) predictor dependence can bias dummy-predictor competition; ii) computation is wasted on recomputing terms shared across experiments. Building on memory-efficient virtual dummies, which sequentially sample projections from their exact conditional law, we estimate the conditional Gaussian law of inactive predictors and map draws onto the remaining sphere radius. A shared lazy Gram cache computes the response product and each requested Gram column once across experiments. Simulations across three covariance structures show that uniform spherical dummies can exceed the target FDR, whereas the proposed method empirically controls FDR. Caching yields more than a sixfold speedup, and the method remains feasible with 100000 predictors, where competing FDR-controlling methods become computationally impractical.

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