A permutation-free family-wise error rate for the moderated top-gene scan under gene correlation

A differential-expression scan reports the genes with the largest moderated t-statistics, so controlling the family-wise error rate means controlling the null distribution of the maximum statistic over genes. Under gene correlation this is widely believed to require permutation, because correlation changes the effective multiplicity and corrupts the empirical-Bayes variance prior behind the moderated t-statistic. We decompose that liberality by an error-budget ablation and show that, within the simulated model class, it reduces principally to an inflation of the empirical-Bayes prior degrees of freedom: substituting the true prior returns the family-wise error to the independent-gene small-sample baseline, so dependence imposes no separate barrier once the prior is correct. Correlation deflates the cross-gene spread of the log sample variances; because the prior degrees of freedom decreases in that spread, the prior is over-estimated and the moderated maximum turns liberal. Dividing the observed spread by one minus the mean squared gene correlation, estimated by a tuning-free spectral U-statistic with an unbiased trace target, reverses the mechanism and holds the family-wise error near the baseline at retained power without permutation. An observable instability index flags when severe co-expression should defer to permutation.