The Cauchy combination test combines p-values by averaging their Cauchy-transformed versions. It's popular because it has a closed-form combined p-value and is asymptotically valid as the significance level α → 0 under broad dependence. But that's the wrong asymptotic question for practice. What happens when α is fixed and the number of combined p-values K grows?
The authors (arXiv:2603.22668) show the raw Cauchy combination test is generally not valid at fixed α when K grows under dependence. Under equicorrelated Gaussian structure with correlation ρ, the test statistic converges to a random latent-factor limit — there is no universal reference distribution. The behavior is governed by a boundary-layer scale s_K = √ρ · (log K)^(3/2). The raw test is exact if and only if ρ_K(log K)³ → 0.
The fix is elegant: the distortion lives entirely in the reference law, not the statistic. The boundary-layer calibrated test (BL-CCT) keeps the test statistic unchanged and replaces only the reference distribution with a Gaussian-smoothed Cauchy family. This extends validity to the weaker condition ρ_K log K → 0.
The through-claim: the error isn't in how the Cauchy test combines evidence — it's in what it compares against. The statistic is fine. The yardstick is wrong. Fixing the reference distribution without touching the test statistic is a fundamentally different approach from existing corrections that modify the combination rule.