The Cauchy combination test (CCT) is widely used because it yields a closed-form combined $p$-value and is known to be asymptotically valid as the nominal level $α\downarrow0$ under broad dependence structures. We study a different asymptotic question: whether the usual Cauchy cutoff remains accurate at an ordinary fixed level when the number $K$ of combined $p$-values grows under dependence. Under a canonical one-factor equicorrelated Gaussian copula model, we show that the raw CCT is generally not asymptotically exact at fixed $α$. With fixed positive correlation, the statistic converges to a random latent-factor limit, so there is no universal fixed-level reference law. When the common correlation $ρ_K$ weakens with $K$, fixed-level behaviour is governed by the boundary-layer scale $s_K=\sqrt{ρ_K}(\log K)^{3/2}$, and the raw CCT is asymptotically exact if and only if $ρ_K(\log K)^3\to0$. Because the size distortion arises entirely from the reference law and not from the statistic, it can be corrected without modifying the test statistic itself. We propose the boundary-layer calibrated CCT (BL-CCT), which replaces the standard Cauchy reference by a one-parameter Gaussian-smoothed Cauchy family. Unlike recent variants that modify the test statistic, BL-CCT leaves the statistic unchanged and corrects only the reference law. BL-CCT is asymptotically exact under the weaker condition $ρ_K\log K\to0$ and provides a useful finite-$K$ approximation on bounded boundary layers. We also conduct several power analyses: although BL-CCT only raises the cutoff, it incurs no first-order power loss relative to the raw CCT on the exactness scale, under local dense, sparse, and dense Gaussian alternatives. Numerical experiments support the calibration theory.

翻译：柯西组合检验（CCT）因能生成闭合形式的组合$p$值，且在宽泛相依结构下当名义水平$α\downarrow0$时渐近有效而得到广泛应用。我们研究一个不同的渐近问题：当组合$p$值个数$K$在相依条件下增长时，通常的柯西阈值在固定水平下是否保持准确性。在标准单因子等相关高斯Copula模型下，我们证明原始CCT在固定$α$下一般不能渐近精确。当存在固定正相关时，统计量收敛至随机潜因子极限，因此不存在普适的固定水平参考分布。当公共相关系数$ρ_K$随$K$减弱时，固定水平行为由边界层尺度$s_K=\sqrt{ρ_K}(\log K)^{3/2}$主导，且原始CCT渐近精确当且仅当$ρ_K(\log K)^3\to0$。由于大小失真完全源于参考分布而非统计量本身，因此可在不修改检验统计量的情况下进行修正。我们提出边界层校准CCT（BL-CCT），用单参数高斯平滑柯西族替代标准柯西参考分布。与近期修改检验统计量的变体不同，BL-CCT保持统计量不变，仅修正参考分布。在更弱条件$ρ_K\log K\to0$下，BL-CCT渐近精确，并在有界边界层上提供有效的有限$K$近似。我们还进行了多项势分析：尽管BL-CCT仅提高阈值，但在局部密集、稀疏和密集高斯备择假设下，相对于原始CCT在精确性尺度上并未产生一阶势损失。数值实验支持校准理论。