Conformal prediction is often calibrated with a single pooled threshold, but this can hide cross-group heterogeneity in score distributions and distort group-wise coverage. We study this phenomenon through the population score distributions underlying split conformal calibration. First, we derive a conservation law and lower bound showing that pooled calibration incurs irreducible group-wise coverage distortion at a scale set by cross-group quantile heterogeneity. Second, we demonstrate that the two leading fairness definitions for conformal prediction, Equalized Coverage and Equalized Set Size, are fundamentally in tension. Third, we quantify the cost of moving between policies which treat groups separately or pool them. Experiments on synthetic and real data confirm the same bidirectional trade-off after finite-sample calibration. Our results show that, for the policy families studied here, calibration choice does not remove cross-group heterogeneity; it determines whether the resulting distortion appears in the coverage or size dimension, providing a principled lens for analyzing fairness-oriented calibration choices in practice.

翻译：共形预测通常使用单一合并阈值进行校准，但这可能掩盖分数分布中的跨组异质性并扭曲组间覆盖率。我们通过总体分数分布（基于分裂共形校准）研究这一现象。首先，我们推导出一个守恒定律和下界，表明合并校准会以跨组分位数异质性设定的尺度，产生不可缩减的组间覆盖率失真。其次，我们证明共形预测的两种主要公平性定义——等化覆盖率和等化集合大小——存在根本性矛盾。第三，我们量化了在分开处理各组或合并处理各组的策略之间移动的成本。在合成数据和真实数据上的实验证实，经有限样本校准后，这一双向权衡依然存在。我们的结果表明，对于所研究的策略族，校准选择并不能消除跨组异质性；它决定了产生的失真出现在覆盖率维度还是集合大小维度，为在实践中分析面向公平性的校准选择提供了原则性视角。