We show that differentiable sorting and ranking operators are structurally incompatible with within-vector rank normalization. We formalize admissibility through monotone invariance (C1), batch independence (C2), and a rank-space stability condition (C3). Gap-sensitive relaxations such as SoftSort violate (C1) by a quantitative margin that depends on the temperature and input scale. Batchwise rank relaxations such as SinkhornSort violate (C2): the same sample can be assigned outputs arbitrarily close to 0 or 1 depending solely on batch context. Condition (C3) implies (C1) under the rank representation used here and should not be read as a third independent failure mode. We also characterize the admissible class: any admissible operator must factor through the rank representation via a Lipschitz function.

翻译：本文证明了可微排序与秩次算子与向量内秩归一化在结构上存在不兼容性。我们通过单调不变性（C1）、批次独立性（C2）以及秩空间稳定性条件（C3）来形式化可容许性。诸如SoftSort等间隙敏感松弛方法会违反条件（C1），其违反程度取决于温度参数和输入尺度。而如SinkhornSort等批次秩松弛方法则违反条件（C2）：同一样本可能仅因批次上下文的不同，被分配任意接近0或1的输出值。条件（C3）在本文使用的秩表示下蕴含条件（C1），不应被解读为第三种独立的失效模式。我们还刻画了可容许算子类：任何可容许算子必须通过Lipschitz函数经由秩表示进行分解。