We study one-shot detection under additive white Cauchy noise (AWCN) using finite constellations, with emphasis on the geometric mechanisms governing symbol-level reliability. Under isotropic Cauchy noise, the maximum-likelihood rule induces the same Euclidean Voronoi decision regions as in the Gaussian case, so the distinction lies not in the decision geometry itself but in how probability mass is distributed over these fixed regions. In the small-noise regime, we derive a reciprocal distance-spectrum upper bound for the symbol error probability, showing that reliability retains a longer-range dependence on the global constellation geometry than under additive white Gaussian noise. In the large-noise regime, we prove that the correct-decision probability converges to a limit determined solely by the angular measure of the associated Voronoi recession cone. These results formalize a regime-dependent transition from distance-based to angle-based reliability descriptors under heavy-tailed noise. The theory is further illustrated through an asymmetric four-point example exhibiting geometric collapse and a standard 4QAM sanity check.

翻译：我们研究了在加性白柯西噪声（AWCN）条件下，利用有限星座进行单次检测的问题，重点分析了控制符号级可靠性的几何机制。在各向同性柯西噪声下，最大似然规则诱导出与高斯情形相同的欧几里得Voronoi判决区域，因此区别不在于决策几何本身，而在于概率质量在这些固定区域上的分布方式。在小噪声区域，我们推导了符号错误概率的倒数距离谱上界，表明相比于加性白高斯噪声，可靠性对全局星座几何结构具有更远距离的依赖性。在大噪声区域，我们证明正确判决概率收敛于一个仅由关联Voronoi退缩锥的角测度决定的极限。这些结果形式化了在重尾噪声下，可靠性描述符从基于距离到基于角度的区域依赖性转变。该理论进一步通过一个呈现几何塌缩的非对称四点示例和标准的4QAM合理性检验加以阐明。