We study one-shot detection under isotropic multivariate Cauchy noise 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, a standard 4QAM sanity check, and finite-$γ$ numerical validation for both asymptotic regimes.

翻译：我们研究有限星座在各向同性多元柯西噪声下的单次检测，重点关注控制符号级可靠性的几何机制。在各向同性柯西噪声条件下，最大似然准则会诱导出与高斯情形相同的欧几里得沃罗诺伊判决区域，因此两者差异并非源于判决几何本身，而在于概率质量在这些固定区域上的分布方式。在小噪声情形下，我们推导出符号错误概率的倒数距离谱上界，表明可靠性对全局星座几何的依赖具有比加性高斯白噪声更长的相关性。在大噪声情形下，我们证明正确判决概率收敛到一个仅由相关沃罗诺伊退缩锥的角度测度决定的极限。这些结果形式化了重尾噪声下基于距离与基于角度的可靠性描述符之间的机制依赖性转变。通过一个表现出几何坍缩的非对称四点示例、标准4QAM合理性校验以及两种渐近情形下的有限−γ数值验证，进一步阐明了该理论。