A test of uniformity on [0,1] is developed for the setting of a single observation recorded with sufficient precision. Although consistency against general alternatives is not attainable with only one draw in the classical large-sample sense, a multiscale harmonic digit expansion provides a framework for structured inference. By aggregating trigonometric components across digit scales at Hadamard-gap frequencies, a quadratic test statistic is constructed whose null distribution converges to a chi-square law via a lacunary central limit theorem. Under departures from uniformity, the statistic is driven by Fourier components induced by digit-scale transformations of the observation, with detectability depending on their coherent accumulation as precision increases. The resulting procedure detects multiscale harmonic structure that remains invisible to classical digit-frequency methods.

翻译：针对单次高精度观测值，本文提出了一种[0,1]区间上的均匀性检验方法。尽管在经典大样本意义下，单次观测无法实现对一般备择假设的一致性检验，但多尺度调和数字展开为结构化推断提供了框架。通过整合Hadamard间隙频率下各数字尺度的三角分量，本文构造了一个二次型检验统计量，其原假设分布通过缺项中心极限定理收敛到卡方分布。当偏离均匀性时，该统计量由观测值数字尺度变换诱导的傅里叶分量驱动，其可检测性取决于这些分量随精度提升的相干累积。所提方法能够检测经典数字频率方法无法识别的多尺度调和结构。