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：针对单次观测在足够精度记录下的均匀性检验问题，本文发展了一种[0,1]区间上的均匀性检验方法。尽管在经典大样本意义下仅凭单次观测无法对一般备择假设实现一致性检验，但通过多尺度谐波数字展开可构建结构化推断框架。通过以Hadamard间隔频率聚合数字尺度上的三角分量，构造了二次型检验统计量，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，该统计量由观测的数字尺度变换所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。本文提出的方法能检测出经典数字频率方法无法识别的多尺度谐波结构。