Detection of a Stable Temporal Scaling Mode (ζ ≈ 1) in Black Hole Time Series Using the Hierarchical Correlation–Lag Exponent Test (HCLET)

Temporal variability in black hole systems often exhibits long-memory behaviour and approximate power-law spectra. However, direct measurements of temporal scaling exponents in the time domain remain limited. In this work we introduce a statistical method designed to detect temporal scaling modes in time series data, which we call the Hierarchical Correlation–Lag Exponent Test (HCLET). The method searches for preferred scaling exponents by scanning a parameter ζ and evaluating improvements in a correlation-based statistic across hierarchical lag structures. Applying HCLET to simulated black hole variability signals, we identify a stable scaling mode near ζ = 1.17 ± 0.13. The detected cluster is supported by a large fraction of independent realizations and persists across multiple noise amplitudes. In the reference configuration (sig = 1.0), the dominant cluster is supported by 22 independent seeds. To evaluate statistical significance, we perform extensive null tests including event permutation, time-shifted realizations, and synthetic baseline models with ζ = 0. Across 60 strict null realizations, no internal peak was detected within the target interval ζ = 1 ± 0.25, corresponding to an empirical upper bound of p < 0.017. The persistence of the ζ ≈ 1 mode across noise levels suggests the presence of a scale-free temporal response structure in the analyzed signals. The HCLET framework therefore provides a new approach for identifying temporal scaling behaviour in astrophysical time series.