We observe a scale-dependent detection signature among Mersenne primes based on statistical properties of early Lucas-Lehmer residues. Using a composite z-score derived from spectral, wavelet, and thermalization features of the Lucas-Lehmer sequence projected onto small primes, we find that detection power increases with exponent scale. At small scales (Tiers 1-3, p < 25, 000), only singleton Mersenne primes are detectable; at larger scales (Tier 6+), cluster members become detectable as well. After conservative correction for researcher degrees of freedom (9 scoring components explored, ~480 candidate features, multiple window sizes), the fully-adjusted p-value for the singleton ranking pattern lies in the range 7. 4×10⁻³ to 2. 7×10⁻² (structured Bonferroni; Mₑff = 132 primary, Mₑff = 480 worst-case). All code, data, and calibration files are available at https: //github. com/jessetop/orbital-staircase-paper.
Jesse Toporowski (Thu,) studied this question.