Local Detection Signatures of Mersenne Primes via Lucas-Lehmer Residue Statistics

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; M_eff = 132 primary, M_eff = 480 worst-case). All code, data, and calibration files are available at https://github.com/jessetop/orbital-staircase-paper.