Retrospective Detection of Seismic Precursors Using Multi-Scale Energy Curvature

Energy curvature controls catastrophic failure in seismic systems. We show this through a self-normalizing logarithmic functional Fω = ω(t)² · log(1 + |ω(t)| / median(|ω|)), where ω is the second derivative of seismic energy release. When F stays bounded, the system remains stable. When F diverges, rupture becomes more probable. Our precursor detection method combines spectral analysis using continuous wavelet transforms, temporal cascade synchronization across 3/7/14/30-day scales with Median Absolute Deviation thresholds, and spatial focusing tracked through Haversine geometry. We tested this retrospectively on seven major earthquakes from M6.3 to M9.1. Every event showed the same four-phase pattern: curvature escalation → multi-scale cascade synchronization → spatial focusing → rupture. Lead times for precursor signals ranged from 2 days (Nepal 2015, fast rupture) to 149 days (Chile 2010, slow nucleation). The events included Japan 2011 (71d lead, SNR=4.97), Turkey 2023 (128d, SNR=12.91), Sumatra 2004 (124d), L'Aquila- Italy 2009 (115d), and Ridgecrest – USA 2019 (94d). Critically, the method correctly distinguishes between months-long slow-slip preparation and days-long rapid nucleation—something most approaches miss. This retrospective study establishes the physics; real-time operational testing is required to evaluate false alarms in prospective deployment.