Golden Ratio Harmonics in X-ray Flares from M87* and Sgr A*: Evidence of Timing Structure

We present evidence that X-ray flares from two supermassive black holes—M87* and Sagittarius A*—exhibit rhythmic behavior governed by golden ratio harmonics. Using over two decades of Chandra data and a hybrid analysis of Lomb-Scargle periodograms, wavelet transforms (Morlet and Mexican Hat), and cross-wavelet comparisons, we identify statistically significant flare periodicities of 56.74 days (M87*) and 85.02 days (Sgr A*). These periods, and their golden harmonics (ϕ⁻¹, ϕ), appear repeatedly across epochs of high activity, suggesting an underlying self-similar structure. Cross-wavelet analysis reveals a shared 70–75 day resonance band, despite the stark contrast in mass and environment between the black holes. We explore the possibility that this resonance reflects a deeper coupling between accretion dynamics, jet instabilities, and harmonic structures embedded in relativistic plasma flows. Flare windows forecast for 2025–2026 offer a timely opportunity to test the model’s predictive capacity. Future multi-wavelength campaigns and magnetohydrodynamic simulations may further clarify whether these harmonic patterns are tied to plasma instabilities, frame-dragging, or jet-disk coupling. To our knowledge, this is the first study to identify golden-ratio-based periodicities in flare activity across two distinct supermassive black holes using long-baseline X-ray datasets. Supporting datasets and materials available at: https://osf.io/kq6ev/