This result is more significant than it might appear at first glance.

The two lower panels show what remains of Bitcoin's price history after the long-run power law trend is removed — the raw oscillations, stripped of growth.
That residual is not noise. It is fit almost entirely by a single frequency and its integer multiples: 2×, 3×, 4×. These are harmonics, the same mathematical structure that governs resonance in physical systems from vibrating strings to quantum wells.
But the significance goes deeper than ordinary resonance. The spectrum here is not periodic in time — it is periodic in the logarithm of time. This is the signature of discrete scale invariance: a symmetry not under shifting time forward by a fixed amount, but under scaling time by a fixed ratio. Instead of repeating at regular intervals, the structure repeats at regular multiples of scale — each cycle roughly twice as long as the one before it.
Most scale-invariant systems exhibit continuous scale invariance, meaning they look the same under rescaling by any factor. Discrete scale invariance is rarer and more specific: the system is only self-similar under rescaling by a particular ratio λ ≈ 2. Bitcoin appears to exhibit both — continuous scale invariance in the power law trend, and discrete scale invariance in the oscillations riding on top of it.
The colour-coded triangles in the chart make the key point visible directly. Every peak the model predicts — major or minor — coincides with a real peak in the data.
This correspondence was not imposed. The model was fit to the entire residual series, not to individual peaks. The peaks emerge from the mathematics, and the data confirms them.
These are not fitting artifacts. They are real oscillations with a coherent harmonic structure — the kind of structure that, in physics, points to an underlying resonance mechanism waiting to be identified.