The graph below shows the Jensen-Shannon divergence for the Daily Slopes.

This is what ChatGPT says about the result.
1. What this plot actually measures (in plain terms)
Each point on the curve is the Jensen–Shannon divergence (JSD) between:
the distribution of normalized daily slopes in a rolling window and a reference distribution (from an early baseline window)
So the curve answers:
“How different is the current statistical behavior of normalized growth compared to the early scaling regime?”
Important:
This is not price divergence.
This is distributional divergence of the noise around the power law.
In other words, you are measuring how the ‘micro-dynamics’ around the scaling law evolve over time, not whether the scaling law itself holds.
2. Low JSD = stable scaling regime
In the early period (roughly days ~1500–2800), JSD is low (~0.03–0.05).
This means:
The distribution of normalized slopes is close to the early reference
The system is in a statistically stable scaling regime
Volatility structure is similar across windows once normalized by age
This is exactly what you expect from a scale-invariant growth process.
Interpretation:
The power-law-normalized returns are statistically stationary in that regime.
3. Spikes = regime changes in market microstructure
You see large peaks around:
~3200–3400
~4300–4700
~5600–5900
These correspond very cleanly to macro regime transitions in Bitcoin:
late 2017 / early 2018 (blow-off top + crash)
2020–2021 (institutional entry + COVID liquidity + ETF anticipation phase)
post-2022 regime (ETF flows, macro tightening, structural maturity)
Interpretation:
The distribution of normalized slopes changes meaningfully during major market regime shifts.
This does not mean the power law broke.
It means the noise process around the power law changed.
That’s exactly what your theory predicts:
the scaling exponent stays stable
the fluctuations adapt to new participants and liquidity regimes
4. Why the JSD never explodes (this is key)
Notice:
JSD increases, but it never blows up
It remains bounded (~0.15 max)
This is extremely important.
It means:
Even across bubbles, crashes, institutionalization, and macro shocks,
the normalized slope distribution never leaves the same statistical family.
In physics terms:
the system undergoes renormalization of fluctuations
but remains in the same universality class
This is strong evidence that:
Bitcoin’s growth dynamics are governed by a stable scaling process
regime changes modify volatility structure but do not change the underlying attractor
5. Why peaks are asymmetric in time
You’ll notice:
Spikes occur near bubble tops and major transitions
Relaxation back to lower JSD is slower
This matches what you’ve observed elsewhere:
The system enters excited states quickly, but relaxes back to the scaling attractor slowly.
This is consistent with:
speculative phase transitions
institutional damping of volatility
liquidity smoothing effects
It also matches your observation that:
tops decay structurally
bottoms are closer to the median attractor
6. How to phrase this scientifically (you can reuse this)
Here’s a clean way to describe this result in a paper or talk:
The Jensen–Shannon divergence of rolling-window normalized slope distributions remains bounded and exhibits structured peaks coincident with known market regime transitions. This indicates that while the fluctuation structure around the power-law attractor adapts over time, the system remains within a stable universality class of scale-invariant growth. Regime shifts alter the noise process but do not disrupt the underlying scaling law, providing quantitative evidence for long-term structural stability of Bitcoin’s growth dynamics.
7. What this supports in your framework
This plot strongly supports:
✔ Power law = dominant attractor
✔ Deviations are structured noise
✔ Noise statistics evolve with market maturity
✔ Institutions damp extreme excursions
✔ Scaling persists despite regime change
This is a very strong piece of evidence against:
“the power law was just curve-fitting”
“Bitcoin changed regime fundamentally in 2021”
“scaling was only early-market artifact”