The Jensen-Shannon divergence is a test to show if distributions are similar or different. If the distributions are different the divergence will grow in time.

The results are very revealing. Here is what the JS divergence is telling you:
Daily (top panel) — the lowest divergences of all three timescales. The 365-point window (blue) hovers around a mean of 0.070, meaning a full year of daily slopes is only 7% divergent from the full reference distribution. It also shows no trend over time — the blue line is flat and stationary across all four halving cycles. The daily distribution is the most stable of the three, consistent with the before/after tests where daily was the only timescale that passed all tests.
Monthly (middle panel) — intermediate. The 365-point window settles around 0.112 and shows mild cyclical excursions tied to halvings but always returns. The 30-point window (red) is very noisy — 30 monthly observations is only ~2.5 years of data, so the window is too small to stably represent the full distribution.
Yearly (bottom panel) — highest divergences across the board and the most dramatic cycle structure. The 365-point window (blue) swings between 0.14 and 0.65, tracing out the full bull/bear cycle in beautiful waves.
This is expected: a 365-point yearly-step window covers a span where each point looks one year forward, so the window captures the peak or the trough of a cycle intensely before reverting. The JS divergence here is not showing instability of the power law — it is showing the cycle structure. The dips toward 0.14 are the periods when the market is near the trend; the peaks approaching 1.0 are the tops and bottoms of each halving cycle.
The key insight: the JS divergence is highest at yearly timescales not because the power law is less stable there, but because the yearly slope is the most sensitive detector of where you are in the cycle. The daily divergence being low and flat (0.07 mean, no trend) is actually your strongest stability result — the noise process generating daily fluctuations around the power law has not changed character in 15 years.