Bitcoin is Two-Phased

A Core Power Law phase (Steady) and a Bubble phase (Turbulent)

Janus, the two faced Roman God (Vatican Museum, CC-BY-3.0)

Two phase analogy with fluids

Like many fluids in nature, Bitcoin has two phases, a steady or laminar phase that adheres to a power law, and a bubble phase that is analogous to turbulent flow in nature. They are well separated, by about a factor of three in terms of the linear price ratio. And the bubble phase, not surprisingly has much higher volatility.

Table 1. Description of Bitcoin’s power law core and bubble phases, in analogy with the laminar and turbulent behavior often found in fluid dynamics. From ChatGPT 4o.

Inside your circulatory system, particularly in your aorta, both laminar and turbulent flow occur, with each heartbeat.

In a previous article https://open.substack.com/pub/stephenperrenod/p/bitcoin-vs-gold-has-two-phases , I showed that Bitcoin’s price history vs. gold demonstrates the two phases, and that they can be modeled reasonably well with two overlapping Gaussian distributions. In this article, I explore in some depth the two phased nature seen in the Bitcoin price history vs. the dollar.

Quantile regression for median power law

The first step is to perform a quantile regression at the 0.5 level, the median level, to determine the power law trend for the full data set, which is a weekly series spanning over 15 years. Then we subtract that median power law in order to determine the log10 residuals, the probability distribution (kernel density estimation) for which is shown in Figure 1 below.

Figure 1. Smoothed probability density distribution of log10 residuals after removing the power law trend; there are over 750 weekly data points in the series.

Gamma distribution fit

We can also plot the histogram of the residuals, seen in Figure 2 (in blue). There is also red curve shown that is the best fit to the residuals as determined for a suitable unimodal distribution, a gamma distribution. The fit is inadequate at the first peak around -0.2 (x-axis value) and also we see that there is a dip between the two peaks at 0.1 and the second peak at around the +0.2 level for the log10 residual values.

Figure 2. Histogram of log10 residuals after removing the power law trend; there are over 750 weekly data points in the series.

Next one can perform a dip test. The dip test statistic measures the significance of the dip located at around 0.1 on the x-axis. The resultant statistic is 0.043, with a p-value 0.00000, and this indicates the data set is likely multimodal (null hypothesis rejected at the 5% level).

Dual Gaussian distribution fit

Then I fit the distribution with two Gaussians, using the GMM or Gaussian Mixture Model method. The two best fit curves are shown in green for the power law core and red for the bubble phase, in the chart above. (The bin interval displayed is different between Figure 2 and Figure 3). The means are considerably separated, by 0.44 in the log10 residual space, and that corresponds to a price ratio of 2.75 times. The standard deviation for points in the right hand bubble phase is 3 times greater (in the log!) than that in the core power law phase.

Figure 3. Histogram of residuals with fit to two Gaussians, the red bubble curve mean residual is higher by 0.44, and its standard deviation larger by a factor of 3.

Here’s what ChatGPT 4o says about the comparison of the unimodal Gamma distribution fit and the bimodal Gaussian fit:

✅ “The Bimodal Gaussian Mixture Model (GMM) is a significantly better fit than the Gamma model.

• It captures both the normal residuals and the bubble regime separately.

• The Gamma model misses the left tail and over-smooths the right tail, meaning it does not fully capture the bubble behavior.”

Quantile regression for the core

One can then redo the quantile regressions using only the data corresponding to the core power law behavior, that is the 47% of the data weight found for the first Gaussian in Figure 3. This is shown below in Figure 4 for a range of levels, and is also compared to an ordinary least squares regression. The OLS (R^2 = 0.998) and median (0.5 level) quantile regression (pseudo R^2 0.997) are quite close with power law indices of 5.89 and 5.92 respectively, and the band defined by the levels is very tight, in the range of 5.84 to 5.96 for their power law indices.

It’s a phenomenally strong indication of the power law nature of Bitcoin, extending as it does for over a dynamic range of a factor of one million in price.

Figure 4. Ordinary least squares and quantile regressions for the lower 47% of the data corresponding to the power law zone. The quantile bands are exceptionally tight and the OLS and median quantile regression are quite close with power law indices of 5.89 and 5.92, respectively. This strong power law behavior extends over a dynamic range of more than a factor of one million in price.

Z-scores each phase

We can break the two zones apart and determine Z-scores separately for each, that is, the number of standard deviations away from the respective mean for each data point. Figure 5 shows these; keep in mind the means (zero levels) are quite different and the standard deviation of the log residuals is about three times higher for the right hand red plot of the bubble zone behavior. And figure 6 shows just the bubble zone along with Bitcoin’s price curve on a log-linear scale.

Figure 5. Respective Z-score scatter plots for power law and bubble zones, relative to their own means and standard deviations. The 2011, 2013, 2017, and 2021 bubbles are evident in the right hand plot and there is early evidence of bubble behavior for 2024/2025.

Figure 6. The Z-score scatter plot for the bubble zone (lowest values truncated), compared to the Bitcoin historical price curve. One clearly sees the maximum absolute Z-scores coming down from one bubble to the next, the decline in volatility.

Bubble zone: Rising or Falling?

Figure 7 shows an attempt to discriminate between the rising portion of the bubble and the falling portion. A six week filter works pretty well, it just asks is the residual higher or lower than it was six weeks ago to determine trend continuation or a change in trend. Both the rising regions and falling regions tend to display persistence, of course some whipsawing between the rising signal and falling signal is unavoidable.

What is our present situation? Based on figures 5 and 6 we are tentatively in the bubble zone. We entered and fell out of it during the first half of 2024 and re-entered it late last year. Follow through would require the Z-score for the bubble phase that is for now less than zero to turn positive. That would require a price increase of another 20% or 25%.

Figure 7. Rising versus falling bubble discrimination using a 6-week filter. A dot is orange (purple) if the log residual from the power law is higher (lower) than 6 weeks prior. The rising portion is shown with weekly orange dots, the falling portion with purple dots. Once the fall begins it tends to be persistent with modest ‘dead cat’ bounces. The rises tend to be sharper than the falls. (These are the residuals, not Z-scores, so not directly comparable with Figure 6).

Bimodal Gaussians by epoch

Now it is also interesting to see how the dual Gaussian fits have evolved over time. To this end, I divided the data into four epochs, prior to the first halving is the first epoch, between the first and second halving is the second epoch, etc. The 9 months of data since the fourth halving has been lumped together with the fourth epoch, so is labelled Fourth Epoch+ in the figure below.

Figure 8 shows the histograms for each epoch and four sets of dual Gaussian best fit curves, labelled with the mean and standard deviation for each component. We see the range of the residuals on the x-axis compress considerably beginning with a range of 1.7 down to about half that as we move through the epochs. And the bubble mean value falls from 0.8 in the first epoch to 0.35 in the fourth epoch (somewhat higher than the prior epoch).

It seems like the core power law component has become more dominant, at least as we move from the second to the fourth epoch and the bubble peaks come down.

Figure 8. The bimodal Gaussian fits for each epoch. The 9 months of data through January 2025 has been included with the fourth epoch, although it properly belongs to the fifth epoch (post fourth-halving). The range compresses from around 1.7 in the log10 (x-axis) down to a span of about 0.9. The means for the core vary little but the means for the bubbles come down considerably with time.

Figure 9 shows how the residual means for the power core portion and the bubble phase have changed with time. The core power law zone (blue line) has been rather flat, in a range from -0.08 to -0.18 centered around -0.13. The power law is very consistent as we have seen. The bubble mean drops dramatically rom 0.81 down to roughly 0.3 for the third and fourth epochs. The difference between the means has dropped in a similar fashion from 0.9 to about half that value.

Figure 9. Graph of the Gaussian fit means for each zone as a function of epoch, in blue for the power core, in green for the bubbles. The difference between the two is in gray. The fourth epoch data point includes 9 months of data from the current fifth epoch. Also shown are the values for the entire 15-year weekly data set.

What is peak value this cycle?

As you have been reading this article you have been asking yourself, ok, but what is the peak $ value for Bitcoin this cycle? Wait no longer, here’s a rough estimate.

We get one lift from the difference in the bubble mean with the power law core as the bubble develops. We get another lift as the bubble moves into its right tail. For the fourth+ epoch the difference in means is 0.43 in the log and the standard deviation is 0.13 in the log.

So for example if we reach Z = 1.5 in the right tail we have a ratio with the core power law that is 0.43 + 1.5 * 0.13 = 0.625, or a linear factor of 10^0.625 = 4.22 x the core power law trend value. Z = 1.5 would correspond to the upper 93% of the data in the bubble, but the bubble phases only contain about half the data, so it is more like a 96.5% level overall, certainly not on the pessimistic side of future peak estimates.

When is this cycle’s peak? Let’s consider July 2025, October 2025, and January 2026 dates, corresponding to ages 16.5, 16.75, 17 for Bitcoin. Now the median quantile regression equation shown in Figure 4 is:

Log Price = -2.383 + 5.924 * log (Age), where these are logs base 10. The respective trend price levels for July, October, January are therefore $67.5K, $73.8K, $80.6K.

Thus if there is a similar type of bubble that reaches Z of 1.5 as seen for the fourth epoch bubble distribution, and the bubble peak is in July, October, or January the price targets are robust at $285K, $311K, or $340K respectively. Using the OLS regression rather than the quantile median regression only lowers these values by about $6K to $8K.

Reaching the values near and above $300K all depends on attaining a Z score of 1.5 rather than topping out at Z of 1, say, which would results in peak values in the $240K to $290K range. And obviously later and longer duration bubbles should reach higher values.