Analyzing from a power law perspective: Why is there no Bitcoin bull run this year, and when will the next bubble occur?

Author: Stephen Perrenod, Technical Analyst

Compiled by: Felix, PANews

The largest bubbles (bull markets) in Bitcoin's history occurred in 2011, 2013, 2017, and 2021. Although many Bitcoin investors firmly believe in a four-year cycle, there was no bubble in 2025.

Why? Because everyone has been brainwashed by the narrative of “bubble according to linear time, once every four years,” expecting peaks to occur after the halving in 2012, 2016, 2020, and 2024. This narrative has been further reinforced by the U.S. business cycle and presidential election years.

Individuals have also been deeply trapped in the hypothesis of a four-year linear cycle.

In the past year, individuals have increasingly focused on modeling bubbles, separating the “core power law zone” (long-term trend) from the “bubble zone” (short-term deviation) for analysis.

The result is becoming clearer: the “energy” of the bubble is diminishing, roughly inversely proportional to the “age” of Bitcoin, or decaying by the power of -0.83 according to age.

People still mistakenly believe that the percentage increase of each bubble will be the same, but in fact, the intensity of the bubble is clearly diminishing. The study also indicates that the intervals between bubbles are lengthening over time.

What about the bubble in 2011? People tend to ignore the big bubble of 2011 as an “early anomaly.” However, the bubble in 2011 is actually just as important as those in 2013 and 2017, and should be treated equally. What if bubbles do not appear in a linear cycle, but rather in a logarithmic cycle? This phenomenon is known as logarithmic cycle behavior.

logarithmic periodic bubble originates from power law

After all, Bitcoin itself is a power-law asset, and its price movements over time exhibit scale invariance. To achieve the same price increase, there must be a proportionate increase in time. For example, if the relationship between price and age is Price ~ Age^k, then in dollars, if k = 5.7, when age doubles, the price will increase 52 times. To achieve the same 52 times price increase, Bitcoin's age needs to grow from 2 years to 4 years, then from 4 years to 8 years, and then from 8 years to 16 years, and so on.

Therefore, it is not surprising that the intervals between the emergence of bubbles are becoming increasingly longer. A logarithmic time interval may correspond to only one major bubble, while the intervals in linear time will become longer and longer.

In fact, as early as 2019, Giovanni Santostasi (who discovered the power law nature of Bitcoin before the second halving) proposed a logarithmic cycle model around the power law trend (the curved/oscillating line in Figure 1). It almost corresponds to the three bubbles in 2011, 2013, and 2017, and even predicts that the next bubble peak will be at x=3.817 (i.e., 10^3.817 ≈ 6561 days, about 18 years), which is around the end of 2026 to the beginning of 2027.

Figure 1

This post from six years ago deserves more likes. Although the amplitude fitting is average, the timing is very accurate, and later we will see that the prediction of the fourth peak may be extremely precise.

Roughly observe the age ratio. The ratios between the first two adjacent bubbles are 2.02 and 1.82, averaging 1.92, and the fourth bubble is about at the age of 17.18 years (early 2026). Of course, this is just a rough preliminary guess**.**

Neither Giovanni's fitted model nor the model I ran after adding 6 and a half years of data on top of it predicted a bubble in 2021 or 2025. This raises a question: was the double bubble in 2021 driven by fundamentals or caused by other reasons?

Complete Logarithmic Cycle Analysis

What needs to be noted next is that the full log-periodic algorithm is much more precise than simply analyzing the interval between two peaks.

It is related to a more complex logarithmic periodic analysis promoted by geophysicist Didier Sornette, which was initially used for earthquakes, other natural phenomena, and financial time series. The complete equation contains seven parameters, including a local power law index and a harmonic (cosine) term with logarithmic time dependence.

In this article, Fourier and wavelet analyses will be performed on the logarithmic time of the complete price history. Bitcoin is measured in ounces of gold, and scale invariance tests were conducted. In the logarithmic price residuals with a base of 10, the power law exponent is 5.30, R² is 0.94, and the standard deviation is 0.314.

Conduct a logarithmic time Fourier transform analysis on the compressed residuals after removing the 5.30 index power law, resulting in the logarithmic period characteristic wavelength parameter λ = 2.07. In the peak interval formula mentioned above, this roughly corresponds to the ratio of the “age” between each bubble, but it is a more global estimate that also takes harmonics into account. Wavelet analysis reveals that the range of λ is between 2.00 and 2.10.

The finalized optimal fitting λ value corresponds to the fundamental mode, which fits well with the bubbles in 2011, 2013, and 2017. At the same time, it predicts that the first fundamental mode bubble after 2017 will appear in 18.4 years, specifically in late May 2027. If we take the lower limit of the wavelet analysis as λ=2.0, the predicted age is 17.9 years, which is about a year later (the end of 2026), closely aligning with Giovanni's prediction in 2019.

“The discrete age ratio model and your LPPL fitting describe the same underlying discrete scale invariance (λ≈2), just in different coordinate systems; once you incorporate ln(t_c − t) along with age t, noise, and half-harmonic considerations, they are completely consistent, both pointing to the fourth fundamental mode bubble appearing around 2026-2027, with the LPPL model leaning slightly later (around 2027) and the rough age ratio method leaning slightly earlier (around 2026).” — GPT5

What is the bubble of 2021?

In this framework, 2021 is not the prototype, but the first harmonic of the prototype (higher frequency but shorter wavelength, because in the LPPL model, the angular frequency of logarithmic periodic behavior satisfies λ = e^{2π/ω}). The corresponding ratio is the square root of 2.07, which is 1.44. 8.95 × 1.44 = 12.89 years, which corresponds exactly to the end of 2021, closely matching the actual double top bubble age of 12.85 years.

What about the year 2025? The logarithmic cycle sequence does not predict a bubble. This year, Bitcoin's performance relative to gold has been weak, which exactly aligns with the expectations of this framework.

Figure 2 shows the complete mapping of the fundamental frequency and the sub-harmonic (first harmonic in frequency, half harmonic in wavelength) patterns. The red vertical lines represent the bubble peaks of the fundamental frequency pattern, while the blue line represents the sub-harmonic. The three red lines match very well, but the blue line also reflects the smaller bubbles of that harmonic quite well. This is not a complete mapping; there are higher-order harmonics, but their contribution is smaller.

The fourth red line corresponds to a prediction of an age of 18.4 years, which indicates the next big bubble of Bitcoin relative to gold. Bitcoin may begin to rise in value against gold starting in early 2026 and peak in the second quarter of 2027.

Figure 2

Why does this behavior occur? Continuous scale invariance + discrete scale invariance.

To understand the price trend of Bitcoin, it is necessary to understand both continuous scale invariance and discrete scale invariance simultaneously. Continuous scale invariance refers to the long-term power-law trend that readers are already familiar with. Discrete scale invariance corresponds to intermittent bubbles, which can be described using the LPPL (Log-Periodic Power Law) model. The inclusion of “Log-Periodic Power Law” in the name can lead to confusion.

The LPPL model is suitable for short to medium-term financial sequences, specifically used in this case to simulate the large fluctuations that deviate from the core power-law trend. Therefore, there are two situations present: the overall power-law trend (continuous) and the high volatility deviations occurring above it (discrete).

In the long term, the adoption rate of Bitcoin follows Metcalfe's Law and exhibits power-law growth.

The emergence of a bubble is triggered by a surge of interest from new participants when the price of Bitcoin breaks through a new magnitude (the price of Bitcoin has risen over 7 magnitudes, or 7 times, in its brief but impactful history).

Bitcoin will only be adopted when it can change the destiny of you (or institutions).

In the early days when prices were low, it mainly attracted crypto punk enthusiasts, retail trading, and investment; over time, it attracted increasingly higher levels of institutional capital: first miners, then exchanges, futures trading, followed by some listed companies, and now sovereign wealth funds, ETFs, and other institutional tools. Currently, institutional capital is a net buyer, and most of their holdings are managed for a large number of clients and shareholders.

When the price and market value of Bitcoin are high enough, pension funds, sovereign wealth funds, and even national treasuries will be tempted to invest. Each subsequent bubble is priced in ounces of gold at levels far higher than the previous one (even higher when priced in fiat currency), leading to a large-scale adoption of Bitcoin by new capital.

Even if the market capitalization reaches 2 trillion USD, Bitcoin is still too cheap for the treasury departments of large enterprises and mid-to-large countries, insufficient to make a significant impact. Even if they could purchase 1 million Bitcoins at around 100 billion USD, it would not “change their fate.”

The power law effect of Bitcoin is driving an influx of increasingly larger participants into the market. This effect originates from the protocol itself: attracting computing power, enhancing security, and the advantages of being an early mover, all of which will ultimately drive up prices.

Therefore, fiat currency is incentivized to flow into Bitcoin, and even fiat currency that might originally flow into gold has turned to Bitcoin. Indeed, Bitcoin's performance in 2025 may be relatively weak compared to gold, but according to the analysis above, this is likely to be a low year for Bitcoin relative to gold.

A new wave of adoption by larger players arrives in the form of LPPL bubbles, which means they must collapse at their local critical time points. Therefore, the price trajectory will revert to long-term power laws.

Due to the power-law relationship P ~ T^k, where k is the exponent and T is the age, and its first derivative δP / P ~ k/T, the spacing of bubbles is expected to show a logarithmic relationship. The relative growth rate decreases with the inverse of the age of Bitcoin, but the emergence of bubbles requires δP / P >> k/T. By definition, a bubble forms when the rate of price increase is far higher than the long-term power-law trend.

This expression can be rewritten as δ (log P) ~ k * δ (logT), therefore it can be expected that large fluctuations in logarithmic price (larger multiplicative ratios) require increasingly longer linear time, that is, they need to increase logarithmically in time proportionally, with an order of magnitude of the 1/k root of the required multiplicative fluctuations.

In fact, when calculated in terms of gold prices, the median peak values of the bubbles in 2011, 2013, and 2017 are 10.1 times relative to the power law trend, with a power law exponent of 5.31. It is expected that λ is approximately 10.1^(1/5.31) = 1.55, which is about three-quarters of the value 2.07 obtained from the logarithmic cycle fitting of peak times. Therefore, empirically, when calculated in terms of gold prices, λ ~ (4/3) F^(1/k), where F is the typical ratio of the peak to the power law trend price.

Summary

The main price movements of Bitcoin consist of the following two aspects:

• Scale invariance can be represented by a power law, with an index of approximately 5.3 when priced in gold and an index of approximately 5.7 when priced in USD.
• Discrete Scale Invariance (DSI) is manifested in intermittent bubble events, as larger new capital tiers will only adopt Bitcoin when prices and market values reach levels sufficient to attract new capital and the regulatory environment permits. These discrete scale invariant events can be described using the LPPL model to represent bubbles and their subsequent crashes.

The 2013 bubble was primarily composed of retail investors, early miners, early Bitcoin startups, and first-generation exchanges. The 2017 bubble attracted participation from hedge funds, family offices, and other high-net-worth individuals, with GBTC trading starting as early as 2015. The peak of the bubble occurred in December 2017 after the Chicago Mercantile Exchange (CME) launched futures, making efficient shorting and hedging possible. The 2021 bubble received recognition from some corporate finance departments and a few small government agencies. Due to increased institutional participation and strengthened related mechanisms, the ability to hedge and short became stronger, leading to the eventual collapse of the bubble and prices returning to a power law trend like a spring.

The next wave of large-scale adoption may focus on 2026 and 2027, when Bitcoin's market capitalization is expected to approach $5 trillion, attracting long-term allocations from large enterprises, sovereign wealth funds, and governments (national pension funds and treasury departments). The phased bubbles of Bitcoin reflect the different tiers of global capital: from retail investors to funds, then to corporations, and finally to sovereign nations, leading to DSI behavior and a decrease in volatility associated with the increase in market capitalization and institutional participation.

“Capital at every level will only be unlocked when Bitcoin reaches a sufficiently large, liquid, and trustworthy scale — due to its returns following a power law growth and k/t decay, this process takes longer and longer over time.” — ChatGPT 5

The fundamental mode wavelength parameter λ of DSI is 2.07 (log time), roughly corresponding to twice the bubble interval. The bubble in 2017 was the last fundamental mode, the first harmonic occurred in 2021, and no bubbles are expected under this framework in 2025. The next peak in Bitcoin priced in gold is expected to occur between the fourth quarter of 2026 (minimum value from wavelet analysis) and the second quarter of 2027 (value from Fourier analysis).

Related reading: Bull or bear? 5 major signals leading the new cycle of Bitcoin.