**Why Bitcoin is Inherently Predictable**

**Introduction**

In the world of finance, predictability is often considered an elusive trait, in fact, an entire fundamental financial theory, the Market Efficiency Hypothesis claims that markets are unpredictable and any observed inefficiency is immediately exploited and canceled out by market forces. Yet, amidst this apparent chaos, Bitcoin — the world’s first and most prominent cryptocurrency — has exhibited a remarkable degree of predictability. This predictability stems from an underlying mathematical concept known as a power law, which has consistently explained Bitcoin’s behavior over the past 15 years, spanning eight orders of magnitude. In addition, Bitcoin demonstrates cyclic periodicity with extremely high precision in comparison with traditional assets, where seasonality, if present is weak and not consistent. Both the long-term behavior and its consistent seasonality make Bitcoin inherently predictable.

**Understanding Power Laws**

A power law is a mathematical relationship where one quantity varies as a power of another:

y = C ⋅ xᵐ

- y and x are the variables.
- C is a constant that scales the relationship.
- m is the exponent. Unlike other relationships that may only apply within a specific range, power laws are
**scale invariant.**This means that the relationship holds true across different scales, whether we examine small or large values. For Bitcoin, this implies that the growth patterns observed in the early years can be extrapolated to much larger scales, providing a robust framework for understanding its future trajectory.

**The Power Law in Bitcoin**

Since its inception in 2009, Bitcoin’s price has followed a power law. This is evident in the logarithmic plots of Bitcoin’s price against time, which show a straight line — indicative of a power law relationship. Over the course of 15 years, Bitcoin’s price has grown in a precise mathematical fashion, adhering to this power law over eight orders of magnitude.

*This consistency is not merely a coincidence; it suggests that there are underlying mechanisms driving Bitcoin’s growth in a predictable manner.*

These mechanisms could include network effects, adoption rates, and feedback loops of all kinds, all of which contribute to its price behavior over time. We already wrote on both these subjects:

How we can derive the main power laws via modeling Bitcoin as a network:

https://bitposeidon.com/the-power-law-theory/f/modeling-bitcoin-growth-with-network-theory

The Power Law Theory is a full fledge Theory of Bitcoin that shows the causative links of all the different on-chain parameters:

**Scale Invariance and Predictability**

The concept of scale invariance is crucial to understanding why Bitcoin’s adherence to a power law makes it inherently predictable. In a scale-invariant system, the same rules apply at every scale.

Let’s analyze this more specifically. Let’s look at the example below. We see the power law. The purple line is the usual long-term relationship. The wavy line is a full model where we add to the general trend also periodic oscillations and even decay of the observed tops. Let’s focus mainly on the general trend or purple line.

The graph presented is a powerful visual representation of Bitcoin’s price behavior over time. What immediately stands out is the triangular pattern formed by the price movements when plotted on a logarithmic scale. This pattern is a hallmark of proportionality (changes the size of a triangle leaves everything proportional).

In the context of this graph, the straight line on a log-log plot signifies that **the price of Bitcoin has increased proportionally with time.** However, this increase is not linear in the traditional sense. Instead, the line represents a relationship where the scale of the price (y-axis) changes in proportion to the scale of time (x-axis). This is the essence of a power law, where one variable varies as the power of another.

**Scale Invariance and Predictability**

Scale invariance is a key concept in understanding why Bitcoin’s behavior might be predictable. In a scale-invariant system, changes in one dimension (like time) require proportional changes in another dimension (like price) to maintain the same relationship. **This means that, as time progresses, the system “scales up” in a predictable way.** For Bitcoin, this has been evident in the way its price has grown over the years, following a consistent pattern that spans several orders of magnitude.

Because the graph shows that Bitcoin’s price has followed this power law relationship so consistently over time, it provides a strong basis for hypothesizing that this behavior will continue in the future. **The straight line in the graph isn’t just an arbitrary fit; it’s a reflection of the underlying scale-invariant nature of the system**. This scale invariance implies that the same rules that governed Bitcoin’s growth in the past will likely continue to apply, even as the system scales up to new magnitudes.

The triangular pattern seen in the graph further emphasizes the idea of scale invariance. As Bitcoin’s price has scaled up, the time intervals between significant price movements have also scaled. This is why the graph expands outward in a triangular fashion — **each step up in price requires a proportionate step up in time.**

Given that Bitcoin has exhibited this scale-invariant behavior across so many orders of magnitude, it is scientifically reasonable to hypothesize that it will continue to do so. The fact that Bitcoin has consistently adhered to this pattern over 15 years, despite various market conditions and external factors, strengthens this hypothesis.

I realize that this is a difficult concept to accept, not much in the logic of it that once understood as “everything is proportional” is not that difficult conceptually. What is difficult for many people is to accept that a financial system that comprises of complex interactions by many agents, in many countries, at different times can express itself in such a simple and predictable pattern. But it is a fact. It is an undeniable fact. It is a most striking aspects of Bitcoin’s price behavior that over the years is how well it adheres to a power law — a simple, yet profoundly powerful mathematical relationship.

What makes this observation so compelling is not its complexity, but rather its simplicity and clarity. **The power law governing Bitcoin’s price is not a theoretical construct or a complex model that requires extensive mathematical gymnastics to understand**. Instead, it is an empirical observation, plainly visible to anyone who examines the data. Many professionals in the field have done so and arrived to the same conclusion. At a minimum, the Bitcoin Power Law is part of the history of Bitcoin and understanding it allows us **to understand better what Bitcoin is.**

**Real-Life Examples of Scale Invariance in Action**

The graph below, taken from a study on the growth of various animal structures such as teeth, shells, and thorns, presents a fascinating example of how power laws can be used to predict future growth based on past trends. In this graph, the growth of elephant tusks is plotted on a log-log scale, with the blue line representing male elephants and the pink line representing female elephants. The y-axis represents the radius of the tusks in millimeters, and the x-axis represents the age of the elephants in years.

https://bmcbiol.biomedcentral.com/articles/10.1186/s12915-021-00990-w

The fact that this graph is presented on a log-log scale indicates that the growth of elephant tusks follows a power law relationship. The straight lines observed in the graph for both male and female tusk growth suggest that the growth of tusks is scale-invariant, meaning that the relationship between age and tusk radius holds true regardless of the scale at which it is observed. One of the key advantages of identifying a power law relationship is its predictive power. Because the growth of elephant tusks follows a power law,** we can make informed predictions about the future size of the tusks based on their past growth.** This is particularly valuable because it means that, given the established pattern of growth, we can confidently extrapolate the tusk size at older ages even if we only have data for younger elephants.

For example, if we observe that an elephant’s tusk grows at a certain rate up to the age of 10 years, and this growth follows a power law, we can predict that the tusk will continue to grow at a consistent rate as the elephant ages. This allows us to forecast the tusk size when the elephant reaches 20, 30, or even 50 years of age, as long as the underlying growth pattern remains unchanged. The reason this predictive approach is valid lies in the nature of power laws. A power law implies that the same relationship between variables will hold across different scales. In the case of elephant tusks, the relationship between age and tusk size has been established as a power law, suggesting that the tusks’ growth rate will remain proportional to the elephant’s age in a consistent manner. This is not a speculative guess but rather an observation grounded in the data we have — data that has consistently followed this pattern across the age range studied.

This concept is crucial when considering how to apply these findings in real-world scenarios. For instance, conservationists or wildlife managers could use this predictive model to estimate the future size of tusks for populations of elephants, helping to better understand and manage their development over time. This information could be even used as a diagnostic tool to see a particular animal is growing in a healthy predictable fashion.

**Bitcoin’s Future Through the Lens of Power Law**

The fact that Bitcoin has followed a power law for 15 years suggests that this trend is likely to continue. While short-term fluctuations are inevitable, the long-term trajectory of Bitcoin is governed by this scale-invariant relationship. This provides a framework for making serious hypotheses about Bitcoin’s future behavior.

For instance, as Bitcoin adoption continues to grow and its network effects become stronger, we can expect its price to continue following the established power law. This does not guarantee specific price points, but it does suggest that Bitcoin’s growth will continue in a particular fashion — one that has been remarkably consistent over its history. Bitcoin’s history is a story of growth that is intimately tied to its context at each stage of its development. From its inception by Satoshi Nakamoto to the global phenomenon it is today, every event that has shaped Bitcoin has done so in a manner that is proportional to where Bitcoin was at that moment in time. This concept of proportionality is not just a philosophical observation — it is a fundamental characteristic of how complex systems evolve, and it is particularly evident in Bitcoin’s trajectory.

**Proportionality in Bitcoin’s Early Days**

When Satoshi Nakamoto released the Bitcoin whitepaper in 2008 and mined the first block in 2009, Bitcoin was nothing more than an idea, a concept shared by a small group of individuals known as cypherpunks. These early adopters were passionate about cryptography, privacy, and the potential for decentralized systems to reshape the financial landscape. At that time, the value of Bitcoin was practically zero, and its influence was limited to a handful of enthusiasts who donated their time, energy, and computational resources to support the network.

The events that occurred during this period — whether it was the mining of early blocks, the exchange of Bitcoin for goods and services, or the development of the first Bitcoin software — were all proportional to Bitcoin’s nascent state. The network was small, the community was tight-knit, and the stakes were relatively low. Yet, these early actions laid the groundwork for everything that was to come.

**Scaling Up: Bitcoin’s Growth in Proportion to Its Size**

As Bitcoin began to grow, so too did the scale of the events that impacted it. The first real-world transaction using Bitcoin, where 10,000 BTC was exchanged for two pizzas, is a famous example of how Bitcoin’s value was still being established. This transaction, while seemingly trivial at the time, was proportional to Bitcoin’s status as a fledgling currency trying to find its footing.

As more people became aware of Bitcoin, the network expanded, and the value of Bitcoin began to increase. With each new participant, the impact of their involvement was proportional to the size and reach of Bitcoin at that moment. This scaling effect continued as Bitcoin moved from the fringes of the internet into mainstream awareness, attracting developers, investors, and eventually, large institutions.

**Proportionality in Major Milestones**

Bitcoin’s major milestones — whether it’s the Mt. Gox exchange hack, the introduction of Bitcoin futures trading, or the increasing regulatory scrutiny from governments — can all be seen as proportional to Bitcoin’s status at the time. When Bitcoin was still relatively small, events like the Mt. Gox collapse were significant within the cryptocurrency community but had little impact on the broader financial system. However, as Bitcoin grew in value and importance, the effects of similar events became far more significant.

For example, the introduction of Bitcoin futures trading in 2017 was a major milestone that signaled Bitcoin’s arrival as a legitimate financial asset. This event was proportional to the level of adoption and the market capitalization that Bitcoin had achieved by that point. Similarly, the growing regulatory attention Bitcoin has faced in recent years is proportional to its increasing influence and the potential it has to disrupt traditional financial systems.

**The Power Law: A Framework for Understanding Proportional Growth**

The concept of proportionality in Bitcoin’s growth is closely related to the power law discussed earlier. Just as the growth of elephant tusks or other natural phenomena can be predicted based on their past behavior, Bitcoin’s future can be understood through the lens of its proportional growth. Each event that affects Bitcoin does so in a way that is consistent with its current state — whether it’s a new technological development, a market fluctuation, or an external regulatory action.

Because Bitcoin has followed this pattern of proportional growth so consistently, it gives us confidence in making predictions about its future. Just as the early contributions of cypherpunks were crucial in Bitcoin’s infancy, the actions of today’s major players — whether they be institutional investors, regulators, or developers — will shape Bitcoin’s future in a way that is proportional to its current status as a global financial asset.

**The Proportional Path: Bitcoin’s Sustainable and Resilient Growth**

Bitcoin’s journey, guided by the principles of proportionality and power laws, represents not just a pattern of growth but an optimal path that is uniquely suited to its purpose of becoming the monetary system of the entire world. This path, characterized by its resilience, sustainability, and balance, is far more than just a trajectory of increasing value — it is the path of least resistance, enabling Bitcoin to grow rapidly while maintaining the stability necessary for long-term success.

**Sustainability Through Proportional Growth**

In contrast to unbridled exponential growth, which often leads to unsustainable bubbles and crashes, Bitcoin’s proportional growth is inherently sustainable. This proportionality ensures that each phase of Bitcoin’s development is matched with the appropriate level of support, infrastructure, and adoption. As Bitcoin grows, the network scales naturally, with the necessary resources and participants coming into place to support its expansion.

This sustainable growth pattern is critical for Bitcoin’s long-term viability. By growing in proportion to its current size and influence, Bitcoin avoids the pitfalls of overextension — where growth outpaces the ability of the network and its participants to support it. Instead, Bitcoin’s growth is balanced, allowing it to expand steadily without losing the foundational support that makes it secure and resilient.

**Resilience to Changes: Adapting to Good and Bad Events**

One of the most remarkable features of Bitcoin’s proportional growth path is its resilience to both positive and negative changes. Whether facing regulatory challenges, market volatility, or technological advancements, Bitcoin has consistently demonstrated an ability to adapt and thrive. This resilience is a direct result of its proportional growth, which ensures that the system remains robust and flexible, able to absorb shocks and continue progressing along its established path.

For example, during periods of significant market corrections or regulatory crackdowns, Bitcoin has experienced temporary setbacks but has always recovered and continued its upward trajectory. This resilience is a testament to the strength of the network and the robustness of its proportional growth model. Similarly, during times of rapid adoption or technological innovation, Bitcoin has been able to integrate these positive changes without destabilizing the network.

This adaptability is crucial for Bitcoin’s goal of becoming the global monetary system. In a world that is constantly changing, a system that can respond to and thrive in the face of both opportunities and challenges is essential. Bitcoin’s proportional path provides this adaptability, ensuring that it can continue to grow and evolve no matter what the future holds.

**The Path of Least Resistance: Optimal Growth for Global Adoption**

Bitcoin’s proportional growth can be seen as the path of least resistance — a trajectory that allows for the fastest possible growth while maintaining balance, sustainability, and robustness. Unlike exponential growth, which can be unsustainable and prone to collapse, proportional growth is inherently balanced. It allows Bitcoin to grow rapidly but in a way that is supported by the underlying infrastructure, community, and market forces.

This optimal growth path is what makes Bitcoin uniquely suited to its purpose of becoming the global monetary system. To fulfill this role, Bitcoin needs to achieve widespread adoption and integration into the global economy. However, this cannot happen overnight; it requires a gradual and sustainable expansion that builds on the existing foundation. Bitcoin’s proportional growth ensures that this expansion happens in a way that is both fast enough to meet demand and stable enough to support long-term adoption.

By following this path, Bitcoin avoids the extremes of both stagnation and unsustainable bubbles. It grows quickly but in a way that is grounded in reality, supported by the ongoing development of the network and the increasing participation of individuals and institutions. This balanced approach is what will enable Bitcoin to continue its journey toward becoming the global standard for money.** The Misconception About Power Laws and Human Behavior**

A common misconception is that power laws, or other mathematical behaviors, do not apply to human activities or social phenomena. Many people find it difficult to reconcile the idea that the seemingly chaotic and unpredictable nature of human behavior could follow simple, mathematical patterns. However, this belief is not only mistaken, but it also overlooks a significant body of empirical evidence demonstrating that power laws are pervasive in various aspects of collective human behavior.

**Power Laws in City Growth and Infrastructure**

One of the most compelling examples of power laws in human systems comes from the work of physicist Geoffrey West, who has extensively studied the growth of cities. West’s research shows that a variety of urban metrics, such as GDP, the number of gas stations, libraries, and even the pace at which people walk in a city, follow power law distributions.

West found that the GDP of a city scales as a power law with its population size. This means that as a city grows, its economic output increases at a predictable, exponential rate relative to its size. Similarly, the number of gas stations, libraries, and other infrastructure components also scales according to power laws. These findings reveal that the growth and organization of cities are not random but follow well-defined mathematical patterns.

**Power Laws in Conflict and Violence**

Power laws are not limited to urban growth; they also apply to more chaotic and seemingly unpredictable events, such as conflicts and violence. The work of researcher Sean Gourley has shown that the distribution of fatalities in wars follows a power law. This means that while most conflicts result in relatively few deaths, there are a few extreme events with a large number of casualties. The same power law distribution applies to the time between wars, suggesting that even the occurrence of wars is governed by predictable, mathematical patterns.

Gourley’s research highlights that power laws are not just academic curiosities; they are deeply embedded in the fabric of human society, influencing even the most extreme forms of human behavior. The fact that such patterns emerge in the context of war — a situation often viewed as chaotic and unpredictable — underscores the power of these mathematical relationships.

**Power Laws in Crime and Terrorism**

Further evidence of power laws in human behavior comes from studies on crime and terrorism. Research has shown that the frequency and severity of terrorist attacks follow a power law distribution. Just as with fatalities in war, most terrorist attacks are relatively small in scale, but a few are devastatingly large. This pattern is consistent across different regions and time periods, indicating that it is a fundamental characteristic of terrorist activity.

Similarly, the occurrence of crimes in cities follows a power law. The majority of crimes are minor, but there are occasional spikes in serious offenses. This distribution reflects the underlying social dynamics and interactions within urban environments, which are governed by the same mathematical principles that apply to other complex systems.

**The Ubiquity of Power Laws in Human Systems**

The examples of city growth, war, crime, and terrorism illustrate that power laws are not only applicable to human behavior but are a fundamental aspect of how human societies function. These patterns are not exceptions; they are the rule. From the distribution of wealth in an economy to the timing of revolutions and the size of companies, power laws describe a wide range of phenomena in human systems.

The difficulty many people have in accepting this idea likely stems from the fact that human behavior is often seen as too complex, too free-willed, or too variable to be captured by simple mathematical formulas. However, the empirical evidence overwhelmingly supports the view that, at the collective level, human actions do indeed follow predictable patterns. These patterns emerge naturally from the interactions of individuals within a larger system, resulting in power law distributions that can be observed and measured.** Conclusion**

Bitcoin’s behavior over the past 15 years has been remarkably consistent with a power law, spanning eight orders of magnitude. This adherence to a scale-invariant relationship makes Bitcoin inherently predictable, offering a robust framework for understanding its future trajectory. While this does not guarantee specific outcomes, it provides a strong basis for making informed hypotheses about Bitcoin’s growth and evolution.

As with other scale-invariant systems, once Bitcoin settled into its power law behavior, it continued to follow this trajectory. This suggests that, despite the volatility and uncertainty often associated with cryptocurrencies, Bitcoin’s long-term growth is governed by predictable and well-established principles. For traders, investors, and analysts, recognizing and understanding this power law relationship is key to navigating the future of Bitcoin. The power law should be celebrated as a sign of Bitcoin's mathematical and philosophical beauty, its uniqueness, and inner intelligence, and the organization should not be despised by ill-informed and myopic investors who want an unpredictable, unbridled, and chaotic growth of Bitcoin for selfish reasons. Bitcoin is a power law, a pattern of nature that makes Bitcoin more similar to planets, galaxies, and living organisms than a mere asset.

**Bitcoin is a Force of Nature.**