One of the best-known instances of power law dynamics occurs in regards to how mammals can be characterized as scaled up versions of each other, according to power laws that roughly determine their size.

Thus, if you know the size of a mammal species, you can use power scaling laws to predict the average efficiency of the species’s circulatory system, as it scales up, precisely based on weight, and hence, approximately estimate how much food it eats per day, what its heart-rate is, how long it will take to mature, average lifespan, etc . For instance, if you compare a mouse, a human and an elephant on a logarithmic chart, you will find that with every doubling of its average weight, a species gets 25% more efficient—and lives 25% longer.

Fundamentally, the issue has to do with the fractal geometry of the networks that supply energy and remove waste from the organism’s body. Hence, as the exponential dimension of any power law makes an organism more efficient, it becomes dominant among competing life forms.

In the case of Bitcoin, as halvings make its coinbase rewards smaller, its power law dynamics keep upgrading its resilience as a “life-form” and hence, its propensity to displace every use-case currently sustaining the utility of the only competing monetary “life form” in existence: fiat currency and its multifarious proxies, including bonds, stocks, financial derivatives, cryptocurrencies and other credit expansion-dependent contraptions.

In this video, Dr. Santostasi’s addresses why, as Bitcoin scales over time, the geometric pattern driving its network effects vs. price, hash rate, addresses, etc. can be best predicted by Power Law dynamics.

As shown in his summary chart (above), Dr. Santostasi found six bitcoin power laws, whose individual trends make Bitcoin’s growing Monetary System ever more stable, secure, and antifragile as their uniquely relevant exponential power sizes (below), evolve in close synergy with each other and become closely interdependent over time :

Power Law I: Price vs Addresses = 2
Power Law II: Price vs Time Difficulty Adjustment= 6
Power Law III: Price vs Hash Rate = 1/2
Power Law IV: Hash Rate vs Time to Block = 12
Power Law V: Hash Rate vs User Growth Rate= 1/4
Power Law VI: User Growth vs Time= 3

Dr. Santostasi explains that Bitcoin’s growth and stability can be best understood through mathematical relationships where one quantity is proportional to a fixed power of another. He demonstrates that Bitcoin’s network value and price increase with the square of its users, aligning with Metcalfe’s Law. This relationship is rooted in Bitcoin’s inherent design mechanisms, such as the mining difficulty adjustment, which self-corrects to maintain system stability. These mechanisms create iterative feedback loops that ensure the network adapts proportionally to changes in user base and mining power.

As Bitcoin’s user base and price grew, it attracted more resources and advanced mining capabilities, further enhancing network security and drawing in more users. Unlike typical S-curve adoption models, Bitcoin’s growth follows a power law due to its unique self-adjusting mechanisms. This ensures a robust and scalable system where interconnected power laws maintain equilibrium, allowing Bitcoin to grow predictably and sustainably.

Dr. Santostasi validates his theory’s predictive value using the following arguments:

1. Introduction to Power Laws:

  • Power laws are mathematical relationships where one quantity is proportional to a fixed power of another.
  • They are common in natural and social phenomena and emerge in iterative processes where the output becomes the new input.

2. Bitcoin and Power Laws:

  • Power laws are prevalent in Bitcoin, not just in price, but also in hash rate and the number of addresses.
  • Observing regularities in Bitcoin’s behavior, Dr. Santostasi discovered a power law between price and time, and subsequently, between other variables.

3. Bitcoin’s Evolution:

  • Initially, Bitcoin’s value and user base grew as more people mined and invested in it.
  • The price increased with the square of its users, following Metcalfe’s Law,
  • As per Metcalfe’s Law, the value of a network is proportional to the square of its number of users. The basic principle linking Bitcoin’s exponential value to network growth.

4. Mining Difficulty Adjustment:

  • Bitcoin’s mining difficulty adjustment mechanism ensures stability by self-adjusting based on the time taken to mine blocks.
  • This mechanism is analogous to a car adjusting its speed to navigate a mountain road safely.

5. Growth and Resource Allocation:

  • As Bitcoin’s price increased, so did the resources allocated to mining, leading to more sophisticated mining operations.
  • The increase in price and hash rate follows a power law relationship.

6. Security and User Growth:

  • Increased hash rate enhances Bitcoin’s security, attracting more users.
  • The user base has been growing with the power of three over time, deviating from typical S-curve adoption models.

7. Feedback Loops and System Stability:

  • Bitcoin’s power laws create interconnected feedback loops where changes in one variable affect others.
  • This system of power laws ensures Bitcoin’s stability and prevents extreme deviations in price, creating a self-correcting mechanism.

8. Outliers and Price Floors:

  • Deviations from power laws, such as bubbles, are corrected by the system.
  • Bitcoin’s price floors are maintained by incentive structures like the cost of mining, preventing significant drops.

9. Robustness and Anti-Bubble Mechanisms:

  • Bitcoin’s mechanisms protect against unhealthy growth, differentiating it from speculative bubbles.
  • The finite supply of Bitcoin amplifies the effects of power laws, ensuring steady and stable growth.

10. Future Insights:

  • Dr. Santostasi plans to further develop and explain the Bitcoin Power Law Theory in upcoming scientific articles and a book.

Overall, Dr. Santostasi’s theory illustrates how Bitcoin’s interconnected power laws create a self-adjusting and stable system, driving its growth and adoption in a predictable manner.
In the early days of Bitcoin, people started mining it on their laptops and invested out of curiosity. As interest grew, more people began participating in discussions and trading Bitcoin seriously. The second point of our theory posits that Bitcoin’s price, and the overall value of the network, increased with the square of its users. This empirical observation is based on the power law relationship between price and the number of addresses, which serve as a proxy for the network’s users. This phenomenon aligns with Metcalfe’s Law.

Although Satoshi Nakamoto likely did not explicitly consider power laws when creating Bitcoin, he designed mechanisms that inherently support these principles. A prime example is Bitcoin’s mining difficulty adjustment mechanism, which self-adjusts to maintain block discovery time around ten minutes. This adjustment is proportional to the deviation from the two-week target for mining 2016 blocks. Think of it like a car navigating a mountain road: it must slow down for curves to avoid danger. Similarly, Bitcoin’s system adjusts during periods of rapid growth or decline in hash rate, demonstrating the iterative self-correcting feedback loops characteristic of power laws.

The third point of our theory highlights that as Bitcoin’s price increased with its growing user base, it attracted more resources, particularly in mining power and capabilities. People developed more sophisticated mining machines, and organized mining operations emerged to maximize efficiency.

The fourth point emphasizes that the rising price shortened the time needed to mine blocks. However, due to the difficulty adjustment mechanism, the hash rate required to mine a block has constantly evolved in an iterative manner. Mining profitability drives this process: the compensation mechanism adjusts proportionally to the increase in price and hash rate, leading to a power law relationship where the total compensation is proportional to the square of the price.

These adaptive mechanisms are crucial for maintaining the system’s stability and scalability amidst Bitcoin’s rapid growth. The fifth point explains that the increased hash rate enhanced the system’s security, attracting more users. While some might argue that security isn’t a primary reason for Bitcoin purchases, the underlying stability it provides encourages investment.

Finally, the sixth point reveals that Bitcoin’s user base has grown with the power of three over time. Contrary to typical S-curve adoption models seen with technologies like TVs and smartphones, Bitcoin follows a power law due to its curbing mechanisms—difficulty adjustment and investment risk. This results in a power law growth of adoption, confirming that Bitcoin’s expanding user base and interconnected power laws ensure its robust and stable development.

“Introduction power laws are mathematical relationships between two quantities such that one is proportional to a fixed power of the other they are ubiquitous in nature but also in Social phenomena and in many parameters related to how a city or a nation grows for example they are so common because it can be shown mathematically and physically that they emerge any time you have some kind of process where the output becomes the new input in an iterative process what does bitcoin’s power law tell us then Watchers of our Channel might answer that it’s a powerful predictive tool that allows us to have a look at bitcoin’s Future price growth what if we told you that power laws are everywhere in Bitcoin not just in the price action from hash rate to addresses Bitcoin has an interconnected array of power laws that give us the ability to predict various future properties of Bitcoin but how do we know all of this well Dr Santostasi started by observing regularities in bitcoin’s behavior and first discovered the power law between price and time over time he discovered more and more power loss as he delve deeper into the subject but when did bitcoin’s power loss come about The Bitcoin Power Law Theory let’s begin with the first point of our Theory we only know that in the beginning there was nothing then Satoshi put his intellectual capacity into creating Bitcoin and people started mining it on their laptops others invested in it out of curiosity as time went on people got more interested and started participating in discussions and trading it more seriously Point number two of our theory asserts that bitcoin’s price but in general the value of the network itself began increasing with the square of it users this empirical observation is based on the power law between price and addresses which are the proxy of the users of the network it is a confirmation of the theoretical result called Metcalfe’s Law. Satoshi probably didn’t think about power laws when he created bitcoin but he definitely created ingenious mechanisms that ensured the stability of bitcoin’s system of power laws a great example of this would be the mining difficulty adjustment mechanism of Bitcoin this mechanism self- adjusts itself and the change in difficulty is in proportion to the amount of time over or under two weeks the previous 2016 blocks took to find imagine a car driving up the mountain the car needs to break quite often because it cannot just keep accelerating all the time without driving off a ledge when it needs to take curves the same is true for when Bitcoin experiences a rapid growth or decline in and hash rate a power lies exactly what you would expect as a result of processes similar to this because iterative self-adjusting feedback loops are a result of power laws influencing each other now to the third point of our Theory the increase in bitcoin’s price after the growth of its user base brought in more resources in particular mining power and capabilities people came up with increasingly sophisticated machines as they competed for their share of the valuable prize organized mining operations were established to ensure the highest possible efficiency Point number four the increase in price decreased the time needed to mine blocks but because of the difficulty adjustment mechanism the hash rate necessary to mine a block has since been constantly changing in an iterative fashion because mining is barely profitable the compensation mechanism needs to be proportional to the increase in price to compensate for the linearly proportional increase in hash rate but also for the price itself so the total compensation is then proportional to P to the^ of 2 this back of the envelope calculation and logical derivation is supported by our empirical observation of a power law in the form of p^ square being proportional to the hash rate in the Bitcoin data adaptive mechanisms like these are necessary to ensure the stability of the system and have to be scaled invariant to keep up with bitcoin’s rapid growth according to the fifth point of her Theory the increase in hash rate brought more security to the system which attracted more users now some people might say that most people do not buy Bitcoin because of security but indirectly they do because if it was not a secure system nobody would invest enormous value in it so yes the security of the system has directly or indirectly brought in more users now to the sixth point of our Theory bitcoin’s users have been growing with the power of three in time this is also a new discovery of our Theory most models of bitcoin’s adoption invoke models depicting s-curves which are typical of the adoption of many Technologies like TVs refrigerators cars cell phones and so on however Bitcoin doesn’t follow an S curve that is initially exponential instead it’s follows a power law of three in time it turns out many phenomena that have an underlying s-curve mechanism of adoption or spread actually become power laws if they have some sort of curbing mechanism in the case of Bitcoin both the difficulty adjustment and the risk involved in any type of investment is that curbing mechanism and this is why we can empirically observe that bitcoin’s growth of adoption is a power law of three in time according to the seventh and final point of our Theory this power law growth of adoption together with previously explained power laws explains why we can observe bitcoin’s various power laws in time such as the ones related to bitcoin’s addresses price and hash Further Discussion rate now coming back to the beginning question of our story when did bitcoin’s power loss come about that’s the thing there’s no certain point in time when they all became established because it was a continuous process what we do know is that bitcoin’s power laws are all tied together and the change in one creates a change in the other in a feedback loop what about the outliers and deviations from the power law let’s take bitcoin’s price action as an example outliers are points that deviate a lot from the average Trends and in this case they are bubbles what we’re left with are the bottoms and they seem to align almost perfectly with a power law all the other power laws are backup systems that create robustness in the system for example crash has happened when bitcoin’s price action has gone up too fast relative to other power laws such as the overall hash rate it is an obvious fact that you cannot scale infrastructure such as mining facilities at the same rate is bitcoin’s price growth during a bubble so the system self-corrects itself however these power laws also protect Bitcoin from dropping too low by creating a price floor that is difficult to get through due to incentive structures created by factors such as the cost of mining Bitcoin in fact we can see that the only time when bitcoin’s bottoms did not line up with the power law was when Bitcoin was new and people did not care about mining Bitcoin at a loss on their laptops Bitcoin has become much more robust and stable ever since and bottoms now line up with moments when mining becomes barely profitable Bitcoin having sex against unbridled unhealthy growth but this is exactly what differentiates Bitcoin from actual speculative bubbles such as the Tulip Mania bitcoin’s anti-bubble mechanisms keep its growth strong and steady in a scale environment way factors such as bitcoin’s finite Supply which is different from scarcity amplify the effects of bitcoin’s power loss in future videos we will also explain what the Bitcoin power law theory says about their origin and nature of the bubbles which are a complimentary phenomenon to the Power All growth of the Bitcoin system in Gnosticism a serpent biting its tail known as the ouroboros symbolized eternity and the soul of the world the soul of Bitcoin is very similar with its protective Loop of power laws Bitcoin will continue to grow in a scale and variant way because this is what Bitcoin is a self-adjusting self-correcting monitor system that is absorbing the wealth of the entire world and reorganizing in an intelligent and anti- anthropic way exactly like a breathing living being at first we started with a heuristic model and now we are developing a full theory of bitcoin’s behavior stay tuned for Dr Santostasi upcoming scientific article and book.”






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