The Physics of Bitcoin

Giovanni Santostasi
23 min readMar 12, 2024


Why Bitcoin is not a common asset but more similar to natural phenomena ruled by universal power laws due to recursive, infinite feedback loops fundamental to the system.

Note: This is a work in progress and a placeholder for my exploration of the significance of the BTC Power Law. Eventually, I will organize this in a more polished work. Mostly, the material below is from my X account: so it is informal in nature until I have time to rewrite and organize all this. All the work below is copyrighted.


Giovanni is a Ph.D. in Astrophysics and he is passionate about Bitcoin and he has 12 years experience in analyzing it and creating price models. He is the discoverer of the BTC Power Law in time. He is also interested in the study of consciousness, intelligence, and memory. He also studies the role of sleep in processing and storing memories. His research areas include computational neuroscience, neural network, and biophysics. If you want to support my work and join our private Discourse server, please subscribe first to my Patreon:

The Patreon membership gives you also access to the Power Law TradingView 1 and Power Law Spiral Trading View 2 indicators (at different levels of membership).



A good introduction to the BTC Power Law and all the related ideas behind the theory can be found in this work in progress. It is not yet organized in a coherent way and it is mostly a place where I’m preserving my many X posts. Eventually, it will become a more organized work that I intend to transform into a book.

Also follow us on YouTube (please remember to like and subscribe as another way to support our BTC Power Law Research team:

BTC Power Law not just a Model but a Powerful Theory of Bitcoin Nature and Behavior

The BTC Power Law in time started as an empirical observation of regularities in Bitcoin price. But it is now evolving in a coherent, powerful Theory of Bitcoin behavior. Recently we found out that Hash Rate, Addresses, Transactions, Price are all power laws of each other (that we already knew from previous work) but also ALL POWER LAWS of TIME. This is incredible because it says the evolution of Bitcoin is driven by scale invariant properties. It also fits completely with what Bitcoin actually is: a self-referential and adaptive network. In particular the Difficulty Adjustment, that was one of the most genius innovations by Satoshi, is the key mechanism behind the Power Law. NO SCARCITY. But the Adaptive Properties of Bitcoin. Power Laws are a mathematical and physical expression of iterative processes where the output becomes the input (hash-rate now influences hash-rate later because of the difficulty adjustment). This is exactly, as I have demonstrated, not just in words but showing the exact math, of it, what produces the power laws in time we observe. So while before The BTC Power Law in time didn’t have causative power now it does in a manner that is much more complete and powerful than any competing model. In fact, we can explain now why we see such powerful barrier at the bottom (it is due to miner capitulation and perfectly reproduced by an iterative model of Bitcoin), we can explain the decoupling of addresses and price from hash-rate during the bulls (hash-rate doesn’t follow the frenzy during bulls because it takes time to invest millions of dollars in the infrastructure and new chip tech to catch up) and so on. Almost all the aspects of Bitcoin behavior can be explained by this Power Law Theory (it is not anymore a model) of Bitcoin behavior. It is a powerful one (pun intended).

The Discovery of the BTC Power Law in Time

I usually do not include these early transactions because were mostly early adopters exchanging BTC for pizza. But when you include them (it is nice to add to the data set the earliest known US to BTC transaction that comes very close to the trend) the difference between current slope and all data slope is almost non existent confirming how good the model is. Dotted line is current model, red one is including earliest transactions.

Power Laws Everywhere!

I mentioned before the power law called Kleiber law. It determines the relationship between the weight of an animal and how much energy it needs to survive. The scale of size goes from a small creature like fruit flies to elephants, a 7 order of magnitude range. That is basically what BTC went through when it went from being traded 10,000 BTC =1 dollar toda

Scale Invariance

Let me give an example from another field of study that I worked on (besides astrophysics), the neuroscience of sleep. Brain waves during sleep in particular. I know it sounds not related but please follow along and you will see it is. I’m specialized in understanding a phase of sleep called slow wave sleep. In this phase we have large and slow waves (in comparison with waking brain waves that are about 30 per second while slow waves are 1 per second) that are considered a hallmark of deep sleep. Larger the waves, deeper the sleep. So it is important when you do a sleep study in a clinic for example to count them or use them to identify how well or deep somebody sleeps. In my work I had to try to understand this phenomenon. Given my training as a physicist I entered the field as a novice but that sometime helps to ask questions nobody usually asks when an insider in the field. For example, the manual sleep doctors use to identify these waves say that only very large waves, above a certain threshold are considered slow waves. A little smaller wave with similar characteristics (like their frequency) is not a slow wave if smaller than this threshold. It sounded so absurd to have an arbitrary threshold that I conducted a research where I collected all the waves with such characteristics (like their frequency range, shape and other things common to all the waves but I excluded size). What I found was: they are scale-invariant! Nobody even knew in the field. There are slow waves small and large and the emphasis of doctors is on large waves that can be easily seen by the naked eye and they didn’t really go in depth about understanding the phenomenon. Most medical doctors have no idea of scale invariance and its relevance. These large waves are the extreme of the range and not representative at all of the phenomenon, they are the large, rare events in the tail. The distribution of these waves has a long, long tail and these events are in the extreme side of the tail. The majority of the slow waves (thousands of them) are much smaller and there is not an artificial threshold to separate large from small, it is made up just randomly by some doctor in the past that decided this how big a slow wave needs to be. But that is not how the brain works. All these slow waves are important. Maybe for young, healthy subject the large ones can be used as a measure of good sleep, but for older patient (the waves become smaller with age and are correlated with cognitive decline) is a very bad measure and can lead to several diagnostic problems. I even showed that was the case in my research. You can publish a paper, get some compliments from other researchers, but nobody cares in the applied medicine field and they don’t change how they do medicine that is a quite sad part of medical research. Anyway, what is the connection with BTC? Well the ETF are the large slow waves. We notice them because they are large, they are at the end of the distribution and our human type of attention get all light up. But BTC went through many, proportional events of this kind from tiny ones, like few more people joining the network when it was small, a website talking about BTC, the first media discussing it and so on. At any points there proportional events. We don’t remember them because there were tiny, will all the other slow waves doctors ignore. If my discovery that BTC is a scale invariant system is true then the ETFs are the event that BTC needed at this time of its evolution to go the next step. Nothing is changed. I know some people take this as bearish news. It is not. The ETF seem to affect current prices and probably they will amplify the bull run that seem to occur after the halving. The peak of the cycle may be larger than what we expect, instead of going down as it had done in the past it could be as large as some of the earlier ones. But these are just “local” effects (they concern changes here and then) but not the long term trajectory. As I said many times, the fact BTC follows a power law it is a sign of health and strength and it we want to root for that. But in terms of understanding the scale invariance properties of BTC, no, they are not going to change BTC long term trajectory because there is no evidence ETFs are different from all the other events when understood from a scale point of view. Maybe I need to do a deep study on that and show that the inflow of capital in BTC has some kind of log-log relationship with BTC price and show it is scale invariant like price itself, but for now this is what my intuition and understanding of scale invariant system suggests.

Growth and Power Law Spirals

Fibonacci spirals are related to Power Law Spirals, they are similar but not identical. There are many growth processes that follow the Power Law Spirals rather than the Fibonacci. For example the growth of teeth and horns in animals or tree rings.

Bitcoin, An Intricate Dance Of Power Laws

Bitcoin is a deep web of power laws, that are showing up everywhere:
price, hash rate, addresses, and transactions are all power laws of each other and all of them are power laws in time.
Even HODLing times and wealth distribution are all power laws (these last ones are power law distributions so slightly different in nature from the other ones).

But transactions go up with the price. They also related with a power law. What is happening that is a self regulating process. It takes higher and higher prices to change the # of transactions by a factor of 10. Power laws sometime are not intuitive but you cannot literally understand BTC if you don’t understand BTC. Every Bitcoiner should catch up and try to understand how power law work. Higher price, more transactions so in a sense more available BTC in the open market. It seems a contradiction but only if you don’t understand power law or use them to make sense of BTC. Notice transactions are in terms of units of BTC not USD so it not just that more USD is traded but the number of BTC.

The Virtuous Circle of Proof of Work

The Math of Iterative Processes is the Math of Power Laws

The power law is the result of an iterative process, we are able to show this: dPrice/dt=5.82 *Price/time. Super cool! You can iterate this formula into the future but of course, as you add more future points it tends to go towards the general Power Law trend. So it is good to tell us what the price is supposed to do 1–2 days in advance. Basically, you can create charts with different weights contributions of these contributors to the Price. Booth strapping Price itself seems to give a good estimate of the current price, not the trending price but the current price (a little conservative during the bubble but still reacting to them). Hash seems to indicate where the bottoms are. Addresses something in between.

Probably most of you have not taken advanced calculus or any calculus at all (or hated it). But for the ones who did or the ones who want to learn a bit here why the Power Law is the result of an iteration. Let me show you the math (a little long but I promise

it is worth it if you have the patience to read through): Let’s consider the equation of the form: dy/dt=n y/t. This is an equation called a differential equation. It is what is used in math and in physics to express a change in some quantity. For example, if I want to express how an object changes in distance y as a function of a constant acceleration a, I can write dy/dt=a t, where dy/dt means the change in distance as a function of time. Here I’m simply saying, that the change in distance is proportional to acceleration and time. It is one of the simplest differential equations. Solving the equation means finding an equation for y that satisfies the equation (that means the right side and left side are equal). Once you find such a y then you know how distance changes with time. There are many methods to solve differential equations. But one of my best professors told me (and I always remember how surprised and perplexed I was at hearing this but it is the wisest advice about math ever) just guess a solution that seems to make sense and test if it works. Simple ! So what is a good guess for this equation? Well, the left side is a derivative and the derivative of t² is 2 t so I propose as a solution of the form y=1/2 a t² so the 2 gets canceled after the derivative and I have a on both sides. You substitute to check if this solution satisfies the equation: Left side dy/dt=d(1/2a t²)dt=1/2 a 2 t=a t that is equal to the right side = a t. Great! So y=1/2 a t² is my solution and it is interesting that the distance grows with the square of time. Note: for the calculus experts of course it is an easy guess given the properties of derivatives but I’m trying to make a point. OK, let’s go back to the original equation we wanted to investigate. dy/t=n y/t. Notice that this equation is peculiar because the quantity we want to solve for is also on the right side, so basically the change in this quantity depends on the quantity itself! Or put in another way the output of the equation is also part of in the input. This is why I’m using it as one of the simplest examples of an iterative or feedback loop process. Let’s make a good guess for a solution of the equation above, what about y=A t^n? Where A is just a constant and it is not that important. Let’s substitute: Left side: dy/dt=d(At^n)=A n t^(n-1), right side: n y/t = n A t^n /t=An t(n-1). Left side = Right side. SOLVED! But what is y=A t^n? That is what a Power Law is! For BTC is Price=At⁵.82 or for Hash=B *t¹¹.4 or Price=C*Hash¹/2 and so on. All power laws or iterative processes (output = new input). So you see now you can take the change dy/dt added it to y and you get the new y that can be used to solve for the change in y and so and so on. So a Power Law is the result of a feedback loop that we know exists because of the miner and user interaction in the network. This is why hash rate, addresses, and price itself are all power laws of each other and all power laws of time. So power laws of value, energy, time, and adoption. How f cool is that????

The Hard Bottom Line

The reason we see so strong and consistent barrier at the bottom is because of “miners capitulation” after the bubble crash. It is the floor under which miners simply lose money by mining. It cannot happen for the system to survive.

Bitcoin a Living Network

Organisms as Networks

Cities and Social Networks

An Unstoppable Time Machine

Scarcity is not what Drives Bitcoin Price

It is a very myopic idea of economic activity. The entire way that modern society works, based on increased productivity through science and technology is overcoming these limitations. For example, right now famine is just a political problem, not a resource one. We have resources to defeat hunger in the world if we really wanted. The fix supply of BTC is more about not dilution of the “stake holders” that creating fake scarcity. It is a very different, and more important property than scarcity. Maltus theory of Scarcity: The theory states that the supply of food cannot keep up with the growth of the human population, inevitably resulting in disease, famine, war, and calamity.

the person that told me that she uses the scarcity argument to tell people to hurry up and get some BTC (good message by the way) also told me “stick with numbers”. I followed her advice (not sure what exactly she meant but again I know she had good intentions). But math is math. Here 2 models, the power law model that has zero assumption on scarcity, just based on the observation the price is going up in a predictable manner. Scarcity it is not part of the model at all. The other, S2F where scarcity is the main driver, in fact, it assumes scarcity increases with time. The power law (with no assumption on scarcity) leads to “hurry up and get some BTC now” . As you can see S2F instead says you can get in at any time and still make the same amount of gains, no matter if you joined 10 year ago (for the same amount of 4 years HODLING time) or 10 years from now (again for the same HODLING time). Both investors will make a 10x return over this time. While you notice for the Power Law model it is important to join as soon as possible. So do you see how people use of the scarcity argument leads to illogical conclusions that are not supported by facts?

I will make this post and then focus for sometime on other stuff given the topic is not so popular for many. But it is very important and I think I got it right when most people are somehow confused about this (that is very surprising to me given it seems almost everybody get it wrong). Thesis: scarcity is needed in Monetary systems only as a security tool against counterfeiting. BTC doesn’t need scarcity because it is already secured against counterfeiting (with one exception). Note: this discussion is around “Monetary Systems” so it doesn’t apply to investments, stamp collecting or other types of valuable things or activities. Only Monetary Systems, so please keep the discussion focused on this. Let explain this using 4 examples. The post is going to be a little long but I think worth reading because it is going to clarify a very important misunderstanding. 1) Casino: A casino can be understood as a monetary system. Customers come in, exchange something “valuable” (like $) and they gets some chips. The chips are exchanged at a precise rate, the red one is worth 10, the blue 50, the yellow 100. The tokens are made of cheap plastic, so something that has nothing to do with scarcity of the material used for example. It is nothing precious like gold. But they are difficult to counterfeit because nobody is going to make more of these chips while in the bathroom. They serve their function of money perfectly. They are not even a fixed system of tokens because as more customers come in more tokens are given away by the cashier but these more tokens do not dilute everybody else tokens. Their value was fixed at the entry and it is not changed during the night while customers play. But the system works perfectly as a monetary system because it has clear rules of exchange, it is pegged on something valuable from outside the system and nobody is counterfeiting the tokens. Scarcity plays zero role in such a system. 2) Monopoly game. Similar arguments. The entire game is supposed to simulate, even if in a very simplified way, a competitive economy, where they are winners and losers. There are very inexpensive tokens (we even use them as a meme for worthless money) that inside the game are meaningful and worth it. The supply is limited but actually grows as the player go around the board and get rewards in the game currency proportional to their luck and/or ability to play the game. Nobody is counterfeiting the money (a cheater could take money from another game, stush their pockets with them and show up to the game and pull them up when nobody watches, but it would not be a non fun way to play the game and also probably easily detectable by the other players). The system is closed; no money in or out of the system besides how it is given by the rules of the game that are fair and understood and agreed by every body. So again no scarcity or even a fixed supply is needed. 3) An ideal, non existent but possible in the imagination society of people that never lie or cheat. These aliens (cannot be human of course) never lie or cheat, their brains are not geared in that way. I know it is difficult to imagine but play along with me. The system is not necessarily closed, it can interact with other systems around this imaginary world (where there are different nations and people but all super fair and respectful of rules and laws). In this case one can use anything absolutely anything that is durable and strong enough to be money. All what you have to do is to mark the “money” with some recognizable sign to indicate that is money or use some identifiable shape or color. Color coded stones painted with a paint that doesn’t chip away or stones with special markings made by some central authority or whatever can be used as money. Scarcity is not needed because again there is no counterfeiting. You can keep also track of this money by having shops and the equivalent of banks keeping track of the total supply and when coins are lost or destroyed by wear and tear others can be supplied in a fair and consistent manner. In this crazy world even the government is fair and honest (ok I know I’m stretching it) so they also do not do any equivalent of money printing (beside as I said to substitute lost or destroyed coins and maintain a constant supply). 4) The Roman Empire using gold and silver coins. This is also an open system. The Roman Empire emits coins that are made of precious metals. The one representing more valuable units are made of a rare, durable metal called gold. The government stamps the coins and guarantees (at least initially) the purity of the metal and each coin is worth its value in weight (meaning even if you melt it you get the value of the metal on the open market). Both metals, but in particular the more valuable one are chosen because they are “scarce” (relative to other metals or even other valuable things). This is done for only one reason given the examples above. To reduce and/or avoid (that is basically impossible in a practical world) counterfeiting. Because gold is difficult to mine (in fact so difficult that during Roman times they basically reached a limit given the mining technology available) it was difficult for bad agents to get some gold and mix it with other less expensive and rarer metals and create more coins. One could have taken existent coins and melt them instead and then dilute them (that by the way this happened quite often in particular in the outer regions of the Empire) but the coins still needed to have at least some large gold % in them to pass as gold coins (in terms of weight and other characteristics typical of gold) and so this processes of counterfeiting was at least curbed by the fact the material used for the money was scarce and valuable. But you can see that this choice of material was all based on a security issue and it had nothing to do with giving value to the money itself. In a society where everybody was honest all what was needed was a somehow fixed supply or even a non fixed one with simple and clear rule of exchange that were fair and agreed by everybody. What about BTC then? BTC fixes all this by: 1) A fixed total supply that cannot be changed. Because it is a digital technology this can be done for the first time in human history. The algo behind BTC guarantees that this supply cannot be changed unless the entire system has consensus to do so and likely this is very difficult to achieve anyway. 2) Before the total fixed supply is reached a very precise, understood agreed upon (by participating in the system you agree implicitly to its rules), transparent method to increase gradually the supply that is supported by an equivalent security measure to “scarcity” that is hashwpower or an exchange of energy and work for coins via mining. You are not getting anything for free but you exchange a valuable thing (energy) with a reward in BTC. This is equivalent to buying chips using $ in the casino, where a “valuable” thing is exchanged for another. It is again a security measure to avoid counterfeiting and the only place where the notion of scarcity (but again one could use abundant solar power to mine coins, so it is not really about scarcity per se but more about exchanging one form of energy into another) comes even remotely into place when we discuss BTC. The number coins, their divisibility, the fact it is fixed supply have nothing to do with scarcity per se. It is all about the rules of the system being clear and based on consensus. The fact the total supply is fixed is a mean to create consensus (by a prior law or rule) around the issue of total monetary supply (that could have been a stumbling block in any other monetary system where a central authority or a democratic process would decide regularly what that supply is). The total supply and supply production over time is chosen by the system itself at the start. This is it. These rules makes things much simpler and this is why is so genius. I hope my examples clarify why scarcity is a bad concept when related to the idea of what gives value to BTC. I believe it creates confusion in the community and even more when we try to explain to others what BTC is all about. I’m not going to insist on this anymore, at least for now, lol and let people think about this and see if it is a useful way to reason on things. Thank you for listening.

Bitcoin is the Pizza that can nourish the entire world (trust an Italian on this

). You know the big debacle about that fund manager lady (not sure about her name, it doesn’t matter) that said something “apparently” stupid about having many sats diluting the value of BTC? She said something stupid, of course, and the internet went crazy and made fun of her for days. But she said this because the emphasis on scarcity confuses people, in particular non-bitcoiners (and bitcoiners too). While the bitcoiners that went after her were not wrong, they were not right either. They were right from a logical and mathematical point of view but wrong from a narrative and educational point of view. One of the favorite examples that people used to show she was wrong was the idea that you cannot solve world hunger by slicing a pizza billions of times. But you know what? With Bitcoin you can. And in fact, it is exactly what we are trying to do. We want BTC to be the world monetary system so in a sense it is like nourishing (or serving a function) for the entire world. We emphasized the wrong concept. This why I insist, at the cost of creating some friction and defending an unpopular idea, that scarcity is the wrong lens and narrative to use with BTC. We missed an enormous opportunity to make the point that yes, there is a way to use 1 pizza to nourish the world by making the pizza very very very nourishing where 1 tiny slice is super nutritious and can meet the needs of each person on earth. If BTC has in its destiny to become the world monetary system it needs to be exactly that magical pizza we said it impossible. We simply shoot ourselves in the foot by focusing on scarcity, it makes us and (the people from outside the community) think incorrectly about the value of BTC and why it is so unique and important. Let’s start to think about BTC from the lens of abundance and infinite utility instead of scarcity. BTC fixes all this, even the hunger of the world for a fair and just monetary system.

The S2F Model

Other Measures of Scarcity in Bitcoin

Power Laws Lesson of Life

I use the word “model” for the Power Law in BTC to curb its significance and being careful in its implication. But it is not a “model” as most people think of math models. If it is true, and I say if, then it tell us something much much bigger about BTC. It is a fundamental property of the system. It implies:

  1. Scale invariance.
  2. Sustainability.
  3. Adaptability.
  4. Order and organization.
  5. Predictability.
  6. Intelligence and information.
  7. BTC is like a city and a living organism.
  8. Long lasting and resilient.
  9. Good things come in time.
  10. Unavoidability of 1 M and even 10 M BTC.

BTC will do what BTC will do. We are all humble observers of this amazing phenomenon. But if what I discovered is even close to truth then you want these properties for BTC, and the math tell us they are not wishful thinking but the mathematical and scientific truth of it.

The Far Future

One of the things to consider when we talk about BTC value is the fact that human productivity and capability are growing, in fact at an exponential pace (while BTC value with a Power Law pace). BTC goal is capture more and more of human productivity value eventually representing the entire monetary system (that keeps growing in value). Given the finite supply the value will continue to grow because human productivity grows. The cap is there not to dilute or have arbitrary forces capture large parts of this value through this arbitrary dilution. But this not scarcity, quite the opposite.