In this conversation, Giovanni Santostasi discusses his new book ‘The Physics of Bitcoin’ and the application of power law analysis to understand Bitcoin’s growth, value, and long-term behavior. The conversation covers the scientific basis of Bitcoin’s patterns, the significance of power laws in natural and social systems, and how this framework challenges common narratives about exponential growth.

Timestamps:

(00:00) – Intro

(02:14) – Why write ‘The Physics of Bitcoin’?

(06:18) – The significance of Bitcoin’s Power Law 

(10:28) – Bitcoin’s growth: Power law vs Exponential models

(21:41) – The nature of Bitcoin as a network effect

(29:46) – How would bitcoin’s power law be falsified?

(37:30) – Bitcoin’s price floor

(41:35) – Can it break to the upside?

(43:14) – Will Bitcoin hit a saturation point?

(52:51) – Sustainable nature of Bitcoin’s growth

Links: 

• https://x.com/Giovann35084111
• http://thephysicsofbitcoin.com 
• https://scientificbitcoininstitute.org/ 

Stephan Livera links:

• Follow me on X: @stephanlivera
• Subscribe to the podcast
• Subscribe to Substack

Transcript:

Stephan Livera (00:00.6)
Hi everyone and welcome back to Stephan Livera Podcast.

Rejoining me on the show today is Giovanni Santostasi. As many of you already know, we spoke earlier on the podcast, but he is the, let’s say the founder or the creator of Bitcoin Power Law Analysis. And he’s back to present and talk to us a little bit about his new book that’s going to be coming out. It’s called The Physics of Bitcoin. He’s also the founder of the Scientific Bitcoin Institute. I, you know, for me, I find this stuff quite interesting. I wouldn’t call myself a Power Law Analyst, but I am, you know, I’m a fan.

sharing and trying to talk about it also because I think it’s an interesting framework for understanding different aspects of Bitcoin whether it’s the adoption, the price, the hash rate and these different explanatory factors of Bitcoin. So yeah, so welcome back to show Giovanni.

Giovanni Santostasi (00:52.549)
Thank so much, it’s my pleasure.

Stephan Livera (00:55.768)
So, yeah, let’s talk a little bit about why you wanted to write this book.

Giovanni Santostasi (00:59.813)
And by the way, it’s funny because I’m doing a lot of AI, using a lot of, you know, it’s an incredible tool. And I was making a plan of how, you know, to let people about the book and explaining the idea. And one of the things he suggested, he said, you should go to Stephan Livera’s podcast.

Stephan Livera (01:22.046)
So I’m in the AI somehow. The AI is telling people to find me, so…

Giovanni Santostasi (01:25.721)
Yeah, like one of the top podcasts where I should go. So I found it funny.

Stephan Livera (01:29.39)
Well, there you go. I think it’s probably because I do often interview Bitcoin authors and so it probably that’s why it comes up in the AIs and whatever so yeah, but yeah, tell us about why you wanted to write this book instead of you know, just doing online podcasts and articles

Giovanni Santostasi (01:35.843)
Yes, yes.

Giovanni Santostasi (01:48.197)
Yeah, so the book is actually kind of it’s going to be almost like a reference book, you know, because it’s a self-contained It has it’s supposed to even to be a book that Maybe introduces Bitcoin to people that never heard about it So there is a an introductory sections where we cover the basics, you know It doesn’t go too much in the technical details or how to use Bitcoin how to get a wallet because there are plenty of books like that, right? And but I do mention them in the references

There are several appendix appendixes that say okay read these read that you know to get these other information But it’s trying to make the big case that is what the power law is all about from the beginning But the coin looks more like a force of nature, right? It’s a similar more to this other process. So basically there are several important assumptions and I have mentioned these assumptions and these points

of use in my post online in my article on medium but i think the time came to put it all together and make a coherent story that somebody that never heard about bitcoin and he thinks it is random it goes up and down and it does have has no value reads these things and gets transformed and say whoa you know hopefully maybe this became one of the most powerful orange peeling mechanisms

Because if you understand, look, is not random, it’s following a very precise pattern. This pattern is a pattern that nature prefers in growing different structures, a tooth and nails. And at a certain point, I mentioned this article that I mentioned before in my tweets about finding it in a book that says, look, nature likes the power law for all these structures. So it’s not just biological structures, but the universe, like galaxies, planets.

things like the language, know, the type of activity that humans do, or cities. know, cities are one of the most important inventions of humankind. know, the entire civilization, the word civilization itself comes from civitas, it means city in Latin, you know. So cities and civilizations are one to one. So all the things that are most fundamental in human life are coming from cities.

Giovanni Santostasi (04:17.783)
you

And they follow power laws. They grow as power laws. once you put Bitcoin in this context, there is this clear picture of what Bitcoin is. And then we can discuss this in more detail. But one of the biggest criticism of Bitcoin is how it has no value. And there is an entire section, entire chapter of the book that takes all this information and tries to build almost a mathematical definition of value. Why Bitcoin is what?

In fact, probably it will become the most valuable invention of humankind or system, whatever you want to call it. And it is connected to this power law theory, you know, all the consequences from the power law theory. So this is why I wanted to write a book. I want to reach a lot of people. want people to use it, maybe to orange peel other people. I want to address the skeptics, you know, because there are a lot of skeptics. Maybe if they read it in a book format, they see the pictures.

They understand the concept that are linked to it because I also build all these concepts, all the foundations that are necessary. talk about self-organized systems. We talk about network theory. We talk about power laws in general. So yeah, there are many reasons why I wanted to write the book.

Stephan Livera (05:42.818)
Yeah. Now, like I said, I’m with you. I’m a fan of the power law, but let’s say a skeptic is looking at this and thinking, well, are you guys just like fitting data to a curve here? Is it just like some kind of weird overfitting thing? And in a few years time, these guys are just kind of come out with a new model because the old one broke and so on that it’s, you know, it’s just a model. Can you explain why there’s something deeper here?

Giovanni Santostasi (06:10.103)
Right. Well, so when I discovered the power load, you it was doing this transformation that I discuss in the book by taking the log of the price and the log of the time, you can…

See the behavior by your height. You don’t even not need to draw a line You can see that something weird happens to the price of the coin. It becomes kind of like a line right that immediately for somebody that is trained to look at this phenomena it Immediately clear what you’re looking at. You’re looking at the process that

it’s proportional in terms of the scale, the scale of price and the scale of time, right? That is a deep, deep insight. Then you can use mathematical tools like regression to draw a line because the line will give you the slope of the curve. it kind of tells you the proportionality relationship between the scale of price and the scale of time.

You can do all kind of tests you can take out of it data and projected in the future you could say well What if I did these? You know the history is of the coin is almost 17 years ago What if I did it eight years ago will I have been able to see the power law remember if it actually I discovered this in 2018 so the power law at a minimum has been predictive of a general trajectory since 2018

And in fact, I saw power laws Not the one in pricing time in Bitcoin since 2012 the one between price and addresses for example That was my first power law and I discovered that in 2012, you know, so this power law have been very consistent and Beyond and you know and today now there are several several

Giovanni Santostasi (08:12.289)
Analyst that are looking at these but they go beyond being just analyst. They are people with mathematics physics engineering background and they did all sort of stability tests to show that actually the power law in time becomes stronger and stronger becomes more stable and not just that for example some of my latest results was to Look at the dynamics. So if you’re looking now Bitcoin changes in time you can

the power law. So it’s beyond just fitting a regression. can look at the changes in time and then study these changes and show that mathematically these changes are consistent with this power law behavior, this scaling variant behavior. So we have done so many sophisticated tests, beyond fitting data to a particular precise pattern is once you understood that we are discussing

Stephan Livera (08:57.697)
Yeah. Got it.

Giovanni Santostasi (09:12.863)
a phenomenon that is called scaling variant. What does it mean? It means that the system is self-similar.

looks like itself at all different scales. And this has many implications, for example, even in understanding adoptions, right? Why there are different agents at different time from the individual investors to small banks to large institution to maybe even nations, right? This is actually foreseen by the power law and is part of a theory that the different stages of the history of Bitcoin will be different agents, but the behavior stays the same. This is why it’s called invariant scale.

The scale is the size of a system at different prices, different times, or different levels of adoption. And it’s the same, it’s self-similar. And only the power law, by the way, as a function, is scaling variant. It’s the only function with the scaling variant. So it’s not by chance. Yeah.

Stephan Livera (10:07.181)
Yeah. So on this, let’s, I love if you could explain for people, because this has been a common thing in Bitcoin for a while, right? Like I came into Bitcoin, you know, early 2013, you were saying you were analyzing things in 2012. Especially at that time, there were a lot of people saying, oh, it’s an exponential, it’s going to grow exponentially. Or the other one is, oh, it’s an S curve, right? That there’s it’s, you know, we’re at the knee of the S curve, and that we’re about to really go through this.

big suddenly moment. Can you explain for us why you disagree with that? Why is Bitcoin not an exponential and it’s not an S curve?

Giovanni Santostasi (10:41.465)
Well, and all this is in the book and if you don’t mind actually this could be an opportunity to show you some of the material in the book. The book is still in a Word version so it’s going to launch soon. But I can show you some pictures and some ideas. So just let me find the right content.

Stephan Livera (10:48.587)
Yeah, sure. Yeah.

Giovanni Santostasi (11:10.245)
So, you know, so for example, it’s actually explained almost immediately at the beginning of a book Where I cover these topics And it’s here it is right so for example

If you show one of the tools and describe these tools in details, okay, here it is. And we can make this bigger. So it’s easier to see. Okay, this one. Are you able to see it? Yes, right. So this is one of the simplest tests that one can do. If you look at the price in a chart where the price is linear,

Stephan Livera (11:46.977)
Yep, I can see it.

Giovanni Santostasi (11:59.413)
Time is linear so you don’t do anything like many people use for example trading view and there is like a little setting where you can choose if you want to see the price in linear scale or if you want to see it in log scale right and they don’t have a Log scale for time. We have only a log scale for the price, right? So Yeah, eventually that yeah

Stephan Livera (12:20.243)
Right, and what you’re talking about is having log on the x-axis and log on the y-axis.

Giovanni Santostasi (12:25.291)
Right, that is the… So we go through this transformation, right? Usually we start with this simple way of describing anything, right? In the book I explain, for example…

like somebody was taking a physics class, right? I taught many of these introductory physics classes. So you’re taking this class and you have some data and you want to understand the nature of the data. And so we suggest in our physics classes as professors to start with the simplest possible assumption, right? The simple possible assumption is that you have some kind of a linear process because in nature there are many of these

Simple relationship between one quantity than another so in that case if you had a linear process You will see a straight line by taking the in this case the price and the time I will look like a straight line and Evidently if you look at the price of the coin it doesn’t look like a straight line in this graph. It looks something like a J Many people don’t even see that because they see these huge big drops is part of a problem There are these huge bubbles. They kind of confuse with it

But if you look at the overall shape, it looks like a J. The problem is not easy to distinguish between different types of J. Here I’m showing two different processes. One is the power law and I show you also the exponential. The exponential is orange. The green is the power law. So you will not be able to distinguish between the two. Now if you go…

and we do what you are able to do in trading view that is taking the log. Okay. If Bitcoin was an exponential, it will be very easy test. Now you can do more sophisticated tests, but one of the things that I love about being a physicist that they taught us to be very, very practical. So the AI can tell a lot of information about data, even before you do all the sophisticated calculation, because that builds intuition. If you don’t have intuition about the data,

Giovanni Santostasi (14:34.535)
You can do all the sophisticated calculation and probably you’re going to do some mistake because you’re not guided by a deep intuition about the data. So if I see, if I do this transformation of taking the log and that is very important with Bitcoin because that is usually.

We do that when we have data that changes a lot of order of magnitude and order of magnitude means ten, hundred, thousands and so on. And Bitcoin went through these changes because if you take the entire history of Bitcoin, it went from few cents all the way to 100,000. So we’re talking about several order of magnitude when we change from ten, hundred, thousands and so on. So the log is actually ideal to look at

a process which changes many order of magnitude. Because what the log does? The log gives you the exponent. So if I write 100 as 10 times 10, or 10 to the 2, the log base 10 is going to give me that 2. If I have 1,000, it’s going to give me 10 times 10 times 10. I can write 1,000 as 10 to the 3. It’s going to give me that 3. I can write 10,000 as 10 to the 4. It’s going to give me that 4, the log.

But this is it’s useful, it’s giving me what is called the scale. So how the price will look like if it was exponential? It will look like a straight line. And you can try that with, for example, your gold, SP500, almost every single asset. If you take the log, it’s going to look over a long period of time like a straight line.

in a log what is called a log linear graph where you keep you do the log of the y-axis and you keep the linear x-axis in this case time yeah

Stephan Livera (16:25.025)
Got it. So just to make sure I’ve got my understanding correct, and I believe Matthew Mazinskis has written and spoken about this idea as well, that kind of most of TradFi, so-called, follows an exponential, but at a low pace exponential. Whereas Bitcoin is different. It’s actually following a power law, which is a different thing altogether. Is that a fair way to summarize?

Giovanni Santostasi (16:39.823)
Correct.

Yeah, yeah, yeah.

Yeah, very different mathematics. They are related. We are related. They are both nonlinear processes, but they are very different at the same time. so, yeah, and these many people associate exponential with fast growing all these other assets to discuss. They are exponential, but they are very slow exponential in comparison with the power law. The power law of Bitcoin is a power with six and that is a huge powerful power law. Imagine a parable of any people’s

say over price went parabolic and actually they mean exponential because many times when you have bubbles the price is exponential, fast exponential in that case, but actually a parabola it’s x square right it’s a power law with power 2. Bitcoin is a power law with power 6 so you can imagine how much faster than a power aula is, very very fast. It’s a very fast moving J. Now this J looks like this curve, this green curve right that is a

simplification. This is a very simple graph, but it doesn’t remind kind of like the curve path that you see in Bitcoin when you plot it in a log linear chart. Everybody’s familiar with that. It looks like this. It looks bent. It doesn’t look straight. Everybody’s familiar with that. Many people call it the rainbow chart or whatever, because it looks almost like half of a rainbow or whatever. That is what happened to a power law. When you take the power law and

Giovanni Santostasi (18:14.893)
do this log linear chart i don’t know why it’s continuing okay you know because white once in a while but anyway so the the problem with this is you know

If you do this to Bitcoin, you know what it is not. It’s not an exponential because otherwise it will look like a straight line. So the question is what it is. And nobody was able to tell what it is because one that is one of the criticism of a power law. It’s so simple of an idea. say, anybody could have done that. Why they didn’t do it? Because most people in finance don’t have this idea of taking the log of time. They don’t need to, because a lot of processes are more like exponentials and they are

power loss. In physics we do that all the time and believe it or not I had almost arguments with financial people in explaining that actually makes sense to take a log of time. They were so unfamiliar with it that in the beginning they even criticized that and there is nothing wrong because you know if I have a thousand days from the Genesis block that is how we measure time in this exercise you know basically measure

time from the creation of Bitcoin, then

You know the log of time would simply be three, right? So log of a thousand is three because it gives me that exponent that we discussed. So you can do it. The question is what does it mean? It simply means I’m emphasizing the scale. I’m looking at the changes in the scale of time. So a change in from a hundred to a thousand, a hundred days to a thousand will be one change of scale. From a thousand to ten thousand will be another change.

Giovanni Santostasi (20:04.423)
scale and so on and they do the with price and if I see a straight line in that graph and this is how the power law will look like like a straight line then I identify a straight power law now it’s not the ultimate test you will not just simply say oh it looks like a straight line goodbye you and you close and and you are happy with that you that is the start but

If the data kind of looks like straight line and you can see that with the price of Bitcoin it does it immediately you can see there is a change the curve that we usually associate with Bitcoin is gone. It looks straight right and once you observe that you know that something important is happening in particular if you know this if you are familiar with power laws

One of the things that you will look at will be that

You’re covering a lot of orders of magnitude, right? And the power law does that. The power law in Bitcoin does exactly that. that people don’t realize how difficult it is for data to behave in that way. You don’t find that often. And in fact, when you find, I mean, on one end you find it often because it’s everywhere in the universe. In the other, just random data doesn’t do that.

You need to have some kind of very precise mechanism and process to create these kind of behaviors.

Stephan Livera (21:32.782)
So I guess the point you’re getting at there is that there’s an underlying…

sort of nature to it, right? Similar as you have explained and Jeffrey West has explained this idea that, you know, even if you plot out the number of calories that an animal consumes and its size, that’s like a power law. Or like the distance of the planets from the sun, again, follows a power law. Or the growth of the cities, that there are different network effects inside of a city, right? Whether it’s the plumbing, the electricity, the roads, the police, the whatever, the hospitals, all these aspects are kind of growing

Giovanni Santostasi (21:38.112)
Right, exactly.

Giovanni Santostasi (21:49.58)
Exactly.

Giovanni Santostasi (21:53.454)
Yes.

Giovanni Santostasi (22:01.283)
Yes.

Stephan Livera (22:06.127)
together in a combined network effect. I guess what you’re getting at to this point is that Bitcoin itself is a network effect. It has a mining hash rate. It has adoption in terms of addresses that people are using when we spend and receive Bitcoin, and also the price of Bitcoin. that’s the underlying, there’s a nature to it, and it’s following this trend of adoption over time.

Giovanni Santostasi (22:10.212)
Yes.

Giovanni Santostasi (22:30.735)
Correct, exactly. So yeah, we could have stopped there in the sense that once we find the power law in time and done a bunch of tests and show that it’s a power law from all kind of point of view, you know, statistics.

regression analysis different type of You could look at half of the daytime and project in the future There are many many different things that people that are professionals in the field know how to do so the question is settled, know, there is it’s and what that is one of the reason also we have a scientific Bitcoin Institute is because now we are in the process of writing serious article that we are going because it’s something that I should have done many years

ago, but I instead I decided to connect with other Bitcoiners. And, you know, the question should have been the other way around. Why I didn’t write scientific articles, books, et cetera, when I discovered this, mostly because I wanted to share my enthusiasm with other Bitcoiners, you know, and I knew that this was an important discovery, but I was also working in other field. I was very busy with my neuroscience research, with my other career that I had at that time.

and I put this on the side. Now it’s time because I’m doing this full time as an educator, as a researcher, and I want to distinguish this type of activity from what people usually do online. I continue to do that, but it’s one thing posting something.

Stephan Livera (24:07.221)
Yeah, so it’s going to be supplemental as well as what you’re doing online. Got it.

Giovanni Santostasi (24:09.175)
Right. So we want to make it a serious endeavor where we want to engage other scientists. We want to bring academia to study this phenomenon. We want to study if this thesis is correct, that Bitcoin is a network and shows network property because the connection between networks and power is explained very well in the book. There is an entire chapter dedicated to network theory. in that chapter, we discuss how

When you look at actually real networks in life, the property of these networks are powerless. They weigh, for example, the different scaling of connections. It’s called a scale-free network. There are big hubs like, for example, you. You are a big huge hub with a lot of connections to the community. And then there are other nodes in the network that have few connections.

But if you do a graph of how many hubs there are with a certain given number of connections. on the y-axis you have how many of these.

Particular hubs you have with a particular number of connections and on the X Axis you have a number of connections it will look like a power law my power law that goes down because you have a lot of Nodes with very few connections with and there are a lot of very very few nodes with a lot of connections, right? So it’s going to look like Power law like this. It’s almost like similar to the wealth of

Let me actually explain one more time. yeah, on the right, it’s a distribution. So on the y-axis you have exactly. Yeah.

Stephan Livera (25:53.077)
As in like wealth distribution in Bitcoin will also follow a power law, is the point you’re saying. That there are very very few whales who have you know a thousand coins or more or whatever and then all the way down to you know shrimp and a smaller number of know people, right?

Giovanni Santostasi (26:04.417)
right exactly so to to make it more clear one more time so on the y-axis you will have how many of these networks or the distributions the frequency right so the percentage also that have a particular

number of nodes and then here the number of connections. So on the right you will have nodes with a lot of connections, but there are few and the ones that have few connections are a lot of them. But there are no, that makes sense, but the fact that it follows a very precise power law is not trivial. And it has been, in fact, the theory, there were theories about networks that didn’t predict at all that

is the type of distribution that you will find. The theory, existing theory before people actually went and

Did the observation, know, fact that the first big observation was from this physicist called Barabazi and Barabazi studied the internet among all the possible network, the network of network that is the internet. And he was very surprised when they actually measured in a very precise empirical experiment, this particular distribution that they follow power law. And then they needed to explain it. They need to come up with a theory that was able to explain this distribution because they were

some theoretical papers in the past previously, they say, given these conditions, you will expect a Gaussian distribution. It is a typical distribution that we observe in-

Giovanni Santostasi (27:47.814)
Height for example, I so if I have height there are very few people that are very short There is an average number of people that are going to have some kind, you know There is a the majority of people are going to have a an average height and then very very few people that are very very tall, right? And as you go away from the average you have less and less people

that have very big height, right? And this type of distribution is very different from a distribution that looks like this. But this is what the theory say. And then when they observe, this is why physics is the queen of sciences because it doesn’t matter your assumption. You go and you do the experiment. You observe first. So it doesn’t matter if you assume that Bitcoin is an exponential or maybe it doesn’t follow any pattern. You go and you do the experiment. You do the measurement.

And when the measurement when the data tells you something you listen, it’s not there is no debate, you know You can debate the technical

quantities right is it this R square calculated in this way we should maybe calculate it in this other way but there is no doubt that bitcoin is a power law you know and we will see when we actually try to publish the paper which kind of feedback we get from the community you know of this is why we want to go through the peer review process you know and see if we actually can publish these papers in journals like you know physics journals etc

Stephan Livera (29:17.483)
Yeah.

Giovanni Santostasi (29:22.119)
But I’m pretty sure that there is no doubt when you look at the data, it’s very difficult to throw away the data saying, no, it’s an artifact or it’s a mistake or you didn’t consider this or consider that.

Stephan Livera (29:34.638)
Yeah. Now, I guess while we’re on this topic, I think the other big question that listeners might be thinking is, how do we understand when or if it has been falsified, right? Like just naively, people might say, well, look, the price right now as we speak, it’s about $71,000 per Bitcoin. And the power law, if you just look at the trend line, might be, call it 120K, something like that.

What is the point at which you would say, the power law has been falsified? I know you touch on this in the book, so maybe you can just explain what would be the correct way to actually assess this.

Giovanni Santostasi (30:09.537)
Stephan, we can edit this, right? Later. Because my headphones are dying. I’m going to substitute with this. Can we pause for a second?

Stephan Livera (30:12.886)
Yes, sir.

Stephan Livera (30:21.837)
Oh right, yeah, yeah, okay. Let’s take it back. Yeah, just, um…

Giovanni Santostasi (30:26.933)
sorry for that i should have charged my batteries but these have a wire i don’t want to do it now before they die so you know because otherwise when i cannot talk to you okay i’m going to turn this off

Stephan Livera (30:30.689)
That’s a good.

Stephan Livera (30:39.627)
Yeah, that’s fine.

Stephan Livera (31:02.163)
And you can probably just end the share as well, so that way it…

Giovanni Santostasi (31:03.607)
Yeah. Yeah. Yes. So, okay. Yeah. Yeah.

Stephan Livera (31:07.263)
Okay.

Ready? Okay, so let me just re-ask and we’ll go from there. So, while we’re talking about this concept of Bitcoin following a power law, right now, the price, as we speak, is about $71,000 and some people are saying, does that mean the power law is broken or it’s falsified? I know in the book, in the draft I looked at, you mentioned certain falsification conditions. So you mentioned a sustained period of daily close below the lower corridor, statistically significant shift in price power law exponent, failure of three law

consistency, sustained collapse of hash rate or adoption reversal. So can you just explain a bit from your perspective what would it mean to actually falsify the power law for Bitcoin?

Giovanni Santostasi (31:47.716)
Yeah, so we talk about falsify because I want people to accept this idea that Bitcoin has this very structured growth over the long period of time. And I want to remind when we are talking about long term, we are talking about years, right? So we’re not talking about changes in the price on the scale of months or days. That is not what the power law is about. It’s about this very long term trajectory.

And it’s important because you know, we don’t know right if I tell you Bitcoin is going to be one million dollars in eight years It’s very different from somebody telling you because it’s going to be a million dollars in two years We never it’s going to reach one million or it will take hundred years, right? So even if these are very long-term trajectory knowing more or less with New number of years. Let’s say it’s a very important contribution. I think this way it give us

and understanding of where Bicon is going. Now, idea of falsifying, so every theory, every scientific theory, and this is a…

And every time I talk with somebody comes to my mind, maybe I should explain what scientific theory is in the book. Maybe we need a little box or that, right? Because I’m trying to be addressing these also with people that maybe are not super familiar with the scientific journal. But differently from how we use theory in our normal language where, yeah, this is just in theory, right? In theory, it should be like this. And many people mean with that, it is kind of like a guess, right?

In physics or in science, a theory is not a guess. It’s like the opposite. It’s actually the apex of our scientific understanding. means it’s the most complete, most structured understanding of something. And the theory usually has some tenet, like, for example, the theory of evolution. You have some criteria that can be used to falsify the theory. And the theory needs to make predictions. It’s fundamental, because if you have a real

Giovanni Santostasi (33:57.44)
deep understanding of something, you should be able to predict something that was never thought before and is a consequence of a theory. And so this is I came up with this falsify criterion. And the idea is that we have to be a little bit careful because on one hand, I think it will be

A misunderstanding that if we don’t follow the power law anymore, the power law was never true. It’s not the case because I think we can come up with a lot of tests to show that up to now Bitcoin has been a power law, has been scaling variant, has followed a very precise pattern. That is very, very difficult to dismiss. What many people are interested, of course, is the future. Can the

power law continue to be valid in time. And so it’s not really falsifying what we have so far, but actually showing that there is something different, right? But I try to simplify and so I use this word falsifying. So it’s very important to make this difference. But yeah, so I think the most important test is this.

because we are talking about a dynamical process. We’re talking about how Bitcoin goes there. So it’s not so much important where we are. So for example, right now we are a little, we are somehow away from the long trajectory, so the long term trajectory that usually when people draw these charts, they show this line, that is kind of in the middle of these big, huge oscillations. Many people think that is the power law.

is just one way of expressing the power law. It’s not really the fundamental power law. The fundamental power law is the process. So it’s basically how we go there. Recently, I made a nice visualization of this. I show all the different slopes that is basically where locally we are moving. So when we have a bull run, for example,

Giovanni Santostasi (36:14.691)
these local slopes, these daily slope, that is the power law calculated in terms of the difference between one day to another, right? They look wild. Some of them, they look very, very, very different from the usual direction of the power law. They look much steeper, for example, and they look like they are pointing up. When we have a bear market, we are pointing down.

and sometimes very, fast. So in a certain sense, it’s almost like you are comparing the position of an object with the velocity. How you go somewhere depends on the velocity, and both the size of a velocity and the direction of a velocity. Guess what? When you average all these white arrows, so I made a beautiful graph where you see all the different arrows. When you average all these

like almost like the wind, like the local wind where the wind is blowing for Bitcoin, you get the power law. And you get it as a dynamical process, not as a position in the chart, Not as a point in the chart.

That quantities also can be used as a test for the validity of Bitcoin. In fact, probably is the most sensitive test because it turns out the slope of these vectors in the average as the total trajectory of the power law.

and we have a precise distribution so it’s actually it’s a beautiful thing because we both show that bitcoin has a long-term trajectory but also describe these changes relative to this long-term trajectory I did test where I not just measure

Giovanni Santostasi (38:07.777)
the average value of n that is oscillating, but is oscillating around the particular average value, but also the shape of this distribution, how big these deviations are. These deviations are stable at least since eight years. And previously, the deviation itself were actually smaller.

Relatively to now something happened in 2017 where the distribution became a little bit wider. So we are actually More volatility in a sense. So many people are associated associate volatility Think about the Bitcoin volatility as decreasing in time, but that is absolute volatility if you measure the volatility Relatively to the slope itself. So basically the change in price Then it’s actually

and has been stable for more than eight years. It’s a really incredible result. actually we have very sophisticated tests to measure if we change this power law. And so far we haven’t, even this distribution is the same. So basically Bitcoin has done the same thing over eight years, at least from a dynamical point of view.

So this is one of the best tests. Now I did mention the floor because the floor is something that it’s easy to understand more and people are very worried that I receive a price going down. They want to have some kind of relief. They want to know that maybe we reach a particular place and we stop there. Right. And historically this is exactly what happened. It is this weird floor. So if you take the general direction and then

It’s very important to also to explain that these applies mostly if you use the dollar, for example. And why the dollar? Because the dollar is what people use worldwide as the currency, right? Like the global currency. They don’t use gold. They don’t trade using gold. Right. So many people mention the same floor using Bitcoin Express in terms of gold.

Giovanni Santostasi (40:22.467)
The Coin Express in terms of gold is still a power law, but that floor is less meaningful. And I will explain you in a moment the meaning of that floor. So there is this floor that historically has never been violated, even by COVID, when we had the COVID crisis, the price arrived to this floor and then bounced back. It’s really fascinating to watch. This floor is about 60 % deviation from the general trajectory.

Now it can be calculated in other ways. For example, you can take ashrate and you can do a model. We know there is a relationship between price and ashrate and we can use that to make a model of the price and you get more or less the floor. And you can also calculate the cost of production. That is a little bit more complicated formula, but it involves the ashrate and guess what? You will get the floor. So the floor seems to be associated with the cost of production.

of bitcoin so this why probably has been never violated because once the price arrive there is already the market is already exhausted the miners stop selling

because they don’t want to lose money. They’d keep Bitcoin at that point, they maybe take loans or whatever. So this is why we basically historically never went below this floor. So I used it as a secondary measure of falsification of the model, right? If we see that the floor is violated, yeah.

Stephan Livera (41:55.501)
Yeah. And just for people to understand the context today, can you give just rough numbers, where would you say the power law is today and where is the flaw today, roughly?

Giovanni Santostasi (42:06.597)
So right now we are 17, price of a floor is more like 55, for example. And so if we violate the floor and it doesn’t have to be like a week, right? Like there is a like a very fast violation and then we go back. That is not, it needs to be an extended one. It needs to be on the order of a sustained.

Stephan Livera (42:12.791)
Yeah.

Stephan Livera (42:29.102)
Right, as in a sustained, we’d have to go below 55 and then stay below 55 for a long period of time. Yeah. Right, to then, yeah.

Giovanni Santostasi (42:33.389)
for months, know, for maybe at least a couple of months, right? And then, yes, then I will say something is wrong here. Something is different. It’s not that the power law is not true because has been true. It’s something different. There is something different. The system, you know, is almost like a patient, right? You have these long-term power laws about the growth of a task, right? In a elephant, for example, you look at this trajectory, you look at all the other

elephants and they behave in this way and there is an elephant that for some reason doesn’t follow anymore this power law it follow up to the point and then it doesn’t follow well

As a scientist, we look at this and say, well, something changed with the elephant. it doesn’t get any more, it’s diet changed. Maybe it’s sick. Maybe there is some kind of illness in the elephant. Something happened in terms of a mechanism that may, it’s not that the idea of it, power law, the task follow power law is wrong. Is this particular elephant that went sick? So I will say this exactly with Bitcoin.

Stephan Livera (43:42.019)
Yeah. And then I guess while we’re there, I guess it’s probably a lot of people will be interested. What about to the upside? Like, is there a similar kind of questionnaire that if Bitcoin just kind of rocketed up crazy high and stayed high, would you make the same argument? That it kind of broke in the other way?

Giovanni Santostasi (43:51.834)
Yeah.

Giovanni Santostasi (43:55.161)
Right.

Right. So again, these are these in that case, we don’t have a floor. So we really love to use something like these are N test, right? These test of a slope. So if we start to see that the average value of the slope goes.

up and up and up instead of being stable and oscillating around a particular average value it starts to go up more and more and more and I actually I should actually add this to the book I will never finish the book if I start to add more and more stuff because I remember I made a post a long time ago where I had different scenarios with this test and I was showing the different type of scenario it was a scenario where we had an exponential growth and you could

it up this test can pick it up almost immediately and it’s actually a visual test you will see this value that right now if you plot it simply oscillates right it goes up it goes down but this oscillates around an average value it is about six if this value starts to go up and it becomes seven eight nine

It’s not a power law because power laws are supposed to have these values stable. You are measuring something else. You are thinking you are dealing with a power law, but now it’s behaving more like an exponential. And so that will be a very sensitive test to show that, yes, Bitcoin is not a power law anymore. It’s becoming something else. And many people…

Stephan Livera (45:18.028)
I see, yeah.

Stephan Livera (45:26.412)
Yeah, and while we’re there, I guess the other question people will have is what about, you know, maybe in the long run, maybe 20 years from now, who knows, does the world eventually hit a quote unquote saturation point? then like basically most people have adopted it and by then, like would things change then? Like let’s say in 20 years time, Bitcoin is, $10 million a coin and most of the world has adopted it by then. Does the power law still stay then or not? Or how do you analyze that?

Giovanni Santostasi (45:44.666)
Yeah.

Giovanni Santostasi (45:54.628)
Yeah, yeah, yeah. So, sorry, I’m losing a little bit my voice, yeah, so that is another important topic, right? And the way to address that is this, so people understand more deeply what I’m talking about.

Giovanni Santostasi (46:14.273)
If you understand how the scaling processes work, one of the reasons why I hope that we continue to behave as a power law, because many people, maybe they don’t like if we break down, right? I saw some people celebrating.

Even the idea that we could break down simply because maybe they simply want to show that I’m wrong or this idea that Bitcoin follows a precise pattern is wrong. So they are right. So if I can go can break down, maybe can break up and but we still want the breaking up, right? Because we hope we become become exponential. And I have insisted so many times saying even that will not be good because most exponentials grow up fast and then they collapse.

That is fiat behavior. Exponentials are associated with fiat behavior.

Stephan Livera (47:01.11)
I see, you’re saying they’re less stable or they’re less kind of robust.

Giovanni Santostasi (47:08.503)
And it’s so beautiful that Bitcoin is not. In fact, have this one of my posts that got more like ever in the history of my post has been this post where I do this scaling test where I show I can even show you this graph if you want in a second where I show how different Bitcoin’s behavior is in terms of the scaling property from NASDAQ, gold, everything else. It’s so visual that really people like that.

I will show you the graph in a moment so we can discuss it. It’s also in the book. yeah, yeah, so let me, yeah, yeah, I’m going there. I’m going there. Believe me, know. Step by step. So the idea is this, right? When you have power law, the essence of a power law is that as we grow, we slow down.

Stephan Livera (47:41.582)
Can you explain a bit more about the saturation idea?

Giovanni Santostasi (48:02.105)
That is the beauty. And this why is also the favorite growth of nature. Because as this elephant grows and he needs to have a bigger and bigger task, if he went exponential, he will consume so much energy, so much resources, that something goes wrong. Just to give you an example about saturation, imagine a process where there are some algae or some bacteria in a lake.

And I caught Saylor doing this example. And I told him, I’m sorry, Saylor. You do a lot of wonderful things for the community, but sometimes you don’t use the right analogy. You’re talking about an exponential process when you should talk about power laws. And what he was talking was this.

Doubling he wanted to show us fastest when angels are and then implying baby coin does that right? So the idea is this imagine I have a a bacteria that is doubling Every five minutes every five minutes you start with one then it becomes two and four then eight and so on right it doubles every five minutes And there is this lake or this container whatever where he leaves right and there are

nourishment, is nutrients for for this living structure, for living being. Then when he arrives, imagine that he took a month, right, or five months or whatever, you know, to feel 50 % of the lake. Well, the question is, how long will it take to feel the rest? So it took one year, let’s say very long time to feel 50. How long will it take to feel feel the rest?

Remember, it doubles every five minutes. Do you know the answer? I don’t want to put you in a bad spot, but it’s a pretty simple, it’s one of these tricky questions. Five minutes, because remember it doubles, right? And it needs to cover the next 50%. So it will take only five minutes. So it took one year to cover 50%. The rest of the 50 % is going to be filled in only five minutes.

Stephan Livera (49:55.032)
No, I don’t know.

Giovanni Santostasi (50:14.819)
That doesn’t happen with power loss. the bad things about filling the lake is that now you don’t have any other space to grow, right? You die out. That’s the only thing you would do. There is no nourishment. There is no place to go. The only thing that can happen is that the colony start to kill itself. And that is what happened with a lot of exponential processes.

They grow fast, they use all the resources. Yeah, they are unsustainable. With the power law, what you will see is that it takes longer and longer to feel the next step because what happened is still growing and relatively fast, but everything slows down. Exactly.

Stephan Livera (50:42.188)
Right, because they’re unsustainable. That’s the point. They grow so fast that they’re unsustainable.

Stephan Livera (50:58.158)
It’s like a diminishing returns kind of idea.

Giovanni Santostasi (51:01.029)
But you know this it looks kind of bad the diminishing returns right because it seems a negative things But actually it allow the system to adapt to the environment and this is what happens with the economy if we want to a Bitcoin that is going to Exist for the next 20 years or longer until it becomes like a million ten million We want exactly to grow as a power law now in terms of your question that you answer So once you start to play with the number you realize for example

that every time we double the size of a system in terms of people that adopt the system or entities that can be even eyes or whatever, the price goes up with the square. And then you say, okay, how long does it take for the price to reach, let’s say,

10 million and the power load gives us an answer is going to be around 20 20 years around 20 years and then you realize that to reach that price of 10 million you and that you know right now let’s say we round it to 100 000 right we are in the neighborhood of 100 000 so that means a change in price of 100 and then you realize because of that square factor between

Adopt you the number of people in the network and the price is a factor of square That you only need ten times more people So we need ten times more people that we have right now to get

And these people can be represented by ETFs, right? And ETFs is like a big, whale that represents, it’s one entity, but it represents thousands of people, right? So the network needs to work by a factor of 10 in terms of adoption to get to a price a hundred times. And then you can ask a question and say, okay, how many people have adopted Bitcoin? Almost everybody agrees that it’s probably around a hundred million. There are many different ways of measuring that, but

Giovanni Santostasi (53:09.671)
close to 100 million. So even if you multiply that by 10, you’re going to get a billion.

that is very, far, at least a factor of 10, relatively to the population now, the population in 20 years probably is going to be even bigger. So we are far away from saturation. And if you talk about maybe AI agents using Bitcoin, et cetera, so you can see the possible people or entity that are using Bitcoin, are very, far away from these numbers, right? One billion users.

users, for example. And that will bring us already to a price of 100.

$10 million Bitcoin and that means if you translate it in terms of a Market cap it will be a market cap of at least 110 Trillions Well you multiply that by another factor of two and you will take just few more years You know, maybe four or five years to get a factor of two more Then you are talking about the wealth of entire world. We are talking about close to 500 Trillions it is close to the wealth of

Everything real estate included, you know So probably a bad point is when i’m expecting something to happen to the power law And it will be something dramatic. It will not even be some kind of a saturation. It’s simply because we are running You know against some of the limitation economic limitation of the world unless something big happens where productivity goes to the stars because Maybe even literally because we start to go to mine asterisk

Giovanni Santostasi (54:54.279)
by that time AI is going to be so much is going to increase productivity so much that these limitations that we have right now we seem you know nothing in comparison with what we can create or produce in 20 years from now you know because people undervalue the revolution of it AI represents right but this is and I discuss this in the book and they say listen if we can predict the behavior of Bitcoin that

According to this reasoning is not going to saturate is not going to saturate at least until it reaches 10 million dollars because we only need a billion people so we are not not running out of people at all about it time

Stephan Livera (55:37.976)
Yeah, and I guess to your point earlier about power law distribution, even at that point, it might be the wealthiest 1 billion people on earth who have, you know what I mean? Like, because it might be a disproportionate, like, even though it’s like not the full 10 billion or whatever at that point, who knows how many people, but it might be the richest 1 billion out of the 10 who have Bitcoin, you know? So that it just represents a larger share of the actual, you know, let’s say people who have wealth.

Giovanni Santostasi (55:49.753)
Bye.

Giovanni Santostasi (55:53.092)
Yeah.

Stephan Livera (56:06.359)
or serious wealth, let’s say. Yeah.

Giovanni Santostasi (56:06.501)
Right, institutions also, right? Because this is where we are switching now a bit from individual investor to large institutions that represents a lot of people, you know, but they became like they behave like whales in the network and so on. So.

Stephan Livera (56:17.069)
Right, and now nation states and central banks and so on, yeah.

Giovanni Santostasi (56:23.691)
We don’t see any signs of saturation because actually some of these are some of the sophisticated mathematical tests that they did because this why I wrote an entire book and the book doesn’t even contain one tenth of all the different tests, the different calculations. So for example, there is this group of scholars that they studied kind of what Barabasi did by measuring all the links in the network. They actually studied other network because at that time

They are mostly the internet as a real example of a large network But now we have many of them like for example, there are these scholars are Chinese scholars with look at Weibo WeChat and many other real-life Networks and they found out with both the nodes and the links in these networks They don’t grow like exponentials below they grow like power loss. Where is that? In fact, you know

And that is one of the recording debates that we have in the community that people mentioned these S curves and I have like entire sections of a book where they explain carefully what is an S curve, how is derived, an S curve is nothing else. So you start with a distribution that looks like a Gaussian distribution, right? Because you have early adopters, have innovators that come later when you have early majority.

late majority laggards, right? And it looks, the typical distribution of adoption looks like a bell curve, like a typical bell curve, right? That we discussed. Similar to what you will get if you look at the distribution of heights in a population. Okay. And if you sum, so the technical term is integrate. If you sum,

all these people in this distribution, guess what? You get the S-curve. This is where the S-curve comes from. And many people use that because they say, this is how the adoption looks like, right? It starts low and when it reaches the majority, the early majority starts to go very fast up and then very saturation. So it looks like an S-curve, right? Bitcoin doesn’t follow that. Because even adoption,

Giovanni Santostasi (58:46.241)
It’s a power law. And these people, they study the real network. I mentioned this paper in the book. They study all these real, for example, it turns out that even the internet exactly works like that. It’s not an S curve because usually this S curve is associated with physical things like telephones, for example, refrigerators, or the car, bicycle and so on.

When you look at social network like internet, because people started to accept the internet because other people are on the internet. Let’s go. It’s a social phenomenon. Social networks, both the links and the nodes follow power laws. And they studied, they even developed a mathematical tool to test.

Basically, they wrote this equation, very general equation, they say, assume that this equation can, if I change slightly this parameter, it can become an exponential. It looks like an exponential. If I change this parameter, it looks like one of these.

curves that saturates like one of these S curves. And if I change this parameter, it looks like a power law. And they run it with the data, and they calculate these parameters, which parameter it is. And they found out over and over again that the parameters that we found show that the process is a power law. I did exactly the same. Yeah.

Stephan Livera (01:00:19.469)
Yeah, that’s a fascinating insight.

I think that’s a really good spot to finish up and have a takeaway for people, which is that Bitcoin is following a power law. And as you said, that’s actually a good thing, paradoxically, because what many people might want is, they want the exponential, they want the S curve because they feel like it’s going to grow faster. But the challenge with that is it’s not sustainable. I think that’s kind of in a simple way of understanding it. That’s what it is. So we’ve hit about an hour. So let’s leave it there. I know there’s a lot of other questions I have, but we’ll keep it manageable.

Giovanni Santostasi (01:00:30.724)
Yes.

Stephan Livera (01:00:51.075)
for listeners but just before we let you go Giovanni where can people find your book and where can they find your scientific institute where they can learn more about it

Giovanni Santostasi (01:01:00.761)
Yeah, so the book, it’s on Amazon, so you can look for our physics of Bitcoin. Now we have a website that is called, it’s all together, vphysicsofbcoin.com. And it’s actually a very nice website because it’s going to be a companion where it be like mini lectures where people can follow, where it be live charts. It’s going to be a very nice reference for anything related to the power law. And for the Institute,

It’s again altogether scientific Bitcoin Institute.org. So they can look for and again, it’s going to be very useful website because we want to be also a hub for learning, maybe maybe courses of a via that we offer to people for free about Bitcoin, all the ideas related to this complex system. So we’re trying to create all kinds of resources for people to understand better.

the beauty, the mathematical beauty and the significance of the studies that we do. So yeah, look at these two websites.

Stephan Livera (01:02:06.371)
Fantastic. Well listeners make sure you check it out. I’ll put the links in the show notes at scientific Bitcoin Institute org and the physics of Bitcoin calm Listeners, make sure you share this episode so more people can learn about Bitcoin as a power law and Giovanni Thank you for joining me today

Giovanni Santostasi (01:02:20.198)
My pleasure, my pleasure.