比特币幂律理论：为何其价格将继续等比例增长？

加密行业的资深OG（老炮，老玩家）们普遍认为，每次减半都将推动下一轮牛市。

Senior OG& #xff08 of the encryption industry; old gun #xff0c; old player #xff09; widely believed xff0c; each time half will drive the next round of cattle.

此处有一个矛盾点，不知道有没有读者发现。本质上，幂律模型否定了S2F。因为在幂律空间中，只有时间（time）维度而并没有供应量（supply）维度。借用科幻小说《三体》里，造成众多物理学家死亡的那个问题，套用过来可以这么说：

There's a paradox here xff0c; it's not clear if the reader finds it. In essence xff0c; the xff0c; in the xff08;time;time;time; xff09; dimension without supply xff08; Suply） dimension. Use of science fiction novels xff0c; the problem that caused the death of a large number of physicists xff0c; this can be said xff1a;

减半不存在了。

There is no more half of it.

曾经有老韭菜绝望地发问：如果说产量减半对BTC有决定性地推动作用，那么为什么它对LTC（莱特币）就没有作用呢？

#xff1a has been asked in despair by old cucumbers; if the reduction of production by half is decisive for BTC #xff0c; then why it is not working for LTC #xff08; Lettcoin #xff09; #xff1f;

减半和牛市，不过是一种虚妄的联想。也许减半牛从来就没有存在过，将来也不会存在。

Halve the bull market #xff0c; it's just an illusion. Maybe half the cow never existed #xff0c; it never will.

如果减半不存在了，那么我们也就完美的解决了文首辨析的问题。答案很简单：既不是减半推动了牛市，也不是牛市领先于减半，二者并没有什么关系。

If the half doesn't exist xff0c; then we also have the perfect answer to the headline question. The answer is simple xff1a; neither half promotes the bull market xff0c; nor is the cow market ahead of the half xff0c; it doesn't matter.

虽然听上去很难接受，但是对于信奉奥卡姆剃刀原理的教链来说，剃除减半的幂律理论，确实更优美。

Although it sounds difficult to accept #xff0c; #xff0c for the church chain that believes in the Ocam Razor doctrine; #xff0c for the ruler that shaves by half; #xff0c; indeed, it's even better.

今天，教链就先把幂律模型的提出者Giovanni Santostasi最近（2024.3.20）的一篇文章《比特币幂律理论》（The Bitcoin Power Law Theory）介绍给各位读者，作为后续继续深入探讨有关话题的基础。

Today xff0c; the church chain began with Giovanni Santostasi, the author of the model law, recently xff08; 2024.3.20） an article, Bitcoin Theory & #xff08; The Bitcoin Power Law Theory #xff09; introduction to readers #xff0c; as a basis for further discussion of the topic.

* * *

图释：比特币类似于自然现象，并非普通资产。这幅美丽的艺术作品归功于 @BainterSAT

与金融资产相比，比特币更像一座城市和一个有机体。

is more like a city and an organism than a financial asset xff0c; Bitcoin is more like a city and an organism.

12 年前，我开始研究比特币。

12 years ago #xff0c; I started studying bitcoin.

我在 Reddit 上发表我的探索，而不是在期刊上发表，主要是因为我想接触更广泛的比特币社区，而不是发表一篇非专业科学家不会阅读的论文。

I published my exploration on Reddit & #xff0c; not in journals & #xff0c; mainly because I wanted to reach out to the wider Bitcoin community & #xff0c; instead of publishing a paper that unprofessional scientists would not read.

我的主要发现是，比特币受幂律支配。其规律性表明，比特币的行为更像一个物理系统，而不是资产。这一直觉是基于比特币价格与时间的关系中观察到的惊人的幂律。

My main finding is that xff0c; bitcoin is dominated by hysteria. Its regularity suggests that xff0c; bitcoin behaves more like a physical system xff0c; not as an asset. This has always been based on the amazing laws observed in the relationship between bitcoin prices and time.

比特币幂律理论。

Theorem of Bitcoin.

我的幂律模型现在已经发展成为一个完整的比特币行为理论，能够以科学连贯和可证伪的方式解释所有主要链上参数，并描述比特币采用率的增长：比特币幂律理论或 PLT。

My model of law has now evolved into a complete bitcoin behavioural theory & #xff0c; capable of interpreting all major chain parameters & #xff0c in a scientifically consistent and perjured manner; and describing the growth of bitcoin adoption & #xff1a; Bitcoin theory or PLT.

下图概括地解释了 PTL 理论，并显示了支持该理论的主要数据。价格、哈希率和地址（我们使用高于阈值的地址来消除灰尘地址）都是彼此和时间的幂律。

The figure below provides an overview of the PTL theory & #xff0c; and shows the main data in support of the theory. Prices, Hashi ratio and address & #xff08; we use addresses above the threshold to remove dust & #xff09; they're the laws of each other and time.

它们在一个连续的反馈循环中相互作用、相互影响。

They interact and interact in a continuous feedback cycle.

幂律是 y=A x^n 形式的数学表达式，在自然界中无处不在，在社会现象以及与城市或国家发展相关的许多参数中也是如此。

The law is y= mathematical expression xff0c in the form of A xn; everywhere in nature xff0c; and also in social phenomena and many parameters related to urban or national development.

它们之所以如此常见，是因为可以用数学和物理学的方法证明，只要有某种过程，输出就会在迭代过程中成为新的输入，它们就会出现。

They are so common xff0c; they can be demonstrated by mathematical and physical methods xff0c; as long as there is a process xff0c; the output becomes a new input xff0c in an iterative process; they appear.

这正是比特币的情况，例如，现在的哈希率会影响以后的哈希率，形成无限循环。因此，比特币的行为具有幂律，这不仅令人惊讶，而且也完全符合比特币的本质。

This is the case of Bitcoin xff0c; e.g. xff0c; the current Hashi rate affects the later Hashi rate xff0c; forming an infinite cycle. So xff0c; Bitcoin's behavior xff0c; it's not only surprising xff0c; it's also perfectly compatible with the nature of Bitcoin.

下图支持这种相互作用，这在社区中是众所周知的。这张图不是我发明的，但我用它来说明理论是如何运作的。

The following graph supports this interaction xff0c; this is well known in the community. This is not my xff0c; but I use it to explain how theory works.

该理论基本上是一种数学表达，基于逻辑、物理和数学的反馈回路。

The theory is essentially a mathematical expression & #xff0c; feedback loop based on logic, physics and mathematics.

1. 最初，比特币需要被中本聪圈子的第一批用户接受和采用。

Initially xff0c; Bitcoin needed to be accepted and adopted by the first users of the central circle.

2. 比特币的“价值”（现在是“价格”，可全天候在线查询）随着用户数量的平方而增加（经验测量值更像是 1.95，但为了简单起见，我们将以下所有幂数四舍五入为整数）。这证实了梅特卡夫定律的理论结果。

Bitcoin's “value” & #xff08; now “price” & #xff0c; full-time online search & #xff09; increased xff08 as the number of users squared; empirically measured values more like 1.95， but for simplicity xff0c; we rounded all of the following to integer xff09; this confirms the theoretical results of Metkaf Maift.

3. 价格上涨带来了更多的资源，尤其是采矿能力。

Higher prices have generated more resources xff0c; in particular, mining capacity.

4. 价格上涨减少了开采一个区块的时间，但由于“难度调整”，开采一个区块所需的哈希率会反复变化。由于挖矿几乎无利可图，因此补偿机制需要与价格的增长成正比，而价格的增长是通过 P=users2 和奖励本身来实现的，因此从逻辑和维度上讲，我们可以得出哈希率=价格2（这正是在幂律的经验值接近 2 或价格=哈希率^1/2 时观察到的结果）。

The price increase has reduced the time for mining a block xff0c; however, because of the “difficult adjustment” xff0c; the Hashi rate required for mining a block will change over and over again. Because mining is almost unprofitable xff0c; thus the compensation mechanism needs to be proportional to price growth xff0c; whereas price growth is achieved through P61; users2 and incentives per se xff0c; thus xff0c in terms of logic and dimensions; we can derive Hashi ratio 61; price 2xff08; this is the result observed at 2 or 61 at 1/2; ff09;

5. 哈希率的增加会给系统带来更多的安全性，从而吸引更多的用户。现在有些读者可能会说，大多数人购买比特币并不是因为“安全”，但他们间接地购买了比特币，因为如果它不是一个安全的系统，就不会有人在它身上投入巨大的价值。所以是的，系统的安全性直接或间接地带来了新用户。

The increase in the Hashi rate will bring more security to the system xff0c; thereby attracting more users. Some readers may now say xff0c; most people do not buy bitcoin because of “safe” xff0c; they buy bitcoin xff0c indirectly; because if it is not a secure system xff0c; no one will put a great deal of value into it. So xff0c; the security of the system brings new users directly or indirectly.

6. 随着时间的推移，用户数量会以 3 的幂次增长。这也是该理论的一个新结果。大多数比特币应用模型都采用 S 型曲线增长。S 曲线是电视、冰箱、汽车、手机等许多技术应用的典型曲线。比特币并不遵循 S 型曲线，它最初是指数型的。它在时间上遵循 3 的幂律。事实证明，许多现象在采用或传播时都有一个潜在的 S 曲线机制（以病毒为例），如果它们有一个抑制机制，它们就会变成幂律。就比特币而言，“难度调整”和任何类型的投资所涉及的风险都是抑制机制，这就是为什么我们通过经验观察到，比特币的采用率增长在时间上呈现 3 的幂律。有大量文献显示，在涉及风险的疾病传播中存在这种抑制现象，例如艾滋病（比特币不是艾滋病，但这些研究表明，如果疾病的传播涉及某种决策，例如与伴侣发生性关系，那么疾病的传播在时间上就会呈现 3 的幂次，而不是 S 曲线或其他类型的逻辑曲线）。

6. Over time xff0c; the number of users will grow exponentially by 3. This is also a new result of the theory. Most bitcoin application models use S curves to grow. S curves are typical of many technology applications, such as television, refrigerators, cars, mobile phones. Bitcoin does not follow S-type curves xff0c; it is initially exponential. It follows the rule of time. xff0c; many phenomena are introduced or transmitted with a potential S curve mechanism xff08; viral cases xff09; xff0c; if they have a containment mechanism xff0c; if they do, they become xxxxx; x#x#xxxxxx#xxxxxx#xxxx#xxxxxxxx#fffffx. #xx.0cx.0; if fffffffffffffft. ; this is used to control disease x.

7. 这种循环无限重复。泡沫是这一循环的重要和必要组成部分，下文的推论将对其进行单独讨论。

This cycle repeats indefinitely. Foams are an important and necessary component of the cycle xff0c; their reasoning will be discussed separately below.

8. 那么，采用率的这种幂律增长（连同之前解释过的幂律）就解释了为什么我们能及时观察到其他幂律：地址=t3，价格=地址2=(t3)2=t?，哈希率=价格2=(t?)2=t12。

So xff0c; this xff08 for application rate; xff09 with previously explained xff09; explains why we were able to observe other xff1a; address 61; t3xff0c; price 61; address 261; t361; xff0c; Hashi ratio 61; price 61; 261; 61; t12.

下面的图表显示了所有相互幂律及其拟议的因果解释。

The chart below shows all cross-colleges and their proposed causal explanations.

幂律理论的后果和预测。

The consequences and predictions of the doctrine of law.

这一理论解释了比特币的长期行为，并产生了许多后果。

This theory explains the long-term behaviour of Bitcoin xff0c; it has many consequences.

其中最令人震惊和最相关的，也是大多数普通比特币投资者经常误解的，就是标度不变性（Scale Invariance）。

The most shocking and relevant of these are xff0c; xff0c, which is often misunderstood by most ordinary bitcoin investors; strong xff08; Scale Invariance） .

标度不变性是物体或定律的一种特性，当长度、能量或其他变量的标度乘以一个公共因子时，它们保持不变。它是物理学、数学和统计学中的一个特征。

Punctuation is a characteristic of an object or law xff0c; it is a feature of physics, mathematics and statistics when the scales of length, energy or other variables are multiplied by a public factor xff0c; they remain unchanged.

标度不变性是受幂律支配的系统的典型特征。

Stereotyping is a typical feature of a system governed by discipline.

从本质上讲，它表示系统在以相同的方式增长时将继续扩大规模（scale），这就是为什么我们可以使用标度不变性来进行预测，鉴于系统的增长已经超过了 9 个数量级，几乎可以肯定的是，其他 1 或 2 个数量级的增长也将继续发生（达到 100 万 BTC 大约需要 10 年时间）。虽然这听起来不可思议，但从长远来看，系统、价格、哈希率和采用率等一切重要因素都是可以预测的。

In essence, xff0c; it means that the system will continue to grow in size as it grows in the same way as xff08; scale） xff0c; this is why we can predict xff0c by using not changing metrics; given that the system has grown by more than 9 orders of magnitude xff0c; almost certainly xff0c; other 1 or 2 levels of volume will also continue to grow xff08; reaching 1 million BTC will take about 10 years xff09; although this sounds incredible xff0c; in the long run, all important factors xff0c; systems, prices, Hashi rates and adoption rates are predictable.

标度不变性还能让我们理解大型机构 ETF 最近对比特币系统的投资流入等事件的作用和重要性。

It also allows us to understand the role and importance of events such as the recent investment inflows into the bitcoin system by large agencies.

标度不变性告诉我们，这些事件不会极大地影响比特币的价格轨迹，相反，它们是比特币系统继续保持标度不变增长所必需的关键事件。

xff0c. The scale is not variable; these events do not significantly affect the price trajectory of Bitcoin xff0c; on the contrary xff0c; they are key events necessary for the Bitcoin system to continue to grow at constant scales.

这也意味着，许多人很难理解，幂律趋势（加上泡沫）就是你所得到的一切。不多也不少。

It also means #xff0c; many people have difficulty understanding #xff0c; #xff08; plus foam #xff09; that's all you get. Not much, not much.

这是该理论最令人震惊的预测。

That is the most alarming prediction of the doctrine.

所有理论都是可证伪的，而这正是证伪该理论（至少是目前形式的理论）的方法之一。

All theories are perjured xff0c; this is one of the methods of perjury xff08; at least in its current form xff09;

未来的理论可以修改，增加斜率变化或相变，但目前的理论认为，比特币的价格路径已经确定，除非发生灾难性事件，否则不会发生改变，尤其是在 1 或 2 个数量级上，而这只是比特币整体历史增长的一小部分。如果比特币在 15 年中保持标度不变，那么在接下来的 10 年（下一个数量级）中，比特币也可能保持标度不变。

Future theories can modify xff0c; increase the change in the tilt rate or the variation xff0c; but the current theory considers xff0c; price paths for bitcoin have been determined xff0c; xff0c; unless catastrophic events occur xff0c; xff0c; especially on 1 or 2 orders of magnitude xff0c; and only a small fraction of the historical growth of bitcoins as a whole. If bitcoin remains constant xff0c over 15 years, then the next 10 years xff08; next level of magnitude xff09; medium xff0c; bitcoin may also remain constant.

顺便提一下，就规模而言，未来 10 年与之前的 15 年是不相容的，因为这只是另一个数量级。对于大多数不熟悉这些概念的人来说，需要花一些时间来理解对数比例的工作原理。

By the way, xff0c; xff0c in terms of size; xff0c in the next 10 years is incompatible with the previous 15 years; because this is just another order of magnitude. xff0c for most people who are not familiar with these concepts; it takes some time to understand the working principle of logarithm.

该理论还有更多的内容（例如，为什么我们会看到如此吻合的底部遵循幂律），但我们将在后续文章中详细阐述。

There is more to the theory xff08; xff0c, for example; why do we see that the bottom of the match follows the rule xff09; xff0c; but we will elaborate on it in a follow-up article.

理论的推论。

Theorem of the theory of

泡沫是如何产生的？

how foams are producedxff1f;

与稀缺无关，与摩尔定律有关。

Not related to scarcity #xff0c; related to Moore's Law.

中本聪知道摩尔定律。这是一个启发式定律，宣称计算能力每两年翻一番。“难度调整”机制确保你需要花费大量的金钱和精力来获得一些额外的比特币。

It's an inspirational law & #xff0c; claims that computing capacity doubles every two years. The "difficult adjustment" mechanism ensures that you spend a lot of money and effort to get some extra bitcoins.

但摩尔定律给了你一个不公平的优势。4 年后，你将拥有 4 倍的哈希值计算能力，基本上与 4 年前的机器（大致相同）的能耗成本相同。由于磨损，你无论如何都需要更新，而机器的成本只是运营成本的一部分。事实证明（我在理论中解释了以何种方式），无论是从逻辑上还是从经验上，我们都可以得出价格（或通常所谓的奖励）=哈希率1/2。因此，基本上 4 倍的哈希率只能带来 2 倍的收益。但减半后，收益减半，增加的收益为零。这一切都是为了让矿工处于盈利的边缘，永远不允许有免费的午餐。这太完美了，不可能是偶然的，我认为中本聪正是这样计划的。

But Moor's law gives you an unfair advantage. Four years later xff0c; you will have four times the capacity to compute Hashi's values xff0c; basically the same energy consumption costs as four years ago xff08; xff09; roughly the same energy consumption costs. Because of wear xff0c; you will need to update xff0c; in any case, machine costs are only part of the operating costs. xff08; I explain in theory how xff09; xfff0c; xff0c; xffoc; we can get a price xff08; 61; #Hahp1/2. So xff0c; basically 4 times the Hashi rate can only yield two times the benefit of the operating cost.

4年，而不是2年或者连续减少奖励，是因为这在供应链方面也是一个好主意，因为芯片行业的更新和进步需要时间，也给了矿工时间去计划更新，让设备自然贬值。这纯属天才之举，与比特币有关的任何事情都极其务实，切中要害。泡沫是“安全吸引更多采用”这一循环的结果。这个循环并不是我发明的，而是其他人发明的，而且它一直被用来解释采用比特币的周期。

Four years xff0c; not two years or consecutive reduction of incentives xff0c; because this is also a good idea in the supply chain xff0c; because it takes time to update and progress in the chip industry xff0c; it also gives time to miners to plan to upgrade xff0c; let equipment depreciate naturally. It's a genius xff0c; everything about bitcoin is very practical xff0c; the point in the equation. The bubble is the result of a cycle that “safe attracts more use”. This cycle is not xff0c; it's xff0c; it's xff0c; and it's been used to explain the cycle of bitco.

这是有道理的，因为直接提高安全性会吸引更多的人，让你对比特币存储价值的能力更有信心。没有这一点，就没有价值。我有一个最好的比喻，当人们搬到一个发展中的城市（就像赛勒（教链注：微策略公司创始人）说的，比特币是数字世界中一个闪亮的城市）时，会出现一阵活动。你想搬进去，因为那里有桥梁、房屋、道路等等。你不一定会直接考虑这些事情，但你会被这些活动所吸引。所有的新事物和好事都在那里发生。这就产生了一种暂时的“FOMO”（教链注：害怕错过），这种“FOMO”是好的“FOMO”，因为它是基于基本面的，而不是一些愚蠢的猜测，也许“FOMO”不是一个最好的词，所以你可以帮我找一个更好的词。但你知道我的意思。

This is a valid xff0c; because a direct increase in security will attract more people xff0c; and make you more confident about the storage value of the currency. There is no such xff0c; there is no value. I have the best metaphor xff0c; when people move to a developing city xff08; like Syll xff08; #xff1a; the founder of micro-strategy xff09; xff0c; Bitcoin is a shiny city in a digital world xff09; xff0c; there is a wave of activity. You want to move in xff0c; because there are bridges, houses, roads, etc.

图释：kuntah ? 是这张神图的作者。

价格上涨很快，几乎呈指数增长。这是价格唯一一次出现这种情况，而不是按幂律增长。事实上，从下图中可以看出，它几乎是完全对称的，价格下跌的速度和上涨的速度一样快（有时更快）。泡沫破灭后，它又会回到平衡状态。这是一种点状进化，是比特币增长的必要条件。

The price rise is fast xff0c; it is almost exponentially increasing. This is the only time that the price goes up xff0c; it does not grow by xff0c. In fact, xff0c; as can be seen from the figure below; it is almost entirely symmetric xff0c; prices fall as fast as they rise xff08; sometimes faster xff09; xff0c; it returns to equilibrium when the bubble collapses. This is a dot-like evolution xff0c; it is a sine qua non for bitcoin growth.

“所谓点状平衡，是指进化是以脉冲的形式发生的，而不是达尔文所说的那种缓慢而稳定的进化。在物种灭绝或新物种出现方面几乎没有活动的长期停滞期，被间歇性的突发活动所打断。”

"The so-called point balance xff0c; evolution is xff0c, which takes the form of a pulse; not the slow and steady evolution that Darwin calls. There is little activity in species extinction or the emergence of new species for a long period of stagnation xff0c; interrupted by intermittent outbreaks."

因此，泡沫也是比特币故事的一部分。它们并不是整个幂律增长的主线，但它们也是其中重要的一部分和必要的一部分。

So xff0c; bubbles are also part of the Bitcoin story. They are not the main line of growth xff0c; they are also an important part and necessary part of it.

我认为，这就完美地解释了整个周期中泡沫外（约 2 年）和泡沫内（约 2 年）的增长。请告诉我你的想法以及这是否合理。

I think xff0c; this is a perfect explanation for the growth of the bubble outside xff08; about 2 years xff09; and within the bubble xff08; about 2 years xff09; ff09;. Please tell me what you think and whether it makes sense.

知名物理学家 D. Sornette 的这篇论文对泡沫的起源和性质持非常相似的立场。

This paper by D. Sornette, a leading physicist, takes a very similar position on the origin and nature of foams.

注释

稀缺性在这一理论中完全不起作用。稀缺性没有任何机制或解释力。

Scarcity does not work at all in this theory. There is no mechanism or interpretation for scarcity.

附录

appendix