# ๐Santostasi's iteration

Core of Bitcoin Power Law Theory aka BPLT

Last updated

Core of Bitcoin Power Law Theory aka BPLT

Last updated

Iterative process is the key

**Price, hash rate, and the number of active addresses** (filtered to exclude dust addresses) **all exhibit power law relationships with each other and over time**.

These variables interact in a **continuous feedback loop**, where **each one influences the others**.

Power laws are mathematical expressions of the form $y = Ax^n$ and are prevalent in both natural and social phenomena, including city and nation growth.

They arise whenever an output becomes a new input in an **iterative process**.

This concept perfectly aligns with Bitcoin's nature, where, for example, the current hash rate affects future hash rates in a self-sustaining cycle.

The theory is supported by the well-known diagram below, illustrating these interactions.\

As we have seen in the concept of scale invariance, fractals are characterized by self-similar patterns that emerge through **iterative processes**.

This concept is mirrored in Bitcoin's market dynamics, where similar patterns of growth and behavior are observed across different time frames.

In fractal geometry, a simple rule is applied repeatedly to create increasingly complex structures. Similarly, **the iterative processes in Bitcoin, as described by Giovanni Santostasi, demonstrate scale invariance through power law relationships between key metrics like hash rate, price, and the number of active addresses**.

In Bitcoin, the iterative nature of its market dynamics can be expressed through the following power law relationships:

This relationship highlights how increased Bitcoin valuation leads to significant growth in network security and processing power.

Just as fractals are generated through iterative rules, Bitcoin undergoes an iterative process where each phase influences the next.

This cyclical feedback loop mirrors the iterative generation of fractal patterns, where each iteration builds upon the previous, maintaining structural consistency at every scale.

This **iterative process** (aka Santostasi's iteration) underpins the scale invariance observed in Bitcoin's metrics, where the same mathematical relationships hold true regardless of the scale of observation.

A critical component of this iterative process is the **difficulty adjustment**, which acts as a **curbing mechanism**.

The difficulty adjustment is a built-in feature of Bitcoin's protocol that **recalibrates the computational difficulty** required to **mine a new block approximately every two weeks**.

This mechanism ensures that blocks are produced at a relatively **stable rate**, regardless of fluctuations in the network's total hash rate:

As the

**price of Bitcoin rises**, mining becomes more profitable,**attracting additional computational power to the network**.**Increase in hash rate**could lead to faster block creation, but the**difficulty adjustment counteracts this by making it harder to mine new blocks**.Conversely, if the

**price drops or miners leave**the network,**the difficulty decreases, making mining easier and maintaining block production time**.

This **adjustment process serves as a** **curbing mechanism** by regulating the rate at which blocks are mined and **maintaining the stability of the Bitcoin network**.

It **prevents runaway scenarios where an unregulated increase in hash rate could lead to excessively rapid block production, thereby distorting the economic incentives within the system**.

In this way, the difficulty adjustment functions similarly to the **curbing mechanisms observed in other natural systems governed by power laws**.

It ensures that the growth of the Bitcoin network, measured by metrics like hash rate, price, and the number of addresses, follows a controlled and sustainable path.

**This regulation is vital for maintaining the integrity and security of the network, making the iterative process of Bitcoin's market dynamics both predictable and resilient across various scales.**

Conclusion

The **Bitcoin Power Law theory**, as proposed by **Giovanni Santostasi**, offers a compelling framework for understanding the intricate and iterative dynamics of the Bitcoin market.

By highlighting the power law relationships between key metrics like price, hash rate, and the number of active addresses, this theory demonstrates the inherent scalability and self-similar nature of Bitcoin's growth.

The iterative feedback loop, bolstered by mechanisms such as the difficulty adjustment, ensures a balanced and sustainable network evolution.

As Bitcoin continues to evolve, this theory provides valuable insights into its potential trajectory, **underlining the importance of mathematical principles in deciphering complex market behaviors**.

$\text{Addresses}=t^3$

This indicates that the **number of active Bitcoin addresses** $(\text{Addresses})$ **grows proportionally to the cube of time** $(t^3)$, suggesting a rapid increase in user adoption over time.

$\text{Price} = \text{Addresses}^2 = (t^3)^2 = t^6$

The **price** $(\text{Price})$ **increases quadratically with the number of addresses** $(\text{Addresses}^2 )$, reflecting that as the network expands, the value derived from Bitcoin grows exponentially.

$\text{Hash Rate} = \text{Price}^2 = (t^6)^2 = t^{12}$

The $\text{Hash Rate}$, representing the total computational power used for mining, **scales with the square of the price** $(\text{Price}^2)$.