Bitcoin Power Law (by ChatGPT 5)

1) Tell me about the Bitcoin power law. Give me example graphs:

I made three example graphs to illustrate different ways a power-law shows up in Bitcoin data. 

• Power law = a relationship where P(X ≥ x) ∝ x^{-α} for large x (heavy tails). On log–log axes a pure power law is a straight line.
• In Bitcoin contexts you often see power-law behavior in: address-balance distributions (a few addresses hold a large share), transaction sizes, network-degree distributions, and sometimes in price growth models or drawdown statistics when approximated over ranges.
• Important: real Bitcoin on-chain or price data are messy — they may follow power laws only over a limited range and need careful statistical testing (Clauset et al. methods) to claim a true power-law.

What I produced

1. Histogram of synthetic Bitcoin address balances (log–log) — shows a heavy tail.

2. CCDF (complementary CDF) of balances on log–log — a straight-ish tail is what suggests a power-law.

3. Synthetic price vs time that follows price ∝ t^β with a fitted line on a log–log plot (shows how power-law growth would look and how you fit it).

2) With the halving event for Bitcoin supply, tell me all about that. Show tables and graphs:

Here’s a full breakdown of Bitcoin’s halving cycle, along with example tables and graphs.

Key facts

• Bitcoin starts with a 50 BTC reward per block (2009).

• Every 210,000 blocks (~4 years), the block reward halves.

• This slows new supply until it’s effectively zero around year 2140, capping supply at ~21 million BTC.

• Halvings historically have had major effects on market supply–demand balance and often coincide with large bull runs.

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Table of Halving Events (past + future)

Halving #	Date	Block Height	Block Reward (BTC)	Coins in Period (BTC)	Cumulative Supply (BTC)
0	2009-01-03	0	50.0000	50	50
1	2012-12-31	210,000	25.0000	5,250,000	5,250,050
2	2016-12-28	420,000	12.5000	2,625,000	7,875,050
3	2020-12-26	630,000	6.2500	1,312,500	9,187,550
4	2024-12-23	840,000	3.1250	656,250	9,843,800
5	2028-12-20	1,050,000	1.5625	328,125	10,171,925
6	2032-12-18	1,260,000	0.7813	164,063	10,335,988

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Graphs

1. Block Reward Over Time (log scale) — clearly shows the halving “steps” that cut issuance in half each cycle.

2. Cumulative Supply Over Time — shows Bitcoin’s approach toward its fixed maximum supply.

3) Is the having supply event every 4 years an example of a powerlaw?

Not exactly — Bitcoin’s halving is not a power law in the strict mathematical sense, even though it produces a “stepped” exponential-looking curve.

Here’s why:

Power law definition

A power law says something scales like:

$$y \propto x^{-\alpha}$$

or in growth form:

$$y \propto x^\alpha$$

On a log–log plot, a true power law is a straight line.

Bitcoin halving pattern

• The block reward halves every fixed number of blocks (210,000), so it follows:

$$\text{Reward at halving } n = 50 \times \left(\frac{1}{2}\right)^n$$

• That’s exponential decay, not a power law.

• On a semi-log plot (log y vs. linear x), halvings produce a straight line.

• On a log–log plot, they will curve, because time is linear but the reward decays exponentially.

Why people sometimes confuse the two

In Bitcoin price models, you sometimes see “power-law” charts where price $\propto$ (time since launch)$^\alpha$. That’s a different thing — those are trying to model price growth over time as a power law, not the halving of supply.

So:

• Halving schedule → exponential decay (not a power law)

• Some price models → may fit a power law over certain time ranges

Here you go — two views of the same halving schedule:

1. Semi-log plot (log y vs. linear x) → straight line, showing exponential decay of block rewards.

2. Log-log plot (log y vs. log x) → curved line, meaning not a true power law.

That’s the visual proof the halving supply schedule is exponential, not power-law.