How does the bitcoin code define its 21 million cap? – Unchained Capital

Many of bitcoin’s staunchest critics have expressed doubt about its 21 million cap, but perhaps the most mindless criticism relates to the fact that the supply limit is not specified in its code in plain language. That may be true, but this part of bitcoin—how exactly the supply cap is defined in its code—is relatively easy to prove.

I’ll just challenge the group to one other thing: how do you know it ends at 21 million? You all read the algorithms? You guys all believe that? I don’t know, I’ve always been a skeptic of stuff like that.

– Jamie Dimon, CEO, JP Morgan Chase, Oct 11, 2021

The bitcoin code uses a mix of consensus rules and simple math agreed upon by everyone who runs a bitcoin node to implicitly establish the limit. Here, we’ll break down the code to verify with certainty that bitcoin’s cap is 21 million. If this is a bit too technical and you’d prefer a higher-level overview of 21 million and its significance, we have that too.

The GetBlockSubsidy function

To understand what the code says about bitcoin’s total coin cap, the first place to look is the function for block subsidies. This part of the code is vital to verifying bitcoin’s fixed supply because all new bitcoin originates from block subsidies (which are included, along with transaction fees, as part of block rewards) paid to miners.

Here’s the function in bitcoin’s code that determines block subsidy, called GetBlockSubsidy():

This section of code defines the amount of the block subsidy and determines the schedule for its reduction over time by way of the halvings. When the block subsidy reaches zero, it signifies bitcoin’s total supply has reached its limit. It will reach zero, eventually, due to 33 pre-programmed halvings.

Line by line, we’ll establish:

1. How the halving epoch is calculated,
2. How the block subsidy is calculated, and finally
3. How the 21 million cap is implied as a summation of the block subsidies.

1. Calculating the halving epoch

Let’s begin by looking at the way bitcoin defines how many halvings have occurred.

CAmount GetBlockSubsidy (int nHeight, const Consensus::Params& consensusParams)

This first line opens the function and specifies that what follows are the parameters for the block subsidy. Next, we get an equation that calculates the number of halvings:

int halvings = nHeight / consensusParams.nSubsidyHalvingInterval

This line says a lot, so let’s break it down:

• int halvings

  specifies the integer called

  halvings

  , rounded down to the nearest whole number.

• nHeight

  is the current number of blocks on the blockchain. When new blocks append to bitcoin’s blockchain, it grows “taller.” As of this writing, bitcoin’s “block height” is 740,805 blocks.

• nSubsidyHalvingInterval

  specifies the number of blocks which must pass before another halving occurs.

That last parameter, nSubsidyHalvingInterval, is a static value set to 210,000 blocks as defined elsewhere in the bitcoin codebase, shown below:

With the values for nHeight (current block height = 740,805) and nSubsidyHalvingInterval (how many blocks must pass before another halving = 210,000) in hand, we can divide the current block height by the subsidy interval. This value, which is the current halving epoch in which we find ourselves, is assigned to the integer halvings: 

int halvings = 740,805 / 210,000 or int halvings = 3.52764286 (rounds to 3)

The end result reveals we are in the epoch after the third halving. Note that before rounding down to the halving integer 3, we have the quotient 3.52764286. This number is useful, too, because it indicates we’re just beyond halfway between the third and fourth halvings.

Next, we have a comment line:

// Force block reward to zero when right shift is undefined,

This comment helps programmers and observers better understand what is happening with the code—in this case defining a bug fix to force the block reward to zero in one circumstance, as described next:

if (halvings >=64) return 0;

These lines often cause confusion because it would seem to define that there will be 64 halvings. That isn’t the case—you can read the technical details about why this bug fix was necessary if you wish.

2. Calculating the block subsidy

To understand how the block subsidy is calculated, you first need to know about COIN. COIN is an agreed-upon value specified elsewhere in the code as “100000000”—or the total smallest divisible units of a bitcoin. Nowadays, we call these satoshis.

Here’s the parameter in the code that specifies the amount of COIN:

CAmount nSubsidy = 50 * COIN;

The next line in the GetBlockSubsidy() function defines a fixed number equal to the block subsidy paid on each of the first set of blocks of bitcoin mined—the 210,000 blocks that occurred before the first halving. In plain English, nSubsidy is equal to 50 times the value of COIN, or 50 bitcoin.

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The binary representation of nSubsidy (or 50 * 100000000) is 33 bits in length, which will be helpful later for understanding how the code divides the block subsidy with each halving:

100101010000001011111001000000000

The next line is another comment.

// Subsidy is cut in half every 210,000 blocks which will occur every 4 years.

Starting with the first block reward of “50 * COIN” (50 bitcoin), the subsidy is subsequently cut in half every 210,000 blocks. The lines below describe the function that divides the subsidy in half based on the current halving, as we first described above.

nSubsidy >>= halvings; return nSubsidy;

The critical element of the code above is the >> or “bitwise right shift”. A bitwise shift is a standard operator in the C++ programming language. Here, it means an “arithmetic shift to the right by one bit”, which is the same as dividing by two. It’s important because it helps illustrate how the supply schedule knows where to cease division.

As calculated earlier, we find ourselves today in the third halving. So, we know three things: 1) nSubsidy = 50 * 100,000,000 (or 50 bitcoin), 2) the operator >> 3;, and 3) return nSubsidy;.

The first value in our calculation above is a fixed number. The second value changes based on where we are in the halving schedule. Here again, is the binary representation for the fixed value of 50 * 100000000 (the initial block subsidy):

100101010000001011111001000000000

A single “bitwise right shift” removes the last digit on the far right from the above string. Converting these binary numbers to their decimal equivalent numbers reveals the subsidy after the first halving:

Before: 100101010000001011111001000000000 = 5,000,000,000 units of COIN (or 50 bitcoin)

After: 10010101000000101111100100000000 = 2,500,000,000 units of COIN (or 25 bitcoin)

A >>3 shifts our initial binary number by three spaces to the right, which drops the three furthest digits and creates another new number. 

Before: 100101010000001011111001000000000

After: 100101010000001011111001000000

When this new binary number is converted to its decimal equivalent, it equals 625,000,000 units of COIN, or 6.25 bitcoin.

You can visit rapidtables.com if you want to play with this operation yourself. Every time you shift by dropping an additional digit from the right side of the initial binary string, the equivalent decimal number cuts in half.

3. Summing it up

The scarcity of newly-issued bitcoin due to these bitwise right shifts, occurring every 210,000 blocks, is eye-opening:

HalvingBinary NumberUnits of COIN (sats)Bitcoin–1001010100000010111110010000000005,000,000,00050 bitcoin1100101010000001011111001000000002,500,000,00025 bitcoin210010101000000101111100100000001,250,000,00012.5 bitcoin3100101010000001011111001000000625,000,0006.25 bitcoin410010101000000101111100100000312,500,0003.125 bitcoin51001010100000010111110010000156,250,0001.5625 bitcoin610010101000000101111100100078,125,00078,125,000 satoshis71001010100000010111110010039,062,50039,062,500 satoshis8100101010000001011111001019,531,25019,531,250 satoshis91001010100000010111110019,765,6259,765,625 satoshis10100101010000001011111004,882,8124,882,812 satoshis1110010101000000101111102,441,4062,441,406 satoshis121001010100000010111111,220,7031,220,703 satoshis1310010101000000101111610,351610,351 satoshis141001010100000010111305,175305,175 satoshis15100101010000001011152,587152,587 satoshis161001010100000010176,29376,293 satoshis17100101010000001038,14638,146 satoshis1810010101000000119,07319,073 satoshis19100101010000009,5369,536 satoshis2010010101000004,7684,768 satoshis211001010100002,3842,384 satoshis22100101010001,1921,192 satoshis231001010100596596 satoshis24100101010298298 satoshis2510010101149149 satoshis2610010107474 satoshis271001013737 satoshis28100101818 satoshis29100199 satoshis3010044 satoshis311022 satoshis32111 satoshi

As you might have guessed, adding all the block subsidies in this table comes in just shy of 21 million, totaling 20,999,999.9769 bitcoin.

Core contributor Pieter Wuille has explained that the real total is a bit lower. While the sum of the block subsidies if collected in their entirety is 20,999,999.9769, after accounting for the coins created in the genesis block not being spendable, early bugs, and miners experimenting with the code (some blocks claimed less than allowed), the total supply is actually less—closer to 20,999,817 bitcoin.