The constant product formula is not an equation, and k does not stay constant

Here is the line, from the second paragraph of the Uniswap v2 whitepaper, describing v1:

maintaining the invariant that the product of the reserves cannot decrease

Cannot decrease. Not stays the same. Every explainer of automated market making opens with x * y = k and then says something like “the product stays constant, which is where the name comes from”. The name comes from there, and the sentence is wrong, and it is wrong in the exact place where the mechanism does its work.

The deployed contract does not check an equality. It checks an inequality, on balances that have already had the fee subtracted, which means k increases on every single trade. That growth is not a rounding artefact. It is how liquidity providers get paid.

What the whitepaper actually specifies

Section 3.2.1, “Adjustment for fee”, gives the v1 rule:

Uniswap v1’s trading fee is applied by reducing the amount paid into the contract by 0.3% before enforcing the constant-product invariant. The contract implicitly enforces the following formula:

(x₁ − 0.003·x_in) · y₁ >= x₀ · y₀

Note the direction of the comparison and note where the 0.003 sits: on the input, inside the product, before the check. For v2, which has to handle flash swaps where both inputs can be non-zero, it generalises and then gets restated in integer arithmetic:

(1000·x₁ − 3·x_in) · (1000·y₁ − 3·y_in) >= 1000000 · x₀ · y₀

That last line is not an illustration. It is the code. From UniswapV2Pair.sol:

uint balance0Adjusted = balance0.mul(1000).sub(amount0In.mul(3));
uint balance1Adjusted = balance1.mul(1000).sub(amount1In.mul(3));
require(balance0Adjusted.mul(balance1Adjusted) >= uint(_reserve0).mul(_reserve1).mul(1000**2), 'UniswapV2: K');

Three lines. Multiply both balances by a thousand, subtract three times each input, require that the product of those adjusted numbers is at least a million times the old product. The error string is 'UniswapV2: K', which is why the whole thing is remembered as an equation about k, and the operator on the line that raises it is >=.

Two things follow that most explanations skip.

The fee is charged on the way in, so it is not 0.3% of what you get. The whitepaper’s own footnote does the arithmetic: relative to the amount withdrawn, the effective fee is 1/(1−0.003) − 1, which is 3/997, or about 0.3009203%. A small difference, and a real one, and the sort of thing that separates a description written from the contract from one written from another description.

The invariant is a floor, not a target. Anyone may hand the pool more than the check requires. The contract does not object, and the surplus becomes part of the reserves, which is to say it belongs to the liquidity providers. That is why the check is an inequality rather than an equality: an equality would forbid overpayment, and overpayment is the mechanism.

Who actually came up with it

The received version is that Vitalik Buterin proposed constant product market makers in 2016 and Uniswap implemented them. Neither half survives the documents, and the person best placed to correct it corrected it twice.

In “On Path Independence”, June 2017, Buterin gives the formula tokenA_balance(p) · tokenB_balance(p) = k and appends, in parentheses, “credit to Martin Koppelmann”. In March 2018, on ethresear.ch, in a post whose title is “Improving front running resistance of x*y=k market makers”, he writes:

Martin Köppelmann from Gnosis suggested a simple approach for doing so, that I call the “x*y=k market maker”

Meanwhile Alan Lu, also at Gnosis, had described the constant product invariant explicitly in a post called “Building a Decentralized Exchange in Ethereum” dated 6 March 2017, three months before the Buterin piece that credits Köppelmann. Uniswap’s own history page goes further than either and states that Lu was the first person to conceive of these market makers on Ethereum, with Köppelmann relaying the idea to Buterin, Buterin publicising it, and Hayden Adams building the implementation.

So the honest sentence is: the idea came out of Gnosis in 2016 and 2017, Buterin analysed and popularised it while explicitly crediting somebody else, and Adams shipped the first version anybody used. Uniswap v1 went to mainnet on 2 November 2018, on the last day of Devcon 4 in Prague, with a 0.3% fee. Uniswap v2 followed in a paper dated March 2020, adding arbitrary ERC20 pairs, a time-weighted price oracle, flash swaps, and a protocol fee switch that has never been the interesting part.

Impermanent loss, and why the name is disputed

A liquidity provider who deposits into a constant product pool and later withdraws can end up with less value than if they had simply kept the two assets. The cause is price divergence: the pool automatically sells whichever asset is appreciating and buys whichever is depreciating, because that is what holding the product constant requires. The larger the divergence, the larger the shortfall against holding.

The name that stuck is “impermanent loss”, and it is contested for a good reason. There is nothing impermanent about it unless prices happen to return to where they started, and there is no mechanism that makes them do so. Naming a loss after the condition under which it would reverse, when that condition is not guaranteed and often does not occur, encourages exactly the reading that gets people into trouble: that the position is fine as long as you do not withdraw. The alternative name in circulation is “divergence loss”, which describes the cause instead of a hypothetical cure, and some venues use it in their own documentation for that reason.

I would like to tell you who coined “impermanent loss” and when. I cannot. The earliest widely-credited use is a 2019 Medium post by an author writing as Pintail, analysing returns for Uniswap liquidity providers, and Medium does not serve to anything that is not a browser. Two different URLs are in circulation for it, sources disagree about whether it is from January or June, and there is a claim I have seen but cannot confirm that the post was later edited to replace “impermanent loss” with “divergence loss”. Every part of that is unverified. So the accurate statement is that the term entered general use through early Uniswap liquidity provider analyses around 2019, with no coiner I can name.

The rigorous literature has largely moved past the argument anyway, toward loss-versus-rebalancing, which measures the shortfall against a strategy that actually rebalances rather than against passive holding.

Five sentences to stop repeating

“The product stays constant.” It does not. It rises on every trade. That is the fee. This is the most common error in the genre and it inverts the mechanism.

“Uniswap invented the AMM.” It did not. The constant product idea came from Gnosis, Uniswap’s own history page says so, and Bancor shipped an on-chain automated market maker before Uniswap existed. What Uniswap built was the first constant product implementation that anybody used at scale, which is a real achievement and a different claim.

“Vitalik Buterin invented x*y=k.” He credits Martin Köppelmann, twice, in writing.

“No order book means no front-running.” Backwards. Buterin’s March 2018 post is titled “Improving front running resistance of x*y=k market makers”, eight months before Uniswap launched. Sandwich attacks are not a later discovery, they are structural: every trade is public before it settles and its price impact is deterministic, which is a better environment for extraction than an order book, not a worse one.

“The loss only becomes permanent if you withdraw.” This is the framing the “divergence loss” camp objects to and they are right. The shortfall against holding is real and continuous from the moment prices diverge. Withdrawing realises it; not withdrawing does not undo it.

What was expanding, what was contracting

What expanded, and it is genuinely one of the most interesting expansions in the period, is the number of assets that could have a market at all. An order book needs somebody to sit on both sides of it, so an asset with no professional market maker has no market. A constant product pool needs nobody to sit on anything. Three lines of arithmetic and two piles of tokens make a price, and the price is continuous, and it is available at three in the morning for a token nobody has heard of.

What contracted was the assumption that a market required a market maker as a role. Not as a function, which is what the pool performs, but as a party with a desk, an inventory policy and a view.

And the thing almost nobody could tell at the time is that the pool takes the other side of every trade at a price set by a formula that does not know anything. It sells what is rising. It buys what is falling. Against an informed counterparty it loses systematically, and the fee is what has to cover that. When the fee covers it the provider is paid for the service. When it does not, the provider has been supplying free optionality to better-informed traders and calling the shortfall impermanent.

That is the trade. It was legible from the arithmetic on day one, and it took the field about three years to name properly.