Concentrated liquidity

Uniswap v2 forces every liquidity provider in a pair to cover every price. The concentrated-liquidity docs give the stablecoin example: if two dollar tokens trade between 0.99 and 1.01, almost all of a v2 pool’s capital is elsewhere. v3 lets the provider deposit only in that interval. The same capital then offers more depth where trades actually happen, and none where they do not.

v4’s architecture page says this capital-efficiency property is inherited. Hooks and the singleton do not replace the range. A hookless v4 pool is still a concentrated-liquidity pool.

Ticks

The whitepaper sets the price at tick i to 1.0001^i. One tick is one basis point. The contracts store the square root of that price as a Q64.96 number, which is what TickMath.getSqrtRatioAtTick returns. The published constants bound the tick at ±887272, corresponding to prices the fixed-point format can represent.

Pools do not initialize every tick. tickSpacing selects a grid. The whitepaper notes the trade: a finer grid is more precise, and each initialized tick a swap crosses costs gas. Fee tiers and spacing are paired so a cheap stablecoin pool can use a tight grid and a volatile pool can use a coarse one.

A position is the liquidity between a lower and an upper tick on that grid. Current liquidity is the sum of positions that contain the current tick. When a swap crosses a tick, the pool adds or removes the liquidity stored at that boundary. The price does not jump over an empty region for free. It walks.

Inside the range, the curve is familiar

For a fixed range, the virtual reserves follow a product curve that hits zero real balance of one token at each edge. Liquidity L is the scale of that curve, playing the role k played as a size parameter in v2. The amount of token0 between two square-root prices is L times the difference of their reciprocals. The amount of token1 is L times the difference of the square-root prices. The whitepaper and SqrtPriceMath are the source of those deltas. A blog primer is not required to use the idea, but it is required not to invent a third formula.

While the price is inside the range, the position earns a share of fees proportional to its L relative to other active liquidity. When the price leaves, the position is entirely token0 or entirely token1. Fee accrual stops. If the price returns, the position becomes active again without a new deposit. The docs are explicit that this single-asset outcome is normal, not a liquidation.

What concentration costs

Capital efficiency is not free yield. The provider is leveraged to the chosen interval. A range that is wrong is a position that no longer earns and that may be entirely the asset which fell. A v2 LP in the same pair would still hold a mix. Managing a range is an extra strategy. The protocol does not rebalance it unless a v4 hook is written to do so, and that hook is someone else’s code.

The receipt differs by version, as the LP share article says. The range does not. An integration that wants “the Uniswap price” still has to decide which pool, which fee tier, and whether the liquidity it is looking at is in range.