# StepCurve

> A pool whose price moves in discrete steps instead of continuously, so there is no infinitesimal arbitrage to take and a quote holds still long enough to be worth quoting.

A production Uniswap v4 hook. Source: https://github.com/nirholas/step-curve. Part of the HookForge catalogue: https://hookforge.pages.dev

## How it works
A constant-product pool changes its price on every swap, by any amount, however small. That is elegant and it has a cost that falls entirely on the people providing the liquidity: the price is always slightly wrong by an amount somebody can capture, and the smaller the increment the more often it is worth capturing. Continuous pricing is what makes an AMM permanently arbitrageable rather than occasionally. Every other market prices in ticks. Equities trade in cents, bonds in thirty-seconds, futures in whatever the exchange decided, and the reason is not tradition: a minimum increment means a quote is worth something for a while, because moving the price at all costs a whole increment rather than a rounding error. This pool prices in increments. The quote is constant across a band of inventory `stepSize` wide and falls by `stepX96` when the band is crossed, so:   price(reserve0) = max(minPriceX96, startPriceX96 - (reserve0 / stepSize) * stepX96) Within a band the pool is a constant-sum market maker at a fixed price, which is to say it fills at exactly the quote with no slippage at all. A swap that would cross bands walks them, filling each at its own price. So a small trade sees a firm quote and no slippage, and a large trade sees exactly the depth the ladder was configured with. The consequence for arbitrage is the point. An external price move smaller than one increment is not tradeable against this pool at all, because moving the price requires consuming a whole band. Providers are exposed to moves larger than the increment and immune to noise below it, which is the trade every quoting venue in the world makes. Liquidity is fungible and proportional; see {ForgeCurveHook}. There are no ticks and no ranges, because the ladder is the range.

## Prior art
Constant-sum hooks (Uniswap's own constant-sum example, the StableSwap and Orbital submissions) replace the curve with a different continuous one. Bancor's Carbon quotes discrete asymmetric ladders, off v4. Discrete tick pricing is universal outside crypto. A v4 custom curve that is constant-sum inside a band and steps between bands, so a quote is firm below the increment and depth is exactly what was configured, is the contribution here.

## Where it does not help
A swap that crosses many bands walks them one at a time, so gas grows with the number of bands crossed and a swap larger than `MAX_STEPS` bands reverts rather than filling partially. Size `stepSize` against the trades the pool expects. The ladder is also fixed at deployment: a pool whose asset moves far outside the configured range runs out of ladder and stops quoting on that side, which is honest but is not the same as a curve that quotes everywhere.

## Facts
Slug: step-curve
Contract: StepCurveHook
Callbacks: none
Parameters: none
Dynamic fee required: no

## Caveats
- Unaudited.
- A deployment with status "deterministic" is a mined CREATE2 address with no code at it yet. Never present one as live.