StableSwap Invariant

The StableSwap invariant is defined as: A * n^n * sum(x_i) + product(x_i) = A * n^n * D + (D / n)^n. This equation hybridizes the constant-sum (sum(x_i) = constant) and constant-product (product(x_i) = constant) invariants. The amplification coefficient (A) controls the weighting between the two models, determining the flatness of the bonding curve.

The amplification coefficient (A) is a tunable parameter that dictates the shape of the bonding curve. A higher A value (e.g., 100-1000) makes the curve flatter within a price range, mimicking a constant-sum market with minimal slippage. A lower A value makes the curve more convex, behaving like a standard constant-product AMM (e.g., Uniswap V2). This allows the invariant to be optimized for different asset pairs.

The primary advantage is dramatically reduced slippage for trades between assets expected to maintain a 1:1 peg, such as different stablecoins (USDC, USDT, DAI). While a constant-product AMM experiences significant slippage even for small trades, StableSwap's near-flat curve allows for large trades with minimal price impact, as long as the pool remains balanced.

To maintain the 1:1 peg and pool balance, StableSwap pools often implement dynamic fees. When the pool becomes imbalanced (e.g., too much USDC, not enough DAI), the fee for trades that worsen the imbalance increases. This incentivizes arbitrageurs to trade in the direction that rebalances the pool, helping to restore the peg and optimal trading conditions.

Liquidity providers (LPs) in StableSwap pools face a different impermanent loss (IL) profile compared to standard AMMs. For perfectly pegged assets, IL is minimal. However, if the peg breaks significantly, LPs can experience losses greater than in a constant-product AMM because the concentrated liquidity amplifies exposure to the depegging asset. The risk/reward is tied directly to the stability of the peg.

While not a direct StableSwap implementation, Uniswap v3's concentrated liquidity tackles a similar problem—reducing slippage—through a different mechanism. Liquidity providers can concentrate capital within custom price ranges, achieving high capital efficiency for stable pairs, effectively creating a piecewise constant sum curve.

The mathematical heart of StableSwap is defined by the equation: A * n^n * sum(x_i) + D = A * n^n * D + D^(n+1) / (n^n * prod(x_i)) Where:

• A is the amplification coefficient, tuning the curve's flatness.
• n is the number of tokens in the pool.
• x_i is the balance of token i.
• D is the total pool liquidity invariant. This creates a curve that is flat (constant sum) near equilibrium for minimal slippage but becomes curved (constant product) at the edges to preserve liquidity.

A crucial, adjustable parameter that controls the pool's behavior. A higher A value makes the curve flatter, ideal for nearly identical assets, leading to lower slippage within a wider price range. A lower A makes the curve more like a Uniswap-style constant product, suitable for correlated but not pegged assets. Protocols can tune this parameter per pool.

Key implementation features built on top of the base invariant:

• Meta-pools: A pool where a new token (e.g., a new stablecoin) is paired against the LP token of an existing base pool. This allows efficient bootstrapping by leveraging the deep liquidity of the base.
• Liquidity Gauges: Smart contracts that measure a user's liquidity contribution over time, used to distribute governance token (CRV) emissions as rewards.

The invariant combines a constant sum (x + y = k) and constant product (x * y = k) formula: A * (x + y) + D = A * D + (D^{n+1})/(n^n * x * y). The amplification coefficient (A) controls the curve's shape. A high A (~100) creates a flatter, low-slippage region, approximating a constant sum. A low A (~1) reverts to a standard constant product curve.

This tunable parameter is the primary economic lever. A higher A:

• Increases capital efficiency within the price peg range.
• Reduces slippage for large trades near parity.
• Increases impermanent loss if the peg breaks, as liquidity is concentrated in a narrow band. Pools can have static A or be dynamically adjusted via governance or algorithms based on pool balance ratios.

The key economic benefit is drastically reduced slippage compared to a constant product AMM for correlated assets. For example, swapping 1 million USDC for USDT might incur <0.01% slippage in a well-tuned StableSwap, versus potentially >0.3% in a Uniswap V2-style pool. This efficiency allows LPs to provide the same depth of liquidity with less capital locked.

Liquidity providers face a unique asymmetric impermanent loss risk. While losses are minimal while assets are pegged, a depeg event can cause significant losses as the pool's composition becomes imbalanced. The concentrated liquidity amplifies the loss compared to a standard AMM curve once the price moves outside the flat region.

The flat section of the curve can be vulnerable to oracle manipulation attacks. An attacker could potentially drain the pool by manipulating an external price oracle to make the pool believe assets are still pegged while executing a large, imbalanced swap on-chain. Robust, time-weighted average price (TWAP) oracles are a critical security mitigation.

Curve Finance is the canonical implementation, using the StableSwap invariant for stablecoin pools (e.g., 3pool: DAI, USDC, USDT). Ellipsis Finance on BSC and Mobius on Celo are other prominent forks. Each implementation may feature different governance for parameter A and fee structures.