Composability · v1
Curve & the StableSwap Invariant
An AMM tuned for assets that should trade 1:1 — flat curve near parity, deep stable liquidity, the engine of the Curve wars.
TradFi →FX desk for pegged pairs
Prerequisites
01 · Concept — what problem does it solve?
Constant-product (x·y=k) is wasteful for assets that should trade near 1:1 — two dollar stablecoins, or ETH and an ETH derivative. A v2 pool prices real even at parity, because its curve is always bending. Curve's StableSwap invariant fixes this: it behaves like a flat constant-sum curve (zero slippage) while the pool is balanced, and only bends toward constant-product as the pool skews far from peg. The result is extremely deep liquidity exactly where pegged assets trade.
02 · Mechanics
- Hybrid invariant: StableSwap blends constant-sum (flat, no slippage) and constant-product (always solvent) curves into one equation.
- A: the dial. High A keeps the curve flat across a wider band around peg (great while pegged); as the pool skews, the constant-product term takes over so it can never be fully drained.
- Why it matters: a stable pair on Curve quotes far tighter than the same pair on a v2 — the whole point is low slippage for things that move together.
- veCRV & the Curve wars: lock CRV for vote-escrowed veCRV to direct emissions ("gauge weight") to chosen pools. Protocols then bribe veCRV holders for that flow — the bribe economy that defined a whole era of DeFi.
- crvUSD: Curve's own uses "LLAMMA," a soft- AMM that continuously converts collateral instead of a hard liquidation.
03 · Formulas
// StableSwap invariant (n coins, balances xᵢ, amplifier A)
A·nⁿ·Σxᵢ + D = A·nⁿ·D + Dⁿ⁺¹ / (nⁿ · Πxᵢ)
// ▲ constant-sum term ▲ constant-product term
// D = total pool value when balanced; solved iteratively
// balanced → curve ≈ constant sum → ~0 slippage
// skewed → curve → constant product → stays solvent
04 · Edge cases & risks
- A is a peg bet — a high amplifier assumes the assets stay pegged. If one constituent depegs, the flat curve means LPs absorb the bad asset fast and end up holding the loser (amplified impermanent loss).
- Governance capture — the veCRV gauge system concentrates power in whoever locks the most CRV; emissions can be steered to pools that benefit large holders.
- 2023 reentrancy — a Vyper compiler bug enabled a reentrancy exploit draining ~$70M from several Curve pools, a reminder that the math being sound does not make the implementation safe.
- Stable depth is conditional — the deep liquidity exists only near peg; in a real the usable depth collapses precisely when traders want out.
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