DEX Automated Market Maker

## ROLE
You are a DeFi protocol developer who has built AMM systems from scratch. You understand the mathematics of Uniswap V2/V3, Curve, and Balancer and can implement efficient trading mechanics with robust security.

## OBJECTIVE
Design an AMM smart contract for [PROTOCOL NAME] implementing [AMM TYPE: constant product, stable swap, concentrated liquidity] for trading [TOKEN PAIRS] on [CHAIN].

## TASK

### Core AMM Mathematics
- Invariant formula: x * y = k (constant product) or StableSwap curve with amplification
- Price calculation: how token prices derive from reserve ratios
- Slippage: price impact formula based on trade size relative to pool depth
- Minimum output: user-specified minimum to prevent sandwich attacks
- TWAP oracle: time-weighted average price accumulator for external consumers

### Liquidity Provision
- Adding liquidity: proportional deposit matching current pool ratio
- Initial liquidity: first depositor sets the price, minimum liquidity locked permanently
- LP tokens: ERC-20 tokens representing share of pool (mint on deposit, burn on withdrawal)
- Imbalanced deposits: single-sided liquidity with automatic rebalancing and penalty/bonus
- Withdrawal: burn LP tokens to receive proportional share of both pool tokens

### Fee System
- Swap fee: configurable fee (e.g., 0.3%) deducted from input amount
- Fee distribution: fees accumulate in the pool, increasing LP token value
- Protocol fee: optional protocol-level fee (e.g., 1/6 of swap fee) sent to treasury
- Dynamic fees: optional volatility-based fee adjustment for impermanent loss mitigation
- Fee accounting: precise tracking to avoid rounding errors over millions of swaps

### Flash Swaps
- Mechanism: borrow tokens from the pool, execute arbitrary logic, repay in same transaction
- Callback pattern: IFlashSwapCallback interface for borrower contracts
- Fee: flash swap fee equal to or greater than standard swap fee
- Validation: verify repayment satisfies invariant after callback completion
- Use cases: arbitrage, liquidations, collateral swaps

### Router Contract
- Multi-hop swaps: route through multiple pools for pairs without direct liquidity
- Path finding: optimal route calculation considering gas and price impact
- Deadline parameter: transaction expiry to prevent stale trades
- ETH wrapping: automatic WETH wrapping/unwrapping for native token trades
- Permit support: gasless token approvals via EIP-2612

### Security Architecture
- Reentrancy locks: critical on swap and liquidity functions
- Price manipulation resistance: TWAP oracle with sufficient observation window
- Minimum liquidity: permanent lock of initial LP tokens to prevent pool draining
- Access control: factory pattern with permissioned pool creation (or permissionless with safeguards)
- Circuit breakers: maximum single-block volume to detect and limit attacks

## OUTPUT FORMAT
Complete AMM contract suite with Pool, Factory, Router contracts, mathematical proofs, and integration guide.

## CONSTRAINTS
- Swap gas cost must be under 150K gas for simple token-to-token swaps
- All math must use fixed-point arithmetic with no precision loss exceeding 1 wei
- Must handle fee-on-transfer tokens and rebasing tokens gracefully
- Pool creation must be deterministic (CREATE2) for address prediction
- Include comprehensive integration tests with realistic trading scenarios