# FHEDEX: An Innovative AMM Architecture to Protect Liquidity Providers' Interests by Mitigating the Incentives for Front-running and MEV Attacks Without Undermining DeFi's Decentralization Goals

### Abstract
This whitepaper introduces **FHEDEX**, a novel AMM architecture that preserves the decentralization values of DEXs while economically mitigating the incentives for front-running and MEV (Miner Extractable Value) attacks, which typically exploit the high slippage inherent in AMM trades. FHEDEX effectively protects the interests of liquidity providers through a unique mechanism.

Inspired by biological systems, the core design of FHEDEX lies in the concept of market clusters, where markets capable of replicating themselves under certain conditions form clusters, and in the dual-layer architecture consisting of the Ethereum mainnet (foreground) where individual markets operate, and the L2 subnet (background) where liquidity exchange occurs among markets within a cluster.

To ensure that this **market cloning mechanism** remains a competitive edge and protect proprietary algorithms, the core function that mutates coefficients characterizing new market is kept confidential. This is achieved by encrypting the function using **CKKS homomorphic encryption scheme**, allowing computations on encrypted data while keeping the function hidden. The **market coefficients** themselves remain transparent to ensure system functionality and user trust.

Each market acts independently like a biological cell, similar to traditional DEXs, yet the market cluster also behaves as a single biological entity, sharing a unified liquidity pool through the background L2 subnet. In other words, individual participants trade within each market as they would in a standard DEX; however, exchange rates are determined not by individual market liquidity pools but by a consolidated liquidity pool. This design enables efficient liquidity utilization akin to centralized exchanges, addressing the high slippage issue often encountered in DEXs and thereby reducing the potential profit from order manipulation attempts.

The FHEDEX architecture achieves full decentralization using a combination of independent, stateless smart contracts without any centralized control or external monitoring. Inspired by the dynamics of osmosis and sodium ion channels, FHEDEX’s reward distribution model for liquidity providers (LPs) dynamically adjusts based on liquidity supply, encouraging external liquidity inflows. This approach leverages Ethereum's L2 solutions to create a robust, scalable, and secure ecosystem.

## 1. Introduction
### 1.1. Background
Decentralized Finance (DeFi) has revolutionized financial markets by enabling permissionless trading and liquidity provision through Automated Market Makers (AMMs). However, existing AMMs face several challenges, including:
- **High Slippage and Impermanent Loss**
- **Liquidity Fragmentation**
- **Front-Running and Miner Extractable Value (MEV) Attacks**
- **Scalability Issues**
- **Inflexible Fee Structures**
- **Inadequate Incentive Mechanisms for Liquidity Providers**

### 1.2. Market Design Inspired by Biological Mechanisms
Without a centralized overseer, FHEDEX borrows concepts from biological systems to model the dynamics of an adaptive, efficient, and self-sustaining market.

- **Cells**: Each market behaves independently like a biological cell, operating under intrinsic rules and aggressively absorbing liquidity.
- **Tissue**: Clusters of markets act as cohesive units, behaving as exclusive entities toward the external environment.
- **Cell Division**: Markets clone themselves dynamically based on liquidity balance, similar to cell division.
- **Genetic Material**: Market characteristics, such as market coefficients, mutate during replication, akin to genetic variation.
- **Osmosis**: Represents the motivational force for external liquidity providers to supply liquidity.
- **Ion Channels**: Represents the limited communication channels between inner-cluster markets and external markets.
- **Nutrition Equilibrium**: Reflects the liquidity equilibrium between markets, signaling the need for market replication within the cluster.

### 1.3. Design Objectives
- **Dual-Layer Architecture**: Unlike traditional DEXs, FHEDEX combines a surface layer of independent markets with a deeper layer connecting these markets via L2 subnets for seamless liquidity exchange.
- **Stateless Smart Contracts**: Utilizing fully independent, stateless smart contracts, FHEDEX fosters complex dynamics without centralized control.
- **Dynamic Market Cloning**: Markets replicate based on liquidity conditions without overarching cluster-level management. The market coefficient, crucial for replication, inherits and mutates within the L2 subnet.
- **Protection of Replication Mechanism**: FHEDEX safeguards its operation by obscuring the replication and mutation functions of market coefficients through CKKS-based homomorphic encryption, making it challenging to mimic the system using public smart contracts alone. High costs and restrictive cloning conditions to prevent potential attacks reconstructing the replication function is also introduced.
- **Unified Liquidity Pool**: By enabling liquidity sharing across all markets in a cluster via the L2 subnet, FHEDEX reduces slippage during individual market transactions, increasing transaction efficiency.
- **External Access Control**: Allows interactions with external liquidity sources on the Ethereum mainnet.
- **Ensured Full Decentralization**: Leverages Ethereum's L2 solutions to eliminate the need for centralized decision-making or monitoring entities.
- **Motivating External Providers**: Utilize osmotic pressure to model benefits and dynamically incentivize external liquidity providers.
- **Transparent Fee Distribution**: Adjust trading fee distribution to LPs based on osmotic pressure, integrating the incentive mechanism into the core protocol.
- **Implementing an L2 Solution with ZKPs**: Enable free and immediate internal liquidity transfers without transaction fees.
- **Developing a New Exchange Rate Mechanism**: Replace the CPMM with a Liquidity-Weighted Exchange Rate model.

## 2. Issues Facing Existing AMMs

### 2.1. High Slippage and Impermanent Loss
- **Slippage**: Significant price impact during large trades due to limited liquidity.
- **Impermanent Loss**: Liquidity providers suffer losses when asset prices change, discouraging participation.

### 2.2. Liquidity Fragmentation
- **Dispersed Liquidity**: Liquidity spread thin across multiple platforms reduces efficiency.
- **Inefficient Price Discovery**: Fragmentation leads to less accurate pricing and higher trading costs.

### 2.3. Front-Running and MEV Attacks
- **Vulnerabilities**: Transparent transactions allow malicious actors to exploit the system.
- **User Losses**: Such attacks erode user trust and lead to financial losses.

### 2.4. Scalability Issues
- **Network Congestion**: High demand leads to increased fees and slower transactions.
- **Limited Throughput**: Existing AMMs struggle with the growing volume of DeFi transactions.

### 2.5. Inflexible Fee Structures
- **Static Fees**: Do not adapt to market conditions, leading to suboptimal incentives.
- **Liquidity Provider Disincentives**: Fixed fees may not adequately compensate providers during volatile periods.

### 2.6. Inadequate Incentive Mechanisms for Liquidity Providers
- **Separate Reward Programs**: Traditional AMMs often rely on external or separate reward programs to incentivize LPs.
- **Lack of Integration**: Incentives are not always integrated into the core protocol, leading to inefficiencies.

## 3. Conceptual Framework

### 3.1. Market Cloning Mechanism

#### 3.1.1. Dynamics of Cloning
- **Natural Equilibrium Trigger**: Cloning occurs when liquidity movements among markets stabilize, indicating equilibrium.
- **Hidden Criteria**: The specific conditions triggering cloning are concealed to prevent manipulation.
- **Cluster Formation**: Cloned markets form a cluster, akin to tumor tissue growth, expanding the ecosystem organically.

#### 3.1.2. Biological Analogy
- **Tumor Growth**: Just as tumor cells divide when nutrients are sufficient, markets clone when liquidity equilibrium is reached.
- **Nutrient Equilibrium**: Liquidity is analogous to nutrients; equilibrium signals readiness for cloning.

### 3.2. Protected Coefficient Generation

#### 3.2.1. Confidential Function
- **Hidden Function**: The function that generates new market coefficients from existing ones is kept confidential.
- **CKKS Encryption**: Use CKKS homomorphic encryption scheme to encrypt the function, allowing computations on encrypted data.

#### 3.2.2. Transparent Coefficients
- **Transparency for Trust**: Market coefficients $\text{Coeff}^{\text{market}}$ are kept transparent to ensure system functionality and user trust.
- **Function Confidentiality**: Protecting the function prevents others from replicating or reverse-engineering proprietary algorithms.

### 3.3. Economic Countermeasures

#### 3.3.1. Expensive MarketFactory Contract
- **High Interaction Costs**: Making `MarketFactory` contract expensive deters attackers from repeatedly triggering cloning processes.
- **Economic Barrier**: Raises the cost of attempting to reconstruct the hidden function through data collection.

#### 3.3.2. Limiting Cloning Conditions
- **Restrictive Criteria**: Cloning conditions are tightly coupled with external liquidity equilibrium.
- **Reduced Cloning Frequency**: Limits the number of cloning events, reducing data available to potential attackers.

### 3.4. Greedy Market Behavior

#### 3.4.1. Markets as Cancer Cells
- **Aggressive Liquidity Acquisition**: Markets aim to maximize their own liquidity, promoting overall cluster growth.
- **Benefit Maximization**: Each market seeks to enhance its benefits, contributing to the cluster's strength.

### 3.5. Free Internal Liquidity Flow

#### 3.5.1. Unified Liquidity Pool
- **Inter-Market Transfer**: Liquidity flows freely within the cluster without restrictions or fees.
- **Lower Slippage**: Unified liquidity reduces price impact for trades within the cluster.

### 3.6. Controlled External Communication

#### 3.6.1. Ion Channel Analogy
- **Selective Permeability**: `PExternalProbe` regulates external liquidity flow, preventing destabilization.
- **Limited Access**: External communication is managed carefully to maintain cluster integrity.

### 3.7. Osmotic Pressure as Motivation for External Providers

#### 3.7.1. Modeling Benefits
- **Internal Liquidity Influence**: Markets use their liquidity levels to model attractive benefits for external providers.
- **Motivational Force**: Osmotic pressure incentivizes external providers to supply liquidity to the cluster.

#### 3.7.2. Fee Distribution Based on Osmotic Pressure
- **Integrated Rewards**: LPs are rewarded through trading fees collected in the tokens they provide.
- **Dynamic Incentives**: The proportion of fees distributed to LPs is adjusted based on osmotic pressure.
- **Smart Contract Implementation**: Use smart contracts to automate and transparently manage fee distribution.

### 3.8. Implementing an L2 Solution with ZKPs

#### 3.8.1. Simplified L2 Infrastructure
- **Purpose**: Facilitate free and immediate internal liquidity transfers without transaction fees.
- **Zero-Knowledge Proofs**: Ensure that liquidity transfers are valid and secure without revealing sensitive information.

## 4. Mathematical Modeling

### 4.1. Osmotic Pressure and Fee Distribution

#### 4.1.1. Osmotic Pressure Equation
Let:
- $L_{\text{desired}}$: Desired internal liquidity level.
- $L_{\text{actual}}$: Current internal liquidity level.
- $P_{\text{osmotic}}$: Osmotic pressure.
$$P_{\text{osmotic}} = \alpha \times \left( \frac{L_{\text{desired}} - L_{\text{actual}}}{L_{\text{desired}}} \right)$$
Where:
- $\alpha$: Osmotic coefficient.

**Fee Distribution Adjustment:**
- **Fee Share to LPs $f_{\text{LP}}$**: The proportion of trading fees distributed to LPs.
$$f_{\text{LP}} = f_{\text{base}} + \beta \times P_{\text{osmotic}}$$
Where:
- $f_{\text{base}}$: Base fee proportion allocated to LPs.
- $\beta$: Coefficient determining the impact of osmotic pressure on fee distribution.

#### 4.1.2. Dynamic Incentive Mechanism
- **Increased Rewards**: As osmotic pressure increases, indicating higher demand for liquidity, the fee share to LPs increases.
- **Equilibrium Adjustment**: As internal liquidity approaches $L_{\text{desired}}$, osmotic pressure decreases, balancing supply and demand.

### 4.2. Protected Coefficient Generation

#### 4.2.1. CKKS Encryption
- **Encrypting the Function**: The function $M$ that computes $\text{Coeff}^{\text{market}}_{i+1} = M(\text{Coeff}^{\text{market}}_{i})$ is encrypted using the CKKS scheme.
- **Homomorphic Computation**: Allows computations on encrypted data without revealing the function.

#### 4.2.2. Transparency of Coefficients
- **Coefficients Remain Public**: $\text{Coeff}^{\text{market}}$ are transparent for system functionality.
- **Function Confidentiality**: The transformation matrices used in $M$ are kept confidential.

### 4.3. Economic Countermeasures

#### 4.3.1. Increasing Cloning Costs
- **High Fees**: The MarketFactory contract requires a significant fee to initiate cloning.
- **Economic Barrier**: Discourages attackers from triggering cloning events for data collection.

#### 4.3.2. Limiting Cloning Conditions
- **Restrictive Criteria**: Cloning only occurs under specific external liquidity equilibrium conditions.
- **Reduced Data Availability**: Limits the number of cloning events, reducing the opportunity for attackers to gather data.

### 4.4. Cloning Criterion Based on Liquidity Equilibrium

#### 4.4.1. Equilibrium Detection
Let:
- $F_{i \rightarrow j}$: Liquidity flow from market $i$ to $j$.
- $F_{\text{net}}$: Net liquidity flow in the cluster.
$$F_{\text{net}} = \sum_{i,j} |F_{i \rightarrow j}|$$

**Cloning Condition:**
Cloning is triggered when $F_{\text{net}}$ stabilizes below a hidden threshold $\varepsilon$ and external liquidity equilibrium conditions are met.

### 4.5. Liquidity Flow Dynamics within the Cluster

#### 4.5.1. Free Flow Equation
- **Assumption**: No friction or fees impede liquidity movement within the cluster.
$$L_{\text{total}} = \sum_{i=1}^{N}L_i$$
Where:
- $L_i$: Liquidity in market $i$.
- $N$: Number of markets in the cluster.

### 4.6. Liquidity-Weighted Exchange Rate Mechanism

#### 4.6.1. Rationale for a New Exchange Mechanism
- **Limitations of CPMM**: The Constant Product Market Maker model ($x \cdot y = k$) is unsuitable due to the unified liquidity pool and dynamic market interactions.
- **Need for a New Model**: Exchange rates must account for total cluster liquidity and free internal liquidity flow.

#### 4.6.2. Liquidity Shares and Token Value
- **Liquidity Share of Token $i$ ($S_i$)**:
$$S_i = \frac{L_i}{L_{\text{total}}}$$
- **Intrinsic Value of Token $i$ ($V_i$)**:
$$V_i = \frac{1}{S_i}$$

#### 4.6.3. Exchange Rate Determination
- **Exchange Rate between Token $i$ and Token $j$**:
$$R_{i \rightarrow j} = \frac{V_j}{V_i} = \frac{S_i}{S_j}$$
- **Adjusted for Trade Size**:
$$\delta_i = \frac{\Delta Q_i}{L_i}$$ $$R_{\text{adjusted}} = R_{i \rightarrow j} \times (1 - \gamma \delta_i)$$
Where:
- $\gamma$: Sensitivity coefficient controlling the impact of trade size on the exchange rate.

### 4.7. Price Impact and Slippage Reduction
- **Lower Slippage**: Larger $L_{\text{total}}$ leads to lower price impact, reducing slippage for users.

### 4.8. Implementing Zero-Knowledge Proofs in L2

#### 4.8.1. Proof Generation and Verification
- **Off-Chain Execution**: Liquidity transfers and protected computations occur on the L2 network.
- **Proof Generation**: ZKPs are generated to prove valid state transitions and computations without revealing details.
- **On-Chain Verification**: Proofs are submitted to mainnet smart contracts for verification.

## 5. Smart Contract Architecture

### 5.1. Overview of Essential Smart Contracts
- **Governance**
- **MarketFactory**
- **Market**
- **MarketCoefficients**
- **ClusterLiquidityManager**
- **EquilibriumDetector**
- **PExternalProbe**
- **ExchangeRouter**
- **L2LiquidityTransfer**
- **MainnetVerifier**

### 5.2. Governance Contract
Manages decentralized decision-making, parameter adjustments, and voting mechanisms.

// SPDX-License-Identifier: MIT
pragma solidity ^0.8.0;

contract Governance {
    address public owner;
    mapping(address => bool) public proposers;
    uint256 public proposalCount;
    uint256 public votingPeriod = 3 days;

    struct Proposal {
        uint256 id;
        address proposer;
        bytes callData;
        uint256 startTime;
        uint256 yesVotes;
        uint256 noVotes;
        bool executed;
        mapping(address => bool) voters;
    }

    mapping(uint256 => Proposal) public proposals;

    event ProposalCreated(uint256 indexed id, address proposer);
    event Voted(uint256 indexed id, address voter, bool support);
    event ProposalExecuted(uint256 indexed id);

    modifier onlyOwner() {
        require(msg.sender == owner, "Not authorized");
        _;
    }

    modifier onlyProposer() {
        require(proposers[msg.sender], "Not a proposer");
        _;
    }

    constructor() {
        owner = msg.sender;
        proposers[msg.sender] = true;
    }

    function addProposer(address proposer) external onlyOwner {
        proposers[proposer] = true;
    }

    function removeProposer(address proposer) external onlyOwner {
        proposers[proposer] = false;
    }

    function createProposal(bytes memory callData) external onlyProposer returns (uint256) {
        proposalCount++;
        Proposal storage p = proposals[proposalCount];
        p.id = proposalCount;
        p.proposer = msg.sender;
        p.callData = callData;
        p.startTime = block.timestamp;

        emit ProposalCreated(proposalCount, msg.sender);
        return proposalCount;
    }

    function vote(uint256 proposalId, bool support) external {
        Proposal storage p = proposals[proposalId];
        require(block.timestamp <= p.startTime + votingPeriod, "Voting period over");
        require(!p.voters[msg.sender], "Already voted");

        p.voters[msg.sender] = true;
        if (support) {
            p.yesVotes++;
        } else {
            p.noVotes++;
        }

        emit Voted(proposalId, msg.sender, support);
    }

    function executeProposal(uint256 proposalId) external {
        Proposal storage p = proposals[proposalId];
        require(block.timestamp > p.startTime + votingPeriod, "Voting period not over");
        require(!p.executed, "Already executed");
        require(p.yesVotes > p.noVotes, "Proposal not approved");

        (bool success, ) = address(this).call(p.callData);
        require(success, "Execution failed");

        p.executed = true;
        emit ProposalExecuted(proposalId);
    }

    // Example function that can be called via governance
    uint256 public alpha; // Osmotic coefficient

    function setAlpha(uint256 _alpha) external {
        require(msg.sender == address(this), "Only callable via governance");
        alpha = _alpha;
    }
}

### 5.3. MarketCoefficients Contract
Acts as a container holding all coefficients for each market. This contract stores the transparent coefficients and interacts with the L2 solution to update coefficients during market cloning.

// SPDX-License-Identifier: MIT
pragma solidity ^0.8.0;

contract MarketCoefficients {
    // Mapping from market address to its coefficients
    mapping(address => Coefficients) public marketCoefficients;

    struct Coefficients {
        uint256[] values;
        uint256 timestamp;
    }

    event CoefficientsUpdated(address indexed market, uint256[] newCoefficients);

    function setCoefficients(address market, uint256[] memory newCoefficients) external {
        // Only authorized contracts (e.g., L2 verifier or MarketFactory) can call this function
        require(msg.sender == address(/* Authorized contract address */), "Not authorized");

        marketCoefficients[market] = Coefficients({
            values: newCoefficients,
            timestamp: block.timestamp
        });

        emit CoefficientsUpdated(market, newCoefficients);
    }

    function getCoefficients(address market) external view returns (uint256[] memory) {
        return marketCoefficients[market].values;
    }
}

### 5.4. MarketFactory Contract
Handles market creation (cloning) and maintains a registry of all markets in the cluster.

// SPDX-License-Identifier: MIT
pragma solidity ^0.8.0;

import "./Market.sol";
import "./MarketCoefficients.sol";

contract MarketFactory {
    address[] public markets;
    address public clusterManager;
    address public marketImplementation;
    uint256 public cloningFee; // High fee to deter attacks
    MarketCoefficients public marketCoefficients;
    address public l2Verifier; // Address of the L2 verifier contract

    event MarketCreated(address indexed market, address token);

    constructor(
        address _clusterManager,
        address _marketImplementation,
        uint256 _cloningFee,
        address _marketCoefficients,
        address _l2Verifier
    ) {
        clusterManager = _clusterManager;
        marketImplementation = _marketImplementation;
        cloningFee = _cloningFee;
        marketCoefficients = MarketCoefficients(_marketCoefficients);
        l2Verifier = _l2Verifier;
    }

    function createMarket(address token) external payable returns (address) {
        require(msg.value >= cloningFee, "Insufficient cloning fee");
        // Cloning conditions based on external liquidity equilibrium
        require(checkCloningConditions(), "Cloning conditions not met");

        // Obtain new coefficients from L2
        uint256[] memory newCoefficients = getNewCoefficientsFromL2();

        address clone = createClone(marketImplementation);
        Market(clone).initialize(token, clusterManager);

        // Set the coefficients for the new market
        marketCoefficients.setCoefficients(clone, newCoefficients);

        markets.push(clone);
        emit MarketCreated(clone, token);
        return clone;
    }

    function createClone(address target) internal returns (address result) {
        // EIP-1167 Minimal Proxy implementation
        bytes20 targetBytes = bytes20(target);
        assembly {
            let clone := mload(0x40)
            mstore(
                clone,
                0x3d602d80600a3d3981f3d82803e903d91602b57fd5bf3
            )
            mstore(
                add(clone, 0x14),
                targetBytes
            )
            mstore(
                add(clone, 0x28),
                0x5af43d82803e903d91602b57fd5bf3ff
            )
            result := create(0, clone, 0x37)
        }
    }

    function checkCloningConditions() internal view returns (bool) {
        // Implement logic to check external liquidity equilibrium conditions
        // For simplicity, returning true
        return true;
    }

    function getNewCoefficientsFromL2() internal returns (uint256[] memory) {
        // Interact with L2 verifier to get new coefficients
        bytes memory proof = requestCoefficientsProof();

        // Verify proof and obtain new coefficients
        uint256[] memory newCoefficients = IL2Verifier(l2Verifier).verifyAndGetCoefficients(proof);

        return newCoefficients;
    }

    function requestCoefficientsProof() internal returns (bytes memory) {
        // Implement the logic to request a proof from L2
        // For simplicity, returning an empty byte array
        return new bytes(0);
    }

    function getMarkets() external view returns (address[] memory) {
        return markets;
    }
}

interface IL2Verifier {
    function verifyAndGetCoefficients(bytes calldata proof) external returns (uint256[] memory);
}

### 5.5. Market Contract
Represents individual markets within the cluster, managing trades and liquidity.

// SPDX-License-Identifier: MIT
pragma solidity ^0.8.0;

import "./ClusterLiquidityManager.sol";
import "@openzeppelin/contracts/token/ERC20/IERC20.sol";

contract Market {
    address public token;
    ClusterLiquidityManager public clusterManager;
    uint256 public totalLiquidity;
    mapping(address => uint256) public lpBalances;
    address[] public lpAddresses;

    bool initialized;

    uint256 public baseFee = 30; // 0.30% in basis points (bps)

    event LiquidityProvided(address indexed lp, uint256 amount);
    event LiquidityRemoved(address indexed lp, uint256 amount);
    event TradeExecuted(address indexed trader, uint256 amountIn, uint256 amountOut);

    function initialize(address _token, address _clusterManager) external {
        require(!initialized, "Already initialized");
        token = _token;
        clusterManager = ClusterLiquidityManager(_clusterManager);
        initialized = true;
    }

    function getCoefficients() public view returns (uint256[] memory) {
        return clusterManager.marketCoefficients().getCoefficients(address(this));
    }

    function provideLiquidity(uint256 amount) external {
        // Transfer tokens from LP to Market
        require(IERC20(token).transferFrom(msg.sender, address(this), amount), "Transfer failed");
        if (lpBalances[msg.sender] == 0) {
            lpAddresses.push(msg.sender);
        }
        lpBalances[msg.sender] += amount;
        totalLiquidity += amount;
        clusterManager.updateLiquidity(address(this), token, int256(amount));

        emit LiquidityProvided(msg.sender, amount);
    }

    function removeLiquidity(uint256 amount) external {
        require(lpBalances[msg.sender] >= amount, "Insufficient balance");
        lpBalances[msg.sender] -= amount;
        totalLiquidity -= amount;
        clusterManager.updateLiquidity(address(this), token, -int256(amount));

        // Transfer tokens from Market to LP
        require(IERC20(token).transfer(msg.sender, amount), "Transfer failed");

        emit LiquidityRemoved(msg.sender, amount);
    }

    function executeTrade(address tokenIn, uint256 amountIn) external {
        // Calculate exchange rate and amountOut using LWER mechanism
        uint256 amountOut = clusterManager.getAmountOut(tokenIn, token, amountIn);

        // Collect trading fee
        uint256 fee = (amountIn * baseFee) / 10000; // baseFee in bps
        uint256 feeToLPs = (fee * getFeeShareToLPs()) / 10000;
        uint256 feeToProtocol = fee - feeToLPs;

        // Update balances
        // Transfer amountIn - fee from trader to Market
        require(IERC20(tokenIn).transferFrom(msg.sender, address(this), amountIn - fee), "Transfer failed");
        // Transfer feeToProtocol to protocol address
        require(IERC20(tokenIn).transferFrom(msg.sender, clusterManager.protocolAddress(), feeToProtocol), "Transfer failed");
        // Distribute feeToLPs among LPs
        distributeFeesToLPs(tokenIn, feeToLPs);

        // Transfer amountOut to trader
        require(IERC20(token).transfer(msg.sender, amountOut), "Transfer failed");

        emit TradeExecuted(msg.sender, amountIn, amountOut);
    }

    function getFeeShareToLPs() public view returns (uint256) {
        uint256 osmoticPressure = clusterManager.calculateOsmoticPressure();
        uint256 feeShare = clusterManager.baseFeeShare() + (clusterManager.beta() * osmoticPressure) / 1e18;
        return feeShare; // In basis points (bps)
    }

    function distributeFeesToLPs(address tokenIn, uint256 feeAmount) internal {
        uint256 totalLPBalance = totalLiquidity;
        for (uint256 i = 0; i < lpAddresses.length; i++) {
            address lp = lpAddresses[i];
            uint256 lpShare = (lpBalances[lp] * feeAmount) / totalLPBalance;
            // Transfer fee share to LP
            require(IERC20(tokenIn).transfer(lp, lpShare), "Fee transfer failed");
        }
    }
}

### 5.6. ClusterLiquidityManager Contract
Manages free liquidity flow within the cluster, including immediate liquidity transfers, and calculates osmotic pressure.

// SPDX-License-Identifier: MIT
pragma solidity ^0.8.0;

contract ClusterLiquidityManager {
    mapping(address => mapping(address => uint256)) public liquidityBalances; // market => token => balance
    uint256 public desiredLiquidity;
    uint256 public alpha; // Osmotic coefficient
    uint256 public beta; // Fee adjustment coefficient
    uint256 public baseFeeShare = 2500; // 25% of fees go to LPs as base

    address public protocolAddress;

    MarketCoefficients public marketCoefficients;

    constructor(
        uint256 _desiredLiquidity,
        uint256 _alpha,
        uint256 _beta,
        address _protocolAddress,
        address _marketCoefficients
    ) {
        // Existing initialization
        marketCoefficients = MarketCoefficients(_marketCoefficients);
    }

    function updateLiquidity(address market, address token, int256 amount) external {
        if (amount >= 0) {
            liquidityBalances[market][token] += uint256(amount);
        } else {
            liquidityBalances[market][token] -= uint256(-amount);
        }
    }

    function calculateOsmoticPressure() public view returns (uint256) {
        uint256 actualLiquidity = getTotalLiquidity();
        if (actualLiquidity >= desiredLiquidity) {
            return 0;
        }
        uint256 osmoticPressure = alpha * ((desiredLiquidity - actualLiquidity) * 1e18 / desiredLiquidity);
        return osmoticPressure;
    }

    function getTotalLiquidity() public view returns (uint256) {
        // Sum total liquidity across all markets and tokens
        uint256 total = 0;
        // Implement logic to sum up liquidityBalances
        // For simplicity, returning a placeholder value
        return total;
    }

    function getAmountOut(address tokenIn, address tokenOut, uint256 amountIn) external view returns (uint256) {
        // Implement LWER mechanism
        // For simplicity, return a dummy value
        return amountIn; // Placeholder
    }
}

### 5.7. EquilibriumDetector Contract
Monitors liquidity movements and triggers cloning when equilibrium is detected.

// SPDX-License-Identifier: MIT
pragma solidity ^0.8.0;

import "./MarketFactory.sol";

contract EquilibriumDetector {
    MarketFactory public marketFactory;
    uint256 public equilibriumThreshold; // Hidden threshold epsilon

    event EquilibriumDetected();
    event MarketCloned(address newMarket);

    constructor(address _marketFactory, uint256 _equilibriumThreshold) {
        marketFactory = MarketFactory(_marketFactory);
        equilibriumThreshold = _equilibriumThreshold;
    }

    function checkEquilibrium() external {
        uint256 netLiquidityFlow = calculateNetLiquidityFlow();
        if (netLiquidityFlow < equilibriumThreshold && checkExternalLiquidityEquilibrium()) {
            emit EquilibriumDetected();
            cloneMarket();
        }
    }

    function calculateNetLiquidityFlow() internal view returns (uint256) {
        // Implement logic to calculate net liquidity flow F_net
        // For simplicity, returning a placeholder value
        return 0;
    }

    function checkExternalLiquidityEquilibrium() internal view returns (bool) {
        // Implement logic to check external liquidity equilibrium
        // For simplicity, returning true
        return true;
    }

    function cloneMarket() internal {
        // Clone a new market
        address newMarket = marketFactory.createMarket{value: marketFactory.cloningFee()}(address(0)); // Specify the token address
        emit MarketCloned(newMarket);
    }
}

### 5.8. PExternalProbe Contract
Controls controlled external communication and liquidity flow, acting as the ion channel.

// SPDX-License-Identifier: MIT
pragma solidity ^0.8.0;

contract PExternalProbe {
    address public clusterManager;
    uint256 public totalLiquidity;
    mapping(address => uint256) public lpBalances;

    event LiquidityProvided(address indexed lp, uint256 amount);
    event LiquidityRemoved(address indexed lp, uint256 amount);

    constructor(address _clusterManager) {
        clusterManager = _clusterManager;
    }

    function provideLiquidity(uint256 amount) external {
        // Transfer tokens from LP to this contract
        // For simplicity, assuming ETH is used
        require(msg.value == amount, "Incorrect amount sent");
        lpBalances[msg.sender] += amount;
        totalLiquidity += amount;

        // Update cluster manager
        // clusterManager.updateExternalLiquidity(amount);

        emit LiquidityProvided(msg.sender, amount);
    }

    function removeLiquidity(uint256 amount) external {
        require(lpBalances[msg.sender] >= amount, "Insufficient balance");
        lpBalances[msg.sender] -= amount;
        totalLiquidity -= amount;

        // Transfer ETH back to LP
        payable(msg.sender).transfer(amount);

        // Update cluster manager
        // clusterManager.updateExternalLiquidity(-int256(amount));

        emit LiquidityRemoved(msg.sender, amount);
    }

    function getTotalLiquidity() external view returns (uint256) {
        return totalLiquidity;
    }
}

### 5.9. ExchangeRouter Contract
Facilitates user trades considering total cluster liquidity.

// SPDX-License-Identifier: MIT
pragma solidity ^0.8.0;

import "./ClusterLiquidityManager.sol";
import "./Market.sol";

contract ExchangeRouter {
    ClusterLiquidityManager public clusterManager;

    constructor(address _clusterManager) {
        clusterManager = ClusterLiquidityManager(_clusterManager);
    }

    function swap(address tokenIn, address tokenOut, uint256 amountIn) external {
        // Determine the best market for the trade
        address bestMarket = findBestMarket(tokenIn, tokenOut);

        // Execute trade on the selected market
        Market(bestMarket).executeTrade(tokenIn, amountIn);
    }

    function findBestMarket(address tokenIn, address tokenOut) internal view returns (address) {
        // Implement logic to find the best market based on liquidity, rates, etc.
        // For simplicity, returning a placeholder address
        return address(0);
    }
}

### 5.10. L2LiquidityTransfer Contract
Manages off-chain liquidity transfers and ZKP generation.

// SPDX-License-Identifier: MIT
pragma solidity ^0.8.0;

contract L2LiquidityTransfer {
    // Off-chain state management and computation
    mapping(address => mapping(address => uint256)) public offChainBalances; // market => token => balance

    event CoefficientsComputed(address indexed market, uint256[] coefficients, bytes proof);

    function computeNewCoefficients(address market, uint256[] memory currentCoefficients) external {
        // Perform off-chain encrypted computation using CKKS
        uint256[] memory newCoefficients = performCKKSComputation(currentCoefficients);

        // Generate ZKP (simplified representation)
        bytes memory proof = generateZKP(market, newCoefficients);

        // Emit event with proof for mainnet submission
        emit CoefficientsComputed(market, newCoefficients, proof);
    }

    function performCKKSComputation(uint256[] memory currentCoefficients) internal returns (uint256[] memory) {
        // Implement CKKS encrypted computation
        // For simplicity, returning dummy values
        uint256[] memory newCoefficients = new uint256[](currentCoefficients.length);
        for (uint256 i = 0; i < currentCoefficients.length; i++) {
            newCoefficients[i] = currentCoefficients[i] + 1; // Placeholder operation
        }
        return newCoefficients;
    }

    function generateZKP(address market, uint256[] memory newCoefficients) internal pure returns (bytes memory) {
        // ZKP generation logic
        // In practice, use a ZKP library or protocol
        return abi.encodePacked(market, newCoefficients);
    }
}

### 5.11. MainnetVerifier Contract
Verifies ZKPs and updates on-chain liquidity states.

// SPDX-License-Identifier: MIT
pragma solidity ^0.8.0;

import "./MarketCoefficients.sol";

contract MainnetVerifier {
    MarketCoefficients public marketCoefficients;

    event CoefficientsVerified(address indexed market, uint256[] coefficients);

    constructor(address _marketCoefficients) {
        marketCoefficients = MarketCoefficients(_marketCoefficients);
    }

    function verifyAndGetCoefficients(bytes calldata proof) external returns (uint256[] memory) {
        // Simplified proof verification
        (address market, uint256[] memory newCoefficients) = abi.decode(proof, (address, uint256[]));

        // Verify the proof (in practice, use actual ZKP verification)
        require(isValidProof(proof), "Invalid proof");

        // Update the coefficients in the MarketCoefficients contract
        marketCoefficients.setCoefficients(market, newCoefficients);

        emit CoefficientsVerified(market, newCoefficients);

        return newCoefficients;
    }

    function isValidProof(bytes memory proof) internal pure returns (bool) {
        // ZKP verification logic
        return true; // Placeholder for actual verification
    }
}

### 5.7. L2 Transaction Process for Getting New Coefficients
**Process Overview**:
- **Trigger Cloning**: When cloning conditions are met, the MarketFactory contract initiates the cloning process.
- **Request Coefficients from L2**: The MarketFactory contract requests new coefficients by interacting with the L2 system.
- **CKKS Computation in L2**: The L2 system performs the encrypted computation using CKKS to generate new coefficients.
- **Proof Generation**: A zero-knowledge proof (ZKP) is generated to prove the correctness of the computation without revealing the function.
- **Verification on Mainnet**: The MainnetVerifier contract verifies the proof and obtains the new coefficients.
- **Update Coefficients**: The MarketCoefficients contract is updated with the new coefficients.
- **Finalize Cloning**: The MarketFactory completes the cloning process, and the new market is initialized with the updated coefficients.

**Sequence Diagram**:

MarketFactory -> L2LiquidityTransfer: Request new coefficients
L2LiquidityTransfer: Perform CKKS computation
L2LiquidityTransfer: Generate ZKP
L2LiquidityTransfer -> MarketFactory: Provide proof
MarketFactory -> MainnetVerifier: Verify proof and get coefficients
MainnetVerifier -> MarketCoefficients: Update coefficients
MarketFactory: Complete cloning with new coefficients

### 5.8. Security and Confidentiality Considerations
- **CKKS Encryption**: The function generating new coefficients is computed using CKKS encryption in the L2 environment.
- **Zero-Knowledge Proofs**: ZKPs are used to prove the correctness of the new coefficients without revealing the proprietary function.
- **Economic Countermeasures**: High fees and restrictive cloning conditions deter attackers from attempting to reverse-engineer the function.
- **Secure L2 Environment**: The L2 environment is secured to prevent unauthorized access to the encrypted function and computations.

## 6. System Dynamics

### 6.1. Market Cloning Dynamics

#### 6.1.1. Ignition for Cloning
- **Liquidity Equilibrium**: Cloning is triggered when liquidity movements stabilize, indicating equilibrium, and external liquidity equilibrium conditions are met.
- **High Cloning Costs**: The expense associated with cloning acts as a deterrent to unauthorized or malicious cloning attempts.

#### 6.1.2. Formation of Cluster
- **Tumor Tissue Analogy**: Cloned markets form a cluster, expanding the ecosystem like tumor cells forming tissue.
- **Organic Growth**: The cluster grows naturally in response to internal dynamics.

### 6.2. Protected Coefficient Generation
- **Confidential Function**: The function generating new market coefficients is encrypted using CKKS and computed within L2.
- **Transparent Coefficients**: Updated coefficients are made transparent to maintain system functionality and user trust.

### 6.3. Economic Countermeasures
- **Deterrence Through Cost**: High fees associated with the MarketFactory contract deter attackers from triggering cloning events to collect data.
- **Restrictive Conditions**: Limiting cloning to specific external liquidity equilibrium conditions reduces the frequency of cloning events.

### 6.4. Greedy Behavior and Liquidity Attraction
- **Benefit Modeling**: Markets model benefits to attract external liquidity, behaving greedily.
- **Cluster Strengthening**: Increased liquidity enhances the cluster's overall capacity.

### 6.5. Free and Immediate Internal Liquidity Transfers
- **L2 Solution with ZKPs**: Enables free and immediate liquidity migration between markets without transaction fees.
- **Unified Liquidity**: Free movement allows the cluster to act as a single liquidity source.
- **Lower Slippage**: Users benefit from reduced price impact due to higher available liquidity.

### 6.6. Controlled External Access
- **Ion Channel Function**: `PExternalProbe` regulates external liquidity inflows.
- **Stability Maintenance**: Controlled access prevents external volatility from affecting the cluster.

### 6.7. Fee Distribution Based on Osmotic Pressure
- **Dynamic Fee Adjustments**: LP rewards are adjusted based on osmotic pressure, incentivizing liquidity provision when needed.
- **Integrated Incentive Mechanism**: Rewards are managed within the core protocol, ensuring transparency and automation.
- **Attracting Liquidity**: As the cluster requires more liquidity, osmotic pressure increases, leading to higher rewards for LPs.

## 7. Addressing Challenges in Existing AMMs

### 7.1. High Slippage and Impermanent Loss
- **Solution**: Unified liquidity and the LWER mechanism reduce slippage.
- **Dynamic Liquidity Distribution**: Free internal liquidity flow mitigates impermanent loss by optimizing liquidity allocation.

### 7.2. Liquidity Fragmentation
- **Solution**: The cluster consolidates liquidity, reducing fragmentation.
- **Market Cloning**: Expands the cluster organically, enhancing overall liquidity depth.

### 7.3. Front-Running and MEV Attacks
- **Solution**: Controlled external access and internal liquidity transfers reduce exposure to external manipulations.
- **L2 Transactions**: Off-chain processing minimizes the risk of front-running.

### 7.4. Scalability Issues
- **Solution**: The L2 solution increases transaction throughput and reduces congestion.
- **Efficient Resource Utilization**: Cloning and free liquidity flow optimize scalability.

### 7.5. Inflexible Fee Structures
- **Solution**: Adjust fee distribution dynamically based on osmotic pressure.
- **Adaptive Incentives**: Align LP rewards with market conditions, incentivizing liquidity provision when needed.

### 7.6. Inadequate Incentive Mechanisms for Liquidity Providers
- **Solution**: Integrate explicit incentive mechanisms based on osmotic pressure directly into the protocol.
- **Dynamic Rewards**: LPs receive rewards that adjust in real-time, ensuring fair compensation and encouraging liquidity provision.
- **Transparency**: Smart contracts manage incentives transparently, building trust with LPs.

## 8. Security Considerations

### 8.1. Protecting Proprietary Functions
- **CKKS Encryption**: The function generating new market coefficients is encrypted, keeping it confidential.
- **Homomorphic Computation**: Allows computations on encrypted data without exposing the function.

### 8.2. Mitigating Reverse Engineering Attempts
- **Economic Countermeasures**: High cloning costs and restrictive conditions deter attackers from gathering data.
- **Limited Data Exposure**: Reducing the frequency of cloning events limits the data available to potential attackers.
- **Monitoring and Detection**: Implementing systems to detect unusual cloning activity or patterns.

### 8.3. Balancing Security and Usability
- **Reasonable Fees**: While fees are high to deter attackers, they are set to avoid prohibitive costs for legitimate users.
- **Transparency in Costs**: Users are informed about the costs associated with cloning.

### 8.4. Additional Security Measures
- **Zero-Knowledge Proofs**: Used to verify computations without revealing sensitive information.
- **Secure L2 Environment**: Protects against side-channel attacks and ensures the encryption keys remain secure.
- **Adaptive Strategies**: Preparedness to adjust functions or introduce additional security measures if needed.

## 9. Simulation of the AMM System

### 9.1. Simulation Objectives
- **Test System Resilience**: Evaluate performance under various market conditions.
- **Identify Potential Issues**: Uncover vulnerabilities or inefficiencies.
- **Optimize Parameters**: Fine-tune system parameters for optimal performance.
- **Demonstrate Viability**: Provide evidence of the system's robustness.

### 9.2. Simulation Approach
- **Mathematical Modeling**: Translate mechanisms into equations.
- **Agent-Based Modeling**: Represent participants like traders, liquidity providers, arbitrageurs.
- **Competitive Analysis**: Include existing AMMs for comparison.
- **Stress Testing**: Simulate extreme market conditions.

### 9.3. Tools and Platforms
- **CADCAD**: For complex adaptive systems modeling.
- **Python Libraries**: Such as SimPy or Mesa for simulations.
- **Blockchain Simulators**: Ganache, Truffle Suite, or Hardhat for smart contract testing.

### 9.4. Analysis and Refinement
- **Performance Metrics**: Measure slippage, liquidity utilization, fee generation.
- **Iterative Refinement**: Adjust parameters based on simulation outcomes.
- **Visualization**: Use graphs and charts to interpret results.

### 9.5. Simulation Scenarios
- **Real-World Competition**: Simulate the AMM's performance against existing AMMs over time.
- **Cloning Dynamics**: Assess how market cloning affects liquidity distribution and system stability.
- **User Behavior**: Model how traders and liquidity providers interact with the system under different conditions.
- **Incentive Mechanisms**: Test how the osmotic pressure-based incentives impact LP participation and overall liquidity.

## 10. Governance Framework

### 10.1. Decentralized Decision-Making
- **Community Involvement**: Stakeholders participate in governance.
- **Adaptive Policies**: Parameters adjust based on consensus and market conditions.

### 10.2. Transparency vs. Confidentiality
- **Balanced Approach**: Keep cloning criteria and proprietary functions hidden while maintaining overall transparency.
- **Assurance Mechanisms**: Use audits and cryptographic proofs to build trust.

### 10.3. Incentive Parameter Adjustment
- **Governance Over Osmotic Parameters**: Community can vote to adjust $\alpha$ and $\beta$ coefficients.
- **Adaptive Incentives**: Ensures that rewards remain aligned with market conditions and LP expectations.

## 11. Deployment Considerations

### 11.1. Layer 2 Solution Implementation
- **L2 Network Setup**: Configure the L2 network focused on liquidity transfers and protected computations.
- **Integration with Mainnet**: Ensure seamless interaction with Ethereum mainnet contracts.
- **Security Measures**: Implement robust security practices for CKKS encryption and ZKP generation and verification.

### 11.2. Ensuring Network Support
- **Node Participation**: Encourage nodes to support the L2 network through incentives other than transaction fees.
- **Simplifying L2 Operations**: Keep the L2 network operations minimalistic to reduce maintenance overhead.

### 11.3. Additional Modules
- **Oracles**: Use decentralized oracles for reliable market data.
- **Monitoring Tools**: Deploy systems to monitor liquidity flows and detect anomalies.
- **Security Audits**: Conduct thorough audits to ensure robustness.

## 12. Conclusion
By integrating biological analogies into market design and implementing a protected coefficient generation mechanism using CKKS scheme, we present a decentralized market cluster that dynamically clones markets based on liquidity equilibrium. This approach addresses key challenges faced by existing AMMs, such as high slippage, liquidity fragmentation, and scalability issues. The architecture promotes a self-sustaining, adaptive ecosystem where markets behave greedily to maximize liquidity, and cloning occurs naturally when conditions are optimal.

Economic countermeasures, such as making `MarketFactory` contract expensive and limiting cloning conditions, protect the proprietary function from potential attacks. Controlled external communication ensures stability, while dynamically adjusted fee distribution based on osmotic pressure motivates external liquidity providers.

By addressing the challenges of zero-fee internal transactions and developing a new exchange rate mechanism, we ensure that the system operates efficiently within the existing blockchain ecosystem. Including all logically required smart contracts ensures a comprehensive framework for implementation. This design balances internal efficiency with controlled external interactions, creating a robust and innovative market ecosystem.

## Appendix

### A. Mathematical Formulas

#### A.1. Osmotic Pressure and Fee Distribution
$$P_{\text{osmotic}} = \alpha \times \left( \frac{L_{\text{desired}} - L_{\text{actual}}}{L_{\text{desired}}} \right)$$
$$f_{\text{LP}} = f_{\text{base}} + \beta \times P_{\text{osmotic}}$$

#### A.2. Protected Coefficient Generation
- **Function $M$**: Confidential function generating $\text{Coeff}^\text{market}_{i+1}$ from $\text{Coeff}^\text{market}_{i}$ using CKKS scheme.

#### A.3. Cloning Criterion
$$F_{\text{net}} = \sum_{i,j} |F_{i \rightarrow j}|$$
Clone when $F_{\text{net}} < \varepsilon$ and external equilibrium conditions are met.

#### A.4. Liquidity-Weighted Exchange Rate
$$R_{i \rightarrow j} = \frac{S_i}{S_j}$$

#### A.5. Adjusted Exchange Rate
$$R_{\text{adjusted}} = R_{i \rightarrow j} \times (1 - \gamma \delta_i)$$

### B. Smart Contract Interfaces
#### B.1. MarketFactory Interface

interface IMarketFactory {
    function createMarket() external returns (address);
    function markets() external view returns (address[] memory);
}

#### B.2. Market Interface

interface IMarket {
    function provideLiquidity(uint256 amount) external;
    function removeLiquidity(uint256 amount) external;
    function executeTrade(address tokenIn, uint256 amountIn) external;
}

#### B.3. EquilibriumDetector Interface

interface IEquilibriumDetector {
    function recordLiquidityMovement(address marketFrom, address marketTo, uint256 amount) external;
}

#### B.4. ClusterLiquidityManager Interface

interface IClusterLiquidityManager {
    function updateLiquidity(address market, address token, int256 amount) external;
    function calculateOsmoticPressure() external view returns (uint256);
    function getAmountOut(address tokenIn, address tokenOut, uint256 amountIn) external view returns (uint256);
}

#### B.5. PExternalProbe Interface

interface IPExternalProbe {
    function provideLiquidity(uint256 amount) external;
    function removeLiquidity(uint256 amount) external;
    function getTotalLiquidity() external view returns (uint256);
}

#### B.6. MarketCoefficients Interface

interface IMarketCoefficients {
    function setCoefficients(address market, uint256[] memory newCoefficients) external;
    function getCoefficients(address market) external view returns (uint256[] memory);
}

#### B.7. IL2Verifier Interface

interface IL2Verifier {
    function verifyAndGetCoefficients(bytes calldata proof) external returns (uint256[] memory);
}

#### B.8. Governance Interface

interface IGovernance {
    function propose(bytes calldata data) external;
    function vote(uint256 proposalId) external;
    function execute(uint256 proposalId) external;
}

### C. Deployment Steps
- **Set Up L2 Infrastructure**: Configure the L2 network and ensure compatibility with mainnet contracts.
- **Deploy Smart Contracts**: Deploy contracts in the correct order, starting with Governance, ClusterLiquidityManager, MarketFactory, and so on.
- **Initialize Contracts**: Set initial parameters and link contracts appropriately.
- **Implement Security Measures**: Ensure CKKS encryption keys and L2 environment are secured.
- **Testing**: Run unit tests and integration tests to ensure all components function correctly.
- **Simulation**: Simulate the system's behavior under various scenarios to refine the design.
- **Security Audits**: Engage with security experts to audit the smart contracts and L2 implementation.
- **Launch**: Deploy the system to the mainnet and L2 network.