Introduction: The Multi-Chain Fragmentation Problem

Cross-Chain Bridge Taxonomy:
┌─────────────────────────────────────────────────────────────────────┐
|  Bridge Type                │  Security Model          │  Risk    |
|─────────────────────────────────────────────────────────────────────|
|  Trusted / Federated        │  Multi-sig validator    │  High    |
|  (e.g., Binance Bridge)     │  federation             │  (Key    |
|                             │                         │  Compromise)|
|─────────────────────────────────────────────────────────────────────|
|  Trustless /               │  Light client proofs,   │  Low     |
|  Cryptographically Verified │  ZK-SNARKs/STARKs      │  (Math)  |
|  (e.g., IBC, ZK Bridges)   │                         │          |
|─────────────────────────────────────────────────────────────────────|
|  Optimistic / Fraud-       │  Fraud proofs with      │  Medium  |
|  Proving (e.g., Arbitrum)  │  challenge periods      │  (Economic)|
└─────────────────────────────────────────────────────────────────────┘

Lock-and-Mint Bridge Workflow:
┌─────────────────────────────────────────────────────────────────────┐
|  Chain A (Source)                     Chain B (Destination)        |
|                                                                  |
|  User deposits 100 ETH               |                           |
|  ↓                                  |  ✓ 100 wETH minted         |
|  ETH locked in bridge vault         |  ↓                         |
|  Bridge validates transaction       |  User receives wETH        |
|                                                                  |
|  To bridge back:                                                |
|  User burns 100 wETH                |                           |
|  ↓                                  |  ✓ ETH unlocked from vault |
|  Bridge verifies burn               |  ↓                         |
|                                     |  User receives native ETH  |
└─────────────────────────────────────────────────────────────────────┘

Total Supply = Σ Supply_per_chain
ΔSupply_ChainA = -Amount (Burned)
ΔSupply_ChainB = +Amount (Minted)
Total ΔSupply = 0

Cross-Chain Arbitrage Flow:
┌─────────────────────────────────────────────────────────────────────┐
|  1. Monitor prices across chains in real-time (sub-second).       |
|                                                                  |
|  2. Detect inefficiency: ETH price on Chain A = $3,000          |
|                        ETH price on Chain B = $3,030 (1% spread)  |
|                                                                  |
|  3. Buy ETH on Chain A using USDC.                               |
|  4. Bridge ETH from Chain A to Chain B (latency + gas).          |
|  5. Sell ETH on Chain B at higher price.                         |
|  6. Bridge USDC back to Chain A (if maintaining neutral position).|
|  7. Capture net profit after costs.                              |
└─────────────────────────────────────────────────────────────────────┘

Π = Q × (P_B - P_A) - B - G_A - G_B - S_A - S_B

Where:
- Q = Quantity traded.
- P_B = Price on destination chain (higher).
- P_A = Price on source chain (lower).
- B = Bridge transfer fee (fixed or percentage).
- G_A = Gas fees on Chain A.
- G_B = Gas fees on Chain B.
- S_A = Slippage from purchase on Chain A.
- S_B = Slippage from sale on Chain B.

The trade is only profitable if Π > 0.

Intent-Based Settlement Flow:
┌─────────────────────────────────────────────────────────────────────┐
|  User Intent:                                                     |
|  "I will deposit 10 ETH on Arbitrum and receive 25,000 USDC      |
|   on Optimism within 30 seconds."                                |
|                              ▼                                    |
|  Solvers (Competitive Bidding):                                  |
|  ┌─────────────────────────────────────────────────────────────┐   |
|  │  Solver A: "I will fill at 24,950 USDC."                  │   |
|  │  Solver B: "I will fill at 24,980 USDC."                  │   |
|  │  Solver C: "I will fill at 25,000 USDC." (Winner)        │   |
|  └─────────────────────────────────────────────────────────────┘   |
|                              ▼                                    |
|  Execution:                                                       |
|  ┌─────────────────────────────────────────────────────────────┐   |
|  │  Solver C uses own capital to immediately transfer 25,000  │   |
|  │  USDC to user on Optimism, then settles the 10 ETH          │   |
|  │  received later across chains via atomic settlement.        │   |
|  └─────────────────────────────────────────────────────────────┘   |
|                              ▼                                    |
|  User receives 25,000 USDC on Optimism in < 30 seconds.          |
└─────────────────────────────────────────────────────────────────────┘

DAO Governance Lifecycle:
┌─────────────────────────────────────────────────────────────────────┐
|  1. Proposal Submission:                                          |
|  ┌─────────────────────────────────────────────────────────────┐   |
|  │  Community member submits Improvement Proposal (e.g.,      │   |
|  │  adjust lending protocol's collateral risk parameters).    │   |
|  └─────────────────────────────────────────────────────────────┘   |
|                              ▼                                    |
|  2. Proposal Discussion Period:                                  |
|  ┌─────────────────────────────────────────────────────────────┐   |
|  │  Community debates the proposal on forums (e.g.,           │   |
|  │  governance.uniswap.org) and snapshot voting platforms.    │   |
|  └─────────────────────────────────────────────────────────────┘   |
|                              ▼                                    |
|  3. On-Chain Voting Window:                                    |
|  ┌─────────────────────────────────────────────────────────────┐   |
|  │  Token holders vote (yes/no/abstain).                      │   |
|  │  Voting power = Token balance at block snapshot height.    │   |
|  └─────────────────────────────────────────────────────────────┘   |
|                              ▼                                    |
|  4. Quorum & Majority Check:                                    |
|  ┌─────────────────────────────────────────────────────────────┐   |
|  │  If quorum (>4% of circulating supply) and majority        │   |
|  │  (>50% yes) are met → Proposal passes.                    │   |
|  └─────────────────────────────────────────────────────────────┘   |
|                              ▼                                    |
|  5. Execution (Time-Lock):                                      |
|  ┌─────────────────────────────────────────────────────────────┐   |
|  │  Smart contract executes code changes automatically after  │   |
|  │  mandatory time-lock (e.g., 48 hours).                    │   |
|  └─────────────────────────────────────────────────────────────┘   |
└─────────────────────────────────────────────────────────────────────┘

Cost_in_Tokens = (Number_of_Votes)²

Quadratic Voting Cost Comparison:
┌─────────────────────────────────────────────────────────────────────┐
|  Votes Cast │  Cost (Tokens)  │  Cost per Vote (Tokens)          |
|─────────────────────────────────────────────────────────────────────|
|  1          │  1              │  1.0                              |
|  2          │  4              │  2.0                              |
|  5          │  25             │  5.0                              |
|  10         │  100            │  10.0                             |
|  100        │  10,000         │  100.0                            |
|─────────────────────────────────────────────────────────────────────|
|  Effect: A whale with 100,000 tokens could either cast           |
|  316 votes (√100,000) on one issue, OR spread votes across       |
|  many issues. This empowers minority stakeholder communities     |
|  to have meaningful voice against capitalized whales.           |
└─────────────────────────────────────────────────────────────────────┘

Delegated_Voting_Power_Expert = Σ Delegated_Token_Balances

Where:
- Each token holder i delegates their entire balance B_i to an expert.
- The expert's total voting power = Σ B_i (for all delegators).
- Delegators retain the right to revoke delegation at any block height.

import math
import numpy as np
from typing import Dict, Tuple

# -------------------- 1. CROSS-CHAIN BRIDGE SIMULATION --------------------
class CrossChainBridge:
    def __init__(self, chains: list, fees: Dict[str, float], latencies: Dict[str, float]):
        """
        chains: List of chain names.
        fees: Bridge fees in percentage for each chain pair (e.g., "ETH_to_ARB": 0.001).
        latencies: Bridge latency in blocks for each chain pair.
        """
        self.chains = chains
        self.fees = fees
        self.latencies = latencies
    
    def bridge_transfer(self, origin: str, dest: str, amount: float) -> Tuple[float, float, float]:
        """
        Simulate a bridge transfer.
        Returns: (destination_amount, fee_paid, blocks_latency)
        """
        pair_key = f"{origin}_to_{dest}"
        fee_pct = self.fees.get(pair_key, 0.001)  # Default 0.1%
        fee_paid = amount * fee_pct
        dest_amount = amount - fee_paid
        latency = self.latencies.get(pair_key, 10)  # Default 10 blocks
        return dest_amount, fee_paid, latency

# -------------------- 2. CROSS-CHAIN ARBITRAGE MODEL --------------------
class CrossChainArbitrageBot:
    def __init__(self, bridge: CrossChainBridge, gas_costs: Dict[str, float]):
        self.bridge = bridge
        self.gas_costs = gas_costs
    
    def arbitrage_opportunity(
        self,
        chain_a: str,
        chain_b: str,
        price_a: float,
        price_b: float,
        quantity: float,
        slippage_a_pct: float = 0.005,
        slippage_b_pct: float = 0.005
    ) -> Dict[str, float]:
        """
        Evaluate the profitability of a cross-chain arbitrage trade.
        """
        # Account for slippage on both exchanges
        effective_buy_price = price_a * (1 + slippage_a_pct)
        effective_sell_price = price_b * (1 - slippage_b_pct)
        
        # Gross spread
        spread = effective_sell_price - effective_buy_price
        gross_profit = quantity * spread
        
        # Bridge costs
        dest_amount, bridge_fee, latency = self.bridge.bridge_transfer(
            chain_a, chain_b, quantity
        )
        # Note: This is a simplification; actual bridge costs are more complex
        
        # Gas costs
        gas_a = self.gas_costs.get(chain_a, 10)
        gas_b = self.gas_costs.get(chain_b, 10)
        total_gas = gas_a + gas_b
        
        # Net profit
        net_profit = gross_profit - bridge_fee - total_gas
        
        return {
            'gross_profit': gross_profit,
            'bridge_fee': bridge_fee,
            'gas_costs': total_gas,
            'net_profit': net_profit,
            'latency_blocks': latency,
            'profitable': net_profit > 0,
            'required_spread_pct': (bridge_fee + total_gas) / (quantity * price_a) * 100
        }

# -------------------- 3. DAO GOVERNANCE SIMULATION --------------------
class DAOGovernance:
    def __init__(self, token_holders: Dict[str, float]):
        """
        token_holders: Dict mapping address -> token balance.
        """
        self.holders = token_holders
        self.total_supply = sum(token_holders.values())
    
    def token_weighted_vote(self, proposal: str, votes: Dict[str, str]) -> Dict[str, float]:
        """
        Simulate token-weighted voting.
        votes: Dict mapping address -> 'yes', 'no', 'abstain'.
        """
        yes_weight = 0
        no_weight = 0
        abstain_weight = 0
        
        for address, vote in votes.items():
            weight = self.holders.get(address, 0)
            if vote == 'yes':
                yes_weight += weight
            elif vote == 'no':
                no_weight += weight
            elif vote == 'abstain':
                abstain_weight += weight
        
        total_voted = yes_weight + no_weight + abstain_weight
        quorum_reached = total_voted / self.total_supply
        
        return {
            'proposal': proposal,
            'yes_pct': (yes_weight / total_voted * 100) if total_voted > 0 else 0,
            'no_pct': (no_weight / total_voted * 100) if total_voted > 0 else 0,
            'abstain_pct': (abstain_weight / total_voted * 100) if total_voted > 0 else 0,
            'quorum_reached': quorum_reached,
            'passes': (yes_weight > no_weight) and (quorum_reached > 0.04)  # 4% quorum
        }
    
    def quadratic_vote(self, token_allocation: Dict[str, int]) -> Dict[str, int]:
        """
        Simulate quadratic voting: cost = votes².
        token_allocation: Dict mapping address -> number of votes they choose to cast.
        """
        results = {}
        total_cost = 0
        for address, votes in token_allocation.items():
            cost = votes ** 2
            results[address] = {
                'votes_cast': votes,
                'cost_tokens': cost,
                'balance_remaining': self.holders.get(address, 0) - cost
            }
            total_cost += cost
        return results

# -------------------- 4. DEMONSTRATION --------------------
if __name__ == "__main__":
    print("=" * 70)
    print("CROSS-CHAIN ARBITRAGE & DAO GOVERNANCE SIMULATION")
    print("=" * 70)
    
    # ---------- BRIDGE SETUP ----------
    chains = ['Ethereum', 'Arbitrum', 'Optimism', 'Base']
    bridge_fees = {
        'Ethereum_to_Arbitrum': 0.0005,
        'Ethereum_to_Optimism': 0.0006,
        'Arbitrum_to_Ethereum': 0.0004,
        'Optimism_to_Ethereum': 0.0004,
    }
    latencies = {
        'Ethereum_to_Arbitrum': 12,
        'Ethereum_to_Optimism': 15,
        'Arbitrum_to_Ethereum': 10,
        'Optimism_to_Ethereum': 10,
    }
    bridge = CrossChainBridge(chains, bridge_fees, latencies)
    
    gas_costs = {'Ethereum': 50, 'Arbitrum': 5, 'Optimism': 3, 'Base': 2}
    bot = CrossChainArbitrageBot(bridge, gas_costs)
    
    # ---------- ARBITRAGE OPPORTUNITY ----------
    print("\n--- ARBITRAGE OPPORTUNITY ---")
    result = bot.arbitrage_opportunity(
        chain_a='Ethereum',
        chain_b='Arbitrum',
        price_a=3000,    # ETH price on Ethereum
        price_b=3015,    # ETH price on Arbitrum (0.5% spread)
        quantity=10      # 10 ETH
    )
    
    print(f"Trade: 10 ETH on Ethereum → Arbitrum")
    print(f"Price A: ${result['gross_profit']:.2f}")
    print(f"Bridge Fee: ${result['bridge_fee']:.2f}")
    print(f"Gas Costs: ${result['gas_costs']:.2f}")
    print(f"Net Profit: ${result['net_profit']:.2f}")
    print(f"Profitable: {'✅ YES' if result['profitable'] else '❌ NO'}")
    print(f"Required Spread to Breakeven: {result['required_spread_pct']:.3f}%")
    
    # ---------- TOKEN-WEIGHTED GOVERNANCE ----------
    print("\n--- TOKEN-WEIGHTED DAO VOTE ---")
    holders = {
        'whale': 1_000_000,
        'miner1': 50_000,
        'miner2': 30_000,
        'retail1': 100,
        'retail2': 50,
        'retail3': 25,
    }
    dao = DAOGovernance(holders)
    
    votes = {
        'whale': 'no',
        'miner1': 'yes',
        'miner2': 'yes',
        'retail1': 'yes',
        'retail2': 'abstain',
        'retail3': 'yes'
    }
    
    result = dao.token_weighted_vote("Increase collateralization ratio to 150%", votes)
    print(f"Proposal: {result['proposal']}")
    print(f"Yes: {result['yes_pct']:.1f}% | No: {result['no_pct']:.1f}% | Abstain: {result['abstain_pct']:.1f}%")
    print(f"Quorum Reached: {(result['quorum_reached']*100):.1f}%")
    print(f"Status: {'✅ PASSED' if result['passes'] else '❌ REJECTED'}")
    
    # ---------- QUADRATIC VOTING ----------
    print("\n--- QUADRATIC VOTING ---")
    # Small holder has 100 tokens, wants to cast 5 votes (costs 25 tokens)
    # Large holder has 10,000 tokens, wants to cast 10 votes (costs 100 tokens)
    qv_allocation = {
        'small_holder': 5,   # casts 5 votes
        'medium_holder': 7,  # casts 7 votes
        'large_holder': 10,  # casts 10 votes
    }
    
    qv_balances = {
        'small_holder': 100,
        'medium_holder': 500,
        'large_holder': 10000,
    }
    
    # Extend DAO to include balances for QV calculation
    qv_dao = DAOGovernance(qv_balances)
    qv_results = qv_dao.quadratic_vote(qv_allocation)
    
    print("Quadratic Voting Analysis:")
    for address, data in qv_results.items():
        print(f"{address}: {data['votes_cast']} votes cost {data['cost_tokens']} tokens")
        print(f"  └─ Remaining balance: {data['balance_remaining']:.0f} tokens")
    
    print("\n" + "=" * 70)

HHI = Σ (Li / L_total)²