# Bonding Curve Mechanics

## Overview

The bonding curve phase implements the **LMSR (Logarithmic Market Scoring Rule)** pricing model, a mathematically rigorous market maker designed specifically for prediction markets. This mechanism enables permissionless market creation without seed liquidity while providing bounded losses and smooth price discovery.

***

## Pricing Model

### LMSR Formula

The protocol uses the Logarithmic Market Scoring Rule with liquidity parameter `b`:

**Cost Function:**

```
C(q) = b × ln(e^(q_yes/b) + e^(q_no/b))
```

**Price Formulas:**

```
Price(YES) = e^(q_yes/b) / (e^(q_yes/b) + e^(q_no/b))
Price(NO)  = e^(q_no/b) / (e^(q_yes/b) + e^(q_no/b))
```

> \[!IMPORTANT] **Key Property**: Price(YES) + Price(NO) = 1.0 **ALWAYS**. This fundamental invariant ensures market coherence.

### Liquidity Parameter (b)

The `b` parameter controls price sensitivity:

* **Higher b** = More liquidity, less price impact per trade
* **Lower b** = Less liquidity, more volatile prices
* **Default**: `b = bondingTarget / 7` (approximately $143 for a $1,000 target)

### Price Impact Examples

Starting from 50/50 prices with b ≈ $143:

| Action       | Shares Bought | YES Price After | NO Price After |
| ------------ | ------------- | --------------- | -------------- |
| $100 YES buy | \~97 shares   | 55.2%           | 44.8%          |
| $200 YES buy | \~189 shares  | 60.1%           | 39.9%          |
| $500 YES buy | \~420 shares  | 73.5%           | 26.5%          |

***

## Trade Execution

### Buy Order Processing (LMSR)

For a YES purchase of USDC amount `A`:

```
1. Calculate current cost: C_before = b × ln(e^(qYes/b) + e^(qNo/b))
2. Binary search for shares where: C(qYes + shares, qNo) - C_before = A
3. shares = MAX(calculated_shares, A)  ← $1 CEILING
4. qYes += shares
5. vault += A
6. Mint shares to user
```

> \[!TIP] **$1 Ceiling Guarantee**: You always receive at least 1 share per $1 spent. If LMSR price exceeds $1, you still get 1:1. This ensures winners always profit.

### Sell Order Processing (LMSR)

For selling `S` YES shares:

```
1. Calculate current cost: C_before = b × ln(e^(qYes/b) + e^(qNo/b))
2. Calculate new cost: C_after = b × ln(e^((qYes-S)/b) + e^(qNo/b))
3. payout = C_before - C_after (capped at vault balance)
4. qYes -= S
5. vault -= payout
6. Burn shares from user
```

***

## LMSR vs CPMM

| Property               | LMSR                  | CPMM                |
| ---------------------- | --------------------- | ------------------- |
| **Price Sum**          | Always = 1.0          | May drift           |
| **Bounded Loss**       | Yes, for market maker | No inherent bound   |
| **Price Sensitivity**  | Adjustable via `b`    | Fixed by pool ratio |
| **Mathematical Basis** | Information theory    | Uniswap-style       |
| **Prediction Markets** | Designed for          | Adapted from DeFi   |

***

## Fee Structure

| Operation | Fee Rate | Notes           |
| --------- | -------- | --------------- |
| Buy       | **0%**   | No trading fees |
| Sell      | **0%**   | No trading fees |

> \[!NOTE] The bonding phase has **zero trading fees**. Solvency is maintained through the natural dynamics of the LMSR curve. Fees are only collected at graduation (2%).

***

## Early Exit Mechanism

Users may liquidate positions at any time during the bonding phase:

### Exit Dynamics

* **LMSR Payout**: Calculated as decrease in cost function when removing shares
* **First-mover advantage**: Earlier sellers when prices are extreme receive better payouts
* **Vault protection**: Payout limited by available reserves

***

## Bonding Phase Termination

The bonding phase ends under one of two conditions:

### Condition 1: Graduation (Success)

```
VaultBalance >= bondingTarget AND Solvency >= 100% AND each side >= 20%
```

Market transitions to Order Book trading with real transferable ERC-20 tokens.

### Condition 2: Deadline Expiry (Failure)

```
block.timestamp > bondingDeadline AND (Solvency < 100% OR Vault < bondingTarget)
```

Refund mode activates; users claim pro-rata share of vault.

***

## Economic Incentives

| Actor           | Incentive       | Mechanism                           |
| --------------- | --------------- | ----------------------------------- |
| Early buyers    | Lower prices    | LMSR starting at 50%                |
| Large buyers    | Price discovery | Exponential price movement          |
| Arbitrageurs    | Balance markets | Cheap side offers profit            |
| OG Participants | Bonus pool      | Early depositors share excess vault |
| Protocol        | Graduation fees | 2% collected at graduation          |

***

## Mathematical Properties

### Solvency Guarantee

LMSR ensures the market maker's maximum loss is bounded:

```
Max Loss = b × ln(2)
```

This property, combined with the $1 ceiling on share prices, ensures the vault can always cover winning payouts.

### Price Smoothness

Unlike CPMM with its hyperbolic curve, LMSR provides smoother price transitions through the entire probability range (0% to 100%), making it ideal for prediction markets where extreme probabilities are meaningful.